leftri rightri

This is PART 28: Centers X(54001) - X(56000)

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

X(54001) = {X(4),X(5)}-HARMONIC CONJUGATE OF X(186)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - 3*a^2*b^4 + 2*b^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) : :
X(54001) = 4 X[5] - X[38448], 5 X[3843] + X[35498]

X(54001) lies on these lines: {2, 3}, {32, 50718}, {49, 18379}, {54, 13851}, {74, 32767}, {112, 39590}, {156, 18430}, {184, 18394}, {265, 11264}, {389, 7722}, {578, 7699}, {946, 31948}, {1112, 12300}, {1147, 18392}, {1173, 11564}, {1199, 18388}, {1614, 18383}, {1986, 10095}, {2914, 3574}, {3043, 10113}, {6152, 11017}, {6153, 6242}, {6241, 23325}, {6344, 14860}, {6696, 12244}, {6746, 45958}, {6748, 18365}, {8537, 18553}, {8744, 43457}, {9707, 18405}, {9820, 12383}, {10110, 32352}, {10312, 39565}, {10632, 42919}, {10633, 42918}, {11438, 11704}, {11464, 34786}, {11561, 12292}, {11572, 14157}, {11597, 22804}, {12133, 22948}, {12254, 32395}, {12289, 18376}, {15012, 43836}, {15031, 44146}, {15081, 26879}, {16223, 46847}, {17854, 32184}, {18488, 46686}, {18504, 46261}, {20417, 34563}, {22330, 32234}, {23292, 43818}, {23324, 34224}, {25739, 43831}, {32171, 52863}, {34545, 43821}, {38140, 41722}, {40640, 44795}, {43846, 46849}, {43865, 52416}

X(54001) = midpoint of X(4) and X(6143)
X(54001) = orthocentroidal-circle-inverse of X(34797)}
X(54001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 34797}, {2, 34797, 17506}, {3, 18567, 10296}, {4, 5, 186}, {4, 235, 26863}, {4, 403, 34484}, {4, 1594, 14865}, {4, 3090, 35471}, {4, 3091, 16868}, {4, 3544, 3147}, {4, 3545, 7505}, {4, 7505, 18559}, {4, 7577, 3520}, {4, 14940, 6240}, {4, 16868, 3518}, {4, 21844, 35480}, {4, 35473, 382}, {4, 35475, 35490}, {4, 35482, 1885}, {4, 35487, 47486}, {4, 37943, 3575}, {4, 44879, 12173}, {4, 44958, 52294}, {4, 52295, 13596}, {5, 3627, 10125}, {5, 3845, 45971}, {5, 6240, 14940}, {5, 18403, 14118}, {5, 18563, 2}, {5, 18567, 3}, {24, 18386, 4}, {378, 3843, 4}, {381, 7507, 35488}, {381, 7547, 4}, {381, 7564, 3832}, {382, 52296, 35473}, {403, 23047, 4}, {427, 7533, 37777}, {546, 1594, 4}, {546, 5066, 13163}, {546, 25402, 5}, {1594, 1885, 35482}, {1656, 35480, 21844}, {1885, 35482, 14865}, {3091, 3832, 7528}, {3091, 7404, 3545}, {3153, 10024, 7512}, {3541, 3839, 4}, {3575, 5066, 35487}, {3575, 35487, 37943}, {3575, 37943, 47486}, {3845, 18560, 4}, {3850, 23047, 403}, {3850, 50137, 3091}, {3851, 18386, 24}, {5094, 35490, 35475}, {6240, 14940, 186}, {7404, 52069, 35500}, {7507, 35488, 4}, {7528, 52295, 3518}, {7547, 35488, 7507}, {10151, 15559, 4}, {10254, 18377, 7488}, {10255, 44263, 22467}, {13163, 48411, 44802}, {13371, 50009, 7464}, {13406, 31724, 23}

X(54002) = {X(2),X(5)}-HARMONIC CONJUGATE OF X(237)

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

X(54002) lies on these lines: {2, 3}, {211, 7867}, {233, 40601}, {1506, 20965}, {3051, 7746}, {3589, 20021}, {3613, 22062}, {3917, 27375}, {7786, 39906}, {11272, 51481}, {11675, 33873}, {14061, 22735}, {18024, 40410}, {18358, 25046}, {21352, 45937}, {31279, 45692}, {40643, 43650}

X(54002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5, 237}, {2, 14957, 140}, {2, 32961, 33734}, {2, 37988, 14096}, {3628, 21531, 2}

X(54003) = {X(3),X(4)}-HARMONIC CONJUGATE OF X(237)

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

X(54003) lies on these lines: {2, 3}, {32, 11424}, {39, 185}, {64, 5013}, {160, 36990}, {216, 12294}, {248, 1970}, {511, 23635}, {570, 34146}, {574, 3331}, {577, 19124}, {578, 34396}, {682, 51869}, {1105, 6394}, {1350, 22062}, {1503, 20775}, {3095, 5889}, {3398, 13434}, {5158, 11470}, {5167, 18860}, {5191, 11430}, {5480, 40981}, {5907, 36212}, {5921, 20794}, {7783, 39355}, {8266, 29181}, {8550, 53246}, {9155, 15030}, {9737, 23098}, {11171, 15072}, {11550, 23195}, {11674, 35002}, {13334, 46850}, {13367, 42671}, {13474, 44437}, {14135, 52006}, {15062, 32464}, {15270, 17845}, {15815, 38297}, {16264, 19189}, {16659, 51255}, {20975, 50649}, {22089, 41167}, {23105, 42660}, {23181, 45303}, {23200, 51739}, {37575, 45932}, {39871, 41008}, {40079, 43278}, {41328, 44882}, {41716, 50645}, {44489, 46327}

X(54003) = crossdifference of every pair of points on line {647, 53345}
X(54003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 4, 237}, {3, 441, 417}, {3, 6660, 7488}, {3, 11479, 37344}, {3, 31952, 20}, {3, 32444, 4}, {4, 37121, 5}, {4, 44893, 235}, {5013, 32445, 43718}, {14118, 37183, 3}, {35934, 47620, 14096}, {42789, 42790, 37124}

X(54004) = {X(3),X(5)}-HARMONIC CONJUGATE OF X(237)

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

X(54004) lies on these lines: {2, 3}, {39, 3289}, {95, 17984}, {160, 10516}, {182, 14575}, {211, 30270}, {216, 2211}, {511, 22062}, {569, 34396}, {570, 44716}, {574, 45938}, {1352, 20775}, {1503, 41328}, {3095, 11412}, {3398, 43651}, {5013, 40805}, {5188, 27375}, {5191, 37513}, {5480, 8266}, {5907, 13334}, {6146, 51869}, {6292, 46094}, {6394, 40448}, {9155, 10170}, {9475, 42441}, {9605, 12160}, {9967, 23635}, {10984, 37479}, {11171, 11459}, {11675, 35002}, {11793, 36212}, {14561, 40981}, {15030, 21163}, {20975, 44479}, {22087, 50648}, {37575, 45937}, {39201, 40550}, {44437, 44870}

X(54004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5, 237}, {3, 6660, 7512}, {3, 11328, 37114}, {3, 31952, 376}, {3, 32444, 20}, {3090, 37114, 11328}, {14096, 35934, 47620}, {37126, 37183, 3}

X(54005) = {X(4),X(5)}-HARMONIC CONJUGATE OF X(237)

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

X(54005) lies on these lines: {2, 3}, {115, 3574}, {217, 5475}, {3331, 43457}, {7697, 11444}, {11197, 47328}, {18424, 45938}, {22682, 27375}, {23635, 39530}, {26883, 40643}, {42862, 44145}

X(54005) = midpoint of X(4) and X(37121)
X(54005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 5, 237}, {4, 44893, 3575}, {3843, 32444, 4}

X(54006) = {X(2),X(3)}-HARMONIC CONJUGATE OF X(2070)

Barycentrics    a^2*(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 + 7*a^2*b^4*c^2 - 2*b^6*c^2 + 7*a^2*b^2*c^4 + 6*b^4*c^4 + 2*a^2*c^6 - 2*b^2*c^6 - c^8) : :
X(54006) = X[3] + 2 X[7550], 3 X[3] - 2 X[44832], 3 X[381] - 4 X[50135], 5 X[1656] - 4 X[37990], 3 X[7550] + X[44832], 3 X[15246] - X[44832]

X(54006) lies on these lines: {2, 3}, {52, 15047}, {54, 32142}, {95, 339}, {113, 32600}, {115, 2963}, {128, 46654}, {143, 12834}, {182, 15087}, {195, 1216}, {216, 52166}, {252, 51255}, {323, 44324}, {389, 12307}, {399, 5092}, {511, 15038}, {567, 3917}, {568, 43650}, {1154, 15037}, {1173, 13421}, {1199, 12316}, {1209, 44862}, {1351, 37283}, {1511, 5888}, {2917, 32767}, {3455, 15561}, {3581, 5892}, {3763, 52990}, {3819, 22115}, {5012, 15067}, {5085, 18445}, {5096, 45923}, {5309, 50660}, {5447, 37472}, {5562, 37471}, {5650, 18475}, {5890, 33533}, {5898, 21357}, {6101, 14627}, {6243, 15004}, {7691, 12006}, {7753, 11063}, {7999, 32046}, {8553, 15484}, {8589, 34866}, {8718, 45958}, {9703, 15066}, {9730, 32608}, {10170, 10540}, {10601, 13321}, {10627, 13434}, {11258, 15563}, {11381, 33539}, {11464, 44299}, {11898, 32621}, {11935, 40913}, {12188, 41328}, {12325, 32165}, {13336, 18436}, {13339, 13754}, {13364, 15107}, {13470, 48675}, {14128, 52525}, {15033, 37496}, {15108, 50708}, {15567, 31843}, {15801, 36153}, {16030, 40631}, {18350, 44110}, {18451, 53094}, {18493, 37557}, {19596, 24206}, {21230, 43808}, {21975, 51477}, {22121, 53026}, {26879, 32333}, {32063, 44883}, {33541, 46850}, {33879, 34513}, {34783, 37515}, {37779, 45969}, {38402, 46267}, {43150, 45730}, {43704, 44325}, {51175, 53019}

X(54006) = midpoint of X(7550) and X(15246)
X(54006) = reflection of X(i) in X(j) for these {i,j}: {3, 15246}, {37349, 5}
X(54006) = X(22454)-isoconjugate of X(44706)
X(54006) = barycentric quotient X(8882)/X(22454)
X(54006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 2070}, {2, 46450, 5}, {3, 5, 13564}, {3, 140, 43809}, {3, 1656, 2937}, {3, 3526, 45735}, {3, 3843, 10323}, {3, 5055, 22}, {3, 5070, 26}, {3, 5899, 6636}, {3, 7393, 1656}, {3, 7395, 382}, {3, 7484, 5054}, {3, 9818, 3534}, {3, 13621, 7512}, {3, 14118, 35498}, {3, 15694, 6644}, {3, 18378, 7525}, {3, 21308, 34006}, {3, 21312, 14093}, {3, 34864, 14130}, {3, 35452, 8703}, {3, 46219, 24}, {3, 49671, 35001}, {5, 6636, 5899}, {22, 5055, 7545}, {23, 547, 21308}, {24, 13154, 46219}, {140, 37126, 3}, {140, 37938, 2}, {182, 23039, 15087}, {186, 1594, 25}, {548, 45308, 3}, {549, 35921, 3}, {1216, 13353, 195}, {1656, 2937, 18369}, {2071, 12100, 3}, {3090, 7525, 18378}, {3520, 15712, 3}, {3524, 18570, 3}, {3530, 14118, 3}, {3628, 7512, 13621}, {3819, 37513, 22115}, {5012, 15067, 50461}, {5054, 30771, 3526}, {5562, 37471, 43845}, {5899, 6636, 13564}, {6101, 43651, 14627}, {6644, 40916, 15694}, {7484, 32216, 40916}, {7485, 7509, 7514}, {7485, 7514, 3}, {7496, 35921, 549}, {7502, 31723, 12083}, {7509, 7516, 3}, {7514, 7516, 7485}, {7527, 8703, 35452}, {7555, 13595, 37956}, {7555, 15699, 13595}, {7574, 37347, 381}, {10170, 22352, 10540}, {10601, 37494, 13321}, {11250, 15717, 3}, {21308, 34006, 23}, {35500, 45308, 548}, {36439, 36457, 37901}, {37848, 37850, 15109}, {44832, 50135, 12083}

X(54007) = {X(4),X(5)}-HARMONIC CONJUGATE OF X(2070)

Barycentrics    a^10 - 4*a^6*b^4 + 2*a^4*b^6 + 3*a^2*b^8 - 2*b^10 + a^6*b^2*c^2 - 3*a^4*b^4*c^2 - 4*a^2*b^6*c^2 + 6*b^8*c^2 - 4*a^6*c^4 - 3*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 4*b^6*c^4 + 2*a^4*c^6 - 4*a^2*b^2*c^6 - 4*b^4*c^6 + 3*a^2*c^8 + 6*b^2*c^8 - 2*c^10 : :
X(54007) = 2 X[4] + X[18364], 4 X[546] + X[35482]

X(54007) lies on these lines: {2, 3}, {49, 48675}, {110, 22804}, {156, 7699}, {265, 3574}, {389, 38724}, {567, 10274}, {578, 18430}, {1568, 15137}, {5562, 6153}, {5895, 18550}, {6145, 43821}, {6288, 50461}, {7703, 32138}, {7728, 11559}, {9630, 38458}, {10095, 14644}, {11430, 52863}, {11591, 32196}, {11801, 11805}, {12902, 37472}, {13851, 43835}, {14128, 41590}, {14676, 18502}, {14860, 14978}, {15033, 18379}, {15038, 32341}, {18424, 38463}, {20299, 43807}, {23325, 37481}, {25739, 43845}, {32340, 44516}, {32608, 34826}, {36753, 40285}

X(54007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 5, 2070}, {4, 3091, 13406}, {4, 10254, 18378}, {4, 18562, 3830}, {4, 18570, 382}, {4, 39504, 14130}, {5, 3153, 34864}, {5, 3627, 34577}, {5, 31724, 3}, {5, 35921, 1656}, {265, 3574, 14627}, {381, 7507, 3}, {546, 33332, 4}, {3851, 18378, 10254}, {5576, 23047, 18403}, {6143, 45971, 37955}, {7547, 7564, 381}, {7574, 13160, 3}, {11563, 50135, 10254}

X(54008) = X(4)X(1903)∩X(9)X(355)

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

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5843.

X(54008) lies on these lines: {4,1903}, {6,1826}, {8,21871}, {9,355}, {10,37062}, {11,3554}, {19,1146}, {34,10366}, {37,5252}, {48,46835}, {56,24005}, {71,3691}, {77,21239}, {80,1743}, {92,5928}, {198,515}, {219,5179}, {281,2182}, {282,5514}, {346,5176}, {391,5086}, {519,21068}, {604,21044}, {610,5787}, {950,10367}, {966,5794}, {1012,11434}, {1212,26063}, {1229,21286}, {1436,6245}, {1449,5722}, {1737,5120}, {1741,37468}, {1836,1899}, {1839,5895}, {1855,3197}, {1856,2192}, {1864,7102}, {1901,5155}, {2082,5090}, {2267,17303}, {2270,5691}, {2276,21860}, {2285,21933}, {2324,5881}, {2994,7381}, {2995,23978}, {3419,3686}, {3553,10950}, {3731,37710}, {4254,10572}, {4863,17362}, {5227,40997}, {5307,13567}, {5786,46878}, {5816,40937}, {5942,21279}, {6611,8808}, {8573,22760}, {11375,50036}, {15817,22758}, {16667,37702}, {17134,25000}, {17299,21801}, {17314,32049}, {18621,28044}, {20927,21277}, {21244,24266}, {21270,30807}, {21853,41687}, {24914,36743}, {46344,51424}

X(54008) = midpoint of X(5942) and X(21279)
X(54008) = reflection of X(i) in X(j) for these {i,j}: {77, 21239}, {198, 20262}
X(54008) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 34411}
X(54008) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 34411}, {46836, 69}
X(54008) = X(i)-Waw conjugate of X(j) for these {i, j}: {4, 1836}
X(54008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {515, 20262, 198}, {2267, 21011, 17303}, {5942, 21279, 34371}

X(54009) = X(9)X(119)∩X(208)X(429)

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

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5843.

X(54009) lies on these lines: {4,1903}, {9,119}, {19,12679}, {48,12678}, {198,6260}, {208,429}, {221,53009}, {226,6611}, {322,329}, {342,10402}, {610,6259}, {1436,20263}, {1532,1741}, {1826,1836}, {1863,1864}, {2182,37414}, {2199,46836}, {2325,8804}, {2331,38357}, {2899,4645}, {6335,34408}, {7101,33650}, {12572,37320}

X(54009) = reflection of X(1436) in X(20263)
X(54009) = X(i)-Dao conjugate of X(j) for these {i, j}: {281, 34408}
X(54009) = X(i)-Ceva conjugate of X(j) for these {i, j}: {6335, 6129}

X(54010) = X(4)X(7149)∩X(223)X(6259)

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

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5843.

X(54010) lies on these lines: {4,7149}, {34,12679}, {73,12678}, {223,6259}, {225,2192}, {1035,20264}, {1118,17832}, {1836,1854}, {1837,1899}, {1895,10361}, {3342,13612}, {7037,47441}, {7103,10374}, {7952,40658}, {8812,10400}

X(54010) = reflection of X(1035) in X(20264)
X(54010) = X(i)-Waw conjugate of X(j) for these {i, j}: {4, 1837}

X(54011) = X(6)X(1885)∩X(393)X(1562)

Barycentrics    (a^2-b^2-c^2)*(a^12-9*a^8*(b^2-c^2)^2+(b^2-c^2)^6+16*a^6*(b^2-c^2)^2*(b^2+c^2)-a^4*(b^2-c^2)^2*(9*b^4+14*b^2*c^2+9*c^4)) : :

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5843.

X(54011) lies on these lines: {6,1885}, {146,45245}, {393,1562}, {647,18890}, {1033,15311}, {1249,5878}, {2331,12940}, {3087,35717}, {3344,13613}, {4846,15851}, {7129,12950}, {15341,15905}, {28783,33546}, {34980,41762}

X(54011) = reflection of X(1033) in X(20265)
X(54011) = X(i)-Waw conjugate of X(j) for these {i, j}: {4, 1899}
X(54011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1562, 14642, 393}, {15311, 20265, 1033}

X(54012) = X(2)X(98)∩X(4)X(373)

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

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5843.

XX(54012) lies on the circumconic {{A, B, C, X(98), X(45088)}} and these lines: {2,98}, {3,16657}, {4,373}, {6,30739}, {20,34417}, {25,44882}, {30,3066}, {51,7386}, {66,19132}, {68,3526}, {69,5650}, {122,5158}, {140,35602}, {141,26869}, {185,6804}, {290,46328}, {343,16419}, {381,5544}, {394,1353}, {427,17825}, {468,5085}, {511,46336}, {574,6388}, {575,37645}, {597,32216}, {599,5646}, {631,11430}, {632,18952}, {858,14561}, {1350,43957}, {1368,10601}, {1370,5943}, {1495,25406}, {1503,11284}, {1656,45303}, {1853,37439}, {1995,46264}, {2549,3124}, {3090,14216}, {3292,14912}, {3524,32225}, {3525,18912}, {3534,20192}, {3542,37515}, {3580,40916}, {3589,5094}, {3618,15812}, {3796,6677}, {3819,6515}, {3832,44300}, {3917,11433}, {4232,35268}, {4846,7728}, {5020,31383}, {5050,11064}, {5054,44569}, {5067,11457}, {5092,7493}, {5159,38110}, {5480,31152}, {5640,16063}, {5810,17529}, {5892,7706}, {6090,8550}, {6353,22352}, {6617,26905}, {6688,6997}, {6816,9729}, {6819,42400}, {7391,11451}, {7392,11550}, {7395,26937}, {7484,13567}, {7499,26958}, {7500,48896}, {7519,10545}, {7539,23332}, {7605,31857}, {7667,17810}, {7734,41588}, {7998,37644}, {8721,37338}, {9815,15028}, {9822,41256}, {10113,50008}, {10170,18917}, {10300,21850}, {10301,48905}, {10519,41586}, {10691,33586}, {10744,30513}, {11245,17811}, {11484,16655}, {11585,15805}, {11793,18916}, {12017,13394}, {12045,18553}, {13363,14791}, {13366,37669}, {14853,51360}, {15018,44493}, {15024,47528}, {15082,34507}, {15106,25329}, {15448,47597}, {15873,37198}, {16111,37470}, {17508,32223}, {17704,37201}, {18440,35283}, {19130,31099}, {20266,26890}, {21015,52424}, {22111,43448}, {23292,31255}, {25738,46219}, {26255,32237}, {26898,45200}, {30771,37649}, {31860,37899}, {31884,47582}, {33879,41724}, {34608,44106}, {34664,37475}, {34944,45979}, {35259,48906}, {38072,47311}, {38136,47315}, {39691,43620}, {40911,50967}, {44210,53094}, {46517,53023}, {47097,47352}

X(54012) = anticomplement of X(16187)
X(54012) = X(i)-Dao conjugate of X(j) for these {i, j}: {16187, 16187}
X(54012) = X(i)-Ceva conjugate of X(j) for these {i, j}: {46326, 2}
X(54012) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {46326, 6327}
X(54012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18911, 1352}, {2, 6776, 5651}, {125, 22112, 2}, {1352, 18911, 1899}, {5640, 16063, 31670}, {7386, 18928, 51}, {7484, 13567, 43653}, {15028, 37444, 9815}, {25406, 40132, 1495}

X(54013) = X(2)X(98)∩X(20)X(7998)

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

See Thanassis Gakopoulos, Antreas Hatzipolakis and Ivan Pavlov, euclid 5843.

X(54013) lies on these lines: {2,98}, {4,15066}, {5,6090}, {6,35283}, {20,7998}, {23,10519}, {24,11487}, {25,48876}, {69,1995}, {107,44134}, {140,26864}, {141,7493}, {193,5640}, {323,14853}, {376,11472}, {394,5480}, {524,3066}, {599,26255}, {631,6800}, {858,47474}, {1216,37122}, {1370,17811}, {1503,46336}, {1993,7392}, {2478,26637}, {2979,6995}, {3060,7398}, {3089,43614}, {3090,6193}, {3167,37439}, {3292,14561}, {3426,46349}, {3522,41462}, {3523,10282}, {3541,44080}, {3545,40112}, {3547,43598}, {3564,11284}, {3580,40132}, {3619,5596}, {3620,4232}, {3763,13394}, {3785,37465}, {3818,31099}, {3819,31383}, {3917,7500}, {3926,37335}, {4549,5891}, {4550,35485}, {4563,11185}, {4576,32815}, {4648,7474}, {5020,6515}, {5032,32127}, {5094,18358}, {5133,37669}, {5646,43273}, {5650,46264}, {5656,15052}, {6353,37636}, {6803,11441}, {6804,14516}, {6815,17814}, {6816,18396}, {7383,10539}, {7426,21356}, {7485,11206}, {7487,11444}, {7494,35264}, {7499,8780}, {7519,33884}, {7550,40913}, {7605,9716}, {7703,30769}, {7999,31305}, {9777,10128}, {10301,33878}, {10516,11064}, {10545,20080}, {10601,12007}, {11402,51732}, {11422,51171}, {11427,37990}, {11821,31304}, {12121,49669}, {12302,44834}, {13416,15060}, {15019,51170}, {15069,37648}, {15106,50008}, {15107,52301}, {15435,20806}, {15533,20192}, {16042,37644}, {17928,18931}, {18440,30739}, {18916,31831}, {18919,26206}, {18928,45968}, {21358,35266}, {25406,40916}, {30221,50149}, {31133,51537}, {31152,39884}, {32113,37980}, {32237,50977}, {32971,46900}, {33926,40680}, {35311,40138}, {35325,41370}, {35486,43586}

X(54013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3410, 23291}, {2, 5921, 18911}, {141, 35259, 7493}, {376, 44833, 21766}, {1352, 5651, 2}, {3619, 35260, 7495}, {22112, 24981, 11179}, {40916, 46818, 25406}

X(54014) = ISOGONAL CONJUGATE OF X(36073)

Barycentrics    Cos[B + Pi/6] - Cos[C + Pi/6] : :

X(54014) lies on these lines: {30, 511}, {1734, 50439}, {4560, 5240}

X(54014) = isogonal conjugate of X(36073)
X(54014) = crossdifference of every pair of points on line {6, 2152}
X(54014) = barycentric quotient X(4488)/X(34380)

X(54015) = ISOGONAL CONJUGATE OF X(36072)

Barycentrics    Cos[B - Pi/6] - Cos[C - Pi/6] : :

X(54015) lies on these lines: {30, 511}, {1734, 50438}, {4560, 5239}

X(54015) = isogonal conjugate of X(36072)
X(54015) = crossdifference of every pair of points on line {6, 2151}

X(54016) = X(102)X(2066)∩X(103)X(6502)

Barycentrics    Sin[A]^2/(Sin[B + Pi/4] - Sin[C + Pi/4]) : :

X(54016) lies on the circumcircle and these lines: {102, 2066}, {103, 6502}, {104, 18460}, {105, 16232}, {163, 39383}, {675, 13390}, {692, 2498}, {1311, 14121}, {1783, 6135}, {2362, 53069}, {9099, 13388}, {13389, 43363}, {26703, 30556}, {32726, 53065}

X(56016) = isogonal conjugate of X(56017)
X(54016) = X(7115)-Ceva conjugate of X(34125)
X(54016) = X(i)-isoconjugate of X(j) for these (i,j): {514, 30557}, {521, 1659}, {522, 13388}, {693, 5414}, {905, 7090}, {1577, 1805}, {2067, 4391}, {2362, 6332}, {3261, 53066}, {4025, 7133}, {6365, 14121}, {35519, 53063}
X(54016) = X(13388)-Dao conjugate of X(15413)
X(54016) = trilinear pole of line {6, 34125}
X(54016) = barycentric product X(i)*X(j) for these {i,j}: {100, 16232}, {101, 13390}, {108, 30556}, {109, 14121}, {651, 42013}, {653, 2066}, {1783, 13389}, {1897, 6502}, {6136, 13388}, {6335, 53064}, {18026, 53065}
X(54016) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 30557}, {1415, 13388}, {1576, 1805}, {2066, 6332}, {6502, 4025}, {8750, 7090}, {13389, 15413}, {13390, 3261}, {14121, 35519}, {16232, 693}, {30556, 35518}, {32674, 1659}, {32739, 5414}, {42013, 4391}, {53063, 6365}, {53064, 905}, {53065, 521}

X(54017) = ISOGONAL CONJUGATE OF X(54016)

Barycentrics    Sin[B + Pi/4] - Sin[C + Pi/4] : :

X(54017) lies on this line: {30, 511}

X(54017) = isogonal conjugate of X(54016)
X(54017) = crossdifference of every pair of points on line {6, 34125}
X(54017) = barycentric quotient X(i)/X(j) for these {i,j}: {11918, 9346}, {14689, 20831}

X(54018) = X(102)X(5414)∩X(103)X(2067)

Barycentrics    Sin[A]^2/(-Sin[B - Pi/4] + Sin[C - Pi/4]) : :

X(54018) lies on the circumcircle and these lines: {102, 5414}, {103, 2067}, {104, 7133}, {105, 2362}, {163, 39384}, {675, 1659}, {692, 2498}, {1311, 7090}, {1783, 6136}, {9098, 13389}, {13388, 43363}, {16232, 53070}, {26703, 30557}, {32726, 53066}

X(54018) = X(7115)-Ceva conjugate of X(34121)
X(54018) = X(i)-isoconjugate of X(j) for these (i,j): {514, 30556}, {521, 13390}, {522, 13389}, {693, 2066}, {905, 14121}, {1577, 1806}, {3261, 53065}, {4025, 42013}, {4391, 6502}, {6332, 16232}, {6364, 7090}, {35519, 53064}
X(54018) = X(13389)-Dao conjugate of X(15413)
X(54018) = trilinear pole of line {6, 34121}
X(54018) = barycentric product X(i)*X(j) for these {i,j}: {100, 2362}, {101, 1659}, {108, 30557}, {109, 7090}, {651, 7133}, {653, 5414}, {1783, 13388}, {1897, 2067}, {6135, 13389}, {6335, 53063}, {18026, 53066}
X(54018) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 30556}, {1415, 13389}, {1576, 1806}, {1659, 3261}, {2067, 4025}, {2362, 693}, {5414, 6332}, {7090, 35519}, {7133, 4391}, {8750, 14121}, {13388, 15413}, {30557, 35518}, {32674, 13390}, {32739, 2066}, {53063, 905}, {53064, 6364}, {53066, 521}

X(54019) = ISOCONAL CONJUGATE OF X(54018)

Barycentrics    Sin[B - Pi/4] - Sin[C - Pi/4] : :

X(54019) lies on these lines: {30, 511}, {4025, 30193}

X(54019) = isogonal conjugate of X(54018)
X(54019) = crossdifference of every pair of points on line {6, 34121}
X(54019) = barycentric product X(20945)*X(41146)

X(54020) = X(105)X(7052)∩X(106)X(7051)

Barycentrics    Sin[A]^2/(-Sin[B + Pi/6] + Sin[C + Pi/6]) : :

X(54020) lies on the circumcircle and these lines: {104, 7126}, {105, 7052}, {106, 7051}, {163, 16806}, {663, 692}, {840, 19373}, {1311, 7043}, {1415, 36072}, {4559, 36073}

X(54020) = isogonal conjugate of X(54021)
X(54020) = X(i)-isoconjugate of X(j) for these (i,j): {514, 5239}, {522, 37772}, {693, 7127}, {3904, 33655}, {3960, 7026}, {4391, 7051}, {4453, 19551}, {23838, 36668}
X(54020) = trilinear pole of line {6, 42624}
X(54020) = barycentric product X(i)*X(j) for these {i,j}: {100, 7052}, {109, 7043}, {651, 7126}, {2222, 5240}, {19373, 51562}
X(54020) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 5239}, {1415, 37772}, {7043, 35519}, {7052, 693}, {7126, 4391}, {19373, 4453}, {32739, 7127}

X(54021) = ISOGONAL CONJUGATE OF X(54020)

Barycentrics    Sin[B + Pi/6] - Sin[C + Pi/6] : :

X(54021) lies on these lines: {30, 511}, {4453, 36668}, {36930, 49274}

X(54021) = isogonal conjugate of X(54020)
X(54021) = crossdifference of every pair of points on line {6, 42624}

X(54022) = X(104)X(19551)∩X(105)X(33655)

Barycentrics    Sin[A]^2/(Sin[B - Pi/6] - Sin[C - Pi/6]) : :

X(54022) lies on the circumcircle and these lines: {104, 19551}, {105, 33655}, {106, 19373}, {163, 16807}, {663, 692}, {840, 7051}, {953, 7127}, {1311, 7026}, {1415, 36073}, {4559, 36072}

X(54022) = X(i)-isoconjugate of X(j) for these (i,j): {514, 5240}, {522, 37773}, {3904, 7052}, {3960, 7043}, {4391, 19373}, {4453, 7126}, {23838, 36669}
X(54022) = barycentric product X(i)*X(j) for these {i,j}: {100, 33655}, {109, 7026}, {651, 19551}, {655, 7127}, {2222, 5239}, {7051, 51562}
X(54022) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 5240}, {1415, 37773}, {7026, 35519}, {7051, 4453}, {7127, 3904}, {19551, 4391}, {33655, 693}

X(54023) = ISOGONAL CONJUGATE OF X(54022)

Barycentrics    Sin[B - Pi/6] - Sin[C - Pi/6] : :

X(54023) lies on these lines: {30, 511}, {4453, 36669}, {36931, 49274}

X(54023) = isogonal conjugate of X(54022)

X(54024) = TRILINEAR POLE OF X(6)X(2154)

Barycentrics    (Sec[A + Pi/6]*Sin[A]^2)/(Cos[B + Pi/6] - Cos[C + Pi/6]) : :

X(54024) lies on the circumcircle and these lines: {14, 10647}, {5995, 32678}, {19305, 36298}

X(54024) = isogonal conjugate of X(54025)
X(54024) = trilinear pole of line {6, 2154}

X(54025) = ISOGONAL CONJUGATE OF X(54024)

Barycentrics    Cos[A + Pi/6]*(Cos[B + Pi/6] - Cos[C + Pi/6]) : :

X(54025) lies on this lines: {30, 511}

X(54025) = isogonal conjugate of X(54024)
X(54025) = crossdifference of every pair of points on line {6, 2154}
X(54025) = barycentric quotient X(52318)/X(3812)

X(54026) = TRILINEAR POLE OF X(6)X(2153)

Barycentrics    (Sec[A - Pi/6]*Sin[A]^2)/(-Cos[B - Pi/6] + Cos[C - Pi/6]) : :

X(54026) lies on the circumcircle and these lines: {13, 10648}, {5994, 32678}, {11080, 42623}, {19304, 36299}

X(54026) = isogonal conjugate of X(54027)
X(54026) = trilinear pole of line {6, 2153}

X(54027) = ISOGONAL CONJUGATE OF X(54026)

Barycentrics    Cos[A - Pi/6]*(-Cos[B - Pi/6] + Cos[C - Pi/6]) : :

X(54027) lies on this line: {30, 511}

X(54027) = isogonal conjugate of X(54026)
X(54027) = crossdifference of every pair of points on line {6, 2153}

X(54028) = ISOGONAL CONJUGATE OF X(39384)

Barycentrics    Tan[B + Pi/4] - Tan[C + Pi/4] : :
Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2 - 2*S) : :

X(54028 lies on these lines: {30, 511}, {647, 14334}, {1328, 2394}, {2501, 14333}, {5664, 13821}, {6563, 14326}, {9131, 13316}, {9979, 13319}, {13807, 42733}, {14237, 43673}, {14325, 31296}, {17431, 47122}, {48539, 48955}, {48540, 48986}

X(54028) = isogonal conjugate of X(39384)
X(54028) = crossdifference of every pair of points on line {6, 3155}
X(54028) = {X(2501),X(17432)}-harmonic conjugate of X(14333)

X(54029) = ISOGONAL CONJUGATE OF X(39383)

Barycentrics    Tan[A + Pi/4]*(Tan[B + Pi/4] - Tan[C + Pi/4])::
Barycentrics    Cot[B + Pi/4] - Cot[C + Pi/4] : :
Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2 + 2*S) : :

X(54029 lies on these lines: {30, 511}, {647, 14333}, {1327, 2394}, {2501, 14334}, {5664, 13701}, {6563, 14325}, {9131, 13317}, {9979, 13320}, {13687, 42733}, {14232, 43673}, {14326, 31296}, {17432, 47122}, {48539, 48956}, {48540, 48987}

X(54029) = isogonal conjugate of X(39383)
X(54029) = crossdifference of every pair of points on line {6, 3156}
X(54029) = barycentric quotient X(33022)/X(7613)
X(54029) = {X(2501),X(17431)}-harmonic conjugate of X(14334)

X(54030) = ISOTOMIC CONJUGATE OF X(54028)

Barycentrics    1/((b^2 - c^2)*(-a^2 + b^2 + c^2 - 2*S)) : :
Barycentrics    Sin[A - B]*Sin[A - C]*(Cos[B] - Sin[B])*(Cos[C] - Sin[C]) : :

X(54030) lies on the Steiner circumellipse and these lines: {99, 39384}, {110, 925}, {290, 6414}, {486, 490}, {488, 13429}, {491, 24245}, {492, 35142}, {1494, 11091}, {1992, 21464}, {3228, 8576}, {5860, 13428}, {6396, 48986}, {8940, 45420}, {10962, 34391}, {16037, 46138}, {18878, 54029}

X(54030) = isotomic conjugate of X(54028)
X(54030) = isotomic conjugate of the isogonal conjugate of X(39384)
X(54030) = X(i)-isoconjugate of X(j) for these (i,j): {31, 54028}, {372, 661}, {491, 798}, {656, 5412}, {810, 1586}, {1924, 45806}, {13461, 51641}, {24006, 26920}
X(54030) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 54028}, {486, 17432}, {5409, 52584}, {9428, 45806}, {10962, 924}, {24245, 523}, {31998, 491}, {33365, 14326}, {36830, 372}, {39062, 1586}, {40596, 5412}
X(54030) = cevapoint of X(i) and X(j) for these (i,j): {2, 54028}, {372, 14326}, {488, 14325}, {523, 615}, {5408, 54029}
X(54030) = trilinear pole of line {2, 371}
X(54030) = barycentric product X(i)*X(j) for these {i,j}: {76, 39384}, {99, 486}, {110, 34392}, {371, 46134}, {492, 925}, {648, 11091}, {670, 8576}, {4563, 41516}, {5408, 30450}, {6331, 6414}, {6528, 26922}, {8940, 35136}, {14570, 16037}, {32734, 45805}
X(54030) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 54028}, {99, 491}, {110, 372}, {112, 5412}, {371, 924}, {486, 523}, {492, 6563}, {645, 13461}, {648, 1586}, {670, 45806}, {925, 485}, {3069, 14326}, {4558, 5409}, {5408, 52584}, {5413, 6753}, {6414, 647}, {8576, 512}, {8911, 30451}, {8940, 3566}, {11091, 525}, {13428, 54029}, {16037, 15412}, {24245, 17432}, {26922, 520}, {32661, 26920}, {32734, 8577}, {34392, 850}, {39383, 44192}, {39384, 6}, {41516, 2501}, {46134, 34391}

X(54031) = ISOTOMIC CONJUGATE OF X(54029)

Barycentrics    1/((b^2 - c^2)*(-a^2 + b^2 + c^2 + 2*S)) : :
Barycentrics    Sin[A - B]*Sin[A - C]*(Cos[B] + Sin[B])*(Cos[C] + Sin[C]) : :

X(54031) lies on the Steiner circumellipse and these lines: {99, 39383}, {110, 925}, {290, 6413}, {485, 489}, {487, 13440}, {491, 35142}, {492, 24246}, {1494, 11090}, {1992, 21463}, {3228, 8577}, {5861, 13439}, {6200, 48987}, {8944, 45421}, {10960, 34392}, {13455, 35144}, {16032, 46138}, {18878, 54028}

X(54031) = isotomic conjugate of X(54029)
X(54031) = isotomic conjugate of the isogonal conjugate of X(39383)
X(54031) = X(i)-isoconjugate of X(j) for these (i,j): {31, 54029}, {371, 661}, {492, 798}, {656, 5413}, {810, 1585}, {1924, 45805}, {8911, 24006}
X(54031) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 54029}, {485, 17431}, {5408, 52584}, {9428, 45805}, {10960, 924}, {24246, 523}, {31998, 492}, {33364, 14325}, {36830, 371}, {39062, 1585}, {40596, 5413}
X(54031) = cevapoint of X(i) and X(j) for these (i,j): {2, 54029}, {371, 14325}, {487, 14326}, {523, 590}, {5409, 54028}
X(54031) = trilinear pole of line {2, 372}
X(54031) = barycentric product X(i)*X(j) for these {i,j}: {76, 39383}, {99, 485}, {110, 34391}, {372, 46134}, {491, 925}, {648, 11090}, {670, 8577}, {4563, 41515}, {4625, 13455}, {5409, 30450}, {6331, 6413}, {8944, 35136}, {14570, 16032}, {32734, 45806}
X(54031) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 54029}, {99, 492}, {110, 371}, {112, 5413}, {372, 924}, {485, 523}, {491, 6563}, {648, 1585}, {670, 45805}, {925, 486}, {3068, 14325}, {4558, 5408}, {5409, 52584}, {5412, 6753}, {6413, 647}, {8577, 512}, {8944, 3566}, {11090, 525}, {13439, 54028}, {13455, 4041}, {16032, 15412}, {24246, 17431}, {26920, 30451}, {32661, 8911}, {32734, 8576}, {34391, 850}, {39383, 6}, {39384, 44193}, {41515, 2501}, {46134, 34392}

X(54032) = ISOGONAL CONJUGATE OF X(33971)

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

X(54032) lies on the cubic K1326 and these lines: {2, 51}, {3, 217}, {4, 276}, {20, 39682}, {52, 22270}, {69, 53174}, {97, 184}, {182, 5481}, {389, 31400}, {394, 418}, {520, 53173}, {577, 17974}, {1073, 6638}, {1092, 28724}, {1214, 3784}, {1216, 14376}, {1217, 15644}, {1297, 3098}, {1350, 40801}, {2706, 6037}, {3346, 13348}, {3522, 38256}, {3796, 34003}, {3926, 5562}, {4549, 15421}, {5171, 43652}, {6243, 22268}, {6389, 42487}, {6784, 7612}, {9418, 37114}, {9730, 46412}, {9821, 51997}, {11206, 32319}, {11427, 37872}, {11821, 46735}, {12122, 15429}, {14489, 33878}, {14938, 37484}, {21843, 31850}, {23039, 34897}, {34579, 52926}, {39683, 51350}, {45301, 46730}

X(54032) = isogonal conjugate of X(33971)
X(54032) = isogonal conjugate of the anticomplement of X(42353)
X(54032) = isogonal conjugate of the polar conjugate of X(42313)
X(54032) = isotomic conjugate of the polar conjugate of X(43718)
X(54032) = X(42313)-Ceva conjugate of X(43718)
X(54032) = X(i)-isoconjugate of X(j) for these (i,j): {1, 33971}, {6, 51315}, {19, 458}, {92, 10311}, {158, 182}, {183, 1096}, {393, 52134}, {823, 3288}, {1973, 44144}, {2190, 39530}, {2207, 3403}, {6784, 23999}, {23878, 24019}
X(54032) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 33971}, {5, 39530}, {6, 458}, {9, 51315}, {1147, 182}, {6337, 44144}, {6338, 20023}, {6503, 183}, {22391, 10311}, {35071, 23878}
X(54032) = trilinear pole of line {520, 42293}
X(54032) = barycentric product X(i)*X(j) for these {i,j}: {3, 42313}, {69, 43718}, {262, 394}, {263, 3926}, {326, 2186}, {327, 577}, {343, 51444}, {3265, 26714}, {5562, 42300}, {6394, 51543}, {15414, 52926}, {17974, 46807}, {35911, 36885}, {37188, 40803}
X(54032) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 51315}, {3, 458}, {6, 33971}, {69, 44144}, {184, 10311}, {216, 39530}, {255, 52134}, {262, 2052}, {263, 393}, {326, 3403}, {327, 18027}, {394, 183}, {520, 23878}, {577, 182}, {2186, 158}, {3402, 1096}, {3926, 20023}, {3998, 42711}, {14585, 34396}, {17974, 46806}, {26714, 107}, {32716, 20031}, {39201, 3288}, {42300, 8795}, {42313, 264}, {43718, 4}, {46319, 2207}, {51386, 51373}, {51394, 51372}, {51444, 275}, {51543, 6530}

X(54033) = X(4)X(276)∩X(20)X(76)

Barycentrics    b^2*c^2*(3*a^8 + 3*a^6*b^2 - 7*a^4*b^4 + a^2*b^6 + 3*a^6*c^2 - 4*a^4*b^2*c^2 - a^2*b^4*c^2 + 2*b^6*c^2 - 7*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(54033) lies on the cubic K1326 and these lines: {2, 15265}, {4, 276}, {20, 76}, {99, 20023}, {183, 47620}, {290, 376}, {1350, 44144}, {6528, 52283}, {8920, 10008}, {10519, 17984}, {11185, 14957}, {11206, 34384}, {18022, 33522}, {44152, 51438}

X(54034) = ISOGONAL CONJUGATE OF X(311)

Barycentrics    a^4*(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(54034) lies on these lines: {3, 54}, {4, 34449}, {5, 96}, {6, 2351}, {25, 8745}, {32, 44077}, {49, 3133}, {50, 23195}, {51, 1576}, {95, 325}, {98, 275}, {137, 50210}, {160, 184}, {228, 2148}, {237, 10547}, {570, 8603}, {878, 2623}, {1141, 50471}, {1370, 43768}, {1410, 52440}, {1994, 14652}, {2167, 35614}, {2200, 52426}, {3051, 14600}, {3425, 19189}, {3456, 42671}, {3575, 8884}, {5576, 36842}, {7399, 19179}, {9792, 45832}, {10558, 14586}, {15958, 42065}, {16032, 49356}, {16037, 49355}, {18420, 19176}, {19161, 21638}, {19188, 37454}, {20975, 34448}, {23286, 34291}, {23292, 51458}, {26864, 33629}, {33581, 44080}, {34093, 38861}, {34986, 39805}, {37347, 40631}, {41205, 45793}

X(54034) = isogonal conjugate of X(311)
X(54034) = isogonal conjugate of the anticomplement of X(570)
X(54034) = isogonal conjugate of the isotomic conjugate of X(54)
X(54034) = isogonal conjugate of the polar conjugate of X(8882)
X(54034) = isotomic conjugate of the isogonal conjugate of X(14573)
X(54034) = polar conjugate of the isotomic conjugate of X(14533)
X(54034) = X(i)-Ceva conjugate of X(j) for these (i,j): {54, 14533}, {933, 2623}, {1166, 6}, {14587, 14586}
X(54034) = X(i)-isoconjugate of X(j) for these (i,j): {1, 311}, {2, 14213}, {4, 18695}, {5, 75}, {19, 28706}, {27, 42698}, {51, 561}, {52, 20571}, {53, 304}, {63, 324}, {76, 1953}, {91, 39113}, {92, 343}, {95, 1087}, {99, 2618}, {158, 52347}, {163, 15415}, {216, 1969}, {264, 44706}, {274, 21011}, {305, 2181}, {310, 21807}, {313, 18180}, {321, 17167}, {326, 13450}, {328, 51801}, {336, 39569}, {662, 18314}, {668, 21102}, {799, 12077}, {811, 6368}, {850, 2617}, {1225, 2216}, {1273, 2166}, {1393, 3596}, {1502, 2179}, {1577, 14570}, {1625, 20948}, {1928, 40981}, {1930, 17500}, {1959, 53245}, {2167, 45793}, {2290, 20573}, {2600, 46405}, {3199, 40364}, {4592, 23290}, {6063, 7069}, {6369, 35174}, {7017, 44708}, {8800, 33808}, {13157, 18750}, {14208, 35360}, {18156, 27364}, {20565, 35194}, {20879, 31610}, {23999, 35442}, {24037, 41221}, {32680, 41078}, {33805, 52945}, {40703, 53174}, {41586, 46277}
X(54034) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 311}, {6, 28706}, {115, 15415}, {206, 5}, {512, 41221}, {1084, 18314}, {1147, 52347}, {1209, 1225}, {3162, 324}, {5139, 23290}, {11597, 1273}, {15259, 13450}, {15261, 27364}, {17423, 6368}, {22391, 343}, {32664, 14213}, {34116, 39113}, {36033, 18695}, {38986, 2618}, {38996, 12077}, {40368, 51}, {40369, 40981}, {40588, 45793}, {46604, 25043}
X(54034) = cevapoint of X(i) and X(j) for these (i,j): {32, 14575}, {184, 52435}, {34952, 41213}
X(54034) = trilinear pole of line {3049, 19627}
X(54034) = crossdifference of every pair of points on line {12077, 18314}
X(54034) = barycentric product X(i)*X(j) for these {i,j}: {1, 2148}, {3, 8882}, {4, 14533}, {6, 54}, {19, 2169}, {25, 97}, {31, 2167}, {32, 95}, {47, 2168}, {48, 2190}, {50, 1141}, {53, 46089}, {64, 33629}, {76, 14573}, {96, 571}, {98, 41270}, {107, 46088}, {110, 2623}, {112, 23286}, {115, 14587}, {163, 2616}, {184, 275}, {186, 11077}, {248, 19189}, {251, 16030}, {252, 2965}, {276, 14575}, {288, 13366}, {393, 19210}, {512, 18315}, {523, 14586}, {570, 1166}, {577, 8884}, {604, 44687}, {647, 933}, {654, 36078}, {661, 36134}, {924, 32692}, {1157, 14579}, {1298, 1971}, {1400, 35196}, {1501, 34384}, {1576, 15412}, {1974, 34386}, {1988, 26887}, {1990, 46090}, {1993, 41271}, {2081, 46966}, {2383, 52968}, {2501, 15958}, {2963, 25044}, {3049, 18831}, {3051, 39287}, {6748, 20574}, {8794, 23606}, {8795, 14585}, {8901, 23357}, {9247, 40440}, {10311, 51444}, {13622, 40633}, {14371, 51936}, {14642, 38808}, {16035, 41890}, {16813, 39201}, {19306, 51804}, {19627, 46138}, {34385, 52436}, {34394, 51275}, {34395, 51268}, {34396, 42300}, {40352, 43768}, {41331, 41488}, {43753, 43917}, {50463, 52418}
X(54034) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 28706}, {6, 311}, {25, 324}, {31, 14213}, {32, 5}, {48, 18695}, {50, 1273}, {51, 45793}, {54, 76}, {95, 1502}, {97, 305}, {184, 343}, {228, 42698}, {275, 18022}, {276, 44161}, {512, 18314}, {523, 15415}, {560, 1953}, {570, 1225}, {571, 39113}, {577, 52347}, {669, 12077}, {798, 2618}, {933, 6331}, {1084, 41221}, {1141, 20573}, {1501, 51}, {1576, 14570}, {1917, 2179}, {1918, 21011}, {1919, 21102}, {1974, 53}, {1976, 53245}, {2148, 75}, {2167, 561}, {2168, 20571}, {2169, 304}, {2179, 1087}, {2190, 1969}, {2205, 21807}, {2206, 17167}, {2207, 13450}, {2211, 39569}, {2489, 23290}, {2616, 20948}, {2623, 850}, {3049, 6368}, {3202, 41480}, {8882, 264}, {8884, 18027}, {8901, 23962}, {9233, 40981}, {9247, 44706}, {9407, 52945}, {9447, 7069}, {9697, 21230}, {11077, 328}, {14270, 41078}, {14533, 69}, {14567, 41586}, {14573, 6}, {14574, 1625}, {14575, 216}, {14585, 5562}, {14586, 99}, {14587, 4590}, {14600, 53174}, {15412, 44173}, {15958, 4563}, {16030, 8024}, {18315, 670}, {19189, 44132}, {19210, 3926}, {19627, 1154}, {23286, 3267}, {25044, 7769}, {27369, 27371}, {32692, 46134}, {33581, 13157}, {33629, 14615}, {34384, 40362}, {34386, 40050}, {34394, 33529}, {34395, 33530}, {34397, 14918}, {35196, 28660}, {36078, 46405}, {36134, 799}, {36417, 14569}, {39287, 40016}, {40373, 217}, {40981, 36412}, {41270, 325}, {41271, 5392}, {44077, 467}, {44162, 3199}, {44687, 28659}, {46088, 3265}, {46089, 34386}, {46288, 17500}, {46680, 27356}, {52435, 52032}, {52436, 52}, {52438, 5891}, {53059, 27364}
X(54034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {54, 97, 16030}, {54, 8883, 3}, {184, 571, 3135}, {23606, 34396, 11402}

X(54035) = X(2)X(29069)∩X(9)X(1746)

Barycentrics    a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 + a^5*c + a^4*b*c - 2*a^3*b^2*c + a*b^4*c - b^5*c + a^4*c^2 - 2*a^3*b*c^2 + 2*a^2*b^2*c^2 - a*b^3*c^2 - a^3*c^3 - a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 - b*c^5 : :
X(54035) = 2 X[72] + X[10454], X[3869] + 2 X[44039]

X(54035) lies on the cubic K and these lines: {2, 29069}, {9, 1746}, {37, 10478}, {40, 1089}, {72, 10454}, {75, 21363}, {92, 21361}, {101, 28950}, {165, 29057}, {190, 21375}, {192, 9535}, {312, 1764}, {321, 573}, {329, 21078}, {515, 5692}, {516, 3971}, {517, 3175}, {572, 26223}, {1215, 10434}, {1490, 30266}, {1699, 29054}, {1766, 21376}, {2051, 28606}, {2292, 50037}, {3219, 13478}, {3718, 10437}, {3731, 10888}, {3869, 22022}, {3882, 20928}, {4415, 19542}, {4647, 9548}, {4656, 10445}, {8680, 28609}, {9554, 21333}, {10246, 19722}, {10440, 29347}, {10444, 30568}, {10446, 41839}, {10465, 19582}, {13244, 32853}, {17134, 28997}, {17861, 28387}, {18229, 24336}, {18750, 21362}, {20927, 29418}, {29311, 32915}, {29369, 37521}, {35635, 41229}

X(54035) = reflection of X(32860) in X(10440)
X(54035) = {X(190),X(23512)}-harmonic conjugate of X(21375)

X(54036) = X(2)X(6030)∩X(30)X(54)

Barycentrics    7*a^6 - a^4*b^2 - 4*a^2*b^4 - 2*b^6 - a^4*c^2 - a^2*b^2*c^2 + 2*b^4*c^2 - 4*a^2*c^4 + 2*b^2*c^4 - 2*c^6 : :
X(54036) = 2 X[2] - 3 X[6030], 2 X[3521] - 5 X[8718], 2 X[69] - 5 X[41464], 4 X[550] - X[16835], 4 X[597] - 5 X[1176], 2 X[1657] + X[43599], 5 X[2916] - 3 X[21358], 3 X[3524] - 2 X[18488], X[3529] + 2 X[44866], X[5059] + 2 X[34563], 4 X[14641] - X[43846], 5 X[15321] - 8 X[34573], 3 X[15689] - X[33541], X[17800] + 2 X[43585], 2 X[22948] - 5 X[52093], X[49139] + 2 X[53779]

X(54036) lies on these lines: {2, 6030}, {20, 45185}, {22, 38397}, {30, 54}, {69, 41464}, {110, 52397}, {376, 11487}, {550, 16835}, {597, 1176}, {1657, 43599}, {2916, 21358}, {3524, 18488}, {3529, 44866}, {3534, 11459}, {5012, 48901}, {5059, 34563}, {5064, 15080}, {6636, 18553}, {7500, 15019}, {7714, 10545}, {9019, 15531}, {9544, 48896}, {14641, 43846}, {14927, 23293}, {15055, 48368}, {15321, 34573}, {15681, 32139}, {15686, 18442}, {15689, 33541}, {17800, 43585}, {20063, 33749}, {22948, 52093}, {26881, 48905}, {31383, 41462}, {49139, 53779}
X(54036) = midpoint of X(15681) and X(52100)
X(54036) = reflection of X(i) in X(j) for these {i,j}: {15062, 376}, {18442, 15686}

X(54037) = X(2)X(5655)∩X(69)X(146)

Barycentrics    a^2*(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 - a^10*b^2*c^2 + 7*a^8*b^4*c^2 - 24*a^6*b^6*c^2 + 25*a^4*b^8*c^2 - 7*a^2*b^10*c^2 - b^12*c^2 - 4*a^10*c^4 + 7*a^8*b^2*c^4 + 19*a^6*b^4*c^4 - 14*a^4*b^6*c^4 - 17*a^2*b^8*c^4 + 9*b^10*c^4 + 5*a^8*c^6 - 24*a^6*b^2*c^6 - 14*a^4*b^4*c^6 + 40*a^2*b^6*c^6 - 7*b^8*c^6 + 25*a^4*b^2*c^8 - 17*a^2*b^4*c^8 - 7*b^6*c^8 - 5*a^4*c^10 - 7*a^2*b^2*c^10 + 9*b^4*c^10 + 4*a^2*c^12 - b^2*c^12 - c^14) : :
X(54037) = 2 X[12162] + X[14094], 2 X[146] + X[12219], X[146] + 2 X[12825], X[12219] - 4 X[12825], 5 X[3091] - 4 X[12099], 4 X[3819] - 3 X[15055], 3 X[3839] - 2 X[45237], 11 X[5056] - 8 X[16270], 2 X[5609] + X[18439], X[5889] - 4 X[38791], 4 X[5907] - X[15054], 4 X[5972] - 3 X[20791], 2 X[17853] - 3 X[20791], 4 X[6053] - X[12270], and many others

X(54037) lies on these lines: {2, 5655}, {54, 7527}, {69, 146}, {74, 5891}, {110, 2071}, {113, 5890}, {323, 45019}, {378, 399}, {403, 7722}, {541, 11459}, {542, 15305}, {1154, 7728}, {1597, 52124}, {1986, 6623}, {2393, 10296}, {2777, 2979}, {2854, 51023}, {3091, 12099}, {3543, 14984}, {3819, 15055}, {3839, 45237}, {5056, 16270}, {5609, 18439}, {5622, 19140}, {5642, 15072}, {5889, 38791}, {5907, 15054}, {5972, 17853}, {6053, 12270}, {6241, 16534}, {9140, 15030}, {9818, 12308}, {9970, 37784}, {10575, 15034}, {10706, 13754}, {10733, 32062}, {10990, 11444}, {11206, 46349}, {11440, 16219}, {11451, 36518}, {11455, 17702}, {11793, 15021}, {11806, 14845}, {12111, 12827}, {12168, 32063}, {12273, 13202}, {12284, 46686}, {12290, 30714}, {12292, 14683}, {12317, 18537}, {14677, 44324}, {14855, 15035}, {15020, 46850}, {15027, 45958}, {15056, 20417}, {15057, 44321}, {15058, 16003}, {15738, 18909}, {15760, 21357}, {38727, 44299}, {41614, 51941}, {43808, 45959}

X(54037) = reflection of X(i) in X(j) for these {i,j}: {74, 5891}, {5890, 113}, {9140, 15030}, {10733, 32062}, {14677, 44324}, {15072, 5642}, {17853, 5972}, {20126, 15060}
X(54037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {146, 12825, 12219}, {5972, 17853, 20791}

X(54038) = X(2)X(34146)∩X(54)X(64)

Barycentrics    a^2*(a^14*b^2 - 3*a^12*b^4 + a^10*b^6 + 5*a^8*b^8 - 5*a^6*b^10 - a^4*b^12 + 3*a^2*b^14 - b^16 + a^14*c^2 - a^12*b^2*c^2 + 7*a^10*b^4*c^2 - 15*a^8*b^6*c^2 - a^6*b^8*c^2 + 17*a^4*b^10*c^2 - 7*a^2*b^12*c^2 - b^14*c^2 - 3*a^12*c^4 + 7*a^10*b^2*c^4 + 8*a^8*b^4*c^4 + 6*a^6*b^6*c^4 - 11*a^4*b^8*c^4 - 13*a^2*b^10*c^4 + 6*b^12*c^4 + a^10*c^6 - 15*a^8*b^2*c^6 + 6*a^6*b^4*c^6 - 10*a^4*b^6*c^6 + 17*a^2*b^8*c^6 + b^10*c^6 + 5*a^8*c^8 - a^6*b^2*c^8 - 11*a^4*b^4*c^8 + 17*a^2*b^6*c^8 - 10*b^8*c^8 - 5*a^6*c^10 + 17*a^4*b^2*c^10 - 13*a^2*b^4*c^10 + b^6*c^10 - a^4*c^12 - 7*a^2*b^2*c^12 + 6*b^4*c^12 + 3*a^2*c^14 - b^2*c^14 - c^16) : :
X(54038) = 2 X[9914] - 5 X[11444], X[12111] + 2 X[46373]

X(54038) lies on these lines: {2, 34146}, {20, 36982}, {54, 64}, {69, 41736}, {110, 1619}, {1899, 12294}, {5878, 5907}, {7391, 41738}, {9914, 11444}, {11459, 15311}, {12058, 41735}, {12085, 34966}, {12111, 46373}, {13567, 32125}, {17811, 34778}

X(54039) = X(2)X(5656)∩X(54)X(1593)

Barycentrics    a^2*(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 + a^10*b^2*c^2 + 8*a^8*b^4*c^2 - 38*a^6*b^6*c^2 + 41*a^4*b^8*c^2 - 11*a^2*b^10*c^2 - 2*b^12*c^2 - 4*a^10*c^4 + 8*a^8*b^2*c^4 + 52*a^6*b^4*c^4 - 36*a^4*b^6*c^4 - 32*a^2*b^8*c^4 + 12*b^10*c^4 + 5*a^8*c^6 - 38*a^6*b^2*c^6 - 36*a^4*b^4*c^6 + 78*a^2*b^6*c^6 - 9*b^8*c^6 + 41*a^4*b^2*c^8 - 32*a^2*b^4*c^8 - 9*b^6*c^8 - 5*a^4*c^10 - 11*a^2*b^2*c^10 + 12*b^4*c^10 + 4*a^2*c^12 - 2*b^2*c^12 - c^14) : :
X(54039) = X[12290] + 2 X[12315], 4 X[64] - 7 X[15056], 2 X[6225] + X[12111], X[6225] + 2 X[36982], X[12111] - 4 X[36982], 4 X[1498] - X[12279], 2 X[1498] + X[36983], X[12279] + 2 X[36983], 8 X[2883] - 5 X[10574], 4 X[7729] - 5 X[10574], 5 X[3522] - 2 X[30443], 4 X[5878] - X[5889], 8 X[6759] - 5 X[52093], X[11412] + 2 X[48672], 5 X[11439] - 2 X[12324], 5 X[11444] - 2 X[12250], 2 X[13093] - 5 X[15058], 8 X[23328] - 9 X[33879]

X(54039) lies on these lines: {2, 5656}, {4, 45088}, {54, 1593}, {64, 15056}, {69, 6225}, {110, 1498}, {235, 6241}, {1503, 15531}, {1619, 13445}, {2781, 50973}, {2883, 7729}, {2979, 15311}, {3522, 30443}, {5878, 5889}, {5895, 44668}, {6759, 52093}, {6776, 11381}, {6823, 18439}, {7691, 9914}, {9968, 41744}, {10996, 11487}, {11412, 48672}, {11439, 12324}, {11444, 12250}, {11455, 12022}, {12086, 18882}, {12174, 43812}, {13093, 15058}, {13380, 43766}, {15683, 34750}, {19149, 52028}, {23328, 33879}, {41468, 46373}

X(54039) = reflection of X(i) in X(j) for these {i,j}: {7729, 2883}, {15072, 5656}, {15683, 34750}
X(54039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1498, 36983, 12279}, {6225, 36982, 12111}

X(54040) = X(2)X(16657)∩X(20)X(64)

Barycentrics    2*a^10 - 5*a^8*b^2 + 4*a^6*b^4 - 2*a^4*b^6 + 2*a^2*b^8 - b^10 - 5*a^8*c^2 + 18*a^6*b^2*c^2 - 10*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 10*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 - 2*a^4*c^6 - 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(54040) = 5 X[3] - 2 X[12370], 5 X[12022] - 4 X[12370], 2 X[20] + X[14516], 4 X[550] - X[34224], 4 X[548] - X[44076], 4 X[1216] - X[18560], 2 X[1657] + X[16659], 2 X[1885] - 5 X[11444], 2 X[5562] + X[52071], 5 X[3091] - 6 X[35283], 5 X[3522] - 2 X[6146], 7 X[3523] - 4 X[12241], X[3529] + 2 X[12134], X[5059] + 2 X[16655], and manyu others

X(54040) lies on these lines: {2, 16657}, {3, 3580}, {4, 15066}, {5, 37477}, {20, 64}, {30, 2979}, {74, 550}, {141, 7527}, {185, 5965}, {343, 2071}, {376, 44665}, {378, 37636}, {394, 44440}, {511, 38323}, {539, 14855}, {548, 44076}, {858, 23325}, {1216, 18560}, {1368, 50435}, {1657, 16659}, {1885, 11444}, {1906, 43614}, {2777, 5562}, {2883, 17847}, {2888, 6247}, {3091, 35283}, {3146, 16654}, {3410, 37944}, {3522, 6146}, {3523, 12241}, {3529, 12134}, {3564, 15072}, {3917, 52069}, {5059, 16655}, {5654, 40112}, {5656, 11441}, {5889, 31829}, {5890, 41628}, {6240, 10625}, {6815, 14853}, {6823, 34148}, {7387, 40914}, {7495, 11430}, {7503, 26156}, {7544, 53023}, {7555, 34153}, {7998, 34664}, {8550, 41617}, {9306, 47096}, {9833, 33524}, {10018, 38793}, {10182, 51394}, {10295, 37478}, {10304, 12024}, {10323, 12118}, {10564, 37118}, {10605, 45794}, {10627, 12358}, {10996, 14912}, {11245, 20791}, {11424, 38317}, {11442, 21312}, {11454, 44683}, {11464, 16618}, {11585, 14644}, {12225, 15644}, {12290, 31831}, {12293, 47528}, {12302, 12359}, {12412, 13564}, {13142, 15043}, {13160, 13346}, {13348, 21659}, {13352, 14389}, {13391, 38321}, {13434, 38110}, {13488, 15056}, {13754, 44458}, {14643, 15761}, {15035, 34351}, {15053, 41588}, {15068, 32111}, {15107, 37458}, {15360, 44273}, {15760, 43574}, {16063, 18396}, {16621, 49135}, {16656, 50691}, {16977, 18466}, {18400, 36987}, {18405, 37444}, {18430, 47341}, {21167, 37126}, {23293, 47090}, {23332, 44450}, {33878, 37196}, {34799, 50693}, {35471, 37486}, {35473, 44201}, {37475, 37644}, {41171, 43576}, {43575, 44682}, {44245, 45731}, {45970, 46853}

X(54040) = reflection of X(i) in X(j) for these {i,j}: {3146, 16654}, {12022, 3}, {41628, 5890}, {52069, 3917}, {52397, 36987}
X(54040) = anticomplement of X(16657)

X(54041) = X(2)X(14845)∩X(3)X(54)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 11*a^4*b^2*c^2 + 9*a^2*b^4*c^2 + b^6*c^2 - 3*a^4*c^4 + 9*a^2*b^2*c^4 + 3*a^2*c^6 + b^2*c^6 - c^8) : :
X(54041) = 2 X[3] + X[2979], 10 X[3] - X[5889], 4 X[3] - X[5890], 7 X[3] + 2 X[6101], 11 X[3] - 2 X[6102], 14 X[3] - 5 X[10574], 5 X[3] + 4 X[10627], 8 X[3] + X[11412], 13 X[3] - 4 X[13630], 5 X[2979] + X[5889], 2 X[2979] + X[5890], 7 X[2979] - 4 X[6101], 11 X[2979] + 4 X[6102], 7 X[2979] + 5 X[10574], 5 X[2979] - 8 X[10627], and many others

X(54041) lies on these lines: {2, 14845}, {3, 54}, {4, 3819}, {5, 44299}, {20, 5447}, {30, 7998}, {51, 631}, {52, 15717}, {140, 9781}, {154, 10323}, {184, 44832}, {185, 21735}, {186, 1974}, {373, 15709}, {376, 3917}, {389, 10299}, {511, 3524}, {548, 12111}, {549, 3060}, {550, 11444}, {568, 12100}, {569, 45308}, {1173, 15805}, {1199, 13347}, {1216, 3522}, {1370, 41171}, {1657, 15056}, {2393, 10519}, {2781, 15035}, {3523, 3567}, {3525, 6688}, {3526, 11592}, {3528, 5562}, {3529, 11793}, {3530, 15043}, {3533, 10110}, {3534, 15067}, {3538, 18950}, {3543, 10170}, {3545, 5650}, {4550, 37944}, {5054, 5640}, {5055, 33879}, {5067, 13598}, {5446, 10303}, {5651, 37925}, {5663, 15688}, {5876, 52093}, {5899, 10546}, {5907, 17538}, {5943, 15702}, {5946, 15693}, {6243, 15712}, {6636, 11464}, {6800, 43572}, {7485, 15033}, {7492, 51393}, {7512, 11202}, {7514, 41462}, {7525, 11449}, {7699, 51392}, {7731, 15051}, {8703, 15072}, {8718, 32063}, {9730, 15692}, {9818, 21766}, {10263, 15028}, {10304, 13754}, {10539, 16661}, {10691, 12022}, {11001, 15030}, {11002, 15708}, {11188, 50977}, {11414, 43598}, {11439, 15704}, {11561, 15042}, {11591, 12279}, {11704, 37452}, {12082, 17811}, {12088, 44082}, {12162, 50693}, {12220, 47090}, {12281, 38726}, {12283, 48876}, {12284, 41673}, {13321, 15707}, {13352, 15246}, {13363, 15701}, {13451, 15713}, {14094, 41463}, {14128, 17800}, {14156, 52300}, {14157, 15066}, {14810, 41716}, {14831, 15715}, {15036, 38446}, {15053, 37494}, {15060, 15681}, {15080, 22115}, {15081, 44321}, {15683, 16194}, {15698, 16836}, {15719, 21969}, {15759, 45956}, {16063, 25739}, {16192, 31738}, {16386, 35254}, {16976, 18438}, {17504, 40280}, {17834, 43597}, {18378, 33542}, {18436, 33923}, {18439, 44245}, {18859, 33533}, {21734, 40647}, {33524, 33543}, {34783, 46853}, {35921, 37480}, {37481, 44682}, {37498, 43651}, {37913, 43586}, {40916, 44413}, {44870, 49138}, {45958, 49137}, {46849, 49140}, {46852, 50690}

X(54041) = midpoint of X(i) and X(j) for these {i,j}: {2979, 20791}, {10304, 33884}
X(54041) = reflection of X(i) in X(j) for these {i,j}: {3545, 5650}, {5640, 5054}, {5890, 20791}, {15045, 3524}, {20791, 3}, {40280, 17504}
X(54041) = anticomplement of X(14845)
X(54041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 2979, 5890}, {3, 6101, 10574}, {3, 10627, 5889}, {20, 5447, 7999}, {20, 5891, 11455}, {20, 7999, 15058}, {376, 3917, 11459}, {549, 13340, 3060}, {550, 11444, 12290}, {550, 44324, 18435}, {1216, 3522, 6241}, {1657, 32142, 15056}, {2979, 5890, 11412}, {3523, 10625, 3567}, {3530, 37484, 15043}, {3534, 15067, 15305}, {3819, 13348, 36987}, {3819, 36987, 4}, {5446, 10303, 11465}, {5891, 11455, 15058}, {7485, 37483, 15033}, {7999, 11455, 5891}, {8703, 23039, 15072}, {10263, 15720, 15028}, {11591, 15696, 12279}, {15066, 35243, 14157}, {18435, 44324, 11444}

X(54042) = X(2)X(13340)∩X(3)X(54)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 8*a^4*b^2*c^2 + 6*a^2*b^4*c^2 + b^6*c^2 - 3*a^4*c^4 + 6*a^2*b^2*c^4 + 3*a^2*c^6 + b^2*c^6 - c^8) : :
X(54042) = 3 X[13340] + 2 X[13364], 7 X[3] - X[5889], 3 X[3] - X[5890], 2 X[3] + X[6101], 4 X[3] - X[6102], 11 X[3] - 5 X[10574], X[3] + 2 X[10627], 5 X[3] + X[11412], 5 X[3] - 2 X[13630], 5 X[3] - 3 X[20791], 7 X[2979] + X[5889], 3 X[2979] + X[5890], 4 X[2979] + X[6102], 11 X[2979] + 5 X[10574], 5 X[2979] - X[11412], and many others

X(54042) lies on these lines: {2, 13340}, {3, 54}, {4, 11017}, {5, 3819}, {20, 11591}, {30, 3917}, {51, 140}, {52, 3530}, {141, 43129}, {143, 631}, {156, 10323}, {185, 33923}, {206, 1511}, {323, 44832}, {373, 10124}, {376, 5663}, {378, 33533}, {381, 7998}, {382, 7999}, {389, 15712}, {511, 549}, {547, 5650}, {548, 5562}, {550, 1216}, {567, 15246}, {568, 3524}, {632, 5446}, {1092, 5944}, {1350, 6644}, {1370, 34514}, {1498, 33543}, {1539, 44440}, {1656, 44299}, {1657, 11444}, {1658, 43652}, {1994, 13339}, {2393, 48876}, {2889, 12325}, {3060, 5054}, {3146, 45958}, {3313, 52262}, {3522, 18436}, {3523, 6243}, {3525, 32205}, {3526, 10095}, {3528, 34783}, {3529, 32137}, {3534, 11459}, {3538, 18951}, {3567, 13421}, {3627, 11793}, {3628, 14845}, {3845, 10170}, {5070, 18874}, {5073, 15056}, {5188, 44221}, {5453, 50597}, {5462, 14869}, {5495, 16287}, {5499, 37536}, {5640, 15694}, {5907, 15704}, {5943, 11539}, {6030, 43572}, {6636, 22115}, {7484, 39522}, {7512, 32171}, {7514, 37483}, {7516, 37498}, {7530, 17811}, {7555, 51393}, {7771, 51440}, {7811, 51383}, {8681, 50978}, {8703, 13754}, {9019, 44287}, {9703, 15080}, {9729, 44682}, {9730, 12100}, {9781, 46219}, {10113, 13416}, {10264, 17710}, {10282, 44544}, {10298, 38898}, {10303, 16982}, {10519, 44441}, {10575, 31834}, {10982, 13154}, {11002, 15702}, {11064, 25337}, {11245, 43934}, {11439, 49137}, {11561, 15051}, {11624, 41943}, {11626, 41944}, {11812, 21969}, {12041, 41673}, {12083, 15066}, {12099, 34128}, {12103, 12162}, {12108, 14449}, {12111, 15696}, {12236, 18438}, {12308, 33544}, {13201, 15040}, {13321, 15701}, {13336, 36153}, {13358, 38728}, {13490, 29181}, {13565, 52295}, {13624, 31737}, {14216, 42021}, {14540, 48366}, {14541, 48365}, {14810, 18475}, {14831, 14891}, {14881, 52042}, {14915, 15686}, {14929, 51386}, {15033, 37496}, {15045, 15693}, {15058, 17800}, {15068, 32063}, {15072, 15688}, {15101, 34153}, {15122, 44201}, {15305, 15681}, {15330, 38793}, {15532, 52104}, {15606, 40647}, {15684, 16261}, {15692, 40280}, {15703, 33879}, {15713, 21849}, {15717, 37481}, {15721, 16981}, {15760, 51391}, {16226, 41983}, {16241, 36978}, {16242, 36980}, {16836, 17504}, {17538, 18439}, {17714, 44082}, {18281, 43653}, {18451, 33532}, {18570, 37480}, {20299, 21230}, {21243, 21357}, {23329, 44668}, {23332, 34826}, {31663, 31738}, {31805, 31836}, {31829, 34798}, {31831, 34750}, {31884, 34513}, {32139, 37198}, {32196, 32348}, {32609, 34006}, {33542, 47748}, {33699, 46847}, {34380, 40673}, {34579, 52926}, {35921, 37477}, {37126, 37495}, {37471, 45308}, {37814, 46728}, {37936, 43586}, {39504, 51360}, {44241, 45118}, {45759, 45956}, {46029, 51392}, {47751, 52100}

X(54042) = midpoint of X(i) and X(j) for these {i,j}: {2, 13340}, {3, 2979}, {20, 18435}, {51, 10625}, {376, 23039}, {1657, 11455}, {3534, 11459}, {3819, 15644}, {5562, 14855}, {5891, 36987}, {15305, 15681}
X(54042) = reflection of X(i) in X(j) for these {i,j}: {5, 3819}, {51, 140}, {2979, 10627}, {3060, 13363}, {3819, 5447}, {3845, 10170}, {5446, 6688}, {5891, 44324}, {5946, 549}, {6101, 2979}, {9730, 12100}, {10263, 51}, {11455, 45959}, {13451, 10124}, {13491, 14855}, {14855, 548}, {15060, 15067}, {15067, 3917}, {16226, 41983}, {18435, 11591}, {33699, 46847}
X(54042) = anticomplement of X(13364)
X(54042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6101, 6102}, {3, 10627, 6101}, {3, 11412, 13630}, {140, 10263, 15026}, {140, 10625, 10263}, {143, 11592, 631}, {376, 33884, 23039}, {382, 7999, 14128}, {548, 5562, 13491}, {550, 1216, 5876}, {631, 37484, 143}, {1092, 7525, 5944}, {1216, 13348, 550}, {1657, 11444, 45959}, {2979, 20791, 11412}, {3060, 5054, 13363}, {3523, 6243, 12006}, {3917, 5891, 44324}, {3917, 36987, 5891}, {5447, 15644, 5}, {5891, 44324, 15067}, {10124, 13451, 373}, {15033, 41462, 54006}, {31834, 44245, 10575}, {34553, 34555, 6101}, {37496, 54006, 15033}

X(54043) = X(4)X(15066)∩X(2979)X(33971)

Barycentrics    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 + 2*a^8*b^2*c^2 + 22*a^6*b^4*c^2 - 24*a^4*b^6*c^2 + 3*a^2*b^8*c^2 - 2*b^10*c^2 - 2*a^8*c^4 + 22*a^6*b^2*c^4 - 26*a^4*b^4*c^4 - 2*a^2*b^6*c^4 + 8*b^8*c^4 + 2*a^6*c^6 - 24*a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 12*b^6*c^6 + a^4*c^8 + 3*a^2*b^2*c^8 + 8*b^4*c^8 - a^2*c^10 - 2*b^2*c^10 : :

X(54043) lies on these lines: {4, 15066}, {2979, 33971}

X(54044) = X(3)X(54)∩X(5)X(11592)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 14*a^4*b^2*c^2 + 12*a^2*b^4*c^2 + b^6*c^2 - 3*a^4*c^4 + 12*a^2*b^2*c^4 + 3*a^2*c^6 + b^2*c^6 - c^8) : :
X(54044) = 3 X[3] + X[2979], 13 X[3] - X[5889], 5 X[3] - X[5890], 5 X[3] + X[6101], 7 X[3] - X[6102], 17 X[3] - 5 X[10574], 2 X[3] + X[10627], 11 X[3] + X[11412], 4 X[3] - X[13630], 7 X[3] - 3 X[20791], 13 X[2979] + 3 X[5889], 5 X[2979] + 3 X[5890], 5 X[2979] - 3 X[6101], 7 X[2979] + 3 X[6102], 17 X[2979] + 15 X[10574], and many others

X(54044) lies on the cubic K and these lines: {3, 54}, {5, 11592}, {20, 14128}, {30, 3819}, {51, 549}, {52, 44682}, {140, 6688}, {143, 3530}, {343, 20379}, {373, 15713}, {376, 15067}, {381, 44299}, {511, 12100}, {548, 5447}, {550, 5891}, {568, 15692}, {631, 10095}, {632, 14845}, {1216, 33923}, {1368, 20304}, {1511, 6636}, {2781, 14810}, {3060, 15693}, {3098, 19136}, {3522, 5876}, {3523, 10263}, {3524, 5946}, {3526, 18874}, {3528, 13491}, {3534, 7998}, {3538, 18952}, {3845, 5650}, {3917, 5663}, {5054, 11451}, {5446, 12108}, {5482, 11277}, {5562, 46853}, {5640, 15701}, {5907, 44245}, {5943, 11812}, {6030, 32609}, {6243, 10299}, {7525, 11202}, {7731, 15042}, {7999, 15696}, {9730, 17504}, {10109, 15082}, {10193, 44668}, {10304, 23039}, {10625, 12006}, {11002, 15719}, {11459, 15688}, {11793, 12103}, {11801, 44321}, {13321, 15718}, {13416, 44249}, {13570, 47478}, {13598, 16239}, {13754, 34200}, {14093, 15072}, {14641, 41981}, {14869, 32205}, {14891, 16836}, {14915, 15690}, {15026, 15720}, {15030, 15686}, {15036, 38898}, {15045, 15700}, {15246, 37477}, {15305, 15689}, {15685, 16261}, {15698, 40280}, {15704, 32062}, {15714, 45956}, {15717, 37484}, {16168, 47509}, {16194, 19710}, {17811, 33532}, {18324, 31884}, {18436, 21735}, {19708, 33884}, {19709, 33879}, {19711, 21969}, {21734, 34783}, {21849, 44580}, {22115, 44832}, {28190, 52796}, {32171, 43652}, {32416, 40670}, {34584, 44458}, {37472, 45308}, {40111, 44108}, {45118, 47114}

X(54044) = midpoint of X(i) and X(j) for these {i,j}: {5, 36987}, {376, 15067}, {548, 44324}, {550, 5891}, {3534, 15060}, {3917, 8703}, {5890, 6101}, {5892, 15644}, {5946, 13340}, {7691, 44325}, {15030, 15686}, {15704, 32062}, {16194, 19710}
X(54044) = reflection of X(i) in X(j) for these {i,j}: {143, 5892}, {5891, 32142}, {5892, 3530}, {5943, 11812}, {11591, 44324}, {11801, 44321}, {13363, 549}, {13364, 140}, {16836, 14891}, {32062, 45958}, {44324, 5447}, {45959, 5891}
X(54044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 10627, 13630}, {548, 5447, 11591}, {550, 32142, 45959}, {3524, 13340, 5946}, {3530, 15644, 143}, {3534, 7998, 15060}, {10625, 12006, 13421}, {10625, 15712, 12006}, {11793, 12103, 32137}, {14869, 45186, 32205}, {34553, 34555, 11412}

X(54045) = X(51)X(47065)∩X(184)X(47064)

Barycentrics    a^2*(a^18*b^2 - 7*a^16*b^4 + 22*a^14*b^6 - 42*a^12*b^8 + 56*a^10*b^10 - 56*a^8*b^12 + 42*a^6*b^14 - 22*a^4*b^16 + 7*a^2*b^18 - b^20 + a^18*c^2 - 13*a^16*b^2*c^2 + 44*a^14*b^4*c^2 - 64*a^12*b^6*c^2 + 40*a^10*b^8*c^2 + 4*a^8*b^10*c^2 - 32*a^6*b^12*c^2 + 36*a^4*b^14*c^2 - 21*a^2*b^16*c^2 + 5*b^18*c^2 - 7*a^16*c^4 + 44*a^14*b^2*c^4 - 78*a^12*b^4*c^4 + 45*a^10*b^6*c^4 + 7*a^8*b^8*c^4 - 19*a^6*b^10*c^4 + 7*a^4*b^12*c^4 + 6*a^2*b^14*c^4 - 5*b^16*c^4 + 22*a^14*c^6 - 64*a^12*b^2*c^6 + 45*a^10*b^4*c^6 - 9*a^8*b^6*c^6 + 9*a^6*b^8*c^6 - 21*a^4*b^10*c^6 + 38*a^2*b^12*c^6 - 20*b^14*c^6 - 42*a^12*c^8 + 40*a^10*b^2*c^8 + 7*a^8*b^4*c^8 + 9*a^6*b^6*c^8 - 30*a^2*b^10*c^8 + 70*b^12*c^8 + 56*a^10*c^10 + 4*a^8*b^2*c^10 - 19*a^6*b^4*c^10 - 21*a^4*b^6*c^10 - 30*a^2*b^8*c^10 - 98*b^10*c^10 - 56*a^8*c^12 - 32*a^6*b^2*c^12 + 7*a^4*b^4*c^12 + 38*a^2*b^6*c^12 + 70*b^8*c^12 + 42*a^6*c^14 + 36*a^4*b^2*c^14 + 6*a^2*b^4*c^14 - 20*b^6*c^14 - 22*a^4*c^16 - 21*a^2*b^2*c^16 - 5*b^4*c^16 + 7*a^2*c^18 + 5*b^2*c^18 - c^20) : :
X(54045) = 2 X[51] - 3 X[47065], 4 X[1141] - X[13505], X[13504] + 2 X[38587], X[11412] + 2 X[38683], 3 X[20791] - 4 X[38618]

X(54045) lies on these lines: {51, 47065}, {184, 47064}, {1141, 1303}, {1154, 13504}, {2979, 25150}, {11202, 34418}, {11412, 38683}, {14073, 44324}, {18435, 32423}, {20791, 38618}

X(54045) = reflection of X(i) in X(j) for these {i,j}: {5890, 1141}, {13505, 5890}, {14073, 44324}

X(54046) = X(3)X(54)∩X(23)X(114)

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

X(54046) lies on these lines: {3, 54}, {23, 114}, {684, 1510}, {930, 32428}, {1291, 1297}, {3518, 16336}, {5965, 14652}, {10594, 35718}, {12107, 14140}, {14981, 37183}, {17714, 18355}, {18875, 44890}

X(54046) = circumcircle-inverse of X(2979)
X(54046) = crossdifference of every pair of points on line {7755, 12077}

X(54047) = X(2)X13451)∩X(3)X(54)

Barycentrics    a^2*(2*a^6*b^2 - 6*a^4*b^4 + 6*a^2*b^6 - 2*b^8 + 2*a^6*c^2 - 13*a^4*b^2*c^2 + 9*a^2*b^4*c^2 + 2*b^6*c^2 - 6*a^4*c^4 + 9*a^2*b^2*c^4 + 6*a^2*c^6 + 2*b^2*c^6 - 2*c^8) : :
X(54047) = 7 X[2] - 4 X[13451], X[3] + 2 X[2979], 11 X[3] - 2 X[5889], 5 X[3] - 2 X[5890], 5 X[3] + 4 X[6101], 13 X[3] - 4 X[6102], 19 X[3] - 10 X[10574], X[3] + 8 X[10627], 7 X[3] + 2 X[11412], 17 X[3] - 8 X[13630], 3 X[3] - 2 X[20791], 11 X[2979] + X[5889], 5 X[2979] + X[5890], 5 X[2979] - 2 X[6101], 13 X[2979] + 2 X[6102], 19 X[2979] + 5 X[10574], and many others

X(54047) lies on these lines: {2, 13451}, {3, 54}, {4, 44324}, {30, 33884}, {51, 3526}, {154, 13564}, {381, 3917}, {382, 5891}, {399, 35243}, {511, 5054}, {568, 15693}, {1216, 1657}, {1350, 2070}, {1656, 3819}, {1986, 15042}, {2781, 32609}, {3060, 15694}, {3098, 22115}, {3534, 6000}, {3830, 15067}, {3843, 7999}, {3851, 32142}, {5055, 7998}, {5070, 13364}, {5073, 11444}, {5079, 45186}, {5562, 15696}, {5663, 15689}, {5892, 6243}, {5899, 15066}, {5907, 49137}, {5943, 15723}, {5946, 15701}, {6636, 9703}, {7484, 15038}, {7492, 40111}, {7514, 37496}, {7545, 17811}, {9730, 15700}, {9914, 32063}, {10263, 11451}, {10303, 14449}, {10620, 41673}, {11002, 11539}, {11455, 11591}, {11459, 15681}, {11465, 16982}, {11592, 15043}, {11935, 14810}, {13348, 14855}, {13421, 15028}, {13754, 15688}, {15045, 15707}, {15058, 49134}, {15060, 15684}, {15072, 15695}, {15305, 15685}, {15606, 34783}, {15706, 40280}, {15709, 16981}, {15716, 16836}, {17538, 31834}, {19708, 45956}, {21766, 39522}, {23327, 44751}, {32062, 49136}, {34380, 43934}, {35434, 46847}, {37486, 43809}, {43957, 45967}, {45959, 49139}, {46114, 52300}

X(54047) = midpoint of X(10625) and X(14845)
X(54047) = reflection of X(i) in X(j) for these {i,j}: {5055, 7998}, {11002, 11539}, {13321, 5054}, {14845, 3819}, {45967, 43957}
X(54047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1216, 36987, 18435}, {2979, 5890, 6101}, {3917, 13340, 381}, {5447, 37484, 3526}, {13364, 44299, 5070}, {18435, 36987, 1657}

X(54048) = X(2)X13321)∩X(3)X(54)

Barycentrics    a^2*(2*a^6*b^2 - 6*a^4*b^4 + 6*a^2*b^6 - 2*b^8 + 2*a^6*c^2 - 7*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + 2*b^6*c^2 - 6*a^4*c^4 + 3*a^2*b^2*c^4 + 6*a^2*c^6 + 2*b^2*c^6 - 2*c^8) : :
X(54048) = 4 X[2] - 3 X[13321], 3 X[2] - 4 X[44324], 9 X[13321] - 16 X[44324], 5 X[3] - 2 X[5889], 3 X[3] - 2 X[5890], X[3] - 4 X[6101], 7 X[3] - 4 X[6102], 13 X[3] - 10 X[10574], 5 X[3] - 8 X[10627], X[3] + 2 X[11412], 11 X[3] - 8 X[13630], 7 X[3] - 6 X[20791], X[195] - 4 X[41590], 5 X[2979] - X[5889], 3 X[2979] - X[5890], 7 X[2979] - 2 X[6102], and many others

X(54048) lies on these lines: {2, 13321}, {3, 54}, {4, 15108}, {6, 54006}, {20, 44748}, {22, 50461}, {49, 46728}, {51, 1216}, {52, 3526}, {69, 31723}, {143, 5070}, {154, 2937}, {155, 13564}, {159, 399}, {323, 7502}, {378, 37496}, {381, 511}, {382, 5562}, {389, 15720}, {394, 2070}, {547, 11002}, {549, 33884}, {568, 3917}, {1350, 18445}, {1351, 15038}, {1482, 31738}, {1657, 5925}, {2393, 11898}, {3060, 5055}, {3090, 14449}, {3146, 31834}, {3313, 39899}, {3519, 18381}, {3525, 16881}, {3534, 13340}, {3567, 32142}, {3830, 11459}, {3843, 11591}, {3851, 10263}, {5071, 13451}, {5072, 5446}, {5073, 5876}, {5076, 5907}, {5079, 11793}, {5447, 14531}, {5640, 15703}, {5650, 15723}, {5663, 15681}, {5899, 15068}, {5946, 7998}, {6090, 51519}, {6800, 34006}, {7485, 15037}, {7509, 14627}, {7512, 9704}, {7545, 33586}, {7555, 9544}, {7556, 40111}, {7574, 34118}, {8681, 51175}, {9019, 50955}, {9641, 11189}, {9730, 15693}, {9781, 13421}, {10170, 21969}, {10304, 45956}, {10564, 35495}, {10606, 18859}, {10620, 21312}, {11202, 22115}, {11381, 49133}, {11403, 33539}, {11649, 15533}, {11935, 18475}, {12111, 17800}, {12162, 49136}, {12290, 49139}, {12308, 13201}, {12325, 18356}, {12702, 31737}, {13169, 14984}, {13432, 21660}, {13512, 20477}, {14070, 32609}, {14130, 37498}, {14269, 15060}, {14791, 45794}, {14831, 15700}, {14855, 15644}, {15030, 38335}, {15033, 33533}, {15040, 41673}, {15045, 15701}, {15047, 37493}, {15072, 15689}, {15305, 15684}, {15706, 16836}, {16261, 35403}, {16644, 36979}, {16645, 36981}, {17538, 45957}, {17834, 45735}, {17853, 38788}, {18350, 44082}, {18438, 18536}, {18439, 49137}, {18451, 37924}, {18534, 41716}, {21230, 52295}, {21357, 39504}, {21850, 50135}, {24474, 31816}, {31180, 38724}, {32139, 47748}, {32359, 34785}, {33542, 37198}, {34864, 36747}, {35264, 37956}, {37347, 48876}, {41597, 44108}

X(54048) = midpoint of X(i) and X(j) for these {i,j}: {2979, 11412}, {18435, 37484}
X(54048) = reflection of X(i) in X(j) for these {i,j}: {3, 2979}, {51, 1216}, {52, 3819}, {381, 23039}, {382, 18435}, {568, 3917}, {2979, 6101}, {3060, 15067}, {3534, 13340}, {3819, 15606}, {3830, 11459}, {5073, 11455}, {6243, 51}, {11455, 5876}, {14855, 15644}, {15684, 15305}, {18435, 5562}, {21969, 10170}, {34783, 14855}
X(54048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12160, 43845}, {3, 12316, 7592}, {143, 7999, 5070}, {323, 7502, 9703}, {394, 37494, 2070}, {568, 3917, 5054}, {1216, 6243, 1656}, {3060, 15067, 5055}, {3567, 32142, 46219}, {5071, 16981, 13451}, {5447, 14531, 37481}, {5562, 37484, 382}, {5889, 10627, 3}, {5946, 7998, 15694}, {6101, 11412, 3}, {10263, 11444, 3851}, {10625, 18436, 1657}, {15644, 34783, 15696}

X(54049) = ISOGONALCONJUGATE OF X(25149)

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

X(54049) lies on the circumcircle and these lines: {2, 35591}, {3, 15907}, {4, 33333}, {55, 44053}, {930, 1510}, {1141, 1154}, {7691, 14979}, {11671, 52110}, {12219, 39431}, {12226, 53959}, {24772, 32142}, {30481, 33643}, {46966, 52603}

X(54049) = reflection of X(i) in X(j) for these {i,j}: {4, 33333}, {11671, 52110}, {15907, 3}, {24772, 32142}
X(54049) = isogonal conjugate of X(25149)
X(54049) = anticomplement of X(35591)
X(54049) = isogonal conjugate of the anticomplement of X(25149)
X(54049) = isogonal conjugate of the complement of X(25149)
X(54049) = Thomson-isogonal conjugate of X(25150)
X(54049) = Collings transform of X(i) for these i: {1154, 1510, 32142, 33333}
X(54049) = X(1)-isoconjugate of X(25149)
X(54049) = cevapoint of X(i) and X(j) for these (i,j): {526, 32142}, {1154, 1510}
X(54049) = trilinear pole of line {6, 39018}
X(54049) = barycentric product X(18315)*X(38899)
X(54049) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 25149}, {35324, 24147}, {38899, 18314}

X(54050) = X(2)X(10606)∩X(4)X(74)

Barycentrics    5*a^10 - 5*a^8*b^2 - 14*a^6*b^4 + 22*a^4*b^6 - 7*a^2*b^8 - b^10 - 5*a^8*c^2 + 36*a^6*b^2*c^2 - 22*a^4*b^4*c^2 - 12*a^2*b^6*c^2 + 3*b^8*c^2 - 14*a^6*c^4 - 22*a^4*b^2*c^4 + 38*a^2*b^4*c^4 - 2*b^6*c^4 + 22*a^4*c^6 - 12*a^2*b^2*c^6 - 2*b^4*c^6 - 7*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(54050) = 3 X[2] - 4 X[23328], 3 X[10606] - 2 X[23328], 4 X[3] - X[6225], 2 X[3] + X[12250], 4 X[3] - 3 X[35260], 2 X[5656] - 3 X[35260], X[6225] + 2 X[12250], X[6225] - 3 X[35260], 2 X[12250] + 3 X[35260], X[4] - 4 X[3357], 5 X[4] - 8 X[20299], X[4] + 2 X[20427], 3 X[4] - 4 X[23325], 5 X[3357] - 2 X[20299], 2 X[3357] + X[20427], and many others

X(54050) lies on the cubic K1327 and these lines: {2, 10606}, {3, 5656}, {4, 74}, {5, 40920}, {20, 64}, {30, 32064}, {140, 48672}, {146, 11598}, {154, 10304}, {185, 14912}, {376, 3917}, {378, 11427}, {459, 34170}, {541, 5654}, {548, 12315}, {550, 13093}, {631, 5878}, {962, 12262}, {1073, 27089}, {1092, 43813}, {1370, 13445}, {1498, 3522}, {1593, 14853}, {1853, 3543}, {1885, 18913}, {1992, 2781}, {2071, 37669}, {2883, 3523}, {2935, 37645}, {3090, 22802}, {3091, 5895}, {3146, 3580}, {3147, 11468}, {3183, 14249}, {3426, 37458}, {3524, 10182}, {3528, 6759}, {3529, 14216}, {3545, 23329}, {3564, 34622}, {3618, 7527}, {3619, 12379}, {3832, 40686}, {3839, 23332}, {4232, 37487}, {4293, 10060}, {4294, 10076}, {4297, 9899}, {5056, 5893}, {5067, 25563}, {5218, 12940}, {5889, 31978}, {5907, 30443}, {6001, 9778}, {6241, 18925}, {6353, 21663}, {6459, 49251}, {6460, 49250}, {6624, 51892}, {6640, 38789}, {6815, 15062}, {7288, 12950}, {7395, 40918}, {7487, 16654}, {7493, 11454}, {7503, 15740}, {7714, 32062}, {8549, 41617}, {8703, 32063}, {8718, 44679}, {8780, 47114}, {9545, 46374}, {9833, 17538}, {9914, 17928}, {9919, 12106}, {10192, 15692}, {10193, 15702}, {10282, 21735}, {10385, 32065}, {10605, 11433}, {10620, 18917}, {10991, 48991}, {11001, 18400}, {11202, 19708}, {11250, 12412}, {11430, 35483}, {11440, 37201}, {11441, 53050}, {11444, 36982}, {11456, 35485}, {11541, 34786}, {12022, 18909}, {12111, 30552}, {12163, 18934}, {12358, 18439}, {12825, 27082}, {12964, 42638}, {12970, 42637}, {13568, 53023}, {14118, 40913}, {14530, 33923}, {14826, 44241}, {15072, 25406}, {15138, 16063}, {15438, 22528}, {15640, 50709}, {15704, 34780}, {15717, 16252}, {16253, 51358}, {16658, 18533}, {16775, 29181}, {17813, 51028}, {17821, 21734}, {17835, 37644}, {18381, 33703}, {18396, 49670}, {20125, 25564}, {20791, 41580}, {22467, 40914}, {23049, 51538}, {23291, 44438}, {23324, 50687}, {23326, 52028}, {26944, 43719}, {31305, 34801}, {32111, 35486}, {32346, 34938}, {32423, 34350}, {33522, 35513}, {34286, 35711}, {34782, 50693}, {34944, 40196}, {35864, 42261}, {35865, 42260}, {36201, 51023}, {36876, 40664}, {37196, 41584}, {37197, 43903}, {37200, 46034}, {38282, 51403}, {41362, 49135}, {49138, 52102}

X(54050) = midpoint of X(i) and X(j) for these {i,j}: {5656, 12250}, {5925, 18405}
X(54050) = reflection of X(i) in X(j) for these {i,j}: {2, 10606}, {146, 15131}, {3146, 18405}, {3543, 1853}, {5656, 3}, {6225, 5656}, {11206, 376}, {15131, 11598}, {18405, 6247}, {32063, 8703}, {51028, 17813}
X(54050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5656, 35260}, {3, 12250, 6225}, {4, 74, 18931}, {4, 18931, 37643}, {20, 64, 12324}, {64, 5894, 20}, {550, 13093, 34781}, {1885, 34469, 18913}, {2883, 8567, 3523}, {3357, 20427, 4}, {5895, 6696, 3091}, {5907, 30443, 36983}, {5925, 6247, 3146}, {6225, 35260, 5656}, {40686, 51491, 3832}

X(54051) = X(2)X(515)∩X(20)X(78)

Barycentrics    5*a^7 - 7*a^6*b - 7*a^5*b^2 + 13*a^4*b^3 - a^3*b^4 - 5*a^2*b^5 + 3*a*b^6 - b^7 - 7*a^6*c + 2*a^5*b*c - a^4*b^2*c - 4*a^3*b^3*c + 7*a^2*b^4*c + 2*a*b^5*c + b^6*c - 7*a^5*c^2 - a^4*b*c^2 + 10*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - 3*a*b^4*c^2 + 3*b^5*c^2 + 13*a^4*c^3 - 4*a^3*b*c^3 - 2*a^2*b^2*c^3 - 4*a*b^3*c^3 - 3*b^4*c^3 - a^3*c^4 + 7*a^2*b*c^4 - 3*a*b^2*c^4 - 3*b^3*c^4 - 5*a^2*c^5 + 2*a*b*c^5 + 3*b^2*c^5 + 3*a*c^6 + b*c^6 - c^7 : :
X(54051) = 4 X[3] - X[9799], X[8] - 4 X[11500], X[20] + 2 X[1490], 2 X[20] + X[6223], 4 X[1490] - X[6223], 2 X[72] + X[9960], 2 X[84] - 5 X[3522], 4 X[548] - X[12684], 4 X[550] - X[12246], 5 X[631] - 2 X[5787], 5 X[631] - 8 X[40262], X[5787] - 4 X[40262], X[962] - 4 X[6261], X[3146] - 4 X[6260], 7 X[3523] - 4 X[6245], 7 X[3528] - 4 X[34862], and many others

X(54051) lies on the cubic K1327 and these lines: {1, 50700}, {2, 515}, {3, 5273}, {4, 4313}, {7, 18446}, {8, 411}, {20, 78}, {30, 5658}, {40, 20007}, {72, 9960}, {84, 3219}, {101, 27382}, {355, 6988}, {376, 971}, {548, 12684}, {550, 12246}, {551, 38150}, {631, 5787}, {912, 28610}, {934, 5932}, {936, 4297}, {938, 944}, {962, 6261}, {997, 43161}, {1006, 18230}, {1012, 36991}, {1060, 3160}, {1064, 4344}, {1210, 13462}, {1385, 6864}, {1750, 4304}, {1807, 36640}, {1895, 18283}, {2287, 7415}, {2800, 34632}, {2947, 22350}, {3146, 6260}, {3485, 6253}, {3487, 20420}, {3488, 19541}, {3523, 6245}, {3528, 34862}, {3529, 6259}, {3601, 37434}, {3616, 6835}, {3868, 9942}, {3876, 12664}, {3940, 5759}, {4305, 5691}, {4323, 21740}, {5049, 7967}, {5080, 6836}, {5126, 5704}, {5175, 6838}, {5328, 6827}, {5435, 5768}, {5550, 6991}, {5603, 8236}, {5720, 6987}, {5734, 40257}, {5748, 6840}, {5758, 6869}, {5811, 6868}, {5815, 17857}, {5817, 16418}, {5842, 9812}, {5927, 11111}, {6001, 9778}, {6256, 6895}, {6705, 15717}, {6764, 22770}, {6796, 7688}, {6828, 18242}, {6831, 10592}, {6849, 24299}, {6855, 18480}, {6865, 18481}, {6880, 31188}, {6894, 40259}, {6904, 10884}, {6909, 10430}, {6918, 34773}, {6985, 12536}, {6986, 12114}, {7971, 20070}, {7992, 12512}, {8166, 18527}, {8726, 17580}, {9776, 18444}, {9910, 33524}, {9948, 35242}, {10304, 52027}, {10580, 22753}, {12528, 12671}, {15704, 48664}, {18391, 44425}, {18525, 52265}, {18623, 46974}, {19067, 43511}, {19068, 43512}, {22792, 33703}, {26062, 35979}, {28381, 48923}, {30332, 37000}, {42637, 49235}, {42638, 49234}, {50702, 52676}

X(54051) = reflection of X(2) in X(52026)
X(54051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 33597, 5703}, {20, 1490, 6223}, {936, 4297, 37423}, {944, 3149, 938}, {5720, 6987, 18228}, {5768, 6905, 5435}, {5787, 40262, 631}, {6869, 37700, 5758}, {18446, 50701, 7}, {34772, 50695, 962}

X(54052) = X(7)X(104)∩X(8)X(20)

Barycentrics    5*a^7 - 3*a^6*b - 11*a^5*b^2 + 5*a^4*b^3 + 7*a^3*b^4 - a^2*b^5 - a*b^6 - b^7 - 3*a^6*c + 18*a^5*b*c - a^4*b^2*c - 12*a^3*b^3*c + 3*a^2*b^4*c - 6*a*b^5*c + b^6*c - 11*a^5*c^2 - a^4*b*c^2 + 10*a^3*b^2*c^2 - 2*a^2*b^3*c^2 + a*b^4*c^2 + 3*b^5*c^2 + 5*a^4*c^3 - 12*a^3*b*c^3 - 2*a^2*b^2*c^3 + 12*a*b^3*c^3 - 3*b^4*c^3 + 7*a^3*c^4 + 3*a^2*b*c^4 + a*b^2*c^4 - 3*b^3*c^4 - a^2*c^5 - 6*a*b*c^5 + 3*b^2*c^5 - a*c^6 + b*c^6 - c^7 : :
X(54052) = 4 X[3] - X[6223], 2 X[3] + X[12246], X[6223] + 2 X[12246], X[4] - 4 X[34862], X[8] - 4 X[1158], X[20] + 2 X[84], 2 X[20] + X[9799], 4 X[84] - X[9799], X[10864] + 2 X[31730], 4 X[140] - X[48664], 2 X[550] + X[12684], 5 X[631] - 2 X[6259], X[962] - 4 X[12114], X[1320] + 2 X[52116], 2 X[1490] - 5 X[3522], 7 X[3090] - 4 X[22792], and many others

X(54052) lies on the cubic K1327 and these lines: {2, 21164}, {3, 5658}, {4, 5435}, {7, 104}, {8, 20}, {21, 10309}, {30, 5770}, {90, 10429}, {140, 48664}, {144, 6282}, {329, 6909}, {376, 971}, {377, 7705}, {516, 34625}, {517, 14646}, {550, 12684}, {631, 6259}, {962, 12114}, {997, 5732}, {1071, 4313}, {1320, 52116}, {1490, 3522}, {1699, 3086}, {1709, 4293}, {1737, 53056}, {1768, 18391}, {2800, 3241}, {2829, 14647}, {2950, 12648}, {3090, 22792}, {3091, 6705}, {3146, 6245}, {3306, 15239}, {3358, 5825}, {3421, 17613}, {3523, 6260}, {3529, 5787}, {3576, 50742}, {3600, 12705}, {3616, 5450}, {3868, 18238}, {3869, 17649}, {3876, 18239}, {3877, 5731}, {4294, 10085}, {4297, 7992}, {4304, 9819}, {4305, 15071}, {4308, 12672}, {4311, 7995}, {4652, 37421}, {5129, 37526}, {5218, 12678}, {5226, 6935}, {5273, 6916}, {5328, 37822}, {5550, 12608}, {5657, 37429}, {5703, 6906}, {5744, 6925}, {5748, 6966}, {5766, 18446}, {5768, 6938}, {5790, 31775}, {5804, 26877}, {5815, 10310}, {5818, 40267}, {5887, 9960}, {6256, 9780}, {6459, 49235}, {6460, 49234}, {6912, 9776}, {6969, 31188}, {6987, 7171}, {7288, 12679}, {8158, 28174}, {8726, 11106}, {9809, 48695}, {9841, 37423}, {9859, 14872}, {9910, 17928}, {9965, 38460}, {10164, 18250}, {10167, 11111}, {10304, 52026}, {10591, 52860}, {10884, 17576}, {10916, 28158}, {11037, 11496}, {11246, 22760}, {12528, 51379}, {12650, 20070}, {12666, 12671}, {12676, 26129}, {16418, 21151}, {21165, 37427}, {21735, 40262}, {24477, 34742}, {24929, 36996}, {25005, 37435}, {26927, 28029}, {30282, 41561}, {35844, 42261}, {35845, 42260}, {37600, 41706}

X(54052) = midpoint of X(5658) and X(12246)
X(54052) = reflection of X(i) in X(j) for these {i,j}: {2, 52027}, {5658, 3}, {6223, 5658}
X(54052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12246, 6223}, {20, 84, 9799}, {1012, 2096, 7}

X(54053) = X(3)X(42452)∩X(20)X(394)

Barycentrics    11*a^16 - 12*a^14*b^2 - 96*a^12*b^4 + 276*a^10*b^6 - 290*a^8*b^8 + 124*a^6*b^10 - 8*a^4*b^12 - 4*a^2*b^14 - b^16 - 12*a^14*c^2 + 208*a^12*b^2*c^2 - 276*a^10*b^4*c^2 - 232*a^8*b^6*c^2 + 460*a^6*b^8*c^2 - 96*a^4*b^10*c^2 - 44*a^2*b^12*c^2 - 8*b^14*c^2 - 96*a^12*c^4 - 276*a^10*b^2*c^4 + 1044*a^8*b^4*c^4 - 584*a^6*b^6*c^4 - 312*a^4*b^8*c^4 + 156*a^2*b^10*c^4 + 68*b^12*c^4 + 276*a^10*c^6 - 232*a^8*b^2*c^6 - 584*a^6*b^4*c^6 + 832*a^4*b^6*c^6 - 108*a^2*b^8*c^6 - 184*b^10*c^6 - 290*a^8*c^8 + 460*a^6*b^2*c^8 - 312*a^4*b^4*c^8 - 108*a^2*b^6*c^8 + 250*b^8*c^8 + 124*a^6*c^10 - 96*a^4*b^2*c^10 + 156*a^2*b^4*c^10 - 184*b^6*c^10 - 8*a^4*c^12 - 44*a^2*b^2*c^12 + 68*b^4*c^12 - 4*a^2*c^14 - 8*b^2*c^14 - c^16 : :
X(54053) = X[4] - 4 X[20329], X[20] + 2 X[3346], 4 X[550] - X[36965], X[3146] - 4 X[33546], 2 X[3183] - 5 X[3522], 7 X[3523] - 4 X[6523], 11 X[5056] - 8 X[51342]

X(54053) lies on the cubic K1327 and these lines: {3, 42452}, {4, 20329}, {20, 394}, {253, 1294}, {376, 15312}, {550, 36965}, {3146, 33546}, {3183, 3522}, {3523, 6523}, {3543, 10714}, {5056, 51342}, {10304, 51877}

X(54053) = reflection of X(42452) in X(3)

X(54054) = X(20)X(78)∩X(189)X(972)

Barycentrics    5*a^12 - 2*a^11*b - 24*a^10*b^2 + 10*a^9*b^3 + 45*a^8*b^4 - 20*a^7*b^5 - 40*a^6*b^6 + 20*a^5*b^7 + 15*a^4*b^8 - 10*a^3*b^9 + 2*a*b^11 - b^12 - 2*a^11*c + 40*a^10*b*c - 6*a^9*b^2*c - 64*a^8*b^3*c - 20*a^7*b^4*c + 16*a^6*b^5*c + 52*a^5*b^6*c - 10*a^3*b^8*c + 8*a^2*b^9*c - 14*a*b^10*c - 24*a^10*c^2 - 6*a^9*b*c^2 + 38*a^8*b^2*c^2 + 40*a^7*b^3*c^2 - 8*a^6*b^4*c^2 - 52*a^5*b^5*c^2 + 4*a^4*b^6*c^2 + 8*a^3*b^7*c^2 - 16*a^2*b^8*c^2 + 10*a*b^9*c^2 + 6*b^10*c^2 + 10*a^9*c^3 - 64*a^8*b*c^3 + 40*a^7*b^2*c^3 + 64*a^6*b^3*c^3 - 20*a^5*b^4*c^3 - 56*a^3*b^6*c^3 + 26*a*b^8*c^3 + 45*a^8*c^4 - 20*a^7*b*c^4 - 8*a^6*b^2*c^4 - 20*a^5*b^3*c^4 - 38*a^4*b^4*c^4 + 68*a^3*b^5*c^4 + 16*a^2*b^6*c^4 - 28*a*b^7*c^4 - 15*b^8*c^4 - 20*a^7*c^5 + 16*a^6*b*c^5 - 52*a^5*b^2*c^5 + 68*a^3*b^4*c^5 - 16*a^2*b^5*c^5 + 4*a*b^6*c^5 - 40*a^6*c^6 + 52*a^5*b*c^6 + 4*a^4*b^2*c^6 - 56*a^3*b^3*c^6 + 16*a^2*b^4*c^6 + 4*a*b^5*c^6 + 20*b^6*c^6 + 20*a^5*c^7 + 8*a^3*b^2*c^7 - 28*a*b^4*c^7 + 15*a^4*c^8 - 10*a^3*b*c^8 - 16*a^2*b^2*c^8 + 26*a*b^3*c^8 - 15*b^4*c^8 - 10*a^3*c^9 + 8*a^2*b*c^9 + 10*a*b^2*c^9 - 14*a*b*c^10 + 6*b^2*c^10 + 2*a*c^11 - c^12 : :
X(54054) = X[20] + 2 X[3345], X[3146] - 4 X[47441], 2 X[3182] - 5 X[3522]

X(54054) lies on the cubic K1327 and these lines: {20, 78}, {189, 972}, {3146, 47441}, {3182, 3522}, {4313, 44696}, {9778, 11206}

X(54055) = X(2)X(52053)∩X(20)X(3413)

Barycentrics    3*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 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 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(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)] - 4*(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]*S + 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)*S : :
X(54055) = 5 X[20] - 2 X[42411], X[20] + 2 X[42412], X[42411] + 5 X[42412], 5 X[3522] - 2 X[40851], X[5059] + 2 X[40852], 3 X[10304] - 2 X[39162]

X(54055) lies on the cubic K1327 and these lines: {2, 52053}, {20, 3413}, {30, 32443}, {376, 39158}, {3522, 40851}, {3543, 39163}, {5059, 40852}, {10304, 39162}

X(54055) = reflection of X(i) in X(j) for these {i,j}: {{2, 52053}, {3543, 39163}, {39158, 376}, {54056, 20}

X(54056) = X(2)X(52054)∩X(20)X(3413)

Barycentrics    3*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 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 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(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)] + 4*(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]*S - 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)*S : :
X(54056) = X[20] + 2 X[42411], 5 X[20] - 2 X[42412], 5 X[42411] + X[42412], 5 X[3522] - 2 X[40852], X[5059] + 2 X[40851], 3 X[10304] - 2 X[39163]

fX(54056) lies on the cubic K1327 and these lines: {2, 52054}, {20, 3413}, {30, 39158}, {376, 39159}, {3522, 40852}, {3543, 39162}, {5059, 40851}, {10304, 39163}

X(54056) = reflection of X(i) in X(j) for these {i,j}: {{2, 52054}, {3543, 39162}, {39159, 376}, {54055, 20}

X(54057) = CIRCUMCIRCLE-INVERSE OF X(250)

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

See HG130623.

X(54057) lies on the cubic K630 and these lines: {3, 250}, {5, 23582}, {249, 1092}, {1968, 23964}, {4230, 5502}, {7750, 18020}, {23109, 39299}, {23110, 39298}

X(54057) = circumcircle inverse of X(250)
X(54057) = X(110)-Ceva conjugate of X(250)
X(54057) = X(i)-isoconjugate of X(j) for these (i,j): {{125, 9390}, {523, 9392}, {3708, 15351}}
X(54057) = X(648)-Dao conjugate of X(850)
X(54057) = barycentric product X(i)*X(j) for these {i,j}: {{110, 39062}, {250, 39352}, {662, 2633}}
X(54057) = barycentric quotient X(i)/X(j) for these {i,j}: {{163, 9392}, {250, 15351}, {2629, 20902}, {2633, 1577}, {19208, 53576}, {39062, 850}, {39352, 339}}

X(54058) = CIRCUMCIRCLE-INVERSE OF X(48)

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

X(54058) lies on these lines: {3, 48}, {101, 2253}, {104, 35182}, {163, 2074}, {652, 663}, {1172, 1731}, {1319, 32660}, {2176, 36033}, {2249, 53925}, {2323, 14963}, {3215, 9310}, {39006, 52407}

X(54058) = circumcircle-inverse of X(48)
X(54058) = crossdifference of every pair of points on line {226, 7649}
X(54058) = barycentric product X(78)*X(5137)
X(54058) = barycentric quotient X(5137)/X(273)

X(54059) = CIRCUMCIRCLE-INVERSE OF X(63)

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

X(54059) lies on these lines: {3, 63}, {35, 5197}, {46, 7163}, {55, 6505}, {100, 914}, {103, 6099}, {501, 1800}, {662, 2073}, {1326, 4575}, {1364, 22128}, {1813, 2078}, {2071, 4511}, {3733, 8646}, {4184, 6061}, {5285, 35980}, {9436, 36003}, {14018, 47106}, {26702, 53952}

X(54059) = reflection of X(51629) in X(51632)
X(54059) = circumcircle-inverse of X(63)
X(54059) = X(43363)-Ceva conjugate of X(63)
X(54059) = crossdifference of every pair of points on line {6591, 16583}
X(54059) = barycentric product X(63)*X(37782)
X(54059) = barycentric quotient X(37782)/X(92)

X(54060) = CIRCUMCIRCLE-INVERSE OF X(66)

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

X(54060) lies on the cubic K039 and these lines: {3, 66}, {22, 5523}, {23, 37801}, {74, 46967}, {186, 1289}, {935, 37978}, {1176, 34137}, {5961, 40079}, {6091, 14909}, {6636, 18018}, {7488, 43678}, {7503, 51940}, {7512, 41377}, {9019, 10317}, {10316, 12220}, {13509, 15080}, {13754, 40080}, {14961, 15388}, {17407, 21213}, {27373, 44884}, {28405, 33802}

X(54060) = isogonal conjugate of X(11605)
X(54060) = circumcircle-inverse of X(66)
X(54060) = isogonal conjugate of the polar conjugate of X(37801)
X(54060) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11605}, {1760, 8791}, {2157, 17907}, {2172, 46105}, {37221, 40938}
X(54060) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11605}, {40583, 17907}
X(54060) = crossdifference of every pair of points on line {2485, 40938}
X(54060) = barycentric product X(i)*X(j) for these {i,j}: {3, 37801}, {23, 14376}, {66, 22151}, {2353, 37804}, {9019, 40404}, {9517, 44766}, {10317, 18018}
X(54060) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11605}, {23, 17907}, {66, 46105}, {2353, 8791}, {8744, 52448}, {9517, 33294}, {10317, 22}, {14376, 18019}, {18374, 8743}, {22151, 315}, {37801, 264}, {37804, 40073}, {42659, 2485}, {46765, 9076}
X(54060) = {X(3),X(2353)}-harmonic conjugate of X(14376)

X(54061) = CIRCUMCIRCLE-INVERSE OF X(68)

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

X(54061) lies on the cubic K039 and these lines: {3, 68}, {24, 16172}, {26, 8906}, {74, 46969}, {131, 50435}, {186, 925}, {847, 22467}, {2071, 5962}, {5392, 45170}, {5963, 7488}, {6644, 14593}, {11589, 15469}, {12095, 44174}, {12364, 13557}, {16178, 37951}, {27087, 32123}, {32734, 51393}, {37814, 46200}, {39986, 40082}, {45781, 52504}

X(54061) = circumcircle-inverse of X(68)
X(54061) = X(45781)-Ceva conjugate of X(68)
X(54061) = X(52557)-Dao conjugate of X(52505)
X(54061) = crossdifference of every pair of points on line {6753, 40939}
X(54061) = barycentric product X(37951)*X(52350)
X(54061) = barycentric quotient X(37951)/X(11547)

X(54062) = CIRCUMCIRCLE-INVERSE OF X(97)

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

X(54062) lies on these lines: {3, 54}, {23, 18315}, {110, 52887}, {323, 933}, {511, 15958}, {1291, 1298}, {1297, 46966}, {10625, 46089}, {11424, 16337}, {18350, 19552}, {34186, 43768}, {37477, 50463}

X(54062) = circumcircle-inverse of X(97)

X(54063) = CIRCUMCIRCLE-INVERSE OF X(116)

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

X(54063) lies on the tangential circle and these lines: {3, 116}, {22, 675}, {24, 917}, {25, 5190}, {199, 34429}, {2079, 47234}, {2931, 8053}, {10117, 20999}, {14729, 23399}, {16681, 39828}, {16873, 39857}, {19165, 23383}

X(54063) = circumcircle-inverse of X(116)
X(54063) = tangential-isogonal conjugate of X(8676)
X(54063) = X(46107)-Ceva conjugate of X(6)
X(54063) = X(32656)-Dao conjugate of X(1331)

X(54064) = CIRCUMCIRCLE-INVERSE OF X(119)

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

X(54064) lies on the tangential circle and these lines: {3, 119}, {22, 1295}, {23, 34550}, {24, 108}, {25, 25640}, {36, 1455}, {40, 2778}, {186, 47149}, {378, 10731}, {1512, 20989}, {2070, 38578}, {2791, 39857}, {2798, 39828}, {2804, 39200}, {2817, 11249}, {2823, 35238}, {2834, 6644}, {2845, 14703}, {2850, 2931}, {2851, 14657}, {2937, 38592}, {6087, 39478}, {6642, 6717}, {7488, 34188}, {7517, 33566}, {9570, 34456}, {10323, 38715}, {10715, 44837}, {12332, 52731}, {13558, 20832}, {13589, 18862}, {15177, 50917}, {17928, 38696}, {47270, 53761}

X(54064) = circumcircle-inverse of X(119)
X(54064) = Stammler-circle-inverse of X(38756)
X(54064) = tangential-isogonal conjugate of X(6001)

X(54065) = CIRCUMCIRCLE-INVERSE OF X(123)

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 - 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 + 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^2*b^2*c^3 + 2*a*b^3*c^3 - b^4*c^3 - a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 - b^3*c^4 + a^2*c^5 - 3*a*b*c^5 + b^2*c^5 + a*c^6 + b*c^6 - c^7) : :
X(54065) = 3 X[9909] - X[13222]

X(54065) lies on the tangential circle and these lines: {3, 119}, {11, 25}, {19, 8428}, {22, 100}, {23, 149}, {24, 104}, {26, 952}, {56, 1866}, {80, 8185}, {153, 7488}, {159, 5848}, {186, 12248}, {378, 10728}, {513, 10016}, {528, 9909}, {659, 14667}, {1145, 8193}, {1317, 8192}, {1387, 11365}, {1484, 37440}, {1593, 52836}, {1603, 17100}, {1610, 6224}, {1768, 3220}, {1993, 3045}, {1995, 31272}, {2070, 12773}, {2079, 21004}, {2217, 20832}, {2771, 2931}, {2783, 39828}, {2787, 39857}, {2800, 3556}, {2802, 49553}, {2828, 14703}, {2830, 14657}, {2831, 3185}, {2915, 23361}, {2932, 2933}, {2937, 12331}, {3032, 9571}, {3036, 9712}, {3517, 20418}, {4185, 9658}, {4186, 9672}, {4996, 11337}, {5020, 6667}, {5083, 22769}, {5096, 38472}, {5541, 9591}, {5840, 7387}, {5854, 12410}, {5899, 48680}, {6264, 9625}, {6326, 9626}, {6642, 6713}, {6644, 38602}, {7484, 31235}, {7502, 11698}, {7517, 10738}, {7526, 22799}, {7529, 23513}, {7530, 22938}, {8276, 13913}, {8277, 13977}, {8674, 10117}, {9570, 34458}, {9659, 12763}, {9673, 13274}, {9714, 26308}, {9715, 10830}, {10058, 13730}, {10090, 37034}, {10323, 34474}, {10711, 44837}, {10776, 46588}, {10831, 10834}, {11249, 51628}, {11414, 24466}, {12088, 13199}, {12329, 14740}, {12751, 15177}, {13205, 20872}, {13558, 23860}, {17928, 38693}, {18378, 51517}, {19459, 51198}, {20095, 37913}, {23304, 35973}, {34189, 48391}, {37123, 38657}, {37485, 51007}

X(54065) = midpoint of X(i) and X(j) for these {i,j}: {3, 9913}, {9798, 9912}
X(54065) = circumcircle-inverse of X(123)
X(54065) = tangential-isogonal conjugate of X(513)
X(54065) = X(4391)-Ceva conjugate of X(6)
X(54065) = X(1415)-Dao conjugate of X(651)
X(54065) = crossdifference of every pair of points on line {13006, 34977}

X(54066) = CIRCUMCIRCLE-INVERSE OF X(126)

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

X(54066) lies on the tangential circle and these lines: {3, 126}, {22, 99}, {23, 7665}, {24, 2374}, {25, 1560}, {157, 13558}, {159, 2930}, {186, 47170}, {1495, 5104}, {1995, 11643}, {2882, 35901}, {2931, 32121}, {2934, 15959}, {5938, 37928}, {8428, 14273}, {9909, 11641}, {14667, 18610}, {14729, 21284}, {16316, 37969}, {19165, 45687}, {34131, 47206}

X(54066) = reflection of X(3) in X(14655)
X(54066) = circumcircle-inverse of X(126)
X(54066) = isogonal conjugate of the isotomic conjugate of X(34518)
X(54066) = tangential-isogonal conjugate of X(2393)
X(54066) = X(44146)-Ceva conjugate of X(6)
X(54066) = X(14908)-Dao conjugate of X(895)
X(54066) = barycentric product X(6)*X(34518)
X(54066) = barycentric quotient X(34518)/X(76)

X(54067) = CIRCUMCIRCLE-INVERSE OF X(128)

Barycentrics    a^2*(a^20 - 5*a^18*b^2 + 9*a^16*b^4 - 6*a^14*b^6 + 6*a^6*b^14 - 9*a^4*b^16 + 5*a^2*b^18 - b^20 - 5*a^18*c^2 + 16*a^16*b^2*c^2 - 12*a^14*b^4*c^2 - 12*a^12*b^6*c^2 + 25*a^10*b^8*c^2 - 15*a^8*b^10*c^2 - 6*a^6*b^12*c^2 + 22*a^4*b^14*c^2 - 18*a^2*b^16*c^2 + 5*b^18*c^2 + 9*a^16*c^4 - 12*a^14*b^2*c^4 - 7*a^12*b^4*c^4 + 14*a^10*b^6*c^4 - 6*a^8*b^8*c^4 + 6*a^6*b^10*c^4 - 17*a^4*b^12*c^4 + 24*a^2*b^14*c^4 - 11*b^16*c^4 - 6*a^14*c^6 - 12*a^12*b^2*c^6 + 14*a^10*b^4*c^6 + 6*a^8*b^6*c^6 - 6*a^6*b^8*c^6 + 2*a^4*b^10*c^6 - 14*a^2*b^12*c^6 + 16*b^14*c^6 + 25*a^10*b^2*c^8 - 6*a^8*b^4*c^8 - 6*a^6*b^6*c^8 + 4*a^4*b^8*c^8 + 3*a^2*b^10*c^8 - 20*b^12*c^8 - 15*a^8*b^2*c^10 + 6*a^6*b^4*c^10 + 2*a^4*b^6*c^10 + 3*a^2*b^8*c^10 + 22*b^10*c^10 - 6*a^6*b^2*c^12 - 17*a^4*b^4*c^12 - 14*a^2*b^6*c^12 - 20*b^8*c^12 + 6*a^6*c^14 + 22*a^4*b^2*c^14 + 24*a^2*b^4*c^14 + 16*b^6*c^14 - 9*a^4*c^16 - 18*a^2*b^2*c^16 - 11*b^4*c^16 + 5*a^2*c^18 + 5*b^2*c^18 - c^20) : :

X(54067) lies on the tangential circle and these lines: {3, 128}, {4, 11587}, {22, 18401}, {24, 933}, {25, 18402}, {26, 53808}, {160, 2934}, {378, 44977}, {399, 2917}, {571, 2079}, {2070, 8157}, {3432, 14367}, {5899, 43919}, {6069, 12383}, {6644, 38616}, {7731, 38897}, {14657, 52036}, {14703, 44809}, {34131, 42731}

X(54067) = reflection of X(38585) in X(8157)
X(54067) = circumcircle-inverse of X(128)
X(54067) = tangential-isogonal conjugate of X(18400)
X(54067) = X(14918)-Ceva conjugate of X(6)

X(54068) = CIRCUMCIRCLE-INVERSE OF X(133)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^16 - 2*a^14*b^2 - 8*a^12*b^4 + 34*a^10*b^6 - 50*a^8*b^8 + 34*a^6*b^10 - 8*a^4*b^12 - 2*a^2*b^14 + b^16 - 2*a^14*c^2 + 17*a^12*b^2*c^2 - 31*a^10*b^4*c^2 - 6*a^8*b^6*c^2 + 64*a^6*b^8*c^2 - 59*a^4*b^10*c^2 + 17*a^2*b^12*c^2 - 8*a^12*c^4 - 31*a^10*b^2*c^4 + 108*a^8*b^4*c^4 - 98*a^6*b^6*c^4 + 28*a^4*b^8*c^4 + 9*a^2*b^10*c^4 - 8*b^12*c^4 + 34*a^10*c^6 - 6*a^8*b^2*c^6 - 98*a^6*b^4*c^6 + 78*a^4*b^6*c^6 - 24*a^2*b^8*c^6 + 16*b^10*c^6 - 50*a^8*c^8 + 64*a^6*b^2*c^8 + 28*a^4*b^4*c^8 - 24*a^2*b^6*c^8 - 18*b^8*c^8 + 34*a^6*c^10 - 59*a^4*b^2*c^10 + 9*a^2*b^4*c^10 + 16*b^6*c^10 - 8*a^4*c^12 + 17*a^2*b^2*c^12 - 8*b^4*c^12 - 2*a^2*c^14 + c^16) : :

X(54068) lies on the tangential circle and these lines: {3, 133}, {22, 5897}, {24, 64}, {25, 50937}, {186, 34178}, {2079, 47228}, {2935, 46587}, {3515, 13558}, {6644, 38624}, {14059, 45735}, {14703, 53255}, {15959, 44879}, {33582, 39857}, {37917, 47215}

X(54068) = circumcircle-inverse of X(133)
X(54068) = tangential isogonal conjugate of X(15311)
X(54068) = X(51358)-Ceva conjugate of X(6)

X(54069) = CIRCUMCIRCLE-INVERSE OF X(136)

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

X(54069) lies on the tangential circle and these lines: {3, 136}, {4, 14769}, {22, 3563}, {24, 110}, {25, 114}, {26, 15478}, {186, 47324}, {2070, 13557}, {2079, 47230}, {2493, 8428}, {3447, 15470}, {3515, 14703}, {7669, 15959}, {10132, 48792}, {10133, 48790}, {13558, 16230}, {14729, 47627}, {18127, 20957}, {19165, 21213}, {34131, 47200}, {37954, 44057}

X(54069) = reflection of X(39119) in X(135)
X(54069) = circumcircle-inverse of X(136)
X(54069) = polar-circle-inverse of X(14769)

X(54070) = CIRCUMCIRCLE-INVERSE OF X(142)

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

X(54070) lies on these lines: {3, 142}, {55, 5011}, {514, 23865}, {758, 23398}, {1308, 41341}, {1323, 1617}, {1324, 20875}, {1621, 5195}, {2175, 4253}, {2195, 3002}, {2942, 5527}, {3322, 5172}, {4251, 21746}, {4262, 23868}, {4314, 20836}, {5030, 17798}, {5088, 23407}, {7742, 53617}, {12651, 20838}, {20988, 36014}, {23850, 23852}, {34179, 40910}

X(54070) = circumcircle-inverse of X(142)
X(54070) = Stammler-circle-inverse of X(31671)
X(54070) = X(19624)-Dao conjugate of X(3935)
X(54070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 51621, 5144}, {20875, 23402, 1324}

X(54071) = CIRCUMCIRCLE-INVERSE OF X(147)

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

X(54071) lies on these lines: {2, 13558}, {3, 147}, {22, 14673}, {23, 3258}, {110, 3917}, {186, 2080}, {511, 19575}, {827, 1297}, {7496, 30789}, {7527, 44943}, {9479, 44822}, {13335, 22467}, {13372, 14652}, {15915, 19165}, {35375, 52128}, {37126, 46654}, {37978, 51458}

X(54071) = circumcircle-inverse of X(147)
X(54071) = 2nd-Brocard-circle-inverse of X(9863)

X(54072) = CIRCUMCIRCLE-INVERSE OF X(148)

Barycentrics    a^2*(a^8 - 3*a^6*b^2 + 3*a^2*b^6 - b^8 - 3*a^6*c^2 + 12*a^4*b^2*c^2 - 8*a^2*b^4*c^2 + 2*b^6*c^2 - 8*a^2*b^2*c^4 + b^4*c^4 + 3*a^2*c^6 + 2*b^2*c^6 - c^8) : :
X(54072) = 2 X[3] + X[14671]

X(54072) lies on these lines: {3, 148}, {23, 31655}, {111, 6636}, {186, 2971}, {3565, 5966}, {5092, 5622}, {5104, 5166}, {5940, 37940}, {7492, 33998}, {7496, 10163}, {7512, 15564}, {7527, 45151}, {22467, 34217}, {31843, 37126}, {44821, 53272}

X(54072) = midpoint of X(3) and X(15567)
X(54072) = reflection of X(14671) in X(15567)
X(54072) = circumcircle-inverse of X(148)

X(54073) = CIRCUMCIRCLE-INVERSE OF X(156)

Barycentrics    a^2*(a^2 - b^2 - c^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 + 6*a^8*b^2*c^2 - 5*a^6*b^4*c^2 + 3*a^4*b^6*c^2 - b^10*c^2 + 2*a^8*c^4 - 5*a^6*b^2*c^4 + a^4*b^4*c^4 - a^2*b^6*c^4 + 4*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 - 3*a^4*c^8 + 4*b^4*c^8 + a^2*c^10 - b^2*c^10) : :
X(54073) = 5 X[110] + X[43578]

X(54073) lies on these lines: {3, 74}, {5, 3043}, {49, 125}, {54, 20304}, {113, 18350}, {182, 32272}, {184, 15061}, {195, 12236}, {265, 1147}, {381, 15463}, {567, 23515}, {1092, 12121}, {1112, 13621}, {1209, 5972}, {1351, 38851}, {1568, 17702}, {1656, 12228}, {1658, 12219}, {1986, 45735}, {2072, 32423}, {2777, 10540}, {2914, 44802}, {2937, 20773}, {3044, 15535}, {3047, 10264}, {3292, 15091}, {3448, 6640}, {3548, 14683}, {3843, 15472}, {5012, 34128}, {5462, 32226}, {5504, 12902}, {6293, 13289}, {6593, 45973}, {6723, 13353}, {6759, 20127}, {7506, 19504}, {7542, 13392}, {7687, 37472}, {7722, 37814}, {7728, 10539}, {9306, 10254}, {9545, 15081}, {9652, 10081}, {9667, 10065}, {9703, 38724}, {9704, 13198}, {9705, 20379}, {9706, 20396}, {9934, 48672}, {10024, 10272}, {10113, 34148}, {10114, 43817}, {10628, 51393}, {11746, 14627}, {11806, 43844}, {11898, 19138}, {12038, 21650}, {12227, 37481}, {12292, 25487}, {12295, 37495}, {12383, 18404}, {12584, 18438}, {12893, 18436}, {13201, 26882}, {13293, 18439}, {13434, 15088}, {13754, 17701}, {14157, 34584}, {14708, 43809}, {15059, 32046}, {15087, 46430}, {15089, 36253}, {15140, 41714}, {16223, 43586}, {18441, 19381}, {18563, 34153}, {19122, 32249}, {19129, 32275}, {19154, 32244}, {21649, 41597}, {22109, 23039}, {22955, 25711}, {25740, 44452}, {32205, 47117}, {32245, 53091}, {33565, 47360}, {35240, 38726}, {44234, 52417}

X(54073) = circumcircle-inverse of X(156)
X(54073) = crossdifference of every pair of points on line {1637, 1879}
X(54073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 110, 156}, {110, 15132, 399}, {1511, 7723, 3}

X(54074) = CIRCUMCIRCLE-INVERSE OF X(157)

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 + 2*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - b^8*c^2 - 2*a^6*c^4 - 2*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + 2*a^4*c^6 - 2*a^2*c^8 - b^2*c^8 + c^10 : :
X(54074) = 2 X[141] - 3 X[34138], 2 X[625] - 3 X[36471], X[193] - 3 X[41363], 5 X[3618] - 3 X[34137]

X(54074) lies on these lines: {2, 13509}, {3, 66}, {113, 625}, {127, 6000}, {193, 41363}, {297, 525}, {325, 1625}, {620, 51393}, {626, 12162}, {2715, 31635}, {2764, 51939}, {3618, 34137}, {3734, 18474}, {3788, 10539}, {6720, 8779}, {7778, 18451}, {13567, 44909}, {14961, 15595}, {15341, 37648}, {18337, 41377}, {25711, 34827}, {35937, 37636}, {44377, 51425}, {44380, 45016}, {45198, 50678}

X(54074) = midpoint of X(i) and X(j) for these {i,j}: {66, 34107}, {2764, 51939}, {18337, 41377}
X(54074) = reflection of X(8779) in X(6720)
X(54074) = complement of X(13509)
X(54074) = circumcircle-inverse of X(157)
X(54074) = complement of the isogonal conjugate of X(34579)
X(54074) = X(i)-complementary conjugate of X(j) for these (i,j): {1953, 138}, {34579, 10}
X(54074) = X(2764)-Ceva conjugate of X(525)
X(54074) = crossdifference of every pair of points on line {184, 2485}

X(54075) = CIRCUMCIRCLE-INVERSE OF X(159)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^8 - 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 - a^2*b^4*c^2 - a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - c^8) : :
X(54075) = X[10317] + 3 X[34897], 5 X[631] - X[41377], 3 X[34366] - X[47286]

X(54075) lies on the cubic K038 and these lines: {2, 5523}, {3, 66}, {20, 51940}, {30, 127}, {36, 18637}, {76, 28697}, {126, 5159}, {131, 36471}, {187, 15526}, {216, 7820}, {230, 339}, {232, 44340}, {325, 15013}, {441, 525}, {524, 10317}, {577, 7801}, {620, 10257}, {625, 10297}, {626, 12605}, {631, 41377}, {858, 935}, {988, 24780}, {1213, 22366}, {1384, 40995}, {1511, 47085}, {1975, 28405}, {2072, 44377}, {2366, 2715}, {2482, 40349}, {2549, 37073}, {2972, 47526}, {3284, 7813}, {3314, 35952}, {3631, 18472}, {3734, 15760}, {3788, 11585}, {3793, 40996}, {3926, 20806}, {3933, 10316}, {3934, 7542}, {5024, 44909}, {6337, 28406}, {6643, 53033}, {6676, 8891}, {6720, 16318}, {6760, 8724}, {7761, 44249}, {7763, 28695}, {7778, 18531}, {7783, 28433}, {7807, 41009}, {7836, 28723}, {7863, 22401}, {8369, 41005}, {9155, 44888}, {10745, 35002}, {12095, 47082}, {13509, 15066}, {14001, 41363}, {28717, 32831}, {28719, 34254}, {28721, 32837}, {34147, 35282}, {34366, 47286}, {40856, 46942}, {44252, 47105}

X(54075) = midpoint of X(i) and X(j) for these {i,j}: {20, 51940}, {858, 935}, {44252, 47105}
X(54075) = reflection of X(i) in X(j) for these {i,j}: {16318, 6720}, {38971, 5159}
X(54075) = complement of X(5523)
X(54075) = circumcircle-inverse of X(159)
X(54075) = complement of the isogonal conjugate of X(18876)
X(54075) = X(i)-complementary conjugate of X(j) for these (i,j): {48, 1560}, {63, 15116}, {1177, 226}, {2373, 20305}, {18876, 10}, {36095, 520}, {37220, 21243}, {41511, 4892}
X(54075) = X(i)-Ceva conjugate of X(j) for these (i,j): {858, 524}, {935, 525}
crossdifference of every pair of points on line {25, 2485}
X(54075) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 23172, 15270}, {441, 6390, 14961}, {3926, 28696, 23115}

X(54076) = CIRCUMCIRCLE-INVERSE OF X(160)

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

X(54076) lies on these lines: {2, 15340}, {3, 66}, {127, 18400}, {128, 36471}, {290, 2367}, {323, 401}, {339, 1971}, {458, 5523}, {626, 11750}, {1625, 15013}, {3331, 40856}, {6720, 51363}, {7512, 36952}, {7816, 10575}, {9418, 46544}, {14767, 19176}, {22121, 51740}, {30737, 32661}, {35474, 47105}, {37124, 41377}, {37200, 51940}, {41334, 41363}

X(54076) = reflection of X(51363) in X(6720)
X(54076) = complement of X(15340)
X(54076) = circumcircle-inverse of X(160)
X(54076) = crossdifference of every pair of points on line {51, 2485}

X(54077) = CIRCUMCIRCLE-INVERSE OF X(185)

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

X(54077) lies on these lines: {3, 49}, {24, 21396}, {30, 13558}, {125, 15781}, {186, 5667}, {378, 2351}, {539, 13496}, {924, 42658}, {933, 32710}, {1516, 5961}, {1593, 21268}, {1624, 37917}, {5890, 52435}, {7722, 8154}, {11410, 44200}, {12085, 15512}, {14917, 42848}, {21659, 45842}, {35225, 37196}

X(54077) = circumcircle-inverse of X(185)

X(54078) = CIRCUMCIRCLE-INVERSE OF X(191)

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

X(54078) lies on these lines: {1, 13204}, {3, 191}, {35, 110}, {100, 21081}, {399, 26285}, {404, 13605}, {1511, 10902}, {2077, 5663}, {2778, 5538}, {2836, 5096}, {2915, 13146}, {3448, 25440}, {3733, 8674}, {3746, 11720}, {4256, 6126}, {5259, 5972}, {5563, 7984}, {5881, 19478}, {6796, 12383}, {7280, 22586}, {7991, 22583}, {9904, 10310}, {11012, 12778}, {11101, 46816}, {11499, 12407}, {11699, 11849}, {14798, 49203}, {15035, 15931}, {15051, 35202}, {17702, 44425}, {19470, 49204}, {32609, 32613}

X(54078) = circumcircle-inverse of X(191)

X(54079) = CIRCUMCIRCLE-INVERSE OF X(198)

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

X(54079) lies on the curve Q071 and these lines: {3, 9}, {41, 46830}, {102, 5011}, {214, 5199}, {515, 5514}, {650, 663}, {820, 836}, {910, 34591}, {1055, 3119}, {1146, 1319}, {1385, 23058}, {1855, 37080}, {2202, 28044}, {2262, 22753}, {2646, 46835}, {3207, 7079}, {3306, 25931}, {5179, 50371}, {5440, 40869}, {5513, 46415}, {6911, 23840}, {10165, 20262}, {14571, 16777}, {14733, 37763}, {17614, 41006}, {25917, 32561}, {26932, 51775}, {34522, 46344}, {35342, 51376}, {38015, 54008}, {40555, 51364}

X(54079) = reflection of X(51364) in X(40555)
X(54079) = circumcircle-inverse of X(198)
X(54079) = Stevanovic-circle-inverse of X(51361)
X(54079) = crossdifference of every pair of points on line {57, 6129}
X(54079) = barycentric product X(7046)*X(52977)
X(54079) = barycentric quotient X(52977)/X(7056)

X(54080) = CIRCUMCIRCLE-INVERSE OF X(206)

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

X(54080) lies on the cubic K904 and these lines: {3, 206}, {25, 35211}, {339, 44089}, {468, 2373}, {647, 8673}, {2070, 13115}, {2072, 6033}, {2936, 14961}, {3162, 5020}, {9909, 34427}, {10316, 23172}, {10547, 26926}, {10749, 34129}, {14376, 20968}

X(54080) = circumcircle-inverse of X(206)
X(54080) = Moses-radical-circle-inverse of X(46243)
X(54080) = X(52513)-Ceva conjugate of X(6)
X(54080) = crossdifference of every pair of points on line {427, 47125}
X(54080) = {X(20993),X(52041)}-harmonic conjugate of X(3)

X(54081) = CIRCUMCIRCLE-INVERSE OF X(214)

Barycentrics    a^2*(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 + 2*b^4*c - a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 + a^2*c^3 - a*b*c^3 - b^2*c^3 + 2*b*c^4 - c^5) : :
X(54081) = 4 X[124] - 3 X[15050]

X(54081) lies on the 2nd Evans circe, the Moses isodynomic circle (cf. X(41183)) these lines: {1, 1283}, {3, 214}, {10, 36558}, {25, 1845}, {28, 52167}, {36, 23205}, {55, 10703}, {56, 106}, {80, 52242}, {100, 6790}, {101, 102}, {104, 1633}, {110, 38568}, {117, 22753}, {124, 958}, {149, 36171}, {399, 2779}, {517, 1324}, {692, 34586}, {859, 5127}, {952, 53279}, {953, 1618}, {956, 13532}, {960, 34868}, {993, 2792}, {995, 2175}, {999, 1486}, {1001, 11734}, {1015, 5019}, {1064, 2317}, {1364, 10966}, {1387, 53302}, {1411, 15906}, {1468, 53542}, {1473, 1795}, {1482, 23843}, {1626, 10246}, {1718, 42753}, {1960, 8677}, {2099, 11334}, {2178, 8649}, {2217, 12699}, {2305, 9259}, {2390, 52407}, {2773, 22586}, {2785, 8301}, {2817, 9798}, {2818, 3556}, {2819, 41186}, {2842, 22148}, {2849, 3960}, {2852, 41184}, {2853, 19162}, {2932, 23832}, {2933, 12702}, {2975, 33650}, {3149, 50899}, {3738, 4491}, {5541, 23858}, {5584, 38691}, {5903, 37259}, {6224, 13589}, {6718, 25524}, {6788, 28083}, {7373, 47115}, {8158, 39600}, {9321, 20672}, {9532, 19159}, {10269, 38607}, {10573, 28077}, {10680, 22654}, {10747, 22758}, {10777, 13744}, {13730, 26437}, {15507, 51506}, {16064, 37525}, {16370, 50918}, {16680, 23402}, {18393, 34300}, {19297, 21781}, {20676, 22744}, {20842, 37567}, {22141, 23344}, {22144, 53290}, {22765, 38579}, {22769, 47038}, {23981, 51236}, {26321, 38780}, {28194, 51637}, {28348, 52129}, {35239, 38600}

X(54081) = circumcircle-inverse of X(214)
X(54081) = Stammler-circle-inverse of X(48667)
X(54081) = isogonal conjugate of the anticomplement of X(15898)
X(54081) = tangential isogonal conjugate of X(1324)
X(54081) = X(3218)-Ceva conjugate of X(6)
X(54081) = X(2161)-Dao conjugate of X(18359)
X(54081) = crossdifference of every pair of points on line {1639, 53522}
X(54081) = barycentric product X(i)*X(j) for these {i,j}: {56, 28829}, {3218, 15898}
X(54081) = barycentric quotient X(i)/X(j) for these {i,j}: {15898, 18359}, {28829, 3596}
X(54081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {692, 53294, 34586}, {3556, 11249, 15654}

X(54082) = CIRCUMCIRCLE-INVERSE OF X(216)

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

X(54082) lies on these lines: {3, 6}, {23, 3331}, {24, 11674}, {25, 5167}, {26, 32445}, {186, 3289}, {217, 7488}, {316, 458}, {401, 3580}, {512, 34983}, {691, 51222}, {1154, 32661}, {1625, 2070}, {1971, 13754}, {2387, 39857}, {2715, 18401}, {3269, 10313}, {5207, 34850}, {5523, 5667}, {5889, 14585}, {6644, 40805}, {7517, 38297}, {8571, 18377}, {10312, 22416}, {10985, 15030}, {10986, 11459}, {13322, 33664}, {19627, 39839}, {20998, 45938}, {22146, 32608}, {34360, 44146}, {35324, 50461}, {35941, 51224}, {37644, 51350}

X(54082) = reflection of X(39849) in X(1971)
X(54082) = circumcircle-inverse of X(216)
X(54082) = isogonal conjugate of the isotomic conjugate of X(44363)
X(54082) = isogonal conjugate of the polar conjugate of X(41203)
X(54082) = crossdifference of every pair of points on line {523, 23292}
X(54082) = barycentric product X(i)*X(j) for these {i,j}: {3, 41203}, {6, 44363}, {99, 42651}
X(54082) = barycentric quotient X(i)/X(j) for these {i,j}: {41203, 264}, {42651, 523}, {44363, 76}
X(54082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 50387, 1691}, {1379, 1380, 216}

X(54083) = CIRCUMCIRCLE-INVERSE OF X(221)

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

X(54083) lies on these lines: {3, 102}, {185, 603}, {1415, 35072}, {1433, 1795}, {1455, 2817}, {1459, 1946}, {1935, 3042}, {2199, 53853}, {2800, 46974}, {2816, 34050}, {2829, 10017}, {2968, 51422}, {10740, 34030}, {11700, 17102}, {34029, 38776}

X(54083) = circumcircle-inverse of X(221)
X(54083) = {X(102),X(109)}-harmonic conjugate of X(221)

X(54084) = CIRCUMCIRCLE-INVERSE OF X(242)

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

X(54084) lies on these lines: {3, 242}, {4, 2077}, {24, 1324}, {25, 5205}, {28, 1792}, {34, 1054}, {35, 37055}, {1829, 37304}, {1851, 4188}, {1870, 24046}, {3518, 45767}, {4874, 23383}, {11383, 37253}, {13739, 19642}, {14192, 30733}, {16066, 52427}, {19842, 37101}

X(54084) = circumcircle-inverse of X(242)
X(54084) = polar-circle-inverse of X(3814)

X(54085) = CIRCUMCIRCLE-INVERSE OF X(246)

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

X(54085) lies on these lines: {3, 74}, {112, 1112}, {125, 1316}, {247, 2777}, {526, 7669}, {1510, 30715}, {1976, 2781}, {2079, 3569}, {3124, 50363}, {3269, 13198}, {3447, 20188}, {3448, 4226}, {5621, 46130}, {5622, 51335}, {7422, 12244}, {9409, 13558}, {13417, 31850}, {14683, 18331}, {14984, 38873}, {15107, 38582}, {15535, 36181}, {15920, 37457}, {24981, 35022}, {39857, 42663}

X(54085) = circumcircle-inverse of X(246)
X(54085) = crossdifference of every pair of points on line {1637, 5972}
X(54085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 110, 246}, {246, 5191, 110}, {5621, 52162, 46130}

X(54086) = CIRCUMCIRCLE-INVERSE OF X(290)

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

X(54086) lies on the cubic K039 and these lines: {3, 76}, {115, 10684}, {186, 22456}, {187, 2966}, {237, 38947}, {287, 5026}, {1976, 39292}, {2076, 39941}, {5661, 40866}, {6037, 35298}, {8289, 46806}, {13586, 48452}, {13754, 17932}, {26613, 37858}, {34536, 35296}, {35297, 51404}, {43187, 47635}, {47388, 52992}

X(54086) = isogonal conjugate of X(52446)
X(54086) = circumcircle-inverse of X(290)
X(54086) = X(1)-isoconjugate of X(52446)
X(54086) = X(3)-Dao conjugate of X(52446)
X(54086) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52446}, {2966, 53603}
X(54086) = {X(98),X(99)}-harmonic conjugate of X(290)

X(54087) = CIRCUMCIRCLE-INVERSE OF X(323)

Barycentrics    a^2*(4*a^10 - 12*a^8*b^2 + 14*a^6*b^4 - 10*a^4*b^6 + 6*a^2*b^8 - 2*b^10 - 12*a^8*c^2 + 23*a^6*b^2*c^2 - 13*a^4*b^4*c^2 - a^2*b^6*c^2 + 3*b^8*c^2 + 14*a^6*c^4 - 13*a^4*b^2*c^4 + 5*a^2*b^4*c^4 - b^6*c^4 - 10*a^4*c^6 - a^2*b^2*c^6 - b^4*c^6 + 6*a^2*c^8 + 3*b^2*c^8 - 2*c^10) : :

X(54087) lies on these lines: {3, 323}, {160, 669}, {325, 6031}, {376, 14731}, {842, 7492}, {5191, 8724}, {5467, 11002}, {5939, 14360}, {5941, 11580}, {9155, 51800}, {10298, 10420}, {11004, 52603}, {14652, 52124}, {34417, 47053}, {40604, 52169}

X(54087) = circumcircle-inverse of X(323)
X(54087) = psi-transform of X(1511)

X(54088) = CIRCUMCIRCLE-INVERSE OF X(325)

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

X(54088) lies on these lines: {3, 315}, {22, 669}, {183, 37930}, {187, 52036}, {323, 353}, {3148, 34245}, {5939, 7669}, {5941, 22329}, {5976, 19165}, {7492, 38940}, {7761, 35936}, {12584, 50567}, {28724, 38525}

X(54088) = circumcircle-inverse of X(325)
X(54088) = 2nd-Brocard-circle-inverse of X(7750)

X(54089) = CIRCUMCIRCLE-INVERSE OF X(339)

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

X(54089) lies on these lines: {3, 76}, {115, 6720}, {148, 4235}, {620, 40484}, {804, 39857}, {5026, 41255}, {6699, 14928}, {7669, 35522}, {8178, 14966}, {9862, 13219}, {23285, 30715}, {24284, 46253}

X(54089) = circumcircle-inverse of X(339)
X(54089) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 99, 339}, {115, 53737, 40856}

X(54090) = CIRCUMCIRCLE-INVERSE OF X(355)

Barycentrics    a^2*(a^8 - a^7*b - 2*a^6*b^2 + 3*a^5*b^3 - 3*a^3*b^5 + 2*a^2*b^6 + a*b^7 - b^8 - a^7*c + 3*a^6*b*c - a^5*b^2*c - 5*a^4*b^3*c + 5*a^3*b^4*c + a^2*b^5*c - 3*a*b^6*c + b^7*c - 2*a^6*c^2 - a^5*b*c^2 + 6*a^4*b^2*c^2 - a^3*b^3*c^2 - 5*a^2*b^4*c^2 + 2*a*b^5*c^2 + b^6*c^2 + 3*a^5*c^3 - 5*a^4*b*c^3 - a^3*b^2*c^3 + 4*a^2*b^3*c^3 - b^5*c^3 + 5*a^3*b*c^4 - 5*a^2*b^2*c^4 - 3*a^3*c^5 + a^2*b*c^5 + 2*a*b^2*c^5 - b^3*c^5 + 2*a^2*c^6 - 3*a*b*c^6 + b^2*c^6 + a*c^7 + b*c^7 - c^8) : :
X(54090) = 5 X[3] - X[35455], 5 X[1324] + X[35455]

X(54090) lies on these lines: {3, 10}, {24, 1785}, {35, 37116}, {36, 1772}, {58, 31760}, {100, 45396}, {186, 45766}, {522, 39200}, {946, 37259}, {1399, 31825}, {2077, 37311}, {2222, 9590}, {2708, 29095}, {2932, 51632}, {3417, 5903}, {3465, 5010}, {5172, 14667}, {7488, 10538}, {8069, 51616}, {14127, 41698}, {24042, 52242}, {35013, 39478}

X(54090) = midpoint of X(3) and X(1324)
X(54090) = circumcircle-inverse of X(355)
X(54090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 2933, 6796}, {3, 23843, 5450}

X(54091) = CIRCUMCIRCLE-INVERSE OF X(389)

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

X(54091) lies on these lines: {3, 6}, {25, 38227}, {186, 47158}, {316, 7395}, {385, 39803}, {842, 37920}, {1513, 9861}, {1607, 7684}, {1608, 7685}, {3425, 20885}, {3515, 21396}, {5899, 30715}, {6642, 14693}, {6644, 38230}, {7503, 14712}, {9876, 37461}, {10003, 53485}, {10323, 43453}, {11479, 13449}, {11676, 39832}, {14575, 19123}, {14853, 37457}, {16188, 37972}, {20854, 34218}, {37928, 47584}, {39231, 44668}, {40947, 52276}

X(54091) = reflection of X(3) in X(32762)
X(54091) = circumcircle-inverse of X(389)
X(54091) = Stammler-circle-inverse of X(6243)
X(54091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1379, 1380, 389}, {38596, 38597, 6243}

X(54092) = CIRCUMCIRCLE-INVERSE OF X(394)

Barycentrics    a^2*(a^2 - b^2 - c^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 - 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(54092) lies on these lines: {3, 49}, {23, 4558}, {25, 44899}, {114, 468}, {122, 11064}, {154, 6503}, {235, 21268}, {669, 684}, {1297, 10420}, {1495, 23181}, {3003, 23357}, {3291, 5941}, {3542, 5962}, {6031, 37668}, {6800, 9723}, {10539, 16391}, {12420, 34853}, {13558, 34382}, {14567, 47406}, {14981, 42671}, {15512, 17836}

X(54092) = circumcircle-inverse of X(394)
X(54092) = crossdifference of every pair of points on line {2501, 3767}
X(54092) = {X(3),X(41615)}-harmonic conjugate of X(47195)

X(54093) = CIRCUMCIRCLE-INVERSE OF X(411)

Barycentrics    a*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8 - 2*a^7*b*c + 2*a^6*b^2*c + 3*a^5*b^3*c - 3*a^4*b^4*c - a*b^7*c + b^8*c - 2*a^7*c^2 + 2*a^6*b*c^2 + 3*a^5*b^2*c^2 + a^4*b^3*c^2 - 2*a^3*b^4*c^2 - 2*a^2*b^5*c^2 + a*b^6*c^2 - b^7*c^2 + 3*a^5*b*c^3 + a^4*b^2*c^3 - 4*a^3*b^3*c^3 + 2*a^2*b^4*c^3 + a*b^5*c^3 - 3*b^6*c^3 - 3*a^4*b*c^4 - 2*a^3*b^2*c^4 + 2*a^2*b^3*c^4 + 3*b^5*c^4 - 2*a^2*b^2*c^5 + a*b^3*c^5 + 3*b^4*c^5 + 2*a^3*c^6 + a*b^2*c^6 - 3*b^3*c^6 - a*b*c^7 - b^2*c^7 - a*c^8 + b*c^8) : :
X(54093) = 2 X[36001] + X[37919], 3 X[37940] - 2 X[51635]

X(54093) lies on these lines: {2, 3}, {100, 2695}, {102, 1290}, {484, 1725}, {517, 5494}, {523, 53277}, {1292, 53926}, {1311, 2691}, {2099, 10149}, {2687, 33637}, {2766, 41904}, {3871, 17479}, {5160, 37567}, {5176, 51629}, {15945, 40582}, {32706, 53952}, {40214, 48897}, {53916, 53941}

X(54093) = reflection of X(i) in X(j) for these {i,j}: {1325, 186}, {3153, 30447}, {37976, 15646}
X(54093) = circumcircle-inverse of X(411)
X(54093) = polar-circle-inverse of X(37368)
X(54093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1113, 1114, 411}

X(54094) = CIRCUMCIRCLE-INVERSE OF X(419)

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

X(54094) lies on these lines: {2, 3}, {107, 43654}, {250, 1974}, {523, 3186}, {571, 38861}, {935, 53704}, {1304, 2698}, {2452, 40981}, {3563, 53937}, {5106, 51324}, {6037, 40118}, {9513, 44127}, {9998, 35325}, {22239, 48259}, {46426, 46432}

X(54094) = circumcircle-inverse of X(419)
X(54094) = polar-circle-inverse of X(21531)
X(54094) = X(i)-isoconjugate of X(j) for these (i,j): {336, 52446}, {656, 53603}
X(54094) = X(40596)-Dao conjugate of X(53603)
X(54094) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 53603}, {2211, 52446}
X(54094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 4230, 4}, {237, 1316, 37918}, {1113, 1114, 419}, {1316, 37918, 37991}, {5000, 5001, 15915}, {21525, 44895, 2}

X(54095) = CIRCUMCIRCLE-INVERSE OF X(442)

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*b*c - 2*a^5*b^2*c + 2*a^2*b^5*c + 2*a*b^6*c - 2*a^6*c^2 - 2*a^5*b*c^2 + a^4*b^2*c^2 - a^2*b^4*c^2 + 2*a*b^5*c^2 + 2*b^6*c^2 - 2*a^2*b^3*c^3 - 2*a*b^4*c^3 - a^2*b^2*c^4 - 2*a*b^3*c^4 - 2*b^4*c^4 + 2*a^2*b*c^5 + 2*a*b^2*c^5 + 2*a^2*c^6 + 2*a*b*c^6 + 2*b^2*c^6 - c^8) : :
X(54095) = 3 X[17532] - 4 X[37982]

X(54095) lies on these lines: {2, 3}, {476, 51760}, {1290, 41345}, {1324, 23860}, {3447, 20872}, {4640, 9591}, {5903, 34435}, {6001, 9625}, {11496, 51883}, {20875, 39857}, {20989, 51632}, {23406, 30715}, {36152, 41495}, {37579, 39751}

X(54095) = reflection of X(7580) in X(37979)
X(54095) = circumcircle-inverse of X(442)
X(54095) = tangential-circle-inverse of X(2915)
X(54095) = tangential-isogonal conjugate of X(2948)
X(54095) = X(19622)-Dao conjugate of X(37783)
X(54095) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 20831, 11101}, {23, 186, 51635}, {23, 37919, 5899}, {186, 36001, 3}, {1113, 1114, 442}, {1325, 37919, 36171}, {2074, 30447, 405}, {2074, 37959, 30447}

X(54096) = CIRCUMCIRCLE-INVERSE OF X(460)

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

X(54096) lies on these lines: {2, 3}, {250, 19118}, {523, 1609}, {1304, 23700}, {2079, 47200}, {2452, 8573}, {2453, 8553}, {3053, 47213}, {11063, 47284}, {12828, 53735}, {14729, 47220}, {20987, 30715}, {41584, 41758}

X(54096) = circumcircle-inverse of X(460)
X(54096) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 4230, 25}, {1113, 1114, 460}





leftri   Perspectors associated with Steiner-circumcevian triangles: X(54097)-X(54113) rightri

Contributed by Clark Kimberling and Peter Moses June 29, 2023.

Let SCE be Steiner circumellipse of a triangle ABC, and Let P = p : q : r be a point not on a sideline of ABC. Let A' be the point, other than A, where the line AP meets line BC. Define B' and C' cyclically. The triangle A'B'C' is here named the Steiner-circumcevian triangle of P, denoted by SCC(P). The vertices of SCC(P) are given by

A' = -q r : q (q + r) : r (q + r)
B' = p (r + p) : - r p : r (r + p)
C' = p (p + q) : q (p + q) : - p q

The triangle SCC(P) is perspective to the anticomplementary. Let SC(P) denote the perspector. Then

SC(P) = P2-Ceva conjugate of X(2)
SC(P) = anticomplement of isotomic conjugate of P2.

The appearance of (i,j) in the following list means that SC(X(i)) = X(j).

(2996, 54097), (330, 54098), (7035, 54099), (276, 54100), (291, 54101), (514, 54102). (8781, 54103), (523, 54104), (40410, 54105), (262, 54106), (333, 54107), (18020, 54108), (314, 54109), (4998, 54110), (253, 54111), (310, 54112), (312, 54113)

The triangle SCC(X(2)) is the triangle Gemini 107, and the triangle SCC(4) is the 9th Brocard triangle.

Let Ta be the line tangent to SCC at A', and define Tb and Tc cyclically. Let A* = Tb ^ Tc and define B* and C* cyclically. Then A*B*C* is perspective to ABC, and the perspector is given by the point T(P) = 1 / (-q*r + r*p + p*q) : : . The transformation T maps curves to curves, as in these examples:

T(Kiepert hyperbola) = Kiepert hyperbola
T(K184) = K007
T(K1023) = K1000
T(K868) = K1002
T(K1014) = K1037)
T(K342a) = K1053a)
T(K342b) = K1053n)

underbar



X(54097) = X(2)X(3)∩X(193)X(44518)

Barycentrics    7*a^4 - 2*a^2*b^2 - 9*b^4 - 2*a^2*c^2 + 14*b^2*c^2 - 9*c^4 : :
X(54097) = 9 X[2] - 8 X[32970], 3 X[2] - 4 X[32972], 3 X[439] - 4 X[32970], 2 X[32970] - 3 X[32972]

X(54097) lies on these lines: {2, 3}, {193, 44518}, {316, 6392}, {543, 32825}, {2996, 20080}, {5023, 39143}, {5254, 51170}, {5395, 18845}, {7748, 32827}, {7760, 43448}, {7781, 32816}, {7825, 32815}, {7842, 32828}, {7848, 32868}, {7873, 46951}, {7898, 32834}, {8589, 32884}, {11185, 32027}, {14023, 39563}, {15301, 32876}, {18424, 32838}, {20094, 32841}, {32829, 43619}, {32883, 39601}, {33684, 39874}, {34505, 41895}, {34803, 44519}, {44377, 51579}

X(54097) = reflection of X(439) in X(32972)
X(54097) = anticomplement of X(439)
X(54097) = X(8769)-anticomplementary conjugate of X(19583)
X(54097) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 19691, 50693}, {2, 33209, 21734}, {2, 50690, 6658}, {2, 50692, 33244}, {3, 52250, 2}, {4, 7841, 32971}, {4, 16045, 11317}, {4, 32974, 32979}, {4, 32982, 2}, {4, 33229, 32974}, {20, 14041, 32980}, {20, 32980, 2}, {377, 33050, 2}, {381, 33238, 32990}, {382, 16041, 32973}, {439, 32972, 2}, {546, 32986, 32987}, {1657, 32969, 35287}, {1657, 37350, 32969}, {2996, 32006, 20080}, {2996, 53419, 38259}, {3091, 33017, 33023}, {3091, 33023, 2}, {3146, 14063, 2}, {3146, 33205, 33193}, {3522, 32966, 2}, {3529, 33228, 32989}, {3543, 5025, 32981}, {3543, 33181, 33280}, {3832, 6655, 2}, {3839, 7791, 32991}, {3850, 5077, 32978}, {5025, 32981, 2}, {5025, 33280, 33181}, {5068, 32965, 2}, {5177, 33051, 2}, {5395, 53418, 18845}, {6658, 33290, 2}, {7791, 14044, 3839}, {7791, 32991, 2}, {7841, 32956, 32974}, {7887, 33703, 35927}, {8597, 16925, 49135}, {11303, 11304, 15702}, {14035, 33200, 2}, {14041, 33279, 20}, {14042, 33251, 33198}, {14045, 33007, 33199}, {14062, 33017, 3091}, {15022, 33004, 2}, {15640, 33203, 33257}, {15717, 32963, 2}, {16044, 33025, 2}, {16044, 33278, 33025}, {17685, 37161, 2}, {20080, 38259, 2996}, {32006, 53419, 2996}, {32963, 33264, 15717}, {32966, 33192, 3522}, {32967, 33271, 10304}, {32971, 32974, 32956}, {32974, 32979, 2}, {32974, 33229, 32982}, {32979, 32982, 32974}, {32993, 32997, 2}, {32993, 33019, 32997}, {32996, 32997, 32993}, {32996, 33019, 2}, {32998, 33267, 15692}, {33006, 33256, 3523}, {33181, 33280, 32981}, {33199, 50691, 33007}, {33200, 50687, 14035}, {33201, 33283, 2}

X(54098) = X(1)X(2)∩X(192)X(17448)

Barycentrics    a^2*b^2 - 6*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2 : :
X(54098) = 9 X[2] - 8 X[27091], 4 X[27091] - 3 X[53675]

X(54098) lies on these lines: {1, 2}, {75, 31999}, {149, 33019}, {192, 17448}, {194, 4788}, {319, 26143}, {330, 1278}, {350, 21219}, {391, 16515}, {495, 33060}, {496, 33061}, {536, 32005}, {956, 16914}, {999, 33062}, {1056, 33030}, {1058, 17685}, {1107, 4704}, {1191, 20158}, {1616, 20142}, {3295, 33063}, {3759, 27291}, {3871, 33004}, {4366, 12513}, {4452, 30662}, {4772, 31997}, {4821, 17143}, {5082, 17565}, {5839, 24761}, {6658, 20076}, {6767, 33047}, {7187, 17158}, {7373, 16917}, {9263, 20081}, {16722, 30939}, {16884, 20146}, {16969, 17349}, {16975, 32026}, {17178, 33296}, {17373, 27106}, {20060, 33018}, {20075, 33260}, {20530, 40598}, {24524, 30998}, {25573, 26135}, {26823, 48333}, {26852, 30941}, {27107, 34063}

X(54098) = anticomplement of X(53675)
X(54098) = anticomplement of the isogonal conjugate of X(53146)
X(54098) = anticomplement of the isotomic conjugate of X(53677)
X(54098) = isotomic conjugate of the isogonal conjugate of X(41397)
X(54098) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7121, 21219}, {7153, 20350}, {32039, 21301}, {53146, 8}, {53677, 6327}, {53678, 69}, {53679, 315}
X(54098) = X(53677)-Ceva conjugate of X(2)
X(54098) = barycentric product X(76)*X(41397)
X(54098) = barycentric quotient X(41397)/X(6)
X(54098) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {239, 16826, 23511}, {330, 17144, 1278}, {1107, 32095, 4704}, {1278, 38247, 330}, {4393, 29570, 5256}, {20055, 26821, 2}

X(54099) = X(2)X(32016)∩X(190)X(17494)

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

X(54099) lies on these lines: {2, 32016}, {190, 17494}, {660, 799}, {668, 20295}, {889, 50520}, {4555, 4608}, {4562, 31290}, {4568, 48082}, {4579, 35356}, {9362, 47763}, {17154, 32030}, {32937, 33798}

X(54099) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6632, 149}, {42372, 320}
X(54099) = {X(3952),X(7192)}-harmonic conjugate of X(7035)

X(54100) = X(3)X(95)∩X(4)X(290)

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

X(54100) lies on these lines: {2, 46394}, {3, 95}, {4, 290}, {5, 16089}, {30, 9291}, {32, 16081}, {76, 37200}, {83, 458}, {297, 34850}, {308, 51252}, {315, 18022}, {324, 401}, {393, 37337}, {1078, 1629}, {1235, 35474}, {3785, 44144}, {6331, 7752}, {8794, 52253}, {14246, 46111}, {16264, 17984}, {18831, 34148}, {36794, 41334}, {37124, 44142}

X(54100) = reflection of X(9291) in X(42368)
X(54100) = isotomic conjugate of X(42487)
X(54100) = anticomplement of X(46394)
X(54100) = isotomic conjugate of the isogonal conjugate of X(1629)
X(54100) = polar conjugate of the isogonal conjugate of X(36794)
X(54100) = X(23582)-Ceva conjugate of X(6528)
X(54100) = X(i)-isoconjugate of X(j) for these (i,j): {31, 42487}, {255, 27375}, {3613, 52430}, {9247, 36952}
X(54100) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 42487}, {850, 15526}, {6523, 27375}, {52591, 3269}
X(54100) = cevapoint of X(1629) and X(36794)
X(54100) = barycentric product X(i)*X(j) for these {i,j}: {76, 1629}, {158, 33764}, {264, 36794}, {276, 30506}, {393, 33769}, {1078, 2052}, {1096, 33778}, {5012, 18027}, {6528, 31296}, {10312, 18022}, {23582, 36901}, {37125, 46104}
X(54100) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 42487}, {264, 36952}, {393, 27375}, {1078, 394}, {1629, 6}, {2052, 3613}, {3050, 39201}, {5012, 577}, {6528, 11794}, {7668, 3269}, {10312, 184}, {18042, 255}, {23582, 27867}, {27010, 1364}, {30506, 216}, {31296, 520}, {33764, 326}, {33769, 3926}, {36794, 3}, {36901, 15526}, {37125, 3917}, {38352, 34980}, {41296, 28724}, {41334, 418}
X(54100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 264, 276}, {4, 18027, 6528}

X(54101) = X(2)X(1978)∩X(37)X(39028)

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

X(54101) lies on these lines: {2, 1978}, {37, 39028}, {42, 19579}, {145, 194}, {190, 25054}, {192, 17794}, {291, 740}, {346, 2998}, {668, 1500}, {812, 23656}, {1015, 17143}, {2276, 52044}, {3802, 30667}, {4360, 52637}, {6542, 40858}, {8264, 17314}, {39786, 40094}

X(54101) = reflection of X(i) in X(j) for these {i,j}: {668, 1500}, {17143, 1015}
X(54101) = anticomplement of the isogonal conjugate of X(51856)
X(54101) = anticomplement of the isotomic conjugate of X(52205)
X(54101) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {292, 20554}, {1911, 20345}, {1922, 17794}, {1927, 30668}, {14598, 33888}, {18267, 2}, {18897, 30667}, {30663, 315}, {40098, 21275}, {51856, 8}, {52205, 6327}
X(54101) = X(52205)-Ceva conjugate of X(2)
X(54101) = {X(17759),X(39925)}-harmonic conjugate of X(291)

X(54102) = X(2)X(1016)∩X(8)X(19950)

Barycentrics    a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3 - b^4 - 2*a^3*c + 8*a^2*b*c - 4*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 - c^4 : :
X(54102) = 3 X[2] - 4 X[6547], 2 X[4440] + X[39349], 5 X[4473] - 4 X[32094], 3 X[17487] - 4 X[32106], 5 X[27191] - 3 X[34024]

X(54102) lies on these lines: {2, 1016}, {8, 19950}, {145, 18343}, {148, 39362}, {149, 21301}, {190, 45213}, {239, 908}, {514, 4440}, {519, 4645}, {1086, 6631}, {1278, 30225}, {1320, 31129}, {4473, 32094}, {6542, 17495}, {6646, 35957}, {9263, 17496}, {17152, 46707}, {17487, 32106}, {20042, 33922}, {26806, 36226}, {27191, 34024}, {32028, 35092}, {39348, 39368}

X(54102) = midpoint of X(4440) and X(6630)
X(54102) = reflection of X(i) in X(j) for these {i,j}: {190, 45213}, {1016, 6547}, {6631, 1086}, {32028, 35092}, {39349, 6630}
X(54102) = anticomplement of X(1016)
X(54102) = anticomplement of the isogonal conjugate of X(1015)
X(54102) = anticomplement of the isotomic conjugate of X(1086)
X(54102) = isotomic conjugate of the isogonal conjugate of X(41395)
X(54102) = anticomplementary isogonal conjugate of X(668)
X(54102) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 668}, {6, 3952}, {11, 21286}, {31, 190}, {56, 21272}, {57, 3888}, {58, 53332}, {81, 53338}, {86, 53363}, {87, 1978}, {163, 50351}, {244, 69}, {292, 23354}, {513, 20295}, {514, 21301}, {560, 46725}, {593, 21295}, {604, 100}, {649, 513}, {663, 4462}, {667, 514}, {693, 21304}, {739, 53340}, {741, 874}, {757, 4576}, {764, 150}, {798, 31290}, {849, 99}, {873, 670}, {875, 812}, {876, 21303}, {1014, 53355}, {1015, 8}, {1019, 512}, {1027, 3766}, {1086, 6327}, {1106, 664}, {1110, 32028}, {1111, 315}, {1178, 799}, {1333, 4427}, {1357, 7}, {1358, 21285}, {1395, 651}, {1397, 4552}, {1398, 4566}, {1400, 3909}, {1402, 3882}, {1408, 17136}, {1416, 883}, {1438, 53358}, {1474, 53349}, {1911, 42720}, {1919, 17494}, {1973, 3732}, {1977, 192}, {1980, 21225}, {2087, 21290}, {2149, 39185}, {2170, 3436}, {2203, 14543}, {2279, 3799}, {2350, 4553}, {2665, 27853}, {2969, 21270}, {3063, 4468}, {3120, 21287}, {3121, 1654}, {3122, 2895}, {3125, 1330}, {3248, 2}, {3249, 9263}, {3271, 329}, {3572, 46403}, {3669, 21302}, {3675, 20552}, {3733, 7192}, {3768, 44008}, {3937, 4329}, {3942, 1370}, {4117, 46714}, {4607, 33917}, {4817, 788}, {6545, 21293}, {6591, 20293}, {7023, 35312}, {7117, 52366}, {7121, 33946}, {7192, 17217}, {7199, 44445}, {7203, 4374}, {8027, 4440}, {8034, 21221}, {8054, 18133}, {9315, 4499}, {9456, 17780}, {16726, 17135}, {16727, 17138}, {17205, 17137}, {17925, 21300}, {18191, 20245}, {21143, 149}, {21762, 41840}, {22096, 6360}, {22383, 20294}, {23345, 21297}, {23349, 47776}, {23892, 891}, {23989, 21275}, {27846, 20345}, {27918, 20554}, {28607, 4781}, {32665, 6550}, {34445, 25310}, {36598, 36863}, {37129, 41314}, {38247, 25312}, {38266, 3699}, {38986, 21219}, {39748, 27808}, {39798, 8050}, {40148, 4033}, {40735, 3807}, {42067, 5905}, {43921, 20347}, {43922, 320}, {43923, 46400}, {43924, 693}, {43925, 7253}, {43929, 53343}, {43932, 46402}, {46289, 33951}, {51866, 660}, {52619, 21305}, {52633, 20355}, {53538, 3434}, {53540, 2893}, {53541, 30660}, {53678, 18830}
X(54102) = X(i)-Ceva conjugate of X(j) for these (i,j): {1086, 2}, {6631, 514}
X(54102) = barycentric product X(i)*X(j) for these {i,j}: {75, 1052}, {76, 41395}
X(54102) = barycentric quotient X(i)/X(j) for these {i,j}: {1052, 1}, {41395, 6}
X(54102) = {X(1016),X(6547)}-harmonic conjugate of X(2)

X(54103) = X(2)X(39764)∩X(20)X(99)

Barycentrics    a^8 + a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + a^6*c^2 - 5*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + b^6*c^2 - 3*a^4*c^4 + 3*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + b^2*c^6 - 2*c^8 : :
X(54103) = 4 X[3767] - 5 X[14061]

X(54103) lies on these lines: {2, 39764}, {20, 99}, {69, 114}, {76, 38383}, {98, 325}, {115, 6392}, {316, 10723}, {491, 19056}, {492, 19055}, {524, 44534}, {538, 671}, {542, 46236}, {620, 3785}, {1007, 6036}, {1078, 3314}, {1494, 53367}, {1569, 7818}, {1975, 10722}, {2023, 9766}, {2782, 7776}, {3329, 7886}, {3767, 7752}, {3933, 6033}, {3964, 9861}, {4027, 7897}, {5149, 7895}, {5152, 7871}, {5182, 12830}, {5319, 6722}, {5468, 30789}, {5976, 6054}, {5989, 45018}, {6337, 38749}, {6390, 38741}, {6721, 34229}, {7750, 21166}, {7763, 34473}, {7767, 15561}, {7773, 14639}, {7778, 12829}, {7779, 36849}, {7811, 8182}, {7856, 37665}, {7925, 36864}, {7946, 39652}, {8290, 33276}, {9862, 32818}, {10754, 50771}, {10991, 32825}, {14651, 32823}, {23234, 37671}, {23698, 32006}, {30786, 31127}, {32815, 39838}, {32828, 36519}, {32829, 38737}, {32954, 41675}, {35136, 39352}, {43150, 46318}

X(54103) = reflection of X(i) in X(j) for these {i,j}: {99, 3926}, {6392, 115}
X(54103) = isotomic conjugate of the isogonal conjugate of X(38873)
X(54103) = barycentric product X(76)*X(38873)
X(54103) = barycentric quotient X(38873)/X(6)
X(54103) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 325, 8781}, {147, 32458, 99}, {147, 37668, 32458}

X(54104) = X(2)X(4590)∩X(99)X(45212)

Barycentrics    a^8 - 2*a^6*b^2 - a^4*b^4 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 8*a^4*b^2*c^2 - 4*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 - c^8 : :
X(54104) = 3 X[2] - 4 X[23991], 2 X[148] + X[39356], X[148] + 2 X[44373], X[39356] - 4 X[44373], 8 X[31644] - 9 X[41135], X[8596] + 2 X[18823], X[20094] - 4 X[23992], 5 X[40429] - 4 X[40553], 4 X[40511] - 3 X[44397]

X(54104) lies on the cubic K079 and these lines: {2, 4590}, {99, 45212}, {115, 31998}, {148, 523}, {385, 858}, {524, 5207}, {892, 31372}, {2854, 32528}, {3221, 31513}, {6625, 24345}, {7665, 36168}, {8596, 18823}, {14588, 44398}, {17162, 20536}, {20094, 23992}, {25051, 39346}, {25054, 39359}, {33915, 45291}, {40429, 40553}, {40511, 44397}

X(54104) = midpoint of X(148) and X(35511)
X(54104) = reflection of X(i) in X(j) for these {i,j}: {99, 45212}, {4590, 23991}, {14588, 44398}, {20094, 33799}, {31372, 892}, {31998, 115}, {33799, 23992}, {35511, 44373}, {39356, 35511}
X(54104) = anticomplement of X(4590)
X(54104) = anticomplement of the isogonal conjugate of X(3124)
X(54104) = anticomplement of the isotomic conjugate of X(115)
X(54104) = isotomic conjugate of the isogonal conjugate of X(33704)
X(54104) = anticomplementary isogonal conjugate of X(4576)
X(54104) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 4576}, {6, 21295}, {10, 53363}, {19, 53350}, {31, 99}, {32, 6758}, {37, 53338}, {42, 53332}, {65, 53355}, {75, 670}, {115, 6327}, {181, 21272}, {213, 4427}, {244, 17143}, {338, 21275}, {512, 7192}, {513, 17159}, {523, 17217}, {560, 14570}, {649, 17166}, {661, 512}, {667, 17161}, {669, 4560}, {756, 668}, {798, 523}, {810, 6563}, {850, 21305}, {872, 190}, {897, 53367}, {923, 5468}, {1015, 17140}, {1084, 192}, {1101, 33799}, {1109, 315}, {1356, 3210}, {1365, 21285}, {1402, 17136}, {1500, 3952}, {1577, 44445}, {1910, 53371}, {1917, 46726}, {1924, 31296}, {1967, 2396}, {1973, 110}, {2170, 35614}, {2171, 3888}, {2179, 50947}, {2333, 53349}, {2489, 7253}, {2501, 21300}, {2643, 69}, {2971, 5905}, {3120, 17137}, {3121, 1}, {3122, 75}, {3123, 34086}, {3124, 8}, {3125, 17135}, {3248, 4360}, {3271, 21273}, {3572, 4155}, {3708, 1370}, {4017, 4374}, {4024, 21301}, {4036, 21304}, {4079, 513}, {4092, 21286}, {4117, 194}, {4516, 20245}, {4705, 20295}, {7063, 3177}, {7143, 35312}, {7148, 1978}, {8029, 21294}, {8754, 21270}, {9406, 14611}, {9427, 17486}, {16732, 17138}, {18070, 688}, {18210, 18659}, {18757, 4610}, {18832, 4609}, {20975, 4329}, {21043, 21287}, {21131, 21293}, {21725, 30660}, {21833, 1330}, {22260, 21221}, {23099, 21220}, {23894, 53365}, {23994, 33796}, {34294, 21278}, {36142, 33919}, {38252, 4563}, {40525, 25295}, {41683, 53366}, {42068, 21216}, {43763, 880}, {46289, 10330}, {50487, 514}, {51641, 4467}, {51906, 17165}, {53540, 20244}, {53581, 17494}
X(54104) = X(i)-Ceva conjugate of X(j) for these (i,j): {115, 2}, {31998, 523}
X(54104) = barycentric product X(i)*X(j) for these {i,j}: {76, 33704}, {99, 13187}
X(54104) = barycentric quotient X(i)/X(j) for these {i,j}: {13187, 523}, {33704, 6}
X(54104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {148, 44373, 39356}, {4590, 23991, 2}

X(54105) = X(2)X(36422)∩X(5)X(95)

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

X(54105) lies on these lines: {2, 36422}, {4, 46724}, {5, 95}, {69, 3855}, {99, 33643}, {233, 40853}, {264, 339}, {311, 15031}, {317, 3091}, {340, 3850}, {467, 19188}, {546, 45198}, {648, 17035}, {1232, 7809}, {3832, 6527}, {3839, 40680}, {3843, 20477}, {3854, 44134}, {3858, 41005}, {5066, 41008}, {5072, 52712}, {7773, 44149}, {14644, 19166}, {32001, 41106}, {36794, 52247}, {40897, 43982}

X(54105) = isotomic conjugate of X(43970)
X(54105) = anticomplement of X(36422)
X(54105) = isotomic conjugate of the isogonal conjugate of X(38848)
X(54105) = X(31)-isoconjugate of X(43970)
X(54105) = X(2)-Dao conjugate of X(43970)
X(54105) = barycentric product X(i)*X(j) for these {i,j}: {76, 38848}, {34987, 42405}
X(54105) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43970}, {34987, 17434}, {38848, 6}
X(54105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 95, 40410}, {5, 32002, 95}, {17035, 36412, 648}

X(54106) = X(2)X(5034)∩X(147)X(193)

Barycentrics    a^8 - 2*a^6*b^2 - 4*a^4*b^4 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 - 10*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 2*b^6*c^2 - 4*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(54106) lies on these lines: {2, 5034}, {147, 193}, {148, 44434}, {194, 315}, {262, 1352}, {385, 9744}, {1992, 43450}, {2548, 7760}, {2896, 10513}, {3767, 39101}, {3832, 6392}, {7615, 19570}, {7796, 7800}, {7837, 20423}, {8721, 20065}, {9890, 39652}, {15993, 51580}

X(54106) = X(3402)-anticomplementary conjugate of X(6194)

X(54107) = X(2)X(6354)∩X(19)X(27)

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

X(54107) lies on these lines: {2, 6354}, {8, 6253}, {9, 20921}, {19, 27}, {40, 52346}, {144, 5739}, {190, 329}, {238, 24218}, {322, 3719}, {347, 37669}, {394, 664}, {489, 46421}, {490, 46422}, {527, 45206}, {653, 15466}, {908, 33116}, {1043, 3869}, {1121, 2994}, {1214, 1944}, {1726, 6996}, {1782, 37088}, {1948, 40152}, {2184, 5931}, {2975, 11101}, {3218, 17862}, {3219, 30807}, {4360, 18662}, {6180, 18663}, {6335, 18736}, {6350, 18134}, {6508, 40882}, {6515, 20078}, {6604, 9965}, {10025, 49514}, {12848, 18928}, {13567, 17950}, {17080, 28950}, {17086, 23292}, {17147, 40571}, {22001, 23512}, {25091, 27420}, {30695, 41821}, {33673, 47848}, {35614, 38480}, {44447, 52365}

X(54107) = anticomplement of X(6354)
X(54107) = anticomplement of the isogonal conjugate of X(7054)
X(54107) = anticomplement of the isotomic conjugate of X(7058)
X(54107) = isotomic conjugate of the isogonal conjugate of X(1630)
X(54107) = polar conjugate of the isogonal conjugate of X(3561)
X(54107) = X(7058)-Ceva conjugate of X(2)
X(54107) = cevapoint of X(1630) and X(3561)
X(54107) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {21, 2893}, {60, 7}, {250, 4566}, {261, 21285}, {283, 2897}, {284, 2475}, {593, 36845}, {757, 6604}, {849, 4452}, {1021, 3448}, {1043, 21287}, {1098, 69}, {1101, 664}, {1253, 46707}, {2150, 145}, {2185, 3434}, {2189, 12649}, {2193, 3152}, {2194, 17778}, {2287, 1330}, {2326, 4}, {2327, 52364}, {2328, 2895}, {4556, 3900}, {4612, 21302}, {4636, 693}, {6061, 329}, {7054, 8}, {7058, 6327}, {7253, 21294}, {21789, 21221}, {23609, 63}, {24000, 18026}, {36069, 36038}, {36421, 5906}, {52379, 21280}, {52914, 46400}, {52935, 46402}
X(54107) = barycentric product X(i)*X(j) for these {i,j}: {75, 411}, {76, 1630}, {264, 3561}, {312, 34035}, {561, 44087}
X(54107) = barycentric quotient X(i)/X(j) for these {i,j}: {411, 1}, {1630, 6}, {3561, 3}, {34035, 57}, {44087, 31}
X(54107) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 92, 333}, {63, 20223, 75}, {63, 45738, 18750}, {394, 6360, 664}

X(54108) = X(99)X(6563)∩X(110)X(685)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(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(54108) lies on these lines: {99, 6563}, {110, 685}, {184, 44155}, {250, 47258}, {316, 33803}, {647, 40866}, {648, 23964}, {892, 4577}, {2966, 23357}, {3047, 23962}, {5641, 9143}, {9514, 23878}, {14480, 41298}, {16077, 18831}, {36830, 36900}

X(54108) = isotomic conjugate of the isogonal conjugate of X(38861)
X(54108) = X(14966)-Dao conjugate of X(11672)
X(54108) = barycentric product X(i)*X(j) for these {i,j}: {76, 38861}, {6331, 13198}, {21525, 43187}
X(54108) = barycentric quotient X(i)/X(j) for these {i,j}: {13198, 647}, {21525, 3569}, {38861, 6}
X(54108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 850, 18020}, {23357, 31296, 2966}

X(54109) = X(2)X(31643)∩X(19)X(27)

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

X(54109) lies on these lines: {2, 31643}, {19, 27}, {261, 2975}, {314, 2995}, {668, 21286}, {894, 24993}, {1409, 1944}, {3436, 3596}, {4329, 14615}, {6335, 18747}, {10447, 12526}, {17137, 20245}, {17143, 21273}, {20345, 20348}, {24547, 38000}

X(54109) = isotomic conjugate of X(42485)
X(54109) = anticomplement of the isogonal conjugate of X(7058)
X(54109) = isotomic conjugate of the anticomplement of X(15267)
X(54109) = isotomic conjugate of the isogonal conjugate of X(1610)
X(54109) = X(31)-isoconjugate of X(42485)
X(54109) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 42485}, {19608, 42550}
X(54109) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {21, 17778}, {60, 3210}, {200, 46707}, {261, 7}, {270, 30699}, {283, 18667}, {314, 2893}, {332, 2897}, {333, 2475}, {593, 17480}, {757, 4452}, {849, 46716}, {873, 6604}, {1021, 148}, {1043, 2895}, {1098, 2}, {1253, 46714}, {1509, 36845}, {1792, 3151}, {1812, 3152}, {2185, 145}, {2287, 1654}, {2326, 193}, {2327, 18666}, {2328, 1655}, {4610, 3900}, {4612, 522}, {4623, 46402}, {4631, 21302}, {4636, 17496}, {6061, 3177}, {6064, 21272}, {6628, 17158}, {7054, 192}, {7058, 8}, {7253, 21221}, {7259, 31290}, {7340, 35312}, {18020, 4566}, {18021, 21285}, {21789, 21220}, {23999, 18026}, {24041, 664}, {46103, 12649}, {52379, 3434}, {52380, 37759}, {52935, 4025}
X(54109) = barycentric product X(i)*X(j) for these {i,j}: {75, 23512}, {76, 1610}
X(54109) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 42485}, {1610, 6}, {23512, 1}, {34267, 34434}
X(54109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2995, 3869, 314}, {20245, 20246, 17137}

X(54110) = X(2)X(31611)∩X(99)X(901)

Barycentrics    (a - b)*(a - 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(54110) lies on these lines: {2, 31611}, {99, 901}, {100, 693}, {101, 48008}, {106, 24191}, {190, 4468}, {320, 50744}, {514, 41405}, {650, 40865}, {664, 4025}, {666, 1252}, {4762, 14589}, {5375, 31150}, {6606, 35157}, {8047, 18821}, {14513, 20295}, {26692, 30610}, {26824, 43986}, {31615, 43991}, {35119, 41395}

X(54110) = isotomic conjugate of X(43974)
X(54110) = isotomic conjugate of the isogonal conjugate of X(1618)
X(54110) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2149, 17036}, {4619, 149}, {31615, 33650}
X(54110) = X(i)-isoconjugate of X(j) for these (i,j): {31, 43974}, {663, 43947}
X(54110) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 43974}, {2284, 6184}
X(54110) = cevapoint of X(100) and X(43991)
X(54110) = barycentric product X(i)*X(j) for these {i,j}: {76, 1618}, {190, 24203}
X(54110) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43974}, {651, 43947}, {1618, 6}, {24203, 514}
X(54110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 693, 4998}, {1252, 17494, 666}, {17494, 51357, 1252}

X(54111) = X(2)X(6)∩X(20)X(40995)

Barycentrics    7*a^8 - 12*a^6*b^2 - 6*a^4*b^4 + 20*a^2*b^6 - 9*b^8 - 12*a^6*c^2 + 28*a^4*b^2*c^2 - 20*a^2*b^4*c^2 + 4*b^6*c^2 - 6*a^4*c^4 - 20*a^2*b^2*c^4 + 10*b^4*c^4 + 20*a^2*c^6 + 4*b^2*c^6 - 9*c^8 : :

X(54111) lies on these lines: {2, 6}, {20, 40995}, {253, 3146}, {264, 50689}, {297, 17037}, {315, 30698}, {317, 17578}, {340, 5059}, {1494, 52443}, {3522, 41005}, {3785, 38437}, {3832, 32000}, {3854, 44134}, {15717, 41008}, {20218, 39352}, {21734, 40680}, {22468, 32830}, {32879, 52347}, {32882, 34007}, {44334, 45245}, {46724, 50693}

X(54111) = anticomplement of X(36413)
X(54111) = isotomic conjugate of the isogonal conjugate of X(1620)
X(54111) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2155, 17037}, {2184, 6225}, {31942, 5905}, {52559, 4329}, {53886, 7192}
X(54111) = X(38253)-Dao conjugate of X(33893)
X(54111) = barycentric product X(76)*X(1620)
X(54111) = barycentric quotient X(1620)/X(6)
X(54111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {253, 32001, 3146}, {253, 40996, 35510}, {1270, 1271, 37669}, {3146, 35510, 253}, {32001, 40996, 253}

X(54112) = X(1)X(17208)∩X(2)X(1258)

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

X(54112) lies on these lines: {1, 17208}, {2, 1258}, {69, 9054}, {310, 2388}, {668, 40007}, {3051, 26810}, {4360, 30941}, {4576, 35614}, {6327, 20554}, {6653, 32863}, {20290, 20561}, {20347, 44139}, {21280, 33796}, {27152, 40728}, {29824, 34020}, {30109, 40586}

X(54112) = anticomplement of X(7109)
X(54112) = isotomic conjugate of the isogonal conjugate of X(23374)
X(54112) = anticomplementary isogonal conjugate of X(46714)
X(54112) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 46714}, {75, 46707}, {86, 1655}, {261, 3177}, {274, 1654}, {310, 2895}, {552, 3210}, {593, 17486}, {757, 194}, {763, 17148}, {799, 31290}, {849, 8264}, {873, 2}, {1019, 25054}, {1098, 46706}, {1509, 192}, {2185, 21218}, {4610, 17494}, {4623, 514}, {4631, 4468}, {6385, 1330}, {6628, 17147}, {7192, 21220}, {7199, 148}, {7304, 41840}, {7307, 32937}, {7340, 4552}, {17206, 18666}, {18021, 329}, {24037, 190}, {24041, 46725}, {30940, 39367}, {34537, 3952}, {46254, 3732}, {52379, 144}, {52612, 513}, {52619, 21221}, {52935, 21225}
X(54112) = barycentric product X(i)*X(j) for these {i,j}: {75, 33792}, {76, 23374}
X(54112) = barycentric quotient X(i)/X(j) for these {i,j}: {23374, 6}, {33792, 1}
X(54112) = {X(8049),X(17135)}-harmonic conjugate of X(310)

X(54113) = X(2)X(1407)∩X(4)X(29958)

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

X(54113) lies on these lines: {2, 1407}, {4, 29958}, {7, 18928}, {8, 12688}, {63, 2183}, {69, 189}, {144, 5739}, {222, 27539}, {223, 40880}, {321, 5942}, {534, 17781}, {651, 27540}, {664, 20211}, {908, 1997}, {1265, 52366}, {2390, 3436}, {2895, 30695}, {5658, 7360}, {5905, 6604}, {6223, 52346}, {6515, 17484}, {10327, 11678}, {12324, 52345}, {14361, 18026}, {17074, 28794}, {20554, 20557}, {21361, 36698}, {27509, 34048}, {30566, 37781}

X(54113) = isotomic conjugate of X(34546)
X(54113) = anticomplement of X(1407)
X(54113) = anticomplement of the isogonal conjugate of X(346)
X(54113) = isotomic conjugate of the anticomplement of X(6609)
X(54113) = isotomic conjugate of the isogonal conjugate of X(1604)
X(54113) = anticomplementary isogonal conjugate of X(4452)
X(54113) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 4452}, {2, 36845}, {6, 17480}, {8, 7}, {9, 145}, {19, 11851}, {21, 3875}, {31, 46716}, {33, 30699}, {55, 3210}, {75, 6604}, {78, 347}, {86, 17158}, {100, 4025}, {190, 3900}, {200, 2}, {210, 17778}, {220, 192}, {281, 12649}, {312, 3434}, {314, 20244}, {333, 3873}, {341, 69}, {345, 52365}, {346, 8}, {480, 3177}, {643, 4467}, {644, 522}, {646, 21302}, {657, 9263}, {668, 46402}, {728, 144}, {765, 664}, {1021, 17154}, {1043, 75}, {1098, 4360}, {1222, 39126}, {1253, 194}, {1257, 3668}, {1260, 6360}, {1261, 57}, {1265, 4329}, {1320, 1266}, {1792, 17134}, {1802, 3164}, {1897, 17896}, {2287, 1}, {2318, 18667}, {2321, 2475}, {2322, 3868}, {2327, 20222}, {2328, 17147}, {3119, 17036}, {3239, 149}, {3596, 21285}, {3680, 4373}, {3689, 30577}, {3692, 20}, {3693, 52164}, {3694, 3152}, {3699, 693}, {3701, 2893}, {3710, 2897}, {3900, 4440}, {3939, 17496}, {3965, 5484}, {4076, 21272}, {4082, 2895}, {4102, 20292}, {4110, 20350}, {4130, 39351}, {4163, 37781}, {4171, 148}, {4183, 3187}, {4397, 150}, {4420, 41808}, {4511, 41803}, {4515, 1654}, {4524, 21220}, {4578, 514}, {4607, 30704}, {4998, 35312}, {5423, 329}, {6061, 18662}, {6065, 4552}, {6555, 8055}, {6556, 21296}, {6558, 513}, {6559, 518}, {6602, 21218}, {6605, 3870}, {6726, 16018}, {6731, 7057}, {6735, 36918}, {7046, 5905}, {7058, 17140}, {7071, 21216}, {7079, 193}, {7080, 5932}, {7101, 4}, {7256, 7192}, {7257, 4374}, {7258, 512}, {7259, 523}, {8641, 21224}, {14427, 39349}, {14827, 17486}, {15742, 4566}, {23617, 36846}, {27398, 20221}, {27538, 20537}, {28071, 239}, {28659, 21280}, {30681, 52366}, {30693, 3436}, {31343, 3676}, {32008, 30628}, {32635, 3879}, {36802, 53357}, {36916, 12648}, {39959, 51351}, {40435, 16465}, {41798, 26015}, {44693, 41804}, {51562, 36038}, {52371, 37759}, {52406, 1370}, {52549, 14923}, {52622, 21293}, {52663, 38460}
X(54113) = X(i)-isoconjugate of X(j) for these (i,j): {31, 34546}, {604, 2123}
X(54113) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 34546}, {3161, 2123}
X(54113) = barycentric product X(i)*X(j) for these {i,j}: {76, 1604}, {2122, 3596}
X(54113) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34546}, {8, 2123}, {1604, 6}, {2122, 56}, {6609, 1407}
X(54113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {189, 329, 312}, {651, 27540, 37669}, {908, 26871, 18141}, {5905, 11433, 6604}

X(54114) = ISOGONAL CONJUGATE OF X(32445)

Barycentrics    (a^6*b^2 - 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 + 2*a^4*c^4 - a^2*b^2*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 - 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(54114) lies on the cubics K146 and K1010 and on these lines: {2, 9291}, {3, 3164}, {4, 14941}, {20, 39682}, {97, 46717}, {194, 17974}, {324, 34287}, {394, 401}, {458, 1073}, {1972, 40815}, {2052, 35061}, {3346, 43981}, {3682, 25252}, {3926, 44137}, {14376, 37337}, {18027, 35071}, {38256, 40896}

X(54114) = reflection of X(4) in X(46033)
X(54114) = isogonal conjugate of X(32445)
X(54114) = isotomic conjugate of X(3164)
X(54114) = polar conjugate of X(3168)
X(54114) = cyclocevian conjugate of X(44175)
X(54114) = anticomplement of the isotomic conjugate of X(40800)
X(54114) = isotomic conjugate of the anticomplement of X(264)
X(54114) = isotomic conjugate of the complement of X(40896)
X(54114) = isotomic conjugate of the isogonal conjugate of X(1988)
X(54114) = isotomic conjugate of the polar conjugate of X(43710)
X(54114) = polar conjugate of the isogonal conjugate of X(40800)
X(54114) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1988, 21270}, {40800, 6327}, {44828, 21300}
X(54114) = X(40800)-Ceva conjugate of X(2)
X(54114) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32445}, {19, 6638}, {31, 3164}, {48, 3168}, {1953, 26887}, {2148, 42453}
X(54114) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3164}, {3, 32445}, {6, 6638}, {216, 42453}, {1249, 3168}
X(54114) = cevapoint of X(i) and X(j) for these (i,j): {2, 40896}, {525, 35071}, {1988, 40800}, {2799, 38974}
X(54114) = trilinear pole of line {520, 6130}
X(54114) = barycentric product X(i)*X(j) for these {i,j}: {69, 43710}, {76, 1988}, {264, 40800}, {850, 44828}
X(54114) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3164}, {3, 6638}, {4, 3168}, {5, 42453}, {6, 32445}, {54, 26887}, {1988, 6}, {40800, 3}, {43710, 4}, {44828, 110}

X(54115) = CEVAPOINT OF X(2) AND X(40900)

Barycentrics    (a^2 + b^2 - 3*c^2 - 2*Sqrt[3]*S)*(a^2 - 3*b^2 + c^2 - 2*Sqrt[3]*S) : :

X(54115) lies on the Kiepert circumhyperbola and these lines: {2, 53463}, {4, 5615}, {13, 3181}, {17, 623}, {69, 5103}, {76, 34540}, {148, 627}, {193, 22235}, {194, 43538}, {298, 11122}, {302, 7783}, {621, 7793}, {626, 40706}, {3091, 43954}, {3180, 43542}, {3391, 33395}, {3392, 33393}, {3552, 11489}, {6658, 53441}, {7862, 11129}, {9886, 42063}, {16001, 16626}, {32961, 34541}, {32993, 40900}, {33477, 42062}, {37172, 43543}, {40693, 50211}, {43447, 47518}

X(54115) = isotomic conjugate of the anticomplement of X(302)
X(54115) = isotomic conjugate of the complement of X(40900)
X(54115) = X(619)-Dao conjugate of X(10616)
X(54115) = cevapoint of X(i) and X(j) for these (i,j): {2, 40900}, {115, 23872}
X(54115) = trilinear pole of line {523, 44384}
X(54115) = barycentric quotient X(395)/X(10616)

X(54116) = CEVAPOINT OF X(2) AND X(40901)

Barycentrics    (a^2 + b^2 - 3*c^2 + 2*Sqrt[3]*S)*(a^2 - 3*b^2 + c^2 + 2*Sqrt[3]*S) : :

X(54116) lies on the Kiepert circumhyperbola and these lines: {2, 53452}, {4, 5611}, {14, 3180}, {18, 624}, {69, 5103}, {76, 34541}, {148, 628}, {193, 22237}, {194, 43539}, {299, 11121}, {303, 7783}, {622, 7793}, {626, 40707}, {3091, 43953}, {3181, 43543}, {3366, 33392}, {3367, 33394}, {3552, 11488}, {6658, 53429}, {7862, 11128}, {9885, 42062}, {16002, 16627}, {32961, 34540}, {32993, 40901}, {33476, 42063}, {37173, 43542}, {40694, 50212}, {43446, 47520}

X(54116) = isotomic conjugate of the anticomplement of X(303)
X(54116) = isotomic conjugate of the complement of X(40901)
X(54116) = X(618)-Dao conjugate of X(10617)
X(54116) = cevapoint of X(i) and X(j) for these (i,j): {2, 40901}, {115, 23873}
X(54116) = trilinear pole of line {523, 44385}
X(54116) = barycentric quotient X(396)/X(10617)

X(54117) = ISOGONAL CONJUGATE OF X(21779)

Barycentrics    (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(54117) lies on the circumconic {{A,.B,C, X(2),X(6)}}, the cubic K132, and these lines: {1, 9403}, {2, 34021}, {6, 2669}, {25, 16998}, {37, 1655}, {42, 894}, {75, 25054}, {111, 53631}, {192, 39926}, {193, 45966}, {194, 941}, {694, 18827}, {1400, 7176}, {1654, 30669}, {2054, 30667}, {2248, 33295}, {2998, 34284}, {3572, 16710}, {6625, 17493}, {6646, 21220}, {8033, 40729}, {8770, 16992}, {9263, 24437}, {25457, 39798}, {27318, 39956}, {37632, 39967}

X(54117) = isogonal conjugate of X(21779)
X(54117) = isotomic conjugate of X(1655)
X(54117) = anticomplement of X(34021)
X(54117) = isotomic conjugate of the anticomplement of X(274)
X(54117) = isotomic conjugate of the complement of X(40908)
X(54117) = isotomic conjugate of the isogonal conjugate of X(40770)
X(54117) = isogonal conjugate of the isotomic conjugate of X(43684)
X(54117) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {40737, 17137}, {40770, 17135}
X(54117) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21779}, {2, 18756}, {6, 1045}, {19, 23079}, {31, 1655}, {32, 51863}, {37, 51330}, {58, 21883}, {213, 39915}, {662, 9402}, {869, 40752}, {1918, 34021}, {7104, 27890}, {40728, 40743}
X(54117) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1655}, {3, 21779}, {6, 23079}, {9, 1045}, {10, 21883}, {1084, 9402}, {6376, 51863}, {6626, 39915}, {32664, 18756}, {40589, 51330}
X(54117) = cevapoint of X(i) and X(j) for these (i,j): {2, 40908}, {513, 1084}, {514, 16592}, {650, 3023}
X(54117) = trilinear pole of line {512, 4369}
X(54117) = barycentric product X(i)*X(j) for these {i,j}: {1, 18298}, {6, 43684}, {75, 40737}, {76, 40770}, {523, 53631}, {18827, 39926}
X(54117) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1045}, {2, 1655}, {3, 23079}, {6, 21779}, {31, 18756}, {37, 21883}, {58, 51330}, {75, 51863}, {86, 39915}, {274, 34021}, {512, 9402}, {870, 40743}, {1909, 27890}, {14621, 40752}, {18298, 75}, {39926, 740}, {40737, 1}, {40770, 6}, {40778, 2276}, {43684, 76}, {53631, 99}
X(54117) = {X(37),X(46714)}-harmonic conjugate of X(1655)

X(54118) = ISOGONAL CONJUGATE OF X(21007)

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

X(54118) lies on these lines: {2, 40216}, {37, 16727}, {100, 17494}, {149, 14947}, {190, 4576}, {192, 13476}, {321, 16728}, {335, 3995}, {693, 26795}, {835, 43076}, {1025, 4552}, {1026, 3952}, {1897, 4238}, {2350, 17027}, {2481, 27190}, {3177, 44005}, {3939, 43190}, {4033, 42720}, {4080, 17244}, {4427, 4613}, {4554, 26985}, {4671, 25242}, {5701, 26846}, {6184, 23989}, {7192, 46148}, {17198, 22032}, {18359, 25244}, {23988, 27009}, {24484, 25049}, {25257, 46802}, {25264, 39698}, {27115, 30610}, {31100, 31125}, {31615, 51357}, {32041, 47869}

X(54118) = isogonal conjugate of X(21007)
X(54118) = isotomic conjugate of X(17494)
X(54118) = anticomplement of X(40619)
X(54118) = isotomic conjugate of the anticomplement of X(693)
X(54118) = isotomic conjugate of the complement of X(26824)
X(54118) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1252, 40007}, {23990, 16552}
X(54118) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21007}, {6, 4040}, {19, 22160}, {31, 17494}, {32, 20954}, {100, 38346}, {109, 38347}, {163, 2486}, {513, 4251}, {593, 21727}, {649, 1621}, {651, 38365}, {657, 38859}, {667, 17277}, {692, 17761}, {1333, 4151}, {1919, 17143}, {1980, 18152}, {2149, 42454}, {3294, 3733}, {8641, 33765}, {14004, 22383}, {27168, 34248}, {32739, 40619}
X(54118) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17494}, {3, 21007}, {6, 22160}, {9, 4040}, {11, 38347}, {37, 4151}, {115, 2486}, {650, 42454}, {1086, 17761}, {5375, 1621}, {6376, 20954}, {6631, 17277}, {8054, 38346}, {9296, 17143}, {32746, 27168}, {38991, 38365}, {39026, 4251}
X(54118) = cevapoint of X(i) and X(j) for these (i,j): {2, 26824}, {37, 514}, {39, 513}, {522, 1212}, {523, 16589}, {525, 18591}, {650, 3058}, {693, 40216}, {918, 6184}
X(54118) = trilinear pole of line {10, 141}
X(54118) = crossdifference of every pair of points on line {38346, 38365}
X(54118) = barycentric product X(i)*X(j) for these {i,j}: {10, 53649}, {100, 40216}, {190, 17758}, {313, 43076}, {668, 13476}, {1018, 40004}, {1978, 2350}, {3952, 39734}, {4033, 39950}
X(54118) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4040}, {2, 17494}, {3, 22160}, {6, 21007}, {10, 4151}, {11, 42454}, {75, 20954}, {100, 1621}, {101, 4251}, {190, 17277}, {194, 27168}, {514, 17761}, {523, 2486}, {649, 38346}, {650, 38347}, {658, 33765}, {663, 38365}, {668, 17143}, {693, 40619}, {756, 21727}, {934, 38859}, {1018, 3294}, {1897, 14004}, {1978, 18152}, {2350, 649}, {3699, 3996}, {3952, 4651}, {4033, 4043}, {4583, 40094}, {6386, 40088}, {13476, 513}, {14549, 46385}, {17494, 26846}, {17496, 26847}, {17758, 514}, {21859, 20616}, {39734, 7192}, {39950, 1019}, {40004, 7199}, {40216, 693}, {40521, 40607}, {43076, 58}, {53649, 86}
X(54118) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 46725, 17494}, {693, 35310, 26795}, {4554, 27134, 26985}, {6184, 23989, 27072}

X(54119) = ISOGONAL CONJUGATE OF X(2305)

Barycentrics    (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(54119) lies on the Kiepert circumhyperbola and these lines: {2, 5110}, {4, 2651}, {8, 42066}, {10, 846}, {63, 148}, {76, 37653}, {81, 6625}, {98, 19642}, {115, 7058}, {226, 1943}, {312, 26081}, {321, 1654}, {1029, 16704}, {1947, 17950}, {2051, 32431}, {2895, 4080}, {2996, 14552}, {3896, 5086}, {4049, 21192}, {4362, 20558}, {4388, 11599}, {19734, 33030}, {21935, 26051}, {24789, 26147}, {26109, 30588}, {31290, 43669}, {40718, 40751}

X(54119) = isogonal conjugate of X(2305)
X(54119) = isotomic conjugate of X(17778)
X(54119) = anticomplement of X(40605)
X(54119) = polar conjugate of X(3144)
X(54119) = isotomic conjugate of the anticomplement of X(333)
X(54119) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1247, 20245}, {36934, 21286}
X(54119) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2305}, {6, 1046}, {31, 17778}, {48, 3144}, {1333, 3178}, {1402, 40605}, {1409, 2907}
X(54119) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17778}, {3, 2305}, {9, 1046}, {37, 3178}, {1249, 3144}
X(54119) = cevapoint of X(i) and X(j) for these (i,j): {6, 20836}, {115, 522}, {513, 16613}, {514, 17058}, {521, 16573}
X(54119) = trilinear pole of line {523, 8045}
X(54119) = barycentric product X(i)*X(j) for these {i,j}: {75, 1247}, {86, 36934}, {850, 53633}
X(54119) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1046}, {2, 17778}, {4, 3144}, {6, 2305}, {10, 3178}, {29, 2907}, {333, 40605}, {1247, 1}, {6740, 36927}, {36934, 10}, {53633, 110}

X(54120) = ISOGONAL CONJUGATE OF X(21008)

Barycentrics    (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(54120) lies on these lines: {8, 1757}, {85, 26806}, {257, 6646}, {312, 3765}, {333, 1931}, {1146, 6645}, {2170, 16044}, {3959, 6650}, {4518, 40794}, {4997, 5718}, {5252, 31359}, {5308, 38255}, {6557, 17316}, {10950, 14942}, {18031, 26541}, {21272, 33820}, {24247, 53675}, {24735, 30608}, {26223, 34527}, {28660, 52137}, {39351, 40845}, {42030, 50048}

X(54120) = isogonal conjugate of X(21008)
X(54120) = isotomic conjugate of X(6646)
X(54120) = isotomic conjugate of the anticomplement of X(894)
X(54120) = isotomic conjugate of the complement of X(31300)
X(54120) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21008}, {6, 17596}, {19, 22161}, {31, 6646}, {32, 20955}, {692, 21212}
X(54120) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6646}, {3, 21008}, {6, 22161}, {9, 17596}, {1086, 21212}, {6376, 20955}
X(54120) = cevapoint of X(i) and X(j) for these (i,j): {2, 31300}, {1146, 3907}, {3805, 53823}
X(54120) = trilinear pole of line {522, 4874}
X(54120) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17596}, {2, 6646}, {3, 22161}, {6, 21008}, {75, 20955}, {514, 21212}

X(54121) = ISOGONAL CONJUGATE OF X(20986)

Barycentrics    b*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(54121) lies on these lines: {1, 2995}, {63, 53081}, {75, 3869}, {92, 18662}, {261, 2975}, {286, 40440}, {311, 313}, {321, 908}, {329, 34262}, {388, 8048}, {693, 41007}, {1441, 4357}, {2171, 4858}, {2517, 44412}, {2861, 53702}, {2997, 3875}, {3112, 11688}, {3436, 30479}, {3596, 11681}, {4360, 14616}, {5253, 31643}, {10447, 11682}, {17861, 40619}, {18698, 40216}, {20060, 52442}, {24220, 40624}, {30807, 49516}

X(54121) = isogonal conjugate of X(20986)
X(54121) = isotomic conjugate of X(2975)
X(54121) = isotomic conjugate of the anticomplement of X(12)
X(54121) = isotomic conjugate of the complement of X(20060)
X(54121) = isotomic conjugate of the isogonal conjugate of X(34434)
X(54121) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {60, 1764}, {20028, 2893}, {46880, 1330}, {52150, 17778}, {53083, 2475}
X(54121) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20986}, {6, 572}, {19, 22118}, {31, 2975}, {32, 14829}, {41, 17074}, {58, 52139}, {184, 11109}, {692, 21173}, {849, 14973}, {1169, 52087}, {1333, 21061}, {2149, 11998}, {2194, 37558}, {2206, 17751}, {4264, 34278}, {7115, 38344}, {8750, 23187}, {14586, 52322}, {17496, 32739}, {23990, 24237}
X(54121) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2975}, {3, 20986}, {6, 22118}, {9, 572}, {10, 52139}, {37, 21061}, {650, 11998}, {693, 26847}, {1086, 21173}, {1214, 37558}, {1577, 34589}, {2051, 23361}, {2092, 46879}, {3160, 17074}, {4075, 14973}, {6376, 14829}, {26932, 23187}, {40603, 17751}, {40619, 17496}, {40622, 51662}, {40628, 38344}
X(54121) = cevapoint of X(i) and X(j) for these (i,j): {2, 20060}, {523, 4858}, {525, 34588}, {6370, 46398}, {16732, 50330}
X(54121) = trilinear pole of line {1577, 3910}
X(54121) = barycentric product X(i)*X(j) for these {i,j}: {75, 2051}, {76, 34434}, {274, 51870}, {313, 53083}, {321, 20028}, {1228, 40453}, {1441, 46880}, {27801, 52150}
X(54121) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 572}, {2, 2975}, {3, 22118}, {6, 20986}, {7, 17074}, {10, 21061}, {11, 11998}, {37, 52139}, {75, 14829}, {92, 11109}, {226, 37558}, {321, 17751}, {514, 21173}, {594, 14973}, {693, 17496}, {905, 23187}, {960, 46879}, {1111, 24237}, {1441, 52358}, {2051, 1}, {2292, 52087}, {2618, 52322}, {4858, 34589}, {6354, 20617}, {6358, 52357}, {7004, 38344}, {7178, 51662}, {16732, 53566}, {20028, 81}, {20906, 27346}, {34267, 1610}, {34387, 40624}, {34434, 6}, {40453, 1169}, {40619, 26847}, {46880, 21}, {51870, 37}, {52150, 1333}, {53083, 58}, {53702, 32641}

X(54122) = ISOGONAL CONJUGATE OF X(5017)

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

X(54122) lies on the Kiepert circumhyperbola and these lines: {2, 12215}, {3, 46323}, {4, 385}, {69, 1916}, {76, 2549}, {83, 3767}, {98, 17008}, {148, 6194}, {193, 14484}, {194, 3399}, {262, 1352}, {376, 9302}, {459, 37187}, {598, 7615}, {671, 7811}, {2052, 17984}, {2996, 6655}, {3314, 40824}, {3407, 7735}, {3424, 37667}, {4019, 43534}, {5152, 9890}, {5304, 5395}, {5485, 32986}, {6392, 37336}, {7394, 30505}, {7612, 17004}, {7777, 14494}, {7783, 16043}, {7795, 10159}, {7828, 43527}, {7875, 18841}, {7897, 35005}, {8587, 23055}, {9740, 41895}, {9755, 37348}, {9770, 10484}, {10155, 17005}, {10302, 52691}, {11177, 32528}, {14492, 20423}, {16925, 38907}, {16987, 32957}, {16988, 32960}, {17006, 53103}, {18842, 32983}, {18845, 33018}, {33019, 38259}, {33279, 53105}, {37242, 47286}

X(54122) = isogonal conjugate of X(5017)
X(54122) = isotomic conjugate of X(7774)
X(54122) = anticomplement of X(51580)
X(54122) = isotomic conjugate of the anticomplement of X(183)
X(54122) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5017}, {31, 7774}, {662, 50550}, {3402, 51580}
X(54122) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7774}, {3, 5017}, {1084, 50550}
X(54122) = cevapoint of X(115) and X(23878)
X(54122) = trilinear pole of line {523, 24284}
X(54122) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7774}, {6, 5017}, {183, 51580}, {512, 50550}
X(54122) = {X(262),X(54106)}-harmonic conjugate of X(7774)

X(54123) = ISOGONAL CONJUGATE OF X(16781)

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

X(54123) lies on the circumconic {{A,B,C,X(1),X(2)}} and these lines: {1, 4899}, {8, 39954}, {28, 20009}, {57, 3208}, {81, 29585}, {88, 29583}, {105, 145}, {192, 279}, {330, 346}, {985, 37588}, {1022, 49277}, {1219, 17280}, {1390, 3622}, {1432, 51058}, {3672, 39724}, {3912, 8056}, {4462, 30694}, {15474, 30699}, {17776, 39696}, {25430, 26626}, {29573, 36603}, {29574, 39980}, {29579, 39963}

X(54123) = isogonal conjugate of X(16781)
X(54123) = isotomic conjugate of the anticomplement of X(344)
X(54123) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16781}, {6, 5272}, {58, 16605}
X(54123) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 16781}, {9, 5272}, {10, 16605}
X(54123) = cevapoint of X(i) and X(j) for these (i,j): {37, 4028}, {1015, 3309}
X(54123) = trilinear pole of line {513, 2977}
X(54123) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 5272}, {6, 16781}, {37, 16605}

X(54124) = ISOGONAL CONJUGATE OF X(3148)

Barycentrics    (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(54124) lies on the Jerabek circumhyperbola and these lines: {2, 248}, {3, 315}, {6, 297}, {66, 264}, {67, 44134}, {68, 76}, {69, 40073}, {70, 1235}, {71, 4150}, {72, 42703}, {83, 46765}, {184, 34405}, {265, 11185}, {290, 1899}, {311, 18124}, {316, 4846}, {327, 1352}, {340, 5486}, {491, 6413}, {492, 6414}, {850, 879}, {1176, 20022}, {1177, 52486}, {1987, 39931}, {5012, 44175}, {5504, 15920}, {5641, 11179}, {7768, 42021}, {7774, 22240}, {7792, 52251}, {8840, 43722}, {9723, 42407}, {9766, 35937}, {11003, 13485}, {11005, 34802}, {14498, 43448}, {15740, 32006}, {16774, 32000}, {16989, 43706}, {17040, 32001}, {18125, 44135}, {32002, 43726}, {32618, 44781}, {32619, 44780}, {34765, 35909}, {37190, 43711}

X(54124) = isogonal conjugate of X(3148)
X(54124) = isotomic conjugate of X(1352)
X(54124) = isotomic conjugate of the anticomplement of X(182)
X(54124) = isotomic conjugate of the complement of X(6776)
X(54124) = isotomic conjugate of the isogonal conjugate of X(3425)
X(54124) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3148}, {6, 16567}, {31, 1352}
X(54124) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1352}, {3, 3148}, {9, 16567}
X(54124) = cevapoint of X(i) and X(j) for these (i,j): {2, 6776}, {125, 23878}
X(54124) = trilinear pole of line {647, 2799}
X(54124) = barycentric product X(76)*X(3425)
X(54124) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16567}, {2, 1352}, {6, 3148}, {3425, 6}

X(54125) = ISOGONAL CONJUGATE OF X(3145)

Barycentrics    (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(54125) lies on the Jerabek circumhyperbola and these lines: {1, 43693}, {2, 40602}, {3, 18134}, {6, 2907}, {8, 43708}, {65, 5174}, {71, 1761}, {72, 1330}, {73, 3152}, {286, 8044}, {349, 2893}, {2475, 52391}, {5764, 26131}, {7108, 24851}, {10693, 33650}, {17515, 34435}, {18123, 51978}, {38535, 52367}

X(54125) = isogonal conjugate of X(3145)
X(54125) = isotomic conjugate of X(2893)
X(54125) = anticomplement of X(40602)
X(54125) = polar conjugate of X(18679)
X(54125) = cyclocevian conjugate of X(92)
X(54125) = isotomic conjugate of the anticomplement of X(284)
X(54125) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3145}, {6, 1762}, {31, 2893}, {48, 18679}, {65, 40602}
X(54125) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2893}, {3, 3145}, {9, 1762}, {1249, 18679}
X(54125) = cevapoint of X(i) and X(j) for these (i,j): {125, 522}, {513, 8286}, {521, 34846}
X(54125) = trilinear pole of line {647, 4458}
X(54125) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1762}, {2, 2893}, {4, 18679}, {6, 3145}, {284, 40602}

X(54126) = ISOGONAL CONJUGATE OF X(12968)

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

X(54126) lies on the Kiepert circumhyperbola and these lines: {2, 53480}, {4, 43133}, {20, 14229}, {69, 7885}, {83, 13829}, {98, 26441}, {148, 488}, {193, 1131}, {385, 12322}, {485, 637}, {486, 11294}, {489, 17008}, {1270, 2996}, {1352, 14245}, {3069, 14035}, {3091, 45101}, {3128, 37892}, {3317, 11292}, {3406, 21737}, {5025, 5491}, {5395, 7586}, {5591, 33283}, {7612, 10851}, {7783, 32805}, {8781, 13653}, {12221, 14244}, {13759, 53101}, {14234, 45510}, {32820, 45472}, {32996, 53512}

X(54126) = isogonal conjugate of X(12968)
X(54126) = isotomic conjugate of the anticomplement of X(492)
X(54126) = X(1)-isoconjugate of X(12968)
X(54126) = X(3)-Dao conjugate of X(12968)
X(54126) = cevapoint of X(115) and X(54029)
X(54126) = trilinear pole of line {523, 44391}
X(54126) = barycentric quotient X(6)/X(12968)

X(54127) = ISOGONAL CONJUGATE OF X(12963)

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

X(54127) lies on the Kiepert circumhyperbola and these lines: {2, 53479}, {4, 43134}, {20, 14244}, {69, 7885}, {83, 13709}, {98, 8982}, {148, 487}, {193, 1132}, {385, 12323}, {485, 11293}, {486, 638}, {490, 17008}, {598, 31411}, {1271, 2996}, {1352, 14231}, {3068, 14035}, {3091, 45102}, {3127, 37892}, {3316, 11291}, {5025, 5490}, {5395, 7585}, {5590, 33283}, {7612, 10852}, {7783, 32806}, {8781, 13773}, {12221, 44368}, {12222, 14229}, {13639, 53101}, {14238, 45511}, {26620, 43536}, {32820, 45473}, {32996, 53515}

X(54127) = isogonal conjugate of X(12963)
X(54127) = isotomic conjugate of the anticomplement of X(491)
X(54127) = X(1)-isoconjugate of X(12963)
X(54127) = X(3)-Dao conjugate of X(12963)
X(54127) = cevapoint of X(115) and X(54028)
X(54127) = trilinear pole of line {523, 44390}
X(54127) = barycentric quotient X(6)/X(12963)

X(54128) = ISOGONAL CONJUGATE OF X(34247)

Barycentrics    (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(54128) lies on the cubics K998 and K1037, and also on these lines: {2, 52657}, {4, 7224}, {7, 1851}, {31, 3212}, {38, 41527}, {63, 194}, {69, 350}, {77, 614}, {81, 31905}, {561, 7155}, {1444, 5324}, {1814, 6654}, {1965, 7033}, {2162, 21138}, {3112, 24349}, {3271, 6063}, {4124, 33930}, {7035, 27538}, {7056, 7195}, {7226, 27807}, {24451, 40087}

X(54128) = isogonal conjugate of X(34247)
X(54128) = isotomic conjugate of X(32937)
X(54128) = anticomplement of X(52657)
X(54128) = isotomic conjugate of the anticomplement of X(982)
X(54128) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34247}, {2, 51949}, {6, 3501}, {31, 32937}, {32, 17786}, {42, 13588}, {100, 23655}, {101, 21348}, {163, 21958}, {662, 22229}, {692, 17072}, {1110, 23772}, {1783, 22443}, {2329, 51986}, {4551, 23864}, {4559, 21388}, {4876, 51956}, {8927, 51928}, {21438, 32739}, {39930, 51858}
X(54128) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 32937}, {3, 34247}, {9, 3501}, {115, 21958}, {514, 23772}, {1015, 21348}, {1084, 22229}, {1086, 17072}, {6376, 17786}, {8054, 23655}, {32664, 51949}, {39006, 22443}, {40592, 13588}, {40619, 21438}, {40625, 21300}, {41771, 51840}
X(54128) = cevapoint of X(i) and X(j) for these (i,j): {513, 21138}, {514, 3271}, {3808, 35119}, {3810, 26932}
X(54128) = trilinear pole of line {812, 905}
X(54128) = crossdifference of every pair of points on line {22229, 23655}
X(54128) = barycentric product X(i)*X(j) for these {i,j}: {75, 3500}, {7249, 39936}, {10030, 43748}, {17170, 30688}, {18033, 51995}
X(54128) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3501}, {2, 32937}, {6, 34247}, {31, 51949}, {75, 17786}, {81, 13588}, {512, 22229}, {513, 21348}, {514, 17072}, {523, 21958}, {649, 23655}, {693, 21438}, {982, 52657}, {1086, 23772}, {1428, 51956}, {1431, 51986}, {1447, 39930}, {1459, 22443}, {1473, 30689}, {3500, 1}, {3662, 51840}, {3737, 21388}, {4560, 21300}, {7167, 8927}, {7195, 28110}, {7252, 23864}, {18155, 21610}, {21138, 5518}, {23189, 23145}, {39914, 14199}, {39936, 7081}, {43748, 4876}, {51995, 7077}
X(54128) = {X(1965),X(18906)}-harmonic conjugate of X(32937)

X(54129) = BARYCENTRIC QUOTIENT X(385)/X(384)

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

X(54129) lies on the cubic K1000 and these lines: {2, 51948}, {4, 2998}, {69, 194}, {710, 6655}, {804, 2514}, {880, 7836}, {1031, 39953}, {1966, 18905}, {3926, 53147}, {4388, 9285}, {5207, 19566}, {7261, 19565}, {7779, 14946}, {7797, 41178}, {18904, 39044}, {20021, 40847}, {20022, 39939}, {33796, 40382}

X(54129) = anticomplement of the isogonal conjugate of X(14946)
X(54129) = anticomplement of the isotomic conjugate of X(51982)
X(54129) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {711, 38844}, {1967, 37889}, {9236, 8782}, {9288, 25332}, {14946, 8}, {40847, 21275}, {51982, 6327}
X(54129) = X(51982)-Ceva conjugate of X(2)
X(54129) = X(i)-isoconjugate of X(j) for these (i,j): {384, 1967}, {694, 1582}, {1581, 1915}, {1916, 1932}, {1925, 8789}, {1927, 9230}, {1965, 9468}, {41517, 51904}
X(54129) = X(i)-Dao conjugate of X(j) for these (i,j): {8290, 384}, {19576, 1915}, {39030, 1925}, {39031, 1932}, {39043, 1582}, {39044, 1965}, {41178, 782}, {53981, 12143}
X(54129) = cevapoint of X(804) and X(41178)
X(54129) = barycentric product X(i)*X(j) for these {i,j}: {385, 9229}, {695, 3978}, {1580, 9239}, {1926, 9288}, {1966, 9285}, {4027, 40847}, {12215, 37892}, {14603, 51948}
X(54129) = barycentric quotient X(i)/X(j) for these {i,j}: {385, 384}, {695, 694}, {732, 4074}, {1580, 1582}, {1691, 1915}, {1926, 1925}, {1933, 1932}, {1966, 1965}, {3505, 3493}, {3978, 9230}, {4027, 16985}, {9229, 1916}, {9236, 1927}, {9239, 1934}, {9285, 1581}, {9288, 1967}, {12215, 37894}, {16985, 36432}, {41178, 35971}, {44089, 11380}, {51318, 51320}, {51903, 51904}, {51948, 9468}, {51982, 41517}

X(54130) = BARYCENTRIC QUOTIENT X(384)/X(385)

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

X(54130) lies on the cubic K1000 and these lines: {2, 3114}, {4, 147}, {6, 20027}, {69, 694}, {292, 7018}, {315, 42061}, {316, 51494}, {334, 893}, {733, 35567}, {805, 14712}, {1031, 14570}, {1581, 4645}, {2896, 8871}, {4496, 43262}, {5207, 19566}, {6655, 47648}, {7245, 43263}, {7736, 8842}, {7777, 45146}, {7779, 14970}, {7787, 8789}, {7791, 47642}, {17970, 20065}, {20021, 39939}, {38382, 45914}
on K1000

X(54130) = anticomplement of the isotomic conjugate of X(41517)
X(54130) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1927, 8782}, {1967, 25332}, {41517, 6327}
X(54130) = X(41517)-Ceva conjugate of X(2)
X(54130) = X(i)-isoconjugate of X(j) for these (i,j): {385, 9288}, {695, 1580}, {1691, 9285}, {1933, 9229}, {1966, 51948}, {3978, 9236}, {9239, 14602}, {51903, 51982}
X(54130) = X(i)-Dao conjugate of X(j) for these (i,j): {9467, 51948}, {35971, 804}, {37895, 385}, {39092, 695}
X(54130) = cevapoint of X(782) and X(35971)
X(54130) = barycentric product X(i)*X(j) for these {i,j}: {384, 1916}, {694, 9230}, {1581, 1965}, {1582, 1934}, {1915, 18896}, {1925, 1967}, {3493, 16101}, {4074, 14970}, {36432, 40847}
X(54130) = barycentric quotient X(i)/X(j) for these {i,j}: {384, 385}, {694, 695}, {1581, 9285}, {1582, 1580}, {1915, 1691}, {1916, 9229}, {1925, 1926}, {1927, 9236}, {1932, 1933}, {1934, 9239}, {1965, 1966}, {1967, 9288}, {3493, 3505}, {4074, 732}, {9230, 3978}, {9468, 51948}, {11380, 44089}, {16985, 4027}, {35971, 41178}, {36432, 16985}, {37894, 12215}, {41517, 51982}, {51320, 51318}, {51904, 51903}
X(54130) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1916, 38947, 148}, {9468, 18896, 2}, {17493, 30669, 1581}

X(54131) = X(2)X(1350)∩X(6)X(30)

Barycentrics    a^6+8*a^4*(b^2+c^2)-2*(b^2-c^2)^2*(b^2+c^2)+a^2*(-7*b^4+6*b^2*c^2-7*c^4) : :
X(54131) = -2*X[2]+X[1350], -2*X[182]+X[3534], -2*X[1482]+X[50790]

See Ivan Pavlov, euclid 5829.

X(54131) lies on these lines: {2, 1350}, {3, 5476}, {4, 524}, {5, 21358}, {6, 30}, {20, 51737}, {25, 5642}, {51, 31152}, {69, 3839}, {113, 5648}, {115, 11173}, {141, 3545}, {182, 3534}, {193, 50687}, {262, 42849}, {317, 42854}, {355, 50783}, {376, 597}, {381, 511}, {382, 576}, {383, 9763}, {403, 47556}, {428, 37672}, {515, 47356}, {518, 31162}, {541, 16010}, {542, 1351}, {547, 38136}, {549, 14561}, {550, 10541}, {551, 38035}, {575, 1657}, {578, 34726}, {631, 48310}, {946, 47358}, {1003, 47619}, {1080, 9761}, {1181, 34613}, {1352, 3845}, {1353, 33699}, {1386, 50811}, {1469, 11238}, {1482, 50790}, {1503, 1992}, {1513, 11184}, {1656, 52987}, {1853, 2781}, {1993, 9143}, {2777, 23048}, {2810, 10710}, {2854, 10706}, {2930, 5655}, {3056, 11237}, {3091, 21356}, {3094, 44422}, {3098, 5054}, {3146, 5032}, {3242, 3656}, {3416, 50796}, {3524, 3589}, {3564, 15687}, {3580, 31105}, {3618, 10304}, {3620, 50982}, {3629, 50974}, {3630, 50958}, {3631, 50960}, {3654, 38087}, {3655, 38315}, {3679, 38144}, {3751, 50865}, {3763, 5055}, {3818, 14269}, {3828, 38146}, {3843, 34507}, {3851, 40107}, {3860, 51189}, {4663, 41869}, {5017, 6034}, {5026, 12117}, {5028, 14537}, {5038, 44519}, {5050, 15681}, {5052, 11648}, {5064, 12294}, {5066, 48876}, {5071, 10519}, {5073, 11482}, {5092, 15688}, {5093, 15684}, {5094, 32225}, {5097, 48904}, {5104, 37637}, {5201, 32444}, {5446, 37473}, {5846, 34627}, {5847, 34648}, {5864, 37333}, {5865, 37332}, {5894, 15741}, {5895, 8549}, {5965, 35403}, {5969, 6054}, {6144, 18440}, {6173, 38143}, {6329, 50971}, {6623, 41585}, {6776, 8584}, {6800, 37901}, {6816, 52518}, {7464, 37827}, {7500, 17809}, {7530, 19596}, {7540, 36747}, {7610, 13860}, {7745, 10542}, {7778, 9993}, {7788, 18906}, {7840, 44434}, {7841, 22486}, {8537, 35490}, {8540, 12943}, {8541, 44438}, {8546, 37946}, {8593, 10723}, {8703, 18583}, {9024, 10711}, {9053, 34631}, {9530, 41145}, {9745, 13192}, {9752, 44401}, {9756, 22329}, {9760, 41070}, {9762, 41071}, {9830, 10753}, {9880, 11646}, {9969, 16072}, {9974, 35821}, {9975, 35820}, {9976, 38790}, {10056, 10387}, {10151, 47551}, {10295, 47544}, {10510, 51993}, {10606, 23327}, {10733, 41720}, {10989, 11002}, {11001, 44882}, {11008, 51215}, {11064, 26255}, {11164, 41137}, {11177, 14614}, {11216, 36201}, {11295, 51017}, {11296, 51019}, {11305, 51753}, {11306, 51754}, {11470, 12173}, {11898, 48889}, {12007, 14927}, {12017, 15689}, {12100, 38079}, {12101, 39884}, {12233, 34621}, {12571, 50787}, {12584, 25566}, {12953, 19369}, {13169, 32274}, {13321, 52989}, {13330, 44518}, {13352, 18374}, {13598, 50649}, {13857, 34417}, {14093, 17508}, {14136, 49947}, {14137, 49948}, {14492, 24256}, {14787, 37486}, {14810, 15693}, {14831, 34146}, {14881, 44453}, {14893, 34380}, {14912, 20583}, {15033, 19127}, {15107, 47596}, {15274, 37200}, {15303, 32233}, {15360, 37638}, {15516, 48896}, {15520, 29323}, {15683, 25406}, {15685, 48898}, {15690, 51732}, {15694, 38317}, {15695, 48885}, {15696, 20190}, {15702, 21167}, {15703, 25565}, {15706, 51137}, {15709, 50966}, {15710, 50969}, {15811, 31802}, {16475, 34628}, {16509, 40927}, {16981, 44555}, {17702, 34319}, {17800, 53092}, {17825, 43957}, {17845, 34117}, {18358, 23046}, {18579, 47453}, {18911, 47314}, {19106, 51203}, {19107, 51200}, {19149, 34603}, {19161, 21849}, {19709, 24206}, {19925, 50781}, {20080, 51214}, {22165, 41099}, {22234, 49137}, {22330, 49136}, {22491, 41016}, {22492, 41017}, {22579, 41023}, {22580, 41022}, {23698, 51798}, {25154, 33517}, {25164, 33518}, {25335, 32273}, {26958, 45311}, {28194, 47359}, {28204, 51000}, {30270, 33237}, {30273, 50779}, {30775, 47296}, {31489, 40248}, {32113, 47332}, {32216, 51360}, {32455, 39874}, {33749, 49134}, {34200, 38110}, {34573, 50970}, {35228, 37940}, {35259, 40112}, {35266, 37645}, {35707, 37925}, {35822, 36719}, {35823, 36733}, {36194, 46124}, {36969, 51207}, {36970, 51206}, {37077, 41614}, {37196, 44102}, {37459, 50571}, {37907, 48912}, {37984, 47473}, {38021, 51003}, {38023, 51705}, {38071, 50964}, {38073, 51151}, {38074, 50949}, {38147, 45310}, {39899, 48884}, {40330, 41106}, {40885, 47740}, {41038, 51483}, {41039, 51482}, {41447, 47454}, {42126, 44498}, {42127, 44497}, {43150, 50954}, {44265, 47581}, {44268, 51734}, {44280, 51742}, {44285, 51744}, {44961, 47448}, {46333, 50975}, {47031, 47457}, {47276, 47336}, {47280, 47309}, {47308, 47458}, {47333, 47455}, {47334, 47450}, {47598, 50980}, {47745, 50789}, {48879, 50664}, {48883, 51677}, {49481, 51044}, {49496, 51065}, {49509, 51038}, {49511, 50802}, {49524, 50810}, {49536, 51120}, {50818, 51147}, {50862, 51196}, {50864, 51192}, {50976, 51171}, {50992, 51537}, {51029, 51170}

X(54131) = midpoint of X(i) in X(j) for these {i,j}: {2, 51212}, {6, 51024}, {69, 51028}, {141, 51166}, {193, 51023}, {1351, 3830}, {1353, 33699}, {1992, 3543}, {3618, 51211}, {3629, 51022}, {3751, 50865}, {6144, 51027}, {6776, 15682}, {8584, 51163}, {8593, 10723}, {10733, 41720}, {11008, 51215}, {11477, 47353}, {12294, 21969}, {14927, 15640}, {15534, 36990}, {18440, 50962}, {20080, 51214}, {20423, 31670}, {32455, 51026}, {43273, 48910}, {44456, 50955}, {48884, 51140}, {49496, 51065}, {49536, 51120}, {50862, 51196}, {50864, 51192}, {51029, 51170}
X(54131) = reflection of X(i) in X(j) for these {i,j}: {2, 5480}, {3, 5476}, {6, 20423}, {20, 51737}, {69, 47354}, {141, 50959}, {193, 51132}, {376, 597}, {599, 381}, {1350, 2}, {1352, 3845}, {1853, 23049}, {2930, 5655}, {3094, 44422}, {3242, 3656}, {3416, 50796}, {3534, 182}, {3589, 51130}, {3620, 51129}, {3630, 50958}, {3631, 50960}, {3763, 50963}, {3830, 48901}, {5085, 14853}, {5648, 113}, {5655, 32271}, {6144, 50962}, {6776, 8584}, {8703, 18583}, {10295, 47544}, {10516, 53023}, {10606, 23327}, {11001, 44882}, {11646, 9880}, {12117, 5026}, {12584, 25566}, {13169, 32274}, {15069, 47353}, {15533, 1352}, {15534, 1351}, {15682, 51163}, {15685, 48898}, {15690, 51732}, {19161, 21849}, {20423, 21850}, {30273, 50779}, {31884, 14561}, {32113, 47332}, {32233, 15303}, {33706, 24256}, {33878, 50977}, {36990, 3830}, {39874, 51136}, {39884, 12101}, {39899, 51140}, {40341, 50955}, {43273, 6}, {44265, 47581}, {44268, 51734}, {44280, 51742}, {44285, 51744}, {46264, 50979}, {47031, 47457}, {47353, 4}, {47355, 51173}, {47358, 946}, {47473, 37984}, {48872, 3534}, {48873, 8703}, {48874, 12100}, {48876, 5066}, {48881, 50983}, {48905, 43273}, {48910, 51024}, {49509, 51038}, {49511, 50802}, {50781, 19925}, {50783, 355}, {50787, 12571}, {50789, 47745}, {50790, 1482}, {50810, 49524}, {50811, 1386}, {50818, 51147}, {50955, 3818}, {50962, 37517}, {50965, 3589}, {50966, 51126}, {50967, 141}, {50968, 3618}, {50970, 34573}, {50971, 6329}, {50973, 69}, {50974, 3629}, {50976, 51171}, {50977, 19130}, {50978, 18358}, {51024, 31670}, {51027, 18440}, {51044, 49481}, {51136, 32455}, {51179, 3630}, {51188, 11898}
X(54131) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2986), X(43273)}}, {{A, B, C, X(4846), X(5485)}}, {{A, B, C, X(14484), X(18575)}}
X(54131) = perspector of circumconic {{A,B,C, X(1302), X(36886)}}
X(54131) = reflection of the anticomplement of X(2) in the Hatzipolakis-Moses image of X(2)
X(54131) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5480, 38072}, {3, 5476, 47352}, {4, 11477, 15069}, {4, 524, 47353}, {6, 30, 43273}, {6, 31670, 48910}, {6, 48910, 48905}, {6, 51024, 30}, {30, 20423, 6}, {30, 21850, 20423}, {30, 31670, 51024}, {30, 43273, 48905}, {30, 50979, 46264}, {30, 51024, 48910}, {69, 3839, 47354}, {141, 50959, 3545}, {182, 14848, 51185}, {193, 50687, 51023}, {376, 14853, 597}, {376, 597, 5085}, {381, 511, 599}, {381, 599, 10516}, {511, 53023, 10516}, {524, 47353, 15069}, {542, 1351, 15534}, {542, 3830, 36990}, {542, 48901, 3830}, {597, 29181, 376}, {599, 53023, 381}, {1350, 38072, 2}, {1351, 3830, 542}, {1351, 48901, 36990}, {1992, 3543, 1503}, {1992, 51538, 3543}, {2781, 23049, 1853}, {3589, 50965, 3524}, {3618, 10304, 50983}, {3818, 44456, 40341}, {3839, 51028, 69}, {5055, 33878, 50977}, {5055, 50963, 19130}, {5055, 50977, 3763}, {5071, 10519, 20582}, {5476, 19924, 3}, {5480, 51212, 1350}, {8703, 38064, 53094}, {10304, 48881, 50968}, {10653, 10654, 18907}, {11477, 47353, 524}, {13857, 34417, 47597}, {14269, 44456, 50955}, {14269, 50955, 3818}, {14853, 29181, 5085}, {15534, 36990, 542}, {18358, 23046, 50956}, {18440, 37517, 6144}, {18583, 48873, 53094}, {19130, 33878, 3763}, {19130, 50977, 5055}, {20423, 31670, 30}, {20423, 51024, 43273}, {21850, 31670, 6}, {23046, 50978, 18358}, {33878, 50963, 5055}, {37517, 38335, 51027}, {37517, 48895, 18440}, {38064, 48873, 8703}, {38079, 48874, 12100}, {38335, 50962, 18440}, {43273, 48910, 30}, {47354, 51028, 50973}, {48881, 50983, 10304}, {50959, 51166, 50967}

X(54132) = X(2)X(51)∩X(4)X(524)

Barycentrics    a^6-13*a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)+a^2*(11*b^4-6*b^2*c^2+11*c^4) : :
X(54132) = -2*X[599]+3*X[3545], -2*X[1352]+3*X[3839], -X[15681]+2*X[48906]

See Ivan Pavlov, euclid 5829.

X(54132) lies on these lines: {2, 51}, {3, 19661}, {4, 524}, {5, 21356}, {6, 376}, {20, 576}, {30, 1351}, {52, 18913}, {69, 381}, {114, 50639}, {141, 5071}, {146, 148}, {182, 10304}, {186, 47544}, {403, 47473}, {528, 10759}, {530, 44459}, {531, 44463}, {541, 895}, {543, 10753}, {544, 10758}, {547, 3619}, {548, 53092}, {549, 3618}, {550, 11482}, {575, 3522}, {597, 1350}, {599, 3545}, {611, 10385}, {631, 47352}, {944, 47356}, {1007, 51438}, {1352, 3839}, {1503, 15534}, {1513, 9770}, {2393, 5656}, {3090, 21358}, {3091, 11178}, {3098, 15053}, {3146, 11645}, {3292, 52301}, {3416, 38074}, {3523, 10168}, {3525, 48310}, {3528, 53093}, {3529, 8550}, {3534, 5093}, {3564, 3830}, {3589, 15702}, {3620, 19130}, {3629, 39874}, {3656, 50999}, {3751, 28194}, {3785, 37345}, {3817, 50787}, {3818, 20080}, {3832, 34507}, {3845, 34380}, {3860, 50957}, {3926, 48673}, {4232, 5642}, {4293, 8540}, {4294, 19369}, {4663, 6361}, {5050, 8703}, {5052, 7739}, {5054, 18583}, {5055, 48876}, {5056, 40107}, {5066, 50963}, {5085, 19708}, {5097, 48873}, {5102, 8584}, {5107, 7737}, {5182, 35927}, {5286, 13330}, {5477, 43618}, {5587, 50781}, {5603, 47358}, {5655, 14984}, {5864, 37172}, {5865, 37173}, {5878, 34788}, {5921, 48901}, {5965, 36324}, {5969, 32474}, {6193, 7540}, {6210, 48830}, {6329, 15715}, {6459, 9974}, {6460, 9975}, {6515, 31133}, {6770, 22580}, {6773, 22579}, {7426, 37645}, {7487, 11470}, {7519, 9143}, {7552, 43841}, {7714, 37672}, {7735, 11173}, {7757, 35439}, {8549, 12250}, {8586, 43453}, {8591, 12177}, {8593, 23698}, {9041, 34631}, {9140, 31099}, {9530, 10766}, {9737, 35287}, {9777, 43957}, {9880, 11161}, {9993, 37668}, {10295, 47545}, {10303, 25555}, {10516, 22165}, {10541, 21735}, {10605, 18919}, {10606, 23326}, {10653, 51207}, {10654, 51206}, {10982, 11821}, {10989, 37644}, {11004, 37901}, {11008, 15687}, {11148, 23235}, {11427, 44210}, {11433, 31152}, {11540, 50981}, {11898, 14269}, {11916, 12257}, {11917, 12256}, {12007, 46333}, {12017, 34200}, {12082, 32621}, {12101, 51174}, {12117, 18800}, {12150, 13355}, {12236, 44441}, {12243, 46034}, {12383, 34319}, {13142, 34725}, {13172, 51798}, {13639, 45511}, {13674, 36719}, {13759, 45510}, {13794, 36733}, {14449, 18281}, {14810, 15705}, {14881, 32828}, {14994, 32874}, {14995, 47076}, {15073, 34621}, {15303, 18533}, {15311, 17813}, {15520, 15697}, {15533, 41099}, {15577, 37939}, {15640, 29012}, {15681, 48906}, {15683, 46264}, {15684, 39899}, {15688, 48874}, {15693, 38110}, {15694, 38079}, {15698, 31884}, {15710, 53094}, {15759, 50987}, {15988, 31156}, {16200, 51089}, {16226, 52520}, {16279, 36163}, {16475, 51705}, {17504, 51732}, {17702, 41720}, {18553, 50689}, {18860, 37809}, {18906, 32836}, {18925, 34726}, {19136, 43574}, {19709, 38136}, {19766, 48936}, {19783, 48939}, {19905, 41135}, {20126, 25320}, {20190, 21734}, {20192, 40132}, {20583, 44882}, {21167, 50970}, {22112, 40911}, {22330, 50693}, {23234, 50567}, {25898, 50407}, {26255, 40112}, {26864, 47312}, {26869, 47311}, {28204, 51192}, {28538, 34627}, {28708, 44211}, {30769, 45311}, {30775, 44569}, {31105, 44555}, {31162, 39898}, {31400, 44453}, {32255, 48679}, {32455, 48905}, {32815, 39099}, {33187, 39141}, {33237, 40268}, {33699, 50986}, {34613, 34781}, {34628, 39870}, {34648, 39885}, {35260, 37904}, {35474, 40138}, {35908, 36890}, {36757, 42511}, {36758, 42510}, {37460, 44102}, {37643, 47097}, {37669, 44212}, {37907, 47581}, {37984, 47551}, {38021, 49511}, {38035, 51003}, {38073, 47595}, {38143, 51151}, {38144, 50949}, {38155, 50786}, {38335, 39884}, {38734, 41895}, {41149, 51136}, {41586, 52284}, {41614, 44413}, {42085, 51200}, {42086, 51203}, {43511, 44657}, {43512, 44656}, {44265, 52238}, {44654, 49039}, {44655, 49038}, {44704, 52282}, {47359, 50810}, {48857, 50600}, {50796, 50950}, {50802, 51004}, {50811, 51005}, {50818, 51000}, {50827, 50953}, {50862, 51197}, {50864, 51001}, {50865, 50952}, {50954, 50985}, {50968, 51138}, {50982, 50993}, {50989, 51129}, {50991, 51130}, {51027, 51187}, {51085, 51153}, {51087, 51146}, {52198, 52450}

X(54132) = midpoint of X(i) in X(j) for these {i,j}: {2, 51028}, {193, 3543}, {381, 44456}, {1992, 51212}, {3830, 50962}, {8584, 51166}, {15534, 51024}, {15682, 50974}, {15684, 39899}, {33699, 50986}, {50862, 51197}, {50864, 51001}, {50865, 50952}, {50992, 51214}, {51027, 51187}
X(54132) = reflection of X(i) in X(j) for these {i,j}: {2, 20423}, {20, 11179}, {69, 381}, {376, 6}, {381, 21850}, {599, 5480}, {944, 47356}, {1350, 597}, {1992, 1351}, {3534, 50979}, {3543, 31670}, {6770, 22580}, {6773, 22579}, {6776, 1992}, {7426, 47571}, {7757, 35439}, {8591, 12177}, {9143, 9970}, {10295, 47545}, {10519, 14853}, {10606, 23326}, {11001, 43273}, {11160, 1352}, {11161, 9880}, {11179, 576}, {11180, 4}, {12117, 18800}, {12383, 34319}, {13172, 51798}, {14912, 5102}, {15533, 47354}, {15534, 51132}, {15681, 48906}, {15682, 51024}, {15683, 46264}, {18440, 15687}, {22165, 50959}, {25406, 5093}, {32247, 9140}, {33878, 549}, {34628, 39870}, {36163, 16279}, {39885, 34648}, {39898, 31162}, {43273, 8584}, {44882, 20583}, {47551, 37984}, {50639, 114}, {50810, 47359}, {50811, 51005}, {50818, 51000}, {50950, 50796}, {50955, 3845}, {50966, 51185}, {50967, 2}, {50973, 22165}, {50974, 15534}, {50978, 5066}, {50989, 51129}, {50990, 50963}, {50991, 51130}, {50992, 50955}, {50994, 51173}, {50999, 3656}, {51004, 50802}, {51023, 3830}, {51136, 41149}, {51179, 15533}, {52987, 10168}
X(54132) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(182), X(15082)}}, {{A, B, C, X(262), X(44556)}}, {{A, B, C, X(263), X(3531)}}, {{A, B, C, X(290), X(50967)}}, {{A, B, C, X(373), X(40803)}}, {{A, B, C, X(5485), X(42313)}}
X(54132) = reflection of the anticomplement of X(6) in the Hatzipolakis-Moses image of X(6)
X(54132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20423, 14853}, {2, 50967, 10519}, {2, 51028, 511}, {2, 511, 50967}, {4, 524, 11180}, {20, 5032, 11179}, {30, 1351, 1992}, {30, 1992, 6776}, {141, 38072, 5071}, {193, 3543, 542}, {511, 14853, 10519}, {511, 20423, 2}, {542, 31670, 3543}, {549, 14848, 3618}, {576, 11179, 5032}, {576, 19924, 11179}, {597, 1350, 3524}, {599, 3545, 40330}, {599, 5480, 3545}, {1351, 51212, 6776}, {1353, 14927, 6776}, {1503, 15534, 50974}, {1503, 51024, 15682}, {1503, 51132, 15534}, {1992, 51212, 30}, {3098, 38064, 15692}, {3534, 25406, 50975}, {3534, 50979, 25406}, {3534, 51172, 5093}, {3564, 3830, 51023}, {3830, 50962, 3564}, {3839, 11160, 1352}, {3845, 34380, 50955}, {5085, 50965, 19708}, {5102, 43273, 8584}, {5102, 51166, 11001}, {8584, 29181, 43273}, {8584, 51166, 29181}, {10516, 50959, 41106}, {11001, 14912, 43273}, {11179, 19924, 20}, {14561, 50977, 2}, {14848, 33878, 549}, {14853, 50967, 2}, {15533, 53023, 47354}, {15534, 51024, 1503}, {15682, 50974, 1503}, {15698, 50966, 31884}, {19709, 51173, 38136}, {20423, 51028, 50967}, {21850, 44456, 69}, {22165, 50959, 10516}, {29181, 43273, 11001}, {31670, 37517, 193}, {31884, 50983, 15698}, {31884, 51185, 50983}, {33750, 50969, 8703}, {34380, 50955, 50992}, {47354, 53023, 41099}, {50962, 51023, 51178}, {50992, 51214, 34380}, {51023, 51538, 3830}, {51024, 51132, 50974}

X(54133) = X(7)X(3428)∩X(55)X(5762)

Barycentrics    3*a^9-5*a^8*(b+c)-2*(b-c)^6*(b+c)^3+a^7*(-7*b^2+6*b*c-7*c^2)+2*a*(b-c)^4*(b+c)^2*(b^2+c^2)+a^2*(b-c)^4*(7*b^3+17*b^2*c+17*b*c^2+7*c^3)+a^6*(13*b^3+3*b^2*c+3*b*c^2+13*c^3)-a^3*(b-c)^2*(5*b^4+4*b^3*c+6*b^2*c^2+4*b*c^3+5*c^4)+a^5*(7*b^4-8*b^3*c+26*b^2*c^2-8*b*c^3+7*c^4)-a^4*(13*b^5-7*b^4*c+10*b^3*c^2+10*b^2*c^3-7*b*c^4+13*c^5) : :

See Ivan Pavlov, euclid 5829.

X(54133) lies on these lines: {7, 3428}, {55, 5762}, {144, 7680}, {517, 4312}, {528, 34617}, {2951, 37569}, {5696, 37625}, {5735, 15733}, {5759, 8255}, {5805, 42014}, {6361, 38454}, {6690, 21168}, {15096, 31671}, {31245, 38107}

X(54133) = reflection of X(i) in X(j) for these {i,j}: {144, 7680}, {3428, 7}, {5759, 8255}, {36999, 31671}, {42014, 5805}
X(54133) = reflection of the anticomplement of X(7) in the Hatzipolakis-Moses image of X(7)

X(54134) = X(4)X(5854)∩X(56)X(952)

Barycentrics    3*a^7+28*a^5*b*c-7*a^6*(b+c)-2*(b-c)^4*(b+c)^3+2*a*(b^2-c^2)^2*(3*b^2-8*b*c+3*c^2)+12*a^4*(b^3-2*b^2*c-2*b*c^2+c^3)-a^2*(b-c)^2*(3*b^3-23*b^2*c-23*b*c^2+3*c^3)-a^3*(9*b^4+12*b^3*c-50*b^2*c^2+12*b*c^3+9*c^4) : :

See Ivan Pavlov, euclid 5829.

X(54134) lies on these lines: {4, 5854}, {8, 6909}, {56, 952}, {145, 7681}, {355, 2098}, {382, 517}, {944, 8256}, {1482, 12611}, {2829, 12531}, {3036, 10785}, {5790, 31246}, {6691, 7967}, {6737, 47745}, {8069, 37706}, {10085, 28204}, {10591, 38156}, {10711, 34710}, {11236, 50907}, {12666, 14923}, {18526, 32612}, {30323, 37712}, {31141, 50798}, {34627, 34706}, {35448, 51515}, {37705, 37821}, {37709, 50196}, {37714, 45035}, {40293, 41684}, {44784, 49163}

X(54134) = reflection of X(i) in X(j) for these {i,j}: {145, 7681}, {944, 8256}, {2098, 355}, {10310, 8}, {18526, 32612}, {31141, 50798}, {36972, 12645}, {37001, 18525}, {37821, 37705}
X(54134) = reflection of the anticomplement of X(8) in the Hatzipolakis-Moses image of X(8)
X(54134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {517, 12645, 36972}, {517, 18525, 37001}, {3632, 5881, 14872}

X(54135) = X(4)X(527)∩X(9)X(1012)

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

See Ivan Pavlov, euclid 5829.

X(54135) lies on these lines: {1, 52684}, {4, 527}, {7, 7682}, {9, 1012}, {57, 971}, {80, 2093}, {84, 5729}, {142, 6969}, {517, 4915}, {999, 30330}, {1490, 10394}, {1532, 6173}, {1709, 41700}, {2951, 3359}, {3339, 6259}, {3452, 5785}, {4312, 41698}, {5220, 12705}, {5732, 6905}, {6692, 21151}, {6830, 38075}, {6844, 52457}, {6848, 43177}, {6930, 50836}, {6950, 21153}, {6968, 38150}, {7489, 37611}, {7956, 38036}, {7991, 34606}, {7992, 36279}, {8727, 30326}, {10388, 15298}, {12848, 36991}, {20196, 38108}, {37787, 52027}

X(54135) = midpoint of X(i) in X(j) for these {i,j}: {2093, 3062}, {12848, 36991}
X(54135) = reflection of X(i) in X(j) for these {i,j}: {7, 7682}, {2951, 3359}, {5732, 8257}, {6282, 9}, {36973, 5779}
X(54135) = reflection of the anticomplement of X(9) in the Hatzipolakis-Moses image of X(9)
X(54135) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {517, 5779, 36973}, {5732, 8257, 21164}

X(54136) = X(4)X(758)∩X(1)X(7683)

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

See Ivan Pavlov, euclid 5829.

X(54136) lies on these lines: {1, 7683}, {4, 758}, {8, 37456}, {10, 3430}, {58, 515}, {355, 511}, {382, 29097}, {1046, 5691}, {2792, 31673}, {2825, 12784}, {2842, 12368}, {3454, 5587}, {3576, 6693}, {4297, 8258}, {5252, 10544}, {7474, 24987}, {10974, 50037}, {12702, 29032}, {13442, 21677}, {15971, 47033}, {18391, 35650}, {18480, 37823}

X(54136) = midpoint of X(i) in X(j) for these {i,j}: {1046, 5691}
X(54136) = reflection of X(i) in X(j) for these {i,j}: {1, 7683}, {3430, 10}, {4297, 8258}, {36974, 355}, {37823, 18480}
X(54136) = reflection of the anticomplement of X(10) in the Hatzipolakis-Moses image of X(10)
X(54136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {355, 511, 36974}

X(54137) = X(4)X(521)∩X(59)X(5840)

Barycentrics    a^12-2*a^11*(b+c)-2*a^9*b*c*(b+c)-(b-c)^6*(b+c)^4*(b^2+c^2)-19*a^6*b*(b-c)^2*c*(b^2+b*c+c^2)+a^10*(b^2+4*b*c+c^2)+a*(b-c)^6*(b+c)^3*(2*b^2+b*c+2*c^2)+5*a^7*(b-c)^2*(2*b^3+3*b^2*c+3*b*c^2+2*c^3)-a^5*(b-c)^4*(10*b^3+21*b^2*c+21*b*c^2+10*c^3)+a^8*(-5*b^4+7*b^3*c-3*b^2*c^2+7*b*c^3-5*c^4)-a^2*(b-c)^4*(b+c)^2*(b^4-b^3*c-b^2*c^2-b*c^3+c^4)-a^3*b*(b-c)^2*c*(5*b^5+5*b^4*c-2*b^3*c^2-2*b^2*c^3+5*b*c^4+5*c^5)+a^4*(b-c)^2*(5*b^6+13*b^5*c+2*b^4*c^2-4*b^3*c^3+2*b^2*c^4+13*b*c^5+5*c^6) : :

See Ivan Pavlov, euclid 5829.

X(54137) lies on these lines: {4, 521}, {59, 5840}, {24466, 33562}, {34474, 40531}

X(54137) = reflection of the anticomplement of X(11) in the Hatzipolakis-Moses image of X(11)

X(54138) = X(2)X(49939)∩X(4)X(532)

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

See Ivan Pavlov, euclid 5829.

X(54138) lies on these lines: {2, 49939}, {3, 16267}, {4, 532}, {13, 14538}, {15, 5472}, {16, 37637}, {30, 22495}, {62, 5309}, {298, 5478}, {396, 5473}, {511, 13103}, {530, 1080}, {576, 41108}, {616, 7684}, {1351, 36970}, {3180, 44666}, {3543, 36327}, {3564, 36961}, {3830, 11477}, {5340, 14540}, {5463, 52266}, {5474, 6783}, {5615, 37835}, {5617, 41036}, {5864, 42813}, {5865, 42431}, {5984, 41023}, {6055, 22571}, {6774, 34755}, {11542, 21156}, {13102, 37517}, {14539, 42155}, {14541, 16965}, {16001, 42973}, {16629, 47066}, {22890, 42152}, {22997, 23698}, {23005, 51206}, {23006, 41406}, {25154, 50855}, {31670, 36962}, {36765, 52194}, {42158, 44250}, {43633, 47068}

X(54138) = reflection of X(i) in X(j) for these {i,j}: {15, 20425}, {298, 5478}, {616, 7684}, {5473, 396}, {5474, 6783}, {14538, 13}, {36969, 13103}, {50855, 25154}
X(54138) = reflection of the anticomplement of X(13) in the Hatzipolakis-Moses image of X(13)
X(54138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {396, 5473, 21158}, {511, 13103, 36969}

X(54139) = X(2)X(49940)∩X(4)X(533)

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

See Ivan Pavlov, euclid 5829.

X(54139) lies on these lines: {2, 49940}, {3, 16268}, {4, 533}, {14, 14539}, {15, 37637}, {16, 5471}, {30, 22496}, {61, 5309}, {299, 5479}, {383, 531}, {395, 5474}, {511, 13102}, {576, 41107}, {617, 7685}, {1351, 36969}, {3181, 44667}, {3543, 35749}, {3564, 36962}, {3830, 11477}, {5339, 14541}, {5464, 52263}, {5473, 6782}, {5611, 37832}, {5613, 41037}, {5864, 42432}, {5865, 42814}, {5984, 41022}, {6055, 22572}, {6771, 34754}, {11543, 21157}, {13103, 37517}, {14538, 42154}, {14540, 16964}, {16002, 42972}, {16628, 47068}, {16963, 44250}, {22843, 42149}, {22998, 23698}, {23004, 51207}, {23013, 41407}, {25164, 50858}, {31670, 36961}, {36329, 44219}, {43632, 47066}

X(54139) = reflection of X(i) in X(j) for these {i,j}: {16, 20426}, {299, 5479}, {617, 7685}, {5473, 6782}, {5474, 395}, {14539, 14}, {36329, 44219}, {36970, 13102}, {50858, 25164}
X(54139) = reflection of the anticomplement of X(14) in the Hatzipolakis-Moses image of X(14)
X(54139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {395, 5474, 21159}, {511, 13102, 36970}

X(54140) = X(4)X(533)∩X(13)X(511)

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

See Ivan Pavlov, euclid 5829.

X(54140) lies on these lines: {3, 3412}, {4, 533}, {5, 21359}, {13, 511}, {14, 1351}, {15, 5473}, {16, 21843}, {17, 5864}, {30, 22495}, {61, 44465}, {62, 5306}, {298, 7684}, {381, 11477}, {383, 34509}, {396, 14538}, {397, 14541}, {524, 41016}, {532, 1080}, {621, 5478}, {634, 51753}, {1350, 42974}, {2080, 6779}, {3180, 41022}, {5092, 43030}, {5102, 42975}, {5463, 52650}, {5474, 9117}, {5611, 36967}, {5615, 16242}, {5858, 41040}, {5865, 16965}, {5965, 41024}, {6321, 25166}, {6771, 16960}, {6778, 13103}, {6780, 23698}, {7809, 51754}, {9733, 35731}, {9735, 42528}, {10653, 14539}, {11127, 44466}, {11486, 21157}, {12155, 35917}, {12817, 16002}, {14540, 40693}, {16808, 44456}, {16809, 20426}, {20416, 42507}, {20423, 50858}, {21158, 42912}, {22843, 42158}, {22890, 36836}, {33458, 41034}, {34380, 41036}, {36770, 52648}, {36776, 36784}, {41107, 44461}, {44223, 50860}, {48656, 48901}

X(54140) = reflection of X(i) in X(j) for these {i,j}: {13, 20425}, {298, 7684}, {621, 5478}, {5473, 15}, {5474, 9117}, {6779, 2080}, {14538, 396}, {19106, 13103}, {22493, 381}, {25166, 6321}, {36967, 5611}
X(54140) = reflection of the anticomplement of X(15) in the Hatzipolakis-Moses image of X(15)
X(54140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {298, 7684, 36765}, {396, 14538, 21156}, {511, 20425, 13}

X(54141) = X(4)X(532)∩X(14)X(511)

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

See Ivan Pavlov, euclid 5829.

X(54141) lies on these lines: {3, 3411}, {4, 532}, {5, 21360}, {13, 1351}, {14, 511}, {15, 21843}, {16, 5474}, {18, 5865}, {30, 22496}, {61, 5306}, {62, 44461}, {299, 7685}, {381, 11477}, {383, 533}, {395, 14539}, {398, 14540}, {524, 41017}, {622, 5479}, {633, 51754}, {1080, 34508}, {1350, 42975}, {2080, 6780}, {3181, 41023}, {5092, 43031}, {5102, 42974}, {5464, 44223}, {5473, 9115}, {5611, 16241}, {5615, 36968}, {5859, 41041}, {5864, 16964}, {5965, 41025}, {6321, 25156}, {6774, 16961}, {6777, 13102}, {6779, 23698}, {7809, 51753}, {9736, 42529}, {10654, 14538}, {11126, 44462}, {11485, 21156}, {12154, 35918}, {12816, 16001}, {14541, 40694}, {16808, 20425}, {16809, 44456}, {20415, 42506}, {20423, 50855}, {21159, 42913}, {22843, 36843}, {22890, 42157}, {33459, 41035}, {34380, 41037}, {35731, 45489}, {36765, 51388}, {36776, 41071}, {41108, 44465}, {48655, 48901}, {50859, 52650}

X(54141) = reflection of X(i) in X(j) for these {i,j}: {14, 20426}, {299, 7685}, {622, 5479}, {5473, 9115}, {5474, 16}, {6780, 2080}, {14539, 395}, {19107, 13102}, {22494, 381}, {25156, 6321}, {36776, 41071}, {36968, 5615}
X(54141) = reflection of the anticomplement of X(16) in the Hatzipolakis-Moses image of X(16)
X(54141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {511, 20426, 14}, {42157, 47066, 22890}

X(54142) = X(17)X(14540)∩X(627)X(5979)

Barycentrics    -3*a^8+2*(b^2-c^2)^4+10*a^6*(b^2+c^2)+a^4*(3*b^4+22*b^2*c^2+3*c^4)-12*a^2*(b^6+c^6)+2*sqrt(3)*(a^6+9*a^4*(b^2+c^2)-2*(b^2-c^2)^2*(b^2+c^2)+a^2*(-8*b^4+6*b^2*c^2-8*c^4))*S : :

See Ivan Pavlov, euclid 5829.

X(54142) lies on these lines: {17, 14540}, {511, 16629}, {627, 5979}, {3627, 36962}, {7684, 44030}, {8259, 22890}, {22832, 44776}

X(54142) = reflection of X(i) in X(j) for these {i,j}: {627, 51753}, {14540, 17}, {22890, 8259}, {44776, 22832}
X(54142) = reflection of the anticomplement of X(17) in the Hatzipolakis-Moses image of X(17)

X(54143) = X(18)X(14541)∩X(628)X(5978)

Barycentrics    3*a^8-2*(b^2-c^2)^4-10*a^6*(b^2+c^2)-a^4*(3*b^4+22*b^2*c^2+3*c^4)+12*a^2*(b^6+c^6)+2*sqrt(3)*(a^6+9*a^4*(b^2+c^2)-2*(b^2-c^2)^2*(b^2+c^2)+a^2*(-8*b^4+6*b^2*c^2-8*c^4))*S : :

See Ivan Pavlov, euclid 5829.

X(54143) lies on these lines: {18, 14541}, {511, 16628}, {628, 5978}, {3627, 36961}, {7685, 44032}, {8260, 22843}, {22831, 44777}

X(54143) = reflection of X(i) in X(j) for these {i,j}: {628, 51754}, {14541, 18}, {22843, 8260}, {44777, 22831}
X(54143) = reflection of the anticomplement of X(18) in the Hatzipolakis-Moses image of X(18)

X(54144) = X(19)X(18446)∩X(4329)X(51755)

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

See Ivan Pavlov, euclid 5829.

X(54144) lies on these lines: {19, 18446}, {4329, 51755}, {21165, 30265}

X(54144) = reflection of X(i) in X(j) for these {i,j}: {4329, 51755}, {18446, 19}, {30265, 34176}
X(54144) = reflection of the anticomplement of X(19) in the Hatzipolakis-Moses image of X(19)
X(54144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30265, 34176, 21165}

X(54145) = X(30)X(65)∩X(191)X(517)

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

See Ivan Pavlov, euclid 5829.

X(54145) lies on these lines: {21, 14110}, {30, 65}, {40, 37292}, {191, 517}, {354, 33858}, {382, 13126}, {758, 4301}, {942, 16132}, {1071, 47319}, {2475, 7686}, {2771, 7728}, {2778, 41723}, {3651, 8261}, {3901, 31162}, {3962, 22798}, {4311, 10122}, {5426, 31786}, {5887, 16160}, {5902, 16143}, {6001, 37433}, {6841, 44782}, {6912, 18259}, {7701, 37625}, {7957, 16139}, {10051, 13129}, {11551, 33668}, {13145, 41853}, {16117, 34339}, {16125, 17653}, {18242, 41550}, {22937, 37585}, {37308, 50371}

X(54145) = midpoint of X(i) in X(j) for these {i,j}: {7701, 37625}
X(54145) = reflection of X(i) in X(j) for these {i,j}: {1071, 47319}, {2475, 7686}, {3651, 8261}, {5887, 16160}, {7957, 16139}, {14110, 21}, {16117, 34339}, {16132, 942}, {17653, 16125}, {37585, 22937}, {44782, 6841}
X(54145) = reflection of the anticomplement of X(21) in the Hatzipolakis-Moses image of X(21)

X(54146) = X(22)X(36989)∩X(30)X(66)

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

See Ivan Pavlov, euclid 5829.

X(54146) lies on these lines: {22, 36989}, {30, 66}, {378, 34177}, {382, 34118}, {1352, 2781}, {1498, 15069}, {1503, 12083}, {1899, 5621}, {7391, 51756}, {11442, 36201}, {23041, 25337}, {34146, 44440}, {39571, 51739}, {49116, 49669}

X(54146) = reflection of X(i) in X(j) for these {i,j}: {378, 34177}, {7391, 51756}, {36989, 22}
X(54146) = reflection of the anticomplement of X(22) in the Hatzipolakis-Moses image of X(22)

X(54147) = X(4)X(9019)∩X(30)X(67)

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

See Ivan Pavlov, euclid 5829.

X(54147) lies on these lines: {4, 9019}, {23, 32233}, {30, 67}, {141, 35484}, {382, 19924}, {511, 7728}, {524, 14094}, {542, 37924}, {1350, 41583}, {1503, 15107}, {1533, 51941}, {5085, 32223}, {5189, 32274}, {7387, 10116}, {7464, 8262}, {8550, 52525}, {9971, 50008}, {10510, 11799}, {10516, 51360}, {10564, 47450}, {10620, 29012}, {10625, 18553}, {15035, 32218}, {15462, 25338}, {16010, 41586}, {16619, 18374}, {19596, 30714}, {25555, 45034}, {35001, 49116}, {37901, 52191}, {51739, 52300}

X(54147) = reflection of X(i) in X(j) for these {i,j}: {1350, 41583}, {5189, 32274}, {7464, 8262}, {10510, 11799}, {16010, 41586}, {32233, 23}, {35001, 49116}, {43576, 141}, {51941, 1533}
X(54147) = reflection of the anticomplement of X(23) in the Hatzipolakis-Moses image of X(23)

X(54148) = X(4)X(45780)∩X(30)X(64)

Barycentrics    (a^2-b^2-c^2)*(a^14-3*a^12*(b^2+c^2)+(b^2-c^2)^6*(b^2+c^2)-a^2*(b^2-c^2)^4*(3*b^4-4*b^2*c^2+3*c^4)+a^10*(3*b^4+16*b^2*c^2+3*c^4)-a^8*(b^6+11*b^4*c^2+11*b^2*c^4+c^6)+a^4*(b^2-c^2)^2*(3*b^6+b^4*c^2+b^2*c^4+3*c^6)-a^6*(b^8+8*b^6*c^2-26*b^4*c^4+8*b^2*c^6+c^8)) : :

See Ivan Pavlov, euclid 5829.

X(54148) lies on these lines: {4, 45780}, {24, 12118}, {30, 64}, {235, 5654}, {3542, 5504}, {6193, 9143}, {6243, 7728}, {7517, 44665}, {9927, 10625}, {11413, 33563}, {11425, 16238}, {17702, 35471}, {18404, 23039}, {19467, 44259}, {21841, 51933}, {31815, 44271}, {37814, 39571}, {44232, 47391}

X(54148) = reflection of X(i) in X(j) for these {i,j}: {11413, 33563}, {12118, 24}, {37444, 9927}, {44752, 235}
X(54148) = reflection of the anticomplement of X(24) in the Hatzipolakis-Moses image of X(24)
X(54148) = reflection of the anticomplement of X(24) in the Hatzipolakis-Moses image of X(24)= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {235, 44752, 5654}

X(54149) = X(4)X(2393)∩X(30)X(69)

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

See Ivan Pavlov, euclid 5829.

X(54149) lies on these lines: {4, 2393}, {25, 6776}, {30, 69}, {542, 6515}, {1352, 1370}, {1368, 40330}, {1495, 11179}, {1503, 10605}, {1596, 10602}, {1992, 18445}, {3542, 8549}, {3564, 18534}, {5050, 44233}, {5622, 6353}, {5921, 7500}, {6403, 41735}, {6644, 25406}, {6815, 43130}, {7530, 32358}, {7728, 14984}, {8263, 10519}, {10249, 35486}, {11487, 14791}, {11898, 44454}, {12244, 36201}, {12324, 37196}, {14157, 41719}, {14912, 19136}, {15069, 16655}, {18909, 37458}, {18928, 44212}, {20423, 51403}, {20987, 37951}, {25739, 36851}, {34621, 53021}, {35260, 40114}, {39898, 44662}, {44276, 51538}

X(54149) = midpoint of X(i) in X(j) for these {i,j}: {5921, 7500}, {11898, 44454}
X(54149) = reflection of X(i) in X(j) for these {i,j}: {1370, 1352}, {6776, 25}, {10602, 1596}, {10605, 41585}, {21312, 8263}
X(54149) = reflection of the anticomplement of X(25) in the Hatzipolakis-Moses image of X(25)
X(54149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1503, 41585, 10605}, {1596, 10602, 14853}, {8263, 21312, 10519}, {39871, 39879, 6776}

X(54150) = X(28)X(1071)∩X(30)X(72)

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

See Ivan Pavlov, euclid 5829.

X(54150) lies on these lines: {28, 1071}, {30, 72}, {5777, 52364}, {10167, 44220}, {12528, 31293}, {12672, 44661}, {18446, 20831}

X(54150) = midpoint of X(i) in X(j) for these {i,j}: {12528, 31293}
X(54150) = reflection of X(i) in X(j) for these {i,j}: {1071, 28}, {52364, 5777}
X(54150) = reflection of the anticomplement of X(28) in the Hatzipolakis-Moses image of X(28)

X(54151) = X(31)X(30273)∩X(18805)X(30269)

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

See Ivan Pavlov, euclid 5829.

X(54151) lies on these lines: {31, 30273}, {18805, 30269}, {29054, 49500}

X(54151) = reflection of X(i) in X(j) for these {i,j}: {30269, 18805}, {30273, 31}
X(54151) = reflection of the anticomplement of X(31) in the Hatzipolakis-Moses image of X(31)

X(54152) = X(4)X(69)∩X(32)X(11257)

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

See Ivan Pavlov, euclid 5829.

X(54152) lies on these lines: {4, 69}, {20, 35430}, {32, 11257}, {39, 9753}, {194, 35436}, {262, 1506}, {1513, 3095}, {2458, 3406}, {2782, 7754}, {3053, 38642}, {3096, 22677}, {5017, 39646}, {6309, 14981}, {6776, 35432}, {7470, 35387}, {7592, 42548}, {7709, 13357}, {7760, 35431}, {7810, 33706}, {7815, 18806}, {7906, 9772}, {7912, 22503}, {9744, 46305}, {10350, 35930}, {11171, 20576}, {12110, 43183}, {12203, 35424}, {13860, 49111}, {32451, 35389}, {36998, 46321}, {38383, 38743}

X(54152) = reflection of X(i) in X(j) for these {i,j}: {20, 35430}, {194, 35436}, {315, 6248}, {6776, 35432}, {11257, 32}, {30270, 18806}, {32451, 35389}, {36998, 46321}
X(54152) = reflection of the anticomplement of X(32) in the Hatzipolakis-Moses image of X(32)
X(54152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {511, 6248, 315}, {18806, 30270, 22712}

X(54153) = X(30)X(37733)∩X(35)X(16113)

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

See Ivan Pavlov, euclid 5829.

X(54153) lies on these lines: {30, 37733}, {35, 16113}, {79, 517}, {2475, 37625}, {3649, 11014}, {4301, 10698}, {5499, 5535}, {7982, 16159}, {11009, 16153}, {11012, 14526}, {13995, 31789}, {16118, 37826}, {16125, 52367}, {16154, 33596}, {33557, 49178}

X(54153) = reflection of X(i) in X(j) for these {i,j}: {11012, 14526}, {11014, 3649}, {16113, 35}, {52367, 16125}
X(54153) = reflection of the anticomplement of X(35) in the Hatzipolakis-Moses image of X(35)

X(54154) = X(1)X(6842)∩X(3)X(5441)

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

See Ivan Pavlov, euclid 5829.

X(54154) lies on these lines: {1, 6842}, {3, 5441}, {4, 758}, {5, 47033}, {8, 13729}, {10, 6920}, {30, 1768}, {36, 10073}, {40, 1728}, {80, 517}, {140, 5426}, {191, 37290}, {200, 3419}, {355, 546}, {484, 5840}, {515, 13279}, {519, 1519}, {912, 41698}, {946, 5086}, {950, 10902}, {1210, 37561}, {1482, 10895}, {1532, 6326}, {1537, 5855}, {1737, 2077}, {1749, 7491}, {2095, 12943}, {2475, 31870}, {3576, 5722}, {3585, 24474}, {3679, 12703}, {4880, 52851}, {5046, 31806}, {5057, 24042}, {5080, 6246}, {5081, 15499}, {5172, 12743}, {5176, 28234}, {5252, 16200}, {5445, 26285}, {5534, 41709}, {5536, 5841}, {5538, 6882}, {5559, 10284}, {5691, 12704}, {5692, 6929}, {5697, 10953}, {5730, 10893}, {5794, 8227}, {5842, 12690}, {5881, 12629}, {5883, 6951}, {5884, 37437}, {5885, 47032}, {5902, 6923}, {5903, 10525}, {5904, 37821}, {6256, 12649}, {6850, 15016}, {6853, 35016}, {6863, 37571}, {6907, 33857}, {6909, 10265}, {6912, 48698}, {6937, 30143}, {6941, 22836}, {6958, 15079}, {6965, 10176}, {6980, 37701}, {7951, 37533}, {10483, 37532}, {10543, 52265}, {10572, 11012}, {10771, 36175}, {10826, 37531}, {10950, 11014}, {11248, 18395}, {11545, 51768}, {11827, 24468}, {12608, 41575}, {12619, 35000}, {12625, 17857}, {12700, 41687}, {14110, 17604}, {14988, 34789}, {15908, 37730}, {18513, 37826}, {18529, 37712}, {22765, 36975}, {35457, 51517}, {37735, 46920}

X(54154) = midpoint of X(i) in X(j) for these {i,j}: {4880, 52851}
X(54154) = reflection of X(i) in X(j) for these {i,j}: {40, 40663}, {2077, 1737}, {5057, 24042}, {5080, 6246}, {5538, 6882}, {6326, 1532}, {6909, 10265}, {12119, 36}, {16113, 1749}, {35000, 12619}, {36975, 22765}
X(54154) = reflection of the anticomplement of X(36) in the Hatzipolakis-Moses image of X(36)
X(54154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 49168, 5693}, {8, 13729, 20117}, {1532, 44669, 6326}, {5538, 37718, 6882}, {5884, 37437, 49178}

X(54155) = X(3)X(12206)∩X(4)X(732)

Barycentrics    -(b^4*c^4*(b^2-c^2)^2)+a^8*(4*b^4+9*b^2*c^2+4*c^4)+a^6*(b^6+16*b^4*c^2+16*b^2*c^4+c^6)-a^4*(4*b^8+b^6*c^2-11*b^4*c^4+b^2*c^6+4*c^8)-a^2*(b^10+6*b^8*c^2+b^6*c^4+b^4*c^6+6*b^2*c^8+c^10) : :

See Ivan Pavlov, euclid 5829.

X(54155) lies on these lines: {3, 12206}, {4, 732}, {5, 42006}, {20, 3095}, {39, 12122}, {76, 6249}, {83, 511}, {98, 41755}, {262, 6292}, {754, 13085}, {1916, 7762}, {5188, 9751}, {6287, 9866}, {6308, 37334}, {6704, 22712}, {7900, 22803}, {9821, 10359}, {10357, 11272}, {12836, 18983}, {12837, 13078}, {13111, 35930}, {22728, 48674}, {31168, 44422}

X(54155) = midpoint of X(i) in X(j) for these {i,j}: {13111, 48673}
X(54155) = reflection of X(i) in X(j) for these {i,j}: {76, 6249}, {5188, 51827}, {6287, 14881}, {9821, 49112}, {12122, 39}, {31168, 44422}, {32476, 3095}
X(54155) = reflection of the anticomplement of X(39) in the Hatzipolakis-Moses image of X(39)
X(54155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5188, 51827, 9751}

X(54156) = X(1)X(104)∩X(40)X(64)

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

See Ivan Pavlov, euclid 5829.

X(54156) lies on these lines: {1, 104}, {3, 7971}, {4, 2093}, {9, 31788}, {10, 5811}, {40, 64}, {57, 12672}, {65, 10396}, {80, 52860}, {84, 517}, {165, 6261}, {392, 37526}, {515, 3529}, {516, 49168}, {518, 17649}, {758, 6769}, {912, 6765}, {936, 3359}, {944, 14646}, {946, 3339}, {952, 52116}, {960, 37560}, {962, 6245}, {971, 12702}, {997, 10270}, {1012, 3340}, {1071, 1697}, {1103, 45269}, {1482, 34862}, {1537, 50443}, {1698, 12608}, {1699, 10598}, {1706, 5777}, {1709, 5903}, {2096, 10106}, {2771, 5534}, {2818, 52117}, {2829, 5881}, {2956, 21147}, {3062, 31673}, {3149, 5128}, {3333, 45776}, {3358, 24474}, {3428, 12330}, {3555, 18238}, {3576, 19535}, {3577, 7285}, {3579, 52026}, {3679, 6256}, {3869, 6282}, {3885, 13243}, {3927, 31798}, {3928, 22770}, {3951, 6223}, {4018, 7982}, {4295, 5715}, {4312, 26332}, {4324, 37000}, {4853, 49171}, {4866, 38127}, {4882, 12666}, {5119, 15071}, {5223, 11362}, {5250, 8726}, {5603, 6705}, {5657, 6260}, {5690, 6259}, {5691, 41684}, {5697, 10085}, {5720, 40266}, {5729, 7686}, {5730, 17613}, {5768, 10624}, {5787, 28174}, {5790, 22792}, {5818, 30326}, {5837, 6916}, {5882, 9819}, {6736, 10309}, {6890, 51423}, {6909, 11682}, {7171, 12687}, {7330, 9623}, {7987, 40257}, {7997, 37714}, {8580, 20117}, {9589, 48482}, {9612, 10599}, {9709, 31821}, {9799, 9804}, {9841, 31786}, {9856, 36279}, {9942, 37551}, {9948, 28194}, {10268, 12520}, {10273, 37234}, {10306, 11523}, {10310, 18237}, {10582, 15016}, {10826, 34789}, {10857, 40249}, {10860, 14110}, {10980, 13464}, {11373, 13226}, {11496, 11529}, {11665, 52851}, {11822, 12457}, {11823, 12456}, {12514, 30503}, {12515, 45770}, {12528, 46685}, {12651, 37625}, {12675, 31393}, {12679, 40663}, {12688, 37567}, {12699, 33899}, {12700, 24392}, {13607, 30337}, {14988, 37531}, {17650, 41539}, {17706, 30330}, {18239, 34790}, {18260, 50190}, {18529, 31871}, {19067, 35775}, {19068, 35774}, {19861, 21164}, {21740, 30282}, {33597, 35445}, {35242, 37837}

X(54156) = midpoint of X(i) in X(j) for these {i,j}: {7991, 7992}, {9799, 20070}, {12245, 12246}
X(54156) = reflection of X(i) in X(j) for these {i,j}: {1, 1158}, {962, 6245}, {1482, 34862}, {1490, 40}, {3555, 18238}, {6259, 5690}, {6261, 40256}, {6765, 49163}, {7971, 3}, {7982, 12114}, {9589, 48482}, {11523, 10306}, {12650, 84}, {12667, 11362}, {12699, 33899}, {13253, 48695}, {18239, 34790}, {43166, 3358}
X(54156) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(104), X(44692)}}, {{A, B, C, X(318), X(52027)}}, {{A, B, C, X(1795), X(38271)}}, {{A, B, C, X(2184), X(34051)}}, {{A, B, C, X(3680), X(15501)}}
X(54156) = reflection of the anticomplement of X(40) in the Hatzipolakis-Moses image of X(40)
X(54156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1158, 52027}, {40, 5693, 200}, {40, 6001, 1490}, {84, 517, 12650}, {1158, 2800, 1}, {2093, 7995, 4}, {2800, 48695, 13253}, {7991, 7992, 515}, {12245, 12246, 515}

X(54157) = X(5)X(51)∩X(30)X(195)

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

See Ivan Pavlov, euclid 5829.

X(54157) lies on circumconics {A,B,C,X(5),X(38006)}, {A,B,C,X(54),X(14128)} and on these lines: {3, 22051}, {4, 12175}, {5, 51}, {30, 195}, {54, 550}, {113, 16982}, {140, 12307}, {235, 6242}, {381, 12325}, {495, 6286}, {496, 7356}, {539, 15687}, {546, 2888}, {549, 7691}, {567, 10203}, {632, 30531}, {1141, 31674}, {1493, 15704}, {1595, 12300}, {1596, 6152}, {2914, 6240}, {2917, 37936}, {3519, 3858}, {3627, 7728}, {3845, 6288}, {3853, 11271}, {5446, 13368}, {5965, 21850}, {6247, 10628}, {6284, 35197}, {6689, 14869}, {6823, 12606}, {7354, 51803}, {7502, 32333}, {7574, 32165}, {8703, 10610}, {9833, 17824}, {9905, 28174}, {10066, 10386}, {10115, 43583}, {10263, 43893}, {10272, 30551}, {10677, 42117}, {10678, 42118}, {11539, 32348}, {11563, 14449}, {11566, 22750}, {11702, 34153}, {11804, 13371}, {11805, 22660}, {12160, 44288}, {12242, 15712}, {12785, 38138}, {12965, 42215}, {12971, 42216}, {13421, 43831}, {13431, 32340}, {13482, 18442}, {15134, 47341}, {15137, 38323}, {15760, 22815}, {16163, 47117}, {16625, 51391}, {18400, 44762}, {19150, 48906}, {19710, 20585}, {20584, 38071}, {22791, 31803}, {26879, 32339}, {27196, 46454}, {27246, 35729}, {31724, 36853}, {32273, 32365}, {32346, 34780}

X(54157) = midpoint of X(i) in X(j) for these {i,j}: {4, 12316}, {11271, 48675}, {13431, 32340}, {15800, 15801}
X(54157) = reflection of X(i) in X(j) for these {i,j}: {3, 22051}, {5, 20424}, {54, 11803}, {550, 54}, {1141, 31674}, {2888, 546}, {3627, 15800}, {7691, 8254}, {12307, 140}, {13368, 5446}, {16163, 47117}, {21230, 3574}, {32352, 44056}, {34153, 11702}, {35729, 27246}, {36966, 195}, {48675, 3853}, {48906, 19150}
X(54157) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(5), X(38006)}}, {{A, B, C, X(54), X(14128)}}
X(54157) = reflection of the anticomplement of X(54) in the Hatzipolakis-Moses image of X(54)
X(54157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 12316, 50708}, {30, 195, 36966}, {1154, 20424, 5}, {1154, 3574, 21230}, {1154, 44056, 32352}, {3574, 21230, 5}, {7691, 8254, 549}, {15800, 15801, 32423}, {15800, 32423, 3627}, {20424, 21230, 3574}

X(54158) = X(4)X(15733)∩X(7)X(517)

Barycentrics    a^9-a^8*(b+c)+4*a^6*(b-c)^2*(b+c)+4*a^2*(b-c)^4*(b+c)^3-(b-c)^6*(b+c)^3+a^7*(-4*b^2+2*b*c-4*c^2)+a*(b-c)^4*(b+c)^2*(b^2+c^2)-2*a^3*(b^2-c^2)^2*(2*b^2+b*c+2*c^2)+2*a^5*(3*b^4+b^3*c+16*b^2*c^2+b*c^3+3*c^4)-2*a^4*(3*b^5-3*b^4*c+8*b^3*c^2+8*b^2*c^3-3*b*c^4+3*c^5) : :

See Ivan Pavlov, euclid 5829.

X(54158) lies on these lines: {4, 15733}, {7, 517}, {55, 5759}, {390, 37533}, {516, 18446}, {528, 10698}, {553, 41338}, {3419, 38149}, {3428, 8255}, {3434, 5805}, {5119, 10059}, {5603, 52457}, {5696, 26332}, {5761, 8543}, {5762, 10679}, {5784, 10532}, {5817, 7680}, {6361, 38454}, {6827, 7671}, {6947, 10177}, {7982, 17647}, {11372, 28609}, {31140, 38073}

X(54158) = reflection of X(i) in X(j) for these {i,j}: {390, 37533}, {3428, 8255}, {3434, 5805}, {5759, 55}, {18446, 41570}, {36976, 10679}, {42014, 7680}
X(54158) = reflection of the anticomplement of X(55) in the Hatzipolakis-Moses image of X(55)
X(54158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {516, 41570, 18446}, {3428, 8255, 21151}, {7680, 42014, 5817}

X(54159) = X(4)X(527)∩X(9)X(374)

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

See Ivan Pavlov, euclid 5829.

X(54159) lies on these lines: {4, 527}, {9, 374}, {57, 5728}, {329, 1699}, {452, 7991}, {516, 2093}, {954, 7962}, {971, 2095}, {1006, 6282}, {1445, 6909}, {1708, 41166}, {1728, 12651}, {1750, 2801}, {2094, 30304}, {3421, 38154}, {4512, 7994}, {5785, 6843}, {6173, 6907}, {8226, 31142}, {9965, 36991}, {10399, 12565}, {24474, 52684}, {30223, 41712}, {30330, 43161}, {37421, 43177}, {37569, 47375}

X(54159) = midpoint of X(i) in X(j) for these {i,j}: {9965, 36991}
X(54159) = reflection of X(i) in X(j) for these {i,j}: {5732, 57}, {6282, 8257}, {52457, 7682}
X(54159) = reflection of the anticomplement of X(57) in the Hatzipolakis-Moses image of X(57)
X(54159) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2093, 10398, 12848}, {6282, 8257, 21153}, {7682, 52457, 38150}

X(54160) = X(4)X(17770)∩X(10)X(511)

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

See Ivan Pavlov, euclid 5829.

X(54160) lies on these lines: {4, 17770}, {10, 511}, {58, 4297}, {387, 24728}, {516, 1046}, {540, 34648}, {758, 4301}, {1330, 19925}, {2392, 31849}, {2792, 51118}, {3430, 8258}, {3817, 7683}, {5691, 20077}, {36974, 38155}

X(54160) = midpoint of X(i) in X(j) for these {i,j}: {5691, 20077}
X(54160) = reflection of X(i) in X(j) for these {i,j}: {1330, 19925}, {3430, 8258}, {4297, 58}
X(54160) = reflection of the anticomplement of X(58) in the Hatzipolakis-Moses image of X(58)
X(54160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3430, 8258, 10164}

X(54161) = X(4)X(758)∩X(21)X(517)

Barycentrics    a*(a^8*(b+c)-(b-c)^6*(b+c)^3-a^7*(2*b^2+5*b*c+2*c^2)+2*a^4*b*c*(-4*b^3+3*b^2*c+3*b*c^2-4*c^3)-2*a^6*(b^3-2*b^2*c-2*b*c^2+c^3)+a*(b^2-c^2)^2*(2*b^4-5*b^3*c+5*b^2*c^2-5*b*c^3+2*c^4)-a^3*(b+c)^2*(6*b^4-17*b^3*c+24*b^2*c^2-17*b*c^3+6*c^4)+a^5*(6*b^4+5*b^3*c-3*b^2*c^2+5*b*c^3+6*c^4)+2*a^2*(b^7-5*b^5*c^2+4*b^4*c^3+4*b^3*c^4-5*b^2*c^5+c^7)) : :

See Ivan Pavlov, euclid 5829.

X(54161) lies on these lines: {4, 758}, {21, 517}, {30, 9961}, {65, 3651}, {411, 33858}, {944, 39772}, {2476, 33592}, {2771, 4018}, {3486, 4302}, {3869, 6841}, {5086, 37230}, {5692, 6873}, {5818, 40661}, {5887, 52269}, {5902, 6876}, {6261, 16126}, {6852, 31806}, {6853, 26725}, {7098, 10058}, {7680, 21677}, {7686, 44782}, {8261, 14110}, {21740, 24474}, {31660, 37533}, {40266, 44258}

X(54161) = reflection of X(i) in X(j) for these {i,j}: {944, 39772}, {3651, 65}, {3869, 6841}, {14110, 8261}, {34195, 24474}, {40266, 44258}, {44782, 7686}
X(54161) = reflection of the anticomplement of X(65) in the Hatzipolakis-Moses image of X(65)
X(54161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8261, 14110, 21161}

X(54162) = X(4)X(524)∩X(23)X(542)

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

See Ivan Pavlov, euclid 5829.

X(54162) lies on these lines: {4, 524}, {5, 22151}, {23, 542}, {30, 32247}, {67, 7464}, {69, 49669}, {186, 8262}, {323, 1352}, {399, 3564}, {511, 10296}, {1503, 12317}, {2071, 49116}, {3580, 6776}, {5921, 7519}, {5965, 32271}, {7527, 34507}, {7552, 8550}, {7565, 18553}, {10295, 47558}, {10510, 32274}, {10540, 27085}, {11064, 40330}, {11179, 52300}, {11645, 15054}, {12383, 32113}, {13169, 34802}, {13754, 41737}, {14118, 40107}, {17702, 41721}, {19510, 43574}, {25321, 47581}, {31861, 50955}, {44267, 48679}, {44961, 45016}

X(54162) = midpoint of X(i) in X(j) for these {i,j}: {5921, 37779}
X(54162) = reflection of X(i) in X(j) for these {i,j}: {323, 1352}, {6776, 3580}, {7464, 67}, {10295, 47558}, {10510, 32274}, {11061, 11799}, {12383, 32113}, {32233, 8262}, {48679, 44267}
X(54162) = reflection of the anticomplement of X(67) in the Hatzipolakis-Moses image of X(67)
X(54162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3564, 11799, 11061}

X(54163) = X(3)X(22533)∩X(30)X(11411)

Barycentrics    (a^2-b^2-c^2)*(3*a^14-8*a^12*(b^2+c^2)+2*(b^2-c^2)^6*(b^2+c^2)-a^2*(b^2-c^2)^4*(5*b^4-4*b^2*c^2+5*c^4)+a^10*(5*b^4+28*b^2*c^2+5*c^4)+4*a^4*(b^2-c^2)^2*(b^6+c^6)+2*a^8*(b^6-11*b^4*c^2-11*b^2*c^4+c^6)-a^6*(3*b^8+4*b^6*c^2-30*b^4*c^4+4*b^2*c^6+3*c^8)) : :

See Ivan Pavlov, euclid 5829.

X(54163) lies on these lines: {3, 22533}, {24, 41587}, {30, 11411}, {68, 11413}, {235, 6193}, {3167, 44235}, {3564, 31725}, {9927, 44752}, {9937, 44269}, {10733, 12111}, {12118, 15078}, {12164, 44271}, {12420, 47096}, {12421, 44440}, {14852, 31282}, {15316, 50435}, {18555, 41619}, {18913, 44240}

X(54163) = reflection of X(i) in X(j) for these {i,j}: {6193, 235}, {11413, 68}, {12118, 33563}, {12164, 44271}, {35490, 12293}, {44752, 9927}
X(54163) = reflection of the anticomplement of X(68) in the Hatzipolakis-Moses image of X(68)
X(54163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12118, 33563, 15078}

X(54164) = X(25)X(3564)∩X(30)X(5921)

Barycentrics    3*a^12-9*a^10*(b^2+c^2)+7*a^2*(b^2-c^2)^4*(b^2+c^2)-2*(b^2-c^2)^4*(b^2+c^2)^2+8*a^8*(b^4+5*b^2*c^2+c^4)+2*a^6*(b^6-17*b^4*c^2-17*b^2*c^4+c^6)+a^4*(-9*b^8+20*b^6*c^2+10*b^4*c^4+20*b^2*c^6-9*c^8) : :

See Ivan Pavlov, euclid 5829.

X(54164) lies on these lines: {25, 3564}, {30, 5921}, {69, 21312}, {110, 47597}, {193, 1596}, {542, 10605}, {1352, 10602}, {1568, 17813}, {2393, 5562}, {5093, 46030}, {5094, 12827}, {5181, 10249}, {5654, 11405}, {6644, 18932}, {6677, 14912}, {6776, 8263}, {10733, 12133}, {11442, 31152}, {11443, 14848}, {14516, 37196}, {18534, 46442}, {20772, 32234}, {23293, 31255}, {33878, 40317}, {34966, 39588}, {39874, 44241}, {44212, 50974}, {44276, 44456}

X(54164) = reflection of X(i) in X(j) for these {i,j}: {193, 1596}, {6776, 8263}, {10602, 1352}, {21312, 69}, {31152, 50955}, {32234, 20772}, {39874, 44241}, {39899, 6644}, {44438, 18440}, {44456, 44276}, {50974, 44212}
X(54164) = reflection of the anticomplement of X(69) in the Hatzipolakis-Moses image of X(69)
X(54164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1352, 10602, 16072}, {14984, 18440, 44438}

X(54165) = X(31)X(29010)∩X(75)X(30269)

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

See Ivan Pavlov, euclid 5829.

X(54165) lies on these lines: {31, 29010}, {75, 30269}, {18805, 30273}, {31134, 51040}

X(54165) = reflection of X(i) in X(j) for these {i,j}: {30269, 75}, {30273, 18805}, {31134, 51040}
X(54165) = reflection of the anticomplement of X(75) in the Hatzipolakis-Moses image of X(75)

X(54166) = X(4)X(18768)∩X(32)X(2782)

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

See Ivan Pavlov, euclid 5829.

X(54166) lies on these lines: {4, 18768}, {32, 2782}, {39, 37071}, {76, 5999}, {194, 14981}, {382, 511}, {3095, 38743}, {6248, 32452}, {6680, 7709}, {7697, 7867}, {7754, 35386}, {7935, 52996}, {9873, 12251}, {11257, 18806}, {13355, 17130}, {35385, 39646}, {38737, 40923}

X(54166) = reflection of X(i) in X(j) for these {i,j}: {11257, 18806}, {30270, 76}, {32452, 6248}, {37004, 13108}
X(54166) = reflection of the anticomplement of X(76) in the Hatzipolakis-Moses image of X(76)
X(54166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {511, 13108, 37004}

X(54167) = X(39)X(550)∩X(83)X(5188)

Barycentrics    4*b^4*c^4*(b^2-c^2)^2+a^10*(b^2+c^2)-31*a^6*b^2*c^2*(b^2+c^2)-11*a^8*(b^2+c^2)^2+a^4*(9*b^8+6*b^6*c^2-22*b^4*c^4+6*b^2*c^6+9*c^8)+a^2*(b^10+14*b^8*c^2-3*b^6*c^4-3*b^4*c^6+14*b^2*c^8+c^10) : :

See Ivan Pavlov, euclid 5829.

X(54167) lies on these lines: {39, 550}, {83, 5188}, {511, 13111}, {6249, 44772}, {6287, 22682}, {12122, 21163}, {20088, 35437}, {29012, 35439}

X(54167) = reflection of X(i) in X(j) for these {i,j}: {5188, 83}, {12122, 51827}, {44772, 6249}
X(54167) = reflection of the anticomplement of X(83) in the Hatzipolakis-Moses image of X(83)
X(54167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12122, 51827, 21163}

X(54168) = X(4)X(525)∩X(249)X(23698)

Barycentrics    a^14-2*a^12*(b^2+c^2)-(b^2-c^2)^6*(b^2+c^2)-a^10*(b^4-8*b^2*c^2+c^4)+2*a^4*(b^2-c^2)^2*(2*b^6+5*b^4*c^2+5*b^2*c^4+2*c^6)+a^8*(7*b^6-9*b^4*c^2-9*b^2*c^4+7*c^6)+a^6*(-9*b^8+5*b^6*c^2+9*b^4*c^4+5*b^2*c^6-9*c^8)+a^2*(b^2-c^2)^2*(b^8-7*b^6*c^2+3*b^4*c^4-7*b^2*c^6+c^8) : :

See Ivan Pavlov, euclid 5829.

X(54168) lies on these lines: {4, 525}, {249, 23698}, {8754, 44769}, {12244, 51258}, {18331, 46982}

X(54168) = reflection of the anticomplement of X(115) in the Hatzipolakis-Moses image of X(115)

X(54169) = X(2)X(1350)∩X(3)X(524)

Barycentrics    4*a^6+5*a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)-2*a^2*(5*b^4+6*b^2*c^2+5*c^4) : :
X(54169) = X[2]+X[1350], -X[4]+3*X[21358], -X[6]+3*X[3524], X[40]+X[47358], X[376]+X[599], -X[381]+2*X[20582], -2*X[575]+5*X[15712], -X[576]+4*X[3530], X[1352]+X[3534], -X[1386]+2*X[50828], X[3242]+X[50810], X[3830]+X[48873], X[6144]+X[51179], X[6776]+X[15533], X[11001]+X[36990], -3*X[14561]+5*X[15694], X[39899]+X[50961], -4*X[48892]+5*X[51134]

See Ivan Pavlov, euclid 5829.

X(54169) lies on these lines: {2, 1350}, {3, 524}, {4, 21358}, {5, 19924}, {6, 3524}, {20, 11164}, {30, 141}, {40, 47358}, {69, 10304}, {74, 5648}, {140, 5476}, {182, 8584}, {187, 20194}, {193, 15705}, {343, 9140}, {376, 599}, {378, 41585}, {381, 20582}, {395, 16940}, {396, 16941}, {511, 549}, {542, 8703}, {548, 34507}, {550, 11645}, {575, 15712}, {576, 3530}, {631, 47352}, {944, 50783}, {1351, 15693}, {1352, 3534}, {1353, 15711}, {1368, 45311}, {1386, 50828}, {1469, 4995}, {1619, 15577}, {1992, 5085}, {2393, 23328}, {2781, 3917}, {3056, 5298}, {3094, 5306}, {3242, 50810}, {3416, 50811}, {3522, 15069}, {3523, 11477}, {3543, 10516}, {3545, 3763}, {3564, 34200}, {3576, 47356}, {3589, 5054}, {3618, 15708}, {3619, 3839}, {3620, 48905}, {3629, 5092}, {3630, 45759}, {3631, 15688}, {3653, 51006}, {3654, 9041}, {3655, 5846}, {3815, 5104}, {3819, 44212}, {3830, 48873}, {3844, 50796}, {3845, 24206}, {4265, 21161}, {4297, 50781}, {5017, 9300}, {5032, 15717}, {5050, 15700}, {5055, 31670}, {5066, 48901}, {5071, 53023}, {5093, 15718}, {5097, 19711}, {5188, 8369}, {5447, 34351}, {5651, 37904}, {5893, 11821}, {5921, 50990}, {5965, 15714}, {5969, 6055}, {5989, 11177}, {6144, 51179}, {6194, 22329}, {6329, 15707}, {6393, 7811}, {6636, 9143}, {6696, 34787}, {6776, 15533}, {7426, 7998}, {7771, 51438}, {8359, 30270}, {8722, 27088}, {9019, 44218}, {9053, 34718}, {9756, 42850}, {9771, 37451}, {10007, 44422}, {10124, 38317}, {10299, 10541}, {10323, 15582}, {10989, 45303}, {11001, 36990}, {11064, 47596}, {11160, 25406}, {11168, 22712}, {11511, 16976}, {11539, 21850}, {11646, 12117}, {11694, 19140}, {11812, 18583}, {12007, 15534}, {12017, 15706}, {12061, 13348}, {12101, 48904}, {12245, 50790}, {12512, 50787}, {13169, 32233}, {13394, 33884}, {13567, 43957}, {13634, 17392}, {13635, 17330}, {14269, 50960}, {14538, 35303}, {14539, 35304}, {14540, 37341}, {14541, 37340}, {14561, 15694}, {14645, 46893}, {14848, 15701}, {14853, 15702}, {14869, 25555}, {14891, 17508}, {14912, 15715}, {14927, 15697}, {14929, 51397}, {15035, 34319}, {15051, 41720}, {15066, 35266}, {15360, 37648}, {15581, 37198}, {15585, 34778}, {15640, 51537}, {15682, 40330}, {15686, 29012}, {15687, 25561}, {15689, 18440}, {15690, 48898}, {15695, 41152}, {15699, 19130}, {15704, 18553}, {15709, 47355}, {15710, 40341}, {15713, 38079}, {15716, 41149}, {15722, 41153}, {15980, 20112}, {16789, 44285}, {17834, 45073}, {18579, 32217}, {19710, 39884}, {20190, 44682}, {21156, 22580}, {21157, 22579}, {21166, 51798}, {21969, 32191}, {22151, 50007}, {22677, 37461}, {23046, 48895}, {23332, 31152}, {25565, 38136}, {28194, 51003}, {28204, 50949}, {28538, 51705}, {29323, 44903}, {29959, 36987}, {30271, 51050}, {30273, 51051}, {30739, 32225}, {32216, 47296}, {33273, 47619}, {33699, 48889}, {34473, 50639}, {34774, 35228}, {36755, 51160}, {36756, 51159}, {37283, 37477}, {37480, 51739}, {37517, 41983}, {38065, 51195}, {38066, 50951}, {38335, 43621}, {39874, 50975}, {39899, 50961}, {40248, 44377}, {40884, 42313}, {41984, 42785}, {42786, 47478}, {44215, 52658}, {44580, 51732}, {47545, 51733}, {48892, 51134}, {49481, 51049}, {49509, 51044}, {49511, 50808}, {49524, 50821}, {50664, 50987}, {50824, 51147}, {50954, 51025}, {51010, 52193}, {51013, 52194}, {51127, 51130}, {51170, 51214}

X(54169) = midpoint of X(i) in X(j) for these {i,j}: {2, 1350}, {6, 50967}, {20, 47353}, {40, 47358}, {69, 43273}, {74, 5648}, {141, 50965}, {193, 50973}, {376, 599}, {944, 50783}, {1352, 3534}, {3094, 33706}, {3098, 50977}, {3242, 50810}, {3416, 50811}, {3589, 50970}, {3620, 50968}, {3630, 51136}, {3631, 50971}, {3763, 50966}, {3830, 48873}, {3845, 48874}, {4297, 50781}, {5476, 52987}, {6144, 51179}, {6776, 15533}, {8703, 48876}, {10519, 31884}, {11001, 36990}, {11646, 12117}, {12245, 50790}, {12512, 50787}, {13169, 32233}, {15682, 48872}, {16789, 44285}, {19710, 39884}, {20423, 33878}, {22165, 44882}, {29959, 36987}, {30271, 51050}, {30273, 51051}, {39874, 51027}, {39899, 50961}, {40341, 50974}, {46264, 50955}, {47354, 48881}, {48905, 51023}, {48906, 50978}, {49509, 51044}, {49511, 50808}
X(54169) = reflection of X(i) in X(j) for these {i,j}: {6, 50983}, {69, 50982}, {141, 50977}, {182, 12100}, {381, 20582}, {597, 549}, {1352, 50991}, {1386, 50828}, {3589, 50984}, {3629, 50979}, {3630, 50978}, {3845, 24206}, {5476, 140}, {5480, 2}, {6329, 51139}, {8550, 51737}, {8584, 182}, {8703, 14810}, {15534, 12007}, {15687, 25561}, {18440, 50958}, {18583, 11812}, {19140, 11694}, {19710, 48885}, {20423, 3589}, {21969, 32191}, {22165, 48876}, {31670, 50959}, {32217, 18579}, {32455, 51138}, {33699, 48889}, {33878, 50970}, {43621, 51026}, {44422, 10007}, {44882, 8703}, {46264, 50971}, {47354, 141}, {48881, 50965}, {48898, 15690}, {48901, 5066}, {48904, 12101}, {49481, 51049}, {49524, 50821}, {50796, 3844}, {50955, 3631}, {50959, 34573}, {50962, 32455}, {50965, 3098}, {50979, 5092}, {51022, 3818}, {51126, 50980}, {51128, 50981}, {51129, 3763}, {51130, 51127}, {51132, 6}, {51133, 3619}, {51136, 48906}, {51147, 50824}, {51163, 3845}, {51166, 21850}, {51732, 44580}, {51737, 3}
X(54169) = complement of X(54131)
X(54169) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(2373), X(51737)}}, {{A, B, C, X(5486), X(14484)}}, {{A, B, C, X(41443), X(43713)}}
X(54169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 51212, 38072}, {3, 524, 51737}, {6, 3524, 50983}, {20, 21356, 47353}, {30, 141, 47354}, {30, 3098, 50965}, {30, 3818, 51022}, {30, 50965, 48881}, {30, 50977, 141}, {69, 10304, 43273}, {140, 5476, 48310}, {141, 3098, 48881}, {141, 50965, 30}, {376, 599, 1503}, {511, 549, 597}, {524, 51737, 8550}, {542, 14810, 8703}, {542, 48876, 22165}, {542, 8703, 44882}, {597, 21167, 549}, {599, 31884, 376}, {1992, 15692, 5085}, {3098, 50977, 30}, {3524, 50967, 6}, {3589, 50984, 5054}, {3763, 51024, 3545}, {5054, 20423, 3589}, {5054, 33878, 20423}, {5055, 31670, 50959}, {8703, 48876, 542}, {10519, 31884, 1503}, {14810, 48876, 44882}, {15688, 46264, 50971}, {15688, 50955, 46264}, {15706, 50962, 12017}, {15708, 51028, 3618}, {17504, 50979, 5092}, {20582, 29181, 381}, {22165, 44882, 542}, {24206, 48874, 51163}, {25561, 29317, 15687}, {33533, 47569, 141}, {34573, 50959, 5055}, {38335, 43621, 51026}, {43621, 50956, 38335}, {45759, 50978, 48906}, {47354, 48881, 30}, {50965, 50977, 47354}, {50967, 50983, 51132}, {50970, 50984, 20423}, {51202, 51205, 22165}

X(54170) = X(30)X(69)∩X(2)X(1350)

Barycentrics    5*a^6+13*a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2+c^2)-a^2*(17*b^4+6*b^2*c^2+17*c^4) : :
X(54170) = -2*X[6]+3*X[10304], -4*X[576]+7*X[3528], -X[1351]+2*X[8703], -X[3751]+2*X[50808], -8*X[3818]+11*X[Z51231], -X[3830]+2*X[48876], -3*X[5050]+4*X[34200], -X[6144]+2*X[51136], -X[11531]+2*X[51089], -6*X[14561]+7*X[15702]

See Ivan Pavlov, euclid 5829.

X(54170) lies on circumconic {{A, B, C, X(14484), X(36889)}} and on these lines: {2, 1350}, {3, 19661}, {4, 7883}, {6, 10304}, {20, 524}, {30, 69}, {141, 3839}, {146, 5648}, {182, 13482}, {193, 43273}, {376, 511}, {381, 10519}, {518, 34632}, {542, 11001}, {549, 14853}, {575, 21735}, {576, 3528}, {597, 15692}, {599, 3543}, {631, 5476}, {962, 47358}, {1351, 8703}, {1352, 15682}, {1353, 15690}, {1370, 9140}, {1469, 10385}, {1503, 11160}, {2781, 9143}, {2794, 50639}, {2854, 37749}, {3091, 21358}, {3098, 3524}, {3146, 47353}, {3242, 50872}, {3416, 50864}, {3522, 5032}, {3523, 47352}, {3529, 11645}, {3534, 6776}, {3545, 3619}, {3564, 15681}, {3589, 15708}, {3620, 47354}, {3629, 50971}, {3630, 51027}, {3631, 51022}, {3751, 50808}, {3763, 50959}, {3830, 48876}, {3845, 40330}, {5050, 34200}, {5054, 21850}, {5059, 15069}, {5092, 15710}, {5093, 14093}, {5104, 7735}, {5188, 22486}, {5642, 37669}, {5691, 50781}, {5731, 47356}, {5847, 34628}, {5921, 15533}, {5969, 11177}, {6054, 32458}, {6090, 47312}, {6144, 51136}, {6194, 42850}, {6225, 34787}, {7710, 7840}, {7714, 12294}, {8550, 50693}, {8593, 38738}, {8722, 47061}, {9770, 37182}, {10303, 48310}, {11008, 46264}, {11147, 41137}, {11482, 33923}, {11531, 51089}, {11898, 15685}, {12017, 45759}, {12100, 14848}, {14532, 52229}, {14540, 37172}, {14541, 37173}, {14561, 15702}, {14645, 47102}, {14810, 15698}, {15107, 26255}, {15360, 37643}, {15534, 15697}, {15640, 22165}, {15686, 34380}, {15688, 44456}, {15689, 48906}, {15693, 18583}, {15699, 50963}, {15700, 38110}, {15701, 38079}, {15703, 38136}, {15705, 50983}, {15711, 51732}, {15721, 21167}, {15933, 24471}, {16051, 32225}, {16163, 41720}, {18358, 38335}, {18906, 33706}, {18919, 21663}, {18925, 37484}, {18928, 43957}, {18931, 37494}, {19876, 38146}, {20080, 48905}, {20582, 53023}, {20583, 33748}, {21734, 53093}, {21969, 52520}, {22151, 37483}, {24206, 41106}, {28194, 50999}, {28198, 39898}, {30270, 32985}, {30775, 51360}, {32216, 47582}, {32220, 47031}, {32455, 50972}, {33703, 34507}, {34379, 34638}, {34608, 41716}, {34803, 40248}, {35260, 40112}, {37952, 47544}, {39874, 46333}, {39899, 51178}, {40341, 51215}, {40884, 42287}, {41099, 48901}, {42090, 51200}, {42091, 51203}, {44280, 47545}, {44497, 52080}, {44498, 52079}, {46853, 53092}, {47333, 52238}, {47355, 50984}, {47599, 50981}, {48883, 50430}, {48892, 51140}, {48895, 50956}, {49496, 51042}, {49509, 51064}, {49511, 50865}, {49536, 50814}, {50787, 51118}, {50811, 51192}, {50815, 51196}, {50957, 51184}, {50958, 51216}, {50968, 51132}, {50980, 51173}, {50982, 51217}, {50991, 51163}, {51050, 51063}, {51126, 51130}

X(54170) = midpoint of X(i) in X(j) for these {i,j}: {11160, 15683}, {11898, 15685}, {14927, 50992}, {15533, 48872}, {39874, 51179}, {48905, 50973}
X(54170) = reflection of X(i) in X(j) for these {i,j}: {2, 1350}, {6, 50965}, {69, 50967}, {141, 50970}, {146, 5648}, {193, 43273}, {962, 47358}, {1351, 8703}, {1353, 15690}, {1992, 376}, {3146, 47353}, {3534, 48874}, {3543, 599}, {3618, 50966}, {3629, 50971}, {3751, 50808}, {3830, 48876}, {5691, 50781}, {5921, 15533}, {6144, 51136}, {6776, 3534}, {8593, 38738}, {11001, 48873}, {11008, 50974}, {11477, 51737}, {11531, 51089}, {14927, 11001}, {15534, 44882}, {15640, 36990}, {15682, 1352}, {18440, 50978}, {18906, 33706}, {20080, 50973}, {20423, 3098}, {21969, 52520}, {22486, 5188}, {31670, 50977}, {32220, 47031}, {32455, 50972}, {36990, 22165}, {41720, 16163}, {43273, 48881}, {44456, 50979}, {48910, 47354}, {49496, 51042}, {49536, 50814}, {50864, 3416}, {50865, 49511}, {50872, 3242}, {50962, 48906}, {50967, 33878}, {50974, 46264}, {51022, 3631}, {51023, 69}, {51024, 141}, {51027, 3630}, {51028, 6}, {51029, 3620}, {51063, 51050}, {51064, 49509}, {51118, 50787}, {51140, 48892}, {51163, 50991}, {51166, 3589}, {51170, 50968}, {51178, 39899}, {51192, 50811}, {51196, 50815}, {51211, 3763}, {51212, 2}, {51214, 193}, {51215, 40341}, {51538, 10519}
X(54170) = anticomplement of X(54131)
X(54170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 50965, 10304}, {30, 33878, 50967}, {30, 50967, 69}, {30, 50978, 18440}, {30, 69, 51023}, {141, 51024, 3839}, {376, 1992, 25406}, {376, 511, 1992}, {542, 11001, 14927}, {542, 48873, 11001}, {597, 31884, 15692}, {599, 29181, 3543}, {3098, 20423, 3524}, {3522, 5032, 51737}, {3524, 20423, 3618}, {3524, 50966, 3098}, {3545, 50977, 3619}, {3620, 50687, 47354}, {10304, 51028, 6}, {11160, 15683, 1503}, {11477, 51737, 5032}, {14810, 38064, 15698}, {14927, 50992, 542}, {15689, 48906, 50975}, {15705, 51171, 50983}, {31670, 50977, 3545}, {46333, 51179, 39874}, {47354, 48910, 50687}, {48910, 50687, 51029}

X(54171) = X(183)X(3524)∩X(458)X(1992)

Barycentrics    (a^6+b^6-13*b^4*c^2+11*b^2*c^4+c^6+a^4*(11*b^2-c^2)-a^2*(13*b^4+6*b^2*c^2+c^4))*(a^6+b^6+11*b^4*c^2-13*b^2*c^4+c^6-a^4*(b^2-11*c^2)-a^2*(b^4+6*b^2*c^2+13*c^4)) : :

See Ivan Pavlov, euclid 5829.

X(54171) lies on these lines: {183, 3524}, {290, 50967}, {458, 1992}, {9740, 46806}

X(54171) = isotomic conjugate of X(54132)
X(54171) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(2), X(183)}}, {{A, B, C, X(4), X(3524)}}, {{A, B, C, X(69), X(671)}}, {{A, B, C, X(76), X(32836)}}, {{A, B, C, X(95), X(18842)}}, {{A, B, C, X(325), X(9740)}}, {{A, B, C, X(393), X(9302)}}, {{A, B, C, X(511), X(50967)}}, {{A, B, C, X(1494), X(5485)}}, {{A, B, C, X(2996), X(5641)}}, {{A, B, C, X(9154), X(43537)}}, {{A, B, C, X(9214), X(38664)}}, {{A, B, C, X(10519), X(20423)}}, {{A, B, C, X(14853), X(50977)}}, {{A, B, C, X(19222), X(52187)}}, {{A, B, C,
X(37665), X(37671)}}

X(54172) = X(182)X(6090)∩X(183)X(3524)

Barycentrics    a^2*(a^6+b^6-13*b^4*c^2+11*b^2*c^4+c^6+a^4*(11*b^2-c^2)-a^2*(13*b^4+6*b^2*c^2+c^4))*(a^6+b^6+11*b^4*c^2-13*b^2*c^4+c^6-a^4*(b^2-11*c^2)-a^2*(b^4+6*b^2*c^2+13*c^4)) : :

See Ivan Pavlov, euclid 5829.

X(54172) lies on these lines: {182, 6090}, {183, 3524}, {1384, 1597}, {4232, 33971}, {9755, 11169}, {9756, 45819}

X(54172) = isogonal conjugate of X(54132)
X(54172) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(2), X(1597)}}, {{A, B, C, X(3), X(111)}}, {{A, B, C, X(6), X(95)}}, {{A, B, C, X(22), X(37935)}}, {{A, B, C, X(25), X(3431)}}, {{A, B, C, X(64), X(7607)}}, {{A, B, C, X(74), X(6090)}}, {{A, B, C, X(251), X(44731)}}, {{A, B, C, X(262), X(14490)}}, {{A, B, C, X(468), X(841)}}, {{A, B, C, X(842), X(8770)}}, {{A, B, C, X(1297), X(40103)}}, {{A, B, C, X(1350), X(38010)}}, {{A, B, C, X(1383), X(5481)}}, {{A, B, C, X(3424), X(3531)}}, {{A, B, C, X(3527), X(47586)}}, {{A, B, C, X(3532), X(14388)}}, {{A, B, C, X(3563), X(20421)}}, {{A, B, C, X(7608), X(22334)}}, {{A, B, C, X(11172), X(40802)}}, {{A, B, C, X(11738), X(53103)}}, {{A, B, C, X(13603), X(14494)}}, {{A, B, C, X(14483), X(39951)}}, {{A, B, C, X(14489), X(29180)}}, {{A, B, C, X(28193), X(39954)}}, {{A, B, C, X(52518), X(53100)}}
X(54172) = X(i)-vertex conjugate of X(j) for these {i, j}: {25, 3426}, {64, 11181}, {1383, 3424}

X(54173) = X(2)X(51)∩X(3)X(524)

Barycentrics    a^6+5*a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)-a^2*(7*b^4+6*b^2*c^2+7*c^4) : :
X(54173) = -X[6]+2*X[549], -2*X[576]+5*X[631], -3*X[5085]+4*X[12100], -3*X[38029]+4*X[50828], -2*X[48904]+5*X[51537]

See Ivan Pavlov, euclid 5829.

X(54173) lies on these lines: {2, 51}, {3, 524}, {4, 7883}, {5, 21358}, {6, 549}, {20, 11180}, {26, 42021}, {30, 599}, {69, 74}, {98, 50639}, {140, 11477}, {141, 381}, {182, 1992}, {183, 51438}, {193, 5092}, {343, 31152}, {394, 44210}, {515, 50781}, {516, 50787}, {517, 47358}, {518, 3654}, {519, 24257}, {523, 21733}, {530, 44461}, {531, 44465}, {541, 5181}, {543, 19905}, {547, 3763}, {550, 15069}, {575, 3523}, {576, 631}, {597, 1351}, {611, 4995}, {613, 5298}, {754, 35387}, {952, 50783}, {1007, 51396}, {1316, 47585}, {1353, 17504}, {1385, 47356}, {1386, 3653}, {1469, 10056}, {1503, 3534}, {1511, 34319}, {1597, 41585}, {1641, 47570}, {2080, 37809}, {2393, 35704}, {2482, 8722}, {2549, 15993}, {2709, 2770}, {2781, 5655}, {2854, 20126}, {3056, 10072}, {3094, 7739}, {3146, 18553}, {3147, 11470}, {3313, 44441}, {3416, 28204}, {3525, 25555}, {3526, 48310}, {3530, 53093}, {3543, 3620}, {3545, 24206}, {3547, 15606}, {3564, 8703}, {3589, 14848}, {3618, 15702}, {3619, 5071}, {3629, 12017}, {3630, 14093}, {3631, 15681}, {3655, 28538}, {3656, 51003}, {3767, 32521}, {3815, 11173}, {3830, 29181}, {3839, 25561}, {3845, 10516}, {4550, 41583}, {5050, 8584}, {5055, 5480}, {5066, 50964}, {5085, 12100}, {5093, 15701}, {5097, 15708}, {5102, 11812}, {5104, 7737}, {5108, 14694}, {5171, 32985}, {5188, 7801}, {5447, 50649}, {5463, 14539}, {5464, 14538}, {5477, 8588}, {5642, 7493}, {5648, 5663}, {5651, 26255}, {5654, 44262}, {5844, 50790}, {5847, 51705}, {5864, 37341}, {5865, 37340}, {5907, 34621}, {5921, 48898}, {5965, 19708}, {5969, 11632}, {6031, 38940}, {6036, 23055}, {6054, 37182}, {6090, 35266}, {6101, 37473}, {6144, 14891}, {6393, 7788}, {6771, 22580}, {6774, 22579}, {6776, 10304}, {7426, 15066}, {7492, 9143}, {7615, 15980}, {7735, 43456}, {7795, 9821}, {7800, 37345}, {7803, 10357}, {7810, 30270}, {7840, 9744}, {8593, 21166}, {9019, 13340}, {9041, 34718}, {9140, 16063}, {9306, 32267}, {9540, 44502}, {9734, 47061}, {9735, 51012}, {9736, 51015}, {9737, 33215}, {9880, 19662}, {9939, 36998}, {9971, 13391}, {10109, 38136}, {10124, 38079}, {10193, 10250}, {10387, 15170}, {10510, 18580}, {10517, 44471}, {10518, 44472}, {10541, 15712}, {10627, 18281}, {10645, 51200}, {10646, 51203}, {10753, 41134}, {10754, 17008}, {11001, 29012}, {11007, 16279}, {11008, 15715}, {11161, 12117}, {11184, 37451}, {11202, 41719}, {11284, 20192}, {11482, 15720}, {11539, 18583}, {11646, 43619}, {11663, 15644}, {11694, 52697}, {11799, 47556}, {11850, 14530}, {11898, 15688}, {12007, 15706}, {12251, 52996}, {13330, 31401}, {13355, 14645}, {13564, 15582}, {13632, 48908}, {13633, 48875}, {13634, 17378}, {13635, 17346}, {13745, 19782}, {13935, 44501}, {14070, 37485}, {14216, 34787}, {14787, 37484}, {14912, 15698}, {14927, 48885}, {14994, 32836}, {15035, 41720}, {15040, 25329}, {15067, 44275}, {15073, 26937}, {15303, 15462}, {15448, 40912}, {15520, 51141}, {15577, 31166}, {15682, 29317}, {15683, 43150}, {15685, 41152}, {15686, 48905}, {15687, 18358}, {15690, 50968}, {15695, 50971}, {15697, 50969}, {15703, 34573}, {15707, 20583}, {15711, 50986}, {15713, 50981}, {15716, 51174}, {15717, 20190}, {15718, 32455}, {15719, 39561}, {15721, 46267}, {15722, 51139}, {15759, 50985}, {15919, 45662}, {16051, 45311}, {16187, 44833}, {16241, 51206}, {16242, 51207}, {16317, 46949}, {17811, 44212}, {18420, 51993}, {18911, 44555}, {19127, 22115}, {19145, 52045}, {19146, 52046}, {19709, 50959}, {19710, 51189}, {19711, 50987}, {22110, 40248}, {22151, 44493}, {23327, 44751}, {25563, 34788}, {26516, 44484}, {26521, 44483}, {26543, 44217}, {28146, 50792}, {28160, 50784}, {28164, 50788}, {28174, 50791}, {28186, 50785}, {28194, 49511}, {28224, 50782}, {28234, 51089}, {28236, 50786}, {28466, 36740}, {29010, 51051}, {29574, 46475}, {31133, 37636}, {31400, 44500}, {32216, 44569}, {32269, 47597}, {32424, 36883}, {33006, 38228}, {33750, 51178}, {33813, 51798}, {34148, 44491}, {34200, 40341}, {34628, 39885}, {34632, 39898}, {34817, 45088}, {35259, 37904}, {35383, 47102}, {35439, 44562}, {35474, 44134}, {35486, 44102}, {35925, 52994}, {35954, 39656}, {37172, 47068}, {37173, 47066}, {37188, 41145}, {37477, 51739}, {37483, 44218}, {37511, 45118}, {37638, 47097}, {37644, 41462}, {37668, 51397}, {37671, 51374}, {37950, 47276}, {38029, 50828}, {38065, 51150}, {38066, 49524}, {38115, 51002}, {38116, 47359}, {38118, 50829}, {38335, 51163}, {40112, 47596}, {41099, 51538}, {41149, 51138}, {41586, 46336}, {43652, 44470}, {44214, 47544}, {44266, 47450}, {44475, 45522}, {44476, 45523}, {44580, 50988}, {46333, 48896}, {48857, 50591}, {48889, 50687}, {48904, 51537}, {50664, 51170}, {50798, 50949}, {50805, 50998}, {50808, 51004}, {50810, 50999}, {50811, 50950}, {50824, 51000}, {50871, 51168}, {50954, 51022}, {50963, 51166}, {50975, 51215}, {51136, 51175}

X(54173) = midpoint of X(i) in X(j) for these {i,j}: {2, 50967}, {20, 11180}, {69, 376}, {98, 50639}, {381, 33878}, {599, 1350}, {3534, 50955}, {6776, 11160}, {8703, 50978}, {9143, 32247}, {11001, 51023}, {11161, 12117}, {11178, 52987}, {15533, 43273}, {15534, 50973}, {15681, 18440}, {22165, 50965}, {34628, 39885}, {34632, 39898}, {50808, 51004}, {50810, 50999}, {50811, 50950}, {50966, 50990}, {50968, 50989}, {50970, 50991}, {50974, 50992}
X(54173) = reflection of X(i) in X(j) for these {i,j}: {2, 50977}, {4, 11178}, {6, 549}, {376, 3098}, {381, 141}, {576, 10168}, {599, 48876}, {1351, 597}, {1352, 599}, {1992, 182}, {3534, 50965}, {3543, 3818}, {3656, 51003}, {3830, 47354}, {5050, 21167}, {5102, 38110}, {5480, 20582}, {7426, 47569}, {8584, 50983}, {9140, 49116}, {9143, 12584}, {9880, 19662}, {9970, 5642}, {10250, 10193}, {11178, 40107}, {11179, 3}, {11180, 34507}, {11799, 47556}, {12177, 2482}, {14912, 17508}, {15533, 50978}, {15534, 50979}, {15681, 48881}, {15683, 48880}, {15687, 18358}, {16279, 11007}, {20423, 2}, {21850, 547}, {22165, 50982}, {22579, 6774}, {22580, 6771}, {31166, 15577}, {31670, 381}, {31958, 22712}, {34319, 1511}, {35439, 44562}, {41149, 51138}, {41719, 11202}, {43273, 8703}, {43621, 3543}, {46264, 376}, {47354, 50991}, {47356, 1385}, {47359, 50821}, {48901, 25561}, {48905, 15686}, {48906, 34200}, {48910, 15687}, {50798, 50949}, {50805, 50998}, {50954, 51142}, {50955, 22165}, {50956, 50993}, {50958, 41152}, {50959, 51143}, {50961, 15533}, {50962, 8584}, {50964, 51186}, {50979, 12100}, {50993, 51184}, {51000, 50824}, {51005, 50828}, {51024, 3845}, {51185, 50980}, {51187, 50986}, {51188, 50985}, {51798, 33813}
X(54173) = complement of X(54132)
X(54173) = anticomplement of X(5476)
X(54173) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(30), X(14907)}}, {{A, B, C, X(74), X(263)}}, {{A, B, C, X(182), X(373)}}, {{A, B, C, X(262), X(1494)}}, {{A, B, C, X(290), X(20423)}}, {{A, B, C, X(2373), X(11179)}}, {{A, B, C, X(2770), X(53764)}}, {{A, B, C, X(7998), X(40803)}}, {{A, B, C, X(10302), X(42313)}}
X(54173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 10519, 50977}, {2, 20423, 14561}, {2, 50967, 511}, {2, 51028, 14853}, {2, 511, 20423}, {3, 524, 11179}, {4, 21356, 11178}, {6, 549, 38064}, {20, 11180, 11645}, {30, 48876, 599}, {30, 599, 1352}, {69, 3098, 46264}, {69, 376, 542}, {141, 33878, 31670}, {376, 542, 46264}, {511, 22712, 31958}, {511, 50977, 2}, {542, 3098, 376}, {547, 21850, 38072}, {599, 1350, 30}, {616, 617, 14907}, {1350, 1352, 48873}, {1350, 36990, 48874}, {1350, 48876, 1352}, {1351, 5054, 597}, {1503, 22165, 50955}, {1503, 50965, 3534}, {1503, 50982, 22165}, {1992, 3524, 182}, {3534, 50955, 1503}, {3564, 15533, 50961}, {3564, 50978, 15533}, {3564, 8703, 43273}, {3631, 48881, 18440}, {3763, 38072, 547}, {3839, 40330, 25561}, {3845, 10516, 50956}, {5050, 50962, 8584}, {5066, 53023, 50964}, {5085, 15534, 50979}, {5085, 50973, 15534}, {5480, 20582, 5055}, {7998, 15360, 2}, {8584, 50983, 5050}, {8703, 50978, 3564}, {10124, 38079, 47355}, {10304, 11160, 6776}, {10516, 51024, 3845}, {10519, 50967, 2}, {11001, 51023, 29012}, {11178, 19924, 4}, {11178, 40107, 21356}, {11178, 52987, 19924}, {11645, 34507, 11180}, {12100, 34380, 50979}, {12100, 50979, 5085}, {14848, 15694, 3589}, {15533, 31884, 43273}, {15533, 43273, 3564}, {15534, 50973, 34380}, {15693, 50962, 5050}, {15694, 44456, 14848}, {15698, 51179, 14912}, {15719, 51214, 39561}, {19708, 50974, 25406}, {19924, 40107, 11178}, {21167, 50983, 15693}, {22165, 50965, 1503}, {25406, 50992, 50974}, {25561, 48901, 3839}, {29181, 47354, 3830}, {29181, 50991, 47354}, {31884, 43273, 8703}, {34380, 50979, 15534}, {38110, 50980, 11812}, {39874, 48892, 46264}, {40107, 52987, 4}, {43273, 50978, 50961}, {50828, 51005, 38029}, {50965, 50982, 50955}, {50966, 50990, 29012}, {50967, 50977, 20423}, {50970, 50991, 29181}, {50974, 50992, 5965}, {50993, 51024, 10516}, {51010, 51013, 69}

X(54174) = X(2)X(51)∩X(20)X(524)

Barycentrics    a^6+23*a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)-a^2*(25*b^4+6*b^2*c^2+25*c^4) : :
X(54174) = -8*X[182]+9*X[15705], -4*X[381]+5*X[3620], -8*X[576]+11*X[15717], -2*X[1351]+3*X[3524], -X[3830]+2*X[50978], -6*X[5050]+7*X[15698], -3*X[5093]+4*X[12100], -3*X[5102]+4*X[50983], -3*X[10516]+4*X[50982], -2*X[37517]+3*X[38064]

See Ivan Pavlov, euclid 5829.

X(54174) lies on these lines: {2, 51}, {3, 5032}, {6, 9542}, {20, 524}, {30, 5921}, {69, 3543}, {147, 50639}, {182, 15705}, {193, 376}, {381, 3620}, {542, 15683}, {549, 44456}, {576, 15717}, {597, 15708}, {599, 3839}, {1350, 1992}, {1351, 3524}, {1352, 50687}, {1353, 15688}, {1503, 50973}, {1699, 50787}, {3091, 21356}, {3098, 51170}, {3146, 11180}, {3522, 11179}, {3523, 11477}, {3534, 34380}, {3545, 48876}, {3564, 11001}, {3618, 15721}, {3619, 38072}, {3830, 50978}, {3832, 11178}, {5050, 15698}, {5056, 21358}, {5059, 11645}, {5068, 40107}, {5071, 21850}, {5085, 51132}, {5093, 12100}, {5102, 50983}, {5999, 9740}, {6776, 48885}, {7809, 10008}, {8359, 40268}, {8584, 31884}, {8703, 14912}, {9737, 11155}, {10109, 51173}, {10299, 11482}, {10303, 47352}, {10516, 50982}, {10753, 52695}, {11008, 48881}, {11173, 37665}, {11412, 34621}, {12017, 15715}, {12101, 50954}, {14848, 15702}, {15069, 49135}, {15078, 37491}, {15531, 36987}, {15533, 15640}, {15534, 25406}, {15681, 39874}, {15682, 50955}, {15685, 51175}, {15686, 39899}, {15690, 50986}, {15697, 43273}, {15701, 51172}, {15707, 51732}, {15709, 18583}, {15716, 50987}, {17504, 53091}, {17578, 34507}, {18906, 32874}, {19708, 50979}, {20583, 53094}, {21167, 51185}, {22165, 51024}, {29012, 50961}, {33751, 35418}, {34638, 39878}, {35513, 44750}, {37483, 37784}, {37488, 37941}, {37517, 38064}, {37668, 51438}, {37712, 50786}, {37907, 47468}, {37952, 47545}, {44280, 47541}, {47354, 50990}, {50801, 51168}, {50808, 50952}, {50811, 51001}, {50815, 51197}, {50864, 50950}, {50865, 51004}, {50872, 50999}, {50958, 50989}, {50959, 50993}, {50960, 51142}, {50969, 51140}, {50991, 51131}, {51077, 51193}, {51136, 51187}, {51174, 51176}

X(54174) = midpoint of X(i) in X(j) for these {i,j}: {11001, 51179}, {15683, 20080}, {15685, 51175}
X(54174) = reflection of X(i) in X(j) for these {i,j}: {2, 50967}, {147, 50639}, {193, 376}, {376, 33878}, {1992, 1350}, {3146, 11180}, {3543, 69}, {3830, 50978}, {5921, 11160}, {8584, 50970}, {11179, 52987}, {15531, 36987}, {15534, 50965}, {15640, 51023}, {15682, 50955}, {39874, 15681}, {39878, 34638}, {39899, 15686}, {44456, 549}, {50864, 50950}, {50865, 51004}, {50872, 50999}, {50952, 50808}, {50962, 8703}, {50974, 3534}, {50986, 15690}, {50992, 50973}, {51001, 50811}, {51023, 15533}, {51024, 22165}, {51028, 2}, {51029, 50989}, {51166, 50991}, {51187, 51136}, {51197, 50815}, {51211, 50990}, {51212, 599}, {51214, 15534}, {51215, 50992}
X(54174) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(182), X(12045)}}, {{A, B, C, X(263), X(14490)}}, {{A, B, C, X(290), X(51028)}}, {{A, B, C, X(15082), X(40803)}}
X(54174) = anticomplement of X(54132)
X(54174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 511, 51028}, {30, 11160, 5921}, {511, 50967, 2}, {599, 51212, 3839}, {1350, 1992, 10304}, {1503, 50973, 50992}, {1503, 50992, 51215}, {3534, 34380, 50974}, {8584, 50970, 31884}, {8703, 50962, 14912}, {10519, 20423, 2}, {11001, 51179, 3564}, {11180, 19924, 3146}, {14853, 50977, 2}, {14912, 50966, 8703}, {15533, 29181, 51023}, {15683, 20080, 542}, {25406, 51214, 15534}, {29181, 51023, 15640}, {50990, 51538, 47354}, {50991, 51166, 53023}

X(54175) = X(9)X(7680)∩X(144)X(3428)

Barycentrics    4*a^9-8*a^8*(b+c)-(b-c)^6*(b+c)^3+a^7*(-7*b^2+8*b*c-7*c^2)+a^5*(b-c)^2*(b^2-8*b*c+c^2)-a*(b-c)^4*(b+c)^2*(b^2+4*b*c+c^2)+a^3*(b^2-c^2)^2*(3*b^2+4*b*c+3*c^2)+a^2*(b-c)^2*(b+c)^3*(7*b^2-10*b*c+7*c^2)+a^6*(21*b^3+11*b^2*c+11*b*c^2+21*c^3)-a^4*(19*b^5+3*b^4*c+10*b^3*c^2+10*b^2*c^3+3*b*c^4+19*c^5) : :

See Ivan Pavlov, euclid 5829.

X(54175) lies on these lines: {9, 7680}, {55, 21168}, {144, 3428}, {517, 51090}, {528, 19914}, {2886, 5762}, {4973, 31657}, {5759, 5842}, {8255, 31658}, {12699, 38454}

X(54175) = midpoint of X(i) in X(j) for these {i,j}: {144, 3428}, {5759, 42014}
X(54175) = reflection of X(i) in X(j) for these {i,j}: {7680, 9}, {8255, 31658}
X(54175) = complement of X(54133)
X(54175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5759, 42014, 5842}

X(54176) = X(1)X(1532)∩X(3)X(5854)

Barycentrics    4*a^7-8*a^6*(b+c)-(b-c)^4*(b+c)^3+a^5*(-3*b^2+32*b*c-3*c^2)-2*a^3*(b-c)^2*(3*b^2+16*b*c+3*c^2)+a*(b^2-c^2)^2*(5*b^2-12*b*c+5*c^2)-6*a^2*(b-c)^2*(b^3-3*b^2*c-3*b*c^2+c^3)+a^4*(15*b^3-23*b^2*c-23*b*c^2+15*c^3) : :

See Ivan Pavlov, euclid 5829.

X(54176) lies on these lines: {1, 1532}, {3, 5854}, {8, 20418}, {46, 7966}, {56, 6942}, {145, 10310}, {355, 3847}, {517, 550}, {529, 4930}, {944, 1317}, {952, 1329}, {1385, 8256}, {3036, 26492}, {3811, 37727}, {3813, 12737}, {5690, 11715}, {6691, 10246}, {6738, 13607}, {7686, 16215}, {11567, 32213}, {12645, 33559}, {12735, 40257}, {18526, 37821}, {31141, 50818}

X(54176) = midpoint of X(i) in X(j) for these {i,j}: {145, 10310}, {944, 2098}, {18526, 37821}, {31141, 50818}
X(54176) = reflection of X(i) in X(j) for these {i,j}: {7681, 1}, {7686, 16215}, {8256, 1385}, {12645, 33559}, {24928, 13607}
X(54176) = complement of X(54134)
X(54176) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {944, 2098, 2829}, {3244, 5882, 12675}

X(54177) = X(20)X(5854)∩X(517)X(3529)

Barycentrics    7*a^7-15*a^6*(b+c)-3*(b-c)^4*(b+c)^3-3*a^5*(b^2-20*b*c+c^2)+a*(b^2-c^2)^2*(11*b^2-28*b*c+11*c^2)-a^2*(b-c)^2*(9*b^3-41*b^2*c-41*b*c^2+9*c^3)+a^4*(27*b^3-47*b^2*c-47*b*c^2+27*c^3)-a^3*(15*b^4+32*b^3*c-102*b^2*c^2+32*b*c^3+15*c^4) : :

See Ivan Pavlov, euclid 5829.

X(54177) lies on these lines: {20, 5854}, {145, 12667}, {517, 3529}, {519, 10085}, {952, 3436}, {2098, 5225}, {3621, 10310}, {3623, 7681}, {12245, 12515}, {12704, 36977}, {18526, 37002}, {25416, 52683}, {28236, 30323}

X(54177) = reflection of X(i) in X(j) for these {i,j}: {3621, 10310}, {37002, 18526}
X(54177) = anticomplement of X(54134)

X(54178) = X(3)X(527)∩X(7)X(6282)

Barycentrics    3*a^8*(b+c)+16*a^6*b*c*(b+c)-4*a*b*(b-c)^4*c*(b+c)^2-(b-c)^6*(b+c)^3-8*a^7*(b^2+b*c+c^2)+8*a^2*(b-c)^4*(b^3+2*b^2*c+2*b*c^2+c^3)-8*a^3*(b-c)^2*(b^4-2*b^3*c-2*b*c^3+c^4)+4*a^5*(4*b^4-5*b^3*c+10*b^2*c^2-5*b*c^3+4*c^4)-2*a^4*(5*b^5+3*b^4*c+8*b^3*c^2+8*b^2*c^3+3*b*c^4+5*c^5) : :

See Ivan Pavlov, euclid 5829.

X(54178) lies on these lines: {3, 527}, {7, 6282}, {57, 21151}, {142, 6907}, {214, 38759}, {516, 6948}, {517, 5542}, {971, 3452}, {3359, 43151}, {5732, 6987}, {5784, 6245}, {5817, 20196}, {6173, 6916}, {6692, 38122}, {6700, 52684}, {6954, 8257}, {7962, 35514}, {12848, 21164}, {30503, 41570}, {31142, 41561}, {36973, 36996}

X(54178) = midpoint of X(i) in X(j) for these {i,j}: {7, 6282}, {5732, 52457}, {7962, 35514}, {36973, 36996}
X(54178) = reflection of X(i) in X(j) for these {i,j}: {3359, 43151}, {7682, 142}
X(54178) = complement of X(54135)

X(54179) = X(7)X(6925)∩X(20)X(527)

Barycentrics    a^9+3*a^8*(b+c)-(b-c)^6*(b+c)^3-3*a*(b^2-c^2)^4-8*a^7*(2*b^2+b*c+2*c^2)+4*a^6*(2*b^3+7*b^2*c+7*b*c^2+2*c^3)-8*a^3*(b-c)^2*(b^4-5*b^3*c-4*b^2*c^2-5*b*c^3+c^4)+a^5*(26*b^4-48*b^3*c+60*b^2*c^2-48*b*c^3+26*c^4)+4*a^2*(b-c)^2*(4*b^5-b^4*c-11*b^3*c^2-11*b^2*c^3-b*c^4+4*c^5)-2*a^4*(13*b^5-b^4*c+4*b^3*c^2+4*b^2*c^3-b*c^4+13*c^5) : :

See Ivan Pavlov, euclid 5829.

X(54179) lies on these lines: {7, 6925}, {20, 527}, {144, 6282}, {329, 971}, {517, 36996}, {2093, 43182}, {2094, 7580}, {5732, 12848}, {5850, 7994}, {6172, 6909}, {6840, 36991}, {27383, 52684}, {37421, 43177}, {37611, 52653}

X(54179) = reflection of X(i) in X(j) for these {i,j}: {144, 6282}, {2093, 43182}, {12848, 5732}, {36991, 52457}
X(54179) = anticomplement of X(54135)

X(54180) = X(1)X(3430)∩X(3)X(758)

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

See Ivan Pavlov, euclid 5829.

X(54180) lies on these lines: {1, 3430}, {3, 758}, {58, 602}, {511, 1385}, {515, 3454}, {540, 51705}, {550, 29097}, {944, 36974}, {1046, 7987}, {1125, 7683}, {1319, 10544}, {1330, 5731}, {2825, 11712}, {2842, 11709}, {3743, 30285}, {4221, 16132}, {4228, 19861}, {4653, 8235}, {5429, 30389}, {5538, 37328}, {6326, 37431}, {6693, 10165}, {9840, 35016}, {10176, 13732}, {11573, 11700}, {11713, 31786}, {18481, 37823}, {19544, 30143}, {22791, 29032}, {24309, 37531}, {31803, 49128}, {37620, 51717}

X(54180) = midpoint of X(i) in X(j) for these {i,j}: {1, 3430}, {944, 36974}, {18481, 37823}
X(54180) = reflection of X(i) in X(j) for these {i,j}: {7683, 1125}
X(54180) = complement of X(54136)

X(54181) = X(20)X(758)∩X(8)X(3430)

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

See Ivan Pavlov, euclid 5829.

X(54181) lies on these lines: {1, 37443}, {8, 3430}, {20, 758}, {40, 50624}, {58, 5731}, {511, 944}, {515, 1330}, {1046, 4297}, {3476, 10544}, {3529, 29097}, {3616, 7683}, {7385, 30143}, {7987, 8258}, {34195, 48890}

X(54181) = reflection of X(i) in X(j) for these {i,j}: {8, 3430}, {1046, 4297}
X(54181) = anticomplement of X(54136)

X(54182) = X(34)X(17520)∩X(65)X(1884)

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

See Ivan Pavlov, euclid 5829.

X(54182) lies on these lines: {34, 17520}, {65, 1884}, {429, 1875}, {1866, 1874}

X(54182) = zosma transform of X(54136)

X(54183) = X(6)X(30)∩X(25)X(182)

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

See Ivan Pavlov, euclid 5829.

X(54183) lies on circumconic {{A, B, C, X(14486), X(34288)}} and on these lines: {3, 2393}, {4, 44503}, {6, 30}, {20, 37784}, {22, 5622}, {25, 182}, {154, 15462}, {155, 14791}, {159, 51393}, {376, 41614}, {394, 542}, {511, 10602}, {524, 18917}, {550, 8548}, {575, 10982}, {576, 46850}, {1181, 44469}, {1350, 5621}, {1352, 1368}, {1370, 1993}, {1503, 18451}, {1593, 44479}, {1596, 14561}, {1660, 8780}, {2386, 13355}, {2790, 12177}, {3534, 39562}, {3818, 16072}, {5050, 18534}, {5085, 6644}, {5967, 36192}, {6000, 11511}, {6102, 11477}, {6800, 37980}, {7500, 34545}, {7530, 53093}, {7728, 19149}, {8538, 10575}, {8550, 36747}, {8681, 37480}, {9019, 37489}, {9715, 19360}, {9813, 16836}, {9976, 37853}, {10168, 47597}, {10541, 12106}, {10594, 43815}, {11178, 32216}, {11180, 15066}, {11216, 14855}, {11413, 15073}, {11414, 44470}, {11416, 15072}, {11456, 22151}, {11745, 36752}, {12084, 15074}, {12085, 50649}, {12283, 40228}, {13248, 44573}, {13352, 32621}, {13598, 44489}, {15069, 32140}, {15113, 30771}, {15760, 23327}, {17710, 44883}, {17825, 38064}, {18438, 34778}, {18533, 25406}, {18914, 37498}, {18919, 35513}, {20806, 39874}, {24206, 31255}, {29012, 44438}, {32599, 53097}, {34117, 51491}, {34514, 47353}, {34777, 37511}, {35259, 40114}, {36753, 45034}, {37196, 44480}, {37458, 37514}, {37506, 51739}, {38029, 51695}, {38110, 44233}, {44275, 47352}, {44276, 53023}, {44454, 53091}

X(54183) = midpoint of X(i) in X(j) for these {i,j}: {1370, 6776}, {10602, 21312}
X(54183) = reflection of X(i) in X(j) for these {i,j}: {25, 182}, {1352, 1368}, {18534, 19136}, {39879, 1660}
X(54183) = complement of X(54149)
X(54183) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5050, 18534, 19136}, {10602, 21312, 511}, {39522, 50979, 6}

X(54184) = X(20)X(2393)∩X(30)X(193)

Barycentrics    3*a^12-10*a^10*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)^2+a^8*(9*b^4+62*b^2*c^2+9*c^4)+4*a^6*(b^6-6*b^4*c^2-6*b^2*c^4+c^6)+6*a^2*(b^2-c^2)^2*(b^6-3*b^4*c^2-3*b^2*c^4+c^6)-a^4*(11*b^8+10*b^4*c^4+11*c^8) : :

See Ivan Pavlov, euclid 5829.

X(54184) lies on these lines: {20, 2393}, {30, 193}, {1353, 44454}, {1370, 5921}, {3060, 6776}, {5032, 15032}, {5656, 11416}, {11179, 34417}, {12244, 14984}, {13445, 50967}, {14912, 18534}, {18919, 47096}, {19136, 33748}, {34621, 37784}

X(54184) = reflection of X(i) in X(j) for these {i,j}: {5921, 1370}, {7500, 6776}, {44454, 1353}
X(54184) = anticomplement of X(54149)

X(54185) = X(28)X(9940)∩X(30)X(553)

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

See Ivan Pavlov, euclid 5829.

X(54185) lies on these lines: {28, 9940}, {30, 553}, {517, 30267}, {1071, 52364}, {5777, 21530}, {11227, 44220}, {12041, 31793}, {18443, 20831}, {31788, 44661}

X(54185) = midpoint of X(i) in X(j) for these {i,j}: {1071, 52364}
X(54185) = reflection of X(i) in X(j) for these {i,j}: {28, 9940}, {5777, 21530}
X(54185) = complement of X(54150)

X(54186) = X(30)X(3868)∩X(1071)X(31293)

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

See Ivan Pavlov, euclid 5829.

X(54186) lies on these lines: {30, 3868}, {1071, 31293}, {12528, 52364}, {18444, 20831}

X(54186) = reflection of X(i) in X(j) for these {i,j}: {12528, 52364}, {31293, 1071}
X(54186) = anticomplement of X(54150)

X(54187) = X(3)X(6)∩X(315)X(11257)

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

See Ivan Pavlov, euclid 5829.

X(54187) lies on these lines: {3, 6}, {315, 11257}, {626, 6248}, {2782, 3933}, {3785, 40923}, {6310, 42061}, {7750, 38642}, {7785, 22503}, {7786, 9753}, {7800, 22677}, {7947, 9772}, {10350, 11676}, {11272, 37451}, {15819, 18806}, {20065, 32522}, {20576, 40108}, {32448, 41651}

X(54187) = midpoint of X(i) in X(j) for these {i,j}: {315, 11257}, {30270, 32452}
X(54187) = reflection of X(i) in X(j) for these {i,j}: {32, 13334}, {6248, 626}, {35430, 3}, {35431, 50652}, {35432, 182}, {35436, 39}, {46321, 13335}
X(54187) = complement of X(54152)
X(54187) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 511, 35430}, {39, 511, 35436}, {182, 511, 35432}, {511, 13334, 32}, {511, 13335, 46321}, {511, 50652, 35431}, {3095, 11171, 9605}, {5188, 21163, 15513}, {13355, 30270, 3}, {21163, 46321, 13335}, {30270, 32452, 511}

X(54188) = X(20)X(185)∩X(32)X(32522)

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

See Ivan Pavlov, euclid 5829.

X(54188) lies on these lines: {20, 185}, {32, 32522}, {1078, 6194}, {2548, 32452}, {3095, 37182}, {3522, 35430}, {5999, 12251}, {7709, 7839}, {7793, 40923}, {9772, 32818}, {14532, 32520}, {22503, 32816}

X(54188) = reflection of X(i) in X(j) for these {i,j}: {20065, 11257}
X(54188) = anticomplement of X(54152)
X(54188) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {511, 11257, 20065}

X(54189) = X(4)X(69)∩X(183)X(3094)

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

See Ivan Pavlov, euclid 5829.

X(54189) lies on these lines: {4, 69}, {98, 50640}, {183, 3094}, {325, 24256}, {384, 35432}, {3098, 38907}, {3917, 25332}, {5052, 7804}, {5092, 34885}, {5152, 14810}, {5969, 37671}, {6393, 49111}, {7795, 46305}, {10007, 37688}, {11161, 14711}, {12212, 39093}, {12215, 13354}, {43461, 51373}

X(54189) = Cundy-Parry Psi of X(54152)

X(54190) = X(21)X(31806)∩X(30)X(51113)

Barycentrics    a*(2*a^9-3*a^8*(b+c)+a^7*(-6*b^2+4*b*c-6*c^2)+3*a*b*c*(b^2-c^2)^2*(b^2-b*c+c^2)-(b-c)^4*(b+c)^3*(b^2+b*c+c^2)+2*a^6*(5*b^3+b^2*c+b*c^2+5*c^3)+a^5*(6*b^4-5*b^3*c+20*b^2*c^2-5*b*c^3+6*c^4)+a^2*(b-c)^2*(6*b^5+8*b^4*c+11*b^3*c^2+11*b^2*c^3+8*b*c^4+6*c^5)-a^4*(12*b^5-5*b^4*c+6*b^3*c^2+6*b^2*c^3-5*b*c^4+12*c^5)-a^3*(2*b^6+2*b^5*c+11*b^4*c^2-6*b^3*c^3+11*b^2*c^4+2*b*c^5+2*c^6)) : :

See Ivan Pavlov, euclid 5829.

X(54190) lies on these lines: {21, 31806}, {30, 51113}, {517, 3647}, {4973, 49107}, {6684, 41542}, {10993, 43174}, {11012, 52126}, {11014, 11684}, {11362, 22937}, {14526, 41547}, {16113, 52367}, {16125, 25639}, {26202, 51118}

X(54190) = midpoint of X(i) in X(j) for these {i,j}: {11012, 52126}, {11014, 11684}, {16113, 52367}
X(54190) = reflection of X(i) in X(j) for these {i,j}: {16125, 25639}
X(54190) = complement of X(54153)

X(54191) = X(517)X(3648)∩X(2475)X(5535)

Barycentrics    3*a^10-4*a^9*(b+c)-2*a*b*(b-c)^4*c*(b+c)^3-(b-c)^6*(b+c)^4+a^8*(-9*b^2+6*b*c-9*c^2)+12*a^7*(b^3+c^3)+a^2*(b^2-c^2)^2*(3*b^4+2*b^3*c-7*b^2*c^2+2*b*c^3+3*c^4)+a^6*(10*b^4-6*b^3*c+33*b^2*c^2-6*b*c^3+10*c^4)+2*a^3*(b-c)^2*(2*b^5+2*b^4*c+5*b^3*c^2+5*b^2*c^3+2*b*c^4+2*c^5)-2*a^5*(6*b^5-5*b^4*c+4*b^3*c^2+4*b^2*c^3-5*b*c^4+6*c^5)-2*a^4*(3*b^6+2*b^5*c+7*b^4*c^2-6*b^3*c^3+7*b^2*c^4+2*b*c^5+3*c^6) : :

See Ivan Pavlov, euclid 5829.

X(54191) lies on these lines: {517, 3648}, {2475, 5535}, {7330, 52126}, {16113, 20066}

X(54191) = reflection of X(i) in X(j) for these {i,j}: {20066, 16113}
X(54191) = anticomplement of X(54153)

X(54192) = X(1)X(6940)∩X(3)X(758)

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

See Ivan Pavlov, euclid 5829.

X(54192) lies on these lines: {1, 6940}, {3, 758}, {30, 21635}, {36, 5083}, {78, 5450}, {140, 35016}, {214, 517}, {404, 31870}, {484, 34474}, {515, 5440}, {519, 11715}, {631, 37571}, {946, 37281}, {1125, 33596}, {1319, 28234}, {1385, 3244}, {1737, 38133}, {2077, 2800}, {2646, 6684}, {3576, 3870}, {3579, 51717}, {3814, 6246}, {3874, 32612}, {3878, 26285}, {3881, 37535}, {3884, 11849}, {3919, 10222}, {4188, 37625}, {4855, 6796}, {4973, 23961}, {5080, 12119}, {5087, 24042}, {5193, 46681}, {5267, 31837}, {5432, 38134}, {5441, 6902}, {5535, 13587}, {5538, 6905}, {5657, 37525}, {5690, 26287}, {5692, 6950}, {5730, 40256}, {5840, 11813}, {5842, 9945}, {5882, 38455}, {6256, 27383}, {6261, 10860}, {6265, 35000}, {6326, 6909}, {6906, 20117}, {6914, 10176}, {6951, 37701}, {6952, 47033}, {6961, 49168}, {10165, 11019}, {10265, 44669}, {10310, 40257}, {11248, 30144}, {11496, 12446}, {11700, 22350}, {12005, 34772}, {12245, 21842}, {12831, 15326}, {13464, 17614}, {14804, 41547}, {14988, 46684}, {24466, 51409}, {26877, 41696}, {27529, 40260}, {31730, 37837}, {33595, 50828}, {35262, 37569}, {40259, 52367}

X(54192) = midpoint of X(i) in X(j) for these {i,j}: {2077, 4511}, {5080, 12119}, {5440, 50371}, {5538, 6905}, {6265, 35000}, {6326, 6909}, {24466, 51409}
X(54192) = reflection of X(i) in X(j) for these {i,j}: {4973, 23961}, {6246, 3814}, {24042, 5087}, {40663, 6684}
X(54192) = complement of X(54154)
X(54192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22836, 5884}, {2077, 4511, 2800}, {5440, 50371, 515}, {5538, 15015, 6905}, {34772, 37561, 12005}

X(54193) = X(1)X(37163)∩X(20)X(758)

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

See Ivan Pavlov, euclid 5829.

X(54193) lies on these lines: {1, 37163}, {8, 6906}, {20, 758}, {30, 9809}, {145, 5884}, {515, 3935}, {517, 6224}, {519, 1768}, {550, 944}, {938, 22768}, {962, 37468}, {1012, 48698}, {1519, 4511}, {3189, 38455}, {3241, 28458}, {3428, 5731}, {3476, 11246}, {3486, 5217}, {3488, 15325}, {3616, 37438}, {5180, 5840}, {5535, 36004}, {5538, 6840}, {5657, 7508}, {5842, 9963}, {6888, 47033}, {6909, 9803}, {6986, 10543}, {8727, 17757}, {10724, 51409}, {11015, 14110}, {12119, 20067}, {12247, 35000}, {15680, 31806}, {22836, 37437}, {31775, 34195}, {34626, 34632}, {37256, 37625}

X(54193) = reflection of X(i) in X(j) for these {i,j}: {6840, 5538}, {9803, 6909}, {10724, 51409}, {12247, 35000}, {20067, 12119}
X(54193) = anticomplement of X(54154)
X(54193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6909, 44669, 9803}

X(54194) = X(34)X(40)∩X(65)X(1884)

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

See Ivan Pavlov, euclid 5829.

X(54194) lies on these lines: {34, 40}, {65, 1884}, {108, 1835}, {1845, 1877}, {1870, 32760}

X(54194) = zosma transform of X(54154)

X(54195) = X(3)X(732)∩X(4)X(2896)

Barycentrics    a^10*(b^2+c^2)+b^2*c^2*(b^4-c^4)^2+a^8*(5*b^4+12*b^2*c^2+5*c^4)+a^6*(-3*b^6+2*b^4*c^2+2*b^2*c^4-3*c^6)-3*a^2*b^2*c^2*(b^6+5*b^4*c^2+5*b^2*c^4+c^6)-a^4*(3*b^8+15*b^6*c^2+22*b^4*c^4+15*b^2*c^6+3*c^8) : :

See Ivan Pavlov, euclid 5829.

X(54195) lies on these lines: {3, 732}, {4, 2896}, {76, 12122}, {83, 22712}, {182, 41755}, {262, 31268}, {511, 6292}, {754, 35430}, {3095, 10519}, {3398, 37455}, {3785, 8725}, {3934, 6249}, {5188, 29012}, {5976, 7767}, {6704, 15819}, {12251, 32476}, {16220, 31950}, {22677, 48673}, {31168, 33706}

X(54195) = midpoint of X(i) in X(j) for these {i,j}: {76, 12122}, {5188, 44772}, {6287, 9821}, {12251, 32476}, {31168, 33706}
X(54195) = reflection of X(i) in X(j) for these {i,j}: {6249, 3934}
X(54195) = complement of X(54155)
X(54195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5188, 44772, 29012}, {41650, 49112, 3398}

X(54196) = X(20)X(732)∩X(83)X(6194)

Barycentrics    a^10*(b^2+c^2)+3*a^8*(3*b^4+7*b^2*c^2+3*c^4)-2*a^6*(b^6-9*b^4*c^2-9*b^2*c^4+c^6)+b^2*c^2*(b^8-b^6*c^2-b^2*c^6+c^8)-a^4*(7*b^8+16*b^6*c^2+11*b^4*c^4+16*b^2*c^6+7*c^8)-a^2*(b^10+9*b^8*c^2+16*b^6*c^4+16*b^4*c^6+9*b^2*c^8+c^10) : :

See Ivan Pavlov, euclid 5829.

X(54196) lies on these lines: {20, 732}, {83, 6194}, {194, 12122}, {511, 2896}, {3091, 42006}, {3146, 12251}, {6249, 31276}, {7893, 8782}, {9821, 12252}, {9866, 45029}, {13111, 32521}

X(54196) = reflection of X(i) in X(j) for these {i,j}: {194, 12122}, {12252, 9821}, {13111, 32521}
X(54196) = anticomplement of X(54155)

X(54197) = X(20)X(6735)∩X(517)X(1394)

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

See Ivan Pavlov, euclid 5829.

X(54197) lies on circumconics {A,B,C,X(1),X(318)}, {A,B,C,X(3),X(15803)}, {A,B,C,X(4),X(34039)}, {A,B,C,X(20),X(28)}, {A,B,C,X(21),X(37417)}, {A,B,C,X(40),X(104)} and on these lines: {20, 6735}, {318, 52027}, {517, 1394}, {1785, 15803}

X(54197) = isogonal conjugate of X(54156)
X(54197) = X(i)-vertex conjugate of X(j) for these {i, j}: {963, 3345}
X(54197) = intersection, other than A, B, C, of these circumconics: {{A, B, C, X(1), X(318)}}, {{A, B, C, X(3), X(15803)}}, {{A, B, C, X(4), X(34039)}}, {{A, B, C, X(20), X(28)}}, {{A, B , C, X(21), X(37417)}} and {{A, B, C, X(40), X(104)}}

X(54198) = X(1)X(10309)∩X(10)X(119)

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

See Ivan Pavlov, euclid 5829.

X(54198) lies on these lines: {1, 10309}, {4, 3340}, {10, 119}, {20, 51423}, {40, 6745}, {65, 7682}, {84, 3296}, {221, 51616}, {226, 12672}, {354, 17649}, {382, 515}, {452, 5924}, {496, 942}, {516, 6261}, {517, 6260}, {519, 6256}, {527, 22770}, {551, 5450}, {944, 9580}, {952, 22792}, {962, 1490}, {971, 22791}, {1071, 1537}, {1125, 1158}, {1159, 18483}, {1210, 1519}, {1385, 43177}, {1420, 2096}, {1532, 4848}, {1697, 15239}, {1699, 5804}, {1709, 11045}, {2093, 6848}, {2098, 12678}, {2099, 12679}, {2801, 49600}, {2829, 4342}, {3057, 12831}, {3359, 6700}, {3452, 31788}, {3485, 12705}, {3555, 18239}, {3616, 52027}, {3667, 42757}, {3817, 12616}, {4297, 40257}, {4304, 21740}, {4356, 50189}, {4847, 5693}, {4861, 9809}, {5045, 18238}, {5128, 6927}, {5493, 6796}, {5542, 7373}, {5768, 9614}, {5795, 37822}, {5811, 9623}, {5837, 6907}, {5853, 12700}, {5886, 6705}, {5901, 34862}, {6223, 12650}, {6361, 52026}, {6738, 26333}, {6847, 7995}, {6916, 15829}, {6925, 11682}, {7956, 31794}, {7967, 41864}, {7982, 12667}, {7992, 11034}, {8196, 12457}, {8203, 12456}, {8227, 14647}, {9843, 34339}, {9955, 33899}, {10085, 11046}, {10164, 40256}, {10247, 48664}, {10384, 36996}, {10572, 34789}, {10595, 12246}, {10624, 18446}, {10698, 46435}, {10914, 13257}, {11047, 12686}, {11048, 12687}, {11362, 18242}, {11496, 18237}, {11500, 28194}, {12005, 21625}, {12330, 22753}, {12675, 16215}, {13227, 14872}, {15071, 30384}, {21620, 45776}, {30196, 52354}, {31419, 31821}, {31730, 37837}, {34123, 52116}, {37001, 37740}, {37561, 50908}, {37615, 51724}, {40266, 51755}, {43182, 51577}, {45636, 49170}, {45637, 49171}

X(54198) = midpoint of X(i) in X(j) for these {i,j}: {4, 7971}, {962, 1490}, {1482, 6259}, {3555, 18239}, {6223, 12650}, {7982, 12667}, {10698, 46435}, {37727, 40267}
X(54198) = reflection of X(i) in X(j) for these {i,j}: {10, 12608}, {1158, 1125}, {4297, 40257}, {5493, 6796}, {6245, 946}, {11362, 18242}, {12114, 13464}, {18238, 5045}, {31730, 37837}, {33899, 9955}, {34862, 5901}
X(54198) = complement of X(54156)
X(54198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {946, 5884, 11019}, {946, 6001, 6245}, {1482, 6259, 515}, {2800, 12608, 10}, {9856, 39542, 946}, {37727, 40267, 515}

X(54199) = X(4)X(7319)∩X(8)X(153)

Barycentrics    a^7+5*a^6*(b+c)+7*a^2*(b-c)^4*(b+c)-(b-c)^4*(b+c)^3-a*(b^2-c^2)^2*(5*b^2-2*b*c+5*c^2)-a^5*(7*b^2+6*b*c+7*c^2)+a^3*(b-c)^2*(11*b^2+26*b*c+11*c^2)+a^4*(-11*b^3+15*b^2*c+15*b*c^2-11*c^3) : :

See Ivan Pavlov, euclid 5829.

X(54199) lies on these lines: {4, 7319}, {7, 12672}, {8, 153}, {20, 7971}, {499, 12767}, {515, 20050}, {517, 6223}, {912, 6764}, {938, 10531}, {944, 30332}, {962, 3868}, {1012, 4323}, {1071, 9785}, {1158, 3616}, {1385, 14646}, {1482, 12246}, {1490, 3935}, {1519, 5704}, {2096, 4308}, {3622, 52027}, {3873, 17649}, {3889, 18238}, {4301, 7992}, {5173, 17650}, {5330, 5731}, {5450, 38314}, {5658, 12702}, {5734, 12114}, {5775, 15908}, {5811, 37562}, {5844, 48664}, {5851, 10912}, {5884, 10580}, {6259, 12245}, {6261, 9778}, {7995, 37434}, {9779, 12616}, {9780, 12608}, {9948, 31162}, {9957, 36996}, {10595, 34862}, {11037, 45776}, {11500, 34632}, {15071, 30305}, {18228, 31788}, {22770, 28610}

X(54199) = reflection of X(i) in X(j) for these {i,j}: {20, 7971}, {7992, 4301}, {9799, 962}, {12245, 6259}, {12246, 1482}, {20070, 1490}
X(54199) = anticomplement of X(54156)
X(54199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {962, 6001, 9799}

X(54200) = X(4)X(11)∩X(34)X(207)

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

See Ivan Pavlov, euclid 5829.

X(54200) lies on circumconic {{A,B,C,X(34),X(104)}} and on these lines: {4, 11}, {29, 5323}, {34, 207}, {57, 21228}, {65, 11436}, {208, 1877}, {318, 3476}, {387, 19365}, {388, 11109}, {393, 604}, {1319, 7952}, {1398, 37226}, {1400, 3087}, {1404, 40138}, {1420, 1785}, {1466, 37028}, {1470, 7412}, {1788, 5081}, {1828, 51399}, {1870, 14257}, {1875, 37566}, {1981, 41785}, {5204, 37410}, {6987, 22341}, {7011, 31789}, {7046, 10944}, {7195, 36118}, {7288, 17555}, {8283, 12667}, {11471, 37550}, {35014, 38517}, {37305, 37579}, {43053, 52283}

X(54200) = X(i)-isoconjugate-of-X(j) for these {i, j}: {78, 945}
X(54200) = barycentric product X(i)*X(j) for these (i, j): {273, 2261}, {278, 944}
X(54200) = barycentric quotient X(i)/X(j) for these (i, j): {608, 945}, {944, 345}, {2261, 78}
X(54200) = zosma transform of X(54156)
X(54200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {207, 1842, 1118}

X(54201) = X(3)X(12325)∩X(5)X(7693)

Barycentrics    2*a^10-a^8*(b^2+c^2)+(b^2-c^2)^4*(b^2+c^2)-2*a^6*(5*b^4+3*b^2*c^2+5*c^4)-a^2*(b^2-c^2)^2*(8*b^4+11*b^2*c^2+8*c^4)+a^4*(16*b^6+5*b^4*c^2+5*b^2*c^4+16*c^6) : :

See Ivan Pavlov, euclid 5829.

X(54201) lies on these lines: {3, 12325}, {4, 21357}, {5, 7693}, {30, 6288}, {54, 3530}, {140, 389}, {195, 549}, {539, 34200}, {546, 1209}, {547, 3574}, {548, 12041}, {550, 2888}, {631, 12316}, {2917, 7555}, {3519, 33923}, {3628, 20424}, {3850, 15800}, {3856, 47582}, {5066, 13565}, {5498, 32338}, {5562, 34577}, {6153, 13391}, {6286, 15325}, {6343, 38706}, {8703, 12254}, {9920, 12324}, {10096, 11591}, {10125, 23039}, {10203, 22115}, {10257, 22815}, {10299, 13432}, {10610, 12100}, {10625, 13368}, {10627, 25563}, {10628, 16252}, {11271, 44682}, {11487, 12106}, {11804, 37452}, {12103, 18400}, {12108, 15801}, {12226, 47090}, {12291, 43607}, {12300, 21841}, {12606, 16196}, {12785, 28224}, {12965, 35255}, {12971, 35256}, {13340, 13423}, {13365, 13451}, {14140, 34804}, {15035, 34483}, {15067, 43581}, {15605, 15690}, {15704, 48675}, {17834, 50136}, {18916, 32333}, {22466, 52073}, {23336, 41590}, {27552, 32358}, {31674, 34837}, {32396, 47599}, {35197, 52793}, {37126, 50476}, {43598, 44264}, {43615, 48876}, {47117, 48378}

X(54201) = midpoint of X(i) in X(j) for these {i,j}: {5, 12307}, {550, 2888}, {7691, 21230}, {10625, 13368}, {12325, 36966}, {15704, 48675}
X(54201) = reflection of X(i) in X(j) for these {i,j}: {54, 3530}, {546, 1209}, {8254, 32348}, {11803, 6689}, {15800, 3850}, {20424, 3628}, {22051, 140}, {31674, 34837}, {32358, 27552}, {47117, 48378}
X(54201) = complement of X(54157)
X(54201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12325, 36966}, {140, 1154, 22051}, {1154, 32348, 8254}, {1154, 6689, 11803}, {7691, 21230, 30}, {8254, 32348, 140}, {12325, 36966, 50708}, {14140, 36837, 34804}

X(54202) = X(3)X(54)∩X(4)X(18551)

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

See Ivan Pavlov, euclid 5829.

X(54202) lies on these lines: {3, 54}, {4, 18551}, {20, 50708}, {30, 12325}, {376, 36966}, {378, 12175}, {381, 20584}, {382, 2888}, {539, 15681}, {550, 13432}, {631, 22051}, {999, 6286}, {1209, 3851}, {1351, 9977}, {1498, 5898}, {1593, 6242}, {1597, 6152}, {1598, 12300}, {1656, 20424}, {1657, 12244}, {2914, 32534}, {2917, 14530}, {3091, 21357}, {3295, 7356}, {3357, 35452}, {3426, 3519}, {3532, 37483}, {3534, 12254}, {3574, 5055}, {3830, 6288}, {3843, 15800}, {5054, 8254}, {5070, 32396}, {5204, 51803}, {5217, 35197}, {5562, 13621}, {5663, 15086}, {5899, 18436}, {5925, 13093}, {5965, 33878}, {6000, 17846}, {6221, 12965}, {6398, 12971}, {6767, 18984}, {7373, 13079}, {7517, 41726}, {8717, 34783}, {9935, 44457}, {10282, 17824}, {10539, 37923}, {10605, 21660}, {10677, 42116}, {10678, 42115}, {10982, 32352}, {11271, 15696}, {11413, 11999}, {11444, 22462}, {11591, 21308}, {11597, 15748}, {11702, 15040}, {11802, 15805}, {11803, 15720}, {12017, 19150}, {12111, 37949}, {12164, 44515}, {12226, 21312}, {12702, 15071}, {13321, 44056}, {13391, 13423}, {13565, 19709}, {13754, 47748}, {14049, 38723}, {14531, 14627}, {14926, 18874}, {15051, 47117}, {15072, 47751}, {15644, 43807}, {15694, 32348}, {15750, 52417}, {17834, 18378}, {23039, 43581}, {23409, 44834}, {32609, 43580}, {37922, 46730}

X(54202) = reflection of X(i) in X(j) for these {i,j}: {3, 12307}, {195, 7691}, {382, 2888}, {5073, 48675}, {12308, 5898}, {12316, 3}, {48675, 3519}
X(54202) = anticomplement of X(54157)
X(54202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1154, 12316}, {195, 12307, 7691}, {195, 7691, 3}, {1154, 12307, 3}, {1154, 7691, 195}, {5898, 10628, 12308}, {6101, 32608, 3}

X(54203) = X(3)X(15348)∩X(9)X(374)

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

See Ivan Pavlov, euclid 5829.

X(54203) lies on these lines: {3, 15348}, {9, 374}, {55, 15299}, {516, 34176}, {528, 3654}, {971, 3428}, {1001, 37533}, {1006, 7671}, {2099, 15298}, {2886, 5805}, {3358, 10860}, {3434, 5759}, {3929, 11372}, {5119, 10384}, {5220, 5887}, {5657, 37787}, {5696, 11012}, {5698, 26921}, {5709, 52682}, {5784, 11249}, {5880, 37532}, {6210, 20601}, {6883, 10177}, {7082, 9580}, {7680, 38108}, {8255, 38122}, {8257, 26446}, {8545, 39542}, {9856, 41229}, {11362, 15297}, {12699, 38454}, {14100, 40292}, {18407, 52835}, {21153, 32613}, {21168, 36976}, {24929, 38031}, {38117, 47373}

X(54203) = midpoint of X(i) in X(j) for these {i,j}: {3428, 42014}, {3434, 5759}, {11372, 41338}
X(54203) = reflection of X(i) in X(j) for these {i,j}: {55, 31658}, {5805, 2886}, {37533, 1001}, {52835, 18407}
X(54203) = complement of X(54158)
X(54203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3428, 42014, 971}

X(54204) = X(20)X(15733)∩X(144)X(517)

Barycentrics    3*a^9-5*a^8*(b+c)-(b-c)^6*(b+c)^3+a^7*(-8*b^2+6*b*c-8*c^2)-a*(b-c)^4*(b+c)^2*(b^2+4*b*c+c^2)-2*a^3*b*(b-c)^2*c*(3*b^2+2*b*c+3*c^2)+16*a^6*(b^3+c^3)+2*a^5*(3*b^4+b^3*c+28*b^2*c^2+b*c^3+3*c^4)-2*a^4*(9*b^5-b^4*c+16*b^3*c^2+16*b^2*c^3-b*c^4+9*c^5)+8*a^2*(b^7-4*b^5*c^2+3*b^4*c^3+3*b^3*c^4-4*b^2*c^5+c^7) : :

See Ivan Pavlov, euclid 5829.

X(54204) lies on these lines: {20, 15733}, {144, 517}, {5759, 20075}, {6992, 7671}, {10572, 30332}, {10679, 21168}, {11372, 17781}, {28610, 41338}, {37569, 52653}

X(54204) = reflection of X(i) in X(j) for these {i,j}: {20075, 5759}
X(54204) = anticomplement of X(54158)

X(54205) = X(3)X(527)∩X(9)X(6935)

Barycentrics    3*a^8*(b+c)+20*a^6*b*c*(b+c)-8*a^7*(b+c)^2-(b-c)^6*(b+c)^3+16*a^5*(b^2+c^2)^2-8*a^3*(b^3-b^2*c+b*c^2-c^3)^2-2*a^4*(5*b^5+7*b^4*c+12*b^3*c^2+12*b^2*c^3+7*b*c^4+5*c^5)+4*a^2*(2*b^7-3*b^6*c+b^5*c^2+b^2*c^5-3*b*c^6+2*c^7) : :

See Ivan Pavlov, euclid 5829.

X(54205) lies on these lines: {3, 527}, {9, 6935}, {57, 3475}, {142, 517}, {329, 5732}, {443, 4301}, {516, 997}, {2095, 38122}, {2801, 21060}, {3452, 8727}, {3576, 41570}, {5220, 6705}, {5735, 6904}, {6881, 7682}, {6992, 7675}, {12848, 21153}, {37407, 43174}, {37611, 43175}

X(54205) = midpoint of X(i) in X(j) for these {i,j}: {329, 5732}, {6282, 52457}
X(54205) = reflection of X(i) in X(j) for these {i,j}: {43175, 37611}
X(54205) = complement of X(54159)
X(54205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6282, 52457, 516}

X(54206) = X(7)X(517)∩X(20)X(527)

Barycentrics    a^9+3*a^8*(b+c)-(b-c)^6*(b+c)^3-8*a^3*(b-c)^4*(b^2+b*c+c^2)-8*a^7*(2*b^2+3*b*c+2*c^2)-a*(b-c)^4*(b+c)^2*(3*b^2-2*b*c+3*c^2)+4*a^6*(2*b^3+9*b^2*c+9*b*c^2+2*c^3)+4*a^2*(b-c)^4*(4*b^3+9*b^2*c+9*b*c^2+4*c^3)+a^5*(26*b^4-8*b^3*c+44*b^2*c^2-8*b*c^3+26*c^4)-2*a^4*(13*b^5+7*b^4*c+12*b^3*c^2+12*b^2*c^3+7*b*c^4+13*c^5) : :

See Ivan Pavlov, euclid 5829.

X(54206) lies on these lines: {7, 517}, {20, 527}, {329, 10431}, {962, 5784}, {1012, 6172}, {2094, 7411}, {2095, 21151}, {2801, 10430}, {3870, 5732}, {5735, 37435}, {5766, 37531}, {5809, 6827}, {6282, 7675}, {6839, 52457}, {7991, 43177}

X(54206) = reflection of X(i) in X(j) for these {i,j}: {9965, 5732}, {12848, 6282}, {36991, 329}
X(54206) = anticomplement of X(54159)

X(54207) = X(19)X(1877)∩X(33)X(42289)

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

See Ivan Pavlov, euclid 5829.

X(54207) lies on these lines: {19, 1877}, {33, 42289}, {65, 28076}, {1119, 15934}, {1876, 4307}, {4196, 11406}

X(54207) = zosma transform of X(54159)

X(54208) = X(3)X(17770)∩X(511)X(1125)

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

See Ivan Pavlov, euclid 5829.

X(54208) lies on these lines: {3, 17770}, {511, 1125}, {516, 3430}, {758, 31788}, {1046, 10164}, {1330, 4297}, {2784, 41014}, {2792, 12512}, {3454, 19925}, {7683, 10171}, {7987, 20077}, {12563, 35650}, {28164, 37823}, {28236, 36974}

X(54208) = midpoint of X(i) in X(j) for these {i,j}: {1330, 4297}
X(54208) = reflection of X(i) in X(j) for these {i,j}: {19925, 3454}
X(54208) = complement of X(54160)

X(54209) = X(1)X(256)∩X(165)X(1046)

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

See Ivan Pavlov, euclid 5829.

X(54209) lies on these lines: {1, 256}, {20, 17770}, {58, 7987}, {72, 18788}, {165, 1046}, {540, 34628}, {758, 6765}, {962, 49458}, {1330, 5691}, {1695, 10884}, {1750, 10381}, {2392, 5538}, {3454, 7989}, {3794, 8583}, {4297, 20077}, {5429, 30389}, {7683, 7988}, {11531, 35665}, {12635, 53097}, {20018, 24728}, {36974, 37712}, {43159, 48878}

X(54209) = reflection of X(i) in X(j) for these {i,j}: {1046, 3430}, {5691, 1330}, {20077, 4297}
X(54209) = anticomplement of X(54160)
X(54209) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1046, 3430, 165}, {8235, 48909, 1}

X(54210) = X(29)X(1876)∩X(1829)X(1874)

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

See Ivan Pavlov, euclid 5829.

X(54210) lies on these lines: {29, 1876}, {1426, 4213}, {1829, 1874}

X(54210) = zosma transform of X(54160)

X(54211) = X(2)X(3357)∩X(20)X(394)

Barycentrics    5*a^10+a^8*(b^2+c^2)+34*a^4*(b^2-c^2)^2*(b^2+c^2)-3*(b^2-c^2)^4*(b^2+c^2)+a^6*(-30*b^4+52*b^2*c^2-30*c^4)-7*a^2*(b^2-c^2)^2*(b^4+6*b^2*c^2+c^4) : :

See Ivan Pavlov, euclid 5829.

X(54211) lies on these lines: {2, 3357}, {4, 3426}, {20, 394}, {25, 32601}, {64, 3091}, {376, 14530}, {541, 34621}, {1503, 11008}, {1559, 3183}, {2071, 32605}, {2777, 5059}, {2883, 3523}, {3090, 35450}, {3146, 5889}, {3522, 5656}, {3529, 12315}, {3543, 5895}, {3832, 22802}, {3839, 6247}, {3854, 20299}, {5056, 15105}, {5261, 10060}, {5274, 10076}, {5894, 10304}, {5921, 34146}, {6523, 51892}, {6616, 36965}, {6622, 34469}, {6696, 7486}, {6759, 50693}, {6815, 11469}, {7408, 13474}, {7488, 9914}, {7691, 52404}, {9833, 15683}, {10303, 10606}, {12950, 14986}, {14216, 17578}, {15072, 30443}, {15682, 34780}, {15692, 16252}, {17538, 32063}, {18381, 50687}, {18400, 50692}, {20725, 27082}, {23249, 35864}, {23259, 35865}, {32064, 50688}, {41736, 50009}

X(54211) = reflection of X(i) in X(j) for these {i,j}: {4, 48672}, {20, 6225}, {3529, 12315}, {5059, 34781}, {12250, 5878}, {12324, 5895}
X(54211) = anticomplement of X(12250)
X(54211) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2777, 34781, 5059}, {5656, 20427, 3522}, {5878, 12250, 2}, {5895, 12324, 3543}, {5925, 11206, 20}, {6225, 15311, 20}, {32064, 51491, 50688}

X(54212) = X(3)X(758)∩X(30)X(5887)

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

See Ivan Pavlov, euclid 5829.

X(54212) lies on these lines: {3, 758}, {30, 5887}, {65, 5719}, {355, 40661}, {442, 517}, {960, 6841}, {1385, 39772}, {2646, 5428}, {2771, 3650}, {3647, 35459}, {3651, 3869}, {3654, 10197}, {3878, 5794}, {5499, 39542}, {6917, 49177}, {8261, 28465}, {11281, 24474}, {13750, 52793}, {21677, 31837}, {22766, 41697}, {26725, 37625}, {31786, 39783}, {37820, 47033}

X(54212) = midpoint of X(i) in X(j) for these {i,j}: {3651, 3869}, {14110, 44782}
X(54212) = reflection of X(i) in X(j) for these {i,j}: {355, 40661}, {6841, 960}, {21677, 31837}, {24474, 11281}, {39772, 1385}
X(54212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14110, 44782, 30}
X(54212) = complement of X(54161)

X(54213) = X(20)X(758)∩X(517)X(2475)

Barycentrics    a*(2*a^8*(b+c)-(b-c)^4*(b+c)^3*(2*b^2-3*b*c+2*c^2)-a^7*(4*b^2+11*b*c+4*c^2)+a^4*b*c*(-19*b^3+11*b^2*c+11*b*c^2-19*c^3)+a^6*(-4*b^3+9*b^2*c+9*b*c^2-4*c^3)+a*(b^2-c^2)^2*(4*b^4-9*b^3*c+11*b^2*c^2-9*b*c^3+4*c^4)+a^5*(12*b^4+13*b^3*c-5*b^2*c^2+13*b*c^3+12*c^4)+a^2*(b-c)^2*(4*b^5+11*b^4*c-3*b^3*c^2-3*b^2*c^3+11*b*c^4+4*c^5)+a^3*(-12*b^6+7*b^5*c+6*b^4*c^2-30*b^3*c^3+6*b^2*c^4+7*b*c^5-12*c^6)) : :

See Ivan Pavlov, euclid 5829.

X(54213) lies on these lines: {20, 758}, {517, 2475}, {3428, 34195}, {3651, 3871}, {3869, 37433}, {4189, 16139}, {5709, 39778}, {6888, 31806}

X(54213) = reflection of X(i) in X(j) for these {i,j}: {37433, 3869}
X(54213) = anticomplement of X(54161)

X(54214) = X(65)X(1884)∩X(407)X(1877)

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

See Ivan Pavlov, euclid 5829.

X(54214) lies on these lines: {65, 1884}, {407, 1877}, {40644, 49745}, {40950, 44840}

X(54214) = zosma transform of X(54161)

X(54215) = X(3)X(524)∩X(4)X(22151)

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

See Ivan Pavlov, euclid 5829.

X(54215) lies on these lines: {3, 524}, {4, 22151}, {6, 50008}, {30, 9970}, {67, 3564}, {155, 14791}, {182, 3580}, {323, 6776}, {325, 52772}, {399, 1503}, {468, 15462}, {511, 1986}, {542, 858}, {576, 38323}, {599, 18580}, {1092, 34507}, {1352, 5094}, {1511, 32113}, {1568, 32250}, {1594, 18553}, {2071, 32247}, {2072, 32274}, {2393, 30714}, {5622, 41724}, {5965, 32285}, {6240, 11470}, {6593, 11799}, {7464, 11061}, {7579, 47354}, {8262, 44214}, {9019, 25711}, {11645, 15063}, {12177, 47526}, {13160, 25555}, {13292, 43810}, {14912, 41617}, {14984, 47280}, {15035, 41721}, {15069, 18281}, {15138, 47337}, {15140, 44267}, {16619, 18374}, {18325, 45016}, {19140, 32111}, {19510, 32275}, {32234, 43574}, {32273, 34470}, {37784, 44480}, {38064, 44569}, {41729, 48873}, {46817, 52697}, {47335, 47546}, {47581, 52699}

X(54215) = midpoint of X(i) in X(j) for these {i,j}: {323, 6776}, {7464, 11061}, {10510, 32233}
X(54215) = reflection of X(i) in X(j) for these {i,j}: {67, 15122}, {1352, 11064}, {3580, 182}, {11799, 6593}, {32111, 19140}, {32113, 1511}, {32275, 19510}, {41721, 47569}
X(54215) = complement of X(54162)
X(54215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3564, 15122, 67}, {10510, 32233, 30}

X(54216) = X(20)X(524)∩X(323)X(5921)

Barycentrics    7*a^12-16*a^10*(b^2+c^2)-3*(b^2-c^2)^4*(b^2+c^2)^2+a^8*(7*b^4+25*b^2*c^2+7*c^4)+a^2*(b^2-c^2)^2*(8*b^6+3*b^4*c^2+3*b^2*c^4+8*c^6)+a^6*(8*b^6+5*b^4*c^2+5*b^2*c^4+8*c^6)-a^4*(11*b^8+7*b^6*c^2+4*b^4*c^4+7*b^2*c^6+11*c^8) : :

See Ivan Pavlov, euclid 5829.

X(54216) lies on these lines: {20, 524}, {323, 5921}, {511, 12270}, {542, 5189}, {576, 34007}, {3091, 22151}, {3153, 10510}, {3564, 7464}, {6776, 7492}, {7592, 37784}, {9970, 52403}, {11160, 35485}, {11180, 31857}, {32275, 43574}, {33532, 39899}, {37952, 47558}, {40112, 52284}

X(54216) = reflection of X(i) in X(j) for these {i,j}: {5921, 323}, {37779, 6776}
X(54216) = anticomplement of X(54162)

X(54217) = X(5)X(5504)∩X(30)X(155)

Barycentrics    (a^2-b^2-c^2)*(4*a^14-2*a^2*(b^2-c^2)^6-11*a^12*(b^2+c^2)+(b^2-c^2)^6*(b^2+c^2)+6*a^10*(b^4+6*b^2*c^2+c^4)+a^4*(b^2-c^2)^2*(3*b^6+5*b^4*c^2+5*b^2*c^4+3*c^6)+a^8*(7*b^6-31*b^4*c^2-31*b^2*c^4+7*c^6)-8*a^6*(b^8-4*b^4*c^4+c^8)) : :

See Ivan Pavlov, euclid 5829.

X(54217) lies on these lines: {3, 12421}, {5, 5504}, {30, 155}, {68, 16196}, {185, 16163}, {235, 1147}, {1092, 11585}, {3167, 31725}, {5654, 44226}, {6193, 11413}, {12038, 33563}, {12420, 21312}, {12901, 43903}, {13292, 37814}, {14516, 43894}, {15078, 18916}, {16238, 39571}, {21841, 51933}, {31804, 44247}, {32263, 43898}, {34148, 45179}

X(54217) = midpoint of X(i) in X(j) for these {i,j}: {6193, 11413}, {12118, 44752}
X(54217) = reflection of X(i) in X(j) for these {i,j}: {68, 16196}, {235, 1147}, {33563, 12038}
X(54217) = complement of X(54163)
X(54217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12118, 44752, 30}

X(54218) = X(6)X(1596)∩X(30)X(1351)

Barycentrics    (a^2-b^2-c^2)*(4*a^10+28*a^6*b^2*c^2-7*a^8*(b^2+c^2)+6*a^4*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^4*(b^2+c^2)-4*a^2*(b^2-c^2)^2*(b^4-3*b^2*c^2+c^4)) : :

See Ivan Pavlov, euclid 5829.

X(54218) lies on these lines: {3, 53021}, {6, 1596}, {25, 14912}, {30, 1351}, {182, 8263}, {184, 35266}, {193, 21312}, {381, 18919}, {389, 2393}, {394, 1368}, {524, 37480}, {549, 5622}, {550, 15073}, {974, 6467}, {1595, 8549}, {2790, 5477}, {5050, 6677}, {5486, 10249}, {5921, 16072}, {6146, 8538}, {6644, 19459}, {6823, 8548}, {10250, 23292}, {11179, 37475}, {12007, 19136}, {13568, 34788}, {13851, 39884}, {15760, 39562}, {16196, 18910}, {18388, 23326}, {18440, 18918}, {18531, 18935}, {18534, 19119}, {18914, 37498}, {19125, 44233}, {23291, 50955}, {31152, 50974}, {32621, 44274}, {33851, 51737}, {39874, 44438}

X(54218) = midpoint of X(i) in X(j) for these {i,j}: {193, 21312}, {6776, 10602}, {18531, 39899}, {31152, 50974}, {39874, 44438}
X(54218) = reflection of X(i) in X(j) for these {i,j}: {1596, 6}, {8263, 182}, {18440, 44920}, {19136, 12007}, {44212, 50979}, {44241, 48906}, {44273, 11179}
X(54218) = complement of X(54164)
X(54218) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6776, 10602, 30}, {14984, 48906, 44241}

X(54219) = X(1370)X(3564)∩X(2393)X(5889)

Barycentrics    7*a^12-20*a^10*(b^2+c^2)-3*(b^2-c^2)^4*(b^2+c^2)^2+a^8*(15*b^4+82*b^2*c^2+15*c^4)+4*a^2*(b^2-c^2)^2*(3*b^6-4*b^4*c^2-4*b^2*c^4+3*c^6)+a^6*(8*b^6-68*b^4*c^2-68*b^2*c^4+8*c^6)+a^4*(-19*b^8+40*b^6*c^2-10*b^4*c^4+40*b^2*c^6-19*c^8) : :

See Ivan Pavlov, euclid 5829.

X(54219) lies on these lines: {1370, 3564}, {1596, 51170}, {2393, 5889}, {5921, 10602}, {6677, 33748}, {9544, 14912}, {14984, 39874}, {18533, 39899}, {20080, 21312}, {31152, 51215}

X(54219) = reflection of X(i) in X(j) for these {i,j}: {5921, 10602}, {18533, 39899}, {20080, 21312}, {51215, 31152}
X(54219) = anticomplement of X(54164)

X(54220) = X(192)X(30269)∩X(2887)X(29010)

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

See Ivan Pavlov, euclid 5829.

X(54220) lies on these lines: {192, 30269}, {2887, 29010}, {31134, 51043}

X(54220) = midpoint of X(i) in X(j) for these {i,j}: {192, 30269}, {31134, 51043}
X(54220) = complement of X(54165)

X(54221) = X(1278)X(30269)∩X(6327)X(29010)

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

See Ivan Pavlov, euclid 5829.

X(54221) lies on these lines: {1278, 30269}, {6327, 29010}, {42058, 51043}

X(54221) = reflection of X(i) in X(j) for these {i,j}: {1278, 30269}, {42058, 51043}
X(54221) = anticomplement of X(54165)

X(54222) = X(32)X(7709)∩X(39)X(1513)

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

See Ivan Pavlov, euclid 5829.

X(54222) lies on these lines: {32, 7709}, {39, 1513}, {76, 11623}, {194, 12203}, {511, 550}, {626, 2782}, {1569, 2794}, {6680, 11171}, {7781, 13355}, {7830, 52996}, {7839, 35386}, {13334, 18806}, {13335, 32516}, {38642, 46283}

X(54222) = midpoint of X(i) in X(j) for these {i,j}: {194, 30270}, {11257, 32452}
X(54222) = reflection of X(i) in X(j) for these {i,j}: {13335, 32516}, {18806, 13334}
X(54222) = complement of X(54165)
X(54222) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11257, 32452, 2794}

X(54223) = X(194)X(8721)∩X(315)X(2782)

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

See Ivan Pavlov, euclid 5829.

X(54223) lies on these lines: {194, 8721}, {315, 2782}, {511, 3529}, {5999, 18768}, {9753, 32448}, {11257, 35430}, {14931, 20081}, {18806, 32522}

X(54223) = reflection of X(i) in X(j) for these {i,j}: {20081, 30270}
X(54223) = anticomplement of X(54165)

X(54224) = X(546)X(3934)∩X(2896)X(5188)

Barycentrics    a^10*(b^2+c^2)+a^8*(13*b^4+28*b^2*c^2+13*c^4)+a^6*(-8*b^6+3*b^4*c^2+3*b^2*c^4-8*c^6)+2*b^2*c^2*(b^8-b^6*c^2-b^2*c^6+c^8)-a^4*(7*b^8+34*b^6*c^2+44*b^4*c^4+34*b^2*c^6+7*c^8)+a^2*(b^10-8*b^8*c^2-25*b^6*c^4-25*b^4*c^6-8*b^2*c^8+c^10) : :

See Ivan Pavlov, euclid 5829.

X(54224) lies on these lines: {546, 3934}, {2896, 5188}, {12122, 44772}, {13111, 15819}

X(54224) = midpoint of X(i) in X(j) for these {i,j}: {2896, 5188}, {12122, 44772}
X(54224) = complement of X(54167)

X(54225) = X(76)X(382)∩X(511)X(7847)

Barycentrics    b^2*c^2*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)+a^8*(12*b^4+25*b^2*c^2+12*c^4)+a^6*(-4*b^6+17*b^4*c^2+17*b^2*c^4-4*c^6)-11*a^2*b^2*c^2*(b^6+b^4*c^2+b^2*c^4+c^6)-a^4*(8*b^8+20*b^6*c^2+11*b^4*c^4+20*b^2*c^6+8*c^8) : :

See Ivan Pavlov, euclid 5829.

X(54225) lies on these lines: {76, 382}, {511, 7847}, {732, 53097}, {754, 9764}, {5188, 20088}, {12252, 22676}, {13111, 22712}

X(54225) = reflection of X(i) in X(j) for these {i,j}: {20088, 5188}
X(54225) = anticomplement of X(54167)

X(54226) = X(40)X(22117)∩X(165)X(7952)

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

See Ivan Pavlov, euclid 5829.

X(54226) lies on these lines: {40, 22117}, {165, 7952}, {7080, 9778}, {7991, 15501}, {7992, 44692}

X(54226) = isogonal conjugate of X(7992)
X(54226) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(1), X(40)}}, {{A, B, C, X(3), X(165)}}, {{A, B, C, X(28), X(53086)}}, {{A, B, C, X(46), X(6282)}}, {{A, B, C, X(58), X(34432)}}, {{A, B, C, X(84), X(972)}}, {{A, B, C, X(102), X(43719)}}, {{A, B, C, X(200), X(775)}}, {{A, B, C, X(267), X(3062)}}, {{A, B, C, X(517), X(7991)}}, {{A, B, C, X(921), X(29374)}}, {{A, B, C, X(1243), X(31793)}}, {{A, B, C, X(1394), X(7992)}}, {{A, B, C, X(2093), X(14110)}}, {{A, B, C, X(2745), X(32899)}}, {{A, B, C, X(3361), X(6244)}}, {{A, B, C, X(3579), X(7987)}}, {{A, B, C, X(6769), X(41338)}}, {{A, B, C, X(9357), X(39946)}}, {{A, B, C, X(9819), X(31798)}}, {{A, B, C, X(10310), X(15803)}}, {{A, B, C, X(11531), X(12702)}}, {{A, B, C, X(16192), X(35242)}}, {{A, B, C, X(16208), X(35239)}}, {{A, B, C, X(16209), X(35238)}}, {{A, B, C, X(30337), X(31797)}}
X(54226) = X(i)-vertex conjugate of X(j) for these {i, j}: {1, 34432}
X(54226) = X(i)-cross conjugate of X(j) for these {i, j}: {1394, 1}

X(54227) = X(1)X(6223)∩X(4)X(3671)

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

See Ivan Pavlov, euclid 5829.

X(54227) lies on these lines: {1, 6223}, {2, 7992}, {3, 43182}, {4, 3671}, {5, 9948}, {10, 5777}, {40, 5658}, {56, 41706}, {84, 1125}, {221, 16870}, {226, 12688}, {329, 12565}, {382, 515}, {516, 1490}, {519, 7971}, {551, 12114}, {581, 4356}, {908, 9961}, {944, 4342}, {946, 971}, {950, 12679}, {993, 18237}, {997, 10309}, {1071, 11019}, {1158, 10164}, {1210, 15071}, {1699, 9799}, {1709, 13411}, {1750, 4295}, {2800, 4067}, {2801, 18239}, {2829, 33337}, {3062, 37434}, {3085, 7995}, {3086, 30304}, {3333, 36996}, {3358, 38059}, {3452, 9943}, {3487, 11372}, {3576, 12246}, {3634, 14647}, {3646, 21151}, {3741, 12547}, {3812, 9842}, {3817, 6245}, {3947, 9949}, {4297, 6261}, {4300, 4656}, {4311, 41690}, {4312, 50700}, {4314, 18446}, {4847, 12528}, {5439, 10863}, {5493, 11500}, {5603, 10864}, {5787, 18483}, {5811, 18250}, {5882, 15172}, {5884, 7682}, {5886, 12684}, {5930, 38357}, {6705, 19862}, {6737, 6925}, {6796, 50808}, {6849, 38151}, {6908, 18249}, {6916, 12447}, {7959, 34048}, {8074, 18913}, {8983, 49234}, {9843, 17649}, {9856, 21620}, {9942, 21616}, {9960, 12047}, {10085, 44675}, {10106, 12678}, {10165, 34862}, {10175, 33899}, {10884, 40998}, {10916, 12666}, {11220, 41012}, {11249, 34646}, {12053, 12680}, {12330, 25440}, {12512, 52026}, {12520, 12572}, {12526, 37421}, {12609, 12664}, {12672, 41543}, {12676, 17647}, {12705, 13405}, {13257, 21075}, {13971, 49235}, {14646, 35242}, {16112, 28628}, {18481, 48664}, {19855, 30326}, {20103, 37560}, {22792, 31673}, {28645, 38454}, {31821, 37424}

X(54227) = midpoint of X(i) in X(j) for these {i,j}: {1, 6223}, {6261, 16127}, {7971, 12667}, {18481, 48664}
X(54227) = reflection of X(i) in X(j) for these {i,j}: {10, 6260}, {84, 1125}, {4297, 6261}, {5493, 11500}, {5787, 18483}, {6245, 12608}, {6260, 18243}, {9948, 5}, {12664, 31871}, {31673, 22792}
X(54227) = complement of X(7992)
X(54227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {226, 12688, 21628}, {946, 12675, 21625}, {6001, 18243, 6260}, {6001, 6260, 10}, {6245, 12608, 3817}

X(54228) = X(2)X(7992)∩X(4)X(5556)

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

See Ivan Pavlov, euclid 5829.

X(54228) lies on these lines: {2, 7992}, {4, 5556}, {7, 10429}, {8, 6001}, {84, 3616}, {144, 12565}, {329, 9961}, {388, 41706}, {392, 10307}, {515, 20050}, {651, 7959}, {938, 15071}, {962, 971}, {1071, 10580}, {1490, 4420}, {1709, 5703}, {2801, 6764}, {3062, 3671}, {3091, 9948}, {3146, 41575}, {3241, 7971}, {3427, 10266}, {4295, 18412}, {4342, 9851}, {5290, 9949}, {5558, 5603}, {5584, 6172}, {5731, 12246}, {5905, 9800}, {5927, 11024}, {6245, 9779}, {6260, 9780}, {7995, 41561}, {9785, 12680}, {9799, 9812}, {9856, 11037}, {9943, 18228}, {9960, 10430}, {10453, 12547}, {10578, 12705}, {10884, 52653}, {11036, 11372}, {12114, 38314}, {14647, 18243}, {14986, 30304}, {16112, 28629}, {16127, 18406}, {28647, 38454}, {40267, 50864}

X(54228) = reflection of X(i) in X(j) for these {i,j}: {8, 6223}
X(54228) = anticomplement of X(7992)
X(54228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9856, 36996, 11037}

X(54229) = ZOSMA TRANSFORM OF X(512)

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

X(54229) lies on these lines: {4, 4444}, {25, 16695}, {34, 51642}, {162, 39054}, {419, 4369}, {513, 1835}, {814, 6591}, {2049, 52599}, {2533, 3287}, {4504, 7009}, {7103, 7216}, {17103, 47736}, {17924, 29051}

X(54229) = zosma transform of X(512)
X(54229) = polar conjugate of X(27805)
X(54229) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(4), X(419)}}, {{A, B, C, X(171), X(18838)}}, {{A, B, C, X(513), X(3287)}}, {{A, B, C, X(804), X(6002)}}, {{A, B, C, X(1876), X(7119)}}, {{A, B, C, X(1877), X(7009)}}, {{A, B, C, X(1884), X(14006)}}, {{A, B, C, X(2533), X(4369)}}, {{A, B, C, X(3805), X(29051)}}, {{A, B, C, X(4128), X(51641)}}, {{A, B, C, X(4504), X(4922)}}, {{A, B, C, X(18155), X(48400)}}, {{A, B, C, X(20981), X(22093)}}
X(54229) = perspector of circumconic {{A,B,C,X(278), X(7009)}}
X(54229) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 3903}, {48, 27805}, {71, 4603}, {78, 29055}, {100, 7015}, {190, 7116}, {219, 37137}, {228, 4594}, {256, 1331}, {257, 906}, {692, 7019}, {874, 17970}, {893, 1332}, {904, 4561}, {1431, 4571}, {1432, 4587}, {2200, 7260}, {3573, 36214}, {3781, 30670}, {4451, 36059}, {4558, 52651}, {4563, 40729}, {4574, 40432}, {7018, 32656}
X(54229) = X(i)-Dao conjugate of X(j) for these {i, j}: {1086, 7019}, {1249, 27805}, {3709, 8611}, {4369, 656}, {5190, 257}, {5521, 256}, {8054, 7015}, {16587, 52609}, {16592, 69}, {20620, 4451}, {36103, 3903}, {40597, 1332}
X(54229) = X(i)-cross conjugate of X(j) for these {i, j}: {20981, 4369}
X(54229) = barycentric product X(i)*X(j) for these (i, j): {4, 4369}, {19, 4374}, {27, 2533}, {92, 4367}, {171, 17924}, {172, 46107}, {264, 20981}, {273, 3287}, {278, 3907}, {419, 4444}, {427, 18111}, {514, 7009}, {653, 4459}, {693, 7119}, {811, 16592}, {894, 7649}, {1119, 4529}, {1215, 17925}, {1237, 43925}, {1824, 16737}, {1826, 17212}, {1840, 7192}, {1847, 4477}, {1897, 7200}, {1909, 6591}, {2052, 22093}, {2501, 17103}, {2969, 18047}, {3064, 7176}, {3572, 17984}, {4128, 6331}, {4922, 6336}, {6649, 8735}, {7175, 44426}, {7178, 14006}, {7196, 18344}, {7234, 44129}, {17787, 43923}, {18200, 41013}, {28006, 40446}
X(54229) = barycentric quotient X(i)/X(j) for these (i, j): {4, 27805}, {19, 3903}, {27, 4594}, {28, 4603}, {34, 37137}, {171, 1332}, {172, 1331}, {286, 7260}, {419, 3570}, {444, 3882}, {514, 7019}, {608, 29055}, {649, 7015}, {667, 7116}, {894, 4561}, {1215, 52609}, {1840, 3952}, {2329, 4571}, {2330, 4587}, {2533, 306}, {3064, 4451}, {3287, 78}, {3572, 36214}, {3907, 345}, {4128, 647}, {4140, 3710}, {4164, 20769}, {4367, 63}, {4369, 69}, {4374, 304}, {4444, 40708}, {4459, 6332}, {4477, 3692}, {4529, 1265}, {4922, 3977}, {6591, 256}, {7119, 100}, {7122, 906}, {7175, 6516}, {7200, 4025}, {7234, 71}, {7649, 257}, {14006, 645}, {16592, 656}, {17103, 4563}, {17212, 17206}, {17924, 7018}, {17925, 32010}, {17984, 27853}, {18111, 1799}, {18200, 1444}, {20964, 4574}, {20981, 3}, {21755, 810}, {22093, 394}, {22373, 822}, {24533, 22370}, {40608, 8611}, {43923, 1432}, {43925, 1178}, {45882, 3781}, {46107, 44187}

X(54230) = X(1)X(6163)∩X(100)X(6161)

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

See Ivan Pavlov, euclid 5829.

X(54230) lies on cubic K661 and these lines: {1, 6163}, {100, 6161}, {513, 6162}, {764, 5376}, {1083, 2975}, {5548, 36238}, {5592, 39185}, {6631, 24093}

X(54230) = eigentransform of X(513)
X(54230) = reflection of X(i) in X(j) for these {i,j}: {100, 6551}
X(54230) = trilinear pole of line {1052, 41395}
X(54230) = X(i)-Dao conjugate of X(j) for these {i, j}: {1016, 668}
X(54230) = barycentric product X(i)*X(j) for these (i, j): {190, 1052}, {668, 41395}
X(54230) = barycentric quotient X(i)/X(j) for these (i, j): {1052, 514}

X(54231) = X(1)X(9323)∩X(101)X(14825)

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

See Ivan Pavlov, euclid 5829.

X(54231) lies on these lines: {1, 9323}, {101, 14825}, {595, 6788}, {21201, 36086}, {23100, 39293}, {24047, 41405}, {31273, 35967}

X(54231) = barycentric product X(i)*X(j) for these (i, j): {190, 38863}
X(54231) = eigentransform of X(514)

X(54232) = X(3)X(101)∩X(4)X(514)

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

See Ivan Pavlov, euclid 5829.

X(54232) lies on cubic K028 and on these lines: {3, 101}, {4, 514}, {76, 18025}, {15634, 17181}, {17170, 44040}, {40116, 41320}

X(54232) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(3), X(514)}}, {{A, B, C, X(4), X(101)}}, {{A, B, C, X(76), X(48381)}}, {{A, B, C, X(103), X(53150)}}, {{A, B, C, X(118), X(50734)}}, {{A, B, C, X(220), X(1736)}}, {{A, B, C, X(847), X(3730)}}, {{A, B, C, X(8608), X(42316)}}
X(54232) = Cundy-Parry Psi of X(514)
X(54232) = perspector of circumconic {A,B,C,X(278), X(52781)}
X(54232) = X(i)-isoconjugate-of-X(j) for these {i, j}: {910, 2989}, {15380, 24014}, {36107, 39470}
X(54232) = X(i)-Dao conjugate of X(j) for these {i, j}: {118, 516}, {39003, 39470}
X(54232) = X(i)-cross conjugate of X(j) for these {i, j}: {916, 103}
X(54232) = barycentric product X(i)*X(j) for these (i, j): {103, 48381}, {916, 52781}, {1736, 36101}, {8608, 18025}
X(54232) = barycentric quotient X(i)/X(j) for these (i, j): {103, 2989}, {916, 26006}, {1736, 30807}, {8608, 516}, {32642, 35182}, {48381, 35517}

X(54233) = X(4)X(101)∩X(3)X(514)

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

See Ivan Pavlov, euclid 5829.

X(54233) lies on the cubic K009 and on these lines: {3, 514}, {4, 101}, {32, 23972}, {1147, 14377}, {2724, 35182}

X(54233) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(3), X(101)}}, {{A, B, C, X(4), X(279)}}, {{A, B, C, X(56), X(44408)}}, {{A, B, C, X(14376), X(26006)}}, {{A, B, C, X(31852), X(41321)}}
X(54233) = Cundy-Parry Phi of X(514)
X(54233) = X(i)-isoconjugate-of-X(j) for these {i, j}: {103, 1736}, {911, 48381}, {916, 36122}, {2253, 52781}, {8608, 36101}
X(54233) = X(i)-Dao conjugate of X(j) for these {i, j}: {516, 118}, {23972, 48381}, {46095, 916}
X(54233) = barycentric product X(i)*X(j) for these (i, j): {516, 2989}, {917, 26006}
X(54233) = barycentric quotient X(i)/X(j) for these (i, j): {516, 48381}, {910, 1736}, {917, 52781}, {2989, 18025}, {23972, 118}, {32699, 40116}, {35182, 677}, {47407, 34335}

X(54234) = X(4)X(9)∩X(25)X(1626)

Barycentrics    (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 5829.

X(54234) lies on these lines: {4, 9}, {25, 1626}, {28, 42326}, {29, 17171}, {33, 1851}, {34, 1458}, {225, 4186}, {513, 1835}, {1119, 19604}, {1465, 33302}, {1633, 1738}, {1785, 5146}, {1827, 1828}, {1836, 17810}, {1838, 4222}, {1848, 14004}, {1856, 11393}, {1886, 2201}, {1891, 5342}, {2310, 2385}, {2969, 23710}, {3008, 20780}, {4196, 5338}, {4207, 24943}, {5101, 53008}, {5307, 6995}, {7071, 11400}, {11109, 25966}, {12053, 33587}, {19372, 28104}, {34823, 36557}, {35993, 52427}, {37168, 40509}

X(54234) = isogonal conjugate of X(1810)
X(54234) = intersection, other than A, B, C, of circumconics:, {{A, B, C, X(9), X(513)}}, {{A, B, C, X(10), X(3008)}}, {{A, B, C, X(19), X(43923)}}, {{A, B, C, X(34), X(7719)}}, {{A, B, C, X(71), X(20780)}}, {{A, B, C, X(281), X(7649)}}, {{A, B, C, X(516), X(6084)}}, {{A, B, C, X(1119), X(15742)}}, {{A, B, C, X(1512), X(51419)}}, {{A, B, C, X(1706), X(51839)}}, {{A, B, C, X(1861), X(36124)}}, {{A, B, C, X(2183), X(48032)}}, {{A, B, C, X(2550), X(52210)}}, {{A, B, C, X(7079), X(18344)}}
X(54234) = zosma transform of X(518)
X(54234) = perspector of circumconic {{A,B,C,X(278), X(1897)}}
X(54234) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 1810}, {3, 1280}, {48, 36807}, {78, 1477}, {212, 35160}, {219, 43760}, {905, 6078}, {1331, 35355}, {4587, 37626}
X(54234) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 1810}, {1249, 36807}, {5521, 35355}, {16593, 69}, {35111, 345}, {36103, 1280}, {39048, 63}, {40837, 35160}
X(54234) = barycentric product X(i)*X(j) for these (i, j): {4, 3008}, {92, 1279}, {273, 2348}, {278, 5853}, {331, 8647}, {1861, 52210}, {1897, 6084}, {2052, 20780}, {6335, 48032}, {16593, 36124}, {36123, 51419}
X(54234) = barycentric quotient X(i)/X(j) for these (i, j): {4, 36807}, {6, 1810}, {19, 1280}, {34, 43760}, {278, 35160}, {608, 1477}, {1279, 63}, {2348, 78}, {3008, 69}, {5853, 345}, {6084, 4025}, {6591, 35355}, {8647, 219}, {8659, 1459}, {8750, 6078}, {20662, 1818}, {20780, 394}, {23704, 4571}, {43923, 37626}, {48032, 905}, {52210, 31637}
X(54234) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1890, 1839}, {4, 242, 1861}, {4, 7713, 1869}, {242, 1861, 8756}, {1878, 1884, 1877}

X(54235) = X(19)X(273)∩X(33)X(92)

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

See Ivan Pavlov, euclid 5829.

X(54235) lies on circumconics {{A,B,C,X(2),X(344)}}, {{A,B,C,X(4),X(31926)}}, {{A,B,C,X(19),X(33)}}, {{A,B,C,X(27),X(14004)}}, {{A,B,C,X(29),X(37389)}}, {{A,B,C,X(75),X(20173)}} and on these lines: {4, 42071}, {19, 273}, {33, 92}, {105, 107}, {242, 52480}, {243, 14197}, {264, 281}, {278, 13149}, {286, 648}, {666, 46133}, {927, 20624}, {1861, 33676}, {3673, 39273}, {5307, 51846}, {6331, 31623}, {7017, 42384}, {8735, 18026}, {8764, 23696}, {13576, 52167}, {16813, 32658}, {46106, 46784}

X(54235) = isogonal conjugate of X(20752)
X(54235) = trilinear pole of line {4, 885}
X(54235) = polar conjugate of X(518)
X(54235) = intersection, other than A, B, C, of these circumconics: {{A, B, C, X(2), X(344)}}, {{A, B, C, X(4), X(31926)}}, {{A, B, C, X(19), X(33)}}, {{A, B, C, X(27), X(14004)}}, {{A, B, C, X(29), X(37389)}}, {{A, B, C, X(75), X(20173)}}, {{A, B, C, X(92), X(264)}}, {{A, B, C, X(105), X(1814)}}, {{A, B, C, X(107), X(648)}}, {{A, B, C, X(158), X(331)}}, {{A, B, C, X(242), X(1861)}}, {{A, B, C, X(274), X(9311)}}, {{A, B, C, X(279), X(24590)}}, {{A, B, C, X(308), X(18825)}}, {{A, B, C, X(318), X(1847)}}, {{A, B, C, X(514), X(1952)}}, {{A, B, C, X(523), X(47212)}}, {{A, B, C, X(525), X(2833)}}, {{A, B, C, X(673), X(14942)}}, {{A, B, C, X(1311), X(37202)}}, {{A, B, C, X(1821), X(18816)}}, {{A, B, C, X(2006), X(3512)}}, {{A, B, C, X(2989), X(40450)}}, {{A, B, C, X(4560), X(34056)}}, {{A, B, C, X(5089), X(42071)}}, {{A, B, C, X(17925), X(36125)}}, {{A, B, C, X(27475), X(31618)}}, {{A, B, C, X(36123), X(46102)}}, {{A, B, C, X(37790), X(37805)}}, {{A, B, C, X(40573), X(46103)}}
X(54235) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 20752}, {3, 672}, {6, 1818}, {31, 25083}, {48, 518}, {63, 2223}, {69, 9454}, {71, 3286}, {78, 52635}, {184, 3912}, {190, 23225}, {212, 241}, {219, 1458}, {222, 2340}, {228, 18206}, {255, 5089}, {292, 20778}, {304, 9455}, {394, 2356}, {577, 1861}, {603, 3693}, {652, 2283}, {665, 1331}, {673, 20776}, {822, 4238}, {906, 2254}, {918, 32656}, {926, 1813}, {1025, 1946}, {1026, 22383}, {1437, 3930}, {1444, 39258}, {1459, 2284}, {1790, 20683}, {1802, 34855}, {1810, 20662}, {1814, 42079}, {1876, 2289}, {2196, 8299}, {2200, 30941}, {3252, 7193}, {3263, 9247}, {3717, 52411}, {4055, 15149}, {4088, 32661}, {4447, 7116}, {4575, 24290}, {4712, 32658}, {5236, 6056}, {6184, 36057}, {6516, 46388}, {9436, 52425}, {9502, 36056}, {20769, 40730}, {31637, 39686}, {32657, 50441}, {32660, 50333}, {37908, 40152}, {46108, 52430}
X(54235) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 25083}, {3, 20752}, {9, 1818}, {105, 20740}, {120, 20728}, {136, 24290}, {1249, 518}, {3162, 2223}, {5190, 2254}, {5521, 665}, {6523, 5089}, {7952, 3693}, {19557, 20778}, {20621, 6184}, {20622, 9502}, {33675, 69}, {36103, 672}, {38966, 52614}, {39048, 20749}, {39053, 1025}, {39060, 883}, {40837, 241}
X(54235) = X(i)-cross conjugate of X(j) for these {i, j}: {105, 2481}, {242, 286}, {5089, 4}, {48408, 668}
X(54235) = barycentric product X(i)*X(j) for these (i, j): {4, 2481}, {19, 18031}, {75, 36124}, {76, 8751}, {92, 673}, {105, 264}, {158, 31637}, {273, 14942}, {278, 36796}, {281, 34018}, {286, 13576}, {294, 331}, {666, 17924}, {885, 18026}, {927, 44426}, {1024, 46404}, {1438, 1969}, {1462, 7017}, {1814, 2052}, {1847, 6559}, {2973, 5377}, {3064, 34085}, {6185, 46108}, {6528, 10099}, {6591, 36803}, {7649, 51560}, {13149, 28132}, {18027, 32658}, {18344, 46135}, {18785, 44129}, {23696, 52938}, {36086, 46107}, {36146, 46110}, {40717, 52030}, {46104, 46149}, {46133, 52456}
X(54235) = barycentric quotient X(i)/X(j) for these (i, j): {1, 1818}, {2, 25083}, {4, 518}, {6, 20752}, {19, 672}, {25, 2223}, {27, 18206}, {28, 3286}, {33, 2340}, {34, 1458}, {92, 3912}, {105, 3}, {107, 4238}, {108, 2283}, {158, 1861}, {238, 20778}, {242, 8299}, {264, 3263}, {273, 9436}, {278, 241}, {281, 3693}, {286, 30941}, {294, 219}, {318, 3717}, {331, 40704}, {393, 5089}, {608, 52635}, {653, 1025}, {666, 1332}, {667, 23225}, {673, 63}, {884, 1946}, {885, 521}, {919, 906}, {927, 6516}, {1024, 652}, {1027, 1459}, {1096, 2356}, {1118, 1876}, {1119, 34855}, {1279, 20749}, {1416, 603}, {1438, 48}, {1462, 222}, {1783, 2284}, {1814, 394}, {1824, 20683}, {1826, 3930}, {1861, 4712}, {1876, 1362}, {1886, 9502}, {1897, 1026}, {1973, 9454}, {1974, 9455}, {2052, 46108}, {2195, 212}, {2223, 20776}, {2333, 39258}, {2356, 42079}, {2402, 24562}, {2481, 69}, {2501, 24290}, {2969, 3675}, {3290, 20728}, {5089, 6184}, {5342, 4684}, {6185, 1814}, {6335, 42720}, {6559, 3692}, {6591, 665}, {6654, 20769}, {7009, 4447}, {7476, 42747}, {7649, 2254}, {8735, 17435}, {8751, 6}, {8756, 14439}, {9503, 1815}, {10099, 520}, {13576, 72}, {14267, 34381}, {14625, 4047}, {14942, 78}, {15149, 16728}, {15344, 34159}, {17924, 918}, {18026, 883}, {18031, 304}, {18344, 926}, {18785, 71}, {23710, 35293}, {24006, 4088}, {28071, 1260}, {31637, 326}, {32658, 577}, {32666, 32656}, {32735, 36059}, {34018, 348}, {34337, 23102}, {36057, 255}, {36086, 1331}, {36118, 41353}, {36123, 36819}, {36124, 1}, {36125, 34230}, {36146, 1813}, {36796, 345}, {36802, 4571}, {39534, 42758}, {40754, 20741}, {41013, 3932}, {41934, 32658}, {42071, 23612}, {43921, 3937}, {43929, 22383}, {44129, 18157}, {44426, 50333}, {46108, 4437}, {46149, 3917}, {51560, 4561}, {51838, 36057}, {51866, 2196}, {52029, 3781}, {52030, 295}, {52456, 912}

X(54236) = X(3)X(518)∩X(4)X(105)

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

See Ivan Pavlov, euclid 5829.

X(54236) lies on circumconics {A,B,C,X(3),X(105)}, {A,B,C,X(4),X(218)}, {A,B,C,X(32),X(8642)}, cubic K009 (Lemoine cubic), and on these lines: {3, 518}, {4, 105}, {1147, 34159}, {4712, 11517}, {28914, 35185}, {39173, 51471}

X(54236) = Cundy-Parry Phi of X(518)
X(54236) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 14268}, {169, 277}, {1292, 21185}, {2191, 3434}, {6601, 34036}
X(54236) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 14268}, {3309, 5511}
X(54236) = barycentric product X(i)*X(j) for these (i, j): {218, 13577}, {344, 3433}, {3870, 44178}, {6604, 40141}, {24562, 26706}
X(54236) = barycentric quotient X(i)/X(j) for these (i, j): {6, 14268}, {218, 3434}, {1617, 37800}, {3309, 26546}, {3433, 277}, {3870, 20927}, {4878, 21073}, {21059, 169}, {40141, 6601}

X(54237) = X(3)X(519)∩X(4)X(106)

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

See Ivan Pavlov, euclid 5829.

X(54237) lies on circumconics {A,B,C,X(1),X(3913)}, {A,B,C,X(3),X(106)}, {A,B,C,X(4),X(145)}, {A,B,C,X(32),X(8643)}, cubic K009 (Lemoine cubic), and on these lines: {3, 519}, {4, 106}, {32, 40621}, {140, 2885}, {572, 3161}, {4487, 4855}, {35186, 44873}

X(54237) = Cundy-Parry Phi of X(519)
X(54237) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 14261}, {3445, 14923}
X(54237) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 14261}, {3667, 5510}, {45036, 14923}
X(54237) = barycentric quotient X(i)/X(j) for these (i, j): {6, 14261}, {1743, 14923}, {33628, 7419}, {40621, 5510}

X(54238) = X(4)X(656)∩X(29)X(8062)

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

See Ivan Pavlov, euclid 5829.

X(54238) lies on these lines: {4, 656}, {28, 23226}, {29, 8062}, {34, 51641}, {513, 1835}, {650, 16228}, {4086, 39585}, {5520, 20620}, {7253, 7518}, {7497, 23189}, {7510, 30212}, {17924, 48340}, {44426, 46385}

X(54238) = zosma transform of X(520)
X(54238) = X(i)-isoconjugate-of-X(j) for these {i, j}: {856, 6080}
X(54238) = X(i)-Dao conjugate of X(j) for these {i, j}: {16595, 69}
X(54238) = barycentric product X(i)*X(j) for these (i, j): {19, 23683}, {16595, 36126}, {26888, 46110}
X(54238) = barycentric quotient X(i)/X(j) for these (i, j): {23683, 304}, {26888, 1813}

X(54239) = X(4)X(522)∩X(19)X(657)

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

See Ivan Pavlov, euclid 5829.

X(54239) lies on these lines: {4, 522}, {19, 657}, {24, 39226}, {25, 39199}, {27, 47785}, {33, 42756}, {34, 1459}, {406, 48186}, {427, 47806}, {469, 47787}, {475, 48228}, {513, 1835}, {514, 16231}, {523, 10151}, {661, 3064}, {1528, 8058}, {1826, 4036}, {1841, 6586}, {1851, 21119}, {2849, 21186}, {2969, 3259}, {3667, 44428}, {4194, 48173}, {4196, 47828}, {4200, 48243}, {4207, 47832}, {4212, 47830}, {4213, 47831}, {5521, 13999}, {6087, 6129}, {6994, 27486}, {6995, 47798}, {7378, 47808}, {7408, 48239}, {7409, 48169}, {7490, 46919}, {20315, 44928}, {20316, 46878}, {23741, 48398}, {28161, 39532}, {40950, 42750}, {42403, 45745}, {46107, 47995}

X(54239) = zosma transform of X(521)
X(54239) = perspector of circumconic {A,B,C,X(158), X(196)}
X(54239) = reflection of X(i) in X(j) for these {i,j}: {7649, 39534}, {20315, 44928}, {44426, 16231}
X(54239) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 13138}, {48, 44327}, {63, 36049}, {69, 32652}, {78, 8059}, {84, 1331}, {100, 1433}, {101, 41081}, {109, 271}, {110, 52389}, {189, 906}, {219, 37141}, {268, 651}, {280, 36059}, {282, 1813}, {285, 23067}, {309, 32656}, {394, 40117}, {662, 41087}, {664, 2188}, {1332, 1436}, {1413, 4571}, {1415, 44189}, {1422, 4587}, {1903, 4558}, {2192, 6516}, {2208, 4561}, {2357, 4592}, {4556, 53010}, {4575, 39130}, {5546, 52037}, {6081, 46974}, {6517, 7008}, {32660, 34404}
X(54239) = X(i)-Dao conjugate of X(j) for these {i, j}: {11, 271}, {57, 6516}, {136, 39130}, {244, 52389}, {281, 190}, {1015, 41081}, {1084, 41087}, {1146, 44189}, {1249, 44327}, {3162, 36049}, {5139, 2357}, {5190, 189}, {5514, 63}, {5521, 84}, {8054, 1433}, {14837, 52616}, {16596, 69}, {20620, 280}, {36103, 13138}, {38991, 268}, {39025, 2188}, {40615, 34400}
X(54239) = X(i)-cross conjugate of X(j) for these {i, j}: {38362, 7952}
X(54239) = barycentric product X(i)*X(j) for these (i, j): {4, 14837}, {19, 17896}, {40, 17924}, {92, 6129}, {196, 522}, {198, 46107}, {208, 4391}, {221, 46110}, {223, 44426}, {273, 14298}, {278, 8058}, {322, 6591}, {329, 7649}, {342, 650}, {347, 3064}, {514, 7952}, {523, 41083}, {653, 38357}, {663, 40701}, {693, 2331}, {905, 47372}, {1577, 3194}, {1817, 24006}, {2360, 14618}, {2501, 8822}, {3195, 3261}, {3209, 35519}, {5514, 36118}, {6087, 52780}, {7149, 8063}, {7192, 53009}, {16596, 36127}, {17925, 21075}, {18344, 40702}, {24002, 40971}
X(54239) = barycentric quotient X(i)/X(j) for these (i, j): {4, 44327}, {19, 13138}, {25, 36049}, {34, 37141}, {40, 1332}, {196, 664}, {198, 1331}, {208, 651}, {221, 1813}, {223, 6516}, {329, 4561}, {342, 4554}, {512, 41087}, {513, 41081}, {522, 44189}, {608, 8059}, {649, 1433}, {650, 271}, {661, 52389}, {663, 268}, {1096, 40117}, {1817, 4592}, {1973, 32652}, {2187, 906}, {2199, 36059}, {2324, 4571}, {2331, 100}, {2360, 4558}, {2489, 2357}, {2501, 39130}, {3063, 2188}, {3064, 280}, {3194, 662}, {3195, 101}, {3209, 109}, {3676, 34400}, {4017, 52037}, {4705, 53010}, {6129, 63}, {6591, 84}, {7011, 6517}, {7074, 4587}, {7649, 189}, {8058, 345}, {8822, 4563}, {10397, 1259}, {14298, 78}, {14837, 69}, {16596, 52616}, {17896, 304}, {17924, 309}, {18344, 282}, {21075, 52609}, {38357, 6332}, {40701, 4572}, {40971, 644}, {41083, 99}, {43923, 1422}, {44426, 34404}, {46107, 44190}, {47372, 6335}, {53009, 3952}
X(54239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {513, 39534, 7649}

X(54240) = X(1)X(8764)∩X(107)X(108)

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

See Ivan Pavlov, euclid 5829.

X(54240) lies on these lines: {1, 8764}, {4, 3270}, {92, 52780}, {107, 108}, {109, 681}, {225, 1896}, {264, 37800}, {273, 4858}, {278, 2052}, {431, 43746}, {450, 41349}, {648, 651}, {653, 1020}, {1068, 1093}, {1415, 16813}, {1441, 6330}, {1577, 39053}, {1758, 41497}, {1813, 1981}, {1880, 16081}, {1897, 4551}, {1948, 22464}, {2405, 52607}, {4552, 6335}, {4554, 6331}, {6528, 32038}, {7952, 14249}, {14165, 37799}, {16080, 40149}, {17924, 23984}, {26704, 52775}, {26705, 52776}, {37798, 46106}, {43035, 52982}, {51358, 51365}

X(54240) = isogonal conjugate of X(36054)
X(54240) = trilinear pole of line {4, 65}
X(54240) = polar conjugate of X(521)
X(54240) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(2), X(2405)}}, {{A, B, C, X(92), X(24035)}}, {{A, B, C, X(107), X(648)}}, {{A, B, C, X(108), X(651)}}, {{A, B, C, X(190), X(1309)}}, {{A, B, C, X(278), X(23706)}}, {{A, B, C, X(523), X(47203)}}, {{A, B, C, X(525), X(2845)}}, {{A, B, C, X(644), X(40117)}}, {{A, B, C, X(650), X(2637)}}, {{A, B, C, X(655), X(26704)}}, {{A, B, C, X(658), X(15455)}}, {{A, B, C, X(681), X(823)}}, {{A, B, C, X(1086), X(39534)}}, {{A, B, C, X(4565), X(36067)}}, {{A, B, C, X(4612), X(30610)}}, {{A, B, C, X(4858), X(24002)}}, {{A, B, C, X(26705), X(27833)}}, {{A, B, C, X(26706), X(32041)}}, {{A, B, C, X(31628), X(46605)}}, {{A, B, C, X(32691), X(37137)}}
X(54240) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 36054}, {3, 652}, {9, 23224}, {21, 822}, {29, 32320}, {32, 52616}, {41, 4131}, {48, 521}, {55, 4091}, {63, 1946}, {71, 23189}, {73, 23090}, {78, 22383}, {101, 1364}, {109, 35072}, {184, 6332}, {212, 905}, {219, 1459}, {255, 650}, {283, 647}, {284, 520}, {326, 3063}, {332, 3049}, {333, 39201}, {394, 663}, {512, 6514}, {513, 2289}, {522, 577}, {651, 2638}, {656, 2193}, {657, 1804}, {664, 39687}, {667, 3719}, {810, 1812}, {906, 7004}, {1021, 22341}, {1092, 3064}, {1264, 1919}, {1331, 7117}, {1415, 24031}, {1433, 10397}, {1437, 8611}, {1794, 52306}, {1795, 52307}, {1813, 3270}, {2175, 30805}, {2194, 24018}, {2287, 51641}, {2299, 52613}, {2318, 7254}, {2968, 32660}, {3239, 7335}, {3269, 4636}, {3682, 7252}, {3737, 3990}, {3900, 7125}, {3937, 4587}, {4025, 52425}, {4041, 18604}, {4055, 4560}, {4100, 44426}, {4391, 52430}, {6507, 18344}, {6517, 14936}, {7016, 22382}, {7065, 52919}, {7128, 23614}, {7183, 8641}, {9247, 35518}, {14331, 14379}, {14395, 35200}, {14585, 35519}, {17434, 35196}, {20752, 23696}, {21789, 40152}, {23606, 46110}, {26932, 32656}, {34591, 36059}, {35071, 52921}, {36055, 46391}, {37754, 52914}
X(54240) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 36054}, {11, 35072}, {123, 47410}, {133, 14395}, {223, 4091}, {226, 52613}, {478, 23224}, {1015, 1364}, {1146, 24031}, {1214, 24018}, {1249, 521}, {3160, 4131}, {3162, 1946}, {5190, 7004}, {5375, 1259}, {5521, 7117}, {6376, 52616}, {6523, 650}, {6631, 3719}, {9296, 1264}, {10001, 326}, {15259, 3063}, {20620, 34591}, {25640, 52307}, {36103, 652}, {38991, 2638}, {39025, 39687}, {39026, 2289}, {39052, 283}, {39053, 63}, {39054, 6514}, {39060, 69}, {39062, 1812}, {40590, 520}, {40593, 30805}, {40596, 2193}, {40611, 822}, {40624, 23983}, {40625, 16731}, {40837, 905}, {47345, 656}, {51221, 46391}
X(54240) = X(i)-cross conjugate of X(j) for these {i, j}: {108, 18026}, {158, 24032}, {278, 23984}, {650, 4}, {3064, 1896}, {6129, 7}, {17924, 2052}, {21348, 1041}, {44426, 273}, {52607, 653}
X(54240) = X(i)-Zayin conjugate of X(j) for these {i, j}: {1, 36054}, {3, 822}, {1745, 652}
X(54240) = X(i)-Ceva conjugate of X(j) for these {i, j}: {823, 653}, {24032, 158}
X(54240) = barycentric product X(i)*X(j) for these (i, j): {1, 52938}, {4, 18026}, {19, 46404}, {65, 6528}, {75, 36127}, {92, 653}, {107, 1441}, {108, 264}, {112, 52575}, {158, 664}, {225, 811}, {226, 823}, {273, 1897}, {278, 6335}, {281, 13149}, {307, 36126}, {318, 36118}, {331, 1783}, {349, 24019}, {393, 4554}, {522, 24032}, {648, 40149}, {651, 2052}, {668, 1118}, {1093, 6516}, {1096, 4572}, {1214, 15352}, {1231, 6529}, {1415, 18027}, {1813, 6521}, {1857, 4569}, {1880, 6331}, {1896, 4566}, {1969, 32674}, {4391, 23984}, {6358, 52919}, {6386, 7337}, {7012, 46107}, {7017, 32714}, {7128, 46110}, {17073, 42389}, {17924, 46102}, {24033, 35519}, {24035, 52780}, {31623, 52607}, {34388, 52920}, {40117, 40701}, {46104, 46152}
X(54240) = barycentric quotient X(i)/X(j) for these (i, j): {4, 521}, {6, 36054}, {7, 4131}, {19, 652}, {25, 1946}, {28, 23189}, {34, 1459}, {56, 23224}, {57, 4091}, {65, 520}, {75, 52616}, {85, 30805}, {92, 6332}, {100, 1259}, {101, 2289}, {107, 21}, {108, 3}, {109, 255}, {112, 2193}, {158, 522}, {162, 283}, {190, 3719}, {225, 656}, {226, 24018}, {264, 35518}, {273, 4025}, {278, 905}, {331, 15413}, {393, 650}, {513, 1364}, {522, 24031}, {608, 22383}, {648, 1812}, {650, 35072}, {651, 394}, {653, 63}, {658, 7183}, {662, 6514}, {663, 2638}, {664, 326}, {668, 1264}, {692, 6056}, {811, 332}, {823, 333}, {934, 1804}, {1020, 40152}, {1042, 51641}, {1093, 44426}, {1096, 663}, {1172, 23090}, {1214, 52613}, {1231, 4143}, {1309, 1809}, {1396, 7254}, {1400, 822}, {1402, 39201}, {1409, 32320}, {1415, 577}, {1441, 3265}, {1461, 7125}, {1783, 219}, {1813, 6507}, {1826, 8611}, {1841, 52306}, {1857, 3900}, {1875, 8677}, {1880, 647}, {1896, 7253}, {1897, 78}, {1981, 6518}, {1990, 14395}, {2052, 4391}, {2207, 3063}, {2331, 10397}, {3063, 39687}, {3064, 34591}, {3270, 23614}, {4036, 7068}, {4077, 17216}, {4391, 23983}, {4551, 3682}, {4552, 3998}, {4554, 3926}, {4559, 3990}, {4560, 16731}, {4565, 18604}, {4566, 52385}, {4569, 7055}, {5317, 7252}, {6059, 8641}, {6335, 345}, {6516, 3964}, {6520, 3064}, {6521, 46110}, {6524, 18344}, {6528, 314}, {6529, 1172}, {6588, 47410}, {6591, 7117}, {7012, 1331}, {7017, 15416}, {7045, 6517}, {7103, 51646}, {7115, 906}, {7120, 22382}, {7128, 1813}, {7337, 667}, {7649, 7004}, {8747, 3737}, {8748, 1021}, {8750, 212}, {8755, 46391}, {8756, 14418}, {13149, 348}, {13437, 6365}, {13459, 6364}, {14571, 52307}, {15352, 31623}, {15742, 4571}, {17924, 26932}, {18026, 69}, {18344, 3270}, {19366, 680}, {21859, 52386}, {23582, 4612}, {23706, 22350}, {23710, 14414}, {23984, 651}, {23985, 1415}, {23987, 46974}, {24000, 4636}, {24019, 284}, {24021, 52921}, {24032, 664}, {24033, 109}, {31623, 15411}, {32230, 52914}, {32660, 4100}, {32674, 48}, {32702, 14578}, {32713, 2194}, {32714, 222}, {36059, 1092}, {36067, 36055}, {36110, 1795}, {36118, 77}, {36123, 37628}, {36124, 23696}, {36126, 29}, {36127, 1}, {36797, 1792}, {39534, 35014}, {40097, 39167}, {40117, 268}, {40149, 525}, {41013, 52355}, {41321, 51376}, {43923, 3937}, {44426, 2968}, {46102, 1332}, {46107, 17880}, {46152, 3917}, {46404, 304}, {47372, 8058}, {52575, 3267}, {52607, 1214}, {52776, 37741}, {52919, 2185}, {52920, 60}, {52921, 1098}, {52938, 75}

X(54241) = X(3)X(108)∩X(4)X(521)

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

See Ivan Pavlov, euclid 5829.

X(54241) lies on circumconics {A,B,C,X(3),X(8)}, {A,B,C,X(4),X(108)}, cubic K028, and on these lines: {3, 108}, {4, 521}, {8, 14249}, {1118, 2745}, {22350, 23706}, {39267, 39268}

X(54241) = Cundy-Parry Psi of X(521)
X(54241) = trilinear pole of line {14571, 52307}
X(54241) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 39175}, {1795, 6001}, {1809, 51662}
X(54241) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 39175}, {2804, 52114}, {25640, 6001}
X(54241) = X(i)-cross conjugate of X(j) for these {i, j}: {517, 1295}
X(54241) = barycentric quotient X(i)/X(j) for these (i, j): {6, 39175}, {1875, 43058}, {14571, 6001}, {32647, 2720}, {36044, 37136}, {42072, 47434}

X(54242) = X(3)X(102)∩X(4)X(522)

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

See Ivan Pavlov, euclid 5829.

X(54242) lies on circumconics {A,B,C,X(3),X(522)}, {A,B,C,X(4),X(109)}, cubic K028, and on these lines: {3, 102}, {4, 522}, {76, 34393}, {953, 35183}

X(54242) = Cundy-Parry Psi of X(522)
X(54242) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2182, 2988}, {14304, 35187}, {15379, 24034}, {32706, 46974}, {36113, 39471}
X(54242) = X(i)-Dao conjugate of X(j) for these {i, j}: {117, 515}
X(54242) = barycentric product X(i)*X(j) for these (i, j): {1735, 36100}, {8607, 34393}
X(54242) = barycentric quotient X(i)/X(j) for these (i, j): {102, 2988}, {8607, 515}, {32643, 35187}, {32667, 36113}

X(54243) = X(3)X(522)∩X(4)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^6-a^5*c-a*(b-c)^2*c^2*(b+c)-a^4*(2*b^2-b*c+c^2)+a^3*c*(b^2-b*c+2*c^2)+(-(b^2*c)+c^3)^2+a^2*(b^4-b^3*c+2*b^2*c^2-b*c^3-c^4))*(a^6-a^5*b-a*b^2*(b-c)^2*(b+c)+a^3*b*(2*b^2-b*c+c^2)-a^4*(b^2-b*c+2*c^2)+(b^3-b*c^2)^2-a^2*(b^4+b^3*c-2*b^2*c^2+b*c^3-c^4)) : :

See Ivan Pavlov, euclid 5829.

X(54243) lies on circumconics {A,B,C,X(3),X(109)}, {A,B,C,X(4),X(280)}, {A,B,C,X(56),X(39199)}, cubic K009 (Lemoine cubic), and on these lines: {3, 522}, {4, 109}, {32, 23986}, {952, 1147}, {2734, 35187}

X(54243) = Cundy-Parry Phi of X(522)
X(54243) = X(i)-isoconjugate-of-X(j) for these {i, j}: {102, 1735}, {8607, 36100}
X(54243) = X(i)-Dao conjugate of X(j) for these {i, j}: {515, 117}
X(54243) = X(i)-cross conjugate of X(j) for these {i, j}: {38554, 515}
X(54243) = barycentric product X(i)*X(j) for these (i, j): {515, 2988}
X(54243) = barycentric quotient X(i)/X(j) for these (i, j): {2182, 1735}, {2988, 34393}, {23986, 117}, {32706, 52780}, {32707, 36067}

X(54244) = X(4)X(6003)∩X(162)X(250)

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

See Ivan Pavlov, euclid 5829.

X(54244) lies on these lines: {4, 6003}, {19, 35347}, {24, 39210}, {25, 3733}, {34, 4017}, {108, 34921}, {162, 250}, {186, 14838}, {340, 18160}, {427, 31946}, {513, 1835}, {656, 14192}, {661, 35993}, {759, 32710}, {798, 44103}, {830, 48340}, {1112, 1830}, {1474, 20981}, {1510, 1831}, {1829, 4132}, {2806, 52355}, {2812, 48303}, {2850, 44409}, {3064, 15313}, {3738, 44426}, {6591, 48026}, {7265, 35057}, {15309, 17925}, {17104, 38936}, {17171, 42327}, {23189, 39212}

X(54244) = zosma transform of X(523)
X(54244) = reflection of X(i) in X(j) for these {i,j}: {7649, 18344}
X(54244) = trilinear pole of line {2611, 47230}
X(54244) = perspector of circumconic {A,B,C,X(278), X(6198)}
X(54244) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(4), X(186)}}, {{A, B, C, X(35), X(18838)}}, {{A, B, C, X(162), X(24006)}}, {{A, B, C, X(445), X(35993)}}, {{A, B, C, X(513), X(2605)}}, {{A, B, C, X(526), X(6003)}}, {{A, B, C, X(1876), X(7282)}}, {{A, B, C, X(1877), X(6198)}}, {{A, B, C, X(1884), X(11107)}}, {{A, B, C, X(2611), X(4017)}}, {{A, B, C, X(2614), X(7178)}}, {{A, B, C, X(3678), X(38938)}}, {{A, B, C, X(35235), X(37964)}}, {{A, B, C, X(37305), X(46468)}}
X(54244) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 6742}, {48, 15455}, {72, 13486}, {78, 26700}, {79, 1331}, {100, 7100}, {110, 52388}, {219, 38340}, {643, 52390}, {664, 8606}, {758, 36061}, {906, 30690}, {1332, 2160}, {1789, 4551}, {1813, 7110}, {3615, 23067}, {3936, 32662}, {4558, 8818}, {4561, 6186}, {4571, 52372}, {4574, 52393}, {4575, 6757}, {4587, 52374}, {6516, 7073}, {8611, 35049}, {20565, 32656}, {36059, 52344}
X(54244) = X(i)-Dao conjugate of X(j) for these {i, j}: {136, 6757}, {244, 52388}, {1015, 52381}, {1249, 15455}, {5190, 30690}, {5521, 79}, {8054, 7100}, {8287, 69}, {14838, 14208}, {16221, 758}, {20620, 52344}, {36103, 6742}, {39025, 8606}
X(54244) = X(i)-Zayin conjugate of X(j) for these {i, j}: {110, 656}
X(54244) = X(i)-Waw conjugate of X(j) for these {i, j}: {4, 1830}
X(54244) = X(i)-Ceva conjugate of X(j) for these {i, j}: {14775, 7649}
X(54244) = barycentric product X(i)*X(j) for these (i, j): {4, 14838}, {19, 4467}, {25, 18160}, {28, 7265}, {35, 17924}, {92, 2605}, {112, 17886}, {162, 8287}, {273, 9404}, {278, 35057}, {319, 6591}, {513, 52412}, {514, 6198}, {648, 2611}, {650, 7282}, {759, 44427}, {811, 20982}, {823, 22094}, {1399, 46110}, {1442, 3064}, {1824, 16755}, {1825, 4560}, {1897, 7202}, {2003, 44426}, {2052, 23226}, {2174, 46107}, {3219, 7649}, {3261, 14975}, {3678, 17925}, {4077, 41502}, {5379, 21141}, {7178, 11107}, {14616, 47230}, {14618, 17104}, {14775, 16585}, {17095, 18344}, {24006, 40214}, {35235, 37140}, {42033, 43923}
X(54244) = barycentric quotient X(i)/X(j) for these (i, j): {4, 15455}, {19, 6742}, {34, 38340}, {35, 1332}, {186, 4585}, {513, 52381}, {608, 26700}, {649, 7100}, {661, 52388}, {1399, 1813}, {1474, 13486}, {1825, 4552}, {2003, 6516}, {2174, 1331}, {2501, 6757}, {2605, 63}, {2611, 525}, {3063, 8606}, {3064, 52344}, {3219, 4561}, {3678, 52609}, {4467, 304}, {6591, 79}, {7180, 52390}, {7202, 4025}, {7252, 1789}, {7265, 20336}, {7282, 4554}, {7649, 30690}, {8287, 14208}, {9404, 78}, {11107, 645}, {14838, 69}, {14975, 101}, {17104, 4558}, {17886, 3267}, {17924, 20565}, {18160, 305}, {18344, 7110}, {20982, 656}, {21741, 23067}, {21824, 4064}, {22094, 24018}, {23226, 394}, {34079, 36061}, {34397, 1983}, {35057, 345}, {40214, 4592}, {41502, 643}, {43923, 52374}, {43925, 52375}, {44427, 35550}, {47230, 758}, {52405, 4571}, {52412, 668}, {52418, 4242}
X(54244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {513, 18344, 7649}

X(54245) = X(4)X(28612)∩X(210)X(430)

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

See Ivan Pavlov, euclid 5829.

X(54245) lies on these lines: {4, 28612}, {25, 21009}, {34, 51655}, {210, 430}, {513, 1835}, {4206, 5338}, {16611, 23230}

X(54245) = zosma transform of X(524)
X(54245) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 51561}, {1444, 34893}, {1790, 34892}
X(54245) = X(i)-Dao conjugate of X(j) for these {i, j}: {16597, 69}, {36103, 51561}
X(54245) = barycentric product X(i)*X(j) for these (i, j): {4, 16611}, {19, 4442}, {92, 39688}, {278, 24394}, {1824, 37756}, {1826, 7292}, {2052, 23230}, {16597, 36128}, {16784, 41013}
X(54245) = barycentric quotient X(i)/X(j) for these (i, j): {19, 51561}, {1824, 34892}, {2333, 34893}, {2832, 15419}, {4442, 304}, {7292, 17206}, {8650, 7254}, {16611, 69}, {16784, 1444}, {23230, 394}, {24394, 345}, {39688, 63}

X(54246) = X(2)X(8877)∩X(6)X(41404)

Barycentrics    a^2*(a^2+b^2-2*c^2)*(a^2-2*b^2+c^2)*(a^10-6*a^8*(b^2+c^2)+a^6*(-7*b^4+52*b^2*c^2-7*c^4)-(b^2+c^2)^3*(b^4-b^2*c^2+c^4)+a^4*(7*b^6-33*b^4*c^2-33*b^2*c^4+7*c^6)+3*a^2*(2*b^8-4*b^6*c^2+15*b^4*c^4-4*b^2*c^6+2*c^8)) : :

See Ivan Pavlov, euclid 5829.

X(54246) lies on cubic K1156 and these lines: {2, 8877}, {6, 41404}, {576, 52474}, {7312, 7313}, {23106, 39296}

X(54246) = eigentransform of X(524)
X(54246) = X(i)-Dao conjugate of X(j) for these {i, j}: {10630, 671}

X(54247) = X(4)X(1577)∩X(19)X(4041)

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

See Ivan Pavlov, euclid 5829.

X(54247) lies on these lines: {4, 1577}, {19, 4041}, {25, 21789}, {28, 14838}, {34, 51640}, {513, 1835}, {830, 17924}, {1891, 3907}, {2299, 21761}, {3064, 8678}, {4198, 4560}, {6591, 50517}, {7497, 39212}, {20620, 46671}, {24019, 32673}, {44705, 47124}

X(54247) = zosma transform of X(525)
X(54247) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1331, 15314}, {4561, 8615}
X(54247) = X(i)-Dao conjugate of X(j) for these {i, j}: {5521, 15314}, {34846, 69}
X(54247) = barycentric product X(i)*X(j) for these (i, j): {4, 16612}, {3064, 4296}, {5279, 7649}, {5285, 17924}, {6591, 7270}, {24019, 34846}
X(54247) = barycentric quotient X(i)/X(j) for these (i, j): {5279, 4561}, {5285, 1332}, {6591, 15314}, {16612, 69}

X(54248) = X(3)X(525)∩X(30)X(14850)

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

See Ivan Pavlov, euclid 5829.

X(54248) lies on these lines: {3, 525}, {30, 14850}, {249, 21166}, {512, 46634}, {524, 35383}, {826, 46633}, {3564, 38723}, {3566, 46987}, {12041, 47293}, {15061, 34953}, {38728, 51258}

X(54249) = X(37)X(513)∩X(241)X(514)

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

See Ivan Pavlov, euclid 5918.

X(54249) lies on these lines {37, 513}, {241, 514}, {512, 4162}, {647, 49293}, {649, 4083}, {654, 1938}, {661, 3777}, {784, 48397}, {1449, 3063}, {1743, 20980}, {3287, 43924}, {3310, 47768}, {3709, 4778}, {3766, 4885}, {3912, 23828}, {4378, 16975}, {4394, 21832}, {4501, 48293}, {4526, 6006}, {4699, 20906}, {4751, 20949}, {4817, 25425}, {4820, 8714}, {4850, 47762}, {4905, 24290}, {4932, 25098}, {4977, 6586}, {5029, 48331}, {6372, 48026}, {6588, 29120}, {6591, 29025}, {7234, 22319}, {8632, 48330}, {8659, 48328}, {14433, 31197}, {14991, 23815}, {16671, 39521}, {16884, 21007}, {17092, 24002}, {17461, 41162}, {17756, 47824}, {20952, 23807}, {21791, 22383}, {21894, 47881}, {23780, 30804}, {24635, 33570}, {28374, 49282}, {29029, 47227}, {30665, 50336}, {40549, 47802}, {44307, 45658}

X(54249) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(37), X(16609)}}, {{A, B, C, X(241), X(3252)}}, {{A, B, C, X(513), X(43041)}}, {{A, B, C, X(649), X(43051)}}, {{A, B, C, X(876), X(3676)}}, {{A, B, C, X(1323), X(4334)}}, {{A, B, C, X(3572), X(3669)}}, {{A, B, C, X(3911), X(17754)}}, {{A, B, C, X(10566), X(21348)}}, {{A, B, C, X(19586), X(39957)}}, {{A, B, C, X(20358), X(39742)}}, {{A, B, C, X(20507), X(24002)}}, {{A, B, C, X(20917), X(40881)}}, {{A, B, C, X(24349), X(43037)}}, {{A, B, C, X(40773), X(40787)}}
X(54249) = reflection of X(i) in X(j) for these {i,j}: {20507, 3676}, {21832, 4394}, {3766, 4885}, {650, 665}
X(54249) = complement of isotomic conjugate of of isogonal conjugate of X(4782)
X(54249) = perspector of circumconic {{A, B, C, X(7), X(87)}}
X(54249) = center of circumconic {{A, B, C, X(4817), X(24720)}}
X(54249) = X(i)-isoconjugate-of-X(j) for these {i, j} {101, 41527}, {651, 7220}
X(54249) = X(i)-Dao conjugate of X(j) for these {i, j} {984, 3807}, {1015, 41527}, {24720, 4762}, {38991, 7220}
X(54249) = X(i)-Zayin conjugate of X(j) for these {i, j}: {43077, 4782}
X(54249) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4817, 513}
X(54249) = X(i)-complementary conjugate of X(j) for these {i, j}: {32739, 27481}, {34475, 53575}, {40735, 11}, {43077, 141}, {51449, 53564}, {52654, 21252}, {53648, 626}
X(54249) = (polar of X(1) wrt inconic with perspector X(2)) ∩ (polar of X(2) wrt inconic with perspector X(1))
X(54249) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {513, 41527}, {663, 7220}, {4334, 664}, {8926, 51614}, {19584, 3807}, {19586, 3799}, {20917, 1978}, {21010, 100}, {21101, 4033}, {22163, 1332}, {24349, 668}, {24720, 75}, {45902, 40785}
X(54249) = barycentric product X(i)*X(j) for these (i, j): {1, 24720}, {1019, 21101}, {4334, 522}, {4458, 8926}, {17754, 514}, {17924, 22163}, {19584, 4817}, {20917, 649}, {21010, 693}, {24349, 513}
X(54249) = barycentric quotient X(i)/X(j) for these (i, j): {513, 41527}, {663, 7220}, {4334, 664}, {8926, 51614}, {17754, 190}, {19584, 3807}, {19586, 3799}, {20917, 1978}, {21010, 100}, {21101, 4033}, {22163, 1332}, {24349, 668}, {24720, 75}, {45902, 40785}
X(54249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {514, 3676, 20507}, {649, 4449, 4435}, {4394, 29226, 21832}, {43060, 48276, 650}

X(54250) = X(6)X(513)∩X(665)X(2516)

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

See Ivan Pavlov, euclid 5918.

X(54250) lies on circumconics {{A, B, C, X(9503), X(51839)}} and on these lines: {6, 513}, {523, 47454}, {650, 4449}, {665, 2516}, {676, 14330}, {3669, 4394}, {6589, 10581}, {6591, 40137}, {33630, 44426}, {37689, 47803}

X(54250) = perspector of circumconic {{A, B, C, X(105), X(3062)}}
X(54250) = (polar of X(1) wrt inconic with perspector X(57)) ∩ (polar of X(57) wrt inconic with perspector X(1))

X(54251) = X(1)X(4785)∩X(42)X(649)

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

See Ivan Pavlov, euclid 5918.

X(54251) lies on these lines: {1, 4785}, {42, 649}, {43, 45313}, {513, 663}, {514, 53553}, {667, 6373}, {798, 9010}, {810, 50515}, {812, 4449}, {1919, 21003}, {3052, 23865}, {3221, 8639}, {3720, 31147}, {3768, 8656}, {3835, 24666}, {4455, 8643}, {4481, 48023}, {4724, 4817}, {9313, 21143}, {16569, 24749}, {20295, 29814}, {21191, 21301}, {23751, 50503}, {24533, 43931}, {25128, 27345}, {25502, 45339}, {25889, 25924}, {28360, 28398}, {29328, 48303}, {29362, 48342}

X(54251) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(875), X(43924)}}, {{A, B, C, X(1319), X(21010)}}, {{A, B, C, X(1458), X(4334)}}, {{A, B, C, X(1463), X(24349)}}, {{A, B, C, X(3572), X(3669)}}, {{A, B, C, X(4017), X(24720)}}, {{A, B, C, X(4785), X(43077)}}, {{A, B, C, X(17754), X(52896)}}, {{A, B, C, X(18108), X(23655)}}, {{A, B, C, X(20459), X(39966)}}, {{A, B, C, X(25426), X(40760)}}
X(54251) = reflection of X(i) in X(j) for these {i,j}: {20979, 667}, {21301, 21191}, {48023, 4481}
X(54251) = perspector of circumconic {{A, B, C, X(57), X(292)}}
X(54251) = X(i)-isoconjugate-of-X(j) for these {i, j}: {100, 41527}, {664, 7220}, {3799, 47647}, {23605, 51614}
X(54251) = X(i)-Dao conjugate of X(j) for these {i, j}: {984, 4505}, {8054, 41527}, {39025, 7220}
X(54251) = X(i)-Zayin conjugate of X(j) for these {i, j}: {32041, 4724}
X(54251) = (polar of X(1) wrt inconic with perspector X(6)) ∩ (polar of X(6) wrt inconic with perspector X(1))
X(54251) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {649, 41527}, {3063, 7220}, {4334, 4554}, {17754, 668}, {19584, 4505}, {19586, 3807}, {19587, 3799}, {20917, 6386}, {21101, 27808}, {22163, 4561}, {24349, 1978}, {24720, 76}
X(54251) = barycentric product X(i)*X(j) for these (i, j): {4334, 650}, {17754, 513}, {19586, 4817}, {20917, 667}, {21010, 514}, {21101, 3733}, {22163, 7649}, {24349, 649}, {24720, 6}
X(54251) = barycentric quotient X(i)/X(j) for these (i, j): {649, 41527}, {3063, 7220}, {17754, 668}, {19584, 4505}, {19586, 3807}, {19587, 3799}, {20917, 6386}, {21010, 190}, {21101, 27808}, {22163, 4561}, {24349, 1978}, {24720, 76}
X(54251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {667, 6373, 20979}, {7234, 50514, 649}, {50517, 51641, 663}

X(54252) = X(38)X(661)∩X(240)X(522)

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

See Ivan Pavlov, euclid 5918.

X(54252) lies on these lines: {38, 661}, {240, 522}, {798, 4083}

X(54252) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(860), X(11328)}}, {{A, B, C, X(7649), X(45907)}}, {{A, B, C, X(18906), X(52651)}}
X(54252) = perspector of circumconic {{A, B, C, X(92), X(1581)}}
X(54252) = X(i)-isoconjugate-of-X(j) for these {i, j}: {99, 47643}, {110, 19222}
X(54252) = X(i)-Dao conjugate of X(j) for these {i, j}: {244, 19222}, {38986, 47643}
X(54252) = (polar of X(1) wrt inconic with perspector X(75)) ∩ (polar of X(75) wrt inconic with perspector X(1))
X(54252) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {661, 19222}, {798, 47643}, {6234, 37134}, {11328, 662}, {18906, 799}, {45907, 1}
X(54252) = barycentric product X(i)*X(j) for these (i, j): {11328, 1577}, {18906, 661}, {19591, 523}, {45907, 75}
X(54252) = barycentric quotient X(i)/X(j) for these (i, j): {661, 19222}, {798, 47643}, {6234, 37134}, {11328, 662}, {18906, 799}, {19591, 99}, {45907, 1}

X(54253) = X(239)X(514)∩X(513)X(1100)

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

See Ivan Pavlov, euclid 5918.

X(54253) lies on these lines: {239, 514}, {513, 1100}, {650, 47922}, {661, 4367}, {663, 2520}, {3250, 4378}, {4024, 17166}, {4079, 16777}, {4435, 4979}, {4826, 48292}, {4893, 14419}, {6372, 8632}, {8672, 20981}, {8678, 50454}, {28195, 50455}, {29212, 47873}, {48019, 48336}, {48026, 48330}, {48266, 48301}

X(54253) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(514), X(9279)}}, {{A, B, C, X(661), X(21196)}}, {{A, B, C, X(16704), X(40750)}}, {{A, B, C, X(18206), X(18791)}}
X(54253) = perspector of circumconic {{A, B, C, X(86), X(1929)}}
X(54253) = X(i)-isoconjugate-of-X(j) for these {i, j}: {100, 40776}
X(54253) = X(i)-Dao conjugate of X(j) for these {i, j}: {8054, 40776}
X(54253) = X(i)-Ceva conjugate of X(j) for these {i, j}: {28840, 4724}
X(54253) = (polar of X(1) wrt inconic with perspector X(81)) ∩ (polar of X(81) wrt inconic with perspector X(1))
X(54253) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {649, 40776}, {9279, 10}, {18791, 32041}, {24342, 668}
X(54253) = barycentric product X(i)*X(j) for these (i, j): {86, 9279}, {18791, 4762}, {24342, 513}, {40750, 514}
X(54253) = barycentric quotient X(i)/X(j) for these (i, j): {649, 40776}, {9279, 10}, {18791, 32041}, {24342, 668}, {40750, 190}
X(54253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 4367, 5029}, {1019, 21832, 649}

X(54254) = X(36)X(238)∩X(43)X(7234)

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

See Ivan Pavlov, euclid 5918.

X(54254) lies on these lines: {36, 238}, {43, 7234}, {649, 2666}, {798, 4367}, {804, 52602}, {812, 50512}, {1125, 4785}, {3741, 45313}, {4762, 4782}, {8631, 48136}, {17018, 50487}, {23506, 50500}, {24948, 50489}, {25502, 29487}, {30116, 30203}

X(54254) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2248), X(3286)}}, {{A, B, C, X(40721), X(52897)}}, {{A, B, C, X(40749), X(52680)}}
X(54254) = midpoint of X(i) in X(j) for these {i,j}: {1019, 4455}, {798, 4367}
X(54254) = perspector of circumconic {{A, B, C, X(81), X(2665)}}
X(54254) = X(i)-isoconjugate-of-X(j) for these {i, j}: {100, 40775}
X(54254) = X(i)-Dao conjugate of X(j) for these {i, j}: {8054, 40775}
X(54254) = (polar of X(1) wrt inconic with perspector X(86)) ∩ (polar of X(86) wrt inconic with perspector X(1))
X(54254) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {649, 40775}, {40721, 668}
X(54254) = barycentric product X(i)*X(j) for these (i, j): {40721, 513}, {40749, 514}
X(54254) = barycentric quotient X(i)/X(j) for these (i, j): {649, 40775}, {40721, 668}, {40749, 190}
X(54254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1019, 4455, 513}

X(54255) = X(36)X(30235)∩X(55)X(650)

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

See Ivan Pavlov, euclid 5918.

X(54255) lies on these lines: {36, 30235}, {55, 650}, {390, 4762}, {497, 45320}, {513, 4162}, {2820, 3669}, {3057, 9443}, {4885, 5274}, {5281, 44567}, {5284, 25925}, {6767, 8760}, {9819, 14077}, {10589, 31250}, {28042, 44842}

X(54255) = perspector of circumconic {{A, B, C, X(294), X(8056)}}
X(54255) = (polar of X(1) wrt inconic with perspector X(9)) ∩ (polar of X(9) wrt inconic with perspector X(1))

X(54256) = X(514)X(661)∩X(523)X(1213)

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

See Ivan Pavlov, euclid 5918.

X(54256) lies on these lines: {514, 661}, {523, 1213}, {649, 48392}, {690, 4931}, {2321, 22044}, {2533, 4024}, {2610, 47873}, {2642, 4777}, {4041, 4838}, {4142, 48277}, {4979, 29150}, {7192, 46192}, {21832, 48393}, {27081, 47792}, {27710, 47659}, {46390, 48120}, {48265, 50522}

X(54256) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(514), X(9279)}}, {{A, B, C, X(693), X(18014)}}, {{A, B, C, X(3936), X(40750)}}, {{A, B, C, X(14210), X(24342)}}, {{A, B, C, X(32679), X(42666)}}
X(54256) = perspector of circumconic {{A, B, C, X(75), X(8818)}}
X(54256) = X(i)-isoconjugate-of-X(j) for these {i, j}: {110, 40776}
X(54256) = X(i)-Dao conjugate of X(j) for these {i, j}: {244, 40776}
X(54256) = (polar of X(2) wrt inconic with perspector X(10)) ∩ (polar of X(10) wrt inconic with perspector X(2))
X(54256) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {661, 40776}, {9279, 1}, {40750, 662}
X(54256) = barycentric product X(i)*X(j) for these (i, j): {75, 9279}, {1577, 40750}, {24342, 523}
X(54256) = barycentric quotient X(i)/X(j) for these (i, j): {661, 40776}, {9279, 1}, {24342, 99}, {40750, 662}

X(54257) = X(216)X(520)∩X(441)X(525)

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

See Ivan Pavlov, euclid 5918.

X(54257) lies on these lines: {216, 520}, {441, 525}, {2485, 5421}, {22089, 32320}, {45907, 52631}

X(54257) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(11064), X(40805)}}, {{A, B, C, X(44436), X(47739)}}
X(54257) = perspector of circumconic {{A, B, C, X(69), X(14941)}}
X(54257) = X(i)-isoconjugate-of-X(j) for these {i, j}: {162, 40815}, {24019, 43711}
X(54257) = X(i)-Dao conjugate of X(j) for these {i, j}: {125, 40815}, {35071, 43711}
X(54257) = (polar of X(2) wrt inconic with perspector X(3)) ∩ (polar of X(3) wrt inconic with perspector X(2))
X(54257) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {520, 43711}, {647, 40815}, {40805, 648}, {47739, 15352}
X(54257) = barycentric product X(i)*X(j) for these (i, j): {40805, 525}, {47739, 52613}
X(54257) = barycentric quotient X(i)/X(j) for these (i, j): {520, 43711}, {647, 40815}, {40805, 648}, {47739, 15352}

X(54258) = X(44)X(513)∩X(1213)X(4806)

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

See Ivan Pavlov, euclid 5918.

X(54258) lies on these lines: {44, 513}, {512, 16589}, {1213, 4806}, {9402, 50487}, {20691, 22320}, {21834, 50483}, {27075, 48049}

X(54258) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(896), X(40749)}}, {{A, B, C, X(2238), X(40721)}}
X(54258) = reflection of X(i) in X(j) for these {i,j}: {661, 46390}
X(54258) = perspector of circumconic {{A, B, C, X(1), X(40749)}}
X(54258) = X(i)-isoconjugate-of-X(j) for these {i, j}: {662, 40775}
X(54258) = X(i)-Dao conjugate of X(j) for these {i, j}: {1084, 40775}
X(54258) = X(i)-Zayin conjugate of X(j) for these {i, j}: {661, 40775}
X(54258) = (polar of X(2) wrt inconic with perspector X(37)) ∩ (polar of X(37) wrt inconic with perspector X(2))
X(54258) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {512, 40775}, {40721, 799}
X(54258) = barycentric product X(i)*X(j) for these (i, j): {40721, 661}, {40749, 523}
X(54258) = barycentric quotient X(i)/X(j) for these (i, j): {512, 40775}, {40721, 799}, {40749, 99}
X(54258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {513, 46390, 661}

X(54259) = X(6)X(523)∩X(525)X(3239)

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

See Ivan Pavlov, euclid 5918.

X(54259) lies on circumconic {{A, B, C, X(5203), X(5921)}} and on these lines: {6, 523}, {525, 3239}, {647, 2506}, {2501, 3566}, {6792, 46982}, {7652, 46425}, {8673, 14346}, {9007, 47138}, {9476, 41254}, {16040, 52588}, {37643, 53374}

X(54259) = perspector of circumconic {{A, B, C, X(98), X(253)}}
X(54259) = X(i)-complementary conjugate of X(j) for these {i, j}: {47735, 21253}
X(54259) = (polar of X(2) wrt inconic with perspector X(4)) ∩ (polar of X(4) wrt inconic with perspector X(2))
X(54259) = barycentric product X(i)*X(j) for these (i, j): {523, 5921}
X(54259) = barycentric quotient X(i)/X(j) for these (i, j): {5921, 99}
X(54259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14333, 14334, 14341}

X(54260) = X(3)X(525)∩X(523)X(4885)

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

See Ivan Pavlov, euclid 5918.

X(54260) lies on these lines: {3, 525}, {523, 4885}, {684, 40494}, {1499, 6334}, {3091, 44705}, {3265, 8057}, {14341, 16230}, {16051, 53383}

X(54260) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(3167), X(21910)}}, {{A, B, C, X(5921), X(35912)}}, {{A, B, C, X(14638), X(53173)}}, {{A, B, C, X(16096), X(34156)}}
X(54260) = reflection of X(i) in X(j) for these {i,j}: {16230, 14341}
X(54260) = perspector of circumconic {{A, B, C, X(287), X(2996)}}
X(54260) = X(i)-Dao conjugate of X(j) for these {i, j}: {6776, 35278}
X(54260) = X(i)-complementary conjugate of X(j) for these {i, j}: {4575, 7710}, {42287, 21253}
X(54260) = (polar of X(2) wrt inconic with perspector X(69)) ∩ (polar of X(69) wrt inconic with perspector X(2))
X(54260) = barycentric product X(i)*X(j) for these (i, j): {525, 5921}
X(54260) = barycentric quotient X(i)/X(j) for these (i, j): {5921, 648}

X(54261) = X(1)X(514)∩X(522)X(676)

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

See Ivan Pavlov, euclid 5918.

X(54261) lies on these lines: {1, 514}, {522, 676}, {650, 28161}, {2254, 4962}, {2496, 6084}, {3239, 47695}, {3617, 4546}, {3667, 3676}, {3960, 30235}, {4765, 47798}, {4778, 39540}, {6362, 52596}, {6608, 21189}, {8058, 21179}, {9780, 44448}, {11019, 24720}, {14350, 48547}, {21104, 28225}, {30723, 34958}, {44551, 50356}, {47800, 48187}, {47801, 53558}, {48239, 48268}

X(54261) = midpoint of X(i) in X(j) for these {i,j}: {3239, 47695}, {3676, 53523}
X(54261) = reflection of X(i) in X(j) for these {i,j}: {30723, 34958}, {7658, 676}
X(54261) = perspector of circumconic {{A, B, C, X(673), X(7319)}}
X(54261) = X(i)-complementary conjugate of X(j) for these {i, j}: {42315, 116}
X(54261) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {26716, 8055}
X(54261) = (polar of X(2) wrt inconic with perspector X(7)) ∩ (polar of X(7) wrt inconic with perspector X(2))
X(54261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {522, 676, 7658}, {3676, 53523, 3667}

X(54262) = X(2)X(3288)∩X(141)X(523)

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

See Ivan Pavlov, euclid 5918.

X(54262) lies on these lines: {2, 3288}, {125, 6071}, {126, 33330}, {141, 523}, {193, 2451}, {297, 525}, {512, 625}, {669, 30217}, {804, 50550}, {1499, 9148}, {1637, 50547}, {2435, 43710}, {2492, 9030}, {3005, 3800}, {3049, 3618}, {3050, 47355}, {5027, 44451}, {6333, 12077}, {7703, 32120}, {9210, 31296}, {11186, 53365}, {14317, 14318}, {17921, 21300}, {23297, 30491}, {23878, 45336}, {30209, 47206}, {31072, 53331}, {32455, 39520}, {37648, 45327}

X(54262) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(141), X(43715)}}, {{A, B, C, X(297), X(11328)}}, {{A, B, C, X(882), X(2501)}}, {{A, B, C, X(18906), X(44146)}}
X(54262) = midpoint of X(i) in X(j) for these {i,j}: {11186, 53365}, {6333, 12077}, {850, 3569}
X(54262) = reflection of X(i) in X(j) for these {i,j}: {14316, 2501}, {24284, 30476}, {45335, 2}, {5027, 44451}
X(54262) = complement of X(3288)
X(54262) = perspector of circumconic {{A, B, C, X(264), X(1916)}}
X(54262) = center of circumconic {{A, B, C, X(39680), X(45907)}}
X(54262) = X(i)-isoconjugate-of-X(j) for these {i, j}: {163, 19222}, {662, 47643}
X(54262) = X(i)-Dao conjugate of X(j) for these {i, j}: {115, 19222}, {1084, 47643}, {45907, 25423}
X(54262) = X(i)-complementary conjugate of X(j) for these {i, j}: {75, 46656}, {262, 8287}, {263, 16592}, {327, 21253}, {662, 15819}, {799, 52658}, {1755, 39009}, {2186, 115}, {3402, 1084}, {6037, 16609}, {26714, 37}, {36132, 230}, {39681, 19563}, {42313, 34846}, {43718, 16573}
X(54262) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {25424, 4329}, {51844, 13219}
X(54262) = (polar of X(2) wrt inconic with perspector X(76)) ∩ (polar of X(76) wrt inconic with perspector X(2))
X(54262) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {512, 47643}, {523, 19222}, {6234, 805}, {11328, 110}, {19591, 662}, {45907, 6}, {51997, 26714}
X(54262) = barycentric product X(i)*X(j) for these (i, j): {1577, 19591}, {11328, 850}, {14295, 6234}, {18906, 523}, {45907, 76}
X(54262) = barycentric quotient X(i)/X(j) for these (i, j): {512, 47643}, {523, 19222}, {6234, 805}, {11328, 110}, {18906, 99}, {19591, 662}, {45907, 6}, {51997, 26714}
X(54262) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {512, 30476, 24284}, {525, 2501, 14316}, {850, 3569, 525}

X(54263) = X(316)X(512)∩X(523)X(3589)

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

See Ivan Pavlov, euclid 5918.

X(54263) lies on these lines: {316, 512}, {523, 3589}, {647, 7950}, {826, 4142}, {2528, 50542}, {3288, 20965}, {3906, 14420}, {7927, 24284}, {23597, 23878}, {45692, 50549}

X(54263) = midpoint of X(i) in X(j) for these {i,j}: {2528, 50542}
X(54263) = perspector of circumconic {{A, B, C, X(308), X(1031)}}
X(54263) = X(i)-complementary conjugate of X(j) for these {i, j}: {43357, 21249}
X(54263) = (polar of X(2) wrt inconic with perspector X(83)) ∩ (polar of X(83) wrt inconic with perspector X(2))
X(54263) = barycentric product X(i)*X(j) for these (i, j): {24273, 523}
X(54263) = barycentric quotient X(i)/X(j) for these (i, j): {24273, 99}

X(54264) = X(142)X(522)∩X(514)X(661)

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

See Ivan Pavlov, euclid 5918.

X(54264) lies on these lines: {2, 45755}, {142, 522}, {144, 46402}, {514, 661}, {594, 45659}, {657, 18230}, {885, 47832}, {1734, 6608}, {3261, 17233}, {3309, 4369}, {3676, 24290}, {3700, 43042}, {3887, 47779}, {3900, 4885}, {4171, 24002}, {4431, 20907}, {4928, 14077}, {4932, 42325}, {5316, 14330}, {6362, 46396}, {7155, 35355}, {8713, 50352}, {17239, 20316}, {17241, 20954}, {22019, 22042}, {22229, 43051}, {26985, 53357}, {28161, 40474}, {29571, 33570}, {31019, 47790}, {40551, 47831}

X(54264) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(514), X(9443)}}, {{A, B, C, X(3835), X(35355)}}
X(54264) = midpoint of X(i) in X(j) for these {i,j}: {3700, 43042}, {4171, 24002}
X(54264) = reflection of X(i) in X(j) for these {i,j}: {21195, 46399}
X(54264) = complement of X(45755)
X(54264) = X(i)-complementary conjugate of X(j) for these {i, j}: {651, 3789}, {1002, 26932}, {2279, 1146}, {8693, 9}, {27475, 124}, {32041, 1329}, {36138, 40869}, {37138, 3452}, {40779, 5514}, {42290, 11}, {42302, 34589}, {51443, 4858}, {51563, 21246}, {52635, 39012}
X(54264) = (polar of X(2) wrt inconic with perspector X(85)) ∩ (polar of X(85) wrt inconic with perspector X(2))
X(54264) = X(9443)-reciprocal conjugate of X(1)
X(54264) = barycentric product X(i)*X(j) for these (i, j): {75, 9443}
X(54264) = barycentric quotient X(i)/X(j) for these (i, j): {9443, 1}
X(54264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {522, 46399, 21195}

X(54265) = X(2)X(4824)∩X(8)X(2533)

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

See Ivan Pavlov, euclid 5918.

X(54265) lies on these lines: {2, 4824}, {8, 2533}, {10, 45332}, {320, 350}, {514, 1125}, {523, 2487}, {649, 48120}, {650, 4802}, {659, 47672}, {661, 4963}, {662, 36239}, {1019, 48393}, {1491, 4379}, {1698, 4705}, {2254, 48253}, {3244, 29298}, {3716, 4977}, {3720, 4724}, {3741, 24720}, {4024, 50342}, {4036, 18154}, {4122, 4789}, {4367, 50457}, {4448, 47969}, {4500, 29078}, {4750, 4777}, {4761, 48291}, {4762, 4782}, {4763, 28179}, {4784, 4804}, {4800, 48021}, {4806, 28840}, {4810, 4979}, {4841, 47799}, {4885, 48030}, {4893, 47928}, {4926, 7659}, {4927, 47989}, {4928, 47992}, {4932, 29328}, {4948, 45691}, {4960, 4983}, {6367, 21192}, {6545, 47968}, {14419, 47683}, {17292, 35352}, {20317, 47922}, {21116, 28195}, {21183, 48007}, {23770, 48276}, {23864, 48382}, {24666, 46385}, {24924, 47827}, {25128, 47843}, {25380, 48233}, {25666, 48002}, {26248, 47797}, {26985, 47945}, {27527, 48209}, {28147, 31286}, {28151, 46915}, {28165, 48254}, {28175, 48000}, {28191, 48214}, {28199, 47773}, {28213, 48001}, {29144, 47123}, {29238, 50515}, {29274, 50517}, {29362, 48399}, {29833, 47691}, {30795, 47810}, {30835, 47909}, {31209, 48176}, {31287, 48194}, {45320, 48027}, {45746, 48227}, {47659, 48241}, {47660, 48326}, {47666, 47822}, {47675, 47804}, {47698, 48185}, {47699, 48177}, {47703, 50340}, {47760, 47953}, {47777, 48608}, {47779, 48010}, {47788, 48047}, {47789, 48062}, {47812, 48153}, {47821, 47946}, {47823, 47975}, {47824, 50341}, {47826, 47910}, {47831, 47996}, {47832, 48024}, {47837, 48407}, {47839, 50449}, {47872, 47918}, {47875, 47959}, {47881, 48088}, {47889, 48131}, {47891, 50348}, {47917, 48162}, {47926, 48226}, {47943, 48414}, {47944, 50522}, {47964, 48197}, {47986, 48547}, {47993, 48183}, {48020, 48167}, {48023, 48184}, {48028, 48202}, {48032, 48251}, {48119, 48578}, {48144, 48392}, {48349, 49283}, {48409, 48569}, {48579, 50359}, {49286, 49296}, {49293, 49295}

X(54265) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(513), X(9279)}}, {{A, B, C, X(523), X(50451)}}, {{A, B, C, X(693), X(18014)}}, {{A, B, C, X(24342), X(30939)}}, {{A, B, C, X(30941), X(40750)}}
X(54265) = midpoint of X(i) in X(j) for these {i,j}: {1019, 48393}, {1491, 48142}, {2533, 17166}, {21146, 47694}, {23770, 48276}, {3716, 49291}, {4010, 7192}, {4024, 50342}, {4367, 50457}, {4369, 49292}, {4724, 48143}, {4761, 48291}, {4782, 48127}, {4784, 4804}, {4810, 4979}, {4932, 48394}, {4960, 4983}, {47660, 48326}, {47703, 50340}, {47704, 48103}, {47944, 50522}, {48024, 48141}, {48029, 48133}, {48119, 50358}, {48144, 48392}, {48153, 50328}, {48349, 49283}, {49286, 49296}, {49293, 49295}, {649, 48120}, {650, 48134}, {659, 47672}, {7662, 43067}
X(54265) = reflection of X(i) in X(j) for these {i,j}: {4948, 45691}, {45676, 2}, {47922, 20317}, {48002, 25666}, {48030, 4885}, {9508, 4369}
X(54265) = complement of X(4824)
X(54265) = perspector of circumconic {{A, B, C, X(79), X(274)}}
X(54265) = X(i)-isoconjugate-of-X(j) for these {i, j}: {101, 40776}
X(54265) = X(i)-Dao conjugate of X(j) for these {i, j}: {1015, 40776}
X(54265) = X(i)-complementary conjugate of X(j) for these {i, j}: {163, 31336}, {25426, 8287}, {27483, 21253}, {28841, 1211}, {30571, 125}
X(54265) = (polar of X(2) wrt inconic with perspector X(86)) ∩ (polar of X(86) wrt inconic with perspector X(2))
X(54265) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {513, 40776}, {9279, 37}, {18791, 37138}, {40750, 100}
X(54265) = barycentric product X(i)*X(j) for these (i, j): {274, 9279}, {24342, 514}, {40750, 693}
X(54265) = barycentric quotient X(i)/X(j) for these (i, j): {513, 40776}, {9279, 37}, {18791, 37138}, {24342, 190}, {40750, 100}
X(54265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {523, 4369, 9508}, {3716, 49291, 4977}, {4369, 49292, 523}, {4379, 48142, 1491}, {4782, 48127, 4762}, {4804, 31148, 4784}, {4932, 48394, 29328}, {7192, 47834, 4010}, {21146, 47694, 513}, {24924, 47934, 47827}, {47672, 47813, 659}, {47694, 47780, 21146}, {47704, 48103, 4802}, {47812, 48153, 50328}, {47832, 48141, 48024}, {48002, 48206, 25666}, {48029, 48133, 28195}, {48119, 48578, 50358}, {48133, 48220, 48029}, {48143, 48234, 4724}

X(54266) = X(522)X(650)∩X(1212)X(3900)

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

See Ivan Pavlov, euclid 5918.

X(54266) lies on circumconic {{A, B, C, X(522), X(9443)}} and on these lines: {522, 650}, {1212, 3900}, {2254, 4171}, {33570, 50356}

X(54266) = reflection of X(i) in X(j) for these {i,j}: {650, 52614}
X(54266) = perspector of circumconic {{A, B, C, X(8), X(14943)}}
X(54266) = (polar of X(2) wrt inconic with perspector X(9)) ∩ (polar of X(9) wrt inconic with perspector X(2))
X(54266) = X(9443)-reciprocal conjugate of X(7)
X(54266) = barycentric product X(i)*X(j) for these (i, j): {8, 9443}
X(54266) = barycentric quotient X(i)/X(j) for these (i, j): {9443, 7}
X(54266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {522, 52614, 650}

X(54267) = X(230)X(231)∩X(694)X(804)

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

See Ivan Pavlov, euclid 5918.

X(54267) lies on these lines: {230, 231}, {684, 7736}, {694, 804}, {2549, 2797}, {2799, 14316}, {2881, 51431}, {3815, 45319}, {5304, 53345}, {6330, 16081}, {9517, 18907}, {21843, 44818}, {23878, 45336}, {37637, 45682}, {46777, 48540}

X(54267) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(232), X(694)}}, {{A, B, C, X(468), X(5999)}}, {{A, B, C, X(2491), X(34212)}}, {{A, B, C, X(6103), X(47737)}}, {{A, B, C, X(16081), X(16318)}}, {{A, B, C, X(38947), X(47110)}}
X(54267) = midpoint of X(i) in X(j) for these {i,j}: {2395, 3569}
X(54267) = complement of isogonal conjugate of X(32716)
X(54267) = complement of isotomic conjugate of X(6037)
X(54267) = perspector of circumconic {{A, B, C, X(4), X(5999)}}
X(54267) = X(i)-isoconjugate-of-X(j) for these {i, j}: {662, 43702}
X(54267) = X(i)-Dao conjugate of X(j) for these {i, j}: {1084, 43702}
X(54267) = X(i)-complementary conjugate of X(j) for these {i, j}: {560, 39009}, {1910, 46656}, {2186, 36471}, {3402, 35088}, {6037, 2887}, {32716, 10}, {36084, 52658}, {36132, 141}, {53196, 21235}
X(54267) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {512, 43702}, {47737, 2966}
X(54267) = (polar of X(2) wrt inconic with perspector X(98)) ∩ (polar of X(98) wrt inconic with perspector X(2))
X(54267) = barycentric product X(i)*X(j) for these (i, j): {523, 5999}, {2799, 47737}
X(54267) = barycentric quotient X(i)/X(j) for these (i, j): {512, 43702}, {5999, 99}, {47737, 2966}

X(54268) = X(184)X(647)∩X(512)X(1570)

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

See Ivan Pavlov, euclid 5918.

X(54268) lies on circumconics {{A, B, C, X(184), X(5921)}} and on these lines: {184, 647}, {512, 1570}, {520, 11589}, {686, 8644}, {1899, 31174}, {2065, 5622}, {6776, 23878}, {23291, 30476}

X(54268) = perspector of circumconic {{A, B, C, X(248), X(1073)}}
X(54268) = (polar of X(3) wrt inconic with perspector X(6)) ∩ (polar of X(6) wrt inconic with perspector X(3))
X(54268) = X(5921)-reciprocal conjugate of X(6331)
X(54268) = barycentric product X(i)*X(j) for these (i, j): {5921, 647}
X(54268) = barycentric quotient X(i)/X(j) for these (i, j): {5921, 6331}

X(54269) = X(30)X(511)∩X(51)X(647)

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

See Ivan Pavlov, euclid 5918.

X(54269) lies on these lines: {30, 511}, {51, 647}, {850, 2979}, {2451, 39201}, {3060, 36900}, {3819, 30476}, {3917, 31174}, {5943, 44560}, {9420, 45907}, {12099, 22264}, {42293, 52590}, {42331, 44173}

X(54269) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(51), X(32428)}}, {{A, B, C, X(520), X(53175)}}, {{A, B, C, X(524), X(40805)}}, {{A, B, C, X(3504), X(3564)}}
X(54269) = perspector of circumconic {{A, B, C, X(2), X(1987)}}
X(54269) = X(i)-isoconjugate-of-X(j) for these {i, j}: {162, 43711}, {662, 40815}
X(54269) = X(i)-Dao conjugate of X(j) for these {i, j}: {125, 43711}, {1084, 40815}
X(54269) = X(i)-complementary conjugate of X(j) for these {i, j}: {40815, 8287}, {43711, 34846}
X(54269) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {40815, 21221}
X(54269) = (polar of X(4) wrt inconic with perspector X(6)) ∩ (polar of X(6) wrt inconic with perspector X(4))
X(54269) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {512, 40815}, {647, 43711}, {47739, 6528}
X(54269) = barycentric product X(i)*X(j) for these (i, j): {40805, 523}, {47739, 520}
X(54269) = barycentric quotient X(i)/X(j) for these (i, j): {512, 40815}, {647, 43711}, {40805, 99}, {47739, 6528}
X(54269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {523, 39469, 512}

X(54270) = X(11)X(1146)∩X(1647)X(4120)

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

See Ivan Pavlov, euclid 5918.

X(54270) lies on these lines: {11, 1146}, {1018, 11814}, {1647, 4120}, {4928, 24228}, {21013, 24003}, {46101, 52338}

X(54270) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(11), X(24131)}}, {{A, B, C, X(1647), X(4124)}}, {{A, B, C, X(2170), X(4919)}}
X(54270) = X(i)-isoconjugate-of-X(j) for these {i, j}: {651, 53682}
X(54270) = X(i)-Dao conjugate of X(j) for these {i, j}: {24188, 30725}, {38991, 53682}
X(54270) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4582, 21132}, {52338, 4530}
X(54270) = (polar of X(8) wrt inconic with perspector X(11)) ∩ (polar of X(11) wrt inconic with perspector X(8))
X(54270) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {663, 53682}, {4530, 6630}, {4919, 5376}, {14122, 7045}, {24131, 7}, {52338, 42555}
X(54270) = barycentric product X(i)*X(j) for these (i, j): {1639, 21204}, {4440, 4530}, {14122, 24026}, {24131, 8}, {52338, 6631}
X(54270) = barycentric quotient X(i)/X(j) for these (i, j): {663, 53682}, {4530, 6630}, {4919, 5376}, {14122, 7045}, {24131, 7}, {52338, 42555}

X(54271) = X(30)X(511)∩X(210)X(650)

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

See Ivan Pavlov, euclid 5918.

X(54271) lies on these lines: {30, 511}, {210, 650}, {354, 45320}, {649, 4477}, {663, 3287}, {667, 3508}, {693, 3873}, {875, 43931}, {905, 24462}, {2488, 3239}, {3158, 16557}, {3669, 53553}, {3681, 31150}, {3709, 52594}, {3740, 44567}, {3742, 4885}, {3794, 39924}, {3868, 47721}, {3877, 47729}, {3892, 48295}, {3894, 47724}, {3898, 48285}, {4024, 50519}, {4134, 48284}, {4164, 48330}, {4374, 52621}, {4430, 47869}, {4502, 50508}, {4512, 8641}, {4524, 4765}, {4661, 17494}, {4820, 50518}, {5902, 50764}, {6050, 17990}, {10030, 21302}, {15280, 24386}, {17072, 21195}, {18081, 48109}, {21003, 48387}, {21183, 30704}, {21260, 40474}, {21388, 21789}, {49285, 53550}

X(54271) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(210), X(740)}}, {{A, B, C, X(516), X(4334)}}, {{A, B, C, X(517), X(21010)}}, {{A, B, C, X(518), X(24349)}}, {{A, B, C, X(522), X(24720)}}, {{A, B, C, X(527), X(17754)}}, {{A, B, C, X(650), X(812)}}, {{A, B, C, X(663), X(4083)}}, {{A, B, C, X(726), X(33676)}}, {{A, B, C, X(732), X(39936)}}, {{A, B, C, X(2344), X(19584)}}, {{A, B, C, X(2784), X(8926)}}, {{A, B, C, X(3287), X(25576)}}, {{A, B, C, X(3709), X(4139)}}, {{A, B, C, X(3794), X(52211)}}, {{A, B, C, X(3808), X(43931)}}, {{A, B, C, X(3810), X(52621)}}, {{A, B, C, X(4162), X(4964)}}, {{A, B, C, X(9025), X(53219)}}, {{A, B, C, X(9443), X(53227)}}, {{A, B, C, X(20917), X(46180)}}
X(54271) = perspector of circumconic {{A, B, C, X(2), X(2319)}}
X(54271) = X(i)-isoconjugate-of-X(j) for these {i, j}: {109, 41527}, {934, 7220}
X(54271) = X(i)-Dao conjugate of X(j) for these {i, j}: {11, 41527}, {14714, 7220}
X(54271) = X(i)-Ceva conjugate of X(j) for these {i, j}: {100, 19584}
X(54271) = X(i)-complementary conjugate of X(j) for these {i, j}: {109, 19584}, {7220, 5514}, {41527, 124}
X(54271) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {41527, 33650}
X(54271) = (polar of X(8) wrt inconic with perspector X(9)) ∩ (polar of X(9) wrt inconic with perspector X(8))
X(54271) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {650, 41527}, {657, 7220}, {4334, 658}, {17754, 664}, {20917, 4572}, {21010, 651}, {22163, 6516}, {24349, 4554}, {24720, 85}
X(54271) = barycentric product X(i)*X(j) for these (i, j): {3239, 4334}, {17754, 522}, {20917, 663}, {21010, 4391}, {21101, 3737}, {22163, 44426}, {24349, 650}, {24720, 9}
X(54271) = barycentric quotient X(i)/X(j) for these (i, j): {650, 41527}, {657, 7220}, {4334, 658}, {17754, 664}, {20917, 4572}, {21010, 651}, {22163, 6516}, {24349, 4554}, {24720, 85}
X(54271) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {513, 29226, 3808}, {518, 4762, 9443}, {522, 926, 3900}

X(54272) = X(30)X(511)∩X(51)X(31174)

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

See Ivan Pavlov, euclid 5918.

X(54272) lies on these lines: {30, 511}, {51, 31174}, {647, 3917}, {850, 3060}, {2524, 3049}, {2979, 36900}, {3819, 44560}, {5943, 30476}, {16695, 23145}

X(54272) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(3), X(32515)}}, {{A, B, C, X(30), X(11328)}}, {{A, B, C, X(511), X(18906)}}, {{A, B, C, X(523), X(45907)}}, {{A, B, C, X(647), X(804)}}, {{A, B, C, X(732), X(3917)}}, {{A, B, C, X(1503), X(1988)}}, {{A, B, C, X(2782), X(6234)}}, {{A, B, C, X(3049), X(3221)}}, {{A, B, C, X(8680), X(19591)}}, {{A, B, C, X(19602), X(43722)}}
X(54272) = perspector of circumconic {{A, B, C, X(2), X(3504)}}
X(54272) = X(i)-isoconjugate-of-X(j) for these {i, j}: {162, 19222}, {811, 47643}
X(54272) = X(i)-Dao conjugate of X(j) for these {i, j}: {125, 19222}, {17423, 47643}
X(54272) = X(i)-Ceva conjugate of X(j) for these {i, j}: {110, 19602}, {25424, 3}
X(54272) = X(i)-complementary conjugate of X(j) for these {i, j}: {162, 19602}, {19222, 34846}, {47643, 16573}
X(54272) = (polar of X(3) wrt inconic with perspector X(69)) ∩ (polar of X(69) wrt inconic with perspector X(3))
X(54272) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {647, 19222}, {3049, 47643}, {11328, 648}, {18906, 6331}, {19591, 811}, {45907, 4}
X(54272) = barycentric product X(i)*X(j) for these (i, j): {11328, 525}, {18906, 647}, {19591, 656}, {24284, 6234}, {45907, 69}
X(54272) = barycentric quotient X(i)/X(j) for these (i, j): {647, 19222}, {3049, 47643}, {11328, 648}, {18906, 6331}, {19591, 811}, {45907, 4}
X(54272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {525, 39469, 520}

X(54273) = X(460)X(512)∩X(882)X(1843)

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

See Ivan Pavlov, euclid 5918.

X(54273) lies on these lines: {460, 512}, {882, 1843}, {5113, 6753}, {17994, 50549}

X(54273) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(460), X(11328)}}, {{A, B, C, X(882), X(2501)}}, {{A, B, C, X(3221), X(5027)}}, {{A, B, C, X(5140), X(18906)}}, {{A, B, C, X(27375), X(51997)}}
X(54273) = polar conjugate of isotomic conjugate of X(45907)
X(54273) = perspector of circumconic {{A, B, C, X(393), X(17980)}}
X(54273) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4592, 19222}
X(54273) = X(i)-Dao conjugate of X(j) for these {i, j}: {5139, 19222}
X(54273) = (polar of X(4) wrt inconic with perspector X(25)) ∩ (polar of X(25) wrt inconic with perspector X(4))
X(54273) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {2489, 19222}, {11328, 4563}, {18906, 52608}, {45907, 69}
X(54273) = barycentric product X(i)*X(j) for these (i, j): {4, 45907}, {11328, 2501}, {18906, 2489}
X(54273) = barycentric quotient X(i)/X(j) for these (i, j): {2489, 19222}, {11328, 4563}, {18906, 52608}, {45907, 69}

X(54274) = X(6)X(512)∩X(184)X(8644)

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

See Ivan Pavlov, euclid 5918.

X(54274) lies on these lines: {6, 512}, {110, 39527}, {184, 8644}, {351, 39689}, {520, 3167}, {523, 8584}, {524, 11183}, {526, 6593}, {542, 18309}, {575, 9175}, {597, 11182}, {599, 45693}, {688, 11205}, {690, 15303}, {804, 8787}, {888, 9135}, {1499, 50979}, {1649, 8030}, {1992, 5652}, {2086, 38988}, {3049, 3051}, {5027, 9023}, {5050, 21733}, {5467, 44814}, {5653, 52699}, {9137, 11422}, {21906, 35507}, {34290, 45690}, {39469, 47405}, {45692, 47352}, {45914, 52721}

X(54274) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(6), X(8030)}}, {{A, B, C, X(110), X(9171)}}, {{A, B, C, X(187), X(17964)}}, {{A, B, C, X(351), X(1649)}}, {{A, B, C, X(512), X(33915)}}, {{A, B, C, X(2482), X(14609)}}, {{A, B, C, X(5095), X(51980)}}, {{A, B, C, X(8430), X(23992)}}, {{A, B, C, X(14443), X(44814)}}, {{A, B, C, X(14567), X(52197)}}, {{A, B, C, X(17414), X(18311)}}, {{A, B, C, X(22260), X(46049)}}, {{A, B, C, X(36792), X(36821)}}, {{A, B, C, X(45143), X(51927)}}
X(54274) = midpoint of X(i) in X(j) for these {i,j}: {1992, 5652}
X(54274) = reflection of X(i) in X(j) for these {i,j}: {11182, 597}, {22260, 9171}, {34290, 45690}, {599, 45693}, {9171, 6}, {9175, 575}, {9208, 9188}
X(54274) = perspector of circumconic {{A, B, C, X(111), X(187)}}
X(54274) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 34574}, {671, 36085}, {691, 46277}, {799, 10630}, {811, 15398}, {892, 897}, {923, 53080}, {1577, 34539}, {4602, 41936}, {18023, 36142}, {20944, 39413}, {23894, 52940}
X(54274) = X(i)-Dao conjugate of X(j) for these {i, j}: {206, 34574}, {524, 670}, {690, 850}, {1648, 76}, {1649, 52632}, {2482, 53080}, {6593, 892}, {17423, 15398}, {21905, 5466}, {23992, 18023}, {38988, 671}, {38996, 10630}, {48317, 46111}
X(54274) = X(i)-Ceva conjugate of X(j) for these {i, j}: {6, 21906}, {110, 187}, {512, 351}, {5095, 23992}
X(54274) = (polar of X(6) wrt inconic with perspector X(187)) ∩ (polar of X(187) wrt inconic with perspector X(6))
X(54274) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {32, 34574}, {187, 892}, {351, 671}, {524, 53080}, {669, 10630}, {690, 18023}, {922, 36085}, {1576, 34539}, {1648, 52632}, {1649, 76}, {2482, 670}, {2642, 46277}, {3049, 15398}, {5095, 6331}, {5467, 52940}, {9426, 41936}, {14273, 46111}, {14443, 338}, {14444, 35522}, {14567, 691}, {21906, 5466}, {23992, 850}, {24038, 4602}, {33915, 3266}, {35507, 17414}, {36792, 4609}, {42081, 799}, {46049, 52628}, {52068, 1978}, {52629, 1502}
X(54274) = barycentric product X(i)*X(j) for these (i, j): {32, 52629}, {110, 23992}, {111, 33915}, {187, 690}, {351, 524}, {1366, 3709}, {1648, 5467}, {1649, 6}, {2482, 512}, {2642, 896}, {3049, 34336}, {5095, 647}, {7067, 7180}, {8030, 9178}, {11183, 18872}, {14273, 3292}, {14417, 44102}, {14419, 21839}, {14443, 249}, {14444, 691}, {14567, 35522}, {16733, 50487}, {17414, 20380}, {20382, 9145}, {21905, 34161}, {21906, 5468}, {24038, 798}, {30454, 6137}, {30455, 6138}, {33921, 48450}, {36792, 669}, {39689, 523}, {42081, 661}, {52038, 9155}, {52068, 649}
X(54274) = barycentric quotient X(i)/X(j) for these (i, j): {32, 34574}, {187, 892}, {351, 671}, {524, 53080}, {669, 10630}, {690, 18023}, {922, 36085}, {1576, 34539}, {1648, 52632}, {1649, 76}, {2482, 670}, {2642, 46277}, {3049, 15398}, {5095, 6331}, {5467, 52940}, {9426, 41936}, {14273, 46111}, {14443, 338}, {14444, 35522}, {14567, 691}, {21906, 5466}, {23992, 850}, {24038, 4602}, {33915, 3266}, {35507, 17414}, {36792, 4609}, {39689, 99}, {42081, 799}, {46049, 52628}, {52068, 1978}, {52629, 1502}
X(54274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 512, 9171}, {512, 9171, 22260}, {526, 9188, 9208}, {9188, 9208, 14428}

X(54275) = X(6)X(4782)∩X(213)X(667)

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

See Ivan Pavlov, euclid 5918.

X(54275) lies on these lines: {6, 4782}, {213, 667}, {649, 854}, {659, 20980}, {798, 50514}, {1919, 8640}, {2484, 50510}, {4501, 50343}, {4784, 16782}, {16969, 48330}, {20979, 23569}, {21389, 50516}, {24512, 24719}

X(54275) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(875), X(43924)}}, {{A, B, C, X(17754), X(51864)}}
X(54275) = perspector of circumconic {{A, B, C, X(56), X(1911)}}
X(54275) = X(i)-isoconjugate-of-X(j) for these {i, j}: {190, 41527}, {3807, 47647}, {4554, 7220}
X(54275) = (polar of X(6) wrt inconic with perspector X(31)) ∩ (polar of X(31) wrt inconic with perspector X(6))
X(54275) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {667, 41527}, {4334, 4572}, {17754, 1978}, {19586, 4505}, {19587, 3807}, {21010, 668}, {24349, 6386}, {24720, 561}
X(54275) = barycentric product X(i)*X(j) for these (i, j): {1919, 20917}, {4334, 663}, {17754, 649}, {19587, 4817}, {21010, 513}, {22163, 6591}, {24349, 667}, {24720, 31}, {40758, 45902}
X(54275) = barycentric quotient X(i)/X(j) for these (i, j): {667, 41527}, {4334, 4572}, {17754, 1978}, {19586, 4505}, {19587, 3807}, {21010, 668}, {24349, 6386}, {24720, 561}

X(54276) = X(6)X(25423)∩X(669)X(881)

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

See Ivan Pavlov, euclid 5918.

X(54276) lies on these lines: {6, 25423}, {512, 1692}, {669, 881}, {804, 2451}, {1613, 45317}, {9426, 9429}, {20965, 31176}, {21001, 44451}

X(54276) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(32), X(33874)}}, {{A, B, C, X(881), X(2489)}}, {{A, B, C, X(11328), X(46522)}}, {{A, B, C, X(18906), X(34238)}}
X(54276) = reflection of X(i) in X(j) for these {i,j}: {9491, 9426}
X(54276) = perspector of circumconic {{A, B, C, X(25), X(9468)}}
X(54276) = X(i)-isoconjugate-of-X(j) for these {i, j}: {799, 19222}, {4602, 47643}
X(54276) = X(i)-Dao conjugate of X(j) for these {i, j}: {38996, 19222}
X(54276) = X(i)-Ceva conjugate of X(j) for these {i, j}: {26714, 51997}
X(54276) = (polar of X(6) wrt inconic with perspector X(32)) ∩ (polar of X(32) wrt inconic with perspector X(6))
X(54276) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {669, 19222}, {9426, 47643}, {11328, 670}, {18906, 4609}, {19591, 4602}, {45907, 76}
X(54276) = barycentric product X(i)*X(j) for these (i, j): {3288, 51997}, {5027, 6234}, {11328, 512}, {18906, 669}, {19591, 798}, {45907, 6}
X(54276) = barycentric quotient X(i)/X(j) for these (i, j): {669, 19222}, {9426, 47643}, {11328, 670}, {18906, 4609}, {19591, 4602}, {45907, 76}
X(54276) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9426, 9429, 9491}

X(54277) = X(44)X(513)∩X(1019)X(6626)

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

See Ivan Pavlov, euclid 5918.

X(54277) lies on these lines: {44, 513}, {512, 18001}, {1019, 6626}, {4079, 7234}, {4369, 18160}, {4826, 8663}, {17731, 29487}, {25820, 25822}, {26983, 27194}

X(54277) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(512), X(9508)}}, {{A, B, C, X(513), X(9279)}}, {{A, B, C, X(2234), X(24342)}}, {{A, B, C, X(2238), X(40750)}}, {{A, B, C, X(6626), X(46195)}}
X(54277) = perspector of circumconic {{A, B, C, X(1), X(2054)}}
X(54277) = X(i)-Dao conjugate of X(j) for these {i, j}: {38986, 40776}
X(54277) = X(i)-Zayin conjugate of X(j) for these {i, j}: {1019, 40776}
X(54277) = (polar of X(6) wrt inconic with perspector X(42)) ∩ (polar of X(42) wrt inconic with perspector X(6))
X(54277) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {798, 40776}, {9279, 75}, {24342, 670}, {40750, 799}
X(54277) = barycentric product X(i)*X(j) for these (i, j): {1, 9279}, {24342, 512}, {40750, 661}
X(54277) = barycentric quotient X(i)/X(j) for these (i, j): {798, 40776}, {9279, 75}, {24342, 670}, {40750, 799}
X(54277) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 46390, 798}

X(54278) = X(9)X(24720)∩X(44)X(513)

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

See Ivan Pavlov, euclid 5918.

X(54278) lies on these lines: {9, 24720}, {44, 513}, {71, 4824}, {514, 3177}, {663, 1475}, {3207, 44408}, {3709, 53539}, {3835, 10025}, {4253, 4794}, {4379, 30988}, {8641, 20980}, {8642, 20981}, {20974, 42771}, {21390, 48073}, {31605, 49296}, {45755, 50356}, {48043, 53395}

X(54278) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(513), X(9443)}}, {{A, B, C, X(2053), X(2348)}}
X(54278) = reflection of X(i) in X(j) for these {i,j}: {649, 46388}
X(54278) = (polar of X(6) wrt inconic with perspector X(57)) ∩ (polar of X(57) wrt inconic with perspector X(6))
X(54278) = X(9443)-reciprocal conjugate of X(75)
X(54278) = barycentric product X(i)*X(j) for these (i, j): {1, 9443}
X(54278) = barycentric quotient X(i)/X(j) for these (i, j): {9443, 75}
X(54278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {513, 46388, 649}

X(54279) = X(36)X(238)∩X(55)X(7234)

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

See Ivan Pavlov, euclid 5918.

X(54279) lies on these lines: {36, 238}, {55, 7234}, {512, 20981}, {649, 2308}, {824, 4378}, {2786, 4367}, {3063, 4834}, {4038, 18200}, {4775, 50459}, {4785, 33682}, {8639, 23467}, {8646, 50454}, {17212, 50451}, {23472, 50509}, {29487, 37604}

X(54279) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(513), X(9279)}}, {{A, B, C, X(18792), X(24342)}}, {{A, B, C, X(40750), X(52897)}}
X(54279) = reflection of X(i) in X(j) for these {i,j}: {4775, 50459}, {50451, 52601}
X(54279) = perspector of circumconic {{A, B, C, X(81), X(2248)}}
X(54279) = (polar of X(6) wrt inconic with perspector X(58)) ∩ (polar of X(58) wrt inconic with perspector X(6))
X(54279) = X(i)-reciprocal conjugate of X(j) for these {i, j}: {667, 40776}, {9279, 321}, {24342, 1978}, {40750, 668}
X(54279) = barycentric product X(i)*X(j) for these (i, j): {81, 9279}, {18791, 4724}, {24342, 649}, {40750, 513}
X(54279) = barycentric quotient X(i)/X(j) for these (i, j): {667, 40776}, {9279, 321}, {24342, 1978}, {40750, 668}
X(54279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3733, 4455, 667}

X(54280) = X(2)X(44)∩X(9)X(69)

Barycentrics    3*a^2 - 2*a*b - b^2 - 2*a*c - c^2 : :
X(54280) = 4 X[3707] - X[42697

X(54280) lies on these lines: {1, 1992}, {2, 44}, {6, 4364}, {7, 17277}, {8, 190}, {9, 69}, {10, 24695}, {37, 193}, {45, 524}, {63, 2183}, {75, 144}, {85, 12848}, {86, 5296}, {141, 16885}, {145, 4664}, {192, 5839}, {198, 1444}, {200, 24708}, {220, 1332}, {239, 4419}, {281, 317}, {302, 30415}, {303, 30414}, {312, 14552}, {319, 346}, {322, 28974}, {329, 333}, {345, 3219}, {348, 651}, {390, 49450}, {491, 30412}, {492, 30413}, {527, 3707}, {536, 20073}, {545, 17119}, {597, 17325}, {599, 4422}, {645, 34016}, {648, 7952}, {666, 46136}, {894, 966}, {948, 17950}, {956, 15507}, {984, 51192}, {1007, 36407}, {1100, 51170}, {1150, 28808}, {1211, 26065}, {1266, 16833}, {1278, 4371}, {1441, 41563}, {1654, 2345}, {1707, 4104}, {1743, 3618}, {1757, 29659}, {1944, 28827}, {1997, 14829}, {2267, 20769}, {2287, 26647}, {2321, 25728}, {2325, 17294}, {2895, 17776}, {2911, 15988}, {3008, 17274}, {3161, 17233}, {3246, 47358}, {3305, 4001}, {3416, 15481}, {3488, 49753}, {3578, 42032}, {3589, 17253}, {3616, 46922}, {3619, 3973}, {3620, 15492}, {3621, 50077}, {3625, 50089}, {3626, 50118}, {3629, 16777}, {3630, 17311}, {3631, 17267}, {3633, 50110}, {3662, 29607}, {3672, 3759}, {3678, 52609}, {3686, 3729}, {3687, 3929}, {3691, 21281}, {3696, 24280}, {3730, 3882}, {3731, 3879}, {3782, 19723}, {3785, 25066}, {3875, 50019}, {3883, 5223}, {3886, 51090}, {3943, 50079}, {3945, 4687}, {4000, 6646}, {4029, 29605}, {4033, 25278}, {4034, 4431}, {4346, 24599}, {4359, 20078}, {4361, 17334}, {4363, 17330}, {4370, 17269}, {4389, 5222}, {4395, 49747}, {4398, 4402}, {4405, 28297}, {4407, 50300}, {4417, 5273}, {4432, 50316}, {4440, 16816}, {4445, 17340}, {4461, 5564}, {4473, 17230}, {4478, 53664}, {4480, 4659}, {4488, 32087}, {4517, 9025}, {4645, 38057}, {4648, 17260}, {4651, 44447}, {4655, 25351}, {4657, 16669}, {4665, 49721}, {4667, 16831}, {4690, 17281}, {4699, 7222}, {4700, 16834}, {4701, 50100}, {4703, 33137}, {4749, 17018}, {4753, 50282}, {4758, 5257}, {4759, 50311}, {4851, 16814}, {4869, 17263}, {4875, 20111}, {4896, 31211}, {4969, 17318}, {5032, 16666}, {5224, 5749}, {5232, 17289}, {5233, 5744}, {5271, 17781}, {5278, 5905}, {5308, 17378}, {5361, 26792}, {5463, 11791}, {5464, 11790}, {5686, 32850}, {5714, 25446}, {5759, 48878}, {6144, 16675}, {6376, 41316}, {6542, 50074}, {6604, 8545}, {6666, 17298}, {7232, 17337}, {7277, 15668}, {7313, 17744}, {7321, 20059}, {7774, 36405}, {9723, 15817}, {9780, 31144}, {9801, 16112}, {9965, 19804}, {11160, 17374}, {11269, 25378}, {11679, 24705}, {12322, 31561}, {12323, 31562}, {12530, 21867}, {14912, 46475}, {15534, 16672}, {16496, 49783}, {16552, 21362}, {16670, 17023}, {16676, 29574}, {16706, 17329}, {16815, 50128}, {16825, 53601}, {16832, 50116}, {16884, 32455}, {17014, 17320}, {17120, 17248}, {17121, 17247}, {17152, 30616}, {17160, 49748}, {17183, 29767}, {17234, 18230}, {17243, 40341}, {17251, 17369}, {17252, 17368}, {17254, 17367}, {17255, 17366}, {17259, 17365}, {17261, 17314}, {17262, 17362}, {17264, 17360}, {17270, 17355}, {17271, 17354}, {17273, 17352}, {17275, 17351}, {17276, 17348}, {17278, 17345}, {17280, 17343}, {17282, 53598}, {17284, 21356}, {17287, 17339}, {17288, 17338}, {17297, 29627}, {17300, 29599}, {17308, 50115}, {17379, 29592}, {17387, 29621}, {17395, 24441}, {17740, 37656}, {17770, 25352}, {18156, 42724}, {18743, 37655}, {19742, 19785}, {20050, 50121}, {20090, 27268}, {20930, 25001}, {21061, 29497}, {24597, 26580}, {24693, 28558}, {26039, 29610}, {26104, 29630}, {28333, 34824}, {29569, 50133}, {29573, 50992}, {29624, 51488}, {30854, 37788}, {30946, 30997}, {31035, 31303}, {31156, 49687}, {32847, 51297}, {36479, 49712}, {36480, 49710}, {41008, 42018}, {45789, 48629}, {49448, 50030}, {49709, 50075}, {49714, 50835}, {49722, 52709}, {49766, 50950}, {49770, 50090}, {50022, 50080}

X(54280) = reflection of X(i) in X(j) for these {i,j}: {4384, 3707}, {4896, 31211}, {17316, 45}, {29605, 4029}, {42697, 4384}
X(54280) = anticomplement of X(4675)
X(54280) = barycentric product X(i)*X(j) for these {i,j}: {75, 35258}, {190, 47785}
X(54280) = barycentric quotient X(i)/X(j) for these {i,j}: {35258, 1}, {47785, 514}
X(54280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4747, 41847}, {2, 20072, 4644}, {6, 4364, 26626}, {6, 17257, 17321}, {6, 17332, 17257}, {8, 190, 50107}, {8, 6172, 190}, {9, 69, 344}, {9, 4416, 69}, {9, 17296, 25101}, {44, 4643, 2}, {141, 16885, 26685}, {144, 391, 75}, {190, 17346, 8}, {239, 4419, 50101}, {239, 17333, 4419}, {319, 17336, 346}, {320, 17335, 2}, {599, 4422, 29579}, {894, 17331, 966}, {894, 29576, 4470}, {966, 4470, 29576}, {1150, 31018, 28808}, {1654, 17350, 2345}, {1743, 4357, 3618}, {3161, 32099, 17233}, {3219, 5739, 345}, {3305, 4001, 18141}, {3416, 15481, 27549}, {3686, 3729, 42696}, {3758, 17256, 2}, {3759, 17258, 3672}, {3973, 17272, 17353}, {4346, 24599, 37756}, {4364, 26626, 17321}, {4419, 37654, 239}, {4480, 50095, 4659}, {4657, 16669, 51171}, {4699, 31300, 7222}, {4795, 16590, 2}, {4798, 52706, 2}, {4969, 17318, 50129}, {4969, 49742, 17318}, {6144, 16675, 17390}, {6172, 17346, 50107}, {6646, 17349, 4000}, {11160, 29583, 17374}, {14829, 18228, 1997}, {15492, 17344, 17279}, {17257, 26626, 4364}, {17260, 17364, 4648}, {17261, 17363, 17314}, {17263, 17361, 4869}, {17264, 17360, 29616}, {17271, 17354, 29611}, {17272, 17353, 3619}, {17277, 17347, 7}, {17279, 17344, 3620}, {17288, 17338, 53665}, {17289, 17328, 5232}, {17333, 37654, 50101}, {17374, 41313, 29583}, {18230, 21296, 17234}, {36480, 49710, 50303}, {49712, 50296, 36479}

X(54281) = X(44)X(57)∩X(45)X(63)

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

X(54283) lies on these lines: {1, 3052}, {2, 7238}, {6, 2243}, {7, 31187}, {44, 57}, {45, 63}, {88, 4383}, {89, 940}, {678, 41711}, {896, 4860}, {1086, 2094}, {1150, 17118}, {1376, 49712}, {1407, 3173}, {1707, 3246}, {2441, 4498}, {3219, 37682}, {3306, 16885}, {3666, 39254}, {3742, 16570}, {3752, 16670}, {3772, 4887}, {3873, 21000}, {3973, 31197}, {3977, 17311}, {3999, 36277}, {4031, 17278}, {4346, 37642}, {4415, 28610}, {4849, 53056}, {4896, 5745}, {5021, 36283}, {5096, 26866}, {5744, 17365}, {5905, 37691}, {9965, 37646}, {15533, 33077}, {16490, 16558}, {16675, 37633}, {17122, 51297}, {17160, 37683}, {17262, 37684}, {17601, 32913}, {17740, 40341}, {18134, 26070}, {19732, 30564}, {20050, 42049}, {20078, 37634}, {26934, 37567}, {29579, 44416}, {30579, 37639}, {33151, 35596}, {38000, 41847}

X(54281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {45, 37520, 37674}, {63, 37520, 45}

X(54282) = X(1)X(3)∩X(2)X(3230)

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

X(54282) lies on these lines: {1, 3}, {2, 3230}, {6, 536}, {31, 10800}, {32, 52134}, {38, 760}, {42, 14839}, {63, 1572}, {75, 2300}, {81, 99}, {83, 213}, {194, 712}, {333, 16829}, {350, 41232}, {519, 37676}, {730, 24259}, {894, 20228}, {981, 41527}, {992, 4967}, {1107, 29529}, {1258, 32017}, {1574, 28254}, {1580, 12194}, {1918, 17445}, {2176, 4384}, {2238, 50095}, {2295, 17023}, {2481, 39940}, {2703, 9081}, {2783, 12177}, {3051, 8621}, {3210, 24282}, {3663, 28369}, {3735, 28606}, {3739, 16685}, {3747, 21352}, {3752, 21888}, {3765, 4721}, {3780, 49770}, {3997, 50114}, {4039, 12263}, {4359, 24254}, {4366, 23660}, {4383, 14535}, {4389, 4503}, {4641, 45751}, {4649, 18794}, {4688, 52897}, {5256, 9620}, {5283, 19735}, {6033, 33106}, {7109, 24592}, {8616, 16497}, {10436, 21769}, {14621, 21760}, {15989, 50167}, {16514, 25368}, {16526, 40750}, {16782, 24326}, {16826, 27002}, {16831, 16969}, {16973, 34377}, {17049, 40934}, {17117, 27644}, {17448, 18206}, {17472, 46903}, {17475, 39714}, {17750, 26626}, {20172, 40728}, {20913, 40886}, {24199, 28350}, {24268, 29069}, {25590, 28365}, {28371, 31198}, {29055, 29352}, {29580, 37633}, {29597, 37674}, {36816, 51987}, {37662, 51390}, {40153, 42051}, {46264, 50629}, {51121, 51122}

X(54282) = crossdifference of every pair of points on line {650, 9010}
X(54282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 20367, 37596}, {1, 37555, 980}, {81, 29584, 16971}, {1429, 5255, 5337}, {10800, 24264, 31}, {25590, 41418, 28365}

X(54283) = X(4)X(9)∩X(37)X(91)

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

X(54283) lies on these lines: {2, 2006}, {4, 9}, {6, 10573}, {8, 2323}, {37, 91}, {78, 2321}, {198, 11499}, {219, 594}, {220, 7359}, {226, 53816}, {345, 4494}, {346, 5552}, {355, 2182}, {499, 8609}, {1146, 17369}, {1352, 24332}, {1737, 8557}, {1899, 21028}, {1944, 3661}, {2171, 5747}, {2172, 2329}, {2359, 10570}, {2915, 9712}, {3247, 13411}, {3436, 21066}, {3939, 28118}, {4000, 24209}, {4007, 6737}, {4363, 26932}, {4671, 28836}, {4873, 6745}, {5227, 21074}, {5285, 7102}, {5554, 5749}, {5745, 19822}, {5746, 15556}, {5750, 19860}, {6708, 32777}, {7085, 7140}, {8257, 26001}, {17075, 40903}, {17281, 46835}, {17286, 27384}, {17303, 19854}, {17314, 22836}, {17359, 34852}, {17754, 26013}, {21091, 24315}, {21871, 31837}, {23676, 33127}, {24149, 45794}, {24958, 25679}, {27059, 28731}, {27382, 52405}, {28125, 52335}, {28796, 31025}, {29611, 52457}, {29828, 50366}

X(54283) = X(7284)-complementary conjugate of X(2886)
X(54283) = X(i)-isoconjugate of X(j) for these (i,j): {57, 3422}, {222, 1061}, {905, 36076}
X(54283) = X(i)-Dao conjugate of X(j) for these (i,j): {5452, 3422}, {38964, 514}
X(54283) = crossdifference of every pair of points on line {1459, 8648}
X(54283) = barycentric product X(i)*X(j) for these {i,j}: {8, 1478}, {10, 11103}, {318, 1060}, {4351, 52409}
X(54283) = barycentric quotient X(i)/X(j) for these {i,j}: {33, 1061}, {55, 3422}, {1060, 77}, {1478, 7}, {4351, 1443}, {8750, 36076}, {11103, 86}
X(54283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {281, 2345, 9}, {17355, 20262, 9}

X(54284) = X(2)X(37)∩X(57)X(92)

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

X(54284) lies on these lines: {2, 37}, {7, 11433}, {11, 20276}, {27, 42467}, {56, 20220}, {57, 92}, {85, 2994}, {239, 394}, {241, 6360}, {242, 1473}, {244, 17871}, {273, 2052}, {314, 24556}, {318, 1210}, {320, 6515}, {341, 25005}, {343, 3662}, {469, 12610}, {614, 4008}, {894, 10601}, {938, 23661}, {982, 26013}, {990, 14004}, {1086, 13567}, {1088, 23989}, {1119, 14361}, {1230, 25003}, {1441, 9776}, {1445, 20223}, {1726, 24618}, {1733, 5272}, {1851, 26929}, {1993, 3759}, {3086, 17869}, {3218, 18750}, {3219, 30854}, {3262, 18141}, {3305, 20879}, {3306, 14213}, {3580, 48629}, {3673, 26001}, {3703, 25973}, {3758, 5422}, {3782, 26005}, {3944, 26010}, {4191, 30273}, {4292, 5342}, {4361, 17811}, {4363, 17825}, {4384, 20882}, {4385, 24982}, {4395, 53415}, {4647, 8583}, {5222, 11427}, {5256, 18690}, {5262, 24537}, {5437, 6358}, {5695, 25893}, {5740, 33672}, {5905, 20921}, {7283, 25875}, {9965, 30807}, {10072, 23580}, {10444, 39592}, {11019, 17860}, {12649, 52346}, {14986, 23528}, {16059, 29010}, {16284, 32863}, {16817, 37228}, {17227, 37636}, {17361, 45794}, {17366, 23292}, {17367, 37649}, {17555, 23537}, {17616, 48878}, {17763, 25938}, {17861, 24177}, {17880, 41081}, {17923, 20266}, {18156, 26639}, {18928, 42697}, {20237, 30567}, {20430, 37355}, {20881, 30568}, {20895, 34255}, {20909, 25955}, {20927, 32939}, {20940, 25954}, {23541, 33131}, {23978, 34404}, {24163, 24186}, {24175, 24209}, {24415, 40688}, {25941, 32860}, {26531, 26550}, {26534, 26565}, {26609, 27792}, {26871, 53994}, {26872, 52457}, {30679, 42050}, {37648, 48627}, {37881, 38487}

X(54284) = isotomic conjugate of the isogonal conjugate of X(3554)
X(54284) = X(649)-complementary conjugate of X(47601)
X(54284) = X(i)-isoconjugate of X(j) for these (i,j): {6, 42019}, {55, 53995}, {837, 4055}, {2175, 34401}
X(54284) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 42019}, {223, 53995}, {3086, 2324}, {3554, 10310}, {24005, 3682}, {38015, 1}, {38357, 14298}, {40593, 34401}, {40650, 3084}, {49171, 6}
X(54284) = cevapoint of X(i) and X(j) for these (i,j): {3086, 53994}, {17869, 24005}
X(54284) = barycentric product X(i)*X(j) for these {i,j}: {75, 3086}, {76, 3554}, {85, 53994}, {86, 17869}, {92, 26871}, {274, 24005}, {1519, 18816}, {6063, 30223}
X(54284) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 42019}, {57, 53995}, {85, 34401}, {309, 34413}, {836, 3990}, {1519, 517}, {3086, 1}, {3554, 6}, {17869, 10}, {19354, 212}, {24005, 37}, {26871, 63}, {26955, 201}, {30223, 55}, {38003, 1124}, {38015, 2324}, {40650, 3083}, {49171, 10310}, {53994, 9}
X(54284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 28605, 26591}, {2, 48380, 312}, {57, 4858, 92}, {1210, 20320, 318}, {3752, 26011, 2}, {4359, 20905, 2}, {20921, 39126, 5905}

X(54285) = X(3)X(37)∩X(6)X(31)

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

X(54285) lies on these lines: {1, 21853}, {3, 37}, {6, 31}, {9, 35}, {19, 37601}, {21, 2345}, {36, 3247}, {44, 4254}, {45, 198}, {48, 1334}, {56, 2171}, {75, 16367}, {100, 966}, {218, 584}, {219, 2278}, {220, 2174}, {284, 2911}, {344, 21511}, {346, 4189}, {405, 17303}, {478, 2197}, {498, 50036}, {572, 10267}, {573, 11248}, {574, 17053}, {579, 24047}, {594, 958}, {595, 5105}, {604, 11510}, {943, 5746}, {956, 17299}, {993, 2321}, {999, 3723}, {1001, 1009}, {1100, 3295}, {1213, 1376}, {1259, 3713}, {1333, 17524}, {1400, 11509}, {1436, 3207}, {1444, 17316}, {1449, 3746}, {1486, 37586}, {1500, 5019}, {1575, 16058}, {1593, 1841}, {1696, 16675}, {1697, 3554}, {1743, 37503}, {2071, 16307}, {2092, 31451}, {2176, 5110}, {2183, 11434}, {2245, 11507}, {2256, 7113}, {2262, 37568}, {2270, 35445}, {2285, 37579}, {2298, 16452}, {2303, 4184}, {2305, 15592}, {2324, 30282}, {2335, 37120}, {2646, 21871}, {2975, 17314}, {3097, 16800}, {3169, 8668}, {3204, 17454}, {3290, 7484}, {3303, 16884}, {3553, 3601}, {3686, 8715}, {3693, 20835}, {3709, 48391}, {3731, 5010}, {3772, 21483}, {3871, 5839}, {3913, 17362}, {3940, 21873}, {3950, 5267}, {3973, 51817}, {4007, 5258}, {4034, 48696}, {4068, 4497}, {4171, 23226}, {4261, 16287}, {4265, 50995}, {4268, 11508}, {4270, 33771}, {4276, 4877}, {4421, 17330}, {4526, 39200}, {4657, 21477}, {4687, 11329}, {4698, 16412}, {5036, 14882}, {5069, 16502}, {5153, 16466}, {5204, 16672}, {5248, 5750}, {5257, 25440}, {5279, 37285}, {5687, 17275}, {5747, 17732}, {5816, 11499}, {5819, 7676}, {6351, 16441}, {6352, 16440}, {6580, 33635}, {6684, 24005}, {7280, 16673}, {7373, 46845}, {7485, 26242}, {7824, 26107}, {8273, 37519}, {8607, 36751}, {8609, 26357}, {8610, 15815}, {9310, 22054}, {9598, 37225}, {9724, 20793}, {10310, 37499}, {10895, 53421}, {10966, 17452}, {11194, 50113}, {11285, 25505}, {11343, 17279}, {11350, 44307}, {11383, 44103}, {12513, 17388}, {13455, 44192}, {13615, 44798}, {16064, 23847}, {16286, 46838}, {16289, 19845}, {16290, 41508}, {16368, 32777}, {16370, 17281}, {16431, 41312}, {16436, 41313}, {16439, 24789}, {16523, 23370}, {16972, 36741}, {17321, 21495}, {17334, 24328}, {17357, 21514}, {17384, 21526}, {17750, 37547}, {17776, 27174}, {18491, 32431}, {19308, 27268}, {20846, 27396}, {21231, 24268}, {21348, 48390}, {21509, 41310}, {21539, 41311}, {21801, 22768}, {22769, 49509}, {25099, 37344}, {31993, 37323}, {35238, 37508}, {40292, 40937}, {41230, 52241}

X(54285) = crossdifference of every pair of points on line {514, 51648}
X(54285) = barycentric product X(1)*X(41229)
X(54285) = barycentric quotient X(41229)/X(75)
X(54285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 37, 2178}, {9, 35, 36744}, {45, 1030, 198}, {71, 2268, 6}, {198, 5217, 1030}, {220, 37504, 2174}, {284, 3730, 2911}, {346, 4189, 38871}, {2267, 2269, 6}, {2268, 41423, 71}, {3295, 5120, 1100}, {5124, 16777, 56}, {16675, 19297, 1696}

X(54286) = X(1)X(88)∩X(4)X(9)

Barycentrics    a*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2 - c^3) : :
X(54286) = 3 X[3359] - X[7171], 3 X[997] - 2 X[5289], 3 X[1376] - X[5289], 3 X[40726] - 2 X[51788]

X(54286) lies on these lines: {1, 88}, {2, 5119}, {3, 5836}, {4, 9}, {5, 37828}, {7, 34619}, {8, 46}, {31, 4695}, {33, 45269}, {35, 19860}, {36, 3872}, {37, 31433}, {55, 3753}, {56, 10914}, {57, 519}, {63, 484}, {65, 3689}, {72, 3711}, {78, 4867}, {145, 3338}, {165, 993}, {191, 16558}, {200, 758}, {210, 5183}, {226, 45701}, {355, 1158}, {377, 10039}, {381, 5123}, {388, 10915}, {392, 4413}, {405, 3698}, {474, 3057}, {495, 5856}, {498, 39599}, {515, 3359}, {517, 997}, {518, 36279}, {528, 5722}, {551, 5437}, {595, 1722}, {612, 4424}, {614, 1739}, {643, 11116}, {655, 51975}, {920, 5086}, {936, 3878}, {942, 3913}, {946, 6944}, {956, 1155}, {958, 3579}, {960, 9709}, {962, 5328}, {975, 37598}, {999, 3880}, {1000, 34711}, {1012, 13528}, {1018, 40131}, {1056, 49626}, {1100, 4646}, {1107, 1571}, {1125, 1697}, {1145, 5252}, {1319, 16371}, {1329, 12699}, {1339, 40151}, {1377, 49226}, {1378, 49227}, {1387, 17564}, {1420, 22837}, {1454, 41687}, {1476, 3361}, {1478, 6735}, {1479, 24982}, {1572, 1575}, {1574, 39248}, {1698, 4193}, {1699, 3814}, {1708, 3419}, {1728, 5175}, {1730, 48863}, {1737, 3434}, {1768, 15863}, {1770, 3436}, {1781, 3692}, {1788, 5082}, {1836, 17757}, {1837, 12690}, {2082, 16549}, {2098, 17614}, {2099, 5440}, {2136, 3244}, {2160, 3704}, {2320, 3612}, {2328, 17519}, {2475, 10827}, {2800, 5720}, {2886, 6882}, {2932, 17636}, {3035, 5886}, {3085, 12609}, {3086, 26062}, {3158, 3919}, {3174, 30329}, {3219, 53620}, {3241, 27003}, {3245, 5692}, {3295, 3812}, {3303, 5439}, {3305, 19875}, {3336, 3632}, {3337, 3633}, {3339, 3874}, {3340, 22836}, {3421, 3474}, {3452, 28194}, {3488, 34607}, {3550, 37817}, {3555, 5221}, {3582, 31224}, {3584, 31266}, {3587, 5745}, {3601, 30147}, {3617, 31295}, {3624, 37563}, {3625, 6762}, {3626, 5128}, {3634, 31435}, {3636, 37556}, {3646, 51073}, {3647, 5234}, {3654, 28452}, {3678, 12526}, {3683, 4731}, {3696, 5774}, {3715, 3921}, {3729, 51284}, {3742, 6767}, {3743, 16673}, {3749, 30117}, {3820, 24703}, {3822, 31434}, {3825, 9614}, {3828, 7308}, {3833, 10582}, {3838, 31479}, {3870, 5902}, {3884, 8583}, {3890, 17531}, {3892, 10980}, {3893, 32636}, {3898, 9819}, {3911, 45700}, {3918, 5248}, {3922, 37080}, {3927, 4662}, {3928, 4669}, {3929, 4745}, {3931, 16777}, {3940, 44663}, {3968, 4512}, {3987, 5264}, {4015, 12446}, {4084, 11523}, {4187, 12701}, {4188, 4861}, {4253, 50022}, {4271, 37150}, {4292, 6736}, {4295, 7080}, {4297, 37560}, {4301, 6700}, {4386, 9620}, {4421, 24929}, {4428, 51787}, {4498, 28591}, {4511, 25415}, {4640, 9708}, {4652, 5258}, {4658, 17207}, {4668, 6763}, {4691, 41348}, {4714, 5271}, {4723, 32933}, {4737, 32939}, {4848, 37550}, {4853, 8666}, {4882, 41228}, {4915, 4973}, {5122, 11194}, {5180, 27131}, {5187, 9780}, {5223, 30353}, {5249, 10056}, {5251, 35258}, {5255, 24440}, {5267, 35242}, {5272, 40091}, {5288, 37524}, {5426, 51817}, {5435, 34625}, {5438, 7982}, {5534, 5884}, {5552, 12047}, {5554, 10572}, {5563, 36846}, {5573, 24168}, {5603, 12703}, {5690, 5794}, {5697, 19861}, {5705, 6943}, {5708, 34791}, {5709, 6885}, {5726, 8545}, {5777, 17646}, {5791, 9710}, {5795, 31730}, {5881, 12247}, {5882, 37534}, {6174, 15950}, {6205, 45751}, {6261, 11499}, {6684, 6891}, {6691, 11373}, {6797, 13205}, {6890, 19843}, {6918, 45776}, {6973, 10175}, {6986, 16208}, {7183, 25719}, {7289, 49529}, {7330, 40256}, {7681, 12700}, {7686, 10306}, {7702, 26482}, {7969, 9679}, {7987, 51111}, {7995, 31871}, {8170, 34862}, {8580, 10176}, {8582, 10624}, {9843, 12575}, {9945, 37728}, {9957, 25524}, {10106, 49169}, {10197, 25525}, {10199, 21630}, {10200, 12053}, {10270, 12650}, {10528, 13407}, {10609, 37740}, {10826, 25005}, {10860, 28164}, {10896, 17619}, {10912, 24928}, {11236, 51362}, {11274, 51767}, {11376, 13747}, {11500, 12520}, {11508, 37282}, {11518, 33815}, {11530, 51570}, {11552, 31164}, {11813, 30827}, {11826, 45080}, {12019, 15297}, {12245, 12704}, {12513, 37582}, {12635, 50193}, {12647, 51433}, {12767, 47320}, {14798, 37301}, {14974, 16605}, {15843, 37401}, {16483, 16610}, {16611, 16970}, {16669, 21896}, {17064, 17734}, {17284, 24590}, {17573, 33895}, {17594, 30116}, {17668, 18908}, {17754, 50287}, {17784, 18391}, {18228, 34632}, {18393, 30852}, {18540, 50796}, {18990, 32049}, {19537, 37605}, {20085, 37711}, {20103, 28228}, {21370, 29673}, {21952, 42657}, {22300, 31778}, {22791, 25681}, {23537, 28402}, {23958, 31145}, {24174, 37588}, {24390, 24914}, {24473, 41711}, {24715, 37716}, {25092, 31426}, {25413, 45770}, {25466, 32157}, {26066, 31419}, {27529, 37692}, {29007, 50736}, {29529, 48812}, {30331, 34639}, {30556, 35610}, {30557, 35611}, {31443, 31449}, {31458, 37551}, {31803, 54156}, {31855, 49500}, {32760, 37300}, {32912, 49984}, {33709, 50443}, {34720, 51463}, {35004, 37700}, {36480, 37555}, {37274, 40863}, {37612, 37727}, {37829, 50239}, {39148, 52140}, {40726, 51788}, {46684, 52027}, {48882, 49734}, {48915, 49728}, {50581, 50633}, {51093, 51786}, {51103, 51779}

X(54286) = midpoint of X(i) and X(j) for these {i,j}: {8, 4293}, {200, 2093}, {3421, 3474}, {5223, 30353}, {17784, 18391}
X(54286) = reflection of X(i) in X(j) for these {i,j}: {997, 1376}, {4342, 1125}, {24703, 3820}
X(54286) = complement of X(30305)
X(54286) = X(i)-Dao conjugate of X(j) for these (i,j): {17595, 17274}, {38962, 514}
X(54286) = crossdifference of every pair of points on line {1459, 1635}
X(54286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5541, 3895}, {8, 4190, 45287}, {10, 40, 12514}, {10, 5493, 12572}, {40, 1706, 10}, {65, 3811, 12559}, {65, 5687, 3811}, {165, 9623, 993}, {404, 14923, 1}, {484, 3679, 63}, {936, 7991, 3878}, {1145, 11112, 5252}, {1276, 1277, 1766}, {1698, 11010, 5250}, {1739, 37610, 614}, {1788, 5082, 10916}, {2136, 3333, 3244}, {2550, 5657, 10}, {3241, 27003, 51816}, {3306, 3895, 1}, {3339, 6765, 3874}, {3698, 37568, 405}, {3754, 8715, 1}, {3885, 5253, 1}, {4188, 4861, 37618}, {4295, 7080, 21077}, {4386, 21888, 9620}, {4853, 15803, 8666}, {5258, 37572, 4652}, {5437, 31393, 551}, {5438, 7982, 30144}, {5657, 48363, 40}, {5883, 25439, 1}, {5902, 48696, 3870}, {6691, 13463, 11373}, {9709, 12702, 960}, {10199, 21630, 37704}, {11499, 37562, 6261}, {11500, 31788, 12520}, {22791, 47742, 25681}, {25005, 52367, 10826}, {30827, 31162, 11813}, {31190, 37704, 10199}, {34612, 40663, 3419}

X(54287) = X(1)X(6)∩X(25)X(35)

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

X(54287) lies on these lines: {1, 6}, {2, 7283}, {3, 44307}, {10, 968}, {21, 975}, {25, 35}, {33, 30733}, {34, 16577}, {36, 27802}, {43, 4204}, {46, 846}, {57, 16290}, {58, 5287}, {75, 37035}, {78, 4653}, {165, 37320}, {192, 16817}, {312, 11110}, {344, 13725}, {386, 3305}, {406, 498}, {429, 7951}, {440, 9612}, {536, 51676}, {612, 5248}, {756, 3811}, {936, 16346}, {940, 31445}, {988, 3624}, {990, 6986}, {993, 27784}, {997, 10448}, {1010, 4687}, {1125, 4011}, {1224, 39954}, {1698, 3712}, {1707, 37559}, {1745, 37324}, {1860, 39585}, {1961, 37316}, {2218, 7322}, {2276, 53387}, {2345, 19857}, {2901, 5271}, {2915, 5010}, {3175, 11357}, {3216, 7308}, {3293, 37553}, {3338, 26102}, {3465, 3612}, {3587, 52524}, {3601, 36011}, {3616, 26223}, {3646, 49979}, {3666, 11108}, {3672, 17554}, {3679, 3695}, {3683, 5711}, {3685, 19853}, {3739, 50044}, {3752, 16842}, {3868, 33761}, {3916, 37674}, {3953, 10582}, {3989, 28082}, {4000, 17552}, {4195, 27268}, {4239, 25440}, {4292, 14021}, {4306, 8545}, {4340, 5308}, {4358, 16342}, {4420, 9330}, {4423, 37592}, {4512, 5264}, {4664, 51595}, {4689, 9709}, {4693, 31327}, {4698, 16458}, {4704, 19851}, {4755, 16394}, {4850, 17536}, {4851, 49716}, {5016, 14020}, {5044, 19765}, {5047, 28606}, {5100, 49746}, {5119, 13724}, {5250, 30116}, {5262, 16859}, {5272, 25542}, {5295, 19732}, {5313, 5506}, {5438, 19287}, {6675, 17720}, {6886, 53599}, {6913, 37528}, {7069, 10393}, {7270, 48814}, {7741, 37315}, {8728, 50065}, {9534, 17260}, {9708, 37548}, {11512, 34595}, {12047, 21062}, {13411, 27413}, {13728, 17279}, {13732, 46475}, {13734, 16389}, {13735, 51488}, {13742, 17321}, {13745, 41313}, {15803, 21483}, {15976, 48883}, {16050, 16831}, {16343, 44417}, {16418, 37539}, {16602, 16854}, {16610, 16853}, {16828, 50314}, {16844, 31993}, {16856, 31197}, {16865, 37817}, {17022, 31424}, {17243, 49728}, {17263, 33833}, {17278, 17590}, {17281, 51679}, {17289, 37039}, {17303, 17514}, {17322, 37036}, {17342, 51680}, {17588, 31035}, {17916, 41227}, {18540, 48897}, {18743, 19270}, {19273, 30818}, {19758, 25066}, {19766, 26685}, {19767, 27065}, {19784, 50290}, {19785, 31259}, {19854, 24210}, {19871, 50126}, {19874, 32929}, {20077, 29569}, {24512, 31442}, {24789, 50067}, {24936, 31053}, {25091, 37224}, {25430, 37322}, {26064, 32858}, {26127, 29680}, {29573, 49723}, {30282, 37052}, {31658, 37537}, {33116, 52258}, {37317, 37552}, {37327, 37603}, {41310, 51677}, {50068, 50202}

X(54287) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 405, 1}, {958, 6051, 1}, {1453, 3247, 1}, {4205, 32777, 1698}, {4698, 50054, 16458}, {5251, 27785, 1}, {17022, 31424, 37522}, {17776, 37314, 10}, {27802, 37246, 36}, {50067, 50205, 24789}

X(54288) = X(8)X(35)∩X(10)X(12)

Barycentrics    (b + c)*(2*a^3 - a^2*b - 2*a*b^2 + b^3 - a^2*c + a*b*c - b^2*c - 2*a*c^2 - b*c^2 + c^3) : :
X(54288) = 3 X[10] - X[226], 2 X[226] - 3 X[3822], X[63] + 3 X[3679], X[1478] - 5 X[3617], 3 X[1478] + X[20078], 15 X[3617] + X[20078], X[5881] + 3 X[21165], X[5905] - 9 X[53620], 9 X[19875] - 5 X[31266], X[31164] - 5 X[51066]

X(54288) lies on these lines: {2, 4867}, {8, 35}, {10, 12}, {63, 484}, {80, 3219}, {191, 5086}, {214, 51113}, {321, 15065}, {333, 759}, {515, 550}, {516, 54175}, {519, 5745}, {527, 4745}, {912, 4662}, {956, 5172}, {960, 3825}, {997, 6681}, {1125, 17706}, {1145, 4669}, {1478, 3617}, {1512, 15064}, {1737, 10176}, {2801, 24393}, {2802, 4847}, {3245, 33110}, {3452, 6702}, {3476, 8666}, {3488, 5248}, {3585, 11684}, {3586, 12514}, {3632, 4917}, {3647, 10572}, {3814, 5692}, {3869, 18393}, {3874, 24987}, {3876, 18395}, {3878, 6734}, {3881, 24391}, {3884, 5837}, {3898, 26015}, {3899, 11680}, {3951, 10827}, {4364, 25390}, {4511, 5444}, {4691, 8256}, {4732, 8680}, {4868, 25080}, {5178, 11010}, {5180, 31159}, {5220, 5790}, {5341, 17275}, {5435, 5775}, {5657, 7688}, {5791, 30147}, {5881, 21165}, {5902, 27186}, {5905, 53620}, {6175, 11552}, {6788, 17123}, {8728, 33815}, {9803, 15931}, {10039, 18389}, {11362, 37585}, {11813, 31165}, {15228, 47033}, {17051, 51108}, {17056, 53114}, {17057, 31053}, {17461, 33141}, {18253, 37730}, {19875, 31266}, {20013, 31452}, {21014, 21078}, {22758, 35000}, {22836, 26066}, {26065, 48826}, {26792, 31160}, {29046, 50308}, {31164, 51066}, {35016, 41575}, {35466, 49682}, {51071, 51463}

X(54288) = midpoint of X(i) and X(j) for these {i,j}: {8, 993}, {11362, 51755}
X(54288) = reflection of X(3822) in X(10)
X(54288) = barycentric product X(321)*X(37525)
X(54288) = barycentric quotient X(37525)/X(81)
X(54288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 65, 3841}, {10, 3919, 3925}, {10, 4067, 12}, {10, 4084, 442}, {10, 4134, 17757}, {10, 4848, 3918}, {3878, 6734, 24387}, {5837, 10916, 3884}

X(54289) = X(1)X(21)∩X(69)X(73)

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

X(54289) lies on these lines: {1, 21}, {3, 6511}, {4, 2000}, {8, 1943}, {20, 36850}, {29, 18750}, {34, 6734}, {40, 11413}, {42, 8895}, {56, 24476}, {69, 73}, {72, 394}, {224, 4303}, {241, 37282}, {270, 1760}, {271, 1257}, {279, 6904}, {377, 1448}, {936, 32782}, {1040, 4652}, {1062, 3916}, {1071, 22129}, {1214, 1259}, {1427, 37229}, {1763, 11337}, {1798, 43708}, {1800, 6507}, {1829, 37581}, {3616, 27509}, {3682, 6505}, {3870, 36706}, {3876, 37659}, {3912, 28769}, {3927, 18447}, {3928, 33178}, {4347, 4847}, {5256, 24609}, {5262, 37666}, {5271, 15149}, {5287, 37169}, {6769, 37048}, {6872, 37782}, {7085, 37613}, {7291, 7520}, {7360, 26027}, {10527, 34036}, {11363, 24320}, {14054, 36742}, {14206, 39585}, {16454, 20235}, {16465, 36746}, {17181, 26120}, {18634, 26167}, {18651, 37179}, {19860, 24570}, {21370, 37231}, {21406, 50314}, {24537, 52345}, {34028, 41228}, {37094, 52346}, {41538, 45729}

X(54289) = X(i)-isoconjugate of X(j) for these (i,j): {393, 45127}, {13395, 18344}
X(54289) = X(405)-Dao conjugate of X(39585)
X(54289) = barycentric product X(i)*X(j) for these {i,j}: {63, 377}, {304, 37538}, {345, 1448}, {4592, 47124}, {17206, 43214}
X(54289) = barycentric quotient X(i)/X(j) for these {i,j}: {255, 45127}, {377, 92}, {1448, 278}, {1813, 13395}, {37538, 19}, {43214, 1826}, {46038, 7040}, {47124, 24006}
X(54289) = {X(63),X(52362)}-harmonic conjugate of X(1)

X(54290) = X(8)X(20)∩X(9)X(46)

Barycentrics    a*(3*a^3 + 3*a^2*b - 3*a*b^2 - 3*b^3 + 3*a^2*c - 2*a*b*c - b^2*c - 3*a*c^2 - b*c^2 - 3*c^3) : :
X(54290) = 5 X[1698] - 3 X[9612], 5 X[1698] - 6 X[26066], 4 X[1125] - 3 X[3485]

X(54290) lies on these lines: {1, 3052}, {3, 12526}, {8, 20}, {9, 46}, {10, 3474}, {21, 11529}, {35, 11523}, {36, 15829}, {55, 41863}, {57, 1125}, {65, 31424}, {72, 165}, {78, 11684}, {100, 3951}, {144, 21075}, {200, 3579}, {329, 6684}, {376, 6737}, {392, 3361}, {405, 3339}, {443, 18249}, {474, 53056}, {484, 1706}, {498, 28609}, {527, 3085}, {595, 3677}, {610, 4047}, {758, 3601}, {908, 31423}, {920, 10396}, {936, 1155}, {942, 4512}, {946, 5744}, {956, 7991}, {958, 2093}, {960, 15803}, {986, 1453}, {993, 3340}, {1046, 17594}, {1071, 10268}, {1210, 5698}, {1259, 7688}, {1420, 3878}, {1697, 3244}, {1722, 7262}, {1768, 9841}, {1788, 12572}, {1836, 5705}, {2136, 11010}, {2270, 3707}, {2292, 37554}, {2975, 7982}, {3146, 5775}, {3158, 5904}, {3218, 3333}, {3219, 37161}, {3243, 3746}, {3247, 31320}, {3306, 3646}, {3336, 5437}, {3359, 10942}, {3421, 43174}, {3428, 12330}, {3496, 16572}, {3555, 53053}, {3576, 3869}, {3587, 16139}, {3623, 31393}, {3624, 51409}, {3633, 5119}, {3648, 18492}, {3652, 18540}, {3670, 7290}, {3678, 46917}, {3679, 10483}, {3680, 5288}, {3683, 5221}, {3697, 15587}, {3701, 25734}, {3712, 10319}, {3753, 5234}, {3811, 35445}, {3847, 24703}, {3868, 35258}, {3874, 10389}, {3895, 20014}, {3899, 37618}, {3911, 25522}, {3925, 31446}, {3940, 31663}, {3962, 5217}, {4293, 5837}, {4294, 24391}, {4295, 5745}, {4302, 12625}, {4330, 41709}, {4333, 47033}, {4480, 6211}, {4654, 10198}, {4677, 34626}, {4691, 41348}, {4847, 6361}, {4853, 12702}, {5057, 5535}, {5082, 5493}, {5084, 51090}, {5175, 28150}, {5204, 31165}, {5223, 5687}, {5231, 5709}, {5247, 16570}, {5248, 11518}, {5262, 36277}, {5264, 7174}, {5267, 13384}, {5325, 19855}, {5436, 5902}, {5438, 5692}, {5440, 16192}, {5530, 24695}, {5552, 17781}, {5560, 6597}, {5657, 12527}, {5693, 52026}, {5708, 10582}, {5730, 7987}, {6734, 41869}, {6738, 11111}, {6743, 50808}, {6745, 37560}, {6765, 37568}, {6974, 12704}, {7308, 51073}, {7330, 18480}, {7580, 7992}, {7701, 52841}, {7962, 8666}, {7971, 11012}, {8227, 11415}, {8583, 37582}, {9580, 10916}, {9588, 17757}, {9589, 24390}, {9623, 37567}, {9624, 51423}, {9965, 21620}, {10404, 37550}, {10461, 52352}, {10527, 31162}, {10624, 24477}, {10882, 23206}, {12635, 30282}, {12717, 16566}, {14110, 52027}, {14450, 31266}, {15172, 31146}, {15446, 30323}, {16209, 41389}, {16418, 31794}, {18398, 38316}, {18493, 37532}, {18499, 37584}, {19535, 53054}, {21616, 31231}, {23085, 37620}, {24467, 34773}, {25728, 46937}, {26364, 31142}, {27065, 46930}, {31053, 31888}, {31249, 34753}, {31445, 36279}

X(54290) = reflection of X(i) in X(j) for these {i,j}: {5229, 10}, {9612, 26066}
X(54290) = X(43533)-Ceva conjugate of X(1)
X(54290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 51576, 16370}, {40, 1158, 10860}, {46, 191, 9}, {57, 12514, 31435}, {484, 41229, 1706}, {986, 1707, 1453}, {3218, 5250, 3333}, {3579, 3927, 200}, {3869, 4652, 3576}, {3929, 5128, 10}, {4018, 16370, 1}, {5119, 6763, 6762}, {5880, 18253, 1698}, {6734, 44447, 41869}

X(54291) = X(1)X(76)∩X(2)X(11)

Barycentrics    a^4*b + a^2*b^3 + a^4*c + a*b^3*c + a*b^2*c^2 + b^3*c^2 + a^2*c^3 + a*b*c^3 + b^2*c^3 : :

X(54291) lies on these lines: {1, 76}, {2, 11}, {8, 2176}, {42, 312}, {43, 3886}, {75, 40934}, {238, 18900}, {291, 3923}, {518, 24514}, {672, 4676}, {740, 33931}, {982, 24259}, {984, 4368}, {1279, 21264}, {1281, 4376}, {1386, 17027}, {1479, 52256}, {1575, 49484}, {1914, 52133}, {2239, 30942}, {2263, 7196}, {2276, 3685}, {3242, 4713}, {3246, 17028}, {3416, 31027}, {3616, 26978}, {3720, 31005}, {3741, 3883}, {3783, 32941}, {3974, 20012}, {4307, 30962}, {4335, 10436}, {4363, 36222}, {4441, 32922}, {4443, 24425}, {4465, 36534}, {4514, 31330}, {4645, 30945}, {4872, 24723}, {5695, 17759}, {6327, 30965}, {7290, 17026}, {10453, 37676}, {15171, 37148}, {15569, 17032}, {16720, 32117}, {17018, 32926}, {17147, 31115}, {17233, 18082}, {17793, 49473}, {21299, 28369}, {21904, 28581}, {24210, 25385}, {24260, 29668}, {24330, 24349}, {24549, 48900}, {29706, 50637}, {30545, 42289}, {30953, 33106}, {30961, 33126}, {30966, 50295}, {30969, 33104}, {30985, 33124}, {31006, 33112}, {31028, 50289}, {36844, 37193}, {37580, 41236}, {41142, 50126}, {41794, 49483}, {48822, 48841}

X(54291) = crossdifference of every pair of points on line {665, 46386}
X(54291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40718, 37632}, {2, 13576, 4429}, {3242, 4713, 17794}, {14942, 32942, 5263}

X(54292) = X(1)X(4)∩X(65)X(81)

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

X(54292) lies on these lines: {1, 4}, {12, 37983}, {56, 5262}, {58, 7098}, {65, 81}, {86, 664}, {109, 4424}, {201, 5247}, {221, 37614}, {227, 37539}, {284, 32674}, {517, 3955}, {534, 2263}, {551, 1421}, {603, 986}, {758, 2003}, {975, 10588}, {1038, 1788}, {1060, 18391}, {1061, 18533}, {1062, 4305}, {1319, 7191}, {1390, 34056}, {1393, 37607}, {1450, 29821}, {1455, 3666}, {1468, 37591}, {1470, 4850}, {1610, 1829}, {1735, 37469}, {1758, 5429}, {1774, 5119}, {1880, 2303}, {1935, 2292}, {1993, 3869}, {2006, 3822}, {2078, 49480}, {2099, 4318}, {2594, 34772}, {2831, 44302}, {3057, 41733}, {3256, 4868}, {3340, 4347}, {3891, 38460}, {3920, 5252}, {3924, 37523}, {4297, 33178}, {4351, 5902}, {4354, 5441}, {4367, 42751}, {4417, 4511}, {4551, 30115}, {4861, 4968}, {5251, 16577}, {5724, 51421}, {5919, 30621}, {8614, 45288}, {9578, 30142}, {9627, 10543}, {17011, 44733}, {17074, 18838}, {17869, 37157}, {18447, 37361}, {24806, 49487}, {30145, 37709}, {34046, 37549}, {37399, 41600}, {37558, 52564}, {37736, 49686}, {39766, 52358}, {41575, 52362}, {47057, 49682}

X(54292) = barycentric product X(226)*X(17512)
X(54292) = barycentric quotient X(17512)/X(333)
X(54292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 34, 3485}, {1, 581, 45230}, {1, 10572, 6198}, {1, 21147, 388}, {4296, 17016, 65}, {4318, 17015, 2099}

X(54293) = X(6)X(19)∩X(11)X(33)

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

X(54293) lies on these lines: {1, 475}, {4, 990}, {6, 19}, {11, 33}, {25, 3752}, {75, 11109}, {318, 32922}, {378, 37817}, {975, 52252}, {1040, 26052}, {1086, 1892}, {1104, 1593}, {1191, 1902}, {1279, 7071}, {1452, 24443}, {1722, 46878}, {1870, 11041}, {1878, 38530}, {1890, 4312}, {2332, 16780}, {2550, 34231}, {2999, 3192}, {3914, 11393}, {4008, 36123}, {4200, 5262}, {4429, 5081}, {5480, 23982}, {5728, 23052}, {5819, 40065}, {7290, 8750}, {9593, 41320}, {11392, 23536}, {16706, 17555}, {17720, 26020}, {24789, 25985}, {24989, 32774}, {37305, 41230}, {40987, 54234}

X(54293) = barycentric product X(i)*X(j) for these {i,j}: {34, 28795}, {92, 22769}
X(54293) = barycentric quotient X(i)/X(j) for these {i,j}: {22769, 63}, {28795, 3718}
X(54293) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1861, 23050}, {6, 1876, 42856}, {2362, 16232, 20613}

X(54294) = X(4)X(9)∩X(8)X(27)

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

X(54294) lies on these lines: {1, 7490}, {4, 9}, {8, 27}, {28, 55}, {65, 278}, {196, 225}, {270, 17126}, {377, 3101}, {380, 5717}, {407, 52082}, {443, 10319}, {469, 9780}, {515, 37379}, {607, 3194}, {938, 37389}, {944, 7554}, {1119, 3339}, {1172, 5711}, {1715, 14647}, {1782, 3474}, {1788, 41342}, {1824, 7952}, {1829, 4196}, {1838, 2093}, {1841, 4646}, {1871, 31788}, {2282, 2357}, {2355, 28076}, {2475, 9536}, {3085, 3198}, {3189, 31900}, {3197, 5706}, {3332, 40660}, {3487, 18673}, {3617, 6994}, {3925, 5142}, {4185, 11406}, {4198, 5174}, {4219, 5584}, {4329, 25015}, {4761, 17926}, {4972, 5125}, {5084, 9816}, {5146, 5183}, {5230, 18678}, {5261, 7282}, {5603, 7543}, {5687, 37377}, {5690, 7534}, {5790, 7546}, {5799, 37381}, {6047, 37567}, {6353, 39586}, {6826, 8251}, {6839, 9537}, {6851, 15941}, {6917, 8141}, {6995, 39570}, {7497, 10306}, {7498, 19859}, {7501, 10902}, {10268, 37028}, {11445, 41723}, {12702, 15762}, {14017, 37601}, {17917, 28629}, {17924, 50499}, {18453, 44229}, {21677, 31902}, {30503, 37417}, {30686, 39579}, {31922, 46883}, {37383, 44695}, {37550, 44696}

X(54294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {19, 1869, 4}, {19, 11471, 7713}, {1824, 37384, 7952}, {4185, 11406, 41227}

X(54295) = X(1)X(3)∩X(10)X(33)

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

X(54295) lies on these lines: {1, 3}, {6, 12711}, {8, 3100}, {9, 607}, {10, 33}, {20, 21147}, {34, 516}, {42, 10393}, {72, 1854}, {73, 12520}, {78, 27379}, {108, 8899}, {200, 3704}, {212, 12514}, {222, 9943}, {223, 12565}, {227, 7580}, {255, 1158}, {355, 36985}, {388, 990}, {390, 5262}, {614, 12053}, {651, 9961}, {728, 2324}, {774, 1708}, {950, 3755}, {962, 34036}, {971, 9370}, {975, 5218}, {997, 22072}, {1074, 26332}, {1103, 1490}, {1253, 2292}, {1394, 10860}, {1448, 3474}, {1455, 37022}, {1486, 44545}, {1616, 17622}, {1698, 9817}, {1699, 19372}, {1709, 1935}, {1717, 10827}, {1721, 9579}, {1722, 9581}, {1724, 30223}, {1745, 12940}, {1763, 3556}, {1766, 20613}, {1852, 41869}, {1870, 6361}, {2331, 8804}, {2817, 36986}, {3208, 7105}, {3465, 17857}, {3554, 5301}, {3617, 9539}, {3679, 9576}, {3702, 27394}, {3868, 8271}, {4200, 45281}, {4296, 9778}, {4300, 45126}, {4313, 17016}, {4318, 20070}, {4329, 8900}, {4342, 30148}, {4347, 5493}, {4354, 10573}, {5179, 17905}, {5272, 50443}, {5657, 6198}, {5687, 51361}, {5690, 8144}, {5930, 30265}, {6001, 7078}, {6261, 22350}, {7191, 9785}, {7719, 25087}, {8256, 9639}, {9643, 11362}, {9906, 12910}, {9907, 12911}, {9911, 11398}, {10167, 34046}, {10703, 11682}, {11495, 15832}, {12527, 45275}, {12688, 34048}, {12699, 37697}, {13724, 15503}, {16870, 21075}, {17903, 21062}, {23528, 27378}, {24430, 41229}, {26446, 37696}, {49553, 52427}

X(54295) = reflection of X(1) in X(1062)
X(54295) = X(i)-Ceva conjugate of X(j) for these (i,j): {345, 9}, {4329, 1763}
X(54295) = X(i)-isoconjugate of X(j) for these (i,j): {7, 7169}, {28, 47344}, {56, 7219}, {57, 7097}, {77, 40169}, {604, 40015}
X(54295) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 7219}, {19, 278}, {3161, 40015}, {5452, 7097}, {40180, 7197}, {40591, 47344}
X(54295) = crossdifference of every pair of points on line {650, 51644}
X(54295) = barycentric product X(i)*X(j) for these {i,j}: {1, 27540}, {8, 1763}, {9, 4329}, {21, 21062}, {55, 20914}, {78, 17903}, {312, 3556}, {318, 22119}, {333, 52359}, {345, 36103}, {644, 21174}, {3718, 21148}
X(54295) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 40015}, {9, 7219}, {41, 7169}, {55, 7097}, {71, 47344}, {607, 40169}, {1763, 7}, {3556, 57}, {4329, 85}, {8900, 7365}, {17903, 273}, {20914, 6063}, {21062, 1441}, {21148, 34}, {21174, 24002}, {22119, 77}, {27540, 75}, {36103, 278}, {40183, 7197}, {52359, 226}
X(54295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40, 8270}, {1, 165, 1038}, {1, 30503, 37558}, {1103, 1490, 4551}, {1697, 33178, 1}, {1854, 7074, 72}, {3556, 52359, 1763}

X(54296) = X(3)X(74)∩X(6)X(31)

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

X(54296) lies on these lines: {3, 74}, {6, 31}, {21, 48917}, {199, 1495}, {228, 52405}, {323, 4184}, {394, 19346}, {405, 12702}, {464, 39874}, {572, 44109}, {573, 34417}, {1985, 33108}, {2194, 17454}, {2249, 28841}, {3098, 16064}, {5235, 46521}, {5278, 49719}, {7430, 12112}, {13632, 14389}, {13738, 35239}, {15018, 37510}, {15032, 37120}, {15107, 20834}, {17524, 51340}, {19732, 32947}, {20835, 33878}, {30944, 37633}, {37499, 41424}

X(54296) = crossdifference of every pair of points on line {514, 1637}
X(54296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1495, 22080, 37508}, {1495, 37508, 199}, {2328, 22080, 199}, {2328, 37508, 1495}

X(54297) = X(3)X(6)∩X(13)X(83)

Barycentrics    a^2*(Sqrt[3]*(a^2*(a^2 - b^2 - c^2) - 2*b^2*c^2) + 2*a^2*S) : :

X(54297) lies on these lines: {2, 34395}, {3, 6}, {5, 46054}, {13, 83}, {14, 14880}, {18, 98}, {76, 22689}, {202, 10799}, {298, 619}, {384, 22687}, {395, 47610}, {397, 47859}, {550, 22522}, {616, 12214}, {627, 7836}, {628, 7779}, {629, 22894}, {636, 6783}, {729, 39637}, {1506, 6771}, {2004, 10601}, {3170, 44109}, {3171, 15066}, {3181, 7793}, {3200, 3203}, {3406, 43539}, {3457, 15018}, {3458, 15080}, {3643, 10351}, {4027, 5980}, {5012, 34394}, {5182, 5463}, {5254, 46855}, {5321, 31703}, {5699, 32115}, {5868, 41041}, {5872, 10104}, {5978, 34508}, {6115, 6694}, {6151, 41472}, {6636, 21462}, {6774, 7749}, {6778, 37825}, {7006, 12835}, {7746, 46053}, {7748, 46854}, {7787, 10653}, {7808, 11306}, {7815, 43274}, {8150, 33482}, {10358, 42813}, {10359, 40693}, {10788, 42151}, {10796, 16965}, {11295, 41107}, {11307, 36766}, {12110, 42158}, {12150, 35931}, {12192, 36209}, {12203, 16964}, {12204, 36967}, {13084, 33274}, {13196, 52194}, {13881, 22891}, {17128, 42675}, {18501, 42155}, {18502, 36969}, {22846, 39565}, {22855, 44777}, {22862, 42099}, {23006, 42990}, {32134, 42148}, {32465, 32467}, {33388, 37334}, {34540, 42089}, {37340, 51159}, {37835, 48655}

X(54297) = midpoint of X(3364) and X(3365)
X(54297) = Brocard-circle-inverse of X(3104)
X(54297) = isogonal conjugate of the polar conjugate of X(16250)
X(54297) = barycentric product X(3)*X(16250)
X(54297) = barycentric quotient X(16250)/X(264)
X(54297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 3104}, {15, 62, 39}, {15, 1691, 36759}, {15, 10646, 36756}, {16, 61, 3105}, {62, 3398, 36759}, {182, 36760, 36759}, {371, 372, 3107}, {1342, 1343, 15}, {1506, 6771, 52642}, {1687, 1688, 62}, {3104, 3107, 32452}, {3107, 39750, 36759}, {3389, 3390, 3106}, {53430, 53464, 5}

X(54298) = X(3)X(6)∩X(14)X(83)

Barycentrics    a^2*(Sqrt[3]*(a^2*(a^2 - b^2 - c^2) - 2*b^2*c^2) - 2*a^2*S) : :

X(54298) lies on these lines: {2, 34394}, {3, 6}, {5, 46053}, {13, 14880}, {14, 83}, {17, 98}, {76, 22687}, {203, 10799}, {299, 618}, {384, 22689}, {396, 47611}, {398, 47860}, {550, 22523}, {617, 12213}, {627, 7779}, {628, 7836}, {630, 22850}, {635, 6782}, {729, 39636}, {1506, 6774}, {2005, 10601}, {2981, 41473}, {3170, 15066}, {3171, 44109}, {3180, 7793}, {3201, 3203}, {3406, 43538}, {3457, 15080}, {3458, 15018}, {3642, 10351}, {4027, 5981}, {5012, 34395}, {5182, 5464}, {5254, 46854}, {5318, 31704}, {5700, 32115}, {5869, 41040}, {5873, 10104}, {5979, 34509}, {6114, 6695}, {6636, 21461}, {6771, 7749}, {6777, 37824}, {7005, 12835}, {7746, 46054}, {7748, 46855}, {7787, 10654}, {7808, 11305}, {7815, 43275}, {8150, 33483}, {10358, 42814}, {10359, 40694}, {10788, 42150}, {10796, 16964}, {11296, 41108}, {12110, 42157}, {12150, 35932}, {12192, 36208}, {12203, 16965}, {12205, 36968}, {13083, 33274}, {13196, 52193}, {13881, 22846}, {17128, 42674}, {18501, 42154}, {18502, 36970}, {22891, 39565}, {22901, 44776}, {22906, 42100}, {23013, 42991}, {32134, 42147}, {32466, 32467}, {33389, 37334}, {34541, 42092}, {36772, 42433}, {37341, 51160}, {37832, 48656}, {41018, 42238}

X(54298) = midpoint of X(3389) and X(3390)
X(54298) = Brocard-circle-inverse of X(3105)
X(54298) = isogonal conjugate of the polar conjugate of X(16249)
X(54298) = barycentric product X(3)*X(16249)
X(54298) = barycentric quotient X(16249)/X(264)
X(54298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 3105}, {15, 62, 3104}, {16, 61, 39}, {16, 1691, 36760}, {16, 10645, 36755}, {61, 3398, 36760}, {182, 36759, 36760}, {371, 372, 3106}, {1342, 1343, 16}, {1506, 6774, 52643}, {1687, 1688, 61}, {3105, 3106, 32452}, {3106, 39750, 36760}, {3364, 3365, 3107}, {53442, 53453, 5}

X(54299) = X(2)X(3)∩X(33)X(72)

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

X(54299) lies on these lines: {2, 3}, {33, 72}, {55, 39585}, {92, 3295}, {281, 5687}, {318, 1260}, {954, 7952}, {1001, 1838}, {1096, 3931}, {1708, 1887}, {1712, 5728}, {1859, 12514}, {1871, 5250}, {1940, 37541}, {3075, 10396}, {3419, 46878}, {3697, 7079}, {3753, 11471}, {4254, 8748}, {5174, 9708}, {7008, 9844}, {7071, 41013}, {9709, 52412}, {10267, 39529}, {11500, 39574}, {12699, 30687}, {36744, 46835}

X(54299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 29, 37393}, {4, 25, 37377}, {4, 406, 442}, {4, 4183, 405}, {4, 4194, 37321}, {4, 7412, 7580}, {4, 16845, 4200}, {4, 30733, 4185}, {21, 7518, 7497}, {29, 1013, 3}, {4186, 11323, 4}, {4219, 7498, 474}, {17516, 37318, 4}, {28044, 37194, 406}

X(54300) = X(2)X(12)∩X(3)X(6)

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

X(54300) lies on these lines: {1, 16287}, {2, 12}, {3, 6}, {8, 35999}, {36, 3216}, {42, 19339}, {55, 16452}, {73, 28274}, {78, 2352}, {238, 23383}, {241, 1410}, {283, 44085}, {333, 404}, {387, 3428}, {474, 5737}, {604, 46879}, {859, 1724}, {956, 5793}, {960, 1402}, {976, 16687}, {978, 20470}, {999, 16286}, {1001, 16289}, {1011, 19765}, {1043, 4203}, {1107, 37609}, {1125, 16288}, {1191, 23853}, {1193, 16678}, {1211, 37255}, {1376, 9534}, {1470, 16295}, {1737, 15232}, {1764, 18178}, {1834, 4192}, {3145, 5347}, {3149, 5786}, {3454, 19258}, {3868, 28936}, {3911, 34831}, {3913, 20018}, {3941, 37552}, {4184, 5331}, {4188, 37652}, {4225, 32911}, {4383, 13738}, {4417, 37030}, {4426, 36025}, {4641, 22345}, {4719, 37575}, {5143, 23844}, {5204, 16451}, {5248, 16300}, {5255, 15621}, {5292, 11249}, {5293, 20990}, {5313, 39578}, {5329, 23843}, {5438, 21384}, {5563, 16296}, {5687, 16400}, {5718, 37225}, {5747, 16848}, {5788, 6911}, {7373, 48855}, {7420, 37732}, {7428, 39748}, {8053, 37573}, {8583, 16878}, {8666, 50605}, {8715, 50588}, {9709, 48852}, {10449, 12513}, {11344, 37538}, {13731, 37662}, {15509, 37264}, {15622, 37570}, {15668, 19518}, {16294, 37579}, {16357, 25526}, {16374, 37522}, {16414, 17749}, {16454, 19769}, {16455, 22766}, {17277, 37442}, {18397, 20803}, {19247, 48867}, {19259, 43531}, {19283, 19701}, {19513, 37646}, {19841, 41258}, {20142, 27656}, {21935, 40109}, {22765, 45939}, {27622, 35466}, {27649, 37681}, {27659, 37694}, {28238, 37663}, {30362, 33096}, {34280, 40453}, {34281, 40153}, {35239, 48857}, {37195, 37537}, {37231, 40980}, {37539, 40956}, {47521, 49745}

X(54300) = Brocard-circle-inverse of X(4267)
X(54300) = crossdifference of every pair of points on line {523, 52326}
X(54300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 4267}, {3, 386, 5132}, {3, 19762, 3286}, {3, 37502, 19760}, {3, 37507, 4252}, {36, 3216, 16453}, {36, 5247, 23361}, {1220, 2975, 958}, {4255, 19759, 3}, {4256, 4278, 3}, {16452, 19767, 55}

X(54301) = X(1)X(6)∩X(35)X(47)

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

X(54301) lies on these lines: {1, 6}, {3, 2003}, {34, 5903}, {35, 47}, {36, 54}, {43, 1771}, {46, 223}, {57, 3157}, {58, 1167}, {78, 1993}, {80, 40950}, {81, 13411}, {155, 5720}, {201, 8555}, {221, 2093}, {222, 15803}, {255, 386}, {394, 936}, {404, 22128}, {498, 5712}, {595, 40958}, {651, 4292}, {692, 42450}, {908, 3193}, {912, 33178}, {937, 42019}, {942, 23071}, {965, 51507}, {970, 3955}, {1046, 1735}, {1155, 8614}, {1181, 1490}, {1210, 3562}, {1399, 2077}, {1427, 3336}, {1451, 5563}, {1466, 23072}, {1498, 1750}, {1745, 1754}, {1772, 6126}, {1785, 3194}, {1870, 15556}, {1936, 37732}, {1994, 34772}, {2078, 5399}, {2183, 2360}, {2361, 2594}, {2964, 32760}, {3072, 4551}, {3075, 3216}, {3100, 41562}, {3145, 23202}, {3340, 44414}, {3601, 36742}, {3746, 14547}, {4303, 13329}, {4383, 41344}, {4641, 17102}, {5044, 22136}, {5219, 5707}, {5285, 5752}, {5312, 11507}, {5313, 8071}, {5396, 52408}, {5398, 37583}, {5537, 38857}, {5584, 38293}, {5703, 37685}, {5704, 14997}, {5706, 9612}, {5711, 31434}, {5713, 7951}, {5716, 12647}, {5717, 10039}, {6149, 52544}, {6282, 37498}, {7066, 11428}, {7193, 29958}, {7592, 18446}, {8726, 37514}, {8757, 9579}, {9370, 9613}, {10573, 34231}, {11010, 15852}, {11429, 40944}, {12161, 37700}, {14557, 40660}, {16577, 26878}, {16853, 22142}, {16948, 17010}, {18443, 36752}, {22072, 37469}, {22076, 26890}, {23070, 37582}, {23154, 26889}, {24929, 36750}, {30282, 36746}, {36747, 37531}, {36749, 37533}, {36753, 37615}, {37530, 37694}, {41227, 52413}

X(54301) = barycentric product X(63)*X(6197)
X(54301) = barycentric quotient X(6197)/X(92)
X(54301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1743, 1728}, {6, 7078, 1}, {73, 580, 36}, {212, 581, 35}, {222, 36745, 15803}, {223, 19349, 34043}, {942, 37509, 52423}, {1124, 1335, 2256}, {2361, 2594, 10902}, {3157, 36754, 57}, {3562, 32911, 1210}, {5706, 34048, 9612}, {23071, 37509, 942}

X(54302) = X(1)X(21)∩X(30)X(84)

Barycentrics    a*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*b*c - 3*a^3*b^2*c + a^2*b^3*c + 3*a*b^4*c - 3*a^4*c^2 - 3*a^3*b*c^2 + a*b^3*c^2 + b^4*c^2 + a^2*b*c^3 + a*b^2*c^3 + 3*a^2*c^4 + 3*a*b*c^4 + b^2*c^4 - c^6) : :
X(54302) = 4 X[8666] - X[16126], 4 X[5428] - X[11523], X[6762] + 2 X[16139]

X(54302) lies on these lines: {1, 21}, {3, 2949}, {7, 11263}, {9, 6675}, {10, 15932}, {20, 1768}, {30, 84}, {35, 16465}, {40, 44238}, {46, 10042}, {56, 44782}, {57, 442}, {72, 37308}, {78, 27086}, {79, 5231}, {224, 7280}, {377, 3336}, {518, 10902}, {936, 1708}, {942, 15823}, {958, 8261}, {1004, 37524}, {1012, 37625}, {1071, 11012}, {1259, 5904}, {1445, 5785}, {1729, 21384}, {1761, 40979}, {1998, 35989}, {2323, 37565}, {2475, 3218}, {2771, 11249}, {3065, 43740}, {3219, 15674}, {3306, 31254}, {3333, 11281}, {3337, 5249}, {3338, 26725}, {3359, 11919}, {3648, 26015}, {3651, 5732}, {3870, 31660}, {3916, 10391}, {3929, 15670}, {4269, 18598}, {4304, 41575}, {5131, 35976}, {5252, 21677}, {5273, 10198}, {5428, 11523}, {5535, 33961}, {5536, 10916}, {5692, 37248}, {5693, 37302}, {5715, 6841}, {5735, 7701}, {5784, 37582}, {5902, 37228}, {6264, 12513}, {6597, 16159}, {6700, 37787}, {6762, 7966}, {6765, 10268}, {9965, 10527}, {10267, 22937}, {10399, 11344}, {10529, 31888}, {10680, 13465}, {12535, 13131}, {12540, 49193}, {12625, 37584}, {12649, 15680}, {12660, 33668}, {12671, 37623}, {12695, 13743}, {13243, 48713}, {15803, 31938}, {16113, 45632}, {16143, 30304}, {17637, 26357}, {18165, 45038}, {18219, 21669}, {18259, 24541}, {22836, 37106}, {24299, 28443}, {24477, 48482}, {25440, 41228}, {28610, 45700}, {31446, 37719}, {34744, 40256}, {37230, 37532}, {37579, 41542}, {37611, 54212}

X(54302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 63, 191}, {21, 39772, 1}, {7701, 12704, 49177}

X(54303) = X(2)X(6)∩X(8)X(77)

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

X(54303) lies on these lines: {2, 6}, {8, 77}, {75, 279}, {269, 4847}, {271, 307}, {309, 321}, {314, 32830}, {326, 20007}, {346, 24635}, {1014, 6904}, {1444, 3522}, {2893, 37421}, {3086, 17272}, {3663, 34625}, {3664, 19843}, {3926, 34282}, {3964, 4189}, {4194, 32001}, {4200, 32000}, {4357, 14986}, {4416, 27508}, {7080, 32099}, {8232, 28797}, {9723, 17548}, {10432, 10446}, {10527, 21296}, {11679, 18623}, {14615, 34284}, {17360, 27525}, {18738, 44147}, {24471, 24477}, {31995, 36595}, {32834, 44139}, {45700, 53598}

X(54303) = isotomic conjugate of the isogonal conjugate of X(37501)
X(54303) = isotomic conjugate of the polar conjugate of X(37276)
X(54303) = barycentric product X(i)*X(j) for these {i,j}: {69, 37276}, {76, 37501}
X(54303) = barycentric quotient X(i)/X(j) for these {i,j}: {37276, 4}, {37501, 6}
X(54303) = {X(1270),X(1271)}-harmonic conjugate of X(5739)

X(54304) = X(7)X(8)∩X(30)X(90)

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

X(54304) lies on these lines: {1, 37438}, {3, 1737}, {7, 8}, {10, 11509}, {30, 90}, {46, 355}, {56, 3419}, {57, 47033}, {72, 18961}, {80, 38761}, {200, 26482}, {224, 41552}, {442, 997}, {517, 10043}, {519, 41540}, {758, 7702}, {950, 52769}, {1155, 6934}, {1158, 37468}, {1210, 22768}, {1319, 10529}, {1454, 11112}, {1470, 6734}, {1478, 41538}, {1788, 4190}, {1836, 5887}, {1858, 6850}, {1898, 6925}, {2099, 12609}, {2245, 54008}, {2646, 3086}, {3358, 5691}, {3476, 5178}, {3486, 37112}, {3612, 5433}, {4863, 37738}, {4930, 17528}, {5260, 25005}, {5704, 37600}, {5787, 15326}, {6833, 17606}, {6897, 18391}, {6984, 17605}, {7098, 17579}, {9579, 41705}, {10044, 31794}, {10573, 34339}, {10826, 37356}, {12666, 41706}, {12701, 14110}, {13273, 18254}, {14988, 41688}, {17728, 22766}, {18838, 49168}, {18962, 41539}, {26475, 37611}, {31231, 37702}, {34447, 41506}, {37730, 44222}

X(54304) = {X(65),X(5794)}-harmonic conjugate of X(5252)

X(54305) = X(1)X(6)∩X(10)X(34)

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

X(54305) lies on these lines: {1, 6}, {3, 1763}, {10, 34}, {20, 5813}, {28, 169}, {33, 12572}, {40, 1593}, {63, 580}, {73, 997}, {78, 581}, {200, 5814}, {212, 12514}, {223, 936}, {241, 16410}, {443, 948}, {474, 1427}, {612, 5717}, {908, 5713}, {910, 37408}, {964, 52345}, {975, 5712}, {976, 40958}, {1040, 31424}, {1060, 5044}, {1062, 31445}, {1260, 37528}, {1773, 5329}, {2000, 2478}, {2267, 18673}, {2551, 34231}, {3073, 42012}, {3198, 37062}, {3305, 52362}, {3421, 5716}, {3488, 15954}, {3682, 45126}, {3811, 14547}, {3920, 5815}, {3929, 33178}, {5130, 5587}, {5285, 7713}, {5687, 15852}, {5705, 19372}, {5720, 5810}, {5752, 37531}, {5783, 30456}, {5791, 37697}, {6282, 14557}, {6554, 7498}, {6986, 24635}, {11396, 26867}, {17441, 37246}, {18443, 47371}, {18607, 37282}, {19843, 34036}, {21060, 30142}, {21370, 37431}, {26893, 37569}

X(54305) = {X(1829),X(7085)}-harmonic conjugate of X(40)

X(54306) = X(2)X(3)∩X(53)X(61)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*(a^2 - b^2 - c^2)^2 + 8*Sqrt[3]*S^3) : :

X(54306) lies on these lines: {2, 3}, {13, 35712}, {53, 61}, {62, 6748}, {389, 5318}, {396, 52670}, {398, 37505}, {578, 5321}, {4994, 51268}, {5334, 11426}, {5335, 11432}, {6116, 42598}, {6117, 42163}, {9786, 42094}, {9833, 41038}, {10662, 36747}, {11425, 42093}, {11430, 42101}, {11438, 42102}, {14216, 41039}, {16001, 16625}, {16002, 35715}, {20415, 35714}, {20416, 31688}, {32002, 52194}

X(54306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 473, 5}

X(54307) = X(2)X(3)∩X(53)X(62)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*(a^2 - b^2 - c^2)^2 - 8*Sqrt[3]*S^3) : :

X(54307) lies on these lines: {2, 3}, {14, 35713}, {53, 62}, {61, 6748}, {389, 5321}, {395, 52671}, {397, 37505}, {578, 5318}, {4994, 51275}, {5334, 11432}, {5335, 11426}, {6116, 42166}, {6117, 42599}, {9786, 42093}, {9833, 41039}, {10661, 36747}, {11425, 42094}, {11430, 42102}, {11438, 42101}, {14216, 41038}, {16001, 35714}, {16002, 16625}, {20415, 31687}, {20416, 35715}, {32002, 52193}

X(54307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 472, 5}

X(54308) = X(1)X(75)∩X(2)X(5105)

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

X(54308) lies on these lines: {1, 75}, {2, 5105}, {6, 980}, {7, 10571}, {9, 27644}, {21, 7290}, {37, 52897}, {42, 3879}, {43, 17270}, {57, 77}, {58, 988}, {69, 386}, {141, 5153}, {142, 16752}, {193, 4270}, {239, 16738}, {261, 1178}, {269, 1434}, {319, 3293}, {333, 2999}, {524, 4272}, {614, 10458}, {664, 31643}, {757, 763}, {940, 16700}, {969, 994}, {995, 17321}, {1019, 21143}, {1100, 16726}, {1193, 4357}, {1386, 3286}, {1429, 18724}, {1442, 17016}, {2092, 3882}, {2300, 3666}, {2663, 18787}, {3187, 27163}, {3216, 5224}, {3240, 32099}, {3305, 27643}, {3589, 5109}, {3629, 4285}, {3663, 17139}, {3664, 16714}, {3672, 17183}, {3677, 5208}, {3759, 29767}, {3786, 7174}, {3945, 6904}, {3946, 17197}, {4184, 16688}, {4264, 21511}, {4267, 4719}, {4281, 17206}, {4384, 27164}, {4393, 17178}, {4596, 4674}, {4967, 10459}, {5037, 37685}, {5110, 5337}, {5145, 18906}, {5222, 16713}, {5269, 13588}, {5283, 27623}, {5287, 5333}, {5313, 17272}, {7032, 17017}, {7146, 18177}, {8025, 17011}, {10461, 37592}, {10477, 50598}, {16589, 28252}, {16666, 18198}, {16667, 18186}, {16704, 17012}, {16710, 17379}, {16711, 50116}, {16712, 17274}, {16736, 37674}, {16742, 28358}, {16750, 40719}, {16753, 37633}, {16755, 21173}, {16831, 25508}, {17013, 26860}, {17014, 26818}, {17019, 31025}, {17022, 25507}, {17167, 19785}, {17173, 33150}, {17174, 33155}, {17179, 17378}, {17202, 17302}, {17212, 48281}, {17296, 30965}, {17322, 49997}, {17594, 38832}, {18171, 20963}, {18190, 18208}, {18204, 18207}, {19717, 39747}, {21796, 29429}, {24557, 25930}, {26042, 30114}, {26149, 26813}, {26819, 45222}, {26979, 29456}, {31855, 32025}, {33297, 50581}, {34020, 37678}, {50178, 53476}

X(54308) = isogonal conjugate of the isotomic conjugate of X(16739)
X(54308) = X(i)-Ceva conjugate of X(j) for these (i,j): {86, 4357}, {664, 7192}, {4610, 1019}, {7260, 18197}, {16705, 17185}, {37215, 52615}
X(54308) = X(i)-isoconjugate of X(j) for these (i,j): {6, 14624}, {37, 2298}, {42, 1220}, {210, 961}, {213, 30710}, {512, 8707}, {523, 32736}, {594, 1169}, {661, 36147}, {756, 2363}, {1240, 1918}, {1500, 14534}, {1791, 1824}, {1798, 7140}, {1826, 2359}, {3700, 8687}, {3709, 6648}, {4041, 36098}, {4557, 4581}, {7109, 40827}, {14973, 40453}
X(54308) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 14624}, {960, 756}, {1193, 14973}, {1211, 10}, {2092, 2321}, {3125, 4024}, {3666, 1089}, {4357, 3963}, {6626, 30710}, {17197, 522}, {17419, 3700}, {34021, 1240}, {36830, 36147}, {38992, 4041}, {39015, 661}, {39054, 8707}, {40589, 2298}, {40592, 1220}, {52087, 37}
X(54308) = cevapoint of X(i) and X(j) for these (i,j): {1193, 3666}, {4267, 40153}
X(54308) = crossdifference of every pair of points on line {798, 4041}
X(54308) = barycentric product X(i)*X(j) for these {i,j}: {1, 16705}, {6, 16739}, {7, 17185}, {21, 3674}, {58, 20911}, {60, 45196}, {75, 40153}, {81, 4357}, {85, 4267}, {86, 3666}, {99, 48131}, {110, 4509}, {274, 1193}, {279, 46877}, {286, 22097}, {310, 2300}, {333, 24471}, {552, 21033}, {593, 18697}, {662, 3004}, {757, 1211}, {763, 20653}, {799, 6371}, {849, 1228}, {873, 2092}, {960, 1434}, {1014, 3687}, {1019, 53332}, {1088, 46889}, {1414, 3910}, {1444, 1848}, {1509, 2292}, {1829, 17206}, {2185, 41003}, {3882, 7192}, {4573, 17420}, {4610, 50330}, {4625, 52326}, {6628, 21810}, {7199, 53280}, {7303, 27697}, {7304, 45197}, {21124, 52935}, {22345, 44129}, {27455, 33296}, {28369, 32010}
X(54308) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14624}, {58, 2298}, {81, 1220}, {86, 30710}, {110, 36147}, {163, 32736}, {274, 1240}, {444, 1840}, {593, 2363}, {662, 8707}, {757, 14534}, {849, 1169}, {873, 40827}, {960, 2321}, {1019, 4581}, {1193, 37}, {1211, 1089}, {1412, 961}, {1414, 6648}, {1434, 31643}, {1437, 2359}, {1634, 35334}, {1682, 21033}, {1790, 1791}, {1829, 1826}, {1848, 41013}, {2092, 756}, {2269, 210}, {2292, 594}, {2300, 42}, {2354, 1824}, {3004, 1577}, {3666, 10}, {3674, 1441}, {3687, 3701}, {3725, 1500}, {3882, 3952}, {3910, 4086}, {3965, 4082}, {4267, 9}, {4357, 321}, {4509, 850}, {4565, 36098}, {4719, 5257}, {6371, 661}, {16705, 75}, {16739, 76}, {17185, 8}, {17420, 3700}, {18235, 4095}, {18697, 28654}, {20911, 313}, {20967, 1334}, {21033, 6057}, {21124, 4036}, {21810, 6535}, {22074, 2318}, {22076, 3949}, {22097, 72}, {22345, 71}, {24471, 226}, {27455, 42027}, {28369, 1215}, {40153, 1}, {41003, 6358}, {41581, 21073}, {41591, 21065}, {41600, 21074}, {45196, 34388}, {45218, 7148}, {46877, 346}, {46889, 200}, {48131, 523}, {50330, 4024}, {52087, 14973}, {52326, 4041}, {53280, 1018}, {53332, 4033}
X(54308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 980, 16574}, {6, 16696, 18206}, {75, 86, 10455}, {81, 25059, 18163}, {86, 274, 10436}, {86, 4360, 30939}, {86, 16709, 17175}, {86, 33296, 314}, {314, 33296, 3875}, {1100, 16726, 18166}, {1449, 18164, 81}, {2092, 28369, 3882}, {3666, 40153, 17185}, {16726, 18166, 17207}, {27644, 40773, 9}

X(54309) = X(1)X(2)∩X(44)X(100)

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

X(54309) lies on these lines: {1, 2}, {44, 100}, {88, 518}, {89, 3751}, {238, 678}, {390, 52429}, {536, 4767}, {1155, 14513}, {1443, 4551}, {1757, 9324}, {2177, 35595}, {3100, 52371}, {3218, 21805}, {3219, 17601}, {3243, 39963}, {3246, 3689}, {3681, 17595}, {3699, 17160}, {3711, 4850}, {4152, 28503}, {4414, 51297}, {4702, 4954}, {4724, 6006}, {4849, 37520}, {4893, 48352}, {4956, 30566}, {9330, 16676}, {9350, 27003}, {14410, 48244}, {15569, 40434}, {16670, 17126}, {17067, 33148}, {21060, 33102}, {21870, 37633}, {24344, 50127}, {24594, 51055}, {28580, 30578}, {37138, 37222}, {52959, 52966}

X(54309) = crossdifference of every pair of points on line {649, 14421}
X(54309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {899, 3935, 7292}, {899, 5524, 3935}, {5212, 49991, 32842}, {5529, 49984, 38460}

X(54310) = X(1)X(89)∩X(6)X(41)

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

X(54310) lies on these lines: {1, 89}, {3, 2177}, {6, 41}, {8, 37608}, {21, 8296}, {31, 999}, {32, 17474}, {35, 16490}, {36, 42}, {57, 49487}, {58, 106}, {81, 37617}, {145, 37603}, {187, 16971}, {213, 8649}, {392, 896}, {404, 3214}, {519, 37639}, {529, 37634}, {551, 8025}, {595, 16489}, {601, 1480}, {602, 16203}, {603, 26437}, {672, 2242}, {750, 956}, {840, 2701}, {940, 11194}, {976, 16496}, {978, 14997}, {993, 3720}, {995, 2308}, {997, 32912}, {1015, 21764}, {1064, 22765}, {1125, 19740}, {1334, 33863}, {1376, 49984}, {1385, 2650}, {1478, 29662}, {1616, 41436}, {1724, 28352}, {2975, 37607}, {3073, 45977}, {3241, 3550}, {3304, 3915}, {3333, 16485}, {3336, 15955}, {3338, 3924}, {3600, 5230}, {3616, 6646}, {3679, 5372}, {3722, 37589}, {3750, 17549}, {3751, 35262}, {3877, 4650}, {3973, 8583}, {3976, 16498}, {4188, 50581}, {4234, 32943}, {4273, 22357}, {4293, 11269}, {4300, 11249}, {4315, 5773}, {4317, 5292}, {4322, 37583}, {4383, 40726}, {4424, 4973}, {4511, 32913}, {4642, 37582}, {4656, 34646}, {4694, 29818}, {5021, 9310}, {5030, 16785}, {5165, 22356}, {5247, 5253}, {5251, 30950}, {5270, 45939}, {5280, 23649}, {5298, 37662}, {5303, 37573}, {5429, 7191}, {5434, 37646}, {5886, 24725}, {6048, 17572}, {7267, 49481}, {8162, 21000}, {8616, 38314}, {8626, 20985}, {8666, 10459}, {9340, 51788}, {9350, 16417}, {9708, 17124}, {11112, 33136}, {11114, 24217}, {11115, 50608}, {12577, 28027}, {15492, 25917}, {15950, 17365}, {16393, 32941}, {16466, 23070}, {16487, 28011}, {16493, 17109}, {16797, 33844}, {17015, 17596}, {17056, 31157}, {17117, 27368}, {17178, 49482}, {17455, 28658}, {17579, 33141}, {18990, 21935}, {21242, 50171}, {22361, 34471}, {24165, 39766}, {24443, 32636}, {25524, 28257}, {29571, 31039}, {33104, 45700}, {33771, 41434}, {34605, 37716}, {37539, 49465}, {37599, 46904}, {37600, 49478}, {37602, 40091}, {37817, 51816}, {41011, 44675}

X(54310) = isogonal conjugate of the isotomic conjugate of X(50116)
X(54310) = crossdifference of every pair of points on line {522, 4120}
X(54310) = barycentric product X(i)*X(j) for these {i,j}: {1, 37520}, {6, 50116}, {101, 47891}, {2163, 27747}
X(54310) = barycentric quotient X(i)/X(j) for these {i,j}: {37520, 75}, {47891, 3261}, {50116, 76}
X(54310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2163, 4257}, {1, 4257, 902}, {31, 999, 1149}, {36, 16474, 4256}, {56, 1468, 1193}, {58, 106, 5315}, {58, 5315, 21747}, {58, 5563, 1201}, {106, 5315, 1201}, {1201, 21747, 5315}, {2067, 6502, 1405}, {3304, 4252, 3915}, {4256, 16474, 42}, {4694, 49480, 29818}, {5247, 5253, 27627}, {5315, 5563, 106}, {7051, 19373, 1400}, {8666, 37522, 10459}

X(54311) = X(2)X(7)∩X(10)X(38)

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

X(54311) lies on these lines: {1, 26034}, {2, 7}, {6, 4001}, {10, 38}, {21, 7293}, {31, 1125}, {42, 49511}, {69, 5256}, {75, 18739}, {81, 5299}, {88, 31247}, {141, 306}, {191, 19881}, {239, 37653}, {312, 4389}, {313, 18136}, {319, 50306}, {321, 3663}, {333, 16706}, {345, 3619}, {354, 4026}, {386, 4101}, {404, 5314}, {405, 1473}, {474, 7085}, {516, 24552}, {519, 33074}, {551, 17469}, {594, 42051}, {614, 50295}, {846, 29637}, {896, 19862}, {899, 4104}, {902, 29686}, {940, 2214}, {942, 13728}, {950, 17676}, {958, 25904}, {964, 4292}, {982, 24163}, {984, 33174}, {993, 19869}, {1009, 22060}, {1038, 1457}, {1086, 31993}, {1150, 32774}, {1210, 5051}, {1211, 3752}, {1266, 28605}, {1698, 33163}, {1707, 3624}, {1738, 31330}, {1764, 12610}, {1999, 17302}, {2239, 43223}, {2308, 29684}, {2321, 17147}, {2887, 6682}, {2895, 17012}, {2999, 5739}, {3008, 5278}, {3011, 26128}, {3120, 31241}, {3175, 17246}, {3187, 3946}, {3210, 3661}, {3220, 37325}, {3416, 17599}, {3419, 11359}, {3488, 51665}, {3586, 50055}, {3589, 4641}, {3616, 37554}, {3617, 41915}, {3634, 26061}, {3664, 19684}, {3672, 34255}, {3687, 4850}, {3703, 3844}, {3717, 7226}, {3720, 50290}, {3739, 40688}, {3741, 3821}, {3744, 44419}, {3755, 17135}, {3763, 3977}, {3772, 37660}, {3782, 4054}, {3840, 4425}, {3879, 17011}, {3883, 7191}, {3891, 4353}, {3912, 28606}, {3916, 17698}, {3920, 33086}, {3944, 29827}, {3966, 49987}, {3969, 29594}, {3989, 4078}, {3998, 37597}, {4000, 5271}, {4028, 33081}, {4030, 49465}, {4035, 31017}, {4138, 33105}, {4205, 5439}, {4358, 4656}, {4360, 50292}, {4364, 44307}, {4383, 4643}, {4384, 24790}, {4392, 29667}, {4398, 42029}, {4414, 24943}, {4415, 30818}, {4416, 32911}, {4429, 25006}, {4431, 50106}, {4438, 30768}, {4640, 35263}, {4650, 25539}, {4652, 37176}, {4655, 25496}, {4660, 29652}, {4667, 19717}, {4675, 19701}, {4683, 32944}, {4684, 17018}, {4712, 24988}, {4847, 4972}, {4851, 20182}, {4855, 37339}, {4863, 48829}, {4886, 17271}, {4966, 37593}, {4970, 49560}, {4980, 53594}, {5119, 48803}, {5121, 25960}, {5192, 12572}, {5222, 14552}, {5224, 19804}, {5235, 26724}, {5241, 16602}, {5252, 48801}, {5263, 33068}, {5283, 29988}, {5287, 17321}, {5302, 25992}, {5550, 36277}, {5722, 50056}, {5737, 17290}, {5743, 16610}, {5847, 17017}, {6057, 49523}, {6536, 30950}, {6685, 33064}, {6703, 37520}, {6734, 16062}, {7174, 10327}, {8025, 21764}, {8362, 25083}, {8616, 29660}, {10468, 11679}, {10479, 23537}, {10856, 37419}, {12436, 16454}, {12514, 19836}, {12609, 19863}, {13369, 50324}, {13405, 33122}, {14213, 26538}, {14829, 17305}, {15315, 50605}, {15523, 46901}, {16060, 20769}, {16349, 25500}, {16368, 18650}, {16570, 34595}, {16815, 26044}, {16818, 18206}, {17020, 37656}, {17045, 37595}, {17046, 26601}, {17056, 48632}, {17123, 24697}, {17126, 29648}, {17127, 29666}, {17165, 26251}, {17185, 17192}, {17197, 27163}, {17227, 18134}, {17238, 17490}, {17247, 41839}, {17249, 18743}, {17253, 37679}, {17273, 33066}, {17278, 19732}, {17284, 17776}, {17285, 42033}, {17288, 17778}, {17289, 32939}, {17293, 50048}, {17307, 19808}, {17308, 19822}, {17320, 34064}, {17323, 50068}, {17325, 37674}, {17348, 49724}, {17355, 32933}, {17367, 37652}, {17376, 37631}, {17383, 37683}, {17392, 37869}, {17449, 29685}, {17526, 31424}, {17591, 32778}, {17592, 33087}, {17593, 33160}, {17594, 33171}, {17596, 32783}, {17598, 33076}, {17600, 32846}, {17811, 26006}, {18044, 19807}, {18139, 21255}, {18229, 23681}, {19857, 52782}, {19864, 21616}, {20043, 32099}, {20106, 33113}, {20582, 50104}, {20879, 26665}, {21075, 26030}, {21529, 23089}, {21620, 26115}, {22031, 26844}, {22230, 23636}, {23790, 47763}, {24175, 24589}, {24178, 31339}, {24210, 30942}, {24231, 32771}, {24239, 25760}, {24470, 50318}, {24564, 31359}, {24598, 29991}, {24723, 32942}, {24987, 37591}, {25058, 30965}, {25065, 42700}, {25101, 33761}, {25354, 25501}, {25881, 25914}, {25896, 25918}, {25958, 29680}, {25959, 29664}, {25982, 26066}, {26015, 32773}, {26091, 41012}, {26098, 29826}, {26104, 37642}, {26563, 45196}, {27162, 28254}, {28516, 48644}, {29596, 33157}, {29633, 32913}, {29650, 32946}, {29663, 32912}, {29819, 49684}, {29821, 33082}, {29828, 33144}, {29833, 37639}, {29841, 37684}, {29960, 40773}, {31136, 50091}, {31237, 50752}, {31264, 32856}, {32772, 33067}, {32775, 32918}, {32859, 53598}, {32917, 33123}, {32920, 50285}, {33078, 49476}, {33089, 39597}, {33091, 49527}, {34573, 44416}, {41711, 47358}

X(54311) = midpoint of X(i) and X(j) for these {i,j}: {17017, 33080}, {24552, 32950}
X(54311) = complement of X(26223)
X(54311) = X(15315)-complementary conjugate of X(141)
X(54311) = X(37218)-Ceva conjugate of X(514)
X(54311) = X(37592)-Dao conjugate of X(2345)
X(54311) = crossdifference of every pair of points on line {663, 50496}
X(54311) = barycentric product X(75)*X(37592)
X(54311) = barycentric quotient X(37592)/X(1)
X(54311) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 63, 5294}, {2, 3219, 17353}, {2, 3662, 5249}, {2, 6646, 27064}, {2, 9965, 5749}, {2, 17184, 226}, {2, 17236, 27184}, {2, 17257, 3305}, {2, 26132, 31266}, {2, 26580, 3452}, {2, 26840, 894}, {2, 27184, 908}, {10, 24177, 4359}, {38, 32781, 10}, {57, 17306, 2}, {141, 3666, 306}, {226, 50092, 17184}, {333, 16706, 26723}, {1150, 32774, 40940}, {2308, 29684, 38049}, {2887, 6682, 29639}, {2999, 17272, 5739}, {3741, 3821, 3914}, {3752, 17237, 1211}, {3782, 44417, 4054}, {3989, 29687, 4078}, {4655, 25496, 41011}, {4850, 32782, 3687}, {4972, 46909, 4847}, {5737, 17290, 24789}, {6646, 27064, 17781}, {7191, 33083, 3883}, {7226, 29679, 3717}, {11679, 17304, 19785}, {14829, 17305, 19786}, {17011, 32863, 3879}, {17165, 26251, 53663}, {17235, 44417, 3782}, {17291, 38000, 2}, {17321, 18141, 5287}, {25914, 25917, 25881}, {26128, 32916, 3011}, {28606, 33172, 3912}, {30942, 32776, 24210}, {31330, 33125, 1738}, {32772, 33067, 50307}, {33081, 46904, 4028}

X(54312) = X(1)X(25)∩X(3)X(42)

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

X(54312) lies on these lines: {1, 25}, {3, 42}, {8, 37090}, {10, 16353}, {22, 17018}, {27, 388}, {31, 37492}, {41, 25941}, {43, 7484}, {55, 63}, {56, 4719}, {81, 1460}, {197, 940}, {284, 1037}, {306, 958}, {405, 33171}, {497, 14004}, {519, 16403}, {899, 16419}, {956, 33088}, {968, 24320}, {993, 4028}, {999, 17017}, {1013, 3486}, {1057, 3478}, {1072, 1860}, {1400, 1617}, {1473, 17594}, {1478, 1889}, {1621, 7083}, {1757, 26867}, {1995, 29814}, {2292, 3295}, {3220, 37553}, {3240, 7485}, {3436, 50400}, {3475, 41230}, {3556, 37548}, {3666, 22769}, {3720, 5020}, {3750, 7295}, {3751, 7085}, {4265, 37577}, {4267, 37579}, {4471, 18613}, {4649, 5329}, {5132, 37578}, {5247, 37246}, {6600, 36559}, {7373, 29819}, {7395, 37699}, {7742, 19763}, {8897, 19860}, {9708, 15523}, {10601, 23638}, {11108, 24943}, {11269, 19544}, {11284, 26102}, {11414, 37529}, {14547, 16541}, {16352, 43223}, {16408, 29663}, {16678, 36744}, {17596, 26866}, {19313, 26037}, {19714, 37079}, {19765, 22654}, {20973, 37503}, {21620, 37396}, {21746, 33586}, {25494, 29839}, {29642, 50715}, {37257, 37607}, {37576, 42042}

X(54312) = crossdifference of every pair of points on line {2522, 29142}
X(54312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {22, 17018, 37580}, {3295, 42461, 2292}, {4649, 5329, 44094}

X(54313) = X(1)X(60)∩X(8)X(21)

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

X(54313) lies on these lines: {1, 60}, {3, 24883}, {8, 21}, {11, 13746}, {28, 1068}, {36, 37405}, {56, 1325}, {58, 5903}, {81, 37227}, {229, 999}, {270, 14015}, {284, 40968}, {355, 24624}, {409, 3616}, {496, 3109}, {499, 37158}, {501, 21842}, {643, 3885}, {976, 4653}, {993, 27368}, {1010, 33108}, {1098, 3877}, {1175, 37533}, {1385, 40214}, {1478, 37369}, {1479, 7424}, {1793, 5727}, {1834, 37311}, {1837, 6740}, {2082, 5546}, {2099, 46441}, {2185, 3897}, {2217, 2363}, {2218, 40430}, {2646, 35195}, {2975, 17512}, {3057, 35193}, {3615, 11376}, {4184, 33137}, {4188, 25459}, {4221, 35239}, {4225, 5230}, {4234, 49719}, {4299, 5196}, {4311, 18653}, {5006, 53165}, {5127, 5697}, {5204, 37294}, {5251, 20653}, {5253, 11116}, {5267, 50755}, {5358, 7419}, {5730, 37783}, {5754, 32911}, {7478, 10072}, {11010, 52680}, {11115, 33110}, {12030, 47274}, {12053, 51382}, {12702, 15952}, {13589, 38511}, {13733, 19767}, {13739, 41227}, {14127, 38497}, {15792, 37525}, {17539, 20095}, {21004, 23903}, {24928, 51420}, {26363, 37152}, {29658, 33325}, {33139, 34868}, {36927, 37740}, {37587, 52375}

X(54313) = crossdifference of every pair of points on line {2610, 7180}
X(54313) = barycentric product X(21)*X(33133)
X(54313) = barycentric quotient X(33133)/X(1441)
X(54313) = {X(1),X(759)}-harmonic conjugate of X(11101)

X(54314) = X(2)X(92)∩X(4)X(75)

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

X(54314) lies on these lines: {2, 92}, {4, 75}, {7, 8048}, {19, 4384}, {25, 26234}, {27, 4359}, {28, 242}, {29, 5262}, {33, 3875}, {34, 10436}, {85, 1119}, {86, 1870}, {239, 1172}, {264, 1969}, {297, 26538}, {313, 1235}, {318, 20895}, {321, 469}, {322, 32000}, {350, 4213}, {406, 17321}, {427, 3263}, {429, 1228}, {451, 17322}, {458, 26665}, {515, 17859}, {857, 26165}, {946, 17858}, {1269, 44146}, {1699, 18691}, {1826, 20236}, {1829, 20911}, {1838, 18698}, {1841, 3739}, {1848, 3687}, {1861, 4967}, {1921, 44129}, {2969, 34336}, {3264, 44142}, {3672, 4194}, {3766, 44428}, {4198, 5342}, {4207, 4441}, {4357, 45196}, {4360, 6198}, {4858, 39039}, {5279, 30807}, {7019, 40717}, {7140, 52787}, {7282, 52442}, {7378, 31130}, {7490, 19804}, {7713, 33945}, {7718, 39731}, {8822, 45797}, {8889, 30758}, {11109, 24993}, {11337, 17134}, {12047, 18692}, {16732, 40941}, {17442, 29960}, {17555, 24547}, {17861, 39585}, {18147, 20926}, {18483, 18699}, {18650, 18690}, {20891, 31909}, {20905, 37448}, {20906, 44426}, {20907, 54239}, {21579, 42714}, {23661, 52364}, {25001, 26003}, {26042, 37337}, {28653, 52252}, {30044, 31916}, {34231, 44735}, {41005, 41007}

X(54314) = isotomic conjugate of X(1791)
X(54314) = polar conjugate of X(2298)
X(54314) = isotomic conjugate of the isogonal conjugate of X(1829)
X(54314) = polar conjugate of the isotomic conjugate of X(20911)
X(54314) = polar conjugate of the isogonal conjugate of X(3666)
X(54314) = X(6331)-Ceva conjugate of X(17924)
X(54314) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2359}, {31, 1791}, {42, 1798}, {48, 2298}, {71, 1169}, {184, 1220}, {212, 961}, {228, 2363}, {652, 8687}, {1240, 14575}, {1459, 32736}, {1946, 36098}, {2200, 14534}, {4581, 32656}, {9247, 30710}, {15420, 32739}, {22383, 36147}
X(54314) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1791}, {9, 2359}, {429, 197}, {960, 228}, {1211, 3}, {1249, 2298}, {2092, 219}, {3125, 647}, {3666, 72}, {17197, 23189}, {17419, 652}, {38992, 1946}, {39015, 22383}, {39053, 36098}, {39060, 6648}, {40592, 1798}, {40619, 15420}, {40837, 961}, {46878, 5285}, {52087, 48}
X(54314) = cevapoint of X(i) and X(j) for these (i,j): {1829, 3666}, {1848, 46878}
X(54314) = barycentric product X(i)*X(j) for these {i,j}: {4, 20911}, {27, 18697}, {28, 1228}, {29, 45196}, {75, 1848}, {76, 1829}, {85, 46878}, {92, 4357}, {264, 3666}, {273, 3687}, {274, 429}, {286, 1211}, {318, 3674}, {331, 960}, {444, 44187}, {561, 2354}, {811, 21124}, {1193, 1969}, {1826, 16739}, {1897, 4509}, {2292, 44129}, {2300, 18022}, {3004, 6335}, {3882, 46107}, {3910, 18026}, {4267, 52575}, {6331, 50330}, {6385, 44092}, {7017, 24471}, {16705, 41013}, {16747, 27067}, {17420, 46404}, {17924, 53332}, {18027, 22345}, {20567, 40976}, {31623, 41003}
X(54314) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2359}, {2, 1791}, {4, 2298}, {27, 2363}, {28, 1169}, {81, 1798}, {92, 1220}, {108, 8687}, {264, 30710}, {278, 961}, {286, 14534}, {331, 31643}, {429, 37}, {444, 172}, {653, 36098}, {693, 15420}, {960, 219}, {1193, 48}, {1211, 72}, {1228, 20336}, {1682, 22074}, {1783, 32736}, {1829, 6}, {1848, 1}, {1897, 36147}, {1969, 1240}, {2092, 228}, {2269, 212}, {2292, 71}, {2300, 184}, {2354, 31}, {3004, 905}, {3666, 3}, {3674, 77}, {3687, 78}, {3704, 3694}, {3725, 2200}, {3882, 1331}, {3910, 521}, {3965, 1260}, {4267, 2193}, {4357, 63}, {4509, 4025}, {6335, 8707}, {6371, 22383}, {16705, 1444}, {16739, 17206}, {17185, 283}, {17420, 652}, {17924, 4581}, {17981, 53689}, {18026, 6648}, {18697, 306}, {20653, 3949}, {20911, 69}, {20967, 52425}, {21033, 2318}, {21124, 656}, {21810, 3690}, {22074, 6056}, {22076, 3990}, {22097, 255}, {22345, 577}, {24471, 222}, {27455, 23086}, {28369, 3955}, {32714, 52928}, {40153, 1437}, {40966, 52370}, {40976, 41}, {41003, 1214}, {41013, 14624}, {41581, 22131}, {41591, 22122}, {41600, 22132}, {41609, 2911}, {41611, 218}, {44092, 213}, {45196, 307}, {45218, 22381}, {46877, 2327}, {46878, 9}, {48131, 1459}, {50330, 647}, {51407, 51379}, {51414, 46974}, {52326, 1946}, {52567, 2197}, {53280, 906}, {53332, 1332}
X(54314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1826, 20883, 46108}, {1841, 3739, 15149}

X(54315) = X(1)X(88)∩X(65)X(82)

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

X(54315) lies on these lines: {1, 88}, {4, 7986}, {8, 141}, {10, 32775}, {21, 986}, {30, 33102}, {36, 49682}, {43, 49454}, {55, 4218}, {65, 82}, {81, 1325}, {222, 18419}, {354, 17015}, {377, 7613}, {386, 34195}, {392, 7292}, {484, 49480}, {495, 33148}, {517, 7191}, {519, 24169}, {614, 3877}, {644, 9620}, {758, 32911}, {846, 16858}, {942, 17016}, {956, 4392}, {964, 7155}, {976, 24440}, {982, 49487}, {988, 3897}, {1086, 5724}, {1159, 17025}, {1201, 5330}, {1203, 4084}, {1403, 4216}, {1478, 33146}, {1482, 19514}, {1621, 4424}, {1722, 3876}, {1724, 11684}, {1737, 33133}, {1739, 30115}, {1963, 35991}, {2292, 5047}, {2722, 53926}, {2975, 3670}, {3120, 17577}, {3125, 5276}, {3210, 49492}, {3241, 17597}, {3419, 33131}, {3496, 39251}, {3616, 17054}, {3677, 3872}, {3735, 33854}, {3751, 3868}, {3752, 4511}, {3753, 3920}, {3782, 5080}, {3944, 37375}, {3953, 15955}, {3961, 4695}, {3980, 51669}, {4217, 24280}, {4393, 24610}, {4427, 13735}, {4720, 32860}, {4906, 5919}, {4968, 26538}, {5082, 36579}, {5086, 23537}, {5256, 11529}, {5300, 50582}, {5563, 24167}, {5657, 26228}, {5687, 36565}, {5692, 37680}, {5697, 30148}, {5721, 9803}, {5722, 33134}, {5725, 31019}, {6175, 17889}, {7226, 9708}, {9593, 25082}, {9791, 14020}, {10176, 37687}, {11043, 28238}, {11113, 33100}, {11114, 24248}, {11533, 17534}, {13407, 26729}, {13605, 36250}, {13740, 17164}, {13741, 25253}, {14829, 39766}, {15934, 16056}, {16485, 35258}, {16519, 21951}, {16821, 46909}, {17061, 40663}, {17074, 18838}, {17126, 36279}, {17531, 24174}, {17537, 44006}, {17549, 17596}, {17679, 50591}, {17757, 33153}, {18343, 36154}, {18391, 19785}, {19860, 26635}, {19869, 32779}, {24982, 34937}, {25248, 33821}, {25270, 33817}, {26446, 29665}, {28082, 37598}, {33067, 38456}, {33107, 39542}, {33143, 37716}, {33155, 37715}, {33815, 37559}, {48696, 49686}

X(54315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4642, 3871}, {1, 5883, 37633}, {1, 24046, 5253}, {1, 24443, 404}, {986, 3924, 21}, {3120, 37717, 17577}, {4424, 30117, 1621}, {9620, 26242, 644}, {17054, 37614, 3616}, {17757, 39544, 33153}

X(54316) = X(6)X(43)∩X(8)X(48)

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

X(54316) lies on these lines: {1, 19285}, {6, 43}, {8, 48}, {9, 35}, {10, 284}, {19, 78}, {37, 1247}, {41, 2345}, {42, 2303}, {55, 965}, {69, 1958}, {71, 100}, {72, 1761}, {75, 18162}, {101, 2321}, {159, 197}, {198, 3713}, {219, 3362}, {220, 7038}, {238, 5301}, {281, 2289}, {306, 1474}, {312, 2201}, {319, 662}, {326, 18161}, {380, 936}, {391, 2267}, {404, 2260}, {480, 5781}, {524, 7175}, {572, 3686}, {573, 6796}, {579, 25440}, {584, 17303}, {594, 2174}, {604, 1788}, {608, 37694}, {908, 1839}, {958, 37504}, {966, 2268}, {992, 1914}, {1018, 52405}, {1100, 3812}, {1107, 5110}, {1172, 3682}, {1213, 6690}, {1248, 3990}, {1259, 15656}, {1333, 5247}, {1429, 4361}, {1441, 24435}, {1449, 37559}, {1582, 3783}, {1630, 6737}, {1731, 25078}, {1762, 3998}, {1765, 2077}, {1781, 22021}, {1802, 27382}, {1826, 2327}, {1935, 1950}, {1940, 7120}, {1943, 6359}, {1953, 4511}, {2173, 3949}, {2182, 3965}, {2256, 3913}, {2257, 5438}, {2271, 34261}, {2278, 17275}, {2280, 26040}, {2294, 34772}, {2302, 6734}, {2304, 9534}, {2326, 52412}, {2330, 15984}, {2893, 20305}, {2911, 3501}, {2975, 22054}, {3033, 11574}, {3204, 17281}, {3216, 16470}, {3434, 27395}, {3579, 4047}, {3694, 38903}, {3826, 16503}, {3841, 24937}, {3912, 40530}, {4000, 25940}, {4053, 5341}, {4251, 5750}, {4254, 5783}, {4861, 17438}, {5086, 21011}, {5432, 5742}, {5440, 40937}, {5552, 26063}, {5564, 18042}, {5776, 10310}, {5778, 11248}, {6700, 40963}, {7113, 17362}, {7119, 7270}, {7145, 8681}, {9310, 17314}, {12513, 37519}, {14543, 45744}, {16488, 49997}, {16548, 21078}, {16685, 37588}, {17117, 27950}, {21061, 35342}, {21384, 36743}, {21388, 48391}, {21933, 44669}, {25993, 26006}, {28604, 40744}, {42696, 52134}

X(54316) = isotomic conjugate of the polar conjugate of X(7076)
X(54316) = X(i)-Ceva conjugate of X(j) for these (i,j): {1943, 1935}, {40406, 1}
X(54316) = X(i)-isoconjugate of X(j) for these (i,j): {7, 7106}, {56, 7108}, {57, 7105}, {273, 7107}, {278, 7016}
X(54316) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 7108}, {5452, 7105}, {8062, 4466}, {16573, 693}
X(54316) = cevapoint of X(43) and X(2939)
X(54316) = trilinear pole of line {21761, 21831}
X(54316) = crossdifference of every pair of points on line {4083, 51648}
X(54316) = barycentric product X(i)*X(j) for these {i,j}: {1, 7283}, {8, 1935}, {9, 1943}, {69, 7076}, {75, 26885}, {78, 1940}, {99, 21831}, {100, 8062}, {101, 17899}, {200, 6359}, {219, 1947}, {281, 7364}, {312, 1950}, {345, 7120}, {668, 21761}, {6335, 22382}
X(54316) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 7108}, {41, 7106}, {55, 7105}, {212, 7016}, {1935, 7}, {1940, 273}, {1943, 85}, {1947, 331}, {1950, 57}, {6359, 1088}, {7076, 4}, {7120, 278}, {7283, 75}, {7364, 348}, {8062, 693}, {16573, 4466}, {17899, 3261}, {21761, 513}, {21831, 523}, {22382, 905}, {26885, 1}, {44096, 1430}, {52425, 7107}
X(54316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 20769, 18162}, {100, 2287, 71}, {200, 610, 5227}, {584, 17303, 41239}, {594, 2174, 2329}, {2173, 3949, 5279}, {4420, 5279, 3949}

X(54317) = X(1)X(39)∩X(6)X(78)

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

X(54317) lies on these lines: {1, 39}, {2, 16968}, {6, 78}, {9, 172}, {31, 39244}, {37, 56}, {40, 3727}, {41, 36404}, {46, 3735}, {55, 39255}, {57, 3721}, {63, 33863}, {72, 5021}, {77, 28391}, {86, 25918}, {169, 5277}, {171, 3061}, {200, 3780}, {213, 997}, {241, 37674}, {304, 16061}, {348, 4648}, {392, 14974}, {474, 16583}, {583, 2215}, {604, 22061}, {612, 1107}, {614, 16604}, {750, 17451}, {936, 2238}, {968, 17798}, {975, 5283}, {976, 1475}, {988, 41269}, {1038, 18591}, {1193, 16972}, {1212, 5275}, {1376, 41015}, {1468, 33299}, {1571, 4424}, {1572, 5264}, {1743, 7296}, {1914, 37552}, {2082, 4386}, {2176, 19861}, {2177, 39247}, {2242, 17742}, {2250, 40401}, {2271, 5440}, {3290, 25524}, {3306, 20271}, {3333, 3726}, {3496, 37603}, {3509, 37608}, {3665, 4675}, {3666, 5013}, {3744, 16781}, {3811, 20963}, {3905, 24631}, {3931, 31448}, {3938, 17474}, {3997, 30144}, {4253, 30115}, {4372, 4384}, {4413, 16605}, {4447, 5268}, {4855, 18755}, {5089, 22479}, {5250, 17735}, {5253, 26242}, {5254, 17720}, {5266, 16502}, {5269, 9575}, {5276, 26690}, {5287, 19715}, {5293, 21384}, {5308, 17081}, {5336, 5750}, {5364, 22065}, {5530, 31497}, {5716, 7736}, {5725, 31460}, {7198, 17276}, {7293, 21771}, {8583, 16970}, {8666, 28594}, {9300, 50070}, {9597, 13161}, {9598, 24210}, {10436, 16720}, {17016, 17756}, {17103, 18055}, {17124, 21921}, {17368, 27954}, {21008, 35262}, {21029, 29662}, {21965, 24914}, {28043, 40133}, {31477, 37548}, {34261, 40937}, {37589, 39254}, {37634, 40997}

X(54317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16549, 9620}, {31, 39244, 39248}, {976, 1475, 16973}, {16604, 16974, 614}

X(54318) = X(1)X(2)∩X(21)X(46)

Barycentrics    a*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - 2*a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2 + c^3) : :
X(54318) = 3 X[1] + X[4915], X[8] + 3 X[15933], X[4915] - 3 X[9623], X[1056] - 3 X[38053], X[1159] + 2 X[15254], X[1159] + 3 X[16857], 2 X[15254] - 3 X[16857], X[31393] - 3 X[38316], X[3577] + 2 X[52769], X[40587] + 2 X[42819], 2 X[6666] + X[14563]

X(54318) lies on these lines: {1, 2}, {3, 3812}, {4, 12520}, {5, 6261}, {9, 758}, {21, 46}, {30, 5880}, {33, 860}, {34, 5136}, {35, 37300}, {36, 3306}, {37, 9620}, {40, 1006}, {41, 21921}, {55, 3753}, {56, 5439}, {57, 993}, {63, 4880}, {65, 405}, {72, 3715}, {86, 998}, {142, 515}, {158, 11109}, {165, 37106}, {169, 41239}, {171, 37817}, {210, 44840}, {214, 13384}, {224, 4197}, {236, 18456}, {281, 1870}, {354, 956}, {355, 6881}, {377, 10572}, {381, 3838}, {388, 34489}, {392, 2099}, {404, 3612}, {442, 1837}, {443, 3486}, {452, 4295}, {474, 2646}, {484, 35258}, {516, 6987}, {517, 1001}, {518, 9708}, {529, 25557}, {535, 6173}, {756, 49454}, {940, 16344}, {942, 958}, {944, 6854}, {946, 6827}, {960, 11108}, {962, 6992}, {968, 4424}, {988, 24046}, {990, 3821}, {996, 2191}, {999, 3742}, {1056, 38053}, {1058, 49600}, {1060, 6703}, {1104, 5711}, {1155, 16370}, {1158, 3560}, {1159, 15254}, {1214, 19727}, {1329, 11281}, {1376, 24929}, {1385, 6911}, {1420, 51111}, {1448, 2647}, {1467, 4298}, {1478, 5249}, {1490, 6843}, {1573, 16973}, {1621, 5119}, {1656, 45770}, {1699, 6840}, {1706, 3918}, {1709, 6912}, {1738, 48837}, {1743, 53114}, {1770, 6872}, {1788, 6857}, {1812, 4658}, {1836, 11113}, {2082, 16783}, {2093, 3919}, {2177, 4695}, {2185, 11116}, {2263, 48866}, {2271, 16605}, {2320, 4881}, {2324, 3986}, {2476, 10826}, {2478, 12047}, {2550, 3488}, {2551, 3487}, {2802, 31393}, {2886, 5722}, {2951, 28158}, {2975, 3338}, {3090, 21740}, {3158, 3968}, {3185, 4245}, {3295, 5836}, {3303, 10914}, {3305, 5425}, {3333, 8666}, {3336, 4652}, {3339, 31424}, {3340, 3878}, {3419, 3925}, {3421, 3475}, {3436, 13407}, {3474, 11111}, {3485, 5084}, {3543, 41860}, {3553, 5257}, {3576, 3833}, {3577, 52769}, {3579, 28466}, {3601, 25440}, {3646, 15829}, {3671, 12572}, {3678, 11523}, {3689, 4731}, {3694, 16777}, {3698, 5687}, {3711, 3921}, {3740, 3940}, {3772, 37715}, {3814, 5219}, {3816, 5886}, {3817, 6844}, {3820, 5719}, {3822, 5587}, {3824, 18480}, {3825, 6830}, {3826, 44669}, {3848, 10246}, {3868, 5260}, {3869, 5047}, {3874, 11518}, {3876, 34195}, {3877, 5284}, {3880, 6600}, {3881, 6762}, {3884, 7982}, {3890, 30323}, {3892, 44841}, {3897, 5253}, {3898, 7962}, {3901, 3951}, {3916, 5221}, {3922, 37568}, {3927, 5302}, {3984, 41696}, {3997, 16970}, {4004, 37567}, {4038, 28920}, {4084, 12526}, {4187, 11375}, {4193, 37692}, {4293, 9776}, {4297, 8726}, {4305, 6904}, {4313, 11024}, {4321, 30275}, {4333, 15680}, {4359, 49492}, {4413, 5440}, {4640, 16418}, {4653, 17594}, {4657, 16608}, {4670, 44664}, {4676, 33309}, {4679, 51409}, {4694, 16499}, {4855, 37571}, {4868, 37553}, {5010, 5426}, {5044, 12635}, {5045, 12513}, {5054, 35459}, {5080, 31019}, {5123, 31479}, {5126, 40726}, {5178, 41709}, {5250, 5259}, {5252, 50208}, {5258, 18398}, {5267, 15803}, {5269, 16485}, {5288, 50190}, {5289, 8167}, {5333, 6505}, {5429, 37604}, {5450, 37534}, {5493, 12651}, {5535, 21165}, {5542, 52457}, {5603, 6947}, {5657, 6878}, {5691, 6839}, {5709, 31870}, {5720, 6858}, {5725, 17056}, {5727, 41867}, {5728, 42014}, {5730, 16842}, {5732, 28164}, {5794, 8728}, {5795, 21620}, {5805, 33558}, {5818, 6877}, {5884, 7330}, {5885, 24467}, {5904, 11520}, {5905, 11551}, {6001, 6913}, {6051, 37614}, {6265, 6667}, {6282, 10164}, {6326, 6702}, {6666, 14563}, {6668, 37733}, {6675, 26066}, {6681, 31190}, {6684, 37531}, {6690, 26446}, {6692, 6954}, {6701, 16132}, {6708, 37697}, {6769, 43174}, {6824, 12616}, {6846, 12617}, {6856, 45230}, {6859, 40257}, {6893, 12608}, {6918, 37837}, {6963, 9624}, {6993, 18444}, {7028, 18454}, {7073, 52388}, {7308, 10176}, {7373, 11260}, {7483, 24914}, {7675, 38052}, {7719, 17442}, {7951, 26725}, {7993, 33812}, {8000, 12756}, {8953, 30557}, {9352, 17549}, {9581, 25639}, {9593, 25092}, {9612, 11263}, {9619, 16604}, {9817, 45272}, {9940, 12114}, {9956, 37700}, {10051, 51432}, {10107, 12702}, {10129, 37375}, {10202, 22758}, {10389, 25439}, {10436, 17861}, {10448, 24443}, {10571, 19372}, {10912, 12260}, {11103, 25526}, {11114, 20292}, {11235, 18527}, {11373, 15842}, {11496, 31788}, {11499, 24299}, {11507, 37248}, {11682, 25542}, {12081, 53034}, {12433, 31419}, {12560, 12848}, {12563, 18250}, {12565, 51118}, {12575, 51724}, {12699, 28459}, {12736, 51506}, {12739, 34122}, {13161, 24159}, {13369, 18761}, {13374, 22770}, {13750, 37228}, {14988, 15297}, {15079, 31262}, {15668, 50317}, {15863, 37736}, {16371, 37600}, {16417, 37606}, {16616, 37411}, {16788, 40131}, {16968, 17750}, {17054, 37592}, {17063, 37617}, {17398, 21933}, {17527, 25681}, {17529, 37724}, {17542, 31165}, {17556, 17605}, {17575, 24954}, {17614, 34471}, {17706, 24391}, {17718, 17757}, {17742, 21808}, {18165, 19259}, {18421, 37787}, {18465, 25507}, {18481, 28452}, {18528, 50796}, {19701, 45126}, {21147, 37523}, {21161, 35242}, {21627, 40270}, {21888, 31433}, {23518, 25017}, {24440, 37573}, {24703, 39542}, {24723, 48814}, {24806, 25496}, {26728, 33144}, {28228, 43166}, {28849, 50290}, {28850, 50302}, {30852, 37701}, {31156, 44447}, {31359, 37035}, {31445, 31794}, {31673, 41854}, {32784, 37796}, {32950, 49735}, {33068, 37038}, {33111, 37717}, {33130, 37716}, {33145, 50066}, {35262, 37525}, {36027, 48900}, {36404, 49758}, {36819, 40437}, {37224, 44547}, {37695, 51421}, {40249, 49170}, {40659, 42871}, {41340, 44545}, {41859, 47033}, {42064, 50441}, {44734, 46883}, {48833, 50116}, {48841, 50080}, {48863, 50314}

X(54318) = midpoint of X(i) and X(j) for these {i,j}: {1, 9623}, {9, 11529}, {2550, 3488}, {6767, 40587}, {9708, 15934}
X(54318) = reflection of X(6767) in X(42819)
X(54318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2, 997}, {1, 10, 3811}, {1, 936, 22836}, {1, 1698, 78}, {1, 1722, 386}, {1, 3624, 19861}, {1, 3679, 3870}, {1, 4853, 3244}, {1, 5268, 30115}, {1, 5272, 995}, {1, 8583, 30144}, {1, 10582, 551}, {1, 12629, 3635}, {2, 18391, 10}, {4, 28629, 12609}, {8, 3616, 10587}, {8, 19855, 10}, {10, 551, 13405}, {10, 1125, 10198}, {10, 3244, 6743}, {10, 6738, 49168}, {10, 10197, 31434}, {10, 13405, 45701}, {10, 30143, 1}, {40, 5436, 5248}, {65, 405, 12514}, {443, 3486, 17647}, {938, 19843, 10916}, {993, 5883, 57}, {1125, 9843, 10200}, {1125, 30147, 1}, {1329, 11281, 11374}, {2099, 4423, 392}, {2551, 3487, 21077}, {3241, 29817, 1}, {3340, 31435, 3878}, {3485, 5084, 21616}, {3560, 34339, 1158}, {3622, 4861, 1}, {3634, 22836, 936}, {3636, 22837, 1}, {3698, 37080, 5687}, {3720, 49487, 1}, {3754, 5248, 40}, {3868, 5260, 41229}, {3872, 4666, 1}, {3897, 5253, 37618}, {3918, 8715, 1706}, {4666, 19860, 3872}, {5136, 40149, 39585}, {5251, 5902, 63}, {5259, 5903, 5250}, {5269, 16485, 49480}, {5587, 25525, 3822}, {5730, 16842, 25917}, {5836, 51715, 3295}, {7951, 26725, 31266}, {8582, 13411, 26364}, {8728, 37730, 5794}, {10459, 28082, 1}, {16418, 36279, 4640}, {17015, 29814, 1}, {17527, 37737, 25681}, {19862, 30144, 8583}, {25440, 35016, 3601}, {30116, 30117, 1}, {38314, 38460, 1}

X(54319) = X(1)X(2)∩X(40)X(106)

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

X(54319) lies on these lines: {1, 2}, {3, 15663}, {40, 106}, {244, 30323}, {517, 3445}, {988, 3898}, {999, 45219}, {1191, 51788}, {1279, 36280}, {1319, 1406}, {1385, 1480}, {1420, 40091}, {1616, 24928}, {2802, 11512}, {3680, 10700}, {4256, 37556}, {4694, 11682}, {5119, 32577}, {7962, 24046}, {8666, 11717}, {9327, 9575}, {10595, 26728}, {12843, 37611}, {15839, 31393}, {16483, 20323}, {37552, 51714}

X(54319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5272, 22837}, {1, 8583, 50637}, {1, 46943, 12629}, {1, 47623, 3811}, {1, 49997, 36846}, {1616, 24928, 37817}, {12629, 46943, 17749}

X(54320) = X(1)X(3)∩X(21)X(34)

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

X(54320) lies on these lines: {1, 3}, {2, 225}, {9, 37694}, {19, 27621}, {20, 40950}, {21, 34}, {33, 411}, {58, 45126}, {63, 73}, {77, 283}, {78, 201}, {212, 3561}, {216, 2277}, {221, 4640}, {222, 3916}, {223, 1935}, {227, 958}, {255, 21165}, {278, 6857}, {348, 6337}, {386, 1708}, {405, 1465}, {631, 1068}, {859, 7713}, {968, 3485}, {970, 19366}, {975, 16577}, {978, 40679}, {993, 21147}, {1042, 4414}, {1072, 7400}, {1074, 6889}, {1076, 6833}, {1254, 10448}, {1368, 26481}, {1410, 22060}, {1441, 16342}, {1448, 18593}, {1449, 2193}, {1451, 5256}, {1452, 4225}, {1455, 15832}, {1457, 5250}, {1490, 24430}, {1728, 37732}, {1745, 7330}, {1785, 6825}, {1825, 20243}, {1829, 52271}, {1838, 6824}, {1870, 6875}, {1877, 6872}, {2006, 7561}, {2067, 19216}, {2197, 5227}, {2594, 3751}, {3011, 7494}, {3149, 9817}, {3193, 17074}, {3523, 23710}, {3524, 38295}, {3911, 5292}, {4000, 7288}, {4189, 4296}, {4267, 18603}, {4292, 5713}, {4331, 18589}, {4551, 41229}, {5159, 47271}, {5248, 34036}, {5253, 26635}, {5433, 11512}, {5705, 18641}, {5745, 5930}, {5752, 20122}, {6198, 6876}, {6349, 24541}, {6350, 24987}, {6502, 19215}, {6509, 30674}, {6675, 37695}, {6734, 34822}, {6988, 7952}, {7004, 10884}, {7289, 18606}, {7386, 29639}, {7508, 32047}, {7741, 37361}, {8609, 36751}, {9816, 28258}, {9843, 43068}, {10257, 11809}, {10527, 17077}, {10538, 26027}, {10571, 12514}, {13323, 19365}, {15829, 34588}, {16272, 16976}, {17321, 22464}, {18446, 44706}, {20277, 22361}, {25490, 39585}, {26066, 51421}, {26377, 37257}, {28034, 54234}, {31445, 34048}, {34033, 51576}, {34977, 51236}, {39751, 44906}

X(54320) = X(11110)-Ceva conjugate of X(3485)
X(54320) = X(i)-isoconjugate of X(j) for these (i,j): {33, 969}, {281, 967}
X(54320) = X(38960)-Dao conjugate of X(44426)
X(54320) = barycentric product X(i)*X(j) for these {i,j}: {63, 3485}, {77, 966}, {348, 968}, {1214, 11110}, {1441, 4288}, {1813, 7650}, {2271, 7182}, {4207, 7183}, {6516, 45745}
X(54320) = barycentric quotient X(i)/X(j) for these {i,j}: {222, 969}, {603, 967}, {966, 318}, {968, 281}, {2271, 33}, {3485, 92}, {4288, 21}, {7650, 46110}, {11110, 31623}, {45745, 44426}, {48099, 3064}
X(54320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1758, 37550}, {1, 15803, 37530}, {3, 1214, 1038}, {3, 17102, 1040}, {3, 37565, 1}, {21, 17080, 34}, {77, 4652, 603}, {223, 31424, 1935}, {405, 1465, 19372}, {6350, 25876, 34823}, {8758, 26357, 1}

X(54321) = X(6)X(31)∩X(28)X(34)

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

X(54321) lies on these lines: {6, 31}, {25, 2260}, {28, 34}, {33, 2257}, {41, 2352}, {48, 5320}, {56, 44098}, {65, 40970}, {73, 1617}, {165, 580}, {171, 26040}, {218, 2318}, {223, 16469}, {238, 5712}, {354, 1104}, {581, 602}, {595, 10389}, {604, 2194}, {609, 38849}, {748, 17056}, {910, 4275}, {1193, 37578}, {1212, 3745}, {1427, 1471}, {1449, 2328}, {1472, 37575}, {1475, 37538}, {1497, 7078}, {1723, 40967}, {1724, 5717}, {1936, 37666}, {2212, 16470}, {2221, 3423}, {3215, 37541}, {3748, 3915}, {4253, 5285}, {5247, 5716}, {5269, 16572}, {7069, 8557}, {7964, 15852}, {10857, 37469}, {16485, 44841}, {16968, 20985}, {22097, 37492}, {28272, 34281}, {28274, 36740}, {34879, 52544}, {41858, 45924}

X(54321) = X(i)-isoconjugate of X(j) for these (i,j): {8, 8814}, {281, 8813}
X(54321) = crossdifference of every pair of points on line {514, 8611}
X(54321) = barycentric product X(57)*X(13615)
X(54321) = barycentric quotient X(i)/X(j) for these {i,j}: {603, 8813}, {604, 8814}, {13615, 312}
X(54321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 31, 212}, {31, 42, 21059}, {58, 1453, 1451}, {2308, 40958, 6}, {5320, 40956, 48}

X(54322) = X(3)X(9)∩X(6)X(31)

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

X(54322) lies on these lines: {1, 5120}, {3, 9}, {6, 31}, {10, 37062}, {19, 1212}, {21, 5749}, {35, 1743}, {36, 3731}, {37, 56}, {40, 2262}, {41, 37504}, {44, 5217}, {45, 1696}, {48, 220}, {100, 391}, {101, 28193}, {144, 24328}, {165, 374}, {197, 44798}, {218, 284}, {219, 572}, {226, 21483}, {228, 26867}, {281, 37305}, {344, 1444}, {346, 2975}, {380, 16572}, {404, 5296}, {405, 5750}, {474, 5257}, {478, 1035}, {480, 15624}, {573, 10310}, {574, 21796}, {579, 16287}, {604, 1334}, {851, 1213}, {859, 4877}, {894, 16367}, {956, 2321}, {958, 2345}, {966, 1376}, {993, 17355}, {999, 3247}, {1001, 41325}, {1012, 10445}, {1014, 5308}, {1030, 16885}, {1100, 3303}, {1125, 21068}, {1214, 6611}, {1259, 16452}, {1400, 1466}, {1449, 3295}, {1473, 5282}, {1474, 37908}, {1500, 5042}, {1723, 40292}, {1766, 3428}, {1778, 4267}, {1826, 4185}, {1901, 37225}, {1953, 34522}, {2092, 31448}, {2099, 21853}, {2183, 15592}, {2223, 16517}, {2245, 11509}, {2250, 34278}, {2260, 5022}, {2264, 37601}, {2277, 5013}, {2278, 2911}, {2287, 4184}, {2297, 4512}, {2300, 14974}, {2303, 3286}, {2323, 10267}, {2324, 3576}, {2646, 3553}, {3057, 3554}, {3207, 22054}, {3304, 16777}, {3305, 11350}, {3361, 47299}, {3523, 27508}, {3587, 23840}, {3683, 20991}, {3686, 5687}, {3693, 5227}, {3746, 16667}, {3781, 37474}, {3913, 5839}, {3925, 4196}, {3950, 8666}, {3973, 5010}, {4130, 23224}, {4192, 5742}, {4263, 31451}, {4268, 11510}, {4287, 17796}, {4357, 21477}, {4421, 37654}, {4423, 17398}, {4856, 25439}, {5105, 16466}, {5242, 21481}, {5243, 21480}, {5249, 16439}, {5273, 15509}, {5279, 25082}, {5283, 34261}, {5285, 13615}, {5294, 16368}, {5303, 38869}, {5314, 20835}, {5563, 16673}, {5657, 53994}, {5710, 20719}, {5745, 16435}, {5746, 13726}, {5747, 16455}, {5817, 36012}, {5819, 11495}, {5830, 49128}, {5838, 7676}, {6666, 37272}, {6684, 20262}, {6971, 15833}, {6986, 27382}, {7071, 22079}, {7083, 37586}, {7114, 41087}, {7129, 40971}, {7308, 37269}, {7484, 22060}, {7573, 27287}, {7738, 28265}, {8557, 26357}, {8609, 10966}, {9310, 37519}, {9605, 16470}, {10434, 15479}, {10979, 23980}, {11329, 17260}, {11340, 27065}, {11343, 17353}, {11349, 18230}, {11517, 37057}, {12513, 17314}, {12572, 37320}, {13006, 36055}, {13733, 17369}, {14021, 28739}, {14379, 30457}, {14439, 20999}, {16058, 17754}, {16370, 50115}, {16431, 50093}, {16440, 30413}, {16441, 30412}, {16518, 23393}, {16548, 35239}, {16669, 37503}, {16675, 21773}, {16970, 37575}, {17134, 25001}, {17257, 21495}, {17306, 21526}, {19762, 25092}, {20471, 38871}, {20780, 53094}, {20818, 52405}, {21233, 24266}, {21488, 26580}, {21511, 26685}, {21811, 52273}, {22769, 50995}, {23397, 23853}, {24005, 24914}, {24612, 27514}, {25068, 37034}, {26036, 37425}, {26059, 37416}, {26063, 37409}, {28627, 37323}, {34820, 51773}

X(54322) = isogonal conjugate of the isotomic conjugate of X(34255)
X(54322) = X(2297)-Ceva conjugate of X(6)
X(54322) = X(25430)-isoconjugate of X(34244)
X(54322) = crossdifference of every pair of points on line {514, 6129}
X(54322) = barycentric product X(i)*X(j) for these {i,j}: {6, 34255}, {8, 34046}, {936, 14551}, {7050, 28616}
X(54322) = barycentric quotient X(i)/X(j) for these {i,j}: {34046, 7}, {34255, 76}
X(54322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 9, 198}, {6, 42316, 71}, {35, 1743, 4254}, {37, 36743, 56}, {45, 2178, 1696}, {45, 5124, 2178}, {55, 20992, 21002}, {71, 2267, 6}, {572, 3730, 219}, {604, 1334, 2256}, {672, 2268, 6}, {1011, 7085, 55}, {1696, 5204, 2178}, {2066, 5414, 7074}, {2178, 5124, 5204}, {8053, 12329, 55}

X(54323) = X(3)X(6)∩X(21)X(90)

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

X(54323) lies on these lines: {1, 1719}, {3, 6}, {10, 37232}, {21, 90}, {35, 283}, {46, 81}, {55, 1437}, {60, 1780}, {65, 1412}, {86, 12609}, {377, 25526}, {501, 2360}, {662, 11110}, {849, 44119}, {859, 22768}, {940, 37063}, {975, 2268}, {993, 6514}, {1010, 2185}, {1098, 37296}, {1408, 11509}, {1812, 12514}, {1819, 30282}, {2174, 31445}, {2194, 17524}, {2327, 31424}, {2328, 17104}, {3193, 5119}, {3746, 23059}, {4190, 19766}, {4300, 44709}, {5358, 17194}, {7414, 48897}, {10458, 37231}, {10572, 11103}, {18165, 52012}, {18178, 34339}, {21616, 31631}, {28251, 37225}, {28620, 28628}

X(54323) = barycentric product X(81)*X(41229)
X(54323) = barycentric quotient X(41229)/X(321)
X(54323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 14868, 997}, {58, 15792, 284}, {60, 4184, 1780}, {501, 4653, 2360}, {2193, 36746, 58}, {2646, 37227, 4653}, {4278, 9275, 58}

X(54324) = X(4)X(9)∩X(37)X(41)

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

X(54324) lies on these lines: {1, 1731}, {2, 1762}, {3, 2173}, {4, 9}, {5, 7359}, {6, 2294}, {25, 40967}, {31, 43214}, {37, 41}, {44, 65}, {45, 55}, {48, 1385}, {57, 1732}, {63, 9816}, {101, 2302}, {142, 16551}, {198, 23843}, {209, 375}, {212, 1859}, {219, 1482}, {220, 21801}, {284, 37571}, {307, 40530}, {379, 8680}, {380, 3731}, {405, 18673}, {579, 1781}, {610, 7987}, {692, 28125}, {748, 40959}, {857, 24682}, {910, 32578}, {984, 41230}, {1108, 20323}, {1212, 2182}, {1213, 27687}, {1253, 21867}, {1633, 24341}, {1698, 1782}, {1723, 2260}, {1760, 28287}, {2171, 2911}, {2175, 21804}, {2225, 5282}, {2265, 19350}, {2266, 2348}, {2277, 28246}, {2289, 33596}, {2315, 3652}, {2332, 25087}, {2886, 24329}, {2939, 13726}, {3011, 7735}, {3101, 27065}, {3198, 3683}, {3219, 24310}, {3305, 10319}, {3306, 31232}, {3576, 22357}, {3707, 21373}, {3925, 17369}, {4209, 27472}, {4266, 5540}, {4370, 34612}, {4470, 36483}, {5279, 24349}, {5325, 21375}, {5506, 18598}, {5750, 19846}, {7522, 53036}, {8558, 24411}, {8609, 9310}, {9028, 25935}, {10246, 23073}, {10536, 26890}, {10914, 52978}, {11428, 26885}, {11529, 16670}, {11683, 17277}, {12329, 21039}, {14021, 24683}, {15492, 21866}, {15656, 45255}, {16192, 18594}, {16305, 33329}, {16561, 17784}, {16666, 44840}, {17330, 21677}, {17438, 20818}, {18161, 37659}, {18162, 24554}, {20785, 24409}, {21811, 36744}, {24435, 25252}, {24591, 37788}, {25885, 41581}, {25917, 52092}, {31261, 41010}, {34048, 37755}

X(54324) = X(2)-isoconjugate of X(3418)
X(54324) = X(32664)-Dao conjugate of X(3418)
X(54324) = crossdifference of every pair of points on line {1459, 3960}
X(54324) = barycentric product X(i)*X(j) for these {i,j}: {1, 3419}, {9, 37695}, {10, 36011}
X(54324) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 3418}, {3419, 75}, {36011, 86}, {37695, 85}
X(54324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1762, 26934}, {2, 14543, 24315}, {2, 24316, 4466}, {9, 19, 71}, {9, 169, 2183}, {9, 16547, 573}, {9, 16548, 3730}, {281, 26063, 21011}, {1212, 2182, 2267}, {6191, 6192, 4}

X(54325) = X(6)X(31)∩X(100)X(109)

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

X(54325) lies on these lines: {6, 31}, {100, 109}, {101, 8693}, {595, 16479}, {663, 2427}, {692, 2874}, {750, 4675}, {765, 3570}, {813, 919}, {1110, 1983}, {1362, 20749}, {1414, 51563}, {2283, 53555}, {4557, 35326}, {4559, 46177}, {4712, 20778}, {6016, 26716}, {17943, 23997}, {20752, 23612}, {28899, 43077}, {51329, 53552}

X(54325) = isogonal conjugate of the isotomic conjugate of X(1026)
X(54325) = X(i)-Ceva conjugate of X(j) for these (i,j): {1110, 42079}, {7012, 9502}, {36086, 101}, {39293, 9310}
X(54325) = X(i)-isoconjugate of X(j) for these (i,j): {7, 885}, {8, 43930}, {11, 927}, {75, 1027}, {76, 43929}, {85, 1024}, {105, 693}, {244, 51560}, {273, 23696}, {277, 2402}, {279, 28132}, {286, 10099}, {294, 24002}, {513, 2481}, {514, 673}, {649, 18031}, {650, 34018}, {666, 1086}, {668, 43921}, {812, 52209}, {884, 6063}, {905, 54235}, {918, 6185}, {919, 23989}, {1015, 36803}, {1111, 36086}, {1358, 36802}, {1416, 35519}, {1438, 3261}, {1462, 4391}, {1814, 17924}, {2170, 34085}, {2195, 52621}, {3271, 46135}, {3669, 36796}, {3676, 14942}, {3766, 52030}, {4014, 14727}, {4025, 36124}, {4444, 6654}, {4858, 36146}, {7192, 13576}, {7199, 18785}, {7649, 31637}, {8751, 15413}, {20907, 51845}, {21132, 39293}, {32735, 34387}, {33676, 43041}, {36057, 46107}
X(54325) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 1027}, {5375, 18031}, {6184, 3261}, {17755, 40495}, {20621, 46107}, {38980, 23989}, {38989, 1111}, {39014, 4858}, {39026, 2481}, {39046, 693}, {39063, 52621}, {40609, 35519}
X(54325) = cevapoint of X(926) and X(20683)
X(54325) = trilinear pole of line {2223, 9454}
X(54325) = crossdifference of every pair of points on line {514, 1111}
X(54325) = barycentric product X(i)*X(j) for these {i,j}: {1, 2284}, {6, 1026}, {9, 2283}, {31, 42720}, {41, 883}, {55, 1025}, {71, 4238}, {99, 39258}, {100, 672}, {101, 518}, {109, 3693}, {110, 3930}, {163, 3932}, {190, 2223}, {220, 41353}, {241, 3939}, {644, 1458}, {651, 2340}, {662, 20683}, {665, 765}, {666, 42079}, {668, 9454}, {677, 9502}, {692, 3912}, {813, 8299}, {901, 14439}, {906, 1861}, {918, 1110}, {919, 4712}, {926, 4564}, {1018, 3286}, {1023, 34230}, {1252, 2254}, {1331, 5089}, {1332, 2356}, {1415, 3717}, {1783, 1818}, {1876, 4587}, {1897, 20752}, {1978, 9455}, {2149, 50333}, {2414, 21059}, {2427, 36819}, {2428, 3870}, {3252, 3573}, {3263, 32739}, {3570, 40730}, {3699, 52635}, {4437, 32666}, {4557, 18206}, {4570, 24290}, {4684, 34074}, {4899, 34080}, {4998, 46388}, {5548, 53531}, {6065, 53544}, {6078, 53552}, {6184, 36086}, {7045, 52614}, {8750, 25083}, {17755, 34067}, {32656, 46108}, {36039, 50441}, {39686, 51560}
X(54325) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1027}, {41, 885}, {59, 34085}, {100, 18031}, {101, 2481}, {109, 34018}, {241, 52621}, {518, 3261}, {560, 43929}, {604, 43930}, {665, 1111}, {672, 693}, {692, 673}, {765, 36803}, {883, 20567}, {906, 31637}, {926, 4858}, {1025, 6063}, {1026, 76}, {1110, 666}, {1252, 51560}, {1253, 28132}, {1458, 24002}, {1818, 15413}, {1919, 43921}, {2149, 927}, {2175, 1024}, {2200, 10099}, {2223, 514}, {2254, 23989}, {2283, 85}, {2284, 75}, {2340, 4391}, {2356, 17924}, {3286, 7199}, {3675, 23100}, {3693, 35519}, {3912, 40495}, {3930, 850}, {3932, 20948}, {3939, 36796}, {4238, 44129}, {4564, 46135}, {5089, 46107}, {8638, 2170}, {8750, 54235}, {9447, 884}, {9454, 513}, {9455, 649}, {18206, 52619}, {18900, 29956}, {20683, 1577}, {20752, 4025}, {21059, 2402}, {23225, 3942}, {23612, 53583}, {23990, 36086}, {24290, 21207}, {32642, 9503}, {32656, 1814}, {32666, 6185}, {32739, 105}, {34067, 52209}, {39258, 523}, {39686, 2254}, {40730, 4444}, {42079, 918}, {42720, 561}, {46388, 11}, {52425, 23696}, {52614, 24026}, {52635, 3676}

X(54326) = X(1)X(3)∩X(25)X(38)

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

X(54326) lies on these lines: {1, 3}, {22, 4392}, {25, 38}, {33, 12721}, {63, 7083}, {105, 5273}, {197, 3242}, {221, 50621}, {222, 3056}, {244, 7484}, {291, 16058}, {613, 3955}, {614, 7085}, {748, 26867}, {756, 11284}, {954, 21321}, {976, 37257}, {984, 5020}, {1036, 3868}, {1037, 17080}, {1350, 1401}, {1395, 1496}, {1407, 10387}, {1473, 5310}, {1621, 35261}, {1995, 7226}, {2330, 52424}, {3509, 4254}, {3688, 17811}, {3752, 12329}, {4220, 4310}, {4294, 26929}, {5324, 11102}, {7337, 23052}, {7580, 36509}, {8543, 44843}, {9335, 40916}, {9708, 33169}, {9709, 33174}, {9812, 44842}, {10544, 34046}, {11108, 32780}, {12589, 26942}, {12595, 20986}, {16049, 36579}, {16419, 17063}, {16556, 19588}, {17017, 44094}, {18183, 37485}, {19544, 33144}, {21342, 22769}, {24477, 41230}, {26040, 33833}, {26241, 38000}, {28082, 37246}, {32913, 37492}, {33115, 50715}, {34247, 37269}, {36559, 37309}, {36574, 37415}, {37499, 41264}

X(54326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 37581, 1460}, {3677, 5285, 56}, {5329, 17598, 999}, {5597, 5598, 37539}

X(54327) = X(19)X(25)∩X(39)X(42)

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

X(54327) lies on these lines: {1, 4191}, {3, 3555}, {10, 37319}, {19, 25}, {31, 16946}, {35, 3961}, {36, 3979}, {39, 42}, {51, 2293}, {100, 3757}, {184, 1253}, {199, 40910}, {200, 1011}, {210, 8053}, {480, 26867}, {516, 21319}, {518, 22060}, {692, 23201}, {851, 13405}, {902, 20967}, {1402, 2177}, {1622, 12333}, {1961, 3746}, {1999, 3871}, {2000, 10267}, {2194, 19624}, {2304, 52370}, {2340, 3690}, {3085, 4196}, {3295, 5287}, {3683, 4557}, {3689, 52139}, {3744, 5132}, {3748, 20470}, {3913, 17156}, {3935, 4184}, {3938, 37575}, {3939, 26890}, {3957, 4210}, {4105, 22388}, {4219, 11491}, {4314, 13724}, {4362, 8715}, {4421, 42051}, {4666, 16059}, {4847, 30944}, {5010, 23205}, {5217, 22344}, {5256, 37590}, {5271, 5687}, {5311, 40638}, {5320, 21059}, {6194, 17147}, {6600, 7085}, {7074, 22079}, {8580, 16373}, {8731, 25006}, {10578, 37262}, {14547, 51377}, {17018, 37609}, {17524, 34790}, {20075, 31394}, {20243, 31395}, {20760, 35258}, {20986, 23202}, {20990, 37593}, {22369, 40952}, {23207, 32078}, {23853, 35289}, {25440, 29651}

X(54327) = isogonal conjugate of the isotomic conjugate of X(34790)
X(54327) = X(46660)-Dao conjugate of X(693)
X(54327) = crossdifference of every pair of points on line {905, 4801}
X(54327) = barycentric product X(i)*X(j) for these {i,j}: {6, 34790}, {37, 17524}, {101, 50338}, {219, 1887}
X(54327) = barycentric quotient X(i)/X(j) for these {i,j}: {1887, 331}, {17524, 274}, {34790, 76}, {50338, 3261}
X(54327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {42, 2223, 40956}, {55, 15624, 228}, {55, 34247, 968}

X(54328) = X(21)X(37)∩X(99)X(101)

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

X(54328) lies on these lines: {21, 37}, {41, 49447}, {56, 24486}, {99, 101}, {163, 4628}, {643, 46148}, {644, 1633}, {660, 2284}, {813, 36081}, {883, 36146}, {1438, 24841}, {2178, 16447}, {2329, 24723}, {2345, 16376}, {2975, 5701}, {4389, 16788}, {4676, 9310}, {5291, 39786}, {13589, 42723}, {16706, 17681}, {23344, 23830}, {51568, 53337}

X(54328) = X(i)-isoconjugate of X(j) for these (i,j): {513, 39979}, {649, 39714}
X(54328) = X(i)-Dao conjugate of X(j) for these (i,j): {5375, 39714}, {39026, 39979}
X(54328) = crossdifference of every pair of points on line {3122, 38363}
X(54328) = barycentric product X(i)*X(j) for these {i,j}: {100, 32922}, {190, 33854}, {765, 46403}, {1252, 20950}, {7035, 21003}
X(54328) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 39714}, {101, 39979}, {20950, 23989}, {21003, 244}, {22155, 3942}, {32922, 693}, {33854, 514}, {46403, 1111}

X(54329) = X(1)X(6)∩X(41)X(43)

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

X(54329) lies on these lines: {1, 6}, {32, 2319}, {36, 20471}, {41, 43}, {101, 978}, {172, 17754}, {190, 3905}, {384, 3729}, {385, 17743}, {609, 16549}, {644, 3915}, {672, 20460}, {728, 3749}, {894, 16822}, {910, 24440}, {966, 19879}, {1018, 7031}, {1334, 8616}, {1423, 7132}, {1429, 4383}, {1438, 39969}, {1698, 26244}, {1707, 18788}, {1740, 9454}, {1783, 1973}, {1914, 3208}, {2053, 18758}, {2108, 39651}, {2271, 42043}, {2276, 37574}, {2295, 7296}, {2321, 5037}, {3052, 19589}, {3169, 16946}, {3496, 9620}, {3502, 3551}, {3663, 17691}, {3912, 37683}, {3923, 49545}, {4195, 17355}, {4251, 7220}, {4513, 37588}, {4641, 7146}, {4859, 17682}, {5039, 50635}, {5255, 30435}, {5305, 37716}, {6210, 12197}, {7766, 17752}, {8056, 37272}, {9310, 21214}, {9593, 17596}, {9605, 37617}, {11321, 25590}, {14829, 17284}, {16913, 17116}, {16914, 17261}, {17286, 33954}, {17304, 33827}, {17349, 30038}, {17350, 17760}, {17367, 49612}, {17750, 37604}, {19812, 29598}, {20970, 52651}, {23681, 50200}, {24982, 40128}, {26036, 33138}, {29573, 41629}, {32911, 52134}, {33950, 49487}

X(54329) = X(i)-Ceva conjugate of X(j) for these (i,j): {1423, 3550}, {7132, 1}
X(54329) = X(514)-isoconjugate of X(28469)
X(54329) = barycentric product X(i)*X(j) for these {i,j}: {100, 28470}, {1255, 41656}, {18098, 41657}
X(54329) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 28469}, {28470, 693}, {41656, 4359}, {41657, 16703}
X(54329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2329, 1}, {32, 3501, 3550}, {172, 17754, 37608}, {218, 5247, 1743}, {609, 16549, 37603}, {2319, 51319, 3550}, {3507, 51291, 3550}, {5280, 16788, 1}, {9310, 33854, 21214}, {16783, 16785, 1}

X(54330) = X(1)X(6)∩X(63)X(101)

Barycentrics    a*(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(54330) lies on these lines: {1, 6}, {8, 5179}, {41, 12514}, {63, 101}, {78, 3730}, {144, 5088}, {165, 35342}, {169, 3869}, {198, 4047}, {200, 1018}, {239, 31018}, {391, 16821}, {517, 37658}, {609, 1707}, {612, 3997}, {644, 3681}, {672, 997}, {728, 4006}, {758, 40131}, {908, 4384}, {936, 16549}, {1334, 3811}, {1482, 4875}, {1759, 12526}, {1766, 2287}, {2082, 3878}, {2238, 9620}, {2348, 31165}, {3207, 3916}, {3216, 9593}, {3295, 4520}, {3419, 17747}, {3501, 50361}, {3684, 5119}, {3686, 21068}, {3693, 3940}, {3729, 27492}, {3899, 5540}, {3958, 42669}, {4051, 30323}, {4251, 5250}, {4253, 19861}, {4262, 35258}, {4512, 5320}, {4513, 34790}, {4559, 8270}, {4855, 24047}, {5022, 17614}, {5030, 35262}, {5219, 5241}, {5271, 22000}, {5440, 42316}, {5687, 21872}, {5739, 21062}, {7176, 41563}, {7719, 41609}, {10436, 46899}, {12047, 26036}, {12559, 21808}, {14829, 30728}, {16819, 27254}, {16833, 31142}, {17143, 20927}, {18206, 26637}, {20236, 32104}, {24578, 52050}, {26074, 27131}

X(54330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 220, 17742}, {72, 41391, 50995}, {220, 50995, 41391}, {3230, 16973, 1}, {5526, 5692, 9}, {41391, 50995, 17742}

X(54331) = X(8)X(31)∩X(10)X(21)

Barycentrics    a^4 + a*b^3 + 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(54331) lies on these lines: {1, 321}, {2, 10448}, {3, 32918}, {4, 25760}, {8, 31}, {9, 10570}, {10, 21}, {12, 29846}, {20, 26034}, {36, 35999}, {42, 1043}, {51, 9565}, {55, 5793}, {56, 16405}, {65, 4418}, {72, 32938}, {75, 3924}, {78, 32931}, {141, 7354}, {145, 4527}, {171, 10457}, {172, 21024}, {213, 24275}, {238, 11319}, {355, 49128}, {377, 25957}, {388, 33171}, {404, 3831}, {405, 31339}, {443, 25961}, {515, 37399}, {740, 17016}, {748, 17697}, {894, 2650}, {936, 27378}, {950, 47511}, {958, 1011}, {960, 32930}, {976, 4385}, {978, 5192}, {986, 32845}, {993, 10479}, {996, 3632}, {997, 25591}, {1010, 10458}, {1089, 30115}, {1104, 32914}, {1125, 33133}, {1193, 13740}, {1201, 32942}, {1203, 48866}, {1215, 34772}, {1222, 37129}, {1329, 37354}, {1376, 28348}, {1428, 50609}, {1441, 2647}, {1468, 10449}, {1698, 16342}, {1738, 25904}, {1834, 29631}, {1891, 4206}, {1909, 33954}, {2049, 19757}, {2268, 2345}, {2292, 7283}, {2298, 2321}, {2309, 5263}, {2475, 2887}, {2478, 25960}, {2551, 30943}, {2646, 44417}, {2975, 3741}, {3057, 43135}, {3419, 36568}, {3454, 3585}, {3601, 29828}, {3616, 37759}, {3701, 5293}, {3704, 5724}, {3714, 17763}, {3822, 25645}, {3840, 5253}, {3846, 5046}, {3868, 32940}, {3869, 3923}, {3925, 49734}, {3961, 4696}, {3962, 17351}, {4189, 32916}, {4201, 32781}, {4216, 25440}, {4292, 33067}, {4642, 32932}, {4712, 9369}, {4972, 19879}, {5016, 32778}, {5174, 14006}, {5178, 29673}, {5230, 37176}, {5262, 32924}, {5271, 50412}, {5285, 40455}, {5295, 27368}, {5434, 48859}, {5691, 19645}, {5695, 37614}, {5711, 16394}, {5716, 33088}, {5772, 12536}, {5794, 13733}, {5835, 10950}, {6284, 32947}, {6327, 50322}, {6645, 31027}, {6734, 33119}, {6737, 17355}, {6872, 50295}, {7270, 14012}, {9350, 26029}, {10404, 33069}, {10483, 48835}, {11321, 29966}, {11354, 16466}, {12437, 53663}, {12514, 21368}, {13161, 32775}, {13741, 27627}, {15338, 44419}, {15680, 33083}, {16062, 19840}, {16393, 37603}, {16785, 21070}, {16824, 21020}, {16827, 33816}, {16915, 24602}, {16919, 24586}, {17033, 17688}, {17122, 19284}, {17128, 31004}, {17155, 37549}, {17531, 46827}, {17541, 29991}, {17647, 37231}, {17676, 32784}, {17686, 29960}, {19133, 49524}, {19271, 29674}, {19784, 48837}, {19808, 27714}, {19860, 26665}, {19869, 23537}, {20060, 33175}, {20172, 30036}, {20911, 24291}, {21674, 33116}, {21677, 44416}, {23536, 33123}, {24549, 34284}, {25466, 29632}, {25524, 30957}, {26035, 41239}, {26115, 37573}, {29611, 37416}, {30175, 33841}, {30969, 52245}, {32776, 50065}, {32920, 36565}, {32929, 37598}, {32949, 49745}, {33086, 37256}, {33169, 36500}, {34605, 50311}, {37542, 48805}, {37583, 52357}, {42031, 49682}

X(54331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 964, 32772}, {1, 4968, 32923}, {8, 4195, 31}, {8, 5247, 32864}, {10, 21, 32917}, {65, 50054, 4418}, {964, 49492, 1}, {976, 4385, 32927}, {1043, 1220, 42}, {1193, 13740, 32944}, {1468, 10449, 32919}, {2292, 7283, 32936}, {3714, 37539, 17763}, {5086, 32779, 10}, {11115, 17751, 171}

X(54332) = X(3)X(6)∩X(76)X(110)

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

X(54332) lies on these lines: {3, 6}, {76, 110}, {184, 538}, {194, 11003}, {691, 38527}, {1975, 40643}, {1976, 13085}, {2387, 35924}, {3203, 7754}, {3734, 9418}, {3934, 5651}, {5012, 7757}, {5969, 19127}, {6248, 46261}, {6683, 22112}, {7801, 36213}, {8369, 51735}, {9306, 9466}, {16187, 31239}, {21766, 52042}, {32217, 36156}, {33706, 43574}, {35704, 37184}, {43650, 44562}

X(54332) = 2nd-Brocard-circle-inverse of X(3001)
X(54332) = {X(1670),X(1671)}-harmonic conjugate of X(3001)

X(54333) = X(2)X(11)∩X(6)X(101)

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

X(54333) lies on these lines: {1, 4557}, {2, 11}, {3, 16686}, {6, 101}, {10, 19244}, {31, 33846}, {35, 19249}, {36, 238}, {43, 18613}, {56, 651}, {214, 16494}, {239, 15571}, {244, 53280}, {536, 33845}, {537, 23343}, {595, 16414}, {748, 16678}, {958, 19238}, {978, 23383}, {993, 19255}, {1054, 23845}, {1086, 15507}, {1120, 20037}, {1125, 4267}, {1193, 3122}, {1421, 23067}, {1646, 5163}, {2093, 45022}, {2223, 3246}, {2308, 40153}, {2836, 3675}, {3052, 16059}, {3185, 5272}, {3248, 16493}, {3295, 19253}, {3720, 18185}, {3742, 20967}, {4216, 8692}, {4432, 4436}, {4649, 37602}, {5010, 8053}, {5096, 51621}, {5217, 19292}, {5264, 16297}, {5400, 15626}, {5563, 16477}, {5701, 18785}, {6767, 19250}, {8168, 49460}, {9355, 53296}, {15254, 37575}, {15621, 16569}, {15668, 25532}, {16468, 19293}, {16484, 19265}, {16495, 53303}, {16560, 53293}, {16602, 37619}, {17123, 52139}, {17259, 19239}, {17277, 18047}, {17278, 31394}, {19242, 32941}, {19243, 24294}, {19550, 35238}, {20468, 36741}, {21214, 23361}, {23844, 24174}, {23853, 37679}, {23981, 43048}, {24841, 52923}, {27623, 27667}, {28365, 28400}, {36740, 38048}

X(54333) = crossdifference of every pair of points on line {37, 665}
X(54333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {238, 20470, 3286}, {238, 49997, 52897}, {859, 52897, 3286}, {1001, 1376, 48805}, {4557, 53307, 1}, {27628, 28393, 11}, {28239, 28353, 3035}, {28250, 28364, 2886}

X(54334) = X(3)X(6)∩X(67)X(69)

Barycentrics    a^2*(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(54334) = 5 X[2] - 4 X[40670], 3 X[9971] - 4 X[16776], 5 X[9971] - 8 X[40670], 5 X[16776] - 6 X[40670], 4 X[3] - X[37473], X[6] + 2 X[3313], X[6] - 4 X[11574], 2 X[52] - 5 X[53093], 4 X[389] - 7 X[10541], 4 X[575] - X[6243], 2 X[576] + X[37484], X[1350] + 2 X[9967], 2 X[1350] + X[44439], 2 X[3098] + X[18438], X[3313] and many others

X(54334) lies on these lines: {2, 9019}, {3, 6}, {22, 18374}, {51, 47352}, {67, 69}, {110, 35707}, {141, 858}, {159, 6090}, {160, 9155}, {206, 2916}, {315, 35549}, {338, 37190}, {373, 9969}, {376, 2781}, {524, 2979}, {542, 23039}, {597, 3060}, {599, 1853}, {888, 14277}, {1154, 11179}, {1176, 19151}, {1205, 2930}, {1216, 15069}, {1249, 53772}, {1352, 14791}, {1368, 16789}, {1503, 11459}, {1576, 46546}, {1594, 3867}, {1843, 3763}, {1974, 21284}, {3589, 5640}, {3618, 11002}, {3619, 41579}, {3631, 12272}, {3818, 7574}, {3819, 21358}, {4549, 5663}, {5596, 12250}, {5890, 51737}, {5891, 47353}, {5946, 38064}, {6144, 32366}, {6403, 37118}, {6467, 9027}, {6593, 7492}, {6636, 19127}, {6787, 36187}, {6800, 20806}, {7467, 18371}, {7493, 40949}, {7499, 51744}, {7502, 15462}, {8177, 46303}, {8546, 23061}, {8550, 11412}, {8584, 44323}, {8681, 15533}, {8717, 51941}, {8889, 51994}, {9220, 15980}, {9306, 19596}, {9822, 15082}, {10170, 10516}, {10295, 48881}, {10300, 47558}, {10323, 34117}, {10519, 44668}, {10627, 15074}, {11443, 15826}, {11451, 48310}, {11645, 18435}, {11649, 50977}, {12294, 37196}, {12367, 15066}, {13201, 25329}, {13391, 20423}, {13451, 38079}, {13754, 43273}, {14173, 44719}, {14179, 44718}, {14915, 48905}, {14984, 54042}, {15030, 36990}, {15035, 35228}, {15045, 50983}, {15072, 16775}, {15122, 47468}, {15138, 34778}, {15140, 19121}, {15270, 22424}, {15302, 30489}, {15534, 40673}, {16051, 32246}, {16111, 35257}, {16285, 23642}, {16981, 51171}, {18358, 47341}, {20987, 26283}, {21969, 51185}, {22078, 23635}, {23326, 54041}, {23327, 34751}, {29181, 38323}, {31670, 50008}, {33879, 34573}, {33962, 43619}, {34118, 36851}, {34725, 46847}, {34990, 37184}, {35921, 51739}, {37440, 43811}, {37978, 52238}, {44280, 50965}, {48906, 54215}

X(54334) = midpoint of X(i) and X(j) for these {i,j}: {11188, 12220}, {15072, 41716}
X(54334) = reflection of X(i) in X(j) for these {i,j}: {568, 182}, {599, 3917}, {1352, 15067}, {3060, 597}, {5890, 51737}, {8584, 44323}, {9971, 2}, {9973, 11188}, {11188, 141}, {15072, 44882}, {15534, 40673}, {19161, 16836}, {29959, 3819}, {34751, 23327}, {36990, 15030}, {40949, 41670}, {47353, 5891}, {52989, 5092}, {54173, 54042}
X(54334) = anticomplement of X(16776)
X(54334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3001, 566}, {141, 12220, 9973}, {1350, 9967, 44439}, {3313, 11574, 6}, {3819, 29959, 21358}, {5092, 52989, 40280}, {6636, 22151, 19127}, {7998, 11188, 141}, {7998, 12220, 11188}, {10625, 44479, 11477}, {15644, 50649, 53097}, {41328, 50645, 13351}

X(54335) = X(1)X(2)∩X(75)X(99)

Barycentrics    a^4 - a^2*b^2 - 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 - b*c^3 : :

X(54335) lies on these lines: {1, 2}, {3, 28612}, {9, 4115}, {21, 4647}, {36, 4359}, {55, 16430}, {56, 16429}, {58, 4697}, {75, 99}, {98, 8691}, {100, 4714}, {191, 17164}, {321, 5251}, {333, 758}, {404, 28611}, {405, 4387}, {442, 36974}, {540, 33097}, {596, 1247}, {740, 4653}, {958, 52531}, {1001, 42713}, {1043, 35016}, {1089, 5260}, {1150, 5902}, {1324, 16678}, {1325, 2975}, {1330, 11263}, {1739, 32918}, {2886, 37346}, {3304, 16425}, {3454, 24161}, {3576, 24435}, {3649, 49716}, {3685, 4717}, {3696, 24929}, {3699, 3956}, {3702, 5259}, {3704, 6675}, {3712, 15670}, {3729, 24074}, {3743, 11110}, {3841, 7270}, {3936, 26725}, {3980, 4257}, {4001, 11551}, {4418, 46895}, {4424, 32917}, {4442, 49735}, {4643, 50273}, {4680, 33108}, {4683, 49723}, {4689, 50083}, {4703, 48839}, {4854, 13745}, {4968, 5258}, {4975, 5284}, {5253, 6533}, {5263, 49480}, {5278, 5692}, {5426, 17163}, {5429, 24342}, {5563, 16424}, {5695, 16418}, {5741, 37701}, {5814, 28628}, {5883, 14829}, {6757, 8666}, {7283, 42031}, {10176, 17277}, {11281, 41014}, {12699, 46975}, {16052, 17070}, {16132, 48877}, {16137, 49718}, {16611, 26244}, {16992, 33936}, {17491, 50215}, {17889, 48835}, {18481, 46617}, {18661, 52709}, {19623, 24325}, {21076, 24937}, {24390, 30447}, {24697, 49729}, {26117, 36250}, {26728, 49511}, {27784, 37035}, {32922, 46912}, {33132, 48843}, {35028, 46800}, {35148, 35957}, {37817, 50314}, {41002, 51409}, {47040, 50086}, {48935, 49177}, {50296, 50886}, {51111, 52244}

X(54335) = midpoint of X(3757) and X(16821)
X(54335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 39766, 1}, {8, 19854, 30172}, {8, 25446, 10}, {8, 25650, 21081}, {10, 1125, 24931}, {10, 50757, 2}, {1125, 21081, 25650}, {46895, 52680, 4418}

X(54336) = X(10)X(31)∩X(32)X(37)

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

X(54336) lies on these lines: {1, 2206}, {10, 31}, {32, 37}, {58, 75}, {65, 1397}, {171, 20083}, {225, 1395}, {595, 31359}, {596, 1468}, {727, 839}, {1106, 3668}, {1125, 1472}, {1714, 17126}, {2218, 49480}, {2901, 44115}, {3915, 42285}, {5271, 39708}, {7121, 42027}, {18833, 52394}, {23604, 40940}, {29645, 34920}, {30148, 51687}, {32774, 37522}

X(54336) = X(i)-isoconjugate of X(j) for these (i,j): {2, 4261}, {3, 5142}, {6, 32782}, {668, 838}
X(54336) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 32782}, {32664, 4261}, {36103, 5142}
X(54336) = trilinear pole of line {661, 1919}
X(54336) = barycentric product X(649)*X(839)
X(54336) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 32782}, {19, 5142}, {31, 4261}, {839, 1978}, {1919, 838}

X(54337) = X(3)X(63)∩X(35)X(42)

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

X(54337) lies on these lines: {3, 63}, {9, 11337}, {10, 16049}, {21, 5285}, {22, 31424}, {35, 42}, {71, 283}, {100, 37402}, {165, 11413}, {205, 3730}, {333, 19842}, {404, 4357}, {579, 2268}, {970, 26890}, {1444, 52396}, {1782, 7688}, {1790, 3682}, {1791, 3710}, {2915, 31445}, {3305, 37034}, {3663, 19850}, {3955, 22076}, {4292, 7465}, {4663, 5217}, {5249, 7523}, {5251, 11101}, {5271, 19845}, {5273, 7520}, {5302, 20989}, {5745, 37231}, {6734, 37399}, {7485, 15803}, {8193, 35258}, {12572, 35996}, {13323, 26893}, {15592, 37577}, {19547, 30852}, {19841, 32939}, {25440, 35980}, {37246, 37581}, {39582, 41229}, {41340, 52362}

X(54337) = isogonal conjugate of the polar conjugate of X(19808)
X(54337) = crossdifference of every pair of points on line {4988, 6591}
X(54337) = barycentric product X(3)*X(19808)
X(54337) = barycentric quotient X(19808)/X(264)
X(54337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3916, 7293}, {3, 7085, 78}

X(54338) = X(7)X(8)∩X(9)X(39)

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

X(54338) lies on these lines: {1, 6007}, {7, 8}, {9, 39}, {78, 21320}, {142, 3831}, {144, 41834}, {726, 3781}, {899, 5220}, {960, 4419}, {1001, 1201}, {1042, 12513}, {1401, 11679}, {2235, 2275}, {3056, 32922}, {3663, 35628}, {3681, 26840}, {3688, 49446}, {3784, 4362}, {3792, 49493}, {3869, 4440}, {4361, 8679}, {4517, 49447}, {5211, 10394}, {5223, 6048}, {5542, 35620}, {5728, 28078}, {5782, 25524}, {7248, 14829}, {10477, 24231}, {15481, 24530}, {17151, 35104}, {17155, 26893}, {21334, 30699}, {25557, 30942}, {26029, 38057}, {26106, 38053}, {26892, 32914}, {29311, 53594}, {49537, 51192}

X(54338) = reflection of X(35892) in X(5542)
X(54338) = crossdifference of every pair of points on line {3063, 50353}

X(54339) = X(1)X(3)∩X(73)X(81)

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

X(54339) lies on these lines: {1, 3}, {2, 1451}, {6, 37694}, {7, 603}, {12, 5247}, {27, 225}, {31, 3485}, {48, 27621}, {58, 226}, {60, 23692}, {73, 81}, {86, 283}, {109, 3671}, {212, 5703}, {238, 11375}, {255, 3487}, {388, 1468}, {404, 1818}, {411, 14547}, {474, 52424}, {580, 13411}, {601, 4295}, {750, 1788}, {962, 52428}, {975, 1708}, {1068, 7554}, {1106, 7365}, {1393, 5262}, {1399, 3649}, {1400, 2303}, {1408, 7175}, {1450, 5253}, {1453, 19372}, {1469, 36508}, {1471, 4648}, {1496, 3475}, {1497, 5603}, {1724, 5219}, {1745, 36742}, {2292, 7098}, {2594, 4649}, {2647, 5429}, {2887, 26363}, {2982, 3682}, {3073, 12047}, {3074, 5398}, {3216, 52423}, {4292, 37469}, {4298, 34050}, {4645, 10527}, {4652, 7190}, {5244, 43045}, {5292, 37093}, {5293, 41538}, {5434, 9363}, {5705, 37660}, {5719, 52408}, {6147, 52407}, {6734, 7270}, {6817, 11269}, {7513, 40950}, {8557, 15656}, {9817, 10396}, {10404, 29658}, {11501, 50581}, {12649, 37684}, {15556, 30115}, {15844, 37646}, {16577, 41547}, {17122, 24914}, {17792, 45728}, {17811, 25524}, {18162, 22345}, {18962, 37716}, {26131, 37797}, {26481, 33140}, {26889, 28349}, {52440, 52783}

X(54339) = barycentric product X(56)*X(19810)
X(54339) = barycentric quotient X(19810)/X(3596)
X(54339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 37591}, {1, 37530, 1936}, {56, 940, 37523}, {57, 37554, 1038}, {58, 226, 1935}, {5398, 11374, 3074}

X(54340) = X(2)X(3)∩X(34)X(81)

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

X(54340) lies on these lines: {2, 3}, {19, 2287}, {34, 81}, {100, 1869}, {225, 33133}, {270, 3194}, {273, 1014}, {1172, 1880}, {1722, 1780}, {1824, 34772}, {1826, 5260}, {1829, 40571}, {1841, 2303}, {1859, 45230}, {1868, 3219}, {1871, 21740}, {1891, 5086}, {2285, 46884}, {2975, 5307}, {5235, 46878}, {41601, 44545}

X(54340) = X(41087)-isoconjugate of X(46014)
X(54340) = barycentric product X(27)*X(19860)
X(54340) = barycentric quotient X(i)/X(j) for these {i,j}: {3194, 46014}, {19860, 306}, {46012, 52389}
X(54340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 28, 21}, {4, 31900, 14016}, {28, 29, 4233}, {28, 4183, 13739}, {28, 4185, 14014}, {28, 31902, 4227}, {29, 37258, 35981}, {7497, 37377, 4198}

X(54341) = X(1)X(2)∩X(6)X(22)

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

X(54341) lies on these lines: {1, 2}, {6, 22}, {23, 20865}, {31, 4283}, {39, 4184}, {51, 50595}, {58, 6636}, {427, 48847}, {579, 21764}, {583, 39673}, {1194, 20970}, {1203, 5310}, {1627, 18755}, {1834, 5133}, {1914, 4261}, {2176, 26911}, {2271, 5359}, {2979, 50591}, {3192, 6995}, {4220, 5396}, {4251, 34482}, {4255, 7485}, {4256, 15246}, {4272, 5276}, {4279, 20966}, {5153, 33854}, {5314, 16470}, {7391, 48837}, {7465, 52544}, {8024, 33296}, {8267, 17499}, {9605, 20835}, {22090, 44435}, {31133, 48842}, {32911, 37325}, {33774, 33863}, {37662, 37990}, {37678, 39998}, {44210, 48861}

X(54341) = crossdifference of every pair of points on line {649, 826}
X(54341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 43, 15523}

X(54342) = X(1)X(7)∩X(55)X(80)

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

X(54342) lies on these lines: {1, 7}, {30, 37703}, {35, 5722}, {55, 80}, {57, 50812}, {226, 5561}, {484, 3488}, {550, 50190}, {950, 18395}, {1000, 3065}, {1387, 3058}, {1479, 6900}, {1621, 9963}, {1698, 50398}, {1737, 51817}, {2320, 21630}, {3295, 5441}, {3434, 5426}, {3486, 37563}, {3582, 30282}, {3583, 5219}, {3584, 3586}, {3601, 4857}, {3612, 37704}, {3746, 5252}, {3911, 5010}, {4870, 18393}, {4894, 52352}, {5218, 37718}, {5443, 9670}, {5444, 11238}, {5531, 6930}, {5697, 10386}, {5719, 6284}, {5902, 15935}, {6767, 36975}, {7741, 31795}, {7951, 52638}, {7967, 13253}, {9668, 37701}, {10246, 14217}, {12433, 37572}, {12690, 17057}, {12953, 37731}, {13405, 18513}, {15171, 15950}, {15172, 21842}, {15228, 15934}, {15338, 18398}, {16173, 37606}, {17601, 24223}, {20066, 30143}, {29638, 48836}, {32844, 47040}

X(54342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4304, 30331, 21578}, {4309, 4313, 1}, {10386, 10543, 5697}, {21578, 30331, 1}

X(54343) = X(2)X(3)∩X(34)X(63)

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

X(54343) lies on these lines: {2, 3}, {8, 41227}, {19, 19860}, {34, 63}, {55, 5174}, {56, 17923}, {65, 1748}, {92, 958}, {219, 608}, {240, 3924}, {243, 22760}, {278, 2975}, {281, 5260}, {993, 1838}, {1259, 5081}, {1844, 30143}, {1848, 24541}, {1852, 2886}, {1870, 1993}, {1888, 4640}, {1891, 24987}, {3486, 41230}, {5208, 44105}, {5251, 39585}, {5253, 17917}, {10528, 38300}, {11471, 35258}, {11681, 37799}

X(54343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 28, 37253}, {3, 5125, 35994}, {4, 21, 1013}, {27, 11109, 37235}, {28, 37305, 37231}, {405, 7497, 29}, {475, 37395, 377}, {3560, 7535, 25516}, {4185, 37228, 37235}, {5047, 17519, 7498}, {5125, 17515, 3}, {7518, 16865, 4183}

X(54344) = X(7)X(8)∩X(37)X(63)

Barycentrics    a*(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(54344) lies on these lines: {7, 8}, {20, 49470}, {21, 757}, {37, 63}, {56, 3964}, {72, 10436}, {86, 960}, {192, 9965}, {193, 2262}, {264, 1887}, {286, 1859}, {317, 1875}, {326, 12635}, {354, 17321}, {478, 23144}, {517, 3879}, {536, 50292}, {740, 4292}, {758, 3664}, {942, 4205}, {1014, 4511}, {1071, 10441}, {1211, 3739}, {1444, 2646}, {1858, 17139}, {1944, 9119}, {2264, 41610}, {3555, 3875}, {3663, 3874}, {3672, 3873}, {3740, 28653}, {3742, 17322}, {3753, 17270}, {3812, 5224}, {3869, 3945}, {3880, 17377}, {3881, 4021}, {3884, 4909}, {3894, 4862}, {3901, 4888}, {4022, 11031}, {4304, 49471}, {4360, 34791}, {4430, 4452}, {4644, 43216}, {4664, 28610}, {4687, 5273}, {4851, 21853}, {4967, 34790}, {5208, 8822}, {5902, 17272}, {5904, 25590}, {5905, 5928}, {6001, 10446}, {6356, 9436}, {9799, 51063}, {10442, 15071}, {10884, 30271}, {10889, 12711}, {11520, 37614}, {12723, 35892}, {16465, 18655}, {17274, 24473}, {17316, 21871}, {17364, 34371}, {17365, 34377}, {17374, 21863}, {17378, 44663}, {17616, 44671}, {17863, 20347}, {18206, 40937}, {18252, 52020}, {18650, 20718}, {21866, 22370}, {21874, 27623}, {22021, 25083}, {29747, 37597}, {37175, 37593}, {44547, 44735}

X(54344) = crossdifference of every pair of points on line {3063, 48099}

X(54345) = X(2)X(3)∩X(8)X(88)

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

X(54345) lies on these lines: {2, 3}, {8, 88}, {386, 17378}, {519, 3976}, {752, 978}, {936, 17274}, {956, 26073}, {975, 17320}, {1125, 49746}, {3419, 27002}, {3877, 44299}, {4255, 17313}, {4256, 17234}, {4257, 17352}, {4302, 25531}, {5015, 11512}, {5044, 17333}, {5293, 50285}, {6361, 25879}, {9668, 26139}, {12436, 50116}, {15172, 26111}, {18481, 25965}, {18990, 26029}, {25524, 48829}, {25914, 48810}, {37592, 50286}, {37607, 50287}

X(54345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 33309}, {2, 4190, 4217}, {2, 4217, 13741}, {2, 6904, 51668}, {2, 16454, 51604}, {2, 36004, 11346}, {2, 51668, 13740}, {17582, 37339, 11110}

X(54346) = X(2)X(34)∩X(10)X(73)

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

X(54346) lies on these lines: {1, 6889}, {2, 34}, {3, 40950}, {5, 1076}, {6, 24914}, {10, 73}, {11, 15852}, {12, 1427}, {33, 6908}, {46, 5713}, {56, 29639}, {65, 17056}, {122, 27687}, {142, 1393}, {201, 226}, {212, 1771}, {222, 5791}, {223, 1698}, {225, 442}, {227, 3925}, {241, 15844}, {405, 1877}, {581, 1737}, {603, 5745}, {631, 34231}, {851, 1869}, {948, 10588}, {1040, 37112}, {1042, 21674}, {1074, 37438}, {1104, 5433}, {1210, 14547}, {1211, 26955}, {1212, 5514}, {1213, 30456}, {1442, 24883}, {1451, 3911}, {1453, 31231}, {1455, 24953}, {1465, 8728}, {1714, 45126}, {1770, 45924}, {1785, 6937}, {1788, 5530}, {1825, 41340}, {1826, 3142}, {1838, 6829}, {1861, 26027}, {1891, 7413}, {1943, 25446}, {3600, 29664}, {4197, 17080}, {4303, 51755}, {4322, 29690}, {4332, 29661}, {5081, 7572}, {5249, 37591}, {5265, 29680}, {5716, 7288}, {6245, 22053}, {6260, 7069}, {6734, 37523}, {6838, 9817}, {6842, 20620}, {7078, 26446}, {7098, 50307}, {8270, 10198}, {10106, 16499}, {11392, 26052}, {17095, 30761}, {19349, 26066}, {19854, 21147}, {22341, 37225}, {24537, 34851}, {24806, 24987}, {27577, 42289}, {37275, 52427}

X(54346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {442, 1214, 225}, {3911, 5717, 1451}, {37438, 37565, 1074}

X(54347) = X(2)X(6)∩X(54)X(67)

Barycentrics    a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 - 6*a^4*b^2*c^2 + a^2*b^4*c^2 - a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + c^8 : :
X(54347) = 5 X[3620] - X[45794], 3 X[13394] - 2 X[19127]

X(54347) lies on these lines: {2, 6}, {4, 34787}, {5, 5181}, {39, 15526}, {53, 37778}, {54, 67}, {66, 19459}, {125, 40673}, {159, 31383}, {182, 10257}, {264, 53477}, {338, 1235}, {378, 1503}, {389, 40107}, {403, 5480}, {427, 2393}, {441, 5063}, {468, 19136}, {511, 15760}, {542, 11430}, {570, 41005}, {571, 41008}, {575, 32257}, {576, 41587}, {578, 34507}, {1154, 19161}, {1352, 9818}, {1594, 15073}, {1899, 32621}, {2071, 44882}, {2076, 35928}, {2854, 12827}, {2892, 3520}, {2930, 7527}, {3260, 41237}, {3541, 8549}, {3548, 44503}, {3549, 44492}, {3564, 52262}, {3867, 9973}, {5094, 5486}, {5117, 45279}, {5133, 11188}, {5449, 32284}, {5476, 47473}, {5621, 23328}, {5648, 41171}, {5972, 41612}, {6143, 32241}, {6146, 34118}, {6193, 11425}, {6247, 18925}, {6467, 23300}, {6623, 53023}, {6697, 32366}, {6776, 10249}, {7403, 43130}, {7542, 44470}, {7687, 25561}, {7706, 47468}, {7753, 45312}, {8541, 51744}, {8542, 37454}, {8548, 15120}, {8681, 21243}, {9722, 14767}, {9967, 51392}, {9969, 41584}, {10169, 21639}, {10192, 18374}, {10516, 18537}, {11178, 18390}, {11412, 11660}, {11438, 50977}, {11585, 44479}, {11746, 20113}, {12294, 51403}, {13371, 15074}, {13394, 16387}, {13403, 18553}, {15118, 52293}, {15531, 23293}, {15581, 16655}, {15585, 20987}, {16043, 40691}, {16511, 19510}, {18560, 38885}, {18583, 44911}, {18642, 36743}, {18919, 52299}, {19118, 31267}, {20192, 47556}, {21850, 41583}, {23330, 25328}, {26864, 31166}, {27376, 51260}, {27377, 53485}, {29181, 44440}, {30522, 39884}, {34417, 47449}, {37855, 53418}, {37962, 47450}, {41593, 46444}, {41729, 52416}, {44441, 54183}, {44458, 48881}, {44668, 45179}, {49672, 50983}, {51163, 52403}, {52281, 53507}

X(54347) = midpoint of X(69) and X(1993)
X(54347) = reflection of X(i) in X(j) for these {i,j}: {6, 23292}, {343, 141}, {8541, 51744}, {41612, 5972}
X(54347) = complement of X(41614)
X(54347) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 8542}, {661, 14672}, {5486, 18589}, {30247, 4369}, {36115, 690}, {37217, 512}, {51831, 21247}
X(54347) = crossdifference of every pair of points on line {512, 41613}
X(54347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 8263, 29959}, {6, 21358, 26958}, {5094, 10602, 23327}, {5181, 29959, 8263}, {5480, 41585, 9971}, {5486, 23327, 10602}, {9971, 32113, 41585}, {11427, 37636, 13567}, {16511, 19510, 30739}

X(54348) = X(2)X(11)∩X(21)X(84)

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

X(54348) lies on these lines: {1, 1167}, {2, 11}, {8, 25875}, {21, 84}, {56, 9965}, {63, 7677}, {238, 25941}, {329, 1617}, {343, 33173}, {392, 1006}, {394, 17127}, {404, 946}, {405, 944}, {411, 41012}, {452, 12667}, {480, 20015}, {516, 35977}, {943, 37244}, {960, 1319}, {962, 37282}, {968, 26635}, {997, 42012}, {1004, 9812}, {1005, 15931}, {1058, 11517}, {1259, 14986}, {1260, 36845}, {1279, 25091}, {1470, 3485}, {1699, 35990}, {1848, 35973}, {2078, 3452}, {2346, 42470}, {2551, 11510}, {3052, 25934}, {3149, 26129}, {3256, 6692}, {3428, 37313}, {3475, 42843}, {3550, 25938}, {3616, 37248}, {3685, 17862}, {3746, 8582}, {3757, 26591}, {3871, 24982}, {3885, 19860}, {3957, 5572}, {4184, 24556}, {4187, 11491}, {4193, 48482}, {4430, 12635}, {4511, 16465}, {4512, 10857}, {5047, 24987}, {5084, 10267}, {5248, 8583}, {5249, 8543}, {5250, 6986}, {5259, 10572}, {5330, 31806}, {5603, 37249}, {5658, 13615}, {5698, 37578}, {5919, 33895}, {6796, 25522}, {6904, 11496}, {6906, 17614}, {6909, 35262}, {6914, 35272}, {6919, 11500}, {6950, 52148}, {8273, 17576}, {8641, 26695}, {9614, 25440}, {9776, 15804}, {9778, 37309}, {10306, 26062}, {10601, 17018}, {10806, 16845}, {11015, 24564}, {11108, 38042}, {11248, 17567}, {11849, 52264}, {13588, 24545}, {15485, 25885}, {17527, 37621}, {17564, 35000}, {18861, 19525}, {20835, 52653}, {25568, 33925}, {25962, 52367}, {25970, 29851}, {26010, 29846}, {26013, 32943}, {26611, 33153}, {31394, 37261}, {34647, 48698}

X(54348) = anticomplement of X(25973)
X(54348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 25893, 2}, {1260, 42884, 36845}, {15931, 40998, 1005}

X(54349) = X(3)X(54)∩X(6)X(41)

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

X(54349) lies on these lines: {1, 21807}, {3, 54}, {6, 41}, {35, 23202}, {51, 2360}, {81, 27622}, {184, 581}, {199, 5752}, {283, 34986}, {386, 44104}, {389, 20838}, {405, 3897}, {474, 17191}, {500, 16064}, {572, 22076}, {580, 13366}, {859, 36750}, {970, 1790}, {1011, 10267}, {1051, 2939}, {1181, 37195}, {1199, 37115}, {1351, 37250}, {1437, 5396}, {1724, 21842}, {1730, 3337}, {1870, 4185}, {1994, 4225}, {2003, 22345}, {2302, 7066}, {2476, 26141}, {2594, 20986}, {3060, 20836}, {3167, 11344}, {3193, 4192}, {3616, 36942}, {3682, 26890}, {3876, 17976}, {4191, 36754}, {4209, 20145}, {4303, 26889}, {5050, 37282}, {5708, 11347}, {6090, 16293}, {7421, 15032}, {7428, 51340}, {7522, 29833}, {9566, 11340}, {9777, 13737}, {11245, 18641}, {11432, 37310}, {13323, 52273}, {14793, 19763}, {14912, 37180}, {15988, 37255}, {16287, 22136}, {16451, 37510}, {16452, 22139}, {16453, 37509}, {19684, 37056}, {19762, 36152}, {19767, 37397}, {20833, 48907}, {20843, 37469}, {27621, 37685}, {28238, 32911}, {28348, 36742}, {35998, 48921}, {37231, 48909}, {44085, 52544}

X(54349) = crossdifference of every pair of points on line {522, 12077}
X(54349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {184, 581, 3145}, {1437, 5396, 37259}

X(54350) = X(1)X(3)∩X(8)X(47)

Barycentrics    a*(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 - 4*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(54350) lies on these lines: {1, 3}, {8, 47}, {10, 2216}, {31, 10573}, {80, 3073}, {109, 45287}, {238, 18395}, {255, 12647}, {498, 33105}, {499, 3915}, {595, 1737}, {952, 1399}, {1201, 10090}, {1331, 10915}, {1935, 37710}, {2190, 5174}, {2361, 5690}, {2964, 5247}, {3085, 33112}, {4317, 9316}, {5398, 41687}, {5790, 7299}, {8070, 33106}, {10944, 52407}, {16473, 50581}, {18360, 18990}, {41686, 49500}

X(54350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 14792, 37617}, {1771, 37610, 1}, {2964, 41684, 5247}, {3075, 37588, 1}, {5264, 37610, 37552}, {5710, 11507, 1}, {22767, 37542, 1}

X(54351) = X(6)X(31)∩X(44)X(58)

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

X(54351) lies on these lines: {6, 31}, {9, 5115}, {35, 4285}, {44, 58}, {45, 1468}, {48, 28607}, {171, 17330}, {213, 5035}, {595, 16666}, {757, 17260}, {1100, 40091}, {1203, 5109}, {1213, 17122}, {1333, 2251}, {1399, 1405}, {1400, 52440}, {1743, 4275}, {2174, 5019}, {2300, 9456}, {3747, 16522}, {4264, 16669}, {4290, 16670}, {4969, 5255}, {5247, 17369}, {5280, 28614}, {5356, 40977}, {15955, 21864}, {16521, 20985}, {17123, 17398}, {17126, 37654}, {19623, 27064}, {28658, 52407}, {37610, 50131}

X(54351) = isogonal conjugate of the isotomic conjugate of X(17021)
X(54351) = barycentric product X(i)*X(j) for these {i,j}: {1, 16474}, {6, 17021}
X(54351) = barycentric quotient X(i)/X(j) for these {i,j}: {16474, 75}, {17021, 76}
X(54351) = {X(213),X(5035)}-harmonic conjugate of X(7113)

X(54352) = X(1)X(21)∩X(43)X(88)

Barycentrics    a*(a^2 + 2*a*b - 2*b^2 + 2*a*c + 2*b*c - 2*c^2) : :
X(54352) = 3 X[750] - 4 X[37520]

X(54352) lies on these lines: {1, 21}, {2, 4407}, {6, 17449}, {7, 33136}, {9, 17450}, {42, 17595}, {43, 88}, {44, 354}, {45, 3720}, {57, 53397}, {89, 171}, {100, 49498}, {145, 32845}, {244, 3751}, {320, 31134}, {518, 750}, {614, 4722}, {678, 3870}, {899, 4860}, {902, 42871}, {982, 17012}, {984, 9345}, {1002, 2239}, {1150, 49479}, {1320, 53115}, {1471, 5083}, {1757, 17125}, {2177, 3218}, {2243, 2280}, {2308, 17597}, {2999, 42040}, {3187, 42055}, {3214, 5708}, {3240, 18201}, {3243, 3722}, {3246, 4641}, {3306, 21805}, {3315, 16468}, {3681, 17124}, {3749, 9340}, {3914, 4887}, {3957, 4650}, {3979, 17782}, {3989, 16672}, {3999, 4663}, {4038, 7226}, {4346, 33145}, {4363, 31136}, {4392, 4649}, {4414, 49478}, {4655, 29835}, {4661, 17122}, {4683, 29843}, {4684, 33156}, {4792, 40215}, {4847, 4896}, {4966, 33161}, {5220, 30950}, {5221, 7248}, {5256, 42038}, {5287, 42039}, {5297, 49503}, {5332, 17599}, {5695, 50001}, {7262, 29817}, {9324, 9352}, {9350, 27003}, {9780, 25961}, {10453, 32940}, {11269, 32856}, {16490, 17461}, {16499, 53114}, {16522, 41269}, {16666, 17017}, {16704, 17146}, {16816, 32864}, {17022, 42041}, {17126, 49675}, {17140, 32853}, {17145, 32941}, {17154, 32921}, {17155, 17160}, {17187, 18198}, {17298, 21026}, {17364, 32844}, {17365, 33104}, {17483, 33141}, {17484, 24217}, {17495, 49497}, {17598, 37685}, {17763, 49499}, {20963, 36283}, {24165, 50018}, {24231, 33128}, {24349, 32919}, {24473, 49487}, {24477, 33105}, {24725, 26015}, {26061, 29596}, {26070, 29839}, {26102, 51297}, {26227, 49491}, {26627, 49457}, {26842, 32865}, {29579, 33163}, {29655, 32859}, {29662, 37691}, {29824, 32935}, {29833, 50285}, {30579, 32934}, {31137, 41242}, {31237, 33069}, {32863, 33169}, {32920, 37639}, {32923, 37683}, {32927, 37684}, {32933, 42057}, {33087, 33170}, {33103, 33142}, {33114, 49676}, {37567, 41682}, {37633, 49448}, {42058, 49700}, {43149, 52434}, {49764, 50105}, {50102, 53601}

X(54352) = barycentric product X(1)*X(17313)
X(54352) = barycentric quotient X(17313)/X(75)
X(54352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {320, 33120, 31134}, {354, 32912, 748}, {3218, 49490, 2177}, {3873, 32913, 31}, {17365, 51463, 33104}, {33069, 33121, 31237}

X(54353) = X(1)X(21)∩X(99)X(109)

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

X(54353) lies on these lines: {1, 21}, {6, 16375}, {55, 3110}, {99, 109}, {100, 43076}, {101, 110}, {162, 37206}, {284, 9319}, {386, 16448}, {579, 24483}, {662, 3939}, {692, 1634}, {1019, 4236}, {1025, 4238}, {1438, 2311}, {1936, 24630}, {2242, 5170}, {2398, 4560}, {2701, 6083}, {3286, 34230}, {3733, 23344}, {4584, 5377}, {6577, 34594}, {16702, 19624}, {17209, 40910}, {17944, 39026}, {18268, 38865}, {28162, 43359}, {30858, 30995}, {53268, 53324}

X(54353) = X(4584)-Ceva conjugate of X(101)
X(54353) = X(i)-isoconjugate of X(j) for these (i,j): {4, 10099}, {10, 1027}, {65, 885}, {105, 523}, {210, 43930}, {225, 23696}, {226, 1024}, {294, 7178}, {321, 43929}, {512, 2481}, {513, 13576}, {514, 18785}, {525, 8751}, {647, 54235}, {656, 36124}, {661, 673}, {666, 3125}, {798, 18031}, {884, 1441}, {919, 16732}, {927, 4516}, {1416, 4086}, {1427, 28132}, {1438, 1577}, {1462, 3700}, {1814, 2501}, {2195, 4077}, {3120, 36086}, {3121, 36803}, {3122, 51560}, {3657, 52456}, {3709, 34018}, {3952, 43921}, {4010, 52030}, {4017, 14942}, {4088, 51838}, {6185, 24290}, {6559, 7216}, {7180, 36796}, {14618, 32658}, {14625, 47915}, {21044, 36146}, {21052, 51845}, {21207, 32666}, {21832, 52209}, {21945, 36041}, {24006, 36057}, {35353, 52902}, {36802, 53540}
X(54353) = X(i)-Dao conjugate of X(j) for these (i,j): {518, 4088}, {5519, 21945}, {6184, 1577}, {17755, 850}, {20621, 24006}, {31998, 18031}, {34961, 14942}, {35094, 21207}, {36033, 10099}, {36830, 673}, {38980, 16732}, {38989, 3120}, {39014, 21044}, {39026, 13576}, {39046, 523}, {39052, 54235}, {39054, 2481}, {39063, 4077}, {40596, 36124}, {40602, 885}, {40609, 4086}
X(54353) = cevapoint of X(1914) and X(53287)
X(54353) = trilinear pole of line {672, 1818}
X(54353) = crossdifference of every pair of points on line {661, 3120}
X(54353) = barycentric product X(i)*X(j) for these {i,j}: {21, 1025}, {58, 42720}, {63, 4238}, {81, 1026}, {86, 2284}, {99, 672}, {100, 18206}, {101, 30941}, {110, 3912}, {162, 25083}, {163, 3263}, {190, 3286}, {241, 643}, {249, 4088}, {284, 883}, {333, 2283}, {518, 662}, {645, 1458}, {648, 1818}, {665, 4600}, {670, 9454}, {692, 18157}, {799, 2223}, {811, 20752}, {918, 4570}, {926, 4620}, {1252, 23829}, {1331, 15149}, {1414, 3693}, {1861, 4558}, {2254, 4567}, {2287, 41353}, {2340, 4573}, {2356, 4563}, {3717, 4565}, {3930, 52935}, {3932, 4556}, {4447, 4603}, {4575, 46108}, {4584, 8299}, {4592, 5089}, {4602, 9455}, {4610, 20683}, {4622, 14439}, {4623, 39258}, {4627, 4684}, {4629, 4966}, {5546, 9436}, {7257, 52635}, {7259, 34855}, {16728, 36086}, {24041, 24290}, {50333, 52378}
X(54353) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 10099}, {99, 18031}, {101, 13576}, {110, 673}, {112, 36124}, {162, 54235}, {163, 105}, {241, 4077}, {284, 885}, {518, 1577}, {643, 36796}, {662, 2481}, {665, 3120}, {672, 523}, {692, 18785}, {883, 349}, {918, 21207}, {926, 21044}, {1025, 1441}, {1026, 321}, {1333, 1027}, {1412, 43930}, {1414, 34018}, {1458, 7178}, {1576, 1438}, {1818, 525}, {1861, 14618}, {2193, 23696}, {2194, 1024}, {2206, 43929}, {2223, 661}, {2254, 16732}, {2283, 226}, {2284, 10}, {2328, 28132}, {2340, 3700}, {2356, 2501}, {3252, 35352}, {3263, 20948}, {3286, 514}, {3693, 4086}, {3912, 850}, {3930, 4036}, {3932, 52623}, {4088, 338}, {4238, 92}, {4558, 31637}, {4567, 51560}, {4570, 666}, {4575, 1814}, {4600, 36803}, {4620, 46135}, {5089, 24006}, {5546, 14942}, {6184, 4088}, {9454, 512}, {9455, 798}, {15149, 46107}, {18157, 40495}, {18206, 693}, {20662, 53558}, {20683, 4024}, {20752, 656}, {20778, 24459}, {23829, 23989}, {24290, 1109}, {25083, 14208}, {30941, 3261}, {32661, 36057}, {32676, 8751}, {34230, 4049}, {37908, 3064}, {39258, 4705}, {41353, 1446}, {42079, 24290}, {42720, 313}, {46388, 4516}, {51329, 7212}, {52378, 927}, {52614, 52335}, {52635, 4017}, {53539, 53545}, {53550, 4466}, {53554, 8287}

X(54354) = X(1)X(21)∩X(9)X(32)

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

X(54354) lies on these lines: {1, 21}, {2, 37603}, {3, 238}, {6, 37573}, {8, 902}, {9, 32}, {10, 3550}, {19, 1247}, {34, 1758}, {35, 43}, {36, 1044}, {40, 49128}, {46, 13733}, {55, 5247}, {57, 28109}, {72, 7262}, {75, 24850}, {78, 2210}, {87, 19762}, {90, 3465}, {100, 6048}, {165, 1722}, {171, 405}, {329, 36573}, {386, 2309}, {404, 748}, {474, 17123}, {497, 22361}, {499, 26091}, {560, 1098}, {579, 40955}, {601, 1006}, {602, 6906}, {609, 3294}, {614, 4652}, {672, 23443}, {750, 5047}, {942, 4650}, {956, 37588}, {958, 3052}, {964, 1698}, {970, 3271}, {976, 3219}, {982, 3916}, {984, 5266}, {986, 1104}, {988, 7290}, {995, 5267}, {1001, 4252}, {1012, 37570}, {1054, 37231}, {1064, 6875}, {1106, 7677}, {1107, 21793}, {1125, 3662}, {1155, 24174}, {1191, 37617}, {1193, 4189}, {1279, 3976}, {1330, 3771}, {1399, 37523}, {1438, 39946}, {1449, 31429}, {1453, 17594}, {1479, 30943}, {1490, 9355}, {1616, 11194}, {1714, 4302}, {1738, 31730}, {1739, 37572}, {1743, 2268}, {1745, 36152}, {1757, 3811}, {1770, 17889}, {1836, 24161}, {1914, 21384}, {1935, 37579}, {1957, 41227}, {1961, 37316}, {2175, 13323}, {2218, 24310}, {2298, 3731}, {2308, 19767}, {2329, 14974}, {2647, 37550}, {2664, 33718}, {2938, 33781}, {3008, 12512}, {3072, 3560}, {3074, 8069}, {3145, 27659}, {3208, 5291}, {3216, 5010}, {3218, 28082}, {3246, 52541}, {3286, 16690}, {3338, 29820}, {3361, 28017}, {3487, 24695}, {3496, 16968}, {3501, 4426}, {3552, 16827}, {3555, 17715}, {3579, 24440}, {3616, 26840}, {3624, 16342}, {3632, 49492}, {3648, 33098}, {3666, 16478}, {3683, 37539}, {3685, 17733}, {3736, 37296}, {3751, 19133}, {3772, 24851}, {3780, 10987}, {3792, 37482}, {3831, 17697}, {3913, 21000}, {3941, 16683}, {3959, 41319}, {3961, 41229}, {3980, 16817}, {4184, 27660}, {4188, 27627}, {4203, 16569}, {4210, 27636}, {4216, 7280}, {4224, 5272}, {4253, 16779}, {4259, 16793}, {4267, 8053}, {4278, 18792}, {4292, 37103}, {4294, 33137}, {4300, 37106}, {4307, 17558}, {4339, 5273}, {4362, 7283}, {4383, 5217}, {4414, 5262}, {4434, 46937}, {4438, 5015}, {4641, 37080}, {4855, 5529}, {4859, 14377}, {5021, 16503}, {5044, 37589}, {5156, 16289}, {5172, 7299}, {5192, 32918}, {5230, 6872}, {5251, 5264}, {5258, 37610}, {5259, 26102}, {5265, 51301}, {5268, 47511}, {5300, 33115}, {5329, 13730}, {5398, 37529}, {5438, 15601}, {5711, 16418}, {5744, 36574}, {5814, 33160}, {6284, 35466}, {6675, 33111}, {6679, 16062}, {6857, 26098}, {7031, 16552}, {7155, 8669}, {7483, 17717}, {7741, 37354}, {8688, 33804}, {8720, 50023}, {10822, 50585}, {11108, 17122}, {11110, 50302}, {11114, 21935}, {11115, 31339}, {11374, 33096}, {12579, 29645}, {12625, 53388}, {13407, 29675}, {13732, 20368}, {13740, 32916}, {14006, 39585}, {14621, 33047}, {15171, 33141}, {15674, 33112}, {16192, 23511}, {16370, 16466}, {16688, 16872}, {16865, 17126}, {16914, 41240}, {17033, 17692}, {17063, 37582}, {17064, 41869}, {17105, 23493}, {17124, 17536}, {17125, 17531}, {17184, 36505}, {17526, 26034}, {17572, 28257}, {17675, 31210}, {17696, 29960}, {17698, 32784}, {19270, 25496}, {19335, 28271}, {19645, 53591}, {19854, 33109}, {20066, 33139}, {20077, 29839}, {21537, 28254}, {24159, 32857}, {24586, 33821}, {24892, 52367}, {25650, 32946}, {26066, 37717}, {26131, 29661}, {26363, 33106}, {27368, 32929}, {27625, 37307}, {28265, 37331}, {29473, 30822}, {32613, 37699}, {33084, 49716}, {33771, 42043}, {35633, 37683}, {37176, 50295}, {37563, 49494}, {37618, 47623}, {37663, 52793}, {37715, 50241}, {39248, 51328}, {49448, 49530}, {49613, 49705}, {50303, 50739}

X(54354) = X(2218)-Ceva conjugate of X(1)
X(54354) = X(2)-isoconjugate of X(45988)
X(54354) = X(32664)-Dao conjugate of X(45988)
X(54354) = crossdifference of every pair of points on line {661, 21348}
X(54354) = barycentric product X(i)*X(j) for these {i,j}: {1, 37652}, {31, 30022}, {63, 37055}
X(54354) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 45988}, {30022, 561}, {37055, 92}, {37652, 75}
X(54354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1707, 1046}, {3, 238, 978}, {9, 37552, 5293}, {21, 31, 1}, {35, 1724, 43}, {36, 1777, 1044}, {55, 5247, 50581}, {58, 5248, 1}, {238, 7295, 1716}, {595, 993, 1}, {958, 3052, 5255}, {964, 32917, 1698}, {1001, 4252, 37607}, {1104, 4640, 986}, {1125, 4257, 37608}, {1468, 1621, 1}, {1621, 16948, 1468}, {2975, 3915, 1}, {4189, 17127, 1193}, {4426, 17735, 3501}, {5172, 7299, 37694}, {5259, 37522, 26102}, {5266, 31445, 984}, {8666, 40091, 1}, {12514, 37817, 1}, {15485, 37608, 1125}, {16342, 32772, 3624}, {16468, 37574, 386}

X(54355) = X(1)X(2)∩X(12)X(81)

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

X(54355) lies on these lines: {1, 2}, {4, 17126}, {5, 33107}, {6, 11681}, {7, 34030}, {12, 81}, {21, 37715}, {31, 5046}, {40, 33134}, {46, 33102}, {58, 5080}, {65, 33133}, {100, 1834}, {149, 5255}, {171, 2475}, {181, 41723}, {238, 37162}, {341, 33114}, {484, 36250}, {601, 37437}, {942, 22321}, {986, 33155}, {1046, 17484}, {1064, 6960}, {1203, 3814}, {1254, 37798}, {1329, 32911}, {1386, 17606}, {1400, 16548}, {1468, 20060}, {1788, 19785}, {2295, 17737}, {2363, 24624}, {2476, 5711}, {2478, 17127}, {2551, 24597}, {2650, 17719}, {2975, 37646}, {3072, 6840}, {3073, 13729}, {3218, 13161}, {3436, 37642}, {3550, 20066}, {3701, 33166}, {3714, 32779}, {3769, 5016}, {3812, 33129}, {3822, 26131}, {3825, 5315}, {3868, 33153}, {3869, 17720}, {4193, 16466}, {4225, 5172}, {4307, 6871}, {4385, 33170}, {4415, 11684}, {4642, 33135}, {4696, 33121}, {4850, 24914}, {5051, 33083}, {5086, 37539}, {5141, 26098}, {5221, 33146}, {5253, 37634}, {5260, 35466}, {5264, 52367}, {5280, 26074}, {5295, 46918}, {5348, 37191}, {5710, 11680}, {5712, 10585}, {5769, 45931}, {10408, 17167}, {11015, 37589}, {13731, 37621}, {16062, 33086}, {17164, 37759}, {17735, 23903}, {17902, 37235}, {18253, 33761}, {19513, 22765}, {23536, 27003}, {24440, 33128}, {24443, 33150}, {25466, 37633}, {26066, 28606}, {31888, 33099}, {33118, 52353}, {37256, 37603}, {37558, 37797}, {37702, 49480}, {38945, 39458}

X(54355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 5292, 33142}, {10, 24883, 33139}, {171, 21935, 2475}, {387, 5552, 3240}, {1468, 37716, 20060}, {2476, 5711, 33112}, {3822, 37559, 26131}

X(54356) = X(1)X(21)∩X(29)X(34)

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

X(54356) lies on these lines: {1, 21}, {2, 581}, {3, 1243}, {4, 17167}, {8, 16713}, {27, 10884}, {28, 1790}, {29, 34}, {33, 3559}, {40, 4184}, {46, 4278}, {51, 13731}, {55, 18178}, {56, 18165}, {65, 3286}, {78, 333}, {270, 2185}, {279, 17169}, {284, 1800}, {285, 1422}, {355, 47515}, {377, 991}, {386, 6910}, {394, 405}, {442, 500}, {452, 3945}, {511, 37225}, {517, 17524}, {572, 37231}, {759, 43345}, {851, 48893}, {855, 48894}, {859, 1385}, {936, 5235}, {940, 11344}, {942, 8021}, {943, 1331}, {946, 14956}, {950, 17197}, {964, 10455}, {970, 30944}, {1010, 19860}, {1011, 10441}, {1014, 1467}, {1043, 3872}, {1064, 24541}, {1100, 46889}, {1104, 40153}, {1412, 34489}, {1437, 36011}, {1464, 11281}, {1730, 16451}, {1745, 31266}, {1754, 37285}, {1764, 16452}, {1778, 3553}, {1789, 4282}, {1816, 3306}, {1817, 8726}, {1838, 4303}, {2360, 4228}, {2475, 17173}, {2478, 17182}, {2594, 6690}, {2646, 4267}, {3136, 48937}, {3145, 37527}, {3191, 3219}, {3560, 18451}, {3576, 4225}, {3601, 18163}, {3612, 4276}, {3615, 17586}, {3616, 10571}, {3720, 41012}, {3736, 37232}, {3737, 21106}, {3924, 17187}, {4185, 37474}, {4216, 10470}, {4296, 37558}, {4337, 12609}, {4511, 46877}, {4720, 4853}, {5046, 17174}, {5047, 37659}, {5256, 37265}, {5396, 7483}, {5400, 7504}, {5439, 52889}, {5453, 6675}, {5482, 16374}, {5495, 5499}, {5706, 20835}, {5707, 37284}, {5721, 47516}, {5752, 16455}, {5886, 37357}, {5943, 28238}, {6176, 13724}, {6198, 52891}, {6668, 45885}, {6734, 14547}, {6883, 36747}, {7032, 28082}, {7190, 8822}, {7289, 41718}, {8025, 51382}, {8227, 14008}, {8583, 17557}, {8731, 22076}, {11103, 25526}, {11110, 18465}, {11518, 18164}, {11553, 17768}, {12111, 33536}, {13323, 13733}, {14953, 27000}, {15149, 25935}, {15680, 52524}, {16053, 25930}, {16132, 37369}, {16287, 37536}, {16696, 37549}, {16699, 34522}, {16700, 17054}, {16704, 34772}, {16705, 17219}, {17168, 37168}, {17171, 26130}, {17175, 17866}, {17188, 17584}, {17202, 26117}, {17440, 23623}, {18185, 37080}, {18444, 37113}, {18446, 25516}, {18646, 30117}, {18653, 31900}, {19684, 27378}, {19767, 37666}, {19782, 37246}, {20846, 37530}, {21319, 29958}, {22080, 48917}, {22769, 41582}, {23154, 23440}, {26102, 37373}, {27412, 31631}, {27506, 46880}, {30503, 37402}, {31156, 48855}, {31660, 53388}, {31902, 41854}, {31938, 40967}, {33586, 37320}, {35258, 37296}, {35262, 37442}, {35466, 52544}, {36746, 37228}, {37292, 45923}, {40937, 46882}, {42443, 53280}

X(54356) = isotomic conjugate of the polar conjugate of X(46884)
X(54356) = X(i)-Ceva conjugate of X(j) for these (i,j): {86, 5249}, {664, 4560}, {4636, 3737}
X(54356) = X(i)-isoconjugate of X(j) for these (i,j): {12, 1175}, {37, 2982}, {55, 52560}, {65, 943}, {71, 40573}, {181, 40412}, {225, 1794}, {226, 2259}, {523, 15439}, {1400, 40435}, {1402, 40422}, {1409, 40447}, {2197, 40395}, {2616, 35320}, {3700, 32651}, {4041, 36048}, {14775, 23067}, {26942, 40570}
X(54356) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 52560}, {442, 10}, {942, 201}, {5249, 40999}, {15607, 4041}, {16585, 1441}, {18591, 226}, {39007, 656}, {40582, 40435}, {40589, 2982}, {40602, 943}, {40605, 40422}, {40937, 6358}, {52544, 40661}
X(54356) = cevapoint of X(i) and X(j) for these (i,j): {942, 4303}, {8021, 46882}, {14547, 40937}
X(54356) = barycentric product X(i)*X(j) for these {i,j}: {21, 5249}, {29, 18607}, {57, 51978}, {69, 46884}, {75, 46882}, {81, 6734}, {85, 8021}, {86, 40937}, {261, 2294}, {274, 14547}, {314, 2260}, {332, 1841}, {333, 942}, {345, 46883}, {442, 2185}, {445, 1789}, {645, 50354}, {811, 52306}, {1234, 2150}, {1509, 40967}, {1812, 1838}, {1859, 17206}, {3615, 16585}, {3718, 46890}, {4303, 31623}, {4612, 23752}, {4625, 33525}, {14597, 44130}, {18021, 40978}, {23207, 44129}, {28660, 40956}, {31938, 52393}, {40952, 52379}, {43740, 46885}
X(54356) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 40435}, {28, 40573}, {29, 40447}, {57, 52560}, {58, 2982}, {163, 15439}, {270, 40395}, {284, 943}, {333, 40422}, {442, 6358}, {500, 16577}, {942, 226}, {1625, 35320}, {1838, 40149}, {1841, 225}, {1859, 1826}, {2150, 1175}, {2185, 40412}, {2193, 1794}, {2194, 2259}, {2260, 65}, {2294, 12}, {4303, 1214}, {4565, 36048}, {5249, 1441}, {6734, 321}, {8021, 9}, {14547, 37}, {14597, 73}, {16585, 40999}, {18591, 201}, {18607, 307}, {23207, 71}, {31938, 3969}, {33525, 4041}, {37993, 2294}, {39791, 37755}, {40937, 10}, {40952, 2171}, {40956, 1400}, {40967, 594}, {40978, 181}, {44095, 1825}, {46882, 1}, {46883, 278}, {46884, 4}, {46887, 41342}, {46890, 34}, {50354, 7178}, {51978, 312}, {52306, 656}
X(54356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 17194, 21}, {21, 81, 283}, {21, 3193, 2328}, {81, 24635, 18206}, {2328, 4658, 3193}, {2646, 18191, 4267}, {4184, 41723, 40}, {8731, 48909, 22076}, {11110, 18465, 19861}

X(54357) = X(2)X(7)∩X(10)X(21)

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

X(54377) lies on these lines: {1, 24597}, {2, 7}, {8, 4917}, {10, 21}, {11, 15254}, {12, 5302}, {20, 5587}, {27, 1268}, {37, 27492}, {40, 6837}, {44, 5718}, {45, 17720}, {55, 25006}, {56, 24564}, {65, 18253}, {69, 28627}, {72, 5719}, {75, 3977}, {78, 6857}, {84, 31423}, {88, 42326}, {119, 11231}, {140, 1071}, {165, 10431}, {169, 24611}, {190, 4054}, {191, 11552}, {209, 18165}, {210, 6690}, {224, 936}, {228, 8731}, {238, 17722}, {306, 319}, {345, 5271}, {377, 1698}, {379, 5179}, {392, 1387}, {405, 1259}, {442, 31445}, {443, 4652}, {464, 26063}, {515, 37106}, {516, 10883}, {518, 37703}, {519, 27754}, {551, 4867}, {594, 50104}, {631, 5720}, {658, 2349}, {748, 24239}, {756, 11031}, {846, 3914}, {896, 50307}, {940, 2911}, {946, 6884}, {950, 16865}, {958, 5252}, {960, 15950}, {968, 33137}, {984, 3011}, {993, 21578}, {1000, 3872}, {1001, 26015}, {1004, 4413}, {1006, 51755}, {1012, 1512}, {1013, 1861}, {1125, 3868}, {1150, 3912}, {1155, 3826}, {1210, 5047}, {1212, 5723}, {1214, 6357}, {1323, 24635}, {1329, 15823}, {1376, 20835}, {1441, 14206}, {1621, 4847}, {1738, 4414}, {1748, 30686}, {1757, 29640}, {1770, 3647}, {1803, 32008}, {1891, 13739}, {2177, 49772}, {2308, 29682}, {2321, 32849}, {2325, 4671}, {2476, 12572}, {2478, 5705}, {2550, 35258}, {2886, 3683}, {2895, 4035}, {2975, 4315}, {3006, 3883}, {3008, 4850}, {3035, 5784}, {3074, 3561}, {3187, 4464}, {3220, 7465}, {3241, 36922}, {3294, 35341}, {3419, 16418}, {3434, 4512}, {3436, 5234}, {3487, 3951}, {3488, 17561}, {3523, 6705}, {3526, 37713}, {3553, 5287}, {3579, 37447}, {3586, 31156}, {3616, 11520}, {3617, 4313}, {3634, 4197}, {3663, 33129}, {3666, 17366}, {3671, 11684}, {3678, 10122}, {3681, 11020}, {3686, 33077}, {3687, 5278}, {3696, 3712}, {3707, 27757}, {3717, 26227}, {3740, 5432}, {3755, 33139}, {3828, 17057}, {3869, 18249}, {3876, 13411}, {3879, 16704}, {3916, 8728}, {3925, 4640}, {3935, 24393}, {3936, 4416}, {3962, 11281}, {3984, 5703}, {3993, 50755}, {4001, 17361}, {4021, 28606}, {4028, 32864}, {4078, 17763}, {4101, 25650}, {4104, 29846}, {4138, 4683}, {4208, 19877}, {4224, 5314}, {4228, 5285}, {4307, 36277}, {4314, 5178}, {4358, 25101}, {4359, 20236}, {4370, 27747}, {4384, 17740}, {4422, 30818}, {4423, 42843}, {4428, 4863}, {4431, 50105}, {4432, 21242}, {4438, 25494}, {4641, 7277}, {4643, 30811}, {4656, 31204}, {4666, 24477}, {4667, 37635}, {4678, 12536}, {4679, 5832}, {4684, 29830}, {4751, 27471}, {4798, 19749}, {4989, 17025}, {4999, 25917}, {5044, 7483}, {5057, 51090}, {5121, 17125}, {5175, 11106}, {5208, 43223}, {5218, 7675}, {5220, 17718}, {5224, 18650}, {5233, 17335}, {5250, 19843}, {5259, 10916}, {5263, 35263}, {5284, 11019}, {5361, 32858}, {5436, 12649}, {5439, 50205}, {5526, 29571}, {5550, 11036}, {5660, 13243}, {5709, 6832}, {5735, 7988}, {5737, 17293}, {5816, 37419}, {5847, 29643}, {6245, 6986}, {6282, 6974}, {6326, 10165}, {6536, 29863}, {6735, 9708}, {6745, 41228}, {6762, 10587}, {6763, 51706}, {6826, 21165}, {6839, 10175}, {6852, 26878}, {6861, 26921}, {6878, 18443}, {6889, 7330}, {7085, 25514}, {7174, 26228}, {7226, 29681}, {7227, 31993}, {7262, 33111}, {7283, 25446}, {7293, 37261}, {7411, 10164}, {7580, 31672}, {7741, 41872}, {8167, 17728}, {8226, 18482}, {8822, 28653}, {9623, 51433}, {9710, 37568}, {9843, 17536}, {9956, 37468}, {10176, 18389}, {10198, 41229}, {10527, 31435}, {11064, 52385}, {11509, 24982}, {11679, 17776}, {11680, 40998}, {11681, 18250}, {12437, 15676}, {12514, 19854}, {14021, 17308}, {14418, 36038}, {15485, 29676}, {15670, 24929}, {15674, 34772}, {15803, 37462}, {16056, 22060}, {16368, 19732}, {16468, 29657}, {16610, 17337}, {16617, 37585}, {16815, 26070}, {17127, 29664}, {17244, 37684}, {17245, 37520}, {17246, 50103}, {17256, 30832}, {17261, 37759}, {17277, 32851}, {17278, 17595}, {17279, 37660}, {17529, 37582}, {17576, 46933}, {17724, 49515}, {17862, 20879}, {18406, 31730}, {18480, 44238}, {18491, 37426}, {18655, 40530}, {18747, 19827}, {19804, 20927}, {19861, 30478}, {19862, 37701}, {20045, 49527}, {20106, 32782}, {20880, 24589}, {21015, 37360}, {21061, 25589}, {21677, 41575}, {22128, 37659}, {22464, 37695}, {23708, 26363}, {24177, 26724}, {24210, 24892}, {24231, 36263}, {24542, 46909}, {24620, 29628}, {24703, 31245}, {24914, 25011}, {25057, 41141}, {25440, 37285}, {25729, 37206}, {25760, 50752}, {26037, 37175}, {26251, 50404}, {26885, 37527}, {27164, 41248}, {27479, 51052}, {27628, 37575}, {29596, 37111}, {29631, 50290}, {29632, 49511}, {29661, 32912}, {29675, 49448}, {29857, 50295}, {29862, 33082}, {29873, 33083}, {30564, 31017}, {30608, 30829}, {30768, 32784}, {31165, 39782}, {31197, 43055}, {31286, 45684}, {31658, 37374}, {32916, 37090}, {33066, 41878}, {33166, 53663}, {35445, 38200}, {37322, 41507}, {37435, 46932}, {37646, 44307}, {37687, 45204}, {37770, 53009}, {42819, 51463}, {47785, 53359}, {49450, 50744}, {49470, 50758}, {49506, 50743}, {49510, 50748}

X(54357) = complement of X(31019)
X(54357) = X(15175)-complementary conjugate of X(141)
X(54357) = crossdifference of every pair of points on line {663, 21828}
X(54357) = barycentric product X(i)*X(j) for these {i,j}: {75, 24929}, {1268, 15670}
X(54357) = barycentric quotient X(i)/X(j) for these {i,j}: {15670, 1125}, {15762, 1838}, {24929, 1}
X(54357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9, 908}, {2, 63, 5249}, {2, 329, 31266}, {2, 3218, 142}, {2, 3219, 226}, {2, 5273, 63}, {2, 5325, 17781}, {2, 5744, 3306}, {2, 5905, 25525}, {2, 18228, 30852}, {2, 27065, 3452}, {2, 31018, 5219}, {2, 35595, 5316}, {9, 5219, 31018}, {45, 31187, 17720}, {84, 31423, 37112}, {226, 3219, 17781}, {226, 5325, 3219}, {333, 33116, 306}, {405, 5791, 6734}, {846, 33138, 3914}, {960, 24953, 24541}, {1698, 31424, 377}, {3634, 4292, 4197}, {3647, 3841, 1770}, {3911, 6666, 2}, {3929, 25525, 5905}, {5044, 7483, 27385}, {5219, 31018, 908}, {5235, 32779, 10}, {5278, 33113, 3687}, {5745, 6666, 3911}, {7262, 33111, 41011}, {29007, 37797, 226}, {31204, 33761, 33133}, {31231, 51780, 2}, {32917, 33115, 10}, {33133, 33761, 4656}

X(54358) = X(1)X(6)∩X(7)X(27)

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

X(54358) lies on these lines: {1, 6}, {3, 2260}, {7, 27}, {11, 5747}, {19, 942}, {31, 4343}, {35, 37500}, {36, 37504}, {42, 6600}, {48, 999}, {55, 579}, {56, 284}, {57, 380}, {71, 3295}, {142, 940}, {144, 37685}, {198, 4251}, {221, 12560}, {226, 40963}, {281, 938}, {347, 17014}, {354, 2264}, {387, 2550}, {388, 5802}, {480, 3190}, {495, 26063}, {497, 5746}, {516, 5706}, {527, 50068}, {572, 3428}, {584, 2178}, {604, 17474}, {607, 5262}, {610, 3333}, {612, 40659}, {946, 5776}, {949, 21346}, {965, 1125}, {971, 36742}, {1126, 2336}, {1202, 15288}, {1214, 1445}, {1249, 17905}, {1400, 1617}, {1409, 34040}, {1433, 7129}, {1475, 2268}, {1479, 1901}, {1498, 11372}, {1714, 3826}, {1754, 11495}, {1765, 11496}, {1781, 18398}, {1826, 5722}, {1861, 5808}, {1953, 19350}, {2214, 5880}, {2262, 44662}, {2271, 2277}, {2285, 5173}, {2287, 3616}, {2294, 3211}, {2298, 6601}, {2303, 22127}, {2334, 2983}, {2335, 2346}, {2354, 37492}, {2393, 18621}, {2982, 12848}, {3059, 3745}, {3173, 8545}, {3174, 5269}, {3197, 11529}, {3304, 23073}, {3668, 3946}, {3686, 10371}, {3694, 3870}, {3713, 4847}, {3873, 5279}, {3920, 34784}, {4047, 5250}, {4289, 21773}, {4321, 34046}, {4326, 7070}, {4360, 25252}, {4361, 18698}, {4383, 6666}, {4640, 41422}, {4667, 6180}, {5045, 22153}, {5119, 21866}, {5257, 37658}, {5271, 19716}, {5542, 5781}, {5563, 37519}, {5707, 5805}, {5710, 5853}, {5732, 36746}, {5738, 16608}, {5742, 10198}, {5749, 24552}, {5755, 10267}, {5778, 5886}, {5779, 36750}, {5783, 19843}, {5798, 48482}, {5930, 12573}, {7190, 23144}, {7191, 11025}, {7373, 20818}, {7672, 17016}, {7676, 17126}, {7678, 33107}, {7742, 36744}, {8232, 34048}, {10443, 43175}, {10580, 27382}, {10980, 18594}, {11019, 40942}, {14828, 25521}, {15733, 20741}, {15851, 38288}, {15931, 37499}, {17189, 18166}, {17321, 23151}, {17366, 24779}, {17391, 26657}, {17398, 19854}, {18230, 32911}, {18734, 39273}, {19133, 37580}, {20182, 25080}, {20195, 37674}, {21153, 36745}, {21617, 37695}, {23146, 48303}, {24512, 33137}, {24937, 31245}, {28082, 40977}, {30456, 34036}, {31658, 36754}, {31671, 45923}, {36743, 40292}, {37559, 38052}, {37672, 39948}, {38107, 45931}

X(54358) = X(i)-isoconjugate of X(j) for these (i,j): {9, 8814}, {33, 8813}
X(54358) = X(478)-Dao conjugate of X(8814)
X(54358) = barycentric product X(7)*X(13615)
X(54358) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 8814}, {222, 8813}, {13615, 8}
X(54358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 219}, {1, 1723, 37}, {1, 2257, 40937}, {6, 37, 218}, {6, 16777, 2911}, {6, 16781, 2300}, {1100, 1108, 1}, {1400, 2280, 4254}, {1475, 2268, 5120}, {17321, 41610, 23151}, {53134, 53135, 72}

X(54359) = X(8)X(9)∩X(19)X(25)

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

X(54359) lies on these lines: {1, 572}, {6, 3057}, {8, 9}, {11, 17303}, {19, 25}, {20, 31325}, {40, 1400}, {41, 380}, {57, 3672}, {63, 192}, {71, 8557}, {169, 3731}, {200, 21033}, {220, 2264}, {226, 4329}, {312, 2339}, {392, 5783}, {497, 2345}, {573, 5119}, {594, 1837}, {608, 1902}, {610, 9310}, {672, 2257}, {941, 989}, {960, 3713}, {1055, 3100}, {1100, 2098}, {1253, 3747}, {1445, 37555}, {1449, 7962}, {1723, 3730}, {1743, 9819}, {1753, 37528}, {1760, 4664}, {1763, 4656}, {1781, 16673}, {2176, 22074}, {2182, 2256}, {2263, 8898}, {2267, 3554}, {2277, 22071}, {2280, 10388}, {2330, 16972}, {2331, 21148}, {2646, 7221}, {2901, 12514}, {3056, 42447}, {3058, 17281}, {3085, 39579}, {3217, 6602}, {3305, 17280}, {3306, 17302}, {3486, 5227}, {3496, 41261}, {3553, 21801}, {3663, 28017}, {3723, 34471}, {3729, 10889}, {3811, 21078}, {3913, 3965}, {3930, 10382}, {3950, 4314}, {4032, 18655}, {4294, 49542}, {4307, 12717}, {4313, 5279}, {4327, 12721}, {4335, 18788}, {4419, 7289}, {5048, 16884}, {5341, 16672}, {5405, 8231}, {5540, 53052}, {5749, 9785}, {5750, 12053}, {5816, 10039}, {5831, 24390}, {7297, 16675}, {7347, 30413}, {7348, 30412}, {7675, 51058}, {8609, 26357}, {8897, 27184}, {9789, 30381}, {9848, 53089}, {10387, 14100}, {10393, 22021}, {10445, 10624}, {10827, 32431}, {10950, 17299}, {10966, 36743}, {11376, 17398}, {11683, 20173}, {12575, 17355}, {16547, 16676}, {16577, 26215}, {16580, 31158}, {17388, 37740}, {17481, 31164}, {17754, 29837}, {18785, 28071}, {20227, 28011}, {21389, 42312}, {25505, 28798}, {25590, 53526}, {26242, 30677}, {26789, 31019}, {26837, 31053}, {28043, 40965}, {33635, 39943}, {37499, 37568}, {39273, 51052}

X(54359) = crossdifference of every pair of points on line {905, 17420}
X(54359) = barycentric product X(i)*X(j) for these {i,j}: {1, 2551}, {19, 23600}, {78, 52082}, {100, 47136}, {281, 10319}
X(54359) = barycentric quotient X(i)/X(j) for these {i,j}: {2551, 75}, {10319, 348}, {23600, 304}, {47136, 693}, {52082, 273}
X(54359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1766, 2285}, {9, 1697, 2269}, {9, 3208, 3692}, {19, 37, 40131}, {37, 910, 1696}, {41, 21809, 2324}, {55, 11997, 4319}, {380, 2324, 41}, {1334, 40968, 9}, {2268, 17452, 1}, {3161, 33950, 9}, {7133, 42013, 612}

X(54360) = X(1)X(24)∩X(42)X(65)

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

X(54360) lies on these lines: {1, 24}, {8, 26090}, {37, 2599}, {42, 65}, {72, 52391}, {201, 41340}, {225, 44661}, {226, 1825}, {603, 18732}, {942, 20277}, {946, 1831}, {950, 1866}, {1193, 28290}, {1214, 22342}, {1400, 18674}, {1439, 52390}, {1451, 40959}, {1830, 6260}, {1835, 5930}, {1905, 2654}, {3215, 26934}, {5903, 18446}, {7114, 18838}, {18210, 22341}, {20280, 49454}, {28787, 40152}

X(54360) = X(i)-isoconjugate of X(j) for these (i,j): {21, 1063}, {29, 7163}, {11103, 18532}
X(54360) = X(i)-Dao conjugate of X(j) for these (i,j): {18588, 317}, {40611, 1063}
X(54360) = barycentric product X(i)*X(j) for these {i,j}: {1, 18588}, {226, 1062}, {1214, 1479}, {5358, 26942}, {17584, 37755}
X(54360) = barycentric quotient X(i)/X(j) for these {i,j}: {1062, 333}, {1400, 1063}, {1409, 7163}, {1479, 31623}, {5358, 46103}, {18588, 75}
X(54360) = {X(65),X(17441)}-harmonic conjugate of X(73)

X(54361) = X(4)X(46)∩X(8)X(11)

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

X(54361) lies on these lines: {1, 3090}, {2, 1837}, {3, 12019}, {4, 46}, {5, 3485}, {7, 10895}, {8, 11}, {10, 497}, {12, 938}, {20, 24914}, {21, 11502}, {33, 1722}, {40, 5225}, {43, 2654}, {55, 5047}, {56, 5704}, {57, 5229}, {65, 3091}, {80, 499}, {116, 277}, {140, 4305}, {145, 11376}, {153, 20118}, {210, 8165}, {226, 7989}, {281, 1731}, {354, 5261}, {355, 3086}, {381, 4295}, {388, 1210}, {390, 17358}, {404, 22760}, {452, 26066}, {496, 5790}, {498, 3488}, {515, 6927}, {516, 41348}, {517, 10591}, {519, 50443}, {546, 36279}, {631, 10572}, {632, 37606}, {942, 10590}, {948, 26012}, {950, 1698}, {960, 6919}, {962, 10896}, {986, 7069}, {999, 18357}, {1056, 10827}, {1058, 10039}, {1111, 24797}, {1118, 7541}, {1125, 5727}, {1155, 3146}, {1159, 5072}, {1317, 38758}, {1376, 5175}, {1387, 12645}, {1478, 3337}, {1479, 5657}, {1482, 10593}, {1610, 37366}, {1633, 17516}, {1656, 37730}, {1699, 4848}, {1732, 1826}, {1836, 3832}, {1854, 23332}, {1857, 5125}, {1858, 6871}, {1864, 3812}, {2093, 18483}, {2099, 7173}, {2345, 21029}, {2348, 27541}, {2362, 31412}, {2475, 16141}, {2476, 10958}, {2550, 24982}, {2551, 6734}, {3057, 3617}, {3085, 5722}, {3189, 5552}, {3295, 38042}, {3340, 3817}, {3419, 17619}, {3421, 10916}, {3434, 25005}, {3436, 24477}, {3487, 7951}, {3525, 3612}, {3545, 12047}, {3582, 34627}, {3583, 6361}, {3586, 6684}, {3600, 17728}, {3601, 3634}, {3614, 5226}, {3616, 7504}, {3621, 5048}, {3622, 37740}, {3626, 7962}, {3632, 50444}, {3679, 12053}, {3683, 18231}, {3689, 27525}, {3698, 17604}, {3753, 31418}, {3814, 49168}, {3826, 5809}, {3828, 4314}, {3851, 39542}, {3869, 5187}, {3911, 5691}, {3947, 11518}, {4000, 20305}, {4208, 10391}, {4292, 18492}, {4293, 18480}, {4294, 26446}, {4297, 31231}, {4302, 5445}, {4304, 31423}, {4313, 5432}, {4316, 5560}, {4333, 15682}, {4342, 4691}, {4511, 6931}, {4654, 38076}, {4662, 17642}, {4731, 9848}, {4853, 24386}, {5046, 5698}, {5055, 37737}, {5056, 11375}, {5067, 37721}, {5068, 17605}, {5071, 37692}, {5123, 7080}, {5128, 51118}, {5176, 10529}, {5204, 7319}, {5219, 6738}, {5231, 5795}, {5252, 14986}, {5260, 26357}, {5281, 46932}, {5298, 50864}, {5400, 10571}, {5433, 5731}, {5435, 7354}, {5550, 34471}, {5554, 11680}, {5603, 7741}, {5687, 34122}, {5690, 9669}, {5693, 12736}, {5714, 5902}, {5716, 37717}, {5730, 17533}, {5748, 12635}, {5768, 18242}, {5770, 37821}, {5804, 7680}, {5806, 41539}, {5815, 31141}, {5825, 17768}, {5828, 6764}, {5881, 44675}, {5887, 6973}, {6256, 10265}, {6261, 6969}, {6554, 21044}, {6702, 26364}, {6736, 24392}, {6737, 30827}, {6745, 12625}, {6788, 24159}, {6843, 44547}, {6844, 7686}, {6856, 45230}, {6912, 11509}, {6920, 11507}, {6946, 22766}, {6981, 45770}, {6982, 34339}, {6984, 13750}, {7004, 24174}, {7486, 37724}, {7967, 37711}, {8256, 11235}, {8287, 27686}, {8582, 26040}, {9352, 31295}, {9578, 11019}, {9580, 43174}, {9613, 50796}, {9614, 11362}, {9655, 34753}, {9656, 52783}, {9778, 12953}, {9785, 11238}, {9812, 37567}, {9947, 17625}, {10072, 37710}, {10106, 37714}, {10157, 12709}, {10303, 37600}, {10385, 19875}, {10392, 38052}, {10525, 12619}, {10580, 15888}, {10592, 15934}, {10595, 23708}, {10707, 34711}, {11037, 11237}, {11230, 37739}, {11415, 34744}, {11522, 30286}, {11681, 12649}, {12245, 30384}, {12246, 41698}, {12247, 39692}, {12433, 31479}, {12447, 20196}, {12528, 18838}, {12607, 36845}, {12647, 37720}, {13226, 40267}, {13384, 19862}, {13405, 37723}, {14011, 51978}, {14100, 40333}, {14257, 32594}, {15016, 41562}, {15017, 41558}, {15299, 38149}, {15325, 18525}, {15803, 31673}, {16232, 42561}, {16569, 22072}, {17531, 22768}, {17541, 28934}, {17567, 17647}, {17603, 37436}, {17784, 37828}, {18228, 21677}, {18397, 31870}, {18908, 50196}, {19065, 44623}, {19066, 44624}, {19872, 53054}, {24430, 24443}, {24987, 26105}, {26358, 53055}, {27383, 44669}, {30852, 41575}, {31399, 31434}, {34700, 45310}, {36574, 37716}, {37001, 54052}, {37709, 38155}, {37734, 38314}

X(54362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10175, 10588}, {2, 1837, 3486}, {4, 1737, 1788}, {4, 1788, 3474}, {4, 14646, 52860}, {5, 18391, 3485}, {8, 26129, 5289}, {10, 9581, 497}, {12, 938, 3475}, {57, 19925, 5229}, {80, 499, 944}, {80, 15079, 499}, {355, 3086, 3476}, {498, 37702, 3488}, {950, 1698, 5218}, {1210, 5587, 388}, {1479, 18395, 5657}, {1737, 10826, 4}, {1837, 17606, 2}, {3036, 10912, 8}, {3617, 5274, 3057}, {3621, 18220, 5048}, {3847, 5289, 26129}, {4313, 19877, 5432}, {5128, 51792, 51118}, {5690, 9669, 30305}, {5722, 9956, 3085}, {7741, 10573, 5603}, {8287, 41501, 27686}, {10593, 11545, 1482}, {10896, 40663, 962}, {11681, 12649, 25568}, {18395, 37718, 1479}

X(54362) = X(2)X(14)∩X(6)X(25)

Barycentrics    a^2*(Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*(a^2 + b^2 + c^2)*S) : :

X(54362) lies on these lines: {2, 14}, {6, 25}, {16, 22}, {18, 7495}, {23, 62}, {32, 3130}, {39, 3129}, {61, 1995}, {111, 16807}, {187, 3132}, {251, 3457}, {395, 44210}, {397, 10301}, {398, 468}, {427, 5321}, {428, 5318}, {462, 5254}, {463, 7745}, {566, 11141}, {574, 3131}, {612, 10638}, {614, 7051}, {842, 40156}, {858, 16964}, {1250, 5310}, {1368, 42117}, {1370, 42085}, {1383, 21461}, {1627, 41409}, {1993, 51207}, {2493, 11088}, {3054, 51546}, {3060, 51206}, {3104, 44718}, {4232, 42999}, {5020, 11485}, {5064, 42093}, {5094, 5339}, {5133, 16809}, {5169, 42814}, {5189, 42432}, {5237, 7492}, {5238, 40916}, {5306, 11081}, {5322, 19373}, {5335, 6995}, {5343, 52284}, {5352, 7496}, {5640, 36757}, {6353, 10632}, {6636, 10646}, {6676, 10634}, {6800, 36758}, {6997, 18582}, {7378, 42133}, {7386, 42119}, {7391, 19107}, {7392, 10643}, {7394, 16808}, {7396, 43466}, {7408, 42134}, {7484, 11480}, {7485, 10645}, {7493, 40694}, {7494, 11489}, {7499, 23303}, {7500, 42086}, {7519, 16965}, {7539, 42095}, {7571, 42914}, {7667, 42087}, {8014, 46342}, {9465, 34395}, {9909, 11486}, {10154, 11267}, {10565, 11420}, {10635, 15818}, {10691, 42122}, {11138, 44529}, {11142, 13338}, {11284, 22236}, {11548, 42143}, {13595, 37776}, {16063, 42157}, {16250, 41254}, {16268, 47596}, {16419, 42116}, {16966, 37990}, {18468, 42816}, {19106, 34603}, {20062, 42100}, {20063, 43633}, {21849, 44498}, {23302, 37439}, {30739, 42147}, {31099, 42160}, {31133, 36970}, {31152, 42154}, {34608, 42120}, {34609, 42126}, {34755, 37913}, {34986, 44497}, {37353, 42918}, {37454, 42163}, {37899, 42148}, {37900, 42158}, {37904, 43229}, {37910, 42924}, {41100, 47313}, {42099, 52397}, {42101, 52285}, {42140, 44442}, {42150, 46336}, {42164, 46517}, {42942, 43957}, {42993, 52300}, {42998, 52301}

X(54362) = isogonal conjugate of the isotomic conjugate of X(11304)
X(54362) = crossdifference of every pair of points on line {525, 6138}
X(54362) = barycentric product X(6)*X(11304)
X(54362) = barycentric quotient X(11304)/X(76)

X(54363) = X(2)X(13)∩X(6)X(25)

Barycentrics    a^2*(Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*(a^2 + b^2 + c^2)*S) : :

X(54363) lies on these lines: {2, 13}, {6, 25}, {15, 22}, {17, 7495}, {23, 61}, {32, 3129}, {39, 3130}, {62, 1995}, {111, 16806}, {187, 3131}, {251, 3458}, {396, 44210}, {397, 468}, {398, 10301}, {427, 5318}, {428, 5321}, {462, 7745}, {463, 5254}, {566, 11142}, {574, 3132}, {612, 1250}, {614, 19373}, {842, 40157}, {858, 16965}, {1368, 42118}, {1370, 42086}, {1383, 21462}, {1627, 41408}, {1993, 51206}, {2493, 11083}, {3054, 51547}, {3060, 51207}, {3105, 44719}, {4232, 42998}, {5020, 11486}, {5064, 42094}, {5094, 5340}, {5133, 16808}, {5169, 42813}, {5189, 42431}, {5237, 40916}, {5238, 7492}, {5306, 11086}, {5310, 10638}, {5322, 7051}, {5334, 6995}, {5344, 52284}, {5351, 7496}, {5640, 36758}, {6353, 10633}, {6636, 10645}, {6676, 10635}, {6800, 36757}, {6997, 18581}, {7378, 42134}, {7386, 42120}, {7391, 19106}, {7392, 10644}, {7394, 16809}, {7396, 43465}, {7408, 42133}, {7484, 11481}, {7485, 10646}, {7493, 40693}, {7494, 11488}, {7499, 23302}, {7500, 42085}, {7519, 16964}, {7539, 42098}, {7571, 42915}, {7667, 42088}, {8015, 46343}, {9465, 34394}, {9909, 11485}, {10154, 11268}, {10565, 11421}, {10634, 15818}, {10691, 42123}, {11139, 44529}, {11141, 13338}, {11284, 22238}, {11548, 42146}, {13595, 37775}, {16063, 42158}, {16249, 41254}, {16267, 47596}, {16419, 42115}, {16967, 37990}, {18470, 42815}, {19107, 34603}, {20062, 42099}, {20063, 43632}, {21849, 44497}, {23303, 37439}, {30739, 42148}, {31099, 42161}, {31133, 36969}, {31152, 42155}, {34608, 42119}, {34609, 42127}, {34754, 37913}, {34986, 44498}, {37353, 42919}, {37454, 42166}, {37899, 42147}, {37900, 42157}, {37904, 43228}, {37910, 42925}, {41101, 47313}, {42100, 52397}, {42102, 52285}, {42141, 44442}, {42151, 46336}, {42165, 46517}, {42943, 43957}, {42992, 52300}, {42999, 52301}

X(54363) = isogonal conjugate of the isotomic conjugate of X(11303)
X(54363) = crossdifference of every pair of points on line {525, 6137}
X(54363) = barycentric product X(6)*X(11303)
X(54363) = barycentric quotient X(11303)/X(76)

X(54364) = X(1)X(514)∩X(31)X(57)

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

X(54364) lies on these lines: {1, 514}, {31, 57}, {33, 92}, {34, 7128}, {43, 1699}, {77, 927}, {294, 3577}, {517, 2427}, {666, 3872}, {908, 35015}, {919, 2717}, {994, 18785}, {997, 35333}, {998, 1438}, {1193, 14267}, {1465, 23981}, {2191, 36041}, {2999, 52210}, {5256, 6654}, {14190, 41162}, {19861, 40724}, {22144, 38575}, {31019, 46784}, {35258, 36086}

X(54364) = isogonal conjugate of X(36819)
X(54364) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36819}, {104, 518}, {241, 52663}, {665, 13136}, {672, 34234}, {909, 3912}, {918, 32641}, {1309, 53550}, {1458, 51565}, {1795, 1861}, {1809, 1876}, {1818, 36123}, {2223, 18816}, {2250, 18206}, {2254, 36037}, {2283, 43728}, {2284, 2401}, {2342, 9436}, {2423, 42720}, {2720, 50333}, {3263, 34858}, {3286, 38955}, {3693, 34051}, {14578, 46108}, {16082, 20752}, {34159, 51832}, {34230, 36944}, {36795, 52635}, {41933, 51390}
X(54364) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36819}, {1145, 3717}, {3259, 2254}, {16586, 3263}, {23980, 3912}, {25640, 1861}, {38981, 50333}, {40613, 518}
X(54364) = cevapoint of X(517) and X(15507)
X(54364) = trilinear pole of line {1769, 2183}
X(54364) = barycentric product X(i)*X(j) for these {i,j}: {75, 51987}, {105, 908}, {294, 22464}, {517, 673}, {666, 1769}, {885, 24029}, {919, 36038}, {927, 46393}, {1027, 2397}, {1438, 3262}, {1457, 36796}, {1462, 6735}, {1465, 14942}, {1785, 1814}, {2183, 2481}, {2804, 36146}, {3310, 51560}, {5377, 42754}, {10015, 36086}, {14571, 31637}, {15507, 52209}, {17139, 18785}, {22350, 54235}, {34085, 53549}, {51381, 52030}, {51390, 51838}
X(54364) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36819}, {105, 34234}, {294, 51565}, {517, 3912}, {673, 18816}, {859, 18206}, {908, 3263}, {919, 36037}, {1024, 43728}, {1027, 2401}, {1416, 34051}, {1438, 104}, {1457, 241}, {1465, 9436}, {1769, 918}, {1785, 46108}, {1875, 5236}, {2183, 518}, {2195, 52663}, {2427, 1026}, {3310, 2254}, {8751, 36123}, {14571, 1861}, {14942, 36795}, {15507, 17755}, {17139, 18157}, {18785, 38955}, {21801, 3932}, {22350, 25083}, {22464, 40704}, {23981, 1025}, {24028, 51390}, {24029, 883}, {32658, 1795}, {32666, 32641}, {32735, 37136}, {36086, 13136}, {36124, 16082}, {42758, 53583}, {46393, 50333}, {51377, 3930}, {51987, 1}, {52480, 46794}

X(54365) = X(1)X(2)∩X(32)X(69)

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

X(54365) lies on these lines: {1, 2}, {32, 69}, {141, 16060}, {384, 1330}, {1043, 6656}, {1654, 4251}, {2271, 5224}, {3430, 6999}, {3620, 4262}, {3936, 17686}, {4417, 7770}, {5741, 17541}, {7789, 17206}, {7807, 14829}, {7819, 41014}, {7832, 33297}, {11321, 18134}, {16930, 26131}, {16931, 26064}, {17234, 33035}, {18266, 33081}, {31023, 52367}, {33033, 41878}, {33172, 33830}, {33181, 37655}

X(54365) = crossdifference of every pair of points on line {649, 2514}
X(54365) = {X(25645),X(29433)}-harmonic conjugate of X(2)

X(54366) = X(2)X(7)∩X(4)X(11)

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

X(54366) lies on these lines: {1, 6908}, {2, 7}, {4, 11}, {12, 19855}, {20, 37583}, {36, 6987}, {46, 5758}, {55, 35514}, {65, 3085}, {72, 1788}, {73, 387}, {145, 18467}, {196, 18687}, {208, 3089}, {218, 52659}, {222, 37642}, {223, 4341}, {241, 17720}, {269, 34050}, {278, 393}, {279, 2006}, {281, 26011}, {342, 17923}, {347, 18593}, {388, 442}, {390, 2078}, {392, 3485}, {405, 7288}, {440, 7011}, {452, 5265}, {496, 37411}, {497, 1617}, {498, 3339}, {499, 3361}, {604, 5802}, {631, 1466}, {651, 24597}, {938, 6838}, {942, 6825}, {943, 11509}, {948, 37695}, {950, 1420}, {954, 5218}, {999, 6907}, {1000, 2099}, {1005, 7677}, {1006, 1470}, {1042, 5230}, {1145, 34619}, {1155, 5759}, {1210, 1467}, {1319, 3488}, {1398, 37376}, {1407, 34032}, {1436, 7490}, {1458, 11269}, {1460, 7413}, {1465, 4000}, {1478, 6843}, {1512, 18391}, {1604, 11347}, {1728, 5811}, {1751, 3451}, {1758, 24248}, {1864, 5658}, {2003, 37666}, {2256, 17056}, {2263, 3011}, {2550, 37240}, {2900, 36845}, {3120, 4331}, {3256, 5281}, {3336, 10320}, {3419, 3476}, {3434, 35990}, {3474, 17613}, {3475, 5173}, {3586, 10072}, {3600, 5177}, {3601, 37108}, {3651, 4294}, {3660, 5728}, {3671, 10198}, {3711, 40663}, {4292, 5715}, {4295, 37550}, {4298, 26363}, {4304, 37427}, {4306, 5292}, {4308, 5175}, {4315, 45700}, {4318, 26228}, {4321, 5231}, {4323, 10587}, {4327, 29639}, {4334, 33140}, {4848, 7080}, {4996, 37313}, {5018, 29658}, {5222, 8776}, {5228, 5718}, {5274, 50696}, {5290, 19854}, {5323, 25516}, {5433, 16845}, {5434, 50741}, {5531, 10573}, {5552, 15556}, {5660, 11570}, {5703, 37112}, {5704, 6953}, {5708, 6863}, {5712, 37543}, {5714, 6832}, {5729, 13257}, {5770, 5777}, {5771, 6147}, {5775, 44848}, {5776, 51365}, {5809, 18450}, {5812, 6891}, {6180, 35466}, {6224, 11240}, {6260, 10396}, {6604, 30828}, {6734, 45039}, {6829, 10590}, {6834, 37566}, {6856, 15844}, {6862, 24470}, {6913, 15325}, {6926, 15803}, {6936, 34880}, {6937, 26437}, {6958, 37545}, {6959, 34753}, {6979, 9964}, {6989, 11374}, {7195, 40615}, {8226, 10589}, {9119, 38015}, {9579, 37434}, {9780, 41824}, {10056, 18421}, {10200, 12572}, {10382, 11019}, {10580, 30284}, {11349, 38900}, {11575, 51489}, {13411, 37407}, {13615, 26105}, {15728, 26007}, {17080, 19785}, {17595, 43056}, {17603, 21151}, {17625, 24477}, {18962, 31410}, {24604, 32625}, {25568, 41539}, {26487, 31794}, {28808, 38468}, {30478, 37224}, {30832, 33298}, {32851, 39126}, {33129, 37800}, {37382, 38461}, {37722, 51773}

X(54366) = X(6350)-Dao conjugate of X(28808)
X(54366) = crossdifference of every pair of points on line {663, 52307}
X(54366) = barycentric product X(i)*X(j) for these {i,j}: {7, 18391}, {85, 8557}, {273, 18446}, {278, 6350}, {331, 19350}
X(54366) = barycentric quotient X(i)/X(j) for these {i,j}: {1512, 6735}, {6350, 345}, {8557, 9}, {18391, 8}, {18446, 78}, {19350, 219}
X(54366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8732, 3911}, {7, 5435, 3218}, {7, 37797, 2}, {226, 1708, 329}, {226, 3911, 9}, {329, 5435, 1708}, {499, 9612, 6846}, {1108, 1427, 43058}, {1427, 3772, 278}, {3487, 6889, 3085}, {3487, 33993, 17718}, {14986, 37421, 950}

X(54367) = X(2)X(3)∩X(10)X(45)

Barycentrics    a^4 - 2*a^3*b - 3*a^2*b^2 - 2*a*b^3 - 2*b^4 - 2*a^3*c - 6*a^2*b*c - 6*a*b^2*c - 2*b^3*c - 3*a^2*c^2 - 6*a*b*c^2 - 2*a*c^3 - 2*b*c^3 - 2*c^4 : :
X(54367) = 3 X[19276] - 2 X[51668]

X(54367) lies on these lines: {2, 3}, {10, 45}, {387, 37654}, {519, 5814}, {551, 5717}, {752, 5711}, {942, 17274}, {975, 50050}, {1125, 48808}, {1330, 17378}, {1698, 17601}, {1834, 17330}, {2901, 50087}, {3017, 48839}, {3175, 3679}, {3295, 49746}, {3876, 27776}, {3927, 17333}, {3931, 5827}, {4357, 5722}, {5015, 50286}, {5143, 31160}, {5225, 19866}, {5263, 9668}, {5292, 49728}, {5484, 7373}, {5725, 50290}, {5743, 48837}, {5774, 37715}, {5808, 50092}, {9708, 32773}, {10449, 17271}, {10896, 19863}, {12572, 50115}, {13161, 50305}, {14555, 48847}, {15934, 27184}, {16589, 22426}, {17251, 50271}, {17301, 50062}, {18440, 25898}, {19765, 27739}, {24723, 36279}, {26625, 51340}, {35652, 50051}, {37674, 48835}, {37679, 48843}, {42044, 50041}, {42051, 50066}, {47037, 50157}, {48803, 49736}, {50047, 50107}, {50067, 50101}

X(54367) = reflection of X(19276) in X(2)
X(54367) = complement of X(51668)
X(54367) = orthocentroidal-circle-inverse of X(37150)
X(54367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 37150}, {2, 377, 51671}, {2, 452, 51673}, {2, 964, 51603}, {2, 4217, 51672}, {2, 11113, 11354}, {2, 11114, 16394}, {2, 14020, 405}, {2, 17528, 50427}, {2, 17537, 964}, {2, 17579, 19290}, {2, 17677, 17528}, {2, 26117, 37038}, {2, 37038, 3}, {2, 37150, 2049}, {2, 37314, 51679}, {2, 48814, 16418}, {2, 48816, 19332}, {2, 48817, 50059}, {2, 49735, 16370}, {2, 50055, 11112}, {2, 50056, 11359}, {2, 50165, 16393}, {2, 50171, 51602}, {2, 51594, 11110}, {2, 51606, 37176}, {2, 51673, 17698}, {2, 51679, 16844}, {4, 4205, 2049}, {5, 13725, 19273}, {442, 37314, 16844}, {442, 51679, 2}, {4205, 37150, 2}, {4205, 52246, 37150}, {4217, 51672, 11354}, {5051, 14020, 2}, {11113, 51672, 4217}, {17556, 51677, 2}, {26117, 52258, 3}, {37038, 52258, 2}, {37144, 37145, 3}, {37146, 37147, 46219}, {37150, 52246, 4}, {37715, 50295, 5774}

X(54368) = X(1)X(19)∩X(4)X(35)

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

X(54368) lies on these lines: {1, 19}, {3, 1838}, {4, 35}, {5, 1852}, {12, 7511}, {21, 39585}, {22, 1074}, {24, 225}, {25, 1324}, {27, 4276}, {29, 5248}, {33, 36009}, {34, 46}, {36, 278}, {40, 1794}, {55, 7497}, {92, 993}, {102, 3362}, {107, 158}, {165, 37305}, {255, 46883}, {270, 9275}, {281, 5251}, {499, 7521}, {516, 37258}, {517, 6056}, {573, 3074}, {758, 1748}, {994, 2190}, {1001, 37393}, {1076, 17928}, {1096, 37817}, {1118, 37583}, {1125, 30687}, {1430, 4257}, {1478, 37395}, {1621, 17519}, {1699, 37380}, {1737, 1751}, {1829, 13750}, {1842, 14017}, {1859, 24929}, {1870, 5902}, {1871, 2646}, {1888, 3579}, {1890, 47042}, {1891, 10039}, {1905, 2355}, {1957, 52680}, {3085, 4198}, {3144, 54084}, {3422, 12047}, {3560, 39574}, {3583, 37372}, {4185, 5530}, {4219, 5010}, {4227, 5307}, {5125, 25440}, {5174, 8715}, {5259, 7498}, {5903, 6197}, {6284, 15763}, {6690, 37321}, {6912, 39531}, {6914, 39529}, {7510, 32613}, {7537, 7741}, {7952, 17562}, {8071, 37245}, {8757, 37489}, {10037, 13161}, {10056, 38300}, {10058, 51282}, {10523, 37376}, {10629, 37392}, {11399, 17523}, {15931, 37028}, {17923, 37304}, {22766, 37236}, {24611, 37231}, {25540, 34851}, {36744, 37377}, {41859, 52252}

X(54368) = barycentric product X(92)*X(5398)
X(54368) = barycentric quotient X(5398)/X(63)
X(54368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 37799, 7951}, {28, 41227, 1}, {92, 17515, 993}, {278, 7501, 36}

X(54369) = X(1)X(2)∩X(48)X(57)

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

X(54369) lies on these lines: {1, 2}, {6, 1214}, {9, 25080}, {31, 22394}, {41, 1763}, {46, 1817}, {48, 57}, {65, 11347}, {81, 6505}, {219, 3666}, {223, 3668}, {226, 3553}, {278, 52033}, {284, 10319}, {440, 10393}, {611, 20588}, {940, 53996}, {990, 2947}, {1006, 1453}, {1442, 37666}, {1468, 36016}, {1723, 32911}, {1780, 27174}, {2003, 34052}, {2324, 4656}, {2328, 17594}, {2331, 40149}, {2646, 21483}, {3752, 37543}, {3755, 40960}, {4272, 18643}, {4383, 40937}, {5249, 24779}, {5706, 15836}, {6198, 18678}, {6261, 19542}, {6349, 53596}, {7146, 21370}, {7190, 23681}, {8557, 16577}, {9121, 50701}, {10397, 21174}, {10572, 37185}, {12514, 16368}, {12520, 37419}, {14110, 16435}, {16466, 37528}, {18734, 23122}, {19788, 24179}, {19790, 24203}, {25252, 27064}, {31266, 37887}, {37181, 48837}, {37662, 37695}

X(54369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2999, 40940}, {1, 50581, 15954}

X(54370) = X(4)X(9)∩X(7)X(90)

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 - 3*b^4*c - 2*a^3*c^2 - 2*a*b^2*c^2 + 4*b^3*c^2 + 2*a^2*c^3 + 4*b^2*c^3 + a*c^4 - 3*b*c^4 - c^5) : :
X(54370) = 3 X[7671] + X[12528], 4 X[15254] - X[43178], 3 X[9] - X[40], X[40] + 3 X[11372], X[2550] - 3 X[5817], 3 X[2550] - 5 X[5818], 9 X[5817] - 5 X[5818], X[35514] - 3 X[38057], 3 X[51090] + X[51118], X[7] - 3 X[38037], 3 X[3062] + 5 X[7987], X[3062] + 2 X[52769], 3 X[5732] - 5 X[7987], 5 X[7987] - 6 X[52769], and many others

X(54370) lies on these lines: {1, 651}, {2, 1709}, {3, 15254}, {4, 9}, {5, 1158}, {7, 90}, {21, 3062}, {36, 8544}, {46, 3091}, {55, 5927}, {57, 1776}, {63, 1699}, {65, 5729}, {72, 42014}, {78, 5696}, {84, 1125}, {109, 9817}, {118, 40131}, {142, 3358}, {144, 11415}, {165, 3305}, {191, 52269}, {200, 15064}, {226, 30223}, {238, 990}, {329, 42012}, {355, 528}, {381, 28534}, {390, 10043}, {405, 12520}, {411, 2951}, {442, 12679}, {499, 30379}, {515, 6930}, {517, 5220}, {518, 1351}, {527, 946}, {920, 1445}, {944, 47357}, {954, 1898}, {958, 9856}, {962, 6172}, {971, 1001}, {997, 1012}, {1005, 1750}, {1006, 50528}, {1071, 10177}, {1376, 10157}, {1490, 5248}, {1698, 6932}, {1708, 1836}, {1721, 13329}, {1723, 3332}, {1728, 4295}, {1736, 2263}, {1737, 6957}, {1741, 5829}, {1768, 3306}, {1770, 6835}, {1858, 5728}, {2346, 36599}, {2475, 52860}, {2476, 38052}, {2950, 6702}, {3100, 15430}, {3218, 9779}, {3219, 9812}, {3333, 43180}, {3359, 6982}, {3361, 9814}, {3434, 20588}, {3485, 5542}, {3486, 10384}, {3587, 28150}, {3616, 10085}, {3634, 37560}, {3646, 9841}, {3652, 5805}, {3671, 10396}, {3678, 6769}, {3683, 7580}, {3685, 48878}, {3740, 6244}, {3754, 54156}, {3811, 5777}, {3826, 6842}, {3869, 4853}, {3878, 54135}, {3884, 12650}, {3895, 37712}, {3911, 10863}, {3913, 9947}, {3928, 50802}, {4067, 7982}, {4294, 5766}, {4297, 11111}, {4301, 36973}, {4413, 17613}, {4423, 10167}, {4466, 6173}, {4640, 19541}, {4672, 48900}, {4676, 13727}, {4679, 37374}, {5047, 9961}, {5086, 38154}, {5119, 30332}, {5219, 21635}, {5250, 5691}, {5259, 10884}, {5284, 11220}, {5437, 10171}, {5693, 12559}, {5709, 18483}, {5715, 12558}, {5762, 40273}, {5811, 21077}, {5843, 20330}, {5851, 5886}, {5853, 47745}, {5903, 41700}, {6001, 6913}, {6259, 25466}, {6260, 10198}, {6282, 10176}, {6666, 6825}, {6684, 9842}, {6838, 18230}, {6846, 12609}, {6847, 21616}, {6852, 20195}, {6856, 38204}, {6857, 38059}, {6860, 17700}, {6872, 36991}, {6888, 37692}, {6893, 12616}, {6915, 30295}, {6939, 12686}, {6945, 30312}, {6985, 11495}, {6988, 43151}, {7069, 8270}, {7098, 38151}, {7171, 10165}, {7308, 10164}, {7411, 41860}, {7491, 31672}, {7675, 37571}, {7680, 37822}, {7705, 7989}, {7971, 30147}, {8167, 11227}, {8255, 11374}, {8581, 20323}, {8715, 47375}, {8727, 24703}, {9778, 27065}, {9809, 31019}, {9943, 11108}, {9955, 24467}, {10179, 30283}, {10398, 12560}, {10582, 30304}, {10595, 51099}, {10826, 13729}, {11376, 38055}, {11662, 36971}, {11729, 25558}, {12526, 54159}, {12635, 31821}, {12699, 38454}, {13257, 17718}, {13743, 45770}, {15296, 34352}, {15481, 51516}, {15803, 30353}, {16209, 17531}, {16617, 31657}, {17728, 41695}, {18450, 37618}, {19862, 37526}, {20116, 30330}, {20117, 37531}, {21740, 38316}, {22758, 42842}, {22793, 26921}, {24248, 53599}, {24430, 34036}, {24723, 36652}, {25524, 34862}, {25917, 37022}, {26202, 32612}, {26333, 51755}, {28236, 31393}, {30340, 51816}, {33179, 42871}, {33596, 42843}, {35258, 44425}, {35664, 39559}, {36663, 52690}, {36996, 38053}, {37360, 42467}, {37624, 42819}, {37695, 38357}, {38036, 41705}

X(54370) = midpoint of X(i) and X(j) for these {i,j}: {4, 5698}, {9, 11372}, {1001, 16112}, {3062, 5732}, {5223, 43166}, {36991, 43161}
X(54370) = reflection of X(i) in X(j) for these {i,j}: {3, 15254}, {5732, 52769}, {5805, 42356}, {5880, 5}, {11495, 31658}, {25558, 11729}, {43177, 1125}, {43178, 3}
X(54370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 5698, 12514}, {200, 30326, 15064}, {390, 29007, 15298}, {405, 12688, 12520}, {1156, 8543, 10394}, {1768, 7988, 3306}, {1836, 7082, 1708}, {3219, 9812, 41338}, {3560, 31937, 6261}, {5057, 10883, 1699}, {5223, 24644, 43166}, {5248, 31871, 1490}, {5777, 11496, 3811}, {5880, 15297, 8257}, {6191, 6192, 169}, {6212, 6213, 5011}, {7308, 10860, 10164}, {8543, 10394, 1}, {10398, 12560, 30329}, {36991, 52653, 43161}

X(54371) = X(3)X(6)∩X(35)X(72)

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

X(54371) lies on these lines: {1, 2915}, {3, 6}, {21, 1211}, {22, 19765}, {28, 17056}, {35, 72}, {36, 4719}, {37, 18598}, {55, 976}, {56, 199}, {229, 37635}, {239, 19841}, {894, 19842}, {896, 5217}, {910, 5277}, {940, 11337}, {958, 32778}, {993, 5814}, {1036, 1191}, {1213, 47512}, {1325, 26131}, {1330, 17512}, {1763, 3601}, {1829, 2646}, {1834, 4220}, {2174, 3682}, {2194, 22076}, {2975, 32842}, {3295, 20918}, {3303, 20851}, {3936, 17521}, {4189, 5739}, {4361, 19844}, {4363, 19845}, {4653, 20831}, {5248, 16686}, {5310, 37080}, {5347, 19767}, {5358, 6675}, {5438, 35342}, {5710, 20847}, {5711, 39582}, {5712, 7520}, {5718, 37231}, {5810, 6914}, {7295, 24697}, {7297, 9895}, {11102, 25650}, {15976, 44517}, {16049, 49745}, {16370, 49723}, {16429, 25526}, {16777, 27802}, {17104, 22136}, {18178, 37527}, {18185, 20844}, {19329, 25524}, {19523, 19744}, {19720, 37151}, {20852, 22654}, {20872, 37573}, {23130, 40214}, {23383, 23868}, {25909, 47296}, {35212, 37594}, {37034, 37674}, {37225, 40980}, {37431, 37662}, {37571, 54095}

X(54371) = X(37539)-Ceva conjugate of X(1191)
X(54371) = crossdifference of every pair of points on line {523, 16612}
X(54371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 386, 5096}, {3, 19762, 5124}, {3, 36740, 4252}, {3, 36744, 19760}, {284, 3430, 52544}

X(54372) = X(2)X(3)∩X(53)X(86)

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

X(54372) lies on these lines: {2, 3}, {53, 86}, {92, 54120}, {264, 17300}, {273, 26806}, {281, 17350}, {317, 1654}, {318, 6542}, {324, 26541}, {393, 17379}, {1249, 37677}, {1785, 16826}, {1897, 29588}, {1990, 46922}, {2052, 6625}, {2202, 40744}, {2322, 27377}, {3087, 17349}, {3945, 43981}, {4393, 34231}, {6646, 7282}, {6748, 17277}, {7046, 20055}, {7952, 29570}, {9308, 20090}, {17343, 32001}, {17375, 32000}

X(54372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 21940, 2}, {297, 11109, 2}, {458, 37448, 2}, {1585, 1586, 4213}, {25986, 37279, 2}

X(54373) = X(1)X(228)∩X(31)X(40)

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

X(54373) lies on these lines: {1, 228}, {2, 2269}, {6, 24310}, {9, 321}, {31, 40}, {43, 26893}, {57, 77}, {63, 194}, {212, 41230}, {226, 4266}, {306, 3169}, {329, 2347}, {380, 41342}, {497, 26013}, {573, 27659}, {978, 22076}, {982, 17441}, {985, 39596}, {986, 1829}, {1193, 9549}, {1278, 3219}, {1423, 19785}, {1621, 1697}, {1745, 50594}, {1763, 36572}, {1764, 2999}, {1851, 24248}, {1914, 10319}, {1999, 21371}, {2262, 3666}, {2270, 5276}, {2286, 45984}, {3057, 25091}, {3061, 3998}, {3208, 17776}, {3501, 5294}, {3772, 4271}, {3882, 25527}, {3914, 6210}, {3929, 50106}, {3969, 4050}, {4000, 22097}, {4254, 37543}, {4384, 10471}, {4640, 40970}, {5119, 8616}, {5222, 28274}, {5230, 9548}, {5250, 16824}, {7308, 53391}, {16503, 19716}, {16572, 36808}, {16609, 19788}, {16779, 19729}, {16826, 39970}, {17451, 28606}, {17595, 18735}, {18178, 37523}, {26723, 27626}, {37400, 40958}

X(54373) = crossdifference of every pair of points on line {4041, 23655}
X(54373) = {X(573),X(40940)}-harmonic conjugate of X(27659)

X(54374) = X(3)X(6)∩X(20)X(66)

Barycentrics    a^2*(a^10 - 4*a^6*b^4 + 2*a^4*b^6 + 3*a^2*b^8 - 2*b^10 - 4*a^6*b^2*c^2 + 4*a^2*b^6*c^2 - 4*a^6*c^4 + 2*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 + 3*a^2*c^8 - 2*c^10) : :
X(54374) = 2 X[3098] + X[37478], X[13352] - 4 X[14810]

X(54374) lies on these lines: {3, 6}, {20, 66}, {22, 34146}, {69, 7691}, {141, 3575}, {159, 45813}, {206, 7488}, {427, 29181}, {1092, 32391}, {1204, 26926}, {1352, 45286}, {1370, 23293}, {1503, 16789}, {2781, 22109}, {3357, 48905}, {3917, 21213}, {5562, 15577}, {5576, 48901}, {5907, 20987}, {6697, 32393}, {7499, 21167}, {7503, 9969}, {7667, 50965}, {7689, 46264}, {9019, 19124}, {9715, 19149}, {9968, 38435}, {10516, 18494}, {10519, 18533}, {11414, 34778}, {11440, 14927}, {12225, 51756}, {12270, 52363}, {14790, 48873}, {15107, 51538}, {20806, 38444}, {23335, 48874}, {29317, 31723}, {34417, 37454}, {38321, 50977}, {41362, 48881}

X(54374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1350, 3313}, {3, 19161, 5157}, {3098, 46728, 1350}, {7488, 41716, 206}, {11440, 41464, 14927}

X(54375) = X(2)X(3)∩X(51)X(97)

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

X(54375) lies on these lines: {2, 3}, {51, 97}, {95, 107}, {110, 216}, {157, 35260}, {323, 30258}, {577, 5640}, {3066, 36748}, {3284, 15019}, {5158, 11422}, {6394, 26235}, {10545, 22052}, {10546, 10979}, {13450, 19179}, {18350, 46025}, {23181, 44180}, {23606, 34545}, {26898, 35264}, {26907, 31626}, {33926, 40680}, {34147, 46832}, {34828, 35283}, {35259, 36751}, {37081, 43651}, {44299, 53852}, {51444, 52153}

X(54375) = X(656)-isoconjugate of X(23232)
X(54375) = X(40596)-Dao conjugate of X(23232)
X(54375) = crossdifference of every pair of points on line {647, 42731}
X(54375) = barycentric quotient X(112)/X(23232)
X(54375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 37068, 26874}, {38431, 38432, 7527}

X(54376) = X(3)X(74)∩X(66)X(67)

Barycentrics    a^2*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10 + a^8*c^2 + a^4*b^4*c^2 - 2*b^8*c^2 - 2*a^6*c^4 + a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 3*b^6*c^4 + 3*b^4*c^6 + 2*a^2*c^8 - 2*b^2*c^8 - c^10) : :
X(54376) = 5 X[74] - X[6241], 3 X[74] + X[12281], 3 X[74] - X[17854], 5 X[110] - 9 X[7998], 3 X[6241] + 5 X[12281], 3 X[6241] - 5 X[17854], 9 X[7998] - 10 X[13416], 4 X[11591] - 5 X[12358], X[12270] - 5 X[15021], 3 X[15055] + X[15100], 2 X[15101] + X[44573], 3 X[51] - 5 X[125], 6 X[51] - 5 X[1112], 9 X[51] - 10 X[11746], and many others

X(54376) lies on these lines: {3, 74}, {5, 43896}, {51, 125}, {66, 67}, {69, 40228}, {113, 7399}, {143, 23294}, {146, 6815}, {184, 15578}, {185, 15151}, {265, 14790}, {468, 34146}, {511, 46517}, {542, 7667}, {974, 10628}, {1205, 2854}, {1370, 3448}, {1899, 44439}, {1986, 3541}, {2393, 47278}, {2777, 3575}, {2979, 11898}, {3269, 35325}, {3357, 44080}, {3917, 24981}, {5157, 6593}, {5462, 38725}, {5576, 18874}, {5621, 13198}, {5622, 19504}, {5972, 7499}, {6000, 37931}, {6053, 11793}, {6101, 11457}, {6699, 25711}, {6723, 37454}, {6746, 20299}, {7505, 44544}, {7687, 16105}, {7728, 18420}, {7731, 46430}, {9140, 13201}, {9826, 15061}, {10113, 31723}, {10117, 21213}, {10264, 23335}, {10627, 34224}, {10721, 18494}, {10938, 35485}, {10990, 15105}, {11432, 19348}, {11557, 20397}, {12006, 43608}, {12058, 32263}, {12236, 20379}, {12244, 12292}, {12824, 15059}, {13289, 44679}, {13391, 25739}, {13630, 14389}, {13754, 47090}, {14708, 52262}, {14915, 47340}, {15081, 15465}, {16111, 44239}, {16223, 38729}, {16981, 31099}, {18435, 54013}, {19208, 54034}, {20126, 44441}, {26926, 32366}, {31152, 41716}, {32125, 32316}, {32226, 34468}, {33884, 39874}, {34783, 37645}, {37453, 41715}, {38356, 44467}, {41580, 52297}, {41671, 45311}

X(54376) = midpoint of X(i) and X(j) for these {i,j}: {7723, 10620}, {10990, 21650}, {12041, 15101}, {12244, 12292}, {12281, 17854}, {12825, 15054}
X(54376) = reflection of X(i) in X(j) for these {i,j}: {110, 13416}, {185, 15151}, {974, 20417}, {1112, 125}, {1986, 16270}, {6053, 11793}, {11557, 20397}, {12133, 15738}, {12236, 20379}, {13148, 974}, {13417, 11746}, {16105, 7687}, {25711, 6699}, {44573, 12041}
X(54376) = circumcircle-inverse of X(13171)
X(54376) = barycentric product X(11832)*X(14919)
X(54376) = barycentric quotient X(11832)/X(46106)
X(54376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 110, 13171}, {74, 12281, 17854}, {125, 1112, 12099}, {125, 13417, 11746}, {5621, 17847, 13198}, {6241, 7998, 26864}, {11746, 13417, 1112}, {13171, 15106, 110}, {32616, 32617, 12174}

X(54377) = X(6)X(19)∩X(44)X(56)

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

X(54377) lies on these lines: {1, 1404}, {6, 19}, {7, 17120}, {9, 604}, {33, 44085}, {37, 1388}, {44, 56}, {45, 1319}, {57, 88}, {374, 37566}, {572, 3612}, {610, 2347}, {612, 1397}, {978, 1400}, {1100, 38296}, {1399, 4290}, {1428, 4327}, {1429, 8545}, {1445, 7175}, {1449, 2171}, {1766, 5697}, {1788, 37654}, {1950, 16946}, {2099, 16666}, {2265, 8557}, {2267, 22061}, {2317, 3553}, {3589, 10401}, {3660, 22163}, {3697, 5783}, {3707, 3911}, {3758, 41245}, {3870, 4579}, {3943, 37738}, {4643, 43053}, {4700, 4848}, {4969, 41687}, {5069, 40590}, {5252, 17369}, {5933, 51170}, {6180, 28017}, {7397, 53020}, {9957, 54359}, {10106, 50115}, {10944, 17281}, {11509, 37503}, {17237, 31230}, {17330, 24914}, {19297, 34880}

X(54377) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3478}, {6332, 9088}
X(54377) = X(i)-Dao conjugate of X(j) for these (i,j): {32664, 3478}, {53838, 4391}
X(54377) = crossdifference of every pair of points on line {521, 4895}
X(54377) = barycentric product X(i)*X(j) for these {i,j}: {1, 3476}, {56, 4737}, {65, 4234}, {108, 9031}, {651, 47766}, {664, 48327}, {4551, 47845}
X(54377) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 3478}, {3476, 75}, {4234, 314}, {4737, 3596}, {9031, 35518}, {47766, 4391}, {47845, 18155}, {48327, 522}
X(54377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2182, 2082}, {57, 16670, 1405}

X(54378) = X(1)X(6)∩X(2)X(13)

Barycentrics    a*(Sqrt[3]*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) + 2*a*S) : :

X(54378) lies on these lines: {1, 6}, {2, 13}, {10, 1250}, {15, 21}, {18, 37162}, {61, 16865}, {62, 5047}, {202, 5284}, {377, 42086}, {396, 15670}, {404, 10646}, {406, 10642}, {442, 5318}, {443, 42120}, {451, 10633}, {452, 5334}, {474, 11481}, {475, 11476}, {993, 7051}, {1125, 19373}, {1276, 54296}, {2306, 3647}, {2475, 19106}, {2476, 16808}, {2478, 18581}, {4187, 23303}, {4189, 10645}, {4190, 42091}, {4193, 16967}, {4197, 16965}, {4208, 43465}, {5046, 16809}, {5084, 11489}, {5141, 42919}, {5154, 42914}, {5177, 42134}, {5187, 42111}, {5237, 17531}, {5248, 10638}, {5260, 7006}, {5277, 19780}, {5278, 40714}, {5321, 11113}, {5351, 17572}, {5352, 17574}, {6175, 36969}, {6675, 11542}, {6856, 42142}, {6857, 11488}, {6871, 42106}, {6872, 42085}, {6910, 42092}, {6933, 42114}, {7483, 23302}, {7504, 42915}, {8728, 42118}, {10654, 31156}, {11095, 32431}, {11097, 37508}, {11108, 11486}, {11111, 42119}, {11112, 42088}, {11114, 19107}, {11480, 16370}, {11485, 16418}, {15671, 16267}, {15672, 16962}, {15673, 42912}, {15674, 16960}, {15677, 36967}, {15680, 42099}, {15988, 51206}, {16408, 42115}, {16773, 17575}, {16842, 22238}, {16862, 36843}, {17525, 42942}, {17527, 42121}, {17528, 42127}, {17529, 42148}, {17530, 42110}, {17532, 42094}, {17536, 34755}, {17556, 42095}, {17561, 37640}, {17571, 42116}, {17579, 42100}, {19526, 22236}, {31295, 42113}, {37325, 54362}, {37375, 42918}, {37462, 42151}, {37675, 41406}, {42096, 50242}, {42097, 50239}, {42112, 50244}, {42117, 50241}, {42145, 50240}, {42155, 44217}

X(54378) = crossdifference of every pair of points on line {513, 6137}
X(54378) = {X(21),X(5362)}-harmonic conjugate of X(15)

X(54379) = X(1)X(6)∩X(2)X(14)

Barycentrics    a*(Sqrt[3]*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) - 2*a*S) : :

X(54379) lies on these lines: {1, 6}, {2, 14}, {10, 10638}, {16, 21}, {17, 37162}, {61, 5047}, {62, 16865}, {203, 5284}, {377, 42085}, {395, 15670}, {404, 10645}, {406, 10641}, {442, 5321}, {443, 42119}, {451, 10632}, {452, 5335}, {474, 11480}, {475, 11475}, {993, 19373}, {1125, 7051}, {1250, 5248}, {1277, 54296}, {2475, 19107}, {2476, 16809}, {2478, 18582}, {3647, 33654}, {4187, 23302}, {4189, 10646}, {4190, 42090}, {4193, 16966}, {4197, 16964}, {4208, 43466}, {5046, 16808}, {5084, 11488}, {5141, 42918}, {5154, 42915}, {5177, 42133}, {5187, 42114}, {5238, 17531}, {5260, 7005}, {5277, 19781}, {5278, 40713}, {5318, 11113}, {5351, 17574}, {5352, 17572}, {6175, 36970}, {6675, 11543}, {6856, 42139}, {6857, 11489}, {6871, 42103}, {6872, 42086}, {6910, 42089}, {6933, 42111}, {7483, 23303}, {7504, 42914}, {8728, 42117}, {10653, 31156}, {11096, 32431}, {11098, 37508}, {11108, 11485}, {11111, 42120}, {11112, 42087}, {11114, 19106}, {11481, 16370}, {11486, 16418}, {15671, 16268}, {15672, 16963}, {15673, 42913}, {15674, 16961}, {15677, 36968}, {15680, 42100}, {15988, 51207}, {16408, 42116}, {16772, 17575}, {16842, 22236}, {16862, 36836}, {17525, 42943}, {17527, 42124}, {17528, 42126}, {17529, 42147}, {17530, 42107}, {17532, 42093}, {17536, 34754}, {17556, 42098}, {17561, 37641}, {17571, 42115}, {17579, 42099}, {19526, 22238}, {31295, 42112}, {37325, 54363}, {37375, 42919}, {37462, 42150}, {37675, 41407}, {42096, 50239}, {42097, 50242}, {42113, 50244}, {42118, 50241}, {42144, 50240}, {42154, 44217}

X(54379) = crossdifference of every pair of points on line {513, 6138}
X(54379) = {X(21),X(5367)}-harmonic conjugate of X(16)

X(54380) = X(2)X(3)∩X(114)X(132)

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)*(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(54380) = 3 X[23234] - X[52094]

X(54380) lies on these lines: {2, 3}, {112, 6033}, {113, 1560}, {114, 132}, {126, 133}, {127, 38553}, {232, 14356}, {265, 8791}, {325, 877}, {542, 6103}, {648, 6054}, {935, 38953}, {1503, 35912}, {1550, 34761}, {1990, 14995}, {3014, 47228}, {3564, 34211}, {5968, 6530}, {9214, 47172}, {10735, 38741}, {11059, 47392}, {13200, 38744}, {16092, 17986}, {16188, 18312}, {16316, 52472}, {16318, 34810}, {18809, 31655}, {23234, 52094}, {23347, 24975}, {23350, 53156}, {30786, 42308}, {31842, 50938}, {34334, 34336}, {34366, 47105}, {36875, 47155}, {41676, 51872}, {46986, 52464}, {47151, 52772}, {53149, 53266}

X(54380) = midpoint of X(4) and X(4235)
X(54380) = polar-circle-inverse of X(36166)
X(54380) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(7473)
X(54380) = X(i)-Ceva conjugate of X(j) for these (i,j): {47105, 542}, {47110, 511}
X(54380) = X(i)-isoconjugate of X(j) for these (i,j): {293, 842}, {656, 53691}, {35200, 53866}, {35909, 36084}, {35911, 36104}
X(54380) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 842}, {133, 53866}, {23967, 287}, {38970, 14223}, {38987, 35909}, {39000, 35911}, {39073, 40080}, {40596, 53691}, {42426, 98}
X(54380) = crossdifference of every pair of points on line {248, 647}
X(54380) = barycentric product X(i)*X(j) for these {i,j}: {297, 542}, {325, 6103}, {877, 1640}, {2247, 40703}, {2799, 7473}, {2967, 46786}, {4230, 18312}, {5191, 44132}, {6333, 35907}, {14618, 42743}, {14999, 16230}, {15595, 47105}, {17986, 51389}, {36790, 52491}, {38552, 46787}
X(54380) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 53691}, {232, 842}, {297, 5641}, {542, 287}, {684, 35911}, {877, 6035}, {1640, 879}, {1990, 53866}, {2247, 293}, {2967, 46787}, {3569, 35909}, {4230, 5649}, {5191, 248}, {6041, 878}, {6103, 98}, {7473, 2966}, {9475, 40080}, {14999, 17932}, {16230, 14223}, {17994, 14998}, {34369, 47388}, {35907, 685}, {38552, 46786}, {42743, 4558}, {47105, 9476}, {47427, 40079}, {51334, 52492}, {51428, 51404}, {52491, 34536}
X(54380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 858, 1650}, {2, 4240, 468}, {114, 132, 2967}, {402, 5159, 2}, {427, 468, 35235}, {5000, 5001, 868}, {16188, 42426, 38552}

X(54381) = X(2)X(3)∩X(53)X(136)

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

X(54381) lies on these lines: {2, 3}, {52, 23307}, {53, 136}, {125, 1205}, {126, 135}, {132, 47208}, {1112, 21850}, {1184, 13854}, {1503, 44080}, {1899, 8549}, {2892, 15106}, {3199, 15820}, {3580, 6403}, {3815, 11062}, {3867, 16776}, {5480, 44084}, {5523, 9465}, {7745, 52905}, {8262, 41585}, {8791, 8882}, {11188, 41584}, {12167, 26869}, {12294, 51360}, {13567, 47328}, {14389, 19128}, {14580, 27376}, {15131, 38851}, {15473, 23306}, {16178, 31655}, {16285, 19595}, {18911, 39588}, {19504, 25321}, {20300, 47296}, {20621, 45946}, {21243, 43130}, {23292, 44077}, {23315, 46682}, {30786, 32002}, {31383, 41602}, {32123, 40909}, {32125, 36990}, {34336, 34338}, {40130, 51434}, {42426, 47222}, {50938, 53832}

X(54381) = complement of X(26283)
X(54381) = polar-circle-inverse of X(37980)
X(54381) = polar conjugate of the isotomic conjugate of X(54347)
X(54381) = X(30251)-complementary conjugate of X(8062)
X(54381) = X(39382)-Ceva conjugate of X(523)
X(54381) = barycentric product X(4)*X(54347)
X(54381) = barycentric quotient X(54347)/X(69)
X(54381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 24, 468}, {2, 858, 11585}, {2, 7401, 11284}, {2, 37977, 10018}, {2, 46336, 7393}, {4, 858, 427}, {4, 35480, 47309}, {25, 37981, 235}, {235, 427, 37981}, {427, 468, 5}, {427, 30739, 5094}, {468, 3575, 25}, {1368, 15809, 427}, {1368, 23335, 858}, {1594, 3147, 7405}, {1594, 7487, 235}, {1995, 16051, 37454}, {1995, 31099, 10297}, {4232, 5169, 403}, {5000, 5001, 15760}, {5159, 16238, 2}, {7576, 45179, 1596}, {12106, 13371, 5}, {15559, 52284, 427}, {34351, 37458, 24}

X(54382) = X(1)X(32)∩X(6)X(19)

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

X(54382) lies on these lines: {1, 32}, {2, 39248}, {6, 19}, {9, 2295}, {10, 9596}, {31, 16968}, {36, 9619}, {37, 5250}, {39, 46}, {40, 2276}, {48, 23623}, {57, 893}, {63, 1107}, {77, 40765}, {78, 4386}, {87, 13610}, {169, 213}, {171, 3061}, {187, 3612}, {191, 31442}, {230, 11375}, {232, 1452}, {257, 14621}, {284, 39598}, {354, 16781}, {386, 5011}, {484, 1571}, {516, 9598}, {612, 20715}, {614, 20271}, {748, 21921}, {750, 39244}, {894, 21281}, {940, 2339}, {942, 16502}, {960, 5275}, {988, 36643}, {997, 5277}, {1015, 3338}, {1046, 21384}, {1100, 37549}, {1155, 5013}, {1159, 43136}, {1191, 3290}, {1210, 9599}, {1333, 54356}, {1400, 23544}, {1454, 43039}, {1468, 2170}, {1500, 5119}, {1573, 41229}, {1707, 21332}, {1722, 21951}, {1737, 2548}, {1766, 31785}, {1770, 2549}, {1788, 7736}, {1836, 5254}, {1837, 7745}, {1899, 41011}, {1905, 2207}, {1909, 52652}, {1965, 21608}, {2093, 9593}, {2176, 40131}, {2280, 2650}, {2646, 3053}, {3306, 16604}, {3340, 7296}, {3474, 7738}, {3485, 7735}, {3550, 39255}, {3579, 31448}, {3751, 3780}, {3752, 24590}, {3767, 12047}, {3772, 5244}, {3815, 24914}, {3868, 16973}, {3869, 5276}, {3915, 21808}, {3916, 31449}, {3924, 21764}, {3970, 37610}, {4000, 27000}, {4136, 4865}, {4252, 34522}, {4264, 5336}, {4275, 17443}, {4292, 9597}, {4295, 5286}, {4333, 7756}, {4338, 7765}, {4383, 16605}, {4426, 19860}, {4641, 4875}, {4644, 6604}, {5021, 43065}, {5023, 37600}, {5128, 9574}, {5228, 40133}, {5256, 18202}, {5280, 5903}, {5282, 10459}, {5283, 12514}, {5299, 5902}, {5305, 39542}, {5322, 21771}, {5332, 11529}, {5445, 31441}, {5475, 10826}, {5657, 31402}, {5697, 16785}, {6684, 31497}, {7737, 10572}, {7746, 37692}, {9331, 37563}, {9592, 15803}, {9605, 36279}, {9650, 10827}, {10039, 31409}, {11010, 31433}, {12526, 16517}, {13881, 17605}, {14974, 16601}, {16466, 16583}, {16552, 49500}, {16606, 27459}, {16716, 40153}, {16784, 18398}, {16974, 21331}, {17365, 30617}, {18907, 37730}, {21029, 33104}, {21387, 32913}, {21874, 37658}, {26066, 37661}, {26446, 31460}, {31422, 37572}, {31429, 54290}, {31459, 49226}, {31477, 37568}, {33950, 36404}, {37588, 51058}, {41826, 50011}

X(54382) = crossdifference of every pair of points on line {521, 1491}
X(54382) = barycentric product X(1)*X(26098)
X(54382) = barycentric quotient X(26098)/X(75)
X(54382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31, 17451, 16968}, {57, 9575, 2275}, {171, 3061, 54317}, {5280, 5903, 9620}

X(54383) = X(6)X(21)∩X(7)X(8)

Barycentrics    a*(2*a^3*b^2 - 2*a*b^4 + 3*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 - a*b*c^3 - b^2*c^3 - 2*a*c^4 - b*c^4) : :
X(54383) = 3 X[2] - 4 X[4260], 3 X[1992] - 2 X[37516]

X(54383) lies on these lines: {1, 28287}, {2, 3786}, {6, 21}, {7, 8}, {20, 185}, {22, 40571}, {42, 63}, {72, 13725}, {78, 1400}, {81, 37090}, {141, 4197}, {145, 9052}, {159, 41739}, {182, 37106}, {209, 345}, {329, 37193}, {386, 10461}, {464, 16465}, {524, 17579}, {674, 51192}, {758, 24248}, {938, 41828}, {942, 37153}, {971, 9962}, {980, 50596}, {991, 18206}, {1010, 51223}, {1012, 1351}, {1210, 29965}, {1284, 12635}, {1350, 7411}, {1352, 6839}, {1423, 11523}, {1714, 35637}, {1812, 37538}, {1843, 4198}, {1992, 37516}, {2269, 7675}, {2287, 19310}, {2810, 9965}, {3056, 4313}, {3098, 37105}, {3218, 35980}, {3242, 9054}, {3564, 37468}, {3620, 4208}, {3781, 17316}, {3794, 37666}, {3876, 5296}, {3901, 32857}, {3940, 19266}, {4189, 5138}, {4292, 34379}, {4304, 51196}, {4335, 12526}, {4343, 5250}, {5032, 50742}, {5249, 31330}, {5480, 10883}, {5728, 37169}, {5751, 36706}, {5757, 13727}, {5767, 37088}, {5847, 25304}, {6007, 24280}, {6403, 37395}, {6646, 17676}, {6837, 14853}, {6884, 14561}, {6993, 40330}, {7102, 37181}, {9024, 9963}, {9047, 11015}, {10431, 51212}, {10444, 29311}, {10459, 11520}, {10519, 37112}, {12294, 37104}, {12649, 37191}, {14054, 37179}, {15988, 37228}, {16704, 50404}, {17153, 36500}, {17558, 51171}, {17576, 51170}, {17778, 37107}, {20080, 37435}, {20683, 27549}, {20835, 37492}, {21850, 37447}, {22277, 26115}, {24391, 30097}, {24473, 50428}, {25494, 32911}, {31670, 37433}, {31775, 34380}, {33088, 52025}, {33878, 37426}, {35628, 52020}, {36740, 37285}, {36741, 37300}, {37103, 37652}, {37109, 45990}, {37180, 45963}, {37467, 37676}, {44238, 48906}, {48013, 50481}

X(54383) = reflection of X(i) in X(j) for these {i,j}: {8, 3779}, {69, 4259}, {10477, 4260}
X(54383) = anticomplement of X(10477)
X(54383) = crossdifference of every pair of points on line {2451, 3063}
X(54383) = {X(4260),X(10477)}-harmonic conjugate of X(2)

X(54384) = X(4)X(70)∩X(30)X(52)

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 + 2*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(54384) = 3 X[51] - 2 X[427], 3 X[3060] - X[7391], 3 X[3917] - 4 X[6676], 3 X[5890] - X[35481], 3 X[5891] - 4 X[46029], 6 X[5943] - 5 X[31236], 3 X[5946] - 2 X[44236], 3 X[9730] - 2 X[18570], 8 X[13413] - 9 X[14845], 2 X[15644] - 3 X[44837], 3 X[16226] - 2 X[44218]

X(54384) lies on these lines: {4, 70}, {6, 34436}, {22, 184}, {25, 15139}, {30, 52}, {51, 125}, {251, 10766}, {378, 389}, {542, 27365}, {568, 10605}, {973, 1595}, {974, 14677}, {1147, 44259}, {1181, 6243}, {1495, 41580}, {1539, 16194}, {1899, 3060}, {1994, 13198}, {2056, 35901}, {3051, 38356}, {3292, 44260}, {3313, 19127}, {3567, 26937}, {3629, 6467}, {3796, 18438}, {3845, 15738}, {3917, 6676}, {5133, 23330}, {5446, 31723}, {5562, 15760}, {5622, 53863}, {5889, 44440}, {5890, 35481}, {5891, 46029}, {5899, 43590}, {5943, 31236}, {5946, 44236}, {6000, 35480}, {6101, 25337}, {6152, 16659}, {6240, 41725}, {6247, 6746}, {6293, 12173}, {6403, 31383}, {6776, 20062}, {7502, 10625}, {9730, 18570}, {9967, 22352}, {10282, 37932}, {11002, 23291}, {11381, 11576}, {11402, 44439}, {11438, 44269}, {11550, 34146}, {12058, 51360}, {12162, 44263}, {12293, 34783}, {13366, 50649}, {13413, 14845}, {13564, 19362}, {13598, 52842}, {13851, 44288}, {15644, 44837}, {16226, 44218}, {16655, 44544}, {18445, 45780}, {19357, 37484}, {19467, 44831}, {20299, 43896}, {21213, 34117}, {21849, 31133}, {21851, 34417}, {23217, 52032}, {27372, 27375}, {32184, 43903}, {32285, 40949}, {34565, 51739}, {37969, 44110}, {41716, 43653}

X(54384) = midpoint of X(i) and X(j) for these {i,j}: {5889, 44440}, {6243, 12083}
X(54384) = reflection of X(i) in X(j) for these {i,j}: {378, 389}, {3313, 19127}, {5562, 15760}, {6101, 25337}, {10625, 7502}, {11550, 47328}, {12162, 44263}, {31133, 21849}, {31723, 5446}, {52842, 13598}
X(54384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {185, 45186, 21659}, {1112, 13567, 51}, {6403, 41715, 31383}, {21213, 34117, 44078}, {34221, 34222, 20300}

X(54385) = X(1)X(19)∩X(37)X(56)

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

X(54385) lies on these lines: {1, 19}, {2, 5227}, {6, 354}, {7, 18589}, {9, 1125}, {34, 2286}, {37, 56}, {57, 71}, {63, 86}, {65, 2256}, {154, 3745}, {169, 1449}, {219, 942}, {226, 1435}, {281, 1056}, {388, 1826}, {474, 3694}, {497, 1839}, {518, 965}, {573, 12704}, {579, 3338}, {604, 21808}, {612, 12329}, {894, 26106}, {936, 3949}, {940, 44101}, {966, 24477}, {968, 2352}, {997, 22021}, {999, 40937}, {1100, 2082}, {1108, 3304}, {1210, 26063}, {1445, 25523}, {1659, 6203}, {1723, 51816}, {1760, 17394}, {1761, 5250}, {1766, 3247}, {1901, 10404}, {1903, 8581}, {2002, 7190}, {2178, 26357}, {2242, 5336}, {2257, 54324}, {2261, 16193}, {2264, 17609}, {2287, 3873}, {2646, 7221}, {3213, 20613}, {3305, 17381}, {3306, 3692}, {3509, 29644}, {3553, 9310}, {3554, 17451}, {3601, 22054}, {3616, 5279}, {3870, 54316}, {3875, 24435}, {3945, 7289}, {4000, 24162}, {4298, 8804}, {4675, 28017}, {5045, 22153}, {5120, 16601}, {5221, 21866}, {5249, 28753}, {5252, 21933}, {5253, 27396}, {5275, 30677}, {5356, 16672}, {5437, 20106}, {5572, 5781}, {5712, 44103}, {5738, 9028}, {5747, 13407}, {5750, 17742}, {5776, 12675}, {6204, 6352}, {7079, 21620}, {7131, 41246}, {8666, 25081}, {9444, 21857}, {9578, 21011}, {11037, 27382}, {11518, 22356}, {15487, 44081}, {15934, 20818}, {16516, 20358}, {16566, 29597}, {17718, 46345}, {17754, 29642}, {18655, 24424}, {21049, 54008}, {24476, 51210}, {27059, 29585}, {28738, 31266}, {30456, 34046}, {32636, 37500}, {37080, 37504}, {37578, 54285}

X(54385) = crossdifference of every pair of points on line {656, 3309}
X(54385) = barycentric product X(i)*X(j) for these {i,j}: {1, 443}, {75, 44094}
X(54385) = barycentric quotient X(i)/X(j) for these {i,j}: {443, 75}, {44094, 1}
X(54385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 2260, 1732}, {9, 3333, 2260}, {16777, 37519, 2646}

X(54386) = X(1)X(6)∩X(31)X(78)

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

X(54386) lies on these lines: {1, 6}, {3, 1707}, {7, 24178}, {8, 989}, {10, 14555}, {31, 78}, {38, 3951}, {40, 43}, {42, 5250}, {46, 2390}, {56, 4641}, {57, 978}, {58, 997}, {63, 988}, {65, 1722}, {145, 42360}, {171, 936}, {191, 5313}, {200, 5255}, {210, 5710}, {329, 13161}, {377, 41011}, {386, 1245}, {387, 24210}, {443, 50307}, {519, 42032}, {579, 23620}, {580, 6261}, {595, 3749}, {602, 18446}, {612, 3876}, {614, 3868}, {651, 4320}, {748, 2650}, {896, 4652}, {908, 5230}, {920, 46016}, {940, 25917}, {942, 5272}, {946, 33137}, {959, 2285}, {968, 19767}, {975, 10176}, {976, 3984}, {979, 43070}, {986, 2999}, {990, 31803}, {1036, 7085}, {1039, 44086}, {1040, 1858}, {1042, 1445}, {1043, 4676}, {1183, 37399}, {1201, 32912}, {1210, 27411}, {1265, 51192}, {1468, 19861}, {1469, 29958}, {1490, 37570}, {1580, 8235}, {1610, 2261}, {1685, 6213}, {1686, 6212}, {1697, 20683}, {1698, 5743}, {1699, 5799}, {1706, 6048}, {1708, 10571}, {1714, 12047}, {1716, 4260}, {1721, 12688}, {1738, 4295}, {1742, 37551}, {1798, 17104}, {1834, 24703}, {1999, 19582}, {2082, 37657}, {2093, 24440}, {2292, 5256}, {3008, 3671}, {3072, 5720}, {3073, 37531}, {3185, 54300}, {3187, 25253}, {3293, 5119}, {3306, 27627}, {3333, 21214}, {3338, 49997}, {3339, 23511}, {3556, 36741}, {3576, 13323}, {3601, 54354}, {3612, 52680}, {3624, 6703}, {3646, 26102}, {3649, 24789}, {3679, 5835}, {3683, 19765}, {3685, 20018}, {3702, 17156}, {3710, 33088}, {3812, 37679}, {3869, 32911}, {3870, 3915}, {3873, 28011}, {3913, 4849}, {3914, 11415}, {3916, 16570}, {3927, 37592}, {3940, 5266}, {3962, 37549}, {4101, 33171}, {4134, 30145}, {4255, 4640}, {4259, 42450}, {4281, 17185}, {4292, 24695}, {4339, 20007}, {4384, 49598}, {4512, 37573}, {4650, 15803}, {4850, 11684}, {4855, 36277}, {5044, 5268}, {5082, 49772}, {5221, 16610}, {5269, 5293}, {5271, 41249}, {5292, 21616}, {5398, 45770}, {5438, 5529}, {5705, 17717}, {5887, 36754}, {5903, 44545}, {5905, 23536}, {6001, 36745}, {6765, 37588}, {6996, 12544}, {7262, 31424}, {7308, 45897}, {7713, 10974}, {7957, 12652}, {8227, 33140}, {8580, 8951}, {8583, 23151}, {9369, 20037}, {9441, 12565}, {9534, 50314}, {9612, 33096}, {9614, 33141}, {10899, 15932}, {11036, 16020}, {11269, 41012}, {11375, 35466}, {11520, 28082}, {11682, 49487}, {12520, 13329}, {12559, 30117}, {12709, 52424}, {13425, 49592}, {13458, 49593}, {15556, 34036}, {16778, 19762}, {16824, 17349}, {17127, 34772}, {17350, 20036}, {17596, 54290}, {17781, 48818}, {19766, 50290}, {19860, 25885}, {22836, 37817}, {24391, 36574}, {24914, 37663}, {24954, 37634}, {25079, 30567}, {25681, 37646}, {26066, 37662}, {27003, 27625}, {27131, 54355}, {27538, 41261}, {28629, 37650}, {30323, 49494}, {31165, 37614}, {34040, 41539}, {35672, 41422}, {37550, 37694}, {39585, 41234}, {39954, 51223}, {46190, 54352}, {50576, 50617}

X(54386) = reflection of X(1) in X(1191)
X(54386) = X(i)-Ceva conjugate of X(j) for these (i,j): {959, 1}, {2285, 17594}
X(54386) = X(514)-isoconjugate of X(28477)
X(54386) = barycentric product X(i)*X(j) for these {i,j}: {100, 28478}, {941, 39773}
X(54386) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 28477}, {28478, 693}, {39773, 34284}
X(54386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1203, 16475}, {1, 1743, 5247}, {1, 3973, 5234}, {1, 5904, 16496}, {1, 16468, 1453}, {1, 16469, 16478}, {6, 960, 1}, {31, 78, 37552}, {57, 978, 11512}, {63, 1193, 988}, {65, 4383, 1722}, {72, 16466, 1}, {386, 12514, 17594}, {595, 3811, 3749}, {978, 1046, 57}, {1104, 12635, 1}, {1203, 5692, 1}, {1616, 34791, 1}, {1695, 2944, 40}, {1714, 12047, 17064}, {2999, 12526, 986}, {3216, 49500, 46}, {3339, 23511, 24174}, {3555, 16483, 1}, {5044, 5711, 5268}, {5315, 5904, 1}, {5529, 37603, 5438}, {7290, 11523, 1}, {12688, 37537, 1721}, {21214, 32913, 3333}

X(54387) = X(1)X(3)∩X(21)X(44)

Barycentrics    a*(4*a^3 - 3*a^2*b - 6*a*b^2 + b^3 - 3*a^2*c - 6*a*b*c - 3*b^2*c - 6*a*c^2 - 3*b*c^2 + c^3) : :
X(54387) = (3*r^2 + 6*r*R + s^2)*X[1] + 6*r^2*X[3]

X(54387) lies on these lines: {1, 3}, {21, 44}, {45, 78}, {89, 17548}, {551, 40688}, {678, 10459}, {1104, 17012}, {1125, 51671}, {1193, 3246}, {3616, 17382}, {3617, 33113}, {4005, 51297}, {4256, 16610}, {4304, 5718}, {4641, 16370}, {4652, 54281}, {4653, 5440}, {4670, 16393}, {4870, 33095}, {5165, 16666}, {5260, 54309}, {5530, 10543}, {5703, 50065}, {13411, 37691}, {17677, 30823}, {19862, 48843}, {27751, 52246}, {30115, 33595}, {32774, 46934}, {37716, 52638}, {39595, 49739}

X(54387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3, 37520}, {1, 17601, 65}, {1, 37574, 17601}, {1, 37599, 3999}, {2646, 37573, 37548}, {3601, 19765, 37539}, {4653, 5440, 44307}

X(54388) = X(3)X(6)∩X(10)X(98)

Barycentrics    a^2*(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(54388) lies on these lines: {1, 20594}, {3, 6}, {5, 53423}, {10, 98}, {41, 43}, {74, 30554}, {83, 2051}, {111, 28564}, {165, 40749}, {181, 10799}, {205, 4239}, {213, 893}, {501, 5213}, {604, 37608}, {699, 28469}, {727, 6010}, {729, 1293}, {733, 28486}, {1078, 14829}, {1174, 53005}, {1213, 50418}, {1385, 17448}, {1682, 12835}, {1695, 10789}, {1764, 25059}, {1790, 19308}, {1973, 4231}, {2174, 21857}, {2175, 23863}, {2238, 9840}, {2268, 37574}, {2304, 2319}, {2317, 22066}, {2323, 22065}, {2328, 16372}, {2330, 18758}, {2360, 19329}, {2388, 15621}, {2698, 2702}, {2705, 5970}, {3029, 12176}, {3031, 12192}, {3032, 12199}, {3203, 9562}, {3509, 22061}, {3651, 47641}, {3687, 20769}, {4027, 34454}, {4653, 15981}, {5182, 21937}, {5277, 19522}, {6685, 7413}, {7787, 9535}, {7793, 37683}, {8715, 32468}, {9310, 37675}, {9553, 10797}, {9554, 10798}, {9956, 25629}, {9959, 21879}, {10359, 36697}, {10791, 50037}, {10796, 49129}, {12195, 44039}, {12203, 13727}, {13193, 34453}, {13194, 34458}, {14880, 36477}, {16589, 48894}, {18904, 31394}, {19514, 24512}, {20777, 26890}, {21753, 50423}, {22267, 39141}, {22366, 41255}, {22449, 26889}, {24309, 32462}, {26243, 52134}, {28246, 33138}, {28841, 53900}

X(54388) = crossdifference of every pair of points on line {523, 20508}
X(54388) = barycentric product X(1)*X(11688)
X(54388) = barycentric quotient X(11688)/X(75)
X(54388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {371, 372, 5145}, {970, 3398, 4279}, {1342, 1343, 572}, {1683, 1684, 573}, {1687, 1688, 58}, {1691, 2092, 4279}, {36759, 36760, 34476}

X(54389) = X(2)X(45)∩X(4)X(9)

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

X(54389) lies on these lines: {1, 2325}, {2, 45}, {4, 9}, {6, 145}, {7, 3834}, {8, 44}, {31, 3974}, {37, 2275}, {69, 17230}, {75, 26685}, {101, 53904}, {141, 144}, {142, 7222}, {192, 3618}, {193, 17233}, {198, 36510}, {220, 5782}, {239, 50107}, {312, 26065}, {320, 17342}, {329, 32777}, {344, 894}, {345, 27064}, {374, 5836}, {390, 49524}, {391, 594}, {497, 16561}, {519, 4873}, {524, 17269}, {527, 17284}, {536, 5222}, {551, 36911}, {597, 17014}, {672, 30942}, {996, 1000}, {997, 36916}, {1018, 4266}, {1100, 20057}, {1125, 16676}, {1213, 46932}, {1215, 25375}, {1219, 1616}, {1265, 4195}, {1266, 3729}, {1376, 1633}, {1441, 28966}, {1449, 3635}, {1575, 36222}, {1731, 5082}, {1732, 24477}, {1743, 2321}, {1897, 40138}, {1978, 44152}, {1992, 6542}, {1999, 42032}, {2161, 3434}, {2173, 24850}, {2178, 38869}, {2182, 30618}, {2245, 27040}, {2265, 24247}, {2267, 2329}, {2295, 27523}, {2999, 42049}, {3008, 4659}, {3052, 7172}, {3061, 21801}, {3210, 46907}, {3217, 54316}, {3241, 4908}, {3247, 3636}, {3285, 17539}, {3589, 3672}, {3617, 17330}, {3619, 6646}, {3620, 17285}, {3621, 4969}, {3622, 16672}, {3623, 50113}, {3624, 3731}, {3629, 17309}, {3661, 54280}, {3679, 3707}, {3685, 36404}, {3686, 3973}, {3710, 5716}, {3739, 7229}, {3751, 49763}, {3758, 17264}, {3763, 17334}, {3770, 29542}, {3790, 51192}, {3871, 37503}, {3912, 4644}, {3932, 4307}, {3945, 17243}, {3948, 41316}, {3986, 19878}, {4007, 4701}, {4011, 26105}, {4034, 4058}, {4069, 41276}, {4072, 4856}, {4081, 28124}, {4082, 5269}, {4188, 19297}, {4357, 25728}, {4360, 51171}, {4361, 4461}, {4371, 4431}, {4402, 4686}, {4416, 17286}, {4418, 26040}, {4429, 24280}, {4432, 36479}, {4439, 50300}, {4452, 17366}, {4480, 17274}, {4488, 17276}, {4512, 53663}, {4641, 34255}, {4643, 6172}, {4664, 26626}, {4667, 29573}, {4670, 5308}, {4671, 24597}, {4675, 29627}, {4687, 49514}, {4693, 50282}, {4702, 47359}, {4727, 20050}, {4740, 29590}, {4741, 21356}, {4747, 17392}, {4748, 17308}, {4754, 27253}, {4755, 4798}, {4869, 17267}, {4871, 17754}, {4896, 41141}, {4942, 17061}, {4982, 51093}, {5218, 14439}, {5225, 36568}, {5232, 17293}, {5263, 27549}, {5273, 44417}, {5296, 16814}, {5325, 18229}, {5712, 17776}, {5744, 30818}, {5772, 52653}, {5905, 33157}, {6541, 50284}, {6666, 25590}, {6687, 52709}, {7046, 8750}, {7227, 17259}, {7228, 17265}, {7232, 20059}, {7277, 17311}, {7288, 25591}, {7321, 17341}, {7397, 29069}, {7735, 37764}, {8557, 26015}, {8609, 26690}, {9780, 24723}, {9791, 26083}, {10436, 25101}, {10589, 33119}, {11008, 17373}, {14039, 30108}, {14543, 24335}, {14953, 30906}, {15254, 39581}, {15492, 17275}, {16020, 49483}, {16086, 48817}, {16669, 17299}, {16675, 17398}, {16720, 27472}, {16989, 33889}, {17054, 37024}, {17116, 17338}, {17118, 17337}, {17119, 24599}, {17120, 17242}, {17132, 31191}, {17229, 32099}, {17231, 21296}, {17232, 31300}, {17246, 47355}, {17250, 17257}, {17254, 29613}, {17255, 34573}, {17258, 17371}, {17261, 17321}, {17266, 50128}, {17268, 17364}, {17277, 20181}, {17278, 31995}, {17292, 17333}, {17295, 20080}, {17302, 25269}, {17323, 51126}, {17348, 32087}, {17349, 42696}, {17362, 20052}, {17367, 50101}, {17377, 51170}, {17378, 29583}, {17395, 47352}, {17484, 30991}, {17756, 53340}, {17790, 28809}, {19822, 27065}, {19867, 51665}, {19998, 37657}, {20078, 33172}, {20106, 28609}, {20471, 38871}, {20927, 26665}, {21076, 21712}, {21689, 27708}, {24248, 33159}, {24331, 38025}, {24411, 40869}, {24485, 36801}, {24695, 29674}, {24817, 36473}, {24821, 29660}, {24864, 53582}, {25243, 26668}, {25734, 54311}, {26098, 33164}, {26244, 42316}, {26799, 27136}, {27013, 53376}, {27058, 27514}, {27334, 28778}, {27382, 46830}, {27508, 34524}, {28739, 41804}, {29585, 46922}, {29679, 44447}, {29713, 34283}, {29834, 32925}, {29860, 33144}, {31018, 32779}, {32034, 33198}, {32847, 50303}, {32938, 33171}, {32941, 49701}, {33068, 44446}, {37640, 37795}, {37641, 37794}, {40859, 48869}, {40940, 42047}, {41241, 50105}, {45789, 48632}, {48864, 52963}, {49458, 49713}, {49462, 49531}, {49491, 51058}, {49688, 49699}, {49712, 50316}, {49756, 50289}, {49768, 51099}, {49770, 50089}, {49772, 50126}, {50019, 50100}

X(54389) = midpoint of X(4873) and X(16670)
X(54389) = reflection of X(i) in X(j) for these {i,j}: {4346, 17290}, {29616, 17269}
X(54389) = complement of X(4346)
X(54389) = anticomplement of X(17290)
X(54389) = X(i)-isoconjugate of X(j) for these (i,j): {57, 3478}, {905, 9088}
X(54389) = X(i)-Dao conjugate of X(j) for these (i,j): {5452, 3478}, {53838, 514}
X(54389) = trilinear pole of line {47766, 48327}
X(54389) = crossdifference of every pair of points on line {1459, 1960}
X(54389) = barycentric product X(i)*X(j) for these {i,j}: {1, 4737}, {8, 3476}, {10, 4234}, {190, 47766}, {668, 48327}, {1897, 9031}, {3952, 47845}
X(54389) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 3478}, {3476, 7}, {4234, 86}, {4737, 75}, {8750, 9088}, {9031, 4025}, {47766, 514}, {47845, 7192}, {48327, 513}
X(54389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 190, 4419}, {2, 4346, 17290}, {2, 4389, 26104}, {2, 4454, 1086}, {2, 17369, 26039}, {2, 20073, 4389}, {6, 346, 17314}, {6, 3943, 145}, {6, 17340, 346}, {7, 17279, 53665}, {8, 44, 37654}, {9, 2345, 966}, {9, 3501, 2183}, {9, 17355, 2345}, {44, 17281, 8}, {45, 17369, 2}, {75, 26685, 37650}, {145, 346, 3943}, {145, 3943, 17314}, {190, 4389, 20073}, {190, 17305, 49748}, {190, 17354, 2}, {312, 26065, 37642}, {320, 17342, 29579}, {344, 894, 4648}, {594, 16885, 391}, {597, 17318, 17014}, {894, 17339, 344}, {1086, 49721, 4454}, {1743, 2321, 5839}, {1743, 3632, 4700}, {2321, 4700, 3632}, {2325, 50115, 1}, {3008, 50118, 4659}, {3161, 5749, 37}, {3589, 17262, 3672}, {3632, 4700, 5839}, {3729, 17353, 4000}, {3758, 17264, 17316}, {3912, 50127, 4644}, {4363, 4422, 2}, {4370, 17369, 45}, {4389, 20073, 4419}, {4419, 26104, 4389}, {4422, 49726, 4363}, {4432, 36479, 47357}, {4432, 50313, 36479}, {4461, 37681, 4361}, {4480, 29596, 17274}, {4643, 17359, 29611}, {4670, 41313, 5308}, {4675, 41310, 29627}, {4727, 50131, 20050}, {4741, 29587, 21356}, {4747, 29621, 17392}, {4969, 50087, 3621}, {5296, 19877, 52706}, {6172, 29611, 4643}, {6646, 17358, 3619}, {7229, 18230, 3739}, {16814, 17303, 5296}, {17230, 17350, 20072}, {17230, 20072, 69}, {17261, 17368, 17321}, {17267, 17365, 4869}, {17279, 17351, 7}, {17280, 17350, 69}, {17280, 20072, 17230}, {17285, 17347, 3620}, {17289, 17336, 17257}, {17293, 17332, 5232}, {17303, 52706, 19877}, {17308, 50093, 4748}, {17776, 26223, 5712}, {29627, 35578, 4675}, {32930, 33163, 497}

X(54390) = X(1)X(2)∩X(44)X(57)

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

X(54390) lies on these lines: {1, 2}, {6, 31197}, {9, 16610}, {38, 30393}, {44, 57}, {45, 3752}, {63, 88}, {77, 31188}, {165, 748}, {210, 5573}, {223, 31231}, {244, 5223}, {269, 3911}, {329, 4887}, {678, 9350}, {750, 16469}, {908, 4859}, {988, 21496}, {1086, 31142}, {1155, 15601}, {1279, 46917}, {1376, 3246}, {1453, 16408}, {1743, 3306}, {2297, 17352}, {2975, 45047}, {3218, 3973}, {3452, 17067}, {3579, 19517}, {3666, 16676}, {3677, 3740}, {3689, 35227}, {3729, 24620}, {3731, 4850}, {3772, 20196}, {3868, 8951}, {3875, 30829}, {3886, 25531}, {4000, 5316}, {4328, 5219}, {4346, 18228}, {4358, 17151}, {4383, 5437}, {4413, 7290}, {4512, 17123}, {4695, 9819}, {4849, 44841}, {4862, 31018}, {4896, 9776}, {4900, 17460}, {4902, 17484}, {5055, 18506}, {5204, 37269}, {5233, 17282}, {5241, 17306}, {5400, 5732}, {5718, 20195}, {8167, 37553}, {8616, 9324}, {10388, 52429}, {10434, 27639}, {10980, 54352}, {12526, 24174}, {16421, 37609}, {16435, 35242}, {16666, 37674}, {16667, 37633}, {16736, 18198}, {16753, 18186}, {16862, 37554}, {17063, 49712}, {17117, 30861}, {17160, 18743}, {17277, 31233}, {17349, 31228}, {17490, 30568}, {17721, 38200}, {21363, 28280}, {21514, 37599}, {21519, 37552}, {21526, 37589}, {21896, 37556}, {24589, 25590}, {24594, 41241}, {24789, 30827}, {25525, 37663}, {25734, 30579}, {26724, 30852}, {28609, 40688}, {31190, 35466}, {36636, 43044}, {37662, 41867}

X(54390) = barycentric product X(75)*X(16486)
X(54390) = barycentric quotient X(16486)/X(1)
X(54390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2999, 17022}, {2, 17020, 5287}, {2, 23511, 2999}, {3306, 37680, 1743}, {4383, 37520, 16670}, {4384, 16831, 16829}, {5256, 17021, 1}, {5272, 16569, 200}, {5287, 17013, 1}, {5437, 16670, 37520}, {16602, 37679, 57}, {17278, 51415, 5219}

X(54391) = X(1)X(21)∩X(8)X(56)

Barycentrics    a*(a^3 - a*b^2 + 3*a*b*c - b^2*c - a*c^2 - b*c^2) : :
X(54391) = 3 X[21] - 2 X[48698], 3 X[2] - 4 X[15325], 4 X[11] - 3 X[37375], 2 X[5080] - 3 X[37375], 4 X[36] - 3 X[13587], 3 X[36] - X[48696], 2 X[100] - 3 X[13587], 3 X[100] - 2 X[48696], 9 X[13587] - 4 X[48696], X[1320] + 2 X[3218], X[36002] + 2 X[38669], 4 X[15326] - 3 X[36005], 2 X[3583] - 3 X[10707], and many others

X(54391) lies on these lines: {1, 21}, {2, 495}, {3, 145}, {4, 10529}, {5, 20060}, {8, 56}, {10, 5253}, {11, 529}, {12, 7504}, {20, 22770}, {22, 19993}, {28, 35058}, {30, 149}, {35, 3244}, {36, 100}, {40, 3885}, {42, 37617}, {44, 5548}, {46, 14923}, {55, 3241}, {57, 3872}, {59, 518}, {65, 4861}, {72, 17624}, {75, 1014}, {78, 1420}, {88, 1739}, {92, 17519}, {101, 45751}, {104, 517}, {105, 666}, {106, 24625}, {108, 5081}, {144, 42884}, {153, 1532}, {165, 3895}, {171, 54310}, {172, 17448}, {190, 15571}, {200, 13462}, {238, 1149}, {239, 11349}, {346, 5120}, {355, 6915}, {376, 20075}, {377, 3600}, {385, 9263}, {388, 2476}, {392, 3219}, {405, 3622}, {411, 944}, {474, 3617}, {484, 2802}, {496, 5046}, {497, 11114}, {499, 11681}, {515, 13279}, {523, 1325}, {527, 51423}, {528, 15326}, {535, 3583}, {550, 20066}, {551, 5251}, {631, 10528}, {644, 672}, {651, 1457}, {759, 39697}, {859, 16704}, {908, 44675}, {912, 12776}, {932, 2382}, {934, 2751}, {952, 6905}, {958, 3304}, {960, 20323}, {961, 4968}, {962, 12114}, {978, 32577}, {982, 49487}, {995, 32911}, {997, 3681}, {1001, 6172}, {1005, 3488}, {1006, 4430}, {1010, 27163}, {1012, 9965}, {1015, 5291}, {1018, 5030}, {1042, 9363}, {1043, 29766}, {1054, 4695}, {1055, 3684}, {1058, 6872}, {1100, 38871}, {1108, 5279}, {1125, 5258}, {1155, 3880}, {1201, 5247}, {1259, 6049}, {1260, 37313}, {1280, 2224}, {1317, 4996}, {1318, 3257}, {1385, 3555}, {1387, 17484}, {1388, 12635}, {1398, 4200}, {1444, 4360}, {1447, 30806}, {1455, 4318}, {1475, 2329}, {1478, 11680}, {1482, 6906}, {1573, 37675}, {1617, 37300}, {1633, 49709}, {1697, 4652}, {1727, 12758}, {1737, 5176}, {1743, 38869}, {1757, 47623}, {1770, 49600}, {1778, 16685}, {1791, 5262}, {1817, 3187}, {1897, 37305}, {1999, 37620}, {2077, 13278}, {2170, 3509}, {2178, 5839}, {2217, 34860}, {2218, 39702}, {2238, 9259}, {2242, 5276}, {2251, 50028}, {2320, 2346}, {2352, 49687}, {2475, 18990}, {2478, 14986}, {2646, 34791}, {2886, 5434}, {3035, 5298}, {3058, 15678}, {3086, 3436}, {3243, 13384}, {3245, 41702}, {3294, 9327}, {3295, 3623}, {3303, 17574}, {3306, 9623}, {3315, 4694}, {3333, 19860}, {3337, 3754}, {3361, 4853}, {3419, 35990}, {3428, 5731}, {3434, 4293}, {3478, 9309}, {3485, 18967}, {3486, 10966}, {3552, 54098}, {3560, 10595}, {3576, 3870}, {3582, 3814}, {3585, 24387}, {3621, 4188}, {3624, 17546}, {3632, 25440}, {3633, 7280}, {3635, 3746}, {3636, 5259}, {3651, 34773}, {3666, 17015}, {3670, 15955}, {3678, 51714}, {3679, 36006}, {3680, 5128}, {3683, 10179}, {3685, 4742}, {3701, 9369}, {3753, 27003}, {3780, 21008}, {3811, 37618}, {3813, 7354}, {3816, 34606}, {3876, 19861}, {3902, 32932}, {3911, 5193}, {3913, 5204}, {3916, 9957}, {3924, 3976}, {3928, 7962}, {3935, 4881}, {3940, 4661}, {3951, 15829}, {3957, 24929}, {4018, 10222}, {4084, 11009}, {4190, 5082}, {4191, 20012}, {4197, 19843}, {4203, 10453}, {4210, 20011}, {4216, 20037}, {4220, 29840}, {4221, 17147}, {4225, 20040}, {4245, 19742}, {4252, 37542}, {4257, 37610}, {4265, 51147}, {4297, 11015}, {4302, 34611}, {4315, 4847}, {4321, 10861}, {4342, 34646}, {4345, 28610}, {4359, 16821}, {4390, 17754}, {4393, 21511}, {4413, 40726}, {4428, 8162}, {4434, 9457}, {4513, 5022}, {4585, 34230}, {4586, 35167}, {4640, 5919}, {4678, 9709}, {4720, 13588}, {4723, 5205}, {4855, 6765}, {4867, 51506}, {4880, 10058}, {4969, 19297}, {4975, 47626}, {4999, 15888}, {5010, 25439}, {5044, 15179}, {5048, 44663}, {5049, 29817}, {5051, 5484}, {5057, 30384}, {5084, 10586}, {5086, 10916}, {5096, 9053}, {5124, 17388}, {5131, 5541}, {5141, 9654}, {5178, 17647}, {5180, 17768}, {5187, 47743}, {5211, 33849}, {5218, 11239}, {5223, 53058}, {5261, 6933}, {5263, 51669}, {5264, 50637}, {5265, 6921}, {5270, 25639}, {5372, 5774}, {5432, 34749}, {5433, 12607}, {5450, 7982}, {5525, 24036}, {5529, 21805}, {5535, 6264}, {5550, 17534}, {5552, 7288}, {5584, 9797}, {5603, 5905}, {5657, 10269}, {5690, 6940}, {5698, 42886}, {5725, 29680}, {5727, 34716}, {5734, 11496}, {5790, 6946}, {5836, 32636}, {5841, 37726}, {5854, 17100}, {5882, 11012}, {5884, 11014}, {5886, 31053}, {5901, 6920}, {5903, 22837}, {5904, 30144}, {6001, 13243}, {6079, 8686}, {6224, 22560}, {6360, 36029}, {6542, 21495}, {6550, 13266}, {6645, 17686}, {6647, 9317}, {6690, 31157}, {6691, 21031}, {6734, 10106}, {6767, 16370}, {6824, 10597}, {6825, 10530}, {6828, 10532}, {6857, 10587}, {6868, 10806}, {6875, 16202}, {6876, 35252}, {6914, 10247}, {6924, 12645}, {6930, 10596}, {6932, 12115}, {6943, 10785}, {6949, 10942}, {6950, 10679}, {7176, 20880}, {7269, 54344}, {7270, 19850}, {7437, 47043}, {7447, 14260}, {7451, 36944}, {7465, 29832}, {7485, 20020}, {7489, 10283}, {7491, 32214}, {7548, 26470}, {7580, 30283}, {7688, 51705}, {7718, 22479}, {7762, 20102}, {7951, 34690}, {8025, 19259}, {8158, 20070}, {8192, 11337}, {8543, 42842}, {8583, 30393}, {8692, 41436}, {9037, 10755}, {9310, 21384}, {9312, 38859}, {9318, 35102}, {9352, 54286}, {9668, 34740}, {9710, 26060}, {9780, 17535}, {9960, 12687}, {9961, 10085}, {9963, 21578}, {10090, 12531}, {10449, 35999}, {10459, 37607}, {10538, 14198}, {10572, 49627}, {10609, 36003}, {10698, 14988}, {10902, 13607}, {10912, 37567}, {10914, 37582}, {11036, 37228}, {11108, 46934}, {11112, 33110}, {11115, 26819}, {11235, 12943}, {11343, 17014}, {11350, 20043}, {11362, 37561}, {11415, 22760}, {11491, 26286}, {11604, 33961}, {11813, 16173}, {12005, 20612}, {12029, 53625}, {12127, 16192}, {12527, 41012}, {12532, 12740}, {12543, 14450}, {12629, 15803}, {12690, 28186}, {12702, 37403}, {12746, 32844}, {13589, 20042}, {13738, 20036}, {13996, 32426}, {15170, 15677}, {15171, 15680}, {15185, 30284}, {15287, 37681}, {15383, 47622}, {15507, 20072}, {15733, 18450}, {15808, 25542}, {15934, 37306}, {16049, 20222}, {16061, 26759}, {16212, 22755}, {16367, 29585}, {16371, 31145}, {16408, 46933}, {16451, 20018}, {16465, 18444}, {16483, 17127}, {16499, 30116}, {16693, 53268}, {16862, 46932}, {16863, 46931}, {16864, 46930}, {17016, 37592}, {17024, 37325}, {17074, 24806}, {17152, 17206}, {17230, 21540}, {17314, 36743}, {17349, 19291}, {17362, 21773}, {17364, 31394}, {17474, 41239}, {17480, 37231}, {17483, 39542}, {17547, 25055}, {17551, 19858}, {17614, 34790}, {17681, 26964}, {17683, 27304}, {17734, 24222}, {17742, 26690}, {17798, 50015}, {18042, 41610}, {18047, 37686}, {18391, 22767}, {18398, 30147}, {18481, 33557}, {18491, 34627}, {19065, 44607}, {19066, 44606}, {19245, 37652}, {19258, 31017}, {19260, 37685}, {19308, 20016}, {19314, 39587}, {19537, 20054}, {19704, 51092}, {19789, 37241}, {20007, 37282}, {20013, 37301}, {20014, 37307}, {20015, 37309}, {20017, 37312}, {20045, 37449}, {20085, 28224}, {20095, 36004}, {20101, 37331}, {20347, 24203}, {20999, 37311}, {21010, 36534}, {21161, 50824}, {21222, 53286}, {21477, 29616}, {21620, 24541}, {21669, 22791}, {21842, 22836}, {22769, 51192}, {23361, 23391}, {23858, 30577}, {23958, 36279}, {23961, 34474}, {24391, 37583}, {24473, 50194}, {24558, 25875}, {24599, 37272}, {24602, 49774}, {24841, 53302}, {24914, 32049}, {24927, 31837}, {25416, 35000}, {25466, 31254}, {26088, 26202}, {26140, 51384}, {26877, 37562}, {26878, 31838}, {27086, 41345}, {27368, 35991}, {28174, 51529}, {28190, 38631}, {28236, 44425}, {28376, 42461}, {28377, 32843}, {28797, 41245}, {28813, 43053}, {29662, 37716}, {29824, 35992}, {30143, 50190}, {30305, 44447}, {30392, 52769}, {31159, 34637}, {31393, 35258}, {33129, 50759}, {33812, 35204}, {33950, 40133}, {33956, 36920}, {34631, 44455}, {34880, 41687}, {35238, 50810}, {35984, 48847}, {36001, 46636}, {36090, 36100}, {37251, 37705}, {37254, 39567}, {37564, 37734}, {37919, 54081}, {38570, 47274}, {38901, 40293}, {40910, 49771}, {41555, 45043}, {49712, 52923}, {50608, 54331}, {51816, 54318}

X(54391) = midpoint of X(i) and X(j) for these {i,j}: {149, 20067}, {3218, 38460}, {3245, 41702}, {5535, 6264}
X(54391) = reflection of X(i) in X(j) for these {i,j}: {8, 40663}, {100, 36}, {153, 1532}, {484, 4973}, {908, 44675}, {1320, 38460}, {3935, 5440}, {4511, 1319}, {5057, 30384}, {5080, 11}, {5176, 1737}, {5440, 5126}, {6163, 238}, {6735, 3911}, {6905, 22765}, {6909, 104}, {11684, 1749}, {12531, 41684}, {17484, 51409}, {17757, 15325}, {35000, 38602}, {36001, 46636}, {45043, 41555}, {51409, 1387}
X(54391) = anticomplement of X(17757)
X(54391) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 153}, {104, 1330}, {909, 2895}, {1412, 36918}, {1795, 52364}, {2401, 21294}, {2423, 21221}, {14578, 3151}, {34051, 2893}, {34234, 21287}, {34858, 1654}
X(54391) = X(9268)-Ceva conjugate of X(100)
X(54391) = X(i)-isoconjugate of X(j) for these (i,j): {6, 14554}, {649, 50039}
X(54391) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 14554}, {1769, 35015}, {5375, 50039}, {34590, 523}
X(54391) = cevapoint of X(517) and X(49997)
X(54391) = trilinear pole of line {21786, 21894}
X(54391) = crossdifference of every pair of points on line {661, 2092}
X(54391) = barycentric product X(i)*X(j) for these {i,j}: {75, 5053}, {99, 21894}, {100, 21222}, {668, 21786}, {4554, 53286}, {5376, 34590}, {6335, 23087}
X(54391) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14554}, {100, 50039}, {5053, 1}, {21222, 693}, {21786, 513}, {21894, 523}, {23087, 905}, {53286, 650}
X(54391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 63, 3877}, {1, 191, 3884}, {1, 993, 1621}, {1, 2975, 21}, {1, 3869, 5330}, {1, 3874, 34195}, {1, 5429, 17469}, {1, 6763, 3878}, {1, 8666, 2975}, {1, 12514, 3890}, {1, 51111, 51683}, {1, 52680, 40091}, {3, 145, 3871}, {8, 56, 404}, {10, 5253, 17531}, {10, 5563, 5253}, {11, 5080, 37375}, {36, 100, 13587}, {40, 36846, 3885}, {56, 12513, 8}, {65, 11260, 4861}, {200, 13462, 35262}, {388, 10527, 2476}, {405, 7373, 3622}, {551, 5251, 5284}, {944, 11249, 411}, {956, 999, 2}, {958, 3304, 3616}, {958, 3616, 5047}, {982, 49487, 54315}, {993, 1621, 21}, {1015, 5291, 33854}, {1125, 5258, 5260}, {1125, 5260, 17536}, {1385, 3555, 34772}, {1420, 6762, 78}, {1478, 11680, 17577}, {1478, 45700, 11680}, {1482, 32153, 6906}, {1621, 2975, 993}, {2242, 16975, 5276}, {3086, 3436, 4193}, {3241, 11194, 17549}, {3428, 5731, 7411}, {3434, 4293, 17579}, {3476, 24477, 8}, {3560, 12001, 10595}, {3582, 3814, 31272}, {3621, 4188, 5687}, {3623, 4189, 3295}, {3633, 7280, 8715}, {3635, 5267, 3746}, {3685, 38475, 4742}, {3813, 7354, 52367}, {3878, 6763, 11684}, {3881, 51111, 1}, {3889, 3897, 1}, {3935, 4881, 5440}, {4293, 34625, 3434}, {4678, 17572, 9709}, {4694, 30117, 3315}, {5010, 51093, 25439}, {5126, 5440, 4881}, {5251, 5284, 16861}, {5251, 37602, 551}, {5265, 7080, 6921}, {5288, 5563, 10}, {5433, 12607, 27529}, {5552, 7288, 17566}, {5603, 22758, 6912}, {5690, 37535, 6940}, {6645, 26801, 17686}, {8158, 37022, 20070}, {9780, 25524, 17535}, {10529, 20076, 4}, {10916, 45287, 5086}, {11240, 34610, 11114}, {11680, 34605, 1478}, {14511, 36037, 1320}, {15325, 17757, 2}, {18967, 22759, 3485}, {18990, 24390, 2475}, {22791, 26321, 21669}, {26286, 37727, 11491}, {34605, 45700, 17577}

X(54392) = X(1)X(2)∩X(21)X(57)

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

X(54392) lies on these lines: {1, 2}, {3, 3306}, {4, 5249}, {5, 18446}, {7, 452}, {9, 3868}, {11, 28628}, {20, 8726}, {21, 57}, {29, 34}, {33, 5125}, {35, 37301}, {36, 20846}, {37, 37549}, {40, 1621}, {46, 5248}, {55, 3812}, {56, 3742}, {63, 405}, {65, 1001}, {72, 3305}, {73, 19372}, {81, 1453}, {84, 6912}, {140, 31224}, {142, 377}, {169, 16783}, {224, 442}, {226, 2478}, {244, 988}, {269, 32086}, {283, 1451}, {307, 6604}, {329, 5129}, {344, 3710}, {354, 958}, {388, 21617}, {392, 11682}, {404, 3601}, {411, 3576}, {443, 3488}, {474, 4855}, {495, 50206}, {497, 28629}, {515, 6835}, {553, 31156}, {631, 37531}, {728, 3247}, {748, 2650}, {750, 37552}, {894, 17697}, {908, 3487}, {940, 1104}, {944, 6864}, {946, 6836}, {956, 5045}, {960, 4423}, {962, 30503}, {964, 10436}, {965, 1100}, {968, 986}, {990, 17304}, {993, 3338}, {999, 16293}, {1006, 5709}, {1010, 19788}, {1012, 9940}, {1038, 27407}, {1040, 2654}, {1043, 19804}, {1054, 37574}, {1060, 7515}, {1062, 18641}, {1071, 6913}, {1086, 50065}, {1158, 15016}, {1220, 2191}, {1229, 4968}, {1257, 25430}, {1259, 37244}, {1266, 15956}, {1279, 5710}, {1329, 17718}, {1376, 13867}, {1385, 3149}, {1392, 45830}, {1393, 54320}, {1394, 17074}, {1420, 3897}, {1442, 31994}, {1446, 4350}, {1448, 17194}, {1449, 2287}, {1478, 51706}, {1479, 12609}, {1482, 31838}, {1490, 3091}, {1656, 37700}, {1697, 38316}, {1699, 6895}, {1706, 3871}, {1723, 25081}, {1728, 18389}, {1750, 3832}, {1790, 54340}, {1834, 24789}, {1837, 25466}, {1870, 7498}, {1883, 25365}, {1891, 26130}, {1895, 11109}, {2082, 16503}, {2098, 10179}, {2263, 41246}, {2280, 21921}, {2320, 30389}, {2339, 40959}, {2475, 3586}, {2476, 9581}, {2551, 3475}, {2646, 25524}, {2647, 4320}, {2975, 3333}, {3090, 5720}, {3146, 5732}, {3174, 40333}, {3189, 26040}, {3218, 16865}, {3219, 16859}, {3295, 3753}, {3303, 3895}, {3339, 4512}, {3340, 3877}, {3419, 8728}, {3428, 13374}, {3436, 21620}, {3485, 26105}, {3522, 10857}, {3523, 6282}, {3555, 9708}, {3560, 10202}, {3612, 35016}, {3615, 43682}, {3646, 34195}, {3648, 5586}, {3649, 24703}, {3662, 26117}, {3666, 17054}, {3671, 11415}, {3680, 51779}, {3681, 41863}, {3691, 51194}, {3698, 3748}, {3746, 54286}, {3750, 24440}, {3752, 19765}, {3754, 5119}, {3758, 32024}, {3816, 11281}, {3817, 6870}, {3822, 10826}, {3824, 17532}, {3825, 37692}, {3833, 25440}, {3834, 50050}, {3838, 10896}, {3869, 5284}, {3873, 5260}, {3874, 41229}, {3876, 7308}, {3884, 25415}, {3885, 37556}, {3889, 6762}, {3890, 7982}, {3898, 30323}, {3911, 6910}, {3916, 5708}, {3918, 25439}, {3927, 16857}, {3928, 16858}, {3929, 16861}, {3940, 16853}, {3984, 5044}, {4002, 4917}, {4004, 12702}, {4101, 14555}, {4187, 11374}, {4188, 30282}, {4189, 15803}, {4190, 4304}, {4193, 5219}, {4197, 37723}, {4198, 18650}, {4202, 17282}, {4204, 10381}, {4208, 5175}, {4217, 50116}, {4233, 7713}, {4252, 37520}, {4255, 16610}, {4292, 6872}, {4296, 27402}, {4297, 50695}, {4303, 7518}, {4313, 6904}, {4328, 32098}, {4357, 5738}, {4389, 32007}, {4428, 37568}, {4533, 51572}, {4640, 5221}, {4648, 5716}, {4653, 24046}, {4657, 18635}, {4658, 40571}, {4662, 41711}, {4675, 49745}, {4687, 32008}, {4863, 9710}, {4966, 10371}, {4999, 17728}, {5016, 18139}, {5046, 9612}, {5051, 25527}, {5080, 5290}, {5122, 19535}, {5177, 10382}, {5226, 6919}, {5251, 18398}, {5258, 50190}, {5259, 5902}, {5276, 16780}, {5281, 26062}, {5294, 13742}, {5314, 37547}, {5422, 54301}, {5426, 7280}, {5438, 17531}, {5440, 16408}, {5534, 5818}, {5542, 12527}, {5587, 6991}, {5603, 6865}, {5687, 50203}, {5691, 6894}, {5692, 12559}, {5715, 6840}, {5719, 17527}, {5728, 37224}, {5731, 50700}, {5736, 27401}, {5744, 17558}, {5750, 51972}, {5768, 6846}, {5787, 8226}, {5804, 6908}, {5806, 7580}, {5880, 6284}, {5886, 6831}, {5901, 6922}, {5905, 12572}, {5930, 37800}, {6147, 31164}, {6173, 8544}, {6245, 6837}, {6260, 6957}, {6261, 6828}, {6326, 31272}, {6505, 41930}, {6667, 12739}, {6675, 41574}, {6684, 37569}, {6690, 24914}, {6692, 6921}, {6705, 6974}, {6767, 10914}, {6832, 51755}, {6838, 7682}, {6855, 21740}, {6860, 40257}, {6883, 24474}, {6906, 37534}, {6909, 37526}, {6911, 24299}, {6914, 37612}, {6915, 51683}, {6918, 10246}, {6920, 7330}, {6943, 9624}, {6985, 13151}, {6988, 37611}, {7078, 10601}, {7171, 21669}, {7269, 32003}, {7270, 17234}, {7293, 13730}, {7489, 24467}, {7532, 37697}, {7590, 8125}, {7741, 26725}, {7991, 21153}, {8000, 12533}, {8082, 8126}, {8167, 12635}, {8666, 51816}, {9352, 35242}, {9575, 40129}, {9778, 12651}, {9812, 12565}, {9816, 18673}, {9961, 11372}, {10106, 51723}, {10129, 16132}, {10175, 10585}, {10177, 12711}, {10404, 25557}, {10430, 18219}, {10451, 10856}, {10589, 45230}, {10624, 51724}, {11018, 19520}, {11024, 17784}, {11031, 25906}, {11106, 21454}, {11115, 26627}, {11227, 37022}, {11230, 45770}, {11517, 50204}, {12109, 26893}, {12513, 17609}, {12560, 52653}, {12564, 42012}, {12625, 20195}, {12650, 54051}, {12699, 37428}, {12701, 49736}, {13323, 26884}, {13369, 37234}, {13373, 22758}, {13614, 17080}, {13725, 54311}, {14020, 17274}, {14923, 31393}, {15071, 54370}, {16048, 40131}, {16284, 17394}, {16368, 19753}, {16370, 37582}, {16484, 37598}, {16485, 37554}, {16749, 17175}, {16845, 54357}, {16968, 24512}, {17048, 25500}, {17272, 26064}, {17379, 27288}, {17534, 51780}, {17557, 46877}, {17566, 31190}, {17570, 27065}, {17571, 37545}, {17594, 24443}, {17619, 31479}, {18483, 50528}, {18634, 33178}, {19283, 35612}, {19309, 20769}, {19665, 51710}, {19684, 30807}, {20070, 43166}, {20171, 24325}, {20292, 41869}, {20330, 31799}, {20905, 23661}, {21031, 37703}, {21165, 37532}, {22128, 36742}, {22345, 28383}, {23542, 25017}, {24174, 37573}, {24391, 31259}, {24470, 50241}, {24590, 37064}, {25015, 32774}, {25992, 38047}, {26229, 36007}, {26729, 33151}, {27378, 34036}, {27384, 27410}, {27413, 30854}, {28381, 48894}, {31053, 37162}, {31880, 53034}, {34824, 49734}, {37189, 40950}, {37225, 39598}, {37231, 51687}, {37246, 37581}, {37522, 37817}, {37539, 37674}, {37605, 40726}, {38028, 52265}, {40836, 41081}, {40942, 41006}, {43531, 43675}

X(54392) = isotomic conjugate of the isogonal conjugate of X(54321)
X(54392) = X(i)-isoconjugate of X(j) for these (i,j): {55, 8814}, {607, 8813}
X(54392) = X(223)-Dao conjugate of X(8814)
X(54392) = cevapoint of X(13615) and X(54358)
X(54392) = barycentric product X(i)*X(j) for these {i,j}: {75, 54358}, {76, 54321}, {85, 13615}
X(54392) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 8814}, {77, 8813}, {13615, 9}, {54321, 6}, {54358, 1}
X(54392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2, 78}, {1, 10, 3870}, {1, 936, 34772}, {1, 1125, 19861}, {1, 1698, 3811}, {1, 1722, 42}, {1, 2999, 19767}, {1, 3624, 997}, {1, 4853, 3241}, {1, 5268, 976}, {1, 5272, 1193}, {1, 6765, 3957}, {1, 8583, 4511}, {1, 9623, 145}, {1, 10582, 3616}, {1, 12629, 3623}, {1, 19860, 3872}, {1, 29820, 28011}, {1, 54318, 19860}, {2, 938, 6734}, {2, 3622, 5703}, {2, 5703, 27385}, {2, 12649, 10}, {2, 34772, 936}, {3, 5439, 3306}, {4, 18443, 10884}, {5, 37615, 18446}, {9, 3868, 3951}, {9, 11518, 3868}, {21, 57, 4652}, {34, 37523, 77}, {46, 5248, 35258}, {57, 5436, 21}, {65, 1001, 5250}, {72, 11108, 3305}, {72, 15934, 11520}, {142, 950, 377}, {145, 29817, 1}, {244, 10448, 988}, {405, 942, 63}, {474, 24929, 4855}, {551, 9843, 13411}, {551, 30147, 1}, {936, 12127, 20007}, {936, 34772, 78}, {1125, 1210, 2}, {1125, 30143, 1}, {1706, 10389, 3871}, {2646, 25524, 35262}, {3091, 18444, 1490}, {3218, 16865, 31424}, {3303, 5836, 3895}, {3305, 11520, 72}, {3485, 26105, 41012}, {3487, 5084, 908}, {3601, 5437, 404}, {3617, 3957, 6765}, {3624, 5705, 2}, {3671, 40998, 11415}, {3698, 3748, 3913}, {3720, 3924, 1}, {3812, 51715, 55}, {3816, 11281, 11375}, {3868, 5047, 9}, {3869, 5284, 31435}, {3876, 17536, 7308}, {4187, 11374, 30852}, {4189, 27003, 15803}, {4304, 12436, 4190}, {4420, 19877, 8580}, {4511, 5550, 8583}, {4666, 19860, 1}, {4666, 54318, 3872}, {4861, 38314, 1}, {5046, 31019, 9612}, {5047, 11518, 3951}, {5129, 11036, 329}, {5248, 5883, 46}, {5259, 5902, 12514}, {5554, 10587, 31397}, {5691, 38150, 6894}, {5708, 16418, 3916}, {5836, 42819, 3303}, {6762, 44841, 3889}, {6918, 10246, 33597}, {7308, 11523, 3876}, {8167, 12635, 25917}, {8582, 13405, 5552}, {8728, 12433, 3419}, {9581, 25525, 2476}, {9843, 13411, 2}, {10449, 16817, 5271}, {11108, 15934, 72}, {11529, 31435, 3869}, {17016, 29814, 1}, {25917, 44840, 12635}, {37246, 37581, 54337}

X(54393) = X(3)X(114)∩X(4)X(69)

Barycentrics    a^8 - a^6*b^2 + a^2*b^6 - b^8 - a^6*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + 2*b^2*c^6 - c^8 : :
X(54393) = 3 X[5] - 2 X[20576], 3 X[32] - 4 X[20576], 3 X[7818] - X[30270], 3 X[7818] + X[36997], 3 X[38317] - 2 X[39750], 4 X[140] - 5 X[7867], 3 X[7841] - X[39646], 5 X[1656] - 4 X[6680], 5 X[3091] - 3 X[9753], 5 X[3091] - X[20065], 3 X[9753] - X[20065], X[5017] - 3 X[10516], 3 X[7697] - 2 X[18806], 5 X[7851] - 3 X[9755], 3 X[14639] - X[36849]

X(54393) lies on these lines: {2, 13335}, {3, 114}, {4, 69}, {5, 32}, {20, 7836}, {30, 7801}, {39, 37242}, {53, 13562}, {83, 3406}, {98, 5025}, {115, 43183}, {125, 41238}, {140, 7867}, {141, 35387}, {147, 6655}, {182, 6656}, {184, 41237}, {187, 37466}, {262, 7785}, {297, 9306}, {325, 9737}, {343, 460}, {355, 760}, {371, 6290}, {372, 6289}, {376, 7870}, {381, 754}, {383, 9989}, {458, 21243}, {517, 4769}, {542, 7841}, {550, 40278}, {575, 7803}, {576, 7762}, {631, 7831}, {736, 3095}, {746, 20430}, {1078, 37446}, {1080, 9988}, {1348, 6178}, {1349, 6177}, {1503, 13355}, {1504, 49356}, {1505, 49355}, {1513, 5171}, {1656, 6680}, {1975, 23698}, {2031, 43620}, {2080, 39603}, {2207, 39569}, {2386, 18531}, {2387, 18474}, {2453, 18375}, {2548, 13357}, {2782, 7748}, {2896, 22712}, {3016, 15068}, {3053, 37071}, {3091, 9753}, {3314, 10722}, {3398, 7834}, {3425, 7503}, {3564, 5028}, {3852, 51756}, {3934, 35430}, {5017, 10516}, {5056, 5395}, {5063, 45921}, {5188, 7873}, {5206, 37459}, {5286, 44499}, {5476, 7812}, {5480, 35389}, {5613, 11304}, {5617, 11303}, {5872, 36252}, {5873, 36251}, {5921, 32982}, {5965, 7754}, {5972, 11331}, {5999, 7885}, {6036, 7887}, {6054, 7833}, {6055, 11318}, {6194, 7929}, {6321, 13108}, {6467, 41757}, {6721, 33233}, {6750, 17814}, {6776, 32974}, {7388, 43120}, {7389, 43121}, {7395, 44200}, {7470, 48898}, {7694, 7800}, {7697, 18806}, {7709, 7847}, {7747, 35930}, {7749, 36519}, {7752, 37334}, {7756, 14981}, {7770, 24206}, {7773, 13860}, {7775, 37345}, {7791, 9744}, {7793, 38227}, {7794, 39838}, {7802, 11676}, {7821, 18860}, {7822, 35385}, {7823, 12110}, {7824, 43461}, {7825, 15980}, {7832, 35925}, {7842, 54187}, {7843, 32189}, {7851, 9755}, {7854, 22505}, {7855, 32515}, {7857, 21445}, {7859, 10359}, {7863, 38738}, {7864, 32467}, {7879, 40107}, {7883, 50977}, {7891, 21166}, {7899, 34473}, {7911, 12177}, {7928, 37455}, {7934, 9862}, {7935, 37479}, {8370, 11178}, {8724, 34504}, {8743, 39604}, {9766, 10983}, {9880, 14645}, {10242, 48663}, {10551, 20022}, {10768, 38521}, {11005, 38523}, {11179, 33190}, {11511, 45279}, {11550, 14957}, {11574, 41761}, {11646, 15069}, {12362, 42353}, {13085, 44422}, {13881, 23514}, {14003, 51372}, {14265, 53174}, {14568, 34623}, {14639, 36849}, {14826, 37174}, {14881, 40250}, {16312, 47339}, {16925, 47113}, {16964, 23013}, {16965, 23006}, {18358, 53418}, {19130, 35431}, {22566, 34510}, {23234, 33274}, {23293, 46571}, {30549, 36245}, {30736, 35894}, {32971, 40330}, {33230, 38064}, {33736, 37527}, {34664, 51611}, {35840, 49087}, {35841, 49086}, {36163, 47213}, {36173, 38528}, {37841, 53797}, {37984, 47577}, {39590, 46321}, {52016, 53477}, {52090, 54222}

X(54393) = midpoint of X(i) and X(j) for these {i,j}: {4, 315}, {30270, 36997}
X(54393) = reflection of X(i) in X(j) for these {i,j}: {3, 626}, {32, 5}, {35387, 141}, {35389, 5480}, {35424, 24206}, {35430, 3934}, {35431, 19130}, {36998, 13335}, {47577, 37984}
X(54393) = complement of X(36998)
X(54393) = anticomplement of X(13335)
X(54393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36998, 13335}, {4, 1352, 6248}, {5, 7745, 10358}, {5, 10104, 7746}, {5, 32151, 10104}, {114, 32152, 3}, {147, 6655, 11257}, {1513, 7750, 5171}, {3091, 20065, 9753}, {5025, 9863, 98}, {6248, 13449, 4}, {7785, 37336, 262}, {7791, 9744, 13334}, {7818, 36997, 30270}, {7823, 13862, 12110}, {10356, 10358, 5}, {20428, 20429, 48901}, {22505, 49111, 40279}, {37824, 37825, 34507}

X(54394) = X(4)X(12)∩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^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) : :

X(54394) lies on these lines: {1, 1859}, {3, 1838}, {4, 12}, {6, 19}, {25, 225}, {27, 4267}, {28, 56}, {29, 1001}, {33, 37080}, {40, 1888}, {52, 8757}, {92, 958}, {108, 17562}, {201, 54324}, {208, 5338}, {222, 18180}, {226, 37377}, {240, 37549}, {273, 1940}, {388, 4198}, {405, 1882}, {407, 11383}, {475, 3925}, {1012, 40946}, {1038, 9816}, {1068, 11399}, {1074, 11414}, {1096, 1104}, {1125, 37393}, {1214, 7535}, {1376, 5125}, {1393, 26934}, {1426, 1452}, {1430, 4252}, {1435, 32636}, {1466, 7490}, {1470, 37245}, {1478, 7511}, {1479, 15763}, {1593, 37601}, {1598, 1785}, {1621, 7518}, {1715, 1777}, {1724, 41342}, {1753, 7957}, {1762, 37591}, {1826, 37318}, {1844, 15934}, {1846, 28353}, {1848, 11375}, {1865, 37225}, {1869, 1877}, {1872, 37569}, {1875, 7713}, {1891, 5252}, {1894, 26378}, {1935, 24310}, {2181, 3924}, {2332, 11553}, {2550, 4200}, {2911, 15443}, {2969, 22479}, {3149, 53850}, {3485, 54340}, {3560, 39529}, {3616, 17519}, {3913, 5174}, {4219, 5217}, {4423, 7498}, {5142, 37799}, {5146, 5172}, {5204, 7501}, {5236, 10404}, {5248, 54299}, {5323, 14014}, {5433, 7521}, {5584, 37305}, {5706, 11428}, {5752, 7066}, {5930, 51687}, {6197, 37567}, {6254, 11436}, {6913, 39574}, {7354, 37395}, {7510, 10267}, {7952, 17602}, {8273, 37028}, {9122, 23207}, {10198, 37321}, {10319, 19372}, {10536, 19365}, {10896, 37372}, {11237, 34666}, {11347, 22341}, {11392, 37398}, {11393, 37368}, {11471, 37391}, {11502, 37381}, {11509, 14018}, {15975, 49745}, {17923, 25524}, {19366, 34032}, {22759, 37790}, {25514, 54320}, {26481, 37362}, {28628, 30687}, {31387, 37117}, {37194, 52427}, {37231, 37800}, {46884, 54358}

X(54394) = polar conjugate of the isotomic conjugate of X(37543)
X(54394) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2335}, {78, 51223}, {345, 2215}, {6332, 36080}
X(54394) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 2335}, {38967, 52355}
X(54394) = crossdifference of every pair of points on line {521, 52306}
X(54394) = barycentric product X(i)*X(j) for these {i,j}: {4, 37543}, {34, 5271}, {57, 39585}, {81, 1882}, {92, 1451}, {108, 23882}, {278, 405}, {331, 5320}, {608, 44140}, {653, 46385}, {1396, 5295}
X(54394) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2335}, {405, 345}, {608, 51223}, {1395, 2215}, {1451, 63}, {1882, 321}, {5271, 3718}, {5320, 219}, {23882, 35518}, {37543, 69}, {39585, 312}, {46385, 6332}
X(54394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 41227, 55}, {19, 34, 65}, {28, 278, 56}, {92, 54343, 958}, {1068, 36009, 11399}, {1426, 2355, 1452}, {1838, 54368, 3}, {2362, 16232, 1841}, {2969, 37238, 22479}, {4214, 11406, 1869}, {17923, 37253, 25524}

X(54395) = X(2)X(99)∩X(4)X(110)

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 - 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(54395) lies on these lines: {2, 99}, {4, 110}, {5, 9155}, {30, 5191}, {69, 48540}, {76, 94}, {83, 13582}, {98, 36163}, {125, 53346}, {141, 311}, {193, 317}, {246, 10264}, {287, 35902}, {297, 525}, {298, 21468}, {299, 21469}, {316, 323}, {385, 40885}, {524, 50187}, {542, 34174}, {625, 36212}, {858, 44420}, {868, 2782}, {1273, 34827}, {1316, 6321}, {1561, 3543}, {1625, 1993}, {1634, 34981}, {1976, 46264}, {1989, 6148}, {2407, 3018}, {2502, 11064}, {2697, 53691}, {2996, 16080}, {3014, 38393}, {3124, 5254}, {3233, 46982}, {3260, 53416}, {3292, 13449}, {3448, 38664}, {3506, 48884}, {4226, 23698}, {5117, 7703}, {5286, 39024}, {5392, 46105}, {5642, 9880}, {5972, 38734}, {7426, 46999}, {7468, 16188}, {7500, 9157}, {7550, 34837}, {7598, 39660}, {7599, 39661}, {7745, 20976}, {7760, 46723}, {7777, 11672}, {7812, 11004}, {7827, 15018}, {7841, 15066}, {8352, 40112}, {8370, 14389}, {8753, 34518}, {8754, 32114}, {8836, 11131}, {8838, 11130}, {9123, 19912}, {9131, 21731}, {9140, 12243}, {9158, 47103}, {9185, 47348}, {9486, 43291}, {10554, 23334}, {10723, 35278}, {11054, 44555}, {11632, 36194}, {14651, 35922}, {14712, 40853}, {14957, 51360}, {15107, 43453}, {15462, 39120}, {17008, 31635}, {18366, 43676}, {19130, 36213}, {19570, 44577}, {20127, 54085}, {20998, 41238}, {21850, 51335}, {22151, 53507}, {22515, 51430}, {25051, 53569}, {25328, 38361}, {31099, 46034}, {31709, 41888}, {31710, 41887}, {34505, 37638}, {35298, 38227}, {36189, 39528}, {37765, 37784}, {38259, 44877}, {39689, 53418}, {39809, 51431}, {48910, 52162}

X(54395) = reflection of X(i) in X(j) for these {i,j}: {2407, 3018}, {3014, 38393}, {4226, 47200}
X(54395) = polar conjugate of X(40118)
X(54395) = isotomic conjugate of the isogonal conjugate of X(2493)
X(54395) = polar conjugate of the isogonal conjugate of X(14984)
X(54395) = X(842)-anticomplementary conjugate of X(4329)
X(54395) = X(5641)-Ceva conjugate of X(38939)
X(54395) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40118}, {163, 51480}, {1910, 40083}, {2159, 51457}, {2642, 35191}
X(54395) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 51480}, {1249, 40118}, {2493, 542}, {3163, 51457}, {11672, 40083}, {16188, 6}, {23967, 51474}
X(54395) = cevapoint of X(2493) and X(14984)
X(54395) = crossdifference of every pair of points on line {184, 351}
X(54395) = barycentric product X(i)*X(j) for these {i,j}: {76, 2493}, {264, 14984}, {325, 34175}, {340, 51847}, {523, 14221}, {850, 7468}, {5641, 16188}, {51481, 52515}
X(54395) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40118}, {30, 51457}, {511, 40083}, {523, 51480}, {542, 51474}, {691, 35191}, {2493, 6}, {7468, 110}, {14221, 99}, {14984, 3}, {16188, 542}, {34175, 98}, {38939, 842}, {51847, 265}, {52515, 2987}
X(54395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {115, 51389, 2}, {141, 53495, 338}, {297, 5523, 46106}, {297, 44146, 14918}, {297, 47286, 3580}, {2592, 2593, 51481}, {3569, 41079, 9979}, {3580, 47286, 51481}, {10723, 35278, 36181}

X(54396) = X(4)X(9)∩X(29)X(33)

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

X(54396) lies on these lines: {1, 5136}, {2, 225}, {4, 9}, {8, 28950}, {25, 29828}, {29, 33}, {34, 92}, {56, 26011}, {63, 34831}, {77, 21588}, {208, 1940}, {219, 5295}, {226, 14257}, {321, 27410}, {377, 34823}, {405, 6708}, {406, 498}, {407, 26066}, {427, 29857}, {429, 1329}, {461, 6745}, {475, 1838}, {551, 38295}, {860, 1698}, {958, 1867}, {960, 1824}, {993, 37117}, {1038, 24537}, {1040, 23661}, {1068, 1125}, {1118, 30686}, {1172, 45032}, {1376, 37194}, {1426, 3812}, {1610, 35635}, {1825, 3869}, {1836, 20306}, {1848, 52082}, {1865, 17303}, {1870, 30147}, {1872, 7524}, {1877, 5554}, {1884, 5090}, {1887, 41538}, {1889, 5302}, {1891, 7102}, {1944, 10449}, {2049, 40937}, {2182, 5786}, {2339, 2478}, {2886, 37368}, {2899, 4194}, {3338, 34589}, {3452, 39579}, {3616, 23710}, {3713, 3714}, {4292, 20205}, {5125, 19808}, {5130, 7140}, {5236, 7103}, {5338, 37390}, {5705, 37381}, {5737, 15823}, {5745, 14018}, {5794, 37239}, {6198, 22836}, {6350, 54346}, {6684, 37414}, {6737, 7046}, {6796, 7412}, {6836, 34822}, {7009, 16066}, {7414, 25440}, {7498, 7952}, {7531, 45766}, {8889, 50752}, {10538, 26091}, {11399, 37393}, {11517, 54299}, {11679, 44734}, {17555, 52412}, {17860, 54295}, {20223, 37591}, {20883, 54293}, {25917, 53861}, {37276, 44696}

X(54396) = polar conjugate of X(44733)
X(54396) = polar conjugate of the isotomic conjugate of X(11679)
X(54396) = polar conjugate of the isogonal conjugate of X(2268)
X(54396) = X(i)-isoconjugate of X(j) for these (i,j): {3, 959}, {48, 44733}, {56, 34259}, {73, 5331}, {77, 2258}, {222, 941}, {603, 31359}, {905, 32693}, {1409, 37870}, {2286, 34260}, {22383, 32038}, {23189, 52931}, {34258, 52411}
X(54396) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 34259}, {958, 1038}, {1249, 44733}, {7952, 31359}, {17417, 905}, {34261, 63}, {36103, 959}
X(54396) = barycentric product X(i)*X(j) for these {i,j}: {4, 11679}, {8, 5307}, {10, 44734}, {27, 3714}, {29, 31993}, {33, 34284}, {92, 958}, {264, 2268}, {273, 3713}, {281, 10436}, {312, 4185}, {318, 940}, {333, 1867}, {1468, 7017}, {1897, 23880}, {6335, 17418}, {15742, 53526}, {36797, 50457}
X(54396) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 44733}, {9, 34259}, {19, 959}, {29, 37870}, {33, 941}, {281, 31359}, {318, 34258}, {607, 2258}, {940, 77}, {958, 63}, {1039, 34260}, {1172, 5331}, {1468, 222}, {1867, 226}, {1897, 32038}, {2268, 3}, {3713, 78}, {3714, 306}, {4185, 57}, {5019, 603}, {5307, 7}, {8672, 51664}, {8750, 32693}, {10436, 348}, {11679, 69}, {17418, 905}, {23880, 4025}, {31993, 307}, {34261, 1038}, {34284, 7182}, {44734, 86}, {50457, 17094}, {53526, 1565}, {53561, 7004}
X(54396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 281, 46878}, {29, 318, 33}, {92, 11109, 34}, {1867, 4185, 5307}, {5136, 41013, 1}, {7090, 14121, 1826}, {7140, 37226, 5130}, {23661, 27378, 1040}, {40950, 53008, 8}

X(54397) = X(4)X(9)∩X(25)X(36)

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

X(54397) lies on these lines: {1, 1828}, {4, 9}, {25, 36}, {28, 39963}, {33, 7962}, {34, 106}, {35, 26378}, {46, 37226}, {57, 1877}, {165, 37391}, {406, 25522}, {428, 5155}, {995, 40982}, {1572, 33853}, {1597, 35238}, {1598, 11249}, {1698, 1883}, {1763, 3586}, {1785, 1851}, {1829, 5697}, {1848, 48803}, {1866, 3340}, {1875, 40961}, {3583, 11390}, {3679, 5101}, {3746, 11400}, {4194, 41012}, {4214, 37572}, {5119, 21361}, {5151, 5541}, {5437, 37168}, {5722, 7289}, {6848, 36986}, {9816, 17532}, {10319, 11113}, {11105, 31435}, {11114, 24611}, {13730, 19372}, {15737, 52427}, {18344, 48111}, {26580, 39579}, {28039, 30117}, {28076, 36572}, {37458, 38761}

X(54397) = crossdifference of every pair of points on line {1459, 14418}
X(54397) = barycentric product X(i)*X(j) for these {i,j}: {19, 50101}, {92, 16483}, {278, 3895}
X(54397) = barycentric quotient X(i)/X(j) for these {i,j}: {3895, 345}, {16483, 63}, {50101, 304}
X(54397) = {X(1828),X(4186)}-harmonic conjugate of X(1)

X(54398) = X(2)X(72)∩X(7)X(10)

Barycentrics    a^4 + 4*a^3*b - 2*a^2*b^2 - 4*a*b^3 + b^4 + 4*a^3*c - 4*a*b^2*c - 2*a^2*c^2 - 4*a*b*c^2 - 2*b^2*c^2 - 4*a*c^3 + c^4 : :
X(54398) = 3 X[2] - 4 X[5791], 5 X[3091] - 4 X[5715], 2 X[4313] - 3 X[50742], 4 X[31424] - 3 X[50742]

X(54398) lies on these lines: {1, 5273}, {2, 72}, {3, 20007}, {4, 144}, {7, 10}, {8, 20}, {9, 938}, {21, 145}, {57, 17580}, {75, 52345}, {78, 3523}, {100, 37105}, {165, 6743}, {169, 391}, {191, 4294}, {200, 10884}, {210, 1788}, {219, 3562}, {224, 4420}, {307, 14256}, {329, 3091}, {333, 37113}, {346, 3730}, {347, 18631}, {377, 3421}, {387, 3672}, {388, 21677}, {390, 12514}, {412, 7046}, {443, 21454}, {452, 3219}, {517, 37434}, {519, 4313}, {527, 50736}, {631, 3940}, {748, 28080}, {758, 19843}, {908, 5056}, {912, 6908}, {936, 5435}, {950, 3929}, {960, 14986}, {962, 4847}, {997, 5265}, {1012, 12245}, {1046, 4307}, {1071, 5657}, {1125, 31446}, {1145, 13243}, {1210, 18228}, {1219, 10461}, {1259, 2975}, {1260, 6986}, {1265, 14829}, {1697, 6764}, {1698, 21060}, {1706, 24393}, {1707, 4339}, {1737, 8165}, {1834, 4419}, {2095, 6864}, {2318, 37523}, {2475, 20078}, {2551, 5220}, {2895, 26054}, {3059, 9943}, {3085, 5904}, {3086, 5692}, {3090, 46873}, {3146, 3419}, {3189, 4640}, {3218, 6904}, {3361, 12447}, {3434, 11684}, {3436, 6839}, {3452, 5704}, {3485, 3962}, {3488, 11106}, {3522, 3916}, {3543, 5175}, {3555, 11020}, {3616, 11520}, {3621, 17576}, {3622, 5730}, {3632, 4304}, {3654, 9859}, {3679, 4292}, {3681, 7080}, {3695, 14021}, {3697, 9954}, {3710, 34255}, {3786, 27334}, {3811, 5281}, {3812, 38057}, {3839, 17781}, {3869, 6837}, {3871, 20015}, {3874, 11038}, {3878, 34625}, {3901, 19854}, {3950, 35629}, {3984, 10303}, {4005, 24914}, {4067, 26363}, {4134, 26364}, {4189, 20013}, {4197, 17757}, {4220, 42461}, {4293, 6763}, {4346, 23537}, {4430, 10587}, {4454, 43533}, {4461, 5295}, {4641, 5716}, {4651, 37109}, {4652, 10304}, {4661, 10528}, {4662, 5784}, {4677, 34639}, {4678, 37435}, {4853, 18219}, {4855, 15692}, {4882, 5732}, {5046, 43740}, {5082, 10431}, {5119, 12632}, {5177, 5905}, {5178, 44447}, {5187, 26792}, {5221, 26040}, {5226, 5705}, {5234, 6738}, {5249, 9780}, {5250, 36845}, {5262, 54305}, {5274, 10916}, {5325, 5436}, {5440, 15717}, {5552, 15016}, {5687, 7411}, {5690, 6916}, {5703, 5745}, {5708, 17582}, {5709, 50700}, {5711, 39587}, {5731, 6737}, {5735, 19925}, {5748, 7486}, {5758, 51755}, {5759, 5787}, {5763, 5789}, {5768, 37423}, {5770, 6926}, {5771, 6988}, {5774, 37088}, {5806, 5817}, {5836, 34744}, {5837, 6762}, {5882, 36922}, {5902, 19855}, {6172, 12572}, {6765, 7675}, {6846, 24474}, {6871, 17484}, {6884, 10527}, {6919, 31018}, {6944, 31835}, {6987, 26921}, {6989, 24475}, {8822, 42696}, {8951, 45204}, {9122, 20212}, {9370, 34035}, {9534, 20367}, {9797, 31393}, {9840, 22149}, {9945, 21735}, {9960, 14872}, {9964, 46685}, {10381, 26125}, {10477, 26059}, {10578, 41863}, {10580, 31435}, {10586, 41389}, {10883, 24390}, {11015, 20052}, {11851, 19851}, {12125, 12671}, {12528, 37421}, {12635, 30478}, {12710, 30628}, {14646, 31777}, {15394, 40417}, {15934, 16845}, {15935, 16866}, {16368, 20043}, {17054, 37650}, {17127, 36579}, {17284, 39589}, {17746, 27541}, {18220, 45700}, {18221, 54318}, {18391, 41229}, {18650, 32099}, {18655, 32087}, {19262, 22458}, {20009, 37683}, {20012, 37175}, {20059, 37161}, {22131, 52058}, {24635, 37528}, {25524, 45085}, {25568, 26066}, {26446, 27525}, {26685, 37024}, {26842, 50237}, {27549, 28287}, {28605, 41013}, {30340, 51706}, {30852, 46936}, {34619, 50835}, {36996, 37424}, {37254, 37547}, {40661, 54366}, {48890, 49716}

X(54398) = reflection of X(i) in X(j) for these {i,j}: {7, 5833}, {3487, 5791}, {4313, 31424}, {5290, 10}
X(54398) = anticomplement of X(3487)
X(54398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5273, 17558}, {2, 3868, 11036}, {4, 3927, 144}, {7, 10, 4208}, {8, 63, 20}, {9, 938, 5129}, {9, 24391, 938}, {10, 3339, 11024}, {10, 5223, 5815}, {40, 9799, 20}, {78, 5744, 3523}, {329, 6734, 3091}, {960, 24477, 14986}, {1071, 5657, 37108}, {1071, 34790, 41228}, {1259, 2975, 37106}, {3219, 12649, 452}, {3487, 5791, 2}, {3488, 31445, 11106}, {3617, 9965, 377}, {3632, 4304, 12536}, {3951, 6734, 329}, {4313, 31424, 50742}, {4847, 12526, 962}, {5745, 11523, 5703}, {5770, 31837, 6926}, {5775, 5815, 10}, {11106, 20008, 3488}, {11520, 54357, 3616}

X(54399) = X(8)X(21)∩X(11)X(60)

Barycentrics    (a + b)*(a - b - c)*(a + c)*(2*a^4 + 2*a^3*b + a*b^3 + b^4 + 2*a^3*c + 2*a^2*b*c - a*b^2*c - a*b*c^2 - 2*b^2*c^2 + a*c^3 + c^4) : :
X(54399) = 3 X[21] - X[1043], X[79] - 3 X[33135]

X(54399) lies on these lines: {5, 9275}, {8, 21}, {11, 60}, {12, 24624}, {30, 58}, {79, 33135}, {81, 3649}, {110, 37722}, {140, 15792}, {191, 18163}, {270, 1852}, {284, 1213}, {442, 6703}, {496, 17104}, {501, 15325}, {662, 6691}, {740, 3647}, {758, 18178}, {759, 37730}, {1503, 37447}, {1761, 40979}, {2185, 4999}, {3058, 35193}, {3109, 37702}, {3286, 3651}, {4225, 5427}, {4276, 5428}, {4653, 15174}, {4658, 16137}, {5127, 15171}, {5358, 44253}, {5433, 40214}, {5441, 52680}, {10122, 18165}, {11263, 17197}, {11684, 16704}, {12433, 37816}, {15670, 49730}, {15674, 26044}, {17637, 18191}, {18653, 32636}, {33857, 54356}

X(54399) = reflection of X(3704) in X(18253)
X(54399) = X(6742)-Ceva conjugate of X(3737)
X(54399) = barycentric product X(4560)*X(14985)
X(54399) = barycentric quotient X(14985)/X(4552)
X(54399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 333, 18253}, {81, 37369, 3649}

X(54400) = X(1)X(104)∩X(6)X(19)

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

X(54400) lies on these lines: {1, 104}, {6, 19}, {8, 28968}, {33, 6001}, {40, 73}, {42, 51660}, {46, 10571}, {56, 15854}, {57, 957}, {63, 24806}, {201, 2324}, {208, 1425}, {212, 30503}, {222, 517}, {223, 2093}, {225, 4295}, {227, 37567}, {388, 50307}, {595, 34489}, {614, 18838}, {758, 8270}, {942, 34040}, {960, 25934}, {991, 1697}, {1038, 3869}, {1042, 37550}, {1060, 14988}, {1191, 37566}, {1319, 3052}, {1393, 3339}, {1394, 3340}, {1406, 4320}, {1420, 4257}, {1435, 32065}, {1455, 2099}, {1465, 36279}, {1572, 52635}, {1708, 49500}, {1737, 34029}, {1771, 6261}, {1836, 51421}, {1877, 18391}, {1935, 19860}, {2122, 13601}, {2199, 3553}, {2654, 12705}, {3057, 34046}, {3157, 37562}, {3195, 51399}, {3359, 22350}, {3660, 16483}, {3753, 34048}, {3877, 17074}, {4084, 4347}, {4337, 5119}, {4424, 45126}, {4551, 54286}, {4559, 40131}, {5250, 37523}, {5252, 5848}, {5711, 12709}, {5727, 18328}, {5836, 9370}, {5886, 43043}, {5902, 34036}, {5903, 21147}, {7078, 31788}, {7191, 18419}, {7355, 11471}, {12047, 34030}, {12514, 37558}, {12672, 41344}, {12940, 52840}, {18421, 34033}, {18915, 46878}, {19366, 42448}, {20306, 26955}, {22072, 37560}, {23070, 25413}, {24914, 51415}, {26888, 44121}, {37696, 40266}, {37740, 51422}

X(54400) = crossdifference of every pair of points on line {521, 46393}
X(54400) = barycentric product X(57)*X(5657)
X(54400) = barycentric quotient X(5657)/X(312)
X(54400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {65, 221, 34}, {5903, 34043, 21147}

X(54401) = X(1)X(2)∩X(9)X(47)

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

X(54401) lies on these lines: {1, 2}, {9, 47}, {11, 7405}, {12, 1060}, {24, 33}, {34, 1594}, {36, 7509}, {37, 921}, {38, 17437}, {46, 201}, {55, 6642}, {56, 7393}, {90, 601}, {91, 6358}, {171, 920}, {240, 23050}, {255, 756}, {750, 17700}, {774, 17699}, {984, 3075}, {988, 10090}, {1038, 1478}, {1062, 5432}, {1385, 21479}, {1479, 7401}, {1490, 4337}, {1870, 10588}, {3147, 5218}, {3337, 4327}, {3583, 7544}, {3585, 37444}, {3761, 28706}, {4296, 10590}, {4302, 7487}, {4319, 44802}, {4320, 5270}, {4324, 31304}, {4682, 44547}, {4995, 44211}, {5010, 7488}, {5217, 14070}, {5250, 54350}, {5348, 26921}, {5707, 41538}, {5818, 54292}, {6796, 16577}, {7280, 37126}, {7741, 14788}, {8144, 34351}, {8270, 12047}, {8965, 44590}, {10127, 15171}, {10592, 32047}, {10827, 21147}, {16238, 37729}, {18397, 37559}, {18447, 31479}, {21077, 54289}, {24431, 24467}, {31423, 33178}, {34036, 37692}, {37034, 40635}

X(54401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5268, 498}, {601, 7069, 90}, {750, 44706, 17700}, {1210, 30142, 1}, {3086, 3920, 1}, {30145, 44675, 1}

X(54402) = X(1)X(6)∩X(52)X(62)

Barycentrics    a^2*(b*c + Sqrt[3]*S) : :

X(54402) lies on these lines: {1, 6}, {11, 40693}, {12, 40694}, {13, 10896}, {14, 10895}, {15, 5204}, {16, 5217}, {35, 22238}, {36, 22236}, {55, 62}, {56, 61}, {203, 3304}, {222, 7344}, {388, 42999}, {395, 498}, {396, 499}, {397, 1479}, {398, 1478}, {497, 42998}, {559, 6191}, {999, 2307}, {1482, 33655}, {2306, 5708}, {3085, 37641}, {3086, 37640}, {3295, 7127}, {3303, 7006}, {3364, 18996}, {3365, 18995}, {3389, 19038}, {3390, 19037}, {3582, 49947}, {3583, 5340}, {3584, 49948}, {3585, 5339}, {3614, 18581}, {4299, 42147}, {4302, 42148}, {4316, 43194}, {4324, 43193}, {5010, 36843}, {5225, 5335}, {5229, 5334}, {5362, 5550}, {5367, 9780}, {5432, 42149}, {5433, 42152}, {5471, 9650}, {5472, 9665}, {6284, 10653}, {7126, 36750}, {7173, 18582}, {7280, 36836}, {7354, 10654}, {7741, 42156}, {7951, 42153}, {8739, 11398}, {8740, 11399}, {9654, 42975}, {9657, 42991}, {9669, 42974}, {9670, 42990}, {10056, 43229}, {10072, 43228}, {10483, 42154}, {10592, 11543}, {10593, 11542}, {10638, 11486}, {11073, 52186}, {11485, 19373}, {12941, 14137}, {12943, 16964}, {12952, 14136}, {12953, 16965}, {15326, 42150}, {15338, 42151}, {31479, 42989}, {33654, 36279}, {37772, 52424}

X(54402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {61, 202, 56}, {62, 7005, 55}

X(54403) = X(1)X(6)∩X(55)X(61)

Barycentrics    a^2*(b*c - Sqrt[3]*S) : :

X(54403) lies on these lines: {1, 6}, {3, 2307}, {11, 40694}, {12, 40693}, {13, 10895}, {14, 10896}, {15, 5217}, {16, 5204}, {35, 22236}, {36, 22238}, {55, 61}, {56, 62}, {202, 3304}, {222, 7345}, {388, 42998}, {395, 499}, {396, 498}, {397, 1478}, {398, 1479}, {497, 42999}, {1082, 6192}, {1250, 11485}, {1482, 7052}, {2306, 36279}, {3085, 37640}, {3086, 37641}, {3303, 7005}, {3364, 19038}, {3365, 19037}, {3389, 18996}, {3390, 18995}, {3582, 49948}, {3583, 5339}, {3584, 49947}, {3585, 5340}, {3614, 18582}, {4299, 42148}, {4302, 42147}, {4316, 43193}, {4324, 43194}, {5010, 36836}, {5225, 5334}, {5229, 5335}, {5362, 9780}, {5367, 5550}, {5432, 42152}, {5433, 42149}, {5471, 9665}, {5472, 9650}, {5708, 33654}, {6284, 10654}, {7051, 11486}, {7173, 18581}, {7280, 36843}, {7354, 10653}, {7741, 42153}, {7951, 42156}, {8739, 11399}, {8740, 11398}, {9654, 42974}, {9657, 42990}, {9669, 42975}, {9670, 42991}, {10056, 43228}, {10072, 43229}, {10483, 42155}, {10592, 11542}, {10593, 11543}, {11072, 52186}, {12942, 14136}, {12943, 16965}, {12951, 14137}, {12953, 16964}, {15326, 42151}, {15338, 42150}, {19551, 36750}, {31479, 42988}, {37773, 52424}

X(54403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {61, 7006, 55}, {62, 203, 56}, {2307, 7127, 3}

X(54404) = X(40)X(75)∩X(57)X(86)

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

X(54404) lies on these lines: {3, 326}, {7, 37550}, {9, 1760}, {19, 28287}, {22, 18615}, {40, 75}, {46, 10436}, {57, 86}, {63, 69}, {77, 283}, {85, 8822}, {191, 17272}, {269, 1758}, {320, 54290}, {348, 7013}, {394, 14597}, {484, 25590}, {1264, 3785}, {1697, 4360}, {1918, 17594}, {2270, 17277}, {2339, 28274}, {3218, 3945}, {3219, 5232}, {3333, 17394}, {3496, 27626}, {3576, 44179}, {3751, 21035}, {3875, 5119}, {3928, 17378}, {3929, 17271}, {4047, 23151}, {4357, 12514}, {4640, 24471}, {4902, 16558}, {4967, 54286}, {5250, 17321}, {5709, 10446}, {6762, 17377}, {11010, 17151}, {16992, 24310}, {17270, 21277}, {17322, 31435}, {17393, 31393}, {18713, 40937}, {19804, 39592}, {21059, 35258}, {27633, 39248}, {33295, 54373}, {34377, 54285}, {43216, 54322}

X(54404) = isotomic conjugate of the polar conjugate of X(5256)
X(54404) = X(10319)-Dao conjugate of X(52082)
X(54404) = barycentric product X(i)*X(j) for these {i,j}: {63, 17321}, {69, 5256}, {77, 14555}, {304, 16466}, {348, 5250}, {1332, 47995}, {3926, 7713}, {3931, 17206}, {4194, 7183}, {4254, 7182}, {4563, 50332}, {4592, 48402}
X(54404) = barycentric quotient X(i)/X(j) for these {i,j}: {3931, 1826}, {4254, 33}, {5250, 281}, {5256, 4}, {7713, 393}, {14555, 318}, {16466, 19}, {17321, 92}, {47995, 17924}, {48402, 24006}, {50332, 2501}
X(54404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 22370, 5227}, {77, 4652, 1444}

X(54405) = X(1)X(19)∩X(2)X(7)

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

X(54405) lies on these lines: {1, 19}, {2, 7}, {3, 37}, {6, 169}, {10, 5227}, {40, 3332}, {44, 5708}, {45, 5356}, {46, 71}, {56, 40937}, {65, 219}, {72, 965}, {75, 16054}, {77, 18726}, {78, 22021}, {86, 1760}, {101, 3553}, {172, 5336}, {192, 37274}, {198, 37034}, {220, 37544}, {222, 30456}, {281, 388}, {282, 951}, {312, 37092}, {344, 37280}, {346, 6904}, {354, 2264}, {355, 21933}, {379, 17863}, {404, 27396}, {443, 2345}, {517, 2256}, {572, 18443}, {573, 5709}, {604, 17451}, {608, 34036}, {612, 5285}, {614, 16470}, {910, 4254}, {912, 5778}, {938, 5802}, {940, 21370}, {948, 1119}, {950, 4198}, {968, 3724}, {969, 1814}, {993, 25081}, {999, 1108}, {1014, 24635}, {1055, 3100}, {1071, 5776}, {1100, 15934}, {1159, 22147}, {1212, 5120}, {1213, 5791}, {1333, 36011}, {1376, 3694}, {1385, 37519}, {1439, 6180}, {1449, 2082}, {1466, 1696}, {1467, 5053}, {1468, 40977}, {1478, 1826}, {1479, 1839}, {1572, 21769}, {1723, 2260}, {1731, 2257}, {1732, 3337}, {1737, 26063}, {1752, 18398}, {1761, 5327}, {1763, 5712}, {1773, 2333}, {1817, 28606}, {1841, 7497}, {1880, 2286}, {1895, 8748}, {1901, 21530}, {2160, 37584}, {2171, 2289}, {2252, 17700}, {2261, 30274}, {2268, 21808}, {2270, 4266}, {2276, 16056}, {2277, 28258}, {2287, 3868}, {2298, 4224}, {2352, 8021}, {2354, 26098}, {3197, 50195}, {3229, 20370}, {3290, 25514}, {3487, 7521}, {3612, 22054}, {3664, 7289}, {3666, 11347}, {3672, 24604}, {3686, 24391}, {3693, 37270}, {3721, 39598}, {3729, 20336}, {3731, 7523}, {3739, 37075}, {3772, 6678}, {3811, 54316}, {3916, 19523}, {3945, 7291}, {4261, 16415}, {4292, 8804}, {4641, 19728}, {4877, 31424}, {4908, 19706}, {5019, 16968}, {5037, 16780}, {5042, 49758}, {5122, 16675}, {5138, 16972}, {5142, 9612}, {5275, 37581}, {5276, 15487}, {5290, 7079}, {5341, 16777}, {5540, 16667}, {5707, 12417}, {5711, 40660}, {5717, 7713}, {5728, 5781}, {5755, 37532}, {5757, 37151}, {5787, 21049}, {5816, 24005}, {6147, 52259}, {6351, 8231}, {7113, 37615}, {7146, 53996}, {7297, 16884}, {7359, 10404}, {7557, 9581}, {7561, 11374}, {7719, 21620}, {8726, 37431}, {8728, 17303}, {9122, 37528}, {9579, 52364}, {9816, 24162}, {10401, 26932}, {10827, 21011}, {10901, 15830}, {11019, 40963}, {12436, 17355}, {12437, 20009}, {13408, 15945}, {13726, 54287}, {13750, 19350}, {14547, 40983}, {15656, 25516}, {15956, 24608}, {16488, 28011}, {16566, 16831}, {16568, 17394}, {16601, 54322}, {16673, 30282}, {16814, 37545}, {16817, 21384}, {16843, 31445}, {17073, 41003}, {17134, 25255}, {17279, 37326}, {17289, 37097}, {17316, 27059}, {17321, 24609}, {17736, 21061}, {17799, 25528}, {18165, 46882}, {18635, 41004}, {18655, 24606}, {18714, 44179}, {19547, 37623}, {19857, 41229}, {20171, 37233}, {21483, 44307}, {21866, 36279}, {23151, 54344}, {24179, 34830}, {24316, 53596}, {24325, 50198}, {26626, 26998}, {32777, 37266}, {37052, 37539}, {37113, 40979}, {37271, 44798}, {37306, 38871}, {37538, 43214}, {42018, 46830}

X(54405) = polar conjugate of the isotomic conjugate of X(54289)
X(54405) = X(i)-isoconjugate of X(j) for these (i,j): {4, 45127}, {650, 13395}
X(54405) = X(i)-Dao conjugate of X(j) for these (i,j): {405, 5271}, {36033, 45127}
X(54405) = crossdifference of every pair of points on line {656, 663}
X(54405) = barycentric product X(i)*X(j) for these {i,j}: {1, 377}, {4, 54289}, {8, 1448}, {75, 37538}, {86, 43214}, {662, 47124}, {5905, 46038}, {28606, 45999}
X(54405) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 45127}, {109, 13395}, {377, 75}, {1448, 7}, {37538, 1}, {43214, 10}, {46038, 2994}, {47124, 1577}, {54289, 69}
X(54405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 610, 284}, {1, 1781, 19}, {1, 18594, 380}, {2, 5279, 9}, {7, 27382, 5746}, {9, 57, 579}, {28, 2303, 284}, {48, 2294, 1}, {226, 40942, 5747}, {354, 2264, 54358}, {1449, 16547, 2082}, {1723, 3338, 2260}, {1880, 2286, 21147}, {2260, 54324, 1723}, {2285, 40131, 9}, {3247, 16548, 54359}, {6203, 6204, 226}, {16777, 37504, 24929}

X(54406) = X(1)X(6)∩X(32)X(78)

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

X(54406) lies on these lines: {1, 6}, {8, 1572}, {10, 9596}, {19, 10822}, {31, 33299}, {32, 78}, {39, 63}, {43, 3496}, {46, 1575}, {58, 54317}, {169, 2238}, {172, 997}, {187, 4855}, {329, 5286}, {574, 4652}, {579, 21744}, {672, 23620}, {748, 21808}, {908, 3767}, {920, 13006}, {936, 5277}, {976, 21764}, {978, 3509}, {1046, 17754}, {1193, 5282}, {1211, 17308}, {1468, 39244}, {1475, 32912}, {1500, 5250}, {1571, 17756}, {1716, 20861}, {1722, 3125}, {1759, 3216}, {1914, 3811}, {2082, 20683}, {2175, 26924}, {2241, 3870}, {2242, 19861}, {2276, 12514}, {2548, 6734}, {2975, 9619}, {3053, 5440}, {3219, 31442}, {3305, 16589}, {3338, 16604}, {3419, 7745}, {3661, 5739}, {3693, 14974}, {3868, 33854}, {3869, 9620}, {3876, 5276}, {3915, 3930}, {3916, 5013}, {3927, 9605}, {3929, 31429}, {3940, 30435}, {3951, 7772}, {3984, 5007}, {4006, 37610}, {4011, 21071}, {4383, 16583}, {4640, 31448}, {4641, 5021}, {5044, 5275}, {5119, 20691}, {5120, 42461}, {5278, 41249}, {5744, 31400}, {5791, 37661}, {5813, 33867}, {6542, 27523}, {7085, 20967}, {7739, 17781}, {7746, 30852}, {9574, 54290}, {9593, 12526}, {9599, 10916}, {12699, 21956}, {15494, 17798}, {16549, 49500}, {17206, 25918}, {17736, 49997}, {18055, 33295}, {24987, 31409}, {25006, 31416}, {26035, 26223}, {26066, 31460}, {31451, 35258}, {33950, 37657}, {34460, 37532}, {49561, 49710}, {50621, 54359}

X(54406) = barycentric product X(1)*X(33088)
X(54406) = barycentric quotient X(33088)/X(75)
X(54406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {44, 21874, 218}, {5299, 5904, 16973}

X(54407) = X(1)X(29)∩X(27)X(33)

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

X(54407) lies on these lines: {1, 19}, {4, 991}, {21, 1038}, {24, 37530}, {25, 940}, {27, 33}, {29, 34}, {37, 46890}, {58, 30733}, {81, 2299}, {112, 2725}, {162, 37131}, {238, 52413}, {241, 1876}, {242, 514}, {278, 38053}, {386, 7521}, {468, 35466}, {500, 15763}, {518, 5089}, {608, 1001}, {648, 14024}, {859, 46974}, {942, 18734}, {1040, 1817}, {1060, 36011}, {1062, 52012}, {1214, 8021}, {1395, 17185}, {1396, 4183}, {1458, 5236}, {1465, 52889}, {1742, 52840}, {1818, 1861}, {1848, 3720}, {1890, 40983}, {1936, 52427}, {2074, 52680}, {2191, 8747}, {2193, 36017}, {2263, 5327}, {2328, 8270}, {3100, 14953}, {3186, 16066}, {3192, 6353}, {3194, 41790}, {3576, 7436}, {3736, 54293}, {4206, 10458}, {4227, 4653}, {4248, 9311}, {4551, 37799}, {4564, 4570}, {5009, 32676}, {5125, 17234}, {5712, 37394}, {5718, 37432}, {5728, 46882}, {6198, 31900}, {7100, 44290}, {7466, 37633}, {7497, 50317}, {7537, 37732}, {11341, 20131}, {11363, 37539}, {11393, 17167}, {11398, 18180}, {14017, 37522}, {14196, 52460}, {15150, 36797}, {16747, 17175}, {17056, 37362}, {17188, 34036}, {17569, 30663}, {19765, 37245}, {20883, 24325}, {26702, 32688}, {31905, 52209}, {32682, 39439}, {37966, 47185}

X(54407) = polar conjugate of the isotomic conjugate of X(18206)
X(54407) = cevapoint of X(i) and X(j) for these (i,j): {1458, 1876}, {2356, 5089}
X(54407) = crossdifference of every pair of points on line {71, 656}
X(54407) = X(i)-isoconjugate of X(j) for these (i,j): {3, 13576}, {10, 36057}, {37, 1814}, {42, 31637}, {63, 18785}, {71, 673}, {72, 105}, {73, 14942}, {100, 10099}, {228, 2481}, {294, 1214}, {306, 1438}, {307, 2195}, {321, 32658}, {525, 919}, {647, 666}, {656, 36086}, {810, 51560}, {885, 23067}, {1409, 36796}, {1416, 3710}, {1439, 28071}, {1462, 3694}, {2200, 18031}, {3049, 36803}, {3682, 36124}, {3990, 54235}, {3998, 8751}, {4551, 23696}, {4580, 46163}, {5377, 18210}, {6559, 52373}, {8611, 36146}, {14208, 32666}, {17094, 52927}, {28132, 52610}, {32735, 52355}, {34018, 52370}, {43929, 52609}
X(54407) = X(i)-Dao conjugate of X(j) for these (i,j): {3162, 18785}, {6184, 306}, {8054, 10099}, {17755, 20336}, {20621, 10}, {35094, 14208}, {36103, 13576}, {36905, 1231}, {38980, 525}, {38989, 656}, {39014, 8611}, {39046, 72}, {39052, 666}, {39062, 51560}, {39063, 307}, {39077, 51366}, {40589, 1814}, {40592, 31637}, {40596, 36086}, {40609, 3710}
X(54407) = barycentric product X(i)*X(j) for these {i,j}: {1, 15149}, {4, 18206}, {19, 30941}, {21, 5236}, {25, 18157}, {27, 518}, {28, 3912}, {29, 241}, {58, 46108}, {81, 1861}, {85, 37908}, {86, 5089}, {92, 3286}, {162, 918}, {274, 2356}, {286, 672}, {333, 1876}, {514, 4238}, {648, 2254}, {665, 811}, {823, 53550}, {1026, 17925}, {1172, 9436}, {1396, 3717}, {1458, 31623}, {1474, 3263}, {1783, 23829}, {2223, 44129}, {2299, 40704}, {2322, 34855}, {8747, 25083}, {16728, 36124}, {17924, 54353}, {17926, 41353}, {22116, 31905}, {36797, 53544}, {44130, 52635}
X(54407) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 13576}, {25, 18785}, {27, 2481}, {28, 673}, {29, 36796}, {58, 1814}, {81, 31637}, {112, 36086}, {162, 666}, {241, 307}, {286, 18031}, {518, 306}, {648, 51560}, {649, 10099}, {665, 656}, {672, 72}, {811, 36803}, {918, 14208}, {926, 8611}, {1026, 52609}, {1172, 14942}, {1333, 36057}, {1458, 1214}, {1474, 105}, {1818, 3998}, {1861, 321}, {1876, 226}, {2203, 1438}, {2204, 2195}, {2206, 32658}, {2223, 71}, {2254, 525}, {2299, 294}, {2332, 28071}, {2340, 3694}, {2356, 37}, {3263, 40071}, {3286, 63}, {3675, 4466}, {3693, 3710}, {3912, 20336}, {3930, 3695}, {3932, 52369}, {4183, 6559}, {4233, 31638}, {4238, 190}, {4447, 4019}, {5089, 10}, {5236, 1441}, {5317, 36124}, {5338, 14625}, {7252, 23696}, {8747, 54235}, {9436, 1231}, {9454, 228}, {9455, 2200}, {9502, 51366}, {15149, 75}, {18157, 305}, {18206, 69}, {20683, 3949}, {20752, 3682}, {23225, 822}, {23829, 15413}, {24290, 4064}, {25083, 52396}, {30941, 304}, {32676, 919}, {35325, 35333}, {37908, 9}, {39258, 3690}, {42071, 3930}, {43925, 1027}, {46108, 313}, {51400, 20235}, {52635, 73}, {52890, 36816}, {53539, 51664}, {53544, 17094}, {53550, 24018}, {54325, 4574}, {54353, 1332}
X(54407) = {X(81),X(4233)}-harmonic conjugate of X(2299)

X(54408) = X(1)X(3)∩X(9)X(11)

Barycentrics    a*(a - b - c)*(a^4 - 2*a^2*b^2 + b^4 - 4*b^3*c - 2*a^2*c^2 + 6*b^2*c^2 - 4*b*c^3 + c^4) : :
X(54408) = (r^2 + 2*r*R - 2*R^2)*X[1] - 2*r*(r + R)*X[3]

X(54408) lies on these lines: {1, 3}, {9, 11}, {33, 38}, {34, 1496}, {48, 53557}, {63, 497}, {84, 6284}, {90, 4857}, {191, 51785}, {200, 11502}, {212, 614}, {244, 1253}, {283, 5324}, {390, 3218}, {496, 26921}, {518, 1998}, {672, 1732}, {984, 9817}, {997, 48713}, {1000, 48363}, {1158, 10624}, {1317, 7966}, {1364, 3056}, {1435, 23710}, {1479, 7330}, {1708, 11019}, {1709, 9580}, {1711, 33141}, {1731, 30224}, {1836, 5735}, {1837, 34606}, {2170, 32578}, {2183, 29639}, {2194, 18163}, {2260, 54359}, {2310, 36263}, {2361, 7290}, {2551, 6734}, {3027, 24469}, {3058, 3928}, {3100, 4392}, {3219, 5274}, {3220, 10833}, {3242, 51361}, {3305, 10589}, {3306, 5218}, {3318, 3321}, {3452, 20588}, {3486, 34610}, {3583, 18540}, {3586, 49176}, {3662, 27542}, {3705, 3719}, {3752, 7074}, {3874, 10393}, {3877, 13279}, {3929, 7082}, {4302, 7171}, {4316, 7284}, {4319, 7004}, {4336, 46901}, {4640, 42842}, {4845, 34925}, {4907, 53524}, {5219, 15298}, {5227, 12589}, {5230, 28272}, {5250, 30478}, {5273, 10527}, {5281, 27003}, {5432, 5437}, {5541, 8275}, {5578, 5579}, {5705, 50206}, {5715, 26481}, {6056, 54373}, {6762, 10950}, {7160, 34485}, {7580, 17625}, {7701, 16142}, {7965, 10957}, {7971, 45288}, {8609, 42316}, {9581, 10953}, {9779, 29007}, {9785, 11240}, {9841, 15338}, {10916, 12572}, {11376, 24953}, {11520, 45230}, {11525, 17636}, {12053, 12514}, {12678, 15239}, {12701, 12705}, {15171, 24467}, {15348, 52457}, {15845, 24703}, {15852, 34046}, {16141, 54302}, {17276, 38357}, {17452, 41423}, {18240, 52769}, {18450, 35986}, {24392, 42012}, {24987, 26040}, {26878, 47743}, {28076, 40950}, {36481, 36483}, {36488, 36540}, {36501, 36504}, {36509, 36572}, {43819, 43856}, {45634, 49170}

X(54408) = X(i)-Ceva conjugate of X(j) for these (i,j): {30237, 513}, {52457, 34526}
X(54408) = X(i)-isoconjugate of X(j) for these (i,j): {57, 34525}, {664, 46006}
X(54408) = X(i)-Dao conjugate of X(j) for these (i,j): {5452, 34525}, {15348, 8}, {39025, 46006}
X(54408) = barycentric product X(i)*X(j) for these {i,j}: {1, 52457}, {7, 34526}
X(54408) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 34525}, {3063, 46006}, {34526, 8}, {52457, 75}
X(54408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 165, 2078}, {1, 5536, 57}, {1, 5709, 37550}, {1, 10268, 11510}, {35, 17437, 37534}, {40, 57, 1155}, {55, 4860, 17603}, {55, 18839, 1}, {63, 497, 30223}, {165, 10388, 55}, {354, 1155, 56}, {1155, 3057, 55}, {1697, 3333, 2646}, {2078, 11012, 37578}, {3333, 24468, 46}, {3576, 35445, 34879}, {3660, 50196, 354}, {5536, 41338, 5709}, {5570, 40292, 18443}, {8186, 8187, 32760}, {32622, 32623, 3359}, {45625, 45626, 10966}

X(54409) = X(3)X(6)∩X(35)X(44)

Barycentrics    a^2*(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) : :

X(54409) lies on these lines: {1, 19297}, {3, 6}, {9, 3467}, {21, 17330}, {23, 5276}, {35, 44}, {36, 16666}, {37, 3746}, {41, 17796}, {45, 55}, {100, 17369}, {141, 21516}, {198, 1953}, {524, 21511}, {546, 50036}, {590, 21553}, {597, 21495}, {599, 11343}, {615, 21492}, {941, 3444}, {966, 16865}, {999, 20997}, {1001, 51550}, {1100, 5563}, {1172, 3518}, {1213, 5047}, {1405, 5172}, {1444, 3629}, {1696, 16674}, {1732, 34879}, {1990, 7412}, {1992, 21508}, {1995, 5275}, {2174, 2269}, {2178, 3304}, {2280, 41341}, {2975, 4969}, {3068, 21565}, {3069, 21568}, {3295, 16672}, {3496, 4053}, {3553, 7991}, {3554, 30389}, {3589, 21540}, {3627, 53421}, {3763, 21496}, {3871, 3943}, {3913, 50087}, {4189, 37654}, {4220, 5306}, {4700, 5267}, {5010, 16670}, {5258, 50082}, {5259, 52706}, {5277, 16427}, {6144, 21517}, {6749, 37305}, {7113, 22357}, {7300, 40937}, {7496, 33854}, {8252, 21549}, {8253, 21546}, {8584, 35276}, {8609, 34486}, {8666, 50131}, {8715, 17281}, {9300, 19649}, {9607, 37328}, {9766, 21485}, {11010, 21864}, {11349, 17392}, {13846, 16433}, {13847, 16432}, {15533, 21509}, {15534, 16436}, {16042, 37675}, {16373, 37673}, {16431, 51185}, {16440, 32788}, {16441, 32787}, {16522, 37609}, {16554, 19302}, {16644, 21481}, {16645, 21480}, {16885, 54285}, {17362, 38871}, {17398, 17531}, {19053, 21567}, {19054, 21566}, {19237, 31144}, {19308, 46922}, {21358, 21514}, {21475, 49948}, {21476, 49947}, {21477, 47352}, {21510, 40341}, {21515, 51186}, {21519, 47355}, {21783, 35216}, {23854, 23868}, {24328, 49747}, {37441, 40138}, {45987, 52555}

X(54409) = isogonal conjugate of the isotomic conjugate of X(37656)
X(54409) = X(39974)-Ceva conjugate of X(6)
X(54409) = crossdifference of every pair of points on line {523, 3960}
X(54409) = barycentric product X(6)*X(37656)
X(54409) = barycentric quotient X(37656)/X(76)
X(54409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 4254, 37503}, {3, 37503, 6}, {6, 1030, 5124}, {6, 4258, 4289}, {6, 36744, 1030}, {6, 37504, 4287}, {32, 4277, 6}, {39, 33882, 6}, {55, 4471, 16686}, {58, 4285, 6}, {61, 62, 36750}, {284, 4271, 6}, {371, 372, 51340}, {386, 4290, 6}, {573, 584, 6}, {1333, 4263, 6}, {1953, 46823, 16777}, {2092, 2220, 6}, {2245, 4251, 6}, {2278, 4266, 6}, {4254, 36744, 6}, {4261, 16946, 6}, {4262, 4266, 2278}, {4264, 4272, 6}, {4270, 4275, 6}, {4273, 4274, 6}, {22236, 22238, 36746}, {36744, 37503, 3}

X(54410) = X(1)X(6)∩X(3)X(75)

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

X(54410) lies on these lines: {1, 6}, {3, 75}, {8, 37225}, {10, 34247}, {21, 192}, {25, 92}, {35, 49474}, {55, 740}, {56, 4032}, {183, 1921}, {228, 5271}, {239, 37502}, {312, 16058}, {321, 1011}, {333, 20760}, {345, 8731}, {346, 52241}, {404, 4699}, {474, 3739}, {495, 4205}, {536, 16370}, {716, 47037}, {726, 993}, {742, 36740}, {894, 37507}, {975, 19518}, {976, 3728}, {1009, 2345}, {1078, 10009}, {1278, 4189}, {1284, 12588}, {1403, 32916}, {1621, 37316}, {1631, 2915}, {1999, 37323}, {2053, 2218}, {2178, 19329}, {2223, 50314}, {2352, 11358}, {2975, 13733}, {3145, 4812}, {3286, 4363}, {3295, 49470}, {3303, 49471}, {3428, 29054}, {3560, 20430}, {3564, 15976}, {3644, 17571}, {3685, 23407}, {3695, 10449}, {3696, 5687}, {3741, 4438}, {3746, 49469}, {3781, 28287}, {3797, 16367}, {3883, 31394}, {3913, 49459}, {3923, 20992}, {3993, 5248}, {4184, 28605}, {4188, 4772}, {4191, 4359}, {4358, 16373}, {4361, 5132}, {4421, 50086}, {4423, 29644}, {4664, 16418}, {4672, 36635}, {4681, 19526}, {4686, 19535}, {4687, 11108}, {4688, 16371}, {4698, 16842}, {4704, 16865}, {4709, 8715}, {4739, 19537}, {4740, 17549}, {4751, 16408}, {4755, 17542}, {4821, 17548}, {4980, 19346}, {5047, 27268}, {5192, 27261}, {5263, 33745}, {5267, 50117}, {5282, 20706}, {5695, 8053}, {5719, 16848}, {5739, 21319}, {5762, 10446}, {6284, 21927}, {7193, 52134}, {7295, 8424}, {8666, 49479}, {10436, 37609}, {10453, 17776}, {10479, 17293}, {10892, 53260}, {11194, 31178}, {11322, 31025}, {11344, 20171}, {11679, 21483}, {12329, 49531}, {13615, 20173}, {13723, 38871}, {14021, 20533}, {16059, 19804}, {16286, 18137}, {16292, 18147}, {16368, 19791}, {16846, 17322}, {16850, 17321}, {16857, 51488}, {16862, 31238}, {16998, 19565}, {17524, 50044}, {17718, 43223}, {17740, 30944}, {17894, 22388}, {18042, 23095}, {19533, 37539}, {19785, 37329}, {21010, 50302}, {21161, 51043}, {21330, 28082}, {23093, 35519}, {23863, 25124}, {24320, 49516}, {24789, 50199}, {25524, 40328}, {26107, 37042}, {27282, 37314}, {27491, 31319}, {28453, 51039}, {28463, 51047}, {30271, 37022}, {30699, 37175}, {36011, 36494}, {36280, 36294}, {36741, 49481}, {37492, 49496}, {51062, 51506}

X(54410) = X(6)-isoconjugate of X(45965)
X(54410) = X(9)-Dao conjugate of X(45965)
X(54410) = barycentric quotient X(1)/X(45965)
X(54410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 21061, 10477}, {2352, 31993, 11358}, {3696, 15624, 5687}, {38871, 41230, 13723}

X(54411) = X(3)X(6)∩X(21)X(73)

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

X(54411) lies on these lines: {1, 5327}, {3, 6}, {21, 73}, {28, 4303}, {29, 34}, {81, 1936}, {212, 4184}, {222, 8021}, {223, 17194}, {1001, 10571}, {1427, 18165}, {1437, 40602}, {1745, 25516}, {1790, 2299}, {1817, 22053}, {1818, 2287}, {1838, 34830}, {2328, 3173}, {2360, 22654}, {2659, 31623}, {3145, 44112}, {3194, 44709}, {3330, 15972}, {4215, 26892}, {5712, 10458}, {5784, 16699}, {7078, 17524}, {7532, 15668}, {7538, 17379}, {7572, 17277}, {10391, 18603}, {15852, 18178}, {17056, 37370}, {17187, 40958}, {18166, 41344}, {27164, 34831}, {36020, 45963}

X(54411) = barycentric product X(i)*X(j) for these {i,j}: {85, 36020}, {333, 45963}
X(54411) = barycentric quotient X(i)/X(j) for these {i,j}: {36020, 9}, {45963, 226}
X(54411) = {X(81),X(35981)}-harmonic conjugate of X(1936)

X(54412) = X(4)X(69)∩X(24)X(99)

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

X(54412) lies on the cubics K518 and K620 and also on these lines: {2, 22401}, {4, 69}, {24, 99}, {25, 305}, {30, 41009}, {32, 15014}, {33, 1909}, {34, 350}, {64, 290}, {75, 46878}, {95, 7395}, {112, 6179}, {183, 1593}, {186, 7782}, {190, 41320}, {193, 21447}, {194, 232}, {235, 325}, {274, 406}, {275, 41231}, {297, 3981}, {310, 4207}, {313, 1869}, {331, 40028}, {339, 382}, {378, 1078}, {384, 10311}, {385, 1968}, {393, 6339}, {403, 7752}, {420, 41259}, {427, 40022}, {458, 4074}, {468, 11059}, {475, 18140}, {538, 3199}, {648, 2207}, {671, 43678}, {850, 46371}, {1007, 6622}, {1093, 6528}, {1118, 18026}, {1172, 34283}, {1228, 4198}, {1230, 6994}, {1321, 34391}, {1322, 34392}, {1494, 52646}, {1596, 3933}, {1861, 6376}, {1885, 7750}, {1974, 12215}, {2052, 2996}, {2211, 32451}, {2481, 43742}, {3053, 37199}, {3087, 26214}, {3088, 32828}, {3089, 3926}, {3091, 26166}, {3144, 30022}, {3146, 30737}, {3172, 14614}, {3192, 33296}, {3266, 4232}, {3520, 7771}, {3541, 32832}, {3542, 7763}, {3575, 32819}, {3934, 33843}, {3972, 10312}, {4194, 34284}, {4196, 18152}, {4200, 18135}, {5186, 5976}, {5200, 45805}, {5286, 17907}, {6337, 6353}, {6390, 21841}, {6623, 32816}, {6823, 45198}, {6995, 8024}, {7378, 39998}, {7487, 28706}, {7505, 7769}, {7547, 15031}, {7738, 37187}, {7745, 27377}, {7748, 40889}, {7757, 39575}, {7760, 8743}, {7767, 13488}, {7770, 36794}, {7773, 37197}, {7802, 18560}, {7805, 14581}, {7812, 37855}, {7814, 44958}, {8149, 33874}, {9217, 14382}, {9464, 52301}, {10604, 40050}, {11414, 46724}, {11470, 39099}, {14063, 26179}, {14265, 22456}, {14457, 54124}, {15149, 30830}, {16089, 18913}, {18018, 34603}, {20477, 39568}, {26164, 32974}, {26235, 52284}, {27376, 47286}, {28660, 37384}, {28809, 37102}, {30716, 37915}, {32581, 52570}, {34505, 52282}, {35477, 43459}, {37765, 41361}, {40009, 46140}, {40680, 52404}, {40890, 52628}, {41584, 51374}, {41760, 44518}, {44228, 47392}, {45806, 52291}, {46105, 53105}, {53197, 53205}

X(54412) = isogonal conjugate of X(40319)
X(54412) = isotomic conjugate of X(6391)
X(54412) = anticomplement of X(22401)
X(54412) = polar conjugate of X(8770)
X(54412) = isotomic conjugate of the isogonal conjugate of X(6353)
X(54412) = isotomic conjugate of the polar conjugate of X(21447)
X(54412) = polar conjugate of the isogonal conjugate of X(193)
X(54412) = X(40413)-anticomplementary conjugate of X(4329)
X(54412) = X(i)-Ceva conjugate of X(j) for these (i,j): {2052, 264}, {34537, 648}
X(54412) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40319}, {3, 38252}, {31, 6391}, {48, 8770}, {63, 53059}, {184, 8769}, {255, 14248}, {560, 6340}, {810, 3565}, {2996, 9247}, {34208, 52430}
X(54412) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6391}, {3, 40319}, {69, 394}, {1249, 8770}, {2489, 3124}, {3162, 53059}, {6353, 9924}, {6374, 6340}, {6388, 520}, {6509, 45199}, {6523, 14248}, {15525, 647}, {36103, 38252}, {39062, 3565}, {51579, 3}
cevapoint of X(i) and X(j) for these (i,j): {4, 6392}, {25, 40318}, {193, 6353}
X(54412) = barycentric product X(i)*X(j) for these {i,j}: {69, 21447}, {76, 6353}, {92, 18156}, {193, 264}, {276, 41588}, {308, 41584}, {683, 40326}, {1502, 19118}, {1707, 1969}, {2052, 6337}, {3053, 18022}, {3167, 18027}, {3566, 6331}, {4028, 44129}, {5139, 34537}, {7017, 17081}, {16081, 51374}, {17876, 46254}, {32459, 46111}, {47733, 51843}
X(54412) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6391}, {4, 8770}, {6, 40319}, {19, 38252}, {25, 53059}, {76, 6340}, {92, 8769}, {193, 3}, {264, 2996}, {324, 27364}, {393, 14248}, {439, 3167}, {648, 3565}, {1611, 53068}, {1707, 48}, {2052, 34208}, {3053, 184}, {3167, 577}, {3566, 647}, {3787, 20775}, {3798, 1459}, {4028, 71}, {5139, 3124}, {6331, 35136}, {6337, 394}, {6353, 6}, {6388, 20975}, {8651, 3049}, {8940, 6414}, {8944, 6413}, {10607, 1092}, {13567, 45199}, {17081, 222}, {17876, 3708}, {18156, 63}, {19118, 32}, {21447, 4}, {21874, 228}, {21970, 5158}, {32459, 3292}, {33632, 10547}, {37174, 40809}, {37199, 9306}, {37778, 5203}, {40318, 15261}, {40326, 6467}, {41584, 39}, {41588, 216}, {47733, 3504}, {51374, 36212}
X(54412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 76, 264}, {4, 3186, 40325}, {4, 32001, 32006}, {4, 44146, 76}, {264, 340, 44133}, {297, 40814, 15466}, {2207, 7754, 648}, {3934, 33843, 37337}, {14615, 44131, 264}, {32001, 44131, 14615}, {37174, 51481, 2052}, {43976, 44132, 264}, {44780, 44781, 14615}

X(54413) = X(2)X729)∩X(6)X(538)

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

X(54413) lies on the cubics K281 and K731 and also on these lines: {2, 729}, {6, 538}, {32, 3231}, {39, 33705}, {182, 14609}, {187, 46319}, {213, 52893}, {574, 9468}, {1084, 30495}, {1918, 52894}, {1974, 5033}, {3016, 14601}, {3224, 7808}, {3288, 22111}, {5034, 39238}, {5970, 6787}, {7787, 42346}, {7815, 36615}, {9431, 15482}, {9463, 12150}, {9490, 17130}, {9516, 42534}, {19137, 41412}, {40354, 52905}

X(54413) = isogonal conjugate of X(7757)
X(54413) = isogonal conjugate of the anticomplement of X(9466)
X(54413) = isogonal conjugate of the isotomic conjugate of X(9462)
X(54413) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7757}, {2, 36289}, {75, 9463}, {662, 5996}, {799, 9009}, {1966, 11654}, {4602, 9489}
X(54413) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 7757}, {206, 9463}, {1084, 5996}, {9467, 11654}, {32664, 36289}, {38996, 9009}
X(54413) = trilinear pole of line {669, 888}
X(54413) = crossdifference of every pair of points on line {5996, 9009}
X(54413) = barycentric product X(i)*X(j) for these {i,j}: {6, 9462}, {512, 9066}
X(54413) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 7757}, {31, 36289}, {32, 9463}, {512, 5996}, {669, 9009}, {9066, 670}, {9426, 9489}, {9462, 76}, {9468, 11654}

X(54414) = X(1)X1864)∩X(9)X(222)

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

X(54414) lies on the cubics K333 and K713 and also on these lines: {1, 1864}, {9, 222}, {57, 23089}, {223, 329}, {226, 2999}, {651, 1422}, {1103, 6260}, {1490, 22350}, {1743, 3173}, {1750, 4551}, {6510, 34032}, {8808, 27508}, {9371, 53087}, {23511, 52659}, {45204, 54366}

X(54414) = X(7)-Ceva conjugate of X(40)
X(54414) = X(2324)-Dao conjugate of X(8)
X(54414) = barycentric product X(347)*X(6765)
X(54414) = barycentric quotient X(6765)/X(280)

X(54415) = ISOGONAL CONJUGATE OF X(2931)

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

X(54415) lies on the cubics K611 and K1169, and on the curve Q066, and on these lines: {2, 38936}, {186, 2931}, {403, 16310}, {3153, 5962}, {3448, 32710}, {5627, 40388}, {10421, 18533}, {12028, 16221}, {14222, 14618}, {51965, 52487}

X(54415) = isogonal conjugate of X(2931)
X(54415) = cyclocevian conjugate of X(94)
X(54415) = isogonal conjugate of the anticomplement of X(23306)
X(54415) = isogonal conjugate of the complement of X(12319)
X(54415) = isotomic conjugate of the anticomplement of X(14910)
X(54415) = X(1)-isoconjugate of X(2931)
X(54415) = X(3)-Dao conjugate of X(2931)
X(54415) = cevapoint of X(i) and X(j) for these (i,j): {512, 39021}, {523, 16221}
X(54415) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2931}, {40388, 40392}

X(54416) = X(1)X(6)∩X(25)X(41)

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

X(54416) lies on these lines: {1, 6}, {2, 31402}, {3, 172}, {5, 9596}, {8, 5276}, {10, 5275}, {11, 2548}, {12, 3767}, {25, 41}, {30, 9598}, {31, 1334}, {32, 55}, {33, 2207}, {35, 609}, {36, 5013}, {39, 56}, {57, 9593}, {58, 3730}, {65, 9620}, {75, 11321}, {81, 7123}, {86, 27248}, {101, 386}, {115, 9650}, {140, 31497}, {165, 31426}, {169, 41015}, {171, 3501}, {183, 27020}, {187, 5217}, {192, 384}, {194, 6645}, {198, 2092}, {217, 19354}, {222, 3497}, {230, 498}, {232, 11399}, {239, 39731}, {304, 894}, {312, 41236}, {315, 26590}, {321, 19281}, {330, 7839}, {344, 33821}, {346, 2298}, {350, 7770}, {371, 31459}, {388, 5286}, {404, 17756}, {406, 1783}, {474, 1575}, {495, 5305}, {496, 9599}, {499, 3815}, {517, 54382}, {574, 5204}, {604, 4322}, {608, 2171}, {612, 1184}, {631, 31462}, {672, 1468}, {728, 5269}, {904, 41268}, {910, 4646}, {940, 3912}, {949, 36018}, {975, 16849}, {976, 3930}, {986, 3509}, {987, 1922}, {993, 25092}, {999, 2275}, {1010, 2303}, {1015, 3304}, {1018, 5264}, {1030, 37557}, {1060, 23115}, {1155, 1571}, {1172, 7718}, {1193, 9310}, {1213, 19784}, {1278, 16913}, {1319, 9619}, {1333, 17524}, {1376, 5277}, {1415, 11509}, {1420, 9592}, {1429, 52424}, {1438, 2334}, {1478, 5254}, {1479, 7745}, {1504, 18996}, {1505, 18995}, {1570, 5194}, {1572, 3057}, {1574, 4413}, {1580, 37327}, {1611, 5268}, {1613, 40790}, {1672, 12051}, {1673, 12050}, {1692, 5148}, {1696, 21796}, {1759, 4424}, {1909, 7754}, {1914, 3295}, {1930, 4363}, {1975, 25264}, {2023, 10069}, {2066, 6424}, {2067, 6422}, {2162, 3494}, {2172, 2174}, {2178, 4261}, {2179, 20665}, {2221, 28606}, {2238, 4205}, {2241, 3303}, {2251, 4258}, {2285, 2286}, {2292, 5282}, {2295, 3695}, {2304, 9454}, {2330, 40825}, {2344, 43073}, {2476, 17737}, {2549, 7354}, {2915, 18755}, {2975, 31449}, {3027, 10797}, {3063, 48327}, {3085, 7735}, {3086, 7736}, {3125, 16550}, {3157, 23128}, {3172, 7071}, {3175, 50060}, {3204, 4272}, {3207, 4255}, {3208, 5255}, {3240, 4239}, {3287, 48290}, {3496, 37598}, {3585, 44518}, {3614, 43620}, {3616, 33854}, {3666, 11343}, {3670, 17736}, {3672, 17691}, {3674, 6180}, {3684, 50581}, {3693, 37539}, {3735, 37614}, {3746, 7031}, {3752, 37272}, {3758, 18156}, {3774, 34247}, {3780, 36479}, {3911, 31396}, {3915, 21764}, {3920, 5359}, {3924, 21808}, {3991, 5266}, {3993, 49545}, {3995, 11320}, {4000, 17682}, {4204, 21753}, {4252, 42316}, {4254, 12410}, {4257, 24047}, {4262, 33771}, {4263, 51436}, {4293, 7738}, {4316, 44519}, {4317, 9607}, {4366, 7787}, {4372, 24326}, {4383, 17023}, {4386, 5687}, {4390, 10459}, {4441, 17686}, {4513, 5710}, {4644, 17170}, {4699, 16911}, {4704, 16914}, {4972, 26085}, {5010, 5023}, {5019, 54322}, {5058, 19038}, {5062, 19037}, {5122, 31430}, {5206, 9341}, {5229, 43448}, {5262, 16048}, {5271, 19725}, {5306, 10056}, {5309, 11237}, {5319, 15888}, {5332, 6767}, {5341, 16545}, {5364, 40978}, {5414, 6423}, {5422, 26639}, {5433, 31401}, {5434, 7739}, {5452, 20970}, {5475, 10896}, {5703, 40129}, {5706, 39591}, {5724, 40997}, {6161, 43929}, {6198, 8743}, {6284, 7737}, {6421, 6502}, {6602, 10460}, {7079, 20310}, {7083, 40969}, {7085, 44119}, {7109, 37316}, {7173, 31415}, {7232, 17192}, {7280, 15815}, {7288, 31400}, {7368, 16283}, {7746, 31476}, {7747, 9664}, {7748, 12943}, {7749, 31501}, {7753, 9665}, {7763, 26686}, {7765, 9651}, {7778, 30103}, {7803, 26561}, {7951, 13881}, {8193, 36744}, {8588, 46846}, {9300, 10072}, {9336, 37602}, {9548, 37320}, {9574, 15803}, {9597, 15048}, {9661, 31463}, {9669, 15484}, {9780, 37675}, {10053, 12829}, {10311, 11398}, {10483, 44526}, {10527, 31466}, {10589, 31404}, {10592, 43291}, {10802, 34870}, {10987, 21309}, {11174, 26959}, {11269, 37315}, {11333, 41318}, {11337, 32758}, {11358, 21877}, {11363, 45786}, {11392, 27376}, {11501, 21859}, {12150, 53680}, {12836, 46305}, {12948, 31472}, {12949, 44622}, {13006, 22766}, {13733, 39686}, {13898, 31481}, {14986, 37665}, {14996, 29583}, {15171, 18907}, {15325, 31406}, {16394, 17281}, {16458, 17303}, {16549, 37522}, {16583, 40131}, {16818, 17259}, {16912, 27268}, {16915, 17759}, {16992, 27255}, {16995, 53675}, {16997, 26752}, {17015, 33950}, {17018, 37325}, {17143, 20172}, {17144, 20179}, {17233, 33954}, {17280, 17688}, {17284, 37674}, {17302, 33827}, {17398, 19836}, {17451, 49487}, {17592, 21775}, {17698, 24512}, {17743, 41240}, {17754, 37607}, {17976, 50598}, {18140, 26687}, {18166, 33953}, {18447, 22120}, {18596, 37241}, {18993, 45582}, {18994, 45583}, {19030, 31411}, {19231, 20136}, {19349, 39643}, {19785, 50200}, {20181, 32104}, {20227, 54385}, {20861, 50591}, {20980, 48332}, {21007, 48324}, {21070, 48863}, {21348, 22157}, {21789, 21837}, {21843, 52793}, {23151, 37676}, {24914, 31398}, {25066, 54317}, {26036, 33137}, {26223, 42707}, {26363, 37661}, {26437, 43039}, {26626, 32911}, {28594, 30142}, {29579, 37633}, {29585, 37685}, {29598, 37679}, {29633, 37673}, {29674, 40750}, {29815, 34482}, {30130, 33937}, {31231, 31428}, {31433, 37568}, {34046, 52635}, {35768, 45512}, {35769, 45513}, {37314, 37657}, {37546, 54409}, {37589, 39255}, {41323, 50592}, {44103, 51686}, {49771, 50028}

X(54416) = isogonal conjugate of the isotomic conjugate of X(2345)
X(54416) = isogonal conjugate of the polar conjugate of X(7102)
X(54416) = polar conjugate of the isotomic conjugate of X(7085)
X(54416) = X(i)-Ceva conjugate of X(j) for these (i,j): {941, 55}, {2285, 1460}, {2303, 612}, {2345, 7085}, {6574, 667}
X(54416) = X(i)-isoconjugate of X(j) for these (i,j): {7, 2339}, {57, 30479}, {75, 2221}, {76, 1472}, {85, 1036}, {274, 1245}, {304, 51686}, {310, 2281}, {348, 1039}, {513, 37215}, {514, 1310}, {4025, 36099}, {10436, 34260}, {15413, 32691}
X(54416) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 2221}, {958, 34284}, {5452, 30479}, {5515, 3261}, {17421, 15413}, {39026, 37215}, {40125, 4000}, {40181, 75}
X(54416) = trilinear pole of line {2484, 8646}
X(54416) = crossdifference of every pair of points on line {513, 3004}
X(54416) = barycentric product X(i)*X(j) for these {i,j}: {1, 612}, {3, 7102}, {4, 7085}, {6, 2345}, {8, 1460}, {9, 2285}, {10, 44119}, {19, 5227}, {31, 4385}, {33, 1038}, {37, 2303}, {42, 1010}, {55, 388}, {56, 3974}, {72, 4206}, {99, 50494}, {100, 8678}, {101, 6590}, {110, 48395}, {190, 2484}, {200, 4320}, {210, 5323}, {220, 7365}, {281, 2286}, {480, 7197}, {663, 14594}, {668, 8646}, {692, 2517}, {941, 34261}, {1184, 30701}, {1260, 7103}, {1474, 3610}, {1783, 2522}, {1918, 44154}, {1973, 19799}, {2287, 8898}, {4557, 47844}, {5286, 7123}, {7070, 10375}, {8750, 23874}, {8816, 30706}, {17742, 40184}
X(54416) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 2221}, {41, 2339}, {55, 30479}, {101, 37215}, {388, 6063}, {560, 1472}, {612, 75}, {692, 1310}, {1010, 310}, {1038, 7182}, {1184, 4000}, {1460, 7}, {1918, 1245}, {1974, 51686}, {2175, 1036}, {2205, 2281}, {2212, 1039}, {2285, 85}, {2286, 348}, {2303, 274}, {2345, 76}, {2484, 514}, {2517, 40495}, {2522, 15413}, {3610, 40071}, {3974, 3596}, {4206, 286}, {4320, 1088}, {4385, 561}, {5227, 304}, {6590, 3261}, {7085, 69}, {7102, 264}, {8646, 513}, {8678, 693}, {8898, 1446}, {14594, 4572}, {19459, 17170}, {19799, 40364}, {30701, 40831}, {34261, 34284}, {36744, 14258}, {40184, 46740}, {44119, 86}, {47844, 52619}, {48395, 850}, {50494, 523}
X(54416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 16502}, {1, 5280, 6}, {1, 5299, 16781}, {1, 17742, 37}, {1, 54329, 41239}, {2, 31402, 31460}, {3, 2276, 31448}, {6, 220, 213}, {6, 2176, 16466}, {6, 2256, 2300}, {6, 16781, 5299}, {31, 1334, 14974}, {32, 1500, 55}, {35, 609, 3053}, {37, 4426, 405}, {39, 2242, 56}, {41, 42, 2271}, {41, 607, 30706}, {41, 51949, 2200}, {115, 9650, 10895}, {172, 2276, 3}, {187, 31451, 5217}, {312, 41258, 41236}, {672, 1468, 5021}, {999, 9605, 2275}, {1124, 1335, 611}, {1914, 7296, 30435}, {2176, 16523, 1}, {2303, 2345, 34261}, {3053, 31477, 35}, {3295, 30435, 1914}, {3767, 31409, 12}, {4386, 20691, 5687}, {5058, 31471, 19038}, {5280, 16785, 1}, {5283, 5291, 958}, {5299, 16781, 16502}, {7031, 9331, 3746}, {7747, 9664, 12953}, {15048, 18990, 9597}

X(54417) = X(3)X(6)∩X(21)X(60)

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

X(54417) lies on these lines: {1, 1437}, {3, 6}, {21, 60}, {28, 1905}, {35, 22276}, {46, 18163}, {55, 283}, {56, 1790}, {65, 81}, {86, 28628}, {184, 19765}, {209, 54337}, {314, 52550}, {333, 26066}, {407, 49745}, {442, 6703}, {501, 34956}, {662, 37442}, {692, 37573}, {859, 22766}, {940, 4185}, {1010, 5794}, {1064, 44709}, {1428, 4719}, {1682, 20959}, {1724, 16455}, {1780, 17524}, {1792, 2330}, {1800, 22768}, {1834, 37527}, {1837, 11103}, {2182, 2303}, {2206, 10457}, {3057, 3193}, {3612, 52680}, {3812, 17518}, {4189, 34259}, {4221, 14110}, {4225, 40214}, {4340, 14018}, {4653, 17104}, {5324, 17603}, {5327, 37422}, {5743, 7483}, {5799, 37468}, {6910, 14555}, {10458, 13733}, {11112, 48845}, {12609, 17197}, {12675, 25713}, {13750, 18180}, {16948, 37600}, {19767, 44085}, {20832, 44092}, {24929, 41608}, {25681, 31631}, {27622, 45897}, {27644, 28275}, {27660, 37225}, {37296, 51290}, {37552, 47373}, {40980, 54356}, {46976, 53421}

X(54417) = isogonal conjugate of the polar conjugate of X(44734)
X(54417) = X(i)-isoconjugate of X(j) for these (i,j): {10, 959}, {12, 5331}, {37, 44733}, {65, 31359}, {225, 34259}, {226, 941}, {522, 52931}, {661, 32038}, {1400, 34258}, {1441, 2258}, {1577, 32693}, {2171, 37870}
X(54417) = X(i)-Dao conjugate of X(j) for these (i,j): {17417, 1577}, {34261, 321}, {36830, 32038}, {40582, 34258}, {40589, 44733}, {40602, 31359}
X(54417) = barycentric product X(i)*X(j) for these {i,j}: {3, 44734}, {21, 940}, {58, 11679}, {60, 31993}, {81, 958}, {86, 2268}, {110, 23880}, {283, 5307}, {284, 10436}, {314, 5019}, {333, 1468}, {593, 3714}, {643, 48144}, {662, 17418}, {1014, 3713}, {1790, 54396}, {1812, 4185}, {2194, 34284}, {4570, 53526}, {4612, 8672}, {4631, 8639}, {4636, 50457}, {5546, 43067}, {16049, 34279}
X(54417) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 34258}, {58, 44733}, {60, 37870}, {110, 32038}, {284, 31359}, {314, 40828}, {940, 1441}, {958, 321}, {1333, 959}, {1415, 52931}, {1468, 226}, {1576, 32693}, {2150, 5331}, {2193, 34259}, {2194, 941}, {2268, 10}, {3713, 3701}, {3714, 28654}, {4185, 40149}, {5019, 65}, {10436, 349}, {11679, 313}, {17418, 1577}, {23880, 850}, {31993, 34388}, {44734, 264}, {48144, 4077}, {52143, 34263}, {53526, 21207}
X(54417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 50597, 33844}, {6, 1030, 50033}, {21, 60, 2194}, {21, 1812, 960}, {58, 284, 4267}, {58, 15792, 4276}, {58, 54323, 3}, {81, 16049, 65}, {333, 37232, 26066}, {1805, 1806, 6}

X(54418) = X(1)X(2)∩X(6)X(19)

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

X(54418) lies on these lines: {1, 2}, {4, 1039}, {6, 19}, {9, 2292}, {12, 3772}, {21, 17594}, {28, 44119}, {31, 40}, {33, 429}, {35, 37817}, {36, 39582}, {46, 58}, {55, 1104}, {56, 197}, {57, 961}, {63, 986}, {75, 1220}, {77, 3212}, {81, 17518}, {100, 37552}, {141, 10371}, {169, 5280}, {171, 24440}, {201, 8557}, {204, 11471}, {208, 17408}, {213, 9620}, {223, 1042}, {238, 5250}, {241, 15832}, {244, 3333}, {294, 5665}, {341, 32926}, {354, 17054}, {377, 1738}, {388, 4000}, {405, 968}, {442, 5725}, {517, 16466}, {518, 37549}, {593, 37405}, {595, 5119}, {601, 3359}, {672, 9593}, {748, 31435}, {750, 37554}, {774, 10396}, {894, 21216}, {896, 54290}, {940, 3812}, {950, 3755}, {956, 19527}, {958, 3666}, {959, 34045}, {960, 4383}, {964, 50314}, {988, 2975}, {990, 5691}, {1001, 37548}, {1038, 1788}, {1040, 3486}, {1046, 21376}, {1054, 37608}, {1056, 23675}, {1062, 37730}, {1086, 10404}, {1155, 4252}, {1191, 3057}, {1203, 5903}, {1253, 1697}, {1279, 3303}, {1329, 17720}, {1334, 16970}, {1376, 37539}, {1386, 5710}, {1394, 9316}, {1402, 13738}, {1403, 22345}, {1448, 3339}, {1451, 37550}, {1455, 1466}, {1458, 1467}, {1469, 16980}, {1478, 23537}, {1497, 24028}, {1575, 54317}, {1616, 5919}, {1706, 4695}, {1716, 26117}, {1721, 3146}, {1724, 4424}, {1739, 37522}, {1743, 12526}, {1751, 2258}, {1757, 3951}, {1763, 35650}, {1764, 34281}, {1772, 17700}, {1854, 1864}, {1891, 5800}, {1902, 3195}, {2092, 5336}, {2093, 2308}, {2099, 40635}, {2170, 9575}, {2176, 52370}, {2177, 16485}, {2191, 2334}, {2197, 2277}, {2274, 37523}, {2276, 16968}, {2280, 16780}, {2295, 16972}, {2300, 10480}, {2475, 33131}, {2476, 17064}, {2478, 24210}, {2550, 5716}, {2646, 4255}, {2650, 11529}, {2901, 42707}, {3052, 37568}, {3058, 34656}, {3120, 9612}, {3304, 52541}, {3306, 24174}, {3338, 24046}, {3340, 34036}, {3361, 54310}, {3436, 13161}, {3485, 19372}, {3553, 26063}, {3589, 5835}, {3610, 17314}, {3612, 4256}, {3646, 17125}, {3663, 12527}, {3664, 41826}, {3671, 5813}, {3677, 6762}, {3685, 17697}, {3691, 16517}, {3698, 3745}, {3702, 5192}, {3704, 32777}, {3727, 39248}, {3729, 17489}, {3735, 54406}, {3736, 37232}, {3743, 54287}, {3744, 3913}, {3749, 3871}, {3751, 3868}, {3753, 5711}, {3780, 16973}, {3813, 17721}, {3869, 32911}, {3880, 37542}, {3891, 4696}, {3895, 37588}, {3946, 5795}, {3987, 5264}, {4195, 32932}, {4217, 28580}, {4260, 39598}, {4295, 41011}, {4296, 37666}, {4298, 24177}, {4300, 30503}, {4332, 10460}, {4339, 17784}, {4348, 4848}, {4361, 5793}, {4414, 31424}, {4429, 7270}, {4652, 17596}, {4656, 18250}, {4673, 32942}, {4674, 54336}, {4749, 6284}, {4868, 5248}, {4917, 16498}, {4972, 5016}, {5018, 23579}, {5046, 33134}, {5252, 17366}, {5253, 11512}, {5255, 16478}, {5260, 28606}, {5266, 5687}, {5275, 16605}, {5290, 23681}, {5310, 8193}, {5315, 5697}, {5322, 9798}, {5338, 44115}, {5429, 37603}, {5436, 37553}, {5584, 15852}, {5587, 21935}, {5706, 7686}, {5712, 28629}, {5718, 28628}, {5724, 5794}, {5727, 7221}, {5838, 12560}, {5880, 49745}, {6051, 11108}, {6203, 18992}, {6204, 18991}, {6210, 23659}, {6261, 37732}, {7174, 21039}, {7194, 39969}, {7354, 34666}, {7986, 40263}, {7991, 16469}, {8715, 49480}, {9548, 27659}, {9605, 43065}, {9619, 53165}, {9643, 37721}, {9817, 54361}, {9895, 37697}, {9957, 16483}, {9958, 48903}, {10315, 52425}, {10375, 41489}, {10436, 20911}, {10448, 46904}, {10572, 48837}, {10822, 26893}, {11114, 50080}, {11236, 50103}, {11319, 32929}, {11354, 50083}, {11375, 37662}, {11523, 49454}, {11681, 33133}, {12607, 17061}, {12652, 20070}, {12709, 34048}, {13407, 24159}, {13601, 34040}, {14014, 46883}, {14110, 36745}, {14529, 44085}, {16451, 16778}, {16473, 53615}, {16474, 18398}, {16583, 40131}, {16600, 17742}, {16610, 25524}, {16974, 20691}, {17164, 26223}, {17185, 27660}, {17277, 31359}, {17279, 25992}, {17301, 34606}, {17597, 34791}, {17602, 21031}, {17863, 27410}, {18178, 40153}, {18446, 37699}, {18623, 41824}, {18732, 23122}, {20060, 33150}, {21075, 34937}, {22119, 41340}, {24161, 31266}, {24474, 44414}, {24789, 25466}, {24914, 37646}, {24954, 51415}, {25681, 37663}, {25917, 37679}, {26066, 35466}, {28076, 40987}, {30380, 31533}, {30381, 31532}, {31426, 41423}, {32860, 54331}, {33094, 41869}, {33781, 50408}, {34046, 37566}, {34339, 36742}, {34720, 50130}, {35258, 54354}, {35635, 51558}, {36752, 37562}, {37314, 50290}, {37529, 42078}, {37558, 45126}, {37615, 37698}, {40941, 54385}, {40959, 52359}, {40985, 44094}, {42051, 48832}, {48812, 50106}, {48821, 50046}, {48826, 50049}, {48827, 49719}, {49732, 50070}, {50055, 50091}

X(54418) = polar conjugate of the isotomic conjugate of X(10319)
X(54418) = X(10319)-Dao conjugate of X(17321)
X(54418) = crossdifference of every pair of points on line {521, 649}
X(54418) = barycentric product X(i)*X(j) for these {i,j}: {4, 10319}, {7, 54359}, {34, 23600}, {57, 2551}, {63, 52082}, {651, 47136}
X(54418) = barycentric quotient X(i)/X(j) for these {i,j}: {2551, 312}, {10319, 69}, {23600, 3718}, {47136, 4391}, {52082, 92}, {54359, 8}
X(54418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10, 612}, {1, 43, 78}, {1, 200, 976}, {1, 614, 28011}, {1, 978, 19861}, {1, 1698, 975}, {1, 1722, 2}, {1, 2999, 1193}, {1, 3216, 997}, {1, 3293, 3811}, {1, 5272, 3616}, {1, 6048, 5293}, {1, 6765, 3938}, {1, 9623, 10459}, {1, 18395, 54401}, {1, 23511, 8583}, {1, 50581, 3870}, {2, 17016, 1}, {6, 3959, 54382}, {6, 41015, 2082}, {8, 5262, 1}, {8, 26965, 31339}, {10, 40940, 5230}, {31, 4642, 40}, {34, 65, 2263}, {40, 1453, 31}, {42, 3924, 1}, {57, 21147, 4320}, {145, 7191, 1}, {238, 37598, 5250}, {388, 4000, 23536}, {405, 3931, 968}, {950, 54295, 4319}, {976, 3214, 200}, {986, 5247, 63}, {995, 15955, 1}, {1104, 4646, 55}, {1193, 49487, 1}, {1386, 5836, 5710}, {1468, 24443, 57}, {1697, 7290, 3915}, {1724, 4424, 12514}, {1788, 54292, 1038}, {2362, 16232, 2285}, {2975, 4850, 988}, {3244, 30148, 1}, {3436, 19785, 13161}, {3616, 17015, 1}, {3869, 32911, 54386}, {3987, 5264, 54286}, {4383, 37614, 960}, {5256, 19860, 1}, {8583, 23511, 27627}, {10459, 17017, 1}, {22836, 49682, 1}, {24174, 37607, 3306}

X(54419) = X(1)X(21)∩X(2)X(41)

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

X(54419) lies on these lines: {1, 21}, {2, 41}, {6, 28287}, {7, 604}, {27, 1973}, {42, 37090}, {48, 86}, {57, 21511}, {101, 16831}, {172, 940}, {239, 2280}, {284, 1958}, {304, 52379}, {379, 2140}, {405, 23151}, {572, 10444}, {584, 3739}, {662, 41847}, {672, 16367}, {750, 18266}, {894, 2268}, {942, 13723}, {1013, 2356}, {1150, 3912}, {1760, 2294}, {2112, 17397}, {2174, 15668}, {2185, 17103}, {2187, 37103}, {2241, 54282}, {2267, 3758}, {2278, 4670}, {2327, 28627}, {2329, 5273}, {3204, 4698}, {3217, 17260}, {3306, 11349}, {3720, 25494}, {3998, 50716}, {4197, 29633}, {4209, 9776}, {4223, 54392}, {4251, 4384}, {4390, 6542}, {4426, 37676}, {4649, 54383}, {5256, 16780}, {5278, 29960}, {5294, 33819}, {5337, 17750}, {5712, 41243}, {5736, 29967}, {6734, 37149}, {7675, 28071}, {9310, 16826}, {9318, 27907}, {9454, 20131}, {13738, 19716}, {16368, 19714}, {16524, 18278}, {16787, 17011}, {17032, 51949}, {17321, 18162}, {17394, 18042}, {17696, 26065}, {17754, 21495}, {20347, 37076}, {20835, 37580}, {23124, 46882}, {24929, 25083}, {26229, 36021}, {29822, 50404}, {29837, 37443}, {30949, 50200}, {30985, 37086}, {34055, 52394}, {37285, 37576}, {37632, 41258}, {38252, 40737}

X(54419) = X(2)-isoconjugate of X(45966)
X(54419) = X(32664)-Dao conjugate of X(45966)
X(54419) = barycentric product X(i)*X(j) for these {i,j}: {1, 16992}, {63, 11341}, {75, 5138}
X(54419) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 45966}, {5138, 1}, {11341, 92}, {16992, 75}
X(54419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 40744, 41}, {284, 10436, 1958}, {1429, 16503, 26626}

X(54420) = X(4)X(9)∩X(6)X(46)

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

X(54410) lies on these lines: {1, 2178}, {3, 2262}, {4, 9}, {6, 46}, {20, 1741}, {30, 54008}, {36, 1609}, {37, 5119}, {48, 12704}, {57, 77}, {63, 3686}, {65, 4254}, {90, 2960}, {142, 24590}, {155, 610}, {198, 517}, {218, 21866}, {219, 910}, {380, 37550}, {484, 1743}, {579, 1195}, {978, 50361}, {1030, 3612}, {1100, 3338}, {1155, 5120}, {1172, 1452}, {1182, 2257}, {1213, 51557}, {1436, 37623}, {1479, 24005}, {1572, 2277}, {1604, 22770}, {1630, 15836}, {1697, 3247}, {1708, 3101}, {1723, 2245}, {1729, 15945}, {1730, 10319}, {1759, 5227}, {1763, 3684}, {1804, 34492}, {1903, 37411}, {2092, 54382}, {2164, 36743}, {2199, 21147}, {2269, 54405}, {2278, 17700}, {2285, 4266}, {2324, 7991}, {2328, 5338}, {2900, 3198}, {2938, 24708}, {3169, 3509}, {3218, 20082}, {3333, 4262}, {3336, 16667}, {3428, 15817}, {3553, 5903}, {3579, 54322}, {3731, 11010}, {3959, 5336}, {4047, 37658}, {4269, 21384}, {4384, 54404}, {4858, 10444}, {5036, 7297}, {5128, 16670}, {5250, 5257}, {5285, 11383}, {7013, 43035}, {7031, 41332}, {7070, 52427}, {7113, 17437}, {7289, 20367}, {7580, 9119}, {7964, 15288}, {10605, 50528}, {11349, 53996}, {11415, 27522}, {12702, 21871}, {15496, 26893}, {16673, 37563}, {16884, 51816}, {17275, 41229}, {19297, 30323}, {20070, 27508}, {21068, 28194}, {21857, 54406}, {22124, 40943}, {25521, 27000}, {31393, 47299}, {36641, 41339}, {37419, 45206}, {37489, 41854}, {37499, 40937}, {37500, 43065}, {50033, 54386}

X(54420) = Bevan-circle-inverse of X(5179)
X(54420) = X(2994)-Ceva conjugate of X(1)
X(54420) = X(2)-isoconjugate of X(34447)
X(54420) = X(i)-Dao conjugate of X(j) for these (i,j): {46, 5905}, {32664, 34447}
X(54420) = crossdifference of every pair of points on line {1459, 4041}
X(54420) = barycentric product X(i)*X(j) for these {i,j}: {1, 11415}, {57, 27522}, {75, 15494}
X(54420) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 34447}, {11415, 75}, {15494, 1}, {27522, 312}
X(54420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {19, 573, 9}, {40, 2270, 9}, {71, 169, 9}, {573, 5011, 19}, {966, 12514, 9}, {1766, 2183, 9}

X(54421) = X(1)X(21)∩X(6)X(19)

Barycentrics    a*(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(54421) lies on these lines: {1, 21}, {2, 54386}, {4, 41011}, {6, 19}, {7, 23536}, {8, 193}, {10, 5739}, {33, 1858}, {40, 42}, {46, 386}, {56, 3185}, {57, 959}, {72, 612}, {73, 37550}, {78, 171}, {86, 969}, {145, 4339}, {172, 52425}, {201, 3553}, {213, 40131}, {222, 4320}, {223, 1254}, {226, 5230}, {238, 54392}, {354, 1191}, {377, 50307}, {387, 1838}, {388, 4644}, {484, 5312}, {517, 36742}, {518, 5710}, {519, 50043}, {524, 5835}, {601, 37531}, {602, 18443}, {604, 23623}, {609, 4291}, {614, 942}, {750, 936}, {757, 44179}, {940, 960}, {946, 11269}, {958, 4641}, {975, 5692}, {976, 5269}, {978, 3306}, {984, 3951}, {986, 5256}, {988, 3218}, {990, 15071}, {995, 3338}, {997, 37522}, {999, 20805}, {1036, 37581}, {1064, 5709}, {1096, 3194}, {1155, 4255}, {1201, 3333}, {1203, 5902}, {1220, 3758}, {1245, 51223}, {1386, 37549}, {1419, 7273}, {1420, 54310}, {1430, 5706}, {1448, 34043}, {1452, 3192}, {1453, 2308}, {1467, 1471}, {1469, 23154}, {1475, 9575}, {1572, 20963}, {1610, 5323}, {1616, 17609}, {1697, 2293}, {1706, 3214}, {1708, 37558}, {1714, 12609}, {1721, 9961}, {1722, 32911}, {1724, 54318}, {1732, 5165}, {1754, 12520}, {1757, 28375}, {1766, 12435}, {1770, 48837}, {1834, 1836}, {1837, 5155}, {1938, 22383}, {2003, 21147}, {2093, 4642}, {2194, 14529}, {2257, 42289}, {2646, 4252}, {2654, 30223}, {2999, 3339}, {3011, 3487}, {3052, 37080}, {3072, 18446}, {3157, 52077}, {3187, 17164}, {3293, 54286}, {3303, 21002}, {3336, 5313}, {3340, 4332}, {3475, 28027}, {3485, 37642}, {3556, 37538}, {3612, 4257}, {3616, 38000}, {3646, 30950}, {3649, 3772}, {3671, 40940}, {3702, 39594}, {3720, 31435}, {3721, 16972}, {3745, 3962}, {3752, 5221}, {3811, 5264}, {3812, 4383}, {3870, 5255}, {3875, 17141}, {3876, 5268}, {3938, 41863}, {3984, 5293}, {3997, 17742}, {4067, 30142}, {4319, 12711}, {4331, 5930}, {4414, 54290}, {4640, 19765}, {4646, 37567}, {4649, 37598}, {4650, 4652}, {4663, 5836}, {4667, 5837}, {4719, 17595}, {4722, 9623}, {4855, 37603}, {4860, 52541}, {5045, 16483}, {5138, 39598}, {5173, 34040}, {5222, 27000}, {5247, 19860}, {5252, 5849}, {5262, 16475}, {5271, 49598}, {5275, 21874}, {5292, 12047}, {5310, 37547}, {5315, 18398}, {5437, 27627}, {5552, 27338}, {5691, 29046}, {5697, 16474}, {5705, 33105}, {5707, 5887}, {5712, 8896}, {5718, 26066}, {5794, 49745}, {5905, 13161}, {6261, 37530}, {6734, 26098}, {7078, 50195}, {7098, 54320}, {7290, 11518}, {8192, 23381}, {8227, 29662}, {8270, 15556}, {8771, 35991}, {9340, 30282}, {9612, 21935}, {9943, 37537}, {10375, 14642}, {10404, 17365}, {10436, 17137}, {10459, 32912}, {10480, 54359}, {10884, 37570}, {11018, 41422}, {11375, 37646}, {11415, 24210}, {11512, 27003}, {11521, 21375}, {11551, 24159}, {12544, 19645}, {12635, 37539}, {12709, 37543}, {13738, 20967}, {14110, 36746}, {15955, 25415}, {16472, 53615}, {16485, 21747}, {16780, 21764}, {16824, 37652}, {16970, 21808}, {17016, 37685}, {17064, 24883}, {17126, 34772}, {17594, 19767}, {17750, 54406}, {17751, 26223}, {17768, 50065}, {18041, 33766}, {18391, 39585}, {19861, 26625}, {20018, 32932}, {22479, 44094}, {24512, 39248}, {24914, 37662}, {25524, 37520}, {25591, 30567}, {25681, 37634}, {25917, 37674}, {28570, 50050}, {28628, 35466}, {30116, 41229}, {30143, 53114}, {31053, 54355}, {32937, 41261}, {34339, 36754}, {34791, 37542}, {35004, 39523}, {35258, 37573}, {35262, 37608}, {36747, 37562}, {36985, 41562}, {37588, 49490}, {37602, 54319}, {37614, 44663}, {37692, 45939}, {40266, 45923}, {40952, 42448}, {44735, 54109}, {50579, 50581}, {50582, 50635}

X(54421) = reflection of X(10371) in X(5835)
X(54421) = X(989)-anticomplementary conjugate of X(1330)
X(54421) = X(5737)-Dao conjugate of X(10447)
X(54421) = crossdifference of every pair of points on line {521, 661}
X(54421) = barycentric product X(i)*X(j) for these {i,j}: {1, 5712}, {28, 8896}, {63, 37384}, {65, 37265}, {225, 23602}
X(54421) = barycentric quotient X(i)/X(j) for these {i,j}: {5712, 75}, {8896, 20336}, {23602, 332}, {37265, 314}, {37384, 92}
X(54421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1046, 63}, {1, 1707, 21}, {1, 12514, 968}, {1, 12526, 2292}, {1, 31424, 10448}, {1, 49500, 12514}, {6, 54382, 2082}, {31, 2650, 1}, {65, 221, 2263}, {72, 5711, 612}, {81, 3869, 1}, {354, 1191, 28011}, {387, 4295, 3914}, {896, 10448, 31424}, {942, 16466, 614}, {1042, 10460, 1193}, {1453, 11529, 3924}, {2308, 3924, 1453}, {2999, 3339, 24443}, {5269, 11523, 976}, {5692, 37559, 975}, {7290, 11518, 28082}, {17126, 34772, 37552}, {21935, 24725, 9612}

X(54422) = X(1)X(21)∩X(7)X(10)

Barycentrics    a*(a^3 + 3*a^2*b - a*b^2 - 3*b^3 + 3*a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 - 3*c^3) : :
X(54422) = 3 X[1] - 4 X[8666], 2 X[21] - 3 X[54302], 2 X[3] - 3 X[3928], 3 X[3928] - X[11523], 4 X[5] - 3 X[28609], 3 X[20] - X[12536], X[20] - 3 X[28610], X[12536] - 9 X[28610], X[12632] - 3 X[34632], 3 X[40] - 2 X[3913], 4 X[3913] - 3 X[6765], X[3680] - 3 X[6762], 2 X[3680] - 3 X[12629], 3 X[165] - 2 X[3811], and many others

X(54422) lies on these lines: {1, 21}, {2, 3951}, {3, 3928}, {4, 527}, {5, 28609}, {7, 10}, {8, 2093}, {9, 942}, {20, 519}, {30, 12625}, {36, 1259}, {40, 518}, {44, 17054}, {46, 200}, {55, 41863}, {56, 3962}, {57, 72}, {65, 9623}, {78, 3218}, {84, 517}, {144, 938}, {145, 4304}, {165, 3811}, {210, 5221}, {223, 37591}, {226, 5705}, {268, 38290}, {269, 52385}, {329, 1210}, {354, 31435}, {376, 12437}, {377, 3679}, {379, 16833}, {380, 1761}, {387, 3663}, {404, 3984}, {405, 3929}, {442, 4654}, {443, 553}, {516, 7992}, {529, 5881}, {537, 37088}, {540, 48890}, {550, 34701}, {551, 17558}, {610, 52012}, {726, 10444}, {908, 6931}, {912, 1490}, {937, 43216}, {946, 24477}, {956, 3340}, {958, 11529}, {960, 3333}, {962, 7995}, {978, 18193}, {982, 54386}, {984, 54344}, {986, 3751}, {997, 3361}, {999, 15829}, {1012, 7982}, {1056, 5837}, {1103, 1735}, {1125, 5273}, {1158, 6769}, {1191, 21342}, {1420, 5730}, {1423, 10381}, {1453, 4641}, {1467, 1708}, {1697, 3555}, {1698, 5249}, {1699, 10916}, {1706, 5784}, {1709, 12651}, {1714, 23681}, {1722, 1757}, {1729, 3509}, {1741, 2324}, {1743, 5279}, {1750, 12528}, {1765, 10441}, {1788, 21075}, {1834, 17276}, {1858, 54408}, {2094, 6904}, {2095, 5777}, {2096, 12245}, {2136, 12702}, {2184, 52889}, {2323, 3157}, {2476, 31164}, {2478, 17781}, {2801, 9960}, {2802, 11519}, {2900, 16117}, {2901, 35629}, {2951, 12669}, {2999, 3670}, {3062, 51118}, {3091, 30326}, {3158, 3579}, {3169, 48917}, {3189, 31730}, {3190, 4303}, {3191, 37523}, {3194, 23052}, {3219, 16859}, {3220, 37547}, {3241, 17576}, {3243, 3295}, {3244, 4313}, {3294, 43220}, {3296, 51723}, {3304, 31165}, {3305, 17534}, {3306, 3876}, {3338, 5692}, {3419, 9579}, {3421, 4848}, {3487, 5745}, {3496, 51194}, {3523, 11407}, {3576, 12635}, {3583, 36599}, {3586, 12649}, {3587, 13369}, {3601, 3916}, {3612, 41696}, {3624, 54357}, {3634, 30393}, {3635, 30337}, {3636, 30343}, {3646, 3742}, {3650, 41864}, {3671, 19843}, {3677, 16466}, {3678, 8580}, {3682, 4306}, {3684, 36643}, {3695, 17296}, {3729, 10449}, {3746, 20835}, {3812, 5220}, {3813, 31162}, {3875, 8822}, {3940, 5438}, {3958, 54385}, {4005, 4413}, {4007, 50042}, {4034, 49718}, {4084, 18421}, {4101, 17740}, {4187, 31142}, {4197, 19875}, {4293, 6737}, {4295, 4847}, {4301, 34625}, {4347, 34033}, {4644, 5717}, {4650, 37552}, {4652, 17548}, {4659, 5295}, {4677, 17579}, {4685, 37109}, {4853, 5903}, {4860, 25917}, {4862, 23537}, {4867, 37618}, {4882, 41228}, {4930, 15178}, {4936, 5525}, {5044, 5437}, {5119, 16465}, {5128, 5687}, {5129, 6172}, {5227, 24476}, {5231, 12047}, {5234, 54318}, {5255, 16496}, {5258, 37228}, {5267, 53054}, {5288, 25415}, {5325, 16845}, {5435, 6700}, {5436, 15934}, {5439, 7308}, {5531, 9964}, {5535, 17857}, {5542, 18249}, {5563, 37248}, {5587, 5852}, {5691, 49168}, {5693, 12704}, {5696, 30353}, {5711, 7174}, {5715, 6866}, {5720, 37251}, {5729, 36973}, {5744, 13411}, {5758, 6245}, {5762, 5787}, {5779, 5806}, {5791, 6147}, {5811, 7682}, {5843, 6259}, {5853, 6361}, {5882, 34610}, {5884, 30503}, {5902, 41229}, {5905, 6734}, {6173, 8728}, {6282, 37403}, {6684, 25568}, {6735, 10940}, {6744, 30330}, {6764, 10430}, {6837, 11522}, {6839, 37714}, {6876, 18446}, {6916, 11362}, {7171, 37585}, {7330, 24474}, {7411, 8715}, {7675, 53053}, {7957, 10860}, {7963, 45763}, {7971, 22770}, {7987, 18444}, {7996, 28526}, {8056, 8951}, {8726, 21153}, {9004, 52359}, {9588, 37112}, {9589, 10431}, {9614, 11415}, {9624, 34647}, {9708, 31794}, {9841, 31793}, {9843, 18228}, {9949, 10429}, {10167, 37551}, {10382, 14054}, {10404, 21677}, {10436, 14007}, {10477, 44421}, {10529, 51423}, {10582, 18398}, {10624, 36845}, {10883, 24387}, {11106, 15933}, {11113, 37723}, {11224, 22837}, {11239, 31436}, {11260, 16200}, {11512, 18201}, {11551, 19854}, {11813, 50444}, {12436, 21454}, {12527, 18391}, {12565, 15071}, {12640, 50810}, {12699, 24392}, {12701, 51463}, {12705, 18219}, {12710, 15185}, {13462, 30144}, {13740, 50127}, {14021, 29573}, {15239, 18239}, {16062, 17274}, {17114, 20683}, {17151, 18655}, {17254, 37164}, {17449, 28011}, {17581, 54405}, {17728, 25522}, {17733, 35621}, {17736, 54330}, {17768, 28646}, {18178, 40979}, {18186, 52564}, {18253, 41870}, {18443, 24475}, {18483, 28647}, {20077, 50582}, {20214, 38271}, {20320, 52345}, {20880, 28612}, {21096, 41325}, {21164, 26877}, {21255, 39589}, {23144, 34043}, {23154, 26893}, {23511, 24046}, {24328, 37320}, {24440, 49712}, {24467, 37531}, {24468, 50528}, {25440, 53056}, {26051, 50128}, {30389, 37106}, {31190, 34753}, {31253, 36835}, {31302, 41261}, {31393, 34791}, {31549, 49592}, {31550, 49593}, {31775, 34742}, {31789, 34695}, {31837, 37534}, {34619, 37108}, {34716, 37727}, {34937, 37642}, {35596, 37256}, {37153, 50116}, {37175, 42042}, {37358, 37720}, {37421, 41561}, {37433, 50865}, {37467, 42043}, {37568, 41711}, {37581, 42461}, {37623, 52026}, {37719, 47516}, {44238, 50811}, {47622, 52181}, {49627, 51785}, {50095, 50735}, {50443, 51409}, {50581, 54383}, {50742, 51071}, {51724, 52653}

X(54422) = midpoint of X(6764) and X(20070)
X(54422) = reflection of X(i) in X(j) for these {i,j}: {4, 24391}, {1490, 5709}, {2136, 12702}, {2900, 37584}, {3189, 31730}, {5691, 49168}, {5758, 6245}, {6765, 40}, {6769, 1158}, {7971, 22770}, {7982, 12513}, {11523, 3}, {12629, 6762}, {37531, 24467}
X(54422) = X(38271)-anticomplementary conjugate of X(1330)
X(54422) = barycentric product X(75)*X(37500)
X(54422) = barycentric quotient X(37500)/X(1)
X(54422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 63, 31424}, {1, 191, 4512}, {1, 16570, 54354}, {7, 5223, 5785}, {7, 54398, 10}, {8, 9965, 4292}, {21, 3868, 11520}, {21, 11520, 1}, {40, 1071, 5732}, {46, 5904, 200}, {57, 72, 936}, {63, 3868, 1}, {63, 11520, 21}, {78, 3218, 15803}, {144, 938, 12572}, {191, 3894, 1}, {405, 24473, 11518}, {942, 3927, 9}, {956, 4018, 3340}, {993, 12559, 1}, {3338, 5692, 8583}, {3339, 5223, 10}, {3868, 39772, 3894}, {3873, 5250, 1}, {3873, 11684, 5250}, {3874, 12514, 1}, {3901, 6763, 1}, {3928, 11523, 3}, {3929, 11518, 405}, {3940, 37582, 5438}, {4641, 37549, 1453}, {4652, 34772, 30282}, {4880, 5904, 46}, {5044, 5708, 5437}, {5261, 5775, 10}, {5273, 11036, 1125}, {5686, 11024, 10}, {5791, 6147, 25525}, {5905, 6734, 9612}, {7991, 30304, 20}, {8056, 8951, 17749}, {11415, 26015, 9614}, {11682, 54391, 1}, {15071, 41338, 12565}, {15934, 31445, 5436}, {24467, 37531, 52027}, {24475, 26921, 18443}, {34790, 36279, 1706}, {41863, 54290, 55}, {43174, 43177, 37108}

X(54423) = X(3)X(6)∩X(37)X(78)

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

X(54423) lies on these lines: {1, 19285}, {2, 45129}, {3, 6}, {37, 78}, {42, 48}, {55, 22074}, {213, 54285}, {387, 37151}, {391, 16347}, {518, 19758}, {940, 1100}, {966, 16342}, {967, 28625}, {980, 16973}, {992, 19283}, {1150, 5839}, {1211, 16350}, {1213, 16343}, {1386, 19761}, {1409, 11509}, {1449, 37522}, {1468, 22054}, {1778, 4189}, {2172, 2174}, {2256, 3190}, {2268, 22072}, {2276, 2911}, {2286, 2594}, {2288, 11434}, {2303, 19767}, {2451, 48382}, {3049, 48391}, {3216, 19523}, {3295, 16685}, {3687, 5737}, {3958, 4414}, {4000, 5736}, {4016, 12635}, {4360, 30882}, {4383, 37323}, {4646, 5706}, {5275, 16352}, {5301, 16466}, {5313, 16470}, {5331, 37091}, {5747, 48837}, {8743, 41502}, {8818, 44518}, {11344, 46889}, {11507, 22134}, {15668, 17023}, {16345, 37673}, {16349, 17277}, {16351, 17330}, {16457, 24931}, {16458, 17398}, {16777, 30115}, {16783, 19286}, {16884, 17054}, {20150, 33035}, {20818, 38903}, {21904, 41243}, {25523, 40940}, {37245, 46890}, {39520, 48384}

X(54423) = crossdifference of every pair of points on line {523, 50501}
X(54423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3053, 4275}, {6, 4255, 4261}, {6, 4258, 2220}, {6, 5013, 583}, {6, 5110, 36743}, {6, 5124, 5021}, {6, 18755, 36744}, {6, 37500, 5165}, {6, 37504, 1333}, {216, 2092, 4261}, {284, 386, 6}, {572, 4270, 6}, {584, 5153, 6}, {965, 19765, 37}, {2278, 4272, 6}, {4251, 5105, 6}, {4261, 4273, 6}, {4263, 5114, 6}, {4268, 4285, 6}, {4275, 17454, 3053}, {5019, 20970, 6}, {5747, 48837, 53417}

X(54424) = X(1)X(19)∩X(9)X(65)

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

X(54424) lies on these lines: {1, 19}, {6, 11529}, {7, 25935}, {9, 65}, {37, 40}, {45, 21866}, {55, 3247}, {57, 40937}, {71, 2093}, {86, 18713}, {196, 226}, {200, 22021}, {219, 3340}, {223, 1880}, {269, 18726}, {282, 2358}, {326, 18714}, {579, 3339}, {938, 40963}, {942, 2257}, {1045, 8769}, {1108, 3333}, {1400, 15830}, {1449, 2264}, {1706, 3694}, {1723, 5902}, {1743, 54324}, {1761, 31424}, {1766, 30503}, {1826, 9612}, {1839, 3586}, {1855, 5290}, {1859, 10382}, {2171, 2324}, {2178, 10902}, {2256, 7982}, {2285, 17451}, {2286, 34039}, {2321, 2550}, {2999, 9816}, {3101, 5287}, {3174, 19589}, {3198, 37553}, {3204, 3553}, {3485, 40942}, {3554, 17443}, {3664, 18725}, {3671, 5746}, {4259, 5785}, {4295, 8804}, {5227, 9623}, {5279, 19860}, {5587, 21933}, {5749, 25904}, {5750, 28629}, {5802, 6738}, {6203, 38004}, {7688, 54285}, {9119, 12709}, {9536, 17019}, {9575, 20227}, {10319, 17022}, {10436, 11683}, {10582, 40959}, {11518, 54358}, {12514, 25081}, {14521, 36973}, {15941, 18506}, {16676, 37567}, {18165, 40979}, {18634, 41003}, {18635, 41010}, {18655, 25255}, {20818, 50194}, {21808, 54359}, {27411, 44733}

X(54424) = barycentric product X(1)*X(5177)
X(54424) = barycentric quotient X(5177)/X(75)
X(54424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 19, 380}, {1, 1781, 610}, {1, 18594, 284}, {19, 2294, 1}, {1953, 54385, 1}, {2093, 3731, 71}, {2171, 40131, 2324}

X(54425) = X(6)X(7)∩X(8)X(34)

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

X(54425) lies on these lines: {1, 5809}, {2, 77}, {6, 7}, {8, 34}, {9, 347}, {73, 3616}, {85, 3618}, {105, 1037}, {142, 1419}, {144, 22464}, {169, 14256}, {212, 9778}, {219, 278}, {222, 9776}, {226, 1449}, {241, 37650}, {269, 3008}, {273, 26668}, {279, 1445}, {281, 36949}, {307, 391}, {342, 1249}, {344, 664}, {348, 17277}, {388, 1386}, {404, 1035}, {452, 5930}, {497, 30621}, {581, 5703}, {658, 30682}, {962, 7078}, {1040, 10430}, {1068, 5811}, {1104, 4308}, {1167, 1777}, {1212, 3160}, {1394, 6904}, {1404, 28081}, {1427, 5435}, {1429, 28015}, {1436, 34813}, {1441, 5749}, {1442, 5308}, {1453, 3600}, {1456, 2550}, {1458, 16020}, {1465, 5744}, {1736, 5825}, {1743, 3668}, {1804, 11349}, {1818, 27383}, {1943, 34255}, {2199, 37274}, {2270, 7013}, {2988, 50442}, {3062, 45275}, {3100, 36991}, {3161, 4552}, {3664, 30275}, {3672, 8545}, {3759, 6604}, {3912, 53997}, {3945, 21617}, {4295, 54301}, {4296, 54305}, {4328, 50114}, {4383, 7365}, {5226, 5712}, {5261, 5717}, {5273, 17080}, {5745, 36636}, {5748, 6510}, {5768, 37697}, {5909, 37417}, {6060, 37104}, {6172, 36640}, {6610, 17278}, {7053, 37272}, {7070, 50696}, {7190, 17014}, {7490, 14557}, {8055, 28996}, {8271, 34036}, {9312, 17353}, {9800, 54295}, {10578, 14547}, {12573, 16469}, {16670, 52819}, {17081, 43062}, {17086, 17257}, {17242, 25726}, {17286, 25719}, {17381, 52422}, {17917, 41883}, {18228, 18624}, {19877, 54346}, {20921, 37669}, {26006, 27508}, {26125, 26626}, {26671, 43045}, {28741, 29627}, {28780, 29611}, {28968, 31995}, {30705, 31638}, {30854, 33673}, {31018, 37798}, {34492, 53996}, {41246, 51171}

X(54425) = X(30705)-Ceva conjugate of X(7)
X(54425) = X(55)-isoconjugate of X(41790)
X(54425) = X(i)-Dao conjugate of X(j) for these (i,j): {223, 41790}, {497, 6554}
X(54425) = cevapoint of X(223) and X(16572)
X(54425) = barycentric product X(i)*X(j) for these {i,j}: {7, 17784}, {651, 25009}, {7131, 41787}, {8817, 43916}
X(54425) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 41790}, {17784, 8}, {25009, 4391}, {43916, 497}
X(54425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 948, 7}, {9, 43035, 347}, {269, 3008, 8732}, {278, 34048, 329}, {279, 37681, 1445}, {651, 37800, 7}, {1743, 3668, 12848}, {4000, 6180, 7}, {4552, 28966, 3161}, {5723, 6180, 4000}, {5819, 39063, 7}, {26668, 30807, 27382}

X(54426) = X(1)X(2)∩X(6)X(25)

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

X(54426) lies on these lines: {1, 2}, {6, 25}, {22, 58}, {31, 579}, {32, 199}, {39, 1011}, {55, 4261}, {228, 2277}, {305, 33296}, {427, 1834}, {430, 5254}, {573, 20966}, {581, 4220}, {942, 24163}, {968, 40934}, {980, 37329}, {991, 17187}, {1180, 4253}, {1184, 2271}, {1196, 20970}, {1245, 51223}, {1368, 48847}, {1370, 48837}, {1468, 5322}, {1627, 4262}, {1724, 37325}, {1848, 3914}, {2082, 40976}, {2176, 3690}, {2177, 16488}, {2221, 37581}, {2223, 19343}, {2258, 39943}, {2273, 26885}, {2275, 40956}, {2300, 26893}, {2332, 3162}, {3052, 4286}, {3060, 50600}, {3198, 3752}, {3291, 20754}, {3736, 37090}, {3917, 50591}, {4204, 5283}, {4207, 5286}, {4251, 5359}, {4255, 7484}, {4256, 7485}, {4257, 6636}, {4259, 40153}, {4270, 5276}, {4272, 5275}, {4281, 19310}, {4749, 5165}, {5105, 33854}, {5153, 37060}, {5156, 5329}, {5396, 19544}, {5721, 37362}, {5943, 50595}, {7453, 9465}, {7494, 37642}, {7499, 37646}, {7772, 47523}, {9605, 13615}, {10319, 22057}, {10565, 37666}, {13588, 24598}, {14547, 20753}, {16368, 19758}, {16466, 37547}, {18147, 32926}, {18697, 32860}, {19309, 19725}, {19792, 32922}, {22021, 22196}, {22090, 47757}, {23536, 37093}, {28606, 37819}, {31152, 48842}, {36000, 37552}, {37439, 37662}, {37678, 40022}, {40941, 43214}, {44212, 48861}, {45962, 54308}

X(54426) = isogonal conjugate of the isotomic conjugate of X(16062)
X(54426) = crossdifference of every pair of points on line {525, 649}
X(54426) = barycentric product X(6)*X(16062)
X(54426) = barycentric quotient X(16062)/X(76)
X(54426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 43, 306}, {2, 54341, 386}, {6, 37538, 44119}, {39, 40984, 1011}, {42, 5311, 41265}, {5285, 16470, 31}

X(54427) = X(1)X(2)∩X(3)X(47)

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

X(54427) lies on these lines: {1, 2}, {3, 47}, {6, 22766}, {35, 1064}, {36, 54}, {46, 10571}, {56, 7130}, {58, 1800}, {65, 1772}, {100, 54350}, {213, 13006}, {222, 40293}, {255, 14793}, {500, 37600}, {581, 3612}, {595, 32760}, {602, 36152}, {920, 46016}, {1042, 3336}, {1066, 1450}, {1079, 4320}, {1191, 11508}, {1319, 5399}, {1385, 2594}, {1409, 50650}, {1457, 5903}, {1464, 37582}, {1468, 16473}, {1470, 3157}, {1478, 37694}, {1725, 17102}, {1745, 4299}, {1807, 9630}, {2003, 37561}, {2635, 10483}, {2646, 5396}, {2654, 7741}, {3073, 10058}, {3075, 10090}, {3468, 4351}, {3585, 6127}, {3914, 39599}, {4253, 8776}, {4255, 11507}, {4261, 22134}, {4300, 5010}, {4303, 7280}, {4551, 45287}, {5348, 6924}, {5540, 40957}, {6149, 14792}, {6265, 33177}, {6326, 33178}, {6914, 7299}, {7078, 8071}, {7428, 42450}, {8068, 21935}, {8069, 16466}, {8578, 22090}, {10087, 37588}, {10269, 36752}, {10572, 37732}, {14529, 20842}, {14547, 37571}, {14803, 37469}, {16453, 37836}, {17605, 48903}, {18480, 45885}, {22768, 36742}, {23070, 52440}, {24431, 31835}, {24443, 53615}, {24475, 53525}, {26437, 44414}, {31870, 43048}, {34471, 37698}, {37564, 52408}

X(54427) = barycentric product X(63)*X(41722)
X(54427) = barycentric quotient X(41722)/X(92)
X(54427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 43, 10573}, {1, 936, 54401}, {1, 978, 499}, {1, 3216, 1737}, {1064, 22072, 35}, {1066, 1450, 5563}, {1193, 22350, 1}, {31397, 50604, 1}

X(54428) = X(1)X(25)∩X(4)X(36)

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

X(54428) lies on these lines: {1, 25}, {3, 9817}, {4, 36}, {10, 35973}, {11, 6756}, {12, 21841}, {24, 33}, {26, 37696}, {28, 1785}, {34, 5563}, {47, 2212}, {55, 3517}, {56, 1598}, {108, 1838}, {172, 3199}, {225, 36009}, {232, 5280}, {235, 3585}, {297, 30103}, {389, 10535}, {406, 5251}, {428, 3582}, {484, 1902}, {496, 7715}, {498, 6353}, {609, 2207}, {613, 7716}, {993, 4194}, {1038, 7387}, {1040, 6642}, {1060, 7517}, {1062, 7506}, {1192, 10060}, {1210, 7466}, {1398, 37587}, {1452, 5903}, {1478, 3089}, {1479, 7487}, {1593, 7280}, {1595, 5433}, {1596, 7354}, {1597, 5204}, {1737, 4231}, {1824, 32760}, {1827, 20837}, {1859, 10902}, {1870, 34484}, {1871, 2078}, {1872, 2077}, {1876, 3337}, {1885, 4316}, {1906, 4325}, {2067, 35765}, {2299, 54301}, {2964, 14975}, {3075, 3220}, {3083, 15187}, {3084, 15188}, {3085, 4232}, {3086, 5322}, {3100, 44802}, {3299, 5412}, {3301, 5413}, {3515, 5010}, {3518, 3746}, {3542, 7951}, {3567, 9638}, {3575, 3583}, {3614, 37942}, {4233, 13411}, {5090, 18395}, {5160, 44272}, {5258, 46878}, {5299, 10311}, {5353, 10642}, {5357, 10641}, {5530, 7438}, {6152, 51803}, {6284, 37458}, {6285, 11438}, {6502, 35764}, {6759, 19366}, {7352, 46261}, {7497, 37583}, {7529, 19372}, {7714, 10072}, {7718, 10573}, {7952, 17562}, {8144, 12106}, {8946, 45613}, {8948, 45614}, {10076, 15811}, {10110, 19365}, {10282, 11429}, {10896, 18494}, {10985, 16784}, {11109, 19864}, {11393, 37122}, {12135, 41684}, {12137, 53616}, {12146, 18244}, {12173, 18514}, {13488, 15326}, {13621, 18455}, {13861, 37697}, {14803, 37117}, {14986, 52301}, {16472, 44105}, {16473, 44086}, {16655, 26955}, {18378, 18447}, {18513, 37197}, {20831, 46974}, {22479, 54397}, {37440, 37729}, {37935, 52793}

X(54428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 33, 35}, {25, 11399, 1}, {25, 11401, 26378}, {25, 26377, 7713}, {1905, 11363, 1}, {3518, 6198, 52427}, {3542, 11392, 7951}, {6198, 52427, 3746}, {6642, 9645, 1040}, {7952, 17562, 54368}

X(54429) = X(2)X(58)∩X(8)X(20)

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

X(54429) lies on these lines: {1, 4001}, {2, 58}, {3, 5739}, {4, 1150}, {8, 20}, {21, 69}, {72, 26892}, {78, 991}, {81, 13725}, {141, 17526}, {145, 31303}, {193, 19767}, {283, 27505}, {306, 31424}, {307, 1394}, {315, 34016}, {333, 377}, {343, 24538}, {376, 3578}, {387, 16704}, {391, 579}, {404, 14555}, {443, 5278}, {452, 37655}, {524, 19765}, {550, 49718}, {631, 5741}, {940, 37314}, {966, 16454}, {1211, 4252}, {1453, 54311}, {1468, 50295}, {1714, 48835}, {1792, 37285}, {1834, 50055}, {1935, 28739}, {2287, 37180}, {2475, 5361}, {2478, 14829}, {2895, 4189}, {2975, 19262}, {3286, 36000}, {3487, 32859}, {3601, 4101}, {3616, 3945}, {3648, 24280}, {3687, 4652}, {3702, 5698}, {3710, 3929}, {3876, 54280}, {3916, 5814}, {3933, 21982}, {3936, 6857}, {4061, 12512}, {4188, 37656}, {4190, 9534}, {4195, 37653}, {4201, 37652}, {4292, 5271}, {4294, 17135}, {4309, 50625}, {4313, 49687}, {4388, 10527}, {4417, 6910}, {4450, 5082}, {4640, 10371}, {4643, 37539}, {4921, 48813}, {5046, 5372}, {5047, 18141}, {5051, 37642}, {5233, 6921}, {5235, 37153}, {5247, 26034}, {5712, 16342}, {5737, 49745}, {5738, 16346}, {6327, 19843}, {6776, 15983}, {6851, 48935}, {6869, 48877}, {6872, 10449}, {7793, 46707}, {8822, 19848}, {10327, 41229}, {10446, 11415}, {13742, 33172}, {14826, 28376}, {16062, 24597}, {16343, 49743}, {16347, 31034}, {16370, 41014}, {16845, 18139}, {16865, 32863}, {16948, 32782}, {17206, 19310}, {17234, 31259}, {17277, 37462}, {17674, 37650}, {17776, 31445}, {19766, 37685}, {19851, 26840}, {24248, 27368}, {24570, 26540}, {25524, 41002}, {25912, 26657}, {26117, 37683}, {28921, 46878}, {32099, 52396}, {33065, 36573}, {33171, 54354}, {34511, 50272}, {37054, 54303}, {37255, 37507}, {47102, 50275}, {48878, 50695}, {48883, 50698}

X(54429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 49716, 5739}, {20, 14552, 8}, {940, 49728, 37314}, {3916, 5814, 17740}, {16704, 17676, 387}, {16948, 32782, 37176}

X(54430) = X(4)X(35)∩X(9)X(21)

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

X(54430) lies on these lines: {1, 201}, {3, 226}, {4, 35}, {9, 21}, {10, 55}, {33, 30733}, {36, 3487}, {56, 954}, {57, 6986}, {72, 993}, {142, 37282}, {198, 37052}, {228, 13733}, {270, 35192}, {329, 4189}, {388, 15931}, {404, 25525}, {411, 5219}, {442, 5432}, {452, 5281}, {497, 5259}, {499, 6878}, {516, 37601}, {551, 10966}, {581, 3074}, {908, 20846}, {936, 37306}, {946, 40292}, {958, 1260}, {968, 54295}, {991, 1935}, {1001, 12053}, {1125, 26357}, {1210, 6883}, {1259, 5745}, {1376, 37224}, {1453, 1612}, {1479, 6832}, {1490, 6906}, {1621, 1697}, {1724, 14547}, {1726, 18673}, {1792, 11679}, {1836, 12511}, {2077, 6908}, {2330, 10477}, {2975, 11523}, {3057, 30147}, {3085, 6987}, {3452, 11344}, {3485, 5759}, {3486, 5251}, {3488, 3746}, {3522, 8232}, {3523, 54366}, {3560, 4304}, {3583, 6990}, {3586, 6920}, {3612, 5450}, {3616, 5766}, {3634, 11502}, {3651, 4333}, {3683, 12711}, {3822, 10953}, {3871, 12625}, {3916, 17603}, {4183, 54396}, {4276, 25516}, {4294, 6846}, {4295, 7688}, {4298, 37578}, {4313, 16865}, {4428, 8170}, {4995, 11113}, {5047, 9581}, {5128, 5665}, {5217, 7580}, {5249, 37301}, {5257, 16346}, {5260, 5727}, {5267, 22768}, {5284, 50443}, {5316, 16293}, {5541, 53053}, {5692, 45230}, {5703, 37106}, {5715, 6905}, {5728, 15837}, {5746, 37297}, {5750, 37065}, {5758, 11012}, {5777, 6914}, {6284, 8226}, {6598, 31660}, {6684, 11507}, {6690, 47510}, {6745, 13615}, {6889, 10320}, {6907, 26285}, {6913, 11499}, {6936, 31452}, {7354, 34879}, {7411, 9579}, {7513, 40573}, {7522, 19760}, {7538, 27287}, {7676, 52835}, {7742, 21620}, {8071, 10165}, {8666, 34471}, {8804, 54285}, {10056, 14798}, {10164, 11509}, {10267, 31397}, {10382, 30393}, {10383, 31424}, {10391, 31445}, {10589, 25542}, {10831, 39475}, {11491, 31434}, {12572, 37284}, {12664, 33597}, {13405, 37579}, {13738, 51687}, {13739, 40395}, {14799, 37719}, {15865, 31789}, {16342, 27394}, {17549, 28609}, {18389, 26921}, {18397, 26878}, {22361, 37522}, {24929, 31837}, {28606, 33178}, {29828, 47511}, {31266, 35979}, {36018, 40131}, {50317, 52408}

X(54430) = crossdifference of every pair of points on line {4017, 43060}
X(54430) = barycentric product X(9)*X(5736)
X(54430) = barycentric quotient X(5736)/X(85)
X(54430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 78, 40661}, {9, 3601, 10393}, {35, 498, 6796}, {55, 405, 950}, {405, 11517, 10}, {405, 19763, 1751}, {943, 1006, 1}, {5010, 9612, 3651}, {5703, 37106, 37583}

X(54431) = X(3)X(6)∩X(25)X(34)

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

X(54431) lies on these lines: {1, 37284}, {3, 6}, {21, 5712}, {24, 3194}, {25, 34}, {31, 73}, {36, 1453}, {42, 37601}, {47, 7078}, {81, 20846}, {193, 1792}, {223, 37583}, {387, 3651}, {393, 8885}, {405, 17056}, {411, 37642}, {593, 26636}, {937, 37561}, {940, 11344}, {993, 5717}, {1036, 16678}, {1191, 1617}, {1193, 37578}, {1211, 37065}, {1212, 5275}, {1259, 4641}, {1399, 19349}, {1402, 3556}, {1408, 44087}, {1451, 1470}, {1460, 23843}, {1468, 14547}, {1472, 3433}, {1490, 8557}, {1612, 3487}, {1724, 37249}, {1834, 7580}, {1838, 3772}, {2911, 3682}, {2975, 5716}, {3145, 37538}, {3149, 37646}, {3428, 15852}, {3560, 5713}, {3755, 12511}, {3767, 53422}, {3915, 33925}, {4185, 40980}, {4340, 37306}, {4383, 37282}, {4646, 5584}, {5265, 36007}, {5292, 6985}, {5324, 27621}, {5438, 16572}, {5563, 16485}, {7083, 23383}, {7742, 16466}, {16293, 37674}, {16344, 25526}, {16346, 19701}, {16410, 37679}, {16968, 37609}, {17337, 50203}, {18603, 19765}, {19309, 25524}, {19767, 37285}, {22760, 40950}, {24597, 35979}, {28466, 48870}, {30478, 37149}, {32911, 37301}, {35466, 37229}, {37228, 49745}, {37234, 45924}

X(54431) = isogonal conjugate of the isotomic conjugate of X(5738)
X(54431) = isogonal conjugate of the polar conjugate of X(37388)
X(54431) = barycentric product X(i)*X(j) for these {i,j}: {3, 37388}, {6, 5738}, {57, 10393}
X(54431) = barycentric quotient X(i)/X(j) for these {i,j}: {5738, 76}, {10393, 312}, {37388, 264}
X(54431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 4254, 19760}, {3, 19762, 36743}, {3, 37492, 4267}, {56, 1035, 1427}, {58, 581, 6}, {19759, 54371, 3}

X(54432) = X(1)X(21)∩X(5)X(57)

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

X(54432) lies on these lines: {1, 21}, {3, 18397}, {5, 57}, {7, 499}, {9, 7483}, {20, 484}, {35, 10391}, {36, 1071}, {40, 10950}, {46, 80}, {56, 5693}, {72, 19525}, {90, 1699}, {91, 267}, {201, 37469}, {224, 15015}, {226, 6852}, {377, 18395}, {405, 30274}, {411, 41562}, {498, 5273}, {515, 7098}, {580, 7004}, {912, 37583}, {942, 7489}, {946, 1776}, {1012, 1727}, {1158, 2093}, {1210, 3218}, {1259, 41686}, {1420, 6265}, {1445, 6915}, {1454, 5587}, {1478, 15932}, {1479, 5536}, {1490, 1708}, {1697, 37728}, {1698, 17700}, {1729, 5540}, {1735, 5247}, {1736, 3075}, {1737, 3336}, {1741, 1743}, {1771, 4650}, {1781, 15656}, {1788, 2096}, {1836, 7701}, {1837, 5535}, {1858, 11012}, {1864, 37623}, {2003, 37565}, {2077, 41538}, {2094, 11023}, {2949, 10393}, {3065, 37433}, {3086, 9965}, {3219, 13411}, {3286, 20803}, {3306, 7504}, {3333, 15950}, {3338, 5443}, {3586, 5709}, {3601, 7508}, {3652, 39542}, {3683, 16193}, {3911, 6949}, {3916, 44547}, {3928, 10396}, {4197, 16763}, {4299, 9799}, {4304, 11010}, {4313, 37563}, {4641, 17102}, {5131, 11220}, {5251, 13750}, {5398, 33178}, {5435, 6979}, {5692, 22766}, {5704, 23958}, {5735, 51768}, {5784, 41700}, {5904, 8069}, {6147, 10021}, {6284, 16113}, {6837, 18393}, {6890, 41563}, {6906, 15556}, {7082, 8227}, {7280, 10884}, {7354, 41697}, {7411, 16767}, {9579, 37230}, {9581, 37532}, {9613, 37550}, {9614, 12704}, {9964, 45764}, {10072, 28610}, {10090, 13243}, {10954, 31434}, {11507, 18412}, {12540, 13089}, {12647, 54398}, {12671, 44425}, {15297, 25522}, {16465, 32760}, {16572, 23972}, {17010, 34772}, {17437, 37358}, {17699, 51784}, {18761, 36279}, {21381, 24432}, {22760, 37625}, {24430, 37530}, {31231, 37612}, {31775, 40663}, {37106, 37616}, {37251, 37582}, {37426, 37572}, {37787, 43177}

X(54432) = barycentric product X(63)*X(7537)
X(54432) = barycentric quotient X(7537)/X(92)
X(54432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 18389, 1}, {57, 7330, 9612}, {58, 44706, 1}, {63, 31424, 191}, {63, 54302, 6763}, {4641, 17102, 54301}, {12704, 30223, 9614}

X(54433) = X(1)X(2)∩X(3)X(345)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 + b^2 + 2*b*c + c^2) : : X(54433) = 5 X[3616] - 4 X[30148]

See HG050723

X(54433) lies on these lines: {1, 2}, {3, 345}, {4, 312}, {5, 28808}, {7, 1930}, {20, 346}, {21, 17776}, {28, 1043}, {30, 42032}, {37, 13725}, {45, 49728}, {56, 3703}, {58, 26065}, {63, 3710}, {69, 72}, {75, 443}, {100, 8193}, {181, 10369}, {189, 51304}, {192, 4201}, {210, 10371}, {315, 33939}, {319, 18156}, {321, 377}, {322, 32000}, {329, 1330}, {332, 14868}, {333, 47512}, {341, 3421}, {344, 405}, {348, 3933}, {376, 42033}, {379, 19838}, {388, 3974}, {404, 17740}, {464, 42706}, {497, 5015}, {631, 32851}, {728, 37551}, {894, 4340}, {942, 18141}, {944, 37431}, {958, 3932}, {960, 3416}, {1010, 2303}, {1058, 4514}, {1089, 1478}, {1104, 13742}, {1191, 5846}, {1228, 44150}, {1229, 6835}, {1257, 18636}, {1376, 3704}, {1453, 17353}, {1468, 33163}, {1479, 4680}, {1724, 26685}, {1770, 24280}, {1792, 11517}, {1801, 11115}, {1959, 26120}, {1973, 54316}, {1997, 4187}, {2289, 2329}, {2292, 26034}, {2321, 54405}, {2327, 16788}, {2475, 4671}, {2478, 4358}, {2551, 46937}, {2899, 36974}, {2901, 48837}, {2975, 32862}, {3058, 48798}, {3061, 53994}, {3159, 48835}, {3161, 17744}, {3175, 48813}, {3189, 41230}, {3191, 22008}, {3303, 4030}, {3434, 3702}, {3436, 3701}, {3487, 18134}, {3610, 5227}, {3685, 4294}, {3693, 36706}, {3712, 5217}, {3714, 5794}, {3729, 4292}, {3751, 41247}, {3767, 34542}, {3790, 4293}, {3797, 7791}, {3820, 5827}, {3869, 33078}, {3876, 5739}, {3883, 31435}, {3940, 21530}, {3949, 18671}, {3951, 4001}, {3966, 25917}, {3977, 4652}, {3984, 4101}, {3995, 17676}, {3998, 37180}, {4000, 33833}, {4019, 52387}, {4037, 9598}, {4082, 12527}, {4123, 6198}, {4133, 8769}, {4188, 33168}, {4189, 32849}, {4195, 17280}, {4198, 49542}, {4202, 19785}, {4252, 44416}, {4295, 4645}, {4296, 28739}, {4299, 7206}, {4320, 8816}, {4329, 51884}, {4359, 37462}, {4387, 6284}, {4388, 19582}, {4417, 5142}, {4513, 37537}, {4664, 51665}, {4673, 5082}, {4684, 41863}, {4869, 11036}, {4894, 4975}, {4901, 6762}, {5044, 5814}, {5084, 18743}, {5088, 32830}, {5175, 7557}, {5253, 33089}, {5280, 5749}, {5423, 5815}, {5434, 48806}, {5687, 12410}, {5690, 19547}, {5716, 13740}, {5788, 19782}, {5839, 16502}, {5847, 54386}, {6057, 7354}, {6327, 11415}, {6604, 37544}, {6857, 33116}, {6910, 33113}, {7046, 52346}, {7230, 7748}, {7386, 19799}, {7520, 52365}, {7523, 14829}, {11111, 17264}, {11112, 50044}, {11359, 50067}, {11374, 30828}, {11523, 17296}, {11681, 37983}, {12572, 30568}, {13728, 17321}, {13736, 54287}, {14210, 32099}, {15170, 48800}, {16284, 20914}, {16454, 19822}, {16466, 51192}, {16781, 17362}, {17095, 32818}, {17181, 37668}, {17263, 17552}, {17281, 50054}, {17289, 37037}, {17299, 40941}, {17342, 51673}, {17350, 20077}, {17359, 51670}, {17526, 33157}, {17559, 30829}, {17582, 19804}, {17678, 42047}, {18719, 20932}, {19844, 37261}, {20237, 20320}, {20928, 41013}, {23537, 30699}, {24701, 33066}, {25516, 51978}, {25527, 34937}, {26117, 41839}, {27509, 34823}, {27539, 46878}, {27549, 41229}, {31359, 34260}, {31993, 37153}, {32777, 37176}, {33079, 37598}, {33167, 37608}, {34791, 49688}, {35652, 50050}, {37093, 44140}, {37162, 46938}, {37231, 49492}, {37655, 54398}, {39731, 42696}, {41313, 50430}, {49716, 54280}, {50073, 51666}

X(54433) = isogonal conjugate of X(51686)
X(54433) = isotomic conjugate of the isogonal conjugate of X(7085)
X(54433) = isotomic conjugate of the polar conjugate of X(2345)
X(54433) = X(i)-isoconjugate of X(j) for these (i,j): {1, 51686}, {4, 1472}, {19, 2221}, {27, 2281}, {28, 1245}, {34, 1036}, {56, 1039}, {513, 32691}, {608, 2339}, {649, 36099}, {1395, 30479}
X(54433) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1039}, {3, 51686}, {6, 2221}, {958, 4185}, {5375, 36099}, {5515, 7649}, {11517, 1036}, {17421, 513}, {23874, 26933}, {36033, 1472}, {39026, 32691}, {40179, 4000}, {40181, 19}, {40591, 1245}
X(54433) = cevapoint of X(23874) and X(26933)
X(54433) = trilinear pole of line {2522, 23874}
X(54433) = barycentric product X(i)*X(j) for these {i,j}: {1, 19799}, {63, 4385}, {69, 2345}, {71, 44154}, {75, 5227}, {76, 7085}, {86, 3610}, {190, 23874}, {304, 612}, {306, 1010}, {312, 1038}, {345, 388}, {348, 3974}, {646, 51644}, {668, 2522}, {1016, 26933}, {1265, 7365}, {1332, 2517}, {2285, 3718}, {2286, 3596}, {2303, 20336}, {3926, 7102}, {4320, 52406}, {4561, 6590}, {4563, 48395}, {6332, 14594}, {7197, 30681}, {7386, 30701}, {40071, 44119}, {47844, 52609}, {50494, 52608}
X(54433) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2221}, {6, 51686}, {9, 1039}, {48, 1472}, {71, 1245}, {78, 2339}, {100, 36099}, {101, 32691}, {219, 1036}, {228, 2281}, {345, 30479}, {388, 278}, {612, 19}, {1010, 27}, {1038, 57}, {1332, 1310}, {1460, 608}, {2285, 34}, {2286, 56}, {2303, 28}, {2345, 4}, {2517, 17924}, {2522, 513}, {3610, 10}, {3974, 281}, {4206, 5317}, {4320, 1435}, {4385, 92}, {4561, 37215}, {5227, 1}, {5286, 1851}, {5323, 1396}, {6590, 7649}, {7085, 6}, {7102, 393}, {7365, 1119}, {7386, 4000}, {8678, 6591}, {8898, 1426}, {14594, 653}, {19459, 16502}, {19799, 75}, {23874, 514}, {26933, 1086}, {34259, 34260}, {34261, 4185}, {44119, 1474}, {44154, 44129}, {47844, 17925}, {48395, 2501}, {50494, 2489}, {51644, 3669}
X(54433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 19836, 3616}, {2, 145, 5262}, {2, 20009, 1}, {3, 3695, 345}, {8, 34255, 10449}, {10, 975, 2}, {10, 17733, 33137}, {20, 346, 7283}, {69, 304, 17170}, {69, 1265, 72}, {145, 33091, 8}, {312, 7270, 4}, {388, 3974, 4385}, {1104, 17279, 13742}, {3702, 5300, 3434}, {3771, 8669, 36573}, {4001, 52354, 3951}, {4358, 5016, 2478}, {4673, 32850, 5082}, {5044, 5814, 14555}, {6327, 25253, 11415}, {10449, 16086, 8}, {17742, 18596, 5279}, {19783, 29585, 1}, {26363, 30172, 30741}, {32777, 37539, 37176}

X(54434) = X(2)X(15032)∩X(5)X(323)

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

See HG020723

X(54434) lies on these lines: {2, 15032}, {3, 7712}, {4, 15066}, {5, 323}, {6, 3090}, {23, 15067}, {52, 10545}, {74, 5907}, {110, 7550}, {128, 47064}, {140, 399}, {141, 7552}, {155, 5067}, {186, 5891}, {195, 35018}, {376, 17811}, {394, 3545}, {546, 37496}, {547, 34545}, {568, 16042}, {631, 11456}, {1181, 3533}, {1199, 1656}, {1209, 2914}, {1216, 15107}, {1495, 7512}, {1498, 10299}, {1511, 14118}, {1594, 18358}, {1614, 5092}, {1993, 5071}, {1994, 5055}, {2071, 15060}, {2888, 50143}, {2979, 52294}, {3055, 45769}, {3098, 7999}, {3431, 7503}, {3518, 10546}, {3520, 4550}, {3524, 18451}, {3525, 11441}, {3526, 43605}, {3528, 35237}, {3542, 3620}, {3544, 36747}, {3581, 11591}, {3619, 7558}, {3628, 15037}, {3819, 14157}, {3917, 37925}, {5056, 11004}, {5068, 16266}, {5097, 22233}, {5449, 12364}, {5609, 13339}, {5651, 11438}, {5654, 14789}, {5888, 52525}, {5899, 44324}, {6126, 20117}, {6832, 14996}, {6852, 37633}, {6920, 51340}, {6983, 14997}, {7464, 15030}, {7486, 12161}, {7488, 33533}, {7505, 11487}, {7509, 26864}, {7530, 33884}, {7556, 35259}, {7565, 51391}, {7691, 47486}, {7998, 46261}, {9306, 11464}, {9729, 43596}, {9781, 37517}, {10109, 15038}, {10303, 32139}, {10539, 15080}, {10540, 15246}, {10564, 14865}, {10594, 33878}, {10625, 26863}, {10821, 34826}, {11064, 44834}, {11178, 22151}, {11412, 34417}, {11430, 35500}, {12111, 37470}, {12383, 34664}, {12812, 14627}, {13565, 15091}, {13595, 23039}, {13754, 43584}, {13861, 48912}, {14002, 37494}, {14094, 16836}, {14643, 51882}, {15022, 36749}, {15028, 15083}, {15081, 52124}, {15087, 15699}, {15350, 21357}, {16261, 37480}, {16534, 52171}, {18350, 37126}, {31831, 43808}, {34507, 41617}, {36753, 46936}, {37636, 37943}, {37945, 54042}, {41106, 44413}, {43576, 46847}, {43651, 44109}, {43844, 50664}, {43845, 48154}

X(54434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15068, 15032}, {3, 15052, 12112}, {110, 10170, 7550}, {547, 50461, 34545}, {1216, 43614, 34484}, {3819, 14157, 44832}, {10546, 11444, 37478}, {10546, 37478, 3518}, {11793, 43598, 7512}

X(54435) = X(1)X(6)∩X(3)X(1250)

Barycentrics    a^2*(Sqrt[3]*b*c - S) : :

X(54435) lies on these lines: {1, 6}, {3, 1250}, {8, 5362}, {11, 18581}, {12, 18582}, {13, 11237}, {14, 11238}, {15, 55}, {16, 56}, {35, 11480}, {36, 11481}, {61, 3303}, {62, 3304}, {115, 10061}, {222, 1082}, {388, 5335}, {395, 10072}, {396, 10056}, {495, 11542}, {496, 11543}, {497, 5334}, {498, 23302}, {499, 23303}, {559, 52424}, {940, 49594}, {999, 7127}, {1069, 10662}, {1277, 37772}, {1478, 5318}, {1479, 5321}, {2306, 12702}, {2307, 3295}, {3058, 10654}, {3085, 11488}, {3086, 11489}, {3157, 10661}, {3582, 16645}, {3583, 42093}, {3584, 16644}, {3585, 42094}, {3614, 42114}, {3616, 5367}, {3638, 5228}, {3639, 6180}, {3746, 22236}, {4293, 42120}, {4294, 42119}, {4299, 42088}, {4302, 42087}, {4309, 42147}, {4317, 42148}, {4325, 43193}, {4330, 43194}, {4383, 53589}, {4857, 5339}, {5204, 10646}, {5217, 10645}, {5225, 42133}, {5229, 42134}, {5270, 5340}, {5432, 42092}, {5433, 42089}, {5434, 10653}, {5471, 10077}, {5472, 10062}, {5563, 22238}, {6114, 12951}, {6115, 12942}, {6284, 42085}, {6782, 12952}, {6783, 12941}, {7173, 42111}, {7354, 42086}, {7741, 42095}, {7951, 42098}, {9654, 42128}, {9655, 42127}, {9656, 42813}, {9657, 16965}, {9668, 42126}, {9669, 42125}, {9670, 16964}, {9671, 42814}, {10483, 42097}, {10590, 42142}, {10591, 42139}, {10592, 42146}, {10593, 42143}, {10641, 11398}, {10642, 11399}, {10895, 16808}, {10896, 16809}, {12943, 19106}, {12953, 19107}, {13075, 23013}, {15171, 42117}, {15325, 42121}, {15326, 42091}, {15338, 42090}, {15888, 40693}, {15934, 33655}, {16772, 31452}, {18972, 22862}, {18974, 23006}, {18990, 42118}, {22860, 31706}, {22904, 31705}, {22906, 22910}, {31479, 42132}, {33653, 51340}, {37719, 42156}, {37720, 42153}, {37722, 40694}

X(54435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5353, 6}, {1, 54403, 54402}, {6, 5353, 54403}, {999, 11486, 19373}, {1124, 1335, 54403}, {1250, 7051, 3}, {2307, 10638, 11485}, {3295, 11485, 10638}, {7127, 19373, 11486}

X(54436) = X(1)X(6)∩X(15)X(56)

Barycentrics    a^2*(Sqrt[3]*b*c + S) : :

X(54436) lies on these lines: {1, 6}, {3, 10638}, {8, 5367}, {11, 18582}, {12, 18581}, {13, 11238}, {14, 11237}, {15, 56}, {16, 55}, {35, 11481}, {36, 11480}, {61, 3304}, {62, 3303}, {115, 10062}, {222, 559}, {388, 5334}, {395, 10056}, {396, 10072}, {495, 11543}, {496, 11542}, {497, 5335}, {498, 23303}, {499, 23302}, {940, 49595}, {999, 7051}, {1069, 10661}, {1082, 52424}, {1250, 3295}, {1251, 51340}, {1276, 37773}, {1478, 5321}, {1479, 5318}, {2307, 7373}, {3058, 10653}, {3085, 11489}, {3086, 11488}, {3157, 10662}, {3582, 16644}, {3583, 42094}, {3584, 16645}, {3585, 42093}, {3614, 42111}, {3616, 5362}, {3638, 6180}, {3639, 5228}, {3746, 22238}, {4293, 42119}, {4294, 42120}, {4299, 42087}, {4302, 42088}, {4309, 42148}, {4317, 42147}, {4325, 43194}, {4330, 43193}, {4383, 53588}, {4857, 5340}, {5204, 10645}, {5217, 10646}, {5225, 42134}, {5229, 42133}, {5270, 5339}, {5432, 42089}, {5433, 42092}, {5434, 10654}, {5471, 10061}, {5472, 10078}, {5563, 22236}, {6114, 12941}, {6115, 12952}, {6284, 42086}, {6767, 7127}, {6782, 12942}, {6783, 12951}, {7052, 15934}, {7173, 42114}, {7354, 42085}, {7741, 42098}, {7951, 42095}, {9654, 42125}, {9655, 42126}, {9656, 42814}, {9657, 16964}, {9668, 42127}, {9669, 42128}, {9670, 16965}, {9671, 42813}, {10483, 42096}, {10590, 42139}, {10591, 42142}, {10592, 42143}, {10593, 42146}, {10641, 11399}, {10642, 11398}, {10895, 16809}, {10896, 16808}, {12702, 33654}, {12943, 19107}, {12953, 19106}, {13076, 23006}, {15171, 42118}, {15325, 42124}, {15326, 42090}, {15338, 42091}, {15888, 40694}, {16773, 31452}, {18973, 22906}, {18975, 23013}, {18990, 42117}, {22859, 31706}, {22862, 22865}, {22905, 31705}, {31479, 42129}, {37719, 42153}, {37720, 42156}, {37722, 40693}

X(54436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5357, 6}, {1, 54402, 54403}, {6, 5357, 54402}, {999, 11485, 7051}, {1124, 1335, 54402}, {3295, 11486, 1250}, {10638, 19373, 3}

---

X(54437) = X(1)X(6)∩X(3)X(202)

Barycentrics    a^2*(2*b*c + Sqrt[3]*S) : :

X(54437) lies on these lines: {1, 6}, {3, 202}, {11, 42156}, {12, 42153}, {13, 9669}, {14, 9654}, {18, 31479}, {35, 36843}, {36, 36836}, {55, 22238}, {56, 22236}, {61, 999}, {62, 3295}, {203, 7373}, {388, 398}, {395, 3085}, {396, 3086}, {397, 497}, {495, 40694}, {496, 40693}, {498, 16645}, {499, 16644}, {1056, 42999}, {1058, 42998}, {1407, 37773}, {1478, 5339}, {1479, 5340}, {2098, 33655}, {2306, 4860}, {2307, 3304}, {3303, 7127}, {3389, 31474}, {3411, 31480}, {3582, 49905}, {3584, 49906}, {3614, 42095}, {3617, 5367}, {4293, 42147}, {4294, 42148}, {4299, 43194}, {4302, 43193}, {5204, 11480}, {5217, 10638}, {5218, 16773}, {5225, 5318}, {5229, 5321}, {5362, 46934}, {5432, 43239}, {5433, 43238}, {6284, 42155}, {6767, 7006}, {7173, 42098}, {7288, 16772}, {7354, 42154}, {9655, 16964}, {9668, 16965}, {10056, 49948}, {10072, 49947}, {10386, 42924}, {10588, 42599}, {10589, 42598}, {10590, 42163}, {10591, 42166}, {10592, 18581}, {10593, 18582}, {10653, 15171}, {10654, 18990}, {11073, 42019}, {14986, 37640}, {15325, 42152}, {33654, 37567}

X(54437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5357, 54402}, {1, 54402, 6}, {202, 7005, 3}, {1124, 1335, 5357}

---

X(54438) = X(1)X(6)∩X(3)X(203)

Barycentrics    a^2*(2*b*c - Sqrt[3]*S) : :

X(54438) lies on these lines: {1, 6}, {3, 203}, {11, 42153}, {12, 42156}, {13, 9654}, {14, 9669}, {17, 31479}, {35, 36836}, {36, 36843}, {55, 2307}, {56, 7127}, {61, 3295}, {62, 999}, {202, 7373}, {388, 397}, {395, 3086}, {396, 3085}, {398, 497}, {495, 40693}, {496, 40694}, {498, 16644}, {499, 16645}, {1056, 42998}, {1058, 42999}, {1250, 5217}, {1407, 37772}, {1478, 5340}, {1479, 5339}, {2098, 7052}, {2306, 37567}, {3364, 31474}, {3412, 31480}, {3582, 49906}, {3584, 49905}, {3614, 42098}, {3617, 5362}, {4293, 42148}, {4294, 42147}, {4299, 43193}, {4302, 43194}, {4860, 33654}, {5204, 7051}, {5218, 16772}, {5225, 5321}, {5229, 5318}, {5367, 46934}, {5432, 43238}, {5433, 43239}, {6284, 42154}, {6767, 7005}, {7173, 42095}, {7288, 16773}, {7354, 42155}, {9655, 16965}, {9668, 16964}, {10056, 49947}, {10072, 49948}, {10386, 42925}, {10588, 42598}, {10589, 42599}, {10590, 42166}, {10591, 42163}, {10592, 18582}, {10593, 18581}, {10653, 18990}, {10654, 15171}, {11072, 42019}, {14986, 37641}, {15325, 42149}

X(54438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5353, 54403}, {1, 54403, 6}, {55, 2307, 22236}, {56, 7127, 22238}, {203, 7006, 3}, {1124, 1335, 5353}

---

X(54439) = X(2)X(99)∩X(3)X(74)

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

X(54439) lies on these lines: {2, 99}, {3, 74}, {6, 2987}, {23, 18860}, {32, 11004}, {39, 2981}, {97, 14586}, {114, 36163}, {187, 249}, {237, 15107}, {247, 12900}, {351, 53247}, {394, 5210}, {476, 46634}, {511, 35298}, {694, 12055}, {842, 7468}, {868, 15561}, {895, 9145}, {1003, 50673}, {1297, 37918}, {1316, 33813}, {1384, 1993}, {1495, 37183}, {1634, 9142}, {1976, 5092}, {1989, 44386}, {1994, 5008}, {2080, 23061}, {2502, 53095}, {2979, 41275}, {3003, 41617}, {3053, 20976}, {3098, 36213}, {3124, 5013}, {3148, 10546}, {3292, 47113}, {3448, 14981}, {3569, 8552}, {3580, 6390}, {3581, 44221}, {3619, 14806}, {3620, 10979}, {4226, 21166}, {4235, 41253}, {5012, 26316}, {5024, 9486}, {5467, 52699}, {5642, 53710}, {5649, 46787}, {5651, 9734}, {5653, 44814}, {5888, 14096}, {6337, 37643}, {6636, 7711}, {7471, 46987}, {7496, 21163}, {7813, 37779}, {7925, 40885}, {8182, 10554}, {8289, 46806}, {8369, 14389}, {8588, 32901}, {8722, 33884}, {8724, 9140}, {8836, 37340}, {8838, 37341}, {9160, 9184}, {9216, 46131}, {9737, 34417}, {9775, 9832}, {10545, 11328}, {10564, 52279}, {10718, 35937}, {10752, 15919}, {10991, 14683}, {11007, 30789}, {11064, 32459}, {11416, 22087}, {13335, 44109}, {14417, 39905}, {14480, 46633}, {14570, 48540}, {14611, 46981}, {14649, 54087}, {14850, 53132}, {15019, 32447}, {15462, 40083}, {15815, 20998}, {17811, 46276}, {23234, 53161}, {23235, 53346}, {27088, 40112}, {31626, 34897}, {32456, 35933}, {32985, 37645}, {33215, 40915}, {33878, 51335}, {34511, 37644}, {34840, 52125}, {36181, 38738}, {36188, 47326}, {37114, 37478}, {37483, 52276}, {38736, 51431}, {38748, 47200}, {39785, 44555}, {40078, 48450}, {40349, 44436}, {40916, 44420}, {43460, 50706}, {51882, 53767}

X(54439) = crossdifference of every pair of points on line {351, 1637}
X(54439) = X(53247)-lineconjugate of X(351)
X(54439) = barycentric product X(99)*X(34291)
X(54439) = barycentric quotient X(34291)/X(523)
X(54439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 9155, 110}, {187, 36212, 323}, {237, 35002, 15107}, {323, 35296, 187}, {620, 51389, 2}, {2981, 6151, 39024}, {9145, 46127, 895}, {11130, 11131, 110}, {32456, 51372, 35933}

X(54440) = X(99)X(109)∩X(100)X(101)

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

X(54440) lies on these lines: {31, 16834}, {55, 14839}, {63, 2809}, {99, 109}, {100, 101}, {110, 6013}, {171, 9881}, {190, 522}, {238, 28580}, {516, 24630}, {536, 19624}, {692, 4436}, {833, 29119}, {835, 6577}, {898, 6014}, {901, 29351}, {932, 1293}, {985, 3802}, {993, 2802}, {1229, 47487}, {1253, 3729}, {1331, 2398}, {1332, 35338}, {1438, 24578}, {1633, 3882}, {2161, 2805}, {2195, 17755}, {2242, 37540}, {2316, 24482}, {2328, 32932}, {3241, 17126}, {3685, 13329}, {3875, 21059}, {3888, 6003}, {3912, 9441}, {4238, 32674}, {4414, 4475}, {4424, 4653}, {4482, 29066}, {4553, 15313}, {4586, 32041}, {4781, 35281}, {5091, 8299}, {8694, 8708}, {9070, 29133}, {17475, 38865}, {22370, 24309}, {23845, 53268}, {25083, 41339}, {28226, 29199}, {28916, 32929}, {29159, 43348}, {30664, 43077}, {43076, 46961}

X(54440) = X(43349)-anticomplementary conjugate of X(21293)
X(54440) = X(4586)-Ceva conjugate of X(101)
X(54440) = X(i)-isoconjugate of X(j) for these (i,j): {244, 37138}, {513, 1002}, {514, 2279}, {523, 51443}, {649, 27475}, {650, 42290}, {661, 42302}, {1015, 32041}, {1086, 8693}, {3122, 51563}, {3669, 40779}
X(54440) = X(i)-Dao conjugate of X(j) for these (i,j): {1001, 47123}, {2276, 824}, {3826, 47704}, {5375, 27475}, {36830, 42302}, {39026, 1002}
X(54440) = cevapoint of X(i) and X(j) for these (i,j): {1001, 4724}, {49478, 50336}
X(54440) = trilinear pole of line {1001, 2280}
X(54440) = crossdifference of every pair of points on line {244, 20974}
X(54440) = barycentric product X(i)*X(j) for these {i,j}: {100, 4384}, {101, 4441}, {109, 28809}, {110, 4044}, {190, 1001}, {644, 40719}, {645, 42289}, {646, 1471}, {651, 3886}, {662, 3696}, {664, 37658}, {668, 2280}, {692, 21615}, {765, 4762}, {1016, 4724}, {1492, 27474}, {1897, 23151}, {3257, 4702}, {3699, 5228}, {3789, 4586}, {4567, 4804}, {4578, 42309}, {4998, 45755}, {37133, 40732}
X(54440) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 27475}, {101, 1002}, {109, 42290}, {110, 42302}, {163, 51443}, {692, 2279}, {765, 32041}, {1001, 514}, {1110, 8693}, {1252, 37138}, {1471, 3669}, {2280, 513}, {3696, 1577}, {3789, 824}, {3886, 4391}, {3939, 40779}, {4044, 850}, {4384, 693}, {4441, 3261}, {4567, 51563}, {4702, 3762}, {4724, 1086}, {4762, 1111}, {4804, 16732}, {5228, 3676}, {21615, 40495}, {23151, 4025}, {28044, 3064}, {28809, 35519}, {37658, 522}, {40719, 24002}, {40732, 3250}, {42289, 7178}, {45755, 11}
X(54440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 644, 1026}, {100, 3573, 101}

X(54441) = X(4)X(11)∩X(72)X(74)

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

X(54441) lies on these lines: {3, 13257}, {4, 11}, {9, 45633}, {72, 74}, {119, 6889}, {153, 6908}, {226, 10058}, {329, 17100}, {405, 38602}, {411, 13243}, {442, 10742}, {950, 10074}, {952, 7580}, {954, 12775}, {971, 5122}, {1006, 37822}, {1035, 38295}, {1490, 1768}, {1728, 10090}, {2800, 5119}, {3149, 13226}, {5450, 37692}, {5658, 18861}, {5851, 21168}, {6223, 52270}, {6713, 6832}, {6878, 21154}, {6906, 11374}, {6976, 10269}, {6990, 31272}, {9945, 37426}, {9946, 10884}, {9957, 10698}, {9963, 33557}, {10393, 11570}, {10936, 12776}, {11523, 25438}, {12690, 12773}, {12750, 48694}, {13615, 34123}, {14803, 48695}, {16127, 36152}, {34122, 37240}, {35979, 40263}, {53252, 53279}

X(54441) = reflection of X(10728) in X(13273)
X(54441) = crossdifference of every pair of points on line {14399, 52307}
X(54441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1490, 1768, 12691}, {12773, 37411, 12690}

X(54442) = X(100)X(110)∩X(107)X(109)

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

X(54442) lies on these lines: {29, 12616}, {46, 13739}, {100, 110}, {107, 109}, {163, 1021}, {270, 1771}, {759, 12736}, {901, 53683}, {1158, 11107}, {1414, 4566}, {1624, 23845}, {1633, 4246}, {1780, 4248}, {2328, 10164}, {4237, 35278}, {6001, 15776}, {6740, 9803}, {11329, 35259}, {24580, 35260}, {24624, 45043}, {53324, 53761}

X(54442) = trilinear pole of line {8557, 18446}
X(54442) = barycentric product X(i)*X(j) for these {i,j}: {99, 8557}, {162, 6350}, {648, 18446}, {662, 18391}, {811, 19350}
X(54442) = barycentric quotient X(i)/X(j) for these {i,j}: {6350, 14208}, {8557, 523}, {18391, 1577}, {18446, 525}, {19350, 656}
X(54442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 3658, 662}, {653, 7452, 107}

X(54443) = X(2)X(304)∩X(75)X(499)

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

X(54443) lies on these lines: {2, 304}, {75, 499}, {76, 17095}, {85, 32832}, {183, 17181}, {312, 7763}, {345, 32829}, {346, 32835}, {348, 20925}, {498, 18156}, {908, 17206}, {1078, 4872}, {1102, 3305}, {1969, 17923}, {3074, 4592}, {3086, 39731}, {3403, 16706}, {3767, 25918}, {3926, 28808}, {4561, 6734}, {5886, 21281}, {7270, 7752}, {7769, 32851}, {7862, 34542}, {11374, 30962}, {17079, 32885}, {17144, 36542}, {17170, 34229}, {17289, 30103}, {17315, 30125}, {17322, 19864}, {18135, 27187}, {18140, 37758}, {18142, 29793}, {24282, 24914}, {26363, 30758}, {27162, 33133}, {32838, 52422}, {44179, 54401}

X(54443) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {348, 32828, 20925}, {7769, 33939, 32851}

X(54444) = X(2)X(2003)∩X(6)X(63)

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

X(54444) lies on these lines: {2, 2003}, {6, 63}, {9, 1993}, {21, 54301}, {31, 50600}, {35, 50593}, {51, 3955}, {57, 5422}, {72, 36750}, {78, 36742}, {81, 908}, {92, 3758}, {182, 7293}, {222, 3306}, {239, 20879}, {275, 1948}, {323, 27065}, {329, 37685}, {394, 3305}, {511, 5314}, {575, 26889}, {576, 26893}, {651, 5249}, {894, 14213}, {914, 5294}, {940, 30852}, {1203, 2975}, {1331, 14547}, {1351, 7085}, {1473, 5050}, {1790, 2183}, {1959, 5280}, {1992, 26872}, {1994, 2323}, {2994, 54283}, {3060, 5285}, {3074, 54356}, {3157, 54392}, {3193, 12572}, {3218, 26740}, {3220, 5012}, {3618, 26871}, {3745, 17615}, {3784, 43650}, {3812, 8614}, {3870, 45729}, {3916, 37509}, {3920, 46685}, {4652, 36754}, {4855, 36746}, {5259, 35197}, {5299, 52134}, {5439, 23070}, {5440, 51340}, {5483, 16585}, {5748, 14996}, {5752, 54337}, {5943, 26884}, {7193, 13366}, {7308, 15066}, {7330, 7592}, {7584, 16028}, {9777, 37581}, {11004, 52405}, {11402, 24320}, {11456, 18540}, {11681, 37559}, {12514, 16473}, {14206, 17120}, {15018, 27003}, {16514, 45843}, {17379, 27287}, {17745, 24635}, {20834, 23202}, {22060, 37510}, {22129, 52424}, {22141, 33595}, {24467, 36753}, {26885, 34986}, {26921, 36749}, {26932, 37649}, {26933, 45298}, {31164, 37543}, {31266, 34048}, {37584, 39522}, {44547, 52362}, {45206, 52351}

X(54444) = reflection of X(5314) in X(26890)
X(54444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2003, 22128}, {182, 26892, 7293}, {222, 10601, 3306}, {1994, 3219, 2323}, {3218, 34545, 52423}

X(54445) = X(2)X(515)∩X(7)X(36)

Barycentrics    7*a^4-2*a^3*(b+c)+2*a*(b-c)^2*(b+c)+a^2*(-8*b^2+4*b*c-8*c^2)+(b^2-c^2)^2 ::
X(54445) = 2*X[1]+7*X[3523], 8*X[3]+X[962], -2*X[4]+11*X[5550], 2*X[40]+7*X[3622], X[165]+2*X[551], -2*X[355]+11*X[3525], X[1482]+8*X[3530], X[3534]+2*X[38034], X[3655]+2*X[11231], -2*X[5881]+11*X[46933], 2*X[7982]+7*Z[15054], 8*X[8715]+X[12541], 2*X[11194]+X[25568], -X[11224]+4*X[51103], -X[12702]+10*X[15712], 2*X[41869]+7*X[50693]

See Ivan Pavlov, Romantics of Geometry 12957.

X(54445) lies on these lines: {1, 3523}, {2, 515}, {3, 962}, {4, 5550}, {7, 36}, {8, 631}, {10, 10303}, {20, 1125}, {21, 10309}, {30, 9779}, {35, 9785}, {40, 3622}, {46, 4323}, {56, 3475}, {104, 6883}, {140, 944}, {145, 6684}, {165, 551}, {214, 9803}, {329, 1006}, {355, 3525}, {376, 5886}, {381, 28190}, {388, 37605}, {390, 30282}, {392, 11227}, {404, 11024}, {405, 5658}, {411, 8273}, {497, 37600}, {499, 4305}, {516, 10304}, {517, 3524}, {519, 15708}, {548, 18493}, {549, 3241}, {572, 5296}, {632, 18525}, {938, 2646}, {946, 3522}, {952, 5054}, {971, 17561}, {991, 49997}, {993, 18228}, {997, 5273}, {999, 10578}, {1000, 25405}, {1001, 6909}, {1012, 5284}, {1056, 5126}, {1151, 13959}, {1152, 13902}, {1319, 5218}, {1388, 52793}, {1478, 5444}, {1479, 37163}, {1482, 3530}, {1483, 12108}, {1698, 38155}, {1737, 31188}, {1788, 34471}, {2094, 21165}, {2320, 6954}, {2771, 31669}, {2807, 20791}, {2975, 5815}, {3085, 4308}, {3086, 3612}, {3090, 18481}, {3091, 3624}, {3146, 8227}, {3244, 9588}, {3361, 11036}, {3474, 15950}, {3476, 5432}, {3485, 5204}, {3486, 5433}, {3488, 15325}, {3526, 5818}, {3528, 12699}, {3529, 9955}, {3533, 9956}, {3534, 38034}, {3543, 3817}, {3545, 28160}, {3579, 10299}, {3600, 13411}, {3601, 14986}, {3617, 5882}, {3620, 39870}, {3621, 13607}, {3623, 11362}, {3654, 15719}, {3655, 11231}, {3656, 15698}, {3679, 15721}, {3753, 10156}, {3816, 6932}, {3828, 37712}, {3839, 7988}, {3845, 50819}, {3869, 9940}, {3876, 12675}, {3890, 31788}, {3892, 15104}, {3897, 6921}, {3904, 44819}, {3911, 13384}, {4188, 10902}, {4189, 37561}, {4293, 5226}, {4295, 7280}, {4300, 21214}, {4301, 16192}, {4304, 5274}, {4311, 5261}, {4342, 31508}, {4345, 5119}, {4423, 6912}, {4511, 5744}, {4539, 38067}, {4666, 6282}, {4677, 51085}, {5010, 30305}, {5047, 12114}, {5049, 33575}, {5055, 28186}, {5056, 5691}, {5059, 18483}, {5067, 18480}, {5068, 31673}, {5080, 6947}, {5092, 39898}, {5175, 6889}, {5180, 23961}, {5250, 24558}, {5298, 15933}, {5304, 9592}, {5333, 7415}, {5428, 14450}, {5436, 37421}, {5450, 6223}, {5690, 15720}, {5758, 26286}, {5768, 13151}, {5770, 28465}, {5840, 32558}, {5881, 46933}, {6001, 35260}, {6049, 21842}, {6176, 30947}, {6224, 6713}, {6245, 24564}, {6256, 37162}, {6693, 54181}, {6796, 17572}, {6853, 26492}, {6857, 17614}, {6872, 26129}, {6875, 11415}, {6897, 52367}, {6904, 24541}, {6925, 26105}, {6940, 10267}, {6943, 25466}, {6960, 10200}, {6967, 27529}, {6972, 10198}, {6989, 10785}, {7406, 29612}, {7411, 22753}, {7486, 19925}, {7492, 9625}, {7586, 9583}, {7984, 48378}, {7989, 19878}, {8583, 10884}, {8715, 12541}, {8726, 19861}, {9615, 13971}, {9624, 21734}, {9708, 38669}, {9746, 11200}, {9799, 37837}, {9802, 33814}, {9809, 38602}, {9933, 20191}, {10124, 38138}, {10167, 33574}, {10186, 28885}, {10247, 15693}, {10283, 12100}, {10519, 38029}, {10527, 37407}, {10580, 24929}, {10590, 21578}, {10624, 18220}, {10916, 12536}, {11001, 50873}, {11019, 53054}, {11038, 21153}, {11194, 25568}, {11224, 51103}, {11500, 17531}, {11522, 12512}, {11539, 28224}, {11735, 15051}, {11812, 38112}, {12104, 16116}, {12245, 15178}, {12246, 19526}, {12263, 32522}, {12528, 25917}, {12571, 50688}, {12702, 15712}, {13405, 13462}, {13464, 20070}, {14890, 38081}, {15022, 18492}, {15177, 15246}, {15626, 19261}, {15640, 30308}, {15676, 16132}, {15688, 28178}, {15689, 28182}, {15690, 50806}, {15694, 34627}, {15696, 40273}, {15701, 50824}, {15705, 28194}, {15709, 28204}, {15710, 28198}, {15713, 50798}, {15722, 50805}, {15726, 38025}, {15759, 50813}, {15837, 51772}, {16370, 21151}, {17127, 37469}, {17183, 37303}, {17504, 28212}, {17538, 22793}, {17576, 41012}, {17777, 38604}, {17923, 37028}, {18357, 46219}, {18391, 37525}, {19003, 42522}, {19004, 42523}, {19708, 51709}, {19875, 28236}, {21164, 35258}, {21167, 38315}, {21168, 38030}, {24386, 34701}, {27625, 37732}, {28150, 38021}, {28172, 50687}, {28216, 45759}, {28466, 38033}, {28858, 53014}, {28866, 44431}, {29648, 50699}, {29666, 50698}, {29817, 37569}, {30332, 30384}, {31145, 38127}, {31399, 46930}, {31657, 51409}, {33748, 34379}, {33923, 48661}, {34474, 38032}, {34862, 54228}, {35202, 37105}, {35239, 45977}, {35271, 38122}, {37306, 52148}, {37557, 45308}, {37714, 51073}, {38053, 38454}, {40333, 43175}, {40998, 50742}, {41869, 50693}, {44299, 52796}, {48893, 50420}, {48923, 50418}, {50808, 51110}, {50814, 51106}, {50818, 51068}, {50829, 51093}, {50865, 51109}, {50872, 51105}, {50977, 51001}, {50983, 50999}, {50984, 51000}, {50998, 51139}, {51045, 51056}, {51049, 51054}, {51071, 51086}, {51137, 51193}, {52705, 53579}

X(54445) = reflection of X(i) in X(j) for these {i,j}: {3839, 7988}, {7988, 19883}
X(54445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3576, 5731}, {2, 51705, 50864}, {3, 3616, 962}, {3, 38028, 5603}, {3, 5603, 9778}, {3, 5901, 6361}, {36, 52769, 37106}, {40, 3622, 5734}, {56, 5703, 11037}, {140, 944, 9780}, {355, 3525, 19877}, {376, 5886, 9812}, {499, 37616, 4305}, {549, 10246, 5657}, {551, 15692, 34632}, {631, 1385, 8}, {631, 7967, 26446}, {1125, 7987, 20}, {1385, 26446, 7967}, {2646, 7288, 938}, {2975, 27383, 5815}, {3085, 37618, 4308}, {3086, 3612, 4313}, {3486, 5433, 5704}, {3522, 46934, 946}, {3524, 3653, 38314}, {3526, 34773, 5818}, {3529, 9955, 10248}, {3576, 10165, 2}, {3576, 5587, 51705}, {3622, 15717, 40}, {3624, 4297, 3091}, {3654, 51084, 15719}, {5603, 38028, 3616}, {5657, 10246, 3241}, {5691, 19862, 5056}, {5886, 17502, 376}, {7988, 28164, 3839}, {7989, 19878, 46936}, {8273, 25524, 411}, {10165, 50828, 3576}, {10299, 10595, 3579}, {11231, 31662, 3655}, {12245, 15178, 20057}, {12512, 15808, 11522}, {13464, 35242, 20070}, {15325, 37606, 3488}, {15712, 51700, 12702}, {19883, 28164, 7988}, {19925, 34595, 7486}, {21151, 38031, 52653}, {30282, 44675, 390}, {30308, 50815, 15640}

X(54446) = X(1482)X(4853)∩X(5902)X(7271)

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

See Ivan Pavlov, Romantics of Geometry 12957.

X(54446) lies on these lines: {1482, 4853}, {5902, 7271}, {10310, 17524}

X(54446) = intersection, other than A, B, C, of these circumonics: {{A*, B, C, X(1), X(3304)}}, {{A, B, C, X(3), X(16615)}}, {{A, B, C, X(4), X(963)}}, {{A, B, C, X(6), X(102)}}, {{A, B, C, X(8), X(3477)}}, {{A, B, C, X(55), X(80)}}, {{A, B, C, X(56), X(1389)}}, {{A, B, C, X(64), X(1126)}}, {{A, B, C, X(65), X(10310)}}, {{A, B, C, X(103), X(14490)}}, {{A, B, C, X(104), X(3531)}}, {{A, B, C, X(939), X(17097)}}, {{A, B, C, X(945), X(1173)}}, {{A, B, C, X(947), X(22334)}}, {{A, B, C, X(953), X(14491)}}, {{A, B, C, X(1000), X(10579)}}, {{A, B, C, X(1057), X(24297)}}, {{A, B, C, X(1243), X(1436)}}, {{A, B, C, X(1392), X(52792)}}, {{A, B, C, X(2099), X(10269)}}, {{A, B, C, X(2217), X(44861)}}, {{A, B, C, X(3062), X(28227)}}, {{A, B, C, X(3426), X(14496)}}, {{A, B, C, X(3431), X(28189)}}, {{A, B, C, X(4900), X(13404)}}, {{A, B, C, X(6583), X(11510)}}, {{A, B, C, X(11509), X(35004)}}, {{A, B, C, X(14483), X(14497)}}, {{A, B, C, X(17098), X(42019)}}, {{A, B, C, X(21398), X(33963)}}

X(54446) = isogonal conjugate of X(54445)

X(54447) = X(2)X(515)∩X(5)X(40)

Barycentrics    a^4+a^3*(b+c)-a*(b-c)^2*(b+c)+4*(b^2-c^2)^2-a^2*(5*b^2+2*b*c+5*c^2) ::
X(54447) = 2*X[3]+7*X[7989], X[4]+8*X[3634], 2*X[355]+7*X[3624], 2*X[382]+7*X[16192], -X[944]+10*X[19862], X[3655]+2*X[38138], X[3656]+2*X[38112], 2*X[11224]+X[50817], 2*X[18525]+7*X[30389]

See Ivan Pavlov, Romantics of Geometry 12957.

X(54447) lies on these lines: {1, 1656}, {2, 515}, {3, 7989}, {4, 3634}, {5, 40}, {8, 7486}, {9, 3814}, {10, 3090}, {11, 31393}, {12, 3333}, {51, 52796}, {57, 6881}, {80, 13384}, {84, 4197}, {115, 9574}, {119, 11219}, {140, 5691}, {165, 381}, {355, 3624}, {382, 16192}, {442, 7701}, {485, 13947}, {486, 13893}, {496, 51784}, {498, 6887}, {499, 9578}, {516, 3545}, {517, 4731}, {547, 3679}, {549, 28190}, {567, 9586}, {569, 9622}, {631, 19925}, {632, 18481}, {944, 19862}, {946, 5056}, {952, 15699}, {962, 15022}, {993, 6946}, {999, 5726}, {1125, 5067}, {1210, 3475}, {1329, 5705}, {1385, 5070}, {1420, 10827}, {1478, 31231}, {1506, 9575}, {1512, 6879}, {1532, 3826}, {1571, 39565}, {1572, 7603}, {1594, 7713}, {1697, 7741}, {1702, 42262}, {1703, 42265}, {1706, 6975}, {1737, 5219}, {1995, 9625}, {2077, 4413}, {2093, 17605}, {2095, 3715}, {2136, 24387}, {2476, 24991}, {2801, 38093}, {2886, 12703}, {2948, 20304}, {3091, 6684}, {3097, 7697}, {3340, 18395}, {3359, 6980}, {3361, 9654}, {3523, 31673}, {3524, 28164}, {3525, 4297}, {3526, 7987}, {3544, 6361}, {3567, 31752}, {3579, 3851}, {3583, 35445}, {3584, 10389}, {3586, 5432}, {3589, 39885}, {3601, 6861}, {3614, 9612}, {3616, 46936}, {3617, 13464}, {3622, 47745}, {3626, 10595}, {3632, 5901}, {3646, 4187}, {3653, 28224}, {3654, 10109}, {3655, 38138}, {3656, 38112}, {3697, 13374}, {3742, 18908}, {3751, 24206}, {3812, 5693}, {3817, 3828}, {3822, 5437}, {3830, 50812}, {3832, 31425}, {3833, 15064}, {3839, 28150}, {3841, 6941}, {3843, 31663}, {3855, 51118}, {3860, 50825}, {3876, 31870}, {3901, 31835}, {3911, 10590}, {3929, 38109}, {4002, 45776}, {4193, 31435}, {4208, 54052}, {4302, 51792}, {4355, 34753}, {4423, 34486}, {4512, 17556}, {4668, 10222}, {4677, 10247}, {4915, 51362}, {5010, 7489}, {5020, 15177}, {5044, 37625}, {5047, 6796}, {5054, 28160}, {5068, 18483}, {5072, 22793}, {5079, 7991}, {5123, 9623}, {5128, 5445}, {5154, 5250}, {5231, 17757}, {5251, 6911}, {5254, 31428}, {5259, 11499}, {5290, 10592}, {5433, 9613}, {5438, 6852}, {5450, 17531}, {5550, 5882}, {5690, 11522}, {5704, 21620}, {5775, 46873}, {5777, 15016}, {5789, 24645}, {5817, 38204}, {5972, 12407}, {6256, 37462}, {6264, 31272}, {6326, 6702}, {6459, 9618}, {6565, 9616}, {6666, 6844}, {6667, 12751}, {6705, 37436}, {6721, 13178}, {6722, 9864}, {6723, 12368}, {6735, 11525}, {6766, 7958}, {6827, 18406}, {6830, 51780}, {6832, 26364}, {6841, 37551}, {6842, 37560}, {6856, 8582}, {6859, 20196}, {6874, 12514}, {6877, 25525}, {6883, 44425}, {6912, 9342}, {6918, 11012}, {6920, 25440}, {6931, 24987}, {6932, 10860}, {6933, 24982}, {6939, 26040}, {6944, 19854}, {6964, 19855}, {6983, 26363}, {6991, 24468}, {7173, 9614}, {7280, 45976}, {7290, 17734}, {7393, 8185}, {7424, 33538}, {7504, 7705}, {7509, 9626}, {7514, 9590}, {7529, 37557}, {7581, 49619}, {7582, 49618}, {7688, 19541}, {7743, 9819}, {7962, 23708}, {8164, 11019}, {8193, 11484}, {8253, 9583}, {8580, 37569}, {8726, 50726}, {8728, 37526}, {8976, 19004}, {9306, 9621}, {9582, 23261}, {9587, 18350}, {9591, 13861}, {9592, 31489}, {9593, 13881}, {9617, 35255}, {9668, 31508}, {9669, 53053}, {9746, 28866}, {9779, 28194}, {9781, 31737}, {9820, 9896}, {9902, 11272}, {9905, 13565}, {9957, 50444}, {10039, 50443}, {10157, 50740}, {10197, 38316}, {10246, 15703}, {10267, 25542}, {10283, 51093}, {10304, 28172}, {10476, 10887}, {10519, 38146}, {10576, 18991}, {10577, 18992}, {10589, 31397}, {10593, 51785}, {10857, 18529}, {10864, 17529}, {10895, 15803}, {10902, 11108}, {11001, 50803}, {11014, 31262}, {11224, 50817}, {11362, 46933}, {11444, 31760}, {11500, 16842}, {11531, 18493}, {11539, 28186}, {12100, 50799}, {12114, 16862}, {12119, 31235}, {12435, 34466}, {12616, 25011}, {12619, 15017}, {12778, 15088}, {12785, 32396}, {12812, 22791}, {12900, 13211}, {13607, 46934}, {13624, 46219}, {13729, 26060}, {13886, 49547}, {13911, 42583}, {13912, 42561}, {13939, 49548}, {13951, 19003}, {13973, 42582}, {13975, 31412}, {14269, 28154}, {14892, 28216}, {15024, 31732}, {15056, 31728}, {15059, 33535}, {15626, 19275}, {15682, 50829}, {15694, 17502}, {15695, 51088}, {15698, 50862}, {15702, 34648}, {15726, 38075}, {15931, 18491}, {16132, 31254}, {16173, 38319}, {16208, 45630}, {16209, 45631}, {16408, 37561}, {16475, 38317}, {16496, 42786}, {16832, 30849}, {17057, 31263}, {17124, 37469}, {18358, 39878}, {18525, 30389}, {19708, 51081}, {19709, 50865}, {19710, 50866}, {19883, 28236}, {20400, 49176}, {21151, 38158}, {21168, 38151}, {23046, 28182}, {24386, 34619}, {24392, 45701}, {24644, 38121}, {24808, 48854}, {26725, 37713}, {28178, 38071}, {28204, 30392}, {28212, 47478}, {28234, 53620}, {30286, 50194}, {30308, 50821}, {31398, 43620}, {31421, 44518}, {31452, 41864}, {31776, 53057}, {34474, 38161}, {34747, 51515}, {34773, 48154}, {35258, 37375}, {37290, 38411}, {37556, 37720}, {37718, 38182}, {38036, 38057}, {38052, 38108}, {38059, 38149}, {38073, 38101}, {38107, 38179}, {38150, 38454}, {38154, 38758}, {38172, 51516}, {38180, 51768}, {41106, 50808}, {41867, 51755}, {44217, 52027}, {48888, 49993}, {48897, 50416}, {50798, 51110}, {50818, 51109}, {51068, 51077}

X(54447) = midpoint of X(7988) and X(19875)
X(54447) = reflection of X(i) in X(j) for these {i,j}: {38021, 7988}, {7988, 5055}
X(54447) = complement of X(54445)
X(54447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 30315, 9956}, {2, 10175, 5587}, {3, 7989, 18492}, {4, 31423, 35242}, {4, 3634, 31423}, {5, 26446, 1699}, {10, 10171, 5603}, {10, 3090, 8227}, {10, 8227, 7982}, {11, 31434, 31393}, {355, 3628, 3624}, {381, 11231, 165}, {517, 5055, 7988}, {547, 38042, 5886}, {1125, 38155, 7967}, {1125, 5818, 5881}, {1385, 5070, 34595}, {1656, 5790, 11230}, {1656, 9956, 1}, {1698, 1699, 26446}, {1699, 26446, 40}, {1737, 5219, 11529}, {3090, 5603, 10171}, {3091, 19877, 6684}, {3091, 6684, 41869}, {3526, 18480, 7987}, {3544, 6361, 12571}, {3614, 24914, 9612}, {3679, 5886, 16200}, {3817, 3828, 5657}, {3817, 5657, 31162}, {4413, 6913, 2077}, {5055, 19875, 38021}, {5055, 38083, 19875}, {5056, 9780, 946}, {5067, 5818, 1125}, {5071, 5657, 3817}, {5818, 7967, 38155}, {5886, 38042, 3679}, {6931, 24987, 25522}, {7486, 31399, 9624}, {7504, 7705, 19860}, {7988, 19875, 517}, {9956, 11230, 5790}, {10165, 50811, 3576}, {10172, 10175, 2}, {15022, 46932, 962}, {18395, 37692, 3340}, {19925, 51073, 631}, {34595, 37714, 1385}, {38182, 38752, 37718}, {51066, 51709, 50817}

X(54448) = X(1)X(5068)∩X(2)X(515)

Barycentrics    5*a^4-4*a^3*(b+c)+4*a*(b-c)^2*(b+c)-7*(b^2-c^2)^2+2*a^2*(b^2+4*b*c+c^2) ::
X(54448) = -4*X[1]+13*X[5068], -8*X[3]+17*X[46932], 4*X[4]+5*X[3617], 4*X[40]+5*X[17578], -2*X[944]+11*X[5056], 2*X[962]+7*X[4678], -2*X[1482]+11*X[3855], X[3632]+8*X[12571], 2*X[3679]+X[9812], X[3830]+2*X[38112], 2*X[5886]+X[34627], 2*X[6361]+7*X[50688], 4*X[7982]+5*X[20052], 2*X[7991]+7*X[10248], -X[11224]+4*X[50802]

See Ivan Pavlov, Romantics of Geometry 12957.

X(54448) lies on these lines: {1, 5068}, {2, 515}, {3, 46932}, {4, 3617}, {5, 3622}, {8, 1699}, {10, 3146}, {20, 5818}, {40, 17578}, {80, 10590}, {119, 20085}, {144, 5080}, {145, 355}, {149, 6957}, {153, 6826}, {165, 15683}, {376, 28190}, {381, 5844}, {516, 50687}, {517, 3839}, {519, 9779}, {546, 12245}, {631, 46930}, {944, 5056}, {946, 3621}, {950, 7319}, {952, 3545}, {962, 4678}, {1056, 12019}, {1131, 19065}, {1132, 19066}, {1385, 7486}, {1478, 21454}, {1482, 3855}, {1483, 5072}, {1698, 15717}, {1837, 3475}, {2551, 6895}, {3090, 18525}, {3241, 3817}, {3436, 6894}, {3522, 5691}, {3523, 9956}, {3524, 28186}, {3534, 50826}, {3543, 5657}, {3544, 5901}, {3579, 49135}, {3600, 17728}, {3616, 7989}, {3623, 5881}, {3632, 12571}, {3679, 9812}, {3698, 9961}, {3753, 50736}, {3830, 38112}, {3850, 12645}, {3851, 10595}, {3858, 8148}, {3868, 9947}, {3877, 10157}, {4297, 19877}, {4309, 5560}, {4661, 18908}, {4677, 50803}, {4691, 9589}, {4731, 15726}, {4745, 51119}, {4886, 45100}, {5055, 28224}, {5059, 31673}, {5066, 10247}, {5067, 34773}, {5071, 10246}, {5086, 6870}, {5154, 24558}, {5177, 5658}, {5226, 5727}, {5229, 11246}, {5252, 5274}, {5265, 17606}, {5686, 38454}, {5704, 9613}, {5726, 10578}, {5734, 20014}, {5787, 37436}, {5794, 8165}, {5886, 34627}, {6246, 20095}, {6361, 50688}, {6684, 50693}, {6835, 20060}, {6839, 9965}, {6884, 10786}, {6920, 18518}, {6946, 18519}, {6990, 10942}, {7377, 24599}, {7384, 29616}, {7982, 20052}, {7988, 28236}, {7991, 10248}, {9654, 11036}, {9708, 36002}, {10283, 19709}, {10303, 18481}, {10304, 28160}, {10580, 51782}, {10591, 37710}, {10826, 14986}, {10883, 17757}, {10944, 18220}, {11038, 11237}, {11111, 38058}, {11224, 50802}, {11231, 15692}, {11235, 32426}, {11239, 38037}, {11500, 16865}, {11522, 20050}, {11551, 18391}, {12111, 23841}, {12114, 17572}, {14269, 28212}, {14646, 15679}, {15056, 16980}, {15626, 19291}, {15640, 50821}, {15708, 28208}, {15721, 17502}, {16200, 20049}, {17547, 38031}, {18444, 18528}, {19875, 28164}, {21734, 31399}, {23249, 35789}, {23259, 35788}, {23675, 28092}, {25005, 37435}, {28178, 38066}, {28216, 38081}, {29621, 36662}, {30308, 50801}, {31730, 50692}, {31888, 37230}, {33697, 49140}, {33699, 50809}, {34632, 38127}, {36926, 39570}, {38034, 41106}, {38158, 52653}, {38176, 50810}, {39885, 51170}, {41099, 50800}, {49524, 51537}, {50865, 51068}

X(54448) = reflection of X(i) in X(j) for these {i,j}: {38314, 7988}, {7988, 38076}
X(54448) = anticomplement of X(54445)
X(54448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 3617, 20070}, {8, 19925, 3832}, {20, 5818, 46933}, {355, 3091, 145}, {355, 38140, 5603}, {944, 5056, 46934}, {962, 18492, 50689}, {1699, 37714, 38155}, {3523, 9956, 46931}, {3616, 7989, 15022}, {3621, 3854, 946}, {3851, 37705, 10595}, {4678, 50689, 962}, {5603, 38140, 3091}, {5691, 9780, 3522}, {5731, 10175, 2}, {5818, 18480, 20}, {7988, 28236, 38314}, {19925, 38155, 1699}, {28236, 38076, 7988}





leftri   Cyclocevian conjugates: X(54449)-X(54459) and X(55019)-X(55037)rightri

Contributed by Clark Kimberling and Peter Moses, July 11, 2023.

As noted in the Glossary, suppose that P = p : q : r (trilinear coordinates, not barycentric) is a point not on a sideline of ABC, and let A'B'C' be the cevian triangle of P. The circumcircle of A'B'C' meets line BC in two points: A' and A"; pairs B', B", and C',C" are obtained cyclically. The lines AA", BB", CC" concur in the cyclocevian conjugate of P. Let

g(a,b,c) = a/[p(qb + rc)] and f(a,b,c) = bc/[g(b,c,a) + g(c,a,b) - g(a,b,c)].

The cyclocevian conjugate of P is given by

f(a,b,c) : f(b,c,a) : f(c,a,b) (trilinears).

The cyclocevian conjugate of a point is the
isotomic conjuguate
  of the anticomplement
    of the isogonal conjugate
      of the complement
        of the isotomic conjugate
          of the point
(Darij Grinberg, January 24, 2003)

Now switching to bartycentric coordinates, suppose that p x + q y + r z = 0 is a line. It's image under cyclocevian conjugationj is the octic curve given by

(c^4*p + c^4*q - c^4*r)*x^4*y^4 + (-2*b^2*c^2*p + 2*c^4*p - 2*a^2*c^2*q + 2*c^4*q - 2*c^4*r)*x^4*y^3*z + (-2*b^2*c^2*p + 2*c^4*p - 2*a^2*c^2*q + 2*c^4*q - 2*c^4*r)*x^3*y^4*z + (-(a^4*p) + b^4*p - 4*b^2*c^2*p + c^4*p + a^4*q - b^4*q - 2*a^2*c^2*q + c^4*q + a^4*r - 2*a^2*b^2*r + b^4*r - c^4*r)*x^4*y^2*z^2 + (-2*a^4*p + 2*b^4*p - 6*b^2*c^2*p + 4*c^4*p + 2*a^4*q - 2*b^4*q - 6*a^2*c^2*q + 4*c^4*q + 2*a^4*r - 4*a^2*b^2*r + 2*b^4*r - 4*c^4*r)*x^3*y^3*z^2 + (-(a^4*p) + b^4*p - 2*b^2*c^2*p + c^4*p + a^4*q - b^4*q - 4*a^2*c^2*q + c^4*q + a^4*r - 2*a^2*b^2*r + b^4*r - c^4*r)*x^2*y^4*z^2 + (2*b^4*p - 2*b^2*c^2*p - 2*b^4*q - 2*a^2*b^2*r + 2*b^4*r)*x^4*y*z^3 + (-2*a^4*p + 4*b^4*p - 6*b^2*c^2*p + 2*c^4*p + 2*a^4*q - 4*b^4*q - 4*a^2*c^2*q + 2*c^4*q + 2*a^4*r - 6*a^2*b^2*r + 4*b^4*r - 2*c^4*r)*x^3*y^2*z^3 + (-4*a^4*p + 2*b^4*p - 4*b^2*c^2*p + 2*c^4*p + 4*a^4*q - 2*b^4*q - 6*a^2*c^2*q + 2*c^4*q + 4*a^4*r - 6*a^2*b^2*r + 2*b^4*r - 2*c^4*r)*x^2*y^3*z^3 + (-2*a^4*p + 2*a^4*q - 2*a^2*c^2*q + 2*a^4*r - 2*a^2*b^2*r)*x*y^4*z^3 + (b^4*p - b^4*q + b^4*r)*x^4*z^4 + (2*b^4*p - 2*b^2*c^2*p - 2*b^4*q - 2*a^2*b^2*r + 2*b^4*r)*x^3*y*z^4 + (-(a^4*p) + b^4*p - 2*b^2*c^2*p + c^4*p + a^4*q - b^4*q - 2*a^2*c^2*q + c^4*q + a^4*r - 4*a^2*b^2*r + b^4*r - c^4*r)*x^2*y^2*z^4 + (-2*a^4*p + 2*a^4*q - 2*a^2*c^2*q + 2*a^4*r - 2*a^2*b^2*r)*x*y^3*z^4 + (-(a^4*p) + a^4*q + a^4*r)*y^4*z^4 = 0

For example, the cyclocevian image of the Euler line passes through A, B, C, and the anticevian triangle of ABC, and through X(i) for these i: 2, 4, 1032, 13580, 13581, 54449.

The appearance of (i,j) in the following list means that the cyclocevian conjugate of X(i) is X(j):

(1,1029, (2,4), (5,54449), (6,1031), (7,7), (8,189), (13,13483), (14,13484), (20,1032), (63,54450), (66,2998), (67,46275), (68,34287), (69,253), (75,8044), (76,41513), (80,8046), (92,54125), (94,54415), (99,13485), (110,15351), (190,44184), (254,13579), (280,54451), (290,9473), (329,1034), (330,7357), (471,19157), (648,13573), (668,8047), (670,35511), (671,13574), (1113,13580), (1114,13581), (1138,13582), (2113,6650), (2184,13583), (2986,50480), (2992,19776), (2993,19777), (2994,7219), (2996,13575), (3223,13584), (3346,6504), (3459,13585), (4373,8048), (5395,39978), (5932,41080), (6339,42484), (6553,34546), (6601,42483), (6625,8049), (6630,8050), (7319,8051), (9510,13576), (10405,13577), (11606,41520), (14355,31907), (14361,14365), (15321,40042), (19712,41898), (19713,41897), (34214,39939), (35058,43712), (36606,52803), (41514,43740), (42427,42428), (44175,54114), (24243,55020), (24244,55021), (36917,55022), (38259,55023), (39695,55024), (39719,55025), (39726,55026), (39748,55027), (39953,55028), (41895,55029), (42361,55030), (44177, 55031), (46270,55032), (46271,55033), (46274,55034), (54117,55035), (54119,55036), (54120, 55037)

If "circumcircle" is replaced by "Steiner circumellipse" in the definition of cyclocevian conjugate, the result is here named the Steiner-cevian conjugate of X. The appearance of (i,j) in the following list means that the Steiner-cevian conjugate of of X(i) is X(j):

(1,13610), (2,2), (4,43710), (6,14370), (7,43750), (8,7155), (69,43714), (75,18298), (148,31998), (192,3212), (194,3186), (513,9267), (514,42555), (523,9293), (1654,17762), (1655,1045), (2896,40035), (4440,6631), (8591,39061), (9263,9296), (17487,9460), (25054,9428), (39350,33675), (39351,10001), (39352,39062), (39354,33678), (39355,39058), (39358,9410), (52637,3499), (54104,13187} If "circumcircle" is replaced by "Kiepert circumhyperbola" in the definition of cyclocevian conjugate, the result is here named the Kiepert-cevian conjugate of X. The appearance of (i,j) in the following list means that the Kiepert-cevian conjugate of of X(i) is X(j):

(1,13486), (2,99), (3,110), (4,35360), (6,13578), (13,36839), (14,36840), (30,476), (399,47053), (616,35314), (617,35315), (5667,4240}

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X(54449) = CYCLOCEVIAN CONJUGATE OF X(5)

Barycentrics    1/(c^2*(a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*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)*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - a^2*b^2 - 2*a^2*c^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*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4)*(-(a^2*b^2) + b^4 - 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 - 3*b^2*c^2 + 2*c^4) - a^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - 3*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 2*c^4)) : :

X(54449) lies on these lines: (),

X(54449) = cyclocevian conjugate of X(5)
X(54449) = isotomic conjugate of the anticomplement of X(288)

X(54450) = CYCLOCEVIAN CONJUGATE OF X(63)

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

X(54450 ) lies on these lines: (377, 1249), , (1231, 2897), , (4296, 5930), , (5279, 8804), , (6839, 14249), , (7270, 52345), , (10152, 37433), , (10431, 33893), , (37163, 38808), , (37456, 40431),

X(54450 ) = isotomic conjugate of X(2897)
X(54450 ) = polar conjugate of X(18687)
X(54450 ) = cyclocevian conjugate of X(63)
X(54450 ) = isotomic conjugate of the anticomplement of X(1172)
X(54450 ) = X(i)-isoconjugate of X(j) for these (i,j): (:31, 2897), , (48, 18687),
X(54450 ) = X(i)-Dao conjugate of X(j) for these (i,j): (:2, 2897), , (1249, 18687),
X(54450 ) = cevapoint of X(i) and X(j) for these (i,j): (:122, 521), , (522, 34846),
X(54450 ) = trilinear pole of line (6587, 16612),
X(54450 ) = barycentric quotient X(i)/X(j) for these (i,j), : (:2, 2897), , (4, 18687),

X(54451) = CYCLOCEVIAN CONJUGATE OF X(280)

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

X(54451) lies on these lines: (40, 3436), , (196, 5905), , (223, 908), , (329, 20928), , (972, 34188), , (3193, 3194), , (3434, 15499), , (4391, 26871),

X(54451) = cyclocevian conjugate of X(280)
X(54451) = isotomic conjugate of the anticomplement of X(222)
X(54451) = X(42464)-anticomplementary conjugate of X(347)
X(54451) = X(i)-isoconjugate of X(j) for these (i,j): (:6, 1158), , (41, 31600), , (55, 34052), , (8609, 10692),
X(54451) = X(i)-Dao conjugate of X(j) for these (i,j): (:9, 1158), , (223, 34052), , (3160, 31600),
X(54451) = cevapoint of X(i) and X(j) for these (i,j): (:9, 5534), , (513, 6506), , (521, 5514),
X(54451) = trilinear pole of line (2804, 6129),
X(54451) = barycentric product X(75)*X(42464)
X(54451) = barycentric quotient X(i)/X(j) for these (i,j), : (:1, 1158), , (7, 31600), , (57, 34052), , (36052, 10692), , (42464, 1),

X(54452) = CYCLOCEVIAN CONJUGATE OF X(903)

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

X(54452) lies on these lines : : {2, 40595}, {150, 35175}, {320, 517}, {859, 34184}, {1443, 1457}, {2183, 3218}, {3264, 21290}, {4389, 14260}

X(54452) = isogonal conjugate of X(23858)
X(54452) = isotomic conjugate of X(21290)
X(54452) = anticomplement of X(40595)
X(54452) = cyclocevian conjugate of X(903)
X(54452) = isotomic conjugate of the anticomplement of X(106)
X(54452) = isotomic conjugate of the complement of X(20098)
X(54452) = isotomic conjugate of the isogonal conjugate of X(34184)
X(54452) = X(34184)-anticomplementary conjugate of X(17495)
X(54452) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23858}, {6, 16561}, {19, 23135}, {31, 21290}, {32, 21600}, {44, 40595}
X(54452) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 21290}, {3, 23858}, {6, 23135}, {9, 16561}, {6376, 21600}
X(54452) = cevapoint of X(i) and X(j) for these (i,j): {2, 20098}, {514, 3259}
X(54452) = trilinear pole of line {3310, 3960}
X(54452) = barycentric product X(76)*X(34184)
X(54452) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16561}, {2, 21290}, {3, 23135}, {6, 23858}, {75, 21600}, {106, 40595}, {34184, 6}

X(54453) = CYCLOCEVIAN CONJUGATE OF X(925)

Barycentrics    (a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 - 2*a^6*c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 2*b^6*c^2 - 5*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 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 2*b^6*c^2 + 2*a^4*c^4 + 3*a^2*b^2*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) : :
X(54453) = X[323] - 4 X[9721]

X(54453) = lies on the cubnic K938 and these lines: {249, 36472}, {323, 9721}, {403, 3564}, {1993, 15538}, {3448, 3566}, {3580, 5866}, {3926, 14253}, {37779, 52504}, {39138, 52451}

X(54453) = reflection of X(249) in X(36472)
X(54453) = isogonal conjugate of X(2079)
X(54453) = antigonal image of X(249)
X(54453) = cyclocevian conjugate of X(925)
X(54453) = isotomic conjugate of the anticomplement of X(4558)
X(54453) = cevapoint of X(i) and X(j) for these (i,j): {6, 12310}, {511, 41181}, {3564, 36472}, {13754, 39021}
X(54453) = trilinear pole of line {5972, 6132}
X(54453) = barycentric quotient X(6)/X(2079)

X(54454) = CYCLOCEVIAN CONJUGATE OF X(2995)

Barycentrics    (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(54454) lies on these lines: {2, 40582}, {69, 4189}, {95, 37291}, {253, 6872}, {264, 5046}, {305, 34282}, {306, 2895}, {307, 1442}, {319, 20336}, {328, 14616}, {1029, 30690}, {1441, 2475}, {1494, 15677}, {2893, 21272}, {3448, 41004}, {5154, 8797}, {6340, 45962}, {7054, 15526}, {15674, 40412}, {17576, 35510}, {17791, 21287}, {20846, 40995}

X(54454) = isotomic conjugate of X(2475)
X(54454) = anticomplement of X(40582)
X(54454) = cyclocevian conjugate of X(2995)
X(54454) = isotomic conjugate of the anticomplement of X(21)
X(54454) = isotomic conjugate of the complement of X(15680)
X(54454) = isotomic conjugate of the isogonal conjugate of X(34435)
X(54454) = X(34435)-anticomplementary conjugate of X(63)
X(54454) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1781}, {25, 52362}, {31, 2475}, {41, 18625}, {42, 229}, {213, 52361}, {1400, 40582}, {1402, 52360}, {1973, 28754}
X(54454) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2475}, {9, 1781}, {3160, 18625}, {6337, 28754}, {6505, 52362}, {6626, 52361}, {40592, 229}, {40605, 52360}
X(54454) = cevapoint of X(i) and X(j) for these (i,j): {2, 15680}, {513, 17058}, {514, 8286}, {521, 15526}, {522, 8287}, {23880, 53829}
X(54454) = trilinear pole of line {525, 14838}
X(54454) = barycentric product X(76)*X(34435)
X(54454) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1781}, {2, 2475}, {7, 18625}, {21, 40582}, {63, 52362}, {69, 28754}, {81, 229}, {86, 52361}, {333, 52360}, {34435, 6}, {37887, 41495}

X(54455) = CYCLOCEVIAN CONJUGATE OF X(3952)

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

X(54455) lies on these lines: {2, 39054}, {148, 7265}, {484, 4645}, {1577, 21221}, {2533, 3448}, {3936, 19308}, {17484, 17789}, {26081, 47318}

X(54455) = isogonal conjugate of X(21004)
X(54455) = isotomic conjugate of X(21221)
X(54455) = anticomplement of X(39054)
X(54455) = cyclocevian conjugate of X(3952)
X(54455) = isotomic conjugate of the anticomplement of X(662)
X(54455) = isotomic conjugate of the complement of X(31297)
X(54455) = X(39137)-anticomplementary conjugate of X(7192)
X(54455) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21004}, {6, 21381}, {19, 22156}, {31, 21221}, {32, 20951}, {58, 21890}, {163, 50574}, {512, 39054}, {692, 21209}, {1333, 21098}
X(54455) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 21221}, {3, 21004}, {6, 22156}, {9, 21381}, {10, 21890}, {37, 21098}, {115, 50574}, {1086, 21209}, {6376, 20951}
X(54455) = cevapoint of X(i) and X(j) for these (i,j): {2, 31297}, {523, 24040}
X(54455) = trilinear pole of line {4458, 6370}
X(54455) = barycentric product X(75)*X(39137)
X(54455) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21381}, {2, 21221}, {3, 22156}, {6, 21004}, {10, 21098}, {37, 21890}, {75, 20951}, {514, 21209}, {523, 50574}, {662, 39054}, {39137, 1}

X(54456) = CYCLOCEVIAN CONJUGATE OF X(7261)

Barycentrics    (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)*(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(54456) lies on the cubic K323 and these lines: {2, 3252}, {6, 6654}, {239, 672}, {350, 518}, {1447, 1458}, {2113, 13576}, {2669, 3286}, {9318, 40721}, {20345, 30941}, {27922, 34230}, {35119, 52205}

X(54456) = reflection of X(52209) in X(35119)
X(54456) = isogonal conjugate of X(2110)
X(54456) = isotomic conjugate of X(17794)
X(54456) = anticomplement of X(36906)
X(54456) = antitomic image of X(52209)
X(54456) = cyclocevian conjugate of X(7261)
X(54456) = isotomic conjugate of the anticomplement of X(291)
X(54456) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2111, 4645}, {33701, 20552}
X(54456) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2110}, {6, 24578}, {19, 20762}, {31, 17794}, {42, 8849}, {55, 52161}, {58, 20694}, {1914, 36906}, {2223, 33674}
X(54456) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17794}, {3, 2110}, {6, 20762}, {9, 24578}, {10, 20694}, {223, 52161}, {40592, 8849}
X(54456) = cevapoint of X(i) and X(j) for these (i,j): {513, 35119}, {514, 38989}, {650, 15615}
X(54456) = trilinear pole of line {665, 812}
X(54456) = barycentric product X(i)*X(j) for these {i,j}: {75, 2111}, {33701, 52209}
X(54456) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 24578}, {2, 17794}, {3, 20762}, {6, 2110}, {37, 20694}, {57, 52161}, {81, 8849}, {291, 36906}, {673, 33674}, {2111, 1}, {33701, 17755}

X(54457) = CYCLOCEVIAN CONJUGATE OF X(7361)

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

X(54457) lies on these lines: {2, 36033}, {8, 34242}, {573, 1759}, {1479, 7253}, {1969, 21270}, {3868, 5081}, {3869, 6327}, {4296, 4511}, {7141, 18474}

X(54457) = isogonal conjugate of X(23843)
X(54457) = isotomic conjugate of X(21270)
X(54457) = anticomplement of X(36033)
X(54457) = polar conjugate of X(17902)
X(54457) = cyclocevian conjugate of X(7361)
X(54457) = isotomic conjugate of the anticomplement of X(48)
X(54457) = isotomic conjugate of the complement of X(20074)
X(54457) = X(7094)-anticomplementary conjugate of X(6360)
X(54457) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23843}, {2, 2908}, {4, 36033}, {6, 1726}, {19, 22130}, {31, 21270}, {32, 20926}, {48, 17902}, {692, 21184}, {1333, 21072}
X(54457) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 21270}, {3, 23843}, {6, 22130}, {9, 1726}, {37, 21072}, {1086, 21184}, {1249, 17902}, {6376, 20926}, {32664, 2908}
X(54457) = cevapoint of X(i) and X(j) for these (i,j): {2, 20074}, {124, 521}, {520, 34846}, {525, 21253}
X(54457) = trilinear pole of line {6589, 16612}
X(54457) = barycentric product X(i)*X(j) for these {i,j}: {75, 7094}, {561, 7139}
X(54457) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1726}, {2, 21270}, {3, 22130}, {4, 17902}, {6, 23843}, {10, 21072}, {31, 2908}, {48, 36033}, {75, 20926}, {514, 21184}, {7094, 1}, {7139, 31}

X(54458) = CYCLOCEVIAN CONJUGATE OF X(9295)

Barycentrics    (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(54458) lies on these lines: {69, 25048}, {141, 27004}, {883, 46152}, {1332, 46163}, {3882, 46148}, {3888, 4576}, {4553, 53332}, {6386, 21301}, {17137, 46158}

X(54458) = isogonal conjugate of X(21005)
X(54458) = isotomic conjugate of X(21301)
X(54458) = cyclocevian conjugate of X(9295)
X(54458) = isotomic conjugate of the anticomplement of X(667)
X(54458) = isotomic conjugate of the complement of X(31291)
X(54458) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21005}, {6, 21389}, {19, 22157}, {31, 21301}, {32, 20952}, {667, 32926}, {692, 21210}, {1333, 21099}, {1973, 28423}
X(54458) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 21301}, {3, 21005}, {6, 22157}, {9, 21389}, {37, 21099}, {1086, 21210}, {6337, 28423}, {6376, 20952}, {6631, 32926}
X(54458) = cevapoint of X(i) and X(j) for these (i,j): {2, 31291}, {141, 513}, {512, 1211}, {514, 2887}, {522, 21244}, {523, 21245}, {812, 20542}, {4083, 21250}, {6371, 51571}, {6373, 20343}
X(54458) = trilinear pole of line {39, 712}
X(54458) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21389}, {2, 21301}, {3, 22157}, {6, 21005}, {10, 21099}, {69, 28423}, {75, 20952}, {190, 32926}, {514, 21210}

X(54459) = CYCLOCEVIAN CONJUGATE OF X(9295)

Barycentrics    (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^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) : :

X(54459) lies on these lines: {2, 8792}, {69, 2916}, {99, 42052}, {253, 7519}, {264, 7533}, {287, 37779}, {305, 1369}, {339, 37349}, {1494, 37901}, {1799, 7664}, {2373, 37760}, {5189, 18019}, {6330, 37766}, {13219, 18018}, {14002, 41896}, {15526, 36415}, {15899, 30786}, {31857, 51884}, {37644, 42287}

X(54459) = isogonal conjugate of X(19596)
X(54459) = isotomic conjugate of X(5189)
X(54459) = anticomplement of X(40583)
X(54459) = cyclocevian conjugate of X(14364)
X(54459) = isotomic conjugate of the anticomplement of X(23)
X(54459) = isotomic conjugate of the complement of X(20063)
X(54459) = isotomic conjugate of the isogonal conjugate of X(34437)
X(54459) = X(i)-isoconjugate of X(j) for these (i,j): {1, 19596}, {6, 16546}, {19, 22121}, {31, 5189}, {32, 20916}, {41, 18627}, {692, 21176}, {896, 8877}, {1333, 21064}, {1964, 38946}, {2157, 40583}
X(54459) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5189}, {3, 19596}, {6, 22121}, {9, 16546}, {37, 21064}, {1086, 21176}, {3160, 18627}, {6376, 20916}, {15899, 8877}, {41884, 38946}
X(54459) = cevapoint of X(i) and X(j) for these (i,j): {2, 20063}, {6292, 9019}, {9517, 15526}
X(54459) = trilinear pole of line {525, 3589}
X(54459) = barycentric product X(76)*X(34437)
X(54459) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16546}, {2, 5189}, {3, 22121}, {6, 19596}, {7, 18627}, {10, 21064}, {23, 40583}, {75, 20916}, {83, 38946}, {111, 8877}, {514, 21176}, {22151, 52363}, {34437, 6}





leftri  H-conics: X(54460) - X(54467)  rightri

This preamble and centers X(54460)-X(54467) were contributed by César Eliud Lozada, July 15, 2023.

Let ABC be a right triangle at A. There exists a lot of finite centers in ETC lying on the hypotenuse BC (produced included), in particular, those having first coordinates with a multiplying factor cos(A) or (-a^2+b^2+c^2).

Let denote by ℋ the set of k such that X(k)-of-ABC lies on the hypotenuse BC. As an example, the subset o ℋ for k≤1000 is:

{3, 48, 49, 63, 68, 69, 71, 72, 73, 77, 78, 97, 122, 123, 125, 127, 130, 131, 155, 184, 185, 201, 212, 216, 217, 219, 222, 224, 228, 248, 255, 265, 268, 271, 283, 287, 293, 295, 296, 304, 305, 306, 307, 326, 328, 332, 336, 337, 339, 343, 345, 348, 394, 408, 417, 418, 426, 440, 441, 454, 464, 465, 466, 487, 488, 563, 577, 603, 606, 647, 652, 656, 682, 684, 686, 810, 820, 822, 828, 836, 852, 856, 878, 879, 895, 905, 906, 914, 974}

Application.

Let ABC be an acute triangle. Build the rectangle BCCaBa such that A lies on CaBa. Two right triangles BaBA and CaCA are obtained. Now, for a given k ∈ ℋ, let B'a = X(k)-of-BaBA and C'a = X(k)-of-CaCA, these centers lying on their hipotenuses AB and AC, respectively. Define C'b, A'b and A'c, B'c cyclically. It is not hard to prove that, for any k∈ℋ, these six points lie on an conic 𝒞( X(k) ), here named the H-conic of X(k) (H stands for hypotenuses).

Depending on the chosen k, 𝒞( X(k) ) can degenerate to two lines or to the line at infinity (as with X(3), X(68) and others). Also, every pair of constructed points on a side of ABC can coincide and the H-conic approaches to a circle, as occurs with X(69), for which the Taylor circle is obtained.

underbar

X(54460) = CENTER OF THE H-CONIC OF X(48)

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

X(54460) lies on these lines: {140, 54465}, {394, 10665}, {492, 40697}, {641, 1583}

X(54460) = center of the H-conic of X(n) for these n: {48, 73, 336, 820, 836}

X(54461) = CENTER OF THE H-CONIC OF X(49)

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

X(54461) lies on these lines: {50, 323}, {252, 32142}, {338, 11140}, {32423, 41590}

X(54461) = center of the H-conic of X(n) for these n: {49, 265}

X(54462) = CENTER OF THE H-CONIC OF X(72)

Barycentrics    a*(a+b-c)*(a-b+c)*(a*b*c-(b+c)*S) : :

X(54462) lies on these lines: {1, 3}, {7, 13386}, {222, 16232}, {482, 46017}, {2262, 51841}, {2362, 52424}, {3083, 52420}, {13390, 39795}, {16608, 30380}, {16663, 23839}

X(54462) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(6348)}} and {{A, B, C, X(3), X(13386)}}
X(54462) = X(52286)-of-intouch triangle, when ABC is acute
X(54462) = center of the H-conic of X(n) for these n: {72, 219}

X(54463) = CENTER OF THE H-CONIC OF X(125)

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

X(54463) lies on these lines: {76, 6504}, {394, 577}, {1216, 12362}, {23292, 46184}

X(54463) = center of the H-conic of X(n) for these n: {125, 130, 184, 185, 217, 287, 296, 686, 974}

X(54464) = CENTER OF THE H-CONIC OF X(222)

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

X(54464) lies on these lines: {3, 6213}, {3718, 13458}

X(54465) = CENTER OF THE H-CONIC OF X(563)

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

X(54465) lies on these lines: {140, 54460}, {394, 10666}, {491, 40697}, {642, 1584}

X(54465) = center of the H-conic of X(n) for these n: {563, 652, 656, 822}

X(54466) = CENTER OF THE H-CONIC OF X(647)

Barycentrics    (-a^2+b^2+c^2)^2*(a^20-5*(b^2+c^2)*a^18+(9*b^4+16*b^2*c^2+9*c^4)*a^16-4*(b^2+c^2)^3*a^14-2*(5*b^8+5*c^8+6*(b^4+b^2*c^2+c^4)*b^2*c^2)*a^12+2*(b^2+c^2)*(9*b^8+9*c^8+2*(b^4+5*b^2*c^2+c^4)*b^2*c^2)*a^10-2*(b^2+c^2)^2*(5*b^8+5*c^8-2*(b^4-b^2*c^2+c^4)*b^2*c^2)*a^8-4*(b^8-c^8)*(b^2-c^2)*(-4*b^2*c^2+(b^2-c^2)^2)*a^6+(b^2-c^2)^4*(9*b^8+9*c^8-2*(4*b^4+5*b^2*c^2+4*c^4)*b^2*c^2)*a^4-(b^2-c^2)^6*(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^2+(b^2-c^2)^8*(b^4+c^4)) : :

X(54466) lies on these lines: {12359, 16196}

X(54467) = CENTER OF THE H-CONIC OF X(684)

Barycentrics    a^16-5*(b^2+c^2)*a^14+2*(5*b^4+7*b^2*c^2+5*c^4)*a^12-(b^2+c^2)*(11*b^4+2*b^2*c^2+11*c^4)*a^10+2*(5*b^8+5*c^8+b^2*c^2*(b^4+c^4))*a^8-(b^4-c^4)*(b^2-c^2)*(11*b^4-2*b^2*c^2+11*c^4)*a^6+2*(b^2-c^2)^2*(5*b^8+5*c^8-3*b^2*c^2*(b^4+c^4))*a^4-(b^4-c^4)*(b^2-c^2)^3*(5*b^4-6*b^2*c^2+5*c^4)*a^2+(b^2-c^2)^6*(b^4+c^4) : :

X(54467) lies on these lines: {5, 578}, {25, 6750}, {26, 5961}, {216, 2165}, {254, 1093}, {7493, 33495}, {8800, 27087}, {9938, 34840}, {13383, 15508}, {16310, 41587}, {23105, 52585}

X(54468) = X(105)X(1336)∩X(1282)X(6212)

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

See Stanley Rabinowitz and Peter Moses, euclid 5951.

X(54468) lies on these lines: {4, 52164}, {100, 13386}, {105, 1336}, {291, 16232}, {1282, 6212}, {8299, 14121}

X(54469) = X(4)X(52164)∩X(100)X(13387)

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

See Stanley Rabinowitz and Peter Moses, euclid 5951.

X(54469) lies on these lines: {4,52164}, {100,13387}, {105,1123}, {291,2362}, {1282,6213}, {7090,8299}

X(54470) = X(6)X(19)∩X(8048)X(13389)

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

See Stanley Rabinowitz and César Lozada, euclid 5956.

X(54470) lies on these lines: {6, 19}, {8048, 13389}, {9042, 23050}

X(54470) = X(13386)-Ceva conjugate of-X(16232)
X(54470) = X(2362)-Dao conjugate of-X(13387)

X(54471) = X(6)X(19)∩X(8048)X(13388)

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

See Stanley Rabinowitz and César Lozada, euclid 5956.

X(54471) lies on these lines: {6, 19}, {7133, 7595}, {8048, 13388}, {9043, 23050}

X(54471) = X(13387)-Ceva conjugate of-X(2362)
X(54471) = X(16232)-Dao conjugate of-X(13386)

X(54472) = X(13)X(511)∩X(51)X(512)

Barycentrics    a^2*(sqrt(3)*(a^2+b^2-c^2)+2*S)*(sqrt(3)*(a^2-b^2+c^2)+2*S)*(-b^4+6*b^2*c^2-c^4+a^2*(b^2+c^2)+2*sqrt(3)*(b^2+c^2)*S) : :
X(54472) = X(13)-3*X(16461)

See Elias Hagos and César Lozada, euclid 5957.

X(54472) lies on these lines: {6, 46342}, {13, 511}, {15, 11142}, {51, 512}, {52, 11555}, {187, 3457}, {265, 11139}, {316, 16770}, {373, 52039}, {396, 15929}, {2380, 5995}, {2393, 22826}, {5611, 21310}, {5640, 21466}, {6104, 13350}, {11080, 16247}, {11537, 11624}, {15609, 47026}, {25178, 53793}, {25219, 34373}, {30439, 36970}, {32761, 54363}, {34325, 36978}, {36208, 44498}, {36755, 41474}

X(54472) = X(15295)-Dao conjugate of-X(34374)
X(54472) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (11060, 34374), (34373, 7799)
X(54472) = barycentric product of X(i) and X(j) for these {i, j}: {1989, 34373}
X(54472) = (X(5995), X(16459))-harmonic conjugate of X(11083)

X(54473) = X(14)X(511)∩X(51)X(512)

Barycentrics    a^2*(sqrt(3)*(a^2+b^2-c^2)-2*S)*(sqrt(3)*(a^2-b^2+c^2)-2*S)*(-b^4+6*b^2*c^2-c^4+a^2*(b^2+c^2)-2*sqrt(3)*(b^2+c^2)*S) : :
X(54473) = X(14)-3*X(16462)

See Elias Hagos and César Lozada, euclid 5957.

X(54473) lies on these lines: {6, 46343}, {14, 511}, {16, 11141}, {51, 512}, {52, 11556}, {187, 3458}, {265, 11138}, {316, 16771}, {373, 52040}, {395, 15930}, {2381, 5994}, {2393, 22827}, {5615, 21311}, {5640, 21467}, {6105, 13349}, {11085, 16248}, {11549, 11626}, {15610, 47027}, {25173, 53793}, {25220, 34375}, {30440, 36969}, {32761, 54362}, {34326, 36980}, {36209, 44497}, {36756, 41475}

X(54473) = X(15295)-Dao conjugate of-X(34376)
X(54473) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (11060, 34376), (34375, 7799)
X(54473) = barycentric product of X(i) and X(j) for these {i, j}: {1989, 34375}
X(54473) = (X(5994), X(16460))-harmonic conjugate of X(11088)



leftri

Orthology centers related to bicevian conics: X(54474)-X(55009)

rightri

This preamble and centers X(54474)-X(55009) were contributed by Ivan Pavlov, July 19, 2023.

Let (c) be the bicevian conic of P={u,v,w} and Q={p,q,r}. Lines AP, BP, CP intersect (c) at six points, three of which form the cevian triangle of P. Denote the other three with A1, B1, and C1. Similarly, using Q, define A2, B2, and C2. The lines A1A2, B1B2, and C1C2 form a triangle TaTbTc, which is always perspective to ABC.

In the cases when a certain fourth degree relation holds, ABC and TaTbTc are also orthologic.

In the particular case when Q=X(2) and P lies on the Kiepert hyperbola, the orthology center of ABC and TaTbTc also lies on the Kiepert hyperbola.

In the particular case when Q=X(4) and P lies on the circumconic with perspector X(4), the configuration is degenerate becasue Ta=Tb=Tc=H. The orthology center (which exists only in the limit) is the isotomic conjugate of (SB u v+SC u w-a^2 v w : :) and lies on the Steiner circumellipse. When the orthology center of ABC and TaTbTc exists it lies on the Euler line.

For more information on how each center arises see the documents attached to Euclid 5932.

Contributed by Peter Moses, July 21, 2023: The appearance of i in the following list means that X(i) is a major center and lies on the Kiepert hyperbola: 54479, 54480, 54534, 54535, 54536, 54537, 54538, 54542, 54543, 54574, 54575, 54576. 54577. 54578, 54579, 54580, 54581, 54591, 54592, 54593, 54594, 54595, 54596, 54597, 54598, 54599, 54634, 54635, 54636

X(54474) = X(1)X(3)∩X(2)X(10186)

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

X(54474) lies on these lines: {1, 3}, {2, 10186}, {37, 1742}, {43, 25075}, {45, 9355}, {77, 9440}, {100, 28125}, {226, 53617}, {515, 44430}, {516, 27475}, {949, 20770}, {954, 5018}, {984, 991}, {1088, 1323}, {1212, 3740}, {1251, 30300}, {1253, 1442}, {1376, 34522}, {1699, 36731}, {1721, 3247}, {1961, 5527}, {1962, 9778}, {2293, 7671}, {2340, 3681}, {2951, 16673}, {3688, 50658}, {3817, 7377}, {3842, 48878}, {3920, 18461}, {3989, 11220}, {4021, 43151}, {4098, 9950}, {4336, 7676}, {4566, 5281}, {4640, 6603}, {4650, 11364}, {4687, 45305}, {5308, 6999}, {5311, 7411}, {5432, 5723}, {5526, 7262}, {5657, 50282}, {5731, 48856}, {5779, 51294}, {5851, 49742}, {5886, 29365}, {5927, 16601}, {6184, 19584}, {6211, 37474}, {6986, 16478}, {7380, 10175}, {7611, 29349}, {8580, 52705}, {8926, 40781}, {9316, 38459}, {9442, 9502}, {10164, 50114}, {11495, 16777}, {14520, 20683}, {15624, 18161}, {16112, 16675}, {16468, 31658}, {16475, 21153}, {16826, 48900}, {17389, 28870}, {17392, 38454}, {24203, 24283}, {24328, 53394}, {24341, 35338}, {25568, 35102}, {26446, 29331}, {28160, 36732}, {28849, 29574}, {29657, 37374}, {29675, 43057}, {30301, 33653}, {31395, 48929}, {38127, 49772}

X(54474) = intersection, other than A, B, C, of circumconics: {{A, B, C, X(1), X(18810)}}, {{A, B, C, X(56), X(28869)}}, {{A, B, C, X(57), X(34521)}}, {{A, B, C, X(1088), X(4860)}}, {{A, B, C, X(5228), X(9442)}}, {{A, B, C, X(9441), X(40779)}}
X(54474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 51300, 41339}

X(54475) = X(2)X(10723)∩X(4)X(41672)

Barycentrics    (5*a^6+5*b^6-12*b^4*c^2+13*b^2*c^4-6*c^6-a^4*(b^2+12*c^2)-a^2*(b^4+4*b^2*c^2-13*c^4))*(5*a^6-6*b^6+13*b^4*c^2-12*b^2*c^4+5*c^6-a^4*(12*b^2+c^2)+a^2*(13*b^4-4*b^2*c^2-c^4)) : :

X(54475) lies on these lines: {2, 10723}, {4, 41672}, {30, 10153}, {115, 43537}, {381, 42011}, {542, 32532}, {2794, 47586}, {3424, 39838}, {5480, 45103}, {5485, 9880}, {5503, 6321}, {7607, 14639}, {10722, 53100}, {38259, 38664}

X(54475) = reflection of X(i) in X(j) for these {i,j}: {43537, 115}
X(54475) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(265), X(39446)}}, {{A, B, C, X(2710), X(14498)}}, {{A, B, C, X(3455), X(52518)}}, {{A, B, C, X(3531), X(6323)}}, {{A, B, C, X(3563), X(10630)}}, {{A, B, C, X(9154), X(10723)}}, {{A, B, C, X(14490), X(52239)}}, {{A, B, C, X(35140), X(38738)}}, {{A, B, C, X(39838), X(45031)}}

X(54476) = X(2)X(5585)∩X(30)X(10155)

Barycentrics    (13*a^2+13*b^2-11*c^2)*(13*a^2-11*b^2+13*c^2) : :

X(54476) lies on these lines: {2, 5585}, {20, 53098}, {30, 10155}, {76, 23334}, {262, 50687}, {381, 53103}, {524, 43681}, {671, 51170}, {3146, 7608}, {3543, 14494}, {3832, 7607}, {3839, 7612}, {3854, 53859}, {5032, 38259}, {5068, 10185}, {5503, 20094}, {7620, 43676}, {8781, 52695}, {10159, 32979}, {11303, 43444}, {11304, 43445}, {11669, 15683}, {14068, 43529}, {17578, 53099}, {32982, 43527}, {32996, 43528}, {33698, 43448}, {38253, 52281}, {41895, 53418}, {43537, 50689}

X(54476) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(11741)}}, {{A, B, C, X(6), X(5585)}}, {{A, B, C, X(428), X(32979)}}, {{A, B, C, X(458), X(50687)}}, {{A, B, C, X(524), X(43726)}}, {{A, B, C, X(1383), X(23334)}}, {{A, B, C, X(3146), X(52281)}}, {{A, B, C, X(3832), X(52282)}}, {{A, B, C, X(3839), X(37174)}}, {{A, B, C, X(5032), X(20080)}}, {{A, B, C, X(5064), X(32982)}}, {{A, B, C, X(7408), X(8370)}}, {{A, B, C, X(7409), X(7841)}}, {{A, B, C, X(11317), X(52301)}}, {{A, B, C, X(13377), X(21765)}}, {{A, B, C, X(14490), X(30535)}}, {{A, B, C, X(46275), X(52223)}}, {{A, B, C, X(52450), X(52695)}}

X(54477) = X(2)X(48884)∩X(30)X(10159)

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

X(54477) lies on these lines: {2, 48884}, {30, 10159}, {76, 3830}, {83, 3845}, {381, 43527}, {383, 10187}, {428, 16080}, {671, 12101}, {1080, 10188}, {1513, 10185}, {2394, 7927}, {3399, 52854}, {5064, 43530}, {5485, 44678}, {6054, 35005}, {7865, 15682}, {9302, 10722}, {10302, 33699}, {14269, 53102}, {14488, 36990}, {15687, 43676}, {18841, 41099}, {43681, 50687}

X(54477) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(25), X(3830)}}, {{A, B, C, X(30), X(428)}}, {{A, B, C, X(74), X(34572)}}, {{A, B, C, X(251), X(13603)}}, {{A, B, C, X(264), X(46204)}}, {{A, B, C, X(381), X(5064)}}, {{A, B, C, X(427), X(3845)}}, {{A, B, C, X(468), X(12101)}}, {{A, B, C, X(1173), X(29316)}}, {{A, B, C, X(1297), X(46848)}}, {{A, B, C, X(1494), X(15321)}}, {{A, B, C, X(1799), X(18550)}}, {{A, B, C, X(1989), X(43458)}}, {{A, B, C, X(3108), X(14487)}}, {{A, B, C, X(3543), X(7714)}}, {{A, B, C, X(4518), X(33696)}}, {{A, B, C, X(5066), X(52285)}}, {{A, B, C, X(6995), X(15682)}}, {{A, B, C, X(7378), X(41099)}}, {{A, B, C, X(7408), X(11001)}}, {{A, B, C, X(7409), X(41106)}}, {{A, B, C, X(7576), X(34603)}}, {{A, B, C, X(7865), X(42037)}}, {{A, B, C, X(10301), X(33699)}}, {{A, B, C, X(11058), X(45819)}}, {{A, B, C, X(11169), X(48911)}}, {{A, B, C, X(11181), X(14490)}}, {{A, B, C, X(11738), X(39955)}}, {{A, B, C, X(13481), X(32085)}}, {{A, B, C, X(14495), X(22334)}}, {{A, B, C, X(16835), X(29322)}}
X(54477) = X(i)-vertex conjugate of X(j) for these {i, j}: {3425, 10185}

X(54478) = X(30)X(10185)∩X(76)X(50989)

Barycentrics    (13*a^2+13*b^2-14*c^2)*(13*a^2-14*b^2+13*c^2) : :

X(54478) lies on these lines: {30, 10185}, {76, 50989}, {98, 12101}, {671, 41149}, {3543, 53859}, {3830, 7607}, {3845, 7608}, {3860, 53108}, {8352, 10159}, {11317, 43527}, {33699, 53104}, {41099, 53098}

X(54478) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(11588)}}, {{A, B, C, X(6), X(50989)}}, {{A, B, C, X(297), X(12101)}}, {{A, B, C, X(428), X(8352)}}, {{A, B, C, X(524), X(41149)}}, {{A, B, C, X(3531), X(20251)}}, {{A, B, C, X(3830), X(52282)}}, {{A, B, C, X(3845), X(52281)}}, {{A, B, C, X(5064), X(11317)}}, {{A, B, C, X(13603), X(32901)}}, {{A, B, C, X(15534), X(51187)}}, {{A, B, C, X(41153), X(50991)}}

X(54479) = X(17)X(3845)∩X(18)X(3830)

Barycentrics    (a-b-c)*(a+b-c)*(a-b+c)*(a+b+c)*(121*a^4-122*(b^2-c^2)^2+a^2*(b^2+c^2-9*sqrt(-3*a^4-3*(b^2-c^2)^2+6*a^2*(b^2+c^2)))) : :
Barycentrics    1 / (9*Sqrt[3]*(-a^2 + b^2 + c^2) - 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] - 27*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54479) lies on these lines: {2, 42112}, {13, 42520}, {14, 12101}, {17, 3845}, {18, 3830}, {30, 10187}, {381, 10188}, {383, 10185}, {3412, 14269}, {3534, 42611}, {3860, 19107}, {3861, 42976}, {5066, 43443}, {5076, 49904}, {5487, 33622}, {5488, 36346}, {8703, 42597}, {11001, 43444}, {11121, 35749}, {11122, 36330}, {12816, 42101}, {12820, 42093}, {12821, 41107}, {14893, 41973}, {15682, 42505}, {15687, 42503}, {16808, 43369}, {16965, 43557}, {19106, 43429}, {19709, 43441}, {19710, 42493}, {22235, 42160}, {33602, 43368}, {33603, 36969}, {33606, 42125}, {33607, 36970}, {33699, 42100}, {36329, 40706}, {36769, 40707}, {36968, 42953}, {38335, 42533}, {41099, 42581}, {41101, 42106}, {41106, 42434}, {41108, 43540}, {41113, 43553}, {41119, 43196}, {41120, 41972}, {41122, 43365}, {42089, 43003}, {42098, 43544}, {42119, 43554}, {42141, 42510}, {42419, 42813}, {42528, 42931}, {42631, 43324}, {42633, 43226}, {42900, 43229}, {43006, 43541}, {43008, 43017}, {43022, 49903}, {43227, 43327}, {43242, 43399}, {43555, 49908}, {44580, 51915}

X(54479) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(471), X(12101)}}, {{A, B, C, X(472), X(3830)}}, {{A, B, C, X(473), X(3845)}}, {{A, B, C, X(7043), X(33696)}}, {{A, B, C, X(8742), X(46204)}}, {{A, B, C, X(18550), X(40712)}}
X(54479) = X(i)-cross conjugate of X(j) for these {i, j}: {43475, 12816}

X(54480) = X(17)X(3830)∩X(18)X(3845)

Barycentrics    (a-b-c)*(a+b-c)*(a-b+c)*(a+b+c)*(121*a^4-122*(b^2-c^2)^2+a^2*(b^2+c^2+9*sqrt(-3*a^4-3*(b^2-c^2)^2+6*a^2*(b^2+c^2)))) : :
Barycentrics    1 / (9*Sqrt[3]*(-a^2 + b^2 + c^2) + 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] + 27*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54480) lies on these lines: {2, 42113}, {13, 12101}, {14, 42521}, {17, 3830}, {18, 3845}, {30, 10188}, {381, 10187}, {1080, 10185}, {3411, 14269}, {3534, 42610}, {3860, 19106}, {3861, 42977}, {5066, 43442}, {5076, 49903}, {5487, 36352}, {5488, 33624}, {8703, 42596}, {11001, 43445}, {11121, 35752}, {11122, 36327}, {12817, 42102}, {12820, 41108}, {12821, 42094}, {14893, 41974}, {15682, 42504}, {15687, 42502}, {16809, 43368}, {16964, 43556}, {19107, 43428}, {19709, 43440}, {19710, 42492}, {22237, 42161}, {33602, 36970}, {33603, 43369}, {33606, 36969}, {33607, 42128}, {33699, 42099}, {35751, 40707}, {36967, 42952}, {38335, 42532}, {40706, 47867}, {41099, 42580}, {41100, 42103}, {41106, 42433}, {41107, 43541}, {41112, 43552}, {41119, 41971}, {41120, 43195}, {41121, 43364}, {42092, 43002}, {42095, 43545}, {42120, 43555}, {42140, 42511}, {42420, 42814}, {42529, 42930}, {42632, 43325}, {42634, 43227}, {42901, 43228}, {43007, 43540}, {43009, 43016}, {43023, 49904}, {43226, 43326}, {43243, 43400}, {43554, 49907}, {44580, 51916}

X(54480) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(470), X(12101)}}, {{A, B, C, X(472), X(3845)}}, {{A, B, C, X(473), X(3830)}}, {{A, B, C, X(7026), X(33696)}}, {{A, B, C, X(8741), X(46204)}}, {{A, B, C, X(18550), X(40711)}}
X(54480) = X(i)-cross conjugate of X(j) for these {i, j}: {43476, 12817}

X(54481) = X(30)X(10290)∩X(1916)X(11645)

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

X(54481) lies on these lines: {30, 10290}, {542, 43688}, {1916, 11645}, {6054, 43529}, {9180, 30217}, {10159, 44224}, {14223, 25423}

X(54481) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(428), X(44224)}}, {{A, B, C, X(512), X(14388)}}, {{A, B, C, X(542), X(25423)}}, {{A, B, C, X(543), X(30217)}}, {{A, B, C, X(804), X(11645)}}, {{A, B, C, X(9830), X(32472)}}

X(54482) = X(2)X(38225)∩X(30)X(10484)

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

X(54482) lies on these lines: {2, 38225}, {30, 10484}, {98, 18424}, {381, 8587}, {1916, 32519}, {5475, 7608}, {5476, 45103}, {7607, 21445}, {7622, 42011}, {9755, 43535}, {11170, 53418}, {14912, 32532}

X(54482) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(38225)}}, {{A, B, C, X(54), X(34154)}}, {{A, B, C, X(592), X(30496)}}, {{A, B, C, X(1173), X(9515)}}, {{A, B, C, X(3425), X(9831)}}, {{A, B, C, X(14356), X(18424)}}, {{A, B, C, X(18550), X(47388)}}

X(54483) = X(4)X(34319)∩X(30)X(10511)

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

X(54483) lies on these lines: {4, 34319}, {30, 10511}, {94, 8352}, {5466, 32228}, {7550, 10185}, {7578, 11317}

X(54483) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(186), X(8352)}}, {{A, B, C, X(265), X(34319)}}, {{A, B, C, X(477), X(36882)}}, {{A, B, C, X(524), X(11564)}}, {{A, B, C, X(5627), X(6094)}}, {{A, B, C, X(6344), X(18818)}}, {{A, B, C, X(7577), X(11317)}}, {{A, B, C, X(7841), X(18559)}}, {{A, B, C, X(22151), X(34802)}}

X(54484) = X(2)X(47610)∩X(76)X(616)

Barycentrics    sqrt(3)*(11*a^8-2*a^6*(b^2+c^2)-14*a^2*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^2*(b^4+16*b^2*c^2+c^4)+2*a^4*(3*b^4-5*b^2*c^2+3*c^4))+6*(5*a^6-a^2*(b^2-c^2)^2-a^4*(b^2+c^2)-3*(b^2-c^2)^2*(b^2+c^2))*S : :

X(54484) lies on these lines: {2, 47610}, {14, 9862}, {17, 6770}, {18, 33420}, {20, 5487}, {30, 11121}, {76, 616}, {83, 22796}, {542, 40706}, {621, 40707}, {1916, 46708}, {2986, 14181}, {3438, 34296}, {3457, 40158}, {5478, 43550}, {5617, 10159}, {6582, 42035}, {6773, 43539}, {9302, 53442}, {10188, 41020}, {10210, 13582}, {11122, 12188}, {11603, 41023}, {36961, 43547}, {39874, 43542}

X(54484) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(34533)}}, {{A, B, C, X(74), X(616)}}, {{A, B, C, X(621), X(1141)}}, {{A, B, C, X(1494), X(8737)}}, {{A, B, C, X(2379), X(11738)}}, {{A, B, C, X(2980), X(11085)}}, {{A, B, C, X(2992), X(11080)}}, {{A, B, C, X(2993), X(34288)}}, {{A, B, C, X(9141), X(34376)}}, {{A, B, C, X(11270), X(23716)}}, {{A, B, C, X(14491), X(34321)}}
X(54484) = X(i)-vertex conjugate of X(j) for these {i, j}: {3439, 9302}

X(54485) = X(2)X(47611)∩X(76)X(617)

Barycentrics    sqrt(3)*(11*a^8-2*a^6*(b^2+c^2)-14*a^2*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^2*(b^4+16*b^2*c^2+c^4)+2*a^4*(3*b^4-5*b^2*c^2+3*c^4))+6*(-5*a^6+a^2*(b^2-c^2)^2+a^4*(b^2+c^2)+3*(b^2-c^2)^2*(b^2+c^2))*S : :

X(54485) lies on these lines: {2, 47611}, {13, 9862}, {17, 33421}, {18, 6773}, {20, 5488}, {30, 11122}, {76, 617}, {83, 22797}, {542, 40707}, {622, 40706}, {1916, 46709}, {2986, 14177}, {3439, 34295}, {3458, 40159}, {5479, 43551}, {5613, 10159}, {6295, 42036}, {6770, 43538}, {9302, 53430}, {10187, 41021}, {11121, 12188}, {11602, 41022}, {36962, 43546}, {39874, 43543}

X(54485) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(34534)}}, {{A, B, C, X(74), X(617)}}, {{A, B, C, X(622), X(1141)}}, {{A, B, C, X(1494), X(8738)}}, {{A, B, C, X(2378), X(11738)}}, {{A, B, C, X(2980), X(11080)}}, {{A, B, C, X(2992), X(34288)}}, {{A, B, C, X(2993), X(11085)}}, {{A, B, C, X(9141), X(34374)}}, {{A, B, C, X(11270), X(23717)}}, {{A, B, C, X(14491), X(34322)}}
X(54485) = X(i)-vertex conjugate of X(j) for these {i, j}: {3438, 9302}

X(54486) = X(2)X(34514)∩X(30)X(11140)

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

X(54486) lies on these lines: {2, 34514}, {30, 11140}, {94, 7540}, {1503, 9221}, {1510, 2394}, {3518, 16080}, {37939, 42410}, {43530, 52295}

X(54486) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(26), X(18559)}}, {{A, B, C, X(30), X(1510)}}, {{A, B, C, X(70), X(34288)}}, {{A, B, C, X(93), X(1989)}}, {{A, B, C, X(186), X(7540)}}, {{A, B, C, X(252), X(30537)}}, {{A, B, C, X(376), X(37122)}}, {{A, B, C, X(381), X(52295)}}, {{A, B, C, X(1138), X(8884)}}, {{A, B, C, X(1141), X(15321)}}, {{A, B, C, X(1179), X(1494)}}, {{A, B, C, X(2980), X(6344)}}, {{A, B, C, X(3520), X(13490)}}, {{A, B, C, X(5627), X(6145)}}, {{A, B, C, X(6240), X(37939)}}, {{A, B, C, X(7512), X(7576)}}, {{A, B, C, X(11738), X(16263)}}, {{A, B, C, X(13489), X(16620)}}, {{A, B, C, X(13596), X(23410)}}, {{A, B, C, X(16837), X(52154)}}, {{A, B, C, X(31181), X(44958)}}, {{A, B, C, X(32085), X(33565)}}, {{A, B, C, X(34797), X(51519)}}, {{A, B, C, X(43908), X(48911)}}
X(54486) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 9221}

X(54487) = X(2)X(5104)∩X(6)X(43535)

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

X(54487) lies on these lines: {2, 5104}, {6, 43535}, {30, 11170}, {76, 7775}, {83, 7833}, {98, 5476}, {381, 43532}, {385, 11167}, {597, 3407}, {598, 3329}, {599, 42006}, {671, 5475}, {1916, 11163}, {3314, 8176}, {3406, 32134}, {3815, 10484}, {5395, 33192}, {5466, 11640}, {5485, 7774}, {5503, 7777}, {7766, 42536}, {7824, 43527}, {7912, 18840}, {8592, 11317}, {10159, 16921}, {10717, 34087}, {11184, 42010}, {11648, 17503}, {17005, 42011}, {18841, 33215}, {22110, 43529}, {32995, 43681}, {33256, 53102}, {37665, 41895}

X(54487) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(5104)}}, {{A, B, C, X(25), X(33013)}}, {{A, B, C, X(251), X(7775)}}, {{A, B, C, X(385), X(9487)}}, {{A, B, C, X(427), X(7833)}}, {{A, B, C, X(428), X(16921)}}, {{A, B, C, X(597), X(3314)}}, {{A, B, C, X(599), X(3329)}}, {{A, B, C, X(1992), X(7774)}}, {{A, B, C, X(3108), X(9515)}}, {{A, B, C, X(5064), X(7824)}}, {{A, B, C, X(5094), X(8597)}}, {{A, B, C, X(5475), X(51541)}}, {{A, B, C, X(5476), X(14356)}}, {{A, B, C, X(6094), X(22336)}}, {{A, B, C, X(7378), X(33215)}}, {{A, B, C, X(7714), X(32962)}}, {{A, B, C, X(7777), X(22329)}}, {{A, B, C, X(7806), X(22110)}}, {{A, B, C, X(7837), X(41624)}}, {{A, B, C, X(7903), X(34572)}}, {{A, B, C, X(7912), X(42037)}}, {{A, B, C, X(8859), X(11184)}}, {{A, B, C, X(8860), X(17005)}}, {{A, B, C, X(8889), X(33192)}}, {{A, B, C, X(9164), X(45108)}}, {{A, B, C, X(9227), X(38005)}}, {{A, B, C, X(9229), X(46645)}}, {{A, B, C, X(10717), X(14609)}}, {{A, B, C, X(11160), X(37665)}}, {{A, B, C, X(14388), X(20251)}}, {{A, B, C, X(18818), X(45819)}}, {{A, B, C, X(30495), X(39389)}}, {{A, B, C, X(45090), X(52395)}}
X(54487) = trilinear pole of line {9208, 523}

X(54488) = X(2)X(52771)∩X(30)X(11172)

Barycentrics    (a^6-3*b^6+11*b^4*c^2-9*b^2*c^4+c^6-a^4*(9*b^2+5*c^2)+a^2*(11*b^4-2*b^2*c^2-5*c^4))*(a^6+b^6-9*b^4*c^2+11*b^2*c^4-3*c^6-a^4*(5*b^2+9*c^2)+a^2*(-5*b^4-2*b^2*c^2+11*c^4)) : :

X(54488) lies on these lines: {2, 52771}, {30, 11172}, {98, 46034}, {262, 43448}, {598, 6776}, {3543, 43535}, {5395, 39646}, {7608, 31400}, {7612, 11676}, {7620, 11167}, {14485, 14853}, {15980, 40824}, {36998, 53100}

X(54488) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(52771)}}, {{A, B, C, X(253), X(48259)}}, {{A, B, C, X(265), X(6776)}}, {{A, B, C, X(393), X(9154)}}, {{A, B, C, X(6530), X(46034)}}, {{A, B, C, X(6620), X(15980)}}, {{A, B, C, X(11676), X(37174)}}, {{A, B, C, X(11738), X(23700)}}, {{A, B, C, X(33971), X(43448)}}

X(54489) = X(2)X(6777)∩X(17)X(542)

Barycentrics    3*sqrt(3)*(a^10-3*a^8*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+3*a^6*(b^4+b^2*c^2+c^4)+a^2*(b^2-c^2)^2*(2*b^4+5*b^2*c^2+2*c^4)-2*a^4*(b^6+c^6))-2*(13*a^8-12*a^6*(b^2+c^2)+(b^2-c^2)^2*(b^4-16*b^2*c^2+c^4)+a^4*(13*b^4-17*b^2*c^2+13*c^4)+a^2*(-15*b^6+16*b^4*c^2+16*b^2*c^4-15*c^6))*S : :

X(54489) lies on these lines: {2, 6777}, {4, 5469}, {13, 52649}, {14, 14830}, {17, 542}, {30, 11602}, {76, 5463}, {83, 5460}, {115, 12816}, {148, 33610}, {530, 11122}, {531, 40707}, {533, 35005}, {543, 49901}, {598, 16809}, {671, 48996}, {2996, 22577}, {3457, 48353}, {5466, 22934}, {5470, 43546}, {6582, 11121}, {6778, 33607}, {7607, 41020}, {11603, 11632}, {14223, 23872}, {16530, 43539}, {22566, 46053}, {22570, 36967}, {33461, 42036}, {33623, 41135}, {36766, 48657}

X(54489) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(74), X(32906)}}, {{A, B, C, X(542), X(23872)}}, {{A, B, C, X(2378), X(3455)}}, {{A, B, C, X(2380), X(46286)}}, {{A, B, C, X(3439), X(34533)}}
X(54489) = midpoint of X(i) in X(j) for these {i,j}: {148, 33610}
X(54489) = reflection of X(i) in X(j) for these {i,j}: {12816, 115}
X(54489) = trilinear pole of line {43228, 523}
X(54489) = X(i)-vertex conjugate of X(j) for these {i, j}: {13, 3439}

X(54490) = X(2)X(6778)∩X(18)X(542)

Barycentrics    3*sqrt(3)*(a^10-3*a^8*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+3*a^6*(b^4+b^2*c^2+c^4)+a^2*(b^2-c^2)^2*(2*b^4+5*b^2*c^2+2*c^4)-2*a^4*(b^6+c^6))+2*(13*a^8-12*a^6*(b^2+c^2)+(b^2-c^2)^2*(b^4-16*b^2*c^2+c^4)+a^4*(13*b^4-17*b^2*c^2+13*c^4)+a^2*(-15*b^6+16*b^4*c^2+16*b^2*c^4-15*c^6))*S : :

X(54490) lies on these lines: {2, 6778}, {4, 5470}, {13, 14830}, {14, 44289}, {18, 542}, {30, 11603}, {76, 5464}, {83, 5459}, {115, 12817}, {148, 33611}, {530, 40706}, {531, 11121}, {532, 35005}, {543, 49902}, {598, 16808}, {671, 48995}, {2996, 22578}, {3458, 48355}, {5466, 22889}, {5469, 43547}, {6295, 11122}, {6777, 33606}, {7607, 41021}, {11602, 11632}, {14223, 23873}, {16529, 43538}, {22566, 46054}, {22568, 36968}, {33460, 42035}, {33625, 41135}

X(54490) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(74), X(32908)}}, {{A, B, C, X(542), X(23873)}}, {{A, B, C, X(2379), X(3455)}}, {{A, B, C, X(2381), X(46286)}}, {{A, B, C, X(3438), X(34534)}}
X(54490) = midpoint of X(i) in X(j) for these {i,j}: {148, 33611}
X(54490) = reflection of X(i) in X(j) for these {i,j}: {12817, 115}
X(54490) = trilinear pole of line {43229, 523}
X(54490) = X(i)-vertex conjugate of X(j) for these {i, j}: {14, 3438}

X(54491) = X(30)X(11608)∩X(226)X(542)

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

X(54491) lies on these lines: {30, 11608}, {226, 542}, {415, 16080}, {522, 14223}, {2394, 2785}, {2796, 43683}, {9180, 28292}

X(54491) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(415)}}, {{A, B, C, X(522), X(542)}}, {{A, B, C, X(543), X(28292)}}, {{A, B, C, X(1311), X(9141)}}, {{A, B, C, X(2796), X(6003)}}

X(54492) = X(10)X(2247)∩X(321)X(542)

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

X(54492) lies on these lines: {10, 2247}, {30, 11611}, {321, 542}, {422, 16080}, {513, 14223}, {2394, 2787}, {9180, 28475}

X(54492) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(422)}}, {{A, B, C, X(105), X(9141)}}, {{A, B, C, X(513), X(542)}}, {{A, B, C, X(543), X(28475)}}

X(54493) = X(30)X(11668)∩X(98)X(38335)

Barycentrics    (10*a^2+10*b^2-11*c^2)*(10*a^2-11*b^2+10*c^2) : :

X(54493) lies on these lines: {30, 11668}, {98, 38335}, {262, 14893}, {381, 53108}, {1657, 10185}, {3627, 7607}, {3843, 7608}, {7827, 18843}, {11303, 43441}, {11304, 43440}, {11669, 23046}, {14044, 43529}, {14066, 43528}, {15684, 53104}, {50691, 53859}

X(54493) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(249), X(14490)}}, {{A, B, C, X(297), X(38335)}}, {{A, B, C, X(458), X(14893)}}, {{A, B, C, X(3431), X(11588)}}, {{A, B, C, X(3627), X(52282)}}, {{A, B, C, X(3843), X(52281)}}, {{A, B, C, X(11741), X(32901)}}

X(54494) = X(6)X(33698)∩X(30)X(11669)

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

X(54494) lies on these lines: {6, 33698}, {30, 11669}, {98, 14269}, {262, 15687}, {381, 53104}, {382, 7608}, {546, 7607}, {597, 53102}, {598, 6329}, {3529, 53098}, {3851, 10185}, {5485, 11008}, {7827, 18845}, {8591, 35005}, {10302, 11317}, {11149, 33257}, {11303, 43442}, {11304, 43443}, {11668, 38071}, {14042, 43529}, {14062, 43528}, {15681, 53108}, {17503, 53418}, {20583, 53105}, {33229, 43527}, {50688, 53099}

X(54494) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(249), X(3531)}}, {{A, B, C, X(297), X(14269)}}, {{A, B, C, X(382), X(52281)}}, {{A, B, C, X(458), X(15687)}}, {{A, B, C, X(546), X(52282)}}, {{A, B, C, X(597), X(31360)}}, {{A, B, C, X(599), X(6329)}}, {{A, B, C, X(1992), X(11008)}}, {{A, B, C, X(3431), X(11741)}}, {{A, B, C, X(5064), X(33229)}}, {{A, B, C, X(7841), X(52285)}}, {{A, B, C, X(10301), X(11317)}}, {{A, B, C, X(10630), X(47060)}}, {{A, B, C, X(11588), X(11736)}}, {{A, B, C, X(11738), X(20251)}}, {{A, B, C, X(20583), X(40341)}}, {{A, B, C, X(33696), X(34914)}}

X(54495) = X(2)X(9717)∩X(74)X(671)

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

X(54495) lies on these lines: {2, 9717}, {30, 12066}, {74, 671}, {76, 36890}, {1494, 8781}, {2394, 15543}, {2433, 5466}, {2986, 9140}, {5627, 39295}, {9180, 14651}

X(54495) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(12065)}}, {{A, B, C, X(74), X(2433)}}, {{A, B, C, X(476), X(1138)}}, {{A, B, C, X(1494), X(18808)}}, {{A, B, C, X(1989), X(9214)}}, {{A, B, C, X(5627), X(12079)}}, {{A, B, C, X(5641), X(40118)}}, {{A, B, C, X(7418), X(52492)}}, {{A, B, C, X(9140), X(33565)}}, {{A, B, C, X(14582), X(35912)}}, {{A, B, C, X(15454), X(15543)}}, {{A, B, C, X(16092), X(52491)}}, {{A, B, C, X(34174), X(53161)}}
X(54495) = trilinear pole of line {6128, 523}
X(54495) = X(i)-cross conjugate of X(j) for these {i, j}: {542, 5627}
X(54495) = barycentric quotient X(i)/X(j) for these (i, j): {2433, 34291}

X(54496) = X(4)X(34986)∩X(317)X(459)

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

X(54496) lies on these lines: {4, 34986}, {30, 13380}, {96, 31180}, {98, 34609}, {317, 459}, {381, 45300}, {671, 37672}, {1368, 7607}, {2052, 27377}, {5020, 7608}, {5392, 14615}, {7396, 43537}, {7398, 53099}, {10159, 41235}, {16072, 40448}, {37874, 53420}

X(54496) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(290), X(34412)}}, {{A, B, C, X(297), X(34609)}}, {{A, B, C, X(317), X(1494)}}, {{A, B, C, X(394), X(43844)}}, {{A, B, C, X(428), X(41235)}}, {{A, B, C, X(467), X(31180)}}, {{A, B, C, X(524), X(37672)}}, {{A, B, C, X(1368), X(52282)}}, {{A, B, C, X(5020), X(52281)}}, {{A, B, C, X(6391), X(27377)}}, {{A, B, C, X(8794), X(40832)}}, {{A, B, C, X(15749), X(37669)}}, {{A, B, C, X(16072), X(52280)}}, {{A, B, C, X(35142), X(41530)}}

X(54497) = X(10)X(35338)∩X(30)X(13576)

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

X(54497) lies on these lines: {10, 35338}, {30, 13576}, {321, 50154}, {537, 43677}, {542, 43671}, {918, 2394}, {1111, 43682}, {2826, 5466}, {15149, 16080}

X(54497) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(918)}}, {{A, B, C, X(74), X(37128)}}, {{A, B, C, X(84), X(36871)}}, {{A, B, C, X(104), X(2991)}}, {{A, B, C, X(274), X(10308)}}, {{A, B, C, X(348), X(34800)}}, {{A, B, C, X(524), X(2826)}}, {{A, B, C, X(537), X(6002)}}, {{A, B, C, X(651), X(1292)}}, {{A, B, C, X(1111), X(7261)}}, {{A, B, C, X(1138), X(9510)}}, {{A, B, C, X(3065), X(4560)}}, {{A, B, C, X(3426), X(39981)}}, {{A, B, C, X(3431), X(39952)}}, {{A, B, C, X(3512), X(17925)}}, {{A, B, C, X(7313), X(17096)}}, {{A, B, C, X(14483), X(39971)}}, {{A, B, C, X(16615), X(32009)}}, {{A, B, C, X(28840), X(28850)}}, {{A, B, C, X(36800), X(39768)}}, {{A, B, C, X(44129), X(48877)}}, {{A, B, C, X(48897), X(52374)}}
X(54497) = trilinear pole of line {354, 523}

X(54498) = X(2)X(15032)∩X(20)X(13582)

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

X(54498) lies on these lines: {2, 15032}, {20, 13582}, {30, 13579}, {376, 6504}, {459, 37943}, {3543, 13585}, {3839, 11538}, {7400, 43681}, {7505, 16080}, {34621, 38259}, {37119, 43530}

X(54498) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(1989)}}, {{A, B, C, X(20), X(37943)}}, {{A, B, C, X(30), X(7505)}}, {{A, B, C, X(54), X(34288)}}, {{A, B, C, X(64), X(52154)}}, {{A, B, C, X(66), X(13597)}}, {{A, B, C, X(68), X(5627)}}, {{A, B, C, X(69), X(6344)}}, {{A, B, C, X(74), X(2165)}}, {{A, B, C, X(93), X(36889)}}, {{A, B, C, X(252), X(14457)}}, {{A, B, C, X(253), X(5900)}}, {{A, B, C, X(254), X(1138)}}, {{A, B, C, X(264), X(45138)}}, {{A, B, C, X(376), X(3542)}}, {{A, B, C, X(381), X(6145)}}, {{A, B, C, X(393), X(3431)}}, {{A, B, C, X(847), X(1494)}}, {{A, B, C, X(1141), X(4846)}}, {{A, B, C, X(1173), X(22270)}}, {{A, B, C, X(1217), X(3459)}}, {{A, B, C, X(2963), X(3426)}}, {{A, B, C, X(3088), X(5071)}}, {{A, B, C, X(3089), X(3524)}}, {{A, B, C, X(3519), X(48911)}}, {{A, B, C, X(3527), X(30537)}}, {{A, B, C, X(3532), X(18317)}}, {{A, B, C, X(3541), X(3545)}}, {{A, B, C, X(3543), X(14940)}}, {{A, B, C, X(3549), X(18559)}}, {{A, B, C, X(3839), X(6143)}}, {{A, B, C, X(6526), X(45736)}}, {{A, B, C, X(7383), X(7714)}}, {{A, B, C, X(7552), X(18533)}}, {{A, B, C, X(7558), X(7576)}}, {{A, B, C, X(10201), X(35471)}}, {{A, B, C, X(11816), X(14542)}}, {{A, B, C, X(13472), X(52187)}}, {{A, B, C, X(14491), X(46952)}}, {{A, B, C, X(14938), X(22334)}}, {{A, B, C, X(16868), X(44441)}}, {{A, B, C, X(17983), X(46259)}}, {{A, B, C, X(18349), X(36612)}}, {{A, B, C, X(18361), X(44157)}}, {{A, B, C, X(20421), X(51316)}}, {{A, B, C, X(21844), X(46451)}}, {{A, B, C, X(22268), X(52518)}}, {{A, B, C, X(33565), X(34285)}}, {{A, B, C, X(34436), X(51761)}}, {{A, B, C, X(34621), X(38282)}}, {{A, B, C, X(43917), X(45838)}}
X(54498) = X(i)-cross conjugate of X(j) for these {i, j}: {11456, 4}

X(54499) = X(10)X(16132)∩X(30)X(13583)

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

X(54499) lies on these lines: {10, 16132}, {30, 13583}

X(54499) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(74), X(1427)}}, {{A, B, C, X(278), X(2349)}}, {{A, B, C, X(376), X(37388)}}, {{A, B, C, X(1138), X(2184)}}, {{A, B, C, X(1903), X(1989)}}, {{A, B, C, X(5627), X(13853)}}, {{A, B, C, X(10308), X(37887)}}, {{A, B, C, X(16132), X(52374)}}, {{A, B, C, X(28609), X(37797)}}

X(54500) = X(2)X(15037)∩X(3)X(13582)

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

X(54500) lies on these lines: {2, 15037}, {3, 13582}, {4, 11063}, {30, 13585}, {94, 7552}, {226, 16763}, {275, 1157}, {376, 13579}, {381, 11538}, {2052, 37943}, {3470, 14940}, {3524, 6504}, {6143, 43530}, {6770, 40104}, {6773, 40105}, {7383, 43681}, {7592, 43666}, {9381, 38542}

X(54500) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(16763)}}, {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(1138)}}, {{A, B, C, X(6), X(15037)}}, {{A, B, C, X(30), X(14940)}}, {{A, B, C, X(54), X(1989)}}, {{A, B, C, X(74), X(2963)}}, {{A, B, C, X(93), X(1494)}}, {{A, B, C, X(95), X(6344)}}, {{A, B, C, X(186), X(7552)}}, {{A, B, C, X(252), X(5627)}}, {{A, B, C, X(264), X(5900)}}, {{A, B, C, X(376), X(7505)}}, {{A, B, C, X(381), X(6143)}}, {{A, B, C, X(631), X(36612)}}, {{A, B, C, X(847), X(13418)}}, {{A, B, C, X(1173), X(22268)}}, {{A, B, C, X(1487), X(6662)}}, {{A, B, C, X(2165), X(3431)}}, {{A, B, C, X(3089), X(15702)}}, {{A, B, C, X(3524), X(3542)}}, {{A, B, C, X(3541), X(5071)}}, {{A, B, C, X(3545), X(37119)}}, {{A, B, C, X(3613), X(13597)}}, {{A, B, C, X(5055), X(35482)}}, {{A, B, C, X(5684), X(16764)}}, {{A, B, C, X(6188), X(20421)}}, {{A, B, C, X(10201), X(21844)}}, {{A, B, C, X(11058), X(17711)}}, {{A, B, C, X(13472), X(22270)}}, {{A, B, C, X(13623), X(20480)}}, {{A, B, C, X(14938), X(16835)}}, {{A, B, C, X(15464), X(46452)}}, {{A, B, C, X(18361), X(18368)}}, {{A, B, C, X(19307), X(34483)}}, {{A, B, C, X(33565), X(45838)}}, {{A, B, C, X(40410), X(45138)}}, {{A, B, C, X(43917), X(53864)}}
X(54500) = X(i)-cross conjugate of X(j) for these {i, j}: {15032, 4}

X(54501) = X(30)X(14223)∩X(542)X(2394)

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

X(54501) lies on these lines: {30, 14223}, {524, 52459}, {542, 2394}, {543, 43673}, {671, 46982}, {1503, 9180}, {2794, 5466}, {7473, 16080}, {10722, 41392}, {11645, 46040}

X(54501) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(99)}}, {{A, B, C, X(316), X(6054)}}, {{A, B, C, X(511), X(11636)}}, {{A, B, C, X(524), X(2794)}}, {{A, B, C, X(543), X(1503)}}, {{A, B, C, X(1302), X(9141)}}, {{A, B, C, X(2782), X(11645)}}, {{A, B, C, X(5627), X(42345)}}, {{A, B, C, X(6033), X(7809)}}, {{A, B, C, X(7799), X(10722)}}, {{A, B, C, X(11006), X(45774)}}, {{A, B, C, X(46982), X(52475)}}
X(54501) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 9180}, {671, 22455}

X(54502) = X(30)X(14228)∩X(591)X(1132)

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

X(54502) lies on the Kiepert hyperbola and on these lines: {30, 14228}, {524, 43566}, {591, 1132}, {1131, 5861}, {5860, 43567}, {12322, 43560}, {14243, 36718}

X(54503) = X(2)X(9675)∩X(76)X(591)

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

X(54503) lies on these lines: {2, 9675}, {30, 14231}, {76, 591}, {381, 14238}, {486, 35948}, {671, 19108}, {3317, 12322}, {5491, 5861}, {6569, 49229}, {10194, 39388}, {13757, 42023}, {13770, 42024}, {15031, 45575}, {19101, 45420}, {35297, 53488}

X(54503) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(9675)}}, {{A, B, C, X(249), X(589)}}, {{A, B, C, X(1016), X(13390)}}, {{A, B, C, X(1509), X(7090)}}, {{A, B, C, X(3069), X(5861)}}, {{A, B, C, X(5860), X(19053)}}, {{A, B, C, X(18820), X(32085)}}

X(54504) = X(30)X(14236)∩X(671)X(13847)

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

X(54504) lies on these lines: {30, 14236}, {637, 34091}, {671, 13847}, {1132, 26618}, {5491, 5860}, {6569, 49213}, {13757, 42024}

X(54504) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(249), X(494)}}, {{A, B, C, X(589), X(32422)}}, {{A, B, C, X(3069), X(5860)}}

X(54505) = X(30)X(14240)∩X(671)X(13846)

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

X(54505) lies on these lines: {30, 14240}, {638, 34089}, {671, 13846}, {1131, 26617}, {5490, 5861}, {6568, 49212}, {13637, 42023}

X(54505) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(249), X(493)}}, {{A, B, C, X(588), X(32420)}}, {{A, B, C, X(3068), X(5861)}}

X(54506) = X(30)X(14243)∩X(1131)X(1991)

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

X(54506) lies on the Kiepert hyperbola and on these lines: {30, 14243}, {524, 43567}, {1131, 1991}, {1132, 5860}, {5861, 43566}, {12323, 43561}, {14228, 36734}

>

X(54507) = X(30)X(14245)∩X(76)X(1991)

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

X(54507) lies on these lines: {30, 14245}, {76, 1991}, {381, 14234}, {485, 35949}, {671, 19109}, {3316, 12323}, {5490, 5860}, {6568, 49228}, {10195, 39387}, {13637, 42024}, {13651, 42023}, {15031, 45574}, {22541, 45421}, {35297, 53487}

X(54507) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(249), X(588)}}, {{A, B, C, X(1016), X(1659)}}, {{A, B, C, X(1509), X(14121)}}, {{A, B, C, X(3068), X(5860)}}, {{A, B, C, X(5861), X(19054)}}, {{A, B, C, X(18819), X(32085)}}

X(54508) = X(30)X(1446)∩X(226)X(15938)

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

X(54508) lies on these lines: {30, 1446}, {226, 15938}, {2394, 3900}, {4183, 16080}

X(54508) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(3900)}}, {{A, B, C, X(4219), X(11113)}}, {{A, B, C, X(8748), X(15938)}}, {{A, B, C, X(17532), X(37372)}}

X(54509) = X(6)X(11167)∩X(98)X(597)

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

X(54509) lies on these lines: {2, 51396}, {6, 11167}, {30, 14485}, {76, 11163}, {83, 8359}, {98, 597}, {262, 42849}, {325, 10302}, {598, 5077}, {671, 3363}, {1506, 18840}, {3329, 8593}, {3424, 11179}, {3815, 5503}, {5461, 9302}, {5466, 44568}, {5485, 7736}, {6054, 43532}, {7737, 18842}, {7840, 42006}, {11648, 32532}, {25555, 43537}, {31489, 42011}, {44401, 53104}

X(54509) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(11163)}}, {{A, B, C, X(183), X(42849)}}, {{A, B, C, X(325), X(597)}}, {{A, B, C, X(427), X(8359)}}, {{A, B, C, X(468), X(3363)}}, {{A, B, C, X(524), X(11169)}}, {{A, B, C, X(599), X(11174)}}, {{A, B, C, X(842), X(30535)}}, {{A, B, C, X(843), X(39389)}}, {{A, B, C, X(1494), X(13377)}}, {{A, B, C, X(1506), X(42037)}}, {{A, B, C, X(1992), X(7736)}}, {{A, B, C, X(3329), X(7840)}}, {{A, B, C, X(3815), X(22329)}}, {{A, B, C, X(4518), X(34914)}}, {{A, B, C, X(5077), X(5094)}}, {{A, B, C, X(7249), X(34892)}}, {{A, B, C, X(7792), X(22110)}}, {{A, B, C, X(8860), X(31489)}}, {{A, B, C, X(9164), X(30537)}}, {{A, B, C, X(9300), X(41624)}}, {{A, B, C, X(9770), X(44556)}}, {{A, B, C, X(18823), X(36897)}}, {{A, B, C, X(23297), X(51224)}}, {{A, B, C, X(35705), X(52094)}}, {{A, B, C, X(36882), X(42286)}}, {{A, B, C, X(37647), X(44401)}}, {{A, B, C, X(43950), X(46316)}}
X(54509) = trilinear pole of line {11186, 523}

X(54510) = X(30)X(14534)∩X(429)X(16080)

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

X(54510) lies on these lines: {30, 14534}, {381, 34258}, {429, 16080}, {3017, 13478}, {4185, 43530}, {37415, 43527}

X(54510) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(429)}}, {{A, B, C, X(65), X(1494)}}, {{A, B, C, X(381), X(4185)}}, {{A, B, C, X(961), X(1168)}}, {{A, B, C, X(1880), X(44835)}}, {{A, B, C, X(1989), X(51870)}}, {{A, B, C, X(3577), X(13610)}}, {{A, B, C, X(5064), X(37415)}}

X(54511) = X(30)X(14554)∩X(226)X(1387)

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

X(54511) lies on these lines: {30, 14554}, {226, 1387}, {2827, 4049}, {4080, 38460}

X(54511) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(51788)}}, {{A, B, C, X(2), X(4)}}, {{A, B, C, X(80), X(1387)}}, {{A, B, C, X(104), X(519)}}, {{A, B, C, X(903), X(46435)}}, {{A, B, C, X(1222), X(16005)}}, {{A, B, C, X(1877), X(2161)}}, {{A, B, C, X(3577), X(39704)}}, {{A, B, C, X(3582), X(6735)}}, {{A, B, C, X(3872), X(7284)}}, {{A, B, C, X(5560), X(51709)}}, {{A, B, C, X(14483), X(53114)}}, {{A, B, C, X(16615), X(43972)}}, {{A, B, C, X(36123), X(36910)}}

X(54512) = X(2)X(7687)∩X(4)X(3163)

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

X(54512) lies on these lines: {2, 7687}, {4, 3163}, {30, 16080}, {275, 3845}, {381, 43530}, {459, 5667}, {2052, 3830}, {2394, 9033}, {10152, 34297}, {10159, 34664}, {11001, 38253}, {12101, 39284}

X(54512) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(3830)}}, {{A, B, C, X(5), X(3845)}}, {{A, B, C, X(6), X(22455)}}, {{A, B, C, X(20), X(15682)}}, {{A, B, C, X(30), X(265)}}, {{A, B, C, X(74), X(15051)}}, {{A, B, C, X(95), X(18550)}}, {{A, B, C, X(140), X(12101)}}, {{A, B, C, X(250), X(34802)}}, {{A, B, C, X(253), X(18847)}}, {{A, B, C, X(287), X(53201)}}, {{A, B, C, X(376), X(3543)}}, {{A, B, C, X(381), X(36430)}}, {{A, B, C, X(382), X(3534)}}, {{A, B, C, X(428), X(34664)}}, {{A, B, C, X(546), X(5066)}}, {{A, B, C, X(547), X(14893)}}, {{A, B, C, X(549), X(15687)}}, {{A, B, C, X(550), X(33699)}}, {{A, B, C, X(895), X(2693)}}, {{A, B, C, X(1093), X(46204)}}, {{A, B, C, X(1105), X(21400)}}, {{A, B, C, X(1217), X(18296)}}, {{A, B, C, X(1300), X(5627)}}, {{A, B, C, X(1302), X(52447)}}, {{A, B, C, X(1513), X(8352)}}, {{A, B, C, X(1551), X(36196)}}, {{A, B, C, X(1989), X(7687)}}, {{A, B, C, X(3091), X(41099)}}, {{A, B, C, X(3146), X(11001)}}, {{A, B, C, X(3524), X(50687)}}, {{A, B, C, X(3529), X(15640)}}, {{A, B, C, X(3545), X(3839)}}, {{A, B, C, X(3627), X(8703)}}, {{A, B, C, X(3832), X(41106)}}, {{A, B, C, X(3843), X(19709)}}, {{A, B, C, X(3853), X(12100)}}, {{A, B, C, X(3861), X(10109)}}, {{A, B, C, X(4846), X(36437)}}, {{A, B, C, X(5054), X(38335)}}, {{A, B, C, X(5055), X(14269)}}, {{A, B, C, X(5073), X(15685)}}, {{A, B, C, X(5076), X(15693)}}, {{A, B, C, X(5505), X(50531)}}, {{A, B, C, X(5667), X(10152)}}, {{A, B, C, X(6110), X(11092)}}, {{A, B, C, X(6111), X(11078)}}, {{A, B, C, X(7576), X(52069)}}, {{A, B, C, X(8431), X(50433)}}, {{A, B, C, X(8597), X(11676)}}, {{A, B, C, X(9909), X(34725)}}, {{A, B, C, X(10127), X(44804)}}, {{A, B, C, X(10201), X(18568)}}, {{A, B, C, X(10297), X(47332)}}, {{A, B, C, X(11317), X(13860)}}, {{A, B, C, X(11564), X(43660)}}, {{A, B, C, X(11738), X(41894)}}, {{A, B, C, X(11744), X(48378)}}, {{A, B, C, X(11812), X(12102)}}, {{A, B, C, X(13603), X(41890)}}, {{A, B, C, X(14093), X(35434)}}, {{A, B, C, X(14487), X(41891)}}, {{A, B, C, X(14490), X(15364)}}, {{A, B, C, X(14860), X(17505)}}, {{A, B, C, X(14892), X(41987)}}, {{A, B, C, X(15318), X(15749)}}, {{A, B, C, X(15319), X(18848)}}, {{A, B, C, X(15679), X(21669)}}, {{A, B, C, X(15681), X(15684)}}, {{A, B, C, X(15686), X(35404)}}, {{A, B, C, X(15694), X(35403)}}, {{A, B, C, X(15698), X(50688)}}, {{A, B, C, X(15718), X(35401)}}, {{A, B, C, X(16263), X(34288)}}, {{A, B, C, X(17578), X(19708)}}, {{A, B, C, X(18323), X(44265)}}, {{A, B, C, X(18361), X(46255)}}, {{A, B, C, X(18377), X(44278)}}, {{A, B, C, X(18405), X(41372)}}, {{A, B, C, X(18572), X(44266)}}, {{A, B, C, X(18850), X(36889)}}, {{A, B, C, X(23046), X(38071)}}, {{A, B, C, X(30247), X(48373)}}, {{A, B, C, X(34613), X(52397)}}, {{A, B, C, X(34621), X(44442)}}, {{A, B, C, X(36490), X(36730)}}, {{A, B, C, X(36551), X(36729)}}, {{A, B, C, X(36720), X(36732)}}, {{A, B, C, X(36721), X(36731)}}, {{A, B, C, X(36722), X(36728)}}, {{A, B, C, X(37904), X(47339)}}, {{A, B, C, X(44262), X(44288)}}, {{A, B, C, X(44263), X(44287)}}, {{A, B, C, X(45301), X(46848)}}, {{A, B, C, X(46429), X(48362)}}, {{A, B, C, X(47097), X(47310)}}
X(54512) = trilinear pole of line {14401, 523}

X(54513) = X(30)X(16277)∩X(2394)X(23881)

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

X(54513) lies on these lines: {30, 16277}, {2394, 23881}, {5392, 7841}, {7488, 10511}, {7509, 7607}, {7608, 14788}, {8370, 40393}

X(54513) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(24), X(7841)}}, {{A, B, C, X(30), X(23881)}}, {{A, B, C, X(265), X(6664)}}, {{A, B, C, X(524), X(3519)}}, {{A, B, C, X(599), X(14528)}}, {{A, B, C, X(1300), X(1502)}}, {{A, B, C, X(1594), X(8370)}}, {{A, B, C, X(6094), X(45195)}}, {{A, B, C, X(6656), X(7576)}}, {{A, B, C, X(6662), X(36882)}}, {{A, B, C, X(7487), X(33190)}}, {{A, B, C, X(7509), X(52282)}}, {{A, B, C, X(8352), X(10018)}}, {{A, B, C, X(11317), X(52296)}}, {{A, B, C, X(11585), X(37855)}}, {{A, B, C, X(14262), X(34505)}}, {{A, B, C, X(14788), X(52281)}}, {{A, B, C, X(20806), X(34801)}}, {{A, B, C, X(33565), X(40405)}}

X(54514) = X(2)X(33708)∩X(30)X(1676)

Barycentrics    a^10+27*a^8*b^2+47*a^6*b^4-37*a^4*b^6+27*a^8*c^2+128*a^6*b^2*c^2+101*a^4*b^4*c^2+47*a^6*c^4+101*a^4*b^2*c^4-37*a^4*c^6-36*a^2*b^4*(b^2-c^2)^2-2*b^6*(b^2-c^2)^2-106*a^2*b^2*c^2*(b^2-c^2)^2-34*b^4*c^2*(b^2-c^2)^2-36*a^2*c^4*(b^2-c^2)^2-34*b^2*c^4*(b^2-c^2)^2-2*c^6*(b^2-c^2)^2+(8*a^8+49*a^6*b^2+9*a^4*b^4+49*a^6*c^2+110*a^4*b^2*c^2+9*a^4*c^4-53*a^2*b^2*(b^2-c^2)^2-13*b^4*(b^2-c^2)^2-53*a^2*c^2*(b^2-c^2)^2-46*b^2*c^2*(b^2-c^2)^2-13*c^4*(b^2-c^2)^2)*sqrt(b^2*c^2+a^2*(b^2+c^2)) : :

X(54514) lies on the Kiepert hyperbola and these lines: {2, 33708}, {30, 1676}, {524, 3818}, {543, 33707}, {1677, 5309}, {11000, 43527}, {16080, 16246}, {16245, 43530}

X(54514) = reflection of X(i) in X(j) for these {i,j}: {33708, 2}

X(54515) = X(2)X(33707)∩X(30)X(1677)

Barycentrics    a^10+27*a^8*b^2+47*a^6*b^4-37*a^4*b^6+27*a^8*c^2+128*a^6*b^2*c^2+101*a^4*b^4*c^2+47*a^6*c^4+101*a^4*b^2*c^4-37*a^4*c^6-36*a^2*b^4*(b^2-c^2)^2-2*b^6*(b^2-c^2)^2-106*a^2*b^2*c^2*(b^2-c^2)^2-34*b^4*c^2*(b^2-c^2)^2-36*a^2*c^4*(b^2-c^2)^2-34*b^2*c^4*(b^2-c^2)^2-2*c^6*(b^2-c^2)^2+(-8*a^8-49*a^6*b^2-9*a^4*b^4-49*a^6*c^2-110*a^4*b^2*c^2-9*a^4*c^4+53*a^2*b^2*(b^2-c^2)^2+13*b^4*(b^2-c^2)^2+53*a^2*c^2*(b^2-c^2)^2+46*b^2*c^2*(b^2-c^2)^2+13*c^4*(b^2-c^2)^2)*sqrt(b^2*c^2+a^2*(b^2+c^2)) : :

X(54515) lies on the Kiepert hyperbola and these lines: {2, 33707}, {30, 1677}, {524, 3818}, {543, 33708}, {1676, 5309}, {10999, 43527}

X(54515) = reflection of X(i) in X(j) for these {i,j}: {33707, 2}

X(54516) = X(30)X(1751)∩X(226)X(381)

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

X(54516) lies on these lines: {29, 43530}, {30, 1751}, {226, 381}, {4080, 12649}, {5125, 16080}

X(54516) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(15934)}}, {{A, B, C, X(2), X(4)}}, {{A, B, C, X(29), X(381)}}, {{A, B, C, X(30), X(5125)}}, {{A, B, C, X(78), X(36599)}}, {{A, B, C, X(80), X(273)}}, {{A, B, C, X(84), X(903)}}, {{A, B, C, X(158), X(36910)}}, {{A, B, C, X(225), X(34288)}}, {{A, B, C, X(307), X(4846)}}, {{A, B, C, X(519), X(12649)}}, {{A, B, C, X(1257), X(10308)}}, {{A, B, C, X(3345), X(36603)}}, {{A, B, C, X(3426), X(4674)}}, {{A, B, C, X(3545), X(7518)}}, {{A, B, C, X(3839), X(7498)}}, {{A, B, C, X(5136), X(52269)}}, {{A, B, C, X(6734), X(10056)}}, {{A, B, C, X(7319), X(36590)}}, {{A, B, C, X(7513), X(17532)}}, {{A, B, C, X(10429), X(36588)}}, {{A, B, C, X(11114), X(37381)}}, {{A, B, C, X(28193), X(39742)}}, {{A, B, C, X(36721), X(37389)}}, {{A, B, C, X(36889), X(39130)}}

X(54517) = X(10)X(28854)∩X(30)X(17758)

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

X(54517) lies on these lines: {10, 28854}, {30, 17758}, {226, 3058}, {1446, 7264}, {2394, 4151}, {3309, 4049}, {3870, 4080}, {10159, 13727}, {14004, 16080}, {36652, 43527}, {36721, 43531}

X(54517) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(3748)}}, {{A, B, C, X(2), X(4)}}, {{A, B, C, X(7), X(30331)}}, {{A, B, C, X(27), X(36722)}}, {{A, B, C, X(30), X(4151)}}, {{A, B, C, X(79), X(3058)}}, {{A, B, C, X(80), X(21453)}}, {{A, B, C, X(428), X(13727)}}, {{A, B, C, X(461), X(3543)}}, {{A, B, C, X(469), X(36721)}}, {{A, B, C, X(514), X(28854)}}, {{A, B, C, X(516), X(4762)}}, {{A, B, C, X(519), X(3309)}}, {{A, B, C, X(903), X(15909)}}, {{A, B, C, X(996), X(42361)}}, {{A, B, C, X(1088), X(5561)}}, {{A, B, C, X(1280), X(16615)}}, {{A, B, C, X(2736), X(53226)}}, {{A, B, C, X(2788), X(2796)}}, {{A, B, C, X(3017), X(27572)}}, {{A, B, C, X(3062), X(39704)}}, {{A, B, C, X(3426), X(53114)}}, {{A, B, C, X(3668), X(34288)}}, {{A, B, C, X(4847), X(10056)}}, {{A, B, C, X(5064), X(36652)}}, {{A, B, C, X(5556), X(51783)}}, {{A, B, C, X(28194), X(29186)}}, {{A, B, C, X(36124), X(52374)}}

X(54518) = X(30)X(18366)∩X(13582)X(18403)

Barycentrics    (7*a^8-4*a^6*(4*b^2+c^2)-(b^2-c^2)^3*(5*b^2+7*c^2)+a^4*(6*b^4+13*b^2*c^2-6*c^4)+a^2*(8*b^6-17*b^4*c^2+13*b^2*c^4-4*c^6))*(7*a^8-4*a^6*(b^2+4*c^2)+(b^2-c^2)^3*(7*b^2+5*c^2)+a^4*(-6*b^4+13*b^2*c^2+6*c^4)+a^2*(-4*b^6+13*b^4*c^2-17*b^2*c^4+8*c^6)) : :

X(54518) lies on these lines: {30, 18366}, {13582, 18403}, {13585, 18566}, {13619, 16080}

X(54518) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(13619)}}, {{A, B, C, X(265), X(1138)}}, {{A, B, C, X(1494), X(6188)}}, {{A, B, C, X(3459), X(17505)}}, {{A, B, C, X(11058), X(16000)}}, {{A, B, C, X(11564), X(18317)}}, {{A, B, C, X(14940), X(18566)}}, {{A, B, C, X(18361), X(45736)}}, {{A, B, C, X(18403), X(37943)}}

X(54519) = X(2)X(41424)∩X(428)X(459)

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

X(54519) lies on these lines: {2, 41424}, {20, 10159}, {30, 18840}, {76, 3543}, {83, 3839}, {381, 18841}, {383, 43446}, {428, 459}, {1080, 43447}, {1503, 43951}, {2394, 3800}, {2996, 50687}, {3091, 43527}, {3424, 5306}, {3830, 5485}, {3845, 18842}, {5304, 14458}, {6776, 14488}, {6811, 43564}, {6813, 43565}, {6995, 16080}, {7000, 10194}, {7374, 10195}, {7378, 43530}, {7714, 38253}, {10302, 15640}, {12101, 32532}, {13860, 53098}, {14269, 18843}, {14484, 36990}, {14893, 18844}, {17578, 43681}, {43676, 50688}

X(54519) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(34572)}}, {{A, B, C, X(20), X(428)}}, {{A, B, C, X(25), X(3543)}}, {{A, B, C, X(30), X(3800)}}, {{A, B, C, X(66), X(18361)}}, {{A, B, C, X(74), X(39955)}}, {{A, B, C, X(251), X(3426)}}, {{A, B, C, X(253), X(11058)}}, {{A, B, C, X(305), X(43699)}}, {{A, B, C, X(376), X(7408)}}, {{A, B, C, X(381), X(7378)}}, {{A, B, C, X(427), X(3839)}}, {{A, B, C, X(1297), X(22334)}}, {{A, B, C, X(1383), X(13603)}}, {{A, B, C, X(1494), X(52223)}}, {{A, B, C, X(1989), X(8801)}}, {{A, B, C, X(3091), X(5064)}}, {{A, B, C, X(3108), X(3531)}}, {{A, B, C, X(3146), X(7714)}}, {{A, B, C, X(3527), X(29180)}}, {{A, B, C, X(3545), X(7409)}}, {{A, B, C, X(3563), X(46851)}}, {{A, B, C, X(3830), X(4232)}}, {{A, B, C, X(3845), X(52284)}}, {{A, B, C, X(5304), X(7788)}}, {{A, B, C, X(5306), X(37668)}}, {{A, B, C, X(5481), X(52518)}}, {{A, B, C, X(6353), X(50687)}}, {{A, B, C, X(6623), X(31133)}}, {{A, B, C, X(7487), X(34603)}}, {{A, B, C, X(7500), X(7576)}}, {{A, B, C, X(7519), X(18559)}}, {{A, B, C, X(9095), X(39732)}}, {{A, B, C, X(9740), X(41624)}}, {{A, B, C, X(10152), X(40174)}}, {{A, B, C, X(10301), X(15640)}}, {{A, B, C, X(13472), X(29316)}}, {{A, B, C, X(13575), X(16251)}}, {{A, B, C, X(14495), X(16835)}}, {{A, B, C, X(15314), X(36916)}}, {{A, B, C, X(15682), X(52301)}}, {{A, B, C, X(22336), X(48911)}}, {{A, B, C, X(32085), X(36889)}}, {{A, B, C, X(43726), X(52188)}}
X(54519) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 43951}

X(54520) = X(2)X(31860)∩X(20)X(43527)

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

X(54520) lies on these lines: {2, 31860}, {20, 43527}, {30, 18841}, {76, 3839}, {83, 3543}, {381, 18840}, {383, 43447}, {459, 5064}, {1080, 43446}, {1513, 53098}, {2996, 7837}, {3091, 10159}, {3424, 53023}, {3830, 18842}, {3845, 5485}, {5395, 50687}, {6811, 43565}, {6813, 43564}, {6995, 43530}, {7000, 10195}, {7374, 10194}, {7378, 16080}, {8796, 42854}, {9300, 14484}, {14492, 37665}, {15687, 18843}, {18844, 38335}, {43681, 50689}, {50688, 53102}

X(54520) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(20), X(5064)}}, {{A, B, C, X(25), X(3839)}}, {{A, B, C, X(30), X(7378)}}, {{A, B, C, X(66), X(52188)}}, {{A, B, C, X(193), X(7837)}}, {{A, B, C, X(251), X(3531)}}, {{A, B, C, X(253), X(43726)}}, {{A, B, C, X(376), X(7409)}}, {{A, B, C, X(381), X(6995)}}, {{A, B, C, X(393), X(43458)}}, {{A, B, C, X(427), X(3543)}}, {{A, B, C, X(428), X(3091)}}, {{A, B, C, X(1297), X(52518)}}, {{A, B, C, X(1383), X(14487)}}, {{A, B, C, X(1494), X(45833)}}, {{A, B, C, X(1799), X(43699)}}, {{A, B, C, X(3087), X(42854)}}, {{A, B, C, X(3108), X(3426)}}, {{A, B, C, X(3527), X(34572)}}, {{A, B, C, X(3545), X(7408)}}, {{A, B, C, X(3830), X(52284)}}, {{A, B, C, X(3832), X(7714)}}, {{A, B, C, X(3845), X(4232)}}, {{A, B, C, X(5481), X(22334)}}, {{A, B, C, X(8801), X(13481)}}, {{A, B, C, X(8889), X(50687)}}, {{A, B, C, X(9300), X(15589)}}, {{A, B, C, X(10002), X(53023)}}, {{A, B, C, X(10304), X(52285)}}, {{A, B, C, X(11058), X(22336)}}, {{A, B, C, X(14483), X(39955)}}, {{A, B, C, X(14490), X(39951)}}, {{A, B, C, X(14583), X(52485)}}, {{A, B, C, X(15321), X(48911)}}, {{A, B, C, X(16251), X(18018)}}, {{A, B, C, X(18361), X(38005)}}, {{A, B, C, X(18575), X(46204)}}, {{A, B, C, X(30537), X(34285)}}, {{A, B, C, X(35512), X(39978)}}, {{A, B, C, X(36889), X(52223)}}, {{A, B, C, X(37665), X(37671)}}, {{A, B, C, X(40174), X(52452)}}, {{A, B, C, X(41099), X(52301)}}
X(54520) = X(i)-vertex conjugate of X(j) for these {i, j}: {3425, 53098}

X(54521) = X(20)X(53102)∩X(30)X(18843)

Barycentrics    (a^4+b^4+10*b^2*c^2-11*c^4+2*a^2*(11*b^2+5*c^2))*(a^4-11*b^4+10*b^2*c^2+c^4+2*a^2*(5*b^2+11*c^2)) : :

X(54521) lies on these lines: {20, 53102}, {30, 18843}, {83, 10304}, {549, 18841}, {598, 15640}, {3091, 43676}, {3534, 18842}, {3543, 53109}, {3839, 53105}, {5055, 18840}, {5066, 5485}, {5306, 43537}, {5395, 15683}, {7000, 43570}, {7374, 43571}, {7486, 10159}, {10303, 43527}, {14458, 37665}, {14853, 53104}, {15684, 18844}

X(54521) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(253), X(18361)}}, {{A, B, C, X(427), X(10304)}}, {{A, B, C, X(428), X(7486)}}, {{A, B, C, X(549), X(7378)}}, {{A, B, C, X(1494), X(52224)}}, {{A, B, C, X(3534), X(52284)}}, {{A, B, C, X(3613), X(52187)}}, {{A, B, C, X(3839), X(37453)}}, {{A, B, C, X(4232), X(5066)}}, {{A, B, C, X(5055), X(6995)}}, {{A, B, C, X(5064), X(10303)}}, {{A, B, C, X(5094), X(15640)}}, {{A, B, C, X(5481), X(43691)}}, {{A, B, C, X(7409), X(15709)}}, {{A, B, C, X(7714), X(15022)}}, {{A, B, C, X(7788), X(37665)}}, {{A, B, C, X(8801), X(11058)}}, {{A, B, C, X(8889), X(15683)}}, {{A, B, C, X(22336), X(46204)}}, {{A, B, C, X(34288), X(45090)}}, {{A, B, C, X(34572), X(40801)}}, {{A, B, C, X(36889), X(45833)}}, {{A, B, C, X(39951), X(43713)}}

X(54522) = X(30)X(18844)∩X(83)X(15692)

Barycentrics    (a^4+b^4-14*b^2*c^2+13*c^4-2*a^2*(13*b^2+7*c^2))*(a^4+13*b^4-14*b^2*c^2+c^4-2*a^2*(7*b^2+13*c^2)) : :

X(54522) lies on these lines: {30, 18844}, {83, 15692}, {547, 18840}, {3424, 9300}, {3543, 53107}, {3839, 53106}, {3860, 32532}, {5054, 18841}, {5485, 19709}, {8703, 18842}, {10159, 46936}, {14853, 53108}, {15681, 18843}

X(54522) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(253), X(45090)}}, {{A, B, C, X(427), X(15692)}}, {{A, B, C, X(428), X(46936)}}, {{A, B, C, X(547), X(6995)}}, {{A, B, C, X(3108), X(44731)}}, {{A, B, C, X(3543), X(52298)}}, {{A, B, C, X(3613), X(45833)}}, {{A, B, C, X(3839), X(52297)}}, {{A, B, C, X(4232), X(19709)}}, {{A, B, C, X(5054), X(7378)}}, {{A, B, C, X(8703), X(52284)}}, {{A, B, C, X(8801), X(30537)}}, {{A, B, C, X(9300), X(37668)}}, {{A, B, C, X(36889), X(52224)}}, {{A, B, C, X(45108), X(52187)}}, {{A, B, C, X(45819), X(46212)}}

X(54523) = X(2)X(44456)∩X(4)X(9606)

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

X(54523) lies on these lines: {2, 44456}, {4, 9606}, {5, 43681}, {30, 18845}, {76, 5071}, {83, 3524}, {376, 5395}, {381, 38259}, {383, 43556}, {598, 11001}, {671, 41106}, {1080, 43557}, {2996, 3545}, {3525, 43527}, {3528, 53102}, {3544, 43676}, {5067, 10159}, {5306, 7612}, {6997, 13582}, {7736, 14458}, {9753, 53108}, {10155, 14853}, {11172, 41624}, {13860, 47586}, {14229, 49263}, {14244, 49260}, {15682, 53101}, {15702, 18841}, {16080, 52299}, {18842, 19708}, {38282, 43530}, {41099, 41895}

X(54523) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(22332)}}, {{A, B, C, X(6), X(44456)}}, {{A, B, C, X(25), X(5071)}}, {{A, B, C, X(30), X(52299)}}, {{A, B, C, X(69), X(11058)}}, {{A, B, C, X(264), X(52188)}}, {{A, B, C, X(376), X(8889)}}, {{A, B, C, X(381), X(38282)}}, {{A, B, C, X(427), X(3524)}}, {{A, B, C, X(428), X(5067)}}, {{A, B, C, X(468), X(41106)}}, {{A, B, C, X(1007), X(5306)}}, {{A, B, C, X(1173), X(14489)}}, {{A, B, C, X(1494), X(46952)}}, {{A, B, C, X(3090), X(7714)}}, {{A, B, C, X(3431), X(39951)}}, {{A, B, C, X(3525), X(5064)}}, {{A, B, C, X(3527), X(36616)}}, {{A, B, C, X(3545), X(6353)}}, {{A, B, C, X(3613), X(17040)}}, {{A, B, C, X(5094), X(11001)}}, {{A, B, C, X(6997), X(37943)}}, {{A, B, C, X(7249), X(18490)}}, {{A, B, C, X(7378), X(15702)}}, {{A, B, C, X(7736), X(7788)}}, {{A, B, C, X(8770), X(14483)}}, {{A, B, C, X(8797), X(34288)}}, {{A, B, C, X(9770), X(41624)}}, {{A, B, C, X(13472), X(40801)}}, {{A, B, C, X(13575), X(45972)}}, {{A, B, C, X(14356), X(36892)}}, {{A, B, C, X(19708), X(52284)}}, {{A, B, C, X(20421), X(39389)}}, {{A, B, C, X(30775), X(35481)}}, {{A, B, C, X(34208), X(52187)}}, {{A, B, C, X(37119), X(44442)}}, {{A, B, C, X(41099), X(52290)}}, {{A, B, C, X(43662), X(52518)}}, {{A, B, C, X(43726), X(52154)}}, {{A, B, C, X(44658), X(48911)}}, {{A, B, C, X(45838), X(52717)}}

X(54524) = X(17)X(47865)∩X(30)X(21845)

Barycentrics    3*(10*a^6-19*a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)+8*a^2*(b^4-3*b^2*c^2+c^4))+sqrt(3)*(-13*a^4+32*b^4-65*b^2*c^2+32*c^4-29*a^2*(b^2+c^2))*S : :

X(54524) lies on these lines: {17, 47865}, {30, 21845}, {530, 43548}, {5459, 43545}, {10188, 35751}, {17503, 49947}, {35749, 43447}, {35752, 43544}, {36768, 43443}, {43554, 51482}

X(54524) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(2981), X(32901)}}

X(54525) = X(18)X(47866)∩X(30)X(21846)

Barycentrics    -3*(10*a^6-19*a^4*(b^2+c^2)+(b^2-c^2)^2*(b^2+c^2)+8*a^2*(b^4-3*b^2*c^2+c^4))+sqrt(3)*(-13*a^4+32*b^4-65*b^2*c^2+32*c^4-29*a^2*(b^2+c^2))*S : :

X(54525) lies on these lines: {18, 47866}, {30, 21846}, {531, 43549}, {5460, 43544}, {10187, 36329}, {17503, 49948}, {36327, 43446}, {36330, 43545}, {43555, 51483}

X(54525) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6151), X(32901)}}

X(54526) = X(10)X(7359)∩X(30)X(226)

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

X(54526) lies on these lines: {2, 45924}, {4, 52956}, {10, 7359}, {29, 16080}, {30, 226}, {381, 1751}, {519, 43683}, {522, 2394}, {542, 11608}, {1446, 15936}, {1784, 40149}, {2785, 14223}, {4049, 6003}, {4080, 34772}, {5125, 43530}, {5466, 28292}, {13576, 18406}, {17758, 37428}, {24624, 52269}, {28580, 43677}

X(54526) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(2341)}}, {{A, B, C, X(2), X(4)}}, {{A, B, C, X(7), X(4304)}}, {{A, B, C, X(29), X(30)}}, {{A, B, C, X(74), X(53114)}}, {{A, B, C, X(78), X(17098)}}, {{A, B, C, X(80), X(5719)}}, {{A, B, C, X(84), X(39704)}}, {{A, B, C, X(225), X(1989)}}, {{A, B, C, X(265), X(307)}}, {{A, B, C, X(273), X(2166)}}, {{A, B, C, X(282), X(44693)}}, {{A, B, C, X(376), X(7518)}}, {{A, B, C, X(381), X(5125)}}, {{A, B, C, X(502), X(5627)}}, {{A, B, C, X(519), X(1389)}}, {{A, B, C, X(524), X(28292)}}, {{A, B, C, X(542), X(2785)}}, {{A, B, C, X(860), X(52269)}}, {{A, B, C, X(996), X(39695)}}, {{A, B, C, X(1065), X(15909)}}, {{A, B, C, X(1257), X(16615)}}, {{A, B, C, X(1494), X(39130)}}, {{A, B, C, X(1861), X(18406)}}, {{A, B, C, X(2287), X(15936)}}, {{A, B, C, X(3345), X(39980)}}, {{A, B, C, X(3543), X(7498)}}, {{A, B, C, X(3584), X(6734)}}, {{A, B, C, X(4674), X(44835)}}, {{A, B, C, X(5556), X(40836)}}, {{A, B, C, X(6002), X(28580)}}, {{A, B, C, X(7513), X(11113)}}, {{A, B, C, X(8747), X(52374)}}, {{A, B, C, X(10308), X(43972)}}, {{A, B, C, X(14004), X(37428)}}, {{A, B, C, X(17577), X(37381)}}, {{A, B, C, X(17677), X(37362)}}, {{A, B, C, X(28840), X(28849)}}, {{A, B, C, X(31155), X(44225)}}, {{A, B, C, X(36722), X(37389)}}, {{A, B, C, X(43917), X(45095)}}
X(54526) = trilinear pole of line {14400, 523}

X(54527) = X(2)X(3233)∩X(30)X(2394)

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

X(54527) lies on these lines: {2, 3233}, {30, 2394}, {524, 43673}, {542, 14223}, {543, 52459}, {671, 10733}, {1503, 5466}, {2794, 9180}, {3081, 12079}, {4240, 9140}, {9141, 34767}, {11645, 43665}, {34761, 53161}

X(54527) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(476)}}, {{A, B, C, X(265), X(9140)}}, {{A, B, C, X(511), X(11645)}}, {{A, B, C, X(524), X(1503)}}, {{A, B, C, X(541), X(17702)}}, {{A, B, C, X(542), X(20404)}}, {{A, B, C, X(543), X(2794)}}, {{A, B, C, X(1494), X(9214)}}, {{A, B, C, X(1989), X(18808)}}, {{A, B, C, X(2697), X(5641)}}, {{A, B, C, X(10733), X(11744)}}, {{A, B, C, X(11050), X(11251)}}, {{A, B, C, X(15454), X(18317)}}, {{A, B, C, X(19924), X(29012)}}, {{A, B, C, X(34765), X(53161)}}, {{A, B, C, X(42308), X(52485)}}
X(54527) = trilinear pole of line {3163, 23967}
X(54527) = X(i)-isoconjugate-of-X(j) for these {i, j}: {35200, 52464}
X(54527) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 5466}
X(54527) = X(i)-Dao conjugate of X(j) for these {i, j}: {133, 52464}, {42426, 52469}
X(54527) = barycentric quotient X(i)/X(j) for these (i, j): {1990, 52464}, {6103, 52469}

X(54528) = X(2)X(6739)∩X(80)X(226)

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

X(54528) lies on these lines: {2, 6739}, {10, 36910}, {30, 24624}, {80, 226}, {321, 52409}, {759, 21161}, {860, 16080}, {1446, 18815}, {1834, 1989}, {2166, 43682}, {2394, 6370}, {4080, 36590}, {5136, 43530}, {5397, 5721}, {10706, 19629}

X(54528) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(1), X(5425)}}, {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(860)}}, {{A, B, C, X(37), X(44835)}}, {{A, B, C, X(74), X(4674)}}, {{A, B, C, X(80), X(2166)}}, {{A, B, C, X(104), X(903)}}, {{A, B, C, X(381), X(5136)}}, {{A, B, C, X(519), X(41558)}}, {{A, B, C, X(655), X(32041)}}, {{A, B, C, X(897), X(2687)}}, {{A, B, C, X(1000), X(14563)}}, {{A, B, C, X(1138), X(5620)}}, {{A, B, C, X(1243), X(39974)}}, {{A, B, C, X(1389), X(40430)}}, {{A, B, C, X(1494), X(38955)}}, {{A, B, C, X(1737), X(39991)}}, {{A, B, C, X(1821), X(35161)}}, {{A, B, C, X(1989), X(45926)}}, {{A, B, C, X(3065), X(4511)}}, {{A, B, C, X(3427), X(36588)}}, {{A, B, C, X(3577), X(3679)}}, {{A, B, C, X(4231), X(17677)}}, {{A, B, C, X(6344), X(15065)}}, {{A, B, C, X(11105), X(28452)}}, {{A, B, C, X(11113), X(37381)}}, {{A, B, C, X(11604), X(33593)}}, {{A, B, C, X(16139), X(41697)}}, {{A, B, C, X(18391), X(36916)}}, {{A, B, C, X(24297), X(36596)}}, {{A, B, C, X(24857), X(34485)}}, {{A, B, C, X(37718), X(51310)}}, {{A, B, C, X(44693), X(52663)}}
X(54528) = trilinear pole of line {17718, 523}
X(54528) = X(i)-isoconjugate-of-X(j) for these {i, j}: {36, 24929}
X(54528) = X(i)-Dao conjugate of X(j) for these {i, j}: {15898, 24929}
X(54528) = X(i)-cross conjugate of X(j) for these {i, j}: {18391, 40437}
X(54528) = barycentric quotient X(i)/X(j) for these (i, j): {2161, 24929}

X(54529) = X(30)X(30505)∩X(275)X(46511)

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

X(54529) lies on these lines: {30, 30505}, {275, 46511}, {6504, 32983}, {13579, 33016}, {13582, 16044}, {16080, 37125}

X(54529) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(5), X(46511)}}, {{A, B, C, X(30), X(37125)}}, {{A, B, C, X(74), X(39968)}}, {{A, B, C, X(308), X(14483)}}, {{A, B, C, X(376), X(37337)}}, {{A, B, C, X(695), X(30537)}}, {{A, B, C, X(1031), X(33565)}}, {{A, B, C, X(1141), X(52395)}}, {{A, B, C, X(1173), X(3228)}}, {{A, B, C, X(2998), X(14491)}}, {{A, B, C, X(3527), X(9462)}}, {{A, B, C, X(3531), X(34816)}}, {{A, B, C, X(3541), X(32986)}}, {{A, B, C, X(3542), X(32983)}}, {{A, B, C, X(7505), X(33016)}}, {{A, B, C, X(15412), X(30535)}}, {{A, B, C, X(16044), X(37943)}}, {{A, B, C, X(33017), X(37119)}}

X(54530) = X(30)X(30588)∩X(2394)X(4777)

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

X(54530) lies on these lines: {30, 30588}, {2394, 4777}, {5466, 28319}

X(54530) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(4777)}}, {{A, B, C, X(524), X(28319)}}, {{A, B, C, X(16615), X(46187)}}, {{A, B, C, X(28840), X(28889)}}

X(54531) = X(4)X(17809)∩X(25)X(53099)

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

X(54531) lies on these lines: {4, 17809}, {25, 53099}, {30, 31363}, {262, 7714}, {376, 13599}, {427, 43537}, {428, 14484}, {459, 3087}, {472, 22235}, {473, 22237}, {1249, 8796}, {1585, 3591}, {1586, 3590}, {2052, 40065}, {2996, 52281}, {3424, 5064}, {3535, 10194}, {3536, 10195}, {3545, 40448}, {5094, 53859}, {5395, 52282}, {5485, 37672}, {6353, 7608}, {7378, 47586}, {7607, 8889}, {10159, 52288}, {38282, 53098}, {43527, 52283}

X(54531) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(17809)}}, {{A, B, C, X(69), X(39286)}}, {{A, B, C, X(70), X(4994)}}, {{A, B, C, X(97), X(13472)}}, {{A, B, C, X(343), X(36809)}}, {{A, B, C, X(394), X(1173)}}, {{A, B, C, X(428), X(52288)}}, {{A, B, C, X(458), X(7714)}}, {{A, B, C, X(1073), X(52518)}}, {{A, B, C, X(1249), X(3087)}}, {{A, B, C, X(1992), X(37672)}}, {{A, B, C, X(2165), X(6748)}}, {{A, B, C, X(3527), X(36609)}}, {{A, B, C, X(3545), X(52280)}}, {{A, B, C, X(5064), X(52283)}}, {{A, B, C, X(5702), X(6749)}}, {{A, B, C, X(6353), X(52281)}}, {{A, B, C, X(8889), X(52282)}}, {{A, B, C, X(13452), X(31626)}}, {{A, B, C, X(15809), X(33190)}}, {{A, B, C, X(36916), X(53817)}}, {{A, B, C, X(39948), X(40396)}}, {{A, B, C, X(39980), X(40397)}}, {{A, B, C, X(42287), X(43726)}}
X(54531) = polar conjugate of X(5056)
X(54531) = X(i)-Dao conjugate of X(j) for these {i, j}: {1249, 5056}

X(54532) = X(30)X(32022)∩X(3839)X(6625)

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

X(54532) lies on these lines: {30, 32022}, {3839, 6625}, {4196, 16080}, {4207, 43530}, {10159, 36670}

X(54532) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(4196)}}, {{A, B, C, X(42), X(3531)}}, {{A, B, C, X(74), X(39965)}}, {{A, B, C, X(381), X(4207)}}, {{A, B, C, X(428), X(36670)}}, {{A, B, C, X(1002), X(10308)}}, {{A, B, C, X(1246), X(52188)}}, {{A, B, C, X(2350), X(3426)}}, {{A, B, C, X(3543), X(4212)}}, {{A, B, C, X(3839), X(4213)}}, {{A, B, C, X(14483), X(39961)}}, {{A, B, C, X(14490), X(39966)}}, {{A, B, C, X(39980), X(45137)}}

X(54533) = X(10)X(2173)∩X(30)X(321)

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

X(54533) lies on these lines: {2, 51420}, {4, 52955}, {10, 2173}, {28, 16080}, {30, 321}, {226, 49744}, {513, 2394}, {535, 43683}, {542, 11611}, {752, 43677}, {2787, 14223}, {5142, 43530}, {5466, 28475}, {10159, 37431}

X(54533) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(28), X(30)}}, {{A, B, C, X(74), X(1169)}}, {{A, B, C, X(79), X(49744)}}, {{A, B, C, X(265), X(20336)}}, {{A, B, C, X(376), X(4198)}}, {{A, B, C, X(381), X(5142)}}, {{A, B, C, X(428), X(37431)}}, {{A, B, C, X(524), X(28475)}}, {{A, B, C, X(542), X(2787)}}, {{A, B, C, X(752), X(6002)}}, {{A, B, C, X(1494), X(43712)}}, {{A, B, C, X(1791), X(34800)}}, {{A, B, C, X(1989), X(41013)}}, {{A, B, C, X(3426), X(46010)}}, {{A, B, C, X(3543), X(7521)}}, {{A, B, C, X(7576), X(37231)}}, {{A, B, C, X(15763), X(31154)}}, {{A, B, C, X(20029), X(34288)}}, {{A, B, C, X(28840), X(28845)}}
X(54533) = trilinear pole of line {14399, 523}

X(54534) = X(2)X(42197)∩X(30)X(3366)

Barycentrics    (a-b-c)*(a+b-c)*(a-b+c)*(a+b+c)*((8+5*sqrt(3))*a^4-2*(5+2*sqrt(3))*(b^2-c^2)^2+a^2*(-((-2+sqrt(3))*b^2)+2*c^2-sqrt(3)*c^2+3*sqrt(-a^4-(b^2-c^2)^2+2*a^2*(b^2+c^2)))) : :
Barycentrics    1 / (3*(2 + Sqrt[3])*(-a^2 + b^2 + c^2) + 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 + 3*(2 + Sqrt[3])*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54534) lies on these lines: {2, 42197}, {13, 32787}, {15, 42639}, {16, 43503}, {17, 18587}, {18, 18585}, {30, 3366}, {381, 3392}, {395, 3845}, {485, 36455}, {486, 16268}, {590, 36967}, {1327, 36970}, {1328, 36450}, {1991, 42036}, {2041, 10195}, {2044, 5351}, {3317, 36465}, {3367, 51854}, {3389, 36436}, {3391, 35822}, {3830, 6221}, {5066, 53439}, {6307, 40706}, {6561, 43475}, {6565, 49948}, {10188, 14814}, {11121, 33441}, {12101, 53438}, {12817, 42284}, {14226, 37641}, {14241, 36446}, {15687, 53459}, {16242, 53444}, {16808, 36466}, {16963, 42235}, {16964, 52217}, {22237, 42248}, {32788, 42507}, {35731, 35786}, {36445, 42256}, {36448, 36968}, {42134, 43567}, {42218, 43542}, {42228, 42813}, {42238, 42587}, {42280, 52214}

X(54534) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(15), X(588)}}, {{A, B, C, X(61), X(5417)}}, {{A, B, C, X(472), X(18585)}}, {{A, B, C, X(473), X(18587)}}, {{A, B, C, X(1585), X(36455)}}, {{A, B, C, X(1659), X(14358)}}, {{A, B, C, X(6221), X(51728)}}

X(54535) = X(2)X(42195)∩X(30)X(3367)

Barycentrics    (a-b-c)*(a+b-c)*(a-b+c)*(a+b+c)*((-8+5*sqrt(3))*a^4-2*(-5+2*sqrt(3))*(b^2-c^2)^2+a^2*(-((2+sqrt(3))*b^2)-2*c^2-sqrt(3)*c^2+3*sqrt(-a^4-(b^2-c^2)^2+2*a^2*(b^2+c^2)))) : :
Barycentrics    1 / (3*(2 - Sqrt[3])*(-a^2 + b^2 + c^2) - 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 - 3*(2 - Sqrt[3])*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54535) lies on these lines: {2, 42195}, {5, 50245}, {6, 50246}, {13, 32788}, {15, 42640}, {16, 43504}, {17, 18586}, {18, 15765}, {30, 3367}, {381, 3391}, {395, 3845}, {485, 16268}, {486, 36437}, {591, 42036}, {615, 36967}, {1327, 36467}, {1328, 36970}, {2042, 10194}, {2043, 5351}, {3316, 36446}, {3366, 51852}, {3390, 36454}, {3392, 35823}, {3830, 6398}, {5066, 53438}, {6303, 40706}, {6560, 43475}, {6564, 49948}, {10188, 14813}, {11121, 33440}, {12101, 53439}, {12817, 42283}, {14226, 36465}, {14241, 37641}, {15687, 53460}, {16242, 53445}, {16808, 36448}, {16963, 42237}, {16964, 52216}, {22237, 42246}, {32787, 42507}, {35731, 36470}, {36463, 42254}, {36466, 36968}, {42134, 43566}, {42220, 43542}, {42227, 42813}, {42236, 42587}, {42281, 52215}

X(54535) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(51728)}}, {{A, B, C, X(15), X(589)}}, {{A, B, C, X(61), X(5419)}}, {{A, B, C, X(472), X(15765)}}, {{A, B, C, X(473), X(18586)}}, {{A, B, C, X(1586), X(36437)}}, {{A, B, C, X(13390), X(14358)}}

X(54536) = X(30)X(3374)∩X(381)X(3387)

Barycentrics    (-8+5*sqrt(2))*a^4-2*(-5+2*sqrt(2))*(b^2-c^2)^2-a^2*((2+sqrt(2))*(b^2+c^2)-6*sqrt(2)*S) : :
Barycentrics    1 / (3*(-1 + Sqrt[2])*(-a^2 + b^2 + c^2) - 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 - 3*(-1 + Sqrt[2])*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54536) lies on the Kiepert hyperbola and these lines: {30, 3374}, {381, 3387}, {3373, 35823}, {3830, 12823}, {3845, 6565}, {6451, 43623}, {10194, 14782}, {10195, 14784}

X(54536) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3830, 42728, 12823}, {6565, 32788, 12822}

X(54537) = X(30)X(3387)∩X(381)X(3374)

Barycentrics    (-8+5*sqrt(2))*a^4-2*(-5+2*sqrt(2))*(b^2-c^2)^2-a^2*((2+sqrt(2))*(b^2+c^2)+6*sqrt(2)*S) : :
Barycentrics    1 / (3*(-1 + Sqrt[2])*(-a^2 + b^2 + c^2) + 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 + 3*(-1 + Sqrt[2])*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54537) lies on the Kiepert hyperbola and these lines: {30, 3387}, {381, 3374}, {3388, 35822}, {3830, 12822}, {3845, 6564}, {6452, 43622}, {10194, 14784}, {10195, 14782}

X(54537) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3830, 42727, 12822}, {6564, 32787, 12823}

X(54538) = X(2)X(42196)∩X(30)X(3392)

Barycentrics    (-8-5*sqrt(3))*a^4+2*(5+2*sqrt(3))*(b^2-c^2)^2+a^2*((-2+sqrt(3))*(b^2+c^2)+6*S) : :
Barycentrics    1 / (3*(2 + Sqrt[3])*(-a^2 + b^2 + c^2) - 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 - 3*(2 + Sqrt[3])*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54538) lies on these lines: {2, 42196}, {14, 32788}, {15, 43504}, {16, 42640}, {17, 18585}, {18, 18587}, {30, 3392}, {381, 3366}, {396, 3845}, {485, 16267}, {486, 36455}, {591, 42035}, {615, 36968}, {1327, 36449}, {1328, 36969}, {2041, 10194}, {2044, 5352}, {3316, 36464}, {3365, 36436}, {3367, 35823}, {3391, 51853}, {3590, 51727}, {3592, 50245}, {3830, 6398}, {5066, 53450}, {6302, 40707}, {6560, 43476}, {6564, 49947}, {10187, 14814}, {11122, 33442}, {12101, 53451}, {12816, 42283}, {14226, 36447}, {14241, 37640}, {15687, 53471}, {16241, 53433}, {16809, 36466}, {16962, 42238}, {16965, 52215}, {22235, 42247}, {32787, 42506}, {33607, 51728}, {35731, 36445}, {36448, 36967}, {42133, 43566}, {42219, 43543}, {42229, 42814}, {42235, 42586}, {42280, 52216}

X(54538) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(16), X(589)}}, {{A, B, C, X(62), X(5419)}}, {{A, B, C, X(472), X(18587)}}, {{A, B, C, X(473), X(18585)}}, {{A, B, C, X(1586), X(36455)}}, {{A, B, C, X(3592), X(51727)}}, {{A, B, C, X(13390), X(14359)}}, {{A, B, C, X(34754), X(51728)}}
X(54538) = X(i)-cross conjugate of X(j) for these {i, j}: {41107, 50246}
X(54538) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16808, 41101, 50246}

X(54539) = X(2)X(8627)∩X(76)X(754)

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

X(54539) lies on these lines: {2, 8627}, {30, 3399}, {76, 754}, {83, 7861}, {98, 48889}, {262, 29012}, {381, 3406}, {384, 10159}, {428, 37892}, {671, 12156}, {732, 43688}, {1916, 12830}, {2896, 14033}, {2996, 20088}, {3849, 10302}, {5025, 43527}, {5306, 43535}, {5503, 8592}, {5999, 7608}, {6292, 14036}, {7607, 13862}, {8290, 8781}, {9751, 53108}, {11057, 14030}, {11606, 12829}, {11645, 14492}, {12206, 18501}, {14042, 43676}, {14062, 53102}, {14068, 43681}, {16041, 18841}, {17766, 34475}, {40163, 42037}, {43534, 50289}

X(54539) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(25), X(11361)}}, {{A, B, C, X(251), X(512)}}, {{A, B, C, X(384), X(428)}}, {{A, B, C, X(385), X(41624)}}, {{A, B, C, X(427), X(14041)}}, {{A, B, C, X(699), X(27375)}}, {{A, B, C, X(732), X(25423)}}, {{A, B, C, X(733), X(8601)}}, {{A, B, C, X(1031), X(32085)}}, {{A, B, C, X(2896), X(3866)}}, {{A, B, C, X(3108), X(14498)}}, {{A, B, C, X(3849), X(12073)}}, {{A, B, C, X(4590), X(22336)}}, {{A, B, C, X(4785), X(17766)}}, {{A, B, C, X(5025), X(5064)}}, {{A, B, C, X(5306), X(7840)}}, {{A, B, C, X(5999), X(52281)}}, {{A, B, C, X(6995), X(14033)}}, {{A, B, C, X(7378), X(16041)}}, {{A, B, C, X(7408), X(14039)}}, {{A, B, C, X(7409), X(33285)}}, {{A, B, C, X(7714), X(14035)}}, {{A, B, C, X(7826), X(34572)}}, {{A, B, C, X(7837), X(14614)}}, {{A, B, C, X(7893), X(46287)}}, {{A, B, C, X(8290), X(12829)}}, {{A, B, C, X(9227), X(45819)}}, {{A, B, C, X(9229), X(15321)}}, {{A, B, C, X(12156), X(52898)}}, {{A, B, C, X(13862), X(52282)}}, {{A, B, C, X(14046), X(52285)}}, {{A, B, C, X(14537), X(51541)}}, {{A, B, C, X(23878), X(29012)}}, {{A, B, C, X(34238), X(46320)}}, {{A, B, C, X(39955), X(44557)}}, {{A, B, C, X(44132), X(48889)}}
X(54539) = reflection of X(i) in X(j) for these {i,j}: {12156, 14537}
X(54539) = trilinear pole of line {14428, 523}

X(54540) = X(30)X(3406)∩X(76)X(7818)

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

X(54540) lies on these lines: {30, 3406}, {76, 7818}, {83, 7748}, {98, 48901}, {381, 3399}, {384, 43527}, {598, 11648}, {671, 7837}, {1916, 9766}, {2996, 7900}, {5025, 10159}, {5064, 37892}, {5999, 7607}, {7608, 13862}, {7914, 14046}, {7938, 16041}, {9302, 41135}, {9765, 10335}, {14033, 18841}, {14042, 53102}, {14062, 43676}, {14614, 43535}, {22728, 38732}, {32996, 43681}

X(54540) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(43098)}}, {{A, B, C, X(25), X(14041)}}, {{A, B, C, X(111), X(18546)}}, {{A, B, C, X(251), X(7818)}}, {{A, B, C, X(384), X(5064)}}, {{A, B, C, X(385), X(9766)}}, {{A, B, C, X(427), X(11361)}}, {{A, B, C, X(428), X(5025)}}, {{A, B, C, X(524), X(7837)}}, {{A, B, C, X(3108), X(30495)}}, {{A, B, C, X(5999), X(52282)}}, {{A, B, C, X(6995), X(16041)}}, {{A, B, C, X(7378), X(14033)}}, {{A, B, C, X(7408), X(33285)}}, {{A, B, C, X(7409), X(14039)}}, {{A, B, C, X(7714), X(14063)}}, {{A, B, C, X(7777), X(13468)}}, {{A, B, C, X(7840), X(9487)}}, {{A, B, C, X(7864), X(9484)}}, {{A, B, C, X(7896), X(34572)}}, {{A, B, C, X(7938), X(42037)}}, {{A, B, C, X(9229), X(43726)}}, {{A, B, C, X(11058), X(18823)}}, {{A, B, C, X(11648), X(42008)}}, {{A, B, C, X(13862), X(52281)}}, {{A, B, C, X(14036), X(52285)}}, {{A, B, C, X(15014), X(31133)}}, {{A, B, C, X(15351), X(18018)}}, {{A, B, C, X(17505), X(51454)}}, {{A, B, C, X(18023), X(45819)}}, {{A, B, C, X(44132), X(48901)}}, {{A, B, C, X(44176), X(53197)}}
X(54540) = X(i)-cross conjugate of X(j) for these {i, j}: {41624, 2}

X(54541) = X(30)X(34087)∩X(888)X(2394)

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

X(54541) lies on these lines: {30, 34087}, {888, 2394}, {14537, 45092}, {16080, 46522}

X(54541) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(888)}}, {{A, B, C, X(1976), X(53221)}}, {{A, B, C, X(30496), X(48911)}}

X(54542) = X(2)X(42538)∩X(20)X(43564)

Barycentrics    12*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+(-143*a^4+145*(b^2-c^2)^2-2*a^2*(b^2+c^2))*S : :
Barycentrics    1 / (6*(-a^2 + b^2 + c^2) + S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 + 12*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54542) lies on these lines: {2, 42538}, {20, 43564}, {30, 34089}, {381, 34091}, {485, 43257}, {1131, 42572}, {1132, 6432}, {2043, 43445}, {2044, 43444}, {3091, 43565}, {3146, 10195}, {3316, 3543}, {3317, 3839}, {3590, 17578}, {3591, 50689}, {3830, 43536}, {3832, 10194}, {7000, 53098}, {8972, 41959}, {14226, 42216}, {15683, 43558}, {19054, 43560}, {23249, 43563}, {42275, 43568}, {42284, 42539}, {42418, 43507}, {43378, 50692}, {43503, 43569}, {43888, 52667}

X(54542) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(371), X(46851)}}, {{A, B, C, X(588), X(14490)}}, {{A, B, C, X(1152), X(6432)}}, {{A, B, C, X(1585), X(50687)}}, {{A, B, C, X(3311), X(6449)}}, {{A, B, C, X(5417), X(46848)}}, {{A, B, C, X(6200), X(13603)}}

X(54543) = X(2)X(42537)∩X(20)X(43565)

Barycentrics    12*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+(143*a^4-145*(b^2-c^2)^2+2*a^2*(b^2+c^2))*S : :
Barycentrics    1 / (6*(-a^2 + b^2 + c^2) - S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (1 - 12*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54543) lies on these lines: {2, 42537}, {20, 43565}, {30, 34091}, {381, 34089}, {486, 43256}, {1131, 6431}, {1132, 42573}, {2043, 43444}, {2044, 43445}, {3091, 43564}, {3146, 10194}, {3316, 3839}, {3317, 3543}, {3590, 31454}, {3591, 17578}, {3832, 10195}, {3845, 43536}, {7374, 53098}, {13941, 41960}, {14241, 42215}, {15683, 43559}, {19053, 43561}, {23259, 43562}, {42276, 43569}, {42283, 42540}, {42417, 43508}, {43379, 50692}, {43504, 43568}, {43887, 52666}

X(54543) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(372), X(46851)}}, {{A, B, C, X(589), X(14490)}}, {{A, B, C, X(1151), X(6431)}}, {{A, B, C, X(1586), X(50687)}}, {{A, B, C, X(3312), X(6450)}}, {{A, B, C, X(5419), X(46848)}}, {{A, B, C, X(6396), X(13603)}}

X(54544) = X(30)X(34258)∩X(381)X(14534)

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

X(54544) lies on these lines: {30, 34258}, {226, 48825}, {381, 14534}, {429, 43530}, {2394, 8672}, {4185, 16080}, {10159, 37415}

X(54544) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(4185)}}, {{A, B, C, X(79), X(48825)}}, {{A, B, C, X(381), X(429)}}, {{A, B, C, X(428), X(37415)}}, {{A, B, C, X(961), X(10308)}}, {{A, B, C, X(1494), X(20029)}}, {{A, B, C, X(1880), X(16263)}}, {{A, B, C, X(51870), X(52154)}}

X(54545) = X(10)X(11645)∩X(30)X(34475)

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

X(54545) lies on these lines: {10, 11645}, {30, 34475}, {2394, 4785}, {3849, 4052}, {5466, 28470}, {9830, 34899}, {16080, 31912}, {28562, 43677}

X(54545) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(4785)}}, {{A, B, C, X(514), X(11645)}}, {{A, B, C, X(524), X(28470)}}, {{A, B, C, X(2789), X(9830)}}, {{A, B, C, X(3667), X(3849)}}, {{A, B, C, X(6002), X(28562)}}

X(54546) = X(10)X(2794)∩X(115)X(3429)

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

X(54546) lies on these lines: {10, 2794}, {30, 34899}, {115, 3429}, {514, 52459}, {542, 4052}, {1503, 11599}, {2394, 2789}, {2784, 43677}, {2786, 43673}, {2792, 43683}, {3667, 14223}, {9180, 28296}, {28470, 46040}, {38309, 39838}

X(54546) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(2789)}}, {{A, B, C, X(511), X(727)}}, {{A, B, C, X(514), X(2794)}}, {{A, B, C, X(542), X(3667)}}, {{A, B, C, X(543), X(28296)}}, {{A, B, C, X(1503), X(2786)}}, {{A, B, C, X(2782), X(28470)}}, {{A, B, C, X(2784), X(6002)}}, {{A, B, C, X(2792), X(6003)}}, {{A, B, C, X(9083), X(9141)}}, {{A, B, C, X(23698), X(28529)}}
X(54546) = reflection of X(i) in X(j) for these {i,j}: {3429, 115}
X(54546) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 11599}

X(54547) = X(2)X(1625)∩X(4)X(7668)

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

X(54547) lies on these lines: {2, 1625}, {4, 7668}, {30, 35098}, {76, 14570}, {96, 32734}, {98, 14157}, {112, 275}, {217, 53576}, {262, 5890}, {2052, 33885}, {7608, 37121}, {13582, 40853}, {13599, 15058}, {16080, 44893}, {32445, 43679}

X(54547) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(50678)}}, {{A, B, C, X(30), X(44893)}}, {{A, B, C, X(74), X(15412)}}, {{A, B, C, X(112), X(1625)}}, {{A, B, C, X(237), X(46511)}}, {{A, B, C, X(276), X(16835)}}, {{A, B, C, X(287), X(45138)}}, {{A, B, C, X(393), X(47383)}}, {{A, B, C, X(1141), X(23582)}}, {{A, B, C, X(1179), X(39454)}}, {{A, B, C, X(1972), X(6344)}}, {{A, B, C, X(2395), X(3455)}}, {{A, B, C, X(3431), X(40815)}}, {{A, B, C, X(4580), X(18876)}}, {{A, B, C, X(7714), X(37186)}}, {{A, B, C, X(14483), X(42300)}}, {{A, B, C, X(14618), X(43917)}}, {{A, B, C, X(14908), X(33885)}}, {{A, B, C, X(17703), X(36952)}}, {{A, B, C, X(27366), X(37125)}}, {{A, B, C, X(34897), X(53246)}}, {{A, B, C, X(37121), X(52281)}}, {{A, B, C, X(37943), X(40853)}}
X(54547) = trilinear pole of line {51, 34093}
X(54547) = X(i)-cross conjugate of X(j) for these {i, j}: {3331, 4}

X(54548) = X(30)X(35353)∩X(536)X(2394)

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

X(54548) lies on these lines: {30, 35353}, {517, 5466}, {536, 2394}, {2783, 9180}, {14223, 35103}

X(54548) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(536)}}, {{A, B, C, X(517), X(524)}}, {{A, B, C, X(542), X(35103)}}, {{A, B, C, X(543), X(2783)}}, {{A, B, C, X(712), X(11645)}}, {{A, B, C, X(2687), X(35155)}}

X(54549) = X(10)X(7262)∩X(30)X(3597)

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

X(54549) lies on these lines: {10, 7262}, {30, 3597}, {226, 17394}, {321, 17363}, {6539, 17350}, {7607, 37360}, {7608, 19544}, {10159, 41236}

X(54549) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(7), X(2985)}}, {{A, B, C, X(314), X(17394)}}, {{A, B, C, X(428), X(41236)}}, {{A, B, C, X(553), X(17350)}}, {{A, B, C, X(1509), X(2994)}}, {{A, B, C, X(7262), X(9277)}}, {{A, B, C, X(11114), X(44734)}}, {{A, B, C, X(17346), X(42028)}}, {{A, B, C, X(17392), X(19723)}}, {{A, B, C, X(19544), X(52281)}}, {{A, B, C, X(19827), X(42029)}}, {{A, B, C, X(37360), X(52282)}}, {{A, B, C, X(39696), X(43733)}}
X(54549) = trilinear pole of line {47820, 523}

X(54550) = X(30)X(37874)∩X(381)X(801)

Barycentrics    (2*a^8-(b^2-c^2)^3*(b^2+2*c^2)-a^6*(5*b^2+2*c^2)+a^4*(3*b^4+17*b^2*c^2)+a^2*(b^6-16*b^4*c^2+17*b^2*c^4-2*c^6))*(2*a^8+(b^2-c^2)^3*(2*b^2+c^2)-a^6*(2*b^2+5*c^2)+a^4*(17*b^2*c^2+3*c^4)+a^2*(-2*b^6+17*b^4*c^2-16*b^2*c^4+c^6)) : :

X(54550) lies on these lines: {30, 37874}, {235, 43530}, {381, 801}, {1593, 16080}, {3839, 43670}, {6812, 10194}, {6814, 10195}, {6823, 43527}, {10159, 11479}

X(54550) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(1593)}}, {{A, B, C, X(235), X(381)}}, {{A, B, C, X(265), X(40032)}}, {{A, B, C, X(428), X(11479)}}, {{A, B, C, X(1105), X(34288)}}, {{A, B, C, X(1494), X(14457)}}, {{A, B, C, X(3531), X(16263)}}, {{A, B, C, X(3839), X(6622)}}, {{A, B, C, X(5064), X(6823)}}, {{A, B, C, X(9307), X(45088)}}, {{A, B, C, X(15740), X(52188)}}, {{A, B, C, X(16657), X(41372)}}, {{A, B, C, X(17703), X(52154)}}, {{A, B, C, X(18361), X(52441)}}, {{A, B, C, X(18848), X(52518)}}, {{A, B, C, X(22336), X(46255)}}, {{A, B, C, X(35512), X(45857)}}

X(54551) = X(30)X(37892)∩X(275)X(14041)

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

X(54551) lies on these lines: {30, 37892}, {275, 14041}, {384, 16080}, {459, 14033}, {2052, 11361}, {3399, 34664}, {5025, 43530}, {14039, 38253}

X(54551) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(11361)}}, {{A, B, C, X(5), X(14041)}}, {{A, B, C, X(20), X(14033)}}, {{A, B, C, X(30), X(384)}}, {{A, B, C, X(265), X(9229)}}, {{A, B, C, X(376), X(14035)}}, {{A, B, C, X(381), X(5025)}}, {{A, B, C, X(382), X(14036)}}, {{A, B, C, X(546), X(14046)}}, {{A, B, C, X(547), X(14044)}}, {{A, B, C, X(549), X(14042)}}, {{A, B, C, X(1502), X(18434)}}, {{A, B, C, X(1657), X(14030)}}, {{A, B, C, X(2998), X(16263)}}, {{A, B, C, X(3091), X(16041)}}, {{A, B, C, X(3146), X(14039)}}, {{A, B, C, X(3524), X(14068)}}, {{A, B, C, X(3534), X(14034)}}, {{A, B, C, X(3543), X(14001)}}, {{A, B, C, X(3545), X(14063)}}, {{A, B, C, X(3830), X(7892)}}, {{A, B, C, X(3832), X(33285)}}, {{A, B, C, X(3839), X(14064)}}, {{A, B, C, X(3845), X(7901)}}, {{A, B, C, X(3858), X(33288)}}, {{A, B, C, X(3860), X(33286)}}, {{A, B, C, X(5054), X(14066)}}, {{A, B, C, X(5055), X(14062)}}, {{A, B, C, X(5066), X(14045)}}, {{A, B, C, X(5071), X(32996)}}, {{A, B, C, X(5999), X(8370)}}, {{A, B, C, X(6145), X(43098)}}, {{A, B, C, X(7833), X(35930)}}, {{A, B, C, X(7841), X(13862)}}, {{A, B, C, X(7924), X(44230)}}, {{A, B, C, X(9516), X(46255)}}, {{A, B, C, X(11001), X(14031)}}, {{A, B, C, X(11744), X(40416)}}, {{A, B, C, X(14032), X(15681)}}, {{A, B, C, X(14037), X(15682)}}, {{A, B, C, X(14038), X(15684)}}, {{A, B, C, X(14043), X(15687)}}, {{A, B, C, X(14047), X(14893)}}, {{A, B, C, X(14065), X(14269)}}, {{A, B, C, X(14067), X(38335)}}, {{A, B, C, X(14069), X(50687)}}, {{A, B, C, X(14498), X(41891)}}, {{A, B, C, X(15014), X(38323)}}, {{A, B, C, X(15980), X(33013)}}, {{A, B, C, X(23046), X(33284)}}, {{A, B, C, X(33283), X(41099)}}, {{A, B, C, X(33289), X(38071)}}, {{A, B, C, X(33290), X(41106)}}

X(54552) = X(30)X(38253)∩X(459)X(3543)

Barycentrics    (13*a^8+4*a^2*(b^2-c^2)^2*(5*b^2-c^2)-4*a^6*(7*b^2+c^2)-(b^2-c^2)^3*(11*b^2+13*c^2)+2*a^4*(3*b^4+14*b^2*c^2-9*c^4))*(13*a^8-4*a^2*(b^2-5*c^2)*(b^2-c^2)^2-4*a^6*(b^2+7*c^2)+(b^2-c^2)^3*(13*b^2+11*c^2)+a^4*(-18*b^4+28*b^2*c^2+6*c^4)) : :

X(54552) lies on these lines: {30, 38253}, {459, 3543}, {2052, 50687}, {3146, 16080}, {3832, 43530}

X(54552) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(50687)}}, {{A, B, C, X(20), X(1494)}}, {{A, B, C, X(30), X(3146)}}, {{A, B, C, X(64), X(34570)}}, {{A, B, C, X(253), X(43699)}}, {{A, B, C, X(265), X(16251)}}, {{A, B, C, X(376), X(17578)}}, {{A, B, C, X(381), X(3832)}}, {{A, B, C, X(382), X(15683)}}, {{A, B, C, X(1217), X(17505)}}, {{A, B, C, X(1294), X(52443)}}, {{A, B, C, X(3091), X(3839)}}, {{A, B, C, X(3426), X(41894)}}, {{A, B, C, X(3522), X(3830)}}, {{A, B, C, X(3545), X(50689)}}, {{A, B, C, X(3613), X(38439)}}, {{A, B, C, X(3845), X(5068)}}, {{A, B, C, X(3854), X(41099)}}, {{A, B, C, X(5059), X(15682)}}, {{A, B, C, X(5076), X(15705)}}, {{A, B, C, X(5896), X(38263)}}, {{A, B, C, X(7391), X(34621)}}, {{A, B, C, X(7408), X(34664)}}, {{A, B, C, X(8801), X(38445)}}, {{A, B, C, X(10152), X(36413)}}, {{A, B, C, X(10304), X(50688)}}, {{A, B, C, X(11001), X(50690)}}, {{A, B, C, X(14269), X(15022)}}, {{A, B, C, X(14490), X(41890)}}, {{A, B, C, X(15077), X(31361)}}, {{A, B, C, X(15640), X(49135)}}, {{A, B, C, X(15687), X(15717)}}, {{A, B, C, X(18296), X(18848)}}, {{A, B, C, X(18550), X(46412)}}, {{A, B, C, X(18846), X(21400)}}, {{A, B, C, X(22466), X(52187)}}, {{A, B, C, X(31942), X(36616)}}

X(54553) = X(10)X(37589)∩X(30)X(38309)

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

X(54553) lies on these lines: {10, 37589}, {30, 38309}, {321, 50118}, {543, 33996}, {597, 2051}, {2482, 34899}, {4049, 29126}, {5485, 37642}, {10302, 14829}, {11611, 50114}

X(54553) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(57), X(37589)}}, {{A, B, C, X(267), X(36603)}}, {{A, B, C, X(333), X(34914)}}, {{A, B, C, X(519), X(29126)}}, {{A, B, C, X(597), X(14829)}}, {{A, B, C, X(897), X(8056)}}, {{A, B, C, X(1992), X(37642)}}, {{A, B, C, X(2363), X(17107)}}, {{A, B, C, X(6703), X(31144)}}, {{A, B, C, X(17763), X(50114)}}, {{A, B, C, X(29574), X(50755)}}, {{A, B, C, X(29594), X(29631)}}, {{A, B, C, X(29600), X(33139)}}

X(54554) = X(2)X(476)∩X(4)X(38395)

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

X(54554) lies on these lines: {2, 476}, {4, 38395}, {30, 39295}, {76, 5641}, {94, 9140}, {98, 1989}, {262, 34370}, {265, 671}, {2394, 5627}, {2986, 51228}, {5466, 14582}, {5476, 7578}, {5485, 51835}, {6344, 46105}, {8781, 52094}, {9180, 14639}, {10412, 43665}, {10722, 41392}, {14223, 34368}, {14355, 14560}, {16080, 18384}, {16092, 51847}, {18316, 48453}, {24624, 36096}, {34365, 51345}, {35909, 43707}

X(54554) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(3258)}}, {{A, B, C, X(74), X(9140)}}, {{A, B, C, X(186), X(18867)}}, {{A, B, C, X(265), X(14582)}}, {{A, B, C, X(476), X(5627)}}, {{A, B, C, X(477), X(34312)}}, {{A, B, C, X(842), X(5641)}}, {{A, B, C, X(1138), X(14731)}}, {{A, B, C, X(1494), X(15328)}}, {{A, B, C, X(1550), X(51227)}}, {{A, B, C, X(1989), X(10412)}}, {{A, B, C, X(2501), X(47220)}}, {{A, B, C, X(6344), X(52449)}}, {{A, B, C, X(6530), X(37765)}}, {{A, B, C, X(8599), X(52192)}}, {{A, B, C, X(9161), X(35364)}}, {{A, B, C, X(9214), X(18808)}}, {{A, B, C, X(12065), X(22104)}}, {{A, B, C, X(14254), X(14583)}}, {{A, B, C, X(16092), X(16188)}}, {{A, B, C, X(34174), X(52094)}}, {{A, B, C, X(52492), X(53177)}}, {{A, B, C, X(52763), X(53158)}}
X(54554) = trilinear pole of line {1989, 14998}
X(54554) = X(i)-isoconjugate-of-X(j) for these {i, j}: {323, 2247}, {542, 6149}, {1101, 53132}, {2624, 14999}
X(54554) = X(i)-Dao conjugate of X(j) for these {i, j}: {523, 53132}, {14993, 542}, {15295, 5191}
X(54554) = X(i)-cross conjugate of X(j) for these {i, j}: {1550, 98}, {1640, 39290}, {43090, 6344}, {53132, 523}
X(54554) = barycentric product X(i)*X(j) for these (i, j): {290, 34370}, {842, 94}, {1577, 36096}, {1989, 5641}, {10412, 5649}, {14223, 476}, {14998, 35139}, {15475, 6035}, {23969, 850}, {35909, 46456}, {51228, 5627}
X(54554) = barycentric quotient X(i)/X(j) for these (i, j): {115, 53132}, {476, 14999}, {842, 323}, {1989, 542}, {5627, 51227}, {5641, 7799}, {5649, 10411}, {10412, 18312}, {11060, 5191}, {14223, 3268}, {14998, 526}, {15475, 1640}, {18384, 6103}, {23969, 110}, {34370, 511}, {35909, 8552}, {36096, 662}, {40355, 48451}, {46787, 51383}, {48453, 1511}, {50942, 45808}, {51228, 6148}

X(54555) = X(4)X(52949)∩X(21)X(16080)

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

X(54555) lies on these lines: {4, 52949}, {21, 16080}, {30, 40149}, {275, 17577}, {459, 11111}, {521, 2394}, {2052, 11114}, {2476, 43530}, {2798, 14223}, {38253, 50739}, {40395, 52269}

X(54555) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(11114)}}, {{A, B, C, X(5), X(17577)}}, {{A, B, C, X(20), X(11111)}}, {{A, B, C, X(21), X(30)}}, {{A, B, C, X(265), X(1441)}}, {{A, B, C, X(376), X(6872)}}, {{A, B, C, X(381), X(2476)}}, {{A, B, C, X(411), X(11113)}}, {{A, B, C, X(442), X(52269)}}, {{A, B, C, X(542), X(2798)}}, {{A, B, C, X(941), X(16263)}}, {{A, B, C, X(1005), X(37428)}}, {{A, B, C, X(1175), X(34570)}}, {{A, B, C, X(1494), X(18123)}}, {{A, B, C, X(3146), X(50739)}}, {{A, B, C, X(3543), X(6857)}}, {{A, B, C, X(3545), X(6871)}}, {{A, B, C, X(3560), X(17579)}}, {{A, B, C, X(3839), X(6856)}}, {{A, B, C, X(6175), X(6841)}}, {{A, B, C, X(6828), X(17532)}}, {{A, B, C, X(6842), X(37375)}}, {{A, B, C, X(6869), X(31156)}}, {{A, B, C, X(6870), X(50741)}}, {{A, B, C, X(6912), X(11112)}}, {{A, B, C, X(6932), X(17556)}}, {{A, B, C, X(7491), X(17549)}}, {{A, B, C, X(7548), X(17530)}}, {{A, B, C, X(8229), X(17677)}}, {{A, B, C, X(10883), X(17528)}}, {{A, B, C, X(13587), X(37290)}}, {{A, B, C, X(15670), X(52841)}}, {{A, B, C, X(15679), X(16617)}}, {{A, B, C, X(26737), X(34578)}}
X(54555) = trilinear pole of line {14395, 523}

X(54556) = X(4)X(51270)∩X(14)X(51254)

Barycentrics    sqrt(3)*(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)*(3*a^14-11*a^12*(b^2+c^2)-51*a^8*b^2*c^2*(b^2+c^2)+a^10*(12*b^4+41*b^2*c^2+12*c^4)-(b^2-c^2)^4*(2*b^6+5*b^4*c^2+5*b^2*c^4+2*c^6)+a^6*(-5*b^8+14*b^6*c^2+70*b^4*c^4+14*b^2*c^6-5*c^8)+a^2*(b^2-c^2)^2*(6*b^8-3*b^6*c^2-14*b^4*c^4-3*b^2*c^6+6*c^8)-a^4*(3*b^10-19*b^8*c^2+32*b^6*c^4+32*b^4*c^6-19*b^2*c^8+3*c^10))+(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)*(-6*a^12+2*b^2*c^2*(b^2-c^2)^4+24*a^10*(b^2+c^2)+4*a^2*b^2*c^2*(b^2-c^2)^2*(b^2+c^2)-2*a^8*(18*b^4+29*b^2*c^2+18*c^4)+4*a^6*(6*b^6+13*b^4*c^2+13*b^2*c^4+6*c^6)-2*a^4*(3*b^8+12*b^6*c^2+2*b^4*c^4+12*b^2*c^6+3*c^8))*S : :

X(54556) lies on these lines: {4, 51270}, {14, 51254}, {30, 40158}, {15441, 23097}, {16080, 40709}

X(54556) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(298)}}, {{A, B, C, X(265), X(300)}}, {{A, B, C, X(302), X(18550)}}, {{A, B, C, X(5627), X(53029)}}, {{A, B, C, X(8445), X(11131)}}, {{A, B, C, X(40709), X(51254)}}

X(54557) = X(4)X(51277)∩X(13)X(51254)

Barycentrics    sqrt(3)*(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)*(3*a^14-11*a^12*(b^2+c^2)-51*a^8*b^2*c^2*(b^2+c^2)+a^10*(12*b^4+41*b^2*c^2+12*c^4)-(b^2-c^2)^4*(2*b^6+5*b^4*c^2+5*b^2*c^4+2*c^6)+a^6*(-5*b^8+14*b^6*c^2+70*b^4*c^4+14*b^2*c^6-5*c^8)+a^2*(b^2-c^2)^2*(6*b^8-3*b^6*c^2-14*b^4*c^4-3*b^2*c^6+6*c^8)-a^4*(3*b^10-19*b^8*c^2+32*b^6*c^4+32*b^4*c^6-19*b^2*c^8+3*c^10))+(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)*(6*a^12-2*b^2*c^2*(b^2-c^2)^4-24*a^10*(b^2+c^2)-4*a^2*b^2*c^2*(b^2-c^2)^2*(b^2+c^2)+a^8*(36*b^4+58*b^2*c^2+36*c^4)-4*a^6*(6*b^6+13*b^4*c^2+13*b^2*c^4+6*c^6)+a^4*(6*b^8+24*b^6*c^2+4*b^4*c^4+24*b^2*c^6+6*c^8))*S : :

X(54557) lies on these lines: {4, 51277}, {13, 51254}, {30, 40159}, {15442, 23097}, {16080, 40710}

X(54557) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(299)}}, {{A, B, C, X(265), X(301)}}, {{A, B, C, X(303), X(18550)}}, {{A, B, C, X(5627), X(53030)}}, {{A, B, C, X(8455), X(11130)}}, {{A, B, C, X(40710), X(51254)}}

X(54558) = X(30)X(40178)∩X(3424)X(34621)

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

X(54558) lies on these lines: {30, 40178}, {3424, 34621}, {6504, 7841}, {7383, 7607}, {7400, 43537}, {13582, 32982}, {47586, 52404}

X(54558) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(141), X(45088)}}, {{A, B, C, X(524), X(42021)}}, {{A, B, C, X(1502), X(52487)}}, {{A, B, C, X(3089), X(33190)}}, {{A, B, C, X(3431), X(6339)}}, {{A, B, C, X(3541), X(8370)}}, {{A, B, C, X(3542), X(7841)}}, {{A, B, C, X(3546), X(37855)}}, {{A, B, C, X(4846), X(6664)}}, {{A, B, C, X(7383), X(52282)}}, {{A, B, C, X(9164), X(44157)}}, {{A, B, C, X(14528), X(34898)}}, {{A, B, C, X(32982), X(37943)}}, {{A, B, C, X(34165), X(34505)}}, {{A, B, C, X(34621), X(52283)}}

X(54559) = X(30)X(40395)∩X(275)X(11113)

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

X(54559) lies on these lines: {30, 40395}, {275, 11113}, {405, 43530}, {442, 16080}, {459, 50741}, {2052, 17532}, {5397, 5796}

X(54559) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(3), X(17532)}}, {{A, B, C, X(5), X(11113)}}, {{A, B, C, X(20), X(50741)}}, {{A, B, C, X(30), X(442)}}, {{A, B, C, X(72), X(1494)}}, {{A, B, C, X(265), X(40412)}}, {{A, B, C, X(376), X(5177)}}, {{A, B, C, X(381), X(405)}}, {{A, B, C, X(452), X(3545)}}, {{A, B, C, X(943), X(5627)}}, {{A, B, C, X(1006), X(17577)}}, {{A, B, C, X(1294), X(1441)}}, {{A, B, C, X(3651), X(6175)}}, {{A, B, C, X(3839), X(16845)}}, {{A, B, C, X(5714), X(7319)}}, {{A, B, C, X(6598), X(34303)}}, {{A, B, C, X(6829), X(11114)}}, {{A, B, C, X(6843), X(11111)}}, {{A, B, C, X(6907), X(11112)}}, {{A, B, C, X(6913), X(17556)}}, {{A, B, C, X(6920), X(37375)}}, {{A, B, C, X(6937), X(17579)}}, {{A, B, C, X(7413), X(17677)}}, {{A, B, C, X(7580), X(17528)}}, {{A, B, C, X(13442), X(16052)}}, {{A, B, C, X(17530), X(31789)}}, {{A, B, C, X(19542), X(37150)}}, {{A, B, C, X(21161), X(46870)}}, {{A, B, C, X(25985), X(34664)}}, {{A, B, C, X(30810), X(36722)}}, {{A, B, C, X(37411), X(44217)}}

X(54560) = X(30)X(4049)∩X(519)X(2394)

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

X(54560) lies on these lines: {30, 4049}, {516, 5466}, {519, 2394}, {2784, 9180}, {2796, 14223}, {16080, 46541}, {17132, 43673}, {28562, 43665}

X(54560) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(519)}}, {{A, B, C, X(511), X(28562)}}, {{A, B, C, X(516), X(524)}}, {{A, B, C, X(542), X(2796)}}, {{A, B, C, X(543), X(2784)}}, {{A, B, C, X(726), X(11645)}}, {{A, B, C, X(1503), X(17132)}}, {{A, B, C, X(2688), X(35153)}}, {{A, B, C, X(17766), X(19924)}}

X(54561) = X(2)X(22797)∩X(13)X(10722)

Barycentrics    sqrt(3)*(a^8+a^4*b^2*c^2+8*a^6*(b^2+c^2)-10*a^2*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^2*(b^4-11*b^2*c^2+c^4))+6*(-4*a^6+2*a^2*(b^2-c^2)^2-a^4*(b^2+c^2)+3*(b^2-c^2)^2*(b^2+c^2))*S : :

X(54561) lies on these lines: {2, 22797}, {13, 10722}, {17, 36962}, {18, 41023}, {30, 40706}, {76, 48656}, {262, 41039}, {462, 16080}, {542, 11121}, {5318, 14492}, {5479, 43547}, {5979, 8781}, {6033, 40707}, {6773, 22237}, {11603, 41022}, {43546, 51753}

X(54561) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(14372)}}, {{A, B, C, X(64), X(23717)}}, {{A, B, C, X(74), X(16460)}}, {{A, B, C, X(842), X(51447)}}, {{A, B, C, X(1494), X(11085)}}, {{A, B, C, X(2378), X(13603)}}, {{A, B, C, X(3426), X(3439)}}, {{A, B, C, X(3441), X(11060)}}, {{A, B, C, X(11080), X(32085)}}, {{A, B, C, X(14483), X(34322)}}
X(54561) = isogonal conjugate of X(36756)

X(54562) = X(2)X(22796)∩X(14)X(10722)

Barycentrics    sqrt(3)*(a^8+a^4*b^2*c^2+8*a^6*(b^2+c^2)-10*a^2*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^2*(b^4-11*b^2*c^2+c^4))+6*(4*a^6-2*a^2*(b^2-c^2)^2+a^4*(b^2+c^2)-3*(b^2-c^2)^2*(b^2+c^2))*S : :

X(54562) lies on these lines: {2, 22796}, {14, 10722}, {17, 41022}, {18, 36961}, {30, 40707}, {76, 48655}, {262, 41038}, {463, 16080}, {542, 11122}, {5321, 14492}, {5478, 43546}, {5978, 8781}, {6033, 40706}, {6770, 22235}, {11602, 41023}, {43547, 51754}

X(54562) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(14373)}}, {{A, B, C, X(64), X(23716)}}, {{A, B, C, X(74), X(16459)}}, {{A, B, C, X(842), X(51446)}}, {{A, B, C, X(1494), X(11080)}}, {{A, B, C, X(2379), X(13603)}}, {{A, B, C, X(3426), X(3438)}}, {{A, B, C, X(3440), X(11060)}}, {{A, B, C, X(11085), X(32085)}}, {{A, B, C, X(14483), X(34321)}}
X(54562) = isogonal conjugate of X(36755)

X(54563) = X(10)X(24705)∩X(30)X(40718)

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

X(54563) lies on these lines: {10, 24705}, {30, 40718}, {226, 50178}, {536, 43677}, {824, 2394}, {5466, 28468}, {6002, 35353}, {16080, 31909}

X(54563) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(824)}}, {{A, B, C, X(79), X(28660)}}, {{A, B, C, X(514), X(20028)}}, {{A, B, C, X(517), X(28840)}}, {{A, B, C, X(524), X(28468)}}, {{A, B, C, X(536), X(6002)}}, {{A, B, C, X(1258), X(16615)}}, {{A, B, C, X(1400), X(47947)}}, {{A, B, C, X(3227), X(28630)}}, {{A, B, C, X(4102), X(43073)}}, {{A, B, C, X(10435), X(36871)}}, {{A, B, C, X(24479), X(35161)}}, {{A, B, C, X(44129), X(48899)}}

X(54564) = X(30)X(4080)∩X(900)X(2394)

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

X(54564) lies on these lines: {30, 4080}, {900, 2394}, {4049, 11125}, {5466, 28294}, {16080, 37168}

X(54564) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(900)}}, {{A, B, C, X(74), X(34079)}}, {{A, B, C, X(524), X(28294)}}, {{A, B, C, X(903), X(45136)}}, {{A, B, C, X(28840), X(28877)}}

X(54565) = X(20)X(43529)∩X(76)X(11180)

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

X(54565) lies on these lines: {20, 43529}, {30, 40824}, {76, 11180}, {1916, 3543}, {3091, 43528}, {3407, 3839}, {5503, 10722}, {6620, 16080}, {37334, 53098}

X(54565) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(6620)}}, {{A, B, C, X(393), X(5641)}}, {{A, B, C, X(419), X(3543)}}, {{A, B, C, X(1976), X(3426)}}, {{A, B, C, X(3839), X(5117)}}, {{A, B, C, X(6531), X(36889)}}, {{A, B, C, X(10630), X(11738)}}, {{A, B, C, X(40708), X(43699)}}

X(54566) = X(30)X(42006)∩X(83)X(11645)

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

X(54566) lies on these lines: {30, 42006}, {76, 19924}, {83, 11645}, {671, 48895}, {7470, 10159}, {10168, 43527}, {11170, 36990}, {14492, 39593}, {43528, 44230}

X(54566) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(428), X(7470)}}, {{A, B, C, X(512), X(13603)}}, {{A, B, C, X(826), X(11645)}}, {{A, B, C, X(3849), X(32473)}}, {{A, B, C, X(5641), X(15321)}}, {{A, B, C, X(11738), X(44557)}}, {{A, B, C, X(14487), X(43950)}}, {{A, B, C, X(28487), X(28562)}}, {{A, B, C, X(44146), X(48895)}}

X(54567) = X(2)X(22505)∩X(30)X(42010)

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

X(54567) lies on these lines: {2, 22505}, {30, 42010}, {115, 53100}, {262, 39838}, {2794, 7607}, {5503, 32479}, {7608, 10722}, {9862, 43537}, {11170, 53017}, {14651, 47586}, {35705, 40824}, {38743, 43529}

X(54567) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(2793), X(32479)}}, {{A, B, C, X(3455), X(16835)}}, {{A, B, C, X(8753), X(39644)}}, {{A, B, C, X(9154), X(38741)}}, {{A, B, C, X(10630), X(29011)}}, {{A, B, C, X(11744), X(53605)}}
X(54567) = reflection of X(i) in X(j) for these {i,j}: {53100, 115}

X(54568) = X(30)X(42011)∩X(381)X(10153)

Barycentrics    (6*a^6+6*b^6-13*b^4*c^2+12*b^2*c^4-5*c^6-a^4*(2*b^2+13*c^2)-2*a^2*(b^4+5*b^2*c^2-6*c^4))*(6*a^6-5*b^6+12*b^4*c^2-13*b^2*c^4+6*c^6-a^4*(13*b^2+2*c^2)+2*a^2*(6*b^4-5*b^2*c^2-c^4)) : :

X(54568) lies on these lines: {30, 42011}, {381, 10153}, {671, 53017}, {1503, 17503}, {2794, 8587}, {5475, 53099}, {47586, 53016}

X(54568) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(64), X(34154)}}, {{A, B, C, X(3425), X(10630)}}, {{A, B, C, X(9515), X(52518)}}
X(54568) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 17503}

X(54569) = X(2)X(9749)∩X(13)X(1503)

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

X(54569) lies on these lines: {2, 9749}, {3, 36761}, {10, 49539}, {13, 1503}, {14, 2794}, {17, 5868}, {18, 41034}, {30, 42035}, {76, 41062}, {83, 41064}, {98, 41044}, {115, 36990}, {262, 41052}, {485, 41050}, {486, 41048}, {530, 5485}, {531, 5503}, {542, 42036}, {671, 41023}, {1327, 49972}, {1328, 49974}, {2394, 27551}, {2782, 36784}, {3391, 13749}, {3392, 13748}, {3543, 22574}, {5334, 14484}, {5335, 47853}, {5478, 43540}, {5869, 43546}, {6108, 53015}, {6114, 22664}, {6115, 7710}, {6770, 43542}, {6776, 9112}, {6777, 43539}, {9750, 36776}, {11121, 44667}, {11602, 19106}, {12816, 41028}, {14223, 27550}, {14539, 40824}, {16964, 22693}, {22796, 38317}, {23006, 41021}, {23698, 41458}, {23870, 43673}, {23871, 52459}, {31710, 46034}, {33602, 41030}, {33604, 41032}, {33607, 41026}, {36772, 41035}, {39838, 42093}, {40707, 44666}

X(54569) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(27551)}}, {{A, B, C, X(64), X(3438)}}, {{A, B, C, X(66), X(8741)}}, {{A, B, C, X(67), X(11080)}}, {{A, B, C, X(74), X(16257)}}, {{A, B, C, X(523), X(8737)}}, {{A, B, C, X(530), X(1499)}}, {{A, B, C, X(531), X(2793)}}, {{A, B, C, X(542), X(27550)}}, {{A, B, C, X(1177), X(8739)}}, {{A, B, C, X(1297), X(51446)}}, {{A, B, C, X(1485), X(10642)}}, {{A, B, C, X(1503), X(23870)}}, {{A, B, C, X(2379), X(3426)}}, {{A, B, C, X(2794), X(23871)}}, {{A, B, C, X(2980), X(8742)}}, {{A, B, C, X(3440), X(34130)}}, {{A, B, C, X(3527), X(34321)}}, {{A, B, C, X(11087), X(18575)}}, {{A, B, C, X(32085), X(41897)}}
X(54569) = reflection of X(i) in X(j) for these {i,j}: {36761, 3}, {36776, 9750}, {36961, 41038}
X(54569) = isogonal conjugate of X(14538)
X(54569) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 13}, {23717, 43532}
X(54569) = X(i)-cross conjugate of X(j) for these {i, j}: {22513, 13}
X(54569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2794, 41038, 36961}

X(54570) = X(2)X(9750)∩X(14)X(1503)

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

X(54570) lies on these lines: {2, 9750}, {3, 41458}, {10, 49541}, {13, 2794}, {14, 1503}, {17, 41035}, {18, 5869}, {30, 42036}, {76, 41063}, {83, 41065}, {98, 41045}, {115, 36990}, {262, 41053}, {485, 41049}, {486, 41051}, {530, 5503}, {531, 5485}, {542, 42035}, {671, 41022}, {1327, 49973}, {1328, 49975}, {2394, 27550}, {3366, 13749}, {3367, 13748}, {3543, 22573}, {5334, 47854}, {5335, 14484}, {5479, 43541}, {5868, 43547}, {6109, 53015}, {6114, 7710}, {6115, 22664}, {6773, 43543}, {6776, 9113}, {6778, 43538}, {11122, 44666}, {11603, 19107}, {12817, 41029}, {14223, 27551}, {14538, 40824}, {16965, 22694}, {22797, 38317}, {23013, 41020}, {23698, 36761}, {23870, 52459}, {23871, 43673}, {31709, 46034}, {33603, 41031}, {33605, 41033}, {33606, 41027}, {39838, 42094}, {40706, 44667}

X(54570) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(27550)}}, {{A, B, C, X(64), X(3439)}}, {{A, B, C, X(66), X(8742)}}, {{A, B, C, X(67), X(11085)}}, {{A, B, C, X(74), X(16258)}}, {{A, B, C, X(523), X(8738)}}, {{A, B, C, X(530), X(2793)}}, {{A, B, C, X(531), X(1499)}}, {{A, B, C, X(542), X(27551)}}, {{A, B, C, X(1177), X(8740)}}, {{A, B, C, X(1297), X(51447)}}, {{A, B, C, X(1485), X(10641)}}, {{A, B, C, X(1503), X(23871)}}, {{A, B, C, X(2378), X(3426)}}, {{A, B, C, X(2794), X(23870)}}, {{A, B, C, X(2980), X(8741)}}, {{A, B, C, X(3441), X(34130)}}, {{A, B, C, X(3527), X(34322)}}, {{A, B, C, X(11082), X(18575)}}, {{A, B, C, X(32085), X(41898)}}
X(54570) = reflection of X(i) in X(j) for these {i,j}: {36962, 41039}, {41458, 3}
X(54570) = isogonal conjugate of X(14539)
X(54570) = X(i)-vertex conjugate of X(j) for these {i, j}: {3, 14}, {23716, 43532}
X(54570) = X(i)-cross conjugate of X(j) for these {i, j}: {22512, 14}
X(54570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2794, 41039, 36962}

X(54571) = X(2)X(9736)∩X(4)X(5472)

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

X(54571) lies on these lines: {2, 9736}, {4, 5472}, {13, 44666}, {14, 5478}, {17, 23005}, {18, 22832}, {30, 42062}, {76, 16629}, {98, 5318}, {532, 42035}, {2996, 22113}, {5965, 11121}, {5982, 8781}, {6115, 16652}, {6770, 43540}, {6772, 10611}, {6776, 31683}, {11602, 22900}, {12816, 41022}, {12820, 36961}, {14458, 41039}, {14853, 43541}, {16001, 16626}, {20377, 22890}, {22235, 22532}, {22237, 41056}, {22892, 43554}, {33388, 36969}, {36782, 43544}, {41020, 43550}, {43546, 52838}, {48666, 53105}

X(54571) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(23716)}}, {{A, B, C, X(54), X(34321)}}, {{A, B, C, X(523), X(11139)}}, {{A, B, C, X(532), X(27551)}}, {{A, B, C, X(2379), X(14483)}}, {{A, B, C, X(3438), X(3527)}}, {{A, B, C, X(5966), X(51446)}}, {{A, B, C, X(11080), X(17983)}}, {{A, B, C, X(41897), X(45857)}}
X(54571) = isogonal conjugate of X(13350)

X(54572) = X(2)X(9735)∩X(4)X(5471)

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

X(54572) lies on these lines: {2, 9735}, {4, 5471}, {13, 5479}, {14, 44667}, {17, 22831}, {18, 23004}, {30, 42063}, {76, 16628}, {98, 5321}, {533, 42036}, {2996, 22114}, {5965, 11122}, {5983, 8781}, {6114, 16653}, {6773, 43541}, {6775, 10612}, {6776, 31684}, {11603, 22856}, {12817, 41023}, {12821, 36962}, {14458, 41038}, {14853, 43540}, {16002, 16627}, {20378, 22843}, {22235, 41057}, {22237, 22531}, {22848, 43555}, {33389, 36970}, {41021, 43551}, {43547, 52839}, {48665, 53105}

X(54572) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(6), X(23717)}}, {{A, B, C, X(54), X(34322)}}, {{A, B, C, X(523), X(11138)}}, {{A, B, C, X(533), X(27550)}}, {{A, B, C, X(2378), X(14483)}}, {{A, B, C, X(3439), X(3527)}}, {{A, B, C, X(5966), X(51447)}}, {{A, B, C, X(11085), X(17983)}}, {{A, B, C, X(41898), X(45857)}}
X(54572) = isogonal conjugate of X(13349)

X(54573) = X(30)X(42410)∩X(6240)X(16080)

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

X(54573) lies on these lines: {30, 42410}, {6240, 16080}, {7547, 43530}

X(54573) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(30), X(6145)}}, {{A, B, C, X(250), X(16835)}}, {{A, B, C, X(381), X(7547)}}, {{A, B, C, X(847), X(38436)}}, {{A, B, C, X(1138), X(15319)}}, {{A, B, C, X(1179), X(21400)}}, {{A, B, C, X(1300), X(13481)}}, {{A, B, C, X(1494), X(16000)}}, {{A, B, C, X(1989), X(38447)}}, {{A, B, C, X(3830), X(32534)}}, {{A, B, C, X(5627), X(8884)}}, {{A, B, C, X(7576), X(12605)}}, {{A, B, C, X(15392), X(15619)}}, {{A, B, C, X(16263), X(43949)}}, {{A, B, C, X(18550), X(45138)}}, {{A, B, C, X(32533), X(34288)}}, {{A, B, C, X(43660), X(46255)}}

X(54574) = X(13)X(43368)∩X(17)X(14893)

Barycentrics    -63*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+2*sqrt(3)*(-661*a^4+662*(b^2-c^2)^2-a^2*(b^2+c^2))*S : :
Barycentrics    1 / (21*Sqrt[3]*(-a^2 + b^2 + c^2) - 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] - 63*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54574) lies on these lines: {13, 43368}, {14, 42683}, {17, 14893}, {18, 38335}, {30, 43440}, {381, 43441}, {3627, 10187}, {3830, 43549}, {3843, 10188}, {3845, 43548}, {12821, 43475}, {15684, 43442}, {19107, 43554}, {22235, 43492}, {22237, 42436}, {23046, 43443}, {33602, 43476}, {33607, 42117}, {42088, 43545}, {42499, 42543}, {43226, 43542}, {43228, 43472}, {43556, 49827}

X(54574) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(472), X(38335)}}, {{A, B, C, X(473), X(14893)}}

X(54575) = X(13)X(42682)∩X(18)X(14893)

Barycentrics    63*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+2*sqrt(3)*(-661*a^4+662*(b^2-c^2)^2-a^2*(b^2+c^2))*S : :
Barycentrics    1 / (21*Sqrt[3]*(-a^2 + b^2 + c^2) + 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] + 63*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54575) lies on these lines: {13, 42682}, {14, 43369}, {17, 38335}, {18, 14893}, {30, 43441}, {381, 43440}, {3627, 10188}, {3830, 43548}, {3843, 10187}, {3845, 43549}, {12820, 43476}, {15684, 43443}, {19106, 43555}, {22235, 42435}, {22237, 43491}, {23046, 43442}, {33603, 43475}, {33606, 42118}, {42087, 43544}, {42498, 42544}, {43227, 43543}, {43229, 43471}, {43557, 49826}

X(54575) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(472), X(14893)}}, {{A, B, C, X(473), X(38335)}}

X(54576) = X(2)X(43367)∩X(4)X(42612)

Barycentrics    -45*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+2*sqrt(3)*(-337*a^4+338*(b^2-c^2)^2-a^2*(b^2+c^2))*S : :
Barycentrics    1 / (15*Sqrt[3]*(-a^2 + b^2 + c^2) - 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] - 45*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54576) lies on these lines: {2, 43367}, {4, 42612}, {13, 42781}, {17, 14269}, {18, 15687}, {30, 43442}, {381, 43443}, {382, 10187}, {546, 10188}, {3830, 43545}, {3845, 43544}, {3861, 42952}, {5351, 43446}, {11480, 43548}, {12101, 33606}, {12816, 43030}, {15681, 43440}, {16242, 43398}, {22235, 49876}, {22237, 41100}, {33604, 36970}, {38071, 43441}, {41107, 43553}, {41108, 43550}, {42093, 42480}, {42105, 43543}, {42589, 42976}, {42947, 43447}, {43229, 43547}, {43418, 43552}

X(54576) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(472), X(15687)}}, {{A, B, C, X(473), X(14269)}}

X(54577) = X(2)X(43366)∩X(4)X(42613)

Barycentrics    -45*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+2*sqrt(3)*(337*a^4-338*(b^2-c^2)^2+a^2*(b^2+c^2))*S : :
Barycentrics    1 / (15*Sqrt[3]*(-a^2 + b^2 + c^2) + 2*S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] + 45*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54577) lies on these lines: {2, 43366}, {4, 42613}, {14, 42782}, {17, 15687}, {18, 14269}, {30, 43443}, {381, 43442}, {382, 10188}, {546, 10187}, {3830, 43544}, {3845, 43545}, {3861, 42953}, {5352, 43447}, {11481, 43549}, {12101, 33607}, {12817, 43031}, {15681, 43441}, {16241, 43397}, {22235, 41101}, {22237, 49875}, {33605, 36969}, {38071, 43440}, {41107, 43551}, {41108, 43552}, {42094, 42481}, {42104, 43542}, {42588, 42977}, {42946, 43446}, {43228, 43546}, {43419, 43553}

X(54577) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(472), X(14269)}}, {{A, B, C, X(473), X(15687)}}

X(54578) = X(4)X(43253)∩X(18)X(50687)

Barycentrics    -36*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+sqrt(3)*(-431*a^4+433*(b^2-c^2)^2-2*a^2*(b^2+c^2))*S : :
Barycentrics    1 / (6*Sqrt[3]*(-a^2 + b^2 + c^2) - S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] - 36*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54578) lies on these lines: {4, 43253}, {18, 50687}, {30, 43444}, {381, 43445}, {3146, 10187}, {3543, 43446}, {3830, 43555}, {3832, 10188}, {3839, 43447}, {3845, 43554}, {12101, 33605}, {15640, 43102}, {15683, 43442}, {17578, 42514}, {33607, 43476}, {42103, 42631}, {42126, 43542}, {42682, 49813}, {42694, 49873}, {43206, 49824}, {43228, 43556}, {43473, 43541}, {43544, 46335}

X(54578) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(472), X(50687)}}

X(54579) = X(4)X(43252)∩X(17)X(50687)

Barycentrics    36*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+sqrt(3)*(-431*a^4+433*(b^2-c^2)^2-2*a^2*(b^2+c^2))*S : :
Barycentrics    1 / (6*Sqrt[3]*(-a^2 + b^2 + c^2) + S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] + 36*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54579) lies on these lines: {4, 43252}, {17, 50687}, {30, 43445}, {381, 43444}, {3146, 10188}, {3543, 43447}, {3830, 43554}, {3832, 10187}, {3839, 43446}, {3845, 43555}, {12101, 33604}, {15640, 43103}, {15683, 43443}, {17578, 42515}, {33606, 43475}, {42106, 42632}, {42127, 43543}, {42683, 49812}, {42695, 49874}, {43205, 49825}, {43229, 43557}, {43474, 43540}, {43545, 46334}

X(54579) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(473), X(50687)}}

X(54580) = X(17)X(3839)∩X(18)X(3543)

Barycentrics    18*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b^2+c^2))+sqrt(3)*(107*a^4-109*(b^2-c^2)^2+2*a^2*(b^2+c^2))*S : :
Barycentrics    1 / (3*Sqrt[3]*(-a^2 + b^2 + c^2) - S) : :    (Peter Moses, July 21, 2023)
Barycentrics    1 / (Sqrt[3] - 18*Cot[A]) : :    (Peter Moses, July 21, 2023)

X(54580) lies on these lines: {2, 42096}, {13, 43364}, {14, 43397}, {17, 3839}, {18, 3543}, {20, 10187}, {30, 43446}, {376, 43444}, {381, 43447}, {1080, 53098}, {2043, 43565}, {2044, 43564}, {3091, 10188}, {3146, 42793}, {3545, 43445}, {3830, 43543}, {3845, 43542}, {5321, 43552}, {5334, 12820}, {10304, 43442}, {12101, 33603}, {12816, 42133}, {12821, 41113}, {15640, 43545}, {15682, 42121}, {15692, 43440}, {17578, 42519}, {22235, 49947}, {22237, 42148}, {33604, 42128}, {33606, 43242}, {33607, 42901}, {33699, 42963}, {35750, 42035}, {36327, 42036}, {41099, 42136}, {41106, 42122}, {41108, 43546}, {41112, 43550}, {41120, 50688}, {42093, 43540}, {42109, 49906}, {42161, 43547}, {42509, 43466}, {42589, 43305}, {42591, 46333}, {42635, 49876}, {42899, 43557}, {42905, 49825}, {42919, 43544}, {42956, 43870}, {43229, 43541}, {43551, 49826}

X(54580) = intersection, other than A, B, C, of these circumonics: {{A, B, C, X(2), X(4)}}, {{A, B, C, X(472), X(3543)}}, {{A, B, C, X(473), X(3839)}}, {{A, B, C, X(31361), X(41901)}}, {{A, B, C, X(40711), X(43699)}}
X(54580) = X(i)-cross conjugate of X(j) for these {i, j}: {43477, 43540}

X(54581) = X(17)X(3543)∩X(18)X(3839)

Barycentrics    -18*a^2*(a^4+(b^2-c^2)^2-2*a^2*(b