leftri rightri

This is PART 27: Centers X(52001) - X(54000)

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

X(52001) = X(1)X(1362)∩X(33)X(42071)

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

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

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

X(52002) = X(1)X(30)∩X(5)X(41496)

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

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

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

X(52003) = X(4)X(974)∩X(6)X(10574)

Barycentrics    a^2*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 4*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 5*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + b^6*c^2 - 4*a^2*b^2*c^4 + 2*a^2*c^6 + b^2*c^6 - c^8) : :
X(52003) = 5 X[10574] - X[11413], X[24] + 3 X[5890], X[5889] + 3 X[15078], 5 X[3567] - X[35490], 3 X[9730] - X[11585], 2 X[13382] + X[21841], 9 X[15045] - 5 X[31282], X[18404] - 5 X[37481]

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

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

X(52004) = X(1)X(926)∩X(101)X(7193)

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

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

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

X(52005) = X(1)X(5)∩X(3)X(513)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*(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 + 3*a^4*b*c - 4*a^2*b^3*c + a*b^4*c + b^5*c - 2*a^4*c^2 + 4*a^2*b^2*c^2 + 2*a^3*c^3 - 4*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(52005) = 5 X[631] - 3 X[47045]

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

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

X(52006) = X(1)X(7095)∩X(2)X(14265)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + a^8*c^2 - 2*a^6*b^2*c^2 + 2*a^4*b^4*c^2 + 2*a^2*b^6*c^2 + b^8*c^2 - 3*a^6*c^4 + 2*a^4*b^2*c^4 - 6*a^2*b^4*c^4 - b^6*c^4 + 3*a^4*c^6 + 2*a^2*b^2*c^6 - b^4*c^6 - a^2*c^8 + b^2*c^8) : :
X(52006) = 5 X[631] - 3 X[47044]

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

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

X(52007) = X(1)X(84)∩X(8)X(44189)

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)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3)*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 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(52007) lies on the cubic K040 and these lines: {1, 84}, {8, 44189}, {40, 1256}, {55, 41081}, {65, 41084}, {105, 6081}, {189, 497}, {271, 960}, {280, 3057}, {518, 13138}, {521, 3239}, {1155, 37141}, {1210, 5908}, {2739, 8059}, {3586, 5923}, {5205, 44327}, {14986, 37744}, {18239, 18283}

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

X(52008) = X(4)X(10117)∩X(6)X(32354)

Barycentrics    (2*a^6 - 3*a^4*b^2 + b^6 - 3*a^4*c^2 - b^4*c^2 - b^2*c^4 + c^6)*(a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + a^6*b^2*c^2 - 4*a^4*b^4*c^2 + a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 4*a^4*b^2*c^4 - 4*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 + a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10) : :
X(52008) = X[54] + 3 X[7576], 3 X[428] - X[32340], X[20424] + 3 X[38322], 3 X[13490] - X[22804], X[15800] + 3 X[38321]

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

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

X(52009) = X(39)X(9469)∩X(98)X(9468)

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

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

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

X(52010) = X(1)X(9405)∩X(5)X(113)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 + a^10*c^2 - a^6*b^4*c^2 - 3*a^4*b^6*c^2 + 4*a^2*b^8*c^2 - b^10*c^2 - 4*a^8*c^4 - a^6*b^2*c^4 + 10*a^4*b^4*c^4 - 5*a^2*b^6*c^4 + 4*b^8*c^4 + 6*a^6*c^6 - 3*a^4*b^2*c^6 - 5*a^2*b^4*c^6 - 6*b^6*c^6 - 4*a^4*c^8 + 4*a^2*b^2*c^8 + 4*b^4*c^8 + a^2*c^10 - b^2*c^10) : :
X(52010) = 3 X[549] - 2 X[47055], 5 X[631] - 3 X[33927]

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

X(52011) = X(4)X(1562)∩X(112)X(376)

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

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

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

X(52012) = X(2)X(3)∩X(46)X(2194)

Barycentrics    a*(a + b)*(a + c)*(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) : :

See Ivan Pavlov, euclid 5507.

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

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

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

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

See Ivan Pavlov, euclid 5509.

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

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

X(52014) = X(3)X(51)∩X(4)X(1609)

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

See Angel Montesdeoca, euclid 5508.

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

X(52015) = X(1)X(41)∩X(2)X(40910)

Barycentrics    a*(a^4 - a^3*b + a^2*b^2 - a*b^3 - a^3*c + 2*a*b^2*c - b^3*c + a^2*c^2 + 2*a*b*c^2 + 2*b^2*c^2 - a*c^3 - b*c^3) : :
X(52015) = 5 X[3616] - X[17170]

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

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

X(52016) = X(3)X(34817)∩X(6)X(1196)

Barycentrics    a^2*(-a^2 + b^2 + c^2)*(a^4 + a^2*b^2 + a^2*c^2 - 2*b^2*c^2) : :
X(52016) = X[6] - 3 X[3167], 3 X[6] - X[6391], 9 X[3167] - X[6391], 3 X[3167] + X[19588], X[6391] + 3 X[19588], 3 X[154] - X[37491], X[576] + 2 X[9925], X[576] - 4 X[41597], X[9925] + 2 X[41597], 2 X[5092] - 3 X[47391], 3 X[5654] - 2 X[19130], 3 X[23042] - 2 X[44470], 2 X[8548] - 3 X[39561], 4 X[9820] - 3 X[38317], X[9924] + 3 X[37672], X[9936] + 2 X[40107], 3 X[10516] - X[12429], 4 X[12038] - 3 X[17508]

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

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

X(52017) = X(10)X(56)∩X(106)X(614)

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

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

X(52018) = X(10)X(55)∩X(58)X(614)

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

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

X(52019) = X(3)X(4549)∩X(22)X(74)

Barycentrics    a^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 - 2*a^10*c^2 - 6*a^8*b^2*c^2 + 2*a^6*b^4*c^2 + 18*a^4*b^6*c^2 - 8*a^2*b^8*c^2 - 4*b^10*c^2 - a^8*c^4 + 2*a^6*b^2*c^4 - 26*a^4*b^4*c^4 + 10*a^2*b^6*c^4 + 7*b^8*c^4 + 4*a^6*c^6 + 18*a^4*b^2*c^6 + 10*a^2*b^4*c^6 - 8*b^6*c^6 - a^4*c^8 - 8*a^2*b^2*c^8 + 7*b^4*c^8 - 2*a^2*c^10 - 4*b^2*c^10 + c^12) : :
X(52019) = 3 X[15041] - X[19376]

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

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

X(52020) = X(1)X(3688)∩X(2)X(35892)

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

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

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

X(52021) = X(2)X(13)∩X(3)X(51012)

Barycentrics    (2*a^2 - b^2 - c^2)*(5*a^2 - b^2 - c^2 + 2*Sqrt[3]*S) : :
X(52021) = 2 X[6669] - 3 X[48314], 4 X[6672] - X[33517], X[299] - 3 X[41134], X[8594] - 3 X[13586], X[6781] + 2 X[44385], X[9114] + 3 X[16530], 3 X[16530] - X[22496], X[9116] + 3 X[16963], 3 X[9167] - 2 X[44382], 5 X[16961] - 3 X[22572], 5 X[16961] - X[22578], 3 X[22572] - X[22578], 3 X[22490] - X[23005], 3 X[26613] - X[37786]

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

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

X(52022) = X(2)X(14)∩X(3)X(51015)

Barycentrics    (2*a^2 - b^2 - c^2)*(5*a^2 - b^2 - c^2 - 2*Sqrt[3]*S) : :
X(52022) = 2 X[6670] - 3 X[48313], 4 X[6671] - X[33518], X[298] - 3 X[41134], X[8595] - 3 X[13586], X[6781] + 2 X[44384], X[9114] + 3 X[16962], X[9116] + 3 X[16529], 3 X[16529] - X[22495], 3 X[9167] - 2 X[44383], 5 X[16960] - 3 X[22571], 5 X[16960] - X[22577], 3 X[22571] - X[22577], 3 X[22489] - X[23004], 3 X[26613] - X[37785]

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

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

X(52023) = X(6)X(7)∩X(11)X(21346)

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

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

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

X(52024) = X(1)X(9551)∩X(7)X(8049)

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

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

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

X(52025) = X(1)X(224)∩X(2)X(272)

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

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

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

X(52026) = X(1)X(227)∩X(2)X(515)

Barycentrics    a*(3*a^6 - 4*a^5*b - 5*a^4*b^2 + 8*a^3*b^3 + a^2*b^4 - 4*a*b^5 + b^6 - 4*a^5*c + 2*a^4*b*c - 4*a^2*b^3*c + 4*a*b^4*c + 2*b^5*c - 5*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 8*a^3*c^3 - 4*a^2*b*c^3 - 4*b^3*c^3 + a^2*c^4 + 4*a*b*c^4 - b^2*c^4 - 4*a*c^5 + 2*b*c^5 + c^6) : :
X(52026) = X[1] + 2 X[11500], X[1] - 4 X[37837], X[11500] + 2 X[37837], 4 X[3] - X[84], 2 X[3] + X[1490], 7 X[3] - X[12684], 5 X[3] - 2 X[34862], X[3] - 4 X[40262], X[84] + 2 X[1490], 7 X[84] - 4 X[12684], 5 X[84] - 8 X[34862], X[84] - 16 X[40262], 7 X[1490] + 2 X[12684], 5 X[1490] + 4 X[34862], X[1490] + 8 X[40262], 4 X[5044] - X[12664], 2 X[5777] + X[12671], 5 X[12684] - 14 X[34862], X[12684] - 28 X[40262], X[34862] - 10 X[40262], X[20] + 2 X[6260], X[40] + 2 X[6261], X[40] - 4 X[6796], 2 X[40] + X[7971], X[6261] + 2 X[6796], 4 X[6261] - X[7971], 8 X[6796] + X[7971], X[72] + 2 X[9942], 4 X[140] - X[5787], 2 X[550] + X[6259], 5 X[631] - 2 X[6245], 2 X[1158] - 5 X[35242], 4 X[1385] - X[12650], X[1657] + 2 X[22792], X[12686] + 2 X[50528], X[2950] - 4 X[33814], 5 X[3522] + X[6223], 7 X[3523] - 4 X[6705], 7 X[3523] - X[9799], 4 X[6705] - X[9799], 7 X[3528] - X[12246], X[5691] - 4 X[18242], X[3868] - 4 X[40249], 5 X[3876] + X[9960], X[7982] - 4 X[40257], 2 X[4297] + X[12667], 5 X[7987] - 2 X[12114], 2 X[6282] + X[15239], 4 X[5450] - X[10864], and many others

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

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

X(52027) = X(1)X(104)∩X(2)X(21164)

Barycentrics    a*(3*a^6 - 2*a^5*b - 7*a^4*b^2 + 4*a^3*b^3 + 5*a^2*b^4 - 2*a*b^5 - b^6 - 2*a^5*c + 10*a^4*b*c - 8*a^2*b^3*c + 2*a*b^4*c - 2*b^5*c - 7*a^4*c^2 + 6*a^2*b^2*c^2 + b^4*c^2 + 4*a^3*c^3 - 8*a^2*b*c^3 + 4*b^3*c^3 + 5*a^2*c^4 + 2*a*b*c^4 + b^2*c^4 - 2*a*c^5 - 2*b*c^5 - c^6) : :
X(52027) = X[1] + 2 X[1158], X[1] - 4 X[5450], 2 X[104] + X[2950], X[1158] + 2 X[5450], X[1768] + 2 X[48695], 2 X[3] + X[84], 4 X[3] - X[1490], 5 X[3] + X[12684], X[3] + 2 X[34862], 7 X[3] - 4 X[40262], 2 X[84] + X[1490], 5 X[84] - 2 X[12684], X[84] - 4 X[34862], 7 X[84] + 8 X[40262], 5 X[1490] + 4 X[12684], X[1490] + 8 X[34862], 7 X[1490] - 16 X[40262], 2 X[3358] + X[5732], 4 X[5044] - X[18239], X[12671] - 4 X[31805], X[12684] - 10 X[34862], 7 X[12684] + 20 X[40262], 7 X[34862] + 2 X[40262], X[4] - 4 X[6705], X[20] + 2 X[6245], X[40] + 2 X[12114], 2 X[40] + X[12650], 4 X[12114] - X[12650], X[72] + 2 X[18238], 4 X[140] - X[6259], 2 X[2096] + X[5924], 2 X[550] + X[5787], 5 X[631] - 2 X[6260], 5 X[631] + X[12246], 2 X[6260] + X[12246], 2 X[960] + X[17649], 4 X[1385] - X[7971], X[5691] - 4 X[12616], 5 X[1656] - 2 X[22792], 5 X[1698] - 2 X[6256], 5 X[3522] + X[9799], 7 X[3523] - X[6223], 3 X[3524] - X[5658], 7 X[3526] - X[48664], 7 X[3624] - 4 X[12608], 4 X[6684] - X[12667], X[5534] - 4 X[26285], X[6257] - 4 X[48748], X[6258] - 4 X[48749], and many others

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

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

X(52028) = X(2)X(154)∩X(6)X(64)

Barycentrics    a^2*(3*a^10 - 5*a^8*b^2 - 2*a^6*b^4 + 6*a^4*b^6 - a^2*b^8 - b^10 - 5*a^8*c^2 + 20*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 13*b^8*c^2 - 2*a^6*c^4 - 6*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 14*b^6*c^4 + 6*a^4*c^6 + 4*a^2*b^2*c^6 + 14*b^4*c^6 - a^2*c^8 - 13*b^2*c^8 - c^10) : :
X(52028) = X[154] - 4 X[10249], 3 X[154] - 4 X[23041], 3 X[5085] - 2 X[23041], 3 X[10249] - X[23041], 4 X[3] - X[9924], 2 X[6] + X[64], X[12294] + 2 X[31978], X[20] + 2 X[15583], X[69] - 4 X[6696], 4 X[182] - X[1498], 8 X[182] - 5 X[19132], 2 X[1498] - 5 X[19132], 4 X[206] - 7 X[10541], 4 X[23300] - X[36990], 2 X[10606] + X[17813], X[895] + 2 X[11598], 4 X[1177] - X[17812], X[1350] + 2 X[8549], 2 X[1350] - 5 X[8567], X[1350] - 4 X[44883], 4 X[8549] + 5 X[8567], X[8549] + 2 X[44883], 5 X[8567] - 8 X[44883], X[1351] + 2 X[3357], 2 X[1352] - 5 X[40686], 4 X[1386] - X[7973], 2 X[2883] - 5 X[3618], X[2935] + 2 X[11579], 2 X[2935] + X[32276], 4 X[11579] - X[32276], 7 X[3523] - 4 X[15585], 4 X[3589] - X[41735], X[3751] + 2 X[12262], 8 X[5092] - 5 X[17821], 4 X[5092] - X[39879], 5 X[17821] - 2 X[39879], 2 X[10250] + X[35450], 4 X[5480] - X[5895], X[5878] - 4 X[18583], 2 X[5894] + X[51212], X[5925] + 2 X[31670], X[6225] - 7 X[51171], 2 X[6247] + X[6776], 2 X[6759] - 5 X[12017], X[17845] + 2 X[36851], X[17845] - 4 X[44882], X[36851] + 2 X[44882], and many others

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

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

X(52029) = X(1)X(41)∩X(8)X(76)

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

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

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

X(52030) = X(1)X(3252)∩X(105)X(238)

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

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

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

X(52031) = X(2)X(514)∩X(57)X(88)

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

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

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

X(52032) = X(2)X(311)∩X(3)X(49)

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

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

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

X(52033) = X(1)X(1826)∩X(6)X(19)

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

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

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

X(52034) = X(3)X(76)∩X(32)X(148)

Barycentrics    a^8 - a^2*b^6 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^2*b^2*c^4 + 3*b^4*c^4 - a^2*c^6 - b^2*c^6 : :
X(52034) = X[99] - 4 X[5939]

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

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

X(52035) = X(4)X(542)∩X(99)X(523)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(52035) = X(52035) = 3 X[9166] - 4 X[14995]

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

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

X(52036) = X(2)X(99)∩X(182)X(1511)

Barycentrics    a^2*(a^8 - 3*a^6*b^2 + a^4*b^4 + 3*a^2*b^6 - 2*b^8 - 3*a^6*c^2 + 13*a^4*b^2*c^2 - 11*a^2*b^4*c^2 + 6*b^6*c^2 + a^4*c^4 - 11*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 6*b^2*c^6 - 2*c^8) : :
X(52036) = X[111] - 3 X[15560], 2 X[111] - 3 X[15563]

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

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

X(52037) = X(1)X(84)∩X(7)X(92)

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

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

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

X(52038) = X(6)X(523)∩X(98)X(843)

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

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

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

X(52039) = X(2)X(13)∩X(14)X(476)

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

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

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

X(52040) = X(2)X(14)∩X(13)X(476)

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

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

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

X(52041) = X(2)X(2138)∩X(3)X(206)

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

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

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

X(52042) = X(2)X(27375)∩X(3)X(3202)

Barycentrics    a^2*(b^2 + c^2)^2*(a^4 - a^2*b^2 - a^2*c^2 - b^2*c^2) : :
X(52042) = X[39] + 3 X[3917], X[76] - 9 X[7998], 3 X[2979] + 5 X[7786], 3 X[3819] - X[3934], X[4173] + 3 X[7810], X[5562] + 3 X[21163], 9 X[5650] - 5 X[31239], X[6101] + 3 X[40108], 7 X[7999] + X[11257], X[32516] + 3 X[44324]

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

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

X(52043) = X(2)X(330)∩X(10)X(38)

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

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

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

X(52044) = X(1)X(668)∩X(2)X(3121)

Barycentrics    b*c*(-(a^3*b^3) + 2*a^4*b*c - a^2*b^2*c^2 - a^3*c^3 + b^3*c^3) : :
X(52044) = 3 X[4479] - 2 X[32035]

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

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

X(52045) = X(2)X(489)∩X(3)X(9680)

Barycentrics    7*a^4 - 8*a^2*b^2 + b^4 - 8*a^2*c^2 - 2*b^2*c^2 + c^4 - 6*a^2*S : :
X(52045) = X[590] + 2 X[6200], 5 X[590] - 2 X[6564], 7 X[590] - 4 X[18538], X[590] - 4 X[35255], 4 X[590] - X[42284], 3 X[590] - 4 X[43211], 5 X[6200] + X[6564], 7 X[6200] + 2 X[18538], X[6200] + 2 X[35255], 8 X[6200] + X[42284], 3 X[6200] + 2 X[43211], 7 X[6564] - 10 X[18538], X[6564] - 10 X[35255], 8 X[6564] - 5 X[42284], 3 X[6564] - 10 X[43211], X[18538] - 7 X[35255], 16 X[18538] - 7 X[42284], 3 X[18538] - 7 X[43211], 16 X[35255] - X[42284], 3 X[35255] - X[43211], 3 X[42284] - 16 X[43211]

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

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

X(52046) = X(2)X(490)∩X(3)X(9681)

Barycentrics    7*a^4 - 8*a^2*b^2 + b^4 - 8*a^2*c^2 - 2*b^2*c^2 + c^4 + 6*a^2*S : :
X(52046) = X[615] + 2 X[6396], 5 X[615] - 2 X[6565], 7 X[615] - 4 X[18762], X[615] - 4 X[35256], 4 X[615] - X[42283], 3 X[615] - 4 X[43212], 5 X[6396] + X[6565], 7 X[6396] + 2 X[18762], X[6396] + 2 X[35256], 8 X[6396] + X[42283], 3 X[6396] + 2 X[43212], 7 X[6565] - 10 X[18762], X[6565] - 10 X[35256], 8 X[6565] - 5 X[42283], 3 X[6565] - 10 X[43212], X[18762] - 7 X[35256], 16 X[18762] - 7 X[42283], 3 X[18762] - 7 X[43212], 16 X[35256] - X[42283], 3 X[35256] - X[43212], 3 X[42283] - 16 X[43212]

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

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

X(52047) = X(2)X(6221)∩X(30)X(371)

Barycentrics    8*a^4 - 7*a^2*b^2 - b^4 - 7*a^2*c^2 + 2*b^2*c^2 - c^4 - 12*a^2*S : :
X(52047) = 7 X[371] - X[3070], 4 X[371] - X[7583], 3 X[371] - X[32787], 13 X[371] - X[35820], 5 X[371] - X[35822], 5 X[371] + X[42258], 11 X[371] + X[42266], 19 X[371] - X[42272], 9 X[371] + X[43210], 4 X[3070] - 7 X[7583], 3 X[3070] - 7 X[32787], 13 X[3070] - 7 X[35820], 5 X[3070] - 7 X[35822], X[3070] + 7 X[41945], 5 X[3070] + 7 X[42258], 11 X[3070] + 7 X[42266], 19 X[3070] - 7 X[42272], 9 X[3070] + 7 X[43210], 3 X[7583] - 4 X[32787], 13 X[7583] - 4 X[35820], 5 X[7583] - 4 X[35822], X[7583] + 4 X[41945], 5 X[7583] + 4 X[42258], 11 X[7583] + 4 X[42266], 19 X[7583] - 4 X[42272], 9 X[7583] + 4 X[43210], 13 X[32787] - 3 X[35820], 5 X[32787] - 3 X[35822], X[32787] + 3 X[41945], 5 X[32787] + 3 X[42258], 11 X[32787] + 3 X[42266], 19 X[32787] - 3 X[42272], 3 X[32787] + X[43210], 5 X[35820] - 13 X[35822], X[35820] + 13 X[41945], 5 X[35820] + 13 X[42258], 11 X[35820] + 13 X[42266], 19 X[35820] - 13 X[42272], 9 X[35820] + 13 X[43210], X[35822] + 5 X[41945], 11 X[35822] + 5 X[42266], 19 X[35822] - 5 X[42272], 9 X[35822] + 5 X[43210], 5 X[41945] - X[42258], and manyu others

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

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

X(52048) = X(2)X(6398)∩X(30)X(372)

Barycentrics    8*a^4 - 7*a^2*b^2 - b^4 - 7*a^2*c^2 + 2*b^2*c^2 - c^4 + 12*a^2*S : :
X(52048) = 7 X[372] - X[3071], 4 X[372] - X[7584], 3 X[372] - X[32788], 13 X[372] - X[35821], 5 X[372] - X[35823], 5 X[372] + X[42259], 11 X[372] + X[42267], 19 X[372] - X[42271], 9 X[372] + X[43209], 4 X[3071] - 7 X[7584], 3 X[3071] - 7 X[32788], 13 X[3071] - 7 X[35821], 5 X[3071] - 7 X[35823], X[3071] + 7 X[41946], 5 X[3071] + 7 X[42259], 11 X[3071] + 7 X[42267], 19 X[3071] - 7 X[42271], 9 X[3071] + 7 X[43209], 3 X[7584] - 4 X[32788], 13 X[7584] - 4 X[35821], 5 X[7584] - 4 X[35823], X[7584] + 4 X[41946], 5 X[7584] + 4 X[42259], 11 X[7584] + 4 X[42267], 19 X[7584] - 4 X[42271], 9 X[7584] + 4 X[43209], 13 X[32788] - 3 X[35821], 5 X[32788] - 3 X[35823], X[32788] + 3 X[41946], 5 X[32788] + 3 X[42259], 11 X[32788] + 3 X[42267], 19 X[32788] - 3 X[42271], 3 X[32788] + X[43209], 5 X[35821] - 13 X[35823], X[35821] + 13 X[41946], 5 X[35821] + 13 X[42259], 11 X[35821] + 13 X[42267], 19 X[35821] - 13 X[42271], 9 X[35821] + 13 X[43209], X[35823] + 5 X[41946], 11 X[35823] + 5 X[42267], 19 X[35823] - 5 X[42271], 9 X[35823] + 5 X[43209], 5 X[41946] - X[42259], and many others

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

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

X(52049) = X(2)X(21902)∩X(10)X(75)

Barycentrics    b*c*(-(a^2*b^2) - a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 + b^2*c^2) : :
X(52049) = X[75] - 4 X[6381]

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

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

X(52050) = X(1)X(908)∩X(9)X(36)

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

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

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

X(52051) = X(2)X(1340)∩X(6)X(30)

Barycentrics    5*a^6 - 5*a^4*b^2 + a^2*b^4 - b^6 - 5*a^4*c^2 - 6*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 + 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :

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

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

X(52052) = X(2)X(1341)∩X(6)X(30)

Barycentrics    5*a^6 - 5*a^4*b^2 + a^2*b^4 - b^6 - 5*a^4*c^2 - 6*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 - 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :

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

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

X(52053) = X(3)X(39162)∩X(20)X(39159)

Barycentrics    (5*a^4 - 4*a^2*b^2 - b^4 - 4*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)] - 2*(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 + 2*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*S : :
X(52053) = 4 X[3] - X[40851], 2 X[3] + X[42412], X[40851] + 2 X[42412], 2 X[20] + X[40852], 4 X[550] - X[42411], 3 X[10304] - X[39158]

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

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

X(52054) = X(3)X(39163)∩X(20)X(39158)

Barycentrics    (5*a^4 - 4*a^2*b^2 - b^4 - 4*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)] + 2*(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 - 2*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*S : :
X(52054) = 4 X[3] - X[40852], 2 X[3] + X[42411], X[40852] + 2 X[42411], 2 X[20] + X[40851], 4 X[550] - X[42412], 3 X[10304] - X[39159]

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

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

X(52055) = X(3)X(7712)∩X(74)X(381)

Barycentrics    a^2*(4*a^8 - 4*a^6*b^2 - 12*a^4*b^4 + 20*a^2*b^6 - 8*b^8 - 4*a^6*c^2 + 31*a^4*b^2*c^2 - 17*a^2*b^4*c^2 - 10*b^6*c^2 - 12*a^4*c^4 - 17*a^2*b^2*c^4 + 36*b^4*c^4 + 20*a^2*c^6 - 10*b^2*c^6 - 8*c^8) : :
X(52055) = 3 X[3] - 2 X[7712], 3 X[3] + 2 X[11738], 2 X[7712] + 3 X[33887], 2 X[11738] - 3 X[33887], 3 X[381] - 4 X[7703], 3 X[381] - 2 X[18550], 3 X[3534] - 4 X[35257], 3 X[1657] - 4 X[41470], 4 X[5092] - 3 X[51797], 4 X[32124] - 5 X[37958]

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

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

X(52056) = X(3)X(3447)∩X(30)X(74)

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)*(3*a^8 - 8*a^6*b^2 + 6*a^4*b^4 - b^8 - 8*a^6*c^2 + 11*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 - 5*a^2*b^2*c^4 - 2*b^4*c^4 + 2*b^2*c^6 - c^8) : :
X(52056) = 4 X[3] - 3 X[14851], 5 X[265] - 6 X[5627], 2 X[265] - 3 X[14993], 3 X[265] - 4 X[34209], X[265] - 3 X[51345], 5 X[476] - 3 X[5627], 4 X[476] - 3 X[14993], 3 X[476] - 2 X[34209], 2 X[476] - 3 X[51345], 4 X[5627] - 5 X[14993], 9 X[5627] - 10 X[34209], 2 X[5627] - 5 X[51345], 9 X[14993] - 8 X[34209], 3 X[20126] - 4 X[46632], 4 X[34209] - 9 X[51345], 2 X[477] - 3 X[38723], 4 X[1553] - 3 X[7728], 2 X[1553] - 3 X[36193], 4 X[3258] - 5 X[38794], 4 X[7471] - 3 X[14643], 3 X[14643] - 2 X[20957], 5 X[14934] - 4 X[41522], 3 X[15061] - 2 X[17511], 3 X[15061] - 4 X[38609], 2 X[16340] - 3 X[38700], 4 X[16340] - 5 X[38728], 6 X[38700] - 5 X[38728]

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

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

X(52057) = X(3)X(9530)∩X(4)X(74)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(4*a^12 - 10*a^10*b^2 + a^8*b^4 + 16*a^6*b^6 - 14*a^4*b^8 + 2*a^2*b^10 + b^12 - 10*a^10*c^2 + 26*a^8*b^2*c^2 - 22*a^6*b^4*c^2 + 2*a^4*b^6*c^2 + 8*a^2*b^8*c^2 - 4*b^10*c^2 + a^8*c^4 - 22*a^6*b^2*c^4 + 24*a^4*b^4*c^4 - 10*a^2*b^6*c^4 + 7*b^8*c^4 + 16*a^6*c^6 + 2*a^4*b^2*c^6 - 10*a^2*b^4*c^6 - 8*b^6*c^6 - 14*a^4*c^8 + 8*a^2*b^2*c^8 + 7*b^4*c^8 + 2*a^2*c^10 - 4*b^2*c^10 + c^12) : :
X(52057) = X[4] - 3 X[107], 2 X[4] - 3 X[133], X[4] + 3 X[5667], 5 X[4] - 3 X[10152], 5 X[107] - X[10152], X[133] + 2 X[5667], 5 X[133] - 2 X[10152], 5 X[5667] + X[10152], 3 X[122] - 4 X[140], 2 X[140] - 3 X[38605], 3 X[376] - X[38686], 2 X[550] - 3 X[3184], 5 X[631] - 3 X[10714], 3 X[1294] - 5 X[3522], 5 X[1656] - 6 X[6716], 5 X[1656] - 3 X[10745], 10 X[1656] - 9 X[36520], 4 X[6716] - 3 X[36520], 2 X[10745] - 3 X[36520], X[1657] - 3 X[23240], X[1657] + 3 X[38577], 3 X[11718] - 2 X[13464], 7 X[3523] - 9 X[23239], 7 X[3523] - 3 X[34186], 7 X[3523] - 6 X[34842], 3 X[23239] - X[34186], 3 X[23239] - 2 X[34842], 4 X[3850] - 3 X[49117], X[5059] + 3 X[34549], X[5073] - 3 X[22337], 3 X[40664] - X[51939], 11 X[21735] - 9 X[38714], 4 X[33923] - 3 X[38621], 3 X[44985] - X[49135]

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

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

X(52058) = X(2)X(6)∩X(4)X(22120)

Barycentrics    a^2*(a^4 - b^4 + a^2*b*c - b^3*c - b*c^3 - c^4)*(a^4 - b^4 - a^2*b*c + b^3*c + b*c^3 - c^4) : :
X(52058) = X[8744] + 2 X[22121]

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

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

X(52059) = X(6)X(909)∩X(32)X(9456)

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

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

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

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

Barycentrics    (b+c-a)*(7*a^2+11*b^2-14*b*c+7*c^2-2*a*(7*b+c))*(7*a^2+7*b^2-14*b*c+11*c^2-2*a*(b+7*c)) : :

See Ivan Pavlov, euclid 5514.

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

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

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

See Ivan Pavlov, euclid 5514.

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

X(52062) = (name pending)

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

See Ivan Pavlov, euclid 5515.

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

X(52063) = X(9)X(222)∩X(963)X(2910)

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

See Ivan Pavlov, euclid 5515.

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

X(52064) = X(9)X(664)∩X(738)X(2125)

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

See Ivan Pavlov, euclid 5516.

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

X(52065) = X(37)X(668)∩X(213)X(904)

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

See Ivan Pavlov, euclid 5516.

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

X(52066) = X(3)X(648)∩X(216)X(2972)

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

See Ivan Pavlov, euclid 5516.

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

X(52067) = X(6)X(99)∩X(39)X(373)

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

See Ivan Pavlov, euclid 5516.

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

X(52068) = X(10)X(190)∩X(67)X(71)

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

See Ivan Pavlov, euclid 5516.

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

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

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4-3*b^2*c^2+c^4)*a^6+4*(b^6+c^6)*a^4-6*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(52069) = 2*X(3)+X(18560), 5*X(4)-2*X(7553), 2*X(4)+X(12225), X(4)+2*X(12605), 3*X(4)-4*X(44804), 4*X(5)-X(6240), 2*X(5)+X(18563), 5*X(5)-2*X(45971), X(20)+2*X(1885), X(20)-4*X(12362), 3*X(381)-2*X(13490), 2*X(428)-3*X(3839)

See Panchapakesan and César Lozada, euclid 5519.

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

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

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

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4-4*b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-6*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(52070) = 5*X(3)-3*X(44458), 5*X(4)-3*X(7540), 5*X(4)-9*X(37077), 5*X(5)-4*X(9825), 7*X(5)-6*X(10127), 13*X(5)-12*X(10128), 3*X(5)-2*X(31833), 2*X(140)-3*X(34664), 3*X(381)-X(6240), 3*X(381)+X(18562), 3*X(381)-2*X(31830), 5*X(381)-3*X(38320), X(382)+3*X(18564), 3*X(428)-4*X(3861), 4*X(546)-3*X(13490), 3*X(549)-2*X(31829)

See Sriram Panchapakesan and César Lozada, euclid 5519.

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

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

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

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^6+4*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4-6*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(52071) = 3*X(2)-4*X(31829), 2*X(3)-3*X(44458), 3*X(4)-4*X(31833), 2*X(4)-3*X(38323), 3*X(376)-2*X(12605), 2*X(382)-3*X(7576), 6*X(428)-5*X(17578)

See Sriram Panchapakesan and César Lozada, euclid 5519.

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

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

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

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

Sriram Panchapakesan and César Lozada, euclid 5519.

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

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

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

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^2-c^2)^2*a^6+2*(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^4-6*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(52073) = 3*X(2)+X(18563), 9*X(2)-X(34797), 3*X(3)+X(18560), 7*X(3)-3*X(44458), 3*X(5)-X(3575), 4*X(5)-3*X(23410), X(5)-3*X(34664), 7*X(5)-3*X(38322), 3*X(381)-X(11819), 3*X(381)+X(12225), 3*X(428)-5*X(3858), 3*X(547)-2*X(9825), 3*X(547)-X(45971)

See Sriram Panchapakesan and César Lozada, euclid 5519.

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

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

X(52074) = X(1)X(24302)∩X(5)X(3244)

Barycentrics    a*(8*a^6 - 18*a^5*b - 6*a^4*b^2 + 36*a^3*b^3 - 12*a^2*b^4 - 18*a*b^5 + 10*b^6 - 18*a^5*c + 64*a^4*b*c - 50*a^3*b^2*c - 43*a^2*b^3*c + 68*a*b^4*c - 21*b^5*c - 6*a^4*c^2 - 50*a^3*b*c^2 + 116*a^2*b^2*c^2 - 50*a*b^3*c^2 - 10*b^4*c^2 + 36*a^3*c^3 - 43*a^2*b*c^3 - 50*a*b^2*c^3 + 42*b^3*c^3 - 12*a^2*c^4 + 68*a*b*c^4 - 10*b^2*c^4 - 18*a*c^5 - 21*b*c^5 + 10*c^6) : :
X(52074) = 5 X[3623] + X[18549]

See Antreas Hatzipolakis and Peter Moses, euclid 5521.

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

X(52075) = (name pending)

Barycentrics    a^2*(a - b - c)*(b - c)^2*(3*a^2 - 3*b^2 + 2*b*c - 3*c^2)*(6*a^3 - 6*a^2*b - 6*a*b^2 + 6*b^3 - 6*a^2*c + 19*a*b*c - 7*b^2*c - 6*a*c^2 - 7*b*c^2 + 6*c^3)*(2*a^5 - 2*a^4*b - 4*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 - 2*b^5 - 2*a^4*c + 5*a^3*b*c + a^2*b^2*c - 5*a*b^3*c + b^4*c - 4*a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 + b^3*c^2 + 4*a^2*c^3 - 5*a*b*c^3 + b^2*c^3 + 2*a*c^4 + b*c^4 - 2*c^5) : :

See Antreas Hatzipolakis and Peter Moses, euclid 5521.

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

X(52076) = X(4)X(512)∩X(98)X(523)

Barycentrics    (b - c)*(b + c)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + b^2*c^2 - c^4)*(-a^4 + b^4 - b^2*c^2 + c^4) : :
X(52076) = 4 X[6130] - 3 X[38227]

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

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

X(52077) = X(3)X(17818)∩X(4)X(155)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 12*a^4*b^2*c^2 - 6*a^2*b^4*c^2 + 4*b^6*c^2 - 6*a^2*b^2*c^4 - 6*b^4*c^4 + 2*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(52077) = 3 X[3167] - 2 X[9306]

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

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

X(52078) = X(1)X(3346)∩X(4)X(57)

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

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

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

X(52079) = X(2)X(42122)∩X(15)X(376)

Barycentrics    3*(9*a^4 - 8*a^2*b^2 - b^4 - 8*a^2*c^2 + 2*b^2*c^2 - c^4) - 8*Sqrt[3]*a^2*S : :

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

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

X(52080) = X(2)X(42123)∩X(16)X(376)

Barycentrics    3*(9*a^4 - 8*a^2*b^2 - b^4 - 8*a^2*c^2 + 2*b^2*c^2 - c^4) + 8*Sqrt[3]*a^2*S : :

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

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

X(52081) = X(2)X(9473)∩X(6)X(264)

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

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

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

X(52082) = X(4)X(8)∩X(19)X(208)

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

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

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

X(52083) = X(2)X(32)∩X(6)X(19585)

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

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

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

X(52084) = X(1)X(41)∩X(9)X(1742)

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

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

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

X(52085) = X(1)X(335)∩X(12)X(85)

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

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

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

X(52086) = X(1)X(33844)∩X(3)X(142)

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

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

X(52087) = X(2)X(573)∩X(37)X(65)

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

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

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

X(52088) = X(6)X(598)∩X(39)X(9878)

Barycentrics    4*a^8 - 2*a^4*b^4 - 2*b^8 - 5*a^4*b^2*c^2 + a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 + a^2*b^2*c^4 + b^4*c^4 + 3*b^2*c^6 - 2*c^8 : :
X(52088) = 3 X[598] - 2 X[671], 2 X[2482] - 3 X[8592], 4 X[5461] - 3 X[43535], 3 X[10033] - 4 X[22566], 3 X[9774] - 2 X[14830]

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

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

X(52089) = X(1)X(28391)∩X(6)X(57)

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

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

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

X(52090) = X(3)X(67)∩X(98)X(140)

Barycentrics    a^8 + 2*a^6*b^2 - 3*a^4*b^4 + 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 - 3*a^4*c^4 + 2*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - c^8 : :
X(52090) = 5 X[3] - 6 X[2482], 2 X[3] - 3 X[8724], 3 X[3] - 2 X[10991], 4 X[3] - 3 X[14830], X[3] - 3 X[48657], 4 X[2482] - 5 X[8724], 9 X[2482] - 5 X[10991], 8 X[2482] - 5 X[14830], 3 X[2482] - 5 X[14981], 2 X[2482] - 5 X[48657], 9 X[8724] - 4 X[10991], 3 X[8724] - 4 X[14981], 8 X[10991] - 9 X[14830], X[10991] - 3 X[14981], 2 X[10991] - 9 X[48657], 3 X[14830] - 8 X[14981], X[14830] - 4 X[48657], 2 X[14981] - 3 X[48657], 3 X[36776] - X[41020], X[4] - 3 X[147], 5 X[4] - 3 X[148], 2 X[4] - 3 X[6033], 4 X[4] - 3 X[6321], 2 X[4] + 3 X[14692], 5 X[4] - 6 X[22505], 7 X[4] - 6 X[22515], 5 X[147] - X[148], 4 X[147] - X[6321], 2 X[147] + X[14692], 5 X[147] - 2 X[22505], 7 X[147] - 2 X[22515], 2 X[148] - 5 X[6033], 4 X[148] - 5 X[6321], 2 X[148] + 5 X[14692], 7 X[148] - 10 X[22515], 5 X[6033] - 4 X[22505], 7 X[6033] - 4 X[22515], X[6321] + 2 X[14692], 5 X[6321] - 8 X[22505], 7 X[6321] - 8 X[22515], 5 X[14692] + 4 X[22505], 7 X[14692] + 4 X[22515], 7 X[22505] - 5 X[22515], 2 X[5] - 3 X[6054], 10 X[5] - 9 X[9166], 4 X[5] - 3 X[11632], 5 X[6054] - 3 X[9166], and many othrs

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

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

X(52091) = X(3)X(2065)∩X(69)X(523)

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

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

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

X(52092) = X(6)X(41)∩X(40)X(78)

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

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

X(52093) = X(2)X(13474)∩X(20)X(185)

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 - 3*b^6*c^2 - 3*a^4*c^4 - 9*a^2*b^2*c^4 + 8*b^4*c^4 + 3*a^2*c^6 - 3*b^2*c^6 - c^8) : :
X(52093) = 9 X[2] - 4 X[13474], 4 X[3] + X[12279], 6 X[3] - X[12290], 13 X[3] - 8 X[14128], 12 X[3] - 7 X[15056], 11 X[3] - 6 X[15060], 8 X[3] - 3 X[15305], 9 X[3] - 4 X[45959], 3 X[12279] + 2 X[12290], 13 X[12279] + 32 X[14128], 3 X[12279] + 7 X[15056], X[12279] + 2 X[15058], 11 X[12279] + 24 X[15060], 2 X[12279] + 3 X[15305], 9 X[12279] + 16 X[45959], 13 X[12290] - 48 X[14128], 2 X[12290] - 7 X[15056], X[12290] - 3 X[15058], 11 X[12290] - 36 X[15060], 4 X[12290] - 9 X[15305], 3 X[12290] - 8 X[45959], 96 X[14128] - 91 X[15056], 16 X[14128] - 13 X[15058], 44 X[14128] - 39 X[15060], 64 X[14128] - 39 X[15305], 18 X[14128] - 13 X[45959], 7 X[15056] - 6 X[15058], 77 X[15056] - 72 X[15060], 14 X[15056] - 9 X[15305], 21 X[15056] - 16 X[45959], 11 X[15058] - 12 X[15060], 4 X[15058] - 3 X[15305], 9 X[15058] - 8 X[45959], 16 X[15060] - 11 X[15305], 27 X[15060] - 22 X[45959], 27 X[15305] - 32 X[45959], 7 X[4] - 12 X[5892], 2 X[4] - 3 X[11451], X[4] + 4 X[14641], 13 X[4] - 18 X[14845], X[4] - 6 X[14855], 8 X[4] - 13 X[15028], 4 X[4] - 9 X[20791], and many others

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

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

X(52094) = X(2)X(1637)∩X(30)X(99)

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

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

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

X(52095) = X(2)X(3558)∩X(4)X(69)

Barycentrics    a^4 - b^4 - c^4 - (a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :
X(52095) = 9 X[2] - 8 X[14632], 3 X[3558] - 4 X[14632], 6 X[1341] - 7 X[3523], 3 X[6194] - 2 X[13326]

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

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

X(52096) = X(2)X(3557)∩X(4)X(69)

Barycentrics    a^4 - b^4 - c^4 + (a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :
X(52096) = 9 X[2] - 8 X[14633], 3 X[3557] - 4 X[14633], 6 X[1340] - 7 X[3523], 3 X[6194] - 2 X[13325]

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

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

X(52097) = X(3)X(77)∩X(4)X(8)

Barycentrics    a*(a^2 - b^2 - c^2)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 2*a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - a*c^3 - c^4) : :
X(52097) = 2 X[1872] - 3 X[5927]

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

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

X(52098) = X(3)X(34437)∩X(4)X(542)

Barycentrics    a^2*(a^10 - 4*a^8*b^2 + 4*a^6*b^4 + 2*a^4*b^6 - 5*a^2*b^8 + 2*b^10 - 4*a^8*c^2 - 5*a^6*b^2*c^2 + 5*a^4*b^4*c^2 + 2*a^2*b^6*c^2 + 2*b^8*c^2 + 4*a^6*c^4 + 5*a^4*b^2*c^4 - 4*a^2*b^4*c^4 - 4*b^6*c^4 + 2*a^4*c^6 + 2*a^2*b^2*c^6 - 4*b^4*c^6 - 5*a^2*c^8 + 2*b^2*c^8 + 2*c^10) : :
X(52098) = 3 X[576] - 2 X[895], X[576] + 2 X[14094], X[895] - 3 X[9970], X[895] + 3 X[14094], 4 X[5095] - 3 X[51140], 5 X[15063] - X[32250], 3 X[20423] - X[32255], X[67] - 3 X[5655], 2 X[67] - 3 X[11178], 2 X[74] - 3 X[17508], 3 X[182] - 4 X[6593], 3 X[182] - 2 X[32305], 2 X[6593] - 3 X[19140], 3 X[19140] - X[32305], 3 X[381] - X[25335], 3 X[399] - X[2930], 3 X[399] + X[48679], X[2930] + 3 X[51941], X[48679] - 3 X[51941], 2 X[575] - 3 X[45016], X[16010] - 3 X[45016], 3 X[5476] - 2 X[25328], 3 X[5621] - 4 X[20190], 3 X[6053] - X[32257], 2 X[9976] - 3 X[15520], 2 X[10264] - 3 X[38317], 3 X[19149] - X[32262], 2 X[11579] - 3 X[39561], 4 X[25556] - 3 X[39561], X[12317] - 3 X[14561], 4 X[13392] - 3 X[21167], 2 X[14810] - 3 X[32609], 3 X[14982] - X[32272], X[15054] - 3 X[15462], 5 X[22234] - 6 X[34155], 4 X[22330] - 3 X[39562], 3 X[25331] - X[39899], 3 X[38789] - 2 X[48889]

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

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

X(52099) = X(3)X(5888)∩X(6)X(3534)

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 - 59*a^4*b^2*c^2 + 49*a^2*b^4*c^2 + 14*b^6*c^2 + 6*a^4*c^4 + 49*a^2*b^2*c^4 - 30*b^4*c^4 - 4*a^2*c^6 + 14*b^2*c^6 + c^8) : :
X(52099) = 3 X[3] - 2 X[5888], 9 X[3] - 2 X[13603], 3 X[3] - X[18551], 3 X[5888] - X[13603], 4 X[5888] - 3 X[14926], 4 X[13603] - 9 X[14926], 2 X[13603] - 3 X[18551], 3 X[14926] - 2 X[18551], 3 X[3534] + 2 X[13623], 4 X[5092] - 3 X[38402]

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

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

X(52100) = X(6)X(382)∩X(20)X(399)

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 - 11*a^4*b^2*c^2 + 9*a^2*b^4*c^2 + 6*b^6*c^2 + 6*a^4*c^4 + 9*a^2*b^2*c^4 - 14*b^4*c^4 - 4*a^2*c^6 + 6*b^2*c^6 + c^8) : :
X(52100) = 5 X[3] - 6 X[6030], 3 X[3] - 2 X[15062], 5 X[3] - 2 X[16835], 3 X[6030] - 5 X[8718], 9 X[6030] - 5 X[15062], 3 X[6030] - X[16835], 12 X[6030] - 5 X[33541], 3 X[8718] - X[15062], 5 X[8718] - X[16835], 4 X[8718] - X[33541], 5 X[15062] - 3 X[16835], 4 X[15062] - 3 X[33541], 4 X[16835] - 5 X[33541], 6 X[6] - 5 X[45034], X[382] - 4 X[44866], 3 X[382] - 4 X[46027], 3 X[3521] - 2 X[46027], 3 X[44866] - X[46027], 4 X[34563] - X[49136], 3 X[3534] - 2 X[18442], 5 X[1656] - 4 X[18488], 2 X[11559] - 3 X[15041], 5 X[2916] - 4 X[14810], X[3529] + 2 X[44755], X[5073] - 4 X[43585], 3 X[11455] - X[15084], X[17800] + 2 X[43599], 5 X[15696] - 4 X[35240]

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

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

X(52101) = X(4)X(541)∩X(5)X(13474)

Barycentrics    2*a^8*b^2 - 7*a^6*b^4 + 9*a^4*b^6 - 5*a^2*b^8 + b^10 + 2*a^8*c^2 + 24*a^6*b^2*c^2 - 7*a^4*b^4*c^2 - 16*a^2*b^6*c^2 - 3*b^8*c^2 - 7*a^6*c^4 - 7*a^4*b^2*c^4 + 42*a^2*b^4*c^4 + 2*b^6*c^4 + 9*a^4*c^6 - 16*a^2*b^2*c^6 + 2*b^4*c^6 - 5*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(52101) = 5 X[4] - 3 X[51993], X[20] - 3 X[4550], X[382] + 3 X[11472], 5 X[631] - 3 X[8717], 3 X[3426] + 5 X[3843], 5 X[3843] - 3 X[7706], 7 X[3526] - 3 X[35237], 7 X[3832] - 3 X[4846], 3 X[4549] + X[33703], 11 X[5070] - 3 X[11820], 5 X[15696] - 9 X[32620], X[18431] - 3 X[23325]

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

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

X(52102) = X(2)X(14862)∩X(3)X(45185)

Barycentrics    2*a^8*b^2 - 7*a^6*b^4 + 9*a^4*b^6 - 5*a^2*b^8 + b^10 + 2*a^8*c^2 + 12*a^6*b^2*c^2 - 9*a^4*b^4*c^2 - 2*a^2*b^6*c^2 - 3*b^8*c^2 - 7*a^6*c^4 - 9*a^4*b^2*c^4 + 14*a^2*b^4*c^4 + 2*b^6*c^4 + 9*a^4*c^6 - 2*a^2*b^2*c^6 + 2*b^4*c^6 - 5*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(53102) = 5 X[5] - 3 X[2883], X[5] - 3 X[6247], 2 X[5] - 3 X[20299], 7 X[5] - 9 X[23332], 5 X[5] - 6 X[32767], X[2883] - 5 X[6247], 2 X[2883] - 5 X[20299], 7 X[2883] - 15 X[23332], 7 X[6247] - 3 X[23332], 5 X[6247] - 2 X[32767], 7 X[20299] - 6 X[23332], 5 X[20299] - 4 X[32767], 15 X[23332] - 14 X[32767], X[20] - 3 X[3357], X[20] + 3 X[14216], 3 X[64] + X[382], 5 X[64] + 3 X[18405], X[382] - 3 X[18381], 5 X[382] - 9 X[18405], 5 X[18381] - 3 X[18405], 3 X[549] - X[44762], 3 X[549] - 2 X[50414], 5 X[631] - 3 X[6759], 10 X[631] - 9 X[10182], 5 X[631] + 3 X[12324], 5 X[631] - 6 X[25563], 35 X[631] - 27 X[35260], 2 X[6759] - 3 X[10182], 7 X[6759] - 9 X[35260], 3 X[10182] + 2 X[12324], 3 X[10182] - 4 X[25563], 7 X[10182] - 6 X[35260], X[12324] + 2 X[25563], 7 X[12324] + 9 X[35260], 14 X[25563] - 9 X[35260], 3 X[1498] - 7 X[3526], X[1498] - 3 X[23329], 7 X[3526] - 9 X[23329], 9 X[1853] - 5 X[3843], 3 X[1853] + X[13093], 3 X[1853] - X[22802], 5 X[3843] + 3 X[13093], 5 X[3843] - 3 X[22802], 7 X[3528] - 3 X[9833], 7 X[3528] - 9 X[11204], X[9833] - 3 X[11204], and many others

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

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

X(52103) = X(4)X(7883)∩X(5)X(1533)

Barycentrics    2*a^8*b^2 - 7*a^6*b^4 + 9*a^4*b^6 - 5*a^2*b^8 + b^10 + 2*a^8*c^2 + 42*a^6*b^2*c^2 + 11*a^4*b^4*c^2 - 52*a^2*b^6*c^2 - 3*b^8*c^2 - 7*a^6*c^4 + 11*a^4*b^2*c^4 + 114*a^2*b^4*c^4 + 2*b^6*c^4 + 9*a^4*c^6 - 52*a^2*b^2*c^6 + 2*b^4*c^6 - 5*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(52103) = X[20] - 3 X[5888], X[382] + 3 X[14926], 7 X[3832] - 3 X[7693], 5 X[3843] + 3 X[18551], 11 X[3855] + 3 X[13603]

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

X(52104) = X(3)X(539)∩X(20)X(68)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6 - b^8 + 2*a^6*c^2 + 4*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 - 5*a^4*c^4 - 4*a^2*b^2*c^4 - 6*b^4*c^4 + 4*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(52104) = 4 X[5] - 3 X[5448], 2 X[5] - 3 X[5449], X[5] - 3 X[12359], 5 X[5] - 3 X[22660], X[5448] - 4 X[12359], 5 X[5448] - 4 X[22660], 5 X[5449] - 2 X[22660], 5 X[12359] - X[22660], X[20] + 3 X[68], X[20] - 3 X[7689], 5 X[16003] + 3 X[32263], 3 X[155] - 7 X[3526], 7 X[3526] - 6 X[43839], X[382] - 3 X[9927], X[382] + 3 X[12163], 5 X[631] - 3 X[1147], 5 X[631] - X[9936], 5 X[631] + 3 X[11411], 5 X[631] - 6 X[20191], 10 X[631] - 3 X[45184], 3 X[1147] - X[9936], X[9936] + 3 X[11411], X[9936] - 6 X[20191], 2 X[9936] - 3 X[45184], X[11411] + 2 X[20191], 2 X[11411] + X[45184], 4 X[20191] - X[45184], 7 X[3528] - 3 X[12118], 4 X[3530] - 3 X[12038], 2 X[3530] - 3 X[44158], 5 X[3843] - 9 X[14852], X[4301] - 3 X[12259], 13 X[5067] - 9 X[5654], 11 X[5070] - 3 X[12164], 3 X[5504] - 7 X[15057], 3 X[6193] - 11 X[15717], 7 X[9588] - 3 X[9928], 3 X[9820] - 4 X[16239], X[9925] - 3 X[50977], 3 X[12293] + X[17800], 3 X[12429] + 5 X[15696], 3 X[12893] - X[23236], 2 X[14862] - 3 X[44278], X[15063] - 3 X[46085], X[16266] - 3 X[23329], 4 X[20396] - 3 X[33547], 3 X[44213] - 2 X[50414]

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

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

X(52105) = X(2)X(34213)∩X(3)X(11178)

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

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

X(52106) = X(39)X(48653)∩X(110)X(187)

Barycentrics    a^2*(a^10*b^2 - 3*a^8*b^4 - 4*a^6*b^6 + 4*a^4*b^8 + 3*a^2*b^10 - b^12 + a^10*c^2 - 10*a^8*b^2*c^2 + 26*a^6*b^4*c^2 - 29*a^4*b^6*c^2 + 11*a^2*b^8*c^2 - 4*b^10*c^2 - 3*a^8*c^4 + 26*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 3*a^2*b^6*c^4 - b^8*c^4 - 4*a^6*c^6 - 29*a^4*b^2*c^6 + 3*a^2*b^4*c^6 + 4*b^6*c^6 + 4*a^4*c^8 + 11*a^2*b^2*c^8 - b^4*c^8 + 3*a^2*c^10 - 4*b^2*c^10 - c^12) : :

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

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

X(52107) = X(2)X(34213)∩X(110)X(599)

Barycentrics    8*a^12 - 32*a^10*b^2 - 20*a^8*b^4 + 52*a^6*b^6 + 4*a^4*b^8 - 20*a^2*b^10 + 8*b^12 - 32*a^10*c^2 - 18*a^8*b^2*c^2 + 25*a^6*b^4*c^2 + 37*a^4*b^6*c^2 + 18*a^2*b^8*c^2 - 8*b^10*c^2 - 20*a^8*c^4 + 25*a^6*b^2*c^4 + 81*a^4*b^4*c^4 + 22*a^2*b^6*c^4 - 8*b^8*c^4 + 52*a^6*c^6 + 37*a^4*b^2*c^6 + 22*a^2*b^4*c^6 + 16*b^6*c^6 + 4*a^4*c^8 + 18*a^2*b^2*c^8 - 8*b^4*c^8 - 20*a^2*c^10 - 8*b^2*c^10 + 8*c^12 : :

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    a^2*(b - c)^2*(b + c)^2*(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 - 2*a^2*c^2 - b^2*c^2 + c^4)^2*(a^12 - 4*a^10*b^2 + 6*a^8*b^4 - 4*a^6*b^6 + a^4*b^8 - 4*a^10*c^2 + 10*a^8*b^2*c^2 - 8*a^6*b^4*c^2 + 3*a^4*b^6*c^2 - 2*a^2*b^8*c^2 + b^10*c^2 + 6*a^8*c^4 - 8*a^6*b^2*c^4 + a^4*b^4*c^4 + 2*a^2*b^6*c^4 - 4*b^8*c^4 - 4*a^6*c^6 + 3*a^4*b^2*c^6 + 2*a^2*b^4*c^6 + 6*b^6*c^6 + a^4*c^8 - 2*a^2*b^2*c^8 - 4*b^4*c^8 + b^2*c^10) : :
X(52110) = X[143] - 3 X[27357], X[15907] - 3 X[47065]

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

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

X(52111) = X(10)X(11)∩X(1364)X(5687)

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

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

X(52112) = X(8)X(2829)∩X(11)X(123)

Barycentrics    (a - b - c)^2*(b - c)^2*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 - a^4*c + 4*a^3*b*c + 2*a^2*b^2*c - 4*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 - 4*a*b*c^3 + 2*b^2*c^3 + a*c^4 - b*c^4 - c^5) : :
X(52112) = 2 X[11798] - 3 X[34122]

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

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

X(52113) = X(11)X(521)∩X(1259)X(6099)

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

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

X(52114) = X(8)X(1309)∩X(521)X(1364)

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

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

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

X(52115) = X(101)X(222)∩X(116)X(5514)

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

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

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

X(52116) = X(8)X(2829)∩X(9)X(119)

Barycentrics    (2*a^4 - a^3*b - 3*a^2*b^2 + a*b^3 + b^4 - a^3*c + 6*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)*(2*a^6 - a^5*b - 5*a^4*b^2 + 2*a^3*b^3 + 4*a^2*b^4 - a*b^5 - b^6 - a^5*c + 2*a^4*b*c + 6*a^3*b^2*c - 4*a^2*b^3*c - 5*a*b^4*c + 2*b^5*c - 5*a^4*c^2 + 6*a^3*b*c^2 - 8*a^2*b^2*c^2 + 6*a*b^3*c^2 + b^4*c^2 + 2*a^3*c^3 - 4*a^2*b*c^3 + 6*a*b^2*c^3 - 4*b^3*c^3 + 4*a^2*c^4 - 5*a*b*c^4 + b^2*c^4 - a*c^5 + 2*b*c^5 - c^6) : :
X(52116) = X[104] - 3 X[14646], 3 X[210] - 2 X[34293], 2 X[1387] - 3 X[52027], 4 X[6705] - 3 X[38038], 4 X[12608] - 5 X[31235]

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

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

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

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

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

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

X(52118) = COMPLEMENT OF X(43346)

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

See Antreas Hatzipolakis and César Lozada, euclid 5523.

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

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

X(52119) = COMPLEMENT OF X(43354)

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

See Antreas Hatzipolakis and César Lozada, euclid 5523.

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

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

X(52120) = COMPLEMENT OF X(46963)

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

See Antreas Hatzipolakis and César Lozada, euclid 5523.

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

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

X(52121) = COMPLEMENT OF X(3465)

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

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

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

X(52122) = COMPLEMENT OF X(3484)

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

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

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

X(52123) = X(1)X(2916)∩X(2)X(16555)

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

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

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

X(52124) = X(4)X(195)∩X(6)X(110)

Barycentrics    a^2*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 11*a^6*b^2*c^2 - 10*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 10*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 5*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10) : :
X(52124) = 4 X[12827] - 5 X[31236]

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

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

X(52125) = X(4)X(110)∩X(111)X(925)

Barycentrics    a^12 - a^10*b^2 - 3*a^8*b^4 + 4*a^6*b^6 + a^4*b^8 - 3*a^2*b^10 + b^12 - a^10*c^2 + 3*a^8*b^2*c^2 - 6*a^4*b^6*c^2 + 9*a^2*b^8*c^2 - 5*b^10*c^2 - 3*a^8*c^4 + 6*a^4*b^4*c^4 - 6*a^2*b^6*c^4 + 11*b^8*c^4 + 4*a^6*c^6 - 6*a^4*b^2*c^6 - 6*a^2*b^4*c^6 - 14*b^6*c^6 + a^4*c^8 + 9*a^2*b^2*c^8 + 11*b^4*c^8 - 3*a^2*c^10 - 5*b^2*c^10 + c^12 : :

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

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

X(52126) = X(8)X(191)∩X(21)X(60)

Barycentrics    a*(a - b - c)*(a^2 - b^2 - b*c - c^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(52126) = 3 X[191] - X[11010], 3 X[21] - 2 X[2646], 3 X[7701] + X[24468], 4 X[6701] - 5 X[31262], 3 X[31159] - X[37005]

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

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

X(52127) = X(6)X(292)∩X(32)X(101)

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

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

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

X(52128) = X(2)X(98)∩X(3)X(1625)

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

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

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

X(52129) = X(4)X(80)∩X(35)X(102)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*b*c - 3*a^5*b^2*c + a^4*b^3*c + 6*a^3*b^4*c - 5*a^2*b^5*c - 3*a*b^6*c + 3*b^7*c + a^6*c^2 - 3*a^5*b*c^2 + 7*a^4*b^2*c^2 - 6*a^3*b^3*c^2 - 5*a^2*b^4*c^2 + 9*a*b^5*c^2 - 3*b^6*c^2 + a^4*b*c^3 - 6*a^3*b^2*c^3 + 14*a^2*b^3*c^3 - 6*a*b^4*c^3 - 3*b^5*c^3 - 3*a^4*c^4 + 6*a^3*b*c^4 - 5*a^2*b^2*c^4 - 6*a*b^3*c^4 + 8*b^4*c^4 - 5*a^2*b*c^5 + 9*a*b^2*c^5 - 3*b^3*c^5 + 3*a^2*c^6 - 3*a*b*c^6 - 3*b^2*c^6 + 3*b*c^7 - c^8) : :
X(52129) = 3 X[1] - 2 X[1364], 3 X[1361] - X[1364], 5 X[1698] - 4 X[3040], 4 X[3042] - 3 X[3679], 4 X[12016] - 5 X[18398]

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

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

X(52130) = X(3)X(74)∩X(4)X(5627)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)*(a^6 - 3*a^2*b^4 + 2*b^6 + 5*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(521) lies on the cubic K846 and these lines: {3, 74}, {4, 5627}, {5, 40630}, {146, 39170}, {185, 34329}, {546, 12079}, {1304, 44879}, {1510, 14380}, {2132, 12290}, {3146, 14989}, {3529, 36875}, {3627, 34150}, {5890, 38933}, {6240, 17986}, {10125, 16243}, {10421, 34797}, {12317, 51254}, {13754, 15786}, {14865, 38937}, {16080, 16868}, {18439, 50467}, {34783, 50464}, {35007, 48451}, {35502, 35908}

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

X(52131) = X(6)X(1345)∩X(110)X(112)

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

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

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

X(52132) = X(6)X(1344)∩X(110)X(112)

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

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

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

X(52133) = X(1)X(257)∩X(2)X(31)

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

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

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

X(52134) = X(1)X(21)∩X(48)X(75)

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

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

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

X(52135) = X(1)X(257)∩X(38)X(256)

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

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

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

X(52136) = X(1)X(257)∩X(6)X(75)

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

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

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

X(52137) = X(1)X(75)∩X(2)X(40776)

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

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

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

X(52138) = X(1)X(75)∩X(6)X(17743)

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

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

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

X(52139) = X(3)X(10)∩X(6)X(31)

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

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

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

X(52140) = X(2)X(514)∩X(63)X(88)

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

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

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

X(52141) = X(2)X(99)∩X(23)X(11643)

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

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

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

X(52142) = X(2)X(691)∩X(25)X(111)

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

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

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

X(52143) = X(1)X(1437)∩X(6)X(25)

Barycentrics    a^3*(a + b)*(a + c)*(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(52143) lies on the cubic K430 and these lines: {1, 1437}, {6, 25}, {21, 1798}, {28, 44545}, {31, 18611}, {58, 3556}, {81, 3827}, {110, 1812}, {197, 22132}, {221, 1408}, {478, 17408}, {692, 5285}, {970, 10282}, {1333, 1402}, {1503, 6703}, {1801, 37577}, {2193, 3185}, {2328, 7169}, {3560, 6759}, {5743, 10192}, {5799, 7511}, {14555, 35260}, {16049, 41600}, {26884, 40959}, {32734, 34858}, {37547, 41608}, {40571, 41607}

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

X(52144) = X(3)X(49)∩X(32)X(51)

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

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

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

X(52145) = X(2)X(647)∩X(3)X(76)

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

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

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

X(52146) = X(1)X(85)∩X(43)X(294)

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

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

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

X(52147) = X(2)X(216)∩X(3)X(107)

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

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

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

X(52148) = X(2)X(104)∩X(9)X(36)

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

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

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

X(52149) = X(2)X(6)∩X(30)X(1272)

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

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

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

X(52150) = X(1)X(859)∩X(3)X(2051)

Barycentrics    a^2*(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(52150) lies on the conic {{A,B,C,X(1),X(6)}}, the cubics K430 and K1057, and these lines: {1, 859}, {3, 2051}, {6, 10457}, {21, 1220}, {28, 36123}, {36, 39949}, {58, 9563}, {86, 4225}, {87, 3286}, {333, 1222}, {958, 996}, {1027, 16695}, {1126, 4281}, {1468, 40496}, {2983, 4269}, {4216, 34262}, {4245, 50605}, {5737, 28383}, {7428, 39748}, {13738, 37674}, {16948, 37129}, {17588, 23375}, {41245, 46502}

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

X(52151) = X(1)X(668)∩X(75)X(141)

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

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

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

X(52152) = X(2)X(523)∩X(3)X(111)

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

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

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

X(52153) = X(3)X(125)∩X(23)X(94)

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

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

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

X(52154) = X(3)X(14579)∩X(6)X(5055)

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

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

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

X(52155) = X(1)X(672)∩X(2)X(165)

Barycentrics    a*(3*a^3*b - 2*a^2*b^2 - a*b^3 + 3*a^3*c + a^2*b*c - 5*a*b^2*c + b^3*c - 2*a^2*c^2 - 5*a*b*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :
X(52155) = X[1] + 2 X[3730], 4 X[3] - X[170], X[20] + 2 X[34848], 4 X[1125] - X[17753], 4 X[2140] - 7 X[3624], 7 X[3523] - 4 X[43158], 5 X[7987] + X[41680]

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

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

X(52156) = X(2)X(658)∩X(8)X(348)

Barycentrics    (a + b - c)*(a - b + c)*(a^3 - a^2*b - a*b^2 + b^3 + a*c^2 + b*c^2 - 2*c^3)*(a^3 + a*b^2 - 2*b^3 - a^2*c + b^2*c - a*c^2 + c^3) : :
X(52156) = 4 X[23058] - 5 X[31640]

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

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

X(52157) = X(2)X(38)∩X(7)X(190)

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

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

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

X(52158) = X(1)X(204)∩X(3)X(64)

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

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

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

X(52159) = X(21)X(572)∩X(31)X(184)

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

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

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

X(52160) = X(1)X(147)∩X(6)X(7)

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

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

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

X(52161) = X(1)X(85)∩X(43)X(57)

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

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

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

X(52162) = X(6)X(157)∩X(22)X(110)

Barycentrics    a^2*(a^8 + a^6*b^2 - 2*a^4*b^4 + a^2*b^6 - b^8 + a^6*c^2 - 3*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 + a^2*c^6 + b^2*c^6 - c^8) : :
X(52162) = X[287] - 3 X[35278], 8 X[40559] - 7 X[47355]

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

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

X(52163) = X(3)X(373)∩X(5)X(599)

Barycentrics    a^2*(a^8 - 8*a^6*b^2 + 18*a^4*b^4 - 16*a^2*b^6 + 5*b^8 - 8*a^6*c^2 + 20*a^4*b^2*c^2 + 36*a^2*b^4*c^2 - 48*b^6*c^2 + 18*a^4*c^4 + 36*a^2*b^2*c^4 + 86*b^4*c^4 - 16*a^2*c^6 - 48*b^2*c^6 + 5*c^8) : :
X(52163) = X[3] + 3 X[3531], X[3] - 3 X[5544], 7 X[3090] - 3 X[44833], 5 X[3091] - 3 X[18489], 13 X[5079] - 3 X[40912], 11 X[27355] - 3 X[44838]

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

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

X(52164) = X(2)X(11)∩X(152)X(516)

Barycentrics    a^5 - 4*a^4*b + 5*a^3*b^2 - 3*a^2*b^3 + 2*a*b^4 - b^5 - 4*a^4*c + 3*a^3*b*c - a*b^3*c + 2*b^4*c + 5*a^3*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 + 2*a*c^4 + 2*b*c^4 - c^5 : :
X(52164) = 3 X[2] - 4 X[50441], 4 X[1146] - 5 X[3617], 7 X[3622] - 8 X[17044], 2 X[10695] - 3 X[38941], 10 X[31640] - 11 X[46933], 16 X[40483] - 17 X[46932]

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

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

X(52165) = X(6)X(30)∩X(66)X(51964)

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

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

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

X(52166) = X(3)X(3163)∩X(6)X(74)

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

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

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

X(52167) = X(1)X(2816)∩X(4)X(80)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^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 - 2*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(52167) lies on the cubics K683 and K1051 and these lines: {1, 2816}, {4, 80}, {7, 1364}, {11, 653}, {19, 41499}, {65, 151}, {92, 1836}, {102, 3485}, {109, 278}, {117, 1788}, {124, 281}, {158, 12699}, {196, 497}, {243, 516}, {329, 3042}, {331, 33867}, {388, 2817}, {499, 8762}, {528, 1897}, {648, 19642}, {946, 1940}, {962, 1118}, {1148, 1479}, {1155, 17923}, {1785, 28194}, {1857, 9812}, {1895, 12701}, {2792, 5307}, {2818, 4295}, {3176, 5225}, {3476, 10696}, {3982, 51808}, {5081, 44663}, {5125, 37567}, {5174, 42379}, {5229, 13532}, {6331, 19643}, {6335, 17777}, {6336, 19636}, {6718, 17917}, {7541, 40663}, {7551, 18393}, {10740, 18391}, {11713, 37028}, {14690, 37417}, {35489, 50148}, {38573, 39542}

X(52168) = X(3)X(1495)∩X(24)X(9064)

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

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

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

X(52169) = X(3)X(74)∩X(184)X(566)

Barycentrics    a^2*(3*a^10 - 7*a^8*b^2 + 4*a^6*b^4 + a^2*b^8 - b^10 - 7*a^8*c^2 + 11*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 - 3*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10) : :
X(52169) = 3 X[381] - 2 X[18576]

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

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

X(52170) = X(3)X(284)∩X(25)X(110)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^10 - 4*a^8*b^2 + 5*a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8 - b^10 - 4*a^8*c^2 + 3*a^6*b^2*c^2 - a^2*b^6*c^2 + 2*b^8*c^2 + 5*a^6*c^4 - 2*a^2*b^4*c^4 - b^6*c^4 - 3*a^4*c^6 - a^2*b^2*c^6 - b^4*c^6 + 2*a^2*c^8 + 2*b^2*c^8 - c^10) : :
X(52170) = 3 X[3167] - 2 X[42065]

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

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

X(52171) = X(2)X(19140)∩X(3)X(74)

Barycentrics    a^2*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 2*a^6*b^2*c^2 - a^4*b^4*c^2 - a^2*b^6*c^2 + 3*b^8*c^2 + 2*a^6*c^4 - a^4*b^2*c^4 + 5*a^2*b^4*c^4 - 4*b^6*c^4 + 2*a^4*c^6 - a^2*b^2*c^6 - 4*b^4*c^6 - 3*a^2*c^8 + 3*b^2*c^8 + c^10) : :
X(52171) = X[74] - 3 X[43578], 3 X[23] - 4 X[40291], 3 X[32235] - 2 X[40291]

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

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

X(52172) = X(4)X(51)∩X(30)X(133)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - a^6*c^2 + 8*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 2*b^6*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 + 8*b^4*c^4 + 5*a^2*c^6 - 2*b^2*c^6 - 2*c^8) : :
X(52172) = 3 X[4] + X[6761], 3 X[4] - X[51892], 5 X[4] + X[51939], X[6761] - 3 X[34170], 5 X[6761] - 3 X[51939], 3 X[34170] + X[51892], 5 X[34170] - X[51939], 5 X[51892] + 3 X[51939], 3 X[381] - X[6760], 4 X[546] - X[34147], 2 X[3627] + X[34109]

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

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

X(52173) = X(4)X(5609)∩X(30)X(13530)

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

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

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

X(52174) = X(2)X(1296)∩X(25)X(187)

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

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

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

X(52175) = X(1)X(7168)∩X(350)X(694)

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

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

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

X(52176) = X(1)X(39917)∩X(10)X(27536)

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

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

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

X(52177) = X(6)X(1987)∩X(51)X(107)

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

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

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

X(52178) = X(1)X(40218)∩X(3)X(8)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + c^3)*(2*a^4 - a^3*b - 3*a^2*b^2 + a*b^3 + b^4 - a^3*c + 6*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(52178) lies on the cubic K654 and K844 and these lines: {1, 40218}, {3, 8}, {4, 6075}, {84, 3667}, {221, 34051}, {936, 36819}, {2401, 21132}, {5265, 37136}, {12672, 15635}, {14986, 40450}

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

X(52179) = X(23)X(110)∩X(67)X(98)

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

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

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

X(52180) = X(1)X(474)∩X(518)X(36603)

Barycentrics    a*(9*a^2-14*(b+c)*a-7*b^2+34*b*c-7*c^2) : :
X(52180) = 7*X(1)-16*X(8688), X(1)-4*X(45047), 4*X(8688)-7*X(45047), 7*X(16192)-4*X(47302)

See Ivan Pavlov and César Lozada, euclid 5531.

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

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

X(52181) = X(1)X(6)∩X(519)X(6552)

Barycentrics    a*(-3*a^3+11*a^2*(b+c)+a*(7*b^2-26*b*c+7*c^2)-(b+c)*(7*b^2-10*b*c+7*c^2)) : :

See Ivan Pavlov and César Lozada, euclid 5531.

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

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

X(52182) = X(1)X(4004)∩X(3973)X(7991)

Barycentrics    a*(3*a^3-21*b^3+33*b^2*c+33*b*c^2-21*c^3+17*a^2*(b+c)-a*(7*b^2+46*b*c+7*c^2)) : :
X(52182) = 2*X(1)-3*X(36603)

See Ivan Pavlov and César Lozada, euclid 5531.

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

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

X(52183) = X(40)X(9519)∩X(986)X(1706)

Barycentrics    a*(3*a^3-7*b^3+15*b^2*c+15*b*c^2-7*c^3+3*a^2*(b+c)-a*(7*b^2+10*b*c+7*c^2)) : :

See Ivan Pavlov and César Lozada, euclid 5531.

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

X(52184) = X(40)X(3973)∩X(101)X(604)

Barycentrics    a* (-a+b+c)*(13*a^3-15*(b+c)*a^2-(7*b-3*c)*(3*b-7*c)*a+7*(b^2-c^2)*(b-c)) : :

See Ivan Pavlov and César Lozada, euclid 5531.

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

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

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

See Ivan Pavlov and César Lozada, euclid 5530.

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

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

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

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

See Ivan Pavlov and César Lozada, euclid 5530.

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

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

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

Barycentrics    (a^4 + 10*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 10*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    Sin[A] / (3*Csc[A] - Sin[A]) : :

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

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

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

Barycentrics    (a^2 - 4*a*b + b^2 - c^2)*(a^2 + 4*a*b + b^2 - c^2)*(a^2 - b^2 - 4*a*c + c^2)*(a^2 - b^2 + 4*a*c + c^2) : :
Barycentrics    Sin[A] / (3*Csc[A] + Sin[A]) : :

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

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

X(52189) = X(2)X(11643)∩X(4)X(542)

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

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

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

X(52190) = X(2)X(98)∩X(67)X(51943)

Barycentrics    (a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :
X(52190) = 5 X[15081] - 2 X[46301]

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

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

X(52191) = X(2)X(67)∩X(4)X(94)

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

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

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

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

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

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

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

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

Barycentrics    (a^2 - b^2 - c^2)*(3*a^2 - 2*Sqrt[3]*S) : :
X(52193) = 3 X[628] - X[51265], 3 X[14] - 5 X[40335], 5 X[40335] - 6 X[44382], X[16] - 3 X[5464], 2 X[16] - 3 X[35303], 3 X[299] - X[622], 3 X[617] + X[622], 3 X[395] - 4 X[6672], 3 X[619] - 2 X[6672], X[42088] - 6 X[47867], 4 X[624] - 3 X[31694], 2 X[5321] - 3 X[31694], 3 X[3107] - X[23009], 5 X[16961] - 3 X[22496], X[19107] + 3 X[36329], X[19107] - 3 X[50858], 3 X[21360] - X[36970], 3 X[22494] + X[42100], X[22856] - 3 X[50860], 3 X[35932] - 2 X[42123]

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

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

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

Barycentrics    (a^2 - b^2 - c^2)*(3*a^2 + 2*Sqrt[3]*S) : :
X(52194) = 3 X[627] - X[51272], 3 X[13] - 5 X[40334], 5 X[40334] - 6 X[44383], X[15] - 3 X[5463], 2 X[15] - 3 X[35304], 3 X[298] - X[621], 3 X[616] + X[621], 3 X[396] - 4 X[6671], 3 X[618] - 2 X[6671], 4 X[623] - 3 X[31693], 2 X[5318] - 3 X[31693], 6 X[36769] - X[42087], 3 X[3106] - X[23000], 3 X[16267] - 5 X[36770], 5 X[16960] - 3 X[22495], X[19106] + 3 X[35751], X[19106] - 3 X[50855], 3 X[21359] - X[36969], 3 X[22493] + X[42099], X[22900] - 3 X[50859], 3 X[35931] - 2 X[42122]

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

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

X(52195) = X(2)X(9309)∩X(56)X(87)

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

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

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

X(52196) = X(2)X(6)∩X(55)X(332)

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

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

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

X(52197) = X(6)X(512)∩X(67)X(524)

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

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

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

X(52198) = X(6)X(691)∩X(316)X(524)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^6 - 6*a^4*b^2 + 9*a^2*b^4 - 2*b^6 - 6*a^4*c^2 + 3*a^2*b^2*c^2 - 3*b^4*c^2 + 9*a^2*c^4 - 3*b^2*c^4 - 2*c^6) : :
X(52198) = 2 X[187] - 3 X[5166], 3 X[14853] - 2 X[46659]

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

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

X(52199) = X(6)X(526)∩X(50)X(237)

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

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

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

X(52200) = X(1)X(30)∩X(5)X(523)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 - 2*a^3*b^2*c + a^2*b^3*c + 2*a*b^4*c - b^5*c + a^4*c^2 - 2*a^3*b*c^2 + 4*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - b*c^5 + c^6) : :
X(52200) = 3 X[50148] + X[51883], 3 X[2072] - X[50144], 3 X[5055] - X[50145], 3 X[5603] + X[38514], 3 X[5886] - X[47270], 5 X[8227] - X[47273], 5 X[10595] - X[36171], 2 X[16305] - 3 X[44282]

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

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

X(52201) = X(13)X(2160)∩X(14)X(2161)

Barycentrics    a*(-a^2 + b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*b*c - 2*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*Sqrt[3]*a*(b + c)*S) : :

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

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

X(52202) = X(13)X(2161)∩X(14)X(2160)

Barycentrics    a*(-a^2 + b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*b*c - 2*a^2*c^2 - 2*b^2*c^2 + c^4 + 2*Sqrt[3]*a*(b + c)*S) : :

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

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

X(52203) = X(5)X(14)∩X(324)X(473)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 + b^2 - c^2 + 2*Sqrt[3]*S)*(a^2 - b^2 + c^2 + 2*Sqrt[3]*S)*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S) : :

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

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

X(52204) = X(5)X(13)∩X(324)X(472)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 + b^2 - c^2 - 2*Sqrt[3]*S)*(a^2 - b^2 + c^2 - 2*Sqrt[3]*S)*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S)) : :

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

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

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

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

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

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

X(52206) = X(2)X(45)∩X(55)X(106)

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

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

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

X(52207) = X(1)X(99)∩X(2)X(694)

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

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

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

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

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

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

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

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

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

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

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

X(52210) = X(2)X(11)∩X(57)X(649)

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

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

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

X(52211) = X(1)X(2162)∩X(7)X(350)

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

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

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

X(52212) = X(1)X(47645)∩X(2)X(655)

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

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

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

X(52213) = X(55)X(103)∩X(57)X(650)

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

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

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

X(52214) = X(2)X(371)∩X(16)X(396)

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

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

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

X(52215) = X(2)X(372)∩X(16)X(396)

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

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

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

X(52216) = X(2)X(372)∩X(14)X(615)

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

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

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

X(52217) = X(2)X(371)∩X(14)X(590)

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

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

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

X(52218) = X(1)X(7125)∩X(6)X(41)

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

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

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

X(52219) = X(4)X(523)∩X(30)X(113)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^12 - 3*a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 + 6*a^4*b^8 - 11*a^2*b^10 + 4*b^12 - 3*a^10*c^2 + 14*a^8*b^2*c^2 - 7*a^6*b^4*c^2 - 25*a^4*b^6*c^2 + 30*a^2*b^8*c^2 - 9*b^10*c^2 - 4*a^8*c^4 - 7*a^6*b^2*c^4 + 38*a^4*b^4*c^4 - 19*a^2*b^6*c^4 + 6*a^6*c^6 - 25*a^4*b^2*c^6 - 19*a^2*b^4*c^6 + 10*b^6*c^6 + 6*a^4*c^8 + 30*a^2*b^2*c^8 - 11*a^2*c^10 - 9*b^2*c^10 + 4*c^12) : :
X(52219) = 3 X[4] - X[34150], X[34150] + 3 X[46045], 5 X[5] - 3 X[47852], 3 X[10151] - X[47146], 3 X[381] - X[46632], 3 X[3845] - X[34209], 5 X[3858] - 3 X[21315], 2 X[12068] - 3 X[36518], 2 X[12295] + X[30221], 5 X[15081] - 3 X[40630], 3 X[15687] - X[21269]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    (a^4 + 6*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 6*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    1 / (2*b^2*c^2 - S^2) : :
Barycentrics    Sin[A] / (2*Csc[A] - Sin[A]) : :

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

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

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

Barycentrics    (a^4 - 2*a^2*b^2 + b^4 - 10*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 10*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    1 / (2*b^2*c^2 + S^2) : :
Barycentrics    Sin[A]/(2*Csc[A] + Sin[A]) : :

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

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

X(52225) = X(1)X(1635)∩X(44)X(3251)

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

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

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

X(52226) = X(1)X(659)∩X(239)X(4375)

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

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

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

X(52227) = X(1)X(41)∩X(919)X(1027)

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

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

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

X(52228) = X(1)X(2254)∩X(85)X(3762)

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

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

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

X(52229) = X(2)X(2418)∩X(30)X(511)

Barycentrics    2*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + 10*b^2*c^2 - c^4 : :

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

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

X(52230) = X(6)X(1296)∩X(32)X(6233)

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

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

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

X(52231) = X(2)X(6)∩X(30)X(11162)

Barycentrics    8*a^6 - 21*a^4*b^2 + 6*a^2*b^4 - b^6 - 21*a^4*c^2 + 30*a^2*b^2*c^2 - 3*b^4*c^2 + 6*a^2*c^4 - 3*b^2*c^4 - c^6 : :
X(52231) = 3 X[37745] - X[37746]

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

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

X(52232) = X(2)X(523)∩X(30)X(111)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(2*a^8 - 6*a^6*b^2 - a^4*b^4 + 6*a^2*b^6 - b^8 - 6*a^6*c^2 + 20*a^4*b^2*c^2 - 10*a^2*b^4*c^2 - a^4*c^4 - 10*a^2*b^2*c^4 + 2*b^4*c^4 + 6*a^2*c^6 - c^8) : :

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

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

X(52233) = X(6)X(512)∩X(111)X(352)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(2*a^6 - 6*a^4*b^2 + 9*a^2*b^4 - b^6 - 6*a^4*c^2 - 3*b^4*c^2 + 9*a^2*c^4 - 3*b^2*c^4 - c^6) : :
X(52233) = X[843] - 3 X[36696]

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

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

X(52234) = X(6)X(2780)∩X(110)X(9027)

Barycentrics    a^2*(2*a^2 - b^2 - c^2)*(a^8 - 2*a^4*b^4 + b^8 - 6*a^6*c^2 + 10*a^4*b^2*c^2 + 10*a^2*b^4*c^2 - 6*b^6*c^2 + a^4*c^4 - 20*a^2*b^2*c^4 + b^4*c^4 + 6*a^2*c^6 + 6*b^2*c^6 - 2*c^8)*(a^8 - 6*a^6*b^2 + a^4*b^4 + 6*a^2*b^6 - 2*b^8 + 10*a^4*b^2*c^2 - 20*a^2*b^4*c^2 + 6*b^6*c^2 - 2*a^4*c^4 + 10*a^2*b^2*c^4 + b^4*c^4 - 6*b^2*c^6 + c^8) : :

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

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

X(52235) = X(2)X(1499)∩X(524)X(46144)

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

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

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

X(52236) = X(2)X(10748)∩X(67)X(39602)

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

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

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

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

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

X(52238) = X(2)X(47455)∩X(6)X(23)

Barycentrics    a^2*(3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6 - a^4*c^2 + 7*a^2*b^2*c^2 - 2*b^4*c^2 - 3*a^2*c^4 - 2*b^2*c^4 + c^6) : :
X(52238) = X[2] - 4 X[47544], X[4] - 4 X[47581], 2 X[6] + X[23], 5 X[6] - 2 X[15826], X[6] + 2 X[32217], 5 X[23] + 4 X[15826], X[23] - 4 X[32217], X[15826] + 5 X[32217], X[69] - 4 X[468], X[69] + 2 X[32220], 2 X[468] + X[32220], 2 X[110] + X[41617], 2 X[141] - 5 X[47453], 4 X[182] - X[7464], X[15035] - 3 X[19128], X[22151] - 4 X[44102], X[193] + 2 X[32113], X[193] + 5 X[37760], X[193] - 4 X[47549], 2 X[32113] - 5 X[37760], X[32113] + 2 X[47549], 5 X[37760] + 4 X[47549], X[323] - 4 X[6593], X[385] - 4 X[47561], 3 X[37907] - 2 X[47450], 8 X[575] + X[37946], 4 X[576] + 5 X[37953], 4 X[597] - X[10989], X[691] - 4 X[2030], 2 X[858] - 5 X[3618], X[858] - 4 X[47457], 5 X[3618] - 8 X[47457], X[895] + 2 X[1495], 2 X[1177] + X[41744], 2 X[1350] - 5 X[37952], X[1351] + 2 X[7575], X[1353] + 2 X[25338], X[1992] + 2 X[7426], X[1992] - 4 X[47545], X[7426] + 2 X[47545], X[2071] - 4 X[51733], X[2452] + 2 X[37906], 2 X[18374] + X[37784], X[3153] - 4 X[51742], 2 X[3580] + X[11061], 8 X[3589] - 5 X[30745], 7 X[3619] - 16 X[47454], X[3629] + 2 X[32218], and many others

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

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

X(52239) = X(6)X(9966)∩X(115)X(599)

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

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

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

X(52240) = X(1)X(647)∩X(2)X(3)

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

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

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

X(52241) = X(1)X(672)∩X(2)X(3)

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

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

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

X(52242) = X(2)X(3)∩X(6)X(661)

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

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

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

X(52243) = X(2)X(3)∩X(75)X(672)

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

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

X(52244) = X(2)X(3)∩X(10)X(501)

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

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

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

X(52245) = X(2)X(3)∩X(10)X(672)

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

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

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

X(52246) = X(2)X(3)∩X(10)X(4370)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    a^4 - 6*a^2*b^2 + 9*b^4 - 6*a^2*c^2 - 14*b^2*c^2 + 9*c^4 : :
X(52250) = 3 X[2] - 4 X[32976]

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

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

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

Barycentrics    a^8 - a^6*b^2 + a^4*b^4 - 3*a^2*b^6 + 2*b^8 - a^6*c^2 + 2*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 4*b^6*c^2 + a^4*c^4 + 3*a^2*b^2*c^4 + 4*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 + 2*c^8 : :
X(52251) = 3 X[2] + X[37174]

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    a^4*b^2 - 2*a^3*b^3 + 2*a*b^5 - b^6 - 2*a^4*b*c + 2*b^5*c + a^4*c^2 + 4*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + b^2*c^4 + 2*a*c^5 + 2*b*c^5 - c^6 : :
X(52254) = 2 X[8728] - 9 X[17533], 2 X[8728] - 3 X[50208], 3 X[17533] - X[50208], 5 X[31259] + 9 X[37375]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    2*a^7 + 2*a^6*b - a^5*b^2 - a^4*b^3 - 2*a^3*b^4 - 2*a^2*b^5 + a*b^6 + b^7 + 2*a^6*c - a^4*b^2*c + 2*a^3*b^3*c - 2*a^2*b^4*c - 2*a*b^5*c + b^6*c - a^5*c^2 - a^4*b*c^2 + 8*a^3*b^2*c^2 + 8*a^2*b^3*c^2 - a*b^4*c^2 - b^5*c^2 - a^4*c^3 + 2*a^3*b*c^3 + 8*a^2*b^2*c^3 + 4*a*b^3*c^3 - b^4*c^3 - 2*a^3*c^4 - 2*a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 - 2*a^2*c^5 - 2*a*b*c^5 - b^2*c^5 + a*c^6 + b*c^6 + c^7 : :
X(52259) = 3 X[2] + X[28], 5 X[2] - X[31154], 9 X[2] - 5 X[31257], 15 X[2] + X[31293], 5 X[28] + 3 X[31154], 3 X[28] + 5 X[31257], 5 X[28] - X[31293], 5 X[631] - X[30267], X[3152] + 3 X[44291], 5 X[21530] - 3 X[31154], 3 X[21530] - 5 X[31257], 5 X[21530] + X[31293], 9 X[31154] - 25 X[31257], 3 X[31154] + X[31293], 25 X[31257] + 3 X[31293], 5 X[31258] + 3 X[44293], X[16100] - 5 X[27195]

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

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

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

Barycentrics    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 - 3*a^4*b^2*c - 2*a^3*b^3*c - 2*a^2*b^4*c + 2*a*b^5*c + 3*b^6*c - a^5*c^2 - 3*a^4*b*c^2 + 4*a^3*b^2*c^2 + 2*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 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 - 5*b^4*c^3 - 4*a^3*c^4 - 2*a^2*b*c^4 - 3*a*b^2*c^4 - 5*b^3*c^4 + 2*a*b*c^5 + b^2*c^5 + 3*a*c^6 + 3*b*c^6 + c^7 : :
X(52260) = 3 X[2] + X[29], 9 X[2] - X[3152], 5 X[2] - X[31155], 9 X[2] - 5 X[31258], 15 X[2] + X[31294], 3 X[29] + X[3152], 5 X[29] + 3 X[31155], 3 X[29] + 5 X[31258], 5 X[29] - X[31294], 5 X[631] - X[30268], X[3152] - 3 X[18641], 5 X[3152] - 9 X[31155], X[3152] - 5 X[31258], 5 X[3152] + 3 X[31294], 5 X[18641] - 3 X[31155], 3 X[18641] - 5 X[31258], 5 X[18641] + X[31294], 9 X[31155] - 25 X[31258], 3 X[31155] + X[31294], 5 X[31257] + 3 X[44291], 25 X[31258] + 3 X[31294]

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

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

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

Barycentrics    2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6 + 2*a^6*c^2 + b^6*c^2 - 3*a^4*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6 : :
X(52261) = 3 X[2] + X[237], 9 X[2] - X[14957], 15 X[2] + X[46518], 3 X[237] + X[14957], X[237] - 3 X[44215], 5 X[237] - X[46518], 5 X[631] - X[47620], X[14957] - 3 X[21531], X[14957] + 9 X[44215], 5 X[14957] + 3 X[46518], X[21531] + 3 X[44215], 5 X[21531] + X[46518], 15 X[44215] - X[46518], 3 X[9155] + X[51481]

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

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

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

Barycentrics    2*a^10 - 5*a^8*b^2 + 2*a^6*b^4 + 4*a^4*b^6 - 4*a^2*b^8 + b^10 - 5*a^8*c^2 + 8*a^6*b^2*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 8*a^2*b^4*c^4 + 2*b^6*c^4 + 4*a^4*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(52262) = 3 X[2] + X[378], 9 X[2] - X[44440], 3 X[3] + X[31723], 3 X[3] - X[44239], X[4] - 5 X[31236], 3 X[5] - X[44263], X[22] - 5 X[631], 4 X[140] - X[16618], 3 X[140] - X[25337], X[378] - 3 X[44218], 3 X[378] + X[44440], 3 X[427] - X[31723], 3 X[427] + X[44239], 3 X[549] - X[7502], 7 X[3090] + X[35481], 5 X[3091] - X[35480], X[3153] + 3 X[37970], 7 X[3523] + X[7391], 7 X[3523] - 3 X[44837], 3 X[3524] + X[31133], 3 X[3524] - X[44261], 9 X[3524] - X[44831], 9 X[5054] - X[12083], 3 X[5054] - X[44210], 3 X[6676] - 2 X[25337], X[6676] + 2 X[44236], X[7391] + 3 X[44837], X[7502] + 3 X[44287], X[7555] - 4 X[12108], 3 X[10257] - X[16387], 13 X[10303] - X[12082], 3 X[11539] - X[44262], X[12083] - 3 X[44210], 5 X[15692] + 3 X[31105], 7 X[15702] - 3 X[47596], X[15760] + 3 X[44218], 3 X[15760] - X[44440], 3 X[16618] - 4 X[25337], X[16618] + 4 X[44236], 3 X[18570] + X[44263], X[25337] + 3 X[44236], 3 X[31133] + X[44831], 5 X[31236] + X[44249], X[37969] - 3 X[44214], 9 X[44218] + X[44440], 3 X[44261] - X[44831], 3 X[10165] - X[51692], X[11605] + 3 X[38699], X[16165] - 3 X[38793], X[18474] + 3 X[39242], X[18474] - 3 X[45303]

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

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

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

Barycentrics    Sqrt[3]*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 2*b^2*c^2 + 3*c^4) - 2*(a^2 + b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(52263) = 3 X[2] + X[383], X[383] - 3 X[44219], 5 X[1656] + X[41034], 3 X[3545] + X[35932], 3 X[5055] - X[31694], X[5474] + 3 X[41037], X[5978] + 3 X[38227], 3 X[21159] + X[36962], 3 X[22511] + X[36776], 3 X[23514] - X[31710]

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

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

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

Barycentrics    2*a^4 - 3*a^2*b^2 + b^4 + 4*a^2*b*c + 4*a*b^2*c - 3*a^2*c^2 + 4*a*b*c^2 - 2*b^2*c^2 + c^4 : :
X(52264) = 3 X[2] + X[404], 9 X[2] - X[5046], 15 X[2] + X[37256], 3 X[404] + X[5046], 5 X[404] - X[37256], 5 X[631] - X[37403], 7 X[3526] + X[37251], 17 X[3533] - X[6903], 3 X[4187] - X[5046], 5 X[4187] + X[37256], 5 X[5046] + 3 X[37256], 3 X[17532] + 5 X[19525]

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

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

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

Barycentrics    2*a^7 - 2*a^6*b - 5*a^5*b^2 + 5*a^4*b^3 + 4*a^3*b^4 - 4*a^2*b^5 - a*b^6 + b^7 - 2*a^6*c + a^4*b^2*c - 2*a^3*b^3*c + 2*a^2*b^4*c + 2*a*b^5*c - b^6*c - 5*a^5*c^2 + a^4*b*c^2 + 4*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a*b^4*c^2 - 3*b^5*c^2 + 5*a^4*c^3 - 2*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 4*a^3*c^4 + 2*a^2*b*c^4 + a*b^2*c^4 + 3*b^3*c^4 - 4*a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - a*c^6 - b*c^6 + c^7 : :
X(52265) = 3 X[2] + X[411], 9 X[2] - X[6895], X[4] - 3 X[17530], X[20] + 3 X[17577], 3 X[411] + X[6895], 5 X[631] - X[6906], 5 X[631] - 3 X[37298], 7 X[3523] - 3 X[17549], 7 X[3523] + X[37437], 3 X[6831] - X[6895], X[6906] - 3 X[37298], 11 X[15717] - 3 X[37299], 3 X[17549] + X[37437], X[35] - 3 X[21155], X[15908] + 3 X[21155]

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

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

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

Barycentrics    Sqrt[3]*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 2*b^2*c^2 + 3*c^4) + 2*(a^2 + b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(52266) = 3 X[2] + X[1080], 5 X[1656] + X[41035], 3 X[3545] + X[35931], 3 X[5055] - X[31693], 3 X[35297] - X[44250], X[15] + 3 X[36765], X[5473] + 3 X[41036], X[5979] + 3 X[38227], X[14538] - 5 X[36770], 3 X[21158] + X[36961], 3 X[23514] - X[31709]

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

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

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

Barycentrics    a^6 - 4*a^4*b^2 - a^2*b^4 + 4*b^6 - 4*a^4*c^2 - 4*b^4*c^2 - a^2*c^4 - 4*b^2*c^4 + 4*c^6 - 2*Sqrt[3]*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*S : :
X(52267) = 5 X[2] - 2 X[15768], 5 X[2] - X[46860], 5 X[2] - 4 X[46862], 2 X[381] + X[46470], 7 X[3832] + 2 X[23721], 4 X[15768] - 5 X[46824], 2 X[15768] + 5 X[46856], X[15768] - 5 X[46858], X[46824] + 2 X[46856], X[46824] - 4 X[46858], 5 X[46824] - 2 X[46860], 5 X[46824] - 8 X[46862], X[46856] + 2 X[46858], 5 X[46856] + X[46860], 5 X[46856] + 4 X[46862], 10 X[46858] - X[46860], 5 X[46858] - 2 X[46862], X[46860] - 4 X[46862]

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

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

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

Barycentrics    a^6 - 4*a^4*b^2 - a^2*b^4 + 4*b^6 - 4*a^4*c^2 - 4*b^4*c^2 - a^2*c^4 - 4*b^2*c^4 + 4*c^6 + 2*Sqrt[3]*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*S : :
X(52268) = 5 X[2] - 2 X[15769], 5 X[2] - X[46861], 5 X[2] - 4 X[46863], 2 X[381] + X[46471], 7 X[3832] + 2 X[23722], 4 X[15769] - 5 X[46825], 2 X[15769] + 5 X[46857], X[15769] - 5 X[46859], X[46825] + 2 X[46857], X[46825] - 4 X[46859], 5 X[46825] - 2 X[46861], 5 X[46825] - 8 X[46863], X[46857] + 2 X[46859], 5 X[46857] + X[46861], 5 X[46857] + 4 X[46863], 10 X[46859] - X[46861], 5 X[46859] - 2 X[46863], X[46861] - 4 X[46863]

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

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

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

Barycentrics    a^7 - a^6*b - 3*a^3*b^4 + 3*a^2*b^5 + 2*a*b^6 - 2*b^7 - a^6*c - 3*a^5*b*c - 2*a^4*b^2*c + a^2*b^4*c + 3*a*b^5*c + 2*b^6*c - 2*a^4*b*c^2 + 2*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - 2*a*b^4*c^2 + 6*b^5*c^2 - 4*a^2*b^2*c^3 - 6*a*b^3*c^3 - 6*b^4*c^3 - 3*a^3*c^4 + a^2*b*c^4 - 2*a*b^2*c^4 - 6*b^3*c^4 + 3*a^2*c^5 + 3*a*b*c^5 + 6*b^2*c^5 + 2*a*c^6 + 2*b*c^6 - 2*c^7 : :
X(52269) = X[3] + 2 X[44258], X[3] - 4 X[46028], 2 X[4] + X[21], X[4] + 2 X[6841], 5 X[4] + 4 X[16617], 4 X[5] - X[3651], 5 X[5] - 2 X[11277], 8 X[5] - 5 X[31254], X[20] - 4 X[6675], X[21] - 4 X[6841], 5 X[21] - 8 X[16617], 4 X[381] - X[6175], X[382] + 2 X[5428], 2 X[442] - 5 X[3091], 4 X[442] - X[33557], 2 X[442] + X[37433], 2 X[546] + X[16160], 8 X[546] + X[21669], 4 X[546] - X[37230], X[2475] - 7 X[3832], X[2475] + 2 X[37447], 10 X[3091] - X[33557], 5 X[3091] + X[37433], X[3146] + 5 X[15674], X[3146] + 2 X[44238], X[3543] + 2 X[15670], X[3627] + 2 X[10021], 4 X[3628] - X[31651], 5 X[3651] - 8 X[11277], 2 X[3651] - 5 X[31254], 4 X[3830] + 5 X[15675], X[3830] + 2 X[44257], 7 X[3832] + 2 X[37447], 5 X[3843] + X[13743], 8 X[3845] + X[15678], 4 X[3845] + X[28461], 4 X[3850] - X[5499], 7 X[3851] - X[16117], 11 X[3855] - 2 X[37401], 10 X[3858] - X[47032], 5 X[5076] + 4 X[12104], 5 X[6841] - 2 X[16617], 16 X[11277] - 25 X[31254], 3 X[14269] + X[28453], 3 X[15671] - 2 X[21161], 9 X[15671] - 8 X[31650], 5 X[15674] - 2 X[44238], 5 X[15675] - 4 X[28443], 5 X[15675] - 8 X[44257], 7 X[15676] + 5 X[17578], X[15679] - 10 X[41099], X[15680] + 11 X[50689], X[15682] + 3 X[31669], X[15682] + 2 X[44255], 2 X[15687] + X[28460], 4 X[16160] - X[21669], 2 X[16160] + X[37230], 3 X[21161] - 4 X[31650], X[21669] + 2 X[37230], 2 X[28463] + 3 X[38335], 3 X[31669] - 2 X[44255], X[33557] + 2 X[37433], X[44258] + 2 X[46028], 4 X[946] - X[34195], X[962] + 2 X[21677], X[11684] + 8 X[18483], X[11684] + 2 X[49177], 4 X[18483] - X[49177], X[5691] + 2 X[35016], X[12528] + 2 X[39772], 4 X[6701] - X[16143], X[7701] + 2 X[16125], 2 X[8261] + X[12688], 4 X[9955] - X[33858], X[10724] + 2 X[35204], X[10733] + 2 X[16164], X[11263] - 4 X[12571], X[16139] + 2 X[22793], X[16159] + 2 X[22798], 4 X[19925] - X[47033], 2 X[31871] + X[47319], X[36990] + 2 X[51729]

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

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

X(52270) = X(2)X(3)∩X(36)X(920)

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

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

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

X(52271) = X(2)X(3)∩X(36)X(1473)

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

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

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

X(52272) = X(2)X(3)∩X(36)X(5693)

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

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

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

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

X(52273) = X(2)X(3)∩X(36)X(846)

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

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

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

X(52274) = X(2)X(3)∩X(32)X(32046)

Barycentrics    a^2*(2*a^8*b^2 - 6*a^6*b^4 + 6*a^4*b^6 - 2*a^2*b^8 + 2*a^8*c^2 - 8*a^6*b^2*c^2 + 3*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - b^8*c^2 - 6*a^6*c^4 + 3*a^4*b^2*c^4 + b^6*c^4 + 6*a^4*c^6 + 4*a^2*b^2*c^6 + b^4*c^6 - 2*a^2*c^8 - b^2*c^8) : :
X(52274) = 5 X[3] - X[31952]

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

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

X(52275) = X(2)X(3)∩X(32)X(1993)

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

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

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

X(52276) = X(2)X(3)∩X(32)X(7592)

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

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

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

X(52277) = X(2)X(3)∩X(32)X(11402)

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

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

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

X(52278) = X(2)X(3)∩X(32)X(12161)

Barycentrics    a^2*(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10 - 5*a^8*c^2 + 12*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 4*a^2*b^6*c^2 + b^8*c^2 + 10*a^6*c^4 - 4*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 10*a^4*c^6 - 4*a^2*b^2*c^6 + 5*a^2*c^8 + b^2*c^8 - c^10) : :
X(52278) = X[3] - 3 X[35302]

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

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

X(52279) = X(2)X(3)∩X(32)X(5890)

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

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

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

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

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

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

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

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

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 4*b^2*c^2 + c^4) : :
X(52281) = X[264] + 2 X[6748], 2 X[264] + X[27377], 4 X[6748] - X[27377]

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

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

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

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 - 2*b^2*c^2 + 2*c^4) : :
X(52282) = 2 X[4] + X[37200], 2 X[53] + X[317], 4 X[53] - X[9308], 2 X[317] + X[9308]

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^2 + 3*b^2 + 3*c^2) : :
X(52285) = 3 X[2] - 5 X[5133], 9 X[2] - 5 X[6636], 6 X[2] - 5 X[7499], 9 X[2] - 10 X[11548], 3 X[4] + X[13596], X[3529] - 5 X[35921], 7 X[3851] - 5 X[37347], 3 X[5133] - X[6636], 3 X[5133] - 2 X[11548], 2 X[6636] - 3 X[7499], 3 X[7499] - 4 X[11548], 5 X[7527] + 7 X[50688]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 5*b^2*c^2 + c^4) : :
X(52294) = X[3] - 7 X[7545], 2 X[3] - 7 X[13595]

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

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

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

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

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

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

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

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - 4*a^4*b^2 + 5*a^2*b^4 - 2*b^6 - 4*a^4*c^2 + 4*a^2*b^2*c^2 + 2*b^4*c^2 + 5*a^2*c^4 + 2*b^2*c^4 - 2*c^6) : :
X(52296) = 8 X[140] - 3 X[38438]

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

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

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

Barycentrics    (4*a^2 - 3*b^2 - 3*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :
X(52297) = 3 X[4] + 7 X[35472]

X(52297) lies on these lines: {2, 3}, {33, 5326}, {34, 7294}, {125, 10192}, {141, 44102}, {184, 47296}, {232, 3054}, {343, 5972}, {590, 13937}, {615, 13884}, {1112, 3917}, {1495, 23332}, {1514, 11204}, {1698, 12135}, {1829, 19862}, {1843, 40670}, {1862, 31235}, {1974, 34573}, {2052, 11668}, {2501, 10190}, {2883, 43903}, {3055, 10311}, {3589, 8541}, {3618, 11405}, {3619, 19118}, {3634, 11363}, {3819, 44084}, {5095, 22165}, {5186, 31274}, {5410, 32786}, {5411, 32785}, {5412, 32790}, {5413, 32789}, {5550, 11396}, {6688, 47328}, {7749, 14581}, {8024, 34336}, {8167, 11383}, {8739, 23302}, {8740, 23303}, {10169, 32113}, {10189, 47217}, {10278, 47627}, {10625, 43823}, {11216, 47279}, {11245, 17809}, {11402, 37643}, {11550, 15448}, {11614, 33842}, {12133, 14855}, {12143, 31239}, {12834, 14389}, {13148, 38795}, {13366, 13567}, {14165, 14569}, {14975, 17122}, {15004, 23292}, {15067, 52000}, {15108, 18947}, {15151, 17853}, {15471, 50991}, {16318, 37637}, {17005, 27377}, {19596, 23300}, {23291, 26864}, {23328, 51403}, {26913, 48906}, {26917, 31804}, {26926, 31267}, {31253, 49542}, {34481, 47298}, {37633, 44097}, {37775, 43329}, {37776, 43328}, {39176, 40559}, {40938, 44467}, {41585, 48310}, {41587, 43839}

X(52297) = complement of X(30744)
X(52297) = polar conjugate of the isotomic conjugate of X(3630)
X(52297) = polar conjugate of the isogonal conjugate of X(15513)
X(52297) = X(15513)-cross conjugate of X(3630)
X(52297) = barycentric product X(i)*X(j) for these {i,j}: {4, 3630}, {264, 15513}
X(52297) = barycentric quotient X(i)/X(j) for these {i,j}: {3630, 69}, {15513, 3}
X(52297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 5159}, {2, 468, 427}, {2, 5020, 37454}, {2, 6353, 5094}, {2, 6676, 30739}, {2, 6677, 37439}, {2, 7493, 30771}, {2, 7494, 31255}, {2, 37453, 468}, {2, 38282, 25}, {2, 40132, 7539}, {5, 34477, 44249}, {5, 44234, 44211}, {25, 5094, 7378}, {25, 7378, 428}, {25, 37453, 38282}, {25, 38282, 468}, {140, 7505, 235}, {140, 37942, 378}, {378, 7505, 37942}, {378, 37942, 235}, {381, 35486, 37931}, {428, 468, 6353}, {428, 5094, 427}, {547, 37935, 4}, {631, 6623, 11410}, {1596, 11539, 37118}, {1656, 3147, 3575}, {2072, 34351, 44239}, {3090, 3515, 23047}, {5000, 5001, 3627}, {5004, 5005, 35497}, {5066, 37934, 35480}, {5071, 37460, 18386}, {5094, 6353, 428}, {6353, 7378, 25}, {6622, 10303, 3516}, {6623, 11410, 1885}, {6639, 16238, 7399}, {6676, 37911, 2}, {7493, 30771, 7667}, {7494, 31255, 43957}, {10018, 14940, 5}, {12100, 37984, 35481}, {12108, 44226, 35477}, {14869, 44960, 3520}, {15699, 37458, 7577}, {16239, 21841, 37119}, {21213, 37454, 427}, {21841, 37119, 1907}, {21844, 35487, 3627}, {34330, 44234, 5}, {37118, 37943, 1596}

X(52298) = BARYCENTRIC PRODUCT X(264)*X(15515)

Barycentrics    (3*a^2 - 4*b^2 - 4*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

X(52298) lies on these lines: {2, 3}, {125, 11402}, {275, 11668}, {599, 11405}, {1112, 11451}, {1180, 44467}, {1506, 3172}, {1698, 11396}, {1853, 26864}, {2904, 15047}, {3167, 23293}, {3763, 12167}, {5050, 26913}, {5090, 19862}, {5095, 51185}, {5326, 11393}, {5410, 8253}, {5411, 8252}, {5422, 15059}, {5550, 12135}, {5650, 47328}, {6090, 21243}, {6403, 44299}, {6697, 19125}, {6723, 12828}, {6746, 7999}, {7140, 17917}, {7294, 11392}, {7703, 35264}, {8541, 21358}, {8791, 39951}, {9308, 17005}, {9777, 26958}, {11363, 34595}, {11408, 43029}, {11409, 43028}, {11426, 26917}, {12174, 13399}, {14389, 40920}, {14975, 17125}, {15153, 19467}, {19118, 47355}, {19347, 23294}, {19357, 32767}, {22112, 44077}, {23292, 26869}, {31489, 45141}, {33565, 44731}, {34397, 43650}, {34469, 43831}, {34573, 41585}, {37679, 44097}, {37682, 44105}

X(52298) = polar conjugate of the isogonal conjugate of X(15515)
X(52298) = barycentric product X(264)*X(15515)
X(52298) = barycentric quotient X(15515)/X(3)
X(52298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 427, 37453}, {2, 5094, 25}, {2, 5159, 31255}, {2, 8889, 468}, {2, 16051, 7499}, {2, 30744, 3}, {2, 30745, 7485}, {2, 30769, 7494}, {2, 30771, 7484}, {2, 30775, 44210}, {2, 31236, 5020}, {3, 7577, 18386}, {140, 7507, 15750}, {381, 37118, 11410}, {427, 468, 6995}, {427, 6995, 5064}, {427, 37453, 25}, {468, 5064, 25}, {468, 8889, 5064}, {1594, 3526, 3515}, {1656, 37119, 1593}, {5000, 5001, 3843}, {5064, 5094, 8889}, {5094, 37453, 427}, {6995, 8889, 427}, {7484, 30771, 32216}, {7577, 13619, 7547}, {7577, 31101, 427}

X(52299) = BARYCENTRIC PRODUCT X(264)*X(15815)

Barycentrics    (3*a^2 - 5*b^2 - 5*c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :

X(52299) lies on these lines: {2, 3}, {112, 15880}, {125, 11427}, {230, 40065}, {262, 38253}, {264, 34803}, {393, 31489}, {459, 14494}, {1007, 32000}, {1235, 8893}, {1249, 3815}, {1285, 15820}, {1829, 19877}, {1853, 39874}, {1870, 5268}, {1899, 44109}, {2052, 10155}, {2356, 25502}, {3087, 37637}, {3624, 7718}, {3763, 41585}, {3819, 6403}, {5090, 5550}, {5272, 6198}, {5622, 15113}, {5702, 9300}, {6036, 20774}, {6340, 7763}, {6688, 12294}, {6776, 23332}, {7713, 51073}, {7716, 51128}, {7998, 47328}, {8280, 13886}, {8281, 13939}, {8541, 21356}, {9342, 11383}, {9744, 41374}, {11160, 11405}, {11396, 46933}, {11430, 18918}, {11433, 34565}, {11550, 35260}, {12135, 46934}, {12828, 15059}, {13854, 39389}, {14826, 45303}, {14853, 26958}, {14912, 23291}, {15153, 18945}, {15302, 40938}, {17811, 39588}, {18388, 18931}, {21243, 37669}, {22110, 38294}, {23293, 37645}, {29639, 38295}, {30792, 44428}, {32001, 34229}, {32064, 44108}, {33522, 51360}, {34595, 49542}, {34986, 50974}, {37687, 44086}, {42391, 44541}

X(52299) = orthocentroidal-circle-inverse of X(38282)
X(52299) = polar-circle-inverse of X(47316)
X(52299) = polar conjugate of X(18845)
X(52299) = isotomic conjugate of the complement of X(41925)
X(52299) = polar conjugate of the isogonal conjugate of X(15815)
X(52299) = X(48)-isoconjugate of X(18845)
X(52299) = X(1249)-Dao conjugate of X(18845)
X(52299) = cevapoint of X(2) and X(41925)
X(52299) = barycentric product X(264)*X(15815)
X(52299) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18845}, {15815, 3}
X(52299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 38282}, {2, 427, 6353}, {2, 858, 7494}, {2, 1368, 631}, {2, 3091, 6677}, {2, 5094, 8889}, {2, 5133, 40132}, {2, 6995, 37453}, {2, 7378, 468}, {2, 7396, 6676}, {2, 8889, 4}, {2, 16419, 3533}, {2, 30744, 16051}, {2, 30769, 1368}, {2, 31074, 7493}, {4, 6143, 3525}, {4, 15702, 186}, {4, 35473, 11001}, {25, 427, 7409}, {125, 11427, 18950}, {378, 3545, 4}, {427, 6353, 4}, {427, 6676, 35480}, {427, 37453, 6995}, {436, 6619, 4}, {468, 7378, 7714}, {631, 1594, 4}, {631, 3090, 7399}, {1656, 3088, 6622}, {3088, 6622, 4}, {3090, 3541, 4}, {3529, 7547, 4}, {3535, 3536, 458}, {5000, 5001, 3832}, {5004, 5005, 38438}, {5094, 5159, 3541}, {6353, 8889, 427}, {6676, 7396, 376}, {6995, 37453, 6353}, {7378, 7714, 4}, {7493, 31074, 44442}, {10565, 34609, 3529}, {23291, 23292, 14912}, {31255, 37454, 2}

X(52300) = BARYCENTRIC PRODUCT X(264)*X(18472)

Barycentrics    2*a^6 - a^4*b^2 - 2*a^2*b^4 + b^6 - a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6 : :
X(52300) = 5 X[632] - 3 X[13629], 5 X[1656] - 3 X[7579], 7 X[3523] - 2 X[35492]

X(52300) lies on these lines: {2, 3}, {17, 37776}, {18, 37775}, {67, 19127}, {76, 7664}, {94, 7607}, {110, 34507}, {111, 7746}, {125, 15080}, {154, 3410}, {184, 41724}, {230, 18573}, {265, 34513}, {323, 44493}, {343, 9544}, {524, 9716}, {542, 38397}, {575, 32225}, {576, 15360}, {599, 11061}, {1209, 26882}, {1352, 35265}, {1383, 7745}, {1495, 18553}, {1503, 7712}, {1648, 39560}, {1649, 44451}, {2888, 9707}, {3053, 9745}, {3066, 7605}, {3448, 6800}, {3580, 8550}, {3620, 41719}, {3934, 10163}, {5012, 43810}, {5467, 7777}, {5480, 48912}, {5486, 37784}, {5640, 25555}, {5642, 40107}, {5971, 7763}, {5972, 7998}, {5987, 52090}, {6031, 7750}, {6032, 7747}, {6103, 22240}, {6689, 9781}, {7578, 7608}, {7703, 29012}, {7711, 7836}, {7749, 10418}, {7752, 26276}, {7755, 9465}, {7768, 26233}, {7806, 46127}, {7810, 9829}, {7812, 51541}, {8547, 25320}, {8743, 40583}, {9076, 14247}, {9143, 15069}, {9159, 22104}, {9464, 32820}, {9717, 16243}, {9970, 50977}, {9999, 32151}, {10192, 37636}, {10545, 38317}, {10546, 24206}, {10990, 11454}, {11002, 14389}, {11064, 33884}, {11412, 44516}, {11422, 41586}, {11459, 16534}, {11464, 30714}, {14169, 40709}, {14170, 40710}, {14643, 33533}, {14683, 26864}, {15056, 32348}, {15066, 17847}, {15072, 20417}, {15074, 45237}, {15513, 39602}, {16981, 47582}, {18312, 47263}, {20481, 44535}, {20791, 44673}, {21243, 26881}, {22352, 26913}, {25563, 52093}, {30516, 44500}, {34504, 42008}, {37688, 44135}, {41939, 44453}, {44010, 47173}, {44386, 47246}

X(52300) = complement of X(31857)
X(52300) = orthoptic-circle-of-Steiner-inellipse-inverse of X(18572)
X(52300) = polar conjugate of the isogonal conjugate of X(18472)
X(52300) = barycentric product X(i)*X(j) for these {i,j}: {264, 18472}, {524, 52189}
X(52300) = barycentric quotient X(i)/X(j) for these {i,j}: {18472, 3}, {52189, 671}
X(52300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 31074}, {2, 23, 5169}, {2, 25, 37353}, {2, 5189, 5094}, {2, 6636, 31101}, {2, 7492, 858}, {2, 7493, 23}, {2, 7494, 15246}, {2, 10565, 7391}, {2, 14002, 5}, {2, 16063, 30745}, {2, 20062, 8889}, {2, 37760, 1995}, {2, 37909, 381}, {2, 37913, 427}, {4, 3523, 14118}, {5, 7426, 14002}, {20, 50007, 3}, {22, 1995, 37972}, {22, 5094, 5189}, {26, 1656, 4}, {468, 6676, 7495}, {468, 7495, 2}, {550, 10024, 4}, {858, 44210, 7492}, {2045, 2046, 7550}, {3580, 13394, 11003}, {5000, 5001, 18559}, {5004, 5005, 18570}, {5068, 31304, 4}, {5094, 5189, 31074}, {6636, 30745, 16063}, {6800, 37638, 3448}, {7485, 37453, 2}, {7505, 47525, 37126}, {7542, 16618, 37118}, {7556, 14002, 23}, {7749, 10418, 39576}, {10018, 34002, 3523}, {14389, 32269, 11002}, {16063, 30745, 31101}, {16618, 37118, 20}, {37347, 44213, 47485}

X(52301) = BARYCENTRIC PRODUCT X(264)*X(21309)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(7*a^2 + b^2 + c^2) : :
X(52301) = 3 X[2] - 4 X[40132]

X(52301) lies on these lines: {2, 3}, {53, 37689}, {98, 16240}, {145, 7713}, {193, 7716}, {232, 14930}, {340, 10513}, {393, 1383}, {1285, 40126}, {1395, 30652}, {1495, 14853}, {1503, 31860}, {1629, 6525}, {1829, 3623}, {1843, 11002}, {1974, 11003}, {1990, 5304}, {2052, 47586}, {2207, 5354}, {2212, 30653}, {2333, 19998}, {3066, 25406}, {3199, 9465}, {3291, 33842}, {3448, 32250}, {3617, 49542}, {3621, 7718}, {4661, 41611}, {5032, 15471}, {5095, 9143}, {5480, 35260}, {5921, 41724}, {6403, 16981}, {6553, 19993}, {6749, 37665}, {6776, 34417}, {6800, 51171}, {7712, 44091}, {7717, 20059}, {7737, 40350}, {8550, 11206}, {8796, 43537}, {9485, 14273}, {9543, 11473}, {10311, 40138}, {11064, 51538}, {11160, 41585}, {11180, 41586}, {11566, 20125}, {12135, 20052}, {12294, 33884}, {14561, 32237}, {14580, 33885}, {14683, 32234}, {14927, 37648}, {15302, 33843}, {16276, 32824}, {16654, 18931}, {18489, 32620}, {20080, 48912}, {21970, 39884}, {22112, 33750}, {23061, 51028}, {26233, 32834}, {31383, 44106}, {32085, 43981}, {32827, 37804}, {35259, 51212}, {35510, 41896}, {36990, 37643}, {37638, 51537}, {43530, 43951}

X(52301) = polar-circle-inverse of X(47311)
X(52301) = polar conjugate of the isogonal conjugate of X(21309)
X(52301) = barycentric product X(264)*X(21309)
X(52301) = barycentric quotient X(i)/X(j) for these {i,j}: {21309, 3}, {44839, 10519}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5059, 16063}, {2, 6995, 7408}, {2, 7408, 7409}, {2, 7492, 15717}, {2, 7519, 3146}, {2, 7533, 5068}, {2, 17578, 31099}, {2, 21734, 7496}, {2, 50689, 5169}, {4, 25, 4232}, {4, 3517, 3523}, {4, 3518, 35486}, {4, 4232, 2}, {4, 5094, 7378}, {4, 6353, 5094}, {4, 7714, 10301}, {4, 10301, 6995}, {4, 18533, 49670}, {4, 21841, 5056}, {4, 35483, 1597}, {4, 35486, 3088}, {4, 37458, 20}, {4, 37984, 3839}, {20, 1995, 2}, {20, 3091, 34664}, {22, 7398, 2}, {25, 428, 6353}, {25, 1596, 26255}, {25, 1598, 1995}, {25, 4186, 7438}, {25, 6756, 7493}, {25, 6995, 2}, {25, 7714, 6995}, {25, 10301, 4}, {25, 18494, 7426}, {382, 44212, 16051}, {428, 5094, 4}, {428, 6353, 7378}, {452, 4239, 2}, {1597, 37934, 35483}, {1598, 37458, 4}, {1995, 37900, 46336}, {3091, 7493, 2}, {3543, 26255, 2}, {3839, 7426, 2}, {4232, 6995, 4}, {5000, 5001, 3524}, {5056, 7495, 2}, {5056, 10565, 7495}, {5177, 26256, 2}, {5480, 41424, 35260}, {6353, 7378, 2}, {6995, 7378, 428}, {6995, 7487, 7519}, {6997, 7495, 5056}, {6997, 10565, 2}, {7500, 13595, 2}, {7500, 16063, 5059}, {7519, 14002, 2}, {11284, 37899, 376}, {18494, 37984, 4}, {26257, 32971, 2}, {30769, 50688, 31133}, {34484, 37122, 3089}, {35483, 37934, 10304}, {37900, 46336, 20}, {42789, 42790, 41463}






leftri  4th tangent line ot two inconics: X(52302) - X(52343)  rightri

This preamble and centers X(52302)-X(52343) were contributed by César Eliud Lozada, November 5, 2022.

Let ABC be a triangle, P1 = x1 : y1 : z1, P2 = x2 : y2 : z2 (barycentrics) two distinct points on the plane of ABC and not on the sidelines of ABC. Let q1, q2 be the inconics of ABC with perspectors P1 and P2, respectively.

There exists a 4th-common tangent line τ to q1 and q2, other than the sidelines of ABC. It can be deduced that:

The preceding coordinates have the following equivalences leading to geometric constructions of τ, T1 and T2:

  1. Q(τ) is the perspector of the circumconic of ABC passing through P1 and P2.

    Consequently, if U' and U" are two given points and q0 is the circumconic {{A,B,C,U',U"}}, then, for any pair of points P', P" on q0, the inconics with perspectors P', P" have the same 4th common tangent line and it is tripolar of the perspector of q0.

  2. T1 is the perspector of the cevian-triangle-of-P1 and the side-triangle-of-(cevian-triangle-of-P1 and cevian-triangle-of-P2), which are always`perspective. (Note: the side-triangle AsBsCs of two triangles A'B'C' and A"B"C" has A-vertex As = B'C' ∩ B"C" and cyclically Bs and Cs).

    Symmetrically, T2 is the perspector of the cevian-triangle-of-P2 and the side-triangle-of-(cevian-triangle-of-P1 and cevian-triangle-of-P2).

    See centers X(41211) to X(41224), which are related to analogous perspectors.
  3. T1 is also the P1-ceva conjugate of-Q(τ) and T2 is also the P2-ceva conjugate of-Q(τ), which means that T1 is the perspector(cevian-of-P1, anticevian-of-Q(τ)) and T2 is the perspector(cevian-of-P2, anticevian-of-Q(τ)).

For shortening, the inconic with perspector U will be denoted in this section as IwP(U).

The appearance of (i, j, m, n, u) in the following (reduced) list indicates that the 4th common tangent line of IwP(X(i)) and IwP(X(j)) touches them at X(m) and X(n), respectively, and has trilinear pole X(u):

(1, 2, 244, 1015, 513), (1, 3, 2638, 3270, 652), (1, 4, 2310, 3270, 650), (1, 5, 41211, 41218, 2600), (1, 6, 3248, 1015, 649), (1, 7, 2310, 3022, 650), (1, 8, 2310, 3271, 650), (1, 10, 2643, 3122, 661), (1, 11, 52302, 52303, 46384), (2, 3, 35071, 2972, 520), (2, 4, 115, 125, 523), (2, 5, 39019, 35442, 6368), (2, 6, 1084, 3124, 512), (2, 7, 1086, 11, 514), (2, 8, 1146, 11, 522), (2, 10, 115, 3120, 523), (2, 11, 35509, 52304, 52305), (3, 4, 20975, 34980, 647), (3, 5, 41212, 41219, 17434), (3, 6, 20975, 3269, 647), (3, 7, 41214, 15616, 52306), (3, 8, 41215, 41220, 52307), (3, 10, 52308, 52309, 52310), (3, 11, 52311, 52312, 52313), (4, 5, 24862, 41221, 12077), (4, 6, 34980, 3269, 647), (4, 7, 3270, 3022, 650), (4, 8, 3270, 3271, 650), (4, 10, 125, 3120, 523), (4, 11, 52314, 52315, 52316), (5, 6, 41213, 41222, 52317), (5, 7, 41216, 31889, 52318), (5, 8, 41217, 41223, 52319), (5, 10, 52320, 52321, 52322), (5, 11, 52323, 52324, 52325), (6, 7, 35505, 15615, 665), (6, 8, 35506, 41224, 52326), (6, 10, 52327, 52328, 42664), (6, 11, 52329, 52330, 52331), (7, 8, 3022, 3271, 650), (7, 10, 31890, 52332, 4841), (7, 11, 31891, 52333, 52334), (8, 10, 4092, 52335, 3700), (8, 11, 52336, 52337, 52338), (10, 11, 52339, 52340, 52341), (13, 14, 30465, 30468, 523), (15, 16, 52342, 52343, 526)

underbar

X(52302) = TOUCHPOINT OF IwP( X(1) ) AND THE 4th-COMMON-TANGENT OF IwP( X(1) ) AND IwP( X(11) )

Barycentrics    a*(-a+b+c)^4*(b-c)^6*(a^2-b^2+b*c-c^2)^2 : :

This 4th-common-tangent touches IwP( X(11) ) at X(52303) and has trilinear pole X(46384).

X(52302) lies on the inconics with perspectors X(n) for n in {1, 1090} and these lines: {2310, 11193}

X(52302) = barycentric product X(1090)*X(35128)
X(52302) = trilinear product X(i)*X(j) for these {i, j}: {11, 52303}, {54, 52323}
X(52302) = trilinear square of X(46384)
X(52302) = crosssum of X(654) and X(2957)
X(52302) = X(1)-Ceva conjugate of-X(46384)

X(52303) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(1) ) AND IwP( X(11) )

Barycentrics    a^2*(b-c)^4*(-a+b+c)^3*(a^2-b^2+b*c-c^2)^2 : :

This 4th-common-tangent touches IwP( X(1) ) at X(52302) and has trilinear pole X(46384).

X(52303) lies on the inconics with perspectors X(n) for n in {11, 59, 3025} and these lines: {1362, 34586}, {1397, 34179}, {3251, 42771}, {3328, 3675}

X(52303) = barycentric product X(i)*X(j) for these {i, j}: {11, 35128}, {1090, 34544}, {1146, 3025}
X(52303) = trilinear product X(i)*X(j) for these {i, j}: {59, 52302}, {215, 1090}, {654, 46384}
X(52303) = X(i)-Ceva conjugate of-X(j) for these (i, j): (11, 46384), (59, 654)

X(52304) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(2) ) AND IwP( X(11) )

Barycentrics    (b-c)^4*((b+c)*a-b^2-c^2)^2*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(2) ) at X(35509) and has trilinear pole X(52305).

X(52304) lies on the inconics with perspectors X(n) for n in {11, 3323, 4998} and these lines: {2, 35313}, {7, 10001}, {8, 6631}, {11, 885}, {56, 18343}, {1259, 47048}, {1317, 50441}, {1566, 3328}, {3126, 3675}, {4996, 8299}, {5532, 21132}, {6068, 16593}, {7336, 42462}, {14393, 35509}, {20504, 42770}, {35967, 52315}

X(52304) = complement of X(35313)
X(52304) = barycentric product X(i)*X(j) for these {i, j}: {11, 35094}, {918, 52305}, {1146, 3323}
X(52304) = trilinear product X(1090)*X(1362)
X(52304) = perspector of the circumconic {{A, B, C, X(918), X(52305)}}
X(52304) = center of the circumconic {{A, B, C, X(11), X(3035)}}
X(52304) = Cevapoint of X(918) and X(4998)
X(52304) = crosssum of X(918) and X(3271)
X(52304) = X(11)-Ceva conjugate of-X(52305)
X(52304) = X(i)-Dao conjugate of-X(j) for these (i, j): (918, 4998), (926, 6066)

X(52305) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(2) ) AND IwP( X(11) )

Barycentrics    (b-c)^3*((b+c)*a-b^2-c^2)*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(2) ) and IwP( X(11) ) at X(35509) and X(52304), respectively.

X(52305) lies on the inconics with perspectors X(n) for n in {885, 43042, 50333} and these lines: {30, 511}, {242, 44428}, {659, 3220}, {693, 40704}, {764, 23760}, {1090, 5532}, {1456, 30725}, {1738, 10015}, {2400, 43930}, {3328, 4542}, {3685, 3904}, {3717, 50333}, {3766, 7112}, {3776, 15280}, {4077, 23742}, {4105, 48117}, {4530, 14393}, {4867, 49276}, {6545, 23615}, {6546, 11124}, {6608, 47930}, {14475, 14476}, {14505, 23100}, {14887, 50351}, {23761, 40166}, {36236, 36802}, {44009, 45290}

X(52305) = complementary conjugate of X(35967)
X(52305) = barycentric product X(i)*X(j) for these {i, j}: {11, 918}, {241, 42455}, {518, 40166}, {665, 34387}, {666, 52304}, {693, 17435}
X(52305) = barycentric quotient X(i)/X(j) for these (i, j): (11, 666), (244, 36146), (514, 39293), (518, 31615), (650, 5377), (665, 59)
X(52305) = trilinear product X(i)*X(j) for these {i, j}: {11, 2254}, {241, 42462}, {244, 50333}, {514, 17435}, {518, 21132}, {522, 3675}
X(52305) = trilinear quotient X(i)/X(j) for these (i, j): (11, 36086), (241, 4619), (244, 32735), (522, 5377), (665, 2149), (693, 39293)
X(52305) = infinite point of tripolar of X(i) for these i: {11, 23989, 31611, 31619, 40166, 46395}
X(52305) = trilinear pole of the line {14393, 35509}
X(52305) = Cevapoint of X(i) and X(j) for these (i, j): {11, 885}, {1086, 2400}
X(52305) = crossdifference of every pair of points on line {X(6), X(59)}
X(52305) = crosspoint of X(i) and X(j) for these (i, j): {2, 35509}, {11, 52304}
X(52305) = crosssum of X(i) and X(j) for these (i, j): {11, 2283}, {125, 7437}, {1086, 2426}
X(52305) = X(1110)-anticomplementary conjugate of-X(14732)
X(52305) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 35509), (4, 35967), (11, 52304), (514, 1566), (660, 46100)
X(52305) = X(i)-complementary conjugate of-X(j) for these (i, j): (1, 35967), (31, 35509), (1110, 1566)
X(52305) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 5377), (513, 32735), (514, 927), (522, 36802), (650, 666)
X(52305) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 36086}, {109, 5377}, {294, 4619}, {666, 2149}, {692, 39293}
X(52305) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (11, 666), (244, 36146), (514, 39293), (518, 31615), (650, 5377)

X(52306) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(7) )

Barycentrics    a^2*((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(-a+b+c)*(-a^2+b^2+c^2)*(b-c) : :

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(7) ) at X(41214) and X(15616), respectively.

X(52306) lies on these lines: {241, 514}, {520, 647}, {649, 17412}, {654, 4282}, {656, 40628}, {822, 8677}, {3709, 46389}, {6586, 14298}, {33525, 50354}

X(52306) = barycentric product X(i)*X(j) for these {i, j}: {78, 50354}, {283, 23752}, {348, 33525}, {442, 23189}, {521, 942}, {525, 46882}
X(52306) = barycentric quotient X(i)/X(j) for these (i, j): (184, 15439), (521, 40422), (603, 36048), (649, 40573), (650, 40447), (652, 40435)
X(52306) = trilinear product X(i)*X(j) for these {i, j}: {77, 33525}, {219, 50354}, {514, 23207}, {520, 46884}, {521, 2260}, {522, 14597}
X(52306) = trilinear quotient X(i)/X(j) for these (i, j): (48, 15439), (216, 35320), (222, 36048), (513, 40573), (521, 40435), (522, 40447)
X(52306) = trilinear pole of the line {15616, 41214}
X(52306) = perspector of the circumconic {{A, B, C, X(3), X(7)}}
X(52306) = tripole of the tangent to incircle at X(15616)
X(52306) = crossdifference of every pair of points on line {X(4), X(12)}
X(52306) = crosspoint of X(i) and X(j) for these (i, j): {3, 41214}, {7, 15616}
X(52306) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 39007), (3, 41214), (7, 15616), (81, 1364), (1214, 7004)
X(52306) = X(31)-complementary conjugate of-X(39007)
X(52306) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 40447), (442, 6335), (942, 4552)
X(52306) = X(i)-isoconjugate-of-X(j) for these {i, j}: {92, 15439}, {100, 40573}, {108, 40435}, {109, 40447}, {275, 35320}
X(52306) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (184, 15439), (521, 40422), (603, 36048), (649, 40573), (650, 40447)
X(52306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (647, 652, 52307), (652, 1459, 36054)

X(52307) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(8) )

Barycentrics    a^2*(-a^2+b^2+c^2)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*(b-c)*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(8) ) at X(41215) and X(41220), respectively.

X(52307) lies on these lines: {101, 109}, {520, 647}, {522, 650}, {1214, 39470}, {1769, 3310}, {2431, 14578}, {2631, 18591}, {6589, 14298}, {7004, 7117}, {7069, 30692}, {7180, 46389}, {22055, 22346}, {43060, 51656}

X(52307) = isogonal conjugate of the polar conjugate of X(2804)
X(52307) = barycentric product X(i)*X(j) for these {i, j}: {3, 2804}, {8, 8677}, {63, 46393}, {78, 1769}, {100, 35014}, {212, 36038}
X(52307) = barycentric quotient X(i)/X(j) for these (i, j): (31, 36110), (32, 32702), (48, 37136), (55, 1309), (101, 39294), (184, 2720)
X(52307) = trilinear product X(i)*X(j) for these {i, j}: {3, 46393}, {9, 8677}, {48, 2804}, {78, 3310}, {101, 35014}, {212, 10015}
X(52307) = trilinear quotient X(i)/X(j) for these (i, j): (3, 37136), (6, 36110), (9, 1309), (31, 32702), (41, 14776), (48, 2720)
X(52307) = trilinear pole of the line {41215, 41220}
X(52307) = perspector of the circumconic {{A, B, C, X(3), X(8)}}
X(52307) = tripole of the tangent to Mandart inellipse at X(41220)
X(52307) = intersection, other than A, B, C, of circumconics {{A, B, C, X(100), X(1364)}} and {{A, B, C, X(101), X(3239)}}
X(52307) = Cevapoint of X(i) and X(j) for these (i, j): {72, 50039}, {219, 5548}
X(52307) = crossdifference of every pair of points on line {X(4), X(11)}
X(52307) = crosspoint of X(i) and X(j) for these (i, j): {3, 41215}, {8, 41220}
X(52307) = crosssum of X(i) and X(j) for these (i, j): {72, 21786}, {212, 44428}, {219, 30725}, {1364, 14571}
X(52307) = X(63)-beth conjugate of-X(39470)
X(52307) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 39004), (3, 41215), (8, 41220), (219, 38353), (906, 47408)
X(52307) = X(31)-complementary conjugate of-X(39004)
X(52307) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 16082), (206, 32702), (1145, 6335)
X(52307) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 36110}, {4, 37136}, {34, 13136}, {57, 1309}, {75, 32702}
X(52307) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 36110), (32, 32702), (48, 37136), (55, 1309), (101, 39294)
X(52307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (647, 652, 52306), (652, 10397, 36054)

X(52308) = TOUCHPOINT OF IwP( X(3) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(10) )

Barycentrics    a^2*(b^2-c^2)^2*(-a*b*c-b^3-c^3+(b+c)*a^2)^2*(-a^2+b^2+c^2) : :

This 4th-common-tangent touches IwP( X(10) ) at X(52309) and has trilinear pole X(52310).

X(52308) lies on the inconics with perspectors X(n) for n in {3, 7141} and these lines: {}

X(52308) = X(3)-Ceva conjugate of-X(52310)

X(52309) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(10) )

Barycentrics    a^4*(b^2-c^2)*(b-c)*(-a^2+b^2+c^2)^2*(-a*b*c-b^3-c^3+(b+c)*a^2)^2 : :

This 4th-common-tangent touches IwP( X(3) ) at X(52308) and has trilinear pole X(52310).

X(52309) lies on the inconic with perspector X(10) and these lines: {}

X(52309) = barycentric product X(1790)*X(52308)
X(52309) = trilinear product X(1437)*X(52308)
X(52309) = X(10)-Ceva conjugate of-X(52310)

X(52310) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(10) )

Barycentrics    a^2*(-a^2+b^2+c^2)*((b+c)*a^2-b*c*a-b^3-c^3)*(b^2-c^2) : :

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(10) ) at X(52308) and X(52309), respectively.

X(52310) lies on these lines: {520, 647}, {523, 661}, {649, 924}, {2294, 6587}, {2501, 46393}, {43708, 52222}

X(52310) = barycentric product X(i)*X(j) for these {i, j}: {72, 21189}, {124, 23067}, {306, 6589}, {512, 51612}, {520, 17555}, {525, 573}
X(52310) = barycentric quotient X(i)/X(j) for these (i, j): (42, 26704), (71, 44765), (228, 36050), (573, 648), (647, 13478), (652, 19607)
X(52310) = trilinear product X(i)*X(j) for these {i, j}: {71, 21189}, {72, 6589}, {521, 40590}, {523, 22134}, {525, 3185}, {573, 656}
X(52310) = trilinear quotient X(i)/X(j) for these (i, j): (37, 26704), (71, 36050), (72, 44765), (228, 32653), (521, 19607), (525, 2995)
X(52310) = trilinear pole of the line {52308, 52309}
X(52310) = perspector of the circumconic {{A, B, C, X(3), X(10)}}
X(52310) = Cevapoint of X(i) and X(j) for these (i, j): {71, 4559}, {656, 6332}
X(52310) = crossdifference of every pair of points on line {X(4), X(58)}
X(52310) = crosspoint of X(i) and X(j) for these (i, j): {3, 52308}, {10, 52309}
X(52310) = crosssum of X(i) and X(j) for these (i, j): {71, 4560}, {656, 32674}
X(52310) = X(i)-Ceva conjugate of-X(j) for these (i, j): (3, 52308), (10, 52309)
X(52310) = X(i)-Dao conjugate of-X(j) for these (i, j): (65, 653), (124, 27), (125, 13478)
X(52310) = X(i)-isoconjugate-of-X(j) for these {i, j}: {27, 36050}, {28, 44765}, {81, 26704}, {108, 19607}, {112, 2995}
X(52310) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (42, 26704), (71, 44765), (228, 36050), (573, 648), (647, 13478)

X(52311) = TOUCHPOINT OF IwP( X(3) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(11) )

Barycentrics    a^2*(-a+b+c)^4*(b-c)^6*((b+c)*a^4-2*(b^3+c^3)*a^2+2*b^2*c^2*a+(b^4-c^4)*(b-c))^2*(-a^2+b^2+c^2) : :

This 4th-common-tangent touches IwP( X(11) ) at X(52312) and has trilinear pole X(52313).

X(52311) lies on the inconic with perspector X(3) and these lines: {}

X(52311) = X(3)-Ceva conjugate of-X(52313)

X(52312) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(11) )

Barycentrics    a^4*(-a^2+b^2+c^2)^2*(b-c)^4*(-a+b+c)^3*((b+c)*a^4-2*(b^3+c^3)*a^2+2*b^2*c^2*a+(b^4-c^4)*(b-c))^2 : :

This 4th-common-tangent touches IwP( X(3) ) at X(52311) and has trilinear pole X(52313).

X(52312) lies on the inconic with perspector X(11) and these lines: {}

X(52312) = X(11)-Ceva conjugate of-X(52313)

X(52313) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(3) ) AND IwP( X(11) )

Barycentrics    a^2*(-a+b+c)^2*((b+c)*a^4-2*(b^3+c^3)*a^2+2*b^2*c^2*a+(b^4-c^4)*(b-c))*(-a^2+b^2+c^2)*(b-c)^3 : :

This 4th-common-tangent touches IwP( X(3) ) and IwP( X(11) ) at X(52311) and X(52312), respectively.

X(52313) lies on these lines: {520, 647}, {4530, 14393}

X(52313) = trilinear pole of the line {52311, 52312}
X(52313) = perspector of the circumconic {{A, B, C, X(3), X(11)}}
X(52313) = crossdifference of every pair of points on line {X(4), X(59)}
X(52313) = crosspoint of X(i) and X(j) for these (i, j): {3, 52311}, {11, 52312}
X(52313) = X(i)-Ceva conjugate of-X(j) for these (i, j): (3, 52311), (11, 52312)

X(52314) = TOUCHPOINT OF IwP( X(4) ) AND THE 4th-COMMON-TANGENT OF IwP( X(4) ) AND IwP( X(11) )

Barycentrics    ((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))^2*(-a^2+b^2+c^2)*(b-c)^6*(-a+b+c)^4 : :

This 4th-common-tangent touches IwP( X(11) ) at X(52315) and has trilinear pole X(52316).

X(52314) lies on the orthic inconic and these lines: {3270, 11193}

X(52314) = trilinear product X(1090)*X(41215)
X(52314) = X(4)-Ceva conjugate of-X(52316)

X(52315) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(4) ) AND IwP( X(11) )

Barycentrics    (b-c)^4*(-a+b+c)^3*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))^2 : :

This 4th-common-tangent touches IwP( X(4) ) at X(52314) and has trilinear pole X(52316).

X(52315) lies on the inconics with perspectors X(n) for n in {11, 3326} and these lines: {11, 2401}, {1360, 1846}, {3259, 3328}, {5532, 23615}, {35967, 52304}

X(52315) = barycentric product X(1146)*X(3326)
X(52315) = X(11)-Ceva conjugate of-X(52316)

X(52316) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(4) ) AND IwP( X(11) )

Barycentrics    (-a+b+c)^2*(b-c)^3*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

This 4th-common-tangent touches IwP( X(4) ) and IwP( X(11) ) at X(52314) and X(52315), respectively.

X(52316) lies on the inconics with perspectors X(n) for n in {2401, 10015} and these lines: {230, 231}, {654, 7297}, {2183, 35013}, {4530, 14393}, {10015, 22464}, {21801, 42763}

X(52316) = complement of the isotomic conjugate of X(43353)
X(52316) = barycentric product X(i)*X(j) for these {i, j}: {11, 2804}, {517, 42455}, {522, 35015}, {523, 14010}, {908, 42462}, {1146, 10015}
X(52316) = barycentric quotient X(1146)/X(13136)
X(52316) = trilinear product X(i)*X(j) for these {i, j}: {11, 46393}, {517, 42462}, {650, 35015}, {661, 14010}, {1021, 42759}, {1090, 2427}
X(52316) = trilinear quotient X(i)/X(j) for these (i, j): (11, 37136), (517, 4619), (1090, 2401), (1146, 36037)
X(52316) = trilinear pole of the line {52314, 52315}
X(52316) = perspector of the circumconic {{A, B, C, X(4), X(11)}}
X(52316) = tripole of the tangent to orthic inconic at X(52314)
X(52316) = Cevapoint of X(11) and X(2401)
X(52316) = crossdifference of every pair of points on line {X(3), X(59)}
X(52316) = crosspoint of X(i) and X(j) for these (i, j): {4, 52314}, {11, 52315}
X(52316) = crosssum of X(11) and X(2427)
X(52316) = X(i)-Ceva conjugate of-X(j) for these (i, j): (4, 52314), (11, 52315), (514, 3259)
X(52316) = X(i)-Dao conjugate of-X(j) for these (i, j): (522, 13136), (1145, 31615)
X(52316) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 37136}, {104, 4619}
X(52316) = X(1146)-reciprocal conjugate of-X(13136)
X(52316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (42462, 46384, 52338), (52334, 52338, 46384)

X(52317) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(6) )

Barycentrics    a^2*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*((b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2) : :
X(52317) = 4*X(6753)-3*X(14397) = 3*X(14397)-2*X(30451)

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(6) ) at X(41213) and X(41222), respectively.

X(52317) lies on the inconic with perspector X(32692) and these lines: {4, 15422}, {6, 47331}, {53, 35361}, {137, 3124}, {187, 237}, {686, 2501}, {924, 6753}, {1510, 2623}, {2081, 2600}

X(52317) = reflection of X(i) in X(j) for these (i, j): (2623, 16040), (30451, 6753)
X(52317) = Gibert-circumtangential conjugate of X(32692)
X(52317) = barycentric product X(i)*X(j) for these {i, j}: {5, 924}, {24, 6368}, {47, 2618}, {51, 6563}, {52, 523}, {136, 23181}
X(52317) = barycentric quotient X(i)/X(j) for these (i, j): (5, 46134), (24, 18831), (32, 32692), (51, 925), (52, 99), (53, 30450)
X(52317) = trilinear product X(i)*X(j) for these {i, j}: {47, 12077}, {52, 661}, {467, 810}, {523, 2180}, {563, 23290}, {571, 2618}
X(52317) = trilinear quotient X(i)/X(j) for these (i, j): (31, 32692), (47, 18315), (51, 36145), (52, 662), (467, 811), (512, 2168)
X(52317) = trilinear pole of the line {41213, 41222}
X(52317) = perspector of the circumconic {{A, B, C, X(5), X(6)}}
X(52317) = tripole of the tangent to Brocard inellipse at X(41222)
X(52317) = intersection, other than A, B, C, of circumconics {{A, B, C, X(5), X(45938)}} and {{A, B, C, X(51), X(52032)}}
X(52317) = Cevapoint of X(i) and X(j) for these (i, j): {6, 32692}, {51, 23181}
X(52317) = crossdifference of every pair of points on line {X(2), X(54)}
X(52317) = crosspoint of X(i) and X(j) for these (i, j): {5, 41213}, {6, 41222}
X(52317) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 41213), (6, 41222), (571, 47421)
X(52317) = X(i)-Dao conjugate of-X(j) for these (i, j): (115, 34385), (134, 1993), (135, 275), (137, 5392), (139, 264)
X(52317) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 32692}, {91, 18315}, {95, 36145}, {96, 662}, {99, 2168}
X(52317) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 46134), (24, 18831), (32, 32692), (51, 925), (52, 99)
X(52317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2081, 12077, 17434), (6753, 30451, 14397)

X(52318) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(7) )

Barycentrics    (b-c)*(-a+b+c)*(2*a^3+(b+c)*a^2-2*(b^2-b*c+c^2)*a-(b^2-c^2)*(b-c))*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(7) ) at X(41216) and X(31889), respectively.

X(52318) lies on these lines: {241, 514}, {2081, 2600}

X(52318) = trilinear pole of the line {31889, 41216}
X(52318) = perspector of the circumconic {{A, B, C, X(5), X(7)}}
X(52318) = tripole of the tangent to incircle at X(31889)
X(52318) = crossdifference of every pair of points on line {X(54), X(55)}
X(52318) = crosspoint of X(i) and X(j) for these (i, j): {5, 41216}, {7, 31889}
X(52318) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 41216), (7, 31889)
X(52318) = {X(2600), X(12077)}-harmonic conjugate of X(52319)

X(52319) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(8) )

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^3-(b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))*(b-c)*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(8) ) at X(41217) and X(41223), respectively.

X(52319) lies on these lines: {522, 650}, {2081, 2600}

X(52319) = trilinear pole of the line {41217, 41223}
X(52319) = perspector of the circumconic {{A, B, C, X(5), X(8)}}
X(52319) = tripole of the tangent to Mandart inellipse at X(41223)
X(52319) = crossdifference of every pair of points on line {X(54), X(56)}
X(52319) = crosspoint of X(i) and X(j) for these (i, j): {5, 41217}, {8, 41223}
X(52319) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 41217), (8, 41223)
X(52319) = {X(2600), X(12077)}-harmonic conjugate of X(52318)

X(52320) = TOUCHPOINT OF IwP( X(5) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(10) )

Barycentrics    (b^2-c^2)^2*(a^3-(b^2-b*c+c^2)*a-(b+c)*b*c)^2*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

This 4th-common-tangent touches IwP( X(10) ) at X(52321) and has trilinear pole X(52322).

X(52320) lies on the inconic with perspector X(5) and these lines: {}

X(52320) = X(5)-Ceva conjugate of-X(52322)

X(52321) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(10) )

Barycentrics    (b^2-c^2)*(b-c)*((b^2+c^2)*a^2-(b^2-c^2)^2)^2*(a^3-(b^2-b*c+c^2)*a-(b+c)*b*c)^2 : :

This 4th-common-tangent touches IwP( X(5) ) at X(52320) and has trilinear pole X(52322).

X(52321) lies on the inconic with perspector X(10) and these lines: {3120, 8902}

X(52321) = X(10)-Ceva conjugate of-X(52322)

X(52322) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(10) )

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^3-(b^2-b*c+c^2)*a-(b+c)*b*c)*(b^2-c^2) : :

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(10) ) at X(52320) and X(52321), respectively.

X(52322) lies on these lines: {523, 661}, {2081, 2600}

X(52322) = barycentric product X(572)*X(18314)
X(52322) = barycentric quotient X(572)/X(18315)
X(52322) = trilinear product X(572)*X(2618)
X(52322) = trilinear quotient X(572)/X(36134)
X(52322) = trilinear pole of the line {52320, 52321}
X(52322) = perspector of the circumconic {{A, B, C, X(5), X(10)}}
X(52322) = crossdifference of every pair of points on line {X(54), X(58)}
X(52322) = crosspoint of X(i) and X(j) for these (i, j): {5, 52320}, {10, 52321}
X(52322) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 52320), (10, 52321)
X(52322) = X(137)-Dao conjugate of-X(2051)
X(52322) = X(572)-reciprocal conjugate of-X(18315)

X(52323) = TOUCHPOINT OF IwP( X(5) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(11) )

Barycentrics    (-a+b+c)^4*(b-c)^6*(a^2-b^2+b*c-c^2)^2*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

This 4th-common-tangent touches IwP( X(11) ) at X(52324) and has trilinear pole X(52325).

X(52323) lies on the inconic with perspector X(5) and these lines: {}

X(52323) = trilinear product X(i)*X(j) for these {i, j}: {5, 52302}, {1090, 41218}
X(52323) = X(5)-Ceva conjugate of-X(52325)

X(52324) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(11) )

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)^2*(b-c)^4*(-a+b+c)^3*(a^2-b^2+b*c-c^2)^2 : :

This 4th-common-tangent touches IwP( X(5) ) at X(52323) and has trilinear pole X(52325).

X(52324) lies on the inconic with perspector X(11) and these lines: {}

X(52324) = trilinear product X(i)*X(j) for these {i, j}: {11, 41211}, {1087, 52303}
X(52324) = X(11)-Ceva conjugate of-X(52325)

X(52325) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(5) ) AND IwP( X(11) )

Barycentrics    (a^2-b^2+b*c-c^2)*(-a+b+c)^2*(b-c)^3*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

This 4th-common-tangent touches IwP( X(5) ) and IwP( X(11) ) at X(52323) and X(52324), respectively.

X(52325) lies on these lines: {2081, 2600}, {4530, 14393}

X(52325) = barycentric product X(11)*X(6369)
X(52325) = trilinear product X(i)*X(j) for these {i, j}: {5, 46384}, {11, 2600}
X(52325) = trilinear pole of the line {52323, 52324}
X(52325) = perspector of the circumconic {{A, B, C, X(5), X(11)}}
X(52325) = crossdifference of every pair of points on line {X(54), X(59)}
X(52325) = crosspoint of X(i) and X(j) for these (i, j): {5, 52323}, {11, 52324}
X(52325) = X(i)-Ceva conjugate of-X(j) for these (i, j): (5, 52323), (11, 52324)

X(52326) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(8) )

Barycentrics    a^2*(b-c)*(-a+b+c)*((b+c)*a+b^2+c^2) : :
X(52326) = X(649)-4*X(7660)

This 4th-common-tangent touches IwP( X(6) ) and IwP( X(8) ) at X(35506) and X(41224), respectively.

X(52326) lies on the inconic with perspector X(8687) and these lines: {187, 237}, {513, 6589}, {514, 25098}, {522, 650}, {652, 3063}, {654, 4282}, {661, 3310}, {784, 45745}, {905, 3798}, {1021, 4435}, {2804, 47135}, {3004, 3666}, {3835, 24782}, {4025, 28374}, {4077, 28116}, {4374, 26114}, {4394, 6586}, {4790, 43060}, {4979, 21828}, {4988, 21132}, {6085, 23751}, {7178, 23723}, {14739, 44416}, {17066, 25511}, {24002, 28025}, {24622, 27293}, {25084, 31286}, {27486, 27648}, {27674, 47785}, {28606, 47660}

X(52326) = reflection of X(7180) in X(6589)
X(52326) = isogonal conjugate of X(6648)
X(52326) = Gibert-circumtangential conjugate of X(8687)
X(52326) = barycentric product X(i)*X(j) for these {i, j}: {1, 17420}, {6, 3910}, {8, 6371}, {9, 48131}, {21, 50330}, {41, 4509}
X(52326) = barycentric quotient X(i)/X(j) for these (i, j): (31, 36098), (32, 8687), (41, 36147), (55, 8707), (513, 31643), (522, 1240)
X(52326) = trilinear product X(i)*X(j) for these {i, j}: {6, 17420}, {9, 6371}, {31, 3910}, {41, 3004}, {55, 48131}, {284, 50330}
X(52326) = trilinear quotient X(i)/X(j) for these (i, j): (6, 36098), (9, 8707), (31, 8687), (41, 32736), (55, 36147), (514, 31643)
X(52326) = trilinear pole of the line {35506, 41224}
X(52326) = perspector of the circumconic {{A, B, C, X(6), X(8)}}
X(52326) = center of the circumconic {{A, B, C, X(48131), X(50353)}}
X(52326) = tripole of the tangent to Brocard inellipse at X(35506)
X(52326) = tripole of the tangent to Mandart inellipse at X(41224)
X(52326) = intersection, other than A, B, C, of circumconics {{A, B, C, X(21), X(1682)}} and {{A, B, C, X(41), X(17108)}}
X(52326) = Cevapoint of X(i) and X(j) for these (i, j): {6, 8687}, {11, 46880}
X(52326) = crossdifference of every pair of points on line {X(2), X(12)}
X(52326) = crosspoint of X(i) and X(j) for these (i, j): {6, 35506}, {8, 41224}
X(52326) = crosssum of X(i) and X(j) for these (i, j): {6, 3910}, {1015, 1999}
X(52326) = X(1021)-beth conjugate of-X(4976)
X(52326) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 38992), (6, 35506), (8, 41224), (21, 3271), (37, 2170)
X(52326) = X(31)-complementary conjugate of-X(38992)
X(52326) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 30710), (206, 8687), (960, 4552), (1015, 31643), (1146, 1240)
X(52326) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 36098}, {7, 36147}, {57, 8707}, {75, 8687}, {85, 32736}
X(52326) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 36098), (32, 8687), (41, 36147), (55, 8707), (513, 31643)
X(52326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (647, 649, 665), (649, 42664, 50511), (2488, 8653, 663)

X(52327) = TOUCHPOINT OF IwP( X(6) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(10) )

Barycentrics    a^2*(b^2-c^2)^2*((b+c)*a+b^2+b*c+c^2)^2 : :

This 4th-common-tangent touches IwP( X(10) ) at X(52328) and has trilinear pole X(42664).

X(52327) lies on the Brocard inellipse, the inconic with perspector X(28654) and these lines: {1015, 20975}, {1977, 20982}

X(52327) = barycentric product X(i)*X(j) for these {i, j}: {313, 52328}, {834, 23282}
X(52327) = trilinear product X(i)*X(j) for these {i, j}: {321, 52328}, {1089, 39016}
X(52327) = barycentric square of X(47842)
X(52327) = perspector of the circumconic {{A, B, C, X(23282), X(42664)}}
X(52327) = touchpoint of the line {52327, 52328} and Brocard inellipse
X(52327) = X(834)-Dao conjugate of-X(593)

X(52328) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(10) )

Barycentrics    a^4*(b^2-c^2)*(b-c)*((b+c)*a+b^2+b*c+c^2)^2 : :

This 4th-common-tangent touches IwP( X(6) ) at X(52327) and has trilinear pole X(42664).

X(52328) lies on the inconics with perspectors X(n) for n in {10, 2206} and these lines: {2170, 3122}, {3120, 7668}

X(52328) = barycentric product X(i)*X(j) for these {i, j}: {10, 39016}, {58, 52327}, {834, 42664}
X(52328) = trilinear product X(i)*X(j) for these {i, j}: {37, 39016}, {834, 50488}, {1333, 52327}
X(52328) = perspector of the circumconic {{A, B, C, X(8637), X(42664)}}
X(52328) = X(834)-Dao conjugate of-X(86)

X(52329) = TOUCHPOINT OF IwP( X(6) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(11) )

Barycentrics    a^2*(-a+b+c)^2*(b-c)^6*((b+c)*a^3-(b+c)^2*a^2-(b^2-c^2)*(b-c)*a+c^4+b^4)^2 : :

This 4th-common-tangent touches IwP( X(11) ) at X(52330) and has trilinear pole X(52331).

X(52329) lies on the Brocard inellipse and these lines: {}

X(52329) = trilinear product X(1090)*X(39017)
X(52329) = touchpoint of the line {52329, 52330} and Brocard inellipse
X(52329) = X(6)-Ceva conjugate of-X(52331)

X(52330) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(11) )

Barycentrics    a^4*(b-c)^4*((b+c)*a^3-(b+c)^2*a^2-(b^2-c^2)*(b-c)*a+c^4+b^4)^2*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(6) ) at X(52329) and has trilinear pole X(52331).

X(52330) lies on the inconic with perspector X(11) and these lines: {}

X(52330) = barycentric product X(i)*X(j) for these {i, j}: {11, 39017}, {59, 52329}, {928, 52331}
X(52330) = X(11)-Ceva conjugate of-X(52331)
X(52330) = X(928)-Dao conjugate of-X(4998)

X(52331) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(6) ) AND IwP( X(11) )

Barycentrics    a^2*(b-c)^3*(-a+b+c)*((b+c)*a^3-(b+c)^2*a^2-(b^2-c^2)*(b-c)*a+c^4+b^4) : :

This 4th-common-tangent touches IwP( X(6) ) and IwP( X(11) ) at X(52329) and X(52330), respectively.

X(52331) lies on these lines: {187, 237}, {654, 1951}, {3310, 8776}, {4530, 14393}

X(52331) = barycentric product X(11)*X(928)
X(52331) = barycentric quotient X(928)/X(4998)
X(52331) = trilinear product X(928)*X(2170)
X(52331) = trilinear quotient X(928)/X(4564)
X(52331) = trilinear pole of the line {52329, 52330}
X(52331) = perspector of the circumconic {{A, B, C, X(6), X(11)}}
X(52331) = tripole of the tangent to Brocard inellipse at X(52329)
X(52331) = crossdifference of every pair of points on line {X(2), X(59)}
X(52331) = crosspoint of X(i) and X(j) for these (i, j): {6, 52329}, {11, 52330}
X(52331) = X(i)-Ceva conjugate of-X(j) for these (i, j): (6, 52329), (11, 52330)
X(52331) = X(929)-isoconjugate-of-X(4564)
X(52331) = X(928)-reciprocal conjugate of-X(4998)

X(52332) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(7) ) AND IwP( X(10) )

Barycentrics    (b^2-c^2)*(b-c)*(3*a+b+c)^2 : :

This 4th-common-tangent touches IwP( X(7) ) at X(31890) and has trilinear pole X(4841).

X(52332) lies on the inconic with perspector X(10) and these lines: {3120, 8287}, {3125, 52335}, {21043, 21950}

X(52332) = barycentric product X(1434)*X(31890)
X(52332) = trilinear product X(1014)*X(31890)
X(52332) = perspector of the circumconic {{A, B, C, X(4841), X(30723)}}

X(52333) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(7) ) AND IwP( X(11) )

Barycentrics    (2*a^2-(b+c)*a-(b-c)^2)^2*(b-c)^4*(-a+b+c)^3 : :

This 4th-common-tangent touches IwP( X(7) ) at X(31891) and has trilinear pole X(52334).

X(52333) lies on the inconics with perspectors X(n) for n in {11, 3328, 5532} and these lines: {55, 40577}, {3328, 30573}, {5532, 42462}, {14393, 52337}

X(52333) = barycentric product X(i)*X(j) for these {i, j}: {11, 35091}, {1146, 3328}, {1275, 31891}
X(52333) = perspector of the circumconic {{A, B, C, X(1638), X(52334)}}
X(52333) = X(11)-Ceva conjugate of-X(52334)

X(52334) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(7) ) AND IwP( X(11) )

Barycentrics    (-a+b+c)^2*(b-c)^3*(2*a^2-(b+c)*a-(b-c)^2) : :
X(52334) = 2*X(1323)-3*X(1638) = 3*X(1639)-4*X(5199)

This 4th-common-tangent touches IwP( X(7) ) and IwP( X(11) ) at X(31891) and X(52333), respectively.

X(52334) lies on the inconic with perspector X(1638) and these lines: {241, 514}, {654, 5011}, {1639, 5199}, {4530, 14393}, {6366, 6603}

X(52334) = barycentric product X(i)*X(j) for these {i, j}: {11, 6366}, {514, 33573}, {527, 42462}, {1111, 14392}, {1146, 1638}, {1155, 42455}
X(52334) = barycentric quotient X(i)/X(j) for these (i, j): (11, 35157), (1055, 4619), (1638, 1275)
X(52334) = trilinear product X(i)*X(j) for these {i, j}: {513, 33573}, {1055, 42455}, {1086, 14392}, {1146, 14413}, {1155, 42462}
X(52334) = trilinear quotient X(i)/X(j) for these (i, j): (11, 37139), (1155, 4619), (1638, 7045)
X(52334) = trilinear pole of the line {31891, 52333}
X(52334) = perspector of the circumconic {{A, B, C, X(7), X(11)}}
X(52334) = tripole of the tangent to incircle at X(31891)
X(52334) = crossdifference of every pair of points on line {X(55), X(59)}
X(52334) = crosspoint of X(i) and X(j) for these (i, j): {7, 31891}, {11, 52333}
X(52334) = X(i)-Ceva conjugate of-X(j) for these (i, j): (7, 31891), (11, 52333), (514, 3328)
X(52334) = X(650)-Dao conjugate of-X(35157)
X(52334) = X(i)-isoconjugate-of-X(j) for these {i, j}: {59, 37139}, {1156, 4619}
X(52334) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (11, 35157), (1055, 4619)
X(52334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4530, 21132, 14393), (46384, 52316, 52338)

X(52335) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(10) )

Barycentrics    (b^2-c^2)*(b-c)*(-a+b+c)^2 : :

This 4th-common-tangent touches IwP( X(8) ) at X(4092) and has trilinear pole X(3700).

X(52335) lies on the inconics with perspectors X(n) for n in {10, 1043, 4082} and these lines: {8, 2648}, {10, 4552}, {210, 6535}, {281, 4336}, {523, 4466}, {594, 21039}, {1042, 39130}, {1109, 2632}, {1146, 2310}, {1441, 21931}, {2254, 23771}, {2293, 23529}, {2321, 4069}, {2784, 14543}, {3119, 42069}, {3122, 21950}, {3125, 52332}, {3271, 4530}, {3939, 36910}, {4092, 4516}, {4858, 17059}, {8013, 40967}, {17216, 17886}, {17874, 21717}, {21014, 23902}, {21033, 21673}, {21043, 21961}, {21252, 42754}, {21677, 52068}, {21686, 42440}, {23978, 24026}, {24411, 37781}, {39350, 48628}

X(52335) = reflection of X(4466) in X(21045)
X(52335) = barycentric product X(i)*X(j) for these {i, j}: {8, 21044}, {10, 1146}, {11, 2321}, {37, 24026}, {42, 23978}, {71, 21666}
X(52335) = barycentric quotient X(i)/X(j) for these (i, j): (8, 4620), (10, 1275), (11, 1434), (37, 7045), (42, 1262), (115, 3668)
X(52335) = trilinear product X(i)*X(j) for these {i, j}: {2, 36197}, {8, 4516}, {9, 21044}, {10, 2310}, {11, 210}, {21, 4092}
X(52335) = trilinear quotient X(i)/X(j) for these (i, j): (10, 7045), (11, 1014), (37, 1262), (42, 24027), (65, 7339), (115, 1427)
X(52335) = perspector of the circumconic {{A, B, C, X(1577), X(3239)}}
X(52335) = Cevapoint of X(i) and X(j) for these (i, j): {657, 2322}, {661, 39130}
X(52335) = crossdifference of every pair of points on line {X(163), X(1461)}
X(52335) = crosssum of X(i) and X(j) for these (i, j): {650, 34043}, {661, 2360}
X(52335) = X(1146)-Ceva conjugate of-X(36197)
X(52335) = X(i)-Dao conjugate of-X(j) for these (i, j): (10, 7045), (11, 1414), (37, 1275), (115, 658), (136, 36118)
X(52335) = X(i)-isoconjugate-of-X(j) for these {i, j}: {21, 7339}, {58, 7045}, {59, 1014}, {81, 1262}, {86, 24027}
X(52335) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 4620), (10, 1275), (11, 1434), (37, 7045), (42, 1262)
X(52335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 4552, 21914), (1146, 4081, 2310), (4092, 4516, 21044), (16732, 21945, 3120)

X(52336) = TOUCHPOINT OF IwP( X(8) ) AND THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(11) )

Barycentrics    (2*a-b-c)^2*(b-c)^6*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(11) ) at X(52337) and has trilinear pole X(52338).

X(52336) lies on the Mandart inellipse, the inconics with perspectors X(n) for n in {7336, 52337} and these lines: {3271, 11193}

X(52336) = barycentric product X(1086)*X(52337)
X(52336) = trilinear product X(244)*X(52337)
X(52336) = touchpoint of the line {52336, 52337} and Mandart inellipse
X(52336) = X(8)-Ceva conjugate of-X(52338)

X(52337) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(11) )

Barycentrics    (2*a-b-c)^2*(b-c)^4*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(8) ) at X(52336) and has trilinear pole X(52338).

X(52337) lies on the inconics with perspectors X(n) for n in {11, 4076, 4542, 7336, 14027} and these lines: {3251, 35092}, {4124, 51402}, {4530, 4542}, {7336, 21132}, {14393, 52333}

X(52337) = barycentric product X(i)*X(j) for these {i, j}: {11, 35092}, {522, 14442}, {900, 52338}, {1016, 52336}, {1086, 4542}, {1146, 14027}
X(52337) = barycentric quotient X(1639)/X(6635)
X(52337) = trilinear product X(i)*X(j) for these {i, j}: {11, 42084}, {244, 4542}, {650, 14442}, {678, 7336}, {764, 4543}, {765, 52336}
X(52337) = perspector of the circumconic {{A, B, C, X(1639), X(14442)}}
X(52337) = X(900)-Dao conjugate of-X(4998)

X(52338) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(8) ) AND IwP( X(11) )

Barycentrics    (2*a-b-c)*(b-c)^3*(-a+b+c) : :
X(52338) = 3*X(1638)-4*X(17067) = 3*X(1639)-2*X(2325)

This 4th-common-tangent touches IwP( X(8) ) and IwP( X(11) ) at X(52336) and X(52337), respectively.

X(52338) lies on the inconics with perspectors X(n) for n in {1639, 30725} and these lines: {9, 24096}, {44, 900}, {142, 24098}, {226, 24115}, {514, 4887}, {522, 650}, {654, 1731}, {665, 8609}, {918, 1266}, {1638, 17067}, {2323, 4435}, {3452, 24113}, {4357, 24141}, {4530, 14393}, {6550, 14442}

X(52338) = barycentric product X(i)*X(j) for these {i, j}: {8, 6550}, {11, 900}, {44, 40166}, {244, 4768}, {514, 4530}, {519, 21132}
X(52338) = barycentric quotient X(i)/X(j) for these (i, j): (8, 6635), (11, 4555), (44, 31615), (55, 6551), (650, 5376), (663, 9268)
X(52338) = trilinear product X(i)*X(j) for these {i, j}: {9, 6550}, {11, 1635}, {44, 21132}, {244, 1639}, {312, 8661}, {513, 4530}
X(52338) = trilinear quotient X(i)/X(j) for these (i, j): (9, 6551), (11, 3257), (312, 6635), (519, 31615), (522, 5376), (646, 42372)
X(52338) = trilinear pole of the line {52336, 52337}
X(52338) = perspector of the circumconic {{A, B, C, X(8), X(11)}}
X(52338) = tripole of the tangent to Mandart inellipse at X(52336)
X(52338) = crossdifference of every pair of points on line {X(56), X(59)}
X(52338) = crosspoint of X(i) and X(j) for these (i, j): {8, 52336}, {11, 52337}
X(52338) = X(i)-Ceva conjugate of-X(j) for these (i, j): (8, 52336), (11, 52337), (522, 4542)
X(52338) = X(i)-Dao conjugate of-X(j) for these (i, j): (11, 5376), (214, 31615), (522, 4582), (650, 4555)
X(52338) = X(i)-isoconjugate-of-X(j) for these {i, j}: {57, 6551}, {59, 3257}, {106, 31615}, {109, 5376}, {604, 6635}
X(52338) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 6635), (11, 4555), (44, 31615), (55, 6551), (650, 5376)
X(52338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (42462, 46384, 52316), (46384, 52316, 52334)

X(52339) = TOUCHPOINT OF IwP( X(10) ) AND THE 4th-COMMON-TANGENT OF IwP( X(10) ) AND IwP( X(11) )

Barycentrics    (b^2-c^2)*(b-c)^5*(-a+b+c)^2*(a^3-(b^2-3*b*c+c^2)*a-(b+c)*b*c)^2 : :

This 4th-common-tangent touches IwP( X(11) ) at X(52340) and has trilinear pole X(52341).

X(52339) lies on the inconic with perspector X(10) and these lines: {}

X(52340) = TOUCHPOINT OF IwP( X(11) ) AND THE 4th-COMMON-TANGENT OF IwP( X(10) ) AND IwP( X(11) )

Barycentrics    (b^2-c^2)^2*(b-c)^2*(a^3-(b^2-3*b*c+c^2)*a-(b+c)*b*c)^2*(-a+b+c) : :

This 4th-common-tangent touches IwP( X(10) ) at X(52339) and has trilinear pole X(52341).

X(52340) lies on the inconic with perspector X(11) and these lines: {}

X(52341) = TRILINEAR POLE OF THE 4th-COMMON-TANGENT OF IwP( X(10) ) AND IwP( X(11) )

Barycentrics    (b^2-c^2)*(b-c)^2*(-a+b+c)*(a^3-(b^2-3*b*c+c^2)*a-(b+c)*b*c) : :
X(52341) = 3*X(4120)-2*X(21801)

This 4th-common-tangent touches IwP( X(10) ) and IwP( X(11) ) at X(52339) and X(52340), respectively.

X(52341) lies on these lines: {523, 661}, {4530, 14393}

X(52341) = trilinear product X(11)*X(21894)
X(52341) = trilinear pole of the line {52339, 52340}
X(52341) = perspector of the circumconic {{A, B, C, X(10), X(11)}}
X(52341) = crossdifference of every pair of points on line {X(58), X(59)}
X(52341) = crosspoint of X(i) and X(j) for these (i, j): {10, 52339}, {11, 52340}
X(52341) = X(i)-Ceva conjugate of-X(j) for these (i, j): (10, 52339), (11, 52340)

X(52342) = TOUCHPOINT OF IwP( X(15) ) AND THE 4th-COMMON-TANGENT OF IwP( X(15) ) AND IwP( X(16) )

Barycentrics    a^2*(2*S+sqrt(3)*(-a^2+b^2+c^2))*(-2*S+sqrt(3)*(-a^2+b^2+c^2))^2*(b^2-c^2)^2 : :

This 4th-common-tangent touches IwP( X(16) ) at X(52343) and has trilinear pole X(526).

X(52342) lies on the inconics with perspectors X(n) for n in {15, 300, 36208} and these lines: {2, 35316}, {16, 323}, {2088, 16186}, {3258, 15609}, {7998, 30261}, {9205, 30468}, {14816, 47055}, {41888, 52039}

X(52342) = complement of X(35316)
X(52342) = barycentric product X(i)*X(j) for these {i, j}: {15, 43962}, {299, 2088}, {300, 18334}, {323, 30468}, {471, 16186}, {526, 23871}
X(52342) = barycentric quotient X(i)/X(j) for these (i, j): (16, 39295), (526, 23896)
X(52342) = trilinear product X(i)*X(j) for these {i, j}: {1094, 30460}, {1095, 30465}
X(52342) = perspector of the circumconic {{A, B, C, X(526), X(17403)}}
X(52342) = center of the circumconic {{A, B, C, X(15), X(300)}}
X(52342) = crossdifference of every pair of points on line {X(476), X(5994)}
X(52342) = X(i)-Ceva conjugate of-X(j) for these (i, j): (15, 526), (300, 23871)
X(52342) = X(526)-Dao conjugate of-X(15)
X(52342) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (16, 39295), (526, 23896)
X(52342) = {X(2088), X(16186)}-harmonic conjugate of X(52343)

X(52343) = TOUCHPOINT OF IwP( X(16) ) AND THE 4th-COMMON-TANGENT OF IwP( X(15) ) AND IwP( X(16) )

Barycentrics    a^2*(-2*S+sqrt(3)*(-a^2+b^2+c^2))*(2*S+sqrt(3)*(-a^2+b^2+c^2))^2*(b^2-c^2)^2 : :

This 4th-common-tangent touches IwP( X(15) ) at X(52342) and has trilinear pole X(526).

X(52343) lies on the inconics with perspectors X(n) for n in {16, 301, 36209} and these lines: {2, 35317}, {15, 323}, {2088, 16186}, {3258, 15610}, {7998, 30260}, {9204, 30465}, {14817, 47055}, {41887, 52040}

X(52343) = complement of X(35317)
X(52343) = barycentric product X(i)*X(j) for these {i, j}: {16, 43961}, {298, 2088}, {301, 18334}, {323, 30465}, {470, 16186}, {526, 23870}
X(52343) = barycentric quotient X(i)/X(j) for these (i, j): (15, 39295), (526, 23895)
X(52343) = trilinear product X(i)*X(j) for these {i, j}: {1094, 30468}, {1095, 30463}
X(52343) = perspector of the circumconic {{A, B, C, X(526), X(17402)}}
X(52343) = center of the circumconic {{A, B, C, X(16), X(301)}}
X(52343) = crossdifference of every pair of points on line {X(476), X(5995)}
X(52343) = X(i)-Ceva conjugate of-X(j) for these (i, j): (16, 526), (301, 23870)
X(52343) = X(526)-Dao conjugate of-X(16)
X(52343) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (15, 39295), (526, 23895)
X(52343) = {X(2088), X(16186)}-harmonic conjugate of X(52342)

X(52344) = X(1)-DAO CONJUGATE OF X(35)

Barycentrics    b*c*(-a + b + c)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2) : :

X(52344) lies on these lines: {2, 36626}, {8, 79}, {10, 94}, {35, 18359}, {75, 3260}, {92, 46468}, {100, 1141}, {265, 5086}, {280, 10527}, {318, 860}, {321, 4420}, {328, 18815}, {333, 52126}, {346, 5552}, {1043, 3615}, {1109, 24851}, {1222, 4968}, {1733, 24883}, {1789, 10538}, {2322, 3559}, {2370, 26700}, {3153, 14213}, {3616, 20320}, {3648, 14206}, {4861, 6742}, {5015, 20887}, {6533, 24026}, {7081, 26266}, {9780, 36624}, {13746, 38336}, {17874, 26131}, {18025, 20880}, {19877, 36625}, {24390, 52200}, {33944, 34387}, {43533, 43682}

X(52344) = isogonal conjugate of X(1399)
X(52344) = isotomic conjugate of X(1442)
X(52344) = isotomic conjugate of the isogonal conjugate of X(7073)
X(52344) = X(52185)-anticomplementary conjugate of X(3648)
X(52344) = X(20565)-Ceva conjugate of X(30690)
X(52344) = X(i)-cross conjugate of X(j) for these (i,j): {2323, 18359}, {3686, 312}, {6734, 8}, {7110, 30690}, {40942, 92}
X(52344) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1399}, {6, 2003}, {28, 22342}, {31, 1442}, {32, 17095}, {35, 56}, {50, 2006}, {57, 2174}, {58, 2594}, {65, 17104}, {77, 14975}, {79, 2477}, {81, 21741}, {108, 23226}, {109, 2605}, {184, 7282}, {319, 1397}, {593, 21794}, {603, 6198}, {604, 3219}, {1042, 35193}, {1106, 4420}, {1333, 16577}, {1400, 40214}, {1408, 3678}, {1410, 11107}, {1411, 6149}, {1415, 14838}, {1427, 35192}, {1437, 1825}, {1461, 9404}, {2149, 7202}, {2206, 40999}, {3969, 16947}, {6186, 7279}, {7206, 7342}
X(52344) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 35}, {2, 1442}, {3, 1399}, {9, 2003}, {10, 2594}, {11, 2605}, {37, 16577}, {442, 500}, {650, 7202}, {1146, 14838}, {1411, 14993}, {2174, 5452}, {2968, 35057}, {3161, 3219}, {4420, 6552}, {4467, 40624}, {6149, 35204}, {6198, 7952}, {6376, 17095}, {7110, 47057}, {7186, 41886}, {9404, 35508}, {17104, 40602}, {21741, 40586}, {22342, 40591}, {23226, 38983}, {40214, 40582}, {40603, 40999}
X(52344) = cevapoint of X(i) and X(j) for these (i,j): {8, 27529}, {4985, 24026}
X(52344) = barycentric product X(i)*X(j) for these {i,j}: {8, 30690}, {9, 20565}, {75, 7110}, {76, 7073}, {79, 312}, {94, 4511}, {314, 8818}, {321, 3615}, {333, 6757}, {522, 15455}, {1043, 43682}, {1969, 8606}, {2160, 3596}, {2166, 32851}, {2361, 20573}, {4391, 6742}, {4397, 38340}, {6186, 28659}, {7017, 7100}
X(52344) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2003}, {2, 1442}, {6, 1399}, {8, 3219}, {9, 35}, {10, 16577}, {11, 7202}, {21, 40214}, {37, 2594}, {42, 21741}, {55, 2174}, {71, 22342}, {75, 17095}, {79, 57}, {92, 7282}, {94, 18815}, {281, 6198}, {284, 17104}, {312, 319}, {314, 34016}, {321, 40999}, {341, 42033}, {346, 4420}, {522, 14838}, {607, 14975}, {650, 2605}, {652, 23226}, {756, 21794}, {1789, 1790}, {1826, 1825}, {1859, 44095}, {1989, 1411}, {2160, 56}, {2166, 2006}, {2174, 2477}, {2287, 35193}, {2321, 3678}, {2322, 11107}, {2323, 6149}, {2328, 35192}, {2361, 50}, {3061, 7186}, {3219, 7279}, {3239, 35057}, {3596, 33939}, {3615, 81}, {3683, 17454}, {3686, 3647}, {3701, 3969}, {3702, 3578}, {3900, 9404}, {4086, 7265}, {4092, 21824}, {4183, 41502}, {4391, 4467}, {4511, 323}, {4516, 20982}, {6186, 604}, {6734, 16585}, {6742, 651}, {6757, 226}, {7073, 6}, {7100, 222}, {7110, 1}, {8606, 48}, {8818, 65}, {13486, 4565}, {15455, 664}, {18155, 16755}, {20565, 85}, {21011, 2599}, {21044, 2611}, {26700, 1461}, {30690, 7}, {33653, 2307}, {34922, 7128}, {35519, 18160}, {38340, 934}, {40937, 500}, {43682, 3668}
X(52344) = {X(79),X(6757)}-harmonic conjugate of X(30690)

X(52345) = X(4)-DAO CONJUGATE OF X(28)

Barycentrics    b*c*(b + c)*(-3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(52345) lies on these lines: {1, 27410}, {2, 51502}, {4, 8}, {10, 307}, {20, 3198}, {37, 6554}, {40, 45738}, {63, 1715}, {69, 21588}, {85, 4208}, {100, 1294}, {169, 21061}, {253, 322}, {306, 857}, {379, 4968}, {387, 17863}, {388, 43214}, {411, 7360}, {668, 35140}, {758, 17869}, {821, 3990}, {942, 20905}, {948, 31993}, {956, 7535}, {1089, 21060}, {1229, 10449}, {1249, 1895}, {1265, 20928}, {1944, 3562}, {2322, 5279}, {2901, 6765}, {2975, 37275}, {3091, 20921}, {3191, 3870}, {3346, 3998}, {3710, 8806}, {3757, 4223}, {3868, 17862}, {3876, 26591}, {3931, 25255}, {4858, 24391}, {5129, 30854}, {5179, 21070}, {5247, 26000}, {5552, 6350}, {5687, 15951}, {5730, 49687}, {7078, 28950}, {7291, 37088}, {7952, 27540}, {9307, 22028}, {12526, 17860}, {14543, 40660}, {15973, 30082}, {16062, 26563}, {17757, 21530}, {18663, 26050}, {20305, 21686}, {21620, 25935}, {22027, 31397}, {24987, 25017}, {26678, 41249}, {28787, 38955}, {28951, 41344}, {30031, 30059}, {30809, 46937}, {38298, 42714}

X(52345) = isotomic conjugate of the isogonal conjugate of X(3198)
X(52345) = polar conjugate of the isotomic conjugate of X(42699)
X(52345) = X(i)-Ceva conjugate of X(j) for these (i,j): {322, 306}, {341, 10}, {18750, 8804}, {20336, 321}
X(52345) = X(i)-isoconjugate of X(j) for these (i,j): {27, 14642}, {28, 19614}, {56, 52158}, {58, 64}, {81, 2155}, {86, 33581}, {253, 2206}, {649, 46639}, {1073, 1474}, {1301, 1459}, {1333, 2184}, {1397, 5931}, {1408, 44692}, {1412, 30457}, {1790, 41489}, {1919, 44326}, {2194, 8809}, {2203, 19611}, {2208, 41082}, {8747, 14379}, {21789, 36079}
X(52345) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 52158}, {4, 28}, {10, 64}, {37, 2184}, {81, 45245}, {122, 513}, {253, 40603}, {269, 1427}, {800, 18603}, {905, 39020}, {1019, 40616}, {1073, 51574}, {1214, 8809}, {1437, 45248}, {2155, 40586}, {3198, 18615}, {5375, 46639}, {5930, 41402}, {8804, 11347}, {9296, 44326}, {18180, 45249}, {19614, 40591}, {30457, 40599}, {33581, 40600}
X(52345) = crosspoint of X(i) and X(j) for these (i,j): {14615, 18750}, {20336, 42699}
X(52345) = crosssum of X(2155) and X(33581)
X(52345) = trilinear pole of line {6587, 17898}
X(52345) = barycentric product X(i)*X(j) for these {i,j}: {4, 42699}, {10, 18750}, {20, 321}, {37, 14615}, {72, 15466}, {75, 8804}, {76, 3198}, {154, 27801}, {190, 17898}, {204, 40071}, {306, 1895}, {312, 5930}, {313, 610}, {341, 36908}, {349, 7070}, {668, 6587}, {1231, 44695}, {1249, 20336}, {1394, 30713}, {1441, 27382}, {2321, 33673}, {3596, 30456}, {3701, 18623}, {3710, 44697}, {3998, 14249}, {4033, 21172}, {4036, 36841}, {4554, 14308}, {6335, 8057}, {37669, 41013}, {38808, 42698}
X(52345) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 52158}, {10, 2184}, {20, 81}, {37, 64}, {42, 2155}, {71, 19614}, {72, 1073}, {100, 46639}, {154, 1333}, {204, 1474}, {210, 30457}, {213, 33581}, {226, 8809}, {228, 14642}, {306, 19611}, {312, 5931}, {321, 253}, {329, 41082}, {610, 58}, {668, 44326}, {1020, 36079}, {1249, 28}, {1394, 1412}, {1562, 18210}, {1783, 1301}, {1824, 41489}, {1895, 27}, {2321, 44692}, {2883, 18603}, {3172, 2203}, {3198, 6}, {3990, 14379}, {3998, 15394}, {5930, 57}, {6525, 5317}, {6587, 513}, {7070, 284}, {7156, 2299}, {8057, 905}, {8804, 1}, {14308, 650}, {14331, 3737}, {14615, 274}, {15466, 286}, {15905, 1437}, {17898, 514}, {18623, 1014}, {18750, 86}, {20336, 34403}, {20580, 4131}, {21172, 1019}, {21871, 41088}, {27382, 21}, {27801, 41530}, {30456, 56}, {33673, 1434}, {35602, 18604}, {36908, 269}, {37669, 1444}, {40933, 1407}, {41013, 459}, {41086, 1436}, {42459, 18180}, {42658, 22383}, {42699, 69}, {44695, 1172}, {44696, 1396}, {44705, 6591}, {52078, 1422}
X(52345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 5080, 5174}, {72, 41013, 321}, {322, 33672, 253}, {20235, 44150, 1446}, {21075, 39130, 306}

X(52346) = X(4)-DAO CONJUGATE OF X(34)

Barycentrics    b*c*(-a + b + c)*(-3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(52346) lies on these lines: {3, 7360}, {4, 20921}, {7, 8}, {20, 3198}, {29, 33}, {100, 41904}, {145, 17862}, {200, 4385}, {280, 341}, {306, 37448}, {309, 326}, {321, 20007}, {331, 35517}, {452, 30854}, {519, 20320}, {653, 8899}, {1034, 1265}, {1097, 37669}, {1103, 24034}, {1259, 10538}, {1260, 7283}, {1427, 26050}, {1895, 15466}, {2057, 3699}, {3100, 27410}, {3146, 30807}, {3436, 37201}, {3661, 26550}, {3687, 37445}, {3718, 5931}, {4673, 23528}, {4692, 4882}, {4737, 6736}, {4858, 12625}, {5552, 33116}, {5930, 14615}, {6527, 33672}, {6734, 19804}, {6735, 44720}, {6737, 17860}, {6745, 46937}, {7081, 19310}, {10446, 43213}, {11679, 16054}, {17880, 21605}, {18743, 27383}, {20013, 48380}, {20928, 34414}, {24218, 24440}, {27385, 30829}

X(52346) = isotomic conjugate of X(8809)
X(52346) = isotomic conjugate of the isogonal conjugate of X(7070)
X(52346) = X(i)-Ceva conjugate of X(j) for these (i,j): {3718, 312}, {14615, 18750}
X(52346) = X(i)-cross conjugate of X(j) for these (i,j): {5930, 8}, {27382, 18750}
X(52346) = X(i)-isoconjugate of X(j) for these (i,j): {7, 33581}, {31, 8809}, {34, 19614}, {56, 64}, {57, 2155}, {222, 41489}, {253, 1397}, {278, 14642}, {604, 2184}, {608, 1073}, {663, 36079}, {1042, 52158}, {1106, 44692}, {1118, 14379}, {1395, 19611}, {1407, 30457}, {1413, 41088}, {1472, 10375}, {6526, 7335}, {7180, 46639}, {7337, 15394}, {8810, 47437}, {41280, 41530}
X(52346) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 64}, {2, 8809}, {4, 34}, {57, 45245}, {122, 4017}, {603, 45248}, {1393, 45249}, {2155, 5452}, {2184, 3161}, {3669, 40616}, {6552, 44692}, {7004, 21172}, {11517, 19614}, {17898, 38357}, {24771, 30457}, {39020, 51640}
X(52346) = barycentric product X(i)*X(j) for these {i,j}: {8, 18750}, {9, 14615}, {20, 312}, {29, 42699}, {75, 27382}, {76, 7070}, {78, 15466}, {154, 28659}, {304, 44695}, {305, 7156}, {314, 8804}, {318, 37669}, {341, 18623}, {345, 1895}, {346, 33673}, {610, 3596}, {645, 17898}, {646, 21172}, {668, 14331}, {799, 14308}, {1249, 3718}, {1265, 44697}, {3198, 28660}, {3719, 14249}, {4086, 36841}, {6587, 7257}
X(52346) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8809}, {8, 2184}, {9, 64}, {20, 57}, {33, 41489}, {41, 33581}, {55, 2155}, {78, 1073}, {154, 604}, {200, 30457}, {204, 608}, {212, 14642}, {219, 19614}, {312, 253}, {318, 459}, {345, 19611}, {346, 44692}, {610, 56}, {643, 46639}, {651, 36079}, {1097, 18623}, {1249, 34}, {1394, 1407}, {1895, 278}, {2287, 52158}, {2289, 14379}, {2324, 41088}, {2345, 10375}, {3079, 3213}, {3172, 1395}, {3198, 1400}, {3213, 1398}, {3718, 34403}, {3719, 15394}, {5930, 1427}, {6060, 610}, {6587, 4017}, {7070, 6}, {7156, 25}, {7257, 44326}, {8057, 51640}, {8804, 65}, {14308, 661}, {14331, 513}, {14615, 85}, {15466, 273}, {15905, 603}, {17898, 7178}, {18623, 269}, {18750, 7}, {21172, 3669}, {27382, 1}, {27398, 41082}, {28659, 41530}, {30456, 1042}, {33673, 279}, {35602, 7125}, {36413, 1394}, {36841, 1414}, {37669, 77}, {40616, 7004}, {41084, 1422}, {42459, 1393}, {42699, 307}, {44695, 19}, {44696, 1435}, {44697, 1119}, {44698, 1396}, {44699, 7128}
X(52346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 23661, 75}, {75, 16284, 1231}, {78, 318, 312}, {280, 7080, 345}, {388, 4012, 8}

X(52347) = X(5)-DAO CONJUGATE OF X(25)

Barycentrics    (a^2 - b^2 - c^2)^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(52347) lies on these lines: {3, 69}, {5, 311}, {30, 44128}, {76, 7399}, {99, 18401}, {141, 570}, {160, 16789}, {183, 7499}, {216, 343}, {264, 305}, {274, 51495}, {315, 20477}, {317, 1975}, {340, 32820}, {394, 6389}, {441, 20806}, {524, 571}, {1368, 20819}, {1370, 6527}, {1591, 34391}, {1592, 34392}, {2871, 3313}, {3001, 23300}, {3148, 13562}, {3260, 6662}, {3265, 43083}, {3541, 32000}, {3631, 14806}, {5562, 6751}, {5576, 44135}, {6148, 32902}, {6515, 8573}, {6656, 28728}, {6815, 32830}, {7667, 7788}, {7750, 46724}, {7776, 14790}, {7796, 14615}, {7802, 38434}, {8362, 28710}, {8901, 34385}, {9722, 44388}, {11585, 20563}, {14570, 42459}, {18494, 32815}, {18533, 32001}, {19553, 45799}, {20208, 28419}, {26906, 40681}, {27364, 41221}, {27377, 44363}, {32002, 32819}, {32821, 44134}, {37183, 46442}, {37688, 37804}, {38321, 52149}, {40981, 41588}, {42487, 51386}

X(52347) = isotomic conjugate of X(8884)
X(52347) = isotomic conjugate of the anticomplement of X(10600)
X(52347) = isotomic conjugate of the isogonal conjugate of X(5562)
X(52347) = isotomic conjugate of the polar conjugate of X(343)
X(52347) = isogonal conjugate of the polar conjugate of X(28706)
X(52347) = X(i)-Ceva conjugate of X(j) for these (i,j): {305, 28706}, {670, 4143}, {28706, 343}
X(52347) = X(i)-cross conjugate of X(j) for these (i,j): {5562, 343}, {10600, 2}
X(52347) = X(i)-isoconjugate of X(j) for these (i,j): {19, 8882}, {25, 2190}, {31, 8884}, {54, 1096}, {163, 15422}, {275, 1973}, {393, 2148}, {560, 8795}, {798, 16813}, {1924, 42405}, {1974, 40440}, {2167, 2207}, {2168, 8745}, {2169, 6524}, {2616, 32713}, {2623, 24019}, {6520, 14533}, {7337, 44687}, {8794, 9247}, {19174, 46289}
X(52347) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8884}, {4, 52032}, {5, 25}, {6, 8882}, {24, 343}, {39, 19174}, {54, 6503}, {95, 6338}, {115, 15422}, {115, 35441}, {130, 669}, {216, 393}, {275, 6337}, {394, 8883}, {512, 2972}, {525, 8901}, {2190, 6505}, {2207, 40588}, {2489, 15450}, {2501, 39019}, {2623, 35071}, {3575, 23292}, {6368, 41221}, {6374, 8795}, {6524, 14363}, {6525, 45249}, {6663, 14569}, {6753, 47421}, {9428, 42405}, {13567, 16035}, {14533, 37867}, {16813, 31998}, {34836, 51887}
X(52347) = crosspoint of X(i) and X(j) for these (i,j): {69, 20563}, {305, 3926}
X(52347) = crosssum of X(i) and X(j) for these (i,j): {25, 44077}, {1974, 2207}
X(52347) = barycentric product X(i)*X(j) for these {i,j}: {3, 28706}, {5, 3926}, {53, 4176}, {63, 18695}, {69, 343}, {76, 5562}, {216, 305}, {217, 40050}, {304, 44706}, {311, 394}, {324, 3964}, {326, 14213}, {418, 1502}, {670, 17434}, {1444, 42698}, {3265, 14570}, {3267, 23181}, {3718, 44708}, {4143, 35360}, {4563, 6368}, {4590, 35442}, {4609, 42293}, {16697, 20336}, {20563, 52032}, {34254, 41168}, {40071, 44709}, {40362, 44088}, {40824, 42353}
X(52347) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8884}, {3, 8882}, {5, 393}, {51, 2207}, {52, 8745}, {53, 6524}, {63, 2190}, {69, 275}, {76, 8795}, {99, 16813}, {141, 19174}, {216, 25}, {217, 1974}, {255, 2148}, {264, 8794}, {304, 40440}, {305, 276}, {311, 2052}, {324, 1093}, {326, 2167}, {343, 4}, {394, 54}, {418, 32}, {520, 2623}, {523, 15422}, {670, 42405}, {1092, 14533}, {1273, 14165}, {1568, 1990}, {1625, 32713}, {1953, 1096}, {2617, 24019}, {3133, 36416}, {3265, 15412}, {3719, 44687}, {3926, 95}, {3964, 97}, {4176, 34386}, {4558, 933}, {4563, 18831}, {5562, 6}, {6368, 2501}, {6503, 8883}, {6507, 2169}, {6509, 16035}, {6514, 35196}, {6751, 42295}, {8798, 41489}, {13157, 6526}, {14213, 158}, {14569, 36434}, {14570, 107}, {14585, 14573}, {15451, 2489}, {15526, 8901}, {16697, 28}, {17167, 8747}, {17434, 512}, {18180, 5317}, {18695, 92}, {23181, 112}, {24018, 2616}, {28706, 264}, {30493, 608}, {35360, 6529}, {35442, 115}, {35602, 33629}, {36212, 19189}, {36412, 14569}, {36831, 32695}, {37669, 38808}, {39019, 41221}, {39113, 11547}, {40981, 36417}, {41168, 13854}, {42293, 669}, {42353, 7735}, {42445, 47328}, {42459, 6525}, {42698, 41013}, {44088, 1501}, {44706, 19}, {44707, 607}, {44708, 34}, {44709, 1474}, {44710, 7115}, {44711, 8739}, {44712, 8740}, {44713, 8737}, {44714, 8738}, {44715, 8749}, {44716, 232}, {45793, 13450}, {46394, 40981}, {46832, 51887}, {52032, 24}
X(52347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 1238, 3933}, {69, 3926, 3964}, {69, 9723, 41008}, {69, 40697, 3}, {69, 50572, 40680}, {311, 1273, 39113}, {311, 39113, 5}, {487, 488, 18925}, {3933, 41005, 69}, {6390, 41008, 9723}

X(52348) = X(6)-DAO CONJUGATE OF X(17)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2 - b^2 - c^2 - 2*Sqrt[3]*S) : :

X(52348) lies on these lines: {2, 13}, {3, 49}, {15, 1993}, {18, 13579}, {22, 14538}, {51, 5615}, {61, 1994}, {62, 5422}, {69, 11516}, {97, 44711}, {249, 40157}, {301, 32037}, {302, 473}, {323, 10645}, {343, 466}, {465, 11064}, {472, 11476}, {511, 3131}, {627, 8839}, {629, 2912}, {1599, 3364}, {1600, 3365}, {2979, 14539}, {3060, 38431}, {3129, 47066}, {3132, 9306}, {3391, 15234}, {3392, 15233}, {3819, 13349}, {4558, 36297}, {5238, 11127}, {5351, 11145}, {6636, 14169}, {7998, 21159}, {9544, 14170}, {10601, 11486}, {10646, 15066}, {11004, 34754}, {11086, 33459}, {11143, 16771}, {11427, 37172}, {11480, 37672}, {11481, 17811}, {11515, 20806}, {13350, 34986}, {14540, 34009}, {17825, 22238}, {18470, 37638}, {22151, 46113}, {26881, 34008}, {34395, 35296}, {34545, 41477}, {37340, 37649}

X(52348) = isogonal conjugate of X(8741)
X(52348) = isotomic conjugate of the polar conjugate of X(61)
X(52348) = isogonal conjugate of the polar conjugate of X(302)
X(52348) = X(i)-Ceva conjugate of X(j) for these (i,j): {302, 61}, {40710, 44719}
X(52348) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8741}, {17, 19}, {92, 21461}, {158, 32585}, {1096, 40712}, {1973, 34389}, {2190, 36300}, {2962, 10641}, {3375, 8738}, {16806, 24006}
X(52348) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8741}, {4, 10640}, {5, 36300}, {6, 17}, {471, 11130}, {1147, 32585}, {6337, 34389}, {6503, 40712}, {21461, 22391}
X(52348) = crossdifference of every pair of points on line {2501, 6137}
X(52348) = barycentric product X(i)*X(j) for these {i,j}: {3, 302}, {18, 44180}, {49, 34390}, {61, 69}, {298, 50468}, {394, 473}, {1994, 40711}, {3926, 10642}, {4558, 23872}, {7769, 32586}, {8838, 44718}, {11126, 40710}, {11132, 36297}, {11146, 40709}, {16771, 44719}, {50466, 52220}
X(52348) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 17}, {6, 8741}, {18, 93}, {49, 62}, {61, 4}, {69, 34389}, {184, 21461}, {216, 36300}, {302, 264}, {394, 40712}, {473, 2052}, {577, 32585}, {1994, 472}, {2965, 10641}, {4558, 32036}, {10642, 393}, {11083, 8737}, {11126, 471}, {11135, 8740}, {11137, 8739}, {11141, 8738}, {11146, 470}, {23872, 14618}, {32037, 38342}, {32586, 2963}, {32661, 16806}, {34390, 20572}, {36296, 11139}, {36297, 11087}, {40711, 11140}, {44180, 303}, {44719, 19779}, {46112, 8603}, {46113, 51890}, {50466, 11600}, {50468, 13}, {51546, 8742}
X(52348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 44718, 44719}, {343, 466, 40710}, {343, 52194, 40712}, {466, 52194, 343}, {627, 19773, 33529}, {5408, 5409, 44718}, {9306, 9736, 3132}, {11126, 11146, 61}, {40710, 40712, 343}

X(52349) = X(6)-DAO CONJUGATE OF X(18)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2 - b^2 - c^2 + 2*Sqrt[3]*S) : :

X(52349) lies on these lines: {2, 14}, {3, 49}, {16, 1993}, {17, 13579}, {22, 14539}, {51, 5611}, {61, 5422}, {62, 1994}, {69, 11515}, {97, 44712}, {249, 40156}, {300, 32036}, {303, 472}, {323, 10646}, {343, 465}, {466, 11064}, {473, 11475}, {511, 3132}, {628, 8837}, {630, 2913}, {1599, 3389}, {1600, 3390}, {2979, 14538}, {3060, 38432}, {3130, 47068}, {3131, 9306}, {3366, 15234}, {3367, 15233}, {3819, 13350}, {4558, 36296}, {5237, 11126}, {5352, 11146}, {6636, 14170}, {7998, 21158}, {9544, 14169}, {10601, 11485}, {10645, 15066}, {11004, 34755}, {11081, 33458}, {11144, 16770}, {11427, 37173}, {11480, 17811}, {11481, 37672}, {11516, 20806}, {13349, 34986}, {14541, 34008}, {17825, 22236}, {18468, 37638}, {22151, 46112}, {26881, 34009}, {34394, 35296}, {34545, 41478}, {37341, 37649}

X(52349) = isogonal conjugate of X(8742)
X(52349) = isotomic conjugate of the polar conjugate of X(62)
X(52349) = isogonal conjugate of the polar conjugate of X(303)
X(52349) = X(i)-Ceva conjugate of X(j) for these (i,j): {303, 62}, {40709, 44718}
X(52349) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8742}, {18, 19}, {92, 21462}, {158, 32586}, {1096, 40711}, {1973, 34390}, {2190, 36301}, {2962, 10642}, {3384, 8737}, {16807, 24006}
X(52349) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8742}, {4, 10639}, {5, 36301}, {6, 18}, {470, 11131}, {1147, 32586}, {6337, 34390}, {6503, 40711}, {21462, 22391}
X(52349) = crossdifference of every pair of points on line {2501, 6138}
X(52349) = barycentric product X(i)*X(j) for these {i,j}: {3, 303}, {17, 44180}, {49, 34389}, {62, 69}, {299, 50469}, {394, 472}, {1994, 40712}, {3926, 10641}, {4558, 23873}, {7769, 32585}, {8836, 44719}, {11127, 40709}, {11133, 36296}, {11145, 40710}, {16770, 44718}, {50465, 52221}
X(52349) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 18}, {6, 8742}, {17, 93}, {49, 61}, {62, 4}, {69, 34390}, {184, 21462}, {216, 36301}, {303, 264}, {394, 40711}, {472, 2052}, {577, 32586}, {1994, 473}, {2965, 10642}, {4558, 32037}, {10641, 393}, {11088, 8738}, {11127, 470}, {11134, 8740}, {11136, 8739}, {11142, 8737}, {11145, 471}, {23873, 14618}, {32036, 38342}, {32585, 2963}, {32661, 16807}, {34389, 20572}, {36296, 11082}, {36297, 11138}, {40712, 11140}, {44180, 302}, {44718, 19778}, {46112, 51891}, {46113, 8604}, {50465, 11601}, {50469, 14}, {51547, 8741}
X(52349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 44719, 44718}, {343, 465, 40709}, {343, 52193, 40711}, {465, 52193, 343}, {628, 19772, 33530}, {5408, 5409, 44719}, {9306, 9735, 3131}, {11127, 11145, 62}, {40709, 40711, 343}

X(52350) = X(6)-DAO CONJUGATE OF X(24)

Barycentrics    (a^2 - b^2 - c^2)^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) : :

X(52350) lies on these lines: {2, 311}, {3, 68}, {69, 97}, {91, 19854}, {96, 631}, {182, 41271}, {276, 7763}, {394, 6389}, {427, 14593}, {441, 52041}, {571, 6515}, {847, 1217}, {925, 1297}, {1073, 11064}, {3135, 23181}, {3546, 40698}, {3548, 32132}, {4558, 45794}, {5449, 15827}, {5962, 18533}, {6388, 10318}, {6459, 13428}, {6460, 13439}, {6511, 26942}, {6512, 26932}, {6815, 45301}, {7667, 40348}, {7836, 46735}, {8906, 11585}, {9723, 37636}, {11427, 50572}, {13567, 47731}, {14376, 36212}, {14790, 34225}, {14919, 37669}, {16032, 45509}, {16037, 45508}, {18420, 34836}, {23335, 46200}, {28724, 37188}, {30450, 46927}, {34385, 37872}, {39113, 52253}

X(52350) = isogonal conjugate of X(8745)
X(52350) = isotomic conjugate of X(11547)
X(52350) = isotomic conjugate of the polar conjugate of X(68)
X(52350) = isogonal conjugate of the polar conjugate of X(20563)
X(52350) = polar conjugate of the isogonal conjugate of X(16391)
X(52350) = X(20563)-Ceva conjugate of X(68)
X(52350) = X(i)-cross conjugate of X(j) for these (i,j): {125, 3265}, {1092, 69}
X(52350) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8745}, {19, 24}, {25, 1748}, {31, 11547}, {47, 393}, {91, 36416}, {92, 44077}, {158, 571}, {162, 6753}, {317, 1973}, {563, 1093}, {823, 34952}, {924, 24019}, {1096, 1993}, {1147, 6520}, {2180, 8884}, {2190, 14576}, {2207, 44179}, {17881, 41937}, {24000, 47421}, {30451, 36126}
X(52350) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 11547}, {3, 8745}, {5, 14576}, {6, 24}, {125, 6753}, {136, 647}, {317, 6337}, {393, 34853}, {467, 52032}, {571, 1147}, {924, 35071}, {1147, 37867}, {1748, 6505}, {1993, 6503}, {2165, 3542}, {2207, 37864}, {3767, 41770}, {6338, 7763}, {14397, 38999}, {22391, 44077}, {30451, 46093}, {34116, 36416}
X(52350) = crosssum of X(6753) and X(47421)
X(52350) = crossdifference of every pair of points on line {6753, 34952}
X(52350) = barycentric product X(i)*X(j) for these {i,j}: {3, 20563}, {68, 69}, {91, 326}, {255, 20571}, {264, 16391}, {304, 1820}, {305, 2351}, {339, 44174}, {394, 5392}, {520, 46134}, {847, 3964}, {925, 3265}, {2165, 3926}, {4176, 14593}, {5562, 34385}, {11090, 11091}, {26922, 34391}, {32132, 40697}
X(52350) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11547}, {3, 24}, {6, 8745}, {63, 1748}, {68, 4}, {69, 317}, {91, 158}, {96, 8884}, {125, 136}, {155, 35603}, {184, 44077}, {216, 14576}, {255, 47}, {326, 44179}, {343, 467}, {394, 1993}, {520, 924}, {571, 36416}, {577, 571}, {647, 6753}, {847, 1093}, {925, 107}, {1092, 1147}, {1636, 14397}, {1820, 19}, {2165, 393}, {2351, 25}, {3265, 6563}, {3269, 47421}, {3519, 14111}, {3926, 7763}, {3964, 9723}, {3998, 42700}, {4100, 563}, {4558, 41679}, {5392, 2052}, {5562, 52}, {6389, 41770}, {6394, 31635}, {6413, 5412}, {6414, 5413}, {11090, 1586}, {11091, 1585}, {13754, 52000}, {14593, 6524}, {15316, 34756}, {16391, 3}, {17879, 17881}, {18604, 18605}, {20563, 264}, {23224, 34948}, {26922, 371}, {30450, 15352}, {32132, 254}, {32320, 30451}, {32734, 32713}, {34385, 8795}, {34853, 3542}, {36145, 24019}, {37802, 14165}, {39201, 34952}, {43083, 43088}, {44174, 250}, {46134, 6528}, {51386, 51439}, {51394, 51393}
X(52350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5392, 2165}, {2, 40697, 52032}, {5392, 37802, 2}, {11090, 11091, 68}, {13430, 13441, 20563}

X(52351) = X(6)-DAO CONJUGATE OF X(36)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(52351) lies on these lines: {2, 2006}, {10, 21}, {63, 343}, {78, 1062}, {94, 7110}, {280, 5552}, {306, 1332}, {333, 39277}, {348, 6350}, {498, 15065}, {655, 908}, {914, 22128}, {1411, 19860}, {1791, 34851}, {2161, 2339}, {2222, 26703}, {3035, 14204}, {3264, 32851}, {3306, 7131}, {6335, 20566}, {6505, 41081}, {6703, 11069}, {6735, 40437}, {6745, 51562}, {6796, 7488}, {10573, 24597}, {14616, 19808}, {16067, 29857}, {16548, 46487}, {16594, 17279}, {17740, 33937}, {17776, 34277}, {24209, 33129}, {26231, 29828}, {27509, 30680}, {30787, 50752}, {30852, 52212}, {32849, 46785}, {35174, 37774}

X(52351) = isotomic conjugate of X(17923)
X(52351) = isotomic conjugate of the polar conjugate of X(80)
X(52351) = isogonal conjugate of the polar conjugate of X(20566)
X(52351) = X(20566)-Ceva conjugate of X(80)
X(52351) = X(i)-cross conjugate of X(j) for these (i,j): {5440, 69}, {14429, 4561}, {22059, 3}
X(52351) = X(i)-isoconjugate of X(j) for these (i,j): {4, 7113}, {6, 1870}, {19, 36}, {25, 3218}, {27, 3724}, {28, 2245}, {31, 17923}, {34, 2323}, {81, 44113}, {108, 654}, {162, 21828}, {186, 2160}, {214, 8752}, {225, 4282}, {278, 2361}, {284, 1835}, {320, 1973}, {604, 5081}, {607, 1443}, {608, 4511}, {649, 4242}, {653, 8648}, {758, 1474}, {860, 1333}, {909, 1845}, {913, 11570}, {1096, 22128}, {1172, 1464}, {1395, 32851}, {1400, 17515}, {1415, 44428}, {1897, 21758}, {1974, 20924}, {1983, 7649}, {2203, 3936}, {2204, 41804}, {2212, 17078}, {2299, 18593}, {3738, 32674}, {3960, 8750}, {4707, 32676}, {8739, 39153}, {8740, 39152}, {8756, 16944}, {13486, 47230}, {17455, 36125}, {30690, 34397}
X(52351) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17923}, {6, 36}, {9, 1870}, {19, 15898}, {37, 860}, {125, 21828}, {226, 18593}, {281, 36909}, {320, 6337}, {654, 38983}, {758, 51574}, {1146, 44428}, {1835, 40590}, {1845, 23980}, {2245, 40591}, {2323, 11517}, {3161, 5081}, {3218, 6505}, {3738, 35072}, {3904, 40626}, {3960, 26932}, {4242, 5375}, {4453, 40618}, {4707, 15526}, {6503, 22128}, {7113, 36033}, {17515, 40582}, {21758, 34467}, {40586, 44113}
X(52351) = cevapoint of X(i) and X(j) for these (i,j): {3, 22123}, {71, 22350}, {306, 3977}
X(52351) = trilinear pole of line {72, 521}
X(52351) = crossdifference of every pair of points on line {8648, 21828}
X(52351) = barycentric product X(i)*X(j) for these {i,j}: {3, 20566}, {35, 328}, {63, 18359}, {69, 80}, {72, 14616}, {75, 1807}, {78, 18815}, {265, 319}, {304, 2161}, {305, 6187}, {306, 24624}, {307, 6740}, {337, 36815}, {345, 2006}, {348, 36910}, {521, 35174}, {525, 47318}, {652, 46405}, {655, 6332}, {759, 20336}, {905, 36804}, {1231, 2341}, {1411, 3718}, {1441, 1793}, {1444, 15065}, {2222, 35518}, {4025, 51562}, {34079, 40071}
X(52351) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1870}, {2, 17923}, {3, 36}, {8, 5081}, {10, 860}, {21, 17515}, {35, 186}, {42, 44113}, {48, 7113}, {63, 3218}, {65, 1835}, {69, 320}, {71, 2245}, {72, 758}, {73, 1464}, {77, 1443}, {78, 4511}, {80, 4}, {100, 4242}, {212, 2361}, {219, 2323}, {228, 3724}, {265, 79}, {304, 20924}, {305, 40075}, {306, 3936}, {307, 41804}, {319, 340}, {328, 20565}, {345, 32851}, {348, 17078}, {394, 22128}, {517, 1845}, {521, 3738}, {522, 44428}, {525, 4707}, {647, 21828}, {652, 654}, {655, 653}, {759, 28}, {905, 3960}, {906, 1983}, {912, 11570}, {1060, 4351}, {1128, 8119}, {1168, 36125}, {1214, 18593}, {1332, 4585}, {1411, 34}, {1565, 4089}, {1793, 21}, {1797, 40215}, {1807, 1}, {1946, 8648}, {2006, 278}, {2161, 19}, {2193, 4282}, {2222, 108}, {2341, 1172}, {3781, 3792}, {3916, 4973}, {3927, 4880}, {3940, 4867}, {3949, 4053}, {3977, 51583}, {4025, 4453}, {4064, 6370}, {4855, 4881}, {5440, 214}, {6187, 25}, {6332, 3904}, {6740, 29}, {7265, 44427}, {10215, 8120}, {14584, 1877}, {14616, 286}, {14628, 37790}, {15065, 41013}, {18359, 92}, {18815, 273}, {20336, 35550}, {20566, 264}, {20769, 27950}, {21054, 35235}, {21912, 51462}, {22123, 40584}, {22350, 34586}, {22356, 17455}, {22383, 21758}, {23071, 6126}, {23224, 22379}, {24624, 27}, {32675, 32674}, {34079, 1474}, {34857, 1824}, {35174, 18026}, {35194, 51801}, {36058, 16944}, {36061, 13486}, {36804, 6335}, {36815, 242}, {36910, 281}, {36927, 2907}, {38938, 5146}, {40437, 36123}, {45926, 1838}, {46405, 46404}, {46974, 11700}, {47318, 648}, {49280, 23884}, {51562, 1897}, {51975, 38462}, {52153, 6186}, {52201, 41225}, {52202, 3179}
X(52351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18359, 2006}, {2, 41226, 18359}, {2006, 36910, 18359}, {18359, 41226, 36910}

X(52352) = X(8)-DAO CONJUGATE OF X(10)

Barycentrics    (a + b)*(a - b - c)*(3*a - b - c)*(a + c) : :
X(52352) = 4 X[1125] - 3 X[24161]

X(52352) lies on these lines: {1, 4234}, {2, 49734}, {8, 21}, {20, 18134}, {29, 4997}, {30, 25650}, {58, 643}, {69, 17576}, {75, 28627}, {81, 3623}, {86, 3445}, {99, 1434}, {100, 7419}, {145, 3052}, {190, 34772}, {284, 2325}, {312, 3601}, {320, 8822}, {386, 13735}, {452, 5233}, {903, 26729}, {950, 32851}, {979, 5331}, {993, 50625}, {1010, 1125}, {1098, 30606}, {1220, 37573}, {1420, 39126}, {1698, 11110}, {1834, 25459}, {1975, 14828}, {2475, 41878}, {2646, 3685}, {2975, 8053}, {3057, 3794}, {3146, 30828}, {3158, 44720}, {3161, 44722}, {3216, 33309}, {3522, 18141}, {3616, 4854}, {3621, 4921}, {3707, 4877}, {3847, 14011}, {3912, 35935}, {3936, 15680}, {3950, 33628}, {3984, 17336}, {4184, 5303}, {4189, 14829}, {4190, 17234}, {4195, 19765}, {4248, 4855}, {4255, 17697}, {4256, 13741}, {4258, 27523}, {4304, 7270}, {4314, 4514}, {4417, 6872}, {4427, 34195}, {4645, 15338}, {4647, 5426}, {4869, 50693}, {4975, 37616}, {5247, 50590}, {5263, 10448}, {5436, 19804}, {5438, 30829}, {7253, 28221}, {7283, 24929}, {7354, 29839}, {7415, 31730}, {7538, 20477}, {9534, 16418}, {9780, 17553}, {10107, 18165}, {10449, 16370}, {10609, 37227}, {11106, 14555}, {13588, 30947}, {14007, 34595}, {15670, 25446}, {15674, 31205}, {15952, 34773}, {16046, 17316}, {16050, 17367}, {16704, 20014}, {16865, 17277}, {17297, 37299}, {17524, 29766}, {17525, 49716}, {17588, 46933}, {17677, 25645}, {18139, 37256}, {19270, 48863}, {19532, 37502}, {19701, 51674}, {19768, 36489}, {24936, 50171}, {26117, 30832}, {27505, 28793}, {28461, 48877}, {34064, 37539}, {41816, 49728}, {49745, 51678}

X(52352) = X(86)-Ceva conjugate of X(333)
X(52352) = X(4162)-cross conjugate of X(30720)
X(52352) = X(i)-isoconjugate of X(j) for these (i,j): {10, 16945}, {37, 40151}, {42, 19604}, {65, 3445}, {213, 27818}, {226, 38266}, {604, 4052}, {661, 38828}, {1042, 3680}, {1293, 4017}, {1400, 8056}, {1402, 4373}, {2333, 27832}, {4849, 16079}, {7178, 34080}, {7180, 27834}, {7250, 31343}, {30572, 36042}
X(52352) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 10}, {65, 45036}, {523, 3756}, {1293, 34961}, {3161, 4052}, {3445, 40602}, {4373, 40605}, {4859, 21949}, {5516, 30572}, {6626, 27818}, {7178, 40621}, {8056, 40582}, {19604, 40592}, {36830, 38828}, {40151, 40589}
X(52352) = cevapoint of X(i) and X(j) for these (i,j): {145, 4855}, {3158, 3161}
X(52352) = crosspoint of X(86) and X(41629)
X(52352) = barycentric product X(i)*X(j) for these {i,j}: {8, 41629}, {21, 18743}, {27, 44722}, {58, 44723}, {81, 44720}, {86, 3161}, {99, 4521}, {145, 333}, {261, 3950}, {274, 3158}, {312, 16948}, {314, 1743}, {345, 4248}, {643, 4462}, {645, 3667}, {799, 4162}, {1043, 5435}, {1434, 6555}, {1790, 44721}, {2287, 39126}, {3052, 28660}, {3596, 33628}, {4394, 7257}, {4404, 4612}, {4534, 4600}, {4546, 4573}, {4560, 43290}, {4567, 4939}, {4610, 44729}, {4620, 4953}, {4631, 4729}, {4848, 7058}, {4855, 31623}, {6064, 21950}, {7192, 30720}, {7256, 30719}, {7258, 51658}, {17197, 44724}, {18653, 44727}, {20818, 44130}
X(52352) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4052}, {21, 8056}, {58, 40151}, {81, 19604}, {86, 27818}, {110, 38828}, {145, 226}, {284, 3445}, {314, 40014}, {333, 4373}, {643, 27834}, {1043, 6557}, {1333, 16945}, {1420, 1427}, {1444, 27832}, {1743, 65}, {2194, 38266}, {2287, 3680}, {3052, 1400}, {3158, 37}, {3161, 10}, {3667, 7178}, {3950, 12}, {4162, 661}, {4248, 278}, {4394, 4017}, {4462, 4077}, {4521, 523}, {4534, 3120}, {4546, 3700}, {4848, 6354}, {4849, 2171}, {4855, 1214}, {4856, 3649}, {4881, 18593}, {4936, 210}, {4939, 16732}, {4943, 14321}, {4953, 21044}, {5435, 3668}, {5546, 1293}, {6555, 2321}, {7259, 31343}, {8643, 7180}, {12640, 4415}, {14425, 30572}, {15519, 3950}, {16948, 57}, {18743, 1441}, {20818, 73}, {21950, 1365}, {30720, 3952}, {30941, 10029}, {33628, 56}, {39126, 1446}, {41629, 7}, {43290, 4552}, {44720, 321}, {44722, 306}, {44723, 313}, {44729, 4024}, {51658, 7216}
X(52352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 1043, 333}, {145, 16948, 41629}, {3712, 10543, 8}

X(52353) = X(8)-DAO CONJUGATE OF X(21)

Barycentrics    b*c*(b + c)*(-3*a + b + c) : :

X(52353) lies on these lines: {1, 4723}, {2, 341}, {8, 392}, {10, 321}, {58, 16729}, {65, 3952}, {75, 46933}, {100, 28029}, {145, 4487}, {210, 17751}, {312, 3617}, {318, 36624}, {344, 10528}, {518, 29982}, {594, 42712}, {946, 30566}, {950, 49991}, {984, 20892}, {1191, 26688}, {1201, 24003}, {1220, 5297}, {1265, 5554}, {1329, 3006}, {1698, 4968}, {1722, 3891}, {1739, 24068}, {1997, 10529}, {2478, 5014}, {2551, 5016}, {2650, 4090}, {2885, 4884}, {2899, 3434}, {2901, 31855}, {2975, 5205}, {3175, 21896}, {3214, 3896}, {3244, 4738}, {3263, 6376}, {3436, 7386}, {3596, 24547}, {3616, 4737}, {3621, 46938}, {3622, 30829}, {3625, 4975}, {3626, 3902}, {3632, 4742}, {3634, 4692}, {3679, 3702}, {3696, 22016}, {3698, 3967}, {3699, 34772}, {3703, 9711}, {3714, 3983}, {3717, 20905}, {3718, 24993}, {3812, 17165}, {3823, 18133}, {3831, 46909}, {3869, 27538}, {3921, 5295}, {3932, 21031}, {3936, 21075}, {3953, 49993}, {3970, 4103}, {3991, 30730}, {3995, 4646}, {4009, 5836}, {4033, 42724}, {4037, 21868}, {4359, 4385}, {4397, 23757}, {4450, 12572}, {4462, 23764}, {4514, 37162}, {4518, 30082}, {4662, 17135}, {4673, 4678}, {4712, 30045}, {4767, 34195}, {4850, 26029}, {4939, 12640}, {5046, 32850}, {5086, 36926}, {5128, 25734}, {5250, 51284}, {5253, 9369}, {5260, 7081}, {5433, 37762}, {5552, 33113}, {6381, 20880}, {7080, 17776}, {8728, 40013}, {9330, 31359}, {9335, 34860}, {10005, 20946}, {11681, 29641}, {12648, 42020}, {14923, 19582}, {17490, 26046}, {17752, 26689}, {17757, 21530}, {17862, 25005}, {18156, 25278}, {19804, 46932}, {20653, 48648}, {20911, 33932}, {20947, 25280}, {21101, 21921}, {21674, 27705}, {23536, 24988}, {24171, 24183}, {24349, 30044}, {24440, 32925}, {25237, 40883}, {25610, 41269}, {25992, 26230}, {25994, 31087}, {26234, 33938}, {26770, 44798}, {28080, 30614}, {28996, 34040}, {30090, 31302}, {30615, 36500}, {30806, 33942}, {30807, 39570}, {31035, 37548}, {32911, 41261}

X(52353) = isotomic conjugate of the isogonal conjugate of X(4849)
X(52353) = X(i)-Ceva conjugate of X(j) for these (i,j): {1441, 321}, {18743, 3950}
X(52353) = X(i)-cross conjugate of X(j) for these (i,j): {4918, 145}, {21950, 4462}
X(52353) = X(i)-isoconjugate of X(j) for these (i,j): {21, 16945}, {58, 3445}, {81, 38266}, {284, 40151}, {1019, 34080}, {1293, 3733}, {1333, 8056}, {1408, 3680}, {2194, 19604}, {2204, 27832}, {2206, 4373}, {6557, 16947}, {7252, 38828}
X(52353) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 21}, {10, 3445}, {37, 8056}, {58, 45036}, {1019, 40621}, {1214, 19604}, {3667, 18211}, {3737, 3756}, {3950, 37679}, {4373, 40603}, {4521, 16726}, {4849, 16688}, {16945, 40611}, {38266, 40586}, {40151, 40590}
X(52353) = crosspoint of X(44720) and X(44721)
X(52353) = trilinear pole of line {4404, 14321}
X(52353) = barycentric product X(i)*X(j) for these {i,j}: {10, 18743}, {65, 44723}, {75, 3950}, {76, 4849}, {145, 321}, {190, 4404}, {226, 44720}, {312, 4848}, {313, 1743}, {349, 3158}, {668, 14321}, {1089, 41629}, {1214, 44721}, {1420, 30713}, {1441, 3161}, {1446, 6555}, {1577, 43290}, {1978, 4729}, {2321, 39126}, {3052, 27801}, {3667, 4033}, {3701, 5435}, {3952, 4462}, {3971, 27496}, {3992, 31227}, {4077, 30720}, {4080, 4487}, {4394, 27808}, {4554, 44729}, {4918, 30710}, {7035, 21950}, {16732, 44724}, {16948, 28654}, {40149, 44722}
X(52353) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 8056}, {37, 3445}, {42, 38266}, {65, 40151}, {145, 81}, {226, 19604}, {307, 27832}, {313, 40014}, {321, 4373}, {1018, 1293}, {1089, 4052}, {1400, 16945}, {1420, 1412}, {1441, 27818}, {1743, 58}, {2321, 3680}, {2885, 16736}, {3052, 1333}, {3158, 284}, {3161, 21}, {3667, 1019}, {3701, 6557}, {3756, 16726}, {3950, 1}, {3952, 27834}, {4058, 10563}, {4162, 7252}, {4394, 3733}, {4404, 514}, {4462, 7192}, {4487, 16704}, {4504, 18200}, {4521, 3737}, {4534, 18191}, {4546, 1021}, {4551, 38828}, {4557, 34080}, {4729, 649}, {4848, 57}, {4849, 6}, {4855, 1790}, {4884, 16696}, {4891, 18166}, {4898, 4658}, {4899, 18206}, {4918, 3666}, {4936, 2328}, {4939, 17197}, {4949, 4840}, {5435, 1014}, {6555, 2287}, {12640, 18163}, {14321, 513}, {16948, 593}, {18743, 86}, {20818, 1437}, {21052, 27837}, {21950, 244}, {23764, 8042}, {30719, 7203}, {30720, 643}, {30730, 31343}, {33628, 849}, {39126, 1434}, {40621, 18211}, {41629, 757}, {43290, 662}, {44720, 333}, {44721, 31623}, {44722, 1812}, {44723, 314}, {44724, 4567}, {44729, 650}
X(52353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 341, 4696}, {8, 46937, 4358}, {10, 3159, 3987}, {10, 3701, 321}, {10, 3971, 4642}, {10, 3992, 3701}, {10, 4066, 4714}, {10, 4075, 4424}, {10, 4125, 4647}, {145, 4487, 4935}, {145, 44720, 4487}, {1698, 4968, 24589}, {2551, 10327, 5016}, {3263, 6376, 26563}, {3263, 26563, 21432}, {3698, 3967, 17164}, {3714, 3983, 4651}, {4009, 5836, 25253}, {4385, 9780, 4359}, {18743, 44720, 145}

X(52354) = X(8)-DAO CONJUGATE OF X(29)

Barycentrics    (3*a - b - c)*(b + c)*(a^2 - b^2 - c^2) : :

X(52354) lies on these lines: {2, 39589}, {8, 3586}, {10, 3120}, {20, 25734}, {40, 49991}, {63, 1265}, {72, 306}, {78, 3977}, {145, 1743}, {344, 11520}, {345, 3984}, {537, 23675}, {976, 35263}, {982, 25881}, {2650, 4078}, {3610, 3958}, {3704, 4005}, {3717, 3869}, {3932, 3962}, {3947, 27690}, {3951, 4001}, {3967, 21677}, {4069, 37558}, {4082, 17751}, {4115, 21073}, {4126, 5836}, {4358, 24391}, {4487, 12640}, {4488, 37435}, {4537, 21081}, {4696, 5837}, {4723, 11362}, {4756, 5086}, {4847, 25253}, {4849, 4918}, {4855, 44722}, {6743, 32929}, {6872, 25728}, {7270, 17781}, {10327, 12526}, {11523, 17776}, {12649, 30568}, {18249, 26227}, {19582, 26015}, {19860, 27549}, {24349, 24564}, {24982, 27538}, {24987, 32937}, {36500, 40998}, {43290, 44727}, {44720, 44721}

X(52354) = isotomic conjugate of the polar conjugate of X(3950)
X(52354) = X(307)-Ceva conjugate of X(306)
X(52354) = X(i)-isoconjugate of X(j) for these (i,j): {27, 38266}, {28, 3445}, {29, 16945}, {1172, 40151}, {1474, 8056}, {2203, 4373}, {2204, 27818}, {2299, 19604}, {17925, 34080}, {27834, 43925}
X(52354) = X(i)-Dao conjugate of X(j) for these (i,j): {8, 29}, {28, 45036}, {226, 19604}, {3445, 40591}, {8056, 51574}, {17925, 40621}
X(52354) = crosspoint of X(44720) and X(44722)
X(52354) = barycentric product X(i)*X(j) for these {i,j}: {69, 3950}, {72, 18743}, {73, 44723}, {145, 306}, {226, 44722}, {304, 4849}, {307, 3161}, {313, 20818}, {321, 4855}, {345, 4848}, {525, 43290}, {1214, 44720}, {1231, 3158}, {1332, 4404}, {1743, 20336}, {3052, 40071}, {3694, 39126}, {3695, 41629}, {3710, 5435}, {4466, 44724}, {4561, 14321}, {17094, 30720}, {40152, 44721}
X(52354) = barycentric quotient X(i)/X(j) for these {i,j}: {71, 3445}, {72, 8056}, {73, 40151}, {145, 27}, {228, 38266}, {306, 4373}, {307, 27818}, {1214, 19604}, {1409, 16945}, {1420, 1396}, {1743, 28}, {3052, 1474}, {3158, 1172}, {3161, 29}, {3667, 17925}, {3694, 3680}, {3695, 4052}, {3710, 6557}, {3950, 4}, {4248, 36419}, {4404, 17924}, {4546, 17926}, {4574, 1293}, {4729, 6591}, {4848, 278}, {4849, 19}, {4855, 81}, {4856, 31900}, {4884, 17171}, {4898, 31902}, {4899, 15149}, {4918, 1848}, {4924, 31922}, {4929, 31925}, {4936, 4183}, {6555, 2322}, {8643, 43925}, {14321, 7649}, {18743, 286}, {20336, 40014}, {20818, 58}, {21950, 2969}, {23067, 38828}, {30720, 36797}, {43290, 648}, {44720, 31623}, {44722, 333}, {44723, 44130}, {44729, 3064}
X(52354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 3695, 4101}, {72, 3710, 306}, {3695, 4101, 306}, {3710, 4101, 3695}

X(52355) = X(11)-DAO CONJUGATE OF X(28)

Barycentrics    (a - b - c)*(b - c)*(b + c)*(a^2 - b^2 - c^2) : :
X(52355) = X[656] - 3 X[14429], X[4064] + 3 X[14429], 2 X[676] - 3 X[48186], X[4581] - 3 X[47809], 3 X[14430] - X[21119], 2 X[21187] - 3 X[41800], 2 X[34958] - 3 X[48209], X[47695] - 3 X[48173]

X(52355) lies on these lines: {37, 2485}, {72, 520}, {100, 2766}, {101, 2769}, {513, 28591}, {521, 6332}, {522, 650}, {523, 1577}, {525, 656}, {676, 48186}, {765, 39189}, {832, 48299}, {905, 20315}, {918, 23800}, {2483, 21390}, {2525, 21107}, {2804, 4397}, {3265, 14208}, {3267, 20336}, {3800, 50332}, {3904, 20293}, {3952, 23067}, {4017, 4088}, {4163, 8058}, {4391, 20294}, {4581, 47809}, {4768, 42337}, {4804, 27588}, {4985, 6362}, {7253, 15776}, {8057, 24018}, {8062, 44409}, {8672, 48047}, {10015, 20316}, {14430, 21119}, {15411, 23189}, {17420, 48278}, {21187, 41800}, {21189, 48272}, {34958, 48209}, {38469, 48290}, {47695, 48173}, {48069, 50495}

X(52355) = midpoint of X(i) and X(j) for these {i,j}: {656, 4064}, {3904, 20293}, {4017, 4088}, {4391, 20294}, {17420, 48278}, {21189, 48272}
X(52355) = reflection of X(i) in X(j) for these {i,j}: {905, 20315}, {10015, 20316}, {44409, 8062}, {48402, 47842}
X(52355) = isotomic conjugate of the polar conjugate of X(3700)
X(52355) = X(i)-complementary conjugate of X(j) for these (i,j): {8615, 11}, {15314, 21252}
X(52355) = X(i)-Ceva conjugate of X(j) for these (i,j): {8, 7068}, {78, 2968}, {100, 3704}, {318, 7358}, {645, 219}, {3718, 26932}, {3952, 72}, {4397, 4086}, {4552, 2321}, {5546, 3703}, {6332, 8611}, {14208, 525}, {36626, 6741}, {36797, 8}
X(52355) = X(i)-cross conjugate of X(j) for these (i,j): {4064, 4086}, {7068, 8}, {8611, 525}, {21044, 40161}
X(52355) = X(i)-isoconjugate of X(j) for these (i,j): {7, 32676}, {19, 4565}, {25, 1414}, {27, 1415}, {28, 109}, {34, 110}, {56, 162}, {57, 112}, {58, 108}, {77, 32713}, {81, 32674}, {99, 1395}, {101, 1396}, {107, 603}, {163, 278}, {222, 24019}, {250, 4017}, {273, 1576}, {284, 32714}, {604, 648}, {607, 4637}, {608, 662}, {643, 1398}, {651, 1474}, {653, 1333}, {658, 2204}, {664, 2203}, {685, 51653}, {811, 1397}, {859, 36110}, {933, 1393}, {934, 2299}, {1014, 8750}, {1019, 7115}, {1020, 2189}, {1106, 36797}, {1118, 4575}, {1172, 1461}, {1301, 1394}, {1304, 51656}, {1408, 1897}, {1412, 1783}, {1416, 4238}, {1417, 46541}, {1426, 4636}, {1435, 5546}, {1437, 36127}, {1451, 36077}, {1813, 5317}, {1835, 36069}, {1880, 4556}, {1973, 4573}, {1974, 4625}, {2149, 17925}, {2194, 36118}, {2206, 18026}, {2212, 4616}, {2332, 4617}, {3194, 8059}, {3213, 46639}, {3733, 7012}, {4183, 6614}, {4564, 43925}, {4570, 43923}, {4592, 7337}, {5323, 32691}, {5338, 5545}, {5379, 43924}, {6335, 16947}, {6357, 36131}, {6529, 7125}, {7128, 7252}, {7335, 36126}, {8747, 36059}, {15439, 46883}, {18020, 51642}, {23189, 24033}, {23964, 51640}, {32230, 51641}, {32651, 46884}, {36046, 43045}, {36104, 43034}, {44770, 51649}
X(52355) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 162}, {4, 6741}, {6, 4565}, {7, 15526}, {10, 108}, {11, 28}, {21, 7358}, {25, 40608}, {27, 1146}, {29, 2968}, {34, 244}, {37, 653}, {56, 125}, {57, 34591}, {58, 38983}, {81, 35072}, {86, 40626}, {107, 7952}, {109, 40591}, {110, 11517}, {112, 5452}, {115, 278}, {122, 44696}, {136, 1118}, {222, 35071}, {226, 934}, {250, 34961}, {273, 4858}, {286, 40624}, {331, 36901}, {521, 23189}, {525, 17094}, {603, 38985}, {608, 1084}, {647, 7178}, {648, 3161}, {650, 17925}, {651, 51574}, {656, 3737}, {859, 39004}, {905, 7192}, {1014, 26932}, {1015, 1396}, {1019, 40628}, {1119, 40622}, {1145, 4246}, {1172, 35508}, {1214, 36118}, {1354, 1650}, {1395, 38986}, {1397, 17423}, {1408, 34467}, {1412, 39006}, {1414, 6505}, {1434, 40618}, {1474, 38991}, {1783, 40599}, {1835, 38982}, {2203, 39025}, {2299, 14714}, {2972, 30493}, {3239, 4560}, {3756, 4248}, {4230, 50440}, {4233, 4904}, {4238, 40609}, {4240, 6739}, {4241, 50441}, {4573, 6337}, {5139, 7337}, {5323, 17421}, {6357, 39008}, {6388, 17081}, {6552, 36797}, {7335, 46093}, {8286, 13739}, {8747, 20620}, {8885, 13612}, {18026, 40603}, {18623, 39020}, {32674, 40586}, {32714, 40590}, {33504, 43045}, {37168, 51402}, {39000, 43034}, {43923, 50330}
X(52355) = cevapoint of X(3700) and X(14308)
X(52355) = crosspoint of X(i) and X(j) for these (i,j): {8, 36797}, {100, 1791}, {307, 4552}, {645, 3596}, {3701, 3952}, {6332, 35518}, {6335, 40422}
X(52355) = crosssum of X(i) and X(j) for these (i,j): {513, 1829}, {1397, 7180}, {1408, 3733}, {2203, 43925}, {2299, 7252}, {22383, 40956}
X(52355) = crossdifference of every pair of points on line {56, 608}
X(52355) = barycentric product X(i)*X(j) for these {i,j}: {8, 525}, {9, 14208}, {10, 6332}, {12, 15411}, {37, 35518}, {55, 3267}, {63, 4086}, {65, 15416}, {69, 3700}, {71, 35519}, {72, 4391}, {75, 8611}, {78, 1577}, {125, 645}, {210, 15413}, {212, 20948}, {219, 850}, {281, 3265}, {304, 4041}, {305, 3709}, {306, 522}, {307, 3239}, {312, 656}, {313, 652}, {318, 24018}, {321, 521}, {332, 4024}, {333, 4064}, {339, 5546}, {341, 51640}, {345, 523}, {346, 17094}, {514, 3710}, {520, 7017}, {643, 20902}, {646, 18210}, {647, 3596}, {648, 7068}, {650, 20336}, {661, 3718}, {663, 40071}, {693, 3694}, {810, 28659}, {905, 3701}, {1018, 17880}, {1214, 4397}, {1231, 3900}, {1259, 14618}, {1264, 2501}, {1265, 7178}, {1459, 30713}, {1565, 30730}, {1812, 4036}, {1857, 4143}, {1946, 27801}, {2318, 3261}, {2321, 4025}, {2968, 4552}, {3049, 40363}, {3682, 46110}, {3692, 4077}, {3695, 4560}, {3699, 4466}, {3703, 4580}, {3704, 15420}, {3708, 7257}, {3712, 14977}, {3719, 24006}, {3949, 18155}, {3952, 26932}, {3998, 44426}, {4033, 7004}, {4092, 4563}, {4103, 17219}, {4140, 7019}, {4171, 7182}, {4518, 24459}, {4561, 21044}, {4571, 16732}, {4574, 34387}, {4587, 21207}, {4997, 14429}, {5547, 45807}, {6057, 15419}, {6333, 15628}, {7117, 27808}, {7253, 26942}, {7359, 34767}, {14308, 34403}, {14638, 44695}, {15526, 36797}, {20294, 40161}, {23067, 23978}, {23090, 34388}, {23189, 28654}, {43728, 51367}
X(52355) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 4565}, {8, 648}, {9, 162}, {10, 653}, {11, 17925}, {33, 24019}, {37, 108}, {41, 32676}, {42, 32674}, {55, 112}, {63, 1414}, {65, 32714}, {69, 4573}, {71, 109}, {72, 651}, {73, 1461}, {77, 4637}, {78, 662}, {125, 7178}, {201, 1020}, {210, 1783}, {212, 163}, {219, 110}, {226, 36118}, {228, 1415}, {281, 107}, {283, 4556}, {304, 4625}, {306, 664}, {307, 658}, {312, 811}, {313, 46404}, {318, 823}, {321, 18026}, {332, 4610}, {345, 99}, {346, 36797}, {348, 4616}, {512, 608}, {513, 1396}, {520, 222}, {521, 81}, {522, 27}, {523, 278}, {525, 7}, {607, 32713}, {644, 5379}, {645, 18020}, {647, 56}, {650, 28}, {652, 58}, {656, 57}, {657, 2299}, {661, 34}, {663, 1474}, {684, 43034}, {798, 1395}, {810, 604}, {822, 603}, {850, 331}, {905, 1014}, {1018, 7012}, {1021, 270}, {1214, 934}, {1231, 4569}, {1259, 4558}, {1260, 5546}, {1264, 4563}, {1265, 645}, {1334, 8750}, {1364, 7254}, {1439, 4617}, {1441, 13149}, {1459, 1412}, {1565, 17096}, {1577, 273}, {1639, 37168}, {1792, 4612}, {1793, 37140}, {1826, 36127}, {1857, 6529}, {1946, 1333}, {2250, 36110}, {2289, 4575}, {2318, 101}, {2321, 1897}, {2325, 46541}, {2327, 4636}, {2335, 36077}, {2489, 7337}, {2501, 1118}, {2522, 5323}, {2525, 3665}, {2610, 1835}, {2631, 51656}, {2632, 51640}, {2968, 4560}, {3049, 1397}, {3063, 2203}, {3064, 8747}, {3125, 43923}, {3239, 29}, {3265, 348}, {3267, 6063}, {3270, 7252}, {3271, 43925}, {3596, 6331}, {3610, 14594}, {3682, 1813}, {3688, 35325}, {3690, 4559}, {3692, 643}, {3693, 4238}, {3694, 100}, {3695, 4552}, {3700, 4}, {3701, 6335}, {3703, 41676}, {3708, 4017}, {3709, 25}, {3710, 190}, {3712, 4235}, {3716, 31905}, {3718, 799}, {3719, 4592}, {3810, 31917}, {3900, 1172}, {3942, 7203}, {3949, 4551}, {3952, 46102}, {3954, 46152}, {3990, 36059}, {3998, 6516}, {4017, 1435}, {4019, 6649}, {4024, 225}, {4025, 1434}, {4036, 40149}, {4041, 19}, {4055, 32660}, {4064, 226}, {4077, 1847}, {4081, 17926}, {4086, 92}, {4088, 5236}, {4092, 2501}, {4105, 2332}, {4120, 1877}, {4130, 4183}, {4140, 7009}, {4143, 7055}, {4148, 14024}, {4157, 46543}, {4163, 2322}, {4171, 33}, {4391, 286}, {4397, 31623}, {4466, 3676}, {4516, 6591}, {4521, 4248}, {4522, 31909}, {4524, 607}, {4529, 14006}, {4551, 7128}, {4557, 7115}, {4561, 4620}, {4563, 7340}, {4571, 4567}, {4574, 59}, {4587, 4570}, {4705, 1880}, {4765, 31903}, {4820, 31902}, {4913, 31904}, {4976, 31900}, {5546, 250}, {6056, 32661}, {6332, 86}, {6587, 44696}, {7004, 1019}, {7017, 6528}, {7065, 32320}, {7068, 525}, {7117, 3733}, {7178, 1119}, {7180, 1398}, {7182, 4635}, {7253, 46103}, {7254, 7341}, {7257, 46254}, {7265, 7282}, {7359, 4240}, {8057, 18623}, {8058, 41083}, {8611, 1}, {8641, 2204}, {9033, 6357}, {9391, 51647}, {10097, 7316}, {10099, 1462}, {10397, 2360}, {14208, 85}, {14298, 3194}, {14308, 1249}, {14331, 44698}, {14395, 51420}, {14401, 1354}, {14417, 7181}, {14429, 3911}, {15411, 261}, {15416, 314}, {15419, 552}, {15526, 17094}, {15627, 1304}, {15628, 685}, {17094, 279}, {17434, 30493}, {17880, 7199}, {17898, 44697}, {18210, 3669}, {18344, 5317}, {20336, 4554}, {20902, 4077}, {20975, 7180}, {21031, 17906}, {21044, 7649}, {21107, 7195}, {21789, 2189}, {21801, 23706}, {21831, 7120}, {22080, 36075}, {22383, 1408}, {23067, 1262}, {23090, 60}, {23189, 593}, {23289, 40574}, {23616, 1367}, {24018, 77}, {24290, 1876}, {24459, 1447}, {26932, 7192}, {26942, 4566}, {30457, 1301}, {30681, 7256}, {30730, 15742}, {32320, 7335}, {34588, 16754}, {34591, 3737}, {35072, 23189}, {35518, 274}, {35519, 44129}, {36054, 1437}, {36197, 18344}, {36797, 23582}, {37754, 51641}, {40071, 4572}, {40161, 1305}, {40869, 4241}, {41087, 8059}, {44707, 1625}, {46382, 1451}, {48278, 17171}, {50333, 15149}, {50347, 31906}, {51640, 269}, {51641, 7099}
X(52355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3265, 14208, 17094}, {4064, 14429, 656}

X(52356) = X(11)-DAO CONJUGATE OF X(36)

Barycentrics    b*(b - c)*c*(-a + b + c)*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :
X(52356) = X[3762] + 3 X[14304], 4 X[10272] - 3 X[35054]

X(52356) lies on the cubic K230 and these lines: {5, 523}, {8, 4086}, {10, 522}, {80, 3738}, {265, 35053}, {281, 17926}, {476, 35056}, {513, 18480}, {656, 18395}, {693, 1441}, {996, 3907}, {1309, 2222}, {1577, 17057}, {2166, 36035}, {2321, 3239}, {2341, 15628}, {2400, 18815}, {2475, 39270}, {2804, 15065}, {3701, 4397}, {3737, 13746}, {4404, 45035}, {7951, 42768}, {8062, 30144}, {8836, 35052}, {8838, 35051}, {10272, 35054}, {14127, 43655}, {14616, 35141}, {15446, 23226}, {17931, 17934}, {23838, 36590}, {28132, 36910}, {35157, 35174}, {36802, 36804}, {37628, 40437}, {37630, 51562}, {41013, 44426}

X(52356) = reflection of X(35050) in X(5)
X(52356) = reflection of X(35050) in the Euler line
X(52356) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {11075, 37781}, {14147, 329}, {26743, 150}, {34921, 6224}
X(52356) = X(i)-Ceva conjugate of X(j) for these (i,j): {35174, 18359}, {36804, 36910}, {51562, 15065}
X(52356) = X(i)-cross conjugate of X(j) for these (i,j): {11, 36590}, {1639, 4391}, {2804, 522}
X(52356) = X(i)-isoconjugate of X(j) for these (i,j): {36, 109}, {50, 38340}, {57, 1983}, {110, 1464}, {163, 18593}, {603, 4242}, {604, 4585}, {651, 7113}, {654, 1262}, {655, 52059}, {692, 1443}, {934, 2361}, {1020, 4282}, {1101, 51645}, {1414, 3724}, {1415, 3218}, {1461, 2323}, {1511, 36064}, {1576, 41804}, {1835, 4575}, {1870, 36059}, {2149, 3960}, {2245, 4565}, {2624, 35049}, {2720, 34586}, {3028, 36069}, {3738, 24027}, {3904, 23979}, {4564, 21758}, {6126, 34921}, {6149, 26700}, {7012, 22379}, {7045, 8648}, {11700, 36040}, {16586, 32669}, {16944, 23703}, {17078, 32739}, {17923, 32660}, {22128, 32674}, {32719, 41801}, {41282, 51562}
X(52356) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 36}, {100, 36909}, {109, 15898}, {115, 18593}, {136, 1835}, {214, 51402}, {244, 1464}, {320, 40624}, {522, 3738}, {523, 51645}, {650, 3960}, {758, 6741}, {1086, 1443}, {1146, 3218}, {1577, 4453}, {1870, 20620}, {1983, 5452}, {2323, 35508}, {2361, 14714}, {2968, 4511}, {3028, 38982}, {3161, 4585}, {3700, 32679}, {3724, 40608}, {3756, 4881}, {4242, 7952}, {4351, 38964}, {4858, 41804}, {7113, 38991}, {8286, 27086}, {8648, 17115}, {10017, 11700}, {14993, 26700}, {17078, 40619}, {22128, 35072}, {34586, 38981}
X(52356) = cevapoint of X(i) and X(j) for these (i,j): {523, 36035}, {4041, 46393}, {4086, 4768}
X(52356) = crosspoint of X(i) and X(j) for these (i,j): {6740, 51562}, {14616, 32680}, {18359, 35174}, {20566, 36804}
X(52356) = crosssum of X(i) and X(j) for these (i,j): {2183, 42657}, {2624, 3724}, {7113, 8648}
X(52356) = trilinear pole of line {1146, 3700}
X(52356) = crossdifference of every pair of points on line {50, 7113}
X(52356) = barycentric product X(i)*X(j) for these {i,j}: {11, 36804}, {80, 4391}, {94, 35057}, {522, 18359}, {650, 20566}, {655, 24026}, {693, 36910}, {850, 2341}, {1146, 35174}, {1577, 6740}, {1793, 14618}, {1807, 46110}, {2006, 4397}, {2161, 35519}, {2222, 23978}, {2310, 46405}, {3239, 18815}, {3700, 14616}, {3762, 36590}, {4086, 24624}, {4560, 15065}, {4858, 51562}, {6741, 32680}, {11107, 14592}, {42033, 43082}
X(52356) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4585}, {11, 3960}, {55, 1983}, {80, 651}, {115, 51645}, {281, 4242}, {476, 35049}, {514, 1443}, {521, 22128}, {522, 3218}, {523, 18593}, {650, 36}, {655, 7045}, {657, 2361}, {661, 1464}, {663, 7113}, {693, 17078}, {759, 4565}, {1146, 3738}, {1411, 1461}, {1577, 41804}, {1639, 214}, {1793, 4558}, {1807, 1813}, {1989, 26700}, {2006, 934}, {2161, 109}, {2166, 38340}, {2222, 1262}, {2310, 654}, {2341, 110}, {2501, 1835}, {2610, 3028}, {2804, 16586}, {3064, 1870}, {3239, 4511}, {3271, 21758}, {3700, 758}, {3709, 3724}, {3716, 27950}, {3762, 41801}, {3900, 2323}, {4041, 2245}, {4086, 3936}, {4092, 2610}, {4391, 320}, {4397, 32851}, {4516, 21828}, {4521, 4881}, {4768, 51583}, {4791, 36589}, {4820, 4880}, {4858, 4453}, {4895, 17455}, {4944, 4867}, {4976, 4973}, {5532, 46384}, {6187, 1415}, {6740, 662}, {6741, 32679}, {7117, 22379}, {8648, 52059}, {9404, 6149}, {10412, 43682}, {11075, 34921}, {11107, 14590}, {14616, 4573}, {14936, 8648}, {15065, 4552}, {17926, 17515}, {18359, 664}, {18815, 658}, {20566, 4554}, {21758, 41282}, {21789, 4282}, {23838, 40215}, {24026, 3904}, {24624, 1414}, {32675, 24027}, {34857, 4559}, {35057, 323}, {35174, 1275}, {35519, 20924}, {36590, 3257}, {36804, 4998}, {36910, 100}, {40166, 4089}, {40437, 37136}, {44426, 17923}, {46384, 3025}, {46393, 34586}, {51562, 4564}

X(52357) = X(12)-DAO CONJUGATE OF X(1)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)^2*(a^3 - a*b^2 + a*b*c - b^2*c - a*c^2 - b*c^2) : :

X(52357) lies on these lines: {8, 4551}, {10, 12}, {56, 50605}, {76, 33298}, {201, 1089}, {221, 5774}, {225, 44143}, {227, 5295}, {307, 34388}, {388, 10479}, {519, 2594}, {596, 51975}, {1091, 1254}, {1210, 37592}, {1214, 3714}, {1420, 30942}, {1450, 19864}, {1737, 3670}, {2197, 2321}, {2975, 34589}, {3741, 10106}, {3778, 21935}, {3831, 3911}, {4357, 24982}, {4968, 6734}, {5219, 31339}, {5795, 26013}, {9578, 31330}, {10895, 48888}, {11679, 21147}, {11681, 41797}, {12527, 34831}, {14973, 20617}, {17751, 37558}, {18961, 48835}, {20718, 51870}, {21674, 21803}, {24310, 50037}, {25005, 27184}, {26125, 46933}, {37579, 48863}

X(52357) = X(75)-Ceva conjugate of X(6358)
X(52357) = X(i)-isoconjugate of X(j) for these (i,j): {21, 52150}, {60, 34434}, {1333, 46880}, {2051, 2150}, {2194, 20028}, {4267, 40453}
X(52357) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 12}, {37, 46880}, {1193, 4267}, {1214, 20028}, {3737, 34589}, {18163, 21796}, {40611, 52150}
X(52357) = crosspoint of X(75) and X(14829)
X(52357) = barycentric product X(i)*X(j) for these {i,j}: {12, 14829}, {85, 14973}, {226, 17751}, {312, 20617}, {321, 37558}, {349, 52139}, {572, 34388}, {1089, 17074}, {1441, 21061}, {2975, 6358}, {4033, 51664}, {11109, 26942}
X(52357) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 46880}, {12, 2051}, {226, 20028}, {572, 60}, {1400, 52150}, {2171, 34434}, {2975, 2185}, {10408, 34262}, {11109, 46103}, {14829, 261}, {14973, 9}, {17074, 757}, {17751, 333}, {20617, 57}, {20986, 2150}, {21061, 21}, {34589, 26856}, {37558, 81}, {51664, 1019}, {52087, 4267}, {52139, 284}
X(52357) = {X(12),X(65)}-harmonic conjugate of X(10408)

X(52358) = X(12)-DAO CONJUGATE OF X(37)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)*(a^3 - a*b^2 + a*b*c - b^2*c - a*c^2 - b*c^2) : :

X(52358) lies on these lines: {2, 7}, {3, 27394}, {8, 73}, {10, 1042}, {12, 19874}, {56, 964}, {65, 22325}, {72, 37154}, {75, 17080}, {77, 11679}, {171, 5764}, {210, 26031}, {222, 1150}, {223, 5271}, {225, 4200}, {241, 16720}, {269, 18229}, {278, 17075}, {310, 4554}, {321, 1214}, {333, 651}, {348, 349}, {475, 7103}, {1211, 40999}, {1254, 49598}, {1407, 37660}, {1412, 27163}, {1427, 1441}, {1442, 1999}, {1458, 3741}, {1465, 4359}, {1471, 25496}, {1617, 24552}, {1758, 4418}, {1788, 26030}, {2003, 16704}, {2006, 39747}, {2051, 20367}, {2275, 3772}, {2295, 17056}, {2975, 11109}, {3187, 45126}, {3757, 4318}, {3868, 26028}, {3936, 26942}, {3947, 16828}, {3995, 16577}, {4077, 26983}, {4101, 7080}, {4192, 20256}, {4298, 19863}, {4306, 10479}, {4322, 50608}, {4417, 17137}, {4551, 4651}, {4640, 26095}, {4652, 27506}, {5261, 19853}, {5278, 34048}, {5290, 19858}, {5737, 6180}, {5830, 43056}, {6354, 41804}, {6358, 18593}, {6708, 30807}, {7175, 16738}, {7288, 17526}, {7677, 32942}, {8055, 28778}, {8270, 26227}, {9316, 32916}, {9364, 32918}, {10478, 17220}, {10888, 18655}, {11115, 37583}, {14829, 17074}, {16586, 20879}, {16698, 18601}, {16705, 17095}, {17084, 44350}, {17625, 46909}, {17751, 37558}, {18097, 27005}, {18134, 28803}, {18662, 48380}, {18743, 25082}, {19684, 37543}, {24778, 45204}, {25000, 41883}, {26126, 32636}, {28734, 28769}, {28755, 28765}, {28761, 28763}, {37759, 41808}, {41539, 46897}, {42289, 43223}, {45206, 48381}

X(52358) = isotomic conjugate of X(46880)
X(52358) = X(274)-Ceva conjugate of X(1441)
X(52358) = X(21061)-cross conjugate of X(17751)
X(52358) = X(i)-isoconjugate of X(j) for these (i,j): {9, 52150}, {31, 46880}, {41, 20028}, {284, 34434}, {2051, 2194}, {2150, 51870}, {2269, 40453}
X(52358) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 46880}, {12, 37}, {478, 52150}, {650, 34589}, {1193, 2269}, {1214, 2051}, {3057, 21796}, {3160, 20028}, {34434, 40590}
X(52358) = cevapoint of X(21061) and X(37558)
X(52358) = barycentric product X(i)*X(j) for these {i,j}: {7, 17751}, {75, 37558}, {85, 21061}, {226, 14829}, {307, 11109}, {314, 20617}, {321, 17074}, {349, 572}, {668, 51664}, {1441, 2975}, {4552, 17496}, {6063, 52139}
X(52358) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 46880}, {7, 20028}, {12, 51870}, {56, 52150}, {65, 34434}, {226, 2051}, {572, 284}, {961, 40453}, {2975, 21}, {11109, 29}, {14829, 333}, {14973, 210}, {17074, 81}, {17496, 4560}, {17751, 8}, {20617, 65}, {20986, 2194}, {21061, 9}, {21173, 3737}, {22118, 2193}, {23187, 23189}, {24237, 17197}, {27346, 27527}, {37558, 1}, {46879, 46889}, {51664, 513}, {52087, 2269}, {52139, 55}
X(52358) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 27339, 17077}, {2, 28739, 28776}, {321, 1214, 4552}, {1427, 31993, 1441}

X(52359) = X(19)-DAO CONJUGATE OF X(27)

Barycentrics    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(52359) lies on these lines: {1, 197}, {3, 44662}, {4, 44670}, {10, 4523}, {12, 431}, {37, 2333}, {40, 3827}, {42, 65}, {46, 18732}, {55, 1829}, {72, 3704}, {78, 41600}, {210, 20653}, {517, 3811}, {692, 40660}, {774, 2183}, {910, 1973}, {976, 3057}, {986, 24476}, {1040, 22654}, {1062, 9798}, {1071, 8679}, {1376, 37613}, {1486, 7713}, {1763, 3556}, {1828, 6284}, {1834, 40965}, {1871, 7680}, {1872, 18242}, {1878, 12953}, {1900, 10895}, {1905, 3185}, {2809, 24391}, {2835, 5493}, {2933, 46974}, {3085, 40635}, {3436, 20243}, {3755, 35650}, {3812, 28600}, {3949, 4515}, {4329, 20914}, {5250, 41581}, {8758, 13738}, {9627, 11363}, {9645, 26309}, {9895, 10198}, {9943, 24683}, {10831, 11383}, {10833, 26378}, {11491, 41722}, {12711, 14557}, {15624, 37528}, {17102, 23361}, {20691, 21853}, {21148, 36103}, {30265, 45188}, {37566, 52218}, {37593, 42443}, {41538, 51377}

X(52359) = reflection of X(1872) in X(18242)
X(52359) = X(i)-Ceva conjugate of X(j) for these (i,j): {306, 37}, {4329, 21062}
X(52359) = X(i)-isoconjugate of X(j) for these (i,j): {58, 7219}, {81, 7097}, {86, 7169}, {270, 47344}, {1333, 40015}, {1444, 40169}
X(52359) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 7219}, {19, 27}, {37, 40015}, {7097, 40586}, {7169, 40600}
X(52359) = crosspoint of X(1763) and X(4329)
X(52359) = crosssum of X(i) and X(j) for these (i,j): {1, 37034}, {7097, 7169}
X(52359) = barycentric product X(i)*X(j) for these {i,j}: {1, 21062}, {10, 1763}, {37, 4329}, {42, 20914}, {65, 27540}, {72, 17903}, {306, 36103}, {321, 3556}, {1018, 21174}, {20336, 21148}, {22119, 41013}
X(52359) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 40015}, {37, 7219}, {42, 7097}, {213, 7169}, {1763, 86}, {2197, 47344}, {2333, 40169}, {3556, 81}, {4329, 274}, {17903, 286}, {20914, 310}, {21062, 75}, {21148, 28}, {21174, 7199}, {22119, 1444}, {27540, 314}, {36103, 27}
X(52359) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9627, 20989, 11363}, {12711, 14557, 42450}, {20691, 21861, 21853}, {20691, 21862, 21861}

X(52360) = X(21)-DAO CONJUGATE OF X(1)

Barycentrics    (a + b)*(a - b - c)*(a + c)*(a^4 - b^4 + a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(52360) lies on these lines: {1, 42005}, {2, 13583}, {8, 13746}, {10, 643}, {11, 21}, {29, 270}, {35, 52244}, {86, 16749}, {99, 17095}, {100, 37158}, {110, 5086}, {229, 2475}, {349, 811}, {409, 2886}, {447, 32939}, {759, 24387}, {950, 2185}, {1010, 1125}, {1043, 3615}, {1212, 27415}, {1220, 36934}, {1479, 17512}, {1621, 37152}, {1631, 37405}, {2905, 11683}, {2975, 7424}, {3109, 24390}, {3152, 28755}, {3582, 4234}, {4861, 6740}, {5263, 17872}, {6739, 37157}, {7253, 17164}, {11101, 11680}, {13740, 25441}, {15680, 40592}, {23665, 32922}, {24541, 40430}, {25639, 37816}

X(52360) = X(75)-Ceva conjugate of X(333)
X(52360) = X(65)-isoconjugate of X(34435)
X(52360) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 21}, {34435, 40602}
X(52360) = cevapoint of X(1) and X(18865)
X(52360) = barycentric product X(i)*X(j) for these {i,j}: {29, 28754}, {75, 40582}, {229, 312}, {314, 1781}, {333, 2475}, {1043, 18625}
X(52360) = barycentric quotient X(i)/X(j) for these {i,j}: {229, 57}, {284, 34435}, {1781, 65}, {2475, 226}, {18625, 3668}, {28754, 307}, {40582, 1}
X(52360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1098, 6734, 333}, {6734, 51382, 1098}

X(52361) = X(21)-DAO CONJUGATE OF X(9)

Barycentrics    (a + b)*(a + c)*(a^4 - b^4 + a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(52361) lies on these lines: {2, 1029}, {12, 35991}, {21, 12615}, {27, 86}, {81, 1086}, {99, 33116}, {110, 20292}, {226, 662}, {229, 2475}, {320, 7058}, {333, 2160}, {354, 19642}, {379, 17103}, {409, 7354}, {501, 11263}, {553, 37756}, {593, 33129}, {673, 40164}, {757, 40940}, {1043, 17647}, {1098, 4292}, {1326, 33130}, {1478, 11116}, {1621, 5196}, {1626, 11101}, {1738, 6043}, {2363, 23536}, {2651, 11246}, {3752, 24617}, {3822, 52244}, {4297, 40430}, {5253, 37369}, {5267, 11110}, {5745, 6626}, {14534, 16706}, {14616, 17862}, {14829, 20913}, {17056, 38814}, {17190, 25507}, {17483, 37783}, {17605, 25533}, {18625, 40582}, {24378, 41245}, {24624, 27003}, {31019, 40214}, {37791, 40688}, {40605, 41878}

X(52361) = X(85)-Ceva conjugate of X(86)
X(52361) = X(1781)-cross conjugate of X(229)
X(52361) = X(37)-isoconjugate of X(34435)
X(52361) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 21}, {34435, 40589}
X(52361) = cevapoint of X(i) and X(j) for these (i,j): {229, 40582}, {1781, 2475}
X(52361) = barycentric product X(i)*X(j) for these {i,j}: {27, 28754}, {75, 229}, {85, 40582}, {86, 2475}, {274, 1781}, {333, 18625}
X(52361) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 34435}, {229, 1}, {1781, 37}, {2475, 10}, {18625, 226}, {28754, 306}, {40582, 9}, {41495, 41501}
X(52361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2185, 5249, 86}, {5249, 18653, 2185}

X(52362) = X(21)-DAO CONJUGATE OF X(29)

Barycentrics    a*(a^2 - b^2 - c^2)*(a^4 - b^4 + a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(52362) lies on these lines: {1, 21}, {3, 18210}, {8, 14544}, {29, 14206}, {34, 2000}, {40, 2071}, {72, 18447}, {73, 18123}, {77, 224}, {78, 1060}, {201, 1331}, {229, 1781}, {307, 1442}, {950, 37782}, {1038, 4855}, {1047, 2617}, {1062, 4652}, {1790, 18673}, {1870, 6734}, {2327, 18675}, {2475, 18625}, {2888, 6326}, {2893, 18631}, {3160, 17136}, {3218, 33178}, {3419, 32047}, {3434, 4347}, {3622, 27509}, {3916, 18455}, {3984, 25909}, {4228, 51698}, {4296, 5930}, {4351, 17647}, {4640, 9627}, {5175, 18624}, {5263, 21406}, {5314, 37613}, {5693, 11441}, {10884, 44243}, {11396, 37581}, {13739, 18721}, {14212, 39585}, {16049, 44661}, {17768, 38336}, {18446, 44665}, {18593, 35979}, {18719, 21376}, {19861, 26543}, {24883, 41557}, {26131, 41550}, {41571, 49743}

X(52362) = isotomic conjugate of the polar conjugate of X(1781)
X(52362) = X(i)-Ceva conjugate of X(j) for these (i,j): {307, 63}, {1442, 6505}, {3879, 8897}, {34772, 224}
X(52362) = X(4)-isoconjugate of X(34435)
X(52362) = X(i)-Dao conjugate of X(j) for these (i,j): {21, 29}, {34435, 36033}
X(52362) = crosssum of X(663) and X(3708)
X(52362) = crossdifference of every pair of points on line {661, 47235}
X(52362) = barycentric product X(i)*X(j) for these {i,j}: {1, 28754}, {63, 2475}, {69, 1781}, {78, 18625}, {229, 306}, {307, 40582}
X(52362) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 34435}, {229, 27}, {1781, 4}, {2475, 92}, {18625, 273}, {28754, 75}, {40582, 29}

X(52363) = X(23)-DAO CONJUGATE OF X(4)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - b^4 + b^2*c^2 - c^4)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6) : :
X(52363) = 3 X[2916] + X[2930], 3 X[2916] - X[34437], X[895] - 3 X[1176]

X(52363) lies on these lines: {3, 18125}, {20, 15582}, {22, 11061}, {23, 6593}, {67, 6636}, {69, 2916}, {110, 3313}, {159, 2892}, {511, 2914}, {542, 7512}, {895, 1176}, {2781, 41590}, {2931, 25406}, {3098, 12281}, {3518, 15462}, {3520, 18553}, {5181, 37929}, {5189, 19596}, {5562, 8718}, {5972, 46026}, {6698, 15246}, {7464, 16163}, {7488, 32233}, {7525, 32306}, {8907, 15581}, {9969, 27866}, {9970, 12088}, {12106, 45034}, {12310, 25320}, {12825, 33851}, {17714, 45016}, {22109, 32305}, {28408, 31099}, {32271, 37925}, {32274, 37126}, {41716, 52098}, {43697, 44260}

X(52363) = midpoint of X(i) and X(j) for these {i,j}: {110, 41464}, {2930, 34437}
X(52363) = reflection of X(46026) in X(5972)
X(52363) = isotomic conjugate of the polar conjugate of X(40583)
X(52363) = X(69)-Ceva conjugate of X(22151)
X(52363) = X(4)-Dao conjugate of X(23)
X(52363) = barycentric product X(i)*X(j) for these {i,j}: {69, 40583}, {316, 22121}, {5189, 22151}, {19596, 37804}
X(52363) = barycentric quotient X(i)/X(j) for these {i,j}: {5189, 46105}, {10317, 34437}, {19596, 8791}, {22121, 67}, {40583, 4}
X(52363) = {X(2916),X(2930)}-harmonic conjugate of X(34437)

X(52364) = X(28)-DAO CONJUGATE OF X(28)

Barycentrics    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 - 2*a^3*b^3*c - a^2*b^4*c - a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 - 2*a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 + a^4*c^3 - 2*a^3*b*c^3 - 2*a^2*b^2*c^3 + 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 - a*b*c^5 + b^2*c^5 - a*c^6 - b*c^6 - c^7 : :
X(52364) = 3 X[2] - 4 X[21530], 9 X[2] - 10 X[31257], 9 X[2] - 8 X[52259], X[28] - 3 X[31154], 3 X[28] - 5 X[31257], 3 X[28] - 4 X[52259], 5 X[631] - 4 X[44220], 5 X[3091] - 4 X[15763], 5 X[15674] - 4 X[44253], 5 X[15692] - 4 X[48370], 2 X[21530] - 3 X[31154], 6 X[21530] - 5 X[31257], 4 X[21530] - X[31293], 3 X[21530] - 2 X[52259], 9 X[31154] - 5 X[31257], 6 X[31154] - X[31293], 9 X[31154] - 4 X[52259], 10 X[31257] - 3 X[31293], 5 X[31257] - 4 X[52259], 3 X[31293] - 8 X[52259], 3 X[44293] - 4 X[52260], 5 X[3616] - 4 X[51698]

X(52364) lies on these lines: {2, 3}, {8, 2893}, {9, 26064}, {10, 3101}, {72, 2895}, {153, 2828}, {226, 4296}, {329, 1330}, {347, 388}, {950, 3100}, {975, 9612}, {1228, 30737}, {1478, 25080}, {1654, 26939}, {1655, 18666}, {1762, 21671}, {1891, 18589}, {1901, 2303}, {2550, 41921}, {2838, 5300}, {2894, 20242}, {3419, 20243}, {3436, 32862}, {3487, 37635}, {3488, 9538}, {3616, 51698}, {3868, 41004}, {5175, 41915}, {5225, 17070}, {5279, 8804}, {5657, 9537}, {5691, 30265}, {6284, 41230}, {6360, 20060}, {7270, 20336}, {9263, 16100}, {9961, 12779}, {10477, 12220}, {13397, 43659}, {17778, 20009}, {17794, 18659}, {18397, 41329}, {18667, 20222}, {32849, 41507}, {39352, 39360}

X(52364) = reflection of X(i) in X(j) for these {i,j}: {2, 31154}, {20, 30267}, {28, 21530}, {9263, 16100}, {31293, 28}, {31294, 44292}, {44294, 18641}
X(52364) = complement of X(31293)
X(52364) = anticomplement of X(28)
X(52364) = orthocentroidal-circle-inverse of X(7557)
X(52364) = deLongchamps-circle-inverse of X(1325)
X(52364) = anticomplement of the isogonal conjugate of X(72)
X(52364) = anticomplement of the isotomic conjugate of X(20336)
X(52364) = isotomic conjugate of the polar conjugate of X(18685)
X(52364) = anticomplementary isogonal conjugate of X(3868)
X(52364) = X(28)-Dao conjugate of X(28)
X(52364) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3868}, {2, 17220}, {3, 1}, {6, 3187}, {9, 92}, {10, 4}, {37, 5905}, {40, 1895}, {42, 193}, {48, 17147}, {59, 14544}, {63, 75}, {65, 12649}, {69, 17135}, {71, 2}, {72, 8}, {73, 145}, {75, 20242}, {77, 3873}, {78, 3869}, {97, 17221}, {100, 7253}, {101, 525}, {184, 17148}, {190, 850}, {201, 2475}, {210, 5942}, {213, 21216}, {219, 63}, {222, 3875}, {228, 192}, {255, 20222}, {268, 20223}, {271, 20220}, {283, 2975}, {295, 740}, {304, 17137}, {305, 17138}, {306, 69}, {307, 3434}, {313, 11442}, {321, 21270}, {326, 20243}, {332, 35614}, {345, 20245}, {348, 20244}, {394, 17134}, {525, 150}, {647, 4440}, {653, 23683}, {656, 149}, {668, 21300}, {692, 17498}, {810, 9263}, {895, 17162}, {906, 4560}, {947, 3562}, {1018, 4391}, {1020, 17896}, {1042, 11851}, {1073, 18655}, {1176, 17150}, {1214, 7}, {1220, 41723}, {1231, 21285}, {1252, 14543}, {1260, 45738}, {1293, 44409}, {1331, 523}, {1332, 7192}, {1334, 30694}, {1400, 30699}, {1409, 3210}, {1410, 17480}, {1439, 36845}, {1444, 17140}, {1459, 17154}, {1751, 2997}, {1790, 4360}, {1791, 17164}, {1794, 21}, {1796, 86}, {1797, 17160}, {1799, 17142}, {1807, 758}, {1812, 21273}, {1813, 4467}, {1826, 6515}, {1897, 520}, {2193, 18662}, {2197, 17778}, {2200, 194}, {2250, 48380}, {2259, 40571}, {2318, 144}, {2332, 46713}, {2333, 6392}, {2359, 81}, {2983, 27}, {3049, 21224}, {3504, 17157}, {3682, 20}, {3690, 1654}, {3692, 18750}, {3694, 329}, {3695, 1330}, {3710, 3436}, {3730, 17911}, {3926, 18659}, {3949, 2895}, {3952, 20293}, {3958, 41821}, {3990, 6360}, {3998, 4329}, {4019, 30660}, {4047, 41915}, {4055, 3164}, {4064, 3448}, {4456, 41361}, {4551, 521}, {4552, 46400}, {4557, 25259}, {4558, 17161}, {4561, 512}, {4563, 17159}, {4570, 110}, {4574, 514}, {4580, 25049}, {4592, 17166}, {6391, 17156}, {6539, 20289}, {7013, 20221}, {7015, 38}, {7019, 17153}, {7066, 3152}, {7100, 3874}, {7123, 19}, {7177, 17158}, {7591, 7057}, {8611, 37781}, {8750, 33294}, {8804, 14361}, {14208, 21293}, {14376, 18656}, {14919, 18661}, {15377, 20090}, {17206, 17143}, {17976, 13174}, {18082, 3060}, {20336, 6327}, {21017, 34163}, {21035, 8878}, {22076, 5484}, {22356, 30579}, {22370, 17149}, {22381, 330}, {23067, 522}, {23086, 17155}, {23604, 43740}, {26942, 2893}, {29014, 17925}, {32656, 31296}, {34055, 17141}, {34897, 18657}, {36057, 32922}, {40071, 315}, {40152, 347}, {40435, 286}, {40442, 34195}, {41013, 5906}, {41014, 2891}, {41087, 9965}, {41267, 10340}, {43072, 2292}, {44189, 20246}, {44717, 17136}, {47487, 1621}, {51366, 152}, {51574, 52025}, {52185, 3193}
X(52364) = X(i)-Ceva conjugate of X(j) for these (i,j): {7270, 8}, {20336, 2}, {21287, 2895}
X(52364) = X(18598)-cross conjugate of X(18632)
X(52364) = crosspoint of X(668) and X(23582)
X(52364) = crosssum of X(667) and X(3269)
X(52364) = barycentric product X(i)*X(j) for these {i,j}: {8, 18632}, {69, 18685}, {75, 18598}
X(52364) = barycentric quotient X(i)/X(j) for these {i,j}: {18598, 1}, {18632, 7}, {18685, 4}
X(52364) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 7557}, {2, 20, 7520}, {2, 3146, 4198}, {2, 15680, 17521}, {2, 31293, 28}, {3, 5142, 2}, {4, 13442, 37456}, {4, 26052, 5177}, {4, 37098, 2475}, {4, 37179, 2}, {5, 7523, 2}, {23, 31103, 2}, {28, 21530, 2}, {28, 31154, 21530}, {28, 31257, 52259}, {440, 13442, 21}, {442, 47512, 2}, {468, 30773, 2}, {851, 28380, 27652}, {851, 30974, 2}, {857, 24606, 2}, {857, 48890, 29}, {858, 26253, 2}, {1375, 30842, 2}, {2915, 30447, 451}, {3151, 3152, 20}, {5002, 5003, 37456}, {14953, 31046, 2}, {21530, 52259, 31257}, {24568, 25014, 2}, {25795, 25869, 2}, {25909, 25987, 2}, {25949, 26022, 2}, {26054, 26120, 2}, {26553, 26681, 2}, {26604, 26646, 2}, {26784, 26831, 2}, {26992, 27122, 2}, {27053, 27175, 2}, {29463, 46499, 29519}, {29725, 29778, 46497}, {30920, 33313, 2}, {31257, 52259, 2}

X(52365) = X(33)-DAO CONJUGATE OF X(33)

Barycentrics    a^6 - 2*a^5*b + a^4*b^2 - a^2*b^4 + 2*a*b^5 - b^6 - 2*a^5*c - 2*a^4*b*c + 2*a^3*b^2*c + 2*a^2*b^3*c + a^4*c^2 + 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 + 2*a^2*b*c^3 - 2*a*b^2*c^3 - a^2*c^4 + b^2*c^4 + 2*a*c^5 - c^6 : :
X(52365) = 3 X[2] - 4 X[34822]

X(52365) lies on these lines: {1, 18652}, {2, 33}, {4, 23661}, {7, 2897}, {8, 20}, {10, 27505}, {22, 100}, {75, 1370}, {92, 10431}, {103, 41906}, {145, 4296}, {152, 329}, {318, 6836}, {347, 3875}, {377, 5174}, {475, 1062}, {497, 17862}, {516, 20223}, {858, 11680}, {944, 37404}, {956, 21312}, {1012, 11406}, {1043, 7169}, {1125, 9643}, {1295, 43347}, {1479, 20320}, {1610, 1619}, {1735, 18391}, {1824, 26118}, {1862, 37366}, {2968, 7580}, {2975, 11413}, {3151, 27484}, {3152, 3210}, {3164, 17759}, {3421, 35513}, {3436, 37201}, {3616, 9538}, {3869, 6225}, {3873, 6604}, {4441, 30737}, {4463, 42699}, {4847, 30265}, {5080, 44440}, {5081, 6925}, {5175, 41915}, {5256, 5807}, {5263, 24543}, {5687, 11414}, {6350, 7411}, {6360, 52164}, {6527, 17134}, {6827, 38462}, {6851, 41013}, {7071, 25907}, {7493, 33116}, {8144, 34120}, {9960, 12324}, {10394, 11433}, {11220, 26871}, {12220, 25304}, {14544, 18623}, {14956, 31623}, {17756, 22240}, {17776, 36018}, {20221, 33673}, {24611, 50698}, {25722, 33075}, {26872, 41228}, {34036, 45281}, {36496, 52082}, {36850, 44661}, {43363, 46964}

X(52365) = reflection of X(i) in X(j) for these {i,j}: {20, 36984}, {33, 34822}, {36985, 10}
X(52365) = anticomplement of X(33)
X(52365) = anticomplement of the isogonal conjugate of X(77)
X(52365) = anticomplement of the isotomic conjugate of X(7182)
X(52365) = isotomic conjugate of the isogonal conjugate of X(18621)
X(52365) = anticomplementary isogonal conjugate of X(5942)
X(52365) = X(33)-Dao conjugate of X(33)
X(52365) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 5942}, {3, 144}, {6, 30694}, {7, 4}, {48, 3177}, {56, 193}, {57, 5905}, {59, 3732}, {63, 329}, {69, 3436}, {73, 1654}, {77, 8}, {81, 92}, {85, 21270}, {109, 25259}, {184, 21218}, {219, 30695}, {222, 2}, {269, 12649}, {273, 5906}, {278, 6515}, {283, 45738}, {304, 21286}, {307, 1330}, {331, 317}, {348, 69}, {603, 192}, {604, 21216}, {608, 6392}, {651, 4391}, {658, 46400}, {664, 20293}, {905, 37781}, {934, 521}, {1014, 3868}, {1214, 2895}, {1231, 21287}, {1262, 651}, {1331, 4468}, {1332, 4462}, {1401, 8878}, {1407, 30699}, {1409, 1655}, {1412, 3187}, {1414, 7253}, {1434, 17220}, {1439, 2475}, {1444, 3869}, {1459, 39351}, {1790, 63}, {1791, 43216}, {1796, 17781}, {1797, 908}, {1798, 11683}, {1803, 9}, {1804, 20}, {1811, 41772}, {1812, 18750}, {1813, 514}, {1814, 30807}, {2189, 46713}, {2197, 46707}, {3937, 17036}, {4025, 33650}, {4131, 34188}, {4565, 525}, {4573, 850}, {4617, 17896}, {4625, 21300}, {6063, 11442}, {6516, 513}, {6517, 20294}, {7011, 20211}, {7013, 6223}, {7023, 11851}, {7053, 145}, {7055, 1370}, {7056, 3434}, {7099, 3210}, {7125, 6360}, {7177, 7}, {7182, 6327}, {7183, 4329}, {7316, 47286}, {7335, 3164}, {7339, 1897}, {8810, 6820}, {13388, 13387}, {13389, 13386}, {17094, 3448}, {17206, 20245}, {18623, 14361}, {22341, 18666}, {23067, 31290}, {23086, 20535}, {27832, 21296}, {30493, 17035}, {30682, 6604}, {32660, 21225}, {34051, 48380}, {34055, 20248}, {34400, 21279}, {36057, 10025}, {36059, 17494}, {40152, 3151}, {40442, 4416}, {40443, 3681}, {41081, 189}, {43034, 40867}, {44717, 190}, {47487, 30625}, {51640, 21221}
X(52365) = X(7182)-Ceva conjugate of X(2)
X(52365) = barycentric product X(i)*X(j) for these {i,j}: {76, 18621}, {305, 21058}
X(52365) = barycentric quotient X(i)/X(j) for these {i,j}: {18621, 6}, {21058, 25}
X(52365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 10430, 189}, {33, 34822, 2}, {1040, 1861, 2}, {1370, 20243, 4329}

X(52366) = X(34)-DAO CONJUGATE OF X(34)

Barycentrics    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 + 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + a^2*b^4*c - 2*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 - 4*a^3*b*c^3 - 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + b^4*c^3 - a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + b^3*c^4 + a^2*c^5 - 2*a*b*c^5 + b^2*c^5 + a*c^6 - b*c^6 - c^7 : :
X(52366) = 3 X[2] - 4 X[34823]

X(52366) lies on these lines: {1, 27505}, {2, 34}, {8, 20}, {21, 6350}, {22, 2975}, {58, 1771}, {69, 3827}, {92, 377}, {100, 11413}, {145, 3100}, {278, 24984}, {281, 24537}, {318, 6925}, {329, 1330}, {333, 16049}, {341, 1370}, {347, 4357}, {406, 1060}, {858, 11681}, {938, 27402}, {956, 11414}, {1220, 24543}, {1222, 35998}, {1305, 2365}, {1396, 24568}, {1398, 25947}, {1654, 3152}, {1783, 23115}, {1936, 7538}, {2840, 21290}, {2897, 32099}, {2968, 37022}, {3164, 21226}, {3194, 27405}, {3241, 9538}, {3244, 9643}, {3434, 20220}, {3486, 36740}, {3868, 26871}, {4292, 20223}, {5080, 37444}, {5081, 6836}, {5082, 35513}, {5130, 26118}, {5174, 10431}, {5176, 42020}, {5303, 38444}, {5342, 6835}, {5484, 6360}, {5657, 37404}, {5687, 21312}, {5703, 27535}, {5744, 7520}, {5758, 5906}, {6737, 30265}, {6850, 41013}, {6916, 23661}, {7102, 15971}, {7718, 28029}, {11415, 33650}, {17080, 25876}, {27378, 34231}, {27379, 46974}

X(52366) = reflection of X(i) in X(j) for these {i,j}: {20, 36986}, {34, 34823}
X(52366) = anticomplement of X(34)
X(52366) = anticomplement of the isogonal conjugate of X(78)
X(52366) = anticomplement of the isotomic conjugate of X(3718)
X(52366) = anticomplementary isogonal conjugate of X(12649)
X(52366) = X(34)-Dao conjugate of X(34)
X(52366) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 12649}, {3, 145}, {6, 30699}, {8, 4}, {9, 5905}, {21, 3868}, {41, 21216}, {48, 3210}, {55, 193}, {56, 11851}, {59, 1897}, {63, 7}, {69, 3434}, {71, 17778}, {72, 2475}, {77, 36845}, {78, 8}, {100, 521}, {190, 46400}, {200, 5942}, {212, 192}, {219, 2}, {220, 30694}, {222, 4452}, {268, 9965}, {271, 962}, {281, 6515}, {283, 1}, {284, 3187}, {304, 21285}, {305, 21280}, {306, 2893}, {312, 21270}, {314, 20242}, {318, 5906}, {332, 17135}, {333, 17220}, {345, 69}, {348, 6604}, {394, 347}, {521, 149}, {603, 17480}, {607, 6392}, {643, 7253}, {644, 4391}, {645, 850}, {648, 23683}, {651, 17896}, {652, 4440}, {895, 4442}, {906, 17496}, {1069, 10529}, {1176, 3891}, {1252, 651}, {1257, 5174}, {1259, 20}, {1260, 144}, {1264, 1370}, {1265, 3436}, {1331, 522}, {1332, 693}, {1444, 3873}, {1789, 3874}, {1790, 3875}, {1791, 65}, {1792, 3869}, {1793, 758}, {1794, 34772}, {1795, 38460}, {1796, 3879}, {1797, 1266}, {1802, 3177}, {1808, 740}, {1809, 517}, {1810, 5853}, {1811, 3880}, {1812, 75}, {1813, 4025}, {1815, 9436}, {1818, 52164}, {1946, 9263}, {2193, 17147}, {2287, 92}, {2289, 6360}, {2318, 1654}, {2327, 63}, {2338, 48381}, {2359, 1999}, {3270, 17036}, {3596, 11442}, {3682, 3152}, {3688, 8878}, {3692, 329}, {3694, 2895}, {3699, 20293}, {3710, 1330}, {3718, 6327}, {3719, 4329}, {3939, 25259}, {3990, 18667}, {3998, 2897}, {4102, 20289}, {4558, 4467}, {4561, 21302}, {4563, 4374}, {4564, 4566}, {4570, 14544}, {4571, 513}, {4587, 514}, {5546, 525}, {5547, 47286}, {5548, 10015}, {6056, 3164}, {6065, 3732}, {6099, 44428}, {6332, 150}, {6514, 17134}, {6516, 3900}, {7015, 29840}, {7017, 317}, {7257, 21300}, {8611, 21221}, {8805, 6820}, {14919, 41804}, {15627, 3580}, {15628, 51481}, {17206, 20244}, {22128, 41803}, {22356, 30577}, {22370, 20537}, {23189, 17154}, {27382, 14361}, {34055, 20247}, {34259, 388}, {35518, 21293}, {36797, 520}, {39167, 20076}, {39947, 39695}, {40399, 273}, {40442, 41575}, {40443, 30628}, {44189, 21279}, {44707, 17035}, {44717, 664}, {44722, 42020}, {45393, 912}, {46102, 18026}, {47487, 3870}, {51379, 153}
X(52366) = X(3718)-Ceva conjugate of X(2)
X(52366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {34, 34823, 2}, {1038, 46878, 2}, {7270, 18750, 3436}, {46421, 46422, 3101}

X(52367) = X(3)-DAO CONJUGATE OF X(34441)

Barycentrics    a^4 - b^4 - a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 - c^4 : :
X(52367) = 3 X[2] - 4 X[25639], 9 X[2] - 10 X[31262], X[35] - 3 X[31159], 3 X[35] - 5 X[31262], X[20066] - 4 X[25639], X[20066] - 6 X[31159], 3 X[20066] - 10 X[31262], 2 X[25639] - 3 X[31159], 6 X[25639] - 5 X[31262], 9 X[31159] - 5 X[31262], 2 X[12] - 3 X[17577], X[3871] - 3 X[17577], 4 X[2646] - 5 X[3616], 5 X[3616] - 2 X[11015], 3 X[3241] - 4 X[11011], 5 X[631] - 4 X[33862], 7 X[9780] - 4 X[37568], 2 X[4324] - 3 X[37299], 4 X[5267] - 3 X[37299], 3 X[4995] - 4 X[6668], 4 X[4999] - 3 X[17549], 2 X[15338] - 3 X[17549], 3 X[7967] - 4 X[33281]

X(52367) lies on these lines: {1, 149}, {2, 35}, {3, 11680}, {4, 8}, {5, 100}, {7, 7702}, {10, 3583}, {11, 404}, {12, 528}, {20, 5450}, {21, 2886}, {30, 2975}, {31, 24883}, {32, 17737}, {36, 24387}, {38, 24851}, {40, 6840}, {55, 2476}, {56, 11235}, {58, 33142}, {60, 19642}, {63, 2894}, {69, 9047}, {76, 20553}, {78, 1699}, {79, 3874}, {83, 13576}, {104, 10943}, {145, 1478}, {146, 2779}, {148, 21226}, {150, 20244}, {153, 5881}, {214, 37735}, {225, 4318}, {312, 5300}, {315, 4441}, {316, 17143}, {377, 497}, {379, 28757}, {381, 5687}, {382, 956}, {386, 33107}, {388, 3241}, {390, 5177}, {405, 9668}, {411, 5842}, {442, 1621}, {443, 5550}, {474, 9669}, {496, 5253}, {498, 5141}, {499, 4188}, {515, 4861}, {516, 6734}, {519, 3585}, {535, 5288}, {546, 17757}, {550, 5303}, {631, 33862}, {662, 3615}, {673, 33839}, {908, 4420}, {938, 13750}, {942, 20292}, {944, 6923}, {946, 4511}, {958, 11114}, {964, 32773}, {976, 3944}, {986, 33094}, {993, 15680}, {999, 50239}, {1001, 4197}, {1054, 28096}, {1056, 20057}, {1058, 38314}, {1089, 33091}, {1125, 4857}, {1145, 18357}, {1154, 48877}, {1193, 33106}, {1259, 10883}, {1278, 7900}, {1320, 10944}, {1329, 34612}, {1330, 2891}, {1352, 25304}, {1376, 4193}, {1385, 6951}, {1468, 33141}, {1484, 37535}, {1698, 37162}, {1714, 17127}, {1724, 11330}, {1770, 3218}, {1836, 3868}, {1839, 5279}, {2077, 6972}, {2292, 33095}, {2320, 4305}, {2478, 2550}, {2548, 17756}, {2599, 4552}, {2802, 37710}, {2893, 17220}, {2901, 33093}, {2932, 45976}, {3006, 7283}, {3035, 7173}, {3058, 6175}, {3085, 6871}, {3086, 4190}, {3091, 5552}, {3153, 14213}, {3244, 5270}, {3295, 17532}, {3454, 33175}, {3476, 18961}, {3496, 21029}, {3501, 26074}, {3521, 38955}, {3576, 37163}, {3579, 6903}, {3586, 19860}, {3600, 11240}, {3614, 6154}, {3632, 18513}, {3670, 33102}, {3678, 26792}, {3679, 18514}, {3685, 7557}, {3701, 32850}, {3702, 7270}, {3705, 37456}, {3746, 3822}, {3754, 37702}, {3811, 31053}, {3813, 7354}, {3816, 17531}, {3817, 27385}, {3826, 17536}, {3829, 5433}, {3831, 32948}, {3832, 7080}, {3838, 37080}, {3854, 27525}, {3870, 9612}, {3872, 5691}, {3876, 24703}, {3877, 5794}, {3878, 47033}, {3885, 5252}, {3889, 10404}, {3895, 9578}, {3913, 10895}, {3914, 5262}, {3916, 28146}, {3923, 36568}, {3925, 5047}, {3935, 21077}, {3957, 13407}, {4000, 26099}, {4189, 4302}, {4200, 11393}, {4202, 32942}, {4209, 28734}, {4214, 12410}, {4292, 26015}, {4293, 10529}, {4295, 12649}, {4299, 45700}, {4304, 24541}, {4309, 10198}, {4324, 5267}, {4347, 37798}, {4355, 31146}, {4366, 33841}, {4413, 9671}, {4429, 5192}, {4514, 4968}, {4642, 37717}, {4645, 24523}, {4652, 5231}, {4847, 51118}, {4855, 8227}, {4872, 20880}, {4894, 33090}, {4972, 13740}, {4995, 6668}, {4996, 5840}, {4999, 15338}, {5010, 37291}, {5051, 5263}, {5084, 19877}, {5134, 16552}, {5154, 26364}, {5218, 6933}, {5223, 7997}, {5229, 20050}, {5247, 33136}, {5250, 9580}, {5255, 21935}, {5260, 11113}, {5261, 11239}, {5266, 33133}, {5274, 6904}, {5283, 9664}, {5284, 8728}, {5291, 7747}, {5292, 17126}, {5432, 7504}, {5434, 15679}, {5439, 18527}, {5440, 6900}, {5540, 26793}, {5556, 6601}, {5587, 13729}, {5603, 6917}, {5657, 6928}, {5705, 35258}, {5731, 6850}, {5734, 10532}, {5744, 6851}, {5748, 6849}, {5787, 9961}, {5805, 25722}, {5818, 6929}, {5880, 7671}, {5884, 49176}, {5886, 6901}, {5901, 10609}, {5904, 17484}, {5906, 48878}, {6246, 39776}, {6253, 36002}, {6327, 10449}, {6653, 16044}, {6655, 26801}, {6735, 19925}, {6737, 51423}, {6745, 12571}, {6796, 6960}, {6818, 26038}, {6821, 26103}, {6825, 37000}, {6826, 7704}, {6828, 11496}, {6830, 11248}, {6835, 9779}, {6836, 9778}, {6842, 11491}, {6845, 45630}, {6853, 32613}, {6872, 19843}, {6888, 45392}, {6902, 26446}, {6909, 11826}, {6911, 11928}, {6915, 7681}, {6921, 10589}, {6932, 11500}, {6937, 10267}, {6941, 11499}, {6943, 10310}, {6944, 10598}, {6945, 10893}, {6948, 10785}, {6952, 13199}, {6965, 9956}, {6980, 32141}, {6982, 10786}, {6985, 18499}, {6990, 11517}, {6999, 28797}, {7191, 23537}, {7280, 36004}, {7319, 30513}, {7377, 28813}, {7382, 34255}, {7394, 10327}, {7548, 7680}, {7613, 28080}, {7743, 17614}, {7745, 21956}, {7748, 16975}, {7785, 17759}, {7812, 37857}, {7951, 8715}, {7967, 33281}, {8256, 12764}, {8257, 9581}, {8666, 10483}, {9342, 17527}, {9397, 48304}, {9579, 24392}, {9613, 36846}, {9614, 19861}, {9655, 34605}, {9709, 17556}, {9809, 12528}, {9963, 15950}, {10129, 11374}, {10200, 17572}, {10266, 24298}, {10479, 33083}, {10528, 10590}, {10580, 16193}, {10584, 17567}, {10588, 34607}, {10593, 13747}, {10624, 24987}, {10742, 12531}, {10782, 18871}, {10826, 25005}, {10942, 38665}, {11109, 23541}, {11238, 25524}, {11322, 19755}, {11682, 31162}, {12047, 34772}, {12247, 25413}, {12513, 12943}, {12526, 50865}, {12532, 31803}, {12559, 41709}, {12572, 25006}, {12682, 12919}, {12690, 37730}, {13278, 26482}, {13589, 23850}, {13727, 23542}, {13743, 48680}, {14956, 51290}, {14986, 37435}, {16062, 24552}, {16086, 25253}, {16160, 48698}, {16370, 31493}, {16478, 33128}, {16547, 21065}, {16865, 19854}, {16910, 27248}, {17522, 31084}, {17533, 47742}, {17541, 26582}, {17550, 20172}, {17616, 50196}, {17647, 30384}, {17671, 24596}, {17680, 27097}, {17686, 26590}, {17751, 21282}, {17753, 21285}, {17889, 28082}, {18393, 22836}, {18398, 26842}, {18802, 38156}, {19767, 26098}, {19855, 31420}, {20076, 34625}, {20084, 37005}, {20085, 37706}, {20247, 33865}, {20293, 23104}, {20533, 28742}, {20919, 41591}, {21279, 31995}, {22791, 37230}, {24026, 34007}, {24042, 51433}, {24443, 24715}, {24850, 33119}, {24916, 25671}, {24936, 33111}, {25439, 37719}, {25525, 41864}, {26064, 31330}, {26105, 37462}, {29662, 37603}, {29817, 51706}, {29958, 38389}, {30171, 33168}, {30172, 32849}, {31164, 41863}, {31737, 38474}, {31777, 37374}, {32558, 35262}, {32949, 35633}, {33086, 50605}, {33089, 50044}, {33105, 37573}, {33771, 37693}, {34195, 39542}, {34773, 47032}, {35614, 48899}, {36175, 47270}, {36944, 38952}, {37371, 41227}, {37579, 37797}, {38460, 45287}, {38903, 52242}, {40273, 51409}, {51886, 51975}

X(52367) = midpoint of X(24468) and X(41869)
X(52367) = reflection of X(i) in X(j) for these {i,j}: {2, 31159}, {8, 5086}, {20, 11012}, {35, 25639}, {145, 11009}, {411, 15908}, {2975, 24390}, {3648, 52126}, {3871, 12}, {4324, 5267}, {6906, 26470}, {11010, 10}, {11015, 2646}, {11491, 6842}, {11849, 5}, {15338, 4999}, {20060, 3585}, {20066, 35}, {20084, 37005}, {34772, 12047}
X(52367) = isogonal conjugate of X(34441)
X(52367) = complement of X(20066)
X(52367) = anticomplement of X(35)
X(52367) = Fuhrmann-circle-inverse of X(5080)
X(52367) = circumcircle-of-anticomplementary-triangle-inverse of X(5180)
X(52367) = anticomplement of the isogonal conjugate of X(79)
X(52367) = anticomplement of the isotomic conjugate of X(20565)
X(52367) = isotomic conjugate of the isogonal conjugate of X(20988)
X(52367) = polar conjugate of the isogonal conjugate of X(22122)
X(52367) = anticomplementary isogonal conjugate of X(3648)
X(52367) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3648}, {57, 41808}, {79, 8}, {94, 21277}, {1870, 12383}, {1989, 17484}, {2160, 2}, {2166, 5080}, {3218, 1272}, {3615, 3869}, {6186, 192}, {6742, 513}, {6757, 1330}, {7073, 144}, {7100, 20}, {7110, 329}, {7113, 18301}, {8818, 2895}, {13486, 523}, {15455, 20295}, {20565, 6327}, {26700, 522}, {30602, 14450}, {30690, 69}, {32678, 49274}, {35049, 17136}, {38340, 693}, {39152, 616}, {39153, 617}, {43682, 2893}
X(52367) = X(20565)-Ceva conjugate of X(2)
X(52367) = X(21065)-cross conjugate of X(20919)
X(52367) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34441}, {2160, 52062}
X(52367) = X(3)-Dao conjugate of X(34441)
X(52367) = cevapoint of X(20988) and X(22122)
X(52367) = barycentric product X(i)*X(j) for these {i,j}: {1, 20919}, {75, 16547}, {76, 20988}, {86, 21065}, {190, 21179}, {264, 22122}, {30710, 41591}
X(52367) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34441}, {35, 52062}, {16547, 1}, {20919, 75}, {20988, 6}, {21065, 10}, {21179, 514}, {22122, 3}, {41591, 3666}, {41592, 16698}
X(52367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36250, 33155}, {2, 20066, 35}, {4, 8, 5080}, {4, 3434, 8}, {4, 5082, 3436}, {4, 12245, 10526}, {5, 100, 27529}, {8, 5180, 3869}, {8, 9812, 11415}, {10, 3583, 5046}, {35, 25639, 2}, {35, 31159, 25639}, {72, 5178, 8}, {72, 22793, 5057}, {79, 3874, 17483}, {149, 2475, 1}, {321, 5015, 8}, {355, 14923, 8}, {377, 497, 3616}, {381, 5687, 11681}, {442, 15171, 1621}, {496, 11112, 5253}, {958, 12953, 11114}, {958, 34706, 12953}, {962, 5175, 8}, {1376, 10896, 4193}, {1770, 10916, 3218}, {1836, 3868, 14450}, {2478, 2550, 9780}, {2550, 5225, 2478}, {2886, 6284, 21}, {2894, 37433, 63}, {3091, 17784, 5552}, {3419, 3869, 8}, {3419, 12699, 3869}, {3434, 3436, 5082}, {3436, 5082, 8}, {3841, 5259, 2}, {3869, 12699, 5180}, {3871, 17577, 12}, {4294, 31418, 2}, {4302, 26363, 4189}, {4324, 5267, 37299}, {4385, 5014, 8}, {4696, 5100, 8}, {4999, 15338, 17549}, {5046, 33110, 10}, {5057, 5178, 72}, {5081, 23528, 8}, {5176, 10914, 8}, {5253, 10707, 496}, {5295, 33075, 8}, {5794, 12701, 3877}, {6653, 16044, 26752}, {6850, 12116, 5731}, {6871, 20075, 3085}, {6952, 13199, 26285}, {7705, 37828, 9780}, {7741, 25440, 2}, {8666, 10483, 20067}, {10525, 37820, 4}, {10529, 31295, 4293}, {10593, 13747, 31272}, {10914, 18480, 5176}, {10944, 13463, 1320}, {11113, 31419, 5260}, {11681, 49719, 5687}, {12699, 18407, 4}, {12953, 31140, 958}, {13271, 13273, 1320}, {26060, 26127, 2}, {31140, 34706, 11114}, {45287, 49600, 38460}

X(52368) = X(36)-DAO CONJUGATE OF X(1)

Barycentrics    a*(a^2 - b^2 + b*c - c^2)*(a^4 - b^4 + a^2*b*c - a*b^2*c - a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(52368) lies on these lines: {1, 149}, {2, 1718}, {8, 14544}, {21, 34977}, {86, 664}, {100, 33649}, {106, 7191}, {109, 283}, {244, 5262}, {323, 758}, {514, 24322}, {515, 3153}, {517, 43574}, {651, 12532}, {860, 1870}, {1125, 38458}, {1325, 12826}, {1421, 3616}, {3218, 4351}, {3869, 22136}, {3904, 6370}, {3920, 24222}, {4658, 17016}, {4861, 6742}, {4996, 11700}, {5080, 37798}, {5086, 18447}, {7286, 17768}, {9627, 11015}, {14988, 50461}, {17015, 23668}, {17080, 51236}, {17100, 24025}, {18689, 18697}, {20932, 49492}, {23669, 49487}, {33961, 39751}

X(52368) = X(75)-Ceva conjugate of X(3218)
X(52368) = X(i)-isoconjugate of X(j) for these (i,j): {80, 34442}, {759, 10693}
X(52368) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 36}, {10693, 34586}
X(52368) = crosspoint of X(75) and X(20920)
X(52368) = barycentric product X(i)*X(j) for these {i,j}: {36, 20920}, {75, 40584}, {320, 16548}, {1325, 3936}, {3218, 5080}, {4511, 37798}, {4585, 21180}, {20924, 20989}
X(52368) = barycentric quotient X(i)/X(j) for these {i,j}: {1325, 24624}, {2245, 10693}, {5080, 18359}, {7113, 34442}, {16548, 80}, {20920, 20566}, {20989, 2161}, {21066, 15065}, {22123, 1807}, {37798, 18815}, {40584, 1}
X(52368) = {X(11700),X(16586)}-harmonic conjugate of X(4996)

X(52369) = X(37)-DAO CONJUGATE OF X(28)

Barycentrics    b*c*(b + c)^2*(-a^2 + b^2 + c^2) : :

X(52369) lies on these lines: {10, 321}, {19, 46738}, {63, 3718}, {92, 341}, {190, 21376}, {226, 42715}, {306, 3610}, {312, 3305}, {329, 2835}, {612, 1215}, {668, 20929}, {1265, 26872}, {1930, 32782}, {2064, 14206}, {2321, 42707}, {2525, 14208}, {3263, 5249}, {3596, 14213}, {3678, 5739}, {3694, 40161}, {3695, 21015}, {3967, 43214}, {3969, 42710}, {4037, 21813}, {4357, 19835}, {4358, 40940}, {4362, 20663}, {4574, 23112}, {4710, 17874}, {4975, 49683}, {6358, 28654}, {7237, 21688}, {19684, 30142}, {19785, 33937}, {19811, 20879}, {20896, 40603}, {42698, 51367}

X(52369) = isotomic conjugate of the isogonal conjugate of X(3949)
X(52369) = isotomic conjugate of the polar conjugate of X(1089)
X(52369) = X(i)-Ceva conjugate of X(j) for these (i,j): {20336, 3695}, {30713, 28654}
X(52369) = X(i)-cross conjugate of X(j) for these (i,j): {125, 14208}, {3708, 4064}, {3949, 1089}, {21678, 10}
X(52369) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36420}, {19, 849}, {25, 593}, {27, 2206}, {28, 1333}, {29, 16947}, {34, 2150}, {56, 2189}, {58, 1474}, {60, 608}, {81, 2203}, {110, 43925}, {112, 3733}, {184, 36419}, {249, 42067}, {250, 1015}, {270, 604}, {281, 7342}, {607, 7341}, {757, 1973}, {1014, 2204}, {1019, 32676}, {1106, 2326}, {1172, 1408}, {1175, 46890}, {1395, 2185}, {1396, 2194}, {1397, 46103}, {1398, 7054}, {1412, 2299}, {1437, 5317}, {1509, 1974}, {1565, 41937}, {1576, 17925}, {1977, 18020}, {2969, 23357}, {2973, 23963}, {3937, 23964}, {7215, 23975}, {7254, 32713}, {8034, 47443}, {22096, 23582}
X(52369) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 2189}, {6, 849}, {10, 1474}, {19, 4075}, {27, 40603}, {28, 37}, {58, 51574}, {226, 1412}, {244, 647}, {244, 43925}, {270, 3161}, {525, 3942}, {593, 6505}, {594, 4222}, {757, 6337}, {1019, 15526}, {1111, 23285}, {1214, 1396}, {1333, 40591}, {1973, 40607}, {2150, 11517}, {2203, 40586}, {2299, 40599}, {2326, 6552}, {3733, 34591}, {4858, 17925}, {36103, 36420}, {40937, 46883}
X(52369) = cevapoint of X(3708) and X(4064)
X(52369) = crosspoint of X(20336) and X(40071)
X(52369) = barycentric product X(i)*X(j) for these {i,j}: {10, 20336}, {12, 3718}, {37, 40071}, {63, 28654}, {69, 1089}, {71, 27801}, {72, 313}, {75, 3695}, {76, 3949}, {78, 34388}, {125, 7035}, {201, 3596}, {304, 594}, {305, 756}, {306, 321}, {307, 3701}, {312, 26942}, {326, 7141}, {339, 765}, {341, 6356}, {345, 6358}, {349, 3694}, {525, 4033}, {561, 3690}, {656, 27808}, {668, 4064}, {872, 40050}, {1016, 20902}, {1018, 3267}, {1214, 30713}, {1231, 2321}, {1441, 3710}, {1500, 40364}, {2197, 28659}, {3708, 31625}, {3952, 14208}, {4036, 4561}, {4103, 15413}, {4574, 20948}, {4601, 21046}, {4605, 15416}, {6057, 7182}, {15742, 17879}, {20618, 30693}
X(52369) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 849}, {8, 270}, {9, 2189}, {10, 28}, {12, 34}, {19, 36420}, {37, 1474}, {42, 2203}, {63, 593}, {69, 757}, {71, 1333}, {72, 58}, {73, 1408}, {77, 7341}, {78, 60}, {92, 36419}, {125, 244}, {181, 1395}, {201, 56}, {210, 2299}, {219, 2150}, {226, 1396}, {228, 2206}, {304, 1509}, {305, 873}, {306, 81}, {307, 1014}, {312, 46103}, {313, 286}, {321, 27}, {339, 1111}, {345, 2185}, {346, 2326}, {442, 46883}, {525, 1019}, {594, 19}, {603, 7342}, {656, 3733}, {661, 43925}, {756, 25}, {762, 2333}, {765, 250}, {872, 1974}, {1018, 112}, {1089, 4}, {1109, 2969}, {1214, 1412}, {1231, 1434}, {1254, 1398}, {1265, 1098}, {1332, 4556}, {1334, 2204}, {1409, 16947}, {1425, 1106}, {1500, 1973}, {1577, 17925}, {1826, 5317}, {2171, 608}, {2197, 604}, {2294, 46890}, {2318, 2194}, {2321, 1172}, {2632, 3937}, {2643, 42067}, {3267, 7199}, {3610, 2303}, {3682, 1437}, {3690, 31}, {3692, 7054}, {3694, 284}, {3695, 1}, {3701, 29}, {3708, 1015}, {3710, 21}, {3718, 261}, {3914, 4211}, {3948, 31905}, {3949, 6}, {3952, 162}, {3977, 30576}, {3992, 37168}, {3998, 1790}, {4001, 30581}, {4013, 36125}, {4024, 6591}, {4033, 648}, {4036, 7649}, {4037, 2201}, {4044, 31926}, {4064, 513}, {4066, 31902}, {4075, 4222}, {4082, 4183}, {4103, 1783}, {4158, 255}, {4424, 4247}, {4466, 16726}, {4515, 2332}, {4557, 32676}, {4571, 4636}, {4574, 163}, {4580, 39179}, {4605, 32714}, {4647, 31900}, {6057, 33}, {6354, 1435}, {6356, 269}, {6358, 278}, {6535, 1824}, {7019, 7303}, {7035, 18020}, {7064, 2212}, {7066, 603}, {7068, 7004}, {7101, 36421}, {7140, 1096}, {7141, 158}, {7182, 552}, {7206, 6198}, {8013, 2355}, {8611, 7252}, {14208, 7192}, {15526, 3942}, {15742, 24000}, {17094, 7203}, {17206, 763}, {17879, 1565}, {20234, 31917}, {20336, 86}, {20618, 738}, {20653, 1829}, {20902, 1086}, {20975, 3248}, {21015, 614}, {21021, 7119}, {21046, 3125}, {21074, 41364}, {21075, 3194}, {21081, 2906}, {21134, 764}, {21671, 1104}, {21674, 40985}, {21675, 1841}, {21678, 40941}, {21810, 2354}, {21859, 32674}, {23994, 2973}, {24018, 7254}, {24020, 7215}, {24459, 50456}, {26942, 57}, {27569, 2905}, {27801, 44129}, {27808, 811}, {28654, 92}, {30591, 46542}, {30713, 31623}, {31625, 46254}, {34388, 273}, {35309, 35325}, {37755, 1407}, {40071, 274}, {40521, 8750}, {41013, 8747}, {41508, 46886}, {42031, 31901}, {42698, 17167}
X(52369) = {X(3718),X(19799)}-harmonic conjugate of X(63)

X(52370) = X(7)-ISOCONJUGATE OF X(27)

Barycentrics    a^3*(a - b - c)*(b + c)*(a^2 - b^2 - c^2) : :

X(52370) lies on these lines: {3, 20752}, {6, 37080}, {10, 21930}, {33, 210}, {37, 41538}, {41, 1253}, {42, 213}, {71, 73}, {72, 4574}, {78, 219}, {212, 1802}, {227, 4559}, {228, 2200}, {283, 1808}, {644, 3701}, {1402, 2198}, {1880, 21871}, {1946, 20781}, {2333, 51377}, {3230, 3924}, {3714, 4513}, {3916, 22164}, {4020, 22344}, {4652, 22163}, {4855, 22127}, {5440, 22126}, {7117, 22072}, {8012, 40957}, {9247, 23202}, {21809, 40977}, {22070, 22350}

X(52370) = isogonal conjugate of the isotomic conjugate of X(3694)
X(52370) = isogonal conjugate of the polar conjugate of X(210)
X(52370) = X(i)-Ceva conjugate of X(j) for these (i,j): {71, 228}, {219, 2318}, {220, 1334}, {4587, 1946}
X(52370) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1434}, {7, 27}, {21, 1847}, {28, 85}, {29, 279}, {34, 274}, {56, 44129}, {57, 286}, {58, 331}, {75, 1396}, {81, 273}, {86, 278}, {92, 1014}, {108, 7199}, {162, 24002}, {225, 1509}, {264, 1412}, {269, 31623}, {270, 1446}, {307, 36419}, {310, 608}, {314, 1435}, {333, 1119}, {348, 8747}, {479, 2322}, {552, 1826}, {648, 3676}, {653, 7192}, {664, 17925}, {757, 40149}, {799, 43923}, {811, 3669}, {873, 1880}, {1019, 18026}, {1041, 16750}, {1088, 1172}, {1118, 17206}, {1395, 6385}, {1398, 28660}, {1407, 44130}, {1408, 1969}, {1414, 17924}, {1440, 41083}, {1474, 6063}, {1896, 7177}, {1897, 17096}, {2203, 20567}, {2969, 4620}, {3064, 4616}, {3668, 46103}, {3733, 46404}, {3737, 13149}, {4183, 23062}, {4248, 27818}, {4560, 36118}, {4565, 46107}, {4572, 43925}, {4573, 7649}, {4625, 6591}, {4626, 17926}, {4635, 18344}, {4637, 44426}, {5317, 7182}, {6331, 43924}, {6335, 7203}, {6628, 8736}, {7012, 16727}, {7056, 8748}, {7195, 40411}, {7233, 31905}, {15419, 36127}, {16947, 18022}, {17205, 46102}, {17219, 23984}, {18155, 32714}, {18600, 40446}
X(52370) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 44129}, {10, 331}, {85, 40591}, {125, 24002}, {206, 1396}, {264, 40599}, {273, 40586}, {274, 11517}, {278, 40600}, {286, 5452}, {1014, 22391}, {1434, 36033}, {1847, 40611}, {3669, 17423}, {6063, 51574}, {6600, 31623}, {7199, 38983}, {17096, 34467}, {17924, 40608}, {17925, 39025}, {24771, 44130}, {38996, 43923}, {40149, 40607}
X(52370) = crosspoint of X(i) and X(j) for these (i,j): {71, 2318}, {210, 3694}, {212, 219}, {220, 1802}
X(52370) = crosssum of X(i) and X(j) for these (i,j): {273, 278}, {279, 1847}, {1014, 1396}, {1172, 4233}, {3676, 17197}, {16727, 17096}
X(52370) = crossdifference of every pair of points on line {7192, 24002}
X(52370) = barycentric product X(i)*X(j) for these {i,j}: {1, 2318}, {3, 210}, {6, 3694}, {8, 228}, {9, 71}, {10, 212}, {21, 3690}, {31, 3710}, {33, 3682}, {37, 219}, {41, 306}, {42, 78}, {48, 2321}, {55, 72}, {63, 1334}, {65, 1260}, {73, 200}, {101, 8611}, {181, 1792}, {184, 3701}, {201, 2328}, {213, 345}, {220, 1214}, {222, 4515}, {226, 1802}, {268, 21871}, {281, 3990}, {283, 756}, {284, 3949}, {295, 4433}, {307, 1253}, {312, 2200}, {318, 4055}, {332, 872}, {346, 1409}, {480, 1439}, {512, 4571}, {521, 4557}, {594, 2193}, {603, 4082}, {607, 3998}, {644, 647}, {646, 3049}, {650, 4574}, {652, 1018}, {656, 3939}, {661, 4587}, {692, 52355}, {810, 3699}, {906, 3700}, {1231, 14827}, {1259, 1824}, {1265, 1402}, {1331, 4041}, {1332, 3709}, {1400, 3692}, {1410, 5423}, {1437, 6057}, {1444, 7064}, {1459, 4069}, {1500, 1812}, {1791, 40966}, {1794, 40967}, {1809, 51377}, {1813, 4171}, {1826, 2289}, {1918, 3718}, {1946, 3952}, {2171, 2327}, {2175, 20336}, {2188, 21075}, {2194, 3695}, {2196, 3985}, {2197, 2287}, {2324, 41087}, {2333, 3719}, {2359, 21033}, {3678, 8606}, {3900, 23067}, {3958, 33635}, {4047, 34820}, {4086, 32656}, {4095, 7116}, {4183, 7066}, {4524, 6516}, {6056, 41013}, {6065, 18210}, {6726, 7591}, {7046, 22341}, {7079, 40152}, {7368, 52037}, {9247, 30713}, {9447, 40071}, {14624, 22074}, {15628, 42702}, {21039, 47487}, {21859, 23090}, {22080, 32635}, {22381, 27538}, {22383, 30730}, {23189, 40521}, {36197, 44717}
X(52370) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 44129}, {32, 1396}, {37, 331}, {41, 27}, {42, 273}, {48, 1434}, {55, 286}, {71, 85}, {72, 6063}, {73, 1088}, {78, 310}, {184, 1014}, {200, 44130}, {210, 264}, {212, 86}, {213, 278}, {219, 274}, {220, 31623}, {228, 7}, {283, 873}, {306, 20567}, {345, 6385}, {644, 6331}, {647, 24002}, {652, 7199}, {669, 43923}, {810, 3676}, {872, 225}, {906, 4573}, {1018, 46404}, {1253, 29}, {1260, 314}, {1265, 40072}, {1331, 4625}, {1334, 92}, {1400, 1847}, {1402, 1119}, {1409, 279}, {1410, 479}, {1437, 552}, {1500, 40149}, {1792, 18021}, {1802, 333}, {1813, 4635}, {1918, 34}, {1946, 7192}, {2175, 28}, {2193, 1509}, {2197, 1446}, {2200, 57}, {2204, 36419}, {2205, 608}, {2212, 8747}, {2289, 17206}, {2318, 75}, {2321, 1969}, {2638, 17219}, {3049, 3669}, {3063, 17925}, {3682, 7182}, {3688, 16747}, {3690, 1441}, {3692, 28660}, {3694, 76}, {3701, 18022}, {3709, 17924}, {3710, 561}, {3939, 811}, {3949, 349}, {3990, 348}, {4041, 46107}, {4055, 77}, {4171, 46110}, {4433, 40717}, {4515, 7017}, {4516, 2973}, {4524, 44426}, {4557, 18026}, {4559, 13149}, {4571, 670}, {4574, 4554}, {4587, 799}, {6056, 1444}, {6066, 5379}, {6602, 2322}, {7064, 41013}, {7071, 1896}, {7109, 1880}, {7117, 16727}, {7124, 16750}, {8611, 3261}, {9247, 1412}, {9447, 1474}, {9448, 2203}, {14575, 1408}, {14827, 1172}, {20336, 41283}, {20665, 31917}, {20753, 33947}, {21871, 40701}, {22061, 7196}, {22074, 16705}, {22079, 17169}, {22341, 7056}, {22363, 7195}, {22364, 7248}, {22368, 18165}, {22369, 4059}, {22383, 17096}, {23067, 4569}, {32656, 1414}, {32660, 4637}, {36054, 15419}, {36059, 4616}, {39258, 5236}, {40972, 17171}, {52355, 40495}
X(52370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {71, 3990, 1409}, {220, 7074, 607}

X(52371) = X(7)-ISOCONJUGATE OF X(36)

Barycentrics    a*(a - b - c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(52371) lies on the cubics K497 and K498 and these lines: {1, 5}, {8, 28829}, {33, 7140}, {42, 1989}, {44, 2342}, {45, 55}, {64, 37567}, {88, 14191}, {89, 40269}, {100, 24433}, {103, 1155}, {200, 4069}, {210, 2328}, {220, 3119}, {528, 26611}, {759, 8694}, {963, 5204}, {997, 16594}, {1043, 3699}, {1168, 5048}, {1725, 18524}, {1793, 5302}, {1864, 40528}, {1936, 49712}, {2173, 20989}, {2265, 51235}, {2340, 42064}, {3021, 7218}, {3058, 3961}, {3214, 10118}, {3293, 38336}, {3465, 40663}, {3617, 27542}, {3716, 52356}, {3935, 14942}, {3938, 11238}, {4511, 4997}, {4792, 10703}, {4845, 41339}, {4860, 52013}, {4895, 46393}, {5160, 5524}, {5293, 10543}, {5297, 7474}, {7026, 36930}, {7043, 36931}, {7072, 7074}, {7220, 36815}, {9355, 33519}, {11075, 17796}, {11502, 17595}, {14629, 15898}, {17601, 24430}, {18360, 40263}, {18815, 21453}, {32286, 34893}, {35459, 38954}, {36737, 39150}, {36738, 39151}, {51805, 52202}, {51806, 52201}

X(52371) = isogonal conjugate of X(1443)
X(52371) = X(i)-Ceva conjugate of X(j) for these (i,j): {80, 2161}, {6740, 36910}, {36590, 9}
X(52371) = X(9629)-cross conjugate of X(7073)
X(52371) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1443}, {6, 17078}, {7, 36}, {56, 320}, {57, 3218}, {58, 41804}, {59, 4089}, {77, 1870}, {81, 18593}, {85, 7113}, {86, 1464}, {106, 41801}, {109, 4453}, {222, 17923}, {269, 4511}, {278, 22128}, {279, 2323}, {604, 20924}, {651, 3960}, {654, 658}, {758, 1014}, {934, 3738}, {1088, 2361}, {1227, 1417}, {1397, 40075}, {1407, 32851}, {1408, 35550}, {1412, 3936}, {1434, 2245}, {1439, 17515}, {1444, 1835}, {1446, 4282}, {1461, 3904}, {1983, 24002}, {2163, 36589}, {3669, 4585}, {3911, 40215}, {4554, 21758}, {4565, 4707}, {4569, 8648}, {4573, 21828}, {4881, 19604}, {5081, 7053}, {16586, 34051}, {18026, 22379}, {20566, 41282}, {37772, 37773}
X(52371) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 320}, {3, 1443}, {7, 15898}, {9, 17078}, {10, 41804}, {11, 4453}, {75, 36909}, {214, 41801}, {1464, 40600}, {3161, 20924}, {3218, 5452}, {3738, 14714}, {3904, 35508}, {3936, 40599}, {3960, 38991}, {4089, 6615}, {4511, 6600}, {5081, 23050}, {18593, 40586}, {24771, 32851}, {36589, 40587}, {38966, 44428}
X(52371) = cevapoint of X(i) and X(j) for these (i,j): {210, 3689}, {2310, 4895}
X(52371) = crosspoint of X(i) and X(j) for these (i,j): {80, 36910}, {2341, 6740}
X(52371) = crosssum of X(i) and X(j) for these (i,j): {1319, 1465}, {1464, 18593}, {3218, 4881}
X(52371) = trilinear pole of line {657, 1334}
X(52371) = crossdifference of every pair of points on line {654, 3960}
X(52371) = barycentric product X(i)*X(j) for these {i,j}: {1, 36910}, {8, 2161}, {9, 80}, {10, 2341}, {33, 52351}, {37, 6740}, {41, 20566}, {44, 36590}, {55, 18359}, {101, 52356}, {200, 2006}, {210, 24624}, {220, 18815}, {281, 1807}, {284, 15065}, {312, 6187}, {333, 34857}, {346, 1411}, {650, 51562}, {655, 3900}, {657, 35174}, {663, 36804}, {759, 2321}, {1168, 2325}, {1334, 14616}, {1793, 1826}, {1989, 4420}, {2222, 3239}, {2316, 51975}, {3701, 34079}, {4041, 47318}, {4397, 32675}, {4876, 36815}, {4997, 40172}, {7026, 7126}, {7043, 19551}, {7073, 41226}, {8641, 46405}
X(52371) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17078}, {6, 1443}, {8, 20924}, {9, 320}, {33, 17923}, {37, 41804}, {41, 36}, {42, 18593}, {44, 41801}, {45, 36589}, {55, 3218}, {80, 85}, {200, 32851}, {210, 3936}, {212, 22128}, {213, 1464}, {220, 4511}, {312, 40075}, {607, 1870}, {650, 4453}, {655, 4569}, {657, 3738}, {663, 3960}, {759, 1434}, {1253, 2323}, {1334, 758}, {1411, 279}, {1793, 17206}, {1807, 348}, {2006, 1088}, {2161, 7}, {2170, 4089}, {2175, 7113}, {2222, 658}, {2321, 35550}, {2325, 1227}, {2332, 17515}, {2333, 1835}, {2341, 86}, {3689, 51583}, {3711, 27757}, {3900, 3904}, {3939, 4585}, {4041, 4707}, {4079, 51645}, {4420, 7799}, {4814, 23884}, {6187, 57}, {6740, 274}, {7064, 4053}, {7079, 5081}, {8641, 654}, {14827, 2361}, {15065, 349}, {18359, 6063}, {20566, 20567}, {32675, 934}, {34079, 1014}, {34857, 226}, {35174, 46406}, {36590, 20568}, {36804, 4572}, {36815, 10030}, {36910, 75}, {40172, 3911}, {47318, 4625}, {51562, 4554}, {52351, 7182}, {52356, 3261}
X(52371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {80, 1807, 1411}, {6187, 34857, 2161}, {7126, 19551, 2161}, {14629, 45272, 15898}, {34857, 40172, 6187}

X(52372) = X(8)-ISOCONJUGATE OF X(35)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(52372) lies on the conic {{A,B,C,X(1),X(6)}}, the cubic K497, and these lines: {1, 30}, {6, 1406}, {12, 5018}, {56, 6186}, {58, 14158}, {65, 1126}, {86, 1443}, {106, 26700}, {582, 6149}, {939, 5217}, {996, 6757}, {1042, 1411}, {1120, 6742}, {1155, 1167}, {1220, 30690}, {1222, 4968}, {1399, 1427}, {1725, 48668}, {2279, 39690}, {2297, 7110}, {2334, 2650}, {2424, 48151}, {2687, 36064}, {3214, 32259}, {3468, 11246}, {3579, 18360}, {3647, 18593}, {4653, 51748}, {5378, 52085}, {6126, 12778}, {9277, 34502}, {13486, 16948}, {36052, 37582}, {36279, 52186}, {37129, 38340}, {37567, 42019}, {39152, 42677}, {39153, 42680}, {43531, 43682}, {51805, 52201}, {51806, 52202}

X(52372) = isogonal conjugate of X(4420)
X(52372) = X(i)-cross conjugate of X(j) for these (i,j): {3125, 3669}, {6186, 2160}, {42753, 43082}
X(52372) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4420}, {6, 42033}, {8, 35}, {9, 3219}, {10, 35193}, {21, 3678}, {41, 33939}, {55, 319}, {60, 7206}, {72, 11107}, {78, 6198}, {100, 35057}, {190, 9404}, {200, 1442}, {220, 17095}, {284, 3969}, {306, 41502}, {312, 2174}, {321, 35192}, {323, 36910}, {341, 1399}, {346, 2003}, {644, 14838}, {943, 31938}, {1043, 2594}, {1250, 40714}, {1260, 7282}, {1334, 34016}, {1792, 1825}, {2287, 16577}, {2321, 40214}, {2323, 41226}, {2328, 40999}, {2341, 42701}, {2605, 3699}, {3578, 33635}, {3647, 32635}, {3701, 17104}, {3718, 14975}, {3939, 4467}, {4102, 17454}, {4570, 6741}, {5546, 7265}, {7058, 21794}, {7161, 52126}, {10638, 40713}, {35194, 44687}, {44688, 46073}, {44689, 46077}
X(52372) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4420}, {9, 42033}, {223, 319}, {478, 3219}, {1442, 6609}, {3160, 33939}, {3678, 40611}, {3969, 40590}, {4467, 40617}, {6741, 50330}, {8054, 35057}, {18160, 40615}, {36908, 40999}
X(52372) = trilinear pole of line {649, 14399}
X(52372) = crossdifference of every pair of points on line {9404, 35057}
X(52372) = barycentric product X(i)*X(j) for these {i,j}: {7, 2160}, {56, 30690}, {57, 79}, {58, 43682}, {85, 6186}, {269, 7110}, {278, 7100}, {279, 7073}, {513, 38340}, {514, 26700}, {554, 2306}, {604, 20565}, {1014, 8818}, {1081, 33654}, {1407, 52344}, {1412, 6757}, {1427, 3615}, {1443, 1989}, {1769, 47317}, {1847, 8606}, {3120, 35049}, {3669, 6742}, {3942, 34922}, {7178, 13486}, {15455, 43924}, {30602, 47057}
X(52372) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 42033}, {6, 4420}, {7, 33939}, {56, 3219}, {57, 319}, {65, 3969}, {79, 312}, {269, 17095}, {604, 35}, {608, 6198}, {649, 35057}, {667, 9404}, {1014, 34016}, {1042, 16577}, {1106, 2003}, {1333, 35193}, {1357, 7202}, {1397, 2174}, {1400, 3678}, {1407, 1442}, {1408, 40214}, {1411, 41226}, {1427, 40999}, {1435, 7282}, {1443, 7799}, {1464, 42701}, {1474, 11107}, {2160, 8}, {2171, 7206}, {2203, 41502}, {2206, 35192}, {2260, 31938}, {2306, 40714}, {3125, 6741}, {3669, 4467}, {3676, 18160}, {4017, 7265}, {6186, 9}, {6742, 646}, {6757, 30713}, {7073, 346}, {7100, 345}, {7110, 341}, {7203, 16755}, {8606, 3692}, {8818, 3701}, {13486, 645}, {16947, 17104}, {20565, 28659}, {26700, 190}, {30690, 3596}, {32636, 3578}, {33654, 40713}, {35049, 4600}, {38340, 668}, {43682, 313}, {43924, 14838}
X(52372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {79, 7100, 7073}, {2306, 33654, 2160}

X(52373) = X(9)-ISOCONJUGATE OF X(29)

Barycentrics    a^2*(a + b - c)^2*(a - b + c)^2*(b + c)*(a^2 - b^2 - c^2) : :

X(52373) lies on these lines: {6, 2155}, {19, 57}, {31, 56}, {48, 222}, {63, 348}, {65, 2357}, {71, 1214}, {73, 228}, {77, 22097}, {81, 934}, {223, 2183}, {226, 1020}, {241, 28274}, {269, 2215}, {279, 2282}, {347, 24310}, {354, 7004}, {658, 1821}, {1398, 1451}, {1400, 1427}, {1458, 2352}, {1461, 2003}, {1730, 43035}, {1826, 8808}, {1901, 13853}, {1943, 24435}, {2148, 33629}, {2360, 34043}, {3173, 22356}, {3182, 11471}, {3333, 3671}, {3937, 23204}, {5706, 37818}, {6354, 40160}, {7013, 10319}, {7114, 19349}, {9316, 37538}, {14547, 45963}, {20277, 40945}, {22053, 23207}, {22153, 23089}, {34032, 46009}, {43058, 46017}

X(52373) = isogonal conjugate of X(2322)
X(52373) = isogonal conjugate of the anticomplement of X(18643)
X(52373) = isotomic conjugate of the polar conjugate of X(1042)
X(52373) = isogonal conjugate of the polar conjugate of X(3668)
X(52373) = X(i)-Ceva conjugate of X(j) for these (i,j): {57, 1427}, {934, 1459}, {1020, 51640}, {1439, 73}, {3668, 1042}, {7053, 1410}, {7177, 1439}
X(52373) = X(i)-cross conjugate of X(j) for these (i,j): {1409, 73}, {1425, 1439}, {23620, 71}, {44093, 3}
X(52373) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2322}, {2, 4183}, {4, 2287}, {8, 1172}, {9, 29}, {10, 2326}, {19, 1043}, {21, 281}, {27, 200}, {28, 346}, {33, 333}, {41, 44130}, {55, 31623}, {58, 7101}, {72, 36421}, {75, 2332}, {78, 8748}, {81, 7046}, {86, 7079}, {92, 2328}, {100, 17926}, {112, 4397}, {158, 2327}, {162, 3239}, {210, 46103}, {219, 1896}, {220, 286}, {270, 2321}, {274, 7071}, {284, 318}, {312, 2299}, {314, 607}, {341, 1474}, {393, 1792}, {643, 3064}, {645, 18344}, {648, 3900}, {650, 36797}, {657, 811}, {1021, 1897}, {1098, 1826}, {1146, 5379}, {1253, 44129}, {1265, 5317}, {1396, 5423}, {1783, 7253}, {1812, 1857}, {1824, 7058}, {1863, 40403}, {2189, 3701}, {2194, 7017}, {2204, 3596}, {2212, 28660}, {2341, 5081}, {3692, 8747}, {4238, 28132}, {4319, 40411}, {4567, 42069}, {4578, 17925}, {4876, 14024}, {5546, 44426}, {5931, 7156}, {6061, 40149}, {6331, 8641}, {6335, 21789}, {6591, 7256}, {7008, 27398}, {7054, 41013}, {7110, 11107}, {7259, 7649}, {8021, 40447}, {13614, 40838}, {15149, 28071}, {15416, 32713}, {17515, 36910}, {17519, 36916}, {18020, 36197}, {36796, 37908}, {41502, 52344}
X(52373) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 2322}, {6, 1043}, {10, 7101}, {27, 6609}, {29, 478}, {92, 36908}, {125, 3239}, {206, 2332}, {223, 31623}, {226, 312}, {281, 40611}, {318, 40590}, {341, 51574}, {346, 40591}, {657, 17423}, {1021, 34467}, {1147, 2327}, {1214, 7017}, {1427, 15466}, {1826, 15267}, {2287, 36033}, {2328, 22391}, {3160, 44130}, {4183, 32664}, {4397, 34591}, {4988, 21666}, {7046, 40586}, {7079, 40600}, {7253, 39006}, {8054, 17926}, {17113, 44129}, {40622, 46110}, {40627, 42069}
X(52373) = cevapoint of X(1409) and X(1410)
X(52373) = crosspoint of X(i) and X(j) for these (i,j): {57, 222}, {63, 1073}, {1214, 52037}, {7053, 7177}
X(52373) = crosssum of X(i) and X(j) for these (i,j): {9, 281}, {19, 1249}, {657, 52335}, {2328, 2332}, {7046, 7079}
X(52373) = trilinear pole of line {810, 42658}
X(52373) = crossdifference of every pair of points on line {3239, 17926}
X(52373) = barycentric product X(i)*X(j) for these {i,j}: {1, 1439}, {3, 3668}, {7, 73}, {10, 7053}, {37, 7177}, {42, 7056}, {48, 1446}, {56, 307}, {57, 1214}, {58, 6356}, {63, 1427}, {65, 77}, {69, 1042}, {71, 279}, {72, 269}, {75, 1410}, {81, 37755}, {85, 1409}, {86, 1425}, {109, 17094}, {201, 1014}, {222, 226}, {223, 52037}, {225, 1804}, {228, 1088}, {273, 22341}, {278, 40152}, {283, 6046}, {284, 20618}, {286, 7138}, {306, 1407}, {321, 7099}, {326, 1426}, {332, 7143}, {348, 1400}, {479, 2318}, {520, 36118}, {525, 1461}, {603, 1441}, {604, 1231}, {647, 658}, {651, 51640}, {656, 934}, {738, 3694}, {810, 4569}, {822, 13149}, {905, 1020}, {1073, 36908}, {1106, 20336}, {1119, 3682}, {1254, 1444}, {1262, 4466}, {1332, 7216}, {1334, 30682}, {1402, 7182}, {1412, 26942}, {1434, 2197}, {1435, 3998}, {1459, 4566}, {1790, 6354}, {1803, 52023}, {1812, 7147}, {1813, 7178}, {1847, 3990}, {1880, 7183}, {2360, 6355}, {3049, 46406}, {3348, 8812}, {3676, 23067}, {3710, 7023}, {4017, 6516}, {4077, 36059}, {4306, 28786}, {4341, 28787}, {4561, 7250}, {4605, 7254}, {4617, 8611}, {6614, 52355}, {7011, 8808}, {7045, 18210}, {7125, 40149}, {7371, 7591}, {8057, 36079}, {10099, 41353}, {14256, 41087}, {18026, 51641}, {19611, 40933}, {23620, 30705}, {24018, 32714}, {40843, 51647}
X(52373) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1043}, {6, 2322}, {7, 44130}, {31, 4183}, {32, 2332}, {34, 1896}, {37, 7101}, {42, 7046}, {48, 2287}, {56, 29}, {57, 31623}, {65, 318}, {71, 346}, {72, 341}, {73, 8}, {77, 314}, {109, 36797}, {184, 2328}, {201, 3701}, {213, 7079}, {222, 333}, {226, 7017}, {228, 200}, {255, 1792}, {269, 286}, {279, 44129}, {307, 3596}, {348, 28660}, {577, 2327}, {603, 21}, {604, 1172}, {608, 8748}, {647, 3239}, {649, 17926}, {656, 4397}, {658, 6331}, {810, 3900}, {906, 7259}, {934, 811}, {1020, 6335}, {1042, 4}, {1106, 28}, {1214, 312}, {1231, 28659}, {1254, 41013}, {1331, 7256}, {1332, 7258}, {1333, 2326}, {1397, 2299}, {1398, 8747}, {1399, 11107}, {1400, 281}, {1402, 33}, {1406, 3559}, {1407, 27}, {1408, 270}, {1409, 9}, {1410, 1}, {1412, 46103}, {1425, 10}, {1426, 158}, {1427, 92}, {1428, 14024}, {1437, 1098}, {1439, 75}, {1446, 1969}, {1459, 7253}, {1461, 648}, {1464, 5081}, {1474, 36421}, {1790, 7058}, {1804, 332}, {1813, 645}, {1918, 7071}, {2197, 2321}, {2200, 220}, {2318, 5423}, {3049, 657}, {3120, 21666}, {3122, 42069}, {3668, 264}, {3682, 1265}, {3690, 4082}, {3694, 30693}, {3990, 3692}, {4017, 44426}, {4055, 1260}, {4091, 15411}, {4303, 51978}, {4466, 23978}, {4574, 6558}, {4848, 44721}, {6356, 313}, {6516, 7257}, {6611, 41083}, {7011, 27398}, {7053, 86}, {7056, 310}, {7066, 3710}, {7099, 81}, {7125, 1812}, {7138, 72}, {7143, 225}, {7147, 40149}, {7177, 274}, {7178, 46110}, {7180, 3064}, {7182, 40072}, {7216, 17924}, {7250, 7649}, {7335, 283}, {7366, 1396}, {7591, 7027}, {16947, 2189}, {17094, 35519}, {18210, 24026}, {20618, 349}, {20975, 52335}, {22341, 78}, {22342, 4420}, {22345, 46877}, {22363, 4319}, {22383, 1021}, {23067, 3699}, {23620, 6554}, {24018, 15416}, {24027, 5379}, {26884, 15146}, {26942, 30713}, {32660, 5546}, {32714, 823}, {36059, 643}, {36118, 6528}, {36908, 15466}, {37755, 321}, {39791, 6734}, {40152, 345}, {40933, 1895}, {40934, 1863}, {51640, 4391}, {51641, 521}, {51642, 18344}, {51647, 1948}, {52037, 34404}
X(52373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 32065, 1457}, {57, 47848, 19}, {57, 51969, 7490}, {222, 7011, 48}, {222, 7053, 7099}, {1214, 1439, 37755}, {1214, 40152, 71}, {1410, 1425, 73}, {22341, 39791, 73}

X(52374) = X(9)-ISOCONJUGATE OF X(35)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :
X(52374) = X[1] + 2 X[41492]

X(52374) lies on the conic {{A,B,C,X(1),X(2)}} and these lines: {1, 30}, {2, 7110}, {7, 25417}, {57, 2160}, {81, 553}, {88, 26723}, {105, 5322}, {226, 1255}, {241, 42326}, {274, 17078}, {278, 18688}, {527, 40399}, {1002, 44661}, {1014, 40143}, {1022, 30726}, {1170, 43035}, {1219, 52344}, {1280, 6742}, {1358, 7313}, {1427, 1989}, {2003, 2982}, {2688, 36064}, {3175, 34892}, {3227, 19796}, {3578, 41804}, {3676, 21141}, {3911, 39962}, {6358, 39722}, {8056, 35466}, {8818, 17056}, {14844, 30571}, {15455, 36805}, {17301, 39948}, {18697, 50043}, {20565, 30710}, {26742, 44794}, {27789, 37635}, {30701, 42032}, {33633, 43261}, {48857, 51223}

X(52374) = isotomic conjugate of X(42033)
X(52374) = X(i)-cross conjugate of X(j) for these (i,j): {2160, 79}, {3120, 3676}, {21102, 36118}, {32636, 7}
X(52374) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4420}, {8, 2174}, {9, 35}, {10, 35192}, {31, 42033}, {37, 35193}, {41, 319}, {55, 3219}, {71, 11107}, {72, 41502}, {100, 9404}, {101, 35057}, {200, 2003}, {210, 40214}, {219, 6198}, {220, 1442}, {284, 3678}, {345, 14975}, {346, 1399}, {644, 2605}, {1043, 21741}, {1098, 21794}, {1253, 17095}, {1802, 7282}, {1825, 2327}, {2150, 7206}, {2175, 33939}, {2194, 3969}, {2259, 31938}, {2287, 2594}, {2321, 17104}, {2322, 22342}, {2328, 16577}, {2361, 41226}, {3647, 33635}, {3939, 14838}, {6065, 7202}, {6149, 36910}, {17454, 32635}
X(52374) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 42033}, {9, 4420}, {35, 478}, {223, 3219}, {319, 3160}, {1015, 35057}, {1214, 3969}, {2003, 6609}, {3678, 40590}, {4467, 40615}, {4988, 6741}, {7265, 40622}, {8054, 9404}, {14838, 40617}, {14993, 36910}, {15267, 21794}, {16577, 36908}, {17095, 17113}, {18591, 31938}, {33939, 40593}, {35193, 40589}
X(52374) = cevapoint of X(i) and X(j) for these (i,j): {1086, 30724}, {2170, 50354}
X(52374) = trilinear pole of line {513, 11125}
X(52374) = barycentric product X(i)*X(j) for these {i,j}: {7, 79}, {56, 20565}, {57, 30690}, {81, 43682}, {85, 2160}, {269, 52344}, {273, 7100}, {279, 7110}, {514, 38340}, {554, 1081}, {693, 26700}, {1014, 6757}, {1088, 7073}, {1434, 8818}, {1443, 2166}, {1565, 34922}, {1989, 17078}, {3615, 3668}, {3669, 15455}, {3676, 6742}, {4077, 13486}, {6063, 6186}, {10015, 47317}, {16732, 35049}, {30602, 41808}
X(52374) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4420}, {2, 42033}, {7, 319}, {12, 7206}, {28, 11107}, {34, 6198}, {56, 35}, {57, 3219}, {58, 35193}, {65, 3678}, {79, 8}, {85, 33939}, {226, 3969}, {269, 1442}, {279, 17095}, {513, 35057}, {553, 3578}, {554, 40713}, {604, 2174}, {649, 9404}, {942, 31938}, {1042, 2594}, {1081, 40714}, {1106, 1399}, {1119, 7282}, {1333, 35192}, {1365, 21054}, {1393, 35194}, {1395, 14975}, {1407, 2003}, {1408, 17104}, {1410, 22342}, {1412, 40214}, {1426, 1825}, {1427, 16577}, {1434, 34016}, {1474, 41502}, {1789, 1792}, {1989, 36910}, {2006, 41226}, {2160, 9}, {3120, 6741}, {3337, 52126}, {3615, 1043}, {3668, 40999}, {3669, 14838}, {3676, 4467}, {6186, 55}, {6742, 3699}, {6757, 3701}, {7073, 200}, {7100, 78}, {7110, 346}, {7178, 7265}, {7248, 7186}, {8606, 1260}, {8818, 2321}, {13486, 643}, {15455, 646}, {17078, 7799}, {17096, 16755}, {18593, 42701}, {20565, 3596}, {24002, 18160}, {26700, 100}, {30690, 312}, {32636, 3647}, {34922, 15742}, {35049, 4567}, {37566, 41562}, {38340, 190}, {39152, 44688}, {39153, 44689}, {43082, 52356}, {43261, 4034}, {43682, 321}, {43924, 2605}, {47317, 13136}, {52344, 341}
X(52374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {554, 1081, 79}, {4654, 47057, 37631}

X(52375) = X(10)-ISOCONJUGATE OF X(35)

Barycentrics    a*(a + b)*(a + c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(52375) lies on these lines: {1, 2940}, {21, 36}, {56, 759}, {57, 1175}, {58, 14158}, {60, 3337}, {81, 24167}, {110, 3336}, {244, 849}, {261, 16709}, {265, 37251}, {267, 2611}, {284, 501}, {354, 15792}, {474, 24931}, {476, 2718}, {519, 35991}, {662, 3874}, {1014, 5358}, {1098, 4973}, {1210, 34301}, {1254, 24027}, {1325, 5563}, {1634, 16414}, {2189, 46883}, {2311, 2503}, {2646, 7073}, {3338, 9275}, {3453, 24046}, {3582, 37369}, {3746, 37294}, {4092, 13514}, {4188, 25650}, {4325, 7424}, {4653, 13624}, {4857, 5196}, {5127, 37582}, {5270, 37158}, {5322, 7469}, {6757, 8666}, {7110, 17581}, {8818, 11108}, {9563, 26884}, {13486, 16944}, {16817, 30690}, {17054, 28607}, {18398, 40214}, {21081, 33078}, {25055, 37032}, {25639, 52361}, {30598, 32014}, {30602, 40143}, {32783, 35983}, {35193, 37524}, {37259, 52153}, {37288, 40605}, {37960, 51883}

X(52375) = isogonal conjugate of X(3678)
X(52375) = isogonal conjugate of the complement of X(3874)
X(52375) = X(2170)-cross conjugate of X(1019)
X(52375) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3678}, {6, 3969}, {8, 2594}, {9, 16577}, {10, 35}, {12, 35193}, {37, 3219}, {42, 319}, {55, 40999}, {58, 7206}, {59, 6741}, {65, 4420}, {72, 6198}, {78, 1825}, {101, 7265}, {201, 11107}, {210, 1442}, {213, 33939}, {312, 21741}, {318, 22342}, {321, 2174}, {333, 21794}, {526, 51562}, {594, 40214}, {765, 2611}, {1016, 20982}, {1018, 14838}, {1089, 17104}, {1110, 17886}, {1252, 8287}, {1334, 17095}, {1399, 3701}, {1400, 42033}, {1500, 34016}, {2003, 2321}, {2161, 42701}, {2245, 41226}, {2318, 7282}, {2599, 44687}, {2605, 3952}, {2624, 36804}, {3615, 7144}, {4467, 4557}, {4551, 35057}, {4552, 9404}, {4567, 21824}, {4570, 21054}, {6149, 15065}, {6358, 35192}, {6539, 17454}, {14975, 20336}, {15742, 22094}, {21065, 52062}, {26942, 41502}
X(52375) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3678}, {9, 3969}, {10, 7206}, {223, 40999}, {319, 40592}, {478, 16577}, {513, 2611}, {514, 17886}, {661, 8287}, {1015, 7265}, {3219, 40589}, {4420, 40602}, {6615, 6741}, {6626, 33939}, {7110, 42710}, {14993, 15065}, {15295, 34857}, {18160, 40620}, {21054, 50330}, {21824, 40627}, {40582, 42033}, {40584, 42701}
X(52375) = cevapoint of X(i) and X(j) for these (i,j): {56, 1393}, {244, 3733}, {2160, 6186}
X(52375) = crosspoint of X(i) and X(j) for these (i,j): {79, 30602}, {1014, 40143}
X(52375) = crosssum of X(i) and X(j) for these (i,j): {1, 2940}, {210, 21873}
X(52375) = trilinear pole of line {4840, 4979}
X(52375) = barycentric product X(i)*X(j) for these {i,j}: {11, 35049}, {27, 7100}, {57, 3615}, {58, 30690}, {60, 43682}, {79, 81}, {86, 2160}, {274, 6186}, {278, 1789}, {476, 3960}, {514, 13486}, {593, 6757}, {757, 8818}, {1014, 7110}, {1019, 6742}, {1333, 20565}, {1412, 52344}, {1434, 7073}, {3733, 15455}, {3737, 38340}, {4453, 32678}, {4560, 26700}, {14158, 21739}, {21758, 35139}, {22379, 46456}, {30602, 40592}
X(52375) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3969}, {6, 3678}, {21, 42033}, {36, 42701}, {37, 7206}, {56, 16577}, {57, 40999}, {58, 3219}, {79, 321}, {81, 319}, {86, 33939}, {244, 8287}, {284, 4420}, {476, 36804}, {513, 7265}, {604, 2594}, {608, 1825}, {757, 34016}, {759, 41226}, {764, 21141}, {849, 40214}, {1014, 17095}, {1015, 2611}, {1019, 4467}, {1086, 17886}, {1333, 35}, {1396, 7282}, {1397, 21741}, {1402, 21794}, {1408, 2003}, {1412, 1442}, {1474, 6198}, {1789, 345}, {1989, 15065}, {2150, 35193}, {2160, 10}, {2170, 6741}, {2189, 11107}, {2206, 2174}, {3122, 21824}, {3125, 21054}, {3248, 20982}, {3615, 312}, {3733, 14838}, {3960, 3268}, {4840, 23883}, {6186, 37}, {6742, 4033}, {6757, 28654}, {7073, 2321}, {7100, 306}, {7110, 3701}, {7192, 18160}, {7252, 35057}, {8606, 3694}, {8818, 1089}, {11060, 34857}, {13486, 190}, {14158, 17484}, {14844, 27569}, {15455, 27808}, {16947, 1399}, {20565, 27801}, {21741, 7144}, {21758, 526}, {22379, 8552}, {26700, 4552}, {30690, 313}, {32678, 51562}, {35049, 4998}, {43261, 42031}, {43682, 34388}, {46882, 31938}, {46883, 445}, {46890, 1844}, {52344, 30713}
X(52375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 79, 51748}, {36, 229, 37816}, {32636, 51420, 58}

X(52376) = X(10)-ISOCONJUGATE OF X(39)

Barycentrics    a*(a + b)*(a^2 + b^2)*(a + c)*(a^2 + c^2) : :

X(52376) lies on the conic {{A,B,C,X(1),X(2)}} these lines: {1, 82}, {2, 32}, {21, 1390}, {57, 7210}, {58, 291}, {81, 18167}, {86, 17192}, {88, 4599}, {105, 827}, {172, 30168}, {274, 1333}, {278, 31906}, {333, 39281}, {593, 39747}, {689, 713}, {757, 39950}, {1010, 1224}, {1022, 39179}, {1176, 5138}, {1255, 18098}, {2006, 18097}, {2224, 34072}, {2915, 51862}, {3227, 4577}, {4593, 32020}, {5291, 39722}, {16054, 18087}, {16082, 42396}, {17285, 33954}, {17500, 51500}, {18089, 26643}, {33955, 39724}

X(52376) = isogonal conjugate of X(3954)
X(52376) = isogonal conjugate of the complement of X(17141)
X(52376) = X(4599)-Ceva conjugate of X(39179)
X(52376) = X(i)-cross conjugate of X(j) for these (i,j): {172, 58}, {7191, 86}, {16735, 274}, {39179, 4599}
X(52376) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3954}, {2, 21035}, {3, 21016}, {6, 15523}, {10, 39}, {37, 38}, {42, 141}, {65, 33299}, {71, 427}, {72, 17442}, {75, 21814}, {76, 41267}, {100, 8061}, {101, 826}, {190, 3005}, {213, 1930}, {226, 3688}, {228, 20883}, {256, 16587}, {257, 40936}, {306, 1843}, {313, 3051}, {321, 1964}, {335, 4093}, {512, 4568}, {513, 35309}, {523, 46148}, {594, 17187}, {661, 4553}, {668, 2084}, {688, 1978}, {756, 16696}, {872, 16703}, {1018, 2530}, {1235, 2200}, {1334, 3665}, {1400, 3703}, {1401, 2321}, {1441, 40972}, {1500, 16887}, {1634, 4024}, {1826, 3917}, {1918, 8024}, {1923, 27801}, {2333, 3933}, {2525, 8750}, {2528, 4628}, {3108, 21038}, {3682, 27376}, {3690, 17171}, {3700, 46153}, {3930, 46149}, {3943, 46150}, {3952, 21123}, {4020, 41013}, {4033, 50521}, {4037, 46159}, {4053, 46160}, {4062, 46154}, {4064, 35325}, {4079, 4576}, {4088, 46163}, {4120, 46162}, {4557, 16892}, {4559, 48278}, {4570, 39691}, {4574, 21108}, {7018, 21752}, {8041, 18082}, {8611, 46152}, {16030, 21011}, {21022, 31613}, {21083, 21355}, {21818, 32010}, {23285, 32739}, {24290, 35333}, {27369, 40071}, {35334, 50330}
X(52376) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3954}, {9, 15523}, {38, 40589}, {82, 21082}, {141, 40592}, {206, 21814}, {321, 41884}, {826, 1015}, {1930, 6626}, {2525, 26932}, {3703, 40582}, {4553, 36830}, {4568, 39054}, {8024, 34021}, {8054, 8061}, {21016, 36103}, {21035, 32664}, {21037, 21249}, {23285, 40619}, {33299, 40602}, {35309, 39026}, {39691, 50330}, {40620, 48084}
X(52376) = cevapoint of X(i) and X(j) for these (i,j): {6, 16687}, {81, 1333}, {82, 251}, {595, 4251}
X(52376) = crosssum of X(37) and X(21880)
X(52376) = trilinear pole of line {513, 1980}
X(52376) = crossdifference of every pair of points on line {3005, 8061}
X(52376) = barycentric product X(i)*X(j) for these {i,j}: {27, 34055}, {28, 1799}, {58, 3112}, {81, 83}, {82, 86}, {99, 18108}, {190, 39179}, {239, 39276}, {251, 274}, {286, 1176}, {308, 1333}, {310, 46289}, {513, 4577}, {514, 4599}, {649, 4593}, {662, 10566}, {667, 689}, {693, 827}, {757, 18082}, {905, 42396}, {1437, 46104}, {1444, 32085}, {1509, 18098}, {1919, 37204}, {1980, 42371}, {2185, 18097}, {2206, 18833}, {3115, 16717}, {3261, 34072}, {4164, 41209}, {4556, 18070}, {4603, 18111}, {4623, 18105}, {4628, 7199}, {4630, 40495}, {5262, 39281}, {6385, 46288}, {7303, 18099}, {7305, 16889}, {18089, 40408}, {18180, 39287}
X(52376) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 15523}, {6, 3954}, {19, 21016}, {21, 3703}, {27, 20883}, {28, 427}, {31, 21035}, {32, 21814}, {58, 38}, {81, 141}, {82, 10}, {83, 321}, {86, 1930}, {101, 35309}, {110, 4553}, {163, 46148}, {172, 16587}, {251, 37}, {274, 8024}, {284, 33299}, {286, 1235}, {308, 27801}, {513, 826}, {560, 41267}, {593, 16696}, {649, 8061}, {662, 4568}, {667, 3005}, {689, 6386}, {693, 23285}, {757, 16887}, {827, 100}, {849, 17187}, {905, 2525}, {1014, 3665}, {1019, 16892}, {1176, 72}, {1333, 39}, {1408, 1401}, {1437, 3917}, {1444, 3933}, {1474, 17442}, {1509, 16703}, {1799, 20336}, {1919, 2084}, {1980, 688}, {2194, 3688}, {2203, 1843}, {2206, 1964}, {2210, 4093}, {2530, 2528}, {3112, 313}, {3125, 39691}, {3733, 2530}, {3737, 48278}, {3803, 3806}, {4577, 668}, {4593, 1978}, {4599, 190}, {4628, 1018}, {4630, 692}, {4911, 17873}, {5007, 21817}, {5299, 17456}, {5317, 27376}, {7122, 40936}, {7191, 21249}, {7192, 48084}, {10547, 228}, {10566, 1577}, {14419, 14424}, {14885, 21880}, {16600, 21037}, {16696, 7794}, {16702, 7813}, {16706, 21425}, {16707, 42554}, {16735, 21248}, {16757, 23881}, {16948, 4884}, {17200, 20898}, {17469, 21038}, {18082, 1089}, {18097, 6358}, {18098, 594}, {18105, 4705}, {18108, 523}, {18167, 16893}, {20022, 42703}, {28724, 3998}, {32085, 41013}, {33632, 21874}, {34055, 306}, {34072, 101}, {38834, 21877}, {39179, 514}, {39276, 335}, {42396, 6335}, {46288, 213}, {46289, 42}, {51369, 51371}, {51420, 51360}, {51508, 3198}, {51906, 21833}

X(52377) = X(11)-ISOCONJUGATE OF X(36)

Barycentrics    a*(a - b)^2*(a - c)^2*(a + b - c)*(a - b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(52377) lies on the Feuerbach circumhyperbola and these lines: {1, 59}, {4, 7012}, {7, 7045}, {8, 765}, {9, 1252}, {21, 4570}, {80, 23592}, {100, 46041}, {104, 1807}, {109, 15343}, {294, 2161}, {314, 4600}, {655, 885}, {901, 2222}, {1000, 34232}, {1110, 15175}, {1262, 7284}, {1320, 1411}, {2006, 3254}, {2320, 4564}, {2481, 18815}, {2648, 34857}, {6187, 9365}, {14888, 21132}, {24298, 45926}, {24457, 40577}, {28780, 43749}, {32675, 34075}, {36037, 43728}

X(52377) = X(i)-cross conjugate of X(j) for these (i,j): {44, 651}, {517, 100}, {1411, 2222}, {1731, 653}, {1807, 51562}, {2161, 655}, {2316, 37136}, {2342, 5548}, {2361, 101}, {12034, 3257}, {45272, 3699}, {51361, 644}
X(52377) = X(i)-isoconjugate of X(j) for these (i,j): {11, 36}, {55, 4089}, {80, 3025}, {106, 51402}, {244, 4511}, {320, 3271}, {513, 3738}, {514, 654}, {649, 3904}, {650, 3960}, {651, 46384}, {663, 4453}, {693, 8648}, {758, 18191}, {909, 46398}, {1015, 32851}, {1086, 2323}, {1111, 2361}, {1443, 2310}, {1459, 44428}, {1870, 7004}, {1983, 40166}, {2006, 35128}, {2170, 3218}, {2245, 17197}, {3937, 5081}, {4282, 16732}, {4391, 21758}, {4530, 40215}, {4560, 21828}, {4707, 7252}, {4858, 7113}, {7117, 17923}, {8735, 22128}, {14936, 17078}, {17219, 44113}, {17515, 18210}, {22379, 44426}
X(52377) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 15898}, {214, 51402}, {223, 4089}, {3738, 39026}, {3904, 5375}, {23980, 46398}, {24026, 36909}, {38991, 46384}
X(52377) = cevapoint of X(i) and X(j) for these (i,j): {1, 23703}, {55, 2427}, {101, 2361}, {109, 1319}, {1411, 2222}
X(52377) = trilinear pole of line {101, 650}
X(52377) = barycentric product X(i)*X(j) for these {i,j}: {59, 18359}, {80, 4564}, {100, 655}, {101, 35174}, {109, 36804}, {190, 2222}, {651, 51562}, {668, 32675}, {692, 46405}, {765, 2006}, {1016, 1411}, {1252, 18815}, {1807, 46102}, {2149, 20566}, {2161, 4998}, {3218, 46649}, {4511, 23592}, {4551, 47318}, {4619, 52356}, {4620, 34857}, {5376, 14584}, {7012, 52351}, {7045, 36910}, {9268, 14628}
X(52377) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 51402}, {57, 4089}, {59, 3218}, {80, 4858}, {100, 3904}, {101, 3738}, {109, 3960}, {517, 46398}, {651, 4453}, {655, 693}, {663, 46384}, {692, 654}, {759, 17197}, {765, 32851}, {1110, 2323}, {1252, 4511}, {1262, 1443}, {1411, 1086}, {1783, 44428}, {1807, 26932}, {2006, 1111}, {2149, 36}, {2161, 11}, {2222, 514}, {2361, 35128}, {4551, 4707}, {4564, 320}, {4998, 20924}, {6187, 2170}, {7012, 17923}, {7045, 17078}, {7113, 3025}, {7115, 1870}, {18359, 34387}, {18815, 23989}, {21859, 6370}, {23592, 18815}, {23990, 2361}, {32660, 22379}, {32675, 513}, {32739, 8648}, {34079, 18191}, {34857, 21044}, {35174, 3261}, {36804, 35519}, {36910, 24026}, {40172, 4530}, {46405, 40495}, {46649, 18359}, {47318, 18155}, {51562, 4391}, {52351, 17880}
X(52377) = {X(23703),X(37630)}-harmonic conjugate of X(2222)

X(52378) = X(11)-ISOCONJUGATE OF X(37)

Barycentrics    a^2*(a - b)^2*(a + b)*(a - c)^2*(a + b - c)*(a + c)*(a - b + c) : :

X(52378) lies on these lines: {59, 3286}, {81, 4619}, {99, 32689}, {109, 691}, {110, 14733}, {163, 1019}, {226, 39295}, {249, 5060}, {250, 1101}, {333, 39294}, {349, 41174}, {648, 32707}, {651, 4556}, {653, 687}, {662, 32669}, {664, 2966}, {1025, 23889}, {1262, 17966}, {1813, 44769}, {2003, 9273}, {2149, 4564}, {2283, 17943}, {2406, 2407}, {2421, 2425}, {4558, 32643}, {4565, 4591}, {4590, 4620}, {4636, 36059}, {5467, 17942}, {7045, 7128}, {23582, 44331}, {32674, 32697}

X(52378) = isogonal conjugate of X(21044)
X(52378) = isogonal conjugate of the complement of X(17136)
X(52378) = isogonal conjugate of the isotomic conjugate of X(4620)
X(52378) = X(i)-cross conjugate of X(j) for these (i,j): {58, 1414}, {73, 664}, {81, 4556}, {284, 110}, {651, 4619}, {1400, 109}, {1412, 4565}, {1790, 662}, {2003, 651}, {2193, 4636}, {4225, 99}, {17209, 4584}, {37583, 934}, {40152, 1813}, {40214, 4629}
X(52378) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21044}, {2, 4516}, {7, 36197}, {8, 3125}, {9, 3120}, {10, 2170}, {11, 37}, {21, 115}, {29, 3708}, {33, 4466}, {41, 21207}, {42, 4858}, {55, 16732}, {57, 52335}, {65, 1146}, {72, 8735}, {81, 4092}, {125, 1172}, {210, 1086}, {213, 34387}, {225, 34591}, {226, 2310}, {244, 2321}, {257, 40608}, {261, 21833}, {270, 21046}, {281, 18210}, {284, 1109}, {312, 3122}, {314, 3124}, {321, 3271}, {333, 2643}, {338, 2194}, {339, 2204}, {512, 4391}, {513, 3700}, {514, 4041}, {521, 2501}, {522, 661}, {523, 650}, {525, 18344}, {594, 18191}, {643, 21131}, {646, 8034}, {647, 44426}, {649, 4086}, {652, 24006}, {656, 3064}, {657, 4077}, {663, 1577}, {693, 3709}, {756, 17197}, {764, 30730}, {798, 35519}, {810, 46110}, {850, 3063}, {885, 24290}, {1015, 3701}, {1018, 21132}, {1020, 23615}, {1024, 4088}, {1084, 40072}, {1111, 1334}, {1214, 42069}, {1358, 4515}, {1365, 2287}, {1400, 24026}, {1402, 23978}, {1409, 21666}, {1427, 4081}, {1441, 14936}, {1446, 3022}, {1812, 8754}, {1824, 26932}, {1826, 7004}, {1880, 2968}, {1896, 3269}, {1903, 38357}, {1946, 14618}, {2160, 6741}, {2185, 21043}, {2193, 2970}, {2250, 35015}, {2299, 20902}, {2333, 17880}, {2358, 7358}, {2489, 35518}, {2611, 7110}, {2632, 8748}, {2969, 3694}, {3119, 3668}, {3121, 3596}, {3239, 4017}, {3248, 30713}, {3270, 40149}, {3615, 21824}, {3676, 4171}, {3680, 21950}, {3737, 4024}, {3900, 7178}, {3954, 18101}, {4036, 7252}, {4049, 4895}, {4069, 6545}, {4079, 18155}, {4120, 23838}, {4163, 7216}, {4397, 7180}, {4415, 40528}, {4435, 35352}, {4451, 16592}, {4518, 39786}, {4524, 24002}, {4526, 35353}, {4530, 4674}, {4542, 30575}, {4551, 42462}, {4557, 40166}, {4559, 42455}, {4560, 4705}, {4612, 8029}, {4631, 22260}, {4843, 47915}, {5317, 7068}, {6057, 16726}, {6385, 7063}, {6591, 52355}, {7064, 16727}, {7073, 8287}, {7117, 41013}, {7649, 8611}, {11998, 51870}, {13576, 17435}, {14395, 18808}, {14432, 23894}, {15232, 38345}, {15320, 38358}, {20975, 31623}, {20982, 52344}, {21809, 40451}, {21828, 52356}, {23893, 30574}, {36795, 42752}
X(52378) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 21044}, {11, 40589}, {115, 40611}, {223, 16732}, {226, 20902}, {338, 1214}, {478, 3120}, {522, 36830}, {850, 10001}, {1109, 40590}, {1146, 40602}, {2970, 47345}, {3064, 40596}, {3160, 21207}, {3239, 34961}, {3700, 39026}, {4086, 5375}, {4092, 40586}, {4391, 39054}, {4516, 32664}, {4858, 40592}, {5452, 52335}, {6626, 34387}, {14618, 39053}, {23978, 40605}, {24026, 40582}, {31998, 35519}, {39052, 44426}, {39062, 46110}
X(52378) = cevapoint of X(i) and X(j) for these (i,j): {35, 101}, {56, 36075}, {58, 163}, {59, 2149}, {63, 3882}, {73, 32660}, {81, 651}, {109, 1400}, {110, 284}, {283, 5546}, {1408, 1415}, {1412, 4565}, {1813, 40152}, {2193, 36059}, {7122, 32739}
X(52378) = crosssum of X(i) and X(j) for these (i,j): {115, 3708}, {4516, 36197}
X(52378) = trilinear pole of line {109, 110}
X(52378) = barycentric product X(i)*X(j) for these {i,j}: {6, 4620}, {7, 4570}, {21, 7045}, {27, 44717}, {42, 7340}, {56, 4600}, {57, 4567}, {58, 4998}, {59, 86}, {65, 24041}, {73, 18020}, {77, 5379}, {81, 4564}, {99, 109}, {100, 1414}, {101, 4573}, {107, 6517}, {108, 4592}, {110, 664}, {162, 6516}, {163, 4554}, {190, 4565}, {226, 249}, {250, 307}, {274, 2149}, {284, 1275}, {314, 24027}, {333, 1262}, {349, 23357}, {604, 4601}, {643, 934}, {644, 4637}, {645, 1461}, {648, 1813}, {651, 662}, {653, 4558}, {658, 5546}, {692, 4625}, {765, 1014}, {799, 1415}, {811, 36059}, {1016, 1412}, {1019, 31615}, {1020, 4612}, {1042, 6064}, {1043, 7339}, {1101, 1441}, {1252, 1434}, {1400, 4590}, {1402, 24037}, {1408, 7035}, {1409, 46254}, {1444, 7012}, {1576, 4572}, {1790, 46102}, {1812, 7128}, {2701, 17933}, {3219, 35049}, {3286, 39293}, {3939, 4616}, {4552, 4556}, {4559, 4610}, {4560, 4619}, {4563, 32674}, {4566, 4636}, {4575, 18026}, {4577, 46153}, {4617, 7259}, {4622, 23703}, {4632, 36075}, {6331, 32660}, {6514, 23984}, {6614, 7256}, {7115, 17206}, {16947, 31625}, {17942, 35154}, {18645, 38809}, {22341, 23999}, {23582, 40152}, {23979, 28660}, {32661, 46404}
X(52378) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 21044}, {7, 21207}, {21, 24026}, {29, 21666}, {31, 4516}, {35, 6741}, {41, 36197}, {42, 4092}, {55, 52335}, {56, 3120}, {57, 16732}, {58, 11}, {59, 10}, {65, 1109}, {73, 125}, {81, 4858}, {86, 34387}, {99, 35519}, {100, 4086}, {101, 3700}, {108, 24006}, {109, 523}, {110, 522}, {112, 3064}, {162, 44426}, {163, 650}, {181, 21043}, {222, 4466}, {225, 2970}, {226, 338}, {249, 333}, {250, 29}, {283, 2968}, {284, 1146}, {307, 339}, {333, 23978}, {349, 23962}, {593, 17197}, {603, 18210}, {604, 3125}, {643, 4397}, {648, 46110}, {651, 1577}, {653, 14618}, {662, 4391}, {664, 850}, {692, 4041}, {765, 3701}, {849, 18191}, {859, 35015}, {906, 8611}, {934, 4077}, {1014, 1111}, {1016, 30713}, {1019, 40166}, {1042, 1365}, {1101, 21}, {1110, 210}, {1214, 20902}, {1252, 2321}, {1262, 226}, {1275, 349}, {1331, 52355}, {1333, 2170}, {1397, 3122}, {1399, 2611}, {1400, 115}, {1402, 2643}, {1408, 244}, {1409, 3708}, {1412, 1086}, {1414, 693}, {1415, 661}, {1434, 23989}, {1437, 7004}, {1441, 23994}, {1442, 17886}, {1444, 17880}, {1457, 42759}, {1461, 7178}, {1474, 8735}, {1576, 663}, {1617, 21945}, {1634, 48278}, {1790, 26932}, {1804, 17216}, {1813, 525}, {1819, 7358}, {2003, 8287}, {2149, 37}, {2193, 34591}, {2194, 2310}, {2197, 21046}, {2206, 3271}, {2283, 4088}, {2299, 42069}, {2328, 4081}, {2360, 38357}, {2420, 14400}, {2594, 21054}, {2701, 18013}, {3285, 4530}, {3682, 7068}, {3733, 21132}, {3737, 42455}, {4225, 124}, {4253, 21946}, {4551, 4036}, {4554, 20948}, {4556, 4560}, {4558, 6332}, {4559, 4024}, {4564, 321}, {4565, 514}, {4567, 312}, {4570, 8}, {4572, 44173}, {4573, 3261}, {4575, 521}, {4590, 28660}, {4592, 35518}, {4600, 3596}, {4601, 28659}, {4619, 4552}, {4620, 76}, {4625, 40495}, {4636, 7253}, {4637, 24002}, {4998, 313}, {5009, 4124}, {5379, 318}, {5467, 14432}, {5546, 3239}, {6065, 4082}, {6514, 23983}, {6516, 14208}, {6517, 3265}, {7012, 41013}, {7045, 1441}, {7098, 23673}, {7115, 1826}, {7122, 40608}, {7128, 40149}, {7180, 21131}, {7252, 42462}, {7253, 23104}, {7339, 3668}, {7340, 310}, {7341, 17205}, {14587, 35196}, {15386, 15232}, {16947, 1015}, {16948, 4939}, {17096, 23100}, {17942, 2785}, {18020, 44130}, {18191, 1090}, {21741, 21824}, {21786, 52341}, {21789, 23615}, {22341, 2632}, {23067, 4064}, {23346, 30574}, {23357, 284}, {23964, 8748}, {23979, 1400}, {23990, 1334}, {23995, 2194}, {24000, 1896}, {24027, 65}, {24037, 40072}, {24041, 314}, {31615, 4033}, {32660, 647}, {32661, 652}, {32674, 2501}, {32676, 18344}, {32739, 3709}, {33628, 4534}, {35049, 30690}, {36059, 656}, {36074, 4838}, {36075, 4988}, {37583, 8286}, {40152, 15526}, {44717, 306}, {46153, 826}, {47390, 283}
X(52378) = {X(5467),X(23346)}-harmonic conjugate of X(17942)

X(52379) = X(12)-ISOCONJUGATE OF X(32)

Barycentrics    b*(a + b)^2*c*(a + c)^2*(-a + b + c) : :

X(52379) lies on these lines: {8, 7257}, {10, 5209}, {21, 261}, {27, 310}, {41, 27958}, {58, 10471}, {75, 757}, {76, 799}, {81, 239}, {86, 1193}, {99, 2975}, {172, 19623}, {270, 14024}, {321, 1931}, {333, 3691}, {348, 4625}, {350, 6626}, {811, 17555}, {1434, 10030}, {1909, 17731}, {2106, 16827}, {3696, 14195}, {3878, 38477}, {4362, 40731}, {5278, 30022}, {6629, 20888}, {8033, 34284}, {11679, 28659}, {14829, 40827}, {16705, 16738}, {17379, 51314}, {17680, 40017}, {19684, 33779}, {24995, 30966}, {25303, 32004}, {27705, 27954}, {31997, 51356}, {33299, 36800}, {33764, 33935}, {33939, 42710}

X(52379) = isotomic conjugate of X(2171)
X(52379) = isotomic conjugate of the anticomplement of X(21233)
X(52379) = isotomic conjugate of the complement of X(21273)
X(52379) = isotomic conjugate of the isogonal conjugate of X(2185)
X(52379) = X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 799}, {46254, 4610}
X(52379) = X(i)-cross conjugate of X(j) for these (i,j): {261, 873}, {314, 18021}, {333, 261}, {4560, 7257}, {4858, 18155}, {17185, 86}, {20882, 75}, {21233, 2}
X(52379) = X(i)-isoconjugate of X(j) for these (i,j): {6, 181}, {7, 7109}, {12, 32}, {25, 2197}, {31, 2171}, {37, 1402}, {41, 1254}, {42, 1400}, {56, 1500}, {57, 872}, {59, 3124}, {65, 213}, {73, 2333}, {109, 4079}, {184, 8736}, {201, 1973}, {220, 7143}, {225, 2200}, {226, 1918}, {228, 1880}, {512, 4559}, {560, 6358}, {594, 1397}, {604, 756}, {607, 1425}, {608, 3690}, {651, 50487}, {667, 21859}, {669, 4552}, {762, 1408}, {798, 4551}, {1016, 1356}, {1018, 51642}, {1037, 21813}, {1042, 1334}, {1084, 4998}, {1253, 7147}, {1275, 7063}, {1365, 23990}, {1395, 3949}, {1403, 6378}, {1407, 7064}, {1409, 1824}, {1415, 4705}, {1441, 2205}, {1501, 34388}, {1922, 7235}, {1974, 26942}, {2149, 2643}, {2175, 6354}, {2207, 7066}, {2212, 37755}, {2489, 23067}, {2971, 44717}, {3027, 51856}, {4092, 23979}, {4557, 7180}, {6046, 14827}, {6186, 21794}, {6535, 16947}, {7104, 7211}, {7115, 20975}, {7148, 41526}, {8687, 42661}, {18097, 41267}, {28654, 41280}, {32675, 42666}, {40147, 52024}
X(52379) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1500}, {2, 2171}, {9, 181}, {11, 4079}, {12, 6376}, {37, 40605}, {42, 40582}, {65, 6626}, {115, 1577}, {201, 6337}, {213, 40602}, {226, 34021}, {650, 2643}, {661, 40625}, {756, 3161}, {872, 5452}, {905, 3708}, {1146, 4705}, {1254, 3160}, {1400, 40592}, {1402, 40589}, {1966, 3027}, {2197, 6505}, {3124, 6615}, {4017, 40620}, {4024, 40624}, {4551, 31998}, {4559, 39054}, {6354, 40593}, {6358, 6374}, {6631, 21859}, {7064, 24771}, {7147, 17113}, {7235, 39028}, {17419, 42661}, {20975, 40628}, {35128, 42666}, {38991, 50487}
X(52379) = cevapoint of X(i) and X(j) for these (i,j): {2, 21273}, {9, 3996}, {75, 14829}, {261, 7058}, {274, 17206}, {314, 333}, {4858, 18155}
X(52379) = trilinear pole of line {3716, 3737}
X(52379) = barycentric product X(i)*X(j) for these {i,j}: {1, 18021}, {8, 873}, {11, 24037}, {21, 310}, {58, 40072}, {60, 561}, {75, 261}, {76, 2185}, {81, 28660}, {85, 7058}, {86, 314}, {99, 18155}, {270, 305}, {274, 333}, {284, 6385}, {286, 332}, {304, 46103}, {312, 1509}, {341, 552}, {514, 4631}, {522, 4623}, {593, 28659}, {645, 7199}, {670, 3737}, {757, 3596}, {763, 30713}, {799, 4560}, {812, 36806}, {849, 40363}, {1098, 6063}, {1111, 6064}, {1444, 44130}, {1502, 2150}, {1812, 44129}, {2170, 34537}, {2189, 40364}, {3061, 7307}, {3261, 4612}, {3701, 6628}, {4391, 4610}, {4590, 4858}, {4601, 17197}, {4602, 7252}, {4625, 7253}, {4636, 40495}, {7054, 20567}, {7192, 7257}, {7258, 17096}, {7304, 27424}, {7340, 24026}, {12836, 14124}, {17185, 40827}, {17206, 31623}, {17880, 18020}, {20568, 30606}, {20882, 31620}, {24041, 34387}, {26932, 46254}, {30940, 36800}
X(52379) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 181}, {2, 2171}, {7, 1254}, {8, 756}, {9, 1500}, {11, 2643}, {21, 42}, {27, 1880}, {29, 1824}, {41, 7109}, {55, 872}, {58, 1402}, {60, 31}, {63, 2197}, {69, 201}, {75, 12}, {76, 6358}, {77, 1425}, {78, 3690}, {81, 1400}, {85, 6354}, {86, 65}, {92, 8736}, {99, 4551}, {190, 21859}, {200, 7064}, {249, 2149}, {261, 1}, {269, 7143}, {270, 25}, {274, 226}, {279, 7147}, {283, 228}, {284, 213}, {285, 2357}, {286, 225}, {304, 26942}, {310, 1441}, {312, 594}, {314, 10}, {318, 7140}, {326, 7066}, {332, 72}, {333, 37}, {341, 6057}, {345, 3949}, {348, 37755}, {350, 7235}, {522, 4705}, {552, 269}, {561, 34388}, {593, 604}, {643, 4557}, {645, 1018}, {646, 4103}, {650, 4079}, {662, 4559}, {663, 50487}, {757, 56}, {763, 1412}, {799, 4552}, {849, 1397}, {873, 7}, {1014, 1042}, {1019, 7180}, {1021, 3709}, {1043, 210}, {1088, 6046}, {1089, 6058}, {1098, 55}, {1111, 1365}, {1172, 2333}, {1434, 1427}, {1444, 73}, {1509, 57}, {1790, 1409}, {1792, 2318}, {1804, 7138}, {1812, 71}, {1909, 7211}, {2082, 21813}, {2150, 32}, {2170, 3124}, {2185, 6}, {2189, 1973}, {2193, 2200}, {2194, 1918}, {2287, 1334}, {2319, 6378}, {2321, 762}, {2326, 607}, {3219, 21794}, {3248, 1356}, {3596, 1089}, {3686, 21816}, {3687, 21810}, {3691, 21820}, {3699, 40521}, {3701, 6535}, {3702, 8013}, {3705, 7237}, {3706, 21699}, {3716, 4155}, {3718, 3695}, {3733, 51642}, {3737, 512}, {3738, 42666}, {3794, 3778}, {3904, 2610}, {3975, 4037}, {3996, 40607}, {4267, 3725}, {4391, 4024}, {4435, 46390}, {4453, 51645}, {4459, 21725}, {4554, 4605}, {4556, 1415}, {4560, 661}, {4573, 1020}, {4590, 4564}, {4592, 23067}, {4610, 651}, {4612, 101}, {4623, 664}, {4625, 4566}, {4631, 190}, {4636, 692}, {4858, 115}, {4985, 6367}, {5324, 40934}, {6061, 1253}, {6064, 765}, {6385, 349}, {6514, 3990}, {6628, 1014}, {6740, 34857}, {7004, 20975}, {7054, 41}, {7058, 9}, {7081, 21803}, {7155, 7148}, {7182, 6356}, {7192, 4017}, {7199, 7178}, {7203, 7250}, {7252, 798}, {7253, 4041}, {7256, 4069}, {7257, 3952}, {7258, 30730}, {7303, 1431}, {7304, 1423}, {7340, 7045}, {7341, 1106}, {8033, 4032}, {8822, 227}, {14024, 862}, {14570, 35307}, {15411, 8611}, {15419, 51640}, {16708, 52023}, {16709, 3649}, {16713, 21808}, {16738, 45208}, {16739, 41003}, {17096, 7216}, {17175, 39793}, {17183, 4642}, {17185, 2092}, {17194, 52020}, {17197, 3125}, {17206, 1214}, {17219, 18210}, {17277, 20616}, {17420, 42661}, {17515, 44113}, {17787, 21021}, {17880, 125}, {18020, 7012}, {18021, 75}, {18155, 523}, {18163, 21796}, {18169, 39780}, {18178, 21936}, {18191, 3122}, {20665, 21815}, {21044, 21833}, {21388, 22229}, {21610, 21958}, {23189, 810}, {24026, 4092}, {24037, 4998}, {24041, 59}, {26856, 2170}, {26932, 3708}, {27398, 21871}, {27527, 21834}, {27958, 2295}, {28659, 28654}, {28660, 321}, {30576, 1404}, {30593, 32636}, {30606, 44}, {30939, 40663}, {30940, 16609}, {31623, 1826}, {31631, 21853}, {32851, 4053}, {33295, 1284}, {34016, 16577}, {34387, 1109}, {34388, 1091}, {35518, 4064}, {35519, 4036}, {36806, 4562}, {37140, 32675}, {39044, 3027}, {39177, 2623}, {40072, 313}, {40166, 21131}, {40214, 21741}, {44129, 40149}, {44130, 41013}, {46103, 19}, {46254, 46102}, {46877, 40966}, {46882, 40978}, {48264, 50538}, {51978, 40967}, {52352, 4849}
X(52379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 34016, 799}, {274, 1509, 873}

X(52380) = X(12)-ISOCONJUGATE OF X(36)

Barycentrics    a*(a + b)^2*(a - b - c)*(a + c)^2*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(52380) lies on the Feuerbach circumhyperbola and these lines: {1, 60}, {4, 162}, {7, 757}, {8, 643}, {9, 1793}, {21, 4636}, {58, 79}, {80, 5127}, {104, 30576}, {256, 24436}, {283, 6598}, {584, 941}, {593, 7284}, {943, 1807}, {1156, 37140}, {1411, 17097}, {1937, 23692}, {2150, 40968}, {2161, 2298}, {2166, 24640}, {2185, 2320}, {2997, 14616}, {4653, 5424}, {6187, 43073}, {7235, 14194}, {7261, 21274}, {10308, 16948}, {18609, 51420}, {23838, 35055}, {39766, 39767}

X(52380) = X(i)-cross conjugate of X(j) for these (i,j): {1731, 333}, {2361, 284}, {45272, 29}
X(52380) = X(i)-isoconjugate of X(j) for these (i,j): {10, 1464}, {12, 36}, {37, 18593}, {42, 41804}, {57, 4053}, {65, 758}, {72, 1835}, {73, 860}, {80, 3028}, {100, 51645}, {109, 6370}, {181, 320}, {201, 1870}, {226, 2245}, {307, 44113}, {651, 2610}, {654, 4605}, {664, 42666}, {756, 1443}, {1254, 4511}, {1400, 3936}, {1402, 35550}, {1411, 4736}, {1425, 5081}, {1441, 3724}, {1500, 17078}, {2006, 35069}, {2171, 3218}, {2197, 17923}, {2323, 6354}, {3960, 21859}, {4552, 21828}, {4559, 4707}, {6358, 7113}, {8736, 22128}, {16186, 34922}
X(52380) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 6370}, {12, 15898}, {758, 40602}, {1089, 36909}, {2610, 38991}, {3936, 40582}, {4053, 5452}, {4736, 35204}, {8054, 51645}, {18593, 40589}, {35550, 40605}, {39025, 42666}, {40592, 41804}
X(52380) = cevapoint of X(i) and X(j) for these (i,j): {58, 51420}, {284, 2361}
X(52380) = trilinear pole of line {284, 650}
X(52380) = crossdifference of every pair of points on line {2610, 51645}
X(52380) = barycentric product X(i)*X(j) for these {i,j}: {21, 24624}, {27, 1793}, {60, 18359}, {80, 2185}, {81, 6740}, {86, 2341}, {261, 2161}, {270, 52351}, {284, 14616}, {314, 34079}, {333, 759}, {522, 37140}, {757, 36910}, {1098, 2006}, {1168, 30606}, {1411, 7058}, {1807, 46103}, {2150, 20566}, {3737, 47318}, {4391, 36069}, {4556, 52356}, {7054, 18815}, {30576, 36590}, {32671, 35519}
X(52380) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 3936}, {55, 4053}, {58, 18593}, {60, 3218}, {80, 6358}, {81, 41804}, {261, 20924}, {270, 17923}, {284, 758}, {333, 35550}, {593, 1443}, {649, 51645}, {650, 6370}, {663, 2610}, {757, 17078}, {759, 226}, {1098, 32851}, {1172, 860}, {1333, 1464}, {1411, 6354}, {1474, 1835}, {1731, 31845}, {1793, 306}, {1807, 26942}, {2150, 36}, {2161, 12}, {2185, 320}, {2189, 1870}, {2194, 2245}, {2204, 44113}, {2222, 4605}, {2323, 4736}, {2326, 5081}, {2341, 10}, {2361, 35069}, {3063, 42666}, {3737, 4707}, {4636, 4585}, {6187, 2171}, {6740, 321}, {7054, 4511}, {7113, 3028}, {14616, 349}, {18178, 51465}, {18359, 34388}, {24624, 1441}, {30576, 41801}, {30606, 1227}, {32671, 109}, {34079, 65}, {35193, 42701}, {36069, 651}, {36910, 1089}, {37140, 664}
X(52380) = {X(6740),X(36927)}-harmonic conjugate of X(8)

X(52381) = X(19)-ISOCONJUGATE OF X(35)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(52381) lies on these lines: {2, 7110}, {21, 36}, {63, 11064}, {77, 6513}, {78, 1060}, {81, 24780}, {94, 2006}, {280, 10527}, {345, 6349}, {499, 15474}, {905, 16697}, {1214, 52351}, {1790, 4466}, {1791, 25909}, {1812, 4001}, {2071, 51883}, {2160, 2339}, {2886, 52002}, {3305, 7131}, {3580, 47057}, {4666, 7073}, {4847, 6742}, {5743, 11076}, {6757, 19854}, {7361, 27339}, {8818, 31266}, {10072, 19785}, {15314, 38822}, {15455, 36795}, {17923, 20565}, {25639, 41496}, {26700, 26703}, {27509, 30679}, {36100, 38340}

X(52381) = isotomic conjugate of the polar conjugate of X(79)
X(52381) = isogonal conjugate of the polar conjugate of X(20565)
X(52381) = X(20565)-Ceva conjugate of X(79)
X(52381) = X(i)-cross conjugate of X(j) for these (i,j): {3916, 69}, {18607, 63}, {22058, 3}, {37565, 77}
X(52381) = X(i)-isoconjugate of X(j) for these (i,j): {2, 14975}, {4, 2174}, {6, 6198}, {19, 35}, {25, 3219}, {29, 21741}, {33, 2003}, {41, 7282}, {65, 41502}, {108, 9404}, {186, 2161}, {225, 35192}, {250, 21824}, {270, 21794}, {281, 1399}, {284, 1825}, {319, 1973}, {607, 1442}, {608, 4420}, {943, 44095}, {1172, 2594}, {1395, 42033}, {1400, 11107}, {1474, 3678}, {1783, 2605}, {1824, 40214}, {1826, 17104}, {1844, 2259}, {1880, 35193}, {1974, 33939}, {2203, 3969}, {2204, 40999}, {2212, 17095}, {2299, 16577}, {5379, 20982}, {7265, 32676}, {8739, 46073}, {8740, 46077}, {8748, 22342}, {8750, 14838}, {8882, 35194}, {18359, 34397}, {32674, 35057}
X(52381) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 35}, {9, 6198}, {186, 40584}, {226, 16577}, {319, 6337}, {445, 16585}, {451, 7110}, {647, 21054}, {1825, 40590}, {1844, 18591}, {2174, 36033}, {2605, 39006}, {3160, 7282}, {3219, 6505}, {3678, 51574}, {4467, 40618}, {7265, 15526}, {9404, 38983}, {11107, 40582}, {14838, 26932}, {14975, 32664}, {35057, 35072}, {40602, 41502}
X(52381) = cevapoint of X(i) and X(j) for these (i,j): {3, 22122}, {905, 4466}
X(52381) = crosssum of X(607) and X(44097)
X(52381) = barycentric product X(i)*X(j) for these {i,j}: {3, 20565}, {36, 328}, {63, 30690}, {69, 79}, {75, 7100}, {77, 52344}, {94, 22128}, {265, 320}, {304, 2160}, {305, 6186}, {307, 3615}, {348, 7110}, {905, 15455}, {1441, 1789}, {1444, 6757}, {1812, 43682}, {4025, 6742}, {6063, 8606}, {6332, 38340}, {7073, 7182}, {8818, 17206}, {13486, 14208}, {26700, 35518}, {40075, 52153}, {40716, 50462}
X(52381) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6198}, {3, 35}, {7, 7282}, {21, 11107}, {31, 14975}, {36, 186}, {48, 2174}, {63, 3219}, {65, 1825}, {69, 319}, {72, 3678}, {73, 2594}, {77, 1442}, {78, 4420}, {79, 4}, {125, 21054}, {222, 2003}, {265, 80}, {283, 35193}, {284, 41502}, {304, 33939}, {306, 3969}, {307, 40999}, {320, 340}, {328, 20566}, {345, 42033}, {348, 17095}, {521, 35057}, {525, 7265}, {603, 1399}, {652, 9404}, {905, 14838}, {942, 1844}, {1062, 4354}, {1071, 41562}, {1214, 16577}, {1409, 21741}, {1437, 17104}, {1459, 2605}, {1789, 21}, {1790, 40214}, {2160, 19}, {2193, 35192}, {2197, 21794}, {2260, 44095}, {2523, 30600}, {3615, 29}, {3695, 7206}, {3708, 21824}, {3784, 7186}, {3916, 3647}, {3942, 7202}, {4001, 3578}, {4025, 4467}, {4292, 46468}, {4303, 500}, {4466, 8287}, {4707, 44427}, {5249, 445}, {6186, 25}, {6742, 1897}, {6757, 41013}, {7073, 33}, {7100, 1}, {7110, 281}, {8606, 55}, {8818, 1826}, {13486, 162}, {14844, 4213}, {15413, 18160}, {15419, 16755}, {15455, 6335}, {17206, 34016}, {17923, 14165}, {18210, 2611}, {18607, 16585}, {20565, 264}, {21828, 47230}, {22054, 17454}, {22128, 323}, {22341, 22342}, {23070, 35197}, {23224, 23226}, {26700, 108}, {30690, 92}, {38340, 653}, {43682, 40149}, {44706, 35194}, {50462, 484}, {52153, 6187}, {52202, 7150}, {52344, 318}, {52351, 41226}
X(52381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 30690, 7110}, {2, 41808, 16585}

X(52382) = X(21)-ISOCONJUGATE OF X(35)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(52382) lies on the cubic K360 and these lines: {1, 30}, {4, 36119}, {7, 40438}, {10, 6757}, {19, 1990}, {37, 8818}, {46, 50462}, {55, 37992}, {56, 759}, {57, 267}, {75, 3260}, {145, 6742}, {265, 18525}, {477, 36064}, {897, 9214}, {1247, 24789}, {1393, 4017}, {1406, 45923}, {1725, 22798}, {1837, 34301}, {1910, 35906}, {2153, 33654}, {2154, 2306}, {2166, 14254}, {2218, 6186}, {2363, 19785}, {2594, 6354}, {3017, 5221}, {3028, 5903}, {3336, 3471}, {3615, 3616}, {3648, 18625}, {4442, 34860}, {4934, 10896}, {6701, 18593}, {7178, 35347}, {8614, 14158}, {8773, 36891}, {11375, 14844}, {13610, 33149}, {15454, 36053}, {18398, 33642}, {18977, 51656}, {20840, 20988}, {23843, 52153}, {30690, 31359}

X(52382) = isogonal conjugate of X(35193)
X(52382) = X(43682)-Ceva conjugate of X(8818)
X(52382) = X(i)-cross conjugate of X(j) for these (i,j): {115, 7178}, {4017, 26700}, {6089, 476}, {42759, 10412}
X(52382) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35193}, {2, 35192}, {3, 11107}, {8, 17104}, {9, 40214}, {21, 35}, {41, 34016}, {58, 4420}, {60, 3678}, {63, 41502}, {110, 35057}, {186, 1793}, {283, 6198}, {284, 3219}, {319, 2194}, {323, 2341}, {332, 14975}, {333, 2174}, {643, 2605}, {662, 9404}, {1043, 1399}, {1098, 2594}, {1101, 6741}, {1175, 31938}, {1333, 42033}, {1442, 2328}, {2003, 2287}, {2150, 3969}, {3467, 35195}, {4282, 41226}, {5546, 14838}, {6149, 6740}, {7054, 16577}, {7058, 21741}, {17190, 33635}, {23226, 36797}, {35194, 35196}
X(52382) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 35193}, {10, 4420}, {35, 40611}, {37, 42033}, {244, 35057}, {319, 1214}, {478, 40214}, {523, 6741}, {1084, 9404}, {1442, 36908}, {2594, 15267}, {3160, 34016}, {3162, 41502}, {3219, 40590}, {4467, 40622}, {6148, 18593}, {6740, 14993}, {11107, 36103}, {16755, 40615}, {32664, 35192}
X(52382) = cevapoint of X(1365) and X(4017)
X(52382) = crosspoint of X(2166) and X(5627)
X(52382) = crosssum of X(1511) and X(6149)
X(52382) = trilinear pole of line {661, 1637}
X(52382) = barycentric product X(i)*X(j) for these {i,j}: {1, 43682}, {7, 8818}, {57, 6757}, {65, 30690}, {79, 226}, {94, 1464}, {349, 6186}, {523, 38340}, {1109, 35049}, {1400, 20565}, {1427, 52344}, {1441, 2160}, {1446, 7073}, {1577, 26700}, {1989, 41804}, {2166, 18593}, {3615, 6354}, {3668, 7110}, {4017, 15455}, {4466, 34922}, {6742, 7178}, {7100, 40149}, {32680, 51645}, {36064, 41079}
X(52382) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35193}, {7, 34016}, {10, 42033}, {12, 3969}, {19, 11107}, {25, 41502}, {31, 35192}, {37, 4420}, {56, 40214}, {65, 3219}, {79, 333}, {115, 6741}, {226, 319}, {512, 9404}, {604, 17104}, {661, 35057}, {1042, 2003}, {1254, 16577}, {1365, 8287}, {1400, 35}, {1402, 2174}, {1427, 1442}, {1441, 33939}, {1464, 323}, {1880, 6198}, {1989, 6740}, {2160, 21}, {2171, 3678}, {2294, 31938}, {3615, 7058}, {3649, 3578}, {3668, 17095}, {3676, 16755}, {4017, 14838}, {4077, 18160}, {6186, 284}, {6354, 40999}, {6742, 645}, {6757, 312}, {7073, 2287}, {7100, 1812}, {7110, 1043}, {7178, 4467}, {7180, 2605}, {8606, 2327}, {8818, 8}, {13486, 4612}, {15455, 7257}, {20565, 28660}, {20982, 3024}, {21773, 35195}, {26700, 662}, {30690, 314}, {32636, 17190}, {35049, 24041}, {36064, 44769}, {38340, 99}, {41804, 7799}, {43682, 75}, {51645, 32679}
X(52382) = {X(79),X(50148)}-harmonic conjugate of X(1)

X(52383) = X(21)-ISOCONJUGATE OF X(36)

Barycentrics    (a + b - c)*(a - b + c)*(b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(52383) lies on the conic {{A,B,C,X(4),X(5)}} and these lines: {1, 5}, {4, 2190}, {10, 15065}, {19, 53}, {37, 21011}, {56, 39270}, {65, 3120}, {75, 311}, {82, 17500}, {109, 13273}, {115, 4559}, {117, 5348}, {158, 13450}, {181, 994}, {201, 41501}, {219, 31141}, {244, 20118}, {267, 1710}, {484, 1263}, {596, 51975}, {655, 897}, {759, 859}, {921, 8800}, {969, 10401}, {1054, 24914}, {1086, 12832}, {1141, 36078}, {1331, 13272}, {1399, 3585}, {1464, 51421}, {1772, 12619}, {1836, 34300}, {1879, 21768}, {1910, 32675}, {2165, 2168}, {2170, 35307}, {2216, 40449}, {2217, 27622}, {2218, 6187}, {2363, 24624}, {2962, 18395}, {4674, 40663}, {5292, 18962}, {6702, 43048}, {6740, 26095}, {7052, 36301}, {8769, 27364}, {8818, 21741}, {9654, 45931}, {14628, 42285}, {17638, 35015}, {18359, 23541}, {18827, 35174}, {18961, 34030}, {19628, 23592}, {20616, 23903}, {21770, 36412}, {22765, 38954}, {23404, 38938}, {25882, 52351}, {33136, 36920}, {33655, 36300}, {36927, 47318}, {37759, 51562}

X(52383) = X(i)-Ceva conjugate of X(j) for these (i,j): {23592, 4559}, {34535, 2171}
X(52383) = X(i)-cross conjugate of X(j) for these (i,j): {2171, 34535}, {3724, 8818}, {4642, 1168}, {6089, 3952}, {40663, 14584}, {42666, 4559}, {42759, 523}
X(52383) = X(i)-isoconjugate of X(j) for these (i,j): {2, 4282}, {3, 17515}, {21, 36}, {58, 4511}, {60, 758}, {81, 2323}, {86, 2361}, {99, 8648}, {110, 3738}, {163, 3904}, {186, 1789}, {215, 14616}, {261, 3724}, {283, 1870}, {284, 3218}, {320, 2194}, {333, 7113}, {645, 21758}, {654, 662}, {759, 4996}, {1098, 1464}, {1172, 22128}, {1333, 32851}, {1437, 5081}, {1443, 2328}, {1983, 4560}, {2150, 3936}, {2185, 2245}, {2193, 17923}, {2311, 27950}, {2316, 17191}, {2600, 18315}, {3615, 6149}, {3960, 5546}, {4242, 23189}, {4575, 44428}, {4585, 7252}, {4612, 21828}, {6369, 36134}, {7054, 18593}, {22379, 36797}, {24624, 34544}
X(52383) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 4511}, {21, 15898}, {36, 40611}, {37, 32851}, {115, 3904}, {136, 44428}, {137, 6369}, {244, 3738}, {320, 1214}, {654, 1084}, {1043, 36909}, {1273, 16577}, {1443, 36908}, {1464, 15267}, {2323, 40586}, {2361, 40600}, {3218, 40590}, {3615, 14993}, {4282, 32664}, {4453, 40622}, {4996, 34586}, {8648, 38986}, {17515, 36103}, {17923, 47345}
X(52383) = cevapoint of X(i) and X(j) for these (i,j): {12, 40663}, {115, 42666}, {523, 45260}, {3120, 30572}, {3724, 21741}
X(52383) = crosspoint of X(i) and X(j) for these (i,j): {1141, 2166}, {2006, 18815}
X(52383) = crosssum of X(i) and X(j) for these (i,j): {1, 34465}, {58, 51966}, {1154, 6149}, {2323, 2361}
X(52383) = trilinear pole of line {661, 2171}
X(52383) = crossdifference of every pair of points on line {654, 34544}
X(52383) = barycentric product X(i)*X(j) for these {i,j}: {10, 2006}, {12, 24624}, {37, 18815}, {57, 15065}, {65, 18359}, {80, 226}, {85, 34857}, {94, 2594}, {225, 52351}, {321, 1411}, {349, 6187}, {512, 46405}, {523, 655}, {661, 35174}, {758, 34535}, {759, 6358}, {850, 32675}, {1020, 52356}, {1400, 20566}, {1441, 2161}, {1577, 2222}, {1807, 40149}, {1989, 40999}, {2166, 16577}, {2171, 14616}, {3668, 36910}, {4017, 36804}, {4080, 14584}, {4674, 14628}, {6354, 6740}, {7178, 51562}, {18314, 36078}, {34079, 34388}, {38955, 52212}
X(52383) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 32851}, {12, 3936}, {19, 17515}, {31, 4282}, {37, 4511}, {42, 2323}, {65, 3218}, {73, 22128}, {80, 333}, {181, 2245}, {213, 2361}, {225, 17923}, {226, 320}, {349, 40075}, {512, 654}, {523, 3904}, {655, 99}, {661, 3738}, {759, 2185}, {798, 8648}, {1254, 18593}, {1284, 27950}, {1319, 17191}, {1400, 36}, {1402, 7113}, {1411, 81}, {1427, 1443}, {1441, 20924}, {1807, 1812}, {1826, 5081}, {1880, 1870}, {1989, 3615}, {2006, 86}, {2161, 21}, {2171, 758}, {2222, 662}, {2245, 4996}, {2341, 1098}, {2501, 44428}, {2594, 323}, {3668, 17078}, {3724, 34544}, {4017, 3960}, {4551, 4585}, {6187, 284}, {6354, 41804}, {6358, 35550}, {6740, 7058}, {7178, 4453}, {8736, 860}, {12077, 6369}, {14584, 16704}, {14628, 30939}, {15065, 312}, {18359, 314}, {18815, 274}, {20566, 28660}, {21741, 6149}, {24624, 261}, {32675, 110}, {34079, 60}, {34535, 14616}, {34857, 9}, {35174, 799}, {36078, 18315}, {36804, 7257}, {36910, 1043}, {40663, 51583}, {40999, 7799}, {42759, 46398}, {46405, 670}, {51562, 645}, {51642, 21758}, {52212, 17139}, {52351, 332}
X(52383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7951, 45944}, {80, 1411, 14584}, {80, 2006, 1411}, {17734, 38945, 5172}

X(52384) = X(21)-ISOCONJUGATE OF X(40)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3) : :

X(52384) lies on these lines: {1, 84}, {4, 43058}, {10, 227}, {19, 56}, {33, 1035}, {34, 7008}, {37, 73}, {57, 37818}, {65, 2357}, {75, 280}, {158, 278}, {189, 31359}, {225, 1427}, {282, 8583}, {285, 40430}, {759, 8059}, {897, 37141}, {958, 30674}, {1104, 7151}, {1212, 7367}, {1319, 2208}, {1410, 1824}, {1426, 18210}, {1451, 7118}, {1468, 2219}, {1838, 7681}, {1880, 7157}, {2182, 7114}, {2188, 2646}, {3198, 22341}, {3668, 52078}, {6705, 17102}, {7335, 10537}, {8769, 9376}, {10200, 37695}, {12688, 51662}, {15836, 34052}, {20991, 22654}, {34434, 42549}

X(52384) = isotomic conjugate of the polar conjugate of X(2358)
X(52384) = polar conjugate of the isotomic conjugate of X(52037)
X(52384) = X(i)-Ceva conjugate of X(j) for these (i,j): {280, 52078}, {1440, 8808}, {8808, 1903}, {39130, 65}
X(52384) = X(i)-cross conjugate of X(j) for these (i,j): {1042, 65}, {1880, 1427}, {2357, 1903}, {23663, 16606}
X(52384) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1819}, {6, 27398}, {8, 2360}, {9, 1817}, {21, 40}, {29, 7078}, {55, 8822}, {58, 7080}, {60, 21075}, {78, 3194}, {81, 2324}, {86, 7074}, {110, 8058}, {196, 2327}, {198, 333}, {208, 1792}, {219, 41083}, {221, 1043}, {223, 2287}, {227, 1098}, {283, 7952}, {284, 329}, {285, 1103}, {314, 2187}, {322, 2194}, {332, 3195}, {347, 2328}, {643, 6129}, {648, 10397}, {662, 14298}, {1434, 7368}, {1444, 40971}, {1812, 2331}, {2185, 21871}, {2322, 7011}, {3342, 13614}, {4183, 7013}, {4570, 38357}, {5546, 14837}, {7070, 41082}
X(52384) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 27398}, {10, 7080}, {40, 40611}, {223, 8822}, {227, 15267}, {244, 8058}, {322, 1214}, {329, 40590}, {347, 36908}, {478, 1817}, {1043, 3341}, {1084, 14298}, {1819, 36033}, {2324, 40586}, {6527, 8808}, {7074, 40600}, {17896, 40622}, {38357, 50330}
X(52384) = cevapoint of X(i) and X(j) for these (i,j): {4017, 18210}, {4516, 7180}
X(52384) = crosspoint of X(i) and X(j) for these (i,j): {1, 3346}, {84, 40836}, {278, 8809}, {1422, 1440}
X(52384) = crosssum of X(i) and X(j) for these (i,j): {1, 1498}, {40, 7078}, {219, 7070}, {2324, 7074}
X(52384) = barycentric product X(i)*X(j) for these {i,j}: {1, 8808}, {4, 52037}, {7, 1903}, {10, 1422}, {21, 13853}, {37, 1440}, {57, 39130}, {65, 189}, {69, 2358}, {84, 226}, {85, 2357}, {225, 41081}, {273, 41087}, {280, 1427}, {282, 3668}, {285, 6354}, {307, 7129}, {309, 1400}, {321, 1413}, {349, 2208}, {523, 37141}, {1042, 34404}, {1172, 6355}, {1214, 40836}, {1231, 7151}, {1402, 44190}, {1426, 44189}, {1433, 40149}, {1436, 1441}, {1439, 7003}, {1446, 2192}, {1577, 8059}, {1817, 7157}, {1824, 34400}, {2184, 52078}, {3701, 6612}, {4017, 44327}, {4077, 36049}, {7178, 13138}, {17094, 40117}
X(52384) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 27398}, {34, 41083}, {37, 7080}, {42, 2324}, {48, 1819}, {56, 1817}, {57, 8822}, {65, 329}, {84, 333}, {181, 21871}, {189, 314}, {213, 7074}, {226, 322}, {268, 1792}, {282, 1043}, {285, 7058}, {309, 28660}, {512, 14298}, {604, 2360}, {608, 3194}, {661, 8058}, {810, 10397}, {1042, 223}, {1400, 40}, {1402, 198}, {1409, 7078}, {1410, 7011}, {1413, 81}, {1422, 86}, {1426, 196}, {1427, 347}, {1433, 1812}, {1436, 21}, {1440, 274}, {1880, 7952}, {1903, 8}, {2171, 21075}, {2188, 2327}, {2192, 2287}, {2208, 284}, {2333, 40971}, {2357, 9}, {2358, 4}, {3125, 38357}, {3668, 40702}, {4017, 14837}, {4516, 5514}, {6355, 1231}, {6612, 1014}, {7008, 2322}, {7118, 2328}, {7129, 29}, {7151, 1172}, {7154, 4183}, {7178, 17896}, {7180, 6129}, {8059, 662}, {8808, 75}, {13138, 645}, {13853, 1441}, {18210, 16596}, {32652, 5546}, {36049, 643}, {37141, 99}, {39130, 312}, {40117, 36797}, {40836, 31623}, {41081, 332}, {41086, 27382}, {41087, 78}, {42549, 17183}, {44190, 40072}, {44327, 7257}, {52037, 69}, {52078, 18750}
X(52384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 84, 2192}, {1, 41402, 1455}, {1, 47848, 221}, {84, 1422, 1413}, {1214, 5930, 227}

X(52385) = X(25)-ISOCONJUGATE OF X(29)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 - c^2)^2 : :

X(52385) lies on these lines: {2, 9119}, {7, 8}, {21, 1442}, {27, 1943}, {63, 77}, {72, 307}, {78, 7013}, {210, 40999}, {226, 17052}, {272, 18191}, {296, 332}, {306, 51368}, {326, 1259}, {329, 1032}, {343, 5249}, {347, 3869}, {379, 2262}, {517, 18655}, {651, 5279}, {664, 8822}, {758, 3668}, {857, 1903}, {912, 41004}, {916, 1071}, {927, 2749}, {1014, 1812}, {1020, 21078}, {1037, 1402}, {1264, 4176}, {1426, 10381}, {1813, 2327}, {1824, 5929}, {1876, 10477}, {1880, 40802}, {2003, 23130}, {2893, 7282}, {3419, 10400}, {3664, 18389}, {3682, 7138}, {3879, 16465}, {3896, 30619}, {3962, 41804}, {3998, 40152}, {4329, 6001}, {4552, 21871}, {5130, 10364}, {5273, 37669}, {5570, 24179}, {5693, 41010}, {5738, 44547}, {5739, 46017}, {5887, 41007}, {6180, 34377}, {7190, 11520}, {9965, 20110}, {14110, 17134}, {14872, 21270}, {15656, 18161}, {16368, 45126}, {28786, 28787}, {30621, 37593}, {34384, 44139}

X(52385) = isotomic conjugate of X(1896)
X(52385) = anticomplement of X(9119)
X(52385) = isotomic conjugate of the isogonal conjugate of X(22341)
X(52385) = isotomic conjugate of the polar conjugate of X(1214)
X(52385) = isogonal conjugate of the polar conjugate of X(1231)
X(52385) = X(i)-Ceva conjugate of X(j) for these (i,j): {69, 307}, {1231, 1214}, {6517, 4131}, {7183, 40152}, {34401, 226}
X(52385) = X(i)-cross conjugate of X(j) for these (i,j): {3682, 3998}, {7066, 40152}, {17216, 4131}, {22341, 1214}
X(52385) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2299}, {6, 8748}, {9, 5317}, {19, 1172}, {21, 1096}, {25, 29}, {27, 607}, {28, 33}, {31, 1896}, {34, 4183}, {55, 8747}, {58, 1857}, {86, 6059}, {92, 2204}, {107, 663}, {112, 3064}, {158, 2194}, {162, 18344}, {270, 1824}, {278, 2332}, {281, 1474}, {283, 6524}, {284, 393}, {286, 2212}, {318, 2203}, {333, 2207}, {522, 32713}, {608, 2322}, {650, 24019}, {652, 6529}, {823, 3063}, {1039, 4206}, {1043, 7337}, {1118, 2328}, {1334, 36419}, {1396, 7079}, {1400, 36421}, {1826, 2189}, {1880, 2326}, {1946, 36126}, {1973, 31623}, {1974, 44130}, {2193, 6520}, {2321, 36420}, {2333, 46103}, {3194, 7008}, {4516, 24000}, {6514, 36434}, {6525, 52158}, {7007, 8885}, {7077, 34856}, {7154, 41083}, {14400, 32695}, {14569, 35196}, {17926, 32674}, {21044, 23964}, {21789, 36127}, {28660, 36417}, {32676, 44426}, {36124, 37908}, {40574, 41320}
X(52385) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1896}, {4, 226}, {6, 1172}, {9, 8748}, {10, 1857}, {21, 6503}, {29, 6505}, {33, 40591}, {63, 3559}, {125, 18344}, {158, 1214}, {223, 8747}, {281, 51574}, {314, 6338}, {393, 40590}, {478, 5317}, {650, 35071}, {663, 38985}, {823, 10001}, {942, 1859}, {1096, 40611}, {1118, 36908}, {1147, 2194}, {1715, 40152}, {1946, 46093}, {2193, 37867}, {2204, 22391}, {2299, 36033}, {3064, 34591}, {4183, 11517}, {6059, 40600}, {6337, 31623}, {6520, 47345}, {6708, 42385}, {15352, 39060}, {15526, 44426}, {17926, 35072}, {36126, 39053}, {36421, 40582}
X(52385) = cevapoint of X(i) and X(j) for these (i,j): {65, 8807}, {3682, 40152}
X(52385) = crosspoint of X(i) and X(j) for these (i,j): {69, 326}, {7055, 7183}
X(52385) = crosssum of X(i) and X(j) for these (i,j): {25, 1096}, {4516, 18344}
X(52385) = crossdifference of every pair of points on line {3063, 18344}
X(52385) = barycentric product X(i)*X(j) for these {i,j}: {3, 1231}, {7, 3998}, {10, 7183}, {37, 7055}, {63, 307}, {65, 3926}, {69, 1214}, {71, 7182}, {72, 348}, {73, 304}, {75, 40152}, {76, 22341}, {77, 306}, {85, 3682}, {108, 4143}, {201, 17206}, {222, 20336}, {225, 1102}, {226, 326}, {255, 349}, {274, 7066}, {305, 1409}, {313, 7125}, {321, 1804}, {332, 37755}, {345, 1439}, {394, 1441}, {520, 4554}, {525, 6516}, {603, 40071}, {651, 3265}, {664, 24018}, {822, 4572}, {1259, 1446}, {1264, 1427}, {1332, 17094}, {1367, 4567}, {1444, 26942}, {1577, 6517}, {1792, 20618}, {1812, 6356}, {1813, 14208}, {1880, 4176}, {1978, 51641}, {2632, 4620}, {3267, 36059}, {3668, 3719}, {3694, 7056}, {3710, 7177}, {3964, 40149}, {3990, 6063}, {4055, 20567}, {4131, 4552}, {4551, 30805}, {4561, 51640}, {4564, 17216}, {7138, 28660}, {7335, 27801}, {15413, 23067}, {18604, 34388}, {27832, 52354}
X(52385) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8748}, {2, 1896}, {3, 1172}, {21, 36421}, {37, 1857}, {48, 2299}, {56, 5317}, {57, 8747}, {63, 29}, {65, 393}, {69, 31623}, {71, 33}, {72, 281}, {73, 19}, {77, 27}, {78, 2322}, {108, 6529}, {109, 24019}, {184, 2204}, {201, 1826}, {212, 2332}, {213, 6059}, {219, 4183}, {222, 28}, {225, 6520}, {226, 158}, {228, 607}, {255, 284}, {283, 2326}, {304, 44130}, {306, 318}, {307, 92}, {326, 333}, {348, 286}, {394, 21}, {520, 650}, {521, 17926}, {525, 44426}, {577, 2194}, {603, 1474}, {647, 18344}, {651, 107}, {653, 36126}, {656, 3064}, {664, 823}, {822, 663}, {828, 11436}, {836, 30223}, {1014, 36419}, {1020, 36127}, {1092, 2193}, {1102, 332}, {1214, 4}, {1231, 264}, {1259, 2287}, {1332, 36797}, {1367, 16732}, {1400, 1096}, {1402, 2207}, {1408, 36420}, {1409, 25}, {1410, 608}, {1415, 32713}, {1425, 1880}, {1427, 1118}, {1429, 34856}, {1437, 2189}, {1439, 278}, {1441, 2052}, {1444, 46103}, {1790, 270}, {1804, 81}, {1813, 162}, {1880, 6524}, {2197, 1824}, {2200, 2212}, {2286, 4206}, {2289, 2328}, {2318, 7079}, {2632, 21044}, {3173, 30733}, {3265, 4391}, {3269, 4516}, {3682, 9}, {3694, 7046}, {3710, 7101}, {3719, 1043}, {3926, 314}, {3964, 1812}, {3990, 55}, {3998, 8}, {4047, 461}, {4055, 41}, {4091, 3737}, {4131, 4560}, {4143, 35518}, {4158, 3694}, {4303, 46884}, {4554, 6528}, {4620, 23999}, {6356, 40149}, {6505, 3559}, {6507, 283}, {6511, 3193}, {6514, 1098}, {6516, 648}, {6517, 662}, {6518, 15146}, {7011, 3194}, {7013, 41083}, {7053, 1396}, {7055, 274}, {7066, 37}, {7125, 58}, {7138, 1400}, {7182, 44129}, {7183, 86}, {7335, 1333}, {8807, 6523}, {14208, 46110}, {16730, 1364}, {17094, 17924}, {17216, 4858}, {18026, 15352}, {18210, 8735}, {18591, 1859}, {18592, 42385}, {18604, 60}, {20336, 7017}, {20752, 37908}, {20769, 14024}, {22057, 2082}, {22128, 17515}, {22129, 17519}, {22341, 6}, {23067, 1783}, {23144, 4233}, {23224, 7252}, {23620, 40987}, {24018, 522}, {26942, 41013}, {30456, 6525}, {30805, 18155}, {32320, 1946}, {32660, 32676}, {36054, 21789}, {36059, 112}, {37755, 225}, {39201, 3063}, {39791, 1841}, {40149, 1093}, {40152, 1}, {41087, 7008}, {41393, 1865}, {44717, 5379}, {51640, 7649}, {51641, 649}, {52037, 40836}
X(52385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 6505, 21482}, {72, 1439, 307}, {326, 7183, 1804}, {2893, 16091, 7282}

X(52386) = X(27)-ISOCONJUGATE OF X(28)

Barycentrics    a^2*(b + c)^2*(a^2 - b^2 - c^2)^2 : :

X(52386) lies on these lines: {2, 40422}, {6, 1260}, {10, 37}, {32, 2911}, {71, 228}, {72, 18591}, {78, 216}, {101, 38868}, {115, 41508}, {201, 2197}, {219, 1794}, {220, 2301}, {306, 18592}, {393, 7080}, {577, 1259}, {836, 3682}, {1018, 2324}, {1331, 22133}, {1818, 14597}, {1865, 17757}, {2281, 7109}, {2289, 35072}, {2295, 3553}, {3969, 25000}, {7368, 30457}, {17796, 41332}, {21031, 21860}, {21075, 21854}, {25091, 25964}

X(52386) = isogonal conjugate of X(36419)
X(52386) = isotomic conjugate of the polar conjugate of X(3690)
X(52386) = isogonal conjugate of the polar conjugate of X(3695)
X(52386) = polar conjugate of the isotomic conjugate of X(4158)
X(52386) = X(i)-Ceva conjugate of X(j) for these (i,j): {1252, 4574}, {3694, 3949}, {3695, 3690}
X(52386) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36419}, {27, 28}, {29, 1396}, {34, 46103}, {75, 36420}, {81, 8747}, {86, 5317}, {107, 1019}, {158, 593}, {162, 17925}, {244, 23582}, {269, 36421}, {270, 278}, {273, 2189}, {286, 1474}, {393, 757}, {811, 43925}, {823, 3733}, {849, 2052}, {873, 2207}, {1014, 8748}, {1015, 23999}, {1086, 24000}, {1096, 1509}, {1111, 23964}, {1118, 2185}, {1119, 2326}, {1412, 1896}, {2203, 44129}, {3942, 32230}, {4211, 40411}, {7192, 24019}, {7199, 32713}, {7254, 36126}, {24033, 26856}, {34856, 37128}, {39179, 46151}, {40395, 46883}, {42067, 46254}
X(52386) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 36419}, {27, 40591}, {125, 17925}, {206, 36420}, {286, 51574}, {393, 40607}, {521, 26856}, {525, 23989}, {593, 1147}, {647, 2973}, {1019, 38985}, {1509, 6503}, {1565, 17434}, {1896, 40599}, {2052, 4075}, {5317, 40600}, {6600, 36421}, {7192, 35071}, {7254, 46093}, {8747, 40586}, {11517, 46103}, {17423, 43925}
X(52386) = crosspoint of X(i) and X(j) for these (i,j): {72, 40161}, {1252, 4574}, {3682, 3998}
X(52386) = crosssum of X(i) and X(j) for these (i,j): {6, 1612}, {1086, 17925}, {2969, 43925}, {5317, 8747}
X(52386) = crossdifference of every pair of points on line {3733, 17925}
X(52386) = barycentric product X(i)*X(j) for these {i,j}: {3, 3695}, {4, 4158}, {8, 7066}, {10, 3682}, {12, 1259}, {37, 3998}, {48, 52369}, {59, 7068}, {63, 3949}, {69, 3690}, {71, 306}, {72, 72}, {73, 3710}, {78, 201}, {181, 1264}, {219, 26942}, {228, 20336}, {255, 1089}, {307, 2318}, {313, 4055}, {321, 3990}, {326, 756}, {341, 7138}, {345, 2197}, {394, 594}, {520, 3952}, {525, 4574}, {577, 28654}, {765, 2632}, {822, 4033}, {1016, 3269}, {1018, 24018}, {1092, 7141}, {1110, 17879}, {1214, 3694}, {1252, 15526}, {1260, 6356}, {1265, 1425}, {1331, 4064}, {1367, 6065}, {1500, 3926}, {1804, 6057}, {2171, 3719}, {2200, 40071}, {2289, 6358}, {2321, 40152}, {2972, 15742}, {3265, 4557}, {3692, 37755}, {3701, 22341}, {3964, 7140}, {4091, 4103}, {4131, 40521}, {6056, 34388}, {7055, 7064}, {23067, 52355}, {23990, 36793}, {27808, 39201}, {40161, 51574}
X(52386) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 36419}, {32, 36420}, {42, 8747}, {71, 27}, {72, 286}, {125, 2973}, {181, 1118}, {201, 273}, {210, 1896}, {212, 270}, {213, 5317}, {219, 46103}, {220, 36421}, {228, 28}, {255, 757}, {306, 44129}, {326, 873}, {394, 1509}, {520, 7192}, {577, 593}, {594, 2052}, {647, 17925}, {756, 158}, {765, 23999}, {822, 1019}, {872, 1096}, {1018, 823}, {1110, 24000}, {1252, 23582}, {1259, 261}, {1264, 18021}, {1334, 8748}, {1409, 1396}, {1425, 1119}, {1500, 393}, {1802, 2326}, {1804, 552}, {2197, 278}, {2200, 1474}, {2289, 2185}, {2318, 29}, {2632, 1111}, {2972, 1565}, {3049, 43925}, {3269, 1086}, {3682, 86}, {3690, 4}, {3694, 31623}, {3695, 264}, {3710, 44130}, {3747, 34856}, {3949, 92}, {3952, 6528}, {3990, 81}, {3998, 274}, {4055, 58}, {4064, 46107}, {4158, 69}, {4557, 107}, {4574, 648}, {6056, 60}, {7064, 1857}, {7065, 1364}, {7066, 7}, {7068, 34387}, {7109, 2207}, {7138, 269}, {7140, 1093}, {7335, 7341}, {15526, 23989}, {18604, 763}, {20727, 31917}, {20975, 2969}, {22080, 31900}, {22341, 1014}, {22363, 4211}, {23990, 23964}, {24018, 7199}, {26942, 331}, {28654, 18027}, {32320, 7254}, {34980, 3937}, {35072, 26856}, {37754, 3942}, {37755, 1847}, {39201, 3733}, {40152, 1434}, {51641, 7203}, {52369, 1969}
X(52386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 21858, 1834}, {72, 51574, 18591}

X(52387) = X(28)-ISOCONJUGATE OF X(28)

Barycentrics    a*(b + c)^2*(a^2 - b^2 - c^2)^2 : :

X(52387) lies on these lines: {1, 1257}, {10, 321}, {31, 3811}, {71, 72}, {78, 1794}, {100, 2939}, {200, 1005}, {201, 3695}, {255, 3719}, {345, 44706}, {872, 1245}, {1215, 3085}, {1259, 24031}, {2198, 3930}, {3556, 4557}, {3670, 20106}, {3682, 3998}, {3693, 44547}, {3842, 19855}, {3932, 21916}, {3952, 7080}, {3965, 45120}, {4073, 4294}, {4158, 7066}, {4303, 25083}, {4574, 7078}, {4712, 5904}, {5295, 40967}, {7283, 44694}

X(52387) = isotomic conjugate of the polar conjugate of X(3949)
X(52387) = isogonal conjugate of the polar conjugate of X(52369)
X(52387) = X(i)-Ceva conjugate of X(j) for these (i,j): {3710, 3695}, {52369, 3949}
X(52387) = X(3269)-cross conjugate of X(24018)
X(52387) = X(i)-isoconjugate of X(j) for these (i,j): {2, 36420}, {6, 36419}, {27, 1474}, {28, 28}, {34, 270}, {58, 8747}, {60, 1118}, {81, 5317}, {107, 3733}, {112, 17925}, {158, 849}, {244, 24000}, {250, 2969}, {261, 7337}, {278, 2189}, {286, 2203}, {393, 593}, {552, 6059}, {608, 46103}, {648, 43925}, {741, 34856}, {757, 1096}, {1015, 23582}, {1019, 24019}, {1086, 23964}, {1172, 1396}, {1407, 36421}, {1408, 1896}, {1412, 8748}, {1435, 2326}, {1509, 2207}, {1857, 7341}, {3248, 23999}, {3937, 32230}, {6529, 7254}, {7192, 32713}, {18020, 42067}, {23985, 26856}, {23989, 41937}, {40395, 46890}
X(52387) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 36419}, {10, 8747}, {27, 51574}, {28, 40591}, {158, 4075}, {270, 11517}, {525, 1111}, {757, 6503}, {849, 1147}, {873, 6338}, {942, 46883}, {1019, 35071}, {1096, 40607}, {3733, 38985}, {3942, 17434}, {5317, 40586}, {8299, 34856}, {8748, 40599}, {17925, 34591}, {24771, 36421}, {32664, 36420}
X(52387) = barycentric product X(i)*X(j) for these {i,j}: {3, 52369}, {10, 3998}, {12, 3719}, {63, 3695}, {69, 3949}, {71, 20336}, {72, 306}, {78, 26942}, {92, 4158}, {201, 345}, {228, 40071}, {255, 28654}, {304, 3690}, {307, 3694}, {312, 7066}, {313, 3990}, {321, 3682}, {326, 594}, {394, 1089}, {520, 4033}, {756, 3926}, {765, 15526}, {822, 27808}, {1016, 2632}, {1018, 3265}, {1102, 7140}, {1110, 36793}, {1214, 3710}, {1231, 2318}, {1252, 17879}, {1259, 6358}, {1264, 2171}, {1265, 37755}, {1332, 4064}, {2197, 3718}, {2289, 34388}, {3269, 7035}, {3692, 6356}, {3701, 40152}, {3952, 24018}, {4055, 27801}, {4103, 4131}, {4564, 7068}, {4574, 14208}, {6057, 7183}, {6507, 7141}, {22341, 30713}, {30805, 40521}
X(52387) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36419}, {31, 36420}, {37, 8747}, {42, 5317}, {71, 28}, {72, 27}, {73, 1396}, {78, 46103}, {200, 36421}, {201, 278}, {210, 8748}, {212, 2189}, {219, 270}, {228, 1474}, {255, 593}, {306, 286}, {326, 1509}, {394, 757}, {520, 1019}, {577, 849}, {594, 158}, {656, 17925}, {756, 393}, {765, 23582}, {810, 43925}, {822, 3733}, {872, 2207}, {1016, 23999}, {1018, 107}, {1089, 2052}, {1110, 23964}, {1252, 24000}, {1259, 2185}, {1260, 2326}, {1425, 1435}, {1500, 1096}, {2171, 1118}, {2197, 34}, {2200, 2203}, {2238, 34856}, {2289, 60}, {2318, 1172}, {2321, 1896}, {2632, 1086}, {2972, 3942}, {3265, 7199}, {3269, 244}, {3682, 81}, {3690, 19}, {3694, 29}, {3695, 92}, {3708, 2969}, {3710, 31623}, {3719, 261}, {3926, 873}, {3949, 4}, {3952, 823}, {3958, 31900}, {3990, 58}, {3998, 86}, {4033, 6528}, {4047, 31903}, {4055, 1333}, {4064, 17924}, {4158, 63}, {4557, 24019}, {4574, 162}, {6056, 2150}, {6356, 1847}, {7066, 57}, {7068, 4858}, {7125, 7341}, {7138, 1407}, {7140, 6520}, {7141, 6521}, {7183, 552}, {15526, 1111}, {17216, 16727}, {17879, 23989}, {18591, 46883}, {20336, 44129}, {20902, 2973}, {21859, 36127}, {22341, 1412}, {23620, 4211}, {24018, 7192}, {24031, 26856}, {26942, 273}, {35309, 46151}, {37754, 3937}, {37755, 1119}, {40152, 1014}, {52369, 264}

X(52388) = X(28)-ISOCONJUGATE OF X(35)

Barycentrics    (b + c)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + c^2) : :

X(52388) lies on these lines: {2, 582}, {3, 125}, {8, 6739}, {9, 46}, {10, 6757}, {40, 30447}, {63, 50462}, {72, 21912}, {78, 1060}, {318, 860}, {355, 36195}, {381, 30436}, {474, 24931}, {476, 36154}, {499, 8286}, {500, 3580}, {517, 27685}, {656, 44706}, {858, 48882}, {956, 26700}, {1063, 44113}, {1376, 34829}, {1479, 8287}, {1836, 14873}, {2166, 18395}, {3419, 34301}, {3579, 27553}, {3679, 50148}, {3708, 23899}, {3753, 27714}, {5657, 27686}, {5690, 36155}, {6361, 27554}, {7420, 20299}, {12699, 27555}, {13747, 25669}, {16297, 24206}, {16451, 23293}, {16452, 26913}, {16453, 21243}, {18749, 20565}, {20840, 32223}, {23555, 51462}, {23674, 33298}, {25015, 30690}, {25442, 37527}, {25646, 46623}, {26446, 27687}, {26958, 37284}, {27529, 37154}, {27721, 50810}, {37648, 50324}, {41586, 48907}, {48928, 51360}

X(52388) = isotomic conjugate of the polar conjugate of X(8818)
X(52388) = X(52344)-Ceva conjugate of X(6757)
X(52388) = X(i)-cross conjugate of X(j) for these (i,j): {3958, 306}, {18675, 226}
X(52388) = X(i)-isoconjugate of X(j) for these (i,j): {4, 17104}, {19, 40214}, {27, 2174}, {28, 35}, {29, 1399}, {34, 35193}, {56, 11107}, {57, 41502}, {58, 6198}, {60, 1825}, {86, 14975}, {107, 23226}, {112, 14838}, {162, 2605}, {186, 759}, {250, 2611}, {270, 2594}, {278, 35192}, {319, 2203}, {1172, 2003}, {1175, 1844}, {1442, 2299}, {1474, 3219}, {1973, 34016}, {2189, 16577}, {2194, 7282}, {2204, 17095}, {4467, 32676}, {9274, 35235}, {14616, 34397}, {16585, 40570}, {21741, 46103}, {22094, 24000}, {32671, 44427}, {37140, 47230}
X(52388) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 11107}, {6, 40214}, {10, 6198}, {35, 40591}, {125, 2605}, {186, 34586}, {226, 1442}, {340, 51583}, {445, 40937}, {500, 942}, {647, 8287}, {1214, 7282}, {3219, 51574}, {4467, 15526}, {5452, 41502}, {6337, 34016}, {7359, 14920}, {11517, 35193}, {14838, 34591}, {14975, 40600}, {16755, 40618}, {17104, 36033}, {23226, 38985}
X(52388) = cevapoint of X(125) and X(656)
X(52388) = crosssum of X(2605) and X(22094)
X(52388) = crossdifference of every pair of points on line {2605, 47230}
X(52388) = barycentric product X(i)*X(j) for these {i,j}: {63, 6757}, {69, 8818}, {71, 20565}, {72, 30690}, {78, 43682}, {79, 306}, {265, 3936}, {307, 7110}, {321, 7100}, {328, 2245}, {349, 8606}, {525, 6742}, {656, 15455}, {1214, 52344}, {1231, 7073}, {1789, 6358}, {2160, 20336}, {3615, 26942}, {6186, 40071}, {38340, 52355}
X(52388) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 40214}, {9, 11107}, {37, 6198}, {48, 17104}, {55, 41502}, {69, 34016}, {71, 35}, {72, 3219}, {73, 2003}, {79, 27}, {125, 8287}, {201, 16577}, {212, 35192}, {213, 14975}, {219, 35193}, {226, 7282}, {228, 2174}, {265, 24624}, {306, 319}, {307, 17095}, {442, 445}, {525, 4467}, {647, 2605}, {656, 14838}, {822, 23226}, {860, 14165}, {1214, 1442}, {1409, 1399}, {1789, 2185}, {1901, 46468}, {2160, 28}, {2171, 1825}, {2197, 2594}, {2245, 186}, {2294, 1844}, {3269, 22094}, {3615, 46103}, {3694, 4420}, {3695, 3969}, {3708, 2611}, {3710, 42033}, {3916, 17190}, {3936, 340}, {3949, 3678}, {3958, 3647}, {4025, 16755}, {4064, 7265}, {6186, 1474}, {6370, 44427}, {6739, 14920}, {6742, 648}, {6757, 92}, {7073, 1172}, {7100, 81}, {7110, 29}, {8606, 284}, {8611, 35057}, {8818, 4}, {14208, 18160}, {14844, 2905}, {15455, 811}, {18210, 7202}, {18591, 500}, {20336, 33939}, {20565, 44129}, {20727, 7186}, {20902, 17886}, {20975, 20982}, {21046, 21054}, {21134, 21141}, {22080, 17454}, {26942, 40999}, {30690, 286}, {32662, 36069}, {36061, 37140}, {40952, 44095}, {41014, 3578}, {42666, 47230}, {43261, 31901}, {43682, 273}, {52153, 34079}, {52344, 31623}

X(52389) = X(28)-ISOCONJUGATE OF X(40)

Barycentrics    a*(b + c)*(a^2 - b^2 - c^2)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3) : :

X(52389) lies on the cubic K345 and these lines: {1, 1073}, {2, 280}, {3, 9}, {10, 227}, {37, 46837}, {58, 36049}, {72, 856}, {78, 271}, {97, 44687}, {123, 6831}, {189, 24635}, {210, 22341}, {226, 8806}, {281, 3346}, {307, 6355}, {405, 7008}, {939, 1256}, {946, 16596}, {960, 46021}, {997, 2192}, {1038, 1422}, {1210, 2968}, {1297, 40117}, {1427, 20309}, {1440, 8813}, {1753, 3149}, {1793, 1800}, {2184, 3182}, {2188, 5440}, {3682, 3694}, {3710, 3998}, {3718, 3926}, {5266, 52041}, {5703, 24553}, {5705, 37565}, {6700, 7515}, {6762, 38290}, {6765, 38288}, {7003, 40937}, {7151, 37065}, {13138, 14919}, {14376, 37592}, {16573, 16583}, {17073, 19843}, {18446, 46881}, {18634, 20208}, {18876, 32652}, {20764, 34790}, {21075, 51368}, {25917, 40946}, {35072, 46830}, {36055, 40396}

X(52389) = isogonal conjugate of X(3194)
X(52389) = complement of X(1895)
X(52389) = complement of the isogonal conjugate of X(19614)
X(52389) = complement of the isotomic conjugate of X(19611)
X(52389) = isotomic conjugate of the anticomplement of X(46837)
X(52389) = isotomic conjugate of the polar conjugate of X(1903)
X(52389) = medial-isogonal conjugate of X(20308)
X(52389) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 20308}, {3, 2883}, {6, 20207}, {25, 20265}, {32, 46829}, {48, 36908}, {64, 5}, {184, 1249}, {253, 21243}, {520, 35968}, {1073, 141}, {1092, 31377}, {1301, 520}, {1974, 20313}, {2155, 226}, {2184, 20305}, {8798, 1209}, {11589, 113}, {14379, 3}, {14642, 2}, {15394, 1368}, {17510, 14390}, {19611, 2887}, {19614, 10}, {30457, 41883}, {33581, 6}, {34403, 626}, {39201, 39020}, {41489, 13567}, {42658, 13613}, {44181, 46185}, {46639, 30476}, {47437, 3343}, {52158, 34831}
X(52389) = X(i)-Ceva conjugate of X(j) for these (i,j): {280, 39130}, {41081, 41087}
X(52389) = X(i)-cross conjugate of X(j) for these (i,j): {37, 1214}, {73, 72}, {18675, 63}, {41087, 52037}, {46837, 2}
X(52389) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3194}, {4, 2360}, {6, 41083}, {19, 1817}, {21, 208}, {25, 8822}, {27, 198}, {28, 40}, {29, 221}, {58, 7952}, {81, 2331}, {86, 3195}, {112, 14837}, {162, 6129}, {196, 284}, {204, 41082}, {223, 1172}, {227, 270}, {286, 2187}, {322, 2203}, {329, 1474}, {333, 3209}, {342, 2194}, {347, 2299}, {608, 27398}, {1014, 40971}, {1118, 1819}, {1396, 2324}, {1437, 47372}, {1896, 7114}, {2199, 31623}, {2204, 40702}, {2322, 6611}, {2332, 14256}, {3342, 8885}, {4570, 38362}, {7011, 8748}, {7078, 8747}, {17896, 32676}, {41088, 44698}
X(52389) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3194}, {6, 1817}, {9, 41083}, {10, 7952}, {29, 3341}, {40, 40591}, {125, 6129}, {196, 40590}, {208, 40611}, {226, 347}, {329, 51574}, {342, 1214}, {2331, 40586}, {2360, 36033}, {3195, 40600}, {3343, 41082}, {6505, 8822}, {14837, 34591}, {15526, 17896}, {38362, 50330}
X(52389) = cevapoint of X(1903) and X(41086)
X(52389) = crosspoint of X(i) and X(j) for these (i,j): {2, 19611}, {271, 280}
X(52389) = crosssum of X(i) and X(j) for these (i,j): {6, 204}, {208, 221}, {2331, 3195}
X(52389) = trilinear pole of line {520, 8611}
X(52389) = barycentric product X(i)*X(j) for these {i,j}: {8, 52037}, {10, 41081}, {63, 39130}, {65, 44189}, {69, 1903}, {71, 309}, {72, 189}, {73, 34404}, {75, 41087}, {78, 8808}, {84, 306}, {210, 34400}, {226, 271}, {228, 44190}, {268, 1441}, {280, 1214}, {282, 307}, {285, 26942}, {304, 2357}, {321, 1433}, {349, 2188}, {525, 13138}, {656, 44327}, {1231, 2192}, {1264, 2358}, {1422, 3710}, {1436, 20336}, {1440, 3694}, {1792, 13853}, {2208, 40071}, {2287, 6355}, {3265, 40117}, {3267, 32652}, {3998, 40836}, {7020, 40152}, {14208, 36049}, {34403, 41086}, {37141, 52355}
X(52389) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 41083}, {3, 1817}, {6, 3194}, {37, 7952}, {42, 2331}, {48, 2360}, {63, 8822}, {65, 196}, {71, 40}, {72, 329}, {73, 223}, {78, 27398}, {84, 27}, {189, 286}, {213, 3195}, {226, 342}, {228, 198}, {268, 21}, {271, 333}, {280, 31623}, {282, 29}, {285, 46103}, {306, 322}, {307, 40702}, {309, 44129}, {525, 17896}, {647, 6129}, {656, 14837}, {1073, 41082}, {1214, 347}, {1334, 40971}, {1400, 208}, {1402, 3209}, {1409, 221}, {1410, 6611}, {1413, 1396}, {1433, 81}, {1436, 28}, {1439, 14256}, {1441, 40701}, {1826, 47372}, {1903, 4}, {2188, 284}, {2192, 1172}, {2197, 227}, {2200, 2187}, {2208, 1474}, {2289, 1819}, {2318, 2324}, {2357, 19}, {2358, 1118}, {3125, 38362}, {3690, 21871}, {3694, 7080}, {3949, 21075}, {3990, 7078}, {6355, 1446}, {7003, 1896}, {7008, 8748}, {7118, 2299}, {7129, 8747}, {7151, 5317}, {7367, 4183}, {8611, 8058}, {8808, 273}, {13138, 648}, {22341, 7011}, {32652, 112}, {34404, 44130}, {36049, 162}, {39130, 92}, {40117, 107}, {40152, 7013}, {41081, 86}, {41086, 1249}, {41087, 1}, {44189, 314}, {44327, 811}, {52037, 7}, {52078, 44697}
X(52389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 280, 40836}, {271, 41081, 1433}

X(52390) = X(29)-ISOCONJUGATE OF X(35)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(52390) lies on the Jerabek circumhyperbola and these lines: {1, 74}, {3, 7100}, {4, 79}, {6, 1406}, {7, 8044}, {54, 3336}, {56, 34435}, {57, 1175}, {64, 7073}, {71, 41393}, {72, 21912}, {656, 14380}, {1020, 2599}, {1789, 4652}, {1903, 8818}, {2457, 3657}, {3426, 7986}, {3431, 18593}, {3485, 14844}, {3615, 37142}, {3649, 27555}, {3754, 6757}, {5903, 50148}, {6742, 14923}, {9275, 43700}, {9405, 13746}, {10693, 27685}, {11101, 13486}, {11270, 37571}, {13369, 34800}, {15453, 51645}, {22342, 37755}, {23070, 43704}, {31794, 44835}, {37732, 43855}, {48857, 51223}

X(52390) = isogonal conjugate of X(11107)
X(52390) = isogonal conjugate of the polar conjugate of X(43682)
X(52390) = X(i)-cross conjugate of X(j) for these (i,j): {3708, 51640}, {39791, 73}
X(52390) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11107}, {2, 41502}, {4, 35193}, {21, 6198}, {28, 4420}, {29, 35}, {92, 35192}, {162, 35057}, {186, 6740}, {250, 6741}, {270, 3678}, {281, 40214}, {314, 14975}, {318, 17104}, {319, 2299}, {607, 34016}, {648, 9404}, {1098, 1825}, {1172, 3219}, {1442, 4183}, {1474, 42033}, {2003, 2322}, {2174, 31623}, {2189, 3969}, {2204, 33939}, {2326, 16577}, {2328, 7282}, {2332, 17095}, {2605, 36797}
X(52390) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11107}, {125, 35057}, {226, 319}, {942, 31938}, {1825, 15267}, {4420, 40591}, {6198, 40611}, {7282, 36908}, {22391, 35192}, {32664, 41502}, {35193, 36033}, {42033, 51574}
X(52390) = crosspoint of X(79) and X(7100)
X(52390) = crosssum of X(i) and X(j) for these (i,j): {35, 6198}, {186, 35201}
X(52390) = trilinear pole of line {647, 2631}
X(52390) = barycentric product X(i)*X(j) for these {i,j}: {3, 43682}, {73, 30690}, {77, 8818}, {79, 1214}, {125, 35049}, {222, 6757}, {226, 7100}, {265, 18593}, {307, 2160}, {525, 26700}, {656, 38340}, {1231, 6186}, {1409, 20565}, {1439, 7110}, {1446, 8606}, {1789, 6354}, {3615, 37755}, {6742, 51640}
X(52390) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11107}, {31, 41502}, {48, 35193}, {71, 4420}, {72, 42033}, {73, 3219}, {77, 34016}, {79, 31623}, {184, 35192}, {201, 3969}, {307, 33939}, {603, 40214}, {647, 35057}, {810, 9404}, {1214, 319}, {1400, 6198}, {1409, 35}, {1410, 2003}, {1425, 16577}, {1427, 7282}, {1439, 17095}, {1789, 7058}, {1835, 14165}, {2160, 29}, {2197, 3678}, {3708, 6741}, {6186, 1172}, {6757, 7017}, {7073, 2322}, {7100, 333}, {8606, 2287}, {8818, 318}, {14582, 52356}, {17094, 18160}, {18591, 31938}, {18593, 340}, {26700, 648}, {30690, 44130}, {35049, 18020}, {37755, 40999}, {38340, 811}, {39791, 16585}, {43682, 264}, {50433, 1793}, {51640, 4467}, {51645, 44427}, {52153, 2341}
X(52390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {79, 34301, 4}, {3649, 40669, 27555}, {7100, 50462, 3}

X(52391) = X(29)-ISOCONJUGATE OF X(36)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(52391) lies on the Jerabek circumhyperbola and these lines: {1, 54}, {3, 201}, {4, 80}, {6, 1411}, {59, 7984}, {64, 37567}, {65, 3120}, {69, 17880}, {73, 18210}, {74, 484}, {290, 35174}, {655, 6740}, {758, 15065}, {759, 1175}, {1243, 45022}, {1246, 18815}, {1903, 21853}, {2006, 5292}, {2341, 46884}, {2595, 13746}, {2992, 36929}, {2993, 36928}, {3145, 14882}, {3431, 16577}, {4551, 43692}, {4552, 6224}, {7066, 43708}, {10260, 15530}, {10457, 46160}, {10950, 51879}, {11270, 37572}, {14528, 34471}, {23071, 43704}, {34259, 52351}, {40663, 50705}, {41329, 43712}, {41538, 43703}

X(52391) = isogonal conjugate of X(17515)
X(52391) = X(2631)-cross conjugate of X(1020)
X(52391) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17515}, {21, 1870}, {27, 2323}, {28, 4511}, {29, 36}, {58, 5081}, {60, 860}, {92, 4282}, {110, 44428}, {112, 3904}, {162, 3738}, {186, 3615}, {261, 44113}, {270, 758}, {284, 17923}, {286, 2361}, {320, 2299}, {648, 654}, {811, 8648}, {933, 6369}, {1098, 1835}, {1172, 3218}, {1443, 4183}, {1474, 32851}, {2189, 3936}, {2204, 20924}, {2245, 46103}, {2326, 18593}, {2332, 17078}, {3737, 4242}, {7113, 31623}, {8748, 22128}
X(52391) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 17515}, {10, 5081}, {29, 15898}, {125, 3738}, {226, 320}, {244, 44428}, {1835, 15267}, {1870, 40611}, {2600, 15450}, {3904, 34591}, {4282, 22391}, {4511, 40591}, {8648, 17423}, {17923, 40590}, {32851, 51574}
X(52391) = crosspoint of X(80) and X(1807)
X(52391) = crosssum of X(i) and X(j) for these (i,j): {36, 1870}, {186, 51801}
X(52391) = trilinear pole of line {647, 2197}
X(52391) = crossdifference of every pair of points on line {2600, 3738}
X(52391) = barycentric product X(i)*X(j) for these {i,j}: {65, 52351}, {71, 18815}, {72, 2006}, {73, 18359}, {80, 1214}, {94, 22342}, {201, 24624}, {222, 15065}, {226, 1807}, {265, 16577}, {306, 1411}, {307, 2161}, {328, 21741}, {348, 34857}, {525, 2222}, {647, 35174}, {655, 656}, {759, 26942}, {810, 46405}, {1231, 6187}, {1409, 20566}, {1439, 36910}, {1793, 6354}, {2197, 14616}, {2341, 6356}, {6740, 37755}, {14208, 32675}, {51562, 51640}
X(52391) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17515}, {37, 5081}, {65, 17923}, {71, 4511}, {72, 32851}, {73, 3218}, {80, 31623}, {184, 4282}, {201, 3936}, {228, 2323}, {307, 20924}, {647, 3738}, {655, 811}, {656, 3904}, {661, 44428}, {759, 46103}, {810, 654}, {1214, 320}, {1231, 40075}, {1400, 1870}, {1409, 36}, {1411, 27}, {1425, 18593}, {1439, 17078}, {1793, 7058}, {1807, 333}, {1825, 14165}, {2006, 286}, {2161, 29}, {2171, 860}, {2197, 758}, {2200, 2361}, {2222, 648}, {2599, 14918}, {3049, 8648}, {4559, 4242}, {6187, 1172}, {15065, 7017}, {15451, 2600}, {16577, 340}, {18359, 44130}, {18815, 44129}, {21741, 186}, {22341, 22128}, {22342, 323}, {23067, 4585}, {26942, 35550}, {32675, 162}, {34079, 270}, {34857, 281}, {35174, 6331}, {37755, 41804}, {50433, 1789}, {51640, 4453}, {52351, 314}
X(52391) = {X(80),X(34300)}-harmonic conjugate of X(4)

X(52392) = X(33)-ISOCONJUGATE OF X(36)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :
X(52392) = 5 X[31266] - 4 X[36949]

X(52392) lies on cubic K660 and these lines: {1, 14516}, {7, 80}, {63, 343}, {69, 17880}, {77, 1060}, {81, 226}, {86, 39277}, {189, 5905}, {265, 1439}, {286, 7282}, {307, 1444}, {320, 18816}, {515, 22464}, {527, 655}, {535, 36589}, {759, 13395}, {969, 10401}, {1411, 7190}, {1813, 4466}, {1814, 9028}, {2161, 5845}, {2222, 9436}, {2250, 24029}, {2283, 24713}, {3738, 36038}, {4318, 29046}, {5219, 16554}, {8048, 34242}, {10446, 34300}, {12831, 14204}, {14584, 30617}, {20078, 41226}, {31164, 52212}, {31266, 36949}, {34393, 51565}

X(52392) = midpoint of X(5905) and X(37781)
X(52392) = reflection of X(i) in X(j) for these {i,j}: {63, 26932}, {651, 226}
X(52392) = isotomic conjugate of X(5081)
X(52392) = isotomic conjugate of the anticomplement of X(46974)
X(52392) = isotomic conjugate of the polar conjugate of X(2006)
X(52392) = X(2006)-anticomplementary conjugate of X(151)
X(52392) = X(14616)-Ceva conjugate of X(18815)
X(52392) = X(i)-cross conjugate of X(j) for these (i,j): {912, 63}, {1807, 52351}, {2850, 651}, {46974, 2}
X(52392) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2361}, {19, 2323}, {21, 44113}, {25, 4511}, {29, 3724}, {31, 5081}, {33, 36}, {41, 17923}, {42, 17515}, {55, 1870}, {186, 7073}, {281, 7113}, {320, 2212}, {607, 3218}, {654, 1783}, {663, 4242}, {692, 44428}, {758, 2299}, {860, 2194}, {1172, 2245}, {1443, 7071}, {1464, 4183}, {1826, 4282}, {1835, 2328}, {1845, 2342}, {1897, 8648}, {1973, 32851}, {1983, 3064}, {2189, 4053}, {2204, 3936}, {2332, 18593}, {3738, 8750}, {34397, 52344}
X(52392) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5081}, {6, 2323}, {33, 15898}, {223, 1870}, {226, 758}, {654, 39006}, {860, 1214}, {1086, 44428}, {1835, 36908}, {2361, 36033}, {3160, 17923}, {3738, 26932}, {3904, 40618}, {4511, 6505}, {6337, 32851}, {7046, 36909}, {8648, 34467}, {17515, 40592}, {40611, 44113}
X(52392) = cevapoint of X(i) and X(j) for these (i,j): {226, 515}, {26932, 39471}
X(52392) = crosssum of X(3724) and X(44113)
X(52392) = trilinear pole of line {905, 1214}
X(52392) = barycentric product X(i)*X(j) for these {i,j}: {7, 52351}, {63, 18815}, {69, 2006}, {77, 18359}, {80, 348}, {85, 1807}, {222, 20566}, {265, 17095}, {304, 1411}, {307, 24624}, {328, 2003}, {655, 4025}, {759, 1231}, {905, 35174}, {1214, 14616}, {1446, 1793}, {1459, 46405}, {2161, 7182}, {2222, 15413}, {7056, 36910}, {17094, 47318}
X(52392) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5081}, {3, 2323}, {7, 17923}, {48, 2361}, {57, 1870}, {63, 4511}, {69, 32851}, {73, 2245}, {77, 3218}, {80, 281}, {81, 17515}, {201, 4053}, {222, 36}, {226, 860}, {265, 7110}, {307, 3936}, {348, 320}, {514, 44428}, {603, 7113}, {651, 4242}, {655, 1897}, {759, 1172}, {905, 3738}, {1214, 758}, {1231, 35550}, {1400, 44113}, {1409, 3724}, {1411, 19}, {1427, 1835}, {1437, 4282}, {1439, 18593}, {1459, 654}, {1465, 1845}, {1793, 2287}, {1804, 22128}, {1807, 9}, {2003, 186}, {2006, 4}, {2161, 33}, {2222, 1783}, {2341, 4183}, {4025, 3904}, {6187, 607}, {6516, 4585}, {6740, 2322}, {7056, 17078}, {7177, 1443}, {7182, 20924}, {7282, 14165}, {14584, 8756}, {14616, 31623}, {14628, 38462}, {17094, 4707}, {17095, 340}, {18359, 318}, {18815, 92}, {20566, 7017}, {22128, 4996}, {22383, 8648}, {23071, 26744}, {24624, 29}, {32675, 8750}, {34079, 2299}, {35174, 6335}, {36059, 1983}, {36910, 7046}, {47318, 36797}, {50433, 8606}, {52212, 1785}, {52351, 8}

X(52393) = X(35)-ISOCONJUGATE OF X(37)

Barycentrics    (a + b)*(a + c)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(52393) lies on these lines: {2, 8818}, {7, 40214}, {21, 36}, {57, 24624}, {60, 24470}, {81, 553}, {86, 17190}, {99, 18139}, {110, 11246}, {265, 28452}, {270, 31900}, {333, 2160}, {354, 5196}, {476, 840}, {497, 41504}, {554, 7051}, {593, 1086}, {662, 17483}, {757, 33150}, {1029, 8287}, {1081, 19373}, {1171, 17366}, {1262, 6354}, {1931, 26724}, {1989, 37646}, {2163, 23681}, {2185, 8025}, {2363, 19785}, {3100, 7073}, {3873, 6742}, {3972, 19684}, {4720, 17647}, {5745, 7110}, {10404, 37405}, {11076, 17012}, {13407, 37294}, {14377, 51311}, {17103, 32774}, {18601, 40432}, {19787, 20565}, {24617, 32911}, {32636, 37369}, {32680, 37222}

X(52393) = isotomic conjugate of X(3969)
X(52393) = X(35049)-Ceva conjugate of X(38340)
X(52393) = X(i)-cross conjugate of X(j) for these (i,j): {11, 7192}, {3337, 757}, {24470, 7}, {29821, 873}, {33135, 32010}
X(52393) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3678}, {8, 21741}, {9, 2594}, {10, 2174}, {12, 35192}, {21, 21794}, {31, 3969}, {35, 37}, {41, 40999}, {42, 3219}, {50, 15065}, {55, 16577}, {71, 6198}, {201, 41502}, {210, 2003}, {213, 319}, {219, 1825}, {281, 22342}, {306, 14975}, {323, 34857}, {594, 17104}, {692, 7265}, {756, 40214}, {765, 20982}, {872, 34016}, {1018, 2605}, {1110, 8287}, {1252, 2611}, {1333, 7206}, {1334, 1442}, {1399, 2321}, {1400, 4420}, {1402, 42033}, {1918, 33939}, {2149, 6741}, {2171, 35193}, {2197, 11107}, {2624, 51562}, {3724, 41226}, {4551, 9404}, {4557, 14838}, {4559, 35057}, {4570, 21824}, {6187, 42701}, {7161, 50657}, {7343, 21864}, {14270, 36804}, {17886, 23990}
X(52393) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3969}, {9, 3678}, {35, 40589}, {37, 7206}, {223, 16577}, {319, 6626}, {478, 2594}, {513, 20982}, {514, 8287}, {650, 6741}, {661, 2611}, {1086, 7265}, {3160, 40999}, {3219, 40592}, {4420, 40582}, {4467, 40620}, {4988, 21054}, {7110, 21081}, {21794, 40611}, {21824, 50330}, {33939, 34021}, {40605, 42033}, {40612, 42701}
X(52393) = cevapoint of X(i) and X(j) for these (i,j): {1, 583}, {79, 2160}, {1019, 1086}
X(52393) = trilinear pole of line {3737, 4960}
X(52393) = barycentric product X(i)*X(j) for these {i,j}: {7, 3615}, {58, 20565}, {79, 86}, {81, 30690}, {273, 1789}, {274, 2160}, {286, 7100}, {310, 6186}, {476, 4453}, {693, 13486}, {757, 6757}, {1014, 52344}, {1019, 15455}, {1434, 7110}, {1509, 8818}, {2185, 43682}, {3960, 32680}, {4560, 38340}, {4858, 35049}, {6742, 7192}, {14158, 40716}, {14844, 40164}, {17219, 34922}, {18155, 26700}
X(52393) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3678}, {2, 3969}, {7, 40999}, {10, 7206}, {11, 6741}, {21, 4420}, {28, 6198}, {34, 1825}, {56, 2594}, {57, 16577}, {58, 35}, {60, 35193}, {79, 10}, {81, 3219}, {86, 319}, {244, 2611}, {270, 11107}, {274, 33939}, {333, 42033}, {476, 51562}, {514, 7265}, {593, 40214}, {603, 22342}, {604, 21741}, {849, 17104}, {1014, 1442}, {1015, 20982}, {1019, 14838}, {1086, 8287}, {1111, 17886}, {1333, 2174}, {1393, 2599}, {1400, 21794}, {1408, 1399}, {1412, 2003}, {1434, 17095}, {1509, 34016}, {1789, 78}, {2150, 35192}, {2160, 37}, {2166, 15065}, {2189, 41502}, {2203, 14975}, {2594, 7144}, {3120, 21054}, {3125, 21824}, {3218, 42701}, {3615, 8}, {3733, 2605}, {3737, 35057}, {3937, 22094}, {3960, 32679}, {4453, 3268}, {4960, 23883}, {5358, 4354}, {6186, 42}, {6545, 21141}, {6742, 3952}, {6757, 1089}, {7073, 210}, {7100, 72}, {7110, 2321}, {7192, 4467}, {7199, 18160}, {7252, 9404}, {8025, 3578}, {8606, 2318}, {8818, 594}, {11076, 21864}, {13486, 100}, {14158, 484}, {14844, 21085}, {15455, 4033}, {16726, 7202}, {18180, 35194}, {20565, 313}, {21758, 2624}, {24624, 41226}, {26700, 4551}, {30581, 17190}, {30602, 502}, {30690, 321}, {32680, 36804}, {35049, 4564}, {38340, 4552}, {43682, 6358}, {46883, 1844}, {46890, 44095}, {52344, 3701}
X(52393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {57, 43682, 38340}, {553, 18653, 81}

X(52394) = X(37)-ISOCONJUGATE OF X(39)

Barycentrics    (a + b)*(a^2 + b^2)*(a + c)*(a^2 + c^2) : :

X(52394) lies on the conic {{A,B,C,X(2),X(7)}} and these lines: {2, 32}, {7, 1397}, {31, 75}, {58, 310}, {81, 335}, {86, 2206}, {199, 51862}, {273, 1395}, {314, 39281}, {673, 4599}, {675, 827}, {689, 727}, {871, 985}, {903, 4577}, {1088, 1106}, {1176, 1246}, {1268, 18082}, {1509, 39734}, {2296, 46289}, {3405, 21376}, {4593, 31002}, {4610, 7303}, {6384, 7121}, {6650, 33150}, {14621, 32774}, {15523, 33954}, {16889, 33730}, {17103, 40038}, {17141, 17150}, {17200, 29654}, {17763, 18099}, {18089, 27483}, {18824, 42371}, {33955, 39728}

X(52394) = isogonal conjugate of X(21035)
X(52394) = isotomic conjugate of X(15523)
X(52394) = polar conjugate of X(21016)
X(52394) = isogonal conjugate of the complement of X(17142)
X(52394) = isotomic conjugate of the anticomplement of X(29654)
X(52394) = isotomic conjugate of the complement of X(17150)
X(52394) = X(4593)-Ceva conjugate of X(39179)
X(52394) = X(i)-cross conjugate of X(j) for these (i,j): {171, 81}, {649, 4610}, {4911, 286}, {16706, 274}, {17200, 86}, {23790, 664}, {23791, 43190}, {24169, 39747}, {29654, 2}, {39179, 4593}
X(52394) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21035}, {2, 21814}, {6, 3954}, {10, 1964}, {31, 15523}, {37, 39}, {38, 42}, {48, 21016}, {65, 3688}, {71, 17442}, {72, 1843}, {75, 41267}, {100, 3005}, {101, 8061}, {141, 213}, {190, 2084}, {210, 1401}, {226, 40972}, {228, 427}, {256, 40936}, {257, 21752}, {291, 4093}, {313, 1923}, {321, 3051}, {512, 4553}, {649, 35309}, {661, 46148}, {668, 688}, {692, 826}, {756, 17187}, {798, 4568}, {872, 16887}, {893, 16587}, {1018, 21123}, {1400, 33299}, {1402, 3703}, {1500, 16696}, {1634, 4705}, {1824, 3917}, {1826, 4020}, {1918, 1930}, {2200, 20883}, {2205, 8024}, {2530, 4557}, {3108, 21817}, {3952, 50521}, {3990, 27376}, {4041, 46153}, {4576, 50487}, {4730, 46162}, {5360, 20021}, {6386, 9494}, {7109, 16703}, {8041, 18098}, {16030, 21807}, {16720, 40729}, {19606, 22028}, {20336, 27369}, {20683, 46149}, {20775, 41013}, {21355, 21880}, {21805, 46150}, {21818, 40432}, {21839, 46154}, {24290, 46163}, {27801, 41331}, {41272, 42713}
X(52394) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 15523}, {3, 21035}, {9, 3954}, {10, 41884}, {38, 40592}, {39, 40589}, {83, 21083}, {141, 6626}, {206, 41267}, {826, 1086}, {1015, 8061}, {1249, 21016}, {1930, 34021}, {2525, 40618}, {3005, 8054}, {3688, 40602}, {3703, 40605}, {4093, 39029}, {4553, 39054}, {4568, 31998}, {4988, 39691}, {5375, 35309}, {6292, 21038}, {16587, 40597}, {16892, 40620}, {21814, 32664}, {33299, 40582}, {36830, 46148}, {40625, 48278}
X(52394) = cevapoint of X(i) and X(j) for these (i,j): {2, 17150}, {6, 16683}, {58, 86}, {82, 83}, {1621, 32911}, {3120, 48101}
X(52394) = crosssum of X(37) and X(21881)
X(52394) = trilinear pole of line {514, 1919}
X(52394) = crossdifference of every pair of points on line {2084, 3005}
X(52394) = barycentric product X(i)*X(j) for these {i,j}: {27, 1799}, {58, 308}, {81, 3112}, {82, 274}, {83, 86}, {99, 10566}, {251, 310}, {261, 18097}, {286, 34055}, {350, 39276}, {513, 4593}, {514, 4577}, {649, 689}, {667, 37204}, {668, 39179}, {693, 4599}, {799, 18108}, {827, 3261}, {873, 18098}, {1176, 44129}, {1333, 18833}, {1509, 18082}, {1790, 46104}, {1919, 42371}, {2206, 40016}, {4025, 42396}, {4107, 41209}, {4594, 18111}, {4620, 18101}, {6385, 46289}, {17167, 39287}, {17168, 39289}, {17200, 40425}, {17206, 32085}, {18089, 40439}, {18091, 40409}, {19786, 39281}, {21110, 33515}, {34072, 40495}
X(52394) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3954}, {2, 15523}, {4, 21016}, {6, 21035}, {21, 33299}, {27, 427}, {28, 17442}, {31, 21814}, {32, 41267}, {58, 39}, {81, 38}, {82, 37}, {83, 10}, {86, 141}, {99, 4568}, {100, 35309}, {110, 46148}, {171, 16587}, {172, 40936}, {251, 42}, {274, 1930}, {284, 3688}, {286, 20883}, {308, 313}, {310, 8024}, {333, 3703}, {513, 8061}, {514, 826}, {593, 17187}, {649, 3005}, {662, 4553}, {667, 2084}, {689, 1978}, {757, 16696}, {827, 101}, {873, 16703}, {1019, 2530}, {1176, 71}, {1333, 1964}, {1412, 1401}, {1434, 3665}, {1437, 4020}, {1474, 1843}, {1509, 16887}, {1790, 3917}, {1799, 306}, {1914, 4093}, {1919, 688}, {2194, 40972}, {2206, 3051}, {3112, 321}, {3120, 39691}, {3261, 23285}, {3589, 21038}, {3733, 21123}, {4025, 2525}, {4556, 1634}, {4560, 48278}, {4565, 46153}, {4577, 190}, {4580, 4064}, {4591, 46162}, {4593, 668}, {4599, 100}, {4610, 4576}, {4628, 4557}, {4630, 32739}, {4750, 14424}, {4972, 21037}, {5299, 20969}, {6629, 7813}, {7122, 21752}, {7191, 17456}, {7192, 16892}, {7199, 48084}, {7293, 22077}, {8747, 27376}, {10547, 2200}, {10566, 523}, {16706, 21249}, {16707, 20898}, {16735, 21336}, {16887, 7794}, {16889, 16886}, {16890, 16894}, {16891, 16893}, {16892, 2528}, {17103, 16720}, {17186, 23208}, {17187, 8041}, {17200, 6292}, {17206, 3933}, {17469, 21817}, {17500, 21011}, {17925, 21108}, {18070, 4036}, {18082, 594}, {18083, 21028}, {18084, 21015}, {18086, 21031}, {18087, 3925}, {18088, 21029}, {18089, 21020}, {18091, 21024}, {18092, 21022}, {18095, 21023}, {18096, 20655}, {18097, 12}, {18098, 756}, {18099, 21021}, {18101, 21044}, {18102, 21025}, {18103, 20484}, {18105, 4079}, {18107, 21051}, {18108, 661}, {18109, 21026}, {18110, 21052}, {18111, 2533}, {18112, 21057}, {18113, 21950}, {18653, 51360}, {18703, 21692}, {18709, 21678}, {18833, 27801}, {20964, 21818}, {21178, 23881}, {27067, 20653}, {28724, 3682}, {32085, 1826}, {33940, 21425}, {33955, 17192}, {34055, 72}, {34072, 692}, {34294, 21043}, {37204, 6386}, {38946, 21064}, {39179, 513}, {39276, 291}, {41284, 18097}, {41629, 4884}, {41884, 21083}, {42396, 1897}, {44129, 1235}, {46288, 1918}, {46289, 213}, {47652, 21125}, {48060, 3806}, {51370, 51371}

X(52395) = X(38)-ISOCONJUGATE OF X(39)

Barycentrics    (a^2 + b^2)^2*(a^2 + c^2)^2 : :

X(52395) lies on these lines: {2, 40425}, {6, 4577}, {32, 26192}, {82, 20964}, {83, 316}, {141, 40000}, {251, 308}, {264, 42396}, {384, 28677}, {428, 32085}, {689, 39449}, {733, 8265}, {827, 12150}, {1031, 9483}, {1176, 42299}, {2980, 7856}, {3329, 46228}, {5480, 8928}, {6664, 7760}, {6665, 35137}, {7787, 9233}, {7827, 38946}, {7878, 14247}, {16095, 18907}, {18082, 21094}, {18097, 41284}, {20981, 39179}, {20990, 36081}, {26195, 28672}

X(52395) = isogonal conjugate of X(8041)
X(52395) = isotomic conjugate of X(7794)
X(52395) = isogonal conjugate of the complement of X(33798)
X(52395) = isotomic conjugate of the anticomplement of X(7829)
X(52395) = isotomic conjugate of the complement of X(7760)
X(52395) = X(i)-cross conjugate of X(j) for these (i,j): {7829, 2}, {18105, 4577}, {31299, 99}
X(52395) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8041}, {31, 7794}, {38, 39}, {41, 41285}, {141, 1964}, {163, 2528}, {427, 4020}, {799, 2531}, {1101, 15449}, {1401, 33299}, {1634, 8061}, {1923, 8024}, {1930, 3051}, {1973, 4175}, {2084, 4576}, {2530, 46148}, {3665, 40972}, {3917, 17442}, {3954, 17187}, {4553, 21123}, {4568, 50521}, {16696, 21035}, {16703, 41267}, {16887, 21814}, {20775, 20883}
X(52395) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7794}, {3, 8041}, {115, 2528}, {141, 41884}, {523, 15449}, {2531, 38996}, {3160, 41285}, {4175, 6337}
X(52395) = cevapoint of X(i) and X(j) for these (i,j): {2, 7760}, {6, 35222}, {32, 5012}, {83, 251}
X(52395) = crosspoint of X(83) and X(40163)
X(52395) = crosssum of X(i) and X(j) for these (i,j): {2, 46715}, {6, 35214}
X(52395) = trilinear pole of line {5027, 7927}
X(52395) = barycentric product X(i)*X(j) for these {i,j}: {8, 41284}, {82, 3112}, {83, 83}, {251, 308}, {689, 18105}, {1176, 46104}, {1799, 32085}, {4580, 42396}, {4599, 18070}, {17500, 39287}, {18833, 46289}, {30505, 41296}, {40016, 46288}, {40163, 41884}
X(52395) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7794}, {6, 8041}, {7, 41285}, {69, 4175}, {82, 38}, {83, 141}, {115, 15449}, {251, 39}, {308, 8024}, {428, 28666}, {523, 2528}, {669, 2531}, {827, 1634}, {1176, 3917}, {1799, 3933}, {3112, 1930}, {4577, 4576}, {4580, 2525}, {4628, 46148}, {5012, 52042}, {7760, 6665}, {8024, 14125}, {10547, 20775}, {10566, 16892}, {16890, 16893}, {18082, 15523}, {18098, 3954}, {18105, 3005}, {18108, 2530}, {20022, 51371}, {22105, 14424}, {32085, 427}, {33632, 3787}, {34294, 39691}, {40163, 33665}, {41284, 7}, {42037, 8362}, {42396, 41676}, {46104, 1235}, {46288, 3051}, {46289, 1964}
X(52395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {83, 41884, 3589}, {83, 46288, 41296}, {40000, 40850, 141}

X(52396) = X(25)-ISOCONJUGATE OF X(28)

Barycentrics    (b + c)*(-a^2 + b^2 + c^2)^2 : :

X(52396) lies on these lines: {10, 75}, {58, 3879}, {63, 69}, {86, 2983}, {101, 2366}, {190, 4150}, {201, 307}, {319, 1043}, {326, 1264}, {332, 1794}, {579, 3912}, {714, 23663}, {1210, 18147}, {1265, 19611}, {1918, 4028}, {2893, 7283}, {3006, 17220}, {3610, 4019}, {3668, 35550}, {3701, 40999}, {3948, 24005}, {4358, 5740}, {4427, 20291}, {5271, 19793}, {5736, 33113}, {5738, 17776}, {6527, 52366}, {6734, 44140}, {7270, 8822}, {17862, 24986}, {18734, 25083}, {41004, 41507}

X(52396) = isotomic conjugate of X(8747)
X(52396) = isotomic conjugate of the isogonal conjugate of X(3682)
X(52396) = isotomic conjugate of the polar conjugate of X(306)
X(52396) = isogonal conjugate of the polar conjugate of X(40071)
X(52396) = X(i)-Ceva conjugate of X(j) for these (i,j): {3718, 20336}, {3926, 3998}, {4600, 4561}, {40071, 306}
X(52396) = X(i)-cross conjugate of X(j) for these (i,j): {3682, 306}, {17216, 3265}
X(52396) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2203}, {6, 5317}, {19, 1474}, {21, 7337}, {25, 28}, {27, 1973}, {29, 1395}, {31, 8747}, {34, 2299}, {37, 36420}, {58, 1096}, {81, 2207}, {107, 667}, {112, 6591}, {158, 2206}, {213, 36419}, {274, 36417}, {278, 2204}, {286, 1974}, {393, 1333}, {513, 32713}, {604, 8748}, {607, 1396}, {608, 1172}, {649, 24019}, {823, 1919}, {1014, 6059}, {1118, 2194}, {1397, 1896}, {1398, 4183}, {1408, 1857}, {1435, 2332}, {1437, 6524}, {1783, 43925}, {1841, 40570}, {1880, 2189}, {1911, 34856}, {1980, 6528}, {3121, 23582}, {3122, 24000}, {3125, 23964}, {3194, 7151}, {4206, 51686}, {5379, 42067}, {6529, 22383}, {7649, 32676}, {14399, 32695}, {16732, 41937}, {18604, 36434}
X(52396) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 8747}, {6, 1474}, {9, 5317}, {10, 1096}, {19, 51574}, {25, 40591}, {27, 6337}, {28, 6505}, {34, 226}, {37, 393}, {58, 6503}, {86, 6338}, {107, 6631}, {158, 40603}, {525, 3120}, {649, 35071}, {667, 38985}, {823, 9296}, {1118, 1214}, {1147, 2206}, {1714, 3998}, {1842, 40940}, {1851, 16583}, {2203, 36033}, {2207, 40586}, {2299, 11517}, {3161, 8748}, {3554, 24005}, {5375, 24019}, {6591, 34591}, {6626, 36419}, {6651, 34856}, {7337, 40611}, {7649, 15526}, {17925, 40618}, {32713, 39026}, {36420, 40589}, {39006, 43925}
X(52396) = cevapoint of X(3265) and X(17216)
X(52396) = crosspoint of X(i) and X(j) for these (i,j): {1264, 3718}, {4561, 4600}
X(52396) = crosssum of X(1395) and X(7337)
X(52396) = trilinear pole of line {3265, 24018}
X(52396) = barycentric product X(i)*X(j) for these {i,j}: {3, 40071}, {10, 3926}, {63, 20336}, {69, 306}, {71, 305}, {72, 304}, {75, 3998}, {76, 3682}, {78, 1231}, {190, 3265}, {226, 1264}, {228, 40364}, {255, 27801}, {307, 345}, {313, 394}, {321, 326}, {332, 26942}, {348, 3710}, {349, 1259}, {520, 1978}, {525, 4561}, {561, 3990}, {668, 24018}, {822, 6386}, {1016, 17216}, {1102, 41013}, {1214, 3718}, {1331, 3267}, {1332, 14208}, {1441, 3719}, {1444, 52369}, {1502, 4055}, {1804, 30713}, {1826, 4176}, {1897, 4143}, {2200, 40050}, {2321, 7055}, {2632, 4601}, {3596, 40152}, {3694, 7182}, {3695, 17206}, {3701, 7183}, {3952, 30805}, {4019, 7019}, {4033, 4131}, {4064, 4563}, {4091, 27808}, {4158, 44129}, {4567, 17879}, {4570, 36793}, {4600, 15526}, {4620, 7068}, {6514, 34388}, {7066, 28660}, {19611, 42699}, {21046, 47389}, {22341, 28659}
X(52396) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 5317}, {2, 8747}, {3, 1474}, {8, 8748}, {10, 393}, {37, 1096}, {42, 2207}, {48, 2203}, {58, 36420}, {63, 28}, {69, 27}, {71, 25}, {72, 19}, {73, 608}, {77, 1396}, {78, 1172}, {86, 36419}, {100, 24019}, {101, 32713}, {190, 107}, {201, 1880}, {212, 2204}, {219, 2299}, {226, 1118}, {228, 1973}, {239, 34856}, {255, 1333}, {283, 2189}, {304, 286}, {305, 44129}, {306, 4}, {307, 278}, {312, 1896}, {313, 2052}, {321, 158}, {326, 81}, {332, 46103}, {345, 29}, {394, 58}, {440, 1842}, {520, 649}, {525, 7649}, {577, 2206}, {656, 6591}, {668, 823}, {822, 667}, {906, 32676}, {1043, 36421}, {1102, 1444}, {1214, 34}, {1231, 273}, {1259, 284}, {1260, 2332}, {1264, 333}, {1265, 2322}, {1331, 112}, {1332, 162}, {1334, 6059}, {1400, 7337}, {1409, 1395}, {1439, 1435}, {1459, 43925}, {1792, 2326}, {1794, 40570}, {1804, 1412}, {1812, 270}, {1826, 6524}, {1897, 6529}, {1918, 36417}, {1978, 6528}, {2200, 1974}, {2289, 2194}, {2318, 607}, {2321, 1857}, {2525, 21108}, {2632, 3125}, {3265, 514}, {3267, 46107}, {3269, 3122}, {3610, 7102}, {3682, 6}, {3690, 2333}, {3692, 4183}, {3694, 33}, {3695, 1826}, {3710, 281}, {3718, 31623}, {3719, 21}, {3926, 86}, {3933, 17171}, {3949, 1824}, {3958, 2355}, {3964, 1790}, {3977, 37168}, {3990, 31}, {3998, 1}, {4001, 31900}, {4019, 7009}, {4025, 17925}, {4047, 5338}, {4055, 32}, {4064, 2501}, {4091, 3733}, {4131, 1019}, {4143, 4025}, {4158, 71}, {4176, 17206}, {4303, 46890}, {4466, 2969}, {4552, 36127}, {4561, 648}, {4567, 24000}, {4568, 46151}, {4570, 23964}, {4574, 8750}, {4600, 23582}, {4601, 23999}, {5227, 4206}, {5489, 21131}, {6335, 36126}, {6507, 1437}, {6514, 60}, {6517, 4565}, {7055, 1434}, {7066, 1400}, {7068, 21044}, {7125, 1408}, {7183, 1014}, {7289, 4211}, {7335, 16947}, {8611, 18344}, {8804, 6525}, {8896, 37384}, {14208, 17924}, {15523, 27376}, {15526, 3120}, {17216, 1086}, {17879, 16732}, {18589, 1851}, {18604, 849}, {18607, 46883}, {20336, 92}, {20580, 21172}, {21011, 14569}, {21046, 8754}, {22057, 16502}, {22076, 2354}, {22341, 604}, {23067, 32674}, {23616, 21134}, {23974, 17216}, {24018, 513}, {26872, 37383}, {26942, 225}, {28787, 46886}, {30805, 7192}, {36793, 21207}, {37669, 44698}, {37755, 1426}, {39201, 1919}, {40071, 264}, {40152, 56}, {41013, 6520}, {41014, 1839}, {41077, 11125}, {41087, 7151}, {42699, 1895}, {42706, 39585}, {42711, 51315}, {51366, 1886}, {51367, 1785}, {51386, 17209}, {51640, 43923}, {52347, 17167}, {52355, 3064}, {52369, 41013}
X(52396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {190, 4150, 8804}, {307, 3710, 20336}, {1264, 3926, 326}

X(52397) = ANTICOMPLEMENT OF X(428)

Barycentrics    2*a^6 + a^4*b^2 - 2*a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4 - c^6 : :
X(52397) = 7 X[2] - 8 X[7734], 9 X[2] - 8 X[10128], 3 X[2] - 4 X[10691], 17 X[2] - 16 X[13361], 5 X[2] - 6 X[43957], 5 X[3] - 2 X[11819], 7 X[3] - 4 X[31830], 2 X[20] + X[12225], 4 X[20] - X[52071], 3 X[376] - X[18559], X[428] - 3 X[7667], 7 X[428] - 12 X[7734], 3 X[428] - 4 X[10128], 17 X[428] - 24 X[13361], 4 X[428] - 3 X[34603], 5 X[428] - 9 X[43957], 4 X[550] - X[6240], 5 X[631] - 2 X[7553], 5 X[631] - 4 X[10127], 5 X[631] - 6 X[43934], 2 X[1657] + X[18560], 2 X[1885] + X[5059], X[3146] - 4 X[12362], 5 X[3522] - 2 X[3575], 7 X[3523] - 4 X[6756], 7 X[3528] - 4 X[31833], X[3529] + 2 X[12605], 3 X[5054] - 2 X[13490], X[5073] - 4 X[52073], X[7553] - 3 X[43934], 5 X[7576] - 4 X[11819], 7 X[7576] - 8 X[31830], 7 X[7667] - 4 X[7734], 9 X[7667] - 4 X[10128], 3 X[7667] - 2 X[10691], 17 X[7667] - 8 X[13361], 4 X[7667] - X[34603], 5 X[7667] - 3 X[43957], 9 X[7734] - 7 X[10128], 6 X[7734] - 7 X[10691], 17 X[7734] - 14 X[13361], 16 X[7734] - 7 X[34603], 20 X[7734] - 21 X[43957], 8 X[9825] - 11 X[15717], 2 X[10127] - 3 X[43934], and many others

X(52397) lies on these lines: {2, 3}, {51, 29317}, {52, 17712}, {98, 20185}, {99, 16275}, {125, 48920}, {141, 15321}, {159, 41736}, {184, 48898}, {305, 7802}, {343, 48881}, {394, 46818}, {395, 11421}, {396, 11420}, {511, 41628}, {524, 12220}, {528, 20243}, {539, 10625}, {542, 12058}, {551, 34657}, {597, 19121}, {612, 10483}, {754, 19568}, {925, 29011}, {930, 1297}, {933, 34168}, {1184, 44526}, {1194, 7756}, {1216, 16659}, {1232, 30737}, {1238, 7788}, {1291, 2697}, {1350, 11442}, {1369, 3933}, {1503, 2979}, {1627, 5254}, {1899, 48873}, {1993, 46264}, {1994, 48906}, {2549, 5359}, {2965, 5306}, {3058, 3100}, {3060, 29181}, {3098, 11550}, {3101, 34612}, {3164, 7837}, {3241, 34634}, {3313, 46442}, {3410, 48876}, {3448, 48874}, {3565, 9076}, {3580, 48880}, {3616, 51719}, {3618, 51745}, {3679, 34668}, {3819, 29323}, {3828, 34633}, {3829, 34655}, {3917, 29012}, {3920, 7354}, {4296, 5434}, {4316, 5322}, {4324, 5310}, {4329, 34611}, {5012, 44882}, {5422, 31670}, {5891, 16658}, {5913, 34481}, {5986, 38741}, {6030, 13394}, {6247, 7691}, {6284, 7191}, {6393, 33796}, {6563, 25423}, {7750, 8024}, {7753, 34661}, {7783, 8878}, {7811, 18018}, {7816, 21248}, {7830, 8891}, {7893, 40904}, {8280, 51911}, {8281, 51910}, {8584, 11416}, {8718, 22660}, {8792, 22121}, {9300, 13351}, {9306, 48896}, {9538, 15170}, {10601, 48910}, {11002, 45298}, {11064, 26881}, {11417, 32787}, {11418, 32788}, {11444, 16655}, {11457, 37486}, {13203, 41602}, {13567, 15107}, {14389, 22352}, {14516, 15644}, {14540, 40712}, {14541, 40711}, {15048, 34482}, {15056, 16621}, {15066, 31383}, {15080, 23292}, {15171, 17024}, {16264, 40684}, {18400, 36987}, {18911, 33586}, {18990, 29815}, {19924, 21969}, {20049, 34667}, {20806, 31166}, {21243, 48885}, {21850, 34545}, {26913, 32269}, {29263, 40998}, {31145, 34656}, {32819, 39998}, {33878, 45794}, {34666, 49732}, {35283, 44299}, {40634, 43768}, {42459, 52058}, {43650, 48901}, {48891, 51360}, {49719, 52365}

X(52397) = midpoint of X(1657) and X(18564)
X(52397) = reflection of X(i) in X(j) for these {i,j}: {2, 7667}, {428, 10691}, {3241, 34634}, {3543, 34664}, {7540, 549}, {7553, 10127}, {7576, 3}, {15683, 34614}, {16658, 5891}, {18560, 18564}, {20049, 34667}, {31145, 34656}, {34603, 2}, {34613, 381}, {34633, 3828}, {34655, 3829}, {34657, 551}, {34661, 7753}, {34666, 49732}, {34668, 3679}, {38320, 10304}, {38321, 8703}, {38322, 34200}, {38323, 376}, {44458, 3534}
X(52397) = anticomplement of X(428)
X(52397) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(44450)
X(52397) = de Longchamps circle inverse of X(20063)
X(52397) = anticomplement of the isogonal conjugate of X(41435)
X(52397) = isotomic conjugate of the polar conjugate of X(41366)
X(52397) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {48, 51860}, {63, 40002}, {3108, 5905}, {7953, 7253}, {10159, 21270}, {35137, 21300}, {41435, 8}
X(52397) = X(7768)-Ceva conjugate of X(141)
X(52397) = X(14378)-Dao conjugate of X(15321)
X(52397) = crosspoint of X(23582) and X(35137)
X(52397) = crosssum of X(3269) and X(8664)
X(52397) = barycentric product X(69)*X(41366)
X(52397) = barycentric quotient X(41366)/X(4)
X(52397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5064, 5133}, {2, 7391, 5064}, {2, 7500, 7714}, {2, 7714, 1995}, {2, 9909, 7426}, {2, 44442, 31133}, {3, 5064, 2}, {3, 7391, 5133}, {4, 20, 33524}, {4, 7485, 37990}, {20, 1370, 22}, {20, 2071, 44239}, {20, 12225, 52071}, {20, 21312, 16386}, {20, 37444, 11414}, {20, 52364, 35998}, {22, 1370, 858}, {22, 30744, 7493}, {25, 1657, 20062}, {25, 20062, 37900}, {140, 52285, 37353}, {376, 44442, 2}, {382, 7484, 7394}, {427, 550, 6636}, {427, 6636, 7495}, {428, 7667, 10691}, {428, 10691, 2}, {548, 47095, 5169}, {550, 5189, 7495}, {1370, 7493, 7396}, {1370, 33524, 37990}, {1657, 16063, 37900}, {2043, 2044, 10323}, {3098, 11550, 37636}, {3529, 7386, 7500}, {3627, 37439, 37349}, {5002, 5003, 5}, {5004, 5005, 3518}, {5189, 6636, 427}, {6676, 46517, 31074}, {7386, 7500, 1995}, {7386, 7714, 2}, {7396, 7493, 30744}, {7396, 30744, 858}, {7485, 33524, 22}, {7488, 44450, 549}, {7492, 31074, 6676}, {7494, 31099, 31236}, {7496, 37349, 37439}, {7553, 43934, 10127}, {7667, 34614, 34658}, {9909, 31152, 2}, {10127, 43934, 631}, {10323, 14790, 13160}, {11112, 11113, 48815}, {11114, 17579, 48813}, {12082, 18531, 47096}, {12083, 14791, 403}, {12103, 46517, 7492}, {16063, 20062, 25}, {17579, 51669, 11112}, {20405, 20406, 5159}, {31101, 37913, 468}

X(52398) = ANTICOMPLEMENT OF X(1598)

Barycentrics    a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 - 20*a^6*b^2*c^2 + 14*a^4*b^4*c^2 + 4*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 + 14*a^4*b^2*c^4 - 10*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 + 4*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(52398) = 5 X[631] - 4 X[6642], 5 X[631] - 3 X[7714], 7 X[3523] - 4 X[7715], 11 X[3525] - 8 X[13861], 4 X[6642] - 3 X[7714], 7 X[37973] - 8 X[44264]

X(53398) lies on these lines: {2, 3}, {69, 14216}, {84, 50861}, {193, 18914}, {389, 41256}, {394, 34781}, {511, 18909}, {578, 25406}, {1056, 4296}, {1058, 3100}, {1092, 11206}, {1192, 48872}, {1217, 33971}, {1249, 23115}, {1350, 6247}, {1352, 13348}, {3421, 52366}, {3426, 11469}, {3618, 37515}, {3926, 20477}, {3933, 6527}, {5082, 52365}, {5447, 11487}, {5562, 12324}, {5709, 26929}, {5800, 36746}, {5921, 34780}, {6193, 37483}, {6696, 48881}, {6759, 37669}, {6776, 37498}, {7330, 26939}, {9729, 31670}, {9786, 29181}, {9815, 17704}, {9833, 14927}, {10110, 18928}, {10625, 11411}, {10629, 37552}, {10984, 11427}, {11412, 12058}, {11425, 44882}, {11433, 45186}, {11438, 41257}, {11677, 30265}, {11745, 48910}, {12163, 12318}, {12250, 15438}, {12256, 12320}, {12257, 12321}, {12319, 16111}, {13203, 16163}, {13336, 19121}, {13346, 18925}, {13347, 14561}, {13391, 18951}, {14826, 16655}, {14853, 37514}, {14912, 36747}, {15583, 17822}, {16621, 17811}, {16836, 51538}, {17834, 18913}, {17845, 34944}, {18488, 41464}, {18917, 37484}, {18919, 44492}, {18931, 46730}, {22528, 41465}, {22660, 35237}, {25712, 34782}, {31383, 43652}, {32064, 36987}, {36844, 36984}, {36988, 41761}, {38738, 39842}, {38749, 39813}, {42329, 44443}

X(52398) = reflection of X(i) in X(j) for these {i,j}: {4, 6643}, {7487, 3}
X(52398) = anticomplement of X(1598)
X(52398) = deLongchamps-circle-inverse of X(37945)
X(52398) = X(46952)-anticomplementary conjugate of X(5905)
X(52398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 11414}, {3, 4, 6803}, {3, 382, 9825}, {3, 1595, 2}, {3, 7404, 631}, {3, 34938, 4}, {4, 20, 35513}, {4, 376, 10996}, {4, 631, 7392}, {4, 3537, 6815}, {4, 3538, 2}, {4, 7386, 6804}, {20, 1370, 4}, {20, 3522, 44239}, {20, 11413, 376}, {20, 12225, 3529}, {20, 30552, 17538}, {20, 37444, 37201}, {376, 3529, 35471}, {382, 18537, 4}, {427, 37198, 7400}, {631, 7714, 6642}, {1368, 39568, 3089}, {1370, 37201, 37444}, {3090, 3529, 37946}, {3146, 6816, 4}, {3146, 16063, 6816}, {3522, 6815, 3537}, {3522, 7391, 6815}, {3529, 47528, 4}, {3541, 10323, 7494}, {3546, 7387, 6353}, {3547, 23335, 8889}, {5002, 5003, 6997}, {6815, 7391, 4}, {9715, 47090, 3523}, {10996, 44442, 4}, {13346, 46264, 18925}, {14216, 15644, 69}, {14784, 14785, 39568}, {17704, 48901, 9815}, {23335, 35243, 3547}, {31152, 34614, 34621}, {37201, 37444, 4}

X(52399) = ANTICOMPLEMENT OF X(2043)

Barycentrics    Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(52399) = 3 X[2] - 4 X[18585], 5 X[2] - 4 X[36457], 2 X[3] - 3 X[36445], 4 X[3] - X[42282], 3 X[4] - 2 X[18586], 4 X[5] - 3 X[36463], X[20] + 2 X[2041], 3 X[376] - 4 X[34551], 2 X[550] + X[42278], 5 X[631] - 4 X[34552], 2 X[2042] - 5 X[17578], X[2043] - 3 X[36455], 5 X[2043] - 6 X[36457], 26 X[2045] - 29 X[46935], 7 X[3090] - 8 X[34562], 5 X[3091] - 4 X[15765], 2 X[3146] + X[35732], X[3146] - 4 X[42280], 7 X[3523] - 4 X[14814], 9 X[3545] - 8 X[34559], 3 X[3839] - 2 X[36437], X[5059] - 4 X[14813], 3 X[10304] - 4 X[36439], X[11001] + 2 X[36466], 10 X[15713] - 11 X[36456], 2 X[18585] - 3 X[36455], 5 X[18585] - 3 X[36457], X[18586] - 3 X[18587], X[35732] + 8 X[42280], 8 X[35738] - 11 X[50689], 11 X[36438] - 12 X[47599], 6 X[36445] - X[42282], 5 X[36455] - 2 X[36457], 4 X[42281] - 7 X[50688]

X(52399) lies on these lines: {2, 3}, {13, 42255}, {14, 42256}, {15, 9541}, {298, 490}, {299, 489}, {395, 42259}, {396, 42258}, {397, 36449}, {398, 36450}, {487, 617}, {488, 616}, {590, 42087}, {615, 42088}, {627, 13678}, {628, 13798}, {1151, 42942}, {1152, 42943}, {1587, 3364}, {1588, 3390}, {3068, 42119}, {3069, 42120}, {3070, 42154}, {3071, 42155}, {3180, 43134}, {3181, 43133}, {3367, 23263}, {3391, 23253}, {5318, 42218}, {5321, 42219}, {5334, 6560}, {5335, 6561}, {5339, 42252}, {5340, 42251}, {5464, 35742}, {5473, 33440}, {5474, 33443}, {5870, 6269}, {5871, 6270}, {6200, 51855}, {6396, 51852}, {6409, 42626}, {6410, 42625}, {6459, 37640}, {6460, 37641}, {6564, 42172}, {6565, 42173}, {6770, 26441}, {6773, 8982}, {7583, 43482}, {7584, 43481}, {7690, 48722}, {7692, 48725}, {8252, 42169}, {8253, 42168}, {9540, 36967}, {13340, 34553}, {13665, 42200}, {13785, 42201}, {13846, 42253}, {13847, 42250}, {13935, 35739}, {16644, 35740}, {16645, 42239}, {16808, 42196}, {16809, 42197}, {16964, 42248}, {16965, 42247}, {19106, 42188}, {19107, 42189}, {22236, 36468}, {22238, 36467}, {23249, 42085}, {23251, 42240}, {23259, 42086}, {23261, 42241}, {23267, 42117}, {23273, 42118}, {32787, 42147}, {32788, 42148}, {35730, 41943}, {35820, 42243}, {35821, 42244}, {35822, 42228}, {35823, 42229}, {36440, 50808}, {36444, 50865}, {36446, 43770}, {36447, 43769}, {36969, 42236}, {36970, 42237}, {36993, 45511}, {36995, 45510}, {40693, 43257}, {40694, 43256}, {42093, 42193}, {42094, 42192}, {42096, 42284}, {42097, 42283}, {42099, 42190}, {42100, 42187}, {42122, 43509}, {42123, 43510}, {42125, 42223}, {42126, 42213}, {42127, 42212}, {42128, 42222}, {42130, 42214}, {42131, 42211}, {42133, 42276}, {42134, 42275}, {42153, 43209}, {42156, 43210}, {42157, 42249}, {42158, 42246}, {42232, 52217}, {42233, 52215}, {42242, 42267}, {42245, 42266}, {42271, 42941}, {42272, 42940}, {42511, 51727}

X(52399) = midpoint of X(2041) and X(2044)
X(52399) = reflection of X(i) in X(j) for these {i,j}: {2, 36455}, {4, 18587}, {20, 2044}, {2043, 18585}, {36454, 3830}
X(52399) = anticomplement of X(2043)
X(52399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2044, 35732}, {3, 2042, 3523}, {3, 2044, 2}, {3, 2045, 15717}, {3, 3534, 34552}, {3, 14813, 631}, {3, 15765, 3524}, {3, 18587, 36455}, {4, 20, 35732}, {4, 376, 2044}, {4, 2041, 42282}, {4, 3529, 2042}, {376, 36445, 2}, {382, 42280, 4}, {2041, 18587, 2}, {2043, 18585, 2}, {2043, 36455, 18585}, {2044, 2046, 36454}, {2044, 36437, 14813}, {2044, 36455, 4}, {2044, 42278, 15683}, {2045, 3861, 3091}, {2046, 15765, 2}, {2046, 33703, 42282}, {3524, 36454, 2}, {3534, 36455, 3839}, {3830, 15765, 4}, {3830, 42278, 15765}, {11001, 36445, 15692}, {14813, 34552, 36437}, {14813, 49134, 3529}, {15764, 17504, 36437}, {18585, 34551, 2044}, {18587, 36436, 42282}, {18587, 36466, 5}, {18587, 42280, 36436}, {19708, 35737, 15764}, {35739, 42254, 13935}, {36436, 36463, 36466}

X(52400) = ANTICOMPLEMENT OF X(2044)

Barycentrics    Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(52400) = 3 X[2] - 4 X[15765], 5 X[2] - 4 X[36439], 4 X[3] - X[35732], 2 X[3] - 3 X[36463], 3 X[4] - 2 X[18587], 4 X[5] - 3 X[36445], X[20] + 2 X[2042], 3 X[376] - 4 X[34552], 2 X[550] + X[42279], 5 X[631] - 4 X[34551], 2 X[2041] - 5 X[17578], X[2044] - 3 X[36437], 5 X[2044] - 6 X[36439], 26 X[2046] - 29 X[46935], 7 X[3090] - 8 X[34559], 5 X[3091] - 4 X[18585], 5 X[3091] - 8 X[35738], X[3146] - 4 X[42281], 2 X[3146] + X[42282], 7 X[3523] - 4 X[14813], 9 X[3545] - 8 X[34562], 3 X[3839] - 2 X[36455], X[5059] - 4 X[14814], 3 X[10304] - 4 X[36457], X[11001] + 2 X[36448], 5 X[15692] - 4 X[15764], 10 X[15713] - 11 X[36438], 2 X[15765] - 3 X[36437], 5 X[15765] - 3 X[36439], 3 X[18586] - X[18587], X[35732] - 6 X[36463], 5 X[36437] - 2 X[36439], 11 X[36456] - 12 X[47599], 4 X[42280] - 7 X[50688], 8 X[42281] + X[42282]

X(52400) lies on these lines: {2, 3}, {13, 42257}, {14, 42254}, {16, 9541}, {298, 489}, {299, 490}, {395, 42258}, {396, 42259}, {397, 36468}, {398, 36467}, {487, 616}, {488, 617}, {590, 42088}, {615, 42087}, {627, 13798}, {628, 13678}, {1131, 51925}, {1132, 51924}, {1151, 42943}, {1152, 42942}, {1587, 3389}, {1588, 3365}, {3068, 42120}, {3069, 42119}, {3070, 42155}, {3071, 42154}, {3180, 43133}, {3181, 43134}, {3366, 23253}, {3392, 23263}, {5318, 42220}, {5321, 42217}, {5334, 6561}, {5335, 6560}, {5339, 42250}, {5340, 42253}, {5473, 33441}, {5474, 33442}, {5870, 6268}, {5871, 6271}, {6200, 51854}, {6396, 51853}, {6409, 42625}, {6410, 42626}, {6459, 37641}, {6460, 37640}, {6564, 42174}, {6565, 42171}, {6770, 8982}, {6773, 26441}, {7583, 43481}, {7584, 43482}, {7690, 48724}, {7692, 48723}, {8252, 42167}, {8253, 42170}, {9540, 36968}, {13340, 34555}, {13665, 42202}, {13785, 42199}, {13846, 42252}, {13847, 42251}, {13935, 36967}, {16644, 42241}, {16645, 42240}, {16808, 42198}, {16809, 42195}, {16964, 42246}, {16965, 42249}, {19106, 42190}, {19107, 42187}, {22236, 36449}, {22238, 36450}, {23249, 35731}, {23251, 35740}, {23259, 42085}, {23261, 42239}, {23267, 42118}, {23273, 42117}, {32787, 42148}, {32788, 42147}, {35739, 42529}, {35820, 42245}, {35821, 42242}, {35822, 42230}, {35823, 42227}, {36458, 50808}, {36462, 50865}, {36464, 43769}, {36465, 43770}, {36969, 42238}, {36970, 42235}, {36993, 45510}, {36995, 45511}, {40693, 43256}, {40694, 43257}, {42093, 42191}, {42094, 42194}, {42096, 42283}, {42097, 42284}, {42099, 42188}, {42100, 42189}, {42122, 43510}, {42123, 43509}, {42125, 42221}, {42126, 42211}, {42127, 42214}, {42128, 42224}, {42130, 42212}, {42131, 42213}, {42133, 42275}, {42134, 42276}, {42153, 43210}, {42156, 43209}, {42157, 42247}, {42158, 42248}, {42231, 52216}, {42234, 52214}, {42243, 42266}, {42244, 42267}, {42271, 42940}, {42272, 42941}

X(52400) = midpoint of X(2042) and X(2043)
X(52400) = reflection of X(i) in X(j) for these {i,j}: {2, 36437}, {4, 18586}, {20, 2043}, {2044, 15765}, {18585, 35738}, {36436, 3830}
X(52400) = anticomplement of X(2044)
X(52400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2043, 42282}, {3, 2041, 3523}, {3, 2043, 2}, {3, 2046, 15717}, {3, 3534, 34551}, {3, 14814, 631}, {3, 18585, 3524}, {3, 18586, 36437}, {4, 20, 42282}, {4, 376, 2043}, {4, 2042, 35732}, {4, 3529, 2041}, {376, 36463, 2}, {381, 15764, 15709}, {382, 42281, 4}, {2042, 18586, 2}, {2043, 2045, 36436}, {2043, 36437, 4}, {2043, 36455, 14814}, {2043, 42279, 15683}, {2044, 15765, 2}, {2044, 36437, 15765}, {2045, 18585, 2}, {2045, 33703, 35732}, {2046, 3861, 3091}, {3524, 36436, 2}, {3534, 36437, 3839}, {3830, 18585, 4}, {3830, 42279, 18585}, {11001, 36463, 15692}, {14814, 34551, 36455}, {14814, 49134, 3529}, {15688, 15764, 376}, {15765, 34552, 2043}, {18586, 36448, 5}, {18586, 36454, 35732}, {18586, 42281, 36454}, {36445, 36454, 36448}

X(52401) = ANTICOMPLEMENT OF X(2045)

Barycentrics    3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*Sqrt[3]*a^2*(a^2 - b^2 - c^2) : :
X(52401) = 10 X[3] + 3 X[36436], 4 X[5] + 9 X[36445], X[20] + 12 X[18585], 20 X[140] - 33 X[36456], 12 X[381] + X[35732], 25 X[631] - 12 X[34552], 2 X[2041] + 11 X[15717], 2 X[2042] - 15 X[3091], 6 X[2044] + 7 X[3832], X[3146] + 12 X[36439], 7 X[3523] + 6 X[36455], 14 X[3523] - X[42282], 7 X[3528] + 6 X[18587], 9 X[3839] + 4 X[14813], 16 X[3850] - 3 X[36454], 17 X[3854] - 4 X[42281], 11 X[3855] - 24 X[34562], 14 X[3857] - X[42279], 22 X[5070] - 9 X[36463], 9 X[10304] + 4 X[42280], 12 X[12100] + X[42278], 19 X[15022] - 6 X[36437], 11 X[15715] + 2 X[36466], 12 X[15764] + X[49135], X[33703] + 12 X[34551], 33 X[36438] - 46 X[41992], 12 X[36455] + X[42282]

X(52401) lies on these lines: {2, 3}, {13, 42246}, {14, 42249}, {15, 9540}, {16, 13935}, {17, 42247}, {18, 42248}, {302, 490}, {303, 489}, {485, 5334}, {486, 5335}, {487, 628}, {488, 627}, {590, 42147}, {615, 42148}, {1151, 16772}, {1152, 16773}, {1587, 3390}, {1588, 3364}, {3070, 42153}, {3071, 42156}, {3304, 36459}, {3316, 42117}, {3317, 42118}, {3366, 42085}, {3367, 18582}, {3391, 18581}, {3392, 42086}, {3592, 36468}, {3594, 36467}, {4857, 36443}, {5270, 36442}, {5318, 42217}, {5321, 42220}, {5339, 42253}, {5340, 42250}, {5870, 33358}, {5871, 33359}, {6201, 51754}, {6202, 51753}, {6409, 42490}, {6410, 42491}, {6453, 36469}, {6454, 36470}, {6459, 11488}, {6460, 11489}, {6564, 42242}, {6565, 42245}, {8252, 42241}, {8253, 42240}, {8960, 52217}, {9541, 42255}, {10576, 42243}, {10577, 42244}, {11522, 36444}, {11542, 23273}, {11543, 23267}, {12974, 49106}, {12975, 49105}, {13678, 33404}, {13798, 33405}, {16644, 42251}, {16645, 42252}, {16808, 42187}, {16809, 42190}, {16964, 42257}, {16965, 42254}, {16966, 42188}, {16967, 42189}, {19106, 42195}, {19107, 42198}, {22236, 31454}, {23251, 42239}, {23261, 35740}, {23302, 42218}, {23303, 42219}, {30389, 36462}, {32785, 42119}, {32786, 42120}, {32789, 42087}, {32790, 42088}, {33416, 42197}, {33417, 42196}, {34089, 43630}, {34091, 43631}, {34555, 37481}, {35739, 42256}, {35786, 42175}, {35787, 42178}, {35812, 42228}, {35813, 42229}, {35820, 42171}, {35821, 42174}, {36440, 43174}, {42093, 42194}, {42094, 42191}, {42095, 42284}, {42098, 42283}, {42125, 42214}, {42126, 42224}, {42127, 42221}, {42128, 42211}, {42129, 42213}, {42132, 42212}, {42133, 42277}, {42134, 42274}, {42167, 42193}, {42170, 42192}, {42179, 42919}, {42180, 42915}, {42181, 42914}, {42182, 42918}, {42231, 43404}, {42234, 43403}, {42235, 42813}, {42236, 42488}, {42237, 42489}, {42238, 42814}

X(52401) = reflection of X(10303) in X(2046)
X(52401) = anticomplement of X(2045)
X(52401) = orthocentroidal-circle-inverse of X(42282)
X(52401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 42282}, {3, 1656, 34551}, {3, 2041, 20}, {3, 2043, 3522}, {3, 2046, 2}, {3, 14814, 376}, {3, 18585, 4}, {3, 42279, 36439}, {4, 631, 2041}, {4, 2044, 35732}, {4, 3090, 2043}, {5, 2041, 2}, {5, 18585, 2041}, {20, 2041, 42282}, {381, 42281, 4}, {546, 18586, 4}, {1656, 42281, 41106}, {2041, 2044, 5}, {2041, 2046, 631}, {2041, 42279, 7486}, {2043, 2044, 36454}, {2043, 3850, 3091}, {2044, 18585, 2}, {2046, 36455, 3}, {3523, 36455, 42282}, {3851, 18585, 3525}, {5079, 18587, 2045}, {14784, 14785, 15765}, {14813, 36454, 35732}, {14814, 19709, 3090}, {18585, 36439, 18587}, {35738, 42278, 36463}, {36439, 36455, 15692}

X(52402) = ANTICOMPLEMENT OF X(2046)

Barycentrics    3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4 + 2*Sqrt[3]*a^2*(a^2 - b^2 - c^2) : :
X(52402) = 10 X[3] + 3 X[36454], 4 X[5] + 9 X[36463], X[20] + 12 X[15765], 20 X[140] - 33 X[36438], 12 X[381] + X[42282], 25 X[631] - 12 X[34551], 2 X[2041] - 15 X[3091], 2 X[2042] + 11 X[15717], 6 X[2043] + 7 X[3832], X[3146] + 12 X[36457], 5 X[3522] + 8 X[35738], 14 X[3523] - X[35732], 7 X[3523] + 6 X[36437], 7 X[3528] + 6 X[18586], 9 X[3839] + 4 X[14814], 16 X[3850] - 3 X[36436], 17 X[3854] - 4 X[42280], 11 X[3855] - 24 X[34559], 14 X[3857] - X[42278], 22 X[5070] - 9 X[36445], 9 X[10304] + 4 X[42281], 12 X[12100] + X[42279], 19 X[15022] - 6 X[36455], 11 X[15715] + 2 X[36448], X[33703] + 12 X[34552], X[35732] + 12 X[36437], 33 X[36456] - 46 X[41992]

X(52402) lies on these lines: {2, 3}, {13, 42248}, {14, 42247}, {15, 13935}, {16, 9540}, {17, 42249}, {18, 42246}, {302, 489}, {303, 490}, {485, 5335}, {486, 5334}, {487, 627}, {488, 628}, {590, 42148}, {615, 42147}, {1151, 16773}, {1152, 16772}, {1587, 3365}, {1588, 3389}, {3070, 42156}, {3071, 42153}, {3303, 36451}, {3304, 36441}, {3316, 42118}, {3317, 42117}, {3366, 18582}, {3367, 42085}, {3391, 42086}, {3392, 18581}, {3592, 36450}, {3594, 36449}, {4857, 36461}, {5270, 36460}, {5318, 42219}, {5321, 42218}, {5339, 42251}, {5340, 42252}, {5870, 33360}, {5871, 33361}, {6201, 51753}, {6202, 51754}, {6409, 42491}, {6410, 42490}, {6453, 36452}, {6454, 36453}, {6459, 11489}, {6460, 11488}, {6564, 42244}, {6565, 42243}, {8252, 42239}, {8253, 35740}, {8960, 52214}, {9541, 42254}, {10576, 42245}, {10577, 42242}, {11522, 36462}, {11542, 23267}, {11543, 23273}, {12974, 49105}, {12975, 49106}, {13678, 33405}, {13798, 33404}, {16644, 42253}, {16645, 42250}, {16808, 42189}, {16809, 42188}, {16964, 42255}, {16965, 42256}, {16966, 42190}, {16967, 42187}, {19106, 42197}, {19107, 42196}, {20070, 34560}, {22238, 31454}, {23251, 42241}, {23261, 42240}, {23302, 42220}, {23303, 42217}, {30389, 36444}, {32785, 42120}, {32786, 42119}, {32789, 42088}, {32790, 42087}, {33416, 42195}, {33417, 42198}, {34089, 43631}, {34091, 43630}, {34553, 37481}, {35731, 42151}, {35786, 42177}, {35787, 42176}, {35812, 42230}, {35813, 42227}, {35820, 42173}, {35821, 42172}, {36458, 43174}, {42093, 42192}, {42094, 42193}, {42095, 42283}, {42098, 42284}, {42125, 42212}, {42126, 42222}, {42127, 42223}, {42128, 42213}, {42129, 42211}, {42132, 42214}, {42133, 42274}, {42134, 42277}, {42168, 42191}, {42169, 42194}, {42179, 42914}, {42180, 42918}, {42181, 42919}, {42182, 42915}, {42232, 43404}, {42233, 43403}, {42235, 42489}, {42236, 42814}, {42237, 42813}, {42238, 42488}, {43540, 51925}, {43541, 51924}

X(52402) = reflection of X(10303) in X(2045)
X(52402) = anticomplement of X(2046)
X(52402) = orthocentroidal-circle-inverse of X(35732)
X(52402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 35732}, {3, 1656, 34552}, {3, 2042, 20}, {3, 2044, 3522}, {3, 2045, 2}, {3, 14813, 376}, {3, 15765, 4}, {3, 42278, 36457}, {4, 631, 2042}, {4, 2043, 42282}, {4, 3090, 2044}, {5, 2042, 2}, {5, 15765, 2042}, {20, 2042, 35732}, {381, 42280, 4}, {546, 18587, 4}, {1656, 42280, 41106}, {2042, 2043, 5}, {2042, 2045, 631}, {2042, 42278, 7486}, {2043, 2044, 36436}, {2043, 15765, 2}, {2044, 3850, 3091}, {2045, 36437, 3}, {3523, 36437, 35732}, {3851, 15765, 3525}, {5079, 18586, 2046}, {12103, 35738, 42281}, {14784, 14785, 18585}, {14813, 19709, 3090}, {14814, 15765, 35738}, {14814, 35738, 2044}, {14814, 36436, 42282}, {15765, 36457, 18586}, {36437, 36457, 15692}

X(52403) = ANTICOMPLEMENT OF X(2071)

Barycentrics    a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + 9*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 7*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 4*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 7*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(52403) = 3 X[2] - 4 X[403], 9 X[2] - 8 X[10257], 15 X[2] - 16 X[44911], 5 X[2] - 8 X[47332], 2 X[3] - 3 X[37943], 3 X[3] - 4 X[44234], 4 X[4] - X[5189], 5 X[4] - 2 X[7574], X[4] + 2 X[18325], 3 X[4] - 2 X[18403], 7 X[4] - 4 X[18572], X[4] - 4 X[44267], 3 X[4] - 4 X[44283], 3 X[4] - X[46450], X[20] - 4 X[11799], 2 X[20] - 5 X[37760], 3 X[20] - 4 X[44246], X[20] - 3 X[46451], 5 X[20] - 8 X[47335], 2 X[23] + X[3146], 3 X[23] - 4 X[47093], 7 X[23] - 4 X[47340], 4 X[186] - 5 X[37760], 3 X[186] - 2 X[44246], 2 X[186] - 3 X[46451], 5 X[186] - 4 X[47335], 3 X[376] - 4 X[15646], 3 X[381] - X[35452], 3 X[381] - 2 X[37938], 3 X[403] - 2 X[10257], 5 X[403] - 4 X[44911], 5 X[403] - 6 X[47332], 8 X[468] - 5 X[3522], 4 X[468] - 3 X[37941], 3 X[468] - 2 X[47114], 4 X[546] - X[35001], 2 X[548] - 3 X[16532], 2 X[550] - 3 X[37955], 5 X[631] - 4 X[34152], 5 X[631] - 8 X[44961], 4 X[858] - 7 X[3832], 3 X[858] - 2 X[47091], X[1657] - 4 X[25338], X[1657] - 3 X[37922], 8 X[2070] - 9 X[37909], 3 X[2071] - 4 X[10257], 5 X[2071] - 8 X[44911], 5 X[2071] - 12 X[47332], and many others

X(52403) lies on these lines: {2, 3}, {51, 13446}, {54, 12897}, {110, 51403}, {113, 43574}, {125, 13445}, {143, 3521}, {146, 7731}, {185, 22533}, {323, 1514}, {388, 10149}, {539, 14094}, {1154, 7728}, {1209, 43613}, {1291, 16337}, {1478, 9539}, {1498, 34799}, {1503, 37784}, {1533, 9934}, {1539, 13391}, {2883, 17824}, {2888, 12162}, {3410, 15305}, {3448, 6000}, {3580, 15311}, {3583, 4351}, {3585, 4354}, {3622, 51713}, {3623, 47471}, {5160, 5229}, {5225, 7286}, {5622, 29012}, {5731, 51701}, {5889, 22802}, {6288, 32137}, {8705, 51538}, {9140, 13399}, {9927, 12290}, {10152, 14918}, {10419, 14989}, {10540, 12383}, {11002, 32411}, {11455, 18474}, {11459, 15108}, {11550, 18392}, {11649, 31670}, {12028, 30529}, {12241, 43838}, {12278, 26883}, {12295, 44407}, {12827, 13202}, {13203, 32269}, {13219, 44146}, {13353, 15807}, {13491, 43808}, {13598, 48914}, {14157, 17702}, {14683, 32111}, {14712, 44969}, {14731, 34549}, {14915, 25739}, {14927, 32217}, {15055, 44673}, {15072, 18390}, {16194, 41171}, {16534, 43572}, {16657, 34545}, {18128, 44866}, {18550, 45972}, {18933, 50434}, {20070, 47321}, {20125, 40111}, {22466, 52003}, {23061, 38791}, {25406, 51733}, {29181, 51998}, {30690, 51883}, {32218, 48872}, {32220, 41735}, {32608, 38790}, {34170, 46106}, {34796, 37489}, {36990, 41614}, {37643, 40196}, {40647, 43816}, {43576, 46686}, {44443, 47282}, {44519, 47186}, {47571, 51170}, {51171, 51742}

X(52403) = midpoint of X(i) and X(j) for these {i,j}: {382, 5899}, {10296, 37945}, {18325, 31726}, {32608, 38790}
X(52403) = reflection of X(i) in X(j) for these {i,j}: {3, 11563}, {4, 31726}, {5, 11558}, {20, 186}, {23, 47096}, {110, 51403}, {186, 11799}, {550, 10096}, {858, 10151}, {1291, 16337}, {2070, 43893}, {2071, 403}, {2072, 47336}, {3153, 4}, {3448, 50435}, {5189, 3153}, {7464, 2072}, {10295, 37971}, {10540, 51548}, {12383, 10540}, {13445, 125}, {13619, 2070}, {16386, 468}, {18403, 44283}, {18859, 5}, {20063, 37945}, {31726, 44267}, {34152, 44961}, {35452, 37938}, {37900, 47094}, {37944, 858}, {37950, 46031}, {43574, 113}, {43576, 51392}, {46450, 18403}, {47090, 37984}, {47094, 47338}, {47337, 44912}, {48914, 13598}, {51392, 46686}
X(52403) = anticomplement of X(2071)
X(52403) = circumcircle-inverse of X(22467)
X(52403) = polar-circle-inverse of X(1885)
X(52403) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(6677)
X(52403) = orthoptic-circle-of-the-Steiner-circumellipse inverse of X(25)
X(52403) = circumcircle-of-anticomplementary-triangle-inverse of X(3146)
X(52403) = deLongchamps-circle-inverse of X(11413)
X(52403) = Stammler-circle-inverse of X(18378)
X(52403) = Stammler-circles-radical-circle-inverse of X(4)
X(52403) = anticomplement of the isogonal conjugate of X(11744)
X(52403) = anticomplement of the isotomic conjugate of X(51967)
X(52403) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 51968}, {11744, 8}, {22239, 7253}, {48373, 7192}, {51967, 6327}
X(52403) = X(51967)-Ceva conjugate of X(2)
X(52403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 381, 50140}, {3, 11563, 37943}, {3, 44271, 4}, {4, 3529, 18404}, {4, 7544, 50689}, {4, 15682, 31723}, {4, 18420, 3839}, {4, 33703, 18569}, {4, 44440, 2}, {4, 46450, 18403}, {4, 50009, 34007}, {20, 3091, 3546}, {20, 6623, 2}, {20, 11799, 37760}, {20, 46451, 186}, {23, 7500, 37901}, {26, 44288, 18531}, {186, 11799, 46451}, {186, 44958, 37943}, {186, 46451, 37760}, {235, 22467, 21451}, {235, 52071, 22467}, {381, 35452, 37938}, {382, 44279, 4}, {403, 2071, 2}, {468, 16386, 37941}, {550, 10096, 37955}, {858, 37962, 2}, {1113, 1114, 22467}, {1596, 38323, 13595}, {2072, 7464, 44450}, {3091, 44450, 2072}, {3543, 7500, 3146}, {3830, 44263, 4}, {3839, 18420, 7533}, {5004, 5005, 37928}, {7464, 47336, 3091}, {7488, 45179, 2}, {7527, 15760, 2}, {7574, 47332, 18537}, {7574, 47335, 47528}, {10295, 37971, 37940}, {10296, 47309, 17578}, {10750, 10751, 3627}, {10750, 15156, 14807}, {10751, 15157, 14808}, {12088, 18563, 20}, {13383, 35491, 38448}, {14807, 14808, 3146}, {15154, 15155, 18378}, {16386, 37941, 3522}, {17578, 20063, 10296}, {18325, 44267, 4}, {18403, 31726, 44283}, {18403, 44246, 18531}, {18403, 44283, 4}, {18403, 46450, 3153}, {30745, 37984, 5068}, {37760, 37901, 37913}, {37948, 44452, 3523}, {44276, 50009, 37349}, {44911, 47332, 403}, {47752, 47753, 3518}

X(52404) = ANTICOMPLEMENT OF X(3088)

Barycentrics    a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 - 20*a^6*b^2*c^2 + 14*a^4*b^4*c^2 + 12*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 14*a^4*b^2*c^4 - 18*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 12*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(52404) = 3 X[2] - 4 X[3547], 17 X[3854] - 16 X[7564]

X(52404) lies on these lines: {2, 3}, {40, 27509}, {69, 1498}, {76, 6527}, {141, 15811}, {193, 1181}, {280, 4385}, {343, 12324}, {347, 3673}, {800, 5286}, {1350, 2883}, {1351, 17040}, {1614, 41619}, {3060, 46363}, {3098, 11821}, {3164, 6392}, {3346, 47633}, {4296, 14986}, {4357, 12705}, {5562, 5656}, {5878, 46728}, {5893, 48881}, {5907, 10519}, {5921, 34781}, {5925, 40196}, {6223, 28739}, {6225, 33522}, {6523, 15466}, {6776, 10112}, {7080, 52365}, {7592, 51170}, {8743, 36413}, {9786, 15740}, {9841, 20266}, {9914, 37485}, {10516, 16656}, {10982, 51171}, {10984, 19121}, {11381, 43653}, {11444, 12058}, {11456, 20080}, {12220, 45186}, {12233, 51212}, {12241, 25406}, {12359, 35237}, {12429, 39874}, {13093, 44683}, {13142, 14912}, {13598, 14853}, {14826, 26883}, {15069, 44762}, {15644, 32605}, {15873, 18928}, {16252, 37669}, {16936, 26958}, {18913, 46850}, {18945, 46264}, {30737, 32834}, {35260, 35602}, {41361, 42458}, {41362, 48905}, {41602, 45813}

X(52404) = reflection of X(3088) in X(3547)
X(52404) = anticomplement of X(3088)
X(52404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1596, 6804}, {3, 3089, 2}, {3, 6677, 631}, {3, 35513, 20}, {4, 376, 12362}, {4, 631, 11479}, {4, 7399, 3091}, {4, 7400, 2}, {4, 11414, 20}, {20, 3091, 1370}, {20, 3523, 11413}, {20, 3543, 12225}, {20, 7488, 3522}, {20, 10304, 30552}, {20, 10565, 3}, {20, 44440, 3146}, {22, 37201, 20}, {235, 37198, 7386}, {550, 44277, 3}, {1370, 7493, 40916}, {1370, 33524, 20}, {1596, 6804, 3091}, {1598, 6803, 7398}, {3088, 3547, 2}, {3529, 44239, 20}, {6823, 39568, 4}, {7400, 34621, 4}, {7493, 11413, 3523}, {9715, 32534, 7488}, {20062, 34007, 3146}, {30552, 38444, 10304}

X(52405) = X(1)X(6)∩X(71)X(74)

Barycentrics    a^2*(a - b - c)*(a^2 - b^2 - b*c - c^2) : :
X(52405) = 2 + 3*Cos[A] + Cos[2*A] : :

X(52405) lies on these lines: {1, 6}, {10, 7110}, {35, 2174}, {48, 3730}, {63, 15066}, {71, 74}, {81, 5325}, {86, 25589}, {198, 35239}, {200, 7072}, {284, 1334}, {323, 1442}, {394, 3929}, {517, 16547}, {527, 37659}, {572, 22356}, {579, 9310}, {583, 5563}, {584, 3746}, {594, 7359}, {610, 3587}, {644, 2287}, {645, 17787}, {672, 41431}, {1201, 4284}, {1332, 4416}, {1495, 3690}, {1731, 17452}, {1781, 21853}, {1802, 4262}, {1944, 17116}, {2175, 4517}, {2269, 41432}, {2289, 6602}, {2303, 3997}, {2318, 2328}, {2330, 7064}, {2338, 33634}, {2340, 41457}, {3098, 3220}, {3197, 41456}, {3217, 4266}, {3452, 26723}, {3579, 16553}, {3620, 27509}, {3631, 26932}, {3678, 6198}, {3713, 4873}, {3915, 5037}, {3928, 17811}, {4007, 4513}, {4034, 4914}, {4858, 17117}, {4890, 8539}, {5086, 21065}, {5092, 7193}, {5228, 20195}, {5273, 14996}, {5314, 15080}, {5341, 21864}, {5707, 31446}, {5745, 37633}, {5750, 24564}, {6173, 25878}, {7085, 26864}, {7293, 41462}, {7330, 37483}, {8693, 26702}, {11009, 17443}, {11464, 26915}, {14997, 18228}, {15018, 27065}, {15032, 26878}, {15068, 26921}, {15817, 37601}, {15988, 50093}, {16548, 21871}, {16603, 25651}, {17274, 26657}, {17296, 23151}, {17781, 52374}, {18253, 37559}, {21059, 41276}, {21236, 30848}, {22052, 35072}, {22054, 24047}, {24320, 33878}, {25006, 40869}, {26872, 37643}, {26890, 44109}, {26893, 34417}, {26939, 39874}, {26942, 47296}, {29574, 41610}, {31445, 51340}, {33774, 38825}, {35192, 35193}, {40659, 41339}, {40910, 41454}

X(52405) = isogonal conjugate of X(52374)
X(52405) = complement of the isotomic conjugate of X(43741)
X(52405) = isogonal conjugate of the isotomic conjugate of X(42033)
X(52405) = X(43741)-complementary conjugate of X(2887)
X(52405) = X(i)-Ceva conjugate of X(j) for these (i,j): {3219, 35}, {4570, 3939}, {32635, 55}
X(52405) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52374}, {2, 52372}, {7, 2160}, {27, 52390}, {34, 52381}, {56, 30690}, {57, 79}, {58, 43682}, {65, 52393}, {81, 52382}, {85, 6186}, {226, 52375}, {269, 7110}, {278, 7100}, {279, 7073}, {513, 38340}, {514, 26700}, {554, 2306}, {604, 20565}, {1014, 8818}, {1081, 33654}, {1396, 52388}, {1407, 52344}, {1412, 6757}, {1427, 3615}, {1443, 1989}, {1769, 47317}, {1847, 8606}, {3120, 35049}, {3669, 6742}, {3942, 34922}, {7178, 13486}, {15455, 43924}, {30602, 47057}
X(52405) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 30690}, {3, 52374}, {10, 43682}, {79, 5452}, {94, 36909}, {1443, 34544}, {3161, 20565}, {3700, 21207}, {6600, 7110}, {6757, 40599}, {8287, 24002}, {11517, 52381}, {17078, 40604}, {24771, 52344}, {32664, 52372}, {38340, 39026}, {40586, 52382}, {40602, 52393}
X(52405) = crosspoint of X(i) and X(j) for these (i,j): {2, 43741}, {3219, 4420}
X(52405) = crosssum of X(i) and X(j) for these (i,j): {1086, 30724}, {2160, 52372}, {2170, 50354}
X(52405) = crossdifference of every pair of points on line {513, 11125}
X(52405) = barycentric product X(i)*X(j) for these {i,j}: {1, 4420}, {6, 42033}, {8, 35}, {9, 3219}, {10, 35193}, {21, 3678}, {41, 33939}, {55, 319}, {60, 7206}, {72, 11107}, {78, 6198}, {100, 35057}, {190, 9404}, {200, 1442}, {220, 17095}, {284, 3969}, {306, 41502}, {312, 2174}, {321, 35192}, {323, 36910}, {341, 1399}, {346, 2003}, {644, 14838}, {943, 31938}, {1043, 2594}, {1250, 40714}, {1260, 7282}, {1334, 34016}, {1792, 1825}, {2287, 16577}, {2321, 40214}, {2323, 41226}, {2328, 40999}, {2341, 42701}, {2605, 3699}, {3578, 33635}, {3647, 32635}, {3701, 17104}, {3718, 14975}, {3939, 4467}, {4102, 17454}, {4570, 6741}, {5546, 7265}, {7058, 21794}, {7161, 52126}, {10638, 40713}, {35194, 44687}, {44688, 46073}, {44689, 46077}
X(52405) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52374}, {8, 20565}, {9, 30690}, {31, 52372}, {35, 7}, {37, 43682}, {41, 2160}, {42, 52382}, {55, 79}, {101, 38340}, {200, 52344}, {210, 6757}, {212, 7100}, {219, 52381}, {220, 7110}, {228, 52390}, {284, 52393}, {319, 6063}, {323, 17078}, {644, 15455}, {692, 26700}, {1250, 554}, {1253, 7073}, {1334, 8818}, {1399, 269}, {1442, 1088}, {2003, 279}, {2174, 57}, {2175, 6186}, {2194, 52375}, {2318, 52388}, {2328, 3615}, {2594, 3668}, {2605, 3676}, {3219, 85}, {3678, 1441}, {3939, 6742}, {3969, 349}, {4420, 75}, {6149, 1443}, {6198, 273}, {6741, 21207}, {7186, 7185}, {7206, 34388}, {9404, 514}, {10638, 1081}, {11107, 286}, {14838, 24002}, {14975, 34}, {16577, 1446}, {17104, 1014}, {17454, 553}, {21741, 1427}, {21794, 6354}, {22342, 1439}, {32641, 47317}, {33939, 20567}, {35057, 693}, {35192, 81}, {35193, 86}, {36910, 94}, {40214, 1434}, {41502, 27}, {42033, 76}, {52371, 2166}
X(52405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3230, 16488}, {9, 219, 2323}, {218, 2256, 1449}, {219, 220, 9}, {644, 2287, 2321}, {2176, 2273, 16470}, {3690, 26885, 5285}, {30556, 30557, 5904}

X(52406) = X(2)X(30701)∩X(8)X(40962)

Barycentrics    b*c*(-a + b + c)^2*(-a^2 + b^2 + c^2) : :
Barycentrics    Cos[A]*Csc[A/2]^4 : :

X(52406) lies on these lines: {2, 30701}, {8, 40962}, {43, 37042}, {75, 23681}, {92, 42709}, {190, 1763}, {200, 341}, {304, 305}, {312, 2321}, {322, 2064}, {345, 3694}, {668, 18750}, {1264, 44189}, {1427, 40883}, {2999, 4360}, {3699, 4123}, {3952, 20243}, {3961, 4737}, {4033, 20928}, {4095, 11679}, {4417, 46738}, {5795, 44720}, {17143, 19814}, {18152, 28659}, {20106, 33937}, {21621, 49773}, {34403, 42699}, {37266, 44153}

X(52406) = isotomic conjugate of X(1435)
X(52406) = isotomic conjugate of the isogonal conjugate of X(3692)
X(52406) = isotomic conjugate of the polar conjugate of X(341)
X(52406) = X(i)-cross conjugate of X(j) for these (i,j): {1265, 3718}, {3692, 341}
X(52406) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1398}, {19, 1106}, {25, 1407}, {31, 1435}, {32, 1119}, {33, 7366}, {34, 604}, {56, 608}, {57, 1395}, {112, 7250}, {222, 7337}, {225, 16947}, {269, 1973}, {278, 1397}, {279, 1974}, {331, 41280}, {560, 1847}, {607, 7023}, {667, 32714}, {738, 2212}, {1042, 1474}, {1096, 7099}, {1262, 42067}, {1333, 1426}, {1357, 7115}, {1396, 1402}, {1408, 1880}, {1410, 5317}, {1413, 3209}, {1415, 43923}, {1425, 36420}, {1427, 2203}, {1919, 36118}, {1980, 13149}, {2189, 7143}, {2207, 7053}, {2969, 23979}, {3195, 6612}, {3248, 7128}, {3937, 23985}, {6611, 7151}, {7056, 36417}, {7216, 32676}, {7342, 8736}, {22096, 23984}, {32674, 43924}
X(52406) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 608}, {2, 1435}, {6, 1106}, {9, 1398}, {19, 6552}, {25, 24771}, {34, 3161}, {37, 1426}, {244, 3239}, {269, 6337}, {345, 28039}, {604, 11517}, {649, 7358}, {656, 1015}, {1042, 51574}, {1119, 6376}, {1146, 43923}, {1357, 40628}, {1395, 5452}, {1396, 40605}, {1407, 6505}, {1847, 6374}, {1973, 6600}, {2207, 23050}, {2968, 6591}, {3669, 40626}, {3772, 36570}, {6338, 7177}, {6503, 7099}, {6631, 32714}, {7216, 15526}, {7250, 34591}, {9296, 36118}, {35072, 43924}, {40618, 43932}
X(52406) = cevapoint of X(i) and X(j) for these (i,j): {306, 42699}, {1265, 30681}
X(52406) = barycentric product X(i)*X(j) for these {i,j}: {8, 3718}, {69, 341}, {75, 1265}, {76, 3692}, {78, 3596}, {85, 30681}, {190, 15416}, {200, 305}, {212, 40363}, {219, 28659}, {220, 40364}, {304, 346}, {312, 345}, {313, 1792}, {314, 3710}, {318, 1264}, {332, 3701}, {348, 30693}, {525, 7258}, {561, 1260}, {646, 6332}, {1043, 20336}, {1253, 40050}, {1502, 1802}, {1812, 30713}, {2287, 40071}, {2318, 40072}, {2327, 27801}, {2968, 7035}, {3267, 7259}, {3694, 28660}, {3699, 35518}, {3719, 7017}, {3926, 7101}, {4033, 15411}, {4076, 17880}, {4397, 4561}, {4571, 35519}, {5423, 7182}, {6558, 15413}, {7058, 52369}, {7256, 14208}, {7257, 52355}, {31625, 34591}
X(52406) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1398}, {2, 1435}, {3, 1106}, {8, 34}, {9, 608}, {10, 1426}, {33, 7337}, {55, 1395}, {63, 1407}, {69, 269}, {72, 1042}, {75, 1119}, {76, 1847}, {77, 7023}, {78, 56}, {190, 32714}, {200, 25}, {201, 7143}, {212, 1397}, {219, 604}, {220, 1973}, {222, 7366}, {271, 1413}, {283, 1408}, {304, 279}, {305, 1088}, {306, 1427}, {312, 278}, {318, 1118}, {326, 7053}, {332, 1014}, {333, 1396}, {341, 4}, {345, 57}, {346, 19}, {348, 738}, {394, 7099}, {480, 2212}, {521, 43924}, {522, 43923}, {525, 7216}, {644, 32674}, {646, 653}, {656, 7250}, {668, 36118}, {728, 607}, {1016, 7128}, {1021, 43925}, {1043, 28}, {1253, 1974}, {1259, 603}, {1260, 31}, {1264, 77}, {1265, 1}, {1332, 1461}, {1792, 58}, {1802, 32}, {1812, 1412}, {1978, 13149}, {2193, 16947}, {2287, 1474}, {2310, 42067}, {2318, 1402}, {2321, 1880}, {2322, 5317}, {2324, 3209}, {2326, 36420}, {2327, 1333}, {2328, 2203}, {2638, 22096}, {2968, 244}, {3059, 40983}, {3239, 6591}, {3270, 3248}, {3596, 273}, {3610, 8898}, {3682, 1410}, {3692, 6}, {3694, 1400}, {3695, 1254}, {3699, 108}, {3701, 225}, {3710, 65}, {3717, 1876}, {3718, 7}, {3719, 222}, {3926, 7177}, {3965, 2354}, {3998, 52373}, {4012, 40987}, {4025, 43932}, {4076, 7012}, {4082, 1824}, {4158, 7138}, {4163, 18344}, {4171, 2489}, {4385, 7103}, {4397, 7649}, {4515, 2333}, {4561, 934}, {4563, 4637}, {4571, 109}, {4578, 8750}, {4587, 1415}, {4723, 1877}, {5423, 33}, {6332, 3669}, {6516, 6614}, {6558, 1783}, {6559, 8751}, {6735, 1875}, {6736, 1828}, {6737, 40985}, {7004, 1357}, {7046, 1096}, {7079, 2207}, {7080, 208}, {7101, 393}, {7182, 479}, {7256, 162}, {7258, 648}, {7259, 112}, {7360, 1430}, {8611, 7180}, {15411, 1019}, {15416, 514}, {15742, 24033}, {17880, 1358}, {19799, 7365}, {20336, 3668}, {23983, 3942}, {24026, 2969}, {24031, 3937}, {26942, 7147}, {27382, 3213}, {27509, 28017}, {28659, 331}, {30681, 9}, {30693, 281}, {30713, 40149}, {34591, 1015}, {35518, 3676}, {40071, 1446}, {41081, 6612}, {42699, 36908}, {44189, 1422}, {44722, 1420}, {51379, 1457}, {51978, 46883}, {52346, 44696}, {52355, 4017}, {52369, 6354}, {52387, 1425}, {52396, 1439}
X(52406) = {X(306),X(19799)}-harmonic conjugate of X(304)

X(52407) = X(1)X(1399)∩X(3)X(73)

Barycentrics    a^3*(a^2 - b^2 - c^2)*(a^2 - b^2 + b*c - c^2) : :
Barycentrics    (1 - 2*Cos[A])*Cos[A]*Sin[A] : :

X(52407) lies on these lines: {1, 1399}, {3, 73}, {5, 1935}, {6, 41442}, {30, 1936}, {31, 999}, {34, 37532}, {35, 5399}, {36, 1464}, {47, 56}, {57, 5398}, {58, 942}, {63, 1060}, {77, 21165}, {109, 517}, {140, 3074}, {171, 495}, {184, 23206}, {221, 11249}, {238, 15325}, {283, 1789}, {355, 1771}, {496, 3073}, {499, 7299}, {580, 1427}, {582, 15803}, {595, 24928}, {601, 1496}, {602, 1106}, {651, 6905}, {750, 31479}, {758, 11700}, {859, 26884}, {906, 20752}, {912, 1795}, {1006, 17074}, {1038, 26921}, {1056, 17126}, {1069, 1433}, {1071, 3561}, {1147, 7335}, {1216, 7066}, {1324, 8679}, {1331, 5440}, {1364, 13754}, {1394, 5709}, {1409, 22118}, {1413, 37498}, {1414, 5088}, {1437, 18604}, {1451, 5708}, {1457, 22765}, {1458, 41345}, {1459, 4091}, {1466, 36754}, {1478, 5348}, {1497, 7373}, {1777, 12699}, {1822, 34593}, {1823, 34592}, {1870, 3218}, {2003, 5396}, {2169, 19210}, {2193, 14597}, {2252, 22123}, {2654, 13743}, {2771, 45272}, {2964, 5563}, {3072, 18990}, {3076, 3311}, {3077, 3312}, {3149, 8757}, {3211, 15905}, {3560, 41344}, {3562, 6906}, {4282, 7113}, {5122, 6610}, {5770, 34231}, {5841, 51421}, {5903, 18360}, {6638, 20764}, {6882, 43043}, {6911, 34048}, {6924, 37694}, {7004, 18455}, {7070, 7171}, {7074, 35238}, {7330, 37696}, {8614, 37564}, {9316, 36279}, {9370, 11499}, {10267, 34046}, {10526, 34030}, {10571, 26286}, {10680, 34040}, {11012, 34043}, {11334, 26892}, {11374, 37522}, {11507, 37698}, {12515, 45269}, {14529, 15654}, {14547, 51340}, {16374, 26890}, {18447, 44706}, {19624, 44858}, {21620, 50749}, {22457, 22458}, {23202, 23205}, {24430, 37729}, {24929, 37469}, {25405, 40091}, {32047, 37591}, {37541, 44414}, {37607, 37737}

X(52407) = reflection of X(1807) in X(46974)
X(52407) = isotomic conjugate of the polar conjugate of X(7113)
X(52407) = isogonal conjugate of the polar conjugate of X(3218)
X(52407) = X(i)-Ceva conjugate of X(j) for these (i,j): {1795, 3}, {1797, 48}, {3218, 7113}
X(52407) = X(i)-isoconjugate of X(j) for these (i,j): {4, 80}, {19, 18359}, {25, 20566}, {28, 15065}, {29, 52383}, {33, 18815}, {35, 6344}, {92, 2161}, {108, 52356}, {158, 1807}, {225, 6740}, {264, 6187}, {273, 52371}, {278, 36910}, {281, 2006}, {286, 34857}, {318, 1411}, {319, 18384}, {393, 52351}, {655, 3064}, {759, 41013}, {1168, 38462}, {1785, 40437}, {1824, 14616}, {1826, 24624}, {1857, 52392}, {1877, 36590}, {1896, 52391}, {2166, 6198}, {2222, 44426}, {2341, 40149}, {2501, 47318}, {6591, 36804}, {7649, 51562}, {18344, 35174}, {32675, 46110}, {36125, 51975}
X(52407) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 18359}, {44, 46109}, {80, 36033}, {92, 40584}, {264, 40612}, {318, 35204}, {1147, 1807}, {2161, 22391}, {5081, 6149}, {6198, 11597}, {6505, 20566}, {7113, 37770}, {15065, 40591}, {34586, 41013}, {35128, 46110}, {38983, 52356}, {38984, 44426}
X(52407) = crosspoint of X(i) and X(j) for these (i,j): {1437, 36058}, {52381, 52392}
X(52407) = crosssum of X(i) and X(j) for these (i,j): {1, 1727}, {1824, 14571}, {38462, 41013}
X(52407) = crossdifference of every pair of points on line {1826, 3064}
X(52407) = barycentric product X(i)*X(j) for these {i,j}: {1, 22128}, {3, 3218}, {36, 63}, {48, 320}, {69, 7113}, {77, 2323}, {184, 20924}, {190, 22379}, {212, 17078}, {214, 1797}, {219, 1443}, {222, 4511}, {255, 17923}, {283, 18593}, {295, 27950}, {307, 4282}, {323, 7100}, {348, 2361}, {394, 1870}, {603, 32851}, {654, 6516}, {758, 1790}, {860, 18604}, {906, 4453}, {1227, 32659}, {1331, 3960}, {1437, 3936}, {1444, 2245}, {1459, 4585}, {1464, 1812}, {1795, 16586}, {1796, 4973}, {1813, 3738}, {1835, 6514}, {1983, 4025}, {2193, 41804}, {3724, 17206}, {3904, 36059}, {3977, 16944}, {4091, 4242}, {4561, 21758}, {4575, 4707}, {4592, 21828}, {5081, 7125}, {5440, 40215}, {6149, 52381}, {8552, 13486}, {9247, 40075}, {17515, 40152}, {22115, 30690}, {34544, 52392}, {36058, 51583}, {39152, 44719}, {39153, 44718}
X(52407) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 18359}, {36, 92}, {48, 80}, {50, 6198}, {63, 20566}, {71, 15065}, {184, 2161}, {212, 36910}, {214, 46109}, {222, 18815}, {255, 52351}, {320, 1969}, {577, 1807}, {603, 2006}, {652, 52356}, {654, 44426}, {906, 51562}, {1331, 36804}, {1409, 52383}, {1437, 24624}, {1443, 331}, {1464, 40149}, {1790, 14616}, {1813, 35174}, {1870, 2052}, {1983, 1897}, {2160, 6344}, {2193, 6740}, {2200, 34857}, {2245, 41013}, {2323, 318}, {2361, 281}, {3218, 264}, {3724, 1826}, {3738, 46110}, {3960, 46107}, {4053, 7141}, {4282, 29}, {4511, 7017}, {4575, 47318}, {6516, 46405}, {7100, 94}, {7113, 4}, {7125, 52392}, {8648, 3064}, {9247, 6187}, {13486, 46456}, {14578, 40437}, {14597, 45926}, {16944, 6336}, {17455, 38462}, {19627, 14975}, {20924, 18022}, {21758, 7649}, {21828, 24006}, {22115, 3219}, {22128, 75}, {22356, 51975}, {22379, 514}, {27950, 40717}, {30690, 18817}, {32659, 1168}, {32660, 2222}, {34544, 5081}, {36059, 655}, {46112, 46077}, {46113, 46073}, {52059, 1870}
X(52407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 23070, 73}, {3, 23071, 22350}, {3, 23072, 3157}, {36, 6149, 2361}, {255, 603, 3}, {601, 1496, 3295}, {1935, 3075, 5}, {4303, 22361, 3}

X(52408) = X(1)X(2361)∩X(3)X(73)

Barycentrics    a^3*(a^2 - b^2 - c^2)*(a^2 - b^2 - b*c - c^2) : :
Barycentrics    Cos[A]*(1 + 2*Cos[A])*Sin[A] : :

X(52408) lies on these lines: {1, 2361}, {3, 73}, {5, 1936}, {9, 37696}, {12, 13408}, {30, 1935}, {31, 3295}, {34, 37584}, {35, 500}, {47, 55}, {57, 582}, {58, 24929}, {63, 1062}, {71, 22123}, {72, 283}, {81, 943}, {109, 227}, {140, 3075}, {171, 49743}, {201, 18447}, {219, 1069}, {221, 35239}, {228, 1437}, {238, 496}, {394, 11517}, {484, 18360}, {495, 3072}, {498, 5348}, {580, 942}, {595, 9957}, {601, 1253}, {602, 999}, {605, 31474}, {643, 7283}, {651, 3651}, {906, 52370}, {1006, 3562}, {1040, 24467}, {1058, 17127}, {1066, 41345}, {1147, 6056}, {1167, 37623}, {1214, 1794}, {1216, 1364}, {1394, 3587}, {1418, 13329}, {1419, 35242}, {1451, 15934}, {1479, 7299}, {1497, 6767}, {1724, 5722}, {1725, 9627}, {1771, 26446}, {1780, 4641}, {2174, 17104}, {2193, 3990}, {2328, 31445}, {2654, 7489}, {2964, 3746}, {3073, 15171}, {3076, 3312}, {3077, 3311}, {3219, 6198}, {3561, 37565}, {3682, 22136}, {3781, 23116}, {4055, 22139}, {4337, 8614}, {5247, 37730}, {5285, 48882}, {5399, 10902}, {5694, 45272}, {5709, 37697}, {5760, 11374}, {6883, 41344}, {6985, 34048}, {7066, 13754}, {7070, 7330}, {7074, 11248}, {7193, 22458}, {7580, 8757}, {7688, 34043}, {8144, 24430}, {8679, 23850}, {9645, 24320}, {11012, 34586}, {14547, 36750}, {15796, 40602}, {16287, 26890}, {16453, 26884}, {18455, 44706}, {18990, 37570}, {20760, 42463}, {20764, 20793}, {22074, 22118}, {22115, 22342}, {22133, 51574}, {22341, 35200}, {22345, 23202}, {22376, 36058}, {22457, 23171}, {25885, 31493}, {31793, 38857}, {31837, 46974}, {37694, 51281}, {40591, 44709}

X(52408) = isotomic conjugate of the polar conjugate of X(2174)
X(52408) = isogonal conjugate of the polar conjugate of X(3219)
X(52408) = X(i)-Ceva conjugate of X(j) for these (i,j): {1794, 3}, {1796, 48}, {3219, 2174}, {35193, 35}
X(52408) = X(22342)-cross conjugate of X(35)
X(52408) = X(i)-isoconjugate of X(j) for these (i,j): {4, 79}, {11, 34922}, {19, 30690}, {25, 20565}, {27, 8818}, {28, 6757}, {29, 52382}, {34, 52344}, {36, 6344}, {92, 2160}, {158, 7100}, {225, 3615}, {264, 6186}, {273, 7073}, {278, 7110}, {281, 52374}, {318, 52372}, {320, 18384}, {393, 52381}, {451, 30602}, {1172, 43682}, {1826, 52393}, {1870, 2166}, {1896, 52390}, {1989, 17923}, {3064, 38340}, {4242, 43082}, {6591, 15455}, {6742, 7649}, {8747, 52388}, {13486, 24006}, {21828, 46456}, {26700, 44426}, {41013, 52375}
X(52408) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 30690}, {79, 36033}, {1147, 7100}, {1870, 11597}, {2160, 22391}, {6344, 15898}, {6505, 20565}, {6757, 40591}, {8287, 46107}, {11517, 52344}, {17923, 34544}
X(52408) = barycentric product X(i)*X(j) for these {i,j}: {3, 3219}, {35, 63}, {48, 319}, {69, 2174}, {72, 40214}, {78, 2003}, {97, 35194}, {184, 33939}, {190, 23226}, {212, 17095}, {219, 1442}, {222, 4420}, {228, 34016}, {283, 16577}, {306, 17104}, {307, 35192}, {323, 1807}, {332, 21741}, {333, 22342}, {345, 1399}, {394, 6198}, {603, 42033}, {906, 4467}, {1214, 35193}, {1331, 14838}, {1332, 2605}, {1437, 3969}, {1790, 3678}, {1794, 16585}, {1796, 3647}, {1812, 2594}, {1813, 35057}, {1825, 6514}, {2193, 40999}, {2289, 7282}, {3926, 14975}, {4567, 22094}, {4575, 7265}, {6149, 52351}, {6516, 9404}, {11107, 40152}, {18160, 32656}, {18359, 22115}, {41502, 52385}, {44718, 46073}, {44719, 46077}
X(52408) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 30690}, {35, 92}, {48, 79}, {50, 1870}, {63, 20565}, {71, 6757}, {73, 43682}, {184, 2160}, {212, 7110}, {219, 52344}, {228, 8818}, {255, 52381}, {319, 1969}, {577, 7100}, {603, 52374}, {906, 6742}, {1331, 15455}, {1399, 278}, {1409, 52382}, {1437, 52393}, {1442, 331}, {1807, 94}, {2003, 273}, {2149, 34922}, {2161, 6344}, {2174, 4}, {2193, 3615}, {2594, 40149}, {2605, 17924}, {3219, 264}, {3990, 52388}, {4420, 7017}, {6149, 17923}, {6198, 2052}, {7202, 2973}, {9247, 6186}, {9404, 44426}, {14838, 46107}, {14975, 393}, {17104, 27}, {18359, 18817}, {21741, 225}, {21824, 2970}, {22094, 16732}, {22115, 3218}, {22342, 226}, {23226, 514}, {32660, 26700}, {32661, 13486}, {33939, 18022}, {35057, 46110}, {35192, 29}, {35193, 31623}, {35194, 324}, {36059, 38340}, {40214, 286}, {41502, 1896}, {46112, 39152}, {46113, 39153}
X(52408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22117, 3157}, {3, 22161, 11573}, {3, 23070, 4303}, {3, 23071, 73}, {35, 2003, 500}, {35, 6149, 1399}, {212, 255, 3}, {283, 1331, 72}, {602, 1496, 999}, {1936, 3074, 5}, {3219, 6198, 35194}, {22350, 22361, 3}

X(52409) = X(8)X(80)∩X(10)X(94)

Barycentrics    b*c*(-a + b + c)*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :
Barycentrics    (Cot[A] + Csc[A]) / (2*Cos[A] - 1) : :

X(52409) lies on these lines: {8, 80}, {10, 94}, {75, 311}, {100, 43655}, {280, 5552}, {318, 7141}, {346, 27546}, {759, 7081}, {1043, 3699}, {1219, 2006}, {1222, 1411}, {1807, 4358}, {2222, 2370}, {3153, 5080}, {3992, 32849}, {4385, 45926}, {4723, 36590}, {4768, 52356}, {5176, 38954}, {7026, 44690}, {7043, 44691}, {7080, 36626}, {12532, 38955}, {18025, 30806}, {21296, 52392}

X(52409) = isotomic conjugate of X(1443)
X(52409) = isotomic conjugate of the isogonal conjugate of X(52371)
X(52409) = X(20566)-Ceva conjugate of X(18359)
X(52409) = X(i)-cross conjugate of X(j) for these (i,j): {2325, 312}, {6735, 8}, {36910, 18359}
X(52409) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1443}, {32, 17078}, {36, 56}, {50, 52374}, {57, 7113}, {58, 1464}, {80, 41282}, {108, 22379}, {214, 1417}, {269, 2361}, {320, 1397}, {603, 1870}, {604, 3218}, {608, 22128}, {651, 21758}, {654, 1461}, {758, 1408}, {934, 8648}, {1014, 3724}, {1106, 4511}, {1319, 16944}, {1333, 18593}, {1404, 40215}, {1407, 2323}, {1410, 17515}, {1412, 2245}, {1415, 3960}, {1427, 4282}, {1437, 1835}, {1983, 3669}, {2006, 52059}, {2206, 41804}, {3936, 16947}, {4565, 21828}, {4881, 16945}, {6149, 52372}, {7051, 19373}, {40075, 41280}
X(52409) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 36}, {1, 36909}, {2, 1443}, {8, 4881}, {10, 1464}, {37, 18593}, {56, 15898}, {654, 35508}, {1145, 34586}, {1146, 3960}, {1577, 4089}, {1870, 7952}, {2245, 40599}, {2323, 24771}, {2361, 6600}, {2968, 3738}, {3161, 3218}, {4453, 40624}, {4511, 6552}, {5452, 7113}, {6376, 17078}, {8648, 14714}, {14993, 52372}, {21758, 38991}, {22379, 38983}, {40603, 41804}
X(52409) = cevapoint of X(i) and X(j) for these (i,j): {3701, 4723}, {4768, 24026}
X(52409) = trilinear pole of line {2321, 3239}
X(52409) = barycentric product X(i)*X(j) for these {i,j}: {8, 18359}, {9, 20566}, {75, 36910}, {76, 52371}, {80, 312}, {94, 4420}, {190, 52356}, {313, 2341}, {318, 52351}, {321, 6740}, {333, 15065}, {341, 2006}, {346, 18815}, {522, 36804}, {655, 4397}, {759, 30713}, {1807, 7017}, {2161, 3596}, {2166, 42033}, {2321, 14616}, {3239, 35174}, {3701, 24624}, {3900, 46405}, {4086, 47318}, {4358, 36590}, {4391, 51562}, {4997, 51975}, {6187, 28659}, {7101, 52392}, {23978, 52377}, {28654, 52380}, {28660, 34857}, {41226, 52344}
X(52409) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1443}, {8, 3218}, {9, 36}, {10, 18593}, {37, 1464}, {55, 7113}, {75, 17078}, {78, 22128}, {80, 57}, {200, 2323}, {210, 2245}, {220, 2361}, {281, 1870}, {312, 320}, {318, 17923}, {321, 41804}, {341, 32851}, {346, 4511}, {522, 3960}, {652, 22379}, {655, 934}, {657, 8648}, {663, 21758}, {759, 1412}, {1320, 40215}, {1334, 3724}, {1411, 1407}, {1793, 1790}, {1807, 222}, {1826, 1835}, {1989, 52372}, {2006, 269}, {2161, 56}, {2166, 52374}, {2222, 1461}, {2316, 16944}, {2321, 758}, {2322, 17515}, {2325, 214}, {2328, 4282}, {2341, 58}, {2361, 52059}, {3161, 4881}, {3239, 3738}, {3596, 20924}, {3685, 27950}, {3686, 4973}, {3689, 17455}, {3699, 4585}, {3701, 3936}, {3900, 654}, {3939, 1983}, {4007, 4880}, {4024, 51645}, {4041, 21828}, {4053, 3028}, {4086, 4707}, {4358, 41801}, {4391, 4453}, {4397, 3904}, {4420, 323}, {4671, 36589}, {4723, 51583}, {4858, 4089}, {4873, 4867}, {6057, 4053}, {6187, 604}, {6735, 16586}, {6740, 81}, {7026, 37772}, {7043, 37773}, {7101, 5081}, {7113, 41282}, {7126, 19373}, {14147, 34921}, {14616, 1434}, {15065, 226}, {18359, 7}, {18815, 279}, {19551, 7051}, {20566, 85}, {23615, 46384}, {24624, 1014}, {28659, 40075}, {30713, 35550}, {34079, 1408}, {34857, 1400}, {35174, 658}, {36590, 88}, {36804, 664}, {36815, 1429}, {36910, 1}, {40172, 1404}, {40437, 34051}, {41226, 1442}, {46405, 4569}, {47318, 1414}, {51562, 651}, {51975, 3911}, {52351, 77}, {52356, 514}, {52371, 6}, {52377, 1262}, {52380, 593}, {52383, 1427}, {52391, 52373}, {52392, 7177}
X(52409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {80, 15065, 18359}, {15065, 51975, 80}

X(52410) = X(6)X(1604)∩X(32)X(604)

Barycentrics    a^4*(a + b - c)^2*(a - b + c)^2 : :
Barycentrics    (Cos[A] - 1)^2*Sin[A]^2 : :

X(52410) lies on these lines: {6, 1604}, {32, 604}, {39, 2286}, {48, 5065}, {56, 478}, {101, 23185}, {109, 21769}, {269, 40746}, {279, 52376}, {560, 1397}, {571, 52059}, {574, 2197}, {577, 7113}, {603, 2300}, {608, 1015}, {713, 934}, {1333, 1407}, {1400, 5042}, {1415, 16946}, {1436, 14936}, {1455, 20227}, {1461, 34077}, {1470, 2092}, {1950, 2241}, {1974, 22096}, {2174, 33871}, {2178, 23980}, {2242, 2285}, {7117, 46432}, {8736, 9651}, {9306, 23086}, {14642, 39687}, {34880, 40590}

X(52410) = isogonal conjugate of the complement of X(46716)
X(52410) = isogonal conjugate of the isotomic conjugate of X(1407)
X(52410) = isogonal conjugate of the polar conjugate of X(1398)
X(52410) = X(1106)-Ceva conjugate of X(1397)
X(52410) = X(i)-isoconjugate of X(j) for these (i,j): {2, 341}, {7, 30693}, {8, 312}, {9, 3596}, {21, 30713}, {41, 40363}, {55, 28659}, {69, 7101}, {75, 346}, {76, 200}, {78, 7017}, {85, 5423}, {92, 1265}, {190, 4397}, {210, 28660}, {220, 561}, {264, 3692}, {273, 30681}, {274, 4082}, {281, 3718}, {304, 7046}, {305, 7079}, {310, 4515}, {313, 2287}, {314, 2321}, {318, 345}, {321, 1043}, {333, 3701}, {480, 20567}, {522, 646}, {523, 7258}, {556, 7027}, {644, 35519}, {645, 4086}, {657, 6386}, {668, 3239}, {670, 4171}, {693, 6558}, {728, 6063}, {765, 23978}, {850, 7259}, {1016, 24026}, {1021, 27808}, {1089, 7058}, {1098, 28654}, {1146, 7035}, {1240, 3965}, {1253, 1502}, {1260, 1969}, {1334, 40072}, {1577, 7256}, {1802, 18022}, {1897, 15416}, {1928, 14827}, {1978, 3900}, {2310, 31625}, {2322, 20336}, {2328, 27801}, {3261, 4578}, {3263, 6559}, {3680, 44723}, {3694, 44130}, {3699, 4391}, {3700, 7257}, {3702, 4102}, {3710, 31623}, {3717, 36796}, {3975, 4518}, {4033, 7253}, {4076, 4858}, {4087, 4876}, {4110, 7155}, {4130, 4572}, {4148, 4583}, {4163, 4554}, {4183, 40071}, {4451, 17787}, {4524, 4602}, {4571, 46110}, {4582, 4768}, {4601, 52335}, {4723, 4997}, {6057, 52379}, {6555, 40014}, {6556, 18743}, {6557, 44720}, {6602, 41283}, {6632, 42455}, {6735, 36795}, {6736, 32017}, {7071, 40364}, {7080, 34404}, {9447, 44159}, {18155, 30730}, {23104, 31615}, {27424, 27538}, {42033, 52344}
X(52410) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 6609}, {206, 346}, {220, 40368}, {223, 28659}, {341, 32664}, {478, 3596}, {513, 23978}, {1265, 22391}, {1502, 17113}, {3160, 40363}, {14827, 40369}, {15267, 28654}, {15416, 34467}, {27801, 36908}, {30713, 40611}
X(52410) = cevapoint of X(1919) and X(22096)
X(52410) = crosspoint of X(i) and X(j) for these (i,j): {1106, 7366}, {1398, 1407}
X(52410) = crosssum of X(i) and X(j) for these (i,j): {6, 1603}, {341, 30693}, {346, 1265}, {1146, 4397}
X(52410) = crossdifference of every pair of points on line {4397, 42337}
X(52410) = barycentric product X(i)*X(j) for these {i,j}: {1, 1106}, {3, 1398}, {6, 1407}, {7, 1397}, {9, 7366}, {12, 7342}, {19, 7099}, {25, 7053}, {28, 1410}, {31, 269}, {32, 279}, {34, 603}, {41, 738}, {48, 1435}, {55, 7023}, {56, 56}, {57, 604}, {58, 1042}, {59, 1357}, {60, 7143}, {65, 1408}, {77, 1395}, {109, 43924}, {110, 7250}, {163, 7216}, {181, 7341}, {184, 1119}, {198, 6612}, {221, 1413}, {222, 608}, {226, 16947}, {244, 24027}, {479, 2175}, {560, 1088}, {649, 1461}, {658, 1919}, {663, 6614}, {667, 934}, {669, 4616}, {692, 43932}, {798, 4637}, {849, 1254}, {1014, 1402}, {1015, 1262}, {1086, 23979}, {1118, 7335}, {1275, 1977}, {1319, 1417}, {1333, 1427}, {1356, 7340}, {1396, 1409}, {1399, 52372}, {1400, 1412}, {1414, 51642}, {1415, 3669}, {1416, 1458}, {1420, 16945}, {1422, 2199}, {1426, 1437}, {1436, 6611}, {1439, 2203}, {1472, 4320}, {1474, 52373}, {1804, 7337}, {1847, 9247}, {1924, 4635}, {1973, 7177}, {1974, 7056}, {1980, 4569}, {2150, 7147}, {2206, 3668}, {3063, 4617}, {3248, 7045}, {3271, 7339}, {3450, 17114}, {4565, 7180}, {4619, 21143}, {6063, 41280}, {6066, 41292}, {7153, 41526}, {9447, 23062}, {14936, 23971}, {22383, 32714}, {36059, 43923}, {41281, 41283}, {41286, 41287}, {41288, 41289}
X(52410) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 40363}, {31, 341}, {32, 346}, {41, 30693}, {56, 3596}, {57, 28659}, {163, 7258}, {184, 1265}, {269, 561}, {279, 1502}, {479, 41283}, {560, 200}, {603, 3718}, {604, 312}, {608, 7017}, {667, 4397}, {738, 20567}, {934, 6386}, {1014, 40072}, {1015, 23978}, {1042, 313}, {1088, 1928}, {1106, 75}, {1119, 18022}, {1262, 31625}, {1356, 4092}, {1357, 34387}, {1395, 318}, {1397, 8}, {1398, 264}, {1400, 30713}, {1402, 3701}, {1407, 76}, {1408, 314}, {1410, 20336}, {1412, 28660}, {1415, 646}, {1427, 27801}, {1428, 4087}, {1435, 1969}, {1461, 1978}, {1501, 220}, {1576, 7256}, {1917, 1253}, {1918, 4082}, {1919, 3239}, {1924, 4171}, {1973, 7101}, {1974, 7046}, {1977, 1146}, {1980, 3900}, {2175, 5423}, {2205, 4515}, {2206, 1043}, {3248, 24026}, {3249, 42462}, {4616, 4609}, {4637, 4602}, {6063, 44159}, {6612, 44190}, {6614, 4572}, {7023, 6063}, {7053, 305}, {7056, 40050}, {7099, 304}, {7143, 34388}, {7177, 40364}, {7216, 20948}, {7250, 850}, {7335, 1264}, {7341, 18021}, {7342, 261}, {7366, 85}, {8027, 42455}, {9233, 14827}, {9247, 3692}, {9426, 4524}, {9447, 728}, {9448, 480}, {14575, 1260}, {16947, 333}, {22096, 2968}, {22383, 15416}, {23979, 1016}, {24027, 7035}, {32739, 6558}, {38859, 40088}, {41280, 55}, {41281, 2175}, {41286, 9448}, {41526, 4110}, {42067, 21666}, {43924, 35519}, {43932, 40495}, {44162, 7071}, {51642, 4086}, {52373, 40071}
X(52410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {48, 5065, 14827}, {56, 478, 17053}, {109, 38855, 21769}, {604, 2199, 20228}

X(52411) = X(1)X(1950)∩X(3)X(2197)

Barycentrics    a^4*(a + b - c)*(a - b + c)*(a^2 - b^2 - c^2) : :
Barycentrics    Sin[A]^4/(1 + Sec[A]) : :

X(52411) lies on these lines: {1, 1950}, {3, 2197}, {6, 909}, {7, 4565}, {31, 2208}, {32, 604}, {41, 5065}, {48, 577}, {56, 608}, {73, 22054}, {172, 2285}, {184, 23197}, {221, 37519}, {222, 1790}, {255, 22074}, {478, 2178}, {571, 7113}, {607, 1436}, {610, 1951}, {906, 20818}, {1319, 5301}, {1395, 40956}, {1397, 2206}, {1400, 5019}, {1404, 16946}, {1405, 5042}, {1455, 1880}, {1968, 7120}, {2171, 2242}, {2174, 5063}, {2289, 20752}, {2327, 22127}, {3215, 3990}, {4282, 10571}, {5193, 16488}, {7053, 32658}, {7124, 15905}, {7354, 8736}, {10607, 23144}, {14575, 22096}, {18954, 46010}, {21794, 31451}, {21933, 51422}, {32659, 32660}, {33882, 38296}, {34046, 37504}

X(52411) = isogonal conjugate of X(7017)
X(52411) = isogonal conjugate of the isotomic conjugate of X(222)
X(52411) = isotomic conjugate of the polar conjugate of X(1397)
X(52411) = isogonal conjugate of the polar conjugate of X(56)
X(52411) = polar conjugate of the isotomic conjugate of X(7335)
X(52411) = X(i)-Ceva conjugate of X(j) for these (i,j): {56, 1397}, {222, 7335}, {603, 184}, {1333, 604}, {1415, 22383}, {44717, 36059}
X(52411) = X(i)-cross conjugate of X(j) for these (i,j): {9247, 184}, {23196, 3}
X(52411) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7017}, {2, 318}, {4, 312}, {7, 7101}, {8, 92}, {9, 264}, {10, 31623}, {19, 3596}, {25, 28659}, {27, 3701}, {28, 30713}, {29, 321}, {33, 76}, {37, 44130}, {41, 18022}, {55, 1969}, {75, 281}, {78, 2052}, {85, 7046}, {100, 46110}, {158, 345}, {190, 44426}, {200, 331}, {210, 44129}, {212, 18027}, {270, 28654}, {273, 346}, {276, 7069}, {278, 341}, {286, 2321}, {304, 1857}, {306, 1896}, {313, 1172}, {314, 1826}, {322, 7003}, {324, 44687}, {329, 7020}, {333, 41013}, {349, 4183}, {393, 3718}, {459, 52346}, {461, 40023}, {470, 44690}, {471, 44691}, {522, 6335}, {561, 607}, {643, 14618}, {644, 46107}, {645, 24006}, {646, 7649}, {648, 4086}, {653, 4397}, {668, 3064}, {811, 3700}, {823, 52355}, {1043, 40149}, {1089, 46103}, {1093, 3719}, {1119, 30693}, {1259, 6521}, {1264, 6520}, {1320, 46109}, {1441, 2322}, {1502, 2212}, {1577, 36797}, {1783, 35519}, {1785, 36795}, {1824, 28660}, {1847, 5423}, {1861, 36796}, {1897, 4391}, {1973, 40363}, {1978, 18344}, {2185, 7141}, {2299, 27801}, {2326, 34388}, {2333, 40072}, {2501, 7257}, {3239, 18026}, {3699, 17924}, {3900, 46404}, {4041, 6331}, {4092, 46254}, {4123, 43678}, {4163, 13149}, {4564, 21666}, {4723, 6336}, {4858, 15742}, {4876, 40717}, {4997, 38462}, {5081, 18359}, {6059, 40364}, {6063, 7079}, {6528, 8611}, {6734, 40447}, {6735, 16082}, {7012, 23978}, {7035, 8735}, {7071, 20567}, {7140, 52379}, {7156, 41530}, {7952, 34404}, {8056, 44721}, {8748, 20336}, {9447, 44161}, {13426, 46745}, {13454, 46744}, {14942, 46108}, {15416, 36127}, {15466, 44692}, {15628, 40703}, {16081, 44694}, {20928, 43742}, {24026, 46102}, {30710, 46878}, {33299, 46104}, {33672, 40838}, {36804, 44428}, {40971, 44190}, {44189, 47372}, {44693, 46106}
X(52411) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 7017}, {6, 3596}, {8, 22391}, {206, 281}, {223, 1969}, {226, 27801}, {264, 478}, {312, 36033}, {318, 32664}, {331, 6609}, {345, 1147}, {607, 40368}, {1264, 37867}, {3160, 18022}, {3700, 17423}, {4391, 34467}, {6337, 40363}, {6505, 28659}, {8054, 46110}, {18027, 40837}, {30713, 40591}, {35519, 39006}, {40589, 44130}
X(52411) = cevapoint of X(3) and X(23159)
X(52411) = crosspoint of X(i) and X(j) for these (i,j): {56, 222}, {603, 7099}, {1415, 23979}, {36059, 44717}
X(52411) = crosssum of X(i) and X(j) for these (i,j): {8, 281}, {318, 7101}, {4391, 23978}, {8735, 44426}
X(52411) = crossdifference of every pair of points on line {2804, 4397}
X(52411) = barycentric product X(i)*X(j) for these {i,j}: {1, 603}, {3, 56}, {4, 7335}, {6, 222}, {7, 184}, {9, 7099}, {19, 7125}, {21, 1410}, {25, 1804}, {28, 22341}, {31, 77}, {32, 348}, {34, 255}, {41, 7177}, {48, 57}, {54, 30493}, {55, 7053}, {58, 73}, {59, 3937}, {60, 1425}, {63, 604}, {65, 1437}, {69, 1397}, {71, 1412}, {72, 1408}, {78, 1106}, {81, 1409}, {84, 7114}, {85, 9247}, {108, 23224}, {109, 1459}, {162, 51641}, {163, 51640}, {201, 849}, {212, 269}, {219, 1407}, {221, 1433}, {228, 1014}, {241, 32658}, {248, 43034}, {268, 6611}, {270, 7138}, {278, 577}, {283, 1042}, {284, 52373}, {293, 51653}, {295, 1428}, {296, 26884}, {305, 41280}, {306, 16947}, {307, 2206}, {326, 1395}, {331, 14585}, {394, 608}, {513, 36059}, {514, 32660}, {560, 7182}, {593, 2197}, {647, 4565}, {649, 1813}, {651, 22383}, {652, 1461}, {667, 6516}, {738, 1802}, {810, 1414}, {905, 1415}, {906, 3669}, {927, 23225}, {934, 1946}, {961, 22345}, {1015, 44717}, {1037, 1473}, {1038, 1472}, {1069, 1406}, {1092, 1118}, {1119, 6056}, {1175, 39791}, {1176, 1401}, {1214, 1333}, {1259, 1398}, {1260, 7023}, {1262, 7117}, {1319, 36058}, {1331, 43924}, {1355, 47388}, {1356, 47389}, {1363, 32230}, {1365, 47390}, {1393, 2169}, {1394, 19614}, {1396, 3990}, {1399, 7100}, {1400, 1790}, {1402, 1444}, {1403, 23086}, {1404, 1797}, {1413, 7078}, {1416, 1818}, {1417, 5440}, {1423, 15373}, {1427, 2193}, {1429, 2196}, {1431, 3955}, {1434, 2200}, {1435, 2289}, {1436, 7011}, {1439, 2194}, {1455, 36055}, {1456, 36056}, {1457, 1795}, {1458, 36057}, {1462, 20752}, {1465, 14578}, {1474, 40152}, {1475, 1803}, {1476, 22344}, {1477, 20780}, {1576, 17094}, {1880, 18604}, {1973, 7183}, {1974, 7055}, {2067, 6502}, {2148, 44708}, {2149, 3942}, {2150, 37755}, {2175, 7056}, {2199, 41081}, {2203, 52385}, {2208, 7013}, {2221, 2286}, {2222, 22379}, {2720, 8677}, {2982, 14597}, {3049, 4573}, {3270, 7339}, {3292, 7316}, {3450, 23154}, {3665, 10547}, {3676, 32656}, {3690, 7341}, {3692, 7366}, {3695, 7342}, {3733, 23067}, {3911, 32659}, {3964, 7337}, {4017, 4575}, {4091, 32674}, {4558, 7180}, {4559, 7254}, {4592, 51642}, {4855, 16945}, {4998, 22096}, {5061, 17971}, {6063, 14575}, {6357, 18877}, {7004, 24027}, {7116, 7175}, {7178, 32661}, {7181, 14908}, {7251, 14376}, {8686, 23205}, {14379, 44696}, {14642, 18623}, {14827, 30682}, {15375, 42461}, {15389, 17082}, {17081, 40319}, {17104, 52390}, {20818, 40151}, {22093, 29055}, {22342, 52375}, {23226, 26700}, {23979, 26932}, {32651, 52306}, {32657, 43035}, {32714, 36054}, {35200, 51656}, {36052, 51651}, {36060, 51655}, {40050, 41281}, {40360, 41286}, {40373, 41283}, {41279, 46089}
X(52411) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3596}, {6, 7017}, {7, 18022}, {31, 318}, {32, 281}, {41, 7101}, {48, 312}, {56, 264}, {57, 1969}, {58, 44130}, {63, 28659}, {69, 40363}, {71, 30713}, {73, 313}, {77, 561}, {181, 7141}, {184, 8}, {212, 341}, {222, 76}, {228, 3701}, {255, 3718}, {278, 18027}, {305, 44159}, {348, 1502}, {560, 33}, {577, 345}, {603, 75}, {604, 92}, {608, 2052}, {649, 46110}, {667, 44426}, {810, 4086}, {906, 646}, {1092, 1264}, {1106, 273}, {1214, 27801}, {1333, 31623}, {1356, 8754}, {1357, 2973}, {1395, 158}, {1397, 4}, {1401, 1235}, {1402, 41013}, {1404, 46109}, {1407, 331}, {1408, 286}, {1409, 321}, {1410, 1441}, {1412, 44129}, {1415, 6335}, {1425, 34388}, {1428, 40717}, {1437, 314}, {1444, 40072}, {1459, 35519}, {1461, 46404}, {1501, 607}, {1576, 36797}, {1790, 28660}, {1802, 30693}, {1804, 305}, {1813, 1978}, {1917, 2212}, {1919, 3064}, {1946, 4397}, {1974, 1857}, {1977, 8735}, {1980, 18344}, {2175, 7046}, {2197, 28654}, {2200, 2321}, {2203, 1896}, {2206, 29}, {2208, 7020}, {3049, 3700}, {3052, 44721}, {3271, 21666}, {3937, 34387}, {4055, 3710}, {4100, 3719}, {4173, 4178}, {4565, 6331}, {4575, 7257}, {6056, 1265}, {6063, 44161}, {6516, 6386}, {6611, 40701}, {7053, 6063}, {7055, 40050}, {7056, 41283}, {7099, 85}, {7114, 322}, {7117, 23978}, {7125, 304}, {7177, 20567}, {7180, 14618}, {7182, 1928}, {7183, 40364}, {7193, 4087}, {7251, 17907}, {7316, 46111}, {7335, 69}, {7337, 1093}, {7366, 1847}, {9247, 9}, {9447, 7079}, {9448, 7071}, {14575, 55}, {14578, 36795}, {14585, 219}, {14600, 15628}, {15373, 27424}, {16947, 27}, {17094, 44173}, {20775, 3703}, {20818, 44723}, {22096, 11}, {22341, 20336}, {22344, 20895}, {22383, 4391}, {23067, 27808}, {23196, 1329}, {23200, 3712}, {23201, 3702}, {23202, 4723}, {23220, 2804}, {23224, 35518}, {23225, 50333}, {23606, 1259}, {23979, 46102}, {26920, 13461}, {30493, 311}, {32656, 3699}, {32658, 36796}, {32659, 4997}, {32660, 190}, {32661, 645}, {34980, 7068}, {36054, 15416}, {36059, 668}, {39201, 52355}, {39791, 1234}, {40152, 40071}, {40373, 2175}, {41280, 25}, {41281, 1974}, {41286, 44162}, {43034, 44132}, {43924, 46107}, {44088, 44707}, {44162, 6059}, {44717, 31625}, {47390, 6064}, {51640, 20948}, {51641, 14208}, {51642, 24006}, {51653, 40703}, {52373, 349}
X(52411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 2286, 2197}, {3, 22132, 22071}, {48, 603, 1409}

X(52412) = X(2)X(92)∩X(10)X(29)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + c^2) : :
Barycentrics    2 + Sec[A] : :

X(52412) lies on these lines: {2, 92}, {4, 2355}, {8, 7498}, {9, 1748}, {10, 29}, {12, 1940}, {19, 469}, {27, 1268}, {28, 5260}, {35, 11107}, {37, 18679}, {40, 52248}, {63, 7079}, {81, 1783}, {100, 4183}, {101, 21072}, {158, 498}, {162, 171}, {165, 39531}, {196, 5226}, {226, 653}, {240, 756}, {242, 427}, {243, 5432}, {286, 28653}, {306, 2322}, {318, 406}, {343, 1944}, {345, 7101}, {355, 7531}, {412, 6684}, {423, 7119}, {445, 3219}, {451, 41013}, {468, 7009}, {475, 5342}, {517, 7551}, {607, 3661}, {608, 17368}, {750, 1957}, {857, 3101}, {894, 25977}, {908, 30686}, {958, 37253}, {993, 37304}, {1013, 1376}, {1096, 5268}, {1118, 10588}, {1146, 23292}, {1148, 11374}, {1430, 17122}, {1585, 7090}, {1586, 14121}, {1698, 5125}, {1762, 20305}, {1767, 8545}, {1784, 3584}, {1824, 4213}, {1838, 3634}, {1844, 3678}, {1848, 8756}, {1851, 8889}, {1855, 26003}, {1857, 5218}, {1859, 3740}, {1861, 14004}, {1867, 3144}, {1877, 5294}, {1891, 4248}, {1895, 3085}, {1943, 11064}, {2331, 5287}, {2333, 31909}, {2994, 37645}, {3176, 5703}, {3452, 30687}, {3535, 13387}, {3536, 13386}, {3820, 37321}, {3925, 37371}, {3969, 4420}, {4359, 46108}, {4413, 35994}, {5047, 41227}, {5081, 5136}, {5090, 37055}, {5130, 16066}, {5235, 44734}, {5249, 25993}, {5251, 17515}, {5261, 44696}, {5281, 44695}, {5743, 41234}, {5744, 37276}, {6335, 20566}, {6353, 7102}, {6354, 47296}, {6513, 44179}, {7017, 32851}, {7110, 40447}, {7359, 26942}, {7466, 26251}, {7510, 38042}, {7518, 46933}, {7541, 10175}, {7719, 17308}, {8232, 40837}, {9708, 37393}, {11231, 39529}, {11341, 29576}, {14006, 32917}, {14024, 15523}, {14165, 16577}, {14571, 44307}, {16585, 17095}, {16706, 20883}, {17605, 52167}, {17902, 17916}, {17906, 33133}, {17911, 17912}, {18589, 30841}, {20205, 34234}, {21016, 31905}, {21028, 26885}, {23521, 33150}, {24603, 37389}, {25647, 41340}, {26066, 37235}, {26872, 27382}, {27132, 52288}, {27287, 46835}, {33108, 37372}, {34591, 45224}, {46110, 47794}

X(52412) = isotomic conjugate of X(52381)
X(52412) = polar conjugate of X(79)
X(52412) = polar conjugate of the isotomic conjugate of X(319)
X(52412) = polar conjugate of the isogonal conjugate of X(35)
X(52412) = X(i)-complementary conjugate of X(j) for these (i,j): {3469, 141}, {34303, 21236}
X(52412) = X(40447)-Ceva conjugate of X(92)
X(52412) = X(i)-cross conjugate of X(j) for these (i,j): {35, 319}, {445, 92}, {6198, 7282}, {16577, 3219}
X(52412) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2160}, {6, 7100}, {31, 52381}, {48, 79}, {57, 8606}, {63, 6186}, {71, 52375}, {184, 30690}, {212, 52374}, {219, 52372}, {222, 7073}, {228, 52393}, {265, 7113}, {284, 52390}, {603, 7110}, {647, 13486}, {652, 26700}, {1333, 52388}, {1400, 1789}, {1409, 3615}, {1437, 8818}, {1870, 50433}, {1946, 38340}, {2193, 52382}, {3218, 52153}, {6742, 22383}, {9247, 20565}, {14395, 36064}, {19302, 50462}, {21828, 36061}, {36296, 39152}, {36297, 39153}, {42623, 52201}
X(52412) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 52381}, {9, 7100}, {35, 22122}, {37, 52388}, {79, 1249}, {905, 8287}, {1100, 3916}, {1789, 40582}, {2160, 36103}, {3162, 6186}, {4466, 14838}, {5249, 18607}, {5452, 8606}, {7110, 7952}, {13486, 39052}, {16221, 21828}, {22058, 25639}, {22128, 40604}, {38340, 39053}, {40590, 52390}, {40837, 52374}, {47345, 52382}
X(52412) = cevapoint of X(281) and X(451)
X(52412) = crosssum of X(3) and X(23165)
X(52412) = trilinear pole of line {7265, 35057}
X(52412) = barycentric product X(i)*X(j) for these {i,j}: {4, 319}, {8, 7282}, {19, 33939}, {27, 3969}, {29, 40999}, {35, 264}, {75, 6198}, {80, 340}, {92, 3219}, {186, 20566}, {273, 4420}, {278, 42033}, {281, 17095}, {286, 3678}, {314, 1825}, {318, 1442}, {349, 41502}, {445, 40435}, {561, 14975}, {648, 7265}, {1441, 11107}, {1783, 18160}, {1826, 34016}, {1844, 40422}, {1897, 4467}, {1969, 2174}, {2003, 7017}, {2594, 44130}, {5379, 17886}, {6335, 14838}, {9404, 46404}, {14165, 52351}, {16577, 31623}, {16585, 40447}, {17923, 41226}, {18020, 21054}, {18026, 35057}, {21824, 46254}, {35194, 40440}, {44427, 47318}
X(52412) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7100}, {2, 52381}, {4, 79}, {10, 52388}, {19, 2160}, {21, 1789}, {25, 6186}, {27, 52393}, {28, 52375}, {29, 3615}, {33, 7073}, {34, 52372}, {35, 3}, {55, 8606}, {65, 52390}, {80, 265}, {92, 30690}, {108, 26700}, {162, 13486}, {186, 36}, {225, 52382}, {264, 20565}, {278, 52374}, {281, 7110}, {318, 52344}, {319, 69}, {323, 22128}, {340, 320}, {445, 5249}, {484, 50462}, {500, 4303}, {653, 38340}, {1399, 603}, {1442, 77}, {1825, 65}, {1826, 8818}, {1844, 942}, {1897, 6742}, {2003, 222}, {2174, 48}, {2594, 73}, {2605, 1459}, {2611, 18210}, {3219, 63}, {3578, 4001}, {3647, 3916}, {3678, 72}, {3969, 306}, {4213, 14844}, {4354, 1062}, {4420, 78}, {4467, 4025}, {6187, 52153}, {6198, 1}, {6335, 15455}, {7150, 52202}, {7186, 3784}, {7202, 3942}, {7206, 3695}, {7265, 525}, {7282, 7}, {8287, 4466}, {9404, 652}, {11107, 21}, {14165, 17923}, {14838, 905}, {14975, 31}, {16577, 1214}, {16585, 18607}, {16755, 15419}, {17095, 348}, {17104, 1437}, {17454, 22054}, {18160, 15413}, {20566, 328}, {21054, 125}, {21741, 1409}, {21794, 2197}, {21824, 3708}, {22342, 22341}, {23226, 23224}, {30600, 2523}, {33939, 304}, {34016, 17206}, {35057, 521}, {35192, 2193}, {35193, 283}, {35194, 44706}, {35197, 23070}, {40149, 43682}, {40214, 1790}, {40999, 307}, {41013, 6757}, {41226, 52351}, {41502, 284}, {41562, 1071}, {42033, 345}, {44095, 2260}, {44427, 4707}, {46468, 4292}, {47230, 21828}
X(52412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 92, 17923}, {2, 281, 92}, {2, 6360, 17073}, {10, 29, 5174}, {171, 7076, 162}, {445, 3219, 7282}, {468, 7140, 7009}, {1698, 39585, 5125}, {4213, 17927, 1824}, {6684, 39574, 412}

X(52413) = X(4)X(29046)∩X(6)X(19)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2) : :
Barycentrics    Sin[A]*(2*Sin[A] - Tan[A]) : :

X(52413) lies on these lines: {4, 29046}, {6, 19}, {9, 1063}, {25, 28658}, {28, 1203}, {33, 44105}, {36, 4282}, {44, 5089}, {47, 579}, {48, 10571}, {58, 1474}, {73, 2302}, {79, 1172}, {81, 1848}, {92, 3758}, {109, 2252}, {216, 1950}, {222, 1435}, {278, 2003}, {320, 17923}, {513, 1430}, {517, 22123}, {613, 23052}, {651, 5236}, {758, 1870}, {894, 20883}, {909, 32667}, {913, 32669}, {971, 36742}, {1395, 3192}, {1400, 2148}, {1404, 32674}, {1415, 8607}, {1464, 7113}, {1783, 8756}, {1842, 3194}, {1861, 5847}, {1871, 36750}, {1877, 8755}, {1951, 3284}, {1973, 5299}, {1990, 8735}, {2077, 22059}, {2149, 2183}, {2210, 32676}, {2245, 2361}, {2299, 2308}, {2355, 44097}, {2356, 8750}, {3002, 32756}, {3271, 51726}, {5142, 37559}, {5280, 17442}, {5338, 44086}, {5410, 34121}, {5411, 34125}, {6748, 8736}, {7583, 16027}, {7584, 16033}, {7719, 16670}, {14936, 40135}, {21758, 46384}, {42067, 44102}

X(52413) = isogonal conjugate of X(52351)
X(52413) = polar conjugate of X(20566)
X(52413) = isogonal conjugate of the isotomic conjugate of X(17923)
X(52413) = polar conjugate of the isotomic conjugate of X(36)
X(52413) = X(i)-Ceva conjugate of X(j) for these (i,j): {17923, 36}, {36125, 25}
X(52413) = X(44113)-cross conjugate of X(1870)
X(52413) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52351}, {2, 1807}, {3, 18359}, {9, 52392}, {48, 20566}, {63, 80}, {69, 2161}, {71, 14616}, {72, 24624}, {77, 36910}, {78, 2006}, {219, 18815}, {226, 1793}, {265, 3219}, {304, 6187}, {306, 759}, {307, 2341}, {328, 2174}, {333, 52391}, {345, 1411}, {348, 52371}, {521, 655}, {652, 35174}, {656, 47318}, {905, 51562}, {1168, 3977}, {1214, 6740}, {1459, 36804}, {1790, 15065}, {1797, 51975}, {1809, 52212}, {1812, 52383}, {1813, 52356}, {1946, 46405}, {2222, 6332}, {4064, 37140}, {7100, 41226}, {7265, 36061}, {17206, 34857}, {20336, 34079}, {26932, 52377}, {26942, 52380}, {32675, 35518}, {33939, 52153}, {40709, 46077}, {40710, 46073}
X(52413) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52351}, {69, 40584}, {80, 3162}, {304, 40612}, {306, 34586}, {345, 35204}, {478, 52392}, {1249, 20566}, {1807, 32664}, {4391, 13999}, {6332, 38984}, {7265, 16221}, {18359, 36103}, {20336, 35069}, {35128, 35518}, {39053, 46405}, {40071, 51583}, {40596, 47318}
X(52413) = crosspoint of X(i) and X(j) for these (i,j): {27, 36123}, {1474, 8752}, {7128, 36110}
X(52413) = crosssum of X(i) and X(j) for these (i,j): {3, 22123}, {71, 22350}, {306, 3977}
X(52413) = trilinear pole of line {8648, 21828}
X(52413) = crossdifference of every pair of points on line {72, 521}
X(52413) = barycentric product X(i)*X(j) for these {i,j}: {1, 1870}, {4, 36}, {6, 17923}, {19, 3218}, {21, 1835}, {25, 320}, {27, 2245}, {28, 758}, {29, 1464}, {33, 1443}, {34, 4511}, {56, 5081}, {58, 860}, {65, 17515}, {79, 186}, {86, 44113}, {92, 7113}, {104, 1845}, {108, 3738}, {109, 44428}, {112, 4707}, {214, 36125}, {273, 2361}, {278, 2323}, {286, 3724}, {340, 6186}, {393, 22128}, {513, 4242}, {607, 17078}, {608, 32851}, {648, 21828}, {653, 654}, {915, 11570}, {1061, 4351}, {1130, 8119}, {1172, 18593}, {1474, 3936}, {1783, 3960}, {1973, 20924}, {1974, 40075}, {1983, 17924}, {2203, 35550}, {2299, 41804}, {2906, 39149}, {3904, 32674}, {4282, 40149}, {4453, 8750}, {4585, 6591}, {5146, 39166}, {6335, 21758}, {6336, 17455}, {8120, 10231}, {8648, 18026}, {8752, 51583}, {8756, 40215}, {11700, 36121}, {16944, 38462}, {20565, 34397}, {34586, 36123}
X(52413) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 20566}, {6, 52351}, {19, 18359}, {25, 80}, {28, 14616}, {31, 1807}, {34, 18815}, {36, 69}, {56, 52392}, {79, 328}, {108, 35174}, {112, 47318}, {186, 319}, {320, 305}, {607, 36910}, {608, 2006}, {653, 46405}, {654, 6332}, {758, 20336}, {860, 313}, {1395, 1411}, {1402, 52391}, {1443, 7182}, {1464, 307}, {1474, 24624}, {1783, 36804}, {1824, 15065}, {1835, 1441}, {1845, 3262}, {1870, 75}, {1973, 2161}, {1974, 6187}, {1983, 1332}, {2194, 1793}, {2203, 759}, {2204, 2341}, {2212, 52371}, {2245, 306}, {2299, 6740}, {2323, 345}, {2361, 78}, {3218, 304}, {3724, 72}, {3738, 35518}, {3936, 40071}, {3960, 15413}, {4053, 52369}, {4242, 668}, {4282, 1812}, {4511, 3718}, {4707, 3267}, {5081, 3596}, {6186, 265}, {7113, 63}, {8648, 521}, {8750, 51562}, {17455, 3977}, {17515, 314}, {17923, 76}, {18344, 52356}, {18593, 1231}, {20924, 40364}, {21758, 905}, {21828, 525}, {22128, 3926}, {22379, 4131}, {32674, 655}, {34397, 35}, {40075, 40050}, {42666, 4064}, {44113, 10}, {44428, 35519}, {47230, 7265}, {52059, 22128}
X(52413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 221, 19350}, {6, 478, 2261}, {6, 608, 19}

X(52414) = X(4)X(3648)∩X(19)X(27)

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2) : :
Barycentrics    4*Cos[A] - Sec[A] : :

X(52414) lies on these lines: {4, 3648}, {19, 27}, {28, 11684}, {29, 191}, {158, 46263}, {162, 240}, {186, 42701}, {342, 1708}, {445, 3219}, {758, 17515}, {822, 1577}, {823, 36102}, {920, 1895}, {1096, 16570}, {1725, 36034}, {1749, 1784}, {1821, 2157}, {1844, 3647}, {1955, 2632}, {1959, 24041}, {2074, 48698}, {2167, 17438}, {2173, 2349}, {2897, 6350}, {3218, 14920}, {3650, 15763}, {4467, 44427}, {5535, 7541}, {5692, 37304}, {6149, 35201}, {14590, 16585}, {15149, 39767}, {16546, 18694}, {17484, 37799}, {17768, 37371}, {17917, 23958}, {18253, 25987}, {19919, 44225}, {23052, 36277}, {36046, 36095}, {36085, 36105}, {46468, 52126}

X(52414) = polar conjugate of X(2166)
X(52414) = polar conjugate of the isogonal conjugate of X(6149)
X(52414) = X(2290)-cross conjugate of X(6149)
X(52414) = X(i)-isoconjugate of X(j) for these (i,j): {2, 52153}, {3, 1989}, {4, 50433}, {5, 11077}, {6, 265}, {13, 36297}, {14, 36296}, {30, 11079}, {32, 328}, {48, 2166}, {53, 50463}, {69, 11060}, {94, 184}, {110, 14582}, {112, 43083}, {216, 1141}, {217, 46138}, {248, 14356}, {323, 14595}, {394, 18384}, {476, 647}, {523, 32662}, {525, 14560}, {577, 6344}, {656, 32678}, {661, 36061}, {810, 32680}, {822, 36129}, {906, 43082}, {1304, 18558}, {1576, 14592}, {1807, 2160}, {1990, 50464}, {2006, 8606}, {2161, 7100}, {2165, 5961}, {2341, 52390}, {2351, 18883}, {2437, 14220}, {3003, 12028}, {3049, 35139}, {3284, 5627}, {3457, 40710}, {3458, 40709}, {4558, 15475}, {5158, 18316}, {6186, 52351}, {6187, 52381}, {8749, 51254}, {9409, 39290}, {10097, 14559}, {10217, 11086}, {10218, 11081}, {10412, 32661}, {11063, 15392}, {11064, 40355}, {11070, 50467}, {11071, 50461}, {11072, 52201}, {11073, 52202}, {11074, 20123}, {11075, 50462}, {11080, 50466}, {11082, 50469}, {11083, 52203}, {11084, 52194}, {11085, 50465}, {11087, 50468}, {11088, 52204}, {11089, 52193}, {14254, 18877}, {14380, 41392}, {14575, 20573}, {14583, 14919}, {14585, 18817}, {14908, 43084}, {14910, 39170}, {16186, 23588}, {18479, 43530}, {18557, 32715}, {20578, 38413}, {20579, 38414}, {20975, 39295}, {21354, 34304}, {23968, 35909}, {30529, 51477}, {32663, 34209}, {34079, 52388}, {36298, 39377}, {36299, 39378}, {39201, 46456}
X(52414) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 34544}, {9, 265}, {48, 11597}, {63, 40604}, {244, 14582}, {328, 6376}, {476, 39052}, {656, 18334}, {661, 16221}, {1249, 2166}, {1953, 18402}, {1989, 36103}, {2314, 12095}, {4858, 14592}, {5190, 43082}, {5664, 20902}, {7100, 40584}, {14206, 14918}, {14213, 14920}, {14356, 39039}, {32664, 52153}, {32678, 40596}, {32680, 39062}, {34591, 43083}, {35069, 52388}, {36033, 50433}, {36061, 36830}, {40612, 52381}
X(52414) = cevapoint of X(i) and X(j) for these (i,j): {1725, 2173}, {2290, 51801}
X(52414) = crosspoint of X(2167) and X(2349)
X(52414) = crosssum of X(i) and X(j) for these (i,j): {1953, 2173}, {2631, 3708}
X(52414) = trilinear pole of line {32679, 35201}
X(52414) = barycentric product X(i)*X(j) for these {i,j}: {1, 340}, {19, 7799}, {27, 42701}, {50, 1969}, {63, 14165}, {75, 186}, {92, 323}, {95, 51801}, {162, 3268}, {264, 6149}, {276, 2290}, {319, 1870}, {320, 6198}, {526, 811}, {561, 34397}, {648, 32679}, {662, 44427}, {799, 47230}, {823, 8552}, {1154, 40440}, {1273, 2190}, {1442, 5081}, {1494, 35201}, {1577, 14590}, {1748, 37802}, {2088, 46254}, {2167, 14918}, {2349, 14920}, {2624, 6331}, {3219, 17923}, {4242, 4467}, {4511, 7282}, {5962, 44179}, {6148, 36119}, {10411, 24006}, {11107, 41804}, {14355, 40703}, {14591, 20948}, {14975, 40075}, {16186, 23999}, {17515, 40999}, {24019, 45792}, {24041, 35235}, {33805, 39176}, {36120, 51383}
X(52414) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 265}, {4, 2166}, {19, 1989}, {31, 52153}, {35, 1807}, {36, 7100}, {47, 5961}, {48, 50433}, {50, 48}, {75, 328}, {92, 94}, {107, 36129}, {110, 36061}, {112, 32678}, {158, 6344}, {162, 476}, {163, 32662}, {186, 1}, {240, 14356}, {323, 63}, {340, 75}, {526, 656}, {562, 2962}, {648, 32680}, {656, 43083}, {661, 14582}, {758, 52388}, {811, 35139}, {823, 46456}, {860, 6757}, {1094, 50466}, {1095, 50465}, {1096, 18384}, {1154, 44706}, {1273, 18695}, {1442, 52392}, {1464, 52390}, {1577, 14592}, {1725, 39170}, {1748, 18883}, {1784, 14254}, {1825, 52383}, {1835, 52382}, {1844, 45926}, {1870, 79}, {1969, 20573}, {1973, 11060}, {1986, 1725}, {2088, 3708}, {2148, 11077}, {2151, 36297}, {2152, 36296}, {2159, 11079}, {2169, 50463}, {2190, 1141}, {2290, 216}, {2361, 8606}, {2594, 52391}, {2624, 647}, {2631, 18558}, {2914, 1749}, {3043, 6149}, {3218, 52381}, {3219, 52351}, {3268, 14208}, {3581, 18477}, {4242, 6742}, {5081, 52344}, {5353, 52201}, {5357, 52202}, {5962, 91}, {6126, 50462}, {6149, 3}, {6198, 80}, {7282, 18815}, {7649, 43082}, {7799, 304}, {8552, 24018}, {8739, 2154}, {8740, 2153}, {10411, 4592}, {11062, 1953}, {11107, 6740}, {14165, 92}, {14270, 810}, {14355, 293}, {14385, 35200}, {14590, 662}, {14591, 163}, {14918, 14213}, {14920, 14206}, {14975, 6187}, {16186, 2632}, {17515, 3615}, {17923, 30690}, {18477, 18478}, {19627, 9247}, {22115, 255}, {24006, 10412}, {32676, 14560}, {32679, 525}, {34210, 36062}, {34397, 31}, {35193, 1793}, {35198, 50469}, {35199, 50468}, {35200, 50464}, {35201, 30}, {35235, 1109}, {36053, 12028}, {36063, 34209}, {36119, 5627}, {36130, 43707}, {38936, 36053}, {39176, 2173}, {40440, 46138}, {41502, 2341}, {42701, 306}, {44067, 1822}, {44068, 1823}, {44427, 1577}, {47230, 661}, {51801, 5}, {51802, 50461}, {51804, 15392}, {51805, 10217}, {51806, 10218}
X(52414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 1748, 92}, {240, 896, 162}, {1844, 3647, 11107}, {2580, 2581, 92}

X(52415) = X(4)X(94)∩X(24)X(136)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
Barycentrics    Cos[2*A]*Csc[3*A]*Sin[A]*Tan[A] : :

X(52415) lies on these lines: {4, 94}, {24, 136}, {52, 14111}, {53, 112}, {107, 34310}, {133, 14583}, {186, 52056}, {378, 14356}, {403, 476}, {1552, 5627}, {1598, 31676}, {2904, 34756}, {5890, 18380}, {6530, 14560}, {7487, 30529}, {8754, 12140}, {8796, 18316}, {10151, 34209}, {10594, 52153}, {10688, 18403}, {11744, 43707}, {12918, 47076}, {13556, 15329}, {14980, 37943}, {16229, 51479}, {35488, 39170}, {35490, 51254}, {44145, 46155}

X(52415) = polar conjugate of X(37802)
X(52415) = polar conjugate of the isotomic conjugate of X(18883)
X(52415) = X(52000)-cross conjugate of X(24)
X(52415) = X(i)-isoconjugate of X(j) for these (i,j): {48, 37802}, {68, 6149}, {91, 22115}, {255, 5962}, {323, 1820}, {8552, 36145}
X(52415) = X(i)-Dao conjugate of X(j) for these (i,j): {68, 14993}, {135, 526}, {139, 41078}, {1249, 37802}, {2351, 15295}, {5962, 6523}, {8552, 39013}, {22115, 34116}
X(52415) = trilinear pole of line {6753, 14576}
X(52415) = barycentric product X(i)*X(j) for these {i,j}: {4, 18883}, {24, 94}, {136, 39295}, {265, 11547}, {317, 1989}, {328, 8745}, {467, 1141}, {571, 18817}, {648, 43088}, {924, 46456}, {1748, 2166}, {1993, 6344}, {2052, 5961}, {6753, 35139}, {7763, 18384}, {10412, 41679}, {14111, 30529}, {14576, 46138}, {20573, 44077}, {40427, 52000}
X(52415) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 37802}, {24, 323}, {94, 20563}, {265, 52350}, {317, 7799}, {393, 5962}, {467, 1273}, {571, 22115}, {924, 8552}, {1989, 68}, {5961, 394}, {6344, 5392}, {6563, 45792}, {6753, 526}, {8745, 186}, {11060, 2351}, {11547, 340}, {14576, 1154}, {18384, 2165}, {18883, 69}, {41679, 10411}, {43088, 525}, {44077, 50}, {46456, 46134}, {47421, 16186}, {50433, 16391}, {52000, 34834}
X(52415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 2970, 44795}, {4, 6344, 265}, {265, 18384, 6344}

X(52416) = X(4)X(49)∩X(24)X(52)

Barycentrics    a^4*(a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
Barycentrics    Cos[2*A]*Sec[A]*Sin[3*A] : :
X(52416) = X[1993] + 2 X[51393], 4 X[23292] - X[25739]

X(52416) lies on these lines: {4, 49}, {24, 52}, {25, 9703}, {30, 15463}, {54, 1594}, {110, 403}, {184, 378}, {186, 323}, {215, 1870}, {232, 9696}, {343, 10018}, {468, 40111}, {511, 37932}, {524, 19128}, {567, 7577}, {569, 52296}, {578, 7547}, {924, 15423}, {1092, 32534}, {1493, 6746}, {1495, 48914}, {1514, 14157}, {1593, 9704}, {1614, 2883}, {2070, 19504}, {2071, 44573}, {2072, 12228}, {2477, 6198}, {3047, 12292}, {3147, 45794}, {3200, 8739}, {3201, 8740}, {3520, 13491}, {3575, 20424}, {5012, 37118}, {5962, 14165}, {6143, 13353}, {6240, 34148}, {6640, 34114}, {6759, 35490}, {9603, 39575}, {9706, 15559}, {10274, 13367}, {10295, 43574}, {10539, 35488}, {11935, 18494}, {13352, 35480}, {13619, 37477}, {13754, 37970}, {14984, 37951}, {15362, 37943}, {16868, 18350}, {18445, 44269}, {22750, 44959}, {23236, 45181}, {32046, 37119}, {32609, 37917}, {34797, 37495}, {35487, 43598}, {37954, 50461}, {38534, 44452}, {38851, 51742}, {44076, 45177}

X(52416) = X(14165)-Ceva conjugate of X(50)
X(52416) = X(i)-isoconjugate of X(j) for these (i,j): {68, 2166}, {91, 265}, {94, 1820}, {14592, 36145}, {20571, 52153}
X(52416) = X(i)-Dao conjugate of X(j) for these (i,j): {68, 11597}, {135, 10412}, {265, 34116}, {14592, 39013}, {20563, 40604}
X(52416) = barycentric product X(i)*X(j) for these {i,j}: {24, 323}, {50, 317}, {186, 1993}, {340, 571}, {526, 41679}, {648, 44808}, {924, 14590}, {1147, 14165}, {1748, 6149}, {3043, 18883}, {6563, 14591}, {6753, 10411}, {7763, 34397}, {7799, 44077}, {11547, 22115}
X(52416) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 94}, {50, 68}, {186, 5392}, {317, 20573}, {323, 20563}, {571, 265}, {924, 14592}, {1993, 328}, {3043, 37802}, {6753, 10412}, {8745, 6344}, {11547, 18817}, {14590, 46134}, {14591, 925}, {19627, 2351}, {22115, 52350}, {30451, 43083}, {34397, 2165}, {34952, 14582}, {36423, 5962}, {41679, 35139}, {44077, 1989}, {44808, 525}
X(52416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {186, 3043, 22115}, {22115, 34397, 186}, {44067, 44068, 1986}, {51393, 52000, 24}

X(52417) = X(4)X(54)∩X(24)X(195)

Barycentrics    a^4*(a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4) : :
Barycentrics    Sec[A]*(Sin[A] - Sin[3*A] + Sin[5*A]) : :
X(52417) = 3 X[54] + X[14157], 3 X[11597] - X[22115]

X(52417) lies on the cubic K934 and these lines: {4, 54}, {24, 195}, {26, 12226}, {49, 143}, {110, 539}, {156, 44958}, {186, 323}, {235, 36966}, {403, 32423}, {468, 50708}, {562, 14918}, {933, 15907}, {1112, 47117}, {1147, 15801}, {1209, 14940}, {1594, 8254}, {1658, 22815}, {2888, 7505}, {2917, 35603}, {3047, 10113}, {3147, 12325}, {3153, 12228}, {3200, 10633}, {3201, 10632}, {3515, 12316}, {3517, 12175}, {3520, 10610}, {3567, 12234}, {3575, 22051}, {5965, 19128}, {6143, 6689}, {6150, 24772}, {6198, 47378}, {6240, 20424}, {6288, 16868}, {6403, 19150}, {6515, 11271}, {7488, 12606}, {7512, 12363}, {7691, 21844}, {7722, 37970}, {9704, 10594}, {9707, 19468}, {10018, 21230}, {10203, 32348}, {10282, 21660}, {10605, 17824}, {10628, 21663}, {10880, 12971}, {10881, 12965}, {11576, 34484}, {11649, 27085}, {11808, 44106}, {12291, 26882}, {12307, 32534}, {13445, 35475}, {13619, 15463}, {14156, 27866}, {15073, 32367}, {15137, 21284}, {15800, 34797}, {18374, 44668}, {19457, 32349}, {19504, 37932}, {20585, 44803}, {21357, 52297}, {32046, 52295}, {32226, 52000}, {32377, 43808}, {33565, 38534}, {35488, 48675}, {35489, 43574}, {44077, 47485}

X(52417) = midpoint of X(i) and X(j) for these {i,j}: {186, 2914}, {10540, 15089}, {10619, 51403}
X(52417) = X(2052)-Ceva conjugate of X(36423)
X(52417) = X(i)-isoconjugate of X(j) for these (i,j): {265, 2962}, {2166, 3519}, {14592, 36148}
X(52417) = X(i)-Dao conjugate of X(j) for these (i,j): {3519, 11597}, {6368, 35591}, {14592, 39018}, {18402, 25043}
X(52417) = crossdifference of every pair of points on line {14582, 17434}
X(52417) = barycentric product X(i)*X(j) for these {i,j}: {49, 14165}, {50, 32002}, {186, 1994}, {323, 3518}, {340, 2965}, {648, 44809}, {1510, 14590}, {3043, 30529}, {7769, 34397}, {10632, 11127}, {10633, 11126}, {14591, 41298}, {14918, 25044}
X(52417) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 3519}, {186, 11140}, {1510, 14592}, {1994, 328}, {2965, 265}, {3518, 94}, {11062, 25043}, {11135, 52204}, {11136, 52203}, {14165, 20572}, {14590, 46139}, {14591, 930}, {19627, 51477}, {32002, 20573}, {34397, 2963}, {36423, 562}, {44809, 525}
X(52417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 195, 6242}, {54, 1614, 10619}, {54, 32379, 12254}, {1986, 11702, 2914}, {2914, 11597, 3043}, {3574, 10619, 13403}, {10610, 12300, 3520}, {44067, 44068, 2914}

X(52418) = X(4)X(6)∩X(23)X(232)

Barycentrics    a^2*(a^2 + b^2 - c^2)^2*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2)^2 : :
Barycentrics    (1 + 2*Cos[2*A])*Tan[A]^2 : :

X(52418) lies on these lines: {4, 6}, {23, 232}, {25, 9407}, {26, 15905}, {50, 186}, {107, 32730}, {112, 3003}, {216, 14118}, {297, 22151}, {323, 340}, {403, 16310}, {419, 46151}, {420, 18371}, {421, 2501}, {566, 3520}, {571, 41758}, {577, 7488}, {648, 37778}, {1177, 43717}, {1180, 33871}, {1825, 21741}, {1968, 5158}, {1994, 11547}, {2052, 7578}, {2493, 37777}, {3260, 41253}, {3289, 41203}, {5063, 39575}, {7505, 46262}, {7517, 38292}, {8739, 34394}, {8740, 34395}, {8749, 14581}, {9308, 44135}, {9465, 10311}, {11063, 16328}, {14570, 44328}, {14591, 38936}, {14865, 41335}, {14910, 37951}, {16318, 34978}, {18300, 50433}, {18563, 42459}, {32713, 34854}, {32715, 40388}, {35296, 41679}, {44228, 45016}, {44375, 44893}, {46432, 51936}, {47162, 47336}

X(52418) = polar conjugate of X(328)
X(52418) = isogonal conjugate of the isotomic conjugate of X(14165)
X(52418) = polar conjugate of the isotomic conjugate of X(186)
X(52418) = polar conjugate of the isogonal conjugate of X(34397)
X(52418) = X(14165)-Ceva conjugate of X(186)
X(52418) = X(i)-cross conjugate of X(j) for these (i,j): {2088, 47230}, {34397, 186}
X(52418) = X(i)-isoconjugate of X(j) for these (i,j): {48, 328}, {63, 265}, {75, 50433}, {94, 255}, {304, 52153}, {326, 1989}, {394, 2166}, {476, 24018}, {520, 32680}, {525, 36061}, {662, 43083}, {822, 35139}, {1102, 18384}, {1807, 52381}, {2349, 51254}, {2632, 39295}, {3265, 32678}, {4100, 18817}, {4575, 14592}, {4592, 14582}, {6344, 6507}, {7100, 52351}, {11077, 18695}, {14206, 50464}, {14208, 32662}, {14213, 50463}, {18557, 36034}
X(52418) = X(i)-Dao conjugate of X(j) for these (i,j): {94, 6523}, {136, 14592}, {206, 50433}, {265, 3162}, {326, 34544}, {328, 1249}, {343, 18402}, {394, 11597}, {525, 16221}, {1084, 43083}, {1989, 15259}, {3258, 18557}, {3265, 18334}, {3926, 40604}, {5139, 14582}, {5664, 36793}, {14920, 28706}
X(52418) = cevapoint of X(i) and X(j) for these (i,j): {2088, 47230}, {14581, 44084}
X(52418) = crosspoint of X(i) and X(j) for these (i,j): {8749, 8882}, {23964, 32695}
X(52418) = crosssum of X(i) and X(j) for these (i,j): {343, 11064}, {1650, 18558}, {15526, 41077}
X(52418) = trilinear pole of line {14270, 47230}
X(52418) = crossdifference of every pair of points on line {520, 5489}
X(52418) = barycentric product X(i)*X(j) for these {i,j}: {4, 186}, {6, 14165}, {24, 5962}, {25, 340}, {50, 2052}, {94, 36423}, {107, 526}, {112, 44427}, {158, 6149}, {250, 35235}, {264, 34397}, {275, 11062}, {323, 393}, {403, 38936}, {470, 8740}, {471, 8739}, {562, 3518}, {648, 47230}, {823, 2624}, {1093, 22115}, {1154, 8884}, {1300, 1986}, {1825, 17515}, {1835, 11107}, {1870, 6198}, {2081, 16813}, {2088, 23582}, {2190, 51801}, {2207, 7799}, {2501, 14590}, {3043, 6344}, {3268, 32713}, {3581, 16263}, {5317, 42701}, {5664, 32695}, {6528, 14270}, {6529, 8552}, {6530, 14355}, {8745, 37802}, {8749, 14920}, {8882, 14918}, {14222, 15329}, {14591, 14618}, {16080, 39176}, {16186, 32230}, {18027, 19627}, {24019, 32679}, {35201, 36119}
X(52418) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 328}, {25, 265}, {32, 50433}, {50, 394}, {107, 35139}, {186, 69}, {323, 3926}, {340, 305}, {393, 94}, {512, 43083}, {526, 3265}, {1093, 18817}, {1096, 2166}, {1154, 52347}, {1495, 51254}, {1637, 18557}, {1974, 52153}, {2052, 20573}, {2088, 15526}, {2207, 1989}, {2489, 14582}, {2501, 14592}, {2624, 24018}, {5962, 20563}, {6149, 326}, {6524, 6344}, {6529, 46456}, {8552, 4143}, {8739, 40710}, {8740, 40709}, {8745, 18883}, {8884, 46138}, {11062, 343}, {14165, 76}, {14270, 520}, {14355, 6394}, {14398, 18558}, {14590, 4563}, {14591, 4558}, {14918, 28706}, {14975, 1807}, {19627, 577}, {22115, 3964}, {23964, 39295}, {24019, 32680}, {32676, 36061}, {32695, 39290}, {32713, 476}, {34397, 3}, {34416, 18479}, {34417, 18478}, {34854, 14356}, {35235, 339}, {36417, 11060}, {36423, 323}, {39176, 11064}, {40352, 50464}, {40354, 11079}, {44077, 5961}, {44084, 39170}, {44113, 52388}, {44427, 3267}, {47230, 525}, {51801, 18695}
X(52418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1990, 15262}, {6, 8743, 40138}, {6, 8745, 393}, {6, 8746, 1249}, {50, 11062, 186}, {3087, 40138, 5286}, {8744, 15262, 1990}, {11062, 39176, 50}, {34854, 44102, 32713}

X(52419) = X(2)X(7)∩X(40)X(176)

Barycentrics    a*(a + b - c)*(a - b + c)*(b*c + S) : :
Barycentrics    (1 + Sin[A]) / (Cot[A] + Csc[A]) : :

X(52419) lies on these lines: {1, 34216}, {2, 7}, {3, 39794}, {40, 176}, {46, 482}, {65, 26504}, {77, 2066}, {109, 9098}, {175, 3333}, {347, 16663}, {481, 3338}, {484, 1371}, {651, 51842}, {1124, 10252}, {1267, 7183}, {1373, 3336}, {1374, 3337}, {1434, 13460}, {1697, 17805}, {1804, 3083}, {3160, 51955}, {5119, 31538}, {5128, 31601}, {5708, 39795}, {6212, 7177}, {7029, 7176}, {7190, 13388}, {11010, 17806}, {31539, 51816}

X(52419) = isogonal conjugate of X(13456)
X(52419) = X(13453)-Ceva conjugate of X(3083)
X(52419) = X(i)-cross conjugate of X(j) for these (i,j): {1124, 3083}, {10252, 13389}
X(52419) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13456}, {6, 13454}, {33, 6213}, {55, 1123}, {200, 13438}, {220, 13437}, {281, 34121}, {607, 13387}, {650, 6135}, {1335, 1857}, {2207, 13458}, {2212, 46745}, {5391, 6059}
X(52419) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 13456}, {9, 13454}, {223, 1123}, {6609, 13438}, {7090, 13388}
X(52419) = cevapoint of X(57) and X(34494)
X(52419) = barycentric product X(i)*X(j) for these {i,j}: {1, 13453}, {7, 3083}, {57, 1267}, {77, 13386}, {85, 1124}, {222, 46744}, {269, 13425}, {326, 13459}, {348, 6212}, {605, 6063}, {664, 6364}, {1336, 7183}, {3926, 13460}, {4564, 22107}, {7182, 34125}, {34401, 38003}
X(52419) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13454}, {6, 13456}, {57, 1123}, {77, 13387}, {109, 6135}, {222, 6213}, {269, 13437}, {326, 13458}, {348, 46745}, {603, 34121}, {605, 55}, {1124, 9}, {1267, 312}, {1407, 13438}, {1804, 3084}, {3083, 8}, {6212, 281}, {6364, 522}, {6502, 7133}, {7125, 1335}, {7183, 5391}, {7335, 606}, {13386, 318}, {13389, 7090}, {13425, 341}, {13453, 75}, {13459, 158}, {13460, 393}, {14440, 14430}, {22107, 4858}, {34125, 33}, {46744, 7017}
X(52419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 34494, 176}, {57, 6204, 1445}

X(52420) = X(2)X(7)∩X(40)X(175)

Barycentrics    a*(a + b - c)*(a - b + c)*(b*c - S) : :
Barycentrics    Cos[A] + Tan[A / 2] - 1 : :

X(52420) lies on these lines: {1, 34215}, {2, 7}, {3, 39795}, {40, 175}, {46, 481}, {65, 26495}, {77, 5414}, {109, 9099}, {176, 3333}, {347, 16662}, {482, 3338}, {484, 1372}, {651, 51841}, {1335, 10253}, {1373, 3337}, {1374, 3336}, {1434, 13438}, {1697, 17802}, {1804, 3084}, {3160, 51957}, {5119, 31539}, {5128, 31602}, {5391, 7183}, {5708, 39794}, {6213, 7177}, {7030, 7176}, {7190, 13389}, {11010, 17803}, {21169, 34494}, {31538, 51816}

X(52420) = isogonal conjugate of X(13427)
X(52420) = X(13436)-Ceva conjugate of X(3084)
X(52420) = X(i)-cross conjugate of X(j) for these (i,j): {1335, 3084}, {10253, 13388}
X(52420) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13427}, {6, 13426}, {33, 6212}, {55, 1336}, {200, 13460}, {220, 13459}, {281, 34125}, {607, 13386}, {650, 6136}, {1124, 1857}, {1267, 6059}, {2207, 13425}, {2212, 46744}
X(52420) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 13427}, {9, 13426}, {223, 1336}, {6609, 13460}, {13389, 14121}
X(52420) = cevapoint of X(57) and X(34495)
X(52420) = barycentric product X(i)*X(j) for these {i,j}: {1, 13436}, {7, 3084}, {57, 5391}, {77, 13387}, {85, 1335}, {222, 46745}, {269, 13458}, {326, 13437}, {348, 6213}, {606, 6063}, {664, 6365}, {1123, 7183}, {3926, 13438}, {4564, 22106}, {7182, 34121}
X(52420) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13426}, {6, 13427}, {57, 1336}, {77, 13386}, {109, 6136}, {222, 6212}, {269, 13459}, {326, 13425}, {348, 46744}, {603, 34125}, {606, 55}, {1335, 9}, {1407, 13460}, {1804, 3083}, {2067, 42013}, {3084, 8}, {5391, 312}, {6213, 281}, {6365, 522}, {7125, 1124}, {7183, 1267}, {7335, 605}, {13387, 318}, {13388, 14121}, {13436, 75}, {13437, 158}, {13438, 393}, {13458, 341}, {14445, 14430}, {22106, 4858}, {34121, 33}, {46745, 7017}
X(52420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 34495, 175}, {57, 6203, 1445}

X(52421) = X(8)X(7055)∩X(63)X(7112)

Barycentrics    b*(-a + b - c)*(a + b - c)*c*(-a^2 + b^2 + b*c + c^2) : :
Barycentrics    (2*Cot[A] + Csc[A]) / (1 + Cos[A]) : :

X(52421) lies on these lines: {8, 7055}, {63, 7112}, {75, 1088}, {76, 18737}, {85, 553}, {226, 27478}, {274, 1231}, {310, 4572}, {321, 4554}, {348, 33935}, {552, 7176}, {883, 4651}, {1494, 4569}, {1943, 4573}, {2064, 46238}, {2985, 20911}, {4359, 40704}, {4980, 37780}, {4998, 7081}, {6358, 7196}, {7056, 42696}, {7799, 16577}, {17163, 35312}, {18033, 33940}, {19804, 21609}, {31627, 42029}, {33298, 33941}

X(52421) = isotomic conjugate of X(7073)
X(52421) = isotomic conjugate of the isogonal conjugate of X(1442)
X(52421) = X(i)-cross conjugate of X(j) for these (i,j): {319, 33939}, {40999, 17095}
X(52421) = X(i)-isoconjugate of X(j) for these (i,j): {25, 8606}, {31, 7073}, {32, 7110}, {41, 2160}, {55, 6186}, {79, 2175}, {560, 52344}, {1253, 52372}, {1918, 3615}, {2212, 7100}, {2323, 11060}, {8641, 26700}, {9447, 30690}, {9448, 20565}, {14827, 52374}
X(52421) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 7073}, {79, 40593}, {223, 6186}, {663, 8287}, {2160, 3160}, {2361, 40604}, {3615, 34021}, {3700, 36197}, {4516, 14838}, {5249, 14547}, {6374, 52344}, {6376, 7110}, {6505, 8606}, {16577, 21807}, {17113, 52372}
X(52421) = cevapoint of X(i) and X(j) for these (i,j): {8, 28755}, {319, 17095}
X(52421) = barycentric product X(i)*X(j) for these {i,j}: {7, 33939}, {35, 20567}, {75, 17095}, {76, 1442}, {85, 319}, {274, 40999}, {304, 7282}, {310, 16577}, {561, 2003}, {664, 18160}, {1088, 42033}, {1399, 1502}, {1441, 34016}, {2174, 41283}, {2594, 6385}, {3219, 6063}, {4467, 4554}, {4572, 14838}, {4620, 17886}, {4625, 7265}, {7799, 18815}, {35057, 46406}
X(52421) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7073}, {7, 2160}, {35, 41}, {57, 6186}, {63, 8606}, {75, 7110}, {76, 52344}, {85, 79}, {274, 3615}, {279, 52372}, {319, 9}, {323, 2361}, {348, 7100}, {349, 6757}, {445, 1859}, {658, 26700}, {1088, 52374}, {1231, 52388}, {1399, 32}, {1411, 11060}, {1434, 52375}, {1441, 8818}, {1442, 6}, {1446, 52382}, {1825, 2333}, {2003, 31}, {2174, 2175}, {2594, 213}, {2605, 3063}, {3219, 55}, {3578, 3683}, {3678, 1334}, {3969, 210}, {4420, 220}, {4467, 650}, {4554, 6742}, {4569, 38340}, {4572, 15455}, {4573, 13486}, {6063, 30690}, {6198, 607}, {6741, 36197}, {7182, 52381}, {7186, 20665}, {7202, 3271}, {7265, 4041}, {7279, 2174}, {7282, 19}, {7799, 4511}, {8287, 4516}, {9404, 8641}, {11107, 2332}, {14838, 663}, {16577, 42}, {16585, 14547}, {16755, 3737}, {17095, 1}, {17206, 1789}, {17886, 21044}, {18160, 522}, {18815, 1989}, {20567, 20565}, {21741, 1918}, {21794, 872}, {22342, 2200}, {33939, 8}, {34016, 21}, {35057, 657}, {40214, 2194}, {40999, 37}, {41226, 52371}, {42033, 200}
X(52421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 7182, 6063}, {75, 20935, 17860}

X(52422) = X(2)X(85)∩X(7)X(12)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 - b^2 - 4*b*c - c^2) : :
Barycentrics    2 + Sec[A/2]^2 : :

X(52422) lies on these lines: {1, 25719}, {2, 85}, {5, 17170}, {7, 12}, {8, 5543}, {10, 6604}, {57, 24603}, {65, 3789}, {69, 6734}, {75, 7080}, {76, 345}, {77, 19372}, {150, 5818}, {226, 17308}, {281, 331}, {304, 28808}, {312, 32834}, {344, 349}, {347, 17322}, {388, 1447}, {474, 6516}, {498, 1111}, {551, 25716}, {631, 5088}, {664, 3616}, {938, 14828}, {966, 41246}, {1125, 9312}, {1210, 14548}, {1231, 21609}, {1323, 19862}, {1434, 5435}, {1441, 17321}, {1565, 1656}, {1698, 9436}, {2345, 10030}, {2476, 41826}, {2898, 8167}, {3085, 3673}, {3090, 17181}, {3091, 4872}, {3160, 5550}, {3212, 3485}, {3306, 7183}, {3476, 36487}, {3598, 5261}, {3619, 21617}, {3634, 10481}, {3672, 17895}, {3674, 5219}, {3926, 20925}, {3947, 10521}, {4059, 24914}, {4328, 4967}, {4400, 5839}, {4403, 31455}, {4461, 40023}, {4470, 40862}, {4643, 14564}, {4644, 9596}, {4748, 17950}, {4751, 8732}, {4798, 6610}, {4911, 10590}, {5226, 33949}, {5260, 38859}, {5283, 43063}, {5433, 7223}, {5552, 20880}, {5657, 17753}, {5714, 33865}, {6063, 18140}, {6666, 42309}, {7110, 30705}, {7176, 7288}, {7179, 7195}, {7190, 42696}, {7198, 11237}, {7209, 27432}, {7264, 10056}, {7270, 15589}, {7278, 10072}, {8232, 17289}, {8582, 10436}, {9776, 50559}, {10527, 30806}, {11375, 43037}, {12848, 17256}, {17084, 17090}, {17227, 30275}, {17327, 52023}, {17675, 20328}, {19872, 21314}, {19877, 32086}, {19878, 43186}, {20911, 28795}, {20924, 32832}, {21044, 26101}, {24477, 36854}, {24701, 36673}, {24774, 41785}, {24805, 30617}, {25055, 25723}, {25583, 26364}, {25718, 38314}, {26150, 41352}, {27816, 27818}, {29603, 43035}, {31598, 41003}, {32007, 46933}, {32830, 32851}, {32874, 42033}, {33116, 33780}, {37990, 39732}, {42032, 46951}, {46932, 51351}

X(52422) = X(3305)-cross conjugate of X(42696)
X(52422) = X(i)-isoconjugate of X(j) for these (i,j): {41, 3296}, {2212, 30679}
X(52422) = X(i)-Dao conjugate of X(j) for these (i,j): {3160, 3296}, {3715, 16777}
X(52422) = cevapoint of X(3305) and X(7190)
X(52422) = barycentric product X(i)*X(j) for these {i,j}: {7, 42696}, {75, 7190}, {85, 3305}, {279, 42032}, {664, 48268}, {3295, 6063}, {4554, 47965}, {4572, 48340}
X(52422) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 3296}, {348, 30679}, {3295, 55}, {3305, 9}, {3697, 210}, {4917, 3158}, {7190, 1}, {42032, 346}, {42696, 8}, {46951, 28808}, {47965, 650}, {48268, 522}, {48340, 663}, {51572, 3715}
X(52422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 85, 348}, {2, 279, 17095}, {2, 30694, 1212}, {7, 9780, 33298}, {10, 40719, 6604}, {85, 348, 17079}, {85, 17078, 43983}, {85, 17095, 279}, {279, 17095, 348}, {304, 32828, 28808}, {3598, 5261, 7247}, {3616, 31994, 664}, {5433, 7223, 17081}, {6706, 46835, 2}, {7195, 10588, 7179}

X(52423) = X(1)X(6883)∩X(6)X(57)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 - b^2 - 3*b*c - c^2) : :
Barycentrics    Cos[A] + Cos[2*A] - 2 : :

X(52423) lies on these lines: {1, 6883}, {2, 2323}, {6, 57}, {9, 10601}, {31, 3256}, {35, 582}, {36, 5396}, {40, 37514}, {42, 2078}, {46, 16472}, {51, 3220}, {56, 5313}, {63, 5422}, {65, 1203}, {81, 3911}, {84, 10982}, {109, 2308}, {181, 1428}, {182, 5285}, {200, 45728}, {219, 7308}, {226, 26723}, {239, 6358}, {241, 47057}, {373, 26885}, {386, 1451}, {394, 5437}, {553, 651}, {559, 5353}, {575, 3955}, {576, 3784}, {611, 3677}, {613, 5269}, {940, 31231}, {942, 23071}, {999, 39523}, {1062, 10399}, {1082, 5357}, {1170, 43035}, {1171, 4565}, {1193, 18772}, {1319, 16474}, {1386, 41539}, {1401, 19369}, {1404, 1412}, {1421, 5173}, {1429, 5280}, {1445, 45126}, {1450, 5193}, {1473, 9777}, {1708, 5256}, {1876, 44097}, {1943, 17121}, {1947, 36794}, {1993, 3306}, {1994, 22128}, {2006, 2982}, {2099, 5315}, {2256, 25430}, {3060, 7293}, {3218, 26740}, {3219, 15018}, {3299, 13388}, {3301, 13389}, {3338, 16473}, {3340, 16466}, {3589, 26942}, {3601, 36745}, {3937, 34565}, {4383, 5219}, {4552, 45222}, {4654, 5228}, {4663, 17625}, {5050, 37581}, {5122, 51340}, {5221, 34043}, {5226, 14997}, {5262, 15556}, {5299, 7146}, {5312, 37579}, {5435, 37685}, {5706, 9581}, {5709, 36752}, {5902, 37697}, {5943, 7193}, {7011, 15851}, {7074, 10389}, {7078, 11518}, {7171, 44413}, {7269, 27065}, {8270, 16475}, {11427, 20266}, {13329, 14547}, {13366, 26884}, {15004, 26892}, {15803, 36742}, {16577, 17011}, {17020, 43048}, {17077, 19717}, {18162, 21361}, {18206, 18602}, {24431, 41700}, {24914, 37559}, {26893, 43650}, {33178, 44547}, {36747, 37534}, {36749, 37612}, {36750, 37582}, {36753, 37532}, {37498, 37526}

X(52423) = X(7269)-Ceva conjugate of X(3746)
X(52423) = X(9)-isoconjugate of X(5557)
X(52423) = X(478)-Dao conjugate of X(5557)
X(52423) = crosssum of X(37) and X(21012)
X(52423) = crossdifference of every pair of points on line {3900, 13401}
X(52423) = barycentric product X(i)*X(j) for these {i,j}: {1, 7269}, {7, 3746}, {56, 5564}, {57, 27065}, {651, 48003}, {664, 48306}, {1014, 4015}
X(52423) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 5557}, {3746, 8}, {4015, 3701}, {5564, 3596}, {7269, 75}, {27065, 312}, {48003, 4391}, {48306, 522}
X(52423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 57, 2003}, {51, 26889, 3220}, {219, 17825, 7308}, {386, 1451, 37583}, {1994, 27003, 22128}, {4383, 37543, 5219}, {5228, 34048, 4654}, {17011, 37787, 16577}

X(52424) = X(2)X(219)∩X(6)X(57)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 - b^2 - 4*b*c - c^2) : :
Barycentrics    (2 + Cos[A])*Sin[A / 2]^2 : :

X(52424) lies on these lines: {1, 5920}, {2, 219}, {3, 1451}, {6, 57}, {7, 32911}, {9, 17825}, {25, 26889}, {31, 37541}, {34, 37544}, {42, 1471}, {48, 37269}, {51, 1473}, {55, 13329}, {56, 181}, {58, 1466}, {63, 10601}, {65, 16466}, {81, 5435}, {109, 41422}, {171, 613}, {182, 37581}, {218, 226}, {220, 7308}, {221, 1203}, {241, 45126}, {278, 607}, {394, 3306}, {458, 1947}, {553, 6180}, {611, 982}, {614, 5173}, {651, 21454}, {748, 42289}, {940, 3911}, {942, 7078}, {950, 37537}, {990, 1864}, {999, 1450}, {1040, 5728}, {1124, 13388}, {1191, 3340}, {1210, 5706}, {1214, 1445}, {1335, 13389}, {1351, 3784}, {1376, 45728}, {1386, 8270}, {1393, 19349}, {1402, 40732}, {1428, 1460}, {1433, 3075}, {1497, 10306}, {1708, 3666}, {1714, 15844}, {1788, 5711}, {1817, 46882}, {1876, 44086}, {1943, 3759}, {1993, 23140}, {2099, 16483}, {2256, 17022}, {2260, 7011}, {2262, 21370}, {2308, 9316}, {2323, 5437}, {2334, 51773}, {2911, 5219}, {2947, 3149}, {3052, 3256}, {3157, 5708}, {3218, 5422}, {3220, 17810}, {3295, 7086}, {3305, 7190}, {3336, 16472}, {3337, 16473}, {3361, 34046}, {3527, 26928}, {3556, 45979}, {3682, 16410}, {3690, 22112}, {3751, 17625}, {3781, 16419}, {3937, 15004}, {3955, 5050}, {4255, 37583}, {4298, 9370}, {4361, 6358}, {4848, 5710}, {5020, 7193}, {5085, 5285}, {5226, 37680}, {5315, 18421}, {5709, 37514}, {5729, 24430}, {5928, 12610}, {5943, 24320}, {6354, 17366}, {6357, 7365}, {6692, 25934}, {7085, 43650}, {7132, 43071}, {7146, 16502}, {7191, 7672}, {7225, 28387}, {7248, 19369}, {7293, 33586}, {7484, 26893}, {7677, 17018}, {8257, 25091}, {8757, 24470}, {9777, 26866}, {10478, 24618}, {11284, 26885}, {11343, 41243}, {11402, 26884}, {11424, 26927}, {11433, 26932}, {12594, 21342}, {13462, 16474}, {14555, 23151}, {14853, 26929}, {15556, 37549}, {15803, 36746}, {15804, 40958}, {15805, 26921}, {16409, 17976}, {16412, 22127}, {16432, 32590}, {16433, 32592}, {16435, 24310}, {16577, 20182}, {17012, 17080}, {17074, 37685}, {17077, 19684}, {17595, 26740}, {17599, 41712}, {18928, 27509}, {20266, 23292}, {20905, 28950}, {21361, 24328}, {22153, 37272}, {23958, 34545}, {26723, 37695}, {28606, 37787}, {28997, 41241}, {31231, 37674}, {34051, 44794}, {36059, 44104}, {36742, 37582}, {36747, 37612}, {36750, 37545}, {36752, 37532}, {37498, 37534}, {37642, 43043}

X(52424) = X(7190)-Ceva conjugate of X(3295)
X(52424) = X(i)-isoconjugate of X(j) for these (i,j): {9, 3296}, {33, 30679}, {3872, 52188}
X(52424) = X(478)-Dao conjugate of X(3296)
X(52424) = crosssum of X(1146) and X(4820)
X(52424) = crossdifference of every pair of points on line {3900, 4976}
X(52424) = barycentric product X(i)*X(j) for these {i,j}: {1, 7190}, {7, 3295}, {56, 42696}, {57, 3305}, {109, 48268}, {651, 47965}, {664, 48340}, {1014, 3697}, {1407, 42032}, {4917, 19604}
X(52424) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 3296}, {222, 30679}, {3295, 8}, {3305, 312}, {3697, 3701}, {4917, 44720}, {7190, 75}, {42696, 3596}, {47965, 4391}, {48268, 35519}, {48340, 522}
X(52424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 57, 222}, {6, 1407, 2003}, {7, 32911, 34048}, {42, 1471, 1617}, {57, 2003, 1407}, {57, 2999, 1465}, {65, 16466, 34040}, {942, 36754, 7078}, {1203, 3339, 221}, {1407, 2003, 222}, {1445, 5256, 1214}, {2323, 5437, 17811}, {4383, 5228, 226}, {5708, 37509, 3157}, {9777, 26866, 26892}

X(52425) = X(1)X(1729)∩X(31)X(32)

Barycentrics    a^4*(a - b - c)*(a^2 - b^2 - c^2) : :
Barycentrics    Cos[A/2]^2*Sin[A]*Sin[2*A] : :

X(52425) lies on these lines: {1, 1729}, {3, 906}, {6, 2197}, {8, 5546}, {31, 32}, {48, 577}, {55, 607}, {101, 4291}, {184, 2200}, {198, 608}, {212, 1802}, {219, 283}, {220, 2342}, {221, 1415}, {222, 1803}, {255, 20752}, {315, 27516}, {571, 2174}, {604, 5065}, {610, 1950}, {692, 34078}, {919, 2724}, {1042, 1055}, {1253, 7118}, {1334, 16283}, {1451, 2266}, {1783, 11491}, {1914, 2082}, {1968, 2202}, {2170, 2241}, {2175, 9448}, {2189, 2335}, {2242, 2650}, {2264, 5301}, {2286, 15905}, {2289, 22074}, {2293, 40957}, {2347, 16946}, {2876, 23852}, {3002, 36152}, {5019, 21748}, {5063, 7113}, {5452, 32561}, {6284, 8735}, {7335, 32657}, {8300, 10801}, {14585, 39687}, {17046, 25886}, {17905, 37000}, {20760, 20799}, {20818, 22118}, {23383, 44081}, {23843, 39690}

X(52425) = isogonal conjugate of X(331)
X(52425) = isogonal conjugate of the isotomic conjugate of X(219)
X(52425) = isotomic conjugate of the polar conjugate of X(2175)
X(52425) = isogonal conjugate of the polar conjugate of X(55)
X(52425) = polar conjugate of the isotomic conjugate of X(6056)
X(52425) = X(i)-Ceva conjugate of X(j) for these (i,j): {48, 184}, {55, 2175}, {219, 6056}, {906, 1946}, {2193, 212}, {7115, 692}, {14578, 23202}
X(52425) = X(i)-cross conjugate of X(j) for these (i,j): {2200, 41}, {22368, 3}
X(52425) = X(i)-isoconjugate of X(j) for these (i,j): {1, 331}, {2, 273}, {4, 85}, {7, 92}, {8, 1847}, {19, 6063}, {25, 20567}, {27, 1441}, {28, 349}, {29, 1446}, {34, 76}, {56, 1969}, {57, 264}, {65, 44129}, {75, 278}, {77, 2052}, {84, 40701}, {86, 40149}, {108, 3261}, {158, 348}, {189, 342}, {196, 309}, {208, 44190}, {225, 274}, {226, 286}, {253, 44697}, {269, 7017}, {276, 1393}, {279, 318}, {281, 1088}, {304, 1118}, {310, 1880}, {312, 1119}, {313, 1396}, {393, 7182}, {459, 33673}, {479, 7101}, {513, 46404}, {514, 18026}, {522, 13149}, {561, 608}, {603, 18027}, {604, 18022}, {648, 4077}, {651, 46107}, {653, 693}, {658, 44426}, {664, 17924}, {811, 7178}, {823, 17094}, {873, 8736}, {903, 37790}, {934, 46110}, {1093, 7183}, {1111, 46102}, {1121, 38461}, {1226, 40397}, {1231, 8747}, {1365, 46254}, {1395, 1502}, {1398, 28659}, {1414, 14618}, {1426, 28660}, {1427, 44130}, {1434, 41013}, {1435, 3596}, {1804, 6521}, {1848, 31643}, {1855, 42311}, {1861, 34018}, {1874, 40017}, {1876, 18031}, {1877, 20568}, {1897, 24002}, {1973, 41283}, {1978, 43923}, {2481, 5236}, {2501, 4625}, {2973, 4564}, {3064, 4569}, {3213, 41530}, {3668, 31623}, {3676, 6335}, {4017, 6331}, {4391, 36118}, {4554, 7649}, {4572, 6591}, {4573, 24006}, {6520, 7055}, {6528, 51640}, {7012, 23989}, {7020, 14256}, {7046, 23062}, {7128, 34387}, {7210, 43678}, {7282, 30690}, {7337, 40364}, {8795, 44708}, {8809, 15466}, {13437, 46744}, {13459, 46745}, {15413, 36127}, {16082, 22464}, {16747, 18097}, {17880, 23984}, {17923, 18815}, {18344, 46406}, {24032, 26932}, {26563, 40446}, {32674, 40495}, {32714, 35519}, {34400, 47372}, {36124, 40704}, {40702, 40836}, {46111, 51655}
X(52425) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 1969}, {3, 331}, {6, 6063}, {7, 22391}, {76, 11517}, {85, 36033}, {206, 278}, {264, 5452}, {273, 32664}, {348, 1147}, {349, 40591}, {608, 40368}, {3161, 18022}, {3261, 38983}, {6331, 34961}, {6337, 41283}, {6505, 20567}, {6600, 7017}, {7055, 37867}, {7178, 17423}, {7952, 18027}, {14618, 40608}, {14714, 46110}, {17924, 39025}, {24002, 34467}, {35072, 40495}, {38991, 46107}, {39026, 46404}, {40149, 40600}, {40602, 44129}, {44132, 50440}
X(52425) = cevapoint of X(3) and X(23078)
X(52425) = crosspoint of X(i) and X(j) for these (i,j): {48, 212}, {55, 219}, {692, 7115}
X(52425) = crosssum of X(i) and X(j) for these (i,j): {1, 1729}, {4, 17905}, {7, 278}, {92, 273}, {693, 26932}, {2973, 17924}, {4077, 4858}, {10015, 38372}, {23989, 24002}, {34387, 46110}
X(52425) = crossdifference of every pair of points on line {693, 17094}
X(52425) = barycentric product X(i)*X(j) for these {i,j}: {1, 212}, {3, 55}, {4, 6056}, {6, 219}, {8, 184}, {9, 48}, {19, 2289}, {21, 228}, {25, 1259}, {29, 4055}, {31, 78}, {32, 345}, {33, 255}, {35, 8606}, {37, 2193}, {40, 2188}, {41, 63}, {42, 283}, {54, 44707}, {56, 1260}, {57, 1802}, {58, 2318}, {59, 3270}, {60, 3690}, {69, 2175}, {71, 284}, {72, 2194}, {73, 2328}, {77, 1253}, {81, 52370}, {100, 1946}, {101, 652}, {163, 8611}, {197, 39167}, {198, 268}, {200, 603}, {210, 1437}, {213, 1812}, {220, 222}, {271, 2187}, {281, 577}, {294, 20752}, {304, 9447}, {305, 9448}, {312, 9247}, {326, 2212}, {332, 1918}, {333, 2200}, {348, 14827}, {394, 607}, {480, 7053}, {521, 692}, {522, 32656}, {560, 3718}, {604, 3692}, {643, 810}, {644, 22383}, {645, 3049}, {647, 5546}, {649, 4587}, {650, 906}, {657, 1813}, {663, 1331}, {667, 4571}, {728, 7099}, {911, 51376}, {943, 23207}, {947, 40945}, {983, 20753}, {1036, 7085}, {1040, 7084}, {1092, 1857}, {1110, 7004}, {1172, 3990}, {1176, 3688}, {1252, 7117}, {1261, 22344}, {1264, 1974}, {1265, 1397}, {1318, 22371}, {1320, 23202}, {1332, 3063}, {1333, 3694}, {1334, 1790}, {1400, 2327}, {1402, 1792}, {1409, 2287}, {1425, 6061}, {1433, 7074}, {1459, 3939}, {1565, 6066}, {1576, 52355}, {1783, 36054}, {1791, 20967}, {1793, 3724}, {1794, 14547}, {1798, 40966}, {1803, 8012}, {1804, 7071}, {1807, 2361}, {1808, 3747}, {1810, 8647}, {1818, 2195}, {1819, 2357}, {1973, 3719}, {2053, 20760}, {2066, 5414}, {2115, 20761}, {2149, 34591}, {2150, 3949}, {2169, 7069}, {2189, 52386}, {2192, 7078}, {2196, 3684}, {2197, 7054}, {2204, 3998}, {2206, 3710}, {2269, 2359}, {2293, 47487}, {2298, 22074}, {2299, 3682}, {2316, 22356}, {2325, 32659}, {2329, 7116}, {2330, 7015}, {2332, 40152}, {2333, 6514}, {2340, 36057}, {2342, 22350}, {2346, 22079}, {2638, 7012}, {3157, 7072}, {3208, 15373}, {3239, 32660}, {3284, 15627}, {3289, 15628}, {3292, 5547}, {3596, 14575}, {3689, 36058}, {3693, 32658}, {3700, 32661}, {3703, 10547}, {3709, 4558}, {3712, 14908}, {3900, 36059}, {3937, 6065}, {3964, 6059}, {4041, 4575}, {4076, 22096}, {4092, 47390}, {4183, 22341}, {4548, 14376}, {4557, 23189}, {4559, 23090}, {4574, 7252}, {5548, 22086}, {6332, 32739}, {6510, 18889}, {6516, 8641}, {6602, 7177}, {7011, 7367}, {7016, 26885}, {7017, 14585}, {7046, 7335}, {7062, 47388}, {7063, 47389}, {7065, 32230}, {7070, 19614}, {7077, 7193}, {7079, 7125}, {7115, 35072}, {7123, 7124}, {7359, 18877}, {8851, 20777}, {10397, 36049}, {10482, 22053}, {14379, 44695}, {14418, 32665}, {14642, 27382}, {14936, 44717}, {15374, 42460}, {15905, 30457}, {19354, 42019}, {20769, 51858}, {21789, 23067}, {22054, 33635}, {22072, 51476}, {22131, 40141}, {22368, 40419}, {23146, 40523}, {23201, 32635}, {23225, 36802}, {23990, 26932}, {32641, 52307}, {32657, 40869}, {34055, 40972}, {34858, 51379}, {36055, 51361}, {36056, 41339}, {36797, 39201}, {39687, 46102}, {40363, 40373}
X(52425) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6063}, {6, 331}, {8, 18022}, {9, 1969}, {31, 273}, {32, 278}, {41, 92}, {48, 85}, {55, 264}, {63, 20567}, {69, 41283}, {71, 349}, {78, 561}, {101, 46404}, {184, 7}, {198, 40701}, {212, 75}, {213, 40149}, {219, 76}, {220, 7017}, {228, 1441}, {255, 7182}, {268, 44190}, {281, 18027}, {283, 310}, {284, 44129}, {305, 41287}, {345, 1502}, {521, 40495}, {560, 34}, {577, 348}, {603, 1088}, {604, 1847}, {607, 2052}, {652, 3261}, {657, 46110}, {663, 46107}, {692, 18026}, {810, 4077}, {906, 4554}, {1092, 7055}, {1253, 318}, {1259, 305}, {1260, 3596}, {1264, 40050}, {1265, 40363}, {1331, 4572}, {1397, 1119}, {1409, 1446}, {1415, 13149}, {1501, 608}, {1792, 40072}, {1802, 312}, {1812, 6385}, {1813, 46406}, {1917, 1395}, {1918, 225}, {1946, 693}, {1974, 1118}, {1980, 43923}, {2175, 4}, {2187, 342}, {2188, 309}, {2193, 274}, {2194, 286}, {2200, 226}, {2205, 1880}, {2212, 158}, {2251, 37790}, {2289, 304}, {2318, 313}, {2327, 28660}, {2328, 44130}, {2638, 17880}, {3022, 21666}, {3049, 7178}, {3063, 17924}, {3270, 34387}, {3271, 2973}, {3596, 44161}, {3688, 1235}, {3690, 34388}, {3692, 28659}, {3694, 27801}, {3709, 14618}, {3718, 1928}, {3719, 40364}, {3955, 7205}, {3990, 1231}, {4055, 307}, {4100, 7183}, {4173, 7217}, {4548, 17907}, {4571, 6386}, {4575, 4625}, {4587, 1978}, {5546, 6331}, {5547, 46111}, {6056, 69}, {6059, 1093}, {6066, 15742}, {6602, 7101}, {7063, 8754}, {7064, 7141}, {7099, 23062}, {7109, 8736}, {7117, 23989}, {7193, 18033}, {7335, 7056}, {8606, 20565}, {8611, 20948}, {8641, 44426}, {9247, 57}, {9447, 19}, {9448, 25}, {9454, 5236}, {9455, 1876}, {9459, 1877}, {10316, 17076}, {10317, 17088}, {14575, 56}, {14585, 222}, {14827, 281}, {15373, 7209}, {20752, 40704}, {20753, 33930}, {20775, 3665}, {22074, 20911}, {22079, 20880}, {22096, 1358}, {22117, 50560}, {22368, 2886}, {22383, 24002}, {23200, 7181}, {23211, 17447}, {23225, 43042}, {23606, 1804}, {23990, 46102}, {32656, 664}, {32657, 52156}, {32658, 34018}, {32660, 658}, {32661, 4573}, {32739, 653}, {34980, 1367}, {36054, 15413}, {36059, 4569}, {39201, 17094}, {39687, 26932}, {40050, 41289}, {40360, 41290}, {40373, 1397}, {40972, 20883}, {41280, 1398}, {44088, 30493}, {44162, 7337}, {44707, 311}, {47390, 7340}, {52355, 44173}, {52370, 321}
X(52425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7124, 7117}, {3, 22131, 22070}, {32, 14827, 41}, {212, 1802, 52370}, {2200, 9247, 184}

X(52426) = X(1)X(4291)∩X(31)X(32)

Barycentrics    a^4*(a - b - c)*(a^2 - b^2 + b*c - c^2) : :
Barycentrics    (Cos[A] + Cos[2*A])*Sin[A]^2 : :

X(52426) lies on these lines: {1, 4291}, {6, 5172}, {31, 32}, {36, 1983}, {48, 571}, {50, 7113}, {112, 2202}, {187, 7117}, {577, 604}, {672, 906}, {1055, 1415}, {1195, 41332}, {1333, 21748}, {1914, 1951}, {1946, 3063}, {2082, 7031}, {2174, 2965}, {2183, 32655}, {2193, 2269}, {2220, 2347}, {2245, 47417}, {2323, 4282}, {3053, 7124}, {3684, 5546}, {5277, 33105}, {5301, 40968}, {8606, 14547}, {9454, 23990}, {19554, 51659}, {20065, 27516}, {25886, 31240}

X(52426) = isogonal conjugate of the isotomic conjugate of X(2323)
X(52426) = X(i)-Ceva conjugate of X(j) for these (i,j): {1983, 8648}, {4282, 2361}, {32655, 31}, {32677, 184}
X(52426) = X(i)-isoconjugate of X(j) for these (i,j): {2, 18815}, {7, 18359}, {57, 20566}, {75, 2006}, {76, 1411}, {80, 85}, {92, 52392}, {94, 1442}, {226, 14616}, {273, 52351}, {274, 52383}, {320, 34535}, {331, 1807}, {349, 759}, {513, 46405}, {514, 35174}, {655, 693}, {658, 52356}, {903, 14628}, {1088, 36910}, {1399, 20573}, {1434, 15065}, {1441, 24624}, {1446, 6740}, {2161, 6063}, {2166, 17095}, {2222, 3261}, {3676, 36804}, {4077, 47318}, {6187, 20567}, {14584, 20568}, {17791, 26743}, {18816, 52212}, {23989, 52377}, {24002, 51562}, {32675, 40495}, {44129, 52391}
X(52426) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 35204}, {206, 2006}, {349, 34586}, {3261, 38984}, {5452, 20566}, {6063, 40584}, {6149, 20924}, {11597, 17095}, {18815, 32664}, {20567, 40612}, {22391, 52392}, {35128, 40495}, {39026, 46405}
X(52426) = crosspoint of X(i) and X(j) for these (i,j): {284, 2342}, {913, 2299}, {2161, 7073}, {2361, 7113}
X(52426) = crosssum of X(i) and X(j) for these (i,j): {7, 37798}, {226, 22464}, {307, 914}, {1442, 3218}, {2006, 52392}, {4707, 21207}, {4858, 36038}, {18359, 18815}
X(52426) = crossdifference of every pair of points on line {693, 1441}
X(52426) = barycentric product X(i)*X(j) for these {i,j}: {1, 2361}, {6, 2323}, {9, 7113}, {21, 3724}, {31, 4511}, {32, 32851}, {36, 55}, {37, 4282}, {41, 3218}, {50, 7110}, {80, 215}, {100, 8648}, {101, 654}, {184, 5081}, {186, 8606}, {212, 1870}, {228, 17515}, {283, 44113}, {284, 2245}, {320, 2175}, {607, 22128}, {644, 21758}, {650, 1983}, {692, 3738}, {758, 2194}, {1253, 1443}, {1464, 2328}, {1946, 4242}, {2150, 4053}, {2161, 34544}, {2316, 17455}, {2342, 34586}, {3063, 4585}, {3689, 16944}, {3904, 32739}, {4089, 6066}, {4636, 42666}, {4996, 6187}, {5546, 21828}, {6149, 7073}, {9447, 20924}, {9448, 40075}, {14827, 17078}, {19302, 26744}, {27950, 51858}, {32656, 44428}, {36910, 52059}
X(52426) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 18815}, {32, 2006}, {36, 6063}, {41, 18359}, {50, 17095}, {55, 20566}, {101, 46405}, {184, 52392}, {215, 320}, {320, 41283}, {560, 1411}, {654, 3261}, {692, 35174}, {1918, 52383}, {1983, 4554}, {2175, 80}, {2194, 14616}, {2245, 349}, {2251, 14628}, {2323, 76}, {2361, 75}, {3218, 20567}, {3724, 1441}, {3738, 40495}, {4282, 274}, {4511, 561}, {4996, 40075}, {5081, 18022}, {7110, 20573}, {7113, 85}, {8606, 328}, {8641, 52356}, {8648, 693}, {9447, 2161}, {9448, 6187}, {9459, 14584}, {14827, 36910}, {19627, 2003}, {21758, 24002}, {32739, 655}, {32851, 1502}, {34397, 7282}, {34544, 20924}, {40075, 41287}, {52059, 17078}
X(52426) = {X(1914),X(1951)}-harmonic conjugate of X(2170)

X(52427) = X(1)X(24)∩X(3)X(34)

Barycentrics    a^2*(a - b - c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2) : :
Barycentrics    (Cot[A/2] - 2*Sin[A])*Sin[A]*Tan[A] : :

X(52427) lies on these lines: {1, 24}, {2, 11393}, {3, 34}, {4, 35}, {11, 468}, {12, 3575}, {19, 25}, {21, 34851}, {22, 1040}, {23, 3100}, {26, 1062}, {28, 4276}, {31, 1182}, {36, 186}, {41, 41320}, {42, 8882}, {51, 11429}, {56, 3515}, {65, 20837}, {92, 37295}, {100, 1861}, {102, 1457}, {108, 2078}, {154, 19354}, {162, 2073}, {184, 11436}, {185, 26888}, {199, 23207}, {204, 1609}, {208, 37579}, {212, 573}, {215, 34397}, {221, 1192}, {225, 7412}, {232, 1914}, {235, 6284}, {242, 243}, {278, 15931}, {284, 2189}, {297, 26629}, {378, 5010}, {390, 4232}, {403, 3583}, {406, 5248}, {427, 5432}, {428, 4995}, {429, 1852}, {444, 21321}, {451, 5259}, {475, 25440}, {495, 37458}, {497, 6353}, {499, 3147}, {607, 4258}, {608, 37499}, {614, 10832}, {650, 1946}, {759, 2074}, {855, 10017}, {860, 13999}, {902, 8750}, {915, 1785}, {919, 9085}, {950, 30733}, {1030, 1841}, {1038, 17928}, {1060, 6644}, {1068, 14798}, {1079, 7163}, {1110, 2356}, {1155, 1876}, {1204, 7355}, {1250, 10641}, {1319, 10016}, {1364, 26884}, {1398, 5204}, {1435, 37578}, {1474, 2269}, {1478, 18533}, {1479, 3542}, {1495, 3270}, {1503, 26956}, {1593, 5217}, {1621, 30687}, {1753, 11248}, {1825, 6197}, {1829, 2646}, {1830, 5537}, {1838, 7414}, {1843, 2330}, {1848, 4231}, {1858, 32126}, {1872, 11849}, {1875, 5172}, {1877, 2077}, {1885, 15338}, {1887, 14882}, {1890, 7466}, {1892, 17718}, {1902, 37568}, {1905, 24929}, {1974, 3056}, {1995, 9817}, {2066, 5413}, {2070, 18455}, {2164, 7008}, {2223, 34179}, {2245, 2361}, {2268, 44103}, {2276, 10311}, {2291, 40116}, {2331, 8573}, {2342, 14776}, {3035, 26020}, {3052, 3195}, {3057, 11363}, {3060, 9637}, {3083, 15204}, {3084, 15203}, {3085, 7487}, {3089, 4294}, {3144, 35206}, {3145, 40946}, {3157, 37489}, {3220, 7004}, {3295, 3517}, {3299, 10881}, {3301, 10880}, {3518, 3746}, {3584, 7576}, {3585, 6240}, {3601, 7713}, {3614, 23047}, {4183, 7110}, {4185, 19760}, {4225, 45231}, {4242, 17923}, {4265, 17603}, {4293, 37460}, {4296, 22467}, {4316, 10295}, {4324, 18560}, {5081, 17515}, {5186, 15452}, {5225, 6622}, {5280, 10312}, {5281, 6995}, {5299, 39575}, {5320, 11435}, {5322, 21213}, {5353, 10632}, {5357, 10633}, {5370, 21284}, {5410, 19037}, {5411, 19038}, {5412, 5414}, {5563, 44879}, {6056, 26893}, {6152, 47378}, {6187, 8756}, {6238, 10539}, {6242, 35197}, {6285, 26883}, {6641, 26904}, {6690, 25985}, {7031, 8743}, {7079, 32561}, {7127, 8740}, {7280, 32534}, {7292, 37977}, {7501, 34231}, {7503, 19372}, {7505, 7741}, {7506, 37696}, {8144, 37440}, {8540, 44102}, {8554, 14936}, {8758, 14667}, {9371, 20872}, {9638, 26882}, {9643, 9645}, {9786, 19349}, {10391, 41611}, {10483, 35471}, {10589, 38282}, {10638, 10642}, {10895, 12173}, {10986, 16785}, {11189, 44082}, {11206, 18922}, {11396, 34471}, {11400, 22479}, {11428, 40952}, {11471, 37601}, {12047, 31384}, {12106, 37729}, {12428, 41587}, {12943, 37196}, {12953, 37197}, {13367, 19365}, {13884, 19030}, {13937, 19029}, {15171, 21841}, {15325, 37935}, {15326, 37931}, {17985, 52167}, {18447, 45735}, {18494, 31479}, {18513, 35480}, {18514, 35488}, {19842, 37087}, {20838, 22341}, {26885, 44707}, {31452, 37122}, {32047, 37814}, {35015, 46588}, {37564, 40985}, {37587, 44878}, {39873, 41584}

X(52427) = isogonal conjugate of X(52392)
X(52427) = Stevanovic-circle-inverse of X(32756)
X(52427) = isogonal conjugate of the isotomic conjugate of X(5081)
X(52427) = polar conjugate of the isotomic conjugate of X(2323)
X(52427) = X(i)-Ceva conjugate of X(j) for these (i,j): {915, 19}, {2766, 650}, {5081, 2323}, {36121, 6}
X(52427) = X(3724)-cross conjugate of X(2361)
X(52427) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52392}, {3, 18815}, {7, 1807}, {57, 52351}, {63, 2006}, {69, 1411}, {73, 14616}, {77, 80}, {86, 52391}, {222, 18359}, {265, 1442}, {307, 759}, {328, 1399}, {348, 2161}, {603, 20566}, {655, 905}, {1214, 24624}, {1231, 34079}, {1439, 6740}, {1444, 52383}, {1459, 35174}, {1565, 52377}, {1793, 3668}, {1797, 14628}, {2222, 4025}, {6187, 7182}, {6356, 52380}, {7056, 52371}, {7177, 36910}, {15413, 32675}, {22128, 34535}, {22383, 46405}, {47318, 51640}
X(52427) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52392}, {69, 35204}, {307, 34586}, {348, 40584}, {693, 13999}, {860, 35516}, {1231, 35069}, {2006, 3162}, {4025, 38984}, {5452, 52351}, {7182, 40612}, {7952, 20566}, {15413, 35128}, {18815, 36103}, {38966, 52356}, {40600, 52391}
X(52427) = cevapoint of X(3724) and X(44113)
X(52427) = crosspoint of X(i) and X(j) for these (i,j): {102, 284}, {7115, 36067}
X(52427) = crosssum of X(i) and X(j) for these (i,j): {226, 515}, {26932, 39471}
X(52427) = crossdifference of every pair of points on line {905, 1214}
X(52427) = barycentric product X(i)*X(j) for these {i,j}: {4, 2323}, {6, 5081}, {9, 1870}, {19, 4511}, {25, 32851}, {29, 2245}, {33, 3218}, {36, 281}, {37, 17515}, {55, 17923}, {92, 2361}, {101, 44428}, {186, 7110}, {270, 4053}, {284, 860}, {318, 7113}, {320, 607}, {333, 44113}, {650, 4242}, {654, 1897}, {758, 1172}, {1443, 7079}, {1464, 2322}, {1783, 3738}, {1835, 2287}, {1857, 22128}, {1983, 44426}, {2204, 35550}, {2212, 20924}, {2299, 3936}, {2332, 41804}, {3724, 31623}, {3904, 8750}, {4183, 18593}, {4282, 41013}, {4585, 18344}, {6335, 8648}, {7071, 17078}, {8606, 14165}, {21828, 36797}
X(52427) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52392}, {19, 18815}, {25, 2006}, {33, 18359}, {36, 348}, {41, 1807}, {55, 52351}, {186, 17095}, {213, 52391}, {215, 22128}, {281, 20566}, {607, 80}, {654, 4025}, {758, 1231}, {860, 349}, {1172, 14616}, {1783, 35174}, {1835, 1446}, {1870, 85}, {1897, 46405}, {1973, 1411}, {1983, 6516}, {2204, 759}, {2212, 2161}, {2245, 307}, {2299, 24624}, {2323, 69}, {2332, 6740}, {2333, 52383}, {2361, 63}, {3218, 7182}, {3724, 1214}, {3738, 15413}, {4242, 4554}, {4282, 1444}, {4511, 304}, {5081, 76}, {7071, 36910}, {7110, 328}, {7113, 77}, {8648, 905}, {8750, 655}, {17515, 274}, {17923, 6063}, {21828, 17094}, {22128, 7055}, {32851, 305}, {34397, 2003}, {44113, 226}, {44428, 3261}
X(52427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11398, 34}, {25, 55, 33}, {25, 11383, 19}, {55, 10833, 4319}, {55, 20989, 51361}, {186, 1870, 36}, {1398, 15750, 5204}, {1495, 3270, 10535}, {3085, 7487, 11392}, {3295, 3517, 11399}, {7412, 41227, 225}

X(52428) = X(1)X(104)∩X(6)X(31)

Barycentrics    a^3*(a - b - c)*(a^2 - b^2 + 4*b*c - c^2) : :
Barycentrics    (2 + Csc[A/2]^2)*Sin[A]^3 : :

X(52428) lies on these lines: {1, 104}, {3, 1057}, {6, 31}, {11, 750}, {33, 1395}, {35, 602}, {40, 1451}, {47, 3746}, {48, 44121}, {58, 1697}, {171, 497}, {238, 5218}, {255, 3295}, {354, 9316}, {390, 1936}, {595, 3601}, {612, 7069}, {748, 5432}, {756, 7082}, {950, 5264}, {976, 1858}, {984, 1776}, {991, 2078}, {999, 1480}, {1001, 25934}, {1058, 3075}, {1064, 8069}, {1106, 3304}, {1155, 1471}, {1200, 5452}, {1201, 22768}, {1279, 17603}, {1386, 9371}, {1399, 1496}, {1457, 34042}, {1458, 33925}, {1468, 3057}, {1617, 22053}, {1777, 21620}, {2187, 20986}, {2192, 7050}, {2324, 4512}, {2646, 3915}, {2654, 5711}, {3010, 21010}, {3058, 5348}, {3072, 4294}, {3073, 3085}, {3076, 35808}, {3077, 35809}, {3306, 35281}, {3486, 5255}, {3744, 10391}, {3816, 25938}, {3920, 24430}, {4300, 37579}, {5266, 12711}, {5281, 17127}, {6690, 25885}, {7084, 14935}, {9310, 35326}, {10459, 22760}, {10535, 21767}, {10589, 17122}, {10624, 37530}, {12053, 37522}, {13329, 35445}, {13384, 40091}, {14100, 51361}, {15845, 37634}, {16283, 20229}, {16466, 22072}, {17602, 38357}, {17720, 35015}, {33587, 42448}, {40967, 42012}

X(52428) = isogonal conjugate of the isotomic conjugate of X(3872)
X(52428) = X(i)-Ceva conjugate of X(j) for these (i,j): {2364, 41}, {36090, 46393}
X(52428) = X(i)-isoconjugate of X(j) for these (i,j): {7, 1000}, {278, 30680}, {279, 36916}, {3676, 51564}, {6063, 34446}
X(52428) = X(75)-Dao conjugate of X(52148)
X(52428) = crosssum of X(i) and X(j) for these (i,j): {2, 12648}, {226, 4424}
X(52428) = crossdifference of every pair of points on line {514, 46393}
X(52428) = barycentric product X(i)*X(j) for these {i,j}: {6, 3872}, {9, 999}, {31, 28808}, {33, 22129}, {41, 42697}, {55, 3306}, {71, 17519}, {284, 3753}, {650, 35281}, {1253, 17079}, {2175, 20925}, {2194, 4054}, {2364, 40587}
X(52428) = barycentric quotient X(i)/X(j) for these {i,j}: {41, 1000}, {212, 30680}, {999, 85}, {1253, 36916}, {3306, 6063}, {3753, 349}, {3872, 76}, {9447, 34446}, {17519, 44129}, {20925, 41283}, {22129, 7182}, {28808, 561}, {35281, 4554}, {42697, 20567}
X(52428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 601, 603}, {31, 55, 212}, {31, 902, 21059}, {31, 1253, 2361}, {55, 2361, 1253}, {390, 17126, 1936}, {612, 30223, 7069}, {902, 2293, 55}, {1253, 2361, 212}, {1399, 3303, 1496}

X(52429) = X(1)X(631)∩X(44)X(55)

Barycentrics    a*(a - b - c)*(a^2 - 4*a*b + b^2 - c^2)*(a^2 - b^2 - 4*a*c + c^2) : :
Barycentrics    (1 + Cos[A])*Sin[A] / (Cos[A] - 2) : :

X(52429) lies on these lines: {1, 631}, {33, 8756}, {42, 52188}, {44, 55}, {64, 37568}, {103, 35445}, {200, 2325}, {212, 51476}, {220, 3689}, {612, 7073}, {678, 1253}, {963, 5217}, {1043, 44720}, {1155, 52013}, {2177, 8557}, {2328, 3158}, {3872, 30608}, {3935, 30680}, {4105, 4543}, {4319, 52371}, {4792, 7962}, {5524, 7220}, {7050, 7074}, {7218, 14556}, {11752, 36738}, {11789, 36737}, {14942, 51564}, {28043, 42064}

X(52429) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17079}, {7, 999}, {56, 42697}, {57, 3306}, {109, 21183}, {269, 3872}, {278, 22129}, {604, 20925}, {1014, 3753}, {1407, 28808}, {1412, 4054}, {1439, 17519}, {2163, 36595}, {3676, 35281}
X(52429) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 42697}, {9, 17079}, {11, 21183}, {3161, 20925}, {3306, 5452}, {3872, 6600}, {4054, 40599}, {24771, 28808}, {36595, 40587}
X(52429) = cevapoint of X(2310) and X(4814)
X(52429) = crosspoint of X(1000) and X(36916)
X(52429) = trilinear pole of line {657, 4895}
X(52429) = barycentric product X(i)*X(j) for these {i,j}: {1, 36916}, {9, 1000}, {33, 30680}, {44, 36596}, {312, 34446}, {650, 51564}
X(52429) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17079}, {8, 20925}, {9, 42697}, {41, 999}, {45, 36595}, {55, 3306}, {200, 28808}, {210, 4054}, {212, 22129}, {220, 3872}, {650, 21183}, {1000, 85}, {1334, 3753}, {2332, 17519}, {30680, 7182}, {34446, 57}, {36596, 20568}, {36916, 75}, {51564, 4554}

X(52430) = X(1)X(1748)∩X(31)X(48)

Barycentrics    a^5*(a^2 - b^2 - c^2)^2 : :
Barycentrics    Cos[A]^2*Sin[A]^3 : :

X(52430) lies on these lines: {1, 1748}, {3, 22069}, {31, 48}, {38, 8766}, {63, 293}, {92, 1954}, {158, 829}, {162, 51299}, {184, 23197}, {204, 32676}, {212, 2638}, {255, 820}, {336, 33764}, {577, 4055}, {603, 7138}, {610, 1096}, {896, 6508}, {1106, 7114}, {1253, 2188}, {1415, 44087}, {1468, 2658}, {2173, 2181}, {2187, 42078}, {2210, 7124}, {2286, 7122}, {2308, 22063}, {4020, 26946}, {8772, 18671}, {15440, 30273}, {17453, 42075}, {22342, 36033}, {23201, 23231}, {36059, 38985}

X(52430) = isogonal conjugate of the anticomplement of X(828)
X(52430) = isogonal conjugate of the isotomic conjugate of X(255)
X(52430) = isotomic conjugate of the polar conjugate of X(9247)
X(52430) = isogonal conjugate of the polar conjugate of X(48)
X(52430) = polar conjugate of the isotomic conjugate of X(4100)
X(52430) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 563}, {48, 9247}, {255, 4100}, {2169, 48}, {4575, 822}, {24000, 163}, {24027, 32660}, {36114, 2631}
X(52430) = X(i)-isoconjugate of X(j) for these (i,j): {2, 2052}, {4, 264}, {5, 8795}, {6, 18027}, {19, 1969}, {25, 18022}, {53, 276}, {63, 6521}, {69, 1093}, {75, 158}, {76, 393}, {93, 32002}, {94, 14165}, {95, 13450}, {107, 850}, {186, 18817}, {225, 44130}, {253, 14249}, {273, 318}, {275, 324}, {278, 7017}, {281, 331}, {286, 41013}, {290, 6530}, {297, 16081}, {304, 6520}, {305, 6524}, {308, 27376}, {309, 47372}, {311, 8884}, {313, 8747}, {317, 847}, {327, 33971}, {338, 23582}, {339, 32230}, {340, 6344}, {342, 7020}, {343, 8794}, {349, 8748}, {427, 46104}, {459, 15466}, {468, 46111}, {523, 6528}, {525, 15352}, {561, 1096}, {648, 14618}, {653, 46110}, {671, 37778}, {811, 24006}, {821, 17858}, {823, 1577}, {1105, 44131}, {1109, 23999}, {1118, 3596}, {1235, 32085}, {1300, 44138}, {1441, 1896}, {1502, 2207}, {1826, 44129}, {1847, 7101}, {1857, 6063}, {1897, 46107}, {1974, 44161}, {2501, 6331}, {2592, 46815}, {2593, 46812}, {2970, 18020}, {2973, 15742}, {2996, 21447}, {3064, 46404}, {3267, 6529}, {3518, 20572}, {3542, 46746}, {5317, 27801}, {5392, 11547}, {6059, 41283}, {6335, 17924}, {6336, 46109}, {6525, 41530}, {6526, 14615}, {6531, 44132}, {7003, 40701}, {7337, 40363}, {9381, 37766}, {12077, 42405}, {14208, 36126}, {14569, 34384}, {14978, 39286}, {15459, 41079}, {15526, 34538}, {16080, 46106}, {16230, 22456}, {16263, 44135}, {16813, 18314}, {17434, 42401}, {17879, 24021}, {17907, 43678}, {17983, 44146}, {18024, 34854}, {18026, 44426}, {18831, 23290}, {20948, 24019}, {23590, 36793}, {23962, 23964}, {23978, 23984}, {23994, 24000}, {24026, 24032}, {28654, 36419}, {31623, 40149}, {32713, 44173}, {34170, 51967}, {34536, 36426}, {35142, 44145}, {35519, 36127}, {36120, 40703}, {36417, 40362}, {36889, 47392}, {37174, 42298}, {37765, 46105}, {39284, 40684}, {40009, 41766}, {40410, 44732}, {42293, 42369}, {44427, 46456}
X(52430) = X(i)-Dao conjugate of X(j) for these (i,j): {6, 1969}, {9, 18027}, {75, 1147}, {92, 22391}, {130, 2618}, {158, 206}, {264, 36033}, {304, 37867}, {520, 17879}, {561, 6503}, {850, 38985}, {1096, 40368}, {1928, 6338}, {2052, 32664}, {3162, 6521}, {6505, 18022}, {14208, 46093}, {17423, 24006}, {20948, 35071}, {34467, 46107}, {40703, 46094}
X(52430) = crosspoint of X(i) and X(j) for these (i,j): {1, 1820}, {48, 255}, {163, 24000}, {577, 7335}, {24027, 32660}
X(52430) = crosssum of X(i) and X(j) for these (i,j): {1, 1748}, {2, 5906}, {4, 17902}, {75, 33808}, {92, 158}, {1577, 2632}, {2618, 20902}, {14618, 21666}, {24026, 46110}
X(52430) = crossdifference of every pair of points on line {1577, 46110}
X(52430) = barycentric product X(i)*X(j) for these {i,j}: {1, 577}, {3, 48}, {4, 4100}, {6, 255}, {9, 7335}, {19, 1092}, {25, 6507}, {31, 394}, {32, 326}, {41, 1804}, {42, 18604}, {55, 7125}, {56, 2289}, {57, 6056}, {58, 3990}, {63, 184}, {68, 563}, {69, 9247}, {71, 1437}, {73, 2193}, {75, 14585}, {81, 4055}, {92, 23606}, {101, 23224}, {109, 36054}, {110, 822}, {162, 32320}, {163, 520}, {212, 222}, {216, 2169}, {219, 603}, {228, 1790}, {250, 37754}, {268, 7114}, {283, 1409}, {284, 22341}, {293, 3289}, {304, 14575}, {418, 2167}, {521, 32660}, {560, 3926}, {604, 1259}, {605, 1335}, {606, 1124}, {610, 14379}, {647, 4575}, {652, 36059}, {656, 32661}, {662, 39201}, {692, 4091}, {810, 4558}, {820, 41890}, {849, 52386}, {905, 32656}, {906, 1459}, {1101, 3269}, {1102, 1974}, {1147, 1820}, {1176, 4020}, {1260, 7099}, {1262, 2638}, {1331, 22383}, {1333, 3682}, {1364, 2149}, {1397, 3719}, {1402, 6514}, {1410, 2327}, {1444, 2200}, {1576, 24018}, {1636, 36034}, {1755, 17974}, {1794, 14597}, {1796, 23201}, {1797, 23202}, {1802, 7053}, {1803, 22079}, {1813, 1946}, {1818, 32658}, {1953, 19210}, {1964, 28724}, {1973, 3964}, {2148, 5562}, {2150, 7066}, {2155, 35602}, {2159, 51394}, {2175, 7183}, {2188, 7011}, {2194, 40152}, {2196, 7193}, {2206, 3998}, {2290, 50463}, {2315, 5504}, {2359, 22345}, {2617, 46088}, {2632, 23357}, {3049, 4592}, {3063, 6517}, {3284, 35200}, {3292, 36060}, {3708, 47390}, {3955, 7116}, {4131, 32739}, {5440, 32659}, {5546, 51641}, {6149, 50433}, {6394, 9417}, {7045, 39687}, {7054, 7138}, {7055, 9447}, {10607, 38252}, {14533, 44706}, {14578, 22350}, {15373, 20760}, {15526, 23995}, {15905, 19614}, {17434, 36134}, {17879, 23963}, {20752, 36057}, {20775, 34055}, {22356, 36058}, {23582, 42080}, {23979, 24031}, {24000, 35071}, {24020, 41937}, {24027, 35072}, {36148, 37084}, {40364, 40373}
X(52430) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18027}, {3, 1969}, {25, 6521}, {31, 2052}, {32, 158}, {48, 264}, {63, 18022}, {163, 6528}, {184, 92}, {212, 7017}, {255, 76}, {304, 44161}, {326, 1502}, {394, 561}, {418, 14213}, {520, 20948}, {560, 393}, {563, 317}, {577, 75}, {603, 331}, {810, 14618}, {822, 850}, {922, 37778}, {1092, 304}, {1102, 40050}, {1259, 28659}, {1437, 44129}, {1501, 1096}, {1576, 823}, {1804, 20567}, {1917, 2207}, {1923, 27376}, {1946, 46110}, {1973, 1093}, {1974, 6520}, {2148, 8795}, {2169, 276}, {2179, 13450}, {2193, 44130}, {2200, 41013}, {2289, 3596}, {2315, 44138}, {2632, 23962}, {2638, 23978}, {3049, 24006}, {3269, 23994}, {3289, 40703}, {3682, 27801}, {3719, 40363}, {3926, 1928}, {3964, 40364}, {3990, 313}, {4020, 1235}, {4055, 321}, {4091, 40495}, {4100, 69}, {4575, 6331}, {6056, 312}, {6507, 305}, {6514, 40072}, {7114, 40701}, {7125, 6063}, {7183, 41283}, {7335, 85}, {9247, 4}, {9417, 6530}, {9447, 1857}, {14533, 40440}, {14574, 24019}, {14575, 19}, {14585, 1}, {14600, 36120}, {17974, 46273}, {18604, 310}, {20775, 20883}, {22341, 349}, {22383, 46107}, {23202, 46109}, {23224, 3261}, {23357, 23999}, {23606, 63}, {23963, 24000}, {23979, 24032}, {23995, 23582}, {24018, 44173}, {28724, 18833}, {32320, 14208}, {32656, 6335}, {32660, 18026}, {32661, 811}, {32676, 15352}, {34396, 51315}, {34980, 20902}, {35071, 17879}, {36054, 35519}, {36059, 46404}, {36060, 46111}, {36134, 42405}, {36433, 255}, {37754, 339}, {39201, 1577}, {39687, 24026}, {40373, 1973}, {41937, 24021}, {42075, 36426}, {42080, 15526}, {42293, 2618}, {44088, 1953}, {46394, 1087}, {47390, 46254}, {51394, 46234}
X(52430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 22130, 22069}, {577, 6056, 4055}, {1954, 1955, 92}

X(52431) = X(6)X(1411)∩X(19)X(34242)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :
Barycentrics    Sin[A]^2 / (2 - Sec[A]) : :

X(524) lies on the cubic K498 and these lines: {6, 1411}, {19, 34242}, {31, 2875}, {37, 14737}, {42, 692}, {48, 216}, {71, 906}, {80, 1172}, {212, 3270}, {219, 1807}, {222, 3942}, {306, 1332}, {519, 2323}, {759, 36080}, {1409, 50433}, {1814, 9028}, {1949, 2253}, {2006, 2982}, {2183, 32675}, {2192, 52371}, {2219, 52383}, {2222, 32726}, {2252, 14578}, {2273, 4263}, {3187, 18359}, {3938, 40141}, {8677, 22086}, {20017, 41226}, {20566, 47318}, {21252, 24892}, {21293, 33137}, {22123, 23071}, {22350, 22356}, {36927, 40582}

X(52431) = isogonal conjugate of X(17923)
X(52431) = isogonal conjugate of the isotomic conjugate of X(52351)
X(52431) = isotomic conjugate of the polar conjugate of X(6187)
X(52431) = isogonal conjugate of the polar conjugate of X(80)
X(52431) = X(i)-Ceva conjugate of X(j) for these (i,j): {80, 6187}, {2341, 2161}
X(52431) = X(23202)-cross conjugate of X(3)
X(52431) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17923}, {2, 1870}, {4, 3218}, {19, 320}, {25, 20924}, {27, 758}, {28, 3936}, {29, 18593}, {33, 17078}, {34, 32851}, {36, 92}, {57, 5081}, {81, 860}, {108, 3904}, {158, 22128}, {162, 4707}, {186, 30690}, {214, 6336}, {226, 17515}, {264, 7113}, {273, 2323}, {274, 44113}, {278, 4511}, {281, 1443}, {286, 2245}, {331, 2361}, {333, 1835}, {340, 2160}, {470, 39153}, {471, 39152}, {514, 4242}, {651, 44428}, {653, 3738}, {654, 18026}, {811, 21828}, {1172, 41804}, {1227, 8752}, {1464, 31623}, {1474, 35550}, {1783, 4453}, {1845, 34234}, {1897, 3960}, {1973, 40075}, {1983, 46107}, {3724, 44129}, {4585, 7649}, {7100, 14165}, {8119, 8126}, {8120, 8125}, {8648, 46404}, {11570, 37203}, {13486, 44427}, {16082, 34586}, {16586, 36123}, {16944, 46109}, {36125, 51583}, {38462, 40215}
X(52431) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 17923}, {6, 320}, {36, 22391}, {92, 15898}, {125, 4707}, {860, 40586}, {1147, 22128}, {1870, 32664}, {3218, 36033}, {3904, 38983}, {3936, 40591}, {3960, 34467}, {4453, 39006}, {5081, 5452}, {6337, 40075}, {6505, 20924}, {7017, 36909}, {11517, 32851}, {17423, 21828}, {35550, 51574}, {38991, 44428}
X(52431) = cevapoint of X(i) and X(j) for these (i,j): {3, 23166}, {71, 22356}, {7117, 22086}
X(52431) = crosspoint of X(i) and X(j) for these (i,j): {80, 52351}, {1795, 1797}
X(52431) = crosssum of X(i) and X(j) for these (i,j): {1785, 8756}, {14400, 35015}
X(52431) = trilinear pole of line {228, 1946}
X(52431) = crossdifference of every pair of points on line {1845, 3738}
X(52431) = barycentric product X(i)*X(j) for these {i,j}: {1, 1807}, {3, 80}, {6, 52351}, {21, 52391}, {35, 265}, {48, 18359}, {55, 52392}, {63, 2161}, {65, 1793}, {69, 6187}, {71, 24624}, {72, 759}, {73, 6740}, {77, 52371}, {78, 1411}, {184, 20566}, {201, 52380}, {212, 18815}, {219, 2006}, {222, 36910}, {228, 14616}, {283, 52383}, {295, 36815}, {306, 34079}, {319, 52153}, {521, 2222}, {647, 47318}, {652, 655}, {1168, 5440}, {1214, 2341}, {1437, 15065}, {1444, 34857}, {1459, 51562}, {1794, 45926}, {1946, 35174}, {4064, 36069}, {6332, 32675}, {7004, 52377}, {7150, 52201}, {7265, 32662}, {9273, 21046}, {22350, 40437}, {22383, 36804}, {36058, 51975}, {36059, 52356}
X(52431) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 320}, {6, 17923}, {31, 1870}, {35, 340}, {42, 860}, {48, 3218}, {55, 5081}, {63, 20924}, {69, 40075}, {71, 3936}, {72, 35550}, {73, 41804}, {80, 264}, {184, 36}, {212, 4511}, {219, 32851}, {222, 17078}, {228, 758}, {265, 20565}, {577, 22128}, {603, 1443}, {647, 4707}, {652, 3904}, {655, 46404}, {663, 44428}, {692, 4242}, {759, 286}, {906, 4585}, {1402, 1835}, {1409, 18593}, {1411, 273}, {1459, 4453}, {1793, 314}, {1807, 75}, {1918, 44113}, {1946, 3738}, {2006, 331}, {2161, 92}, {2194, 17515}, {2200, 2245}, {2222, 18026}, {2341, 31623}, {3049, 21828}, {3937, 4089}, {5440, 1227}, {6187, 4}, {6740, 44130}, {9247, 7113}, {18359, 1969}, {20566, 18022}, {22356, 51583}, {22383, 3960}, {23201, 4973}, {23202, 214}, {24624, 44129}, {32659, 40215}, {32675, 653}, {34079, 27}, {34857, 41013}, {36815, 40717}, {36910, 7017}, {40172, 38462}, {46160, 16747}, {47318, 6331}, {50433, 52381}, {52153, 79}, {52351, 76}, {52371, 318}, {52391, 1441}, {52392, 6063}

X(52432) = X(4)X(54)∩X(24)X(52)

Barycentrics    a^4*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)^2 : :
Barycentrics    Cos[2*A]^2*Tan[A] : :

X(52432) lies on these lines: {3, 18532}, {4, 54}, {24, 52}, {25, 49}, {69, 3147}, {70, 125}, {110, 3542}, {156, 235}, {182, 37119}, {185, 44269}, {186, 1092}, {206, 9707}, {215, 11398}, {378, 10575}, {403, 9927}, {421, 847}, {427, 32046}, {567, 7507}, {569, 1594}, {576, 1974}, {933, 45135}, {1112, 20773}, {1598, 9704}, {1986, 12893}, {2477, 11399}, {3043, 15034}, {3053, 41759}, {3088, 11003}, {3089, 9544}, {3135, 19210}, {3146, 15472}, {3515, 22115}, {3517, 9703}, {3520, 10984}, {3541, 5012}, {3580, 46443}, {5094, 13353}, {5651, 14940}, {6143, 43650}, {6146, 45179}, {6240, 13352}, {6243, 11597}, {7487, 9545}, {7502, 12363}, {7505, 9306}, {7592, 45110}, {7689, 37970}, {7713, 9622}, {8745, 34756}, {8901, 34449}, {9677, 11473}, {9706, 37122}, {10018, 37636}, {10282, 44078}, {10540, 37197}, {10594, 44080}, {11449, 45172}, {11456, 36982}, {11457, 44679}, {12173, 37472}, {12228, 18569}, {13198, 14216}, {13336, 37118}, {13346, 15463}, {13371, 34114}, {15575, 21397}, {16266, 21213}, {16868, 41171}, {18374, 34787}, {18436, 37954}, {18533, 34148}, {19141, 40318}, {19154, 46444}, {21284, 37484}, {21844, 43652}, {32138, 44281}, {32534, 43898}, {35488, 46261}, {35503, 37480}, {37196, 37495}, {37478, 41590}

X(52432) = isotomic conjugate of the polar conjugate of X(36416)
X(52432) = X(i)-Ceva conjugate of X(j) for these (i,j): {11547, 571}, {18020, 41679}
X(52432) = X(i)-cross conjugate of X(j) for these (i,j): {3133, 24}, {39013, 15423}
X(52432) = X(i)-isoconjugate of X(j) for these (i,j): {68, 91}, {1820, 5392}, {2351, 20571}
X(52432) = X(i)-Dao conjugate of X(j) for these (i,j): {68, 34116}, {125, 924}, {134, 6368}, {577, 52350}
X(52432) = crosspoint of X(18020) and X(41679)
X(52432) = barycentric product X(i)*X(j) for these {i,j}: {24, 1993}, {47, 1748}, {69, 36416}, {110, 15423}, {249, 34338}, {275, 3133}, {317, 571}, {924, 41679}, {1147, 11547}, {4590, 6754}, {7763, 44077}, {8745, 9723}, {18020, 39013}
X(52432) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 5392}, {571, 68}, {1147, 52350}, {1748, 20571}, {1993, 20563}, {3133, 343}, {6754, 115}, {8745, 847}, {15423, 850}, {34338, 338}, {35603, 39116}, {36416, 4}, {39013, 125}, {41679, 46134}, {44077, 2165}
X(52432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 2904, 52}, {24, 35603, 52000}, {54, 1614, 18925}, {578, 6759, 21659}, {578, 10274, 184}, {1147, 44077, 24}, {15463, 35471, 13346}

X(52433) = X(3)X(6)∩X(53)X(10312)

Barycentrics    a^4*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4) : :
Barycentrics    (3 - 2*Cos[2*A])*Sin[A]^2 : :

X(52433) lies on these lines: {3, 6}, {53, 10312}, {112, 6748}, {193, 45795}, {230, 251}, {232, 8792}, {1501, 9604}, {1627, 3815}, {1879, 7755}, {1989, 13490}, {1990, 8882}, {2174, 21743}, {2493, 40583}, {3018, 40136}, {3567, 37813}, {3767, 18353}, {4558, 32455}, {5304, 20062}, {5306, 34603}, {7394, 7735}, {7669, 9969}, {8744, 14577}, {9722, 18907}, {9971, 40947}, {18374, 40981}

X(52433) = isogonal conjugate of the isotomic conjugate of X(34545)
X(52433) = isogonal conjugate of the polar conjugate of X(34484)
X(52433) = X(34567)-Ceva conjugate of X(184)
X(52433) = X(i)-isoconjugate of X(j) for these (i,j): {92, 34483}, {1577, 20189}, {20879, 34110}
X(52433) = X(22391)-Dao conjugate of X(34483)
X(52433) = crosspoint of X(34484) and X(34545)
X(52433) = crosssum of X(i) and X(j) for these (i,j): {2, 15108}, {6, 13564}
X(52433) = barycentric product X(i)*X(j) for these {i,j}: {3, 34484}, {6, 34545}, {54, 10095}, {110, 20188}, {140, 39168}, {186, 31676}, {1173, 36153}, {34567, 46084}
X(52433) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 34483}, {1576, 20189}, {10095, 311}, {20188, 850}, {31676, 328}, {34484, 264}, {34545, 76}, {36153, 1232}, {39168, 40410}
X(52433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 32, 2965}, {6, 1609, 566}, {6, 2965, 50}, {6, 8553, 13351}, {6, 11063, 570}, {6, 13345, 13338}, {6, 15109, 5421}, {6, 33886, 13345}, {32, 5008, 10317}, {32, 13338, 50}, {32, 13345, 6}, {32, 33886, 13338}, {187, 5421, 15109}, {371, 372, 37484}, {570, 5007, 6}, {577, 33872, 6}, {1609, 30435, 6}, {2965, 13338, 6}, {5063, 13342, 6}

X(52434) = X(31)X(184)∩X(110)X(238)

Barycentrics    a^4*(a^2 - b^2 + b*c - c^2) : :
Barycentrics    (1 - 2*Cos[A])*Sin[A]^3 : :

X(52434) lies on these lines: {1, 21326}, {6, 20962}, {31, 184}, {36, 22128}, {38, 3955}, {42, 2317}, {54, 3072}, {55, 4287}, {110, 238}, {171, 5012}, {181, 13366}, {182, 750}, {215, 2361}, {244, 1428}, {323, 3792}, {602, 1147}, {667, 838}, {692, 902}, {748, 9306}, {756, 26890}, {849, 9563}, {896, 7193}, {1106, 7335}, {1193, 1437}, {1203, 17104}, {1254, 19365}, {1395, 44077}, {1400, 44117}, {1404, 6187}, {1458, 36059}, {1460, 11402}, {1495, 3271}, {1614, 3073}, {1962, 17440}, {1977, 14567}, {1993, 5329}, {2003, 5322}, {2194, 2260}, {2206, 18892}, {2212, 44080}, {2223, 23202}, {2225, 19554}, {2310, 10535}, {3120, 5137}, {3924, 14529}, {4358, 5150}, {4579, 32927}, {4850, 5197}, {5363, 11422}, {5371, 21757}, {5651, 17125}, {6636, 7186}, {6800, 7295}, {7083, 26864}, {7299, 9652}, {9454, 32739}, {9459, 32719}, {9544, 17127}, {10448, 13323}, {11003, 17126}, {17124, 43650}, {17977, 32848}, {20665, 32664}, {20958, 51377}, {20959, 40952}, {20967, 23201}, {33105, 37527}, {34148, 37570}, {35505, 39689}, {40958, 44087}

X(52434) = isogonal conjugate of X(20566)
X(52434) = isogonal conjugate of the isotomic conjugate of X(36)
X(52434) = X(i)-Ceva conjugate of X(j) for these (i,j): {9456, 32}, {16944, 52059}, {34858, 31}
X(52434) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20566}, {2, 18359}, {8, 18815}, {10, 14616}, {75, 80}, {76, 2161}, {85, 36910}, {86, 15065}, {92, 52351}, {94, 3219}, {264, 1807}, {300, 46077}, {301, 46073}, {310, 34857}, {312, 2006}, {313, 759}, {314, 52383}, {318, 52392}, {319, 2166}, {321, 24624}, {328, 6198}, {334, 36815}, {349, 2341}, {514, 36804}, {522, 35174}, {561, 6187}, {650, 46405}, {655, 4391}, {664, 52356}, {693, 51562}, {903, 51975}, {1168, 3264}, {1411, 3596}, {1441, 6740}, {1577, 47318}, {1989, 33939}, {2174, 20573}, {2222, 35519}, {3262, 40437}, {4997, 14628}, {6063, 52371}, {7265, 32680}, {27801, 34079}, {30690, 41226}, {32851, 34535}, {34387, 52377}, {34388, 52380}, {35194, 46138}, {36795, 52212}, {40422, 45926}, {44130, 52391}
X(52434) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 20566}, {76, 40584}, {80, 206}, {313, 34586}, {319, 11597}, {561, 40612}, {3596, 35204}, {6187, 40368}, {7113, 21587}, {15065, 40600}, {18359, 32664}, {21237, 21411}, {22391, 52351}, {27801, 35069}, {33939, 34544}, {35519, 38984}, {39025, 52356}
X(52434) = crosspoint of X(i) and X(j) for these (i,j): {6, 34442}, {58, 909}, {1411, 2160}, {9456, 16944}, {24027, 32669}
X(52434) = crosssum of X(i) and X(j) for these (i,j): {1, 21368}, {2, 5080}, {8, 32849}, {10, 908}, {37, 22288}, {75, 17791}, {80, 52351}, {313, 3264}, {3219, 4511}, {4358, 51975}
X(52434) = crossdifference of every pair of points on line {321, 4391}
X(52434) = barycentric product X(i)*X(j) for these {i,j}: {1, 7113}, {6, 36}, {25, 22128}, {31, 3218}, {32, 320}, {41, 1443}, {44, 16944}, {48, 1870}, {50, 79}, {56, 2323}, {57, 2361}, {58, 2245}, {65, 4282}, {80, 52059}, {81, 3724}, {100, 21758}, {106, 17455}, {109, 654}, {110, 21828}, {184, 17923}, {214, 9456}, {215, 2006}, {284, 1464}, {323, 6186}, {513, 1983}, {560, 20924}, {604, 4511}, {651, 8648}, {667, 4585}, {692, 3960}, {758, 1333}, {849, 4053}, {902, 40215}, {909, 34586}, {1397, 32851}, {1409, 17515}, {1411, 34544}, {1415, 3738}, {1501, 40075}, {1576, 4707}, {1783, 22379}, {1790, 44113}, {1835, 2193}, {1845, 14578}, {1911, 27950}, {2151, 39153}, {2152, 39152}, {2160, 6149}, {2175, 17078}, {2194, 18593}, {2206, 3936}, {2624, 13486}, {3792, 40746}, {4089, 23990}, {4242, 22383}, {4453, 32739}, {4556, 42666}, {4867, 28607}, {4880, 34819}, {4881, 38266}, {4973, 28615}, {5353, 42623}, {6126, 19302}, {7127, 19373}, {11570, 32655}, {11700, 32677}, {16586, 34858}, {19619, 51235}, {19627, 20565}, {21773, 51803}, {32660, 44428}, {34397, 52381}, {34442, 40584}, {36910, 41282}
X(52434) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 20566}, {31, 18359}, {32, 80}, {36, 76}, {50, 319}, {79, 20573}, {109, 46405}, {184, 52351}, {213, 15065}, {215, 32851}, {320, 1502}, {560, 2161}, {604, 18815}, {654, 35519}, {692, 36804}, {758, 27801}, {1333, 14616}, {1397, 2006}, {1415, 35174}, {1443, 20567}, {1464, 349}, {1501, 6187}, {1576, 47318}, {1870, 1969}, {1983, 668}, {2175, 36910}, {2205, 34857}, {2206, 24624}, {2245, 313}, {2251, 51975}, {2323, 3596}, {2361, 312}, {3063, 52356}, {3218, 561}, {3724, 321}, {3960, 40495}, {4282, 314}, {4511, 28659}, {4585, 6386}, {4707, 44173}, {6149, 33939}, {6186, 94}, {7113, 75}, {8648, 4391}, {9247, 1807}, {9447, 52371}, {14270, 7265}, {14599, 36815}, {16944, 20568}, {17078, 41283}, {17455, 3264}, {17923, 18022}, {19627, 35}, {20924, 1928}, {21758, 693}, {21828, 850}, {22128, 305}, {22379, 15413}, {27950, 18891}, {32739, 51562}, {32851, 40363}, {40075, 40362}, {41282, 17078}, {52059, 320}
X(52434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 21368, 21326}, {6, 20989, 20962}, {184, 1397, 31}, {1428, 26884, 244}, {1977, 14567, 14599}, {2361, 7113, 3724}, {20986, 44085, 42}

X(52435) = X(2)X(51458)∩X(3)X(19362)

Barycentrics    a^6*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
Barycentrics    Cos[A]*Cos[2*A]*Sin[A]^3 : :

X(52435) lies on these lines: {2, 51458}, {3, 19362}, {6, 2351}, {25, 1576}, {39, 13366}, {50, 3135}, {54, 15653}, {155, 16391}, {184, 418}, {237, 20968}, {571, 44077}, {933, 2052}, {1501, 9419}, {1993, 51776}, {2055, 19467}, {2120, 3567}, {3051, 22391}, {3060, 13558}, {3133, 34116}, {3148, 42442}, {3167, 6503}, {5392, 41205}, {5654, 13557}, {7577, 8154}, {10132, 10665}, {10133, 10666}, {15257, 52277}, {15512, 37493}, {19210, 19357}, {31381, 32046}, {34986, 47195}, {37893, 43653}, {40820, 43754}

X(52435) = isogonal conjugate of the isotomic conjugate of X(1147)
X(52435) = isogonal conjugate of the polar conjugate of X(571)
X(52435) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 14585}, {933, 52317}, {1166, 14533}, {1576, 34952}, {44174, 32661}
X(52435) = X(i)-isoconjugate of X(j) for these (i,j): {4, 20571}, {75, 847}, {91, 264}, {92, 5392}, {158, 20563}, {561, 14593}, {1577, 30450}, {1820, 18027}, {1969, 2165}, {6521, 52350}, {24006, 46134}
X(52435) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 577}, {206, 847}, {264, 34116}, {1147, 20563}, {1216, 1225}, {5392, 22391}, {14593, 40368}, {15415, 47421}, {20571, 36033}
X(52435) = crosspoint of X(i) and X(j) for these (i,j): {6, 24}, {571, 1147}, {14586, 47390}, {32661, 44174}
X(52435) = crosssum of X(i) and X(j) for these (i,j): {2, 68}, {136, 14618}, {847, 5392}, {2970, 18314}, {34391, 34392}
X(52435) = crossdifference of every pair of points on line {6334, 14618}
X(52435) = barycentric product X(i)*X(j) for these {i,j}: {1, 563}, {3, 571}, {6, 1147}, {24, 577}, {32, 9723}, {47, 48}, {50, 5961}, {52, 14533}, {110, 30451}, {184, 1993}, {228, 18605}, {237, 51776}, {317, 14585}, {371, 26920}, {372, 8911}, {394, 44077}, {906, 34948}, {924, 32661}, {1092, 8745}, {2169, 2180}, {4558, 34952}, {7763, 14575}, {9247, 44179}, {11547, 23606}, {14576, 19210}, {14600, 51439}, {15958, 52317}, {16391, 36416}, {18877, 51393}, {32662, 44808}, {39013, 44174}, {39201, 41679}, {47390, 47421}
X(52435) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 18027}, {32, 847}, {47, 1969}, {48, 20571}, {184, 5392}, {563, 75}, {571, 264}, {577, 20563}, {1147, 76}, {1501, 14593}, {1576, 30450}, {1993, 18022}, {5961, 20573}, {7763, 44161}, {8911, 34392}, {9247, 91}, {9723, 1502}, {14533, 34385}, {14575, 2165}, {14585, 68}, {19627, 5962}, {23606, 52350}, {26920, 34391}, {30451, 850}, {32661, 46134}, {34952, 14618}, {36433, 16391}, {44077, 2052}, {51776, 18024}
X(52435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {184, 577, 23195}, {14575, 23606, 184}

X(52436) = X(6)X(26)∩X(24)X(47421)

Barycentrics    a^6*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
Barycentrics    Cos[2*A]*Sin[A]^4 : :

X(52436) lies on these lines: {6, 26}, {24, 47421}, {25, 41271}, {32, 184}, {50, 9603}, {155, 3053}, {159, 40825}, {187, 5562}, {230, 12134}, {571, 1147}, {1352, 1691}, {1609, 9908}, {1692, 1843}, {1915, 2548}, {1970, 7737}, {1971, 3767}, {2030, 43130}, {2353, 14600}, {2715, 14265}, {2909, 27369}, {3202, 14573}, {3574, 7747}, {5889, 39839}, {6423, 44192}, {6424, 44193}, {8553, 42445}, {9233, 36433}, {9418, 20968}, {9700, 20859}, {9781, 40633}, {10539, 45938}, {14574, 44162}, {32379, 32445}, {35007, 43844}, {42295, 44527}

X(52436) = isogonal conjugate of the isotomic conjugate of X(571)
X(52436) = isogonal conjugate of the polar conjugate of X(44077)
X(52436) = X(25)-Ceva conjugate of X(14575)
X(52436) = X(i)-isoconjugate of X(j) for these (i,j): {2, 20571}, {68, 1969}, {75, 5392}, {76, 91}, {92, 20563}, {304, 847}, {561, 2165}, {925, 20948}, {1577, 46134}, {1820, 18022}, {14208, 30450}, {14213, 34385}, {14593, 40364}, {36145, 44173}
X(52436) = X(i)-Dao conjugate of X(j) for these (i,j): {76, 34116}, {206, 5392}, {305, 577}, {2165, 40368}, {20563, 22391}, {20571, 32664}, {39013, 44173}
X(52436) = crosspoint of X(i) and X(j) for these (i,j): {25, 8745}, {571, 44077}
X(52436) = crosssum of X(i) and X(j) for these (i,j): {2, 44128}, {69, 52350}, {3267, 23962}, {5392, 20563}
X(52436) = crossdifference of every pair of points on line {850, 15415}
X(52436) = barycentric product X(i)*X(j) for these {i,j}: {3, 44077}, {6, 571}, {19, 563}, {24, 184}, {25, 1147}, {31, 47}, {32, 1993}, {110, 34952}, {112, 30451}, {213, 18605}, {317, 14575}, {560, 44179}, {577, 8745}, {692, 34948}, {924, 1576}, {1501, 7763}, {1748, 9247}, {1974, 9723}, {2148, 2180}, {2211, 51776}, {3049, 41679}, {3133, 41271}, {5412, 8911}, {5413, 26920}, {5961, 34397}, {6563, 14574}, {6753, 32661}, {6754, 44174}, {9418, 31635}, {11547, 14585}, {14397, 32640}, {14533, 14576}, {14560, 44808}, {14573, 39113}, {14586, 52317}, {14601, 51439}, {18883, 19627}, {23357, 47421}, {40352, 51393}
X(52436) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 18022}, {31, 20571}, {32, 5392}, {47, 561}, {184, 20563}, {317, 44161}, {560, 91}, {563, 304}, {571, 76}, {924, 44173}, {1147, 305}, {1501, 2165}, {1576, 46134}, {1974, 847}, {1993, 1502}, {7763, 40362}, {8745, 18027}, {9723, 40050}, {14573, 96}, {14574, 925}, {14575, 68}, {14585, 52350}, {18605, 6385}, {19627, 37802}, {30451, 3267}, {34948, 40495}, {34952, 850}, {40373, 2351}, {44077, 264}, {44162, 14593}, {44179, 1928}, {47421, 23962}, {52317, 15415}
X(52436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 19627, 14585}, {1501, 14585, 32}, {9418, 40373, 20968}

X(52437) = X(3)X(69)∩X(23)X(325)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2) : :
Barycentrics    (1 + 2*Cos[2*A])*Cot[A] : :
X(52437) = 2 X[11063] - 3 X[35296]

X(52437) lies on these lines: {3, 69}, {23, 325}, {26, 32825}, {50, 323}, {76, 7550}, {95, 252}, {99, 477}, {141, 50660}, {183, 7496}, {186, 340}, {264, 14865}, {298, 11146}, {299, 11145}, {311, 35500}, {316, 37946}, {317, 3518}, {319, 7279}, {320, 4996}, {328, 12028}, {524, 11063}, {895, 43705}, {1007, 1995}, {1078, 44148}, {1272, 2071}, {1494, 37948}, {1599, 32810}, {1600, 32811}, {1609, 11008}, {1634, 37183}, {1975, 7527}, {1992, 52275}, {3001, 9145}, {3003, 41617}, {3265, 15414}, {3268, 15470}, {3284, 4558}, {3292, 47390}, {3520, 44134}, {3631, 15109}, {5201, 35298}, {6503, 37669}, {6636, 7788}, {6644, 32837}, {7488, 32821}, {7492, 37668}, {7512, 7796}, {7514, 32836}, {7526, 32824}, {7530, 32816}, {7556, 32818}, {7782, 44133}, {7809, 37925}, {7811, 44832}, {7814, 34484}, {8553, 20080}, {8681, 14060}, {9146, 15365}, {9155, 52238}, {9737, 11188}, {10411, 14385}, {10607, 20806}, {11284, 34803}, {12082, 32006}, {12086, 44136}, {12088, 44128}, {12584, 51397}, {13337, 15018}, {14118, 32820}, {14570, 40888}, {14919, 43755}, {15246, 37671}, {17271, 37293}, {22087, 22143}, {23061, 51439}, {26863, 32002}, {31861, 32815}, {32000, 35475}, {32001, 44879}, {32833, 35921}, {34229, 40916}, {34990, 37784}, {35520, 46423}, {36163, 45921}, {36829, 37980}, {36896, 44769}, {39113, 44802}, {44377, 44533}, {44718, 46112}, {44719, 46113}, {45198, 45799}

X(52437) = isogonal conjugate of X(18384)
X(52437) = isotomic conjugate of X(6344)
X(52437) = isotomic conjugate of the isogonal conjugate of X(22115)
X(52437) = isotomic conjugate of the polar conjugate of X(323)
X(52437) = isogonal conjugate of the polar conjugate of X(7799)
X(52437) = X(7799)-Ceva conjugate of X(323)
X(52437) = X(i)-cross conjugate of X(j) for these (i,j): {16186, 8552}, {22115, 323}
X(52437) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18384}, {19, 1989}, {25, 2166}, {31, 6344}, {92, 11060}, {94, 1973}, {158, 52153}, {162, 15475}, {265, 1096}, {512, 36129}, {560, 18817}, {798, 46456}, {1141, 2181}, {1784, 40355}, {2153, 8738}, {2154, 8737}, {2489, 32680}, {2501, 32678}, {6520, 50433}, {8750, 43082}, {10412, 32676}, {14560, 24006}, {14582, 24019}, {14583, 36119}
X(52437) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6344}, {3, 18384}, {4, 40604}, {6, 1989}, {19, 34544}, {25, 11597}, {94, 6337}, {125, 15475}, {265, 6503}, {323, 37943}, {328, 6338}, {403, 3580}, {462, 33527}, {463, 33526}, {1147, 52153}, {1511, 14583}, {1990, 3284}, {2088, 47236}, {2166, 6505}, {2501, 18334}, {2970, 5664}, {6374, 18817}, {8737, 40581}, {8738, 40580}, {10412, 15526}, {11060, 22391}, {13450, 14920}, {14569, 18402}, {14582, 35071}, {15595, 43089}, {17433, 51513}, {26932, 43082}, {31998, 46456}, {36129, 39054}, {37867, 50433}
X(52437) = cevapoint of X(8552) and X(16186)
X(52437) = crosspoint of X(34384) and X(40832)
X(52437) = crosssum of X(462) and X(463)
X(52437) = crossdifference of every pair of points on line {2489, 3199}
X(52437) = barycentric product X(i)*X(j) for these {i,j}: {3, 7799}, {50, 305}, {69, 323}, {76, 22115}, {97, 1273}, {99, 8552}, {110, 45792}, {186, 3926}, {287, 51383}, {298, 44719}, {299, 44718}, {304, 6149}, {319, 22128}, {340, 394}, {525, 10411}, {526, 4563}, {1154, 34386}, {1444, 42701}, {2088, 47389}, {3265, 14590}, {3268, 4558}, {3964, 14165}, {4590, 16186}, {4592, 32679}, {6148, 14919}, {6393, 14355}, {9723, 37802}, {11128, 50466}, {11129, 50465}, {19627, 40050}, {45807, 51478}
X(52437) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6344}, {3, 1989}, {6, 18384}, {15, 8738}, {16, 8737}, {50, 25}, {63, 2166}, {69, 94}, {76, 18817}, {97, 1141}, {99, 46456}, {184, 11060}, {186, 393}, {305, 20573}, {323, 4}, {340, 2052}, {394, 265}, {441, 43089}, {520, 14582}, {525, 10412}, {526, 2501}, {577, 52153}, {647, 15475}, {662, 36129}, {905, 43082}, {1092, 50433}, {1154, 53}, {1273, 324}, {1511, 1990}, {2081, 51513}, {2088, 8754}, {2290, 2181}, {3265, 14592}, {3268, 14618}, {3284, 14583}, {3926, 328}, {4558, 476}, {4563, 35139}, {4575, 32678}, {4592, 32680}, {6148, 46106}, {6149, 19}, {6390, 43084}, {7799, 264}, {8552, 523}, {9723, 18883}, {10411, 648}, {11062, 14569}, {11064, 14254}, {11145, 46925}, {11146, 46926}, {14165, 1093}, {14270, 2489}, {14355, 6531}, {14385, 8749}, {14417, 51479}, {14590, 107}, {14591, 32713}, {14918, 13450}, {14919, 5627}, {16186, 115}, {17402, 36309}, {17403, 36306}, {18877, 40355}, {19210, 11077}, {19294, 463}, {19295, 462}, {19627, 1974}, {22115, 6}, {22128, 79}, {32661, 14560}, {32679, 24006}, {34386, 46138}, {34397, 2207}, {34834, 403}, {36212, 14356}, {37802, 847}, {39371, 51965}, {40604, 37943}, {41078, 23290}, {42701, 41013}, {43704, 11071}, {44180, 30529}, {44718, 14}, {44719, 13}, {44808, 6753}, {44814, 14273}, {45792, 850}, {46112, 3458}, {46113, 3457}, {50433, 14595}, {50465, 11080}, {50466, 11085}, {51383, 297}
X(52437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 9723, 44180}, {340, 1273, 52149}, {340, 7799, 1273}, {487, 488, 18917}, {1238, 41008, 69}, {1273, 6148, 340}, {3964, 9723, 69}, {4558, 36212, 22151}, {6148, 7799, 52149}, {14368, 14369, 186}

X(52438) = X(6)X(1511)∩X(32)X(184)

Barycentrics    a^6*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) : :
Barycentrics    (2 + Cos[2*A])*Sin[A]^4 : :

X(52438) lies on these lines: {6, 1511}, {32, 184}, {39, 51394}, {249, 7757}, {378, 11653}, {682, 2909}, {1691, 11179}, {1692, 40673}, {1970, 3767}, {1971, 7737}, {3003, 18475}, {3202, 44162}, {5052, 44102}, {5309, 32761}, {5642, 7753}, {7745, 51425}, {7747, 51403}, {8573, 14533}, {9604, 41336}, {10540, 18373}, {14574, 14575}, {19626, 40823}, {32621, 40825}

X(52438) = isogonal conjugate of the isotomic conjugate of X(5063)
X(52438) = isogonal conjugate of the polar conjugate of X(44080)
X(52438) = X(i)-isoconjugate of X(j) for these (i,j): {75, 34289}, {561, 34288}, {1302, 20948}, {1969, 4846}, {36149, 44173}
X(52438) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 34289}, {34288, 40368}
X(52438) = crosspoint of X(5063) and X(44080)
X(52438) = barycentric product X(i)*X(j) for these {i,j}: {3, 44080}, {6, 5063}, {32, 15066}, {110, 42660}, {184, 378}, {237, 11653}, {1501, 32833}, {1576, 8675}, {10564, 40352}, {14574, 30474}, {14575, 44134}
X(52438) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34289}, {378, 18022}, {1501, 34288}, {5063, 76}, {8675, 44173}, {11653, 18024}, {14574, 1302}, {14575, 4846}, {15066, 1502}, {32833, 40362}, {42660, 850}, {44080, 264}, {44134, 44161}
X(52438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 19627, 1501}, {1501, 14585, 19627}

X(52439) = X(3)X(6530)∩X(25)X(393)

Barycentrics    a^2*(a^2 + b^2 - c^2)^3*(a^2 - b^2 + c^2)^3 : :
Barycentrics    Sin[A]^2*Tan[A]^3 : :

X(52439) lies on these lines: {3, 6530}, {4, 46741}, {6, 34854}, {25, 393}, {107, 2374}, {159, 1990}, {264, 5020}, {297, 37491}, {648, 19588}, {1093, 1598}, {1249, 19459}, {1593, 10002}, {1974, 2207}, {1995, 43981}, {3172, 14575}, {3199, 33578}, {3964, 15143}, {8743, 19125}, {8745, 19118}, {8746, 19153}, {9909, 37765}, {10790, 41365}, {15259, 41762}, {15905, 44096}, {16264, 18535}, {18534, 34334}, {20975, 41489}, {34208, 37777}, {37488, 39569}, {37921, 47158}, {39879, 41204}

X(52439) = isogonal conjugate of X(4176)
X(52439) = isogonal conjugate of the isotomic conjugate of X(6524)
X(52439) = isogonal conjugate of the polar conjugate of X(36434)
X(52439) = polar conjugate of the isotomic conjugate of X(2207)
X(52439) = polar conjugate of the isogonal conjugate of X(36417)
X(52439) = X(6524)-Ceva conjugate of X(2207)
X(52439) = X(36417)-cross conjugate of X(2207)
X(52439) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4176}, {2, 1102}, {63, 3926}, {69, 326}, {75, 3964}, {76, 6507}, {77, 1264}, {78, 7055}, {255, 305}, {304, 394}, {332, 52385}, {336, 51386}, {345, 7183}, {348, 3719}, {561, 1092}, {577, 40364}, {662, 4143}, {873, 4158}, {1231, 6514}, {1259, 7182}, {1332, 30805}, {1444, 52396}, {1502, 4100}, {1804, 3718}, {1928, 23606}, {2632, 47389}, {2972, 24037}, {3265, 4592}, {3998, 17206}, {4131, 4561}, {4563, 24018}, {4602, 32320}, {6517, 35518}, {7035, 7215}, {18020, 24020}, {18604, 40071}, {34537, 37754}
X(52439) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4176}, {69, 15259}, {206, 3964}, {305, 6523}, {512, 2972}, {1084, 4143}, {1092, 40368}, {1102, 32664}, {3162, 3926}, {3265, 5139}, {14713, 44141}, {23606, 40369}
X(52439) = crosspoint of X(6524) and X(36434)
X(52439) = barycentric product X(i)*X(j) for these {i,j}: {3, 36434}, {4, 2207}, {6, 6524}, {19, 1096}, {25, 393}, {31, 6520}, {32, 1093}, {107, 2489}, {125, 23975}, {158, 1973}, {264, 36417}, {278, 6059}, {281, 7337}, {512, 6529}, {560, 6521}, {607, 1118}, {608, 1857}, {669, 15352}, {798, 36126}, {1824, 5317}, {1974, 2052}, {2333, 8747}, {2501, 32713}, {2970, 41937}, {2971, 23582}, {3124, 32230}, {3172, 6526}, {3199, 8884}, {3708, 24022}, {6525, 41489}, {6531, 34854}, {7140, 36420}, {8745, 14593}, {8754, 23964}, {8794, 40981}, {8882, 14569}, {17994, 20031}, {18027, 44162}, {20975, 23590}, {23985, 42069}, {40144, 41766}
X(52439) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 4176}, {25, 3926}, {31, 1102}, {32, 3964}, {158, 40364}, {393, 305}, {512, 4143}, {560, 6507}, {607, 1264}, {608, 7055}, {1084, 2972}, {1093, 1502}, {1096, 304}, {1395, 7183}, {1501, 1092}, {1917, 4100}, {1973, 326}, {1974, 394}, {1977, 7215}, {2052, 40050}, {2207, 69}, {2211, 51386}, {2212, 3719}, {2333, 52396}, {2489, 3265}, {2971, 15526}, {3080, 22401}, {3199, 52347}, {4117, 37754}, {6059, 345}, {6520, 561}, {6521, 1928}, {6524, 76}, {6529, 670}, {7109, 4158}, {7337, 348}, {8029, 23107}, {8754, 36793}, {9233, 23606}, {9426, 32320}, {9427, 34980}, {14569, 28706}, {15352, 4609}, {18027, 40360}, {20975, 23974}, {22260, 23616}, {23216, 35071}, {23964, 47389}, {23975, 18020}, {24022, 46254}, {32230, 34537}, {32713, 4563}, {34854, 6393}, {36126, 4602}, {36417, 3}, {36434, 264}, {42068, 3269}, {42295, 44141}, {44162, 577}
X(52439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 1033, 40947}, {8745, 32713, 19118}

X(52440) = X(3)X(7163)∩X(31)X(56)

Barycentrics    a^3*(a + b - c)*(a - b + c)*(a^2 - b^2 + b*c - c^2) : :
Barycentrics    Cos[3*A/2]*Sin[A/2]^3 : :

X(52440) lies on these lines: {1, 18360}, {3, 7163}, {12, 17122}, {31, 56}, {36, 1464}, {50, 7113}, {58, 14158}, {65, 15955}, {109, 1319}, {171, 5434}, {222, 1470}, {238, 5298}, {244, 51656}, {255, 5204}, {354, 37469}, {601, 3304}, {604, 28607}, {692, 23205}, {750, 11237}, {849, 1408}, {1193, 8614}, {1394, 5573}, {1411, 1455}, {1414, 1447}, {1415, 2251}, {1428, 3733}, {1454, 4320}, {1458, 3446}, {1468, 5221}, {1469, 7236}, {1496, 5217}, {1777, 11376}, {1914, 36075}, {1935, 5433}, {1936, 15326}, {2099, 9316}, {3075, 7354}, {3157, 40293}, {3230, 36074}, {3417, 26914}, {3649, 37607}, {3880, 35281}, {4292, 51751}, {4293, 5348}, {4303, 37564}, {4973, 18593}, {7286, 18201}, {7288, 7299}, {9363, 10944}, {9364, 40663}, {9456, 32669}, {10404, 37522}, {10571, 34880}, {11509, 34046}, {11510, 21000}, {19373, 42623}, {25938, 31141}, {32668, 34068}, {51236, 51651}

X(52440) = isogonal conjugate of the isotomic conjugate of X(1443)
X(52440) = X(i)-isoconjugate of X(j) for these (i,j): {2, 36910}, {8, 80}, {9, 18359}, {10, 6740}, {21, 15065}, {55, 20566}, {75, 52371}, {100, 52356}, {200, 18815}, {210, 14616}, {281, 52351}, {312, 2161}, {314, 34857}, {318, 1807}, {321, 2341}, {341, 1411}, {346, 2006}, {519, 36590}, {522, 51562}, {650, 36804}, {655, 3239}, {657, 46405}, {759, 3701}, {1043, 52383}, {1089, 52380}, {1168, 4723}, {1320, 51975}, {1793, 41013}, {1989, 42033}, {2166, 4420}, {2222, 4397}, {2321, 24624}, {3596, 6187}, {3700, 47318}, {3900, 35174}, {4518, 36815}, {6735, 40437}, {7026, 7043}, {7046, 52392}, {7110, 41226}, {24026, 52377}, {30713, 34079}, {36926, 36935}, {36927, 36934}, {36930, 36937}, {36931, 36938}, {44690, 46077}, {44691, 46073}
X(52440) = X(i)-Dao conjugate of X(j) for these (i,j): {206, 52371}, {223, 20566}, {312, 40584}, {341, 35204}, {478, 18359}, {3596, 40612}, {3701, 34586}, {4397, 38984}, {4420, 11597}, {6609, 18815}, {8054, 52356}, {15065, 40611}, {30713, 35069}, {32664, 36910}, {34544, 42033}
X(52440) = crosspoint of X(i) and X(j) for these (i,j): {1014, 34051}, {1408, 1417}
X(52440) = crosssum of X(i) and X(j) for these (i,j): {3701, 4723}, {4768, 24026}
X(52440) = crossdifference of every pair of points on line {2321, 3239}
X(52440) = barycentric product X(i)*X(j) for these {i,j}: {6, 1443}, {7, 7113}, {31, 17078}, {34, 22128}, {36, 57}, {56, 3218}, {58, 18593}, {81, 1464}, {109, 3960}, {222, 1870}, {269, 2323}, {279, 2361}, {320, 604}, {323, 52372}, {603, 17923}, {653, 22379}, {654, 934}, {658, 8648}, {664, 21758}, {758, 1412}, {1014, 2245}, {1106, 32851}, {1319, 40215}, {1333, 41804}, {1397, 20924}, {1407, 4511}, {1408, 3936}, {1414, 21828}, {1415, 4453}, {1417, 51583}, {1434, 3724}, {1461, 3738}, {1790, 1835}, {1983, 3676}, {2149, 4089}, {3668, 4282}, {3911, 16944}, {4053, 7341}, {4556, 51645}, {4585, 43924}, {4881, 40151}, {5081, 7099}, {6149, 52374}, {7051, 37773}, {9456, 41801}, {16947, 35550}, {17515, 52373}, {18359, 41282}, {18815, 52059}, {19373, 37772}, {28607, 36589}, {34051, 34586}
X(52440) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 36910}, {32, 52371}, {36, 312}, {50, 4420}, {56, 18359}, {57, 20566}, {109, 36804}, {320, 28659}, {603, 52351}, {604, 80}, {649, 52356}, {654, 4397}, {758, 30713}, {934, 46405}, {1106, 2006}, {1333, 6740}, {1397, 2161}, {1399, 41226}, {1400, 15065}, {1404, 51975}, {1407, 18815}, {1408, 24624}, {1412, 14616}, {1415, 51562}, {1443, 76}, {1461, 35174}, {1464, 321}, {1870, 7017}, {1983, 3699}, {2206, 2341}, {2245, 3701}, {2323, 341}, {2361, 346}, {3218, 3596}, {3724, 2321}, {3960, 35519}, {4282, 1043}, {4881, 44723}, {6149, 42033}, {7099, 52392}, {7113, 8}, {8648, 3239}, {9456, 36590}, {16944, 4997}, {16947, 759}, {17078, 561}, {17455, 4723}, {18593, 313}, {20924, 40363}, {21758, 522}, {21828, 4086}, {22128, 3718}, {22379, 6332}, {23979, 52377}, {27950, 4087}, {41282, 3218}, {41804, 27801}, {46384, 23104}, {52059, 4511}, {52372, 94}
X(52440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 603, 1399}, {603, 1106, 56}, {1455, 18838, 1411}

X(52441) = ISOGONAL CONJUGATE OF X(26883)

Barycentrics    (a^6*b^2-3*a^4*b^4+3*a^2*b^6-b^8-4*a^6*c^2-a^4*b^2*c^2+2*a^2*b^4*c^2+3*b^6*c^2+8*a^4*c^4-a^2*b^2*c^4-3*b^4*c^4-4*a^2*c^6+b^2*c^6)*(4*a^6*b^2-8*a^4*b^4+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+3*a^4*c^4-2*a^2*b^2*c^4+3*b^4*c^4-3*a^2*c^6-3*b^2*c^6+c^8) : :

See Ivan Pavlov, euclid 5545.

X(52441) lies on these lines: {216, 3523}, {376, 5562}, {3146, 8798}

X(52441) = isogonal conjugate of X(26883)

X(52442) = X(1442)X(4357)∩X(3219)X(3687)

Barycentrics    (a^4-2*a^2*b^2+b^4+a^2*b*c+a*b^2*c-3*a*b*c^2-c^4)*(a^4-b^4+a^2*b*c-3*a*b^2*c-2*a^2*c^2+a*b*c^2+c^4) : :

See Ivan Pavlov, euclid 5545.

X(52442) lies on these lines: {1442, 4357}, {3219, 3687}

X(52442) = cross conjugate of X(2975) and X(2)

X(52443) = X(264)X(3854)∩X(307)X(25718)

Barycentrics    (5*a^4-10*a^2*b^2+5*b^4+6*a^2*c^2+6*b^2*c^2-11*c^4)*(5*a^4+6*a^2*b^2-11*b^4-10*a^2*c^2+6*b^2*c^2+5*c^4) : :

See Ivan Pavlov, euclid 5545.

X(52443) lies on these lines: {264, 3854}, {307, 25718}

X(52443) = cross conjugate of X(3146) and X(2)






leftri  A locus related to K025: X(52444) - X(52447)   rightri

This preamble and centers X(52444)-X(52447) were contributed by César Eliud Lozada, November 15, 2022.

Let ABC be a triangle and P a point on its plane. Let A'B'C' be the cevian triangle of P, A" the reflection of A' in A. Let A* be the orthogonal projection of A" in BC and build B*, C* cyclically.

Then the locus of P such that ABC, A*B*C are perspective is the cubic K025 and the locus of the perspectors Q(P) is also K025.

The appearance of (i, j) in the following list means that Q( X(i) ) = X(j):

(4, 4), (30, 265), (265, 30), (316, 671), (671, 316), (1263, 19552), (1300, 5962), (5080, 11604), (5134, 39993), (5203, 41521), (5523, 11605), (5962, 1300), (10152, 34170), (11604, 5080), (11605, 5523), (11703, 52444), (19552, 1263), (34150, 39985), (34169, 34171), (34170, 10152), (34171, 34169), (34173, 47104), (34174, 34175), (34175, 34174), (34239, 34240), (34240, 34239), (38945, 39992), (38946, 52445), (38947, 52446), (38949, 39990), (38952, 39991), (39158, 39159), (39159, 39158), (39985, 34150), (39990, 38949), (39991, 38952), (39992, 38945), (39993, 5134), (41521, 5203), (42809, 42810), (42810, 42809), (47103, 52447), (47104, 34173), (47105, 47110), (47110, 47105), (52444, 11703), (52445, 38946), (52446, 38947), (52447, 47103)

Some additional properties:

underbar

X(52444) = X(4)X(2889)∩X(30)X(20189)

Barycentrics    (a^4-(2*b^2+3*c^2)*a^2+(b^2-c^2)^2)*(a^8-2*(b^2+c^2)*a^6-3*b^2*c^2*a^4+(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^2-c^2)^4)*(a^4-(3*b^2+2*c^2)*a^2+(b^2-c^2)^2) : :

See César Lozada, euclid 5551.

X(52444) lies on the cubic K025 and these lines: {4, 2889}, {30, 20189}, {140, 34110}, {10627, 11538}, {34174, 39989}

X(52444) = anticomplement of the circumperp conjugate of X(33542)
X(52444) = antigonal conjugate of X(5899)
X(52444) = inverse of X(11817) in polar circle
X(52444) = orthoassociate of X(11817)

X(52445) = X(4)X(18125)∩X(30)X(1287)

Barycentrics    (a^6+(b^2-c^2)*a^4+(b^4-b^2*c^2-c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^6-(b^2-c^2)*a^4-(b^4+b^2*c^2-c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^4-b^4-b^2*c^2-c^4) : :

See César Lozada, euclid 5551.

X(52445) lies on the cubic K025 and these lines: {4, 18125}, {30, 1287}, {827, 5133}, {1263, 34174}, {6636, 46654}, {19552, 34175}

X(52445) = reflection of X(i) in X(j) for these (i, j): (827, 5133), (6636, 46654)
X(52445) = antigonal conjugate of X(6636)
X(52445) = syngonal image of X(5133)

X(52446) = X(4)X(18321)∩X(30)X(2698)

Barycentrics    a^2*(c^2*a^6-(b^4-b^2*c^2+2*c^4)*a^4+(b^4-b^2*c^2+c^4)*c^2*a^2+(b^2-c^2)^2*b^2*c^2)*(b^2*a^6-(2*b^4-b^2*c^2+c^4)*a^4+(b^4-b^2*c^2+c^4)*b^2*a^2+(b^2-c^2)^2*b^2*c^2)*((b^2+c^2)*a^2-b^4-c^4) : :

See César Lozada, euclid 5551.

X(52446) lies on the cubic K025 and these lines: {4, 18321}, {30, 2698}, {32, 2086}, {237, 2679}, {446, 34157}, {805, 21531}, {1316, 40077}, {2387, 37841}, {14251, 44114}, {14957, 41520}

X(52446) = reflection of X(i) in X(j) for these (i, j): (237, 2679), (805, 21531)
X(52446) = antigonal conjugate of X(237)
X(52446) = syngonal image of X(21531)

X(52447) = X(4)X(541)∩X(30)X(9060)

Barycentrics   (a^6-(b^2-2*c^2)*a^4-(b^4-6*b^2*c^2+7*c^4)*a^2+(b^2+4*c^2)*(b^2-c^2)^2)*(a^6+(2*b^2-c^2)*a^4-(7*b^4-6*b^2*c^2+c^4)*a^2+(4*b^2+c^2)*(b^2-c^2)^2)*(5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(52447) = 3*X(3839)-2*X(50935)

See César Lozada, euclid 5551.

X(52447) lies on the cubic K025 and these lines: {4, 541}, {30, 9060}, {316, 1494}, {381, 1302}, {671, 39985}, {3839, 50935}, {6128, 52187}, {11605, 47111}, {34175, 38951}, {39263, 44750}

X(52447) = reflection of X(1302) in X(381)
X(52447) = isogonal conjugate of the circumperp conjugate of X(11820)
X(52447) = antigonal conjugate of X(376)
X(52447) = X(376)-reciprocal conjugate of-X(40112)
X(52447) = syngonal image of X(381)
X(52447) = barycentric quotient X(376)/X(40112)






leftri  A locus related to K616: X(52448) - X(52456)  rightri

This preamble and centers X(52448)-X(52456) were contributed by César Eliud Lozada, November 15, 2022.

Let ABC be a triangle and P a point on its plane. Let A'B'C' be the cevian triangle of P, A" the midpoint of AA'. Let A* be the orthogonal projection of A" in BC and build B*, C* cyclically.

Then the locus of P such that ABC, A*B*C are perspective is the cubic K616 and the locus of the perspectors Q(P) is the cubic q' = anticomplement-of-K1265, given by:

  q': ∑[(SB*(S^2+(2*SB+SC)*SC)*y - SC*(S^2+(SB+2*SC)*SB)*z)*y*z] + (SB-SC)*(SC-SA)*(SA-SB) *x*y*z = 0

q' circumscribes ABC and passes through X(n) for these n: {2, 4, 3434, 4846, 11185, 14593, 30513, 34029, 41370, 52448, 52449, 52450, 52451, 52452, 52453, 52454, 52455, 52456}.

The appearance of (i, j) in the following list means that Q( X(i) ) = X(j):

(4, 4), (69, 2), (376, 4846), (1249, 52448), (3421, 30513), (5485, 11185), (6601, 3434), (9214, 52449), (34208, 14593), (36874, 52450), (36875, 52451), (36876, 52