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This is PART 22: Centers X(42001) - X(44000)

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(42001) = X(13)X(5916)∩X(62)X(21466)

Barycentrics    (S*(Sqrt[3]*a^2 + S) + 3*SB*SC)*(Sqrt[3]*S*(-a^2 + 2*SA) + 3*(a^2*SA - 2*SB*SC))^2 : :

X(42001) lies on the 1st Simmons inconic and these lines: {13, 5916}, {62, 21466}, {546, 11555}, {8014, 18777}, {11537, 30467}, {20578, 30466}, {23283, 30454}, {30460, 31945}

X(42001) = X(i)-Ceva conjugate of X(j) for these (i,j): {13, 11537}, {36839, 9200}
X(42001) = crosspoint of X(13) and X(11537)
X(42001) = barycentric product X(i)*X(j) for these {i,j}: {13, 13, 520}, {530, 11537}, {11078, 30469}
X(42001) = barycentric quotient X(30469)/X(11092)

X(42002) = X(14)X(5917)∩X(61)X(21467)

Barycentrics    (S*(Sqrt[3]*a^2 - S) - 3*SB*SC)*(Sqrt[3]*S*(-a^2 + 2*SA) - 3*(a^2*SA - 2*SB*SC))^2 : :

X(42002) lies on the 2nd Simmons inconic and these lines: {14, 5917}, {61, 21467}, {546, 11556}, {8015, 18776}, {11549, 30470}, {20579, 30469}, {23284, 30455}, {30463, 31945}

X(42002) = X(i)-Ceva conjugate of X(j) for these (i,j): {14, 11549}, {36840, 9201}
X(42002) = crosspoint of X(14) and X(11549)
X(42002) = barycentric product X(i)*X(j) for these {i,j}: {14, 14, 530}, {531, 11549}, {11092, 30466}
X(42002) = barycentric quotient X(30466)/X(11078)

X(42003) = X(13)X(11600)∩X(323)X(532)

Barycentrics    (a^2*SA + Sqrt[3]*S*(-a^2 + 2*SA) - 2*SB*SC)^2*(S*(Sqrt[3]*a^2 + S) + 3*SB*SC) : :

X(42003) lies on the 1st Simmons inconic and these lines: {13, 11600}, {323, 532}, {11542, 30465}, {14446, 18803}, {30452, 34325}

X(42003) = barycentric product X(i)*X(j) for these {i,j}: {13, 13, 532}, {11078, 30462}
X(42003) = barycentric quotient X(30462)/X(11092)

X(42004) = X(14)X(11601)∩X(323)X(533)

Barycentrics    (a^2*SA - Sqrt[3]*S*(-a^2 + 2*SA) - 2*SB*SC)^2(S*(Sqrt[3]*a^2 - S) - 3*SB*SC) : :

X(42004) lies on the 2nd Simmons inconic and these lines: {14, 11601}, {323, 533}, {11543, 30468}, {14447, 18804}, {30453, 34326}

X(42004) = barycentric product X(i)*X(j) for these {i,j}: {14, 14, 532}, {11079, 30459}
X(42004) = barycentric quotient X(30459)/X(11078)

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Perpsectors associated with inverse triangles: X(42005)-X(42066)

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This preamble is contributed by Clark Kimberling and Peter Moses, March 18, 2021.

Suppose that A'B'C' is a triangle with vertices A', B', C' represented in normalized barycentric coordinates. Let M be the matrix representation of A'B'C', and let inv(M) denote the inverse of M. Then the rows of inv(M), interpreted as vertices of a triangle, define the inverse triangle of A'B'C'. The row sums of the inverse triangle are all 1, so that this triangle is "automatically" normalized.

Let P = p : q : r be a point, and let A'B'C' be the cevian triangle of P, given by A ' = 0 : q : r, with B' and C' determined cycllically from A'. The inverse triangle of A'B'C' is the triangle A''B''C'' given by A'' = - q - r : q + r : p + q, with B'' and C'' determined cyclically from A''; thus, A''B''C'' is the anticevian triangle of the point q + r : r + p : p + q.

Here, let A'B'C' be the cevian triangle of P. The inverse of A'B'C' is the triangle A''B''C'' given by A'' = 0 : p - q + r : p + q - r, with B'' and C'' determined cyclically from A''; thus A''B''C'' is the cevian triangle of the point - p + q + r ; p - q + r : p + q - r.

A triangle A'B'C' is inscribed in ABC if and only if its inverse circumscribes ABC.

See the preambles just before X(43280), X(43344), and X(46961).

X(42005) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-FEUERBACH

Barycentrics    b*c*(b + 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(42005) lies on these lines: {8, 79}, {10, 1109}, {12, 1089}, {75, 267}, {274, 20951}, {321, 21081}, {349, 23994}, {523, 24390}, {1125, 20886}, {1325, 2975}, {1365, 34829}, {1577, 23105}, {1631, 2915}, {1733, 24640}, {1930, 20437}, {2611, 24387}, {2643, 23626}, {3992, 27690}, {4858, 4999}, {4975, 37737}, {5506, 18151}, {17874, 31880}, {17886, 20880}, {18698, 19854}, {20236, 25650}, {20634, 20913}, {25446, 28611}

X(42005) = Feuerbach-to-ABC barycentric image of X(12)

X(42006) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-2ND-NEUBERG

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

Let A' be the apex of the isosceles triangle BCA' constructed inward on BC such that angle(A'BC) = angle(A'CB) = ω. Define B' and C' cyclically. Let KA be the symmedian point of BCA', and define KB and KC cyclically. The lines AKA, BKB, CKC concur in X(42006). (Randy Hutson, May 31, 2021)

The trilinear polar of X(42006) meets the line at infinity at X(523).

X(42006) lies on the Kiepert hyperbola and these lines: {2, 732}, {4, 2896}, {6, 33686}, {10, 33891}, {39, 10159}, {76, 4045}, {83, 385}, {98, 5092}, {141, 1916}, {183, 3407}, {194, 18840}, {262, 3314}, {321, 21817}, {384, 6308}, {511, 14492}, {538, 10302}, {598, 754}, {671, 7924}, {1447, 17741}, {1799, 40163}, {2023, 35005}, {2782, 9302}, {3329, 14994}, {3399, 32515}, {3406, 9755}, {3620, 14484}, {5395, 15589}, {5466, 31950}, {5976, 11606}, {6248, 12122}, {6704, 7755}, {7608, 7925}, {7766, 40332}, {7770, 12206}, {7787, 41650}, {7794, 32190}, {7836, 40108}, {7879, 22728}, {7885, 22682}, {7904, 22676}, {7931, 8781}, {8024, 31630}, {8587, 11168}, {8992, 19092}, {9866, 24256}, {12216, 32149}, {12263, 12783}, {13983, 19091}, {14458, 22712}, {14603, 39998}, {16987, 31239}, {16989, 18841}, {22706, 30998}, {26235, 34087}, {33278, 41895}, {40022, 40162}

X(42006) = isogonal conjugate of X(12212)
X(42006) = isotomic conjugate of X(3329)

X(42007) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-2ND-BROCARD

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

X(42007) lies on these lines: {6, 110}, {53, 17983}, {69, 31125}, {141, 30786}, {187, 2393}, {316, 524}, {511, 9186}, {526, 5505}, {574, 8542}, {576, 30498}, {599, 8288}, {691, 5104}, {2444, 6088}, {2871, 11173}, {3569, 9023}, {5024, 15268}, {5107, 9027}, {5585, 6091}, {5648, 41939}, {6784, 22111}, {8430, 8675}, {8546, 10485}, {8681, 9132}, {8869, 17710}, {9019, 41404}, {9225, 15398}, {9486, 20975}, {9872, 10510}, {10417, 32260}, {10602, 21448}, {12367, 32729}, {13192, 41617}, {13330, 14246}, {14609, 30495}, {15993, 16092}, {18023, 33769}, {20977, 41936}, {25052, 31128}, {25322, 41909}

X(42007) = isogonal conjugate of isotomic conjugate of X(42008)
X(42007) = trilinear pole of line X(574)X(17414)
X(42007) = perspector of ABC and cross-triangle of ABC and 2nd Ehrmann triangle
X(42007) = barycentric product X(6)*X(42008)
X(42007) = barycentric quotient X(42008)/X(76)

X(42008) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-4TH-BROCARD

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

X(42008) lies on these lines: {2, 99}, {23, 32479}, {67, 524}, {110, 9830}, {316, 41404}, {325, 9187}, {381, 9775}, {427, 8753}, {523, 34206}, {542, 10554}, {598, 9515}, {599, 8288}, {625, 10630}, {691, 3849}, {850, 10562}, {892, 7840}, {897, 25383}, {1153, 7496}, {1641, 11646}, {2770, 36196}, {2782, 9759}, {2799, 5466}, {3266, 18023}, {3268, 14977}, {5094, 11165}, {5169, 8176}, {5189, 10416}, {5468, 11161}, {5485, 16051}, {5968, 11184}, {5971, 17964}, {6032, 11163}, {6321, 14694}, {7610, 30542}, {7775, 14246}, {8182, 16063}, {8430, 31174}, {8597, 26276}, {8877, 31074}, {9133, 41133}, {9148, 9178}, {9213, 23878}, {9214, 9770}, {9741, 30775}, {9766, 11058}, {9829, 35955}, {10130, 15810}, {10302, 31078}, {10488, 39689}, {10557, 30789}, {10561, 30476}, {11148, 30769}, {11318, 14263}, {11645, 32729}, {12036, 19662}, {13857, 32583}, {14908, 31152}, {15398, 30745}, {16509, 30739}, {18818, 23297}, {23334, 31099}, {32216, 40727}, {34169, 37350}, {34806, 37746}, {39061, 41136}

X(42008) = isotomic conjugate of isogonal conjugate of X(42007)
X(42008) = trilinear pole of line X(599)X(3906) (the line through X(599) parallel to the trilinear polar of X(599))
X(42008) = barycentric product X(76)*X(42007)
X(42008) = barycentric quotient X(42007)/X(6)

d

X(42009) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-LUCAS-TANGENT

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

X(42009) lies on these lines: {2, 13880}, {20, 488}, {32, 591}, {69, 485}, {76, 6228}, {99, 32437}, {193, 19103}, {298, 6305}, {299, 6304}, {315, 32421}, {325, 35685}, {371, 492}, {486, 10008}, {491, 6118}, {511, 6289}, {599, 626}, {638, 6250}, {1078, 13088}, {1504, 7888}, {3620, 13834}, {3788, 8997}, {3933, 32497}, {5591, 32955}, {6337, 13701}, {6680, 13847}, {8180, 35812}, {9893, 9939}, {10519, 14229}, {12222, 32814}, {13651, 20080}, {13873, 32458}, {32436, 35820}

X(42010) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-McCAY

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

X(42010) lies on these lines: {4, 8591}, {76, 5461}, {98, 7840}, {99, 8786}, {114, 14488}, {148, 32532}, {524, 8587}, {543, 17503}, {598, 2482}, {625, 671}, {1916, 22110}, {3314, 11167}, {3407, 11163}, {5503, 7925}, {5969, 15814}, {7607, 40107}, {7608, 15850}, {7931, 10302}, {8289, 9770}, {8592, 10811}, {8596, 41895}, {8859, 10153}, {10484, 10487}

X(42010) = isotomic conjugate of X(8859)

X(42011) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-ANTI-McCAY

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

X(42011) lies on these lines: {4, 7618}, {6, 10153}, {76, 16509}, {98, 8600}, {262, 9771}, {524, 7607}, {598, 11149}, {671, 11165}, {1007, 11172}, {2996, 11148}, {5485, 39785}, {7612, 9770}, {7777, 8587}, {11167, 22110}, {11668, 15597}, {12040, 17503}

X(42011) = isotomic conjugate of X(8860)

X(42012) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-APUS

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

X(420) lies on these lines: {1, 394}, {2, 15299}, {8, 90}, {9, 55}, {10, 1728}, {31, 1723}, {40, 1726}, {46, 6734}, {56, 18251}, {57, 2886}, {63, 516}, {72, 11496}, {78, 5248}, {84, 3428}, {405, 12710}, {517, 3927}, {528, 3929}, {674, 5227}, {758, 3872}, {936, 8069}, {956, 6001}, {958, 12711}, {968, 3190}, {1617, 5784}, {1621, 41228}, {1698, 17699}, {1708, 2550}, {1763, 6210}, {1824, 12549}, {1836, 5857}, {2099, 6762}, {2308, 28125}, {2328, 4319}, {2968, 3966}, {2975, 10085}, {3305, 6745}, {3338, 10044}, {3358, 10860}, {3436, 12617}, {3650, 7701}, {3680, 6597}, {3719, 3886}, {3740, 15297}, {3870, 15298}, {3928, 31140}, {4295, 9965}, {4314, 5250}, {4430, 4861}, {4652, 12511}, {5172, 5438}, {5173, 12560}, {5220, 17658}, {5269, 8557}, {5271, 17860}, {5437, 31245}, {5696, 15931}, {5709, 37820}, {6067, 11246}, {6690, 7308}, {6736, 18249}, {7411, 25722}, {8257, 26040}, {10582, 11018}, {10679, 34790}, {11679, 17738}, {12513, 12709}, {12527, 21628}, {12704, 24390}, {15503, 26885}, {15587, 37270}, {17742, 21369}, {18499, 37584}, {24929, 31435}, {31424, 40292}

X(42013) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-2ND-PAMFILOS-ZHOU

Barycentrics    a (-a+b+c)*((2*a^4-4*(b+c)*a^3-4*(b+c)^2*a^2+4*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2)*S-(a+b-c)*(a-b+c)*(a^4+2*(b+c)*a^3-2*(b+c)^2*a^2-2*(b+c)^3*a+(b^2-c^2)^2)) : :
Barycentrics    a*(a*(a - b - c)*(a + b + c) + 2*(b + c)*S) : :
Barycentrics    Sin[A]/(1 + Cot[A] - Csc[A]) : :
Trilinears    1/(1 - tan(A/2)) : :
Trilinears    (tan A) (1 + cot(A/2)) : :

X(42013) lies on the Feuerbach circumhyperbola, the cubics K233 and K631, and these lines: {1, 371}, {4, 1336}, {6, 9043}, {7, 13389}, {8, 14121}, {9, 5414}, {19, 25}, {21, 1805}, {44, 19037}, {57, 30279}, {65, 13460}, {79, 30426}, {80, 30432}, {84, 6502}, {90, 372}, {104, 18460}, {177, 21465}, {193, 13386}, {256, 30361}, {281, 13454}, {497, 6352}, {1100, 19038}, {1108, 18999}, {1124, 36742}, {1703, 38271}, {1721, 6203}, {1826, 41516}, {1904, 13973}, {3062, 30289}, {3295, 8965}, {3296, 30342}, {3297, 34046}, {3302, 7741}, {3303, 38487}, {3553, 5415}, {4194, 7090}, {5218, 6351}, {5405, 7595}, {5416, 8557}, {7162, 35809}, {7284, 35769}, {13388, 16441}

X(42013) = isogonal conjugate of X(13388)
X(42013) = isogonal conjugate of the complement of X(13386)
X(42013) = polar conjugate of the isotomic conjugate of X(30556)
X(42013) = X(i)-Ceva conjugate of X(j) for these (i,j): {281, 7133}, {6136, 650}, {13390, 16232}
X(42013) = X(i)-cross conjugate of X(j) for these (i,j): {6, 7133}, {650, 6136}, {30376, 7}
X(42013) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13388}, {2, 2067}, {3, 1659}, {7, 5414}, {57, 30557}, {63, 2362}, {77, 7133}, {222, 7090}, {226, 1805}, {1335, 13390}, {3084, 16232}, {6213, 13389}, {6502, 13387}
X(42013) = cevapoint of X(6) and X(34125)
X(42013) = crosspoint of X(i) and X(j) for these (i,j): {1, 15891}, {281, 13426}, {13390, 14121}
X(42013) = crosssum of X(2067) and X(5414)
X(42013) = crossdifference of every pair of points on line {905, 6365}
X(42013) = barycentric product X(i)*X(j) for these {i,j}: {1, 14121}, {4, 30556}, {8, 16232}, {9, 13390}, {92, 2066}, {281, 13389}, {318, 6502}, {1336, 30557}, {1806, 41013}, {6212, 7090}, {7133, 13386}, {13388, 13426}
X(42013) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 13388}, {19, 1659}, {25, 2362}, {31, 2067}, {33, 7090}, {41, 5414}, {55, 30557}, {607, 7133}, {1806, 1444}, {2066, 63}, {2194, 1805}, {5414, 3084}, {6502, 77}, {7133, 13387}, {13388, 13436}, {13389, 348}, {13390, 85}, {13427, 14121}, {14121, 75}, {16232, 7}, {30556, 69}, {30557, 5391}, {34125, 13389}
X(42013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6212, 16232}, {1, 32556, 2067}, {19, 33, 7133}, {37, 55, 7133}, {4319, 40131, 7133}, {5089, 7071, 7133}, {5275, 11997, 7133}

X(42014) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-4TH-MIXTILINEAR

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

X(42014) lies on these lines: {2, 8255}, {3, 5696}, {6, 28125}, {7, 2886}, {8, 190}, {9, 55}, {10, 5729}, {44, 28043}, {45, 2340}, {56, 5784}, {63, 15726}, {78, 15254}, {100, 32076}, {142, 17728}, {144, 3434}, {220, 2310}, {329, 7965}, {391, 4012}, {516, 3419}, {517, 4915}, {518, 2099}, {527, 1836}, {956, 2801}, {958, 10394}, {971, 3428}, {1001, 4511}, {1376, 37787}, {1445, 15587}, {2098, 4516}, {2550, 12848}, {3035, 18801}, {3062, 41338}, {3189, 5766}, {3679, 41700}, {3927, 41869}, {3928, 30353}, {3957, 30628}, {3962, 4853}, {4312, 4880}, {4413, 8257}, {4416, 30620}, {4423, 10177}, {4666, 5572}, {4860, 5231}, {5221, 5880}, {5228, 24341}, {5759, 5842}, {5762, 37820}, {5817, 7680}, {6690, 18230}, {8543, 12635}, {10059, 15298}, {10387, 40968}, {10527, 25557}, {11018, 30330}, {11194, 18450}, {11260, 30318}, {11495, 25722}, {16885, 19624}, {17275, 23529}, {17330, 28118}, {17335, 28058}, {17718, 41570}, {18407, 31671}, {21168, 37000}, {28124, 37654}, {29007, 34784}

X(42014) = isogonal conjugate of X(7)-vertex conjugate of X(55)
X(42014) = isotomic conjugate of X(18810)
X(42014) = anticomplement of X(8255)

X(42015) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-HUTSON-EXTOUCH

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

X(42015) lies on the Feuerbach hyperbola and these lines: {1, 3059}, {4, 5223}, {7, 4847}, {8, 9898}, {21, 4326}, {84, 2951}, {104, 6575}, {200, 2346}, {480, 30393}, {516, 10429}, {518, 5665}, {1389, 12654}, {3062, 41338}, {3296, 12864}, {3577, 4915}, {4853, 11526}, {4882, 7160}, {5220, 33576}, {6067, 10980}, {6737, 7320}, {7091, 15587}, {8001, 9874}, {10390, 15185}

X(42016) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-3RD-HATZIPOLAKIS

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

X(42016) lies on these lines: {3, 40632}, {4, 22972}, {54, 2929}, {195, 3521}, {1154, 22549}, {2888, 23308}, {3574, 22971}, {10619, 19460}, {11744, 14049}, {12307, 22978}, {14483, 22750}, {16665, 22962}, {18550, 36747}, {22533, 33565}, {22538, 22585}

X(42017) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-INNER-TANGENTIAL-MID-ARC

Barycentrics    Cos[A/2]*(Sin[B/2] + Sin[C/2]) : :

X(42017) lies on the Feuerbach circumhyperbola and these lines: {1, 188}, {3, 10023}, {4, 12694}, {7, 2091}, {8, 7027}, {9, 6731}, {84, 164}, {104, 3659}, {177, 178}, {3577, 11528}, {7707, 16016}, {8372, 17623}

X(42017) = midpoint of X(7057) and X(11691)
X(42017) = reflection of X(i) in X(j) for these {i,j}: {177, 178}, {188, 18258}
X(42017) = X(i)-Ceva conjugate of X(j) for these (i,j): {188, 15997}, {2090, 16016}, {7048, 2090}
X(42017) = crosspoint of X(i) and X(j) for these (i,j): {8, 188}, {7028, 7048}
X(42017) = crosssum of X(56) and X(266)
X(42017) = trilinear pole of line {650, 6730}
X(42017) = barycentric product X(i)*X(j) for these {i,j}: {8, 16015}, {178, 7028}, {188, 2090}, {312, 16011}, {346, 2091}, {556, 15997}, {3659, 4391}, {7027, 41799}, {7048, 16016}
X(42017) = barycentric quotient X(i)/X(j) for these {i,j}: {177, 18886}, {2090, 4146}, {2091, 279}, {3659, 651}, {7707, 2089}, {15997, 174}, {16011, 57}, {16012, 173}, {16015, 7}, {16016, 7057}, {18887, 177}, {41799, 7371}

X(42018) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-2ND-HATZIPOLAKIS

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

X(42018) lies on these lines: {1, 15851}, {2, 1119}, {3, 9}, {5, 281}, {19, 19541}, {44, 15905}, {45, 216}, {144, 25932}, {219, 1807}, {344, 41005}, {346, 2968}, {440, 18228}, {441, 26685}, {577, 16885}, {1062, 2324}, {1214, 7308}, {1249, 15252}, {1743, 38292}, {1863, 33306}, {2257, 38288}, {3452, 17279}, {3715, 23207}, {3731, 17102}, {3912, 40995}, {4422, 6389}, {5158, 16777}, {5273, 7536}, {6554, 21530}, {6666, 17073}, {7079, 11108}, {7515, 27382}, {8558, 37500}, {10254, 15833}, {11374, 40942}, {15526, 17267}, {17350, 21940}, {20226, 37694}, {20263, 21068}, {20818, 37700}, {21482, 27065}, {21499, 28731}, {25087, 30457}, {34524, 35072}

X(42019) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-MANDART-EXCIRCLES

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

X(42019) lies on these lines: {1, 10601}, {6, 1497}, {8, 36123}, {34, 517}, {46, 269}, {56, 1066}, {58, 8069}, {86, 3085}, {200, 1256}, {219, 7129}, {221, 24028}, {255, 1413}, {521, 5687}, {837, 3436}, {1124, 37885}, {1411, 2098}, {1474, 22132}, {2829, 9370}, {5534, 19354}, {11507, 41442}, {22117, 35448}, {24928, 36754}

X(42019) = isogonal conjugate of X(3086)
X(42019) = cevapoint of X(1124) and X(1335)
X(42019) = perspector of ABC and unary cofactor triangle of Mandart-excircles triangle

X(42020) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-INNER-MIXTILINEAR-TANGENTS

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

X(42020) lies on these lines: {2, 1222}, {4, 10744}, {8, 210}, {10, 11512}, {69, 3264}, {145, 3699}, {345, 6736}, {668, 6604}, {1330, 3421}, {1654, 4461}, {1997, 36846}, {3617, 5484}, {3621, 6555}, {3913, 4571}, {4487, 12649}, {4738, 10573}, {4853, 28808}, {4997, 18220}, {5554, 20905}, {6735, 34823}, {7774, 39354}, {12607, 30828}, {12640, 30568}, {14548, 24524}, {32850, 41772}

X(42021) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-ANTI-INCIRCLE-CIRCLES

Barycentrics    (a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(42021) = (a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(42021) = Cot[A]/(2 + Csc[A]^2) : :

X(42021) lies on the Jerabek circumhyperbola, the cubics K471 and K917, and these lines: {2, 1173}, {3, 9936}, {4, 1216}, {6, 140}, {20, 16835}, {30, 22334}, {54, 3523}, {64, 550}, {65, 10044}, {66, 14864}, {68, 3917}, {69, 5447}, {70, 16063}, {71, 24467}, {74, 3522}, {265, 41673}, {343, 38260}, {376, 13452}, {394, 34002}, {599, 23335}, {631, 13472}, {633, 41897}, {634, 41898}, {895, 3546}, {1092, 40441}, {1147, 1176}, {1352, 10627}, {1656, 3527}, {1657, 3426}, {1899, 3519}, {3431, 10299}, {3521, 23039}, {3531, 3851}, {3532, 33923}, {3564, 34817}, {3800, 35364}, {3854, 14487}, {4846, 5562}, {5056, 14483}, {5504, 38727}, {5921, 17712}, {6145, 14791}, {6391, 12359}, {6643, 15077}, {6776, 41435}, {6816, 18555}, {7525, 34437}, {9813, 40107}, {9927, 15749}, {11270, 21735}, {11457, 15108}, {11821, 17702}, {11850, 12164}, {12118, 34801}, {12325, 18368}, {13421, 22336}, {13623, 34783}, {13754, 15740}, {14528, 15712}, {14861, 18436}, {15577, 34207}, {15606, 18420}, {17505, 18404}, {18531, 32533}, {18532, 32534}, {27866, 38534}, {32142, 39571}

X(42021) = isogonal conjugate of X(10594)
X(42021) = isotomic conjugate of the anticomplement of X(10979)
X(42021) = X(10979)-cross conjugate of X(2)
X(42021) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10594}, {19, 5422}, {92, 13345}, {1973, 32832}
X(42021) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 5422}, {6, 10594}, {69, 32832}, {184, 13345}

X(42022) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-LUCAS-ANTIPODAL

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

X(42022) lies on these lines: {155, 1351}, {371, 19442}, {487, 19464}, {494, 3167}, {1147, 26507}, {1161, 12164}, {1993, 26374}, {3564, 5491}, {5200, 26503}, {6193, 24243}, {6391, 10666}, {6406, 19461}, {8681, 9975}, {17839, 17843}, {26461, 30435}

X(42023) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-3RD-TRI-SQUARES

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

X(42023) lies on these lines: {2, 13834}, {4, 5860}, {30, 14232}, {83, 19102}, {262, 9768}, {485, 1991}, {488, 1132}, {524, 1327}, {543, 13712}, {591, 1328}, {598, 12159}, {599, 42024}, {637, 1131}, {641, 3317}, {671, 32808}, {5861, 14241}, {6118, 34089}, {6279, 14238}, {6289, 36723}, {10195, 11313}, {12124, 14229}, {13468, 13835}, {13850, 40727}, {14237, 36719}, {14244, 32419}, {18842, 19053}, {22563, 22645}

X(42023) = isogonal conjugate of X(41411)

X(42024) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-4TH-TRI-SQUARES

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

X(42024) lies on these lines: {2, 13711}, {4, 5861}, {30, 14237}, {83, 19105}, {262, 9767}, {486, 591}, {487, 1131}, {524, 1328}, {543, 13835}, {598, 12158}, {599, 42023}, {638, 1132}, {642, 3316}, {671, 32809}, {1327, 1991}, {5860, 14226}, {6119, 34091}, {6280, 14234}, {6290, 36726}, {10194, 11314}, {12123, 14244}, {13468, 13712}, {13932, 40727}, {14229, 32421}, {14232, 36733}, {18842, 19054}, {22562, 22616}

X(42024) = isogonal conjugate of X(41410)

X(42025) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-11

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

X(42025) lies on these lines: {1, 39711}, {2, 6}, {21, 551}, {58, 17553}, {63, 17207}, {99, 28330}, {274, 29584}, {314, 4980}, {519, 25526}, {553, 1014}, {593, 41311}, {894, 1255}, {903, 17587}, {1010, 3241}, {1281, 3315}, {1408, 4870}, {1412, 4654}, {1509, 30581}, {1790, 6173}, {1817, 17168}, {3175, 4670}, {3187, 41847}, {3218, 37869}, {3219, 28639}, {3616, 16948}, {3656, 4221}, {3679, 4658}, {3828, 17551}, {3929, 18164}, {4034, 28651}, {4184, 4428}, {4225, 40726}, {4234, 38314}, {4361, 25417}, {4418, 5625}, {4496, 7303}, {4954, 18792}, {5253, 35206}, {5284, 33682}, {5287, 41242}, {16046, 16712}, {16052, 26131}, {16700, 25060}, {16753, 25059}, {16826, 32090}, {16834, 17175}, {17045, 26842}, {17169, 35935}, {17376, 41850}, {17557, 28620}, {17589, 31145}, {18185, 35983}, {18200, 31147}, {18601, 25058}, {19290, 19767}, {19796, 37095}, {26643, 33955}, {26840, 41820}, {28194, 37402}, {29570, 32933}, {29574, 33953}, {29586, 33947}, {29597, 40773}, {29615, 33770}, {30599, 30939}, {34914, 40143}

X(42025) = isotomic conjugate of polar conjugate of X(31901)

X(42026) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-14

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

X(42026) lies on these lines: {2, 45}, {89, 40833}, {106, 38314}, {519, 4674}, {551, 4781}, {679, 20072}, {812, 6548}, {1320, 9945}, {2094, 2316}, {2226, 4615}, {3218, 21372}, {3241, 4792}, {3257, 35596}, {4049, 4750}, {4555, 40891}, {4850, 39704}, {6336, 7490}, {6549, 41140}, {10707, 19636}, {14953, 16088}, {16590, 24589}, {17488, 24620}, {17537, 24046}, {24184, 30564}, {25055, 26627}, {31145, 36593}

X(42027) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-16

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

X(42027) lies on these lines: {1, 87}, {2, 17038}, {10, 3728}, {19, 2319}, {37, 714}, {42, 25295}, {65, 740}, {75, 982}, {76, 18832}, {82, 34252}, {244, 20892}, {256, 1221}, {291, 17787}, {313, 3122}, {314, 18827}, {321, 22167}, {335, 1581}, {518, 34434}, {519, 994}, {522, 876}, {536, 13476}, {537, 31165}, {700, 40881}, {730, 21746}, {756, 27438}, {759, 932}, {897, 4598}, {899, 25277}, {984, 7275}, {1278, 4365}, {1400, 4039}, {1655, 25421}, {1740, 24351}, {1910, 34071}, {1999, 13610}, {2053, 2218}, {2162, 2214}, {2217, 23086}, {2228, 18040}, {2321, 21100}, {3121, 22218}, {3596, 17065}, {3644, 39739}, {3663, 39712}, {3701, 22220}, {3840, 20891}, {3842, 27432}, {3948, 21095}, {3997, 21752}, {4043, 22045}, {4044, 22214}, {4135, 22016}, {4377, 21238}, {4664, 39737}, {4674, 4709}, {4704, 32925}, {4735, 28593}, {4788, 17146}, {4871, 20923}, {6385, 23824}, {7209, 39126}, {8769, 11679}, {9902, 21299}, {10009, 33789}, {10479, 39708}, {16571, 24621}, {16888, 21927}, {17063, 30090}, {17279, 24653}, {17793, 28366}, {18785, 21061}, {18833, 21443}, {20688, 22036}, {20876, 23853}, {20917, 41886}, {21219, 26069}, {21278, 23633}, {21435, 23680}, {21759, 40747}, {22316, 40504}, {23051, 29652}, {24225, 39714}, {24325, 27455}, {25120, 28244}, {25121, 30473}, {27450, 31323}, {27478, 40775}, {28358, 30982}, {28522, 35633}, {32039, 35143}, {33296, 40409}, {36494, 40783}

X(42027) = isogonal conjugate of X(38832)
X(42027) = isotomic conjugate of X(33296)

X(42028) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-21

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

X(42028) lies on these lines: {1, 4234}, {2, 6}, {21, 3304}, {27, 39704}, {57, 17207}, {58, 551}, {63, 17394}, {99, 17223}, {110, 9105}, {190, 17019}, {191, 41815}, {274, 16834}, {354, 3794}, {519, 1010}, {552, 553}, {593, 30593}, {846, 5625}, {894, 3175}, {999, 19247}, {1043, 3241}, {1444, 2094}, {1449, 4771}, {1961, 4096}, {1999, 4670}, {3210, 16884}, {3284, 25908}, {3286, 4428}, {3616, 4831}, {3622, 16948}, {3664, 19786}, {3679, 25526}, {3699, 4682}, {3758, 5287}, {3829, 14009}, {3879, 19808}, {3928, 17185}, {3929, 18206}, {4001, 17322}, {4038, 32942}, {4052, 14534}, {4102, 40438}, {4267, 40726}, {4349, 4514}, {4421, 13588}, {4641, 16826}, {4649, 4685}, {4667, 33066}, {4785, 18200}, {4980, 30599}, {5271, 41847}, {5294, 17317}, {8822, 17320}, {9534, 19332}, {11110, 25055}, {14007, 19875}, {16046, 17103}, {16054, 33955}, {16477, 25501}, {16696, 25058}, {16700, 25059}, {16723, 25536}, {16833, 17175}, {17045, 26840}, {17326, 41850}, {17377, 19822}, {17389, 33954}, {17391, 32777}, {18172, 29580}, {18601, 25060}, {18602, 18603}, {18646, 37265}, {19336, 19767}, {19819, 37095}, {19883, 28620}, {23140, 25521}, {23812, 33135}, {24841, 29816}, {26626, 33947}, {29573, 33953}, {29574, 33770}, {29584, 33296}, {29615, 32004}, {30606, 37756}, {31162, 37422}, {34632, 37402}, {37869, 38000}, {37870, 39980}, {39914, 39915}

X(42028) = isotomic conjugate of polar conjugate of X(31903)

X(42029) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-23

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

X(42029) lies on these lines: {2, 37}, {8, 1836}, {10, 33154}, {76, 30713}, {85, 4102}, {92, 1839}, {190, 5271}, {226, 4431}, {274, 29597}, {304, 29574}, {319, 5905}, {322, 31164}, {329, 4886}, {333, 3729}, {519, 17789}, {527, 20920}, {553, 39126}, {594, 27184}, {940, 17116}, {1089, 19875}, {1930, 29573}, {1999, 4363}, {2321, 18134}, {3187, 3758}, {3241, 4673}, {3247, 25507}, {3661, 3782}, {3679, 4385}, {3681, 17163}, {3696, 32937}, {3702, 38314}, {3706, 24349}, {3757, 4428}, {3759, 26223}, {3769, 4418}, {3773, 17889}, {3790, 3925}, {3828, 4066}, {3875, 41823}, {3928, 4659}, {3969, 31019}, {3994, 26037}, {4009, 26038}, {4052, 34258}, {4054, 4417}, {4361, 27064}, {4365, 32771}, {4383, 17117}, {4384, 32088}, {4387, 16823}, {4415, 4665}, {4421, 32932}, {4442, 29667}, {4656, 4967}, {4669, 4737}, {4676, 32914}, {4677, 4692}, {4693, 29651}, {4762, 21438}, {4785, 20952}, {4935, 31145}, {5249, 17233}, {5256, 17160}, {5278, 17336}, {5564, 5739}, {6535, 25957}, {7206, 41859}, {7283, 16418}, {10436, 34064}, {10447, 30710}, {10449, 24473}, {14213, 20921}, {14555, 32087}, {16708, 40493}, {16817, 16857}, {16833, 32104}, {16834, 17143}, {17019, 41847}, {17144, 19722}, {17184, 17228}, {17227, 33146}, {17240, 18139}, {17241, 27186}, {17261, 19732}, {17276, 37653}, {17286, 23681}, {17299, 17778}, {17310, 20432}, {17319, 19701}, {17351, 37652}, {17354, 26723}, {17360, 32859}, {17361, 17483}, {17389, 17762}, {17393, 19684}, {17592, 28522}, {18141, 31995}, {19876, 28611}, {20236, 31142}, {20237, 28609}, {20888, 20917}, {20909, 31147}, {20913, 29577}, {20919, 33941}, {20943, 29593}, {21020, 32925}, {21085, 33101}, {21605, 34284}, {21611, 31150}, {21615, 40087}, {25385, 32855}, {26792, 41821}, {28654, 30596}, {29580, 31997}, {30599, 30939}

X(42029) = isotomic conjugate of X(39948)

X(42030) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-24

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

X(42030) lies on these lines: {2, 319}, {8, 3683}, {9, 4102}, {75, 3578}, {85, 553}, {92, 1839}, {257, 29617}, {312, 3686}, {333, 4034}, {345, 30711}, {519, 31359}, {1121, 3929}, {1220, 3679}, {1311, 8652}, {3687, 30608}, {4384, 32015}, {4997, 11679}, {6557, 14555}, {17281, 34527}, {17294, 32008}, {17363, 37631}, {17743, 19723}, {26860, 28651}, {34234, 37211}

X(42030) = isotomic conjugate of X(4654)

X(42031) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-26

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

X(42031) lies on these lines: {1, 4365}, {3, 29347}, {8, 3585}, {10, 321}, {75, 1125}, {192, 19858}, {226, 21081}, {306, 11263}, {312, 3634}, {341, 4745}, {519, 17789}, {551, 3702}, {594, 22036}, {758, 5295}, {1269, 40034}, {1698, 4671}, {1909, 31013}, {2321, 12609}, {3244, 4968}, {3263, 39580}, {3625, 4692}, {3626, 4385}, {3635, 4673}, {3671, 6358}, {3678, 3696}, {3695, 3841}, {3704, 3822}, {3706, 3874}, {3714, 3754}, {3739, 27784}, {3743, 31993}, {3840, 24176}, {3919, 17751}, {3967, 4015}, {3995, 16828}, {4084, 17164}, {4133, 18697}, {4134, 17163}, {4358, 28611}, {4359, 19862}, {4461, 19843}, {4519, 5439}, {4669, 4696}, {4686, 37592}, {4720, 41696}, {4737, 4746}, {4975, 15808}, {5248, 5695}, {6533, 19883}, {6743, 17860}, {8714, 23685}, {10447, 17733}, {12447, 20320}, {16825, 32104}, {17019, 41812}, {17147, 19863}, {17495, 19864}, {18743, 31253}, {19789, 19836}, {19804, 19878}, {19881, 33150}, {20888, 33930}, {21071, 24044}, {24160, 33160}, {24167, 30942}, {33066, 41814}

X(42032) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-35

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

X(42032) lies on these lines: {2, 37}, {8, 3058}, {69, 17781}, {85, 32869}, {190, 34255}, {304, 17079}, {306, 36889}, {329, 17233}, {333, 3161}, {348, 32833}, {376, 7283}, {381, 3695}, {497, 3790}, {553, 3729}, {1043, 36624}, {1089, 10056}, {1479, 7206}, {2321, 14555}, {2325, 5325}, {2899, 3704}, {3543, 7270}, {3685, 3974}, {3687, 4873}, {3703, 11238}, {3706, 27549}, {3782, 29579}, {3886, 4082}, {3912, 4654}, {3969, 31018}, {3994, 33171}, {3996, 5423}, {4135, 33144}, {4415, 17269}, {4431, 7308}, {4656, 17286}, {4754, 17316}, {4854, 9780}, {4942, 4966}, {5233, 8055}, {5309, 7230}, {5712, 17242}, {5749, 34064}, {6541, 26098}, {14552, 17336}, {17078, 32830}, {17314, 27064}, {17340, 26065}, {17361, 20214}, {20925, 32892}, {28809, 30713}, {29616, 33066}

X(42032) = X(i)-isoconjugate of X(j) for these (i,j): {604, 3296}, {1395, 30679}
X(42032) = barycentric product X(i)*X(j) for these {i,j}: {312, 3305}, {314, 3697}, {341, 7190}, {3295, 3596}
X(42032) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 3296}, {345, 30679}, {3295, 56}, {3305, 57}, {3697, 65}, {4917, 1420}, {7190, 269}
X(42032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 346, 42033}, {2, 42033, 345}, {304, 32836, 17079}, {312, 345, 28808}, {312, 346, 345}, {312, 42033, 2}, {2321, 30568, 14555}, {4387, 6057, 8}, {17264, 42034, 2}, {17279, 22034, 30699}, {17281, 35652, 2}

X(42033) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-37

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

X(42033) lies on these lines: {2, 37}, {8, 3683}, {9, 4886}, {30, 3695}, {35, 7206}, {55, 3790}, {63, 17233}, {81, 17315}, {85, 32836}, {171, 6541}, {190, 306}, {304, 17078}, {319, 3219}, {320, 32858}, {333, 2321}, {341, 36626}, {519, 595}, {553, 3912}, {644, 1812}, {646, 30713}, {726, 33124}, {728, 3719}, {740, 33118}, {846, 3773}, {894, 37631}, {940, 17242}, {1043, 3710}, {1089, 3584}, {1211, 17261}, {1999, 3943}, {2185, 2329}, {2325, 3687}, {2363, 4234}, {2901, 3017}, {3058, 3685}, {3161, 14555}, {3262, 20919}, {3692, 17346}, {3699, 4082}, {3705, 4387}, {3712, 4995}, {3717, 3996}, {3729, 4654}, {3896, 33166}, {3923, 33073}, {3932, 32932}, {3950, 34064}, {3961, 4439}, {3971, 33160}, {3977, 14829}, {3986, 41817}, {3993, 32780}, {3994, 29846}, {4001, 17295}, {4011, 32855}, {4054, 41878}, {4062, 32938}, {4135, 17719}, {4360, 5294}, {4365, 33115}, {4383, 17339}, {4385, 10056}, {4427, 33078}, {4432, 32866}, {4442, 29873}, {4513, 17389}, {4564, 7364}, {4641, 6542}, {4656, 30832}, {4676, 33088}, {4693, 29673}, {4872, 7788}, {4873, 11679}, {4970, 33159}, {5233, 30568}, {5256, 17354}, {5278, 5564}, {5695, 29641}, {5739, 17336}, {6535, 32917}, {7321, 18139}, {7799, 16577}, {11648, 34542}, {15523, 24723}, {17079, 21605}, {17160, 26723}, {17229, 37653}, {17231, 26840}, {17256, 33761}, {17258, 32782}, {17262, 27184}, {17266, 40688}, {17299, 37652}, {17314, 26065}, {17340, 27064}, {17347, 25734}, {17351, 17778}, {17361, 20078}, {17368, 20182}, {17600, 24295}, {19723, 29617}, {21070, 24053}, {28516, 33147}, {28522, 33132}, {29574, 33770}, {29674, 32934}, {29687, 32845}, {32848, 32930}, {32850, 32862}, {32915, 33121}, {32925, 33126}

X(42033) = X(i)-Ceva conjugate of X(j) for these (i,j): {4600, 3699}, {33939, 319}
X(42033) = X(4420)-cross conjugate of X(319)
X(42033) = X(i)-isoconjugate of X(j) for these (i,j): {56, 2160}, {57, 6186}, {79, 604}, {608, 7100}, {649, 26700}, {667, 38340}, {1106, 7110}, {1397, 30690}, {1407, 7073}, {1408, 8818}, {1435, 8606}, {1443, 11060}, {3122, 35049}, {6757, 16947}, {7180, 13486}, {14399, 36064}
X(42033) = crossdifference of every pair of points on line {667, 23751}
X(42033) = barycentric product X(i)*X(j) for these {i,j}: {8, 319}, {9, 33939}, {35, 3596}, {75, 4420}, {261, 7206}, {312, 3219}, {313, 35193}, {314, 3678}, {333, 3969}, {341, 1442}, {346, 17095}, {644, 18160}, {645, 7265}, {646, 14838}, {668, 35057}, {1043, 40999}, {1265, 7282}, {1978, 9404}, {2174, 28659}, {2321, 34016}, {3578, 4102}, {3699, 4467}, {3718, 6198}, {4600, 6741}, {6064, 21054}, {7799, 36910}, {11107, 20336}, {16755, 30730}, {27801, 35192}, {30713, 40214}, {31938, 40422}, {32851, 41226}, {40071, 41502}, {40713, 40714}
X(42033) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 79}, {9, 2160}, {35, 56}, {55, 6186}, {78, 7100}, {100, 26700}, {190, 38340}, {200, 7073}, {312, 30690}, {319, 7}, {346, 7110}, {643, 13486}, {646, 15455}, {1043, 3615}, {1260, 8606}, {1399, 1106}, {1442, 269}, {1792, 1789}, {1825, 1426}, {2003, 1407}, {2174, 604}, {2321, 8818}, {2594, 1042}, {3219, 57}, {3578, 553}, {3596, 20565}, {3647, 32636}, {3678, 65}, {3699, 6742}, {3701, 6757}, {3969, 226}, {4420, 1}, {4467, 3676}, {4567, 35049}, {6198, 34}, {6741, 3120}, {7186, 7248}, {7206, 12}, {7265, 7178}, {7282, 1119}, {7799, 17078}, {9404, 649}, {11107, 28}, {14838, 3669}, {14975, 1395}, {15742, 34922}, {16577, 1427}, {16755, 17096}, {17095, 279}, {17104, 1408}, {18160, 24002}, {21054, 1365}, {22342, 1410}, {31938, 942}, {33939, 85}, {34016, 1434}, {35057, 513}, {35192, 1333}, {35193, 58}, {35194, 1393}, {36910, 1989}, {40214, 1412}, {40713, 554}, {40714, 1081}, {40999, 3668}, {41226, 2006}, {41502, 1474}, {41562, 37566}
X(42033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 346, 42032}, {2, 42032, 312}, {190, 306, 33066}, {192, 32777, 19786}, {304, 32833, 17078}, {312, 345, 32851}, {321, 32849, 33116}, {345, 346, 312}, {345, 42032, 2}, {1278, 24789, 19820}, {3219, 3969, 319}, {3685, 3703, 4514}, {3695, 7283, 7270}, {3712, 6057, 7081}, {3923, 33092, 33073}, {15523, 32936, 24723}, {17147, 33157, 16706}, {29674, 32934, 33068}, {32848, 32930, 33071}, {32858, 32933, 320}, {32862, 32929, 32850}, {32915, 33161, 33121}, {32925, 33156, 33126}

X(42034) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-40

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

X(42034) lies on these lines: {2, 37}, {8, 3967}, {76, 4052}, {92, 4102}, {190, 3929}, {226, 17233}, {304, 29573}, {314, 41629}, {319, 329}, {320, 34255}, {333, 17336}, {341, 1089}, {519, 4066}, {984, 4135}, {1999, 3758}, {2064, 17378}, {2321, 4417}, {2886, 3790}, {2999, 17160}, {3187, 41242}, {3241, 3702}, {3452, 4431}, {3661, 4415}, {3685, 4428}, {3696, 27538}, {3705, 3829}, {3706, 32937}, {3729, 3928}, {3740, 4903}, {3757, 4387}, {3759, 27064}, {3769, 3923}, {3773, 3944}, {3782, 17227}, {3869, 14973}, {3966, 17777}, {3969, 31053}, {3974, 32850}, {3994, 31330}, {4044, 6376}, {4054, 17240}, {4125, 4745}, {4362, 4676}, {4365, 32931}, {4421, 5695}, {4442, 29679}, {4519, 10453}, {4647, 19875}, {4654, 17297}, {4656, 5224}, {4659, 30567}, {4669, 4717}, {4677, 4737}, {4693, 29670}, {4696, 31145}, {4734, 28484}, {4762, 21611}, {4886, 31018}, {4968, 38314}, {5249, 17241}, {5256, 41823}, {5271, 17335}, {5287, 41847}, {5564, 14555}, {5712, 17315}, {5737, 17261}, {5905, 17361}, {6057, 29641}, {6382, 21615}, {6535, 25760}, {6541, 33111}, {7283, 16370}, {7321, 18141}, {8055, 32087}, {16284, 17294}, {16817, 17542}, {16833, 17143}, {16834, 17144}, {17056, 17242}, {17116, 37674}, {17117, 37679}, {17158, 33941}, {17228, 27184}, {17262, 38000}, {17277, 30568}, {17283, 23681}, {17285, 25527}, {17310, 17789}, {17329, 37653}, {17351, 37683}, {17354, 40940}, {17360, 33066}, {17369, 29841}, {17386, 17778}, {17394, 34064}, {17591, 28516}, {17781, 18750}, {17786, 27792}, {18145, 33935}, {18150, 36791}, {18156, 29574}, {20888, 29600}, {20911, 33780}, {20913, 29582}, {20917, 29577}, {20927, 31142}, {20952, 31147}, {21093, 33084}, {21600, 39996}, {25385, 33092}, {29597, 31997}, {30710, 39948}, {30854, 33938}, {32017, 36603}, {34284, 40023}

X(42034) = isotomic conjugate of X(39980)

X(42035) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-7th-FERMAT-DAO

Barycentrics    1/(-5*a^2 + b^2 + c^2 - 2*Sqrt[3]*S) : :

Let A' be the orthocenter of triangle BCX(13), and define B' and C' cyclically. Then X(42035) is the orthocenter of triangle A'B'C'. (Randy Hutson, May 31, 2021)

X(42035) lies on these lines: {2, 22574}, {4, 530}, {13, 524}, {14, 543}, {17, 9763}, {18, 9885}, {98, 531}, {99, 40672}, {115, 599}, {262, 9762}, {298, 671}, {538, 21359}, {598, 12155}, {633, 22235}, {1992, 9112}, {2482, 16645}, {3424, 41022}, {5464, 7610}, {5466, 23870}, {5469, 5969}, {5470, 14645}, {5472, 15534}, {5858, 12816}, {5859, 33607}, {5862, 33602}, {5863, 33604}, {6114, 9877}, {6115, 9770}, {6777, 9830}, {7607, 13083}, {7612, 21156}, {7620, 31710}, {8587, 8594}, {9113, 18800}, {9114, 9886}, {9166, 40706}, {9180, 23871}, {9741, 38412}, {9760, 36776}, {10754, 22573}, {11121, 14904}, {11122, 41135}, {11180, 41045}, {12817, 33459}, {18842, 37641}, {22570, 36967}, {33603, 35690}, {33605, 35691}, {33606, 35696}, {35304, 36772}, {36316, 40709}

X(42035) = isogonal conjugate of X(41406)
X(42035) = isotomic conjugate of X(37786)
X(42035) = {X(599),X(40727)}-harmonic conjugate of X(42036)

X(42036) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-8th-FERMAT-DAO

Barycentrics    1/(-5*a^2 + b^2 + c^2 + 2*Sqrt[3]*S) : :

Let A' be the orthocenter of triangle BCX(14), and define B' and C' cyclically. Then X(42036) is the orthocenter of triangle A'B'C'. (Randy Hutson, May 31, 2021)

X(42036) lies on these lines: {2, 22573}, {4, 531}, {13, 543}, {14, 524}, {17, 9886}, {18, 9761}, {98, 530}, {99, 40671}, {115, 599}, {262, 9760}, {299, 671}, {538, 21360}, {598, 12154}, {634, 22237}, {1992, 9113}, {2482, 16644}, {3424, 41023}, {5463, 7610}, {5466, 23871}, {5469, 14645}, {5470, 5969}, {5471, 15534}, {5858, 33606}, {5859, 12817}, {5862, 33605}, {5863, 33603}, {6114, 9770}, {6115, 9877}, {6778, 9830}, {7607, 13084}, {7612, 21157}, {7620, 31709}, {8587, 8595}, {9112, 18800}, {9116, 9885}, {9166, 40707}, {9180, 23870}, {10754, 22574}, {11121, 41135}, {11122, 14905}, {11180, 41044}, {12816, 33458}, {18842, 37640}, {22568, 36968}, {33602, 35694}, {33604, 35695}, {33607, 35692}, {36317, 40710}

X(42036) = isogonal conjugate of X(41407)
X(42036) = isotomic conjugate of X(37785)
X(42036) = {X(599),X(40727)}-harmonic conjugate of X(42035)

X(42037) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-42

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

X(42037) lies on these lines: {2, 32}, {6, 16276}, {22, 7878}, {82, 17264}, {308, 14614}, {384, 19568}, {428, 32085}, {598, 16277}, {1176, 34608}, {3108, 35929}, {5007, 16950}, {7394, 7856}, {7760, 16932}, {7827, 34603}, {8667, 18092}, {9870, 34482}, {10191, 35277}, {16275, 18907}, {17409, 36794}, {30435, 40022}, {38817, 39927}

X(42038) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-49

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

X(42038) lies on these lines: {1, 4757}, {2, 38}, {31, 3677}, {42, 4003}, {201, 5298}, {354, 1962}, {519, 3670}, {524, 18183}, {536, 4022}, {551, 2292}, {614, 3929}, {678, 3938}, {750, 18193}, {846, 3315}, {896, 7191}, {976, 16371}, {986, 3241}, {1086, 29690}, {1089, 6534}, {1201, 31165}, {1254, 5434}, {1393, 11237}, {1647, 4415}, {1739, 4745}, {2310, 11238}, {2650, 24473}, {3058, 7004}, {3218, 9340}, {3666, 17449}, {3679, 24443}, {3681, 36634}, {3720, 3999}, {3722, 17596}, {3728, 4688}, {3741, 4980}, {3742, 3989}, {3752, 21805}, {3782, 3829}, {3828, 24167}, {3840, 3994}, {3873, 17591}, {3920, 18201}, {3957, 17593}, {3976, 38314}, {3987, 34641}, {4414, 4428}, {4640, 29818}, {4650, 17024}, {4664, 21330}, {4669, 4695}, {4683, 5211}, {4697, 29823}, {4722, 29821}, {4860, 5311}, {4884, 29687}, {5293, 36006}, {5573, 17125}, {7174, 17124}, {9345, 10980}, {11194, 37549}, {12782, 29577}, {16418, 28082}, {17446, 37756}, {17450, 28606}, {17483, 17722}, {17716, 23958}, {17721, 33098}, {19875, 24046}, {20942, 32925}, {21020, 24165}, {24231, 33105}, {24239, 32856}, {24477, 33128}, {24627, 32923}, {25557, 29682}, {26015, 33145}, {26840, 32844}, {29668, 32933}, {29676, 33146}, {29680, 33103}, {29816, 37520}, {29840, 33067}, {29844, 32950}, {31148, 40471}, {33595, 37599}

X(42038) = {X(2),X(38)}-harmonic conjugate of X(42039)

X(42039) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-50

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

X(42039) lies on these lines: {1, 4127}, {2, 38}, {10, 6534}, {31, 3929}, {44, 29819}, {63, 9340}, {201, 5434}, {518, 1962}, {519, 2292}, {536, 3728}, {612, 3928}, {678, 3961}, {726, 4980}, {846, 3722}, {896, 3920}, {976, 16370}, {1254, 11237}, {2310, 3058}, {3175, 31136}, {3219, 17469}, {3666, 21805}, {3670, 3828}, {3677, 17125}, {3679, 4642}, {3717, 32781}, {3741, 3994}, {3938, 4428}, {3953, 19883}, {3967, 31241}, {3987, 38098}, {4022, 4755}, {4357, 33162}, {4364, 29685}, {4389, 33117}, {4414, 4421}, {4415, 29690}, {4419, 33094}, {4424, 4669}, {4641, 29816}, {4643, 32854}, {4661, 17592}, {4664, 22167}, {4695, 4745}, {4703, 29832}, {4850, 36634}, {4921, 35623}, {4971, 23928}, {4995, 7004}, {5220, 17017}, {5293, 13587}, {6646, 33072}, {7069, 11238}, {7262, 29815}, {7322, 17124}, {10385, 24430}, {15170, 35194}, {15254, 29818}, {16830, 32940}, {16857, 28082}, {17133, 23668}, {17163, 28516}, {17184, 21026}, {17258, 32947}, {17261, 32943}, {17301, 21039}, {17598, 27065}, {17722, 26792}, {18183, 20582}, {19875, 24443}, {19876, 24046}, {20942, 30942}, {21342, 30950}, {21806, 28606}, {24697, 33090}, {25006, 33145}, {28840, 40471}, {29664, 33101}, {31145, 37598}, {32927, 38000}, {32933, 36480}

X(42039) = {X(2),X(38)}-harmonic conjugate of X(42038)

X(42040) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-53

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

X(42040) lies on these lines: {1, 9352}, {2, 38}, {31, 18193}, {42, 3999}, {57, 17469}, {88, 3961}, {519, 3953}, {536, 21330}, {551, 3670}, {597, 18183}, {614, 896}, {748, 3929}, {750, 3677}, {774, 10072}, {899, 21342}, {902, 4906}, {976, 16417}, {986, 38314}, {1155, 29818}, {1193, 24473}, {1393, 5434}, {1401, 21969}, {1646, 22199}, {1647, 3782}, {1739, 4669}, {1962, 17591}, {2292, 25055}, {3120, 3829}, {3241, 3976}, {3315, 17596}, {3666, 17450}, {3679, 24046}, {3720, 4003}, {3722, 4421}, {3752, 17449}, {3924, 11194}, {3994, 20942}, {4022, 4688}, {4428, 17595}, {4677, 4695}, {4740, 22167}, {4745, 24168}, {4860, 17017}, {4980, 24165}, {5211, 33067}, {5272, 36263}, {6384, 20889}, {7004, 11238}, {7191, 18201}, {9340, 23958}, {9350, 16496}, {11019, 33145}, {12782, 29582}, {16370, 28082}, {17301, 21346}, {17533, 28096}, {17593, 29817}, {17598, 27003}, {17722, 26842}, {17728, 33143}, {17872, 37756}, {18173, 18601}, {21805, 36634}, {24177, 33136}, {24440, 31145}, {25557, 29688}, {27002, 32927}, {29690, 40688}, {29819, 37520}, {37549, 40726}

X(42040) = {X(2),X(38)}-harmonic conjugate of X(42041)

X(42041) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-54

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

X(42041) lies on these lines: {1, 3988}, {2, 38}, {9, 17469}, {10, 4980}, {44, 29816}, {45, 3938}, {201, 11237}, {210, 3989}, {519, 14020}, {612, 896}, {748, 7174}, {750, 3928}, {774, 10056}, {976, 16418}, {1962, 3681}, {2292, 3679}, {2308, 15481}, {2310, 10385}, {3058, 7069}, {3688, 21969}, {3715, 17017}, {3722, 4428}, {3728, 4664}, {3828, 24443}, {3829, 29690}, {3961, 33761}, {3967, 30970}, {3971, 4981}, {3994, 31330}, {4009, 31241}, {4078, 33081}, {4104, 32848}, {4113, 4681}, {4126, 4364}, {4422, 29686}, {4424, 4745}, {4656, 33136}, {4722, 5220}, {4755, 21330}, {4995, 24431}, {5268, 36263}, {5293, 17549}, {6376, 20889}, {10459, 31165}, {16675, 41711}, {16830, 32938}, {17257, 33074}, {17258, 32948}, {17260, 32923}, {17261, 32945}, {17542, 28082}, {17598, 35595}, {19346, 34247}, {21020, 32925}, {21026, 27184}, {21805, 28606}, {24697, 33091}, {31136, 35652}

X(42041) = {X(2),X(38)}-harmonic conjugate of X(42040)

X(42042) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-59

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

X(42042) lies on these lines: {1, 2}, {6, 3750}, {30, 37529}, {35, 19346}, {55, 4649}, {81, 2177}, {85, 25721}, {87, 40433}, {100, 37604}, {165, 1002}, {171, 4421}, {192, 36645}, {291, 5269}, {381, 37699}, {518, 17592}, {527, 4335}, {536, 3510}, {553, 4334}, {581, 28194}, {672, 16667}, {846, 3751}, {902, 37685}, {968, 1757}, {984, 37593}, {986, 24473}, {1011, 3746}, {1051, 9052}, {1100, 17716}, {1126, 5248}, {1376, 4038}, {1386, 17715}, {1449, 2276}, {1468, 17549}, {1575, 16884}, {1621, 16468}, {1962, 3681}, {1985, 37721}, {2108, 3722}, {2238, 3247}, {2334, 19765}, {2356, 7714}, {2667, 4664}, {3136, 37719}, {3158, 35104}, {3242, 17600}, {3303, 16058}, {3304, 16059}, {3475, 33147}, {3656, 5396}, {3689, 37595}, {3723, 37673}, {3729, 40721}, {3731, 37657}, {3737, 4948}, {3755, 17889}, {3829, 17717}, {3873, 17591}, {3875, 37632}, {3896, 4980}, {3913, 11358}, {3928, 17594}, {3989, 4661}, {3993, 32937}, {4021, 30946}, {4026, 33084}, {4085, 18134}, {4090, 41839}, {4191, 5563}, {4192, 7982}, {4199, 11523}, {4255, 40726}, {4300, 34632}, {4343, 6172}, {4360, 4479}, {4383, 16484}, {4512, 40774}, {4650, 4689}, {4658, 8715}, {4660, 17778}, {4663, 7262}, {4734, 24165}, {4785, 23655}, {4849, 15569}, {4851, 33079}, {4854, 33101}, {4870, 37694}, {4883, 17063}, {4921, 10458}, {4954, 18792}, {4966, 33174}, {4970, 24349}, {5247, 16418}, {5264, 16395}, {5331, 39969}, {5710, 16396}, {5712, 33109}, {5718, 33141}, {7196, 25716}, {7991, 37400}, {9909, 37580}, {10107, 31503}, {10222, 19540}, {10385, 14547}, {11518, 16056}, {11520, 37467}, {11522, 22392}, {13587, 37608}, {15485, 32911}, {15621, 18185}, {16189, 19647}, {16370, 37573}, {16371, 37607}, {16496, 37676}, {16777, 21904}, {17056, 32865}, {17317, 24760}, {17379, 36646}, {17393, 24766}, {17718, 33135}, {18140, 25287}, {18169, 41629}, {18173, 25059}, {19684, 32945}, {20182, 41711}, {21223, 25264}, {21746, 21849}, {21806, 28606}, {21838, 24528}, {23638, 39543}, {24217, 37662}, {24524, 31008}, {25074, 40133}, {27804, 32925}, {31034, 32947}, {31165, 37548}, {32921, 41823}, {33158, 38047}, {33771, 37603}, {34612, 37631}, {37355, 37722}, {37365, 37727}, {37370, 37724}, {37732, 38021}

X(42042) = {X(2),X(42)}-harmonic conjugate of X(42043)
X(42042) = {X(42),X(17018)}-harmonic conjugate of X(1)

X(42043) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-60

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

X(42043) lies on these lines: {1, 2}, {6, 3550}, {9, 21904}, {30, 37699}, {39, 24528}, {55, 16468}, {100, 28523}, {165, 511}, {192, 4090}, {210, 17592}, {238, 4428}, {291, 39980}, {381, 37529}, {518, 17591}, {549, 37698}, {672, 31508}, {726, 4734}, {872, 4096}, {984, 4849}, {1002, 36603}, {1126, 25440}, {1376, 4649}, {1449, 1575}, {1468, 13587}, {1743, 2276}, {1757, 3929}, {2177, 8616}, {2238, 3731}, {2258, 18793}, {2663, 16571}, {3052, 16477}, {3158, 3795}, {3247, 37673}, {3304, 16409}, {3501, 20970}, {3510, 41142}, {3654, 5396}, {3666, 21870}, {3689, 17716}, {3711, 20182}, {3736, 18192}, {3746, 16058}, {3750, 4383}, {3751, 3928}, {3755, 3944}, {3829, 33141}, {3875, 4479}, {3896, 32931}, {3973, 37657}, {3984, 11533}, {3993, 27538}, {3996, 25496}, {4038, 4413}, {4085, 4417}, {4192, 7991}, {4203, 8715}, {4234, 4281}, {4255, 11194}, {4272, 17281}, {4335, 6172}, {4551, 4654}, {4641, 17601}, {4650, 4663}, {4689, 7262}, {4857, 6818}, {4863, 17722}, {4866, 16850}, {4921, 18169}, {4970, 32937}, {4980, 32860}, {5010, 19346}, {5247, 16370}, {5264, 16396}, {5270, 6817}, {5400, 30308}, {5563, 16059}, {5718, 32865}, {5881, 37365}, {7196, 25721}, {7714, 40976}, {7982, 19540}, {9342, 9345}, {9350, 37633}, {9548, 14636}, {9909, 37576}, {16189, 19546}, {16371, 37608}, {16417, 37607}, {16418, 37573}, {16484, 37679}, {16667, 17754}, {17379, 39972}, {17598, 41711}, {17718, 33132}, {21805, 28606}, {21849, 23638}, {24217, 37663}, {24524, 34020}, {25075, 40133}, {25264, 41840}, {25286, 30964}, {25287, 31008}, {25568, 33152}, {25590, 37632}, {31034, 32948}, {31162, 37732}, {31165, 37598}, {32926, 41823}, {33160, 38047}, {36646, 37677}, {37355, 37720}, {37370, 37721}

X(42043) = {X(2),X(42)}-harmonic conjugate of X(42042)
X(42043) = {X(42),X(43)}-harmonic conjugate of X(1)

X(42044) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-68

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

X(42044) lies on these lines: {2, 37}, {43, 3994}, {81, 3729}, {190, 3187}, {210, 28484}, {306, 33151}, {354, 28555}, {519, 3869}, {528, 34603}, {538, 17389}, {553, 28301}, {726, 3873}, {740, 3681}, {984, 4365}, {1255, 10436}, {1717, 3811}, {1836, 33093}, {1999, 32933}, {2171, 4654}, {2292, 3679}, {2321, 32782}, {2325, 26723}, {2895, 17299}, {2901, 3868}, {3058, 28503}, {3120, 33092}, {3159, 3876}, {3219, 17262}, {3240, 3967}, {3247, 5333}, {3305, 17151}, {3416, 33100}, {3578, 17333}, {3663, 33172}, {3685, 3891}, {3703, 33134}, {3706, 7226}, {3712, 29665}, {3760, 20889}, {3769, 4427}, {3773, 32776}, {3782, 3943}, {3790, 4972}, {3875, 32911}, {3896, 32937}, {3912, 33146}, {3914, 32862}, {3920, 5695}, {3923, 32928}, {3929, 4921}, {3932, 33131}, {3938, 4693}, {3944, 32848}, {3950, 5249}, {3969, 27184}, {3971, 28522}, {3993, 32771}, {4011, 32924}, {4062, 33101}, {4102, 17271}, {4110, 40603}, {4135, 4970}, {4360, 26223}, {4361, 27065}, {4362, 32936}, {4363, 17019}, {4387, 7191}, {4415, 33077}, {4418, 9347}, {4430, 28582}, {4439, 33117}, {4442, 29641}, {4659, 5287}, {4661, 28581}, {4676, 17150}, {4851, 17483}, {4854, 29667}, {4956, 11235}, {4967, 28651}, {5057, 33088}, {5256, 41242}, {5271, 33761}, {5278, 17261}, {5905, 17314}, {6057, 29679}, {6535, 32784}, {6541, 25957}, {6542, 32859}, {9352, 29649}, {10129, 29671}, {11238, 21333}, {11246, 28556}, {15523, 33154}, {16727, 40493}, {17011, 17318}, {17144, 31036}, {17155, 28516}, {17175, 29597}, {17184, 17233}, {17242, 18139}, {17243, 27186}, {17276, 32863}, {17319, 19684}, {17336, 19742}, {17351, 37685}, {17361, 31011}, {17393, 19717}, {17763, 32934}, {18145, 28659}, {19738, 29584}, {19875, 39708}, {20017, 33066}, {20691, 31060}, {24210, 33089}, {24248, 33078}, {24703, 32842}, {25269, 37652}, {29674, 33145}, {29687, 33149}, {30568, 37680}, {32846, 33098}, {32847, 33094}, {32852, 33099}, {32854, 33095}, {32921, 32930}, {32926, 32929}, {33128, 33164}, {33135, 33161}, {33143, 33158}, {33152, 33156}

X(42044) = anticomplement of X(42051)

X(42045) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-69

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

X(42045) lies on these lines: {1, 540}, {2, 6}, {30, 944}, {72, 10108}, {306, 4667}, {320, 17011}, {321, 3879}, {376, 41810}, {445, 648}, {511, 3873}, {519, 2650}, {538, 17389}, {551, 41815}, {553, 41801}, {754, 29584}, {894, 3969}, {903, 41823}, {1100, 17184}, {1171, 25536}, {1230, 30939}, {1449, 32774}, {1959, 3970}, {1962, 17770}, {2092, 18601}, {2308, 24542}, {3664, 4359}, {3679, 41812}, {3758, 32858}, {3759, 27186}, {3782, 24724}, {3909, 40952}, {3989, 17771}, {3995, 17390}, {4026, 20290}, {4038, 32843}, {4062, 4697}, {4357, 41820}, {4360, 17483}, {4363, 20017}, {4393, 33146}, {4421, 41811}, {4442, 33097}, {4450, 17018}, {4478, 6539}, {4644, 32933}, {4649, 4972}, {4658, 5051}, {4722, 29653}, {4754, 6542}, {4851, 26223}, {4854, 17491}, {4938, 21085}, {4981, 34379}, {5625, 6536}, {13745, 38314}, {14544, 41571}, {16477, 29851}, {17019, 33066}, {17020, 24183}, {17120, 33157}, {17121, 26724}, {17147, 17365}, {17317, 27065}, {17344, 37869}, {17364, 28606}, {17377, 28605}, {17484, 34064}, {17768, 27804}, {20072, 33761}, {21020, 23812}, {23731, 28840}, {23905, 31064}, {25056, 27782}, {26580, 37595}, {33081, 33682}

X(42046) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-72

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

X(42046) lies on these lines: {1, 190}, {2, 292}, {31, 4586}, {37, 20141}, {239, 20331}, {664, 36276}, {716, 1966}, {2279, 3226}, {3768, 28840}, {4465, 16826}, {16522, 20176}, {16526, 17261}, {16827, 28283}, {17475, 29584}, {29580, 39916}

X(42046) = reflection of X(43270) in X(2)

X(42047) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-76

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

X(42047) lies on these lines: {2, 37}, {4, 519}, {8, 4415}, {165, 28557}, {226, 17314}, {329, 5839}, {527, 10442}, {545, 19645}, {726, 24477}, {740, 25568}, {940, 7222}, {964, 38314}, {966, 4656}, {1266, 30567}, {1766, 3928}, {1999, 4644}, {2298, 35578}, {2901, 3487}, {3241, 5716}, {3452, 17151}, {3474, 17763}, {3475, 32915}, {3729, 37642}, {3769, 24280}, {3782, 34255}, {3914, 3974}, {3950, 25525}, {3971, 38057}, {4054, 5712}, {4080, 20017}, {4220, 4421}, {4361, 18228}, {4362, 5698}, {4371, 14555}, {4402, 8055}, {4419, 11679}, {4442, 10327}, {4659, 39595}, {4873, 20106}, {4891, 11038}, {4916, 17778}, {5016, 31145}, {5241, 41915}, {5273, 17262}, {5658, 29016}, {5743, 32087}, {5846, 9812}, {6703, 7229}, {9778, 28530}, {9779, 28472}, {11194, 37399}, {11235, 28503}, {17233, 26132}, {17276, 37655}, {17351, 37666}, {17390, 41825}, {21949, 39570}, {25055, 37037}, {28580, 34607}, {30568, 37650}, {31995, 37674}

X(42048) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-77

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

X(42048) lies on these lines: {2, 85}, {7, 1146}, {196, 1855}, {220, 31994}, {344, 18159}, {481, 7090}, {482, 14121}, {514, 5603}, {527, 1478}, {544, 34627}, {1111, 4000}, {1360, 5434}, {1699, 2391}, {1737, 7960}, {3241, 14942}, {3474, 28118}, {3476, 9318}, {3732, 5819}, {4419, 31397}, {4421, 9305}, {4644, 18391}, {7223, 40127}, {10481, 23058}, {21258, 32086}, {24477, 35102}, {28609, 29594}, {28610, 28638}

X(42049) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-78

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

X(42049) lies on these lines: {2, 37}, {8, 4884}, {40, 376}, {57, 17314}, {63, 5839}, {333, 4371}, {524, 28610}, {726, 25568}, {740, 24477}, {1699, 28557}, {2325, 23511}, {3161, 37679}, {3241, 5710}, {3474, 32845}, {3475, 17155}, {3687, 4419}, {3875, 37642}, {3913, 37328}, {3929, 37654}, {3950, 5437}, {4035, 4862}, {4361, 5273}, {4398, 26132}, {4421, 28503}, {4641, 20043}, {4644, 32939}, {4851, 21454}, {4852, 37666}, {5325, 16833}, {5698, 32934}, {5712, 7222}, {5737, 32087}, {5745, 17151}, {5846, 9778}, {7228, 41825}, {9812, 28530}, {10445, 17132}, {10856, 17133}, {16046, 41629}, {17056, 31995}, {17262, 18228}, {17299, 37655}, {17595, 34255}, {19732, 41915}, {22145, 37672}, {24165, 38053}, {24248, 32855}, {24280, 33071}, {24621, 24654}

X(42050) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-79

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

X(42050) lies on these lines: {1, 527}, {2, 85}, {7, 34522}, {9, 1323}, {77, 34526}, {142, 21314}, {144, 6603}, {165, 2391}, {220, 651}, {277, 34578}, {514, 5657}, {518, 11200}, {544, 944}, {1121, 33298}, {1642, 38941}, {2094, 5228}, {2124, 3929}, {3474, 28125}, {3496, 3928}, {4971, 6764}, {5088, 5819}, {5222, 17595}, {5723, 5744}, {5731, 5845}, {6173, 10481}, {7181, 40127}, {11201, 15726}, {16020, 26273}, {17301, 40133}, {17762, 25242}, {25568, 35102}, {26658, 32024}

X(42051) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-80

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

X(42051) lies on these lines: {1, 19276}, {2, 37}, {8, 33068}, {38, 3696}, {43, 4706}, {44, 32933}, {57, 17151}, {63, 4361}, {65, 519}, {81, 4852}, {210, 726}, {239, 4641}, {244, 4365}, {306, 1086}, {314, 16700}, {319, 26840}, {333, 17117}, {354, 740}, {518, 17155}, {527, 14557}, {528, 7667}, {535, 34666}, {537, 4685}, {545, 17781}, {551, 4065}, {614, 5695}, {899, 3967}, {940, 3875}, {966, 41915}, {980, 32104}, {982, 3706}, {986, 3679}, {1155, 4362}, {1211, 3663}, {1266, 3687}, {1269, 21857}, {1279, 32929}, {1386, 4418}, {1738, 3703}, {1812, 16759}, {1999, 17160}, {2321, 24177}, {2895, 17345}, {2901, 5439}, {2999, 4659}, {3006, 21949}, {3058, 28580}, {3219, 17348}, {3305, 17262}, {3670, 5295}, {3681, 28582}, {3683, 16825}, {3689, 32920}, {3697, 24068}, {3704, 23536}, {3714, 24443}, {3729, 4383}, {3740, 28555}, {3741, 4003}, {3742, 28484}, {3744, 32922}, {3745, 3980}, {3757, 4689}, {3773, 24169}, {3823, 32862}, {3834, 32858}, {3838, 29849}, {3840, 4519}, {3844, 33125}, {3873, 28581}, {3886, 17597}, {3896, 17140}, {3912, 40688}, {3928, 24310}, {3929, 16552}, {3966, 24248}, {3969, 17231}, {3971, 28516}, {3991, 14746}, {3999, 10453}, {4001, 17362}, {4009, 16569}, {4052, 14554}, {4054, 37662}, {4096, 28554}, {4135, 24003}, {4360, 37595}, {4363, 5256}, {4371, 14552}, {4387, 5272}, {4395, 26723}, {4398, 27184}, {4440, 33066}, {4496, 7019}, {4527, 24200}, {4640, 32845}, {4644, 20043}, {4646, 4968}, {4647, 37592}, {4656, 5241}, {4660, 4914}, {4663, 32940}, {4670, 17011}, {4682, 32928}, {4693, 29820}, {4696, 21896}, {4716, 32913}, {4762, 16892}, {4849, 17165}, {4860, 39594}, {4884, 25006}, {4886, 6646}, {4906, 32943}, {4970, 24325}, {5249, 7263}, {5271, 17119}, {5287, 17318}, {5294, 17366}, {5325, 21233}, {5564, 37653}, {5712, 31995}, {5739, 17276}, {5839, 9965}, {5847, 11246}, {5880, 33088}, {5918, 28850}, {6603, 28951}, {7283, 33309}, {7321, 17778}, {9776, 17314}, {10167, 29016}, {10436, 20182}, {11679, 17595}, {15254, 32936}, {16834, 20963}, {16885, 25734}, {17020, 41242}, {17045, 41850}, {17135, 21342}, {17143, 37596}, {17144, 24621}, {17229, 33172}, {17235, 32782}, {17254, 41816}, {17351, 32911}, {17372, 32863}, {17374, 20017}, {17376, 26842}, {17600, 24342}, {17733, 32636}, {17889, 32855}, {19701, 25590}, {20292, 32842}, {20508, 23878}, {20691, 20913}, {23681, 30811}, {24715, 32866}, {25125, 31060}, {28557, 40998}, {29584, 33296}, {32778, 33149}, {32857, 32861}, {33075, 33102}, {33077, 33146}, {33089, 33131}, {33132, 33167}, {33147, 33160}, {34728, 34742}, {36871, 39948}, {37631, 39774}

X(42051) = complement of X(42044)
X(42051) = anticomplement of X(35652)

X(42052) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-83

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

Let LA be the radical axis of the circumcircle and reflected A-Neuberg circle, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. A'B'C' is homothetic to ABC at X(251). X(42052) = X(2)-of-A'B'C'. (Randy Hutson, May 31, 2021)

X(42052) lies on these lines: {2, 32}, {69, 41464}, {311, 34603}, {428, 7767}, {524, 23642}, {1180, 7893}, {1225, 30737}, {2979, 3852}, {3108, 33021}, {3266, 10691}, {6636, 7768}, {7750, 8024}, {7779, 38862}, {7788, 9723}, {7794, 35929}, {7826, 8267}, {7854, 16932}, {7860, 37353}, {20063, 40002}

X(42053) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-89

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

X(42053) lies on these lines: {2, 38}, {10, 21342}, {354, 740}, {519, 942}, {545, 25371}, {551, 3743}, {553, 752}, {614, 4672}, {726, 3742}, {1086, 29655}, {1125, 39544}, {3175, 28554}, {3306, 32920}, {3315, 4418}, {3678, 6532}, {3679, 28611}, {3741, 3999}, {3757, 18201}, {3829, 20256}, {3846, 24231}, {3848, 28582}, {3980, 17597}, {4085, 24177}, {4090, 16602}, {4359, 4732}, {4362, 4860}, {4363, 29668}, {4432, 29820}, {4434, 27003}, {4666, 32934}, {4685, 22295}, {4688, 24182}, {4697, 7191}, {4883, 4970}, {4891, 28522}, {4974, 32913}, {5211, 33097}, {5272, 32935}, {5625, 17600}, {5880, 29844}, {7292, 32940}, {7321, 33106}, {11246, 28494}, {17147, 17450}, {17155, 28516}, {17595, 29651}, {17726, 23812}, {17767, 40998}, {18193, 32916}, {25351, 29673}, {25557, 29671}, {26842, 32844}, {29817, 32845}, {29843, 33149}

X(42053) = complement of X(42054)

X(42054) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-91

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

X(42054) lies on these lines: {1, 33309}, {2, 38}, {8, 4703}, {9, 32920}, {10, 3782}, {37, 22199}, {45, 29651}, {63, 4434}, {72, 519}, {190, 3961}, {200, 32934}, {210, 726}, {306, 4439}, {329, 4865}, {518, 3971}, {536, 4685}, {545, 24336}, {551, 27784}, {612, 4697}, {740, 3681}, {752, 17781}, {1757, 3791}, {2667, 4664}, {2796, 34612}, {2886, 21093}, {2887, 3717}, {3187, 4753}, {3219, 32927}, {3242, 4011}, {3666, 4090}, {3678, 24068}, {3679, 4385}, {3699, 17596}, {3706, 4135}, {3715, 16825}, {3718, 17274}, {3740, 24165}, {3741, 3967}, {3749, 25728}, {3750, 17261}, {3790, 33084}, {3840, 4009}, {3874, 4075}, {3891, 4974}, {3920, 4672}, {3932, 33064}, {3935, 32936}, {3938, 4432}, {3994, 17135}, {4003, 6686}, {4113, 4709}, {4362, 5220}, {4415, 29673}, {4421, 20760}, {4422, 29672}, {4655, 10327}, {4660, 30615}, {4661, 32915}, {4679, 29844}, {4683, 33091}, {4732, 28605}, {4756, 32930}, {4849, 4970}, {4871, 21342}, {4892, 29641}, {4899, 24210}, {5223, 32853}, {5297, 32940}, {6646, 33079}, {7174, 25496}, {8616, 17336}, {9954, 21084}, {16496, 30568}, {17147, 21805}, {17330, 40085}, {17350, 17716}, {17484, 33072}, {20834, 24820}, {20942, 31137}, {24821, 32939}, {24841, 29820}, {25351, 33146}, {26580, 33162}, {26792, 32844}, {27065, 32923}, {27184, 28595}, {28516, 32860}, {29819, 41241}, {32847, 33066}, {32850, 33099}, {32862, 33065}, {33117, 33151}, {33118, 33152}, {33126, 33164}

X(42054) = anticomplement of X(42053)

X(42055) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-92

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

X(42055) lies on these lines: {1, 4234}, {2, 38}, {7, 4865}, {10, 40688}, {57, 4434}, {65, 519}, {75, 39742}, {190, 29820}, {320, 32866}, {321, 17449}, {354, 726}, {518, 4685}, {527, 24180}, {536, 13476}, {545, 24424}, {551, 6051}, {614, 32935}, {740, 3873}, {894, 17598}, {903, 40038}, {976, 19336}, {1086, 29673}, {1836, 29844}, {1930, 17179}, {2275, 28592}, {2796, 3058}, {2887, 24231}, {3218, 32923}, {3241, 17480}, {3242, 3980}, {3315, 32930}, {3662, 28595}, {3677, 25496}, {3705, 4892}, {3741, 21342}, {3742, 3971}, {3782, 29655}, {3791, 32913}, {3816, 21093}, {3840, 3999}, {3923, 17597}, {3957, 32845}, {3961, 24841}, {3967, 4871}, {3993, 4883}, {3995, 17450}, {4003, 6685}, {4052, 11019}, {4090, 16610}, {4363, 29652}, {4430, 32860}, {4432, 32933}, {4440, 33095}, {4514, 32857}, {4672, 7191}, {4860, 29649}, {4884, 25557}, {4891, 28555}, {4906, 17351}, {4921, 32914}, {4974, 32912}, {4980, 31136}, {4987, 17362}, {5211, 33096}, {7081, 18201}, {7292, 32938}, {7321, 33109}, {8666, 37227}, {8669, 32636}, {8720, 37080}, {11246, 17766}, {11346, 28082}, {16711, 17141}, {16825, 19723}, {17017, 19738}, {17146, 17147}, {17483, 32844}, {17595, 29670}, {17599, 19722}, {20834, 24826}, {25351, 33117}, {26840, 33076}, {26842, 33072}, {27003, 32927}, {28516, 32915}, {29817, 32936}, {29835, 33145}, {29840, 33097}, {29843, 33154}, {33067, 33090}, {33069, 33089}, {33120, 33146}, {33121, 33147}, {33124, 33167}

X(42055) = anticomplement of X(4096)

X(42056) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-94

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

X(42056) lies on these lines: {2, 38}, {9, 4434}, {10, 4009}, {43, 4664}, {210, 392}, {312, 3679}, {321, 4937}, {536, 3740}, {551, 4090}, {3626, 4519}, {3706, 4669}, {3715, 29649}, {3826, 21093}, {3828, 31993}, {3967, 4688}, {3985, 17281}, {4023, 6541}, {4104, 29594}, {4113, 34641}, {4126, 29655}, {4358, 31136}, {4533, 35633}, {4671, 4732}, {4672, 5297}, {4685, 35652}, {4697, 5268}, {4740, 26038}, {4865, 18228}, {5293, 13735}, {7308, 32920}, {7322, 25496}, {8580, 32934}, {16833, 30393}, {17271, 20947}, {17274, 30758}, {17338, 17725}, {18743, 31137}, {19875, 27798}, {21060, 29600}, {21805, 31035}, {25351, 33151}, {28554, 32925}, {29577, 33084}, {32927, 35595}

X(42057) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-99

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

X(42057) lies on these lines: {1, 2}, {30, 12545}, {38, 3993}, {81, 32943}, {149, 32949}, {244, 3896}, {320, 33095}, {333, 16484}, {350, 3879}, {354, 740}, {376, 10476}, {497, 32946}, {515, 39550}, {516, 10439}, {518, 3971}, {527, 35645}, {528, 38484}, {536, 13476}, {537, 3175}, {553, 7248}, {672, 3950}, {726, 3873}, {752, 3058}, {902, 37639}, {940, 32941}, {982, 4970}, {1001, 32853}, {1011, 8666}, {1015, 21877}, {1269, 4479}, {1279, 3791}, {1575, 17388}, {1621, 32919}, {2321, 24512}, {2796, 5208}, {2887, 4966}, {2901, 3881}, {3136, 24387}, {3304, 11358}, {3315, 32924}, {3550, 37684}, {3663, 30941}, {3664, 4441}, {3685, 32913}, {3706, 4883}, {3742, 28581}, {3750, 14829}, {3751, 4011}, {3755, 24169}, {3769, 17715}, {3875, 30962}, {3886, 3980}, {3913, 16059}, {3946, 30945}, {3995, 17145}, {3996, 17122}, {4038, 5263}, {4090, 4358}, {4135, 17165}, {4191, 8715}, {4192, 5882}, {4359, 4709}, {4360, 17598}, {4365, 17140}, {4387, 32935}, {4417, 24217}, {4421, 23853}, {4430, 32925}, {4432, 4641}, {4514, 32846}, {4649, 32942}, {4667, 24330}, {4684, 24210}, {4688, 25124}, {4693, 32939}, {4715, 24705}, {4780, 24177}, {4849, 24003}, {4851, 4865}, {4852, 4906}, {4856, 37657}, {4889, 20530}, {5247, 33309}, {5284, 32864}, {5563, 13588}, {6682, 37593}, {8616, 37683}, {8731, 24391}, {10222, 37365}, {10441, 28194}, {10465, 34628}, {10471, 17180}, {11246, 17764}, {12437, 16056}, {12513, 16058}, {12536, 37110}, {12607, 37355}, {15485, 37652}, {17056, 21242}, {17147, 17449}, {17151, 30350}, {17155, 28522}, {17231, 28595}, {17233, 33169}, {17234, 32865}, {17300, 33109}, {17314, 17754}, {17315, 37686}, {17318, 24691}, {17377, 30963}, {17390, 21264}, {17448, 21838}, {17597, 32921}, {17778, 33106}, {17781, 35614}, {18059, 21443}, {18134, 21241}, {18139, 33136}, {19540, 37727}, {20162, 24586}, {20963, 21071}, {22214, 24513}, {24552, 33682}, {24692, 33094}, {27269, 41912}, {28562, 34611}, {30986, 37730}, {32773, 33087}, {32863, 32947}, {32945, 37633}, {33069, 33134}, {33121, 33158}, {33124, 33135}, {34379, 40998}, {34612, 35626}, {38832, 41629}

X(42058) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-GEMINI-103

Barycentrics    5*a^3 - b^3 - c^3 : :

X(42058) lies on these lines: {2, 31}, {209, 34607}, {519, 32912}, {545, 3891}, {674, 1992}, {744, 4740}, {758, 3241}, {896, 29832}, {902, 31034}, {1617, 41801}, {1621, 17378}, {2094, 2835}, {2390, 11206}, {3006, 36277}, {3187, 28580}, {3744, 4715}, {3920, 17333}, {4217, 17751}, {4651, 37654}, {4655, 29831}, {4660, 21747}, {5055, 20575}, {5711, 14020}, {7357, 33767}, {10304, 30269}, {16704, 21283}, {17342, 33078}, {17382, 32950}, {17486, 39347}, {17491, 26228}, {20045, 24695}, {21282, 24597}, {26065, 28599}, {27754, 33073}, {28494, 33128}, {28498, 33156}, {28503, 32933}, {28508, 33143}, {28512, 33161}, {28566, 33114}, {28570, 33122}, {31303, 32941}, {34611, 41629}

X(42058) = anticomplement of X(31134)

X(42059) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-HATZIPOLAKIS-MOSES

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 + 5*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 5*a^6*b^2*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 + 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(42059) lies on the Jerabek hyperbola and these lines: {4, 17824}, {68, 10224}, {74, 6799}, {195, 265}, {1199, 38006}, {1498, 18550}, {2917, 5944}, {3519, 14076}, {3521, 18400}, {3574, 22466}, {5900, 17823}, {6143, 13418}, {10274, 13621}, {10628, 11559}, {13423, 37932}, {21400, 36749}, {32351, 33565}

X(42059) = isogonal conjugate of anticomplement of X(6143)

X(42060) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-2ND-INNER-VECTEN

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

X(42060) lies on these lines: {2, 13921}, {20, 487}, {32, 1991}, {69, 486}, {76, 6229}, {99, 32434}, {193, 19104}, {298, 6301}, {299, 6300}, {315, 32419}, {325, 35684}, {372, 491}, {485, 10008}, {492, 6119}, {511, 6290}, {599, 626}, {637, 6251}, {1078, 13087}, {1505, 7888}, {3620, 13711}, {3788, 13989}, {3933, 32494}, {5590, 32955}, {6337, 13821}, {6680, 13846}, {8184, 35813}, {9891, 9939}, {10519, 14244}, {13770, 20080}, {13926, 32458}, {32433, 35821}

X(42061) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-3RD-BROCARD

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

X(42061) lies on these lines: {32, 8789}, {39, 695}, {76, 115}, {511, 694}, {736, 18829}, {737, 805}, {2458, 9467}, {3094, 19602}, {3199, 17980}, {3721, 3865}, {5167, 16068}, {11654, 18872}

X(42061) = X(i)-isoconjugate of X(j) for these (i,j): {385, 3113}, {1580, 3114}, {1926, 18898}, {1966, 3407}
X(42061) = barycentric product X(i)*X(j) for these {i,j}: {694, 3094}, {1581, 3116}, {1916, 3117}, {3314, 9468}, {3862, 3863}, {5117, 17970}, {17415, 18829}, {18896, 18899}
X(42061) = barycentric quotient X(i)/X(j) for these {i,j}: {694, 3114}, {1967, 3113}, {3094, 3978}, {3116, 1966}, {3117, 385}, {3314, 14603}, {8789, 18898}, {9006, 5027}, {9468, 3407}, {14251, 8840}, {17415, 804}, {17938, 33514}, {18829, 9063}, {18899, 1691}
X(42061) = {X(9468),X(17970)}-harmonic conjugate of X(32)

X(42062) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-4TH-INNER-FERMAT-DAO-NHI

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

X(42062) lies on these lines: {2, 5472}, {4, 20415}, {13, 35931}, {14, 5459}, {17, 530}, {18, 22489}, {76, 9763}, {99, 40671}, {396, 671}, {524, 40707}, {531, 11602}, {543, 11122}, {597, 11161}, {5466, 9194}, {8595, 23302}, {8838, 36316}, {12821, 35019}, {33475, 40706}, {33602, 33623}, {33604, 33610}

X(42062) = isotomic conjugate of complement of X(37786)

X(42063) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-4TH-OUTER-FERMAT-DAO-NHI

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

X(42063) lies on these lines: {2, 5471}, {4, 20416}, {13, 5460}, {14, 35932}, {17, 22490}, {18, 531}, {76, 9761}, {99, 40672}, {395, 671}, {524, 40706}, {530, 11603}, {543, 11121}, {597, 11161}, {5466, 9195}, {8594, 23303}, {8836, 36317}, {12820, 35020}, {33474, 40707}, {33603, 33625}, {33605, 33611}

X(42063) = isotomic conjugate of complement of X(37785)

X(42064) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-K721-OSCULATING

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

X(42064) lies on these lines: {1, 528}, {33, 8735}, {55, 2170}, {103, 517}, {200, 1146}, {220, 2310}, {664, 3957}, {1121, 3935}, {1212, 3939}, {1642, 2161}, {2342, 41339}, {4511, 14942}, {4666, 17044}, {4845, 6603}, {5527, 38454}

X(42064) = isogonal conjugate of X(38459)

X(42065) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-7TH-BROCARD

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

X(42065) lies on these lines: {25, 110}, {32, 1147}, {49, 10547}, {98, 325}, {155, 2353}, {184, 23217}, {394, 2351}, {487, 6402}, {488, 6401}, {511, 39644}, {684, 878}, {1092, 40319}, {1402, 36051}, {1570, 34382}, {3289, 14600}, {3455, 13754}, {3456, 41597}, {5504, 9517}, {6193, 32816}, {8884, 18831}, {9149, 9925}, {10723, 33803}, {14601, 23098}, {14908, 22115}, {17702, 39838}, {33581, 41619}, {35456, 41533}, {40352, 41615}

X(42065) = isogonal conjugate of X(44145)
X(42065) = trilinear pole of line X(577)X(3049)
X(42065) = cevapoint of X(184) and X(3289)
X(42065) = crosspoint of X(2987) and X(43705)
X(42065) = crosssum of X(230) and X(460)
X(42065) = trilinear product X(i)*X(j) for these {i,j}: {3, 36051}, {31, 43705}, {48, 2987}, {63, 32654}, {184, 8773}, {255, 3563}, {293, 34157}, {810, 10425}, {822, 32697}, {4575, 35364}, {8781, 9247}, {36105, 39201}

X(42066) = PERSPECTOR OF THESE TRIANGLES: ABC AND INVERSE-OF-APOLLONIUS

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

X(42066) lies on these lines: {1, 21}, {2, 20360}, {42, 23928}, {65, 6042}, {181, 756}, {192, 20536}, {210, 2643}, {321, 1109}, {740, 4388}, {2611, 21333}, {2632, 4094}, {3725, 4016}, {3728, 26893}, {5202, 26885}, {17476, 21342}, {21020, 25760}, {21254, 33124}, {21805, 24048}, {23913, 26037}

X(42067) = X(4)X(6335)∩X(19)X(1843)

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

X(42067) lies on the orthic inconic and these lines: {4, 6335}, {19, 1843}, {25, 3052}, {28, 17946}, {34, 9432}, {125, 5521}, {607, 8541}, {608, 1974}, {1015, 22096}, {1086, 2969}, {1331, 24822}, {1474, 17962}, {1829, 2836}, {1880, 2880}, {2082, 40673}, {2310, 14935}, {2393, 7297}, {3125, 3271}, {3270, 11918}, {5139, 5517}, {5190, 5509}, {5341, 9969}, {6212, 6291}, {6213, 6406}, {7300, 32366}, {8735, 8754}, {9822, 27059}, {11574, 26998}, {14936, 20975}

X(42067) = polar conjugate of X(31625)
X(42067) = isogonal conjugate of the isotomic conjugate of X(2969)
X(42067) = polar conjugate of the isotomic conjugate of X(1015)
X(42067) = polar conjugate of the isogonal conjugate of X(1977)
X(42067) = orthic-isogonal conjugate of X(6591)
X(42067) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 6591}, {19, 2489}, {915, 3310}, {2969, 1015}
X(42067) = X(1977)-cross conjugate of X(1015)
X(42067) = X(i)-isoconjugate of X(j) for these (i,j): {3, 7035}, {48, 31625}, {59, 3718}, {63, 1016}, {69, 765}, {71, 4601}, {72, 4600}, {77, 4076}, {78, 4998}, {100, 4561}, {190, 1332}, {201, 6064}, {304, 1252}, {305, 1110}, {306, 4567}, {326, 15742}, {345, 4564}, {646, 1813}, {664, 4571}, {668, 1331}, {799, 4574}, {905, 6632}, {906, 1978}, {1018, 4563}, {1264, 7012}, {1265, 7045}, {1275, 3692}, {3690, 24037}, {3694, 4620}, {3695, 24041}, {3699, 6516}, {3949, 4590}, {3952, 4592}, {3977, 5376}, {4033, 4558}, {4158, 23999}, {4554, 4587}, {4570, 20336}, {4575, 27808}, {4619, 15416}, {5383, 22370}, {6065, 7182}, {6332, 31615}, {6386, 32656}, {7257, 23067}, {23990, 40364}
X(42067) = crosspoint of X(4) and X(6591)
X(42067) = crosssum of X(i) and X(j) for these (i,j): {3, 1332}, {69, 4561}, {100, 17776}, {345, 4571}
X(42067) = crossdifference of every pair of points on line {1332, 4561}
X(42067) = barycentric product X(i)*X(j) for these {i,j}: {4, 1015}, {6, 2969}, {11, 608}, {19, 244}, {25, 1086}, {27, 3122}, {28, 3125}, {32, 2973}, {34, 2170}, {56, 8735}, {92, 3248}, {125, 36420}, {264, 1977}, {278, 3271}, {281, 1357}, {286, 3121}, {393, 3937}, {512, 17925}, {513, 6591}, {593, 8754}, {607, 1358}, {648, 8034}, {649, 7649}, {667, 17924}, {764, 1783}, {1096, 3942}, {1111, 1973}, {1118, 7117}, {1119, 14936}, {1146, 1398}, {1365, 2189}, {1395, 4858}, {1396, 4516}, {1435, 2310}, {1436, 38362}, {1474, 3120}, {1509, 2971}, {1565, 2207}, {1647, 8752}, {1824, 16726}, {1880, 18191}, {1897, 21143}, {1974, 23989}, {2052, 22096}, {2087, 36125}, {2203, 16732}, {2333, 17205}, {2423, 39534}, {2489, 7192}, {2501, 3733}, {3669, 18344}, {3675, 8751}, {5317, 18210}, {6335, 8027}, {6545, 8750}, {7115, 7336}, {7154, 38374}, {7215, 36434}, {7250, 17926}, {7337, 26932}, {14571, 15635}, {20975, 36419}, {21132, 32674}
X(42067) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 31625}, {19, 7035}, {25, 1016}, {28, 4601}, {244, 304}, {607, 4076}, {608, 4998}, {649, 4561}, {667, 1332}, {669, 4574}, {764, 15413}, {1015, 69}, {1084, 3690}, {1086, 305}, {1111, 40364}, {1356, 2197}, {1357, 348}, {1395, 4564}, {1398, 1275}, {1474, 4600}, {1919, 1331}, {1973, 765}, {1974, 1252}, {1977, 3}, {1980, 906}, {2170, 3718}, {2189, 6064}, {2203, 4567}, {2207, 15742}, {2489, 3952}, {2501, 27808}, {2969, 76}, {2971, 594}, {2973, 1502}, {3022, 30681}, {3063, 4571}, {3120, 40071}, {3121, 72}, {3122, 306}, {3124, 3695}, {3125, 20336}, {3248, 63}, {3249, 1459}, {3271, 345}, {3733, 4563}, {3937, 3926}, {4128, 4019}, {6591, 668}, {7117, 1264}, {7649, 1978}, {8027, 905}, {8034, 525}, {8735, 3596}, {8750, 6632}, {8754, 28654}, {14936, 1265}, {17924, 6386}, {17925, 670}, {18344, 646}, {21143, 4025}, {21762, 20760}, {22096, 394}, {23560, 22149}, {23989, 40050}, {36420, 18020}, {38986, 22370}

X(42068) = X(4)X(6331)∩X(25)X(694)

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

X(42068) lies on the orthic inconic and these lines: {4, 6331}, {25, 694}, {125, 5139}, {136, 2679}, {512, 6388}, {1084, 23216}, {1112, 1843}, {1974, 32740}, {2872, 6784}, {2971, 3124}, {5167, 32269}, {6786, 41360}, {6995, 25046}, {11332, 23584}, {16240, 40325}, {34383, 36898}, {34417, 40951}, {34980, 38356}

X(42068) = isogonal conjugate of the isotomic conjugate of X(2971)
X(42068) = polar conjugate of X(44168)
X(42068) = polar conjugate of the isotomic conjugate of X(1084)
X(42068) = polar conjugate of the isogonal conjugate of X(9427)
X(42068) = orthic-isogonal conjugate of X(2489)
X(42068) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 2489}, {2971, 1084}, {3563, 2491}
X(42068) = X(9427)-cross conjugate of X(1084)
X(42068) = X(i)-isoconjugate of X(j) for these (i,j): {63, 34537}, {69, 24037}, {249, 40364}, {304, 4590}, {305, 24041}, {670, 4592}, {799, 4563}, {1101, 40050}, {3718, 7340}, {4176, 23999}, {4558, 4602}, {4561, 4623}, {4575, 4609}, {4601, 17206}, {6064, 7182}, {14208, 31614}, {23995, 40360}
X(42068) = crosspoint of X(4) and X(2489)
X(42068) = crosssum of X(3) and X(4563)
X(42068) = crossdifference of every pair of points on line {4563, 24284}
X(42068) = polar conjugate of barycentric square of X(670)
X(42068) = pole wrt polar circle of line X(670)X(888) (the tangent to the Steiner circumellipse at X(670))
X(42068) = barycentric product X(i)*X(j) for these {i,j}: {4, 1084}, {6, 2971}, {25, 3124}, {32, 8754}, {92, 4117}, {112, 22260}, {115, 1974}, {125, 36417}, {232, 15630}, {264, 9427}, {278, 7063}, {281, 1356}, {470, 41993}, {471, 41994}, {512, 2489}, {648, 23099}, {669, 2501}, {1501, 2970}, {1824, 3121}, {1924, 24006}, {1973, 2643}, {1977, 7140}, {2052, 23216}, {2086, 17980}, {2203, 21833}, {2207, 20975}, {2333, 3122}, {2422, 17994}, {2969, 7109}, {6331, 23610}, {8753, 21906}, {9426, 14618}, {27369, 34294}, {34980, 36434}
X(42068) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 34537}, {115, 40050}, {338, 40360}, {669, 4563}, {1084, 69}, {1356, 348}, {1924, 4592}, {1973, 24037}, {1974, 4590}, {2489, 670}, {2501, 4609}, {2643, 40364}, {2970, 40362}, {2971, 76}, {3124, 305}, {4117, 63}, {7063, 345}, {8754, 1502}, {9426, 4558}, {9427, 3}, {22260, 3267}, {23099, 525}, {23216, 394}, {23610, 647}, {36417, 18020}, {41993, 40709}, {41994, 40710}

X(42069) = X(3)X(7040)∩X(4)X(653)

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

X(42069) lies on the orthic inconic and these lines: {3, 7040}, {4, 653}, {11, 2969}, {25, 1857}, {29, 17947}, {33, 7140}, {118, 4605}, {125, 20620}, {158, 235}, {243, 468}, {281, 1863}, {430, 1859}, {1086, 7649}, {1118, 37197}, {1146, 3270}, {1398, 40836}, {1461, 18328}, {1824, 1856}, {1826, 1827}, {1885, 1940}, {1892, 39531}, {1984, 40616}, {2310, 8735}, {2355, 16240}, {2501, 2643}, {2968, 14010}, {3022, 4092}, {3120, 38362}, {3271, 5532}, {4081, 5514}, {5095, 40950}, {5190, 15607}, {6335, 34337}, {6555, 7046}, {9669, 31387}, {12138, 36123}, {32706, 38554}

X(42069) = polar conjugate of X(1275)
X(42069) = isogonal conjugate of the isotomic conjugate of X(21666)
X(42069) = polar conjugate of the isotomic conjugate of X(1146)
X(42069) = polar conjugate of the isogonal conjugate of X(14936)
X(42069) = orthic-isogonal conjugate of X(3064)
X(42069) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 3064}, {158, 2501}, {1857, 18344}, {7040, 650}, {21666, 1146}, {40836, 6591}
X(42069) = X(i)-cross conjugate of X(j) for these (i,j): {4516, 2310}, {14936, 1146}
X(42069) = pole wrt polar circle of trilinear polar of X(1275) (line X(100)X(658))
X(42069) = trilinear pole, wrt orthic triangle, of line X(1)X(4)
X(42069) = crosspoint of X(i) and X(j) for these (i,j): {4, 3064}, {90, 1021}, {393, 7649}, {3900, 41509}
X(42069) = crosssum of X(i) and X(j) for these (i,j): {3, 1813}, {46, 1020}, {222, 36059}, {394, 1331}, {651, 3562}, {1461, 4306}
X(42069) = crossdifference of every pair of points on line {906, 1813}
X(42069) = X(i)-isoconjugate of X(j) for these (i,j): {3, 7045}, {48, 1275}, {59, 77}, {63, 1262}, {69, 24027}, {78, 7339}, {108, 6517}, {109, 6516}, {222, 4564}, {249, 37755}, {304, 23979}, {348, 2149}, {394, 7128}, {603, 4998}, {651, 1813}, {658, 906}, {664, 36059}, {765, 7053}, {905, 4619}, {934, 1331}, {1016, 7099}, {1020, 4558}, {1092, 24032}, {1101, 6356}, {1102, 23985}, {1110, 7056}, {1252, 7177}, {1260, 24013}, {1332, 1461}, {1409, 4620}, {1410, 4600}, {1414, 23067}, {1425, 24041}, {1439, 4570}, {1802, 23586}, {1804, 7012}, {3692, 23971}, {3964, 24033}, {4554, 32660}, {4566, 4575}, {4569, 32656}, {4571, 6614}, {4574, 4637}, {4587, 4617}, {6507, 23984}, {7115, 7183}, {7138, 18020}
X(42069) = barycentric product X(i)*X(j) for these {i,j}: {4, 1146}, {6, 21666}, {8, 8735}, {11, 281}, {19, 24026}, {25, 23978}, {29, 21044}, {33, 4858}, {92, 2310}, {125, 36421}, {158, 34591}, {220, 2973}, {244, 7101}, {264, 14936}, {273, 3119}, {278, 4081}, {286, 36197}, {318, 2170}, {331, 3022}, {346, 2969}, {393, 2968}, {522, 3064}, {523, 17926}, {607, 34387}, {653, 23615}, {1021, 24006}, {1086, 7046}, {1093, 35072}, {1109, 2326}, {1111, 7079}, {1119, 23970}, {1847, 24010}, {1857, 26932}, {2052, 3270}, {2322, 3120}, {2332, 21207}, {2501, 7253}, {2638, 6521}, {2970, 7054}, {3239, 7649}, {3271, 7017}, {3900, 17924}, {4183, 16732}, {4391, 18344}, {4397, 6591}, {4516, 31623}, {5514, 40836}, {6506, 7040}, {6520, 24031}, {6524, 23983}, {6526, 40616}, {7003, 38357}, {7058, 8754}, {7071, 23989}, {7140, 26856}, {8755, 15633}, {14618, 21789}, {23104, 32674}
X(42069) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1275}, {11, 348}, {19, 7045}, {25, 1262}, {29, 4620}, {33, 4564}, {115, 6356}, {244, 7177}, {281, 4998}, {607, 59}, {608, 7339}, {650, 6516}, {652, 6517}, {657, 1331}, {663, 1813}, {1015, 7053}, {1021, 4592}, {1086, 7056}, {1096, 7128}, {1119, 23586}, {1146, 69}, {1358, 30682}, {1365, 20618}, {1398, 23971}, {1435, 24013}, {1847, 24011}, {1973, 24027}, {1974, 23979}, {2170, 77}, {2212, 2149}, {2310, 63}, {2322, 4600}, {2326, 24041}, {2332, 4570}, {2501, 4566}, {2638, 6507}, {2643, 37755}, {2968, 3926}, {2969, 279}, {3022, 219}, {3063, 36059}, {3064, 664}, {3119, 78}, {3121, 1410}, {3124, 1425}, {3125, 1439}, {3239, 4561}, {3248, 7099}, {3270, 394}, {3271, 222}, {3709, 23067}, {3900, 1332}, {4081, 345}, {4092, 26942}, {4105, 4587}, {4130, 4571}, {4183, 4567}, {4516, 1214}, {4524, 4574}, {4858, 7182}, {5532, 26932}, {6059, 7115}, {6520, 24032}, {6524, 23984}, {6591, 934}, {7004, 7183}, {7046, 1016}, {7071, 1252}, {7079, 765}, {7101, 7035}, {7117, 1804}, {7253, 4563}, {7649, 658}, {8641, 906}, {8735, 7}, {8750, 4619}, {8754, 6354}, {14936, 3}, {17924, 4569}, {17925, 4616}, {17926, 99}, {18344, 651}, {21044, 307}, {21666, 76}, {21789, 4558}, {23615, 6332}, {23970, 1265}, {23978, 305}, {23983, 4176}, {24010, 3692}, {24012, 1802}, {24026, 304}, {24031, 1102}, {26932, 7055}, {34591, 326}, {35072, 3964}, {35508, 1260}, {36197, 72}, {36421, 18020}, {38362, 14256}, {39014, 20776}, {39687, 1092}
X(42069) = {X(281),X(1863)}-harmonic conjugate of X(7071)

X(42070) = X(4)X(145)∩X(25)X(23858)

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

X(42070) lies on the orthic inconic and these lines: {4, 145}, {25, 23858}, {33, 7140}, {125, 1834}, {430, 8754}, {468, 5205}, {1830, 17660}, {1864, 3270}, {3689, 3943}, {4120, 4895}, {4370, 22371}, {4742, 38462}, {7071, 34446}, {14191, 41556}

X(42070) = isogonal conjugate of isotomic conjugate of polar conjugate of X(2226)
X(42070) = pole wrt polar circle of line X(900)X(903)
X(42070) = polar conjugate of the isotomic conjugate of X(4370)
X(42070) = polar conjugate of the isogonal conjugate of X(1017)
X(42070) = orthic-isogonal conjugate of X(8756)
X(42070) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 8756}, {32704, 14425}
X(42070) = X(1017)-cross conjugate of X(4370)
X(42070) = cevapoint of X(3251) and X(4542)
X(42070) = crosspoint of X(4) and X(8756)
X(42070) = crosssum of X(3) and X(1797)
X(42070) = crossdifference of every pair of points on line {1797, 22086} (the tangent to the MacBeath circumconic at X(1797))
X(42070) = X(i)-isoconjugate of X(j) for these (i,j): {3, 679}, {63, 2226}, {77, 1318}, {88, 1797}, {304, 41935}, {903, 36058}, {905, 4638}, {1459, 4618}, {1790, 30575}, {20568, 32659}
X(42070) = barycentric product X(i)*X(j) for these {i,j}: {4, 4370}, {19, 4738}, {25, 36791}, {44, 38462}, {92, 678}, {264, 1017}, {278, 4152}, {281, 1317}, {286, 21821}, {519, 8756}, {653, 4543}, {1824, 16729}, {1877, 2325}, {1897, 6544}, {2052, 22371}, {3251, 6335}, {3689, 37790}, {3943, 37168}, {6336, 8028}, {15742, 35092}
X(42070) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 679}, {25, 2226}, {607, 1318}, {678, 63}, {902, 1797}, {1017, 3}, {1317, 348}, {1783, 4618}, {1824, 30575}, {1974, 41935}, {2251, 36058}, {3251, 905}, {4152, 345}, {4370, 69}, {4542, 26932}, {4543, 6332}, {4738, 304}, {6544, 4025}, {8028, 3977}, {8750, 4638}, {8756, 903}, {9459, 32659}, {21821, 72}, {22371, 394}, {35092, 1565}, {36791, 305}, {38462, 20568}
X(42070) = {X(1862),X(1897)}-harmonic conjugate of X(2969)

X(42071) = X(6)X(3270)∩X(19)X(1843)

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

X(42071) lies on the orthic inconic and these lines: {6, 3270}, {19, 1843}, {25, 20468}, {31, 14935}, {125, 15904}, {184, 40141}, {518, 1861}, {692, 1974}, {926, 20455}, {1783, 5185}, {1814, 2808}, {1824, 1830}, {1826, 1827}, {1865, 2870}, {1897, 32029}, {2356, 20683}, {2807, 3751}, {2875, 8541}, {4437, 34337}, {5095, 8674}, {6184, 20776}, {13476, 21252}, {16973, 23050}, {18026, 36215}

X(42071) = reflection of X(3270) in X(6)
X(42071) = isogonal conjugate of the isotomic conjugate of X(34337)
X(42071) = polar conjugate of the isotomic conjugate of X(6184)
X(42071) = polar conjugate of the isogonal conjugate of X(39686)
X(42071) = orthic-isogonal conjugate of X(5089)
X(42071) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 5089}, {34337, 6184}
X(42071) = X(39686)-cross conjugate of X(6184)
X(42071) = X(i)-isoconjugate of X(j) for these (i,j): {63, 6185}, {105, 31637}, {304, 41934}, {673, 1814}, {927, 23696}, {2481, 36057}, {18031, 32658}
X(42071) = crosspoint of X(4) and X(5089)
X(42071) = crosssum of X(3) and X(1814)
X(42071) = crossdifference of every pair of points on line {1814, 39470}
X(42071) = barycentric product X(i)*X(j) for these {i,j}: {4, 6184}, {6, 34337}, {19, 4712}, {25, 4437}, {264, 39686}, {281, 1362}, {518, 5089}, {672, 1861}, {1783, 3126}, {1824, 16728}, {1876, 3693}, {2052, 20776}, {2340, 5236}, {2356, 3912}, {4238, 24290}, {8751, 23102}, {15149, 20683}, {15742, 35505}
X(42071) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 6185}, {672, 31637}, {1362, 348}, {1861, 18031}, {1876, 34018}, {1974, 41934}, {2223, 1814}, {2356, 673}, {3126, 15413}, {4437, 305}, {4712, 304}, {5089, 2481}, {6184, 69}, {9454, 36057}, {9455, 32658}, {15615, 7117}, {20776, 394}, {23612, 25083}, {34337, 76}, {35505, 1565}, {39014, 3270}, {39686, 3}

X(42072) = X(4)X(957)∩X(33)X(51)

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

X(42072) lies on the orthic inconic and these lines: {4, 957}, {25, 3052}, {33, 51}, {108, 3937}, {125, 429}, {184, 3195}, {208, 1425}, {225, 1828}, {1824, 1856}, {3259, 20621}, {21664, 26611}, {34980, 40952}

X(42072) = isogonal conjugate of the isotomic conjugate of X(21664)
X(42072) = polar conjugate of the isotomic conjugate of X(23980)
X(42072) = orthic-isogonal conjugate of X(14571)
X(42072) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 14571}, {108, 3310}, {21664, 23980}
X(42072) = X(i)-isoconjugate of X(j) for these (i,j): {304, 41933}, {1795, 18816}
X(42072) = crosspoint of X(4) and X(14571)
X(42072) = barycentric product X(i)*X(j) for these {i,j}: {4, 23980}, {6, 21664}, {19, 24028}, {25, 26611}, {281, 1361}, {517, 14571}, {1785, 2183}, {2427, 39534}, {3326, 7115}, {6591, 15632}, {23984, 41215}
X(42072) = barycentric quotient X(i)/X(j) for these {i,j}: {1361, 348}, {1974, 41933}, {14571, 18816}, {21664, 76}, {23980, 69}, {24028, 304}, {26611, 305}, {41215, 23983}

X(42073) = X(25)X(105)∩X(125)X(430)

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

X(42073) lies on the orthic inconic and these lines: {25, 105}, {125, 430}, {1565, 26705}, {1827, 2262}, {34980, 40954}

X(42073) = isogonal conjugate of the isotomic conjugate of X(21665)
X(42073) = polar conjugate of the isotomic conjugate of X(23972)
X(42073) = orthic-isogonal conjugate of X(1886)
X(42073) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 1886}, {21665, 23972}, {26705, 676}
X(42073) = X(i)-isoconjugate of X(j) for these (i,j): {1815, 36101}, {18025, 36056}
X(42073) = crosspoint of X(4) and X(1886)
X(42073) = crosssum of X(3) and X(1815)
X(42073) = crossdifference of every pair of points on the tangent to the MacBeath circumconic at X(1815)
X(42073) = barycentric product X(i)*X(j) for these {i,j}: {4, 23972}, {6, 21665}, {19, 24014}, {281, 1360}, {516, 1886}, {676, 41321}, {3234, 7649}
X(42073) = barycentric quotient X(i)/X(j) for these {i,j}: {1360, 348}, {1886, 18025}, {3234, 4561}, {21665, 76}, {23972, 69}, {24014, 304}

leftri

Points on the Hofstadter inellipse: X(42074)-X(42084)

rightri

These points are contributed by Peter Moses, March 20, 2021.

The Hofstadter inellipse is introduced at X(359), where it is denoted by E(1/2).

The Hofstadter inellipse is also the incentral inellipse, the trilinear square of the antiorthic axis, the X(1)-Ceva conjugate of the antiorthic axis, the barycentric product X(1)*[Steiner inellipse], the barycentric product X(9)*[incircle], and the locus of trilinear poles, wrt the incentral triangle, of lines passing through X(1). (Randy Hutson, May 31, 2021)

X(42074) = X(1)-CEVA CONJUGATE OF X(2173)

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

X(42074) lies on the Hofstadter inellipse and these lines: {1, 162}, {31, 2153}, {204, 32676}, {244, 1104}, {2308, 2310}, {2631, 14399}, {2638, 14547}, {3163, 6062}


X(42074) = isogonal conjugate of the isotomic conjugate of X(1099)
X(42074) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2173}, {162, 2631}
X(42074) = X(i)-isoconjugate of X(j) for these (i,j): {2, 40384}, {6, 31621}, {74, 1494}, {76, 40353}, {525, 34568}, {1304, 34767}, {2159, 33805}, {9139, 36890}, {14264, 40423}, {14380, 16077}, {14919, 16080}, {16076, 41433}
X(42074) = crosspoint of X(i) and X(j) for these (i,j): {1, 2173}, {1354, 3163}
X(42074) = crosssum of X(1) and X(2349)
X(42074) = crossdifference of every pair of points on line {2349, 2631}
X(42074) = trilinear square of X(2173)
X(42074) = trilinear pole, wrt incentral triangle, of line X(1)X(656)
X(42074) = barycentric product X(i)*X(j) for these {i,j}: {1, 3163}, {6, 1099}, {9, 1354}, {19, 16163}, {30, 2173}, {31, 36789}, {48, 34334}, {57, 6062}, {63, 16240}, {75, 9408}, {162, 14401}, {610, 38956}, {661, 3233}, {1495, 14206}, {1553, 36151}, {1784, 3284}, {2159, 23097}, {2349, 3081}, {2420, 36035}, {2631, 4240}, {3260, 9406}, {9409, 24001}, {24000, 39008}
X(42074) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 31621}, {30, 33805}, {31, 40384}, {560, 40353}, {1099, 76}, {1354, 85}, {1495, 2349}, {2173, 1494}, {2631, 34767}, {3081, 14206}, {3163, 75}, {3233, 799}, {6062, 312}, {9406, 74}, {9407, 2159}, {9408, 1}, {14401, 14208}, {14581, 36119}, {16163, 304}, {16240, 92}, {32676, 34568}, {34334, 1969}, {36435, 1099}, {36789, 561}, {39008, 17879}

X(42075) = X(1)-CEVA CONJUGATE OF X(1755)

Barycentrics    a^5*(a^2*b^2 - b^4 + a^2*c^2 - c^4)^2 : :
Trilinears    a^2 cos^2(A + ω) : :

X(42075) lies on the Hofstadter inellipse and these lines: {1, 1821}, {31, 1927}, {38, 2632}, {240, 1959}, {244, 8850}, {560, 563}, {1953, 1964}, {2260, 3248}, {2269, 2638}, {2309, 2310}, {7062, 11672}, {16725, 23996}

X(42075) = isogonal conjugate of the isotomic conjugate of X(23996)
X(42075) = X(1)-Ceva conjugate of X(1755)
X(42075) = X(i)-isoconjugate of X(j) for these (i,j): {2, 34536}, {76, 41932}, {98, 290}, {287, 16081}, {336, 36120}, {850, 41173}, {879, 22456}, {1976, 18024}, {14265, 40428}, {14295, 18858}, {14382, 36897}
X(42075) = crosspoint of X(i) and X(j) for these (i,j): {1, 1755}, {1355, 11672}
X(42075) = crosssum of X(1) and X(1821)
X(42075) = trilinear square of X(1755)
X(42075) = trilinear pole, wrt incentral triangle, of line X(1)X(810)
X(42075) = barycentric product X(i)*X(j) for these {i,j}: {1, 11672}, {6, 23996}, {9, 1355}, {31, 36790}, {42, 16725}, {48, 2967}, {57, 7062}, {75, 9419}, {163, 41167}, {237, 1959}, {240, 3289}, {325, 9417}, {511, 1755}, {560, 32458}, {561, 36425}, {798, 15631}, {1821, 23611}, {1910, 23098}, {3569, 23997}, {5360, 17209}, {17462, 34157}, {23995, 35088}
X(42075) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 34536}, {237, 1821}, {560, 41932}, {1355, 85}, {1755, 290}, {1959, 18024}, {2211, 36120}, {2967, 1969}, {3289, 336}, {7062, 312}, {9417, 98}, {9418, 1910}, {9419, 1}, {11672, 75}, {14966, 36036}, {15631, 4602}, {16725, 310}, {23611, 1959}, {23996, 76}, {32458, 1928}, {36425, 31}, {36790, 561}, {41167, 20948}
{X(1),X(39342)}-harmonic conjugate of X(1821)

X(42076) = X(1)-CEVA CONJUGATE OF X(2182)

Barycentrics    a*(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 : :
Trilinears    a^2 ((b + c) sec A - b sec B - c sec C)^2 : :

X(42076) lies on the Hofstadter inellipse and these lines: {1, 36100}, {31, 33}, {42, 2638}, {56, 244}, {2632, 2650}, {3248, 40958}, {24031, 36050}

X(42076) = isogonal conjugate of the isotomic conjugate of X(24034)
X(42076) = X(1)-Ceva conjugate of X(2182)
X(42076) = X(102)-isoconjugate of X(34393)
X(42076) = crosspoint of X(i) and X(j) for these (i,j): {1, 2182}, {1359, 23986}
X(42076) = crosssum of X(1) and X(36100)
X(42076) = trilinear square of X(2182)
X(42076) = trilinear pole, wrt incentral triangle, of line X(1)X(521)
X(42076) = barycentric product X(i)*X(j) for these {i,j}: {1, 23986}, {6, 24034}, {9, 1359}, {19, 38554}, {515, 2182}, {2425, 14304}
X(42076) = barycentric quotient X(i)/X(j) for these {i,j}: {1359, 85}, {2182, 34393}, {23986, 75}, {24034, 76}, {38554, 304}

X(42077) = X(1)-CEVA CONJUGATE OF X(910)

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

X(42077) lies on the Hofstadter inellipse and these lines: {1, 36101}, {6, 2310}, {31, 57}, {42, 24012}, {678, 14392}, {756, 3195}, {1253, 2331}, {1783, 24010}, {1962, 2632}, {2293, 2638}, {2643, 40977}, {3248, 20978}, {4617, 34033}, {16469, 24644}

X(42077) = isogonal conjugate of the isotomic conjugate of X(24014)
X(42077) = X(1)-Ceva conjugate of X(910)
X(42077) = X(i)-isoconjugate of X(j) for these (i,j): {103, 18025}, {677, 2400}
X(42077) = crosspoint of X(i) and X(j) for these (i,j): {1, 910}, {1360, 23972}
X(42077) = crosssum of X(1) and X(36101)
X(42077) = trilinear square of X(910)
X(42077) = trilinear pole, wrt incentral triangle, of line X(1)X(905)
X(42077) = barycentric product X(i)*X(j) for these {i,j}: {1, 23972}, {6, 24014}, {9, 1360}, {48, 21665}, {513, 3234}, {516, 910}, {1456, 40869}
X(42077) = barycentric quotient X(i)/X(j) for these {i,j}: {910, 18025}, {1360, 85}, {3234, 668}, {21665, 1969}, {23972, 75}, {24014, 76}

X(42078) = X(1)-CEVA CONJUGATE OF X(2183)

Barycentrics    a^3*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)^2 : :
Trilinears    ((a - c) cos B + (a - b) cos C)^2 : :

X(42078) lies on the Hofstadter inellipse and these lines: {1, 34234}, {31, 692}, {42, 1864}, {55, 2638}, {65, 244}, {221, 7138}, {872, 34857}, {2177, 24012}, {2292, 2632}, {2643, 3725}, {3878, 24025}, {4088, 23757}, {14571, 21801}

X(42078) = isogonal conjugate of the isotomic conjugate of X(24028)
X(42078) = X(1)-Ceva conjugate of X(2183)
X(42078) = X(i)-isoconjugate of X(j) for these (i,j): {76, 41933}, {104, 18816}, {2401, 13136}, {34051, 36795}
X(42078) = crosspoint of X(i) and X(j) for these (i,j): {1, 2183}, {1361, 23980}
X(42078) = crosssum of X(1) and X(34234)
X(42078) = crossdifference of every pair of points on line {3762, 24618}
X(42078) = trilinear square of X(2183)
X(42078) = trilinear pole, wrt incentral triangle, of line X(1)X(522)
X(42078) = barycentric product X(i)*X(j) for these {i,j}: {1, 23980}, {6, 24028}, {9, 1361}, {31, 26611}, {48, 21664}, {517, 2183}, {649, 15632}, {859, 21801}, {909, 23101}, {1769, 2427}, {2149, 3326}, {7128, 41215}, {14571, 22350}
X(42078) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 41933}, {1361, 85}, {2183, 18816}, {15632, 1978}, {21664, 1969}, {23980, 75}, {24028, 76}, {26611, 561}

X(42079) = X(1)-CEVA CONJUGATE OF X(672)

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

X(42079) lies on the Hofstadter inellipse and these lines: {1, 673}, {6, 292}, {37, 2293}, {42, 244}, {48, 692}, {55, 20995}, {75, 39775}, {101, 2195}, {200, 14829}, {241, 518}, {560, 21059}, {664, 35961}, {665, 926}, {756, 38358}, {900, 20681}, {922, 19624}, {984, 991}, {1026, 17755}, {1088, 36905}, {1279, 3009}, {1962, 14746}, {2223, 9454}, {2294, 2643}, {3010, 6603}, {3725, 4117}, {4712, 16728}, {21805, 38980}, {23612, 39686}, {33701, 39044}

X(42079) = reflection of X(2310) in X(37)
X(42079) = isogonal conjugate of the isotomic conjugate of X(4712)
X(42079) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 672}, {3252, 39258}
X(42079) = X(i)-isoconjugate of X(j) for these (i,j): {2, 6185}, {76, 41934}, {105, 2481}, {294, 34018}, {885, 927}, {1024, 34085}, {1438, 18031}, {1462, 36796}, {31637, 36124}
X(42079) = crosspoint of X(i) and X(j) for these (i,j): {1, 672}, {1362, 6184}, {2223, 40730}, {9436, 17758}
X(42079) = crosssum of X(i) and X(j) for these (i,j): {1, 673}, {2195, 4251}
X(42079) = crossdifference of every pair of points on line {673, 812}
X(42079) = Hofstadter-inellipse antipode of X(2310)
X(42079) = trilinear square of X(672)
X(42079) = trilinear pole, wrt incentral triangle, of line X(1)X(514)
X(42079) = barycentric product X(i)*X(j) for these {i,j}: {1, 6184}, {6, 4712}, {9, 1362}, {31, 4437}, {42, 16728}, {48, 34337}, {75, 39686}, {92, 20776}, {101, 3126}, {241, 2340}, {518, 672}, {665, 1026}, {673, 23612}, {765, 35505}, {926, 1025}, {1110, 35094}, {1438, 23102}, {1458, 3693}, {1818, 5089}, {1861, 20752}, {2223, 3912}, {2254, 2284}, {2356, 25083}, {3252, 8299}, {3263, 9454}, {3286, 3930}, {7084, 17060}, {14439, 34230}, {17464, 34159}, {17755, 40730}, {18206, 20683}, {30941, 39258}, {33570, 37138}, {33700, 39341}
X(42079) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 6185}, {518, 18031}, {560, 41934}, {672, 2481}, {1026, 36803}, {1362, 85}, {1458, 34018}, {2223, 673}, {2283, 34085}, {2340, 36796}, {3126, 3261}, {4437, 561}, {4712, 76}, {6184, 75}, {8638, 1024}, {9454, 105}, {9455, 1438}, {15615, 2170}, {16728, 310}, {20752, 31637}, {20776, 63}, {23612, 3912}, {34337, 1969}, {35505, 1111}, {39014, 2310}, {39258, 13576}, {39686, 1}
X(42079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 39341, 673}, {673, 37138, 39341}

X(42080) = X(1)-CEVA CONJUGATE OF X(823)

Barycentrics    a^5*(b - c)^2*(b + c)^2*(a^2 - b^2 - c^2)^4 : :
Trilinears    1/(sec^2 B - sec^2 C)^2 : :
Trilinears    1/(csc B/2 - csc C/2)^2 : :
Trilinears    1/(csc 2B - csc 2C)^2 : :

X(42080) lies on the Hofstadter inellipse and these lines: {1, 823}, {73, 2660}, {244, 38985}, {836, 4094}, {2638, 38344}, {7065, 35071}

X(42080) = isogonal conjugate of the polar conjugate of X(37754)
X(42080) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 822}, {1248, 652}
X(42080) = X(i)-isoconjugate of X(j) for these (i,j): {2, 34538}, {75, 24021}, {76, 23590}, {107, 6528}, {158, 23999}, {264, 32230}, {561, 24022}, {648, 15352}, {811, 36126}, {1093, 18020}, {1502, 23975}, {2052, 23582}, {6331, 6529}, {18027, 23964}, {34537, 36434}
X(42080) = crosspoint of X(i) and X(j) for these (i,j): {1, 822}, {1363, 35071}
X(42080) = crosssum of X(i) and X(j) for these (i,j): {1, 823}, {6521, 36126}
X(42080) = trilinear square of X(823)
X(42080) = trilinear pole, wrt incentral triangle, of line X(1)X(29)
X(42080) = barycentric product X(i)*X(j) for these {i,j}: {1, 35071}, {3, 37754}, {9, 1363}, {32, 24020}, {42, 16730}, {48, 2972}, {57, 7065}, {63, 34980}, {125, 4100}, {255, 3269}, {520, 822}, {560, 23974}, {577, 2632}, {656, 32320}, {823, 23613}, {1092, 3708}, {2167, 41219}, {6507, 20975}, {7138, 35072}, {14585, 17879}, {20902, 23606}, {23103, 24019}, {23994, 36433}, {24018, 39201}
X(42080) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 34538}, {32, 24021}, {560, 23590}, {577, 23999}, {810, 15352}, {822, 6528}, {1363, 85}, {1501, 24022}, {1917, 23975}, {2632, 18027}, {2972, 1969}, {3049, 36126}, {4100, 18020}, {4117, 36434}, {7065, 312}, {9247, 32230}, {14585, 24000}, {16730, 310}, {20975, 6521}, {23613, 24018}, {23974, 1928}, {24020, 1502}, {32320, 811}, {34980, 92}, {35071, 75}, {36433, 1101}, {37754, 264}, {39201, 823}, {41219, 14213}

X(42081) = X(1)-CEVA CONJUGATE OF X(897)

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

X(42081) lies on the Hofstadter inellipse and these lines: {1, 662}, {44, 39256}, {48, 2157}, {214, 238}, {244, 1100}, {501, 6042}, {560, 4575}, {678, 9508}, {896, 922}, {1193, 3248}, {1964, 4117}, {2173, 17462}, {2310, 2646}, {2482, 7067}, {2642, 14419}, {4094, 38348}, {5550, 26081}, {16733, 24038}

X(42081) = isogonal conjugate of the isotomic conjugate of X(24038)
X(42081) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 896}, {662, 2642}, {24041, 23889}
X(42081) = X(i)-isoconjugate of X(j) for these (i,j): {2, 10630}, {4, 15398}, {76, 41936}, {111, 671}, {115, 34539}, {523, 34574}, {691, 5466}, {892, 9178}, {895, 17983}, {5968, 9154}, {8753, 30786}, {9139, 9214}, {9979, 39413}, {10415, 14246}, {18023, 32740}, {23894, 36085}
X(42081) = crosspoint of X(i) and X(j) for these (i,j): {1, 896}, {1366, 2482}, {23889, 24041}
X(42081) = crosssum of X(i) and X(j) for these (i,j): {1, 897}, {2643, 23894}
X(42081) = crossdifference of every pair of points on line {897, 2642}
X(42081) = trilinear square of X(897)
X(42081) = trilinear pole, wrt incentral triangle, of line X(1)X(661)
X(42081) = barycentric product X(i)*X(j) for these {i,j}: {1, 2482}, {6, 24038}, {9, 1366}, {31, 36792}, {42, 16733}, {48, 34336}, {57, 7067}, {63, 5095}, {75, 39689}, {187, 14210}, {351, 24039}, {524, 896}, {662, 1649}, {690, 23889}, {897, 8030}, {922, 3266}, {923, 23106}, {2642, 5468}, {4062, 16702}, {6629, 21839}, {17466, 34161}, {20380, 36263}, {23992, 24041}, {33915, 36085}
X(42081) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 10630}, {48, 15398}, {163, 34574}, {187, 897}, {351, 23894}, {560, 41936}, {896, 671}, {922, 111}, {1101, 34539}, {1366, 85}, {1649, 1577}, {2482, 75}, {2642, 5466}, {5095, 92}, {5467, 36085}, {7067, 312}, {8030, 14210}, {14210, 18023}, {14567, 923}, {16733, 310}, {23200, 36060}, {23889, 892}, {23992, 1109}, {24038, 76}, {34336, 1969}, {36792, 561}, {39689, 1}
X(42081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 39339, 897}, {662, 897, 39339}, {662, 17467, 2643}

X(42082) = X(1)-CEVA CONJUGATE OF X(1155)

Barycentrics    a*(2*a^2 - b^2 + 2*b*c - c^2 - a*(b + c))^2 : :
Trilinears    (2 cos A - cos B - cos C)^2 : :

X(42082) lies on the Hofstadter inellipse and these lines: {1, 651}, {6, 244}, {44, 9502}, {73, 2638}, {106, 16469}, {109, 1253}, {580, 1106}, {678, 2254}, {1155, 6610}, {1201, 3248}, {1254, 3157}, {1458, 5126}, {2293, 24012}, {2632, 18675}, {2643, 2650}, {3245, 6126}, {3945, 17719}, {4349, 24222}, {4585, 4712}, {6068, 35110}, {6594, 35293}, {15346, 28125}, {24980, 41801}

X(42082) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 1155}, {7045, 23890}
X(42082) = crosspoint of X(i) and X(j) for these (i,j): {1, 1155}, {3321, 35110}, {7045, 23890}
X(42082) = crosssum of X(i) and X(j) for these (i,j): {1, 1156}, {2310, 23893}
X(42082) = crossdifference of every pair of points on line {1156, 3887}
X(42082) = trilinear square of X(1155)
X(42082) = trilinear pole, wrt incentral triangle, of line X(1)X(650)
X(42082) = X(i)-isoconjugate of X(j) for these (i,j): {1121, 2291}, {23351, 35157}, {23893, 37139}, {34056, 41798}
X(42082) = barycentric product X(i)*X(j) for these {i,j}: {1, 35110}, {9, 3321}, {57, 6068}, {527, 1155}, {1055, 30806}, {1323, 6603}, {3328, 4564}, {5528, 15729}, {6366, 23890}, {6510, 23710}, {6610, 6745}, {7045, 35091}
X(42082) = barycentric quotient X(i)/X(j) for these {i,j}: {1055, 1156}, {1155, 1121}, {3321, 85}, {3328, 4858}, {6068, 312}, {6139, 23893}, {23346, 37139}, {23890, 35157}, {35091, 24026}, {35110, 75}

X(42083) = X(1)-CEVA CONJUGATE OF X(899)

Barycentrics    a*(a*b + a*c - 2*b*c)^2 : :
X(42083) = 3 X[4664] - X[41683], 5 X[4704] - X[17154]

X(42083) lies on the inellipse centered at X(24003) (the trilinear square of the Nagel line).

X(42083) lies on the Hofstadter inellipse and these lines: {1, 190}, {37, 244}, {44, 17475}, {75, 24003}, {88, 24419}, {192, 872}, {292, 16672}, {518, 17460}, {536, 899}, {644, 36267}, {659, 678}, {726, 34587}, {740, 4738}, {891, 3768}, {900, 20681}, {984, 2802}, {1964, 17262}, {2234, 39916}, {2292, 2643}, {2310, 3057}, {2632, 18674}, {2667, 4117}, {3123, 40521}, {3799, 24338}, {4088, 23757}, {4094, 4145}, {4319, 24012}, {4370, 27846}, {4704, 17154}, {5550, 26076}, {6533, 25079}, {8683, 34247}, {16676, 24578}, {17160, 39044}, {17261, 17445}, {17464, 21801}, {17487, 24722}, {17793, 24004}, {19582, 25268}

X(42083) = midpoint of X(192) and X(3952)
X(42083) = reflection of X(i) in X(j) for these {i,j}: {75, 24003}, {244, 37}, {2667, 14752}
X(42083) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 899}, {190, 3768}, {7035, 23891}
X(42083) = X(i)-isoconjugate of X(j) for these (i,j): {739, 3227}, {889, 23349}, {4607, 23892}
X(42083) = crosspoint of X(i) and X(j) for these (i,j): {1, 899}, {7035, 23891}
X(42083) = crosssum of X(i) and X(j) for these (i,j): {1, 37129}, {3248, 23892}
X(42083) = crossdifference of every pair of points on line {3768, 23892}
X(42083) = antipode of X(75) in the inellipse centered at X(24003)
X(42083) = antipode of X(244) in the Hofstadter inellipse
X(42083) = trilinear square of X(899)
X(42083) = trilinear pole, wrt incentral triangle, of line X(1)X(649)
X(42083) = barycentric product X(i)*X(j) for these {i,j}: {1, 13466}, {190, 14434}, {536, 899}, {891, 23891}, {3230, 6381}, {3768, 41314}, {4728, 23343}, {7035, 39011}, {8031, 37129}
X(42083) = barycentric quotient X(i)/X(j) for these {i,j}: {536, 31002}, {890, 23892}, {899, 3227}, {3230, 37129}, {8031, 6381}, {13466, 75}, {14434, 514}, {14441, 21143}, {23343, 4607}, {23891, 889}, {39011, 244}

X(42084) = X(1)-CEVA CONJUGATE OF X(1635)

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

X(42084) lies on the Hofstadter inellipse and these lines: {1, 3257}, {31, 40172}, {44, 678}, {214, 238}, {244, 513}, {512, 2643}, {518, 17460}, {663, 3248}, {679, 9325}, {765, 9282}, {984, 24482}, {2087, 3251}, {2310, 4162}, {3122, 4983}, {4542, 33922}, {7208, 28886}, {7290, 36267}, {16507, 38348}, {17145, 20072}, {19945, 38989}

X(42084) = midpoint of X(17145) and X(20072)
X(42084) = reflection of X(21805) in X(44)
X(42084) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 1635}, {244, 2087}, {678, 3251}, {4738, 6544}, {9282, 44}, {9325, 513}, {39771, 14442}, {40172, 1960}
X(42084) = crosspoint of X(i) and X(j) for these (i,j): {1, 1635}, {44, 513}, {244, 2087}, {678, 3251}, {4738, 6544}, {14027, 35092}
X(42084) = crosssum of X(i) and X(j) for these (i,j): {1, 3257}, {88, 100}, {679, 4618}, {765, 5376}
X(42084) = crossdifference of every pair of points on line {1022, 1023}
X(42084) = trilinear square of X(1635)
X(42084) = trilinear pole, wrt incentral triangle, of line X(1)X(88)
X(42084) = X(i)-isoconjugate of X(j) for these (i,j): {88, 5376}, {100, 4618}, {190, 4638}, {679, 765}, {901, 4555}, {903, 9268}, {1016, 2226}, {1318, 4998}, {4567, 30575}, {6548, 6551}, {6635, 23345}, {17780, 39414}, {31625, 41935}
X(42084) = barycentric product X(i)*X(j) for these {i,j}: {1, 35092}, {9, 14027}, {44, 1647}, {57, 4542}, {100, 14442}, {244, 4370}, {513, 6544}, {514, 3251}, {519, 2087}, {650, 39771}, {678, 1086}, {900, 1635}, {1015, 4738}, {1017, 1111}, {1022, 33922}, {1023, 6550}, {1317, 2170}, {1319, 4530}, {1960, 3762}, {3122, 16729}, {3248, 36791}, {3669, 4543}, {3942, 42070}, {4895, 30725}, {8661, 24004}, {17205, 21821}
X(42084) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 4618}, {667, 4638}, {678, 1016}, {902, 5376}, {1015, 679}, {1017, 765}, {1023, 6635}, {1635, 4555}, {1647, 20568}, {1960, 3257}, {2087, 903}, {2251, 9268}, {3122, 30575}, {3248, 2226}, {3251, 190}, {4370, 7035}, {4542, 312}, {4543, 646}, {4738, 31625}, {4895, 4582}, {6544, 668}, {8661, 1022}, {14027, 85}, {14442, 693}, {14637, 1635}, {14835, 4370}, {33922, 24004}, {35092, 75}, {39771, 4554}
X(42084) = {X(1),X(39343)}-harmonic conjugate of X(3257)

leftri

Gibert (i,j,k) points: X(42085)-X(42284)

rightri

This preamble is contributed by Peter Moses, March 22, 2021.

Bernard Gibert has noted a set of points in connecton with the cubic K1191. These points are given by the combo

SW*X[6]*i + Sqrt[3]*S*(j*X[4] + k*X[3]) where (i,j,k) are constants or other 0-degree functions of a,b,c.

See K1191 .

X(42085) = GIBERT(1,-1,1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 6*SB*SC : :

X(42085) is the intersection of the tangents to conic {{X(4),X(13),X(14),X(15),X(16)}} at X(13) and X(16). (Randy Hutson, May 31, 2021)

X(42085) lies on these lines: {2, 10645}, {3, 5321}, {4, 15}, {5, 11480}, {6, 30}, {13, 3543}, {14, 376}, {16, 20}, {18, 3522}, {61, 3146}, {62, 3529}, {69, 531}, {141, 11295}, {146, 10657}, {193, 530}, {302, 22491}, {381, 23302}, {382, 5318}, {393, 6110}, {395, 3534}, {396, 3830}, {397, 5073}, {398, 1657}, {532, 11008}, {533, 20080}, {546, 36836}, {550, 5339}, {616, 6777}, {617, 7898}, {619, 37171}, {621, 34540}, {622, 40901}, {623, 37172}, {631, 16967}, {1249, 6111}, {1250, 4302}, {1478, 10638}, {1479, 7051}, {1656, 5349}, {2041, 35820}, {2042, 35821}, {2043, 6396}, {2044, 6200}, {3090, 5352}, {3091, 5238}, {3098, 22512}, {3180, 19569}, {3389, 35732}, {3523, 5365}, {3524, 37835}, {3545, 16241}, {3589, 11296}, {3619, 3642}, {3620, 3643}, {3627, 11542}, {3767, 19781}, {3839, 37832}, {3843, 16772}, {3845, 16644}, {4299, 19373}, {5059, 41973}, {5237, 16961}, {5357, 10483}, {5362, 11114}, {5367, 17579}, {5473, 6782}, {5474, 6114}, {5878, 10675}, {6108, 37689}, {6115, 36772}, {6411, 34551}, {6412, 34552}, {6564, 36455}, {6565, 36437}, {6770, 23005}, {6783, 36962}, {7519, 37776}, {7735, 41409}, {8588, 25164}, {8703, 16645}, {9736, 16002}, {9833, 10676}, {10304, 16242}, {10633, 35471}, {10642, 18533}, {10643, 18537}, {10658, 12383}, {10662, 12118}, {10678, 12254}, {11001, 34755}, {11297, 34573}, {11409, 37196}, {12103, 36843}, {12816, 33604}, {14538, 33518}, {15640, 41107}, {15682, 36969}, {15696, 16773}, {15704, 22238}, {16063, 37775}, {16635, 18424}, {16802, 16804}, {16960, 17578}, {16965, 33703}, {19708, 41122}, {22531, 22856}, {22843, 31706}, {23013, 39874}, {23334, 36775}, {32785, 36445}, {32786, 36463}, {33517, 36994}, {35695, 36330}, {41035, 41038}

X(42085) = reflection of X(42086) in X(6)

X(42086) = GIBERT(1,1,-1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 6*SB*SC : :

X(42086) is the intersection of the tangents to conic {{X(4),X(13),X(14),X(15),X(16)}} at X(14) and X(15). (Randy Hutson, May 31, 2021)

X(42086) lies on these lines: {2, 10646}, {3, 5318}, {4, 16}, {5, 11481}, {6, 30}, {13, 376}, {14, 3543}, {15, 20}, {17, 3522}, {61, 3529}, {62, 3146}, {69, 530}, {141, 11296}, {146, 10658}, {193, 531}, {303, 22492}, {381, 23303}, {382, 5321}, {393, 6111}, {395, 3830}, {396, 3534}, {397, 1657}, {398, 5073}, {532, 20080}, {533, 11008}, {546, 36843}, {550, 5340}, {616, 7898}, {617, 6778}, {618, 37170}, {621, 40900}, {622, 34541}, {624, 37173}, {631, 16966}, {1249, 6110}, {1250, 1478}, {1479, 19373}, {1656, 5350}, {2041, 35821}, {2042, 35820}, {2043, 6200}, {2044, 6396}, {3090, 5351}, {3091, 5237}, {3098, 22513}, {3181, 19569}, {3365, 35732}, {3523, 5366}, {3524, 37832}, {3545, 16242}, {3589, 11295}, {3619, 3643}, {3620, 3642}, {3627, 11543}, {3767, 19780}, {3839, 37835}, {3843, 16773}, {3845, 16645}, {4299, 7051}, {4302, 10638}, {5059, 41974}, {5238, 16960}, {5353, 10483}, {5362, 17579}, {5367, 11114}, {5473, 6115}, {5474, 6783}, {5878, 10676}, {6109, 37689}, {6114, 36962}, {6411, 34552}, {6412, 34551}, {6564, 36437}, {6565, 36455}, {6773, 23004}, {6782, 36961}, {7519, 37775}, {7735, 41408}, {8588, 25154}, {8703, 16644}, {9735, 16001}, {9833, 10675}, {10304, 16241}, {10632, 35471}, {10641, 18533}, {10644, 18537}, {10657, 12383}, {10661, 12118}, {10677, 12254}, {11001, 34754}, {11298, 34573}, {11408, 37196}, {12103, 36836}, {12817, 33605}, {14136, 36772}, {14539, 33517}, {15640, 41108}, {15682, 36970}, {15696, 16772}, {15704, 22236}, {16063, 37776}, {16634, 18424}, {16803, 16805}, {16961, 17578}, {16964, 33703}, {19708, 41121}, {22532, 22900}, {22890, 31705}, {23006, 39874}, {23249, 35731}, {32785, 36463}, {32786, 36445}, {33518, 36992}, {35691, 35752}, {41034, 41039}

X(42086) = reflection of X(42085) in X(6)

X(42087) = GIBERT(1,-1,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA - 6*SB*SC : :

X(42087) lies on these lines: {3, 5321}, {4, 11480}, {5, 10645}, {6, 20}, {13, 15}, {14, 8703}, {16, 398}, {18, 33923}, {61, 15704}, {62, 12103}, {140, 5349}, {376, 395}, {382, 16772}, {397, 1657}, {531, 15300}, {546, 5352}, {548, 10646}, {549, 16967}, {621, 35931}, {623, 35304}, {1250, 15338}, {1503, 36993}, {1885, 10641}, {2883, 30402}, {3146, 11488}, {3479, 35725}, {3522, 5339}, {3529, 5335}, {3530, 33416}, {3534, 10654}, {3543, 16644}, {3575, 11475}, {3627, 5238}, {3845, 16241}, {4316, 5357}, {4324, 5353}, {5059, 5340}, {5073, 5350}, {5254, 19781}, {5305, 41409}, {5343, 21735}, {5351, 16961}, {5362, 15680}, {5365, 10299}, {5367, 37256}, {5474, 22512}, {5895, 17826}, {6284, 7051}, {6671, 31693}, {7354, 10638}, {8175, 40668}, {8550, 36995}, {10187, 15712}, {10295, 10633}, {10304, 16645}, {10632, 18560}, {10653, 15681}, {10658, 34153}, {10675, 15311}, {10676, 34782}, {11515, 31829}, {11812, 12817}, {12100, 37835}, {15326, 19373}, {15683, 37640}, {15686, 36968}, {15687, 37832}, {15690, 41108}, {15695, 41113}, {15696, 40694}, {15759, 41122}, {15768, 30465}, {16242, 34200}, {16965, 34754}, {17538, 22238}, {17800, 40693}, {19708, 33603}, {19710, 41101}, {33518, 36755}, {35404, 41943}, {37776, 37900}, {41035, 41056}, {41977, 41981}

X(42087) = {X(6),X(20)}-harmonic conjugate of X(42088)

X(42088) = GIBERT(1,1,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA + 6*SB*SC : :

X(42088) lies on these lines: {3, 5318}, {4, 11481}, {5, 10646}, {6, 20}, {13, 8703}, {14, 16}, {15, 397}, {17, 33923}, {61, 12103}, {62, 15704}, {140, 5350}, {376, 396}, {382, 16773}, {398, 1657}, {530, 15300}, {546, 5351}, {548, 10645}, {549, 16966}, {622, 35932}, {624, 35303}, {1250, 7354}, {1503, 36995}, {1885, 10642}, {2883, 30403}, {3146, 11489}, {3480, 35726}, {3522, 5340}, {3529, 5334}, {3530, 33417}, {3534, 10653}, {3543, 16645}, {3575, 11476}, {3627, 5237}, {3845, 16242}, {4316, 5353}, {4324, 5357}, {5059, 5339}, {5073, 5349}, {5254, 19780}, {5305, 41408}, {5344, 21735}, {5352, 16960}, {5362, 37256}, {5366, 10299}, {5367, 15680}, {5473, 22513}, {5895, 17827}, {6284, 19373}, {6672, 31694}, {7051, 15326}, {8174, 40667}, {8550, 36993}, {10188, 15712}, {10295, 10632}, {10304, 16644}, {10633, 18560}, {10638, 15338}, {10654, 15681}, {10657, 34153}, {10675, 34782}, {10676, 15311}, {11516, 31829}, {11812, 12816}, {12100, 37832}, {15683, 37641}, {15686, 36967}, {15687, 37835}, {15690, 41107}, {15695, 41112}, {15696, 40693}, {15759, 41121}, {15769, 30468}, {16241, 34200}, {16964, 34755}, {17538, 22236}, {17800, 40694}, {19708, 33602}, {19710, 41100}, {33517, 36756}, {35404, 41944}, {37775, 37900}, {41034, 41057}, {41978, 41981}

X(42088) = {X(6),X(20)}-harmonic conjugate of X(42087)

X(42089) = GIBERT(-1,1,3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA - 6*SB*SC : :

X(42089) lies on these lines: {2, 13}, {3, 5321}, {4, 10187}, {5, 11481}, {6, 140}, {14, 3524}, {15, 631}, {17, 3533}, {18, 3523}, {20, 16809}, {61, 10303}, {62, 3525}, {376, 19107}, {395, 5054}, {396, 15694}, {398, 15720}, {498, 19373}, {499, 1250}, {549, 10654}, {574, 40921}, {623, 37173}, {627, 34541}, {628, 40900}, {629, 22907}, {632, 11542}, {1656, 5318}, {2045, 6200}, {2046, 6396}, {2548, 19780}, {3090, 5237}, {3091, 5351}, {3147, 10641}, {3526, 11486}, {3541, 10642}, {3542, 11476}, {3545, 36968}, {3546, 11516}, {3548, 10635}, {3618, 6671}, {3628, 36843}, {5024, 40922}, {5067, 16965}, {5071, 36969}, {5339, 15712}, {5362, 17566}, {6114, 21157}, {6640, 18470}, {6674, 22862}, {6782, 21156}, {7404, 10644}, {8252, 34552}, {8253, 34551}, {10304, 36970}, {10633, 37119}, {11268, 18281}, {11539, 16644}, {12100, 41120}, {12108, 36836}, {14216, 30403}, {14869, 22236}, {15692, 36967}, {15693, 41113}, {15698, 33603}, {15702, 16241}, {15708, 16268}, {15709, 16963}, {15717, 16964}, {15719, 41108}, {17827, 40686}, {19781, 21843}, {21158, 33518}, {33884, 36981}

X(42089) = {X(3),X(5321)}-harmonic conjugate of X(42090)
X(42089) = {X(4),X(10646)}-harmonic conjugate of X(42091)
X(42089) = {X(6),X(140)}-harmonic conjugate of X(42092)

X(42090) = GIBERT(1,-1,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA - 6*SB*SC : :

X(42090) lies on these lines: {2, 12821}, {3, 5321}, {4, 10188}, {6, 550}, {13, 11001}, {14, 10304}, {15, 20}, {16, 376}, {17, 5059}, {18, 21735}, {30, 11480}, {61, 17538}, {382, 23302}, {395, 15688}, {396, 15681}, {548, 11481}, {631, 16809}, {1657, 5318}, {2041, 35786}, {2042, 35787}, {2549, 19781}, {3091, 33417}, {3146, 5352}, {3522, 5334}, {3523, 16967}, {3524, 33416}, {3528, 11489}, {3529, 5238}, {3534, 10653}, {3543, 16241}, {4299, 10638}, {4302, 7051}, {5286, 41409}, {5339, 33923}, {5349, 15720}, {5878, 30402}, {5925, 17826}, {8703, 11543}, {10632, 35481}, {10633, 35503}, {10657, 12244}, {10675, 20427}, {11267, 34350}, {11475, 18533}, {11486, 15696}, {11542, 15704}, {12103, 22236}, {12817, 15719}, {15682, 37832}, {15683, 36969}, {15685, 41119}, {15692, 37835}, {15697, 41101}, {16242, 19708}, {16645, 34200}, {16772, 17800}, {19710, 41112}, {22843, 22861}

X(42090) = {X(3),X(5321)}-harmonic conjugate of X(42089)
X(42090) = {X(4),X(10645)}-harmonic conjugate of X(42092)
X(42090) = {X(6),X(550)}-harmonic conjugate of X(42091)

X(42091) = GIBERT(1,1,-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA + 6*SB*SC : :

X(42091) lies on these lines: {2, 12820}, {3, 5318}, {4, 10187}, {6, 550}, {13, 10304}, {14, 11001}, {15, 376}, {16, 20}, {17, 21735}, {18, 5059}, {30, 11481}, {62, 17538}, {382, 23303}, {395, 15681}, {396, 15688}, {548, 11480}, {631, 16808}, {1250, 4299}, {1657, 5321}, {2041, 35787}, {2042, 35786}, {2549, 19780}, {3091, 33416}, {3146, 5351}, {3522, 5335}, {3523, 16966}, {3524, 33417}, {3528, 11488}, {3529, 5237}, {3534, 10654}, {3543, 16242}, {4302, 19373}, {5286, 41408}, {5340, 33923}, {5350, 15720}, {5878, 30403}, {5925, 17827}, {8703, 11542}, {10632, 35503}, {10633, 35481}, {10658, 12244}, {10676, 20427}, {11268, 34350}, {11476, 18533}, {11485, 15696}, {11543, 15704}, {12103, 22238}, {12816, 15719}, {13939, 35739}, {15682, 37835}, {15683, 36970}, {15685, 41120}, {15692, 37832}, {15697, 41100}, {16241, 19708}, {16644, 34200}, {16773, 17800}, {19710, 41113}, {22890, 22907}

X(42091) = {X(3),X(5318)}-harmonic conjugate of X(42092)
X(42091) = {X(4),X(10646)}-harmonic conjugate of X(42089)
X(42091) = {X(6),X(550)}-harmonic conjugate of X(42090)

X(42092) = GIBERT(1,1,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA + 6*SB*SC : :

X(42092) lies on these lines: {2, 14}, {3, 5318}, {4, 10188}, {5, 11480}, {6, 140}, {13, 3524}, {16, 631}, {17, 3523}, {18, 3533}, {20, 16808}, {61, 3525}, {62, 10303}, {376, 19106}, {395, 15694}, {396, 5054}, {397, 15720}, {498, 7051}, {499, 10638}, {549, 10653}, {574, 40922}, {624, 37172}, {627, 40901}, {628, 34540}, {630, 22861}, {632, 11543}, {1656, 5321}, {2045, 6396}, {2046, 6200}, {2548, 19781}, {3090, 5238}, {3091, 5352}, {3147, 10642}, {3526, 11485}, {3541, 10641}, {3542, 11475}, {3545, 36967}, {3546, 11515}, {3548, 10634}, {3618, 6672}, {3628, 36836}, {5024, 40921}, {5067, 16964}, {5071, 36970}, {5340, 15712}, {5367, 17566}, {6108, 36764}, {6115, 21156}, {6640, 18468}, {6673, 22906}, {6770, 36766}, {6782, 36770}, {6783, 21157}, {7404, 10643}, {8252, 34551}, {8253, 34552}, {10304, 36969}, {10632, 37119}, {11267, 18281}, {11539, 16645}, {12100, 41119}, {12108, 36843}, {14216, 30402}, {14869, 22238}, {15692, 36968}, {15693, 41112}, {15698, 33602}, {15702, 16242}, {15708, 16267}, {15709, 16962}, {15717, 16965}, {15719, 41107}, {17826, 40686}, {19780, 21843}, {21159, 33517}, {33884, 36979}

X(42092) = {X(3),X(5318)}-harmonic conjugate of X(42091)
X(42092) = {X(4),X(10645)}-harmonic conjugate of X(42090)
X(42092) = {X(6),X(140)}-harmonic conjugate of X(42089)

X(42093) = GIBERT(-1,2,0) POINT

Barycentrics    Sqrt[3]*a^2*S - 12*SB*SC : :

X(42093) lies on these lines: {3, 16809}, {4, 6}, {5, 11480}, {13, 12821}, {14, 3830}, {15, 381}, {16, 382}, {18, 5073}, {20, 23303}, {30, 11481}, {62, 5076}, {115, 36961}, {187, 36992}, {383, 37637}, {395, 3543}, {396, 3839}, {462, 41424}, {463, 31860}, {472, 26958}, {546, 18582}, {599, 621}, {622, 40341}, {623, 11295}, {1080, 31489}, {1250, 12953}, {1351, 16002}, {1656, 10645}, {1657, 10646}, {2043, 8252}, {2044, 8253}, {3091, 23302}, {3146, 11489}, {3366, 35821}, {3367, 35820}, {3412, 3843}, {3426, 11138}, {3531, 11139}, {3534, 37835}, {3627, 11543}, {3763, 11304}, {3832, 11488}, {3845, 10654}, {3851, 16966}, {3853, 40694}, {3855, 16772}, {3861, 40693}, {5024, 41024}, {5055, 33417}, {5072, 5238}, {5079, 5352}, {5093, 16001}, {5210, 41034}, {5353, 18514}, {5357, 18513}, {5471, 36962}, {5479, 16942}, {6409, 35732}, {6411, 14813}, {6412, 14814}, {6425, 35740}, {6777, 13103}, {7051, 10896}, {7507, 11475}, {9735, 14162}, {10516, 20428}, {10632, 35488}, {10633, 35480}, {10638, 10895}, {10641, 37197}, {10642, 12173}, {10653, 15687}, {10657, 38789}, {10658, 12902}, {10662, 12293}, {11302, 33561}, {11646, 41060}, {12101, 41113}, {12943, 19373}, {13881, 19781}, {15069, 20429}, {15305, 36980}, {15681, 16242}, {15684, 36968}, {16241, 19709}, {16773, 33703}, {17845, 30403}, {18584, 41040}, {19364, 21659}, {22615, 35738}, {22795, 22906}, {22797, 23013}, {22971, 22974}, {25164, 33518}, {31133, 37775}, {33699, 41120}, {36969, 38335}

X(42093) = {X(4),X(6)}-harmonic conjugate of X(42094)
X(42093) = {X(42095),X(42096)}-harmonic conjugate of X(3)
X(42093) = {X(42175),X(42176)}-harmonic conjugate of X(3)

X(42094) = GIBERT(1,2,0) POINT

Barycentrics    Sqrt[3]*a^2*S + 12*SB*SC : :

X(42094) lies on these lines: {3, 16808}, {4, 6}, {5, 11481}, {13, 3830}, {14, 12820}, {15, 382}, {16, 381}, {17, 5073}, {20, 23302}, {30, 11480}, {61, 5076}, {115, 36962}, {187, 36994}, {383, 31489}, {395, 3839}, {396, 3543}, {462, 31860}, {463, 41424}, {473, 26958}, {546, 18581}, {599, 622}, {621, 40341}, {624, 11296}, {1080, 37637}, {1250, 10895}, {1351, 16001}, {1656, 10646}, {1657, 10645}, {2043, 8253}, {2044, 8252}, {3091, 23303}, {3146, 11488}, {3391, 35821}, {3392, 35820}, {3411, 3843}, {3426, 11139}, {3531, 11138}, {3534, 37832}, {3627, 11542}, {3763, 11303}, {3832, 11489}, {3845, 10653}, {3851, 16967}, {3853, 40693}, {3855, 16773}, {3861, 40694}, {5024, 41025}, {5055, 33416}, {5072, 5237}, {5079, 5351}, {5093, 16002}, {5210, 41035}, {5353, 18513}, {5357, 18514}, {5472, 36961}, {5478, 16943}, {6409, 35740}, {6410, 35732}, {6411, 14814}, {6412, 14813}, {6778, 13102}, {7051, 12943}, {7507, 11476}, {9736, 14162}, {10516, 20429}, {10632, 35480}, {10633, 35488}, {10638, 12953}, {10641, 12173}, {10642, 37197}, {10654, 15687}, {10657, 12902}, {10658, 38789}, {10661, 12293}, {10896, 19373}, {11301, 33560}, {11646, 41061}, {12101, 41112}, {13881, 19780}, {15069, 20428}, {15305, 36978}, {15681, 16241}, {15684, 36967}, {16242, 19709}, {16772, 33703}, {17845, 30402}, {18584, 41041}, {19363, 21659}, {22644, 35738}, {22794, 22862}, {22796, 23006}, {22971, 22975}, {25154, 33517}, {31133, 37776}, {33699, 41119}, {36970, 38335}

X(42094) = {X(4),X(6)}-harmonic conjugate of X(42093)
X(42094) = {X(42097),X(42098)}-harmonic conjugate of X(3)
X(42094) = {X(42177),X(42178)}-harmonic conjugate of X(3)

X(42095) = GIBERT(-1,2,2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA - 12*SB*SC : :

X(42095) lies on these lines: {2, 5321}, {3, 16809}, {4, 11481}, {5, 6}, {13, 16961}, {14, 5055}, {15, 1656}, {16, 381}, {18, 3851}, {61, 5079}, {62, 5072}, {115, 36765}, {382, 10646}, {395, 3545}, {396, 5071}, {397, 5068}, {398, 5056}, {546, 36843}, {547, 10654}, {599, 624}, {622, 9761}, {623, 3763}, {1250, 10896}, {1853, 10676}, {2041, 6412}, {2042, 6411}, {3090, 5334}, {3091, 5318}, {3523, 5349}, {3526, 10645}, {3533, 5365}, {3628, 36836}, {3830, 16242}, {3832, 16773}, {3843, 19106}, {3854, 5350}, {5050, 20416}, {5054, 36970}, {5066, 10653}, {5070, 16964}, {5076, 5351}, {5094, 11475}, {5141, 5367}, {5154, 5362}, {5617, 33517}, {6114, 31489}, {6144, 34509}, {6409, 35738}, {6560, 34562}, {6561, 34559}, {6670, 11298}, {6672, 11296}, {7486, 16772}, {7507, 10642}, {7547, 10633}, {7685, 41040}, {8252, 18587}, {8253, 18586}, {9113, 10612}, {10109, 41120}, {10658, 38724}, {10895, 19373}, {11297, 33561}, {11301, 16942}, {11476, 37197}, {12817, 15701}, {14269, 36968}, {15694, 36967}, {15703, 16241}, {19781, 31706}, {32395, 32398}, {34508, 40341}, {36990, 41041}, {37464, 41038}, {37832, 41122}

X(42095) = {X(3),X(42093)}-harmonic conjugate of X(42096)
X(42095) = {X(4),X(11481)}-harmonic conjugate of X(42097)
X(42095) = {X(5),X(6)}-harmonic conjugate of X(42098)

X(42096) = GIBERT(1,-2,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA - 12*SB*SC : :

X(42096) lies on these lines: {3, 16809}, {4, 11480}, {6, 30}, {13, 15684}, {14, 15681}, {15, 382}, {16, 1657}, {20, 5321}, {376, 23303}, {381, 10645}, {395, 11001}, {396, 15682}, {398, 5059}, {530, 6144}, {531, 40341}, {550, 18581}, {1546, 30402}, {2043, 6412}, {2044, 6411}, {3146, 5318}, {3522, 5349}, {3529, 5334}, {3534, 10646}, {3543, 11488}, {3627, 18582}, {3763, 11295}, {3830, 16644}, {3843, 16966}, {3851, 33417}, {5073, 5340}, {5076, 5238}, {5335, 33703}, {5895, 10675}, {7051, 12953}, {10632, 35490}, {10638, 12943}, {10642, 37196}, {10657, 38790}, {10676, 17845}, {11475, 12173}, {11486, 16964}, {11543, 15704}, {12817, 15695}, {14269, 16241}, {15640, 37640}, {15685, 36968}, {15688, 37835}, {15689, 16242}, {16772, 17578}, {32789, 36445}, {32790, 36463}, {34754, 36969}, {35434, 41943}, {36761, 39838}, {36993, 41039}, {37832, 38335}

X(42096) = reflection of X(42097) in X(6)
X(42096) = {X(3),X(42093)}-harmonic conjugate of X(42095)
X(42096) = {X(4),X(11480)}-harmonic conjugate of X(42098)

X(42097) = GIBERT(1,2,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA + 12*SB*SC : :

X(42097) lies on these lines: {3, 16808}, {4, 11481}, {6, 30}, {13, 15681}, {14, 15684}, {15, 1657}, {16, 382}, {20, 5318}, {376, 23302}, {381, 10646}, {395, 15682}, {396, 11001}, {397, 5059}, {530, 40341}, {531, 6144}, {550, 18582}, {1250, 12943}, {1545, 30403}, {2043, 6411}, {2044, 6412}, {3146, 5321}, {3522, 5350}, {3529, 5335}, {3534, 10645}, {3543, 11489}, {3627, 18581}, {3763, 11296}, {3830, 16645}, {3843, 16967}, {3851, 33416}, {5073, 5339}, {5076, 5237}, {5334, 33703}, {5895, 10676}, {10633, 35490}, {10641, 37196}, {10658, 38790}, {10675, 17845}, {11476, 12173}, {11485, 16965}, {11542, 15704}, {12816, 15695}, {12953, 19373}, {14269, 16242}, {15640, 37641}, {15685, 36967}, {15688, 37832}, {15689, 16241}, {16773, 17578}, {32789, 36463}, {32790, 36445}, {34755, 36970}, {35434, 41944}, {36995, 41038}, {37835, 38335}, {39838, 41458}

X(42097) = reflection of X(42096) in X(6)
X(42097) = {X(3),X(42094)}-harmonic conjugate of X(42098)
X(42097) = {X(4),X(11481)}-harmonic conjugate of X(42095)

X(42098) = GIBERT(1,2,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA + 12*SB*SC : :

X(42098) lies on these lines: {2, 5318}, {3, 16808}, {4, 11480}, {5, 6}, {13, 5055}, {14, 16960}, {15, 381}, {16, 1656}, {17, 3851}, {61, 5072}, {62, 5079}, {115, 36771}, {382, 10645}, {395, 5071}, {396, 3545}, {397, 5056}, {398, 5068}, {546, 36836}, {547, 10653}, {599, 623}, {621, 9763}, {624, 3763}, {1853, 10675}, {2041, 6411}, {2042, 6412}, {3090, 5335}, {3091, 5321}, {3523, 5350}, {3526, 10646}, {3533, 5366}, {3628, 36843}, {3830, 16241}, {3832, 16772}, {3843, 19107}, {3854, 5349}, {5050, 20415}, {5054, 36969}, {5066, 10654}, {5070, 16965}, {5076, 5352}, {5094, 11476}, {5141, 5362}, {5154, 5367}, {5472, 36765}, {5613, 33518}, {6115, 31489}, {6144, 34508}, {6410, 35738}, {6560, 34559}, {6561, 34562}, {6669, 11297}, {6671, 11295}, {7051, 10895}, {7486, 16773}, {7507, 10641}, {7547, 10632}, {7684, 41041}, {8252, 18586}, {8253, 18587}, {9112, 10611}, {10109, 41119}, {10638, 10896}, {10657, 38724}, {11298, 33560}, {11302, 16943}, {11475, 37197}, {12816, 15701}, {13103, 36766}, {14269, 36967}, {15694, 36968}, {15703, 16242}, {19780, 31705}, {22892, 36772}, {32395, 32397}, {34509, 40341}, {36990, 41040}, {37463, 41039}, {37835, 41121}

X(42098) = {X(3),X(42094)}-harmonic conjugate of X(42097)
X(42098) = {X(4),X(11480)}-harmonic conjugate of X(42096)
X(42098) = {X(5),X(6)}-harmonic conjugate of X(42095)

X(42099) = GIBERT(1,-2,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA - 12*SB*SC : :

X(42099) lies on these lines: {3, 16809}, {4, 10188}, {6, 1657}, {13, 15}, {14, 3534}, {16, 20}, {17, 5073}, {18, 550}, {61, 3529}, {62, 15704}, {74, 8173}, {376, 16242}, {381, 33417}, {382, 11480}, {395, 15686}, {398, 34755}, {531, 35751}, {548, 23303}, {621, 5463}, {623, 35931}, {1250, 4324}, {2777, 10657}, {3146, 5238}, {3206, 8718}, {3543, 37832}, {3627, 5352}, {3830, 16241}, {4316, 19373}, {5059, 5335}, {5237, 11543}, {5254, 41409}, {5339, 16961}, {5349, 33923}, {5351, 11489}, {5473, 6777}, {6240, 11475}, {6781, 19780}, {7748, 19781}, {8703, 12817}, {10483, 10638}, {10633, 13619}, {10641, 18560}, {10642, 35471}, {10653, 15683}, {10654, 11001}, {10658, 12121}, {10676, 34785}, {11476, 35481}, {11485, 16965}, {11486, 15681}, {11488, 33703}, {12821, 15707}, {15072, 36981}, {15640, 41121}, {15684, 16644}, {15685, 41101}, {15689, 16645}, {15690, 41122}, {15691, 41944}, {18468, 18565}, {19710, 41108}, {20063, 37776}, {22802, 30402}, {22843, 36756}, {22855, 36993}, {25166, 33518}, {25236, 41023}, {35304, 40334}, {36766, 36961}, {36992, 41024}

X(42099) = {X(6),X(1657)}-harmonic conjugate of X(42100)

X(42100) = GIBERT(1,2,-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA + 12*SB*SC : :

X(42100) lies on these lines: {3, 16808}, {4, 10187}, {6, 1657}, {13, 3534}, {14, 16}, {15, 20}, {17, 550}, {18, 5073}, {61, 15704}, {62, 3529}, {74, 8172}, {376, 16241}, {381, 33416}, {382, 11481}, {396, 15686}, {397, 34754}, {530, 36329}, {548, 23302}, {622, 5464}, {624, 35932}, {1250, 10483}, {2777, 10658}, {3146, 5237}, {3205, 8718}, {3543, 37835}, {3627, 5351}, {3830, 16242}, {4316, 7051}, {4324, 10638}, {5059, 5334}, {5238, 11542}, {5254, 41408}, {5340, 16960}, {5350, 33923}, {5352, 11488}, {5474, 6778}, {6240, 11476}, {6781, 19781}, {7748, 19780}, {8703, 12816}, {10632, 13619}, {10641, 35471}, {10642, 18560}, {10653, 11001}, {10654, 15683}, {10657, 12121}, {10675, 34785}, {11475, 35481}, {11485, 15681}, {11486, 16964}, {11489, 33703}, {12820, 15707}, {15072, 36979}, {15640, 41122}, {15684, 16645}, {15685, 41100}, {15689, 16644}, {15690, 41121}, {15691, 41943}, {18470, 18565}, {19710, 41107}, {20063, 37775}, {22802, 30403}, {22890, 36755}, {22901, 36995}, {25156, 33517}, {25235, 41022}, {35303, 40335}, {36994, 41025}

X(42100) = {X(6),X(1657)}-harmonic conjugate of X(42099)

X(42101) = GIBERT(-1,3,0) POINT

Barycentrics    Sqrt[3]*a^2*S - 18*SB*SC : :

X(42101) lies on these lines: {3, 42103}, {4, 6}, {5, 10645}, {13, 14893}, {14, 12821}, {15, 546}, {16, 3627}, {18, 41977}, {30, 10646}, {62, 12102}, {381, 23302}, {382, 16773}, {383, 3054}, {395, 3830}, {396, 3845}, {548, 33416}, {550, 16967}, {621, 3631}, {622, 3630}, {1080, 3055}, {2043, 32790}, {2044, 32789}, {3091, 11480}, {3146, 11481}, {3411, 3853}, {3543, 11489}, {3832, 16772}, {3839, 11488}, {3843, 18582}, {3850, 16966}, {3857, 5238}, {3861, 11542}, {5066, 36967}, {5076, 11486}, {5352, 12811}, {6221, 35740}, {6411, 35732}, {8588, 41034}, {8589, 41035}, {10151, 10641}, {10653, 38335}, {10654, 14269}, {11138, 13603}, {11139, 14487}, {11304, 34573}, {11475, 23047}, {12101, 12817}, {15682, 16645}, {16194, 36980}, {16241, 38071}, {16539, 32062}, {16644, 41099}, {16962, 41987}, {18323, 18470}, {18358, 20428}, {18424, 41017}, {21850, 25164}, {22512, 41408}, {23046, 37832}, {33699, 36968}

X(42101) = {X(4),X(6)}-harmonic conjugate of X(42102)
X(42101) = {X(42103),X(42104)}-harmonic conjugate of X(3)
X(42101) = {X(42107),X(42108)}-harmonic conjugate of X(3)
X(42101) = {X(42183),X(42184)}-harmonic conjugate of X(3)

X(42102) = GIBERT(1,3,0) POINT

Barycentrics    Sqrt[3]*a^2*S + 18*SB*SC : :

X(42102) lies on these lines: {3, 42105}, {4, 6}, {5, 10646}, {13, 12820}, {14, 14893}, {15, 3627}, {16, 546}, {17, 41978}, {30, 10645}, {61, 12102}, {381, 23303}, {382, 16772}, {383, 3055}, {395, 3845}, {396, 3830}, {548, 33417}, {550, 16966}, {621, 3630}, {622, 3631}, {1080, 3054}, {2043, 32789}, {2044, 32790}, {3091, 11481}, {3146, 11480}, {3412, 3853}, {3543, 11488}, {3832, 16773}, {3839, 11489}, {3843, 18581}, {3850, 16967}, {3857, 5237}, {3861, 11543}, {5066, 36968}, {5076, 11485}, {5351, 12811}, {6200, 35740}, {6412, 35732}, {8588, 41035}, {8589, 41034}, {10151, 10642}, {10653, 14269}, {10654, 38335}, {11138, 14487}, {11139, 13603}, {11303, 34573}, {11476, 23047}, {12101, 12816}, {15682, 16644}, {16194, 36978}, {16242, 38071}, {16538, 32062}, {16645, 41099}, {16963, 41987}, {18323, 18468}, {18358, 20429}, {18424, 41016}, {21850, 25154}, {22513, 41409}, {23046, 37835}, {33699, 36967}

X(42102) = {X(4),X(6)}-harmonic conjugate of X(42101)
X(42102) = {X(42105),X(42106)}-harmonic conjugate of X(3)
X(42102) = {X(42109),X(42110)}-harmonic conjugate of X(3)
X(42102) = {X(42113),X(42114)}-harmonic conjugate of X(3)
X(42102) = {X(42185),X(42186)}-harmonic conjugate of X(3)

X(42103) = GIBERT(-1,3,1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 18*SB*SC : :

X(42103) lies on these lines: {2, 12821}, {3,42102}, {4, 16}, {5, 11480}, {6, 546}, {13, 33603}, {14, 3839}, {15, 3091}, {17, 3854}, {20, 16967}, {376, 33416}, {381, 396}, {382, 23303}, {395, 14269}, {622, 22491}, {3090, 10645}, {3146, 10646}, {3543, 37835}, {3544, 5238}, {3545, 16966}, {3627, 11481}, {3832, 5334}, {3843, 5318}, {3845, 10653}, {3851, 5349}, {3855, 11488}, {3857, 22236}, {3858, 5339}, {3860, 41119}, {5056, 33417}, {5071, 36967}, {5352, 15022}, {7394, 37776}, {11268, 18568}, {12102, 36843}, {12811, 36836}, {12817, 37832}, {15682, 16242}, {15687, 16645}, {16644, 38071}, {16961, 36969}, {22795, 22907}, {33517, 41042}, {33518, 41036}, {33607, 41108}

X(42103) = {X(3),X(42101)}-harmonic conjugate of X(42104)
X(42103) = {X(4),X(16)}-harmonic conjugate of X(42105)
X(42103) = {X(6),X(546)}-harmonic conjugate of X(42106)

X(42104) = GIBERT(1,-3,1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 18*SB*SC : :

X(42104) lies on these lines: {3, 42101), {4, 15}, {6, 3627}, {14, 15682}, {16, 3146}, {20, 16809}, {30, 11481}, {376, 16967}, {382, 5321}, {395, 15684}, {396, 38335}, {546, 11480}, {1657, 23303}, {3091, 10645}, {3522, 33416}, {3529, 10646}, {3543, 5334}, {3545, 33417}, {3830, 5318}, {3832, 16966}, {3839, 36967}, {3843, 23302}, {3853, 40693}, {5073, 5349}, {5076, 11485}, {5237, 11541}, {5335, 16964}, {5344, 41973}, {7391, 37775}, {11001, 37835}, {11489, 33703}, {11542, 15687}, {12102, 22236}, {12817, 15640}, {14893, 16644}, {15683, 16242}, {16241, 41099}, {16960, 41119}, {33625, 36326}, {33699, 41113}

X(42104) = {X(3),X(42101)}-harmonic conjugate of X(42103)
X(42104) = {X(4),X(15)}-harmonic conjugate of X(42106)
X(42104) = {X(6),X(3627)}-harmonic conjugate of X(42105)

X(42105) = GIBERT(1,3,-1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 18*SB*SC : :

X(42105) lies on these lines: {3,42102}, {4, 16}, {6, 3627}, {13, 15682}, {15, 3146}, {20, 16808}, {30, 11480}, {376, 16966}, {382, 5318}, {395, 38335}, {396, 15684}, {546, 11481}, {1657, 23302}, {3091, 10646}, {3522, 33417}, {3529, 10645}, {3543, 5335}, {3545, 33416}, {3830, 5321}, {3832, 16967}, {3839, 36968}, {3843, 23303}, {3853, 40694}, {5073, 5350}, {5076, 11486}, {5238, 11541}, {5334, 16965}, {5343, 41974}, {7391, 37776}, {11001, 37832}, {11488, 33703}, {11543, 15687}, {12102, 22238}, {12816, 15640}, {14893, 16645}, {15683, 16241}, {16242, 41099}, {16961, 41120}, {33623, 36324}, {33699, 41112}

X(42105) = {X(3),X(42102)}-harmonic conjugate of X(42106)
X(42105) = {X(4),X(16)}-harmonic conjugate of X(42103)
X(42105) = {X(6),X(3627)}-harmonic conjugate of X(42104)

X(42106) = GIBERT(1,3,1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 18*SB*SC : :

X(42106) lies on these lines: {2, 12820}, {3,42102}, {4, 15}, {5, 11481}, {6, 546}, {13, 3839}, {14, 33602}, {16, 3091}, {18, 3854}, {20, 16966}, {376, 33417}, {381, 395}, {382, 23302}, {396, 14269}, {621, 22492}, {3090, 10646}, {3146, 10645}, {3543, 37832}, {3544, 5237}, {3545, 16967}, {3627, 11480}, {3832, 5335}, {3843, 5321}, {3845, 10654}, {3851, 5350}, {3855, 11489}, {3857, 22238}, {3858, 5340}, {3860, 41120}, {5056, 33416}, {5071, 36968}, {5351, 15022}, {7394, 37775}, {11267, 18568}, {12102, 36836}, {12811, 36843}, {12816, 37835}, {15682, 16241}, {15687, 16644}, {16645, 38071}, {16960, 36970}, {22794, 22861}, {33517, 41037}, {33518, 41043}, {33606, 41107}

X(42106) = {X(3),X(42102)}-harmonic conjugate of X(42105)
X(42106) = {X(4),X(15)}-harmonic conjugate of X(42104)
X(42106) = {X(6),X(546)}-harmonic conjugate of X(42103)

X(42107) = GIBERT(-1,3,2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA - 18*SB*SC : :

X(42107) lies on these lines: {3,42101}, {4, 11481}, {5, 15}, {6, 3091}, {13, 38071}, {14, 5066}, {16, 546}, {18, 3858}, {30, 16967}, {61, 12811}, {62, 3857}, {140, 19107}, {381, 395}, {396, 3545}, {397, 3850}, {398, 3851}, {547, 33417}, {550, 33416}, {623, 31694}, {1656, 5349}, {3090, 11480}, {3544, 22236}, {3614, 10638}, {3627, 10646}, {3628, 10645}, {3832, 11489}, {3839, 16645}, {3843, 16773}, {3845, 19106}, {3854, 5340}, {3855, 5335}, {3856, 16965}, {3859, 16961}, {5068, 5339}, {5072, 11485}, {5133, 37775}, {5238, 12812}, {5351, 12102}, {6777, 20252}, {7051, 7173}, {10109, 16241}, {10151, 11476}, {10297, 10635}, {10632, 35487}, {10642, 23047}, {10654, 19709}, {10658, 11801}, {11737, 37832}, {14893, 36968}, {15022, 36836}, {15687, 16242}, {15699, 36967}, {16626, 18358}, {22513, 22847}, {23046, 36969}, {30403, 41362}, {32789, 35732}, {33561, 37352}, {34755, 41991}

X(42107) = {X(3),X(42101)}-harmonic conjugate of X(42108)
X(42107) = {X(4),X(11481)}-harmonic conjugate of X(42109)
X(42107) = {X(6),X(3091)}-harmonic conjugate of X(42110)

X(42108) = GIBERT(1,-3,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA - 18*SB*SC : :

X(42108) lies on these lines: {3, 42101}, {4, 11480}, {6, 3146}, {13, 33699}, {14, 16}, {15, 3627}, {20, 23303}, {382, 5318}, {396, 3543}, {397, 19106}, {398, 5073}, {546, 10645}, {548, 16967}, {550, 16809}, {622, 36331}, {1657, 5349}, {3529, 11481}, {3830, 18582}, {3845, 16966}, {3850, 33417}, {3853, 16772}, {5059, 11489}, {5238, 12102}, {5334, 33703}, {5335, 15682}, {5350, 11542}, {5893, 30402}, {6108, 41151}, {8703, 33416}, {10646, 15704}, {10654, 15684}, {11488, 17578}, {11541, 22238}, {12101, 37832}, {14893, 16241}, {15683, 16645}, {15686, 37835}, {15687, 36967}, {16242, 19710}, {16773, 17800}, {34603, 37776}, {35404, 36969}

X(42108) = {X(3),X(42101)}-harmonic conjugate of X(42107)
X(42108) = {X(4),X(11480)}-harmonic conjugate of X(42110)
X(42108) = {X(6),X(3146)}-harmonic conjugate of X(42109)

X(42109) = GIBERT(1,3,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA + 18*SB*SC : :

X(42109) lies on these lines: {3,42102}, {4, 11481}, {6, 3146}, {13, 15}, {14, 33699}, {16, 3627}, {20, 23302}, {382, 5321}, {395, 3543}, {397, 5073}, {398, 19107}, {546, 10646}, {548, 16966}, {550, 16808}, {621, 35750}, {623, 36768}, {1657, 5350}, {3529, 11480}, {3830, 18581}, {3845, 16967}, {3850, 33416}, {3853, 16773}, {5059, 11488}, {5237, 12102}, {5334, 15682}, {5335, 33703}, {5349, 11543}, {5893, 30403}, {6109, 41151}, {8703, 33417}, {10645, 15704}, {10653, 15684}, {11489, 17578}, {11541, 22236}, {12101, 37835}, {14893, 16242}, {15683, 16644}, {15686, 37832}, {15687, 36968}, {16241, 19710}, {16772, 17800}, {34603, 37775}, {35404, 36970}

X(42109) = {X(3),X(42102)}-harmonic conjugate of X(42110)
X(42109) = {X(4),X(11481)}-harmonic conjugate of X(42107)
X(42109) = {X(6),X(3146)}-harmonic conjugate of X(42108)

X(42110) = GIBERT(1,3,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA + 18*SB*SC : :

X(42110) lies on these lines: {3,42102}, {4, 11480}, {5, 16}, {6, 3091}, {13, 5066}, {14, 38071}, {15, 546}, {17, 3858}, {30, 16966}, {61, 3857}, {62, 12811}, {140, 19106}, {381, 396}, {395, 3545}, {397, 3851}, {398, 3850}, {547, 33416}, {550, 33417}, {624, 31693}, {1250, 3614}, {1656, 5350}, {3090, 11481}, {3544, 22238}, {3627, 10645}, {3628, 10646}, {3832, 11488}, {3839, 16644}, {3843, 16772}, {3845, 19107}, {3854, 5339}, {3855, 5334}, {3856, 16964}, {3859, 16960}, {5068, 5340}, {5072, 11486}, {5133, 37776}, {5237, 12812}, {5352, 12102}, {6778, 20253}, {7173, 19373}, {10109, 16242}, {10151, 11475}, {10297, 10634}, {10633, 35487}, {10641, 23047}, {10653, 19709}, {10657, 11801}, {11737, 37835}, {14893, 36967}, {15022, 36843}, {15687, 16241}, {15699, 36968}, {16627, 18358}, {22512, 22893}, {23046, 36970}, {30402, 41362}, {32790, 35732}, {33560, 37351}, {34754, 41991}

X(42110) = {X(3),X(42102)}-harmonic conjugate of X(42109)
X(42110) = {X(4),X(11480)}-harmonic conjugate of X(42108)
X(42110) = {X(6),X(3091)}-harmonic conjugate of X(42107)

X(42111) = GIBERT(-1,3,3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA - 18*SB*SC : :

X(42111) lies on these lines: {2, 10645}, {3,42101}, {4, 10187}, {5, 6}, {14, 5071}, {15, 3090}, {16, 3091}, {18, 5068}, {20, 33416}, {61, 15022}, {62, 3544}, {303, 22491}, {381, 23303}, {395, 19709}, {396, 41120}, {546, 11481}, {623, 3619}, {624, 3620}, {631, 19107}, {1656, 5321}, {3545, 10653}, {3618, 5460}, {3628, 11480}, {3832, 19106}, {3839, 16242}, {3851, 5318}, {3857, 36843}, {5055, 10654}, {5056, 5334}, {5066, 16645}, {5067, 33417}, {5072, 11486}, {5079, 11485}, {5339, 35018}, {6411, 35738}, {6670, 37170}, {7486, 16964}, {10109, 16644}, {10658, 15081}, {11008, 34509}, {11306, 34573}, {12811, 22238}, {12812, 22236}, {18586, 32789}, {18587, 32790}, {20080, 34508}, {22907, 41409}, {31706, 41407}, {32785, 35731}, {36968, 41099}, {36969, 41106}, {37640, 41122}, {37641, 41119}

X(42111) = {X(3),X(42101)}-harmonic conjugate of X(42112)
X(42111) = {X(4),X(10646)}-harmonic conjugate of X(42113)
X(42111) = {X(5),X(6)}-harmonic conjugate of X(42114)

X(42112) = GIBERT(1,-3,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA - 18*SB*SC : :

X(42112) lies on these lines: {3,42101}, {4, 10188}, {6, 30}, {14, 15683}, {15, 3146}, {16, 3529}, {20, 10646}, {61, 11541}, {376, 16809}, {382, 16772}, {395, 15685}, {530, 11008}, {531, 20080}, {1657, 5321}, {3522, 16967}, {3528, 33416}, {3534, 23303}, {3543, 16808}, {3627, 11480}, {3830, 23302}, {3832, 33417}, {5059, 5334}, {5073, 5318}, {5343, 16961}, {6110, 33630}, {11001, 11489}, {11295, 34573}, {11481, 15704}, {11488, 15682}, {15640, 36969}, {16644, 33699}, {16645, 19710}, {17800, 40694}, {19106, 33703}, {36968, 41113}

X(42112) = reflection of X(42113) in X(6)
X(42112) = {X(3),X(42101)}-harmonic conjugate of X(42111)
X(42112) = {X(4),X(10645)}-harmonic conjugate of X(42114)

X(42113) = GIBERT(1,3,-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA + 18*SB*SC : :

X(42113) lies on these lines: {3,42102, {4, 10187}, {6, 30}, {13, 15683}, {15, 3529}, {16, 3146}, {20, 10645}, {62, 11541}, {376, 16808}, {382, 16773}, {396, 15685}, {530, 20080}, {531, 11008}, {1657, 5318}, {3522, 16966}, {3528, 33417}, {3534, 23302}, {3543, 16809}, {3627, 11481}, {3830, 23303}, {3832, 33416}, {5059, 5335}, {5073, 5321}, {5344, 16960}, {6111, 33630}, {11001, 11488}, {11296, 34573}, {11480, 15704}, {11489, 15682}, {15640, 36970}, {16644, 19710}, {16645, 33699}, {17800, 40693}, {19107, 33703}, {36967, 41112}

X(42113) = reflection of X(42112) in X(6)
X(42113) = {X(3),X(42102)}-harmonic conjugate of X(42114)
X(42113) = {X(4),X(10646)}-harmonic conjugate of X(42111)

X(42114) = GIBERT(1,3,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA + 18*SB*SC : :

X(42114) lies on these lines: {2, 10646}, {3, 42102}, {4, 10188}, {5, 6}, {13, 5071}, {15, 3091}, {16, 3090}, {17, 5068}, {20, 33417}, {61, 3544}, {62, 15022}, {302, 22492}, {381, 23302}, {395, 41119}, {396, 19709}, {546, 11480}, {623, 3620}, {624, 3619}, {631, 19106}, {1656, 5318}, {3545, 10654}, {3618, 5459}, {3628, 11481}, {3832, 19107}, {3839, 16241}, {3851, 5321}, {3857, 36836}, {5055, 10653}, {5056, 5335}, {5066, 16644}, {5067, 33416}, {5072, 11485}, {5079, 11486}, {5340, 35018}, {6412, 35738}, {6669, 37171}, {7486, 16965}, {10109, 16645}, {10657, 15081}, {11008, 34508}, {11305, 34573}, {12811, 22236}, {12812, 22238}, {18586, 32790}, {18587, 32789}, {20080, 34509}, {22861, 41408}, {31705, 41406}, {36967, 41099}, {36970, 41106}, {37640, 41120}, {37641, 41121}

X(42114) = {X(3),X(42102)}-harmonic conjugate of X(42113)
X(42114) = {X(4),X(10645)}-harmonic conjugate of X(42112)
X(42114) = {X(5),X(6)}-harmonic conjugate of X(42111)

X(42115) = GIBERT(-2,0,3) POINT

Barycentrics    a^2*(2*Sqrt[3]*S - 9*SA) : :
Barycentrics    a^2*(2*Sqrt[3]*S - 9*SA) : :
Barycentrics    (3*Sqrt[3]*Cos[A] - 2*Sin[A])*Sin[A] : :

X(42115) lies on these lines: {2, 33602}, {3, 6}, {4, 42121}, {5, 42120}, {13, 15694}, {14, 15681}, {18, 5073}, {20, 11543}, {30, 11489}, {69, 35303}, {140, 5335}, {186, 11408}, {302, 11296}, {376, 42117}, {378, 11409}, {381, 23303}, {382, 16773}, {395, 3534}, {396, 15693}, {397, 15720}, {398, 42090}, {466, 37643}, {546, 42141}, {548, 42119}, {549, 11488}, {550, 5334}, {616, 11302}, {631, 11542}, {999, 1250}, {1593, 10633}, {1597, 10642}, {1598, 11476}, {1656, 5318}, {1657, 5321}, {2041, 18762}, {2042, 18538}, {2043, 42225}, {2044, 42226}, {2045, 42202}, {2046, 42201}, {3068, 15764}, {3070, 42198}, {3071, 42196}, {3090, 42138}, {3091, 42137}, {3132, 26864}, {3146, 42135}, {3295, 19373}, {3426, 32586}, {3522, 42122}, {3523, 42124}, {3526, 18582}, {3529, 42136}, {3544, 42591}, {3618, 35304}, {3619, 37341}, {3620, 37173}, {3627, 42139}, {3628, 42142}, {3763, 5463}, {3830, 16645}, {3843, 19106}, {3851, 16967}, {3858, 42473}, {5054, 10653}, {5055, 16242}, {5070, 16965}, {5072, 42106}, {5076, 42103}, {5079, 42110}, {5204, 5353}, {5217, 5357}, {5339, 16961}, {5340, 16966}, {5362, 16371}, {5366, 35018}, {5367, 16370}, {5464, 40341}, {6000, 17827}, {8703, 37641}, {9541, 36457}, {10605, 21648}, {10632, 15750}, {10654, 15688}, {10675, 14530}, {10676, 13093}, {11202, 17826}, {11244, 35450}, {11268, 12085}, {11300, 34540}, {11421, 21312}, {12100, 37640}, {12812, 42493}, {14269, 37835}, {14813, 23249}, {14814, 23259}, {14869, 42627}, {14891, 42633}, {15684, 41944}, {15685, 36970}, {15686, 42497}, {15689, 16963}, {15695, 36967}, {15696, 40694}, {15701, 16644}, {15704, 42140}, {15707, 16241}, {15714, 42517}, {15723, 42501}, {15765, 32785}, {17538, 42585}, {17800, 19107}, {18424, 22862}, {18585, 32786}, {19709, 36969}, {21475, 37633}, {21476, 37680}, {23267, 42224}, {23273, 42222}, {30403, 32063}, {33417, 42156}, {34200, 42634}, {35255, 36437}, {35256, 36455}, {35400, 42429}, {35434, 42513}, {35732, 42213}, {36995, 41041}, {42104, 42163}, {42105, 42107}, {42108, 42159}, {42187, 42193}, {42188, 42250}, {42189, 42191}, {42190, 42252}, {42192, 42246}, {42194, 42248}, {42195, 42256}, {42197, 42254}, {42211, 42282}, {42217, 42280}, {42219, 42281}, {42498, 42505}

X(42115) = isogonal conjugate of X(33603)
X(42115) = isogonal conjugate of the anticomplement of X(33618)
X(42115) = X(1)-isoconjugate of X(33603)
X(42115) = Brocard-circle-inverse of X(42116)
X(42115) = Schoute-circle-inverse of X(22238)
X(42115) = barycentric quotient X(6)/X(33603)
X(42115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 42118, 42128}, {3, 6, 42116}, {3, 16, 11486}, {3, 11486, 11485}, {4, 42121, 42129}, {4, 42123, 42131}, {5, 42120, 42127}, {6, 10646, 3}, {6, 11480, 34754}, {6, 11481, 10646}, {6, 41408, 21309}, {6, 42116, 11485}, {14, 42625, 15681}, {15, 16, 22238}, {16, 5237, 11481}, {16, 5351, 15}, {16, 10645, 34755}, {16, 10646, 6}, {16, 11481, 3}, {18, 42100, 42093}, {20, 11543, 42126}, {61, 35739, 1152}, {62, 34754, 6}, {140, 5335, 42132}, {550, 5334, 42130}, {1350, 21159, 3}, {3365, 35739, 6454}, {3627, 42628, 42139}, {5237, 36843, 3}, {5318, 42089, 1656}, {5321, 42091, 1657}, {5351, 22238, 3}, {6200, 6396, 11481}, {6200, 6445, 42116}, {6221, 6398, 11486}, {6221, 6451, 42116}, {6396, 6446, 42116}, {6398, 6452, 42116}, {6410, 17851, 42116}, {10645, 34755, 6}, {10646, 34755, 10645}, {11481, 36843, 16}, {11486, 42116, 6}, {15655, 33878, 42116}, {16242, 42155, 5055}, {16645, 36968, 3830}, {16645, 42097, 16809}, {16773, 42088, 18581}, {16809, 36968, 42097}, {16809, 42097, 3830}, {16961, 42099, 5339}, {16963, 42528, 42154}, {16965, 33416, 42098}, {16965, 42491, 5070}, {16967, 42094, 3851}, {16967, 42158, 42094}, {18581, 42088, 382}, {18581, 42113, 42101}, {19106, 42095, 3843}, {23303, 42086, 381}, {23303, 42102, 42111}, {33416, 42098, 5070}, {42086, 42111, 42102}, {42088, 42101, 42113}, {42089, 42151, 5318}, {42091, 42149, 5321}, {42093, 42100, 5073}, {42098, 42491, 33416}, {42101, 42113, 382}, {42102, 42111, 381}, {42103, 42109, 5076}, {42109, 42599, 42103}, {42121, 42123, 4}, {42121, 42145, 42143}, {42123, 42143, 42145}, {42129, 42131, 4}, {42135, 42584, 3146}, {42143, 42145, 4}, {42153, 42433, 17800}, {42154, 42528, 15689}, {42195, 42256, 42284}, {42195, 42284, 42279}, {42197, 42254, 42283}, {42197, 42283, 42278}, {42221, 42223, 4}

X(42116) = GIBERT(2,0,3) POINT

Barycentrics    a^2*(2*Sqrt[3]*S + 9*SA) : :
Barycentrics    a^2*(2*Sqrt[3]*S + 9*SA) : :
Barycentrics    Sin[A]*(3*Sqrt[3]*Cos[A] + 2*Sin[A]) : :

X(42116) lies on these lines: {2, 33603}, {3, 6}, {4, 42122}, {5, 42119}, {13, 15681}, {14, 15694}, {17, 5073}, {20, 11542}, {30, 11488}, {69, 35304}, {140, 5334}, {186, 11409}, {303, 11295}, {376, 42118}, {378, 11408}, {381, 23302}, {382, 16772}, {395, 15693}, {396, 3534}, {397, 42091}, {398, 15720}, {465, 37643}, {546, 42140}, {548, 42120}, {549, 11489}, {550, 5335}, {617, 11301}, {631, 11543}, {999, 10638}, {1593, 10632}, {1597, 10641}, {1598, 11475}, {1656, 5321}, {1657, 5318}, {2041, 18538}, {2042, 18762}, {2043, 42226}, {2044, 42225}, {2045, 42199}, {2046, 42200}, {3069, 15764}, {3070, 42197}, {3071, 42195}, {3090, 42135}, {3091, 42136}, {3131, 26864}, {3146, 42138}, {3295, 7051}, {3426, 32585}, {3522, 42123}, {3523, 42121}, {3526, 18581}, {3529, 42137}, {3544, 42590}, {3618, 35303}, {3619, 37340}, {3620, 37172}, {3627, 42142}, {3628, 42139}, {3763, 5464}, {3830, 16644}, {3843, 19107}, {3851, 16966}, {3858, 42472}, {5054, 10654}, {5055, 16241}, {5070, 16964}, {5072, 42103}, {5076, 42106}, {5079, 42107}, {5204, 5357}, {5217, 5353}, {5339, 16967}, {5340, 16960}, {5362, 16370}, {5365, 35018}, {5367, 16371}, {5463, 40341}, {6000, 17826}, {6771, 36772}, {8703, 37640}, {9541, 36439}, {10605, 21647}, {10633, 15750}, {10653, 15688}, {10675, 13093}, {10676, 14530}, {11202, 17827}, {11243, 35450}, {11267, 12085}, {11299, 34541}, {11420, 21312}, {12100, 37641}, {12812, 42492}, {14269, 37832}, {14813, 23259}, {14814, 23249}, {14869, 42628}, {14891, 42634}, {15684, 41943}, {15685, 36969}, {15686, 42496}, {15689, 16962}, {15695, 36968}, {15696, 40693}, {15701, 16645}, {15704, 42141}, {15707, 16242}, {15714, 42516}, {15723, 42500}, {15765, 32786}, {17538, 42584}, {17800, 19106}, {18424, 22906}, {18585, 32785}, {19709, 36970}, {21475, 37680}, {21476, 37633}, {23267, 42223}, {23273, 42221}, {30402, 32063}, {33416, 42153}, {34200, 42633}, {35255, 36455}, {35256, 36437}, {35400, 42430}, {35434, 42512}, {35732, 42212}, {36993, 41040}, {42104, 42110}, {42105, 42166}, {42109, 42162}, {42187, 42251}, {42188, 42194}, {42189, 42253}, {42190, 42192}, {42191, 42247}, {42193, 42249}, {42196, 42257}, {42198, 42255}, {42214, 42282}, {42218, 42281}, {42220, 42280}, {42499, 42504}

X(42116) = reflection of X(42116) in Brocard axis
X(42116) = isogonal conjugate of X(33602)
X(42116) = isogonal conjugate of the anticomplement of X(33619)
X(42116) = Brocard-circle-inverse of X(42115)
X(42116) = Schoute-circle-inverse of X(22236)
X(42116) = X(1)-isoconjugate of X(33602)
X(42116) = barycentric quotient X(6)/X(33602)
X(42116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 42117, 42125}, {3, 6, 42115}, {3, 15, 11485}, {3, 11485, 11486}, {4, 42122, 42130}, {4, 42124, 42132}, {5, 42119, 42126}, {6, 10645, 3}, {6, 11480, 10645}, {6, 11481, 34755}, {6, 41409, 21309}, {6, 42115, 11486}, {13, 42626, 15681}, {15, 16, 22236}, {15, 5238, 11480}, {15, 5352, 16}, {15, 10645, 6}, {15, 10646, 34754}, {15, 11480, 3}, {17, 42099, 42094}, {20, 11542, 42127}, {61, 34755, 6}, {140, 5334, 42129}, {550, 5335, 42131}, {1350, 21158, 3}, {3627, 42627, 42142}, {5238, 36836, 3}, {5318, 42090, 1657}, {5321, 42092, 1656}, {5352, 22236, 3}, {6200, 6396, 11480}, {6200, 6445, 42115}, {6221, 6398, 11485}, {6221, 6451, 42115}, {6396, 6446, 42115}, {6398, 6452, 42115}, {6410, 17851, 42115}, {10645, 34754, 10646}, {10646, 34754, 6}, {11480, 36836, 15}, {11485, 42115, 6}, {15655, 33878, 42115}, {16241, 42154, 5055}, {16644, 36967, 3830}, {16644, 42096, 16808}, {16772, 42087, 18582}, {16808, 36967, 42096}, {16808, 42096, 3830}, {16960, 42100, 5340}, {16962, 42529, 42155}, {16964, 33417, 42095}, {16964, 42490, 5070}, {16966, 42093, 3851}, {16966, 42157, 42093}, {18582, 42087, 382}, {18582, 42112, 42102}, {19107, 42098, 3843}, {23302, 42085, 381}, {23302, 42101, 42114}, {33417, 42095, 5070}, {42085, 42114, 42101}, {42087, 42102, 42112}, {42090, 42152, 5318}, {42092, 42150, 5321}, {42094, 42099, 5073}, {42095, 42490, 33417}, {42101, 42114, 381}, {42102, 42112, 382}, {42106, 42108, 5076}, {42108, 42598, 42106}, {42122, 42124, 4}, {42122, 42146, 42144}, {42124, 42144, 42146}, {42130, 42132, 4}, {42138, 42585, 3146}, {42144, 42146, 4}, {42155, 42529, 15689}, {42156, 42434, 17800}, {42196, 42257, 42284}, {42196, 42284, 42278}, {42198, 42255, 42283}, {42198, 42283, 42279}, {42222, 42224, 4}

X(42117) = GIBERT(2,-1,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA - 6*SB*SC : :

X(42117) is the intersection of the tangents to the Evans conic at X(14) and X(15). (Randy Hutson, May 31, 2021)

X(42117) lies on these lines: {2, 33603}, {3, 5334}, {4, 11408}, {5, 15}, {6, 30}, {13, 12820}, {14, 549}, {16, 398}, {17, 3858}, {18, 15712}, {20, 11486}, {53, 6110}, {61, 3627}, {62, 15704}, {69, 11295}, {140, 5339}, {141, 531}, {235, 10632}, {302, 35304}, {303, 31694}, {381, 11488}, {382, 5335}, {395, 8703}, {396, 3845}, {397, 19106}, {495, 10638}, {496, 7051}, {530, 3629}, {533, 3630}, {546, 18582}, {548, 11481}, {617, 37351}, {618, 35022}, {621, 37340}, {632, 5238}, {1546, 21647}, {1595, 11475}, {1596, 10641}, {1656, 5343}, {2043, 6398}, {2044, 6221}, {3054, 6109}, {3534, 37641}, {3618, 11296}, {3619, 11297}, {3628, 36836}, {3631, 3643}, {3642, 34573}, {3830, 37640}, {3851, 5365}, {3853, 40693}, {5066, 16644}, {5352, 14869}, {5353, 6284}, {5357, 7354}, {5362, 11113}, {5367, 11112}, {5617, 36772}, {6200, 34551}, {6396, 34552}, {6823, 10634}, {8972, 36445}, {10299, 22237}, {10301, 37776}, {10642, 37458}, {10678, 36966}, {11299, 34540}, {11304, 34541}, {11409, 18533}, {11539, 37835}, {11812, 41120}, {12100, 16645}, {12103, 22238}, {13350, 16002}, {13665, 36455}, {13785, 36437}, {13941, 36463}, {15686, 34755}, {15699, 16241}, {15713, 41122}, {15714, 41944}, {15760, 18468}, {15765, 18762}, {16242, 17504}, {16773, 16961}, {16962, 23046}, {18538, 18585}, {18586, 23259}, {18587, 23249}, {19710, 36968}, {22512, 22906}, {30739, 37775}, {33699, 36969}, {35255, 36439}, {35256, 36457}, {36993, 41034}, {37832, 38071}, {41943, 41971}

X(42117) = reflection of X(42118) in X(6)

X(42118) = GIBERT(2,1,-1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA + 6*SB*SC : :

X(42118) is the intersection of the tangents to the Evans conic at X(13) and X(16). (Randy Hutson, May 31, 2021)

X(42118) lies on these lines: {2, 33602}, {3, 5335}, {4, 11409}, {5, 16}, {6, 30}, {13, 549}, {14, 12821}, {15, 397}, {17, 15712}, {18, 3858}, {20, 11485}, {53, 6111}, {61, 15704}, {62, 3627}, {69, 11296}, {140, 5340}, {141, 530}, {235, 10633}, {302, 31693}, {303, 35303}, {381, 11489}, {382, 5334}, {395, 3845}, {396, 8703}, {398, 19107}, {495, 1250}, {496, 19373}, {531, 3629}, {532, 3630}, {546, 18581}, {548, 11480}, {616, 37352}, {619, 35022}, {622, 37341}, {632, 5237}, {1545, 21648}, {1595, 11476}, {1596, 10642}, {1656, 5344}, {2043, 6221}, {2044, 6398}, {3054, 6108}, {3365, 35740}, {3534, 37640}, {3618, 11295}, {3619, 11298}, {3628, 36843}, {3631, 3642}, {3643, 34573}, {3830, 37641}, {3851, 5366}, {3853, 40694}, {5066, 16645}, {5351, 14869}, {5353, 7354}, {5357, 6284}, {5362, 11112}, {5367, 11113}, {6200, 34552}, {6396, 34551}, {6823, 10635}, {8972, 36463}, {10299, 22235}, {10301, 37775}, {10641, 37458}, {10677, 36966}, {11300, 34541}, {11303, 34540}, {11408, 18533}, {11539, 37832}, {11812, 41119}, {12100, 16644}, {12103, 22236}, {13349, 16001}, {13665, 36437}, {13785, 36455}, {13941, 36445}, {15686, 34754}, {15699, 16242}, {15713, 41121}, {15714, 41943}, {15760, 18470}, {15765, 18538}, {16241, 17504}, {16772, 16960}, {16963, 23046}, {18585, 18762}, {18586, 23249}, {18587, 23259}, {19710, 36967}, {22513, 22862}, {30739, 37776}, {33699, 36970}, {35255, 36457}, {35256, 36439}, {36995, 41035}, {37835, 38071}, {41944, 41972}

X(42118) = reflection of X(42117) in X(6)

X(42119) = GIBERT(2,-1,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 3*SB*SC : :

X(42119) lies on these lines: {2, 5321}, {3, 5334}, {4, 15}, {6, 20}, {13, 15682}, {14, 3524}, {16, 376}, {18, 10299}, {30, 5335}, {61, 3529}, {62, 17538}, {140, 5343}, {185, 18929}, {382, 11542}, {388, 10638}, {395, 10304}, {396, 3543}, {397, 5059}, {398, 3522}, {497, 7051}, {550, 11486}, {616, 5862}, {617, 7865}, {621, 37172}, {622, 3849}, {631, 10645}, {1656, 5365}, {1885, 11408}, {2041, 23249}, {2042, 23259}, {2044, 9541}, {2883, 17826}, {3090, 5238}, {3091, 23302}, {3146, 5318}, {3523, 5339}, {3525, 5352}, {3528, 10646}, {3545, 16966}, {3832, 16772}, {3839, 16644}, {4190, 5367}, {4299, 5357}, {4302, 5353}, {5067, 33417}, {5068, 5349}, {5071, 16241}, {5073, 5344}, {5350, 22235}, {5362, 6872}, {6225, 10675}, {6770, 33517}, {6773, 23013}, {7735, 19781}, {10653, 11001}, {10676, 11206}, {10996, 11515}, {11420, 37201}, {12820, 16962}, {14912, 36995}, {15692, 16645}, {15698, 16242}, {15702, 37835}, {15710, 16268}, {15719, 41120}, {16773, 21734}, {16961, 19708}, {18930, 19467}, {19106, 33703}, {19772, 37643}, {21735, 41973}, {36327, 36521}, {36771, 41030}, {36772, 41022}, {36968, 41972}, {37832, 41099}, {41107, 41971}

X(42119) = {X(6),X(20)}-harmonic conjugate of X(42120)

X(42120) = GIBERT(2,1,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 3*SB*SC : :

X(42120) lies on these lines: {2, 5318}, {3, 5335}, {4, 16}, {6, 20}, {13, 3524}, {14, 15682}, {15, 376}, {17, 10299}, {30, 5334}, {61, 17538}, {62, 3529}, {140, 5344}, {185, 18930}, {382, 11543}, {388, 1250}, {395, 3543}, {396, 10304}, {397, 3522}, {398, 5059}, {497, 19373}, {550, 11485}, {616, 7865}, {617, 5863}, {621, 3849}, {622, 37173}, {631, 10646}, {1656, 5366}, {1885, 11409}, {2041, 23259}, {2042, 23249}, {2043, 9541}, {2883, 17827}, {3090, 5237}, {3091, 23303}, {3146, 5321}, {3523, 5340}, {3525, 5351}, {3528, 10645}, {3545, 16967}, {3832, 16773}, {3839, 16645}, {4190, 5362}, {4299, 5353}, {4302, 5357}, {5067, 33416}, {5068, 5350}, {5071, 16242}, {5073, 5343}, {5349, 22237}, {5367, 6872}, {6225, 10676}, {6770, 23006}, {6773, 33518}, {7735, 19780}, {10654, 11001}, {10675, 11206}, {10996, 11516}, {11421, 37201}, {12821, 16963}, {14912, 36993}, {15692, 16644}, {15698, 16241}, {15702, 37832}, {15710, 16267}, {15719, 41119}, {16772, 21734}, {16960, 19708}, {18929, 19467}, {19107, 33703}, {19773, 37643}, {21735, 41974}, {31412, 35740}, {35749, 36521}, {36967, 41971}, {37835, 41099}, {41108, 41972}

X(42120) = {X(6),X(20)}-harmonic conjugate of X(42119)

X(42121) = GIBERT(-2,1,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA - 6*SB*SC : :

X(42121) lies on these lines: {2, 11486}, {3, 5334}, {4,42115}, {5, 16}, {6, 140}, {13, 15699}, {14, 8703}, {15, 395}, {18, 550}, {30, 11481}, {61, 14869}, {62, 632}, {141, 6672}, {302, 37341}, {396, 11539}, {397, 16966}, {398, 10645}, {427, 10633}, {495, 19373}, {496, 1250}, {546, 36843}, {547, 10653}, {590, 34551}, {597, 6671}, {615, 34552}, {619, 33459}, {621, 35303}, {623, 3849}, {631, 11485}, {1368, 10635}, {1595, 10642}, {1596, 11476}, {1656, 5335}, {2045, 6221}, {2046, 6398}, {3054, 14139}, {3147, 11408}, {3411, 16772}, {3526, 11488}, {3530, 11480}, {3541, 11409}, {3627, 5237}, {3628, 18582}, {3845, 19106}, {5054, 37641}, {5339, 33923}, {5340, 35018}, {5351, 15704}, {5353, 5433}, {5357, 5432}, {5362, 13747}, {5367, 7483}, {6114, 22848}, {6774, 36755}, {10124, 16644}, {10654, 12100}, {11267, 34351}, {11268, 23335}, {11308, 34540}, {11585, 18470}, {12108, 22236}, {14216, 17827}, {14813, 18538}, {14814, 18762}, {15686, 36970}, {15687, 36968}, {15690, 41120}, {15694, 37640}, {15711, 41108}, {15713, 16241}, {15759, 41113}, {15765, 35255}, {16239, 40693}, {16268, 17504}, {18585, 35256}, {18907, 19780}, {18914, 19364}, {18930, 26944}, {19710, 41122}, {19711, 41101}, {20252, 23006}, {21735, 22237}, {22847, 23005}, {35734, 36446}, {36969, 38071}, {41974, 41977}

X(42121) = {X(3),X(5334)}-harmonic conjugate of X(42122)
X(42121) = {X(4),X(42115)}-harmonic conjugate of X(42123)
X(42121) = {X(6),X(140)}-harmonic conjugate of X(42124)

X(42122) = GIBERT(2,-1,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA - 6*SB*SC : :

X(42122) lies on these lines: {3, 5334}, {4,42116}, {5, 11480}, {6, 550}, {13, 15}, {14, 12100}, {16, 548}, {17, 41978}, {20, 11485}, {61, 12103}, {140, 5321}, {376, 11486}, {382, 11488}, {395, 34200}, {397, 34754}, {398, 10646}, {531, 36521}, {546, 5238}, {547, 33417}, {549, 18581}, {621, 35304}, {1657, 5335}, {1885, 10632}, {3530, 16964}, {3627, 18582}, {3628, 5352}, {3850, 16966}, {3853, 16772}, {5066, 16241}, {5305, 19781}, {5339, 15712}, {5343, 15720}, {5349, 35018}, {5353, 15338}, {5357, 15326}, {5878, 17826}, {6756, 11475}, {7051, 15171}, {8703, 10654}, {10633, 37931}, {10634, 31829}, {10638, 18990}, {10641, 13488}, {10653, 15686}, {11812, 37835}, {12108, 33416}, {14891, 16242}, {14893, 37832}, {15681, 37640}, {15687, 16644}, {15688, 37641}, {15690, 41101}, {15691, 36968}, {15698, 33605}, {15704, 22236}, {15711, 41113}, {15759, 41108}, {16645, 17504}, {16961, 41973}, {16963, 41982}, {19711, 41120}, {34755, 41981}, {36993, 41035}, {37776, 37899}

X(42122) = {X(3),X(5334)}-harmonic conjugate of X(42121)
X(42122) = {X(4),X(42116)}-harmonic conjugate of X(42124)
X(42122) = {X(6),X(550)}-harmonic conjugate of X(42123)

X(42123) = GIBERT(2,1,-3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA + 6*SB*SC : :

X(42123) lies on these lines: {3, 5335}, {4,42115}, {5, 11481}, {6, 550}, {13, 12100}, {14, 16}, {15, 548}, {18, 41977}, {20, 11486}, {62, 12103}, {140, 5318}, {376, 11485}, {382, 11489}, {396, 34200}, {397, 10645}, {398, 34755}, {530, 36521}, {546, 5237}, {547, 33416}, {549, 18582}, {622, 35303}, {1250, 18990}, {1657, 5334}, {1885, 10633}, {3530, 16965}, {3627, 18581}, {3628, 5351}, {3850, 16967}, {3853, 16773}, {5066, 16242}, {5305, 19780}, {5340, 15712}, {5344, 15720}, {5350, 35018}, {5353, 15326}, {5357, 15338}, {5878, 17827}, {6756, 11476}, {8703, 10653}, {10632, 37931}, {10635, 31829}, {10642, 13488}, {10654, 15686}, {11812, 37832}, {12108, 33417}, {14891, 16241}, {14893, 37835}, {15171, 19373}, {15681, 37641}, {15687, 16645}, {15688, 37640}, {15690, 41100}, {15691, 36967}, {15698, 33604}, {15704, 22238}, {15711, 41112}, {15759, 41107}, {16644, 17504}, {16960, 41974}, {16962, 41982}, {19711, 41119}, {34754, 41981}, {36995, 41034}, {37775, 37899}

X(42123) = {X(3),X(5335)}-harmonic conjugate of X(42124)
X(42123) = {X(4),X(42115)}-harmonic conjugate of X(42121)
X(42123) = {X(6),X(550)}-harmonic conjugate of X(42122)

X(42124) = GIBERT(2,1,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA + 6*SB*SC : :

X(42124) lies on these lines: {2, 11485}, {3, 5335}, {4,42116}, {5, 15}, {6, 140}, {13, 8703}, {14, 15699}, {16, 396}, {17, 550}, {30, 11480}, {61, 632}, {62, 14869}, {141, 6671}, {303, 37340}, {395, 11539}, {397, 10646}, {398, 16967}, {427, 10632}, {495, 7051}, {496, 10638}, {546, 36836}, {547, 10654}, {590, 34552}, {597, 6672}, {615, 34551}, {618, 33458}, {622, 35304}, {624, 3849}, {631, 11486}, {1368, 10634}, {1595, 10641}, {1596, 11475}, {1656, 5334}, {2045, 6398}, {2046, 6221}, {3054, 14138}, {3147, 11409}, {3412, 16773}, {3526, 11489}, {3530, 11481}, {3541, 11408}, {3627, 5238}, {3628, 18581}, {3845, 19107}, {5054, 37640}, {5339, 35018}, {5340, 33923}, {5352, 15704}, {5353, 5432}, {5357, 5433}, {5362, 7483}, {5367, 13747}, {6115, 22892}, {6771, 36756}, {10124, 16645}, {10653, 12100}, {11267, 23335}, {11268, 34351}, {11307, 34541}, {11585, 18468}, {12108, 22238}, {14216, 17826}, {14813, 18762}, {14814, 18538}, {15686, 36969}, {15687, 36967}, {15690, 41119}, {15694, 37641}, {15711, 41107}, {15713, 16242}, {15759, 41112}, {15765, 35256}, {16239, 40694}, {16267, 17504}, {18585, 35255}, {18907, 19781}, {18914, 19363}, {18929, 26944}, {19710, 41121}, {19711, 41100}, {20253, 23013}, {21735, 22235}, {22513, 36763}, {22893, 23004}, {35734, 36447}, {36970, 38071}, {41973, 41978}

X(42124) = {X(3),X(5335)}-harmonic conjugate of X(42123)
X(42124) = {X(4),X(42116)}-harmonic conjugate of X(42122)
X(42124) = {X(6),X(140)}-harmonic conjugate of X(42121)

X(42125) = GIBERT(-2,2,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA - 12*SB*SC : :

X(42125) lies on these lines: {2, 33603}, {3, 5321}, {4, 11409}, {5, 5334}, {6, 13}, {15, 1656}, {16, 382}, {18, 1657}, {30, 11489}, {61, 5072}, {69, 31694}, {140, 5343}, {141, 22491}, {302, 11295}, {395, 3830}, {396, 19709}, {398, 3851}, {403, 11408}, {546, 5335}, {550, 5365}, {617, 31684}, {621, 11306}, {1250, 9668}, {1384, 37332}, {2043, 18762}, {2044, 18538}, {3091, 11542}, {3526, 11480}, {3534, 10646}, {3618, 31693}, {3619, 37351}, {3620, 37171}, {3629, 22492}, {3843, 5318}, {3845, 37641}, {5024, 37333}, {5054, 10645}, {5055, 10654}, {5066, 37640}, {5073, 5349}, {5076, 16961}, {5079, 16966}, {5094, 37775}, {5353, 10896}, {5357, 10895}, {5362, 17556}, {5367, 17532}, {5611, 33518}, {5869, 22831}, {6114, 13102}, {6144, 22496}, {6221, 18586}, {6398, 18587}, {6437, 35731}, {6782, 13103}, {7685, 41038}, {8972, 36454}, {9655, 19373}, {9761, 35697}, {10187, 15720}, {10612, 22862}, {10633, 12173}, {10638, 31479}, {10642, 18494}, {10653, 14269}, {10662, 12429}, {10676, 34780}, {11165, 22568}, {11304, 34540}, {11516, 18536}, {12817, 36968}, {13941, 36436}, {15688, 16242}, {15693, 36967}, {15765, 23259}, {16268, 34755}, {16628, 31706}, {16644, 34754}, {16773, 17800}, {16942, 22861}, {17827, 18400}, {18396, 21648}, {18585, 23249}, {32785, 36439}, {32786, 36457}, {32789, 36456}, {32790, 36438}, {33417, 36836}, {35749, 37785}, {36990, 41037}, {41041, 41054}

X(42125) = {X(3),X(5321)}-harmonic conjugate of X(42126)
X(42125) = {X(4),X(11486)}-harmonic conjugate of X(42127)
X(42125) = {X(6),X(381)}-harmonic conjugate of X(42128)

X(42126) = GIBERT(2,-2,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA - 12*SB*SC : :

X(42126) lies on these lines: {3, 5321}, {4, 11408}, {6, 382}, {13, 38335}, {14, 3534}, {15, 381}, {16, 1657}, {20, 11543}, {30, 5334}, {61, 5076}, {140, 5365}, {395, 15681}, {396, 14269}, {398, 5073}, {546, 11488}, {550, 5343}, {621, 11295}, {1656, 11480}, {2043, 35256}, {2044, 35255}, {3526, 10645}, {3627, 5335}, {3830, 5318}, {3843, 18582}, {3851, 5349}, {5054, 16967}, {5072, 16966}, {5079, 5238}, {5340, 41973}, {5353, 12953}, {5357, 12943}, {6240, 11409}, {7051, 9669}, {9654, 10638}, {10632, 37197}, {10633, 37196}, {10646, 15696}, {10653, 15684}, {12817, 16241}, {13102, 22512}, {14093, 16242}, {15685, 41113}, {15687, 37640}, {15688, 16645}, {15693, 37835}, {15695, 41120}, {16808, 22236}, {16960, 41101}, {16961, 36843}, {17800, 40694}, {17827, 34785}, {22796, 36772}, {31152, 37775}, {36993, 41041}

X(42126) = {X(3),X(5321)}-harmonic conjugate of X(42125)
X(42126) = {X(4),X(11485)}-harmonic conjugate of X(42128)
X(42126) = {X(6),X(382)}-harmonic conjugate of X(42127)

X(42127) = GIBERT(2,2,-1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA + 12*SB*SC : :

X(42127) lies on these lines: {3, 5318}, {4, 11409}, {6, 382}, {13, 3534}, {14, 38335}, {15, 1657}, {16, 381}, {20, 11542}, {30, 5335}, {62, 5076}, {140, 5366}, {395, 14269}, {396, 15681}, {397, 5073}, {546, 11489}, {550, 5344}, {622, 11296}, {1250, 9654}, {1656, 11481}, {2043, 35255}, {2044, 35256}, {3526, 10646}, {3627, 5334}, {3830, 5321}, {3843, 18581}, {3851, 5350}, {5054, 16966}, {5072, 16967}, {5079, 5237}, {5339, 41974}, {5353, 12943}, {5357, 12953}, {6240, 11408}, {9669, 19373}, {10632, 37196}, {10633, 37197}, {10645, 15696}, {10654, 15684}, {12816, 16242}, {13103, 22513}, {14093, 16241}, {15685, 41112}, {15687, 37641}, {15688, 16644}, {15693, 37832}, {15695, 41119}, {16809, 22238}, {16960, 36836}, {16961, 41100}, {17800, 40693}, {17826, 34785}, {31152, 37776}, {36995, 41040}

X(42127) = {X(3),X(5318)}-harmonic conjugate of X(42128)
X(42127) = {X(4),X(11486)}-harmonic conjugate of X(42125)
X(42127) = {X(6),X(382)}-harmonic conjugate of X(42126)

X(42128) = GIBERT(2,2,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA + 12*SB*SC : :

X(42128) lies on these lines: {2, 33602}, {3, 5318}, {4, 11408}, {5, 5335}, {6, 13}, {15, 382}, {16, 1656}, {17, 1657}, {30, 11488}, {62, 5072}, {69, 31693}, {140, 5344}, {141, 22492}, {303, 11296}, {395, 19709}, {396, 3830}, {397, 3851}, {403, 11409}, {546, 5334}, {550, 5366}, {616, 31683}, {622, 11305}, {1250, 31479}, {1384, 37333}, {2043, 18538}, {2044, 18762}, {3091, 11543}, {3526, 11481}, {3534, 10645}, {3618, 31694}, {3619, 37352}, {3620, 37170}, {3629, 22491}, {3843, 5321}, {3845, 37640}, {5024, 37332}, {5054, 10646}, {5055, 10653}, {5066, 37641}, {5073, 5350}, {5076, 16960}, {5079, 16967}, {5094, 37776}, {5353, 10895}, {5357, 10896}, {5362, 17532}, {5367, 17556}, {5615, 33517}, {5868, 22832}, {6115, 13103}, {6144, 22495}, {6221, 18587}, {6398, 18586}, {6409, 35730}, {6411, 35731}, {6783, 13102}, {7051, 9655}, {7684, 41039}, {8972, 36436}, {9668, 10638}, {9763, 35693}, {10188, 15720}, {10611, 22906}, {10632, 12173}, {10641, 18494}, {10654, 14269}, {10661, 12429}, {10675, 34780}, {11165, 22570}, {11303, 34541}, {11515, 18536}, {12816, 36967}, {13941, 36454}, {15688, 16241}, {15693, 36968}, {15765, 23249}, {16267, 34754}, {16629, 31705}, {16645, 34755}, {16772, 17800}, {16943, 22907}, {17826, 18400}, {18396, 21647}, {18585, 23259}, {32785, 36457}, {32786, 36439}, {32789, 36438}, {32790, 36456}, {33416, 36843}, {36327, 37786}, {36990, 41036}, {41040, 41055}

X(42128) = {X(3),X(5318)}-harmonic conjugate of X(42127)
X(42128) = {X(4),X(11485)}-harmonic conjugate of X(42126)
X(42128) = {X(6),X(381)}-harmonic conjugate of X(42125)

X(42129) = GIBERT(-2,2,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA - 12*SB*SC : :

X(42129) lies on these lines: {2, 11485}, {3, 5321}, {4,42115}, {5, 5335}, {6, 17}, {14, 5054}, {15, 3526}, {16, 381}, {62, 5079}, {140, 5334}, {302, 11306}, {382, 11481}, {395, 5055}, {396, 15703}, {547, 37641}, {599, 40335}, {618, 35697}, {624, 9761}, {1250, 9669}, {1594, 11409}, {1657, 10646}, {2045, 18762}, {2046, 18538}, {3090, 11542}, {3533, 22237}, {3534, 12817}, {3628, 11488}, {3843, 16773}, {3851, 5318}, {5070, 23302}, {5072, 16808}, {5076, 5237}, {5339, 10645}, {5340, 34755}, {5343, 15712}, {5365, 33923}, {5460, 9886}, {7505, 11408}, {7507, 10633}, {7615, 33474}, {9541, 15765}, {9654, 19373}, {10612, 23013}, {10632, 37453}, {10653, 19709}, {10654, 15694}, {11297, 30472}, {14813, 32785}, {14814, 32786}, {15300, 22578}, {15688, 36970}, {15693, 41122}, {15699, 37640}, {15700, 36967}, {15701, 41120}, {15723, 16241}, {16268, 16644}, {17827, 18381}, {19106, 36843}, {22236, 33417}, {36968, 38335}, {41108, 41971}

X(42129) = {X(3),X(5321)}-harmonic conjugate of X(42130)
X(42129) = {X(4),X(42115)}-harmonic conjugate of X(42131)
X(42129) = {X(6),X(1656)}-harmonic conjugate of X(42132)

X(42130) = GIBERT(2,-2,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA - 12*SB*SC : :

X(42130) lies on these lines: {3, 5321}, {4,42116}, {6, 1657}, {14, 15688}, {15, 382}, {16, 3534}, {20, 11486}, {30, 5335}, {376, 11543}, {381, 11480}, {395, 15689}, {396, 15684}, {548, 11489}, {550, 5334}, {1656, 10645}, {3146, 11542}, {3526, 16809}, {3627, 11488}, {3830, 18582}, {3843, 23302}, {5054, 36970}, {5072, 5352}, {5073, 5318}, {5076, 16808}, {5079, 33417}, {5339, 10646}, {5340, 34754}, {5343, 33923}, {5365, 15712}, {7051, 9668}, {9655, 10638}, {10653, 15685}, {10654, 15681}, {11408, 18560}, {11409, 35471}, {11475, 18494}, {11481, 15696}, {14093, 16645}, {15686, 37641}, {15693, 33416}, {15700, 37835}, {15720, 16967}, {16644, 38335}, {17826, 22802}, {19106, 22236}, {34755, 41973}

X(42130) = {X(3),X(5321)}-harmonic conjugate of X(42129)
X(42130) = {X(4),X(42116)}-harmonic conjugate of X(42132)
X(42130) = {X(6),X(1657)}-harmonic conjugate of X(42131)

X(42131) = GIBERT(2,2,-3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA + 12*SB*SC : :

X(42131) lies on these lines: {3, 5318}, {4,42115}, {6, 1657}, {13, 15688}, {15, 3534}, {16, 382}, {20, 11485}, {30, 5334}, {376, 11542}, {381, 11481}, {395, 15684}, {396, 15689}, {548, 11488}, {550, 5335}, {1250, 9655}, {1656, 10646}, {3146, 11543}, {3526, 16808}, {3627, 11489}, {3830, 18581}, {3843, 23303}, {5054, 36969}, {5072, 5351}, {5073, 5321}, {5076, 16809}, {5079, 33416}, {5339, 34755}, {5340, 10645}, {5344, 33923}, {5366, 15712}, {9668, 19373}, {10653, 15681}, {10654, 15685}, {11408, 35471}, {11409, 18560}, {11476, 18494}, {11480, 15696}, {14093, 16644}, {15686, 37640}, {15693, 33417}, {15700, 37832}, {15720, 16966}, {16645, 38335}, {17827, 22802}, {19107, 22238}, {34754, 41974}

X(42131) = {X(3),X(5318)}-harmonic conjugate of X(42132)
X(42131) = {X(4),X(42115)}-harmonic conjugate of X(42129)
X(42131) = {X(6),X(1657)}-harmonic conjugate of X(42130)

X(42132) = GIBERT(2,2,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA + 12*SB*SC : :

X(42132) lies on these lines: {2, 11486}, {3, 5318}, {4,42116}, {5, 5334}, {6, 17}, {13, 5054}, {15, 381}, {16, 3526}, {61, 5079}, {115, 36763}, {140, 5335}, {303, 11305}, {382, 11480}, {395, 15703}, {396, 5055}, {547, 37640}, {599, 40334}, {619, 35693}, {623, 9763}, {1594, 11408}, {1657, 10645}, {2045, 18538}, {2046, 18762}, {3090, 11543}, {3533, 22235}, {3534, 12816}, {3628, 11489}, {3843, 16772}, {3851, 5321}, {5070, 23303}, {5072, 16809}, {5076, 5238}, {5339, 34754}, {5340, 10646}, {5344, 15712}, {5366, 33923}, {5459, 9885}, {6771, 36771}, {7051, 9654}, {7505, 11409}, {7507, 10632}, {7615, 33475}, {9541, 18585}, {9669, 10638}, {10611, 23006}, {10633, 37453}, {10653, 15694}, {10654, 19709}, {11298, 30471}, {14813, 32786}, {14814, 32785}, {15300, 22577}, {15688, 36969}, {15693, 41121}, {15699, 37641}, {15700, 36968}, {15701, 41119}, {15723, 16242}, {16267, 16645}, {17826, 18381}, {19107, 36836}, {22238, 33416}, {36967, 38335}, {41107, 41972}

X(42132) = {X(3),X(5318)}-harmonic conjugate of X(42131)
X(42132) = {X(4),X(42116)}-harmonic conjugate of X(42130)
X(42132) = {X(6),X(1656)}-harmonic conjugate of X(42129)

X(42133) = GIBERT(-2,3,0) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*SB*SC : :

X(42133) lies on these lines: {2, 10645}, {3,42135}, {4, 6}, {14, 3543}, {15, 3091}, {16, 3146}, {18, 5059}, {20, 10646}, {30, 11489}, {193, 22575}, {376, 23303}, {381, 11488}, {382, 11543}, {383, 16942}, {395, 15682}, {396, 41099}, {472, 37643}, {546, 11485}, {621, 3620}, {622, 20080}, {2043, 32786}, {2044, 32785}, {3090, 11480}, {3523, 16967}, {3529, 11481}, {3544, 36836}, {3545, 23302}, {3619, 11304}, {3627, 11486}, {3830, 37641}, {3832, 16964}, {3839, 10654}, {3843, 11542}, {3845, 37640}, {5068, 16966}, {5238, 15022}, {5471, 31684}, {5479, 37689}, {6200, 35732}, {6437, 35740}, {6451, 14813}, {6452, 14814}, {6623, 10641}, {7051, 10591}, {7486, 33417}, {9541, 36454}, {10151, 11408}, {10304, 37835}, {10590, 10638}, {11001, 16645}, {11138, 11738}, {11541, 36843}, {15640, 36968}, {15655, 41034}, {15717, 33416}, {16002, 37517}, {16644, 41106}, {16960, 41973}, {16961, 22237}, {17578, 19106}, {25164, 31670}, {31099, 37775}, {36969, 41113}

X(42133) = {X(4),X(6)}-harmonic conjugate of X(42134)
X(42133) = {X(42135),X(42136)}-harmonic conjugate of X(3)
X(42133) = {X(42139),X(42140)}-harmonic conjugate of X(3)

X(42134) = GIBERT(2,3,0) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*SB*SC : :

X(42134) lies on these lines: {2, 10646}, {3,42137}, {4, 6}, {13, 3543}, {15, 3146}, {16, 3091}, {17, 5059}, {20, 10645}, {30, 11488}, {193, 22576}, {376, 23302}, {381, 11489}, {382, 11542}, {395, 41099}, {396, 15682}, {473, 37643}, {546, 11486}, {621, 20080}, {622, 3620}, {1080, 16943}, {1250, 10590}, {2043, 32785}, {2044, 32786}, {3090, 11481}, {3523, 16966}, {3529, 11480}, {3544, 36843}, {3545, 23303}, {3619, 11303}, {3627, 11485}, {3830, 37640}, {3832, 16965}, {3839, 10653}, {3843, 11543}, {3845, 37641}, {5068, 16967}, {5237, 15022}, {5472, 31683}, {5478, 37689}, {6396, 35732}, {6411, 35740}, {6451, 14814}, {6452, 14813}, {6623, 10642}, {7486, 33416}, {9541, 36436}, {10151, 11409}, {10304, 37832}, {10591, 19373}, {11001, 16644}, {11139, 11738}, {11541, 36836}, {15640, 36967}, {15655, 41035}, {15717, 33417}, {16001, 37517}, {16645, 41106}, {16960, 22235}, {16961, 41974}, {17578, 19107}, {25154, 31670}, {31099, 37776}, {36970, 41112}

X(42134) = {X(4),X(6)}-harmonic conjugate of X(42133)
X(42134) = {X(42137),X(42138)}-harmonic conjugate of X(3)
X(42134) = {X(42141),X(42142)}-harmonic conjugate of X(3)

X(42135) = GIBERT(-2,3,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA - 18*SB*SC : :

X(42135) lies on these lines: {3,42133}, {4, 11409}, {5, 15}, {6, 546}, {13, 23046}, {14, 3845}, {16, 3627}, {30, 11481}, {61, 3857}, {381, 5334}, {382, 11489}, {395, 15687}, {396, 38071}, {398, 3858}, {427, 37775}, {549, 16967}, {550, 5349}, {621, 31694}, {632, 10645}, {1656, 5365}, {3091, 11485}, {3628, 11480}, {3843, 5335}, {3850, 5339}, {3851, 5343}, {3856, 40693}, {3860, 41113}, {3861, 40694}, {5066, 10654}, {7051, 10593}, {8703, 12817}, {9541, 18586}, {10592, 10638}, {10646, 15704}, {10653, 14893}, {11539, 36967}, {11737, 16644}, {12101, 41120}, {12102, 22238}, {12811, 22236}, {12812, 36836}, {14269, 37641}, {15686, 16242}, {15699, 33417}, {15712, 33416}, {16960, 41108}, {20253, 22512}, {33699, 41122}, {35404, 36968}

X(42135) = {X(3),X(42133)}-harmonic conjugate of X(42136)
X(42135) = {X(4),X(11486)}-harmonic conjugate of X(42137)
X(42135) = {X(6),X(546)}-harmonic conjugate of X(42138)

X(42136) = GIBERT(2,-3,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA - 18*SB*SC : :

X(42136) lies on these lines: {3,42133}, {4, 11408}, {5, 11480}, {6, 3627}, {13, 12101}, {14, 16}, {15, 546}, {61, 12102}, {140, 5349}, {382, 5334}, {396, 14893}, {398, 19106}, {428, 37776}, {547, 36967}, {548, 23303}, {550, 18581}, {622, 36327}, {1657, 5365}, {3146, 11486}, {3530, 16967}, {3628, 10645}, {3830, 5335}, {3843, 11488}, {3845, 18582}, {3850, 23302}, {3853, 5318}, {3856, 16772}, {3857, 36836}, {3860, 37832}, {3861, 16808}, {5066, 16966}, {5073, 5343}, {5238, 12811}, {5350, 41973}, {5352, 12812}, {10151, 10632}, {10646, 12103}, {10653, 33699}, {10654, 15687}, {11481, 15704}, {11737, 16241}, {12100, 12817}, {15684, 37641}, {15686, 16645}, {15690, 16242}, {16644, 23046}, {33417, 35018}, {34200, 37835}, {37640, 38335}

X(42136) = {X(3),X(42133)}-harmonic conjugate of X(42135)
X(42136) = {X(4),X(11485)}-harmonic conjugate of X(42138)
X(42136) = {X(6),X(3627)}-harmonic conjugate of X(42137)

X(42137) = GIBERT(2,3,-1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA + 18*SB*SC : :

X(42137) lies on these lines: {3,42134}, {4, 11409}, {5, 11481}, {6, 3627}, {13, 15}, {14, 12101}, {16, 546}, {62, 12102}, {140, 5350}, {382, 5335}, {395, 14893}, {397, 19107}, {428, 37775}, {547, 36968}, {548, 23302}, {550, 18582}, {621, 35749}, {623, 36769}, {1657, 5366}, {3146, 11485}, {3530, 16966}, {3628, 10646}, {3830, 5334}, {3843, 11489}, {3845, 18581}, {3850, 23303}, {3853, 5321}, {3856, 16773}, {3857, 36843}, {3860, 37835}, {3861, 16809}, {5066, 16967}, {5073, 5344}, {5237, 12811}, {5349, 41974}, {5351, 12812}, {10151, 10633}, {10645, 12103}, {10653, 15687}, {10654, 33699}, {11480, 15704}, {11737, 16242}, {12100, 12816}, {15684, 37640}, {15686, 16644}, {15690, 16241}, {16645, 23046}, {33416, 35018}, {34200, 37832}, {37641, 38335}

X(42137) = {X(3),X(42134)}-harmonic conjugate of X(42138)
X(42137) = {X(4),X(11486)}-harmonic conjugate of X(42135)
X(42137) = {X(6),X(3627)}-harmonic conjugate of X(42136)

X(42138) = GIBERT(2,3,1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA + 18*SB*SC : :

X(42138) lies on these lines: {3,42134}, {4, 11408}, {5, 16}, {6, 546}, {13, 3845}, {14, 23046}, {15, 3627}, {30, 11480}, {62, 3857}, {381, 5335}, {382, 11488}, {395, 38071}, {396, 15687}, {397, 3858}, {427, 37776}, {549, 16966}, {550, 5350}, {622, 31693}, {632, 10646}, {1250, 10592}, {1656, 5366}, {3091, 11486}, {3628, 11481}, {3843, 5334}, {3850, 5340}, {3851, 5344}, {3856, 40694}, {3860, 41112}, {3861, 40693}, {5066, 10653}, {8703, 12816}, {9541, 18587}, {10593, 19373}, {10645, 15704}, {10654, 14893}, {11539, 36968}, {11737, 16645}, {12101, 41119}, {12102, 22236}, {12811, 22238}, {12812, 36843}, {14269, 37640}, {15686, 16241}, {15699, 33416}, {15712, 33417}, {16961, 41107}, {20252, 22513}, {33699, 41121}, {35404, 36967}

X(42138) = {X(3),X(42134)}-harmonic conjugate of X(42137)
X(42138) = {X(4),X(11485)}-harmonic conjugate of X(42136)
X(42138) = {X(6),X(546)}-harmonic conjugate of X(42135)

X(42139) = GIBERT(-2,3,2) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 9*SB*SC : :

X(42139) lies on these lines: {2, 5321}, {3,42133}, {4, 16}, {5, 5334}, {6, 3091}, {13, 33605}, {14, 3545}, {15, 3090}, {20, 23303}, {61, 3544}, {140, 5365}, {376, 19107}, {381, 5335}, {395, 3839}, {397, 3854}, {398, 5068}, {546, 11486}, {621, 625}, {622, 7615}, {624, 22491}, {631, 16967}, {1250, 5225}, {1656, 5343}, {3146, 11481}, {3522, 5349}, {3524, 33416}, {3525, 10645}, {3529, 10646}, {3543, 16645}, {3832, 5318}, {3851, 11542}, {3855, 16808}, {3858, 5344}, {5056, 5339}, {5067, 16964}, {5071, 10654}, {5187, 5362}, {5229, 19373}, {5351, 11541}, {5367, 6871}, {5863, 33627}, {6622, 10641}, {6997, 37776}, {7051, 10589}, {8889, 11475}, {10588, 10638}, {10653, 12816}, {10676, 32064}, {11001, 16242}, {11008, 22114}, {11299, 16942}, {11409, 23047}, {12817, 19708}, {15022, 22236}, {15702, 36967}, {16773, 17578}, {16963, 41972}, {17827, 41362}, {18586, 35255}, {18587, 35256}, {18945, 19364}, {22856, 41094}, {23013, 31684}, {32785, 35732}, {33412, 37178}, {33603, 41101}, {37832, 41113}

X(42139) = {X(3),X(42133)}-harmonic conjugate of X(42140)
X(42139) = {X(4),X(16)}-harmonic conjugate of X(42141)
X(42139) = {X(6),X(3091)}-harmonic conjugate of X(42142)

X(42140) = GIBERT(2,-3,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 9*SB*SC : :

X(42140) lies on these lines: {3,42133}, {4, 15}, {6, 3146}, {14, 11001}, {16, 3529}, {20, 5321}, {30, 5334}, {62, 11541}, {376, 16242}, {382, 5335}, {395, 15683}, {550, 5365}, {631, 16809}, {1327, 36446}, {1328, 36465}, {1370, 37775}, {1657, 5343}, {3090, 10645}, {3091, 11480}, {3522, 23303}, {3523, 5349}, {3524, 16967}, {3543, 5318}, {3544, 5352}, {3545, 36967}, {3627, 11485}, {3830, 11542}, {3832, 23302}, {3855, 16966}, {5059, 5339}, {5071, 33417}, {5225, 7051}, {5229, 10638}, {5367, 31295}, {5862, 36352}, {5893, 17826}, {10299, 33416}, {10646, 17538}, {10654, 15682}, {12817, 15698}, {16241, 41106}, {16964, 33703}, {18930, 21659}, {19708, 37835}, {33442, 36360}, {33443, 36361}

X(42140) = {X(3),X(42133)}-harmonic conjugate of X(42139)
X(42140) = {X(4),X(15)}-harmonic conjugate of X(42142)
X(42140) = {X(6),X(3146)}-harmonic conjugate of X(42141)

X(42141) = GIBERT(2,3,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 9*SB*SC : :

X(42141) lies on these lines: {3,42134}, {4, 16}, {6, 3146}, {13, 11001}, {15, 3529}, {20, 5318}, {30, 5335}, {61, 11541}, {376, 16241}, {382, 5334}, {396, 15683}, {550, 5366}, {631, 16808}, {1250, 5229}, {1327, 36464}, {1328, 36447}, {1370, 37776}, {1657, 5344}, {3090, 10646}, {3091, 11481}, {3522, 23302}, {3523, 5350}, {3524, 16966}, {3543, 5321}, {3544, 5351}, {3545, 36968}, {3627, 11486}, {3830, 11543}, {3832, 23303}, {3855, 16967}, {5059, 5340}, {5071, 33416}, {5225, 19373}, {5362, 31295}, {5863, 36346}, {5893, 17827}, {10299, 33417}, {10645, 17538}, {10653, 15682}, {12816, 15698}, {16242, 41106}, {16965, 33703}, {18929, 21659}, {19708, 37832}, {32785, 35740}, {33440, 36353}, {33441, 36355}

X(42141) = {X(3),X(42134)}-harmonic conjugate of X(42142)
X(42141) = {X(4),X(16)}-harmonic conjugate of X(42139)
X(42141) = {X(6),X(3146)}-harmonic conjugate of X(42140)

X(42142) = GIBERT(2,3,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 9*SB*SC : :

X(42142) lies on these lines: {2, 5318}, {3,42134}, {4, 15}, {5, 5335}, {6, 3091}, {13, 3545}, {14, 33604}, {16, 3090}, {20, 23302}, {62, 3544}, {140, 5366}, {376, 19106}, {381, 5334}, {396, 3839}, {397, 5068}, {398, 3854}, {546, 11485}, {621, 7615}, {622, 625}, {623, 22492}, {631, 16966}, {1250, 10588}, {1656, 5344}, {3146, 11480}, {3522, 5350}, {3524, 33417}, {3525, 10646}, {3529, 10645}, {3543, 16644}, {3832, 5321}, {3851, 11543}, {3855, 16809}, {3858, 5343}, {5056, 5340}, {5067, 16965}, {5071, 10653}, {5187, 5367}, {5225, 10638}, {5229, 7051}, {5352, 11541}, {5362, 6871}, {5478, 36771}, {5862, 33626}, {6622, 10642}, {6997, 37775}, {8889, 11476}, {10589, 19373}, {10654, 12817}, {10675, 32064}, {11001, 16241}, {11008, 22113}, {11300, 16943}, {11408, 23047}, {12816, 19708}, {15022, 22238}, {15702, 36968}, {16772, 17578}, {16962, 41971}, {17826, 41362}, {18586, 35256}, {18587, 35255}, {18945, 19363}, {22900, 41098}, {23006, 31683}, {32786, 35732}, {33413, 37177}, {33602, 41100}, {37835, 41112}

X(42142) = {X(3),X(42134)}-harmonic conjugate of X(42141)
X(42142) = {X(4),X(15)}-harmonic conjugate of X(42140)
X(42142) = {X(6),X(3091)}-harmonic conjugate of X(42139)

X(42143) = GIBERT(-2,3,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA - 18*SB*SC : :

X(42143) lies on these lines: {2, 33603}, {5, 6}, {13, 11737}, {14, 547}, {15, 3628}, {16, 546}, {18, 3850}, {30, 10646}, {61, 12812}, {62, 12811}, {140, 5321}, {187, 31706}, {302, 31694}, {381, 11489}, {395, 5066}, {396, 10109}, {397, 16961}, {398, 16966}, {548, 19107}, {623, 34573}, {624, 3631}, {632, 11480}, {1656, 5334}, {2041, 6452}, {2042, 6451}, {3055, 6114}, {3090, 11485}, {3091, 11486}, {3530, 33416}, {3589, 5460}, {3614, 5357}, {3619, 11306}, {3627, 11481}, {3630, 34508}, {3845, 16645}, {3851, 5335}, {3857, 22238}, {3859, 16965}, {3860, 36969}, {3861, 16773}, {5055, 11488}, {5237, 12102}, {5349, 33923}, {5353, 7173}, {5362, 17533}, {5365, 15720}, {5367, 17530}, {5459, 6329}, {6200, 35738}, {6669, 35020}, {6782, 20252}, {6783, 10612}, {10297, 18470}, {10633, 23047}, {10641, 37942}, {10653, 38071}, {10654, 15699}, {11812, 36967}, {12100, 36970}, {12101, 36968}, {12817, 15759}, {14892, 16268}, {16239, 16964}, {16626, 41407}, {16644, 41120}, {18586, 32785}, {18587, 32786}, {19709, 37641}, {37454, 37775}

X(42143) = {X(5),X(6)}-harmonic conjugate of X(42146)

X(42144) = GIBERT(2,-3,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA - 18*SB*SC : :

X(42144) lies on these lines: {5, 10645}, {6, 30}, {13, 35404}, {14, 15686}, {15, 3627}, {16, 15704}, {18, 550}, {20, 11543}, {382, 11542}, {395, 19710}, {396, 12816}, {531, 3630}, {546, 11480}, {548, 18581}, {549, 16809}, {1657, 5334}, {2043, 6452}, {2044, 6451}, {3146, 11485}, {3529, 11486}, {3534, 11489}, {3619, 11295}, {3830, 11488}, {3845, 23302}, {3853, 18582}, {3857, 5352}, {3858, 16966}, {5073, 5335}, {5318, 34754}, {5349, 15712}, {5350, 16960}, {8703, 23303}, {11481, 12103}, {12101, 16644}, {12102, 36836}, {12817, 15711}, {15685, 37641}, {15687, 16808}, {15690, 16645}, {16241, 23046}, {16964, 34755}

X(42144) = reflection of X(42145) in X(6)

X(42145) = GIBERT(2,3,-3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA + 18*SB*SC : :

X(42145) lies on these lines: {5, 10646}, {6, 30}, {13, 15686}, {14, 35404}, {15, 15704}, {16, 3627}, {17, 550}, {20, 11542}, {382, 11543}, {395, 12817}, {396, 19710}, {530, 3630}, {546, 11481}, {548, 18582}, {549, 16808}, {1657, 5335}, {2043, 6451}, {2044, 6452}, {3146, 11486}, {3529, 11485}, {3534, 11488}, {3619, 11296}, {3830, 11489}, {3845, 23303}, {3853, 18581}, {3857, 5351}, {3858, 16967}, {5073, 5334}, {5321, 34755}, {5349, 16961}, {5350, 15712}, {8703, 23302}, {11480, 12103}, {12101, 16645}, {12102, 36843}, {12816, 15711}, {15685, 37640}, {15687, 16809}, {15690, 16644}, {16242, 23046}, {16965, 34754}

X(42145) = reflection of X(42144) in X(6)

X(42146) = GIBERT(2,3,3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA + 18*SB*SC : :

X(42146) lies on these lines: {2, 33602}, {5, 6}, {13, 547}, {14, 11737}, {15, 546}, {16, 3628}, {17, 3850}, {30, 10645}, {61, 12811}, {62, 12812}, {140, 5318}, {187, 31705}, {303, 31693}, {381, 11488}, {395, 10109}, {396, 5066}, {397, 16967}, {398, 16960}, {548, 19106}, {623, 3631}, {624, 34573}, {632, 11481}, {1656, 5335}, {2041, 6451}, {2042, 6452}, {3055, 6115}, {3090, 11486}, {3091, 11485}, {3530, 33417}, {3589, 5459}, {3614, 5353}, {3619, 11305}, {3627, 11480}, {3630, 34509}, {3845, 16644}, {3851, 5334}, {3857, 22236}, {3859, 16964}, {3860, 36970}, {3861, 16772}, {5055, 11489}, {5238, 12102}, {5350, 33923}, {5357, 7173}, {5362, 17530}, {5366, 15720}, {5367, 17533}, {5460, 6329}, {6396, 35738}, {6670, 35019}, {6782, 10611}, {6783, 20253}, {10297, 18468}, {10632, 23047}, {10642, 37942}, {10653, 15699}, {10654, 38071}, {11812, 36968}, {12100, 36969}, {12101, 36967}, {12816, 15759}, {14892, 16267}, {16239, 16965}, {16627, 41406}, {16645, 41119}, {18586, 32786}, {18587, 32785}, {19709, 37640}, {37454, 37776}

X(42146) = {X(5),X(6)}-harmonic conjugate of X(42143)

X(42147) = GIBERT(3,-1,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA - 2*SB*SC : :

X(42147) lies on these lines: {2, 5339}, {3, 395}, {4, 396}, {5, 15}, {6, 20}, {13, 3627}, {14, 140}, {16, 548}, {17, 546}, {18, 549}, {30, 61}, {62, 550}, {203, 15171}, {298, 32820}, {376, 22238}, {381, 5349}, {382, 5318}, {383, 22532}, {389, 36980}, {531, 635}, {617, 11290}, {621, 11307}, {628, 11304}, {631, 5334}, {632, 37835}, {633, 11299}, {1327, 18587}, {1328, 18586}, {1657, 10653}, {1885, 8740}, {1906, 10641}, {1907, 11475}, {2041, 3070}, {2042, 3071}, {2043, 41946}, {2044, 41945}, {2307, 6284}, {2883, 11243}, {3090, 5343}, {3091, 16644}, {3104, 32448}, {3106, 32516}, {3146, 5340}, {3411, 10646}, {3412, 3853}, {3522, 36843}, {3523, 16645}, {3526, 18581}, {3528, 11481}, {3530, 10645}, {3545, 5365}, {3628, 16241}, {3832, 11488}, {3843, 18582}, {3845, 16962}, {3850, 37832}, {3861, 16808}, {4325, 5357}, {4330, 5353}, {5054, 41113}, {5066, 41943}, {5237, 8703}, {5305, 41407}, {5335, 33703}, {5344, 15682}, {5351, 33923}, {5460, 6674}, {5471, 37512}, {5479, 22893}, {5480, 36993}, {6109, 16002}, {6561, 35740}, {6694, 37352}, {6772, 41020}, {7005, 18990}, {7051, 37722}, {7127, 15338}, {7685, 10616}, {8259, 10613}, {8260, 22843}, {8359, 12154}, {8918, 11549}, {8929, 11586}, {10617, 21158}, {10638, 15888}, {11137, 34148}, {11486, 15696}, {11489, 15717}, {11539, 41122}, {11626, 16836}, {12007, 36995}, {12100, 16268}, {12103, 36968}, {12817, 38071}, {13567, 19772}, {14138, 22795}, {14892, 41978}, {14893, 41121}, {15684, 41112}, {15686, 41100}, {15687, 16267}, {15694, 41120}, {15712, 16242}, {16239, 16967}, {16963, 34200}, {17504, 41944}, {19773, 23292}, {20415, 31710}, {20429, 22906}, {21156, 22847}, {21850, 36757}, {22114, 33459}, {33387, 36330}, {34153, 36209}, {34508, 35304}, {38335, 41119}, {41021, 41746}

X(42147) = {X(6),X(20)}-harmonic conjugate of X(42148)

X(42148) = GIBERT(3,1,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA + 2*SB*SC : :

X(42148) lies on these lines: {2, 5340}, {3, 396}, {4, 395}, {5, 16}, {6, 20}, {13, 140}, {14, 3627}, {15, 548}, {17, 549}, {18, 546}, {30, 62}, {61, 550}, {202, 15171}, {299, 32820}, {376, 22236}, {381, 5350}, {382, 5321}, {389, 36978}, {530, 636}, {616, 11289}, {622, 11308}, {627, 11303}, {631, 5335}, {632, 37832}, {634, 11300}, {1080, 22531}, {1250, 15888}, {1327, 18586}, {1328, 18587}, {1657, 10654}, {1885, 8739}, {1906, 10642}, {1907, 11476}, {2041, 3071}, {2042, 3070}, {2043, 41945}, {2044, 41946}, {2307, 15326}, {2883, 11244}, {3090, 5344}, {3091, 16645}, {3105, 32448}, {3107, 32516}, {3146, 5339}, {3411, 3853}, {3412, 10645}, {3522, 36836}, {3523, 16644}, {3526, 18582}, {3528, 11480}, {3530, 10646}, {3545, 5366}, {3628, 16242}, {3832, 11489}, {3843, 18581}, {3845, 16963}, {3850, 37835}, {3861, 16809}, {4325, 5353}, {4330, 5357}, {5054, 41112}, {5066, 41944}, {5238, 8703}, {5305, 41406}, {5334, 33703}, {5343, 15682}, {5352, 33923}, {5459, 6673}, {5472, 37512}, {5478, 22847}, {5480, 36995}, {6108, 16001}, {6695, 37351}, {6775, 41021}, {7006, 18990}, {7127, 7354}, {7684, 10617}, {8259, 22890}, {8260, 10614}, {8359, 12155}, {8919, 11537}, {8930, 15743}, {10616, 21159}, {11134, 34148}, {11485, 15696}, {11488, 15717}, {11539, 41121}, {11624, 16836}, {12007, 36993}, {12100, 16267}, {12103, 36967}, {12816, 38071}, {13567, 19773}, {14139, 22794}, {14892, 41977}, {14893, 41122}, {15684, 41113}, {15686, 41101}, {15687, 16268}, {15694, 41119}, {15712, 16241}, {16239, 16966}, {16962, 34200}, {17504, 41943}, {19373, 37722}, {19772, 23292}, {20416, 31709}, {20428, 22862}, {21157, 22893}, {21850, 36758}, {22113, 33458}, {33386, 35752}, {34153, 36208}, {34509, 35303}, {38335, 41120}, {41020, 41745}

X(42148) = {X(6),X(20)}-harmonic conjugate of X(42147)

X(42149) = GIBERT(-3,1,3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 2*SB*SC : :

X(42149) lies on these lines: {2, 17}, {3, 395}, {4, 16}, {5, 5340}, {6, 140}, {13, 3090}, {14, 20}, {15, 3523}, {30, 36843}, {39, 22714}, {61, 631}, {69, 6672}, {202, 3085}, {203, 7288}, {298, 11308}, {302, 315}, {371, 2045}, {372, 2046}, {376, 5351}, {381, 5350}, {396, 3526}, {397, 1656}, {471, 11547}, {485, 14813}, {486, 14814}, {499, 7127}, {547, 41119}, {549, 22236}, {550, 5339}, {616, 22511}, {617, 22114}, {618, 37177}, {621, 39554}, {628, 3181}, {632, 16644}, {633, 7793}, {635, 37178}, {1657, 5321}, {2041, 35813}, {2042, 35812}, {2043, 36452}, {2044, 36470}, {3068, 3390}, {3069, 3389}, {3086, 7006}, {3091, 5366}, {3104, 6194}, {3106, 12251}, {3146, 36968}, {3147, 8740}, {3205, 11003}, {3364, 9540}, {3365, 13935}, {3522, 5334}, {3524, 5238}, {3525, 37640}, {3528, 36967}, {3529, 36970}, {3530, 36836}, {3533, 11488}, {3541, 8739}, {3543, 41122}, {3545, 41100}, {3618, 6694}, {3832, 36969}, {3851, 5318}, {5054, 16772}, {5055, 41112}, {5056, 5335}, {5059, 5365}, {5067, 37832}, {5068, 5344}, {5071, 41107}, {5073, 5349}, {5206, 5471}, {5218, 7005}, {5352, 15717}, {5463, 36251}, {5611, 10617}, {5613, 22848}, {5862, 33386}, {5868, 41034}, {6114, 41021}, {6770, 16530}, {6775, 20416}, {6782, 41020}, {7803, 11290}, {9744, 37464}, {9753, 37463}, {9761, 37341}, {10187, 22235}, {10299, 10645}, {10303, 16241}, {10304, 41108}, {10359, 36759}, {10614, 37825}, {11244, 14216}, {11302, 33459}, {11305, 33474}, {11480, 15712}, {11485, 15720}, {11626, 13340}, {12154, 35287}, {13084, 37172}, {13103, 22847}, {13846, 34551}, {13847, 34552}, {14136, 36770}, {14137, 14541}, {15444, 36300}, {15445, 36305}, {15692, 41101}, {15702, 16962}, {15709, 41943}, {16529, 40898}, {16630, 33412}, {20081, 32466}, {20415, 41745}, {21735, 41973}, {22491, 35230}, {33389, 39647}, {34508, 37173}, {36980, 37484}

X(42149) = {X(6),X(140)}-harmonic conjugate of X(42152)
X(42149) = {X(3),X(398)}-harmonic conjugate of X(42150)
X(42149) = {X(4),X(16)}-harmonic conjugate of X(42151)

X(42150) = GIBERT(3,-1,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 2*SB*SC : :

X(42150) lies on these lines: {2, 5238}, {3, 395}, {4, 15}, {5, 36836}, {6, 550}, {13, 3146}, {14, 631}, {16, 3522}, {18, 3523}, {20, 61}, {30, 5340}, {62, 376}, {140, 5339}, {203, 4294}, {381, 16772}, {382, 396}, {397, 1657}, {531, 37172}, {546, 16644}, {548, 22238}, {549, 41113}, {634, 9939}, {1656, 5321}, {2307, 4302}, {3090, 16241}, {3091, 36970}, {3104, 7709}, {3106, 32522}, {3364, 6460}, {3365, 6459}, {3389, 9541}, {3411, 21734}, {3412, 33703}, {3524, 41108}, {3525, 37835}, {3528, 5237}, {3529, 16965}, {3530, 16645}, {3533, 16967}, {3543, 16962}, {3642, 37177}, {3832, 37832}, {3839, 41943}, {3851, 5349}, {4293, 7005}, {5054, 41120}, {5056, 5365}, {5059, 5335}, {5068, 16966}, {5073, 5318}, {5286, 41407}, {5344, 19106}, {5351, 10304}, {5471, 15515}, {5862, 22844}, {5868, 8721}, {5878, 11243}, {6111, 15005}, {6560, 14814}, {6561, 14813}, {6695, 37173}, {8703, 36843}, {9754, 37464}, {10299, 11489}, {10646, 21735}, {11481, 33923}, {11543, 15712}, {12154, 33215}, {12244, 36208}, {15682, 16267}, {15683, 41107}, {15692, 16268}, {15698, 41944}, {15702, 41122}, {15717, 16242}, {15720, 23303}, {16960, 22235}, {16963, 19708}, {17538, 36968}, {36772, 41020}, {36980, 37481}

X(42150) = {X(6),X(550)}-harmonic conjugate of X(42151)
X(42150) = {X(3),X(398)}-harmonic conjugate of X(42149)
X(42150) = {X(4),X(15)}-harmonic conjugate of X(42152)

X(42151) = GIBERT(3,1,-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 2*SB*SC : :

X(42151) lies on these lines: {2, 5237}, {3, 396}, {4, 16}, {5, 36843}, {6, 550}, {13, 631}, {14, 3146}, {15, 3522}, {17, 3523}, {20, 62}, {30, 5339}, {61, 376}, {140, 5340}, {202, 4294}, {381, 16773}, {382, 395}, {398, 1657}, {530, 37173}, {546, 16645}, {548, 22236}, {549, 41112}, {633, 9939}, {1656, 5318}, {3090, 16242}, {3091, 36969}, {3105, 7709}, {3107, 32522}, {3364, 9541}, {3389, 6460}, {3390, 6459}, {3411, 33703}, {3412, 21734}, {3524, 41107}, {3525, 37832}, {3528, 5238}, {3529, 16964}, {3530, 16644}, {3533, 16966}, {3543, 16963}, {3643, 37178}, {3832, 37835}, {3839, 41944}, {3851, 5350}, {4293, 7006}, {4299, 7127}, {5054, 41119}, {5056, 5366}, {5059, 5334}, {5068, 16967}, {5073, 5321}, {5286, 41406}, {5343, 19107}, {5352, 10304}, {5472, 15515}, {5863, 22845}, {5869, 8721}, {5878, 11244}, {6110, 15005}, {6560, 14813}, {6561, 14814}, {6694, 37172}, {8703, 36836}, {9754, 37463}, {10299, 11488}, {10645, 21735}, {11480, 33923}, {11542, 15712}, {12155, 33215}, {12244, 36209}, {15682, 16268}, {15683, 41108}, {15692, 16267}, {15698, 41943}, {15702, 41121}, {15717, 16241}, {15720, 23302}, {16961, 22237}, {16962, 19708}, {17538, 36967}, {36978, 37481}

X(42151) = {X(6),X(550)}-harmonic conjugate of X(42150)
X(42151) = {X(3),X(397)}-harmonic conjugate of X(42152)
X(42151) = {X(4),X(16)}-harmonic conjugate of X(42149)

X(42152) = GIBERT(3,1,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 2*SB*SC : :

X(42152) lies on these lines: {2, 18}, {3, 396}, {4, 15}, {5, 5339}, {6, 140}, {13, 20}, {14, 3090}, {16, 3523}, {30, 36836}, {39, 22715}, {62, 631}, {69, 6671}, {202, 7288}, {203, 3085}, {299, 11307}, {303, 315}, {371, 2046}, {372, 2045}, {376, 5352}, {381, 5349}, {395, 3526}, {398, 1656}, {470, 11547}, {485, 14814}, {486, 14813}, {498, 2307}, {547, 41120}, {549, 22238}, {550, 5340}, {616, 22113}, {617, 22510}, {619, 37178}, {622, 39555}, {627, 3180}, {632, 16645}, {634, 7793}, {636, 37177}, {1657, 5318}, {2041, 35812}, {2042, 35813}, {2043, 36453}, {2044, 36469}, {3068, 3365}, {3069, 3364}, {3086, 7005}, {3091, 5365}, {3105, 6194}, {3107, 12251}, {3146, 36967}, {3147, 8739}, {3206, 11003}, {3389, 9540}, {3390, 13935}, {3522, 5335}, {3524, 5237}, {3525, 37641}, {3528, 36968}, {3529, 36969}, {3530, 36843}, {3533, 11489}, {3541, 8740}, {3543, 41121}, {3545, 41101}, {3618, 6695}, {3832, 36970}, {3851, 5321}, {5054, 16773}, {5055, 41113}, {5056, 5334}, {5059, 5366}, {5067, 37835}, {5068, 5343}, {5071, 41108}, {5073, 5350}, {5206, 5472}, {5218, 7006}, {5351, 15717}, {5464, 36252}, {5615, 10616}, {5617, 22892}, {5863, 33387}, {5869, 41035}, {6115, 41020}, {6772, 20415}, {6773, 16529}, {6783, 41021}, {7803, 11289}, {9744, 37463}, {9753, 37464}, {9763, 37340}, {10188, 22237}, {10299, 10646}, {10303, 16242}, {10304, 41107}, {10359, 36760}, {10613, 37824}, {11243, 14216}, {11301, 33458}, {11306, 33475}, {11481, 15712}, {11486, 15720}, {11624, 13340}, {12155, 35287}, {13083, 37173}, {13102, 22893}, {13846, 34552}, {13847, 34551}, {14136, 14540}, {15444, 36304}, {15445, 36301}, {15692, 41100}, {15702, 16963}, {15709, 41944}, {16530, 40899}, {16631, 33413}, {20081, 32465}, {20416, 41746}, {21735, 41974}, {22492, 35229}, {33388, 39647}, {34509, 37172}, {36763, 41022}, {36978, 37484}

X(42152) = {X(6),X(140)}-harmonic conjugate of X(42149)
X(42152) = {X(3),X(397)}-harmonic conjugate of X(42151)
X(42152) = {X(4),X(15)}-harmonic conjugate of X(42150)

X(42153) = GIBERT(-3,2,2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA - 4*SB*SC : :

X(42153) lies on these lines: {2, 398}, {3, 14}, {4, 395}, {5, 6}, {13, 3851}, {15, 3526}, {16, 382}, {17, 5055}, {20, 5321}, {30, 36843}, {61, 1656}, {62, 381}, {140, 10654}, {202, 9654}, {299, 22114}, {302, 1975}, {376, 5343}, {383, 5868}, {396, 3090}, {397, 3091}, {546, 10653}, {549, 41113}, {569, 11137}, {599, 636}, {619, 11310}, {621, 11308}, {623, 11311}, {624, 33464}, {627, 9761}, {629, 11301}, {631, 5334}, {634, 5858}, {635, 3763}, {1151, 2042}, {1152, 2041}, {1250, 9670}, {1657, 5237}, {3106, 13108}, {3146, 5349}, {3364, 8976}, {3365, 13951}, {3366, 35812}, {3367, 35813}, {3389, 13785}, {3390, 13665}, {3411, 3843}, {3412, 16966}, {3529, 5365}, {3534, 5351}, {3643, 35020}, {3830, 16963}, {3832, 5318}, {3839, 5350}, {3855, 5335}, {5054, 5238}, {5056, 37640}, {5067, 23302}, {5070, 11485}, {5073, 36968}, {5079, 37832}, {5352, 15720}, {5366, 41099}, {5464, 33386}, {5471, 7746}, {5617, 36251}, {5865, 20426}, {5869, 6773}, {6114, 37637}, {6425, 35738}, {6695, 11298}, {6770, 22847}, {7005, 31479}, {7006, 9669}, {7127, 10896}, {7486, 11488}, {7507, 8739}, {7685, 36990}, {8260, 41039}, {8836, 37638}, {8919, 18777}, {9657, 19373}, {9781, 36978}, {10539, 11134}, {10613, 40334}, {10614, 41037}, {10617, 36993}, {10646, 15696}, {11315, 33444}, {11316, 33445}, {11412, 36980}, {11459, 11626}, {11737, 41119}, {12817, 15684}, {13103, 16530}, {13846, 18586}, {13847, 18587}, {14137, 37825}, {14269, 41100}, {15534, 34509}, {15694, 41101}, {15702, 33605}, {15703, 16962}, {17800, 19107}, {18440, 36758}, {19780, 20428}, {20429, 31706}, {21360, 36368}, {22491, 37341}, {22845, 33415}, {33459, 37171}, {33474, 37172}, {36209, 38724}, {36436, 41951}, {36454, 41952}, {38071, 41112}

X(42153) = {X(3),X(5339)}-harmonic conjugate of X(42154)
X(42153) = {X(4),X(22238)}-harmonic conjugate of X(42155)
X(42153) = {X(5),X(6)}-harmonic conjugate of X(42156)
X(42153) = {X(14),X(16645)}-harmonic conjugate of X(42154)

X(42154) = GIBERT(3,-2,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA - 4*SB*SC : :

X(42154) lies on these lines: {2, 5321}, {3, 14}, {4, 396}, {5, 36836}, {6, 30}, {13, 3830}, {15, 381}, {16, 3534}, {17, 3843}, {20, 398}, {61, 382}, {62, 1657}, {203, 9668}, {298, 1975}, {376, 395}, {383, 36993}, {397, 3146}, {530, 15534}, {531, 599}, {532, 6144}, {533, 40341}, {542, 23013}, {549, 18581}, {550, 36843}, {590, 36445}, {615, 36463}, {616, 5858}, {617, 7784}, {619, 11306}, {621, 11299}, {622, 5859}, {623, 11301}, {631, 5343}, {1080, 41038}, {1151, 2044}, {1152, 2043}, {1525, 11243}, {1656, 5238}, {2307, 12953}, {3090, 5365}, {3091, 5349}, {3107, 22728}, {3180, 7823}, {3522, 16773}, {3524, 23303}, {3526, 5352}, {3543, 5318}, {3545, 23302}, {3627, 40693}, {3642, 3763}, {3818, 22906}, {3839, 11488}, {3845, 18582}, {5054, 10645}, {5055, 16241}, {5064, 11475}, {5073, 16965}, {5077, 12154}, {5237, 15696}, {5335, 15682}, {5350, 17578}, {5463, 15300}, {5473, 9115}, {5479, 41041}, {5890, 36980}, {5978, 7778}, {5979, 9766}, {6108, 19781}, {6109, 25164}, {6409, 34551}, {6410, 34552}, {6770, 41039}, {6772, 36990}, {7005, 9655}, {7051, 11238}, {8703, 11543}, {8739, 37196}, {9117, 41043}, {9749, 41060}, {9761, 35931}, {10304, 11489}, {10613, 36992}, {10638, 11237}, {10646, 15688}, {11092, 37638}, {11137, 13352}, {11486, 15681}, {11542, 15687}, {12100, 41120}, {12101, 41119}, {12817, 16966}, {14093, 41944}, {14269, 16808}, {15684, 19106}, {15685, 41100}, {15689, 16963}, {15693, 41122}, {15694, 16967}, {15695, 16961}, {15701, 33416}, {15703, 33417}, {16267, 34754}, {16960, 35403}, {18586, 23261}, {18587, 23251}, {19708, 33605}, {21467, 36185}, {21734, 22237}, {22491, 35304}, {22532, 22893}, {22891, 35229}, {33517, 36383}, {33699, 41112}, {36208, 38790}, {36772, 41042}, {36962, 41020}, {41023, 41746}, {41121, 41971}

X(42154) = reflection of X(42155) in X(6)
X(42154) = {X(3),X(5339)}-harmonic conjugate of X(42153)
X(42154) = {X(4),X(22236)}-harmonic conjugate of X(42156)
X(42154) = {X(14),X(16645)}-harmonic conjugate of X(42153)

X(42155) = GIBERT(3,2,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA + 4*SB*SC : :

X(42155) lies on these lines: {2, 5318}, {3, 13}, {4, 395}, {5, 36843}, {6, 30}, {14, 3830}, {15, 3534}, {16, 381}, {18, 3843}, {20, 397}, {61, 1657}, {62, 382}, {202, 9668}, {299, 1975}, {376, 396}, {383, 41039}, {398, 3146}, {530, 599}, {531, 15534}, {532, 40341}, {533, 6144}, {542, 23006}, {549, 18582}, {550, 36836}, {590, 36463}, {615, 36445}, {616, 7784}, {617, 5859}, {618, 11305}, {621, 5858}, {622, 11300}, {624, 11302}, {631, 5344}, {1080, 36995}, {1151, 2043}, {1152, 2044}, {1250, 11237}, {1524, 11244}, {1656, 5237}, {3090, 5366}, {3091, 5350}, {3106, 22728}, {3181, 7823}, {3522, 16772}, {3524, 23302}, {3526, 5351}, {3543, 5321}, {3545, 23303}, {3627, 40694}, {3643, 3763}, {3818, 22862}, {3839, 11489}, {3845, 18581}, {5054, 10646}, {5055, 16242}, {5064, 11476}, {5073, 16964}, {5077, 12155}, {5238, 15696}, {5334, 15682}, {5349, 17578}, {5464, 15300}, {5474, 9117}, {5478, 41040}, {5890, 36978}, {5978, 9766}, {5979, 7778}, {6108, 25154}, {6109, 19780}, {6409, 34552}, {6410, 34551}, {6773, 41038}, {6775, 36990}, {7006, 9655}, {7127, 12943}, {8703, 11542}, {8740, 37196}, {9115, 41042}, {9750, 41061}, {9763, 35932}, {10304, 11488}, {10614, 36994}, {10645, 15688}, {11078, 37638}, {11134, 13352}, {11238, 19373}, {11485, 15681}, {11543, 15687}, {12100, 41119}, {12101, 41120}, {12816, 16967}, {14093, 41943}, {14269, 16809}, {15684, 19107}, {15685, 41101}, {15689, 16962}, {15693, 41121}, {15694, 16966}, {15695, 16960}, {15701, 33417}, {15703, 33416}, {16268, 34755}, {16961, 35403}, {18586, 23251}, {18587, 23261}, {19708, 33604}, {21466, 36186}, {21734, 22235}, {22492, 35303}, {22531, 22847}, {22846, 35230}, {33518, 36382}, {33699, 41113}, {36209, 38790}, {36961, 41021}, {41022, 41745}, {41122, 41972}

X(42155) = reflection of X(42154) in X(6)
X(42155) = {X(3),X(5340)}-harmonic conjugate of X(42156)
X(42155) = {X(4),X(22238)}-harmonic conjugate of X(42153)
X(42155) = {X(13),X(16644)}-harmonic conjugate of X(42156)

X(42156) = GIBERT(3,2,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA + 4*SB*SC : :

X(42156) lies on these lines: {2, 397}, {3, 13}, {4, 396}, {5, 6}, {14, 3851}, {15, 382}, {16, 3526}, {18, 5055}, {20, 5318}, {30, 36836}, {61, 381}, {62, 1656}, {140, 10653}, {203, 9654}, {298, 22113}, {303, 1975}, {376, 5344}, {395, 3090}, {398, 3091}, {546, 10654}, {549, 41112}, {569, 11134}, {599, 635}, {615, 35740}, {618, 11309}, {622, 11307}, {623, 33465}, {624, 11312}, {628, 9763}, {630, 11302}, {631, 5335}, {633, 5859}, {636, 3763}, {1080, 5869}, {1151, 2041}, {1152, 2042}, {1657, 5238}, {2307, 10895}, {3107, 13108}, {3146, 5350}, {3364, 13785}, {3365, 13665}, {3389, 8976}, {3390, 13951}, {3391, 35812}, {3392, 35813}, {3411, 16967}, {3412, 3843}, {3529, 5366}, {3534, 5352}, {3642, 35019}, {3830, 16962}, {3832, 5321}, {3839, 5349}, {3855, 5334}, {5054, 5237}, {5056, 37641}, {5067, 23303}, {5070, 11486}, {5073, 36967}, {5079, 37835}, {5351, 15720}, {5365, 41099}, {5463, 33387}, {5472, 7746}, {5613, 36252}, {5864, 20425}, {5868, 6770}, {6115, 37637}, {6426, 35738}, {6694, 11297}, {6773, 22893}, {7005, 9669}, {7006, 31479}, {7051, 9657}, {7486, 11489}, {7507, 8740}, {7684, 36990}, {8259, 41038}, {8838, 37638}, {8918, 18776}, {9670, 10638}, {9781, 36980}, {10539, 11137}, {10613, 41036}, {10614, 40335}, {10616, 36995}, {10645, 15696}, {11315, 33446}, {11316, 33447}, {11412, 36978}, {11459, 11624}, {11737, 41120}, {12816, 15684}, {13102, 16529}, {13846, 18587}, {13847, 18586}, {14136, 37824}, {14269, 41101}, {15534, 34508}, {15694, 41100}, {15702, 33604}, {15703, 16963}, {17800, 19106}, {18440, 36757}, {19781, 20429}, {20428, 31705}, {21359, 36366}, {22492, 37340}, {22844, 33414}, {33458, 37170}, {33475, 37173}, {36208, 38724}, {36436, 41952}, {36454, 41951}, {38071, 41113}

X(42156) = {X(3),X(5340)}-harmonic conjugate of X(42155)
X(42156) = {X(4),X(22236)}-harmonic conjugate of X(42154)
X(42156) = {X(5),X(6)}-harmonic conjugate of X(42153)
X(42156) = {X(13),X(16644)}-harmonic conjugate of X(42155)

X(42157) = GIBERT(3,-2,3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 4*SB*SC : :

X(42157) lies on these lines: {2, 5352}, {3, 14}, {4, 15}, {5, 5238}, {6, 1657}, {13, 382}, {16, 398}, {20, 62}, {30, 61}, {140, 5321}, {202, 4299}, {203, 6284}, {376, 5237}, {381, 36836}, {395, 548}, {396, 3627}, {531, 633}, {546, 16772}, {616, 22844}, {617, 636}, {628, 21360}, {631, 37835}, {635, 11299}, {1607, 3439}, {1614, 3201}, {1656, 11480}, {2777, 36208}, {2794, 5869}, {3104, 11257}, {3105, 22696}, {3146, 5344}, {3205, 34148}, {3364, 6560}, {3365, 6561}, {3389, 14813}, {3390, 14814}, {3411, 15696}, {3522, 5334}, {3523, 5343}, {3524, 41122}, {3529, 10653}, {3534, 22238}, {3543, 12816}, {3830, 16962}, {3843, 16644}, {3845, 41943}, {3850, 23302}, {3851, 16966}, {3855, 12821}, {4302, 7006}, {4324, 7127}, {4857, 7051}, {5055, 12817}, {5059, 41974}, {5073, 5340}, {5254, 41407}, {5270, 10638}, {5318, 34754}, {5350, 11542}, {5464, 11304}, {5470, 9880}, {5473, 5864}, {6102, 36981}, {6694, 11303}, {6777, 13188}, {6780, 37825}, {7005, 7354}, {7755, 19781}, {7814, 30471}, {7833, 12154}, {8172, 10263}, {8175, 15445}, {8703, 16268}, {8739, 35471}, {8740, 18560}, {10187, 15720}, {10304, 41113}, {10619, 10678}, {10658, 30714}, {11082, 32535}, {11137, 37495}, {11243, 22802}, {11295, 36329}, {11489, 21735}, {11543, 33923}, {12121, 36209}, {12203, 36760}, {13630, 36980}, {14862, 30402}, {15681, 41100}, {15687, 41121}, {15692, 41120}, {15712, 23303}, {16529, 36962}, {17538, 37641}, {19778, 40712}, {22796, 36782}, {31670, 36757}, {33703, 37640}, {34508, 35931}

X(42157) = {X(6),X(1657)}-harmonic conjugate of X(421

X(42158) = GIBERT(3,2,-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 4*SB*SC : :

X(42158) lies on these lines: {2, 5351}, {3, 13}, {4, 16}, {5, 5237}, {6, 1657}, {14, 382}, {15, 397}, {20, 61}, {30, 62}, {140, 5318}, {202, 6284}, {203, 4299}, {376, 5238}, {381, 36843}, {395, 3627}, {396, 548}, {530, 634}, {546, 16773}, {616, 635}, {617, 22845}, {627, 21359}, {631, 37832}, {636, 11300}, {1250, 5270}, {1608, 3438}, {1614, 3200}, {1656, 11481}, {2307, 4316}, {2777, 36209}, {2794, 5868}, {3104, 22695}, {3105, 11257}, {3146, 5343}, {3206, 34148}, {3364, 14814}, {3365, 14813}, {3366, 35740}, {3389, 6560}, {3390, 6561}, {3412, 15696}, {3522, 5335}, {3523, 5344}, {3524, 41121}, {3529, 10654}, {3534, 22236}, {3543, 12817}, {3830, 16963}, {3843, 16645}, {3845, 41944}, {3850, 23303}, {3851, 16967}, {3855, 12820}, {4302, 7005}, {4857, 19373}, {5055, 12816}, {5059, 41973}, {5073, 5339}, {5254, 41406}, {5321, 34755}, {5349, 11543}, {5463, 11303}, {5469, 9880}, {5474, 5865}, {6102, 36979}, {6695, 11304}, {6778, 13188}, {6779, 37824}, {7006, 7354}, {7127, 10483}, {7755, 19780}, {7814, 30472}, {7833, 12155}, {8173, 10263}, {8174, 15444}, {8703, 16267}, {8739, 18560}, {8740, 35471}, {10188, 15720}, {10304, 41112}, {10619, 10677}, {10657, 30714}, {11087, 32535}, {11134, 37495}, {11244, 22802}, {11296, 35751}, {11488, 21735}, {11542, 33923}, {12121, 36208}, {12203, 36759}, {13630, 36978}, {14862, 30403}, {15681, 41101}, {15687, 41122}, {15692, 41119}, {15712, 23302}, {16530, 36961}, {17538, 37640}, {19779, 40711}, {31670, 36758}, {33352, 38400}, {33703, 37641}, {34504, 36775}, {34509, 35752}, {35256, 35739}

X(42158) = {X(6),X(1657)}-harmonic conjugate of X(42157)

X(42159) = GIBERT(-3,3,1) POINT

Barycentrics    Sqrt[3]*a^2*S - a^2*SA - 6*SB*SC : :

X(42159) lies on these lines: {2, 5238}, {3, 5321}, {4, 14}, {5, 5339}, {6, 546}, {13, 3832}, {15, 3090}, {16, 3146}, {17, 3545}, {18, 20}, {30, 36843}, {61, 3091}, {202, 5229}, {203, 10591}, {376, 41122}, {381, 398}, {382, 395}, {396, 3851}, {397, 3843}, {550, 16645}, {621, 7938}, {622, 22114}, {623, 37177}, {629, 11147}, {631, 37835}, {632, 11480}, {636, 37171}, {1327, 18585}, {1328, 15765}, {1657, 16773}, {3364, 31412}, {3366, 9540}, {3367, 13935}, {3389, 23259}, {3390, 23249}, {3411, 17578}, {3522, 16242}, {3523, 36967}, {3525, 5352}, {3529, 5237}, {3543, 12817}, {3544, 11488}, {3627, 11543}, {3628, 36836}, {3839, 41112}, {3845, 5340}, {3855, 37640}, {3857, 11542}, {5055, 16772}, {5056, 41973}, {5067, 16241}, {5068, 37832}, {5071, 41101}, {5072, 11485}, {5076, 11486}, {5079, 23302}, {5225, 7006}, {5350, 14269}, {5366, 41107}, {5460, 37173}, {5868, 41017}, {6561, 35738}, {7005, 10590}, {7685, 33420}, {10303, 10645}, {10646, 17538}, {11001, 41944}, {11304, 22491}, {11481, 15704}, {12154, 32984}, {14539, 31706}, {15022, 16966}, {15682, 16963}, {16267, 41106}, {16627, 16635}, {18436, 36980}, {18586, 41945}, {18587, 41946}, {20416, 22512}, {22113, 22492}, {25164, 40921}, {33703, 36968}, {36251, 41042}

X(42159) = {X(3),X(5321)}-harmonic conjugate of X(42160)
X(42159) = {X(4),X(62)}-harmonic conjugate of X(42161)
X(42159) = {X(6),X(546)}-harmonic conjugate of X(42162)

X(42160) = GIBERT(3,-3,1) POINT

Barycentrics    Sqrt[3]*a^2*S + a^2*SA - 6*SB*SC : :

X(42160) lies on these lines: {2, 5352}, {3, 5321}, {4, 13}, {5, 36836}, {6, 3627}, {14, 20}, {15, 3091}, {16, 3529}, {17, 3832}, {18, 376}, {30, 5339}, {62, 3146}, {203, 5225}, {381, 5349}, {382, 398}, {395, 1657}, {396, 3843}, {397, 3830}, {546, 18582}, {548, 16645}, {631, 36967}, {3090, 5238}, {3364, 23249}, {3365, 23259}, {3523, 10187}, {3525, 10645}, {3528, 16242}, {3534, 16773}, {3543, 16965}, {3544, 16966}, {3545, 12817}, {3628, 11480}, {3839, 41101}, {3850, 16644}, {3851, 16772}, {3853, 5340}, {3855, 37832}, {5056, 16241}, {5059, 36968}, {5072, 23302}, {5076, 5318}, {5229, 7005}, {5351, 11489}, {6694, 37170}, {8596, 22114}, {8721, 41038}, {9541, 35732}, {10303, 16967}, {10304, 41122}, {11001, 16268}, {11481, 12103}, {11543, 15704}, {14269, 41119}, {14540, 22861}, {14927, 36758}, {15683, 16963}, {15687, 41112}, {16002, 22512}, {16962, 41099}, {17578, 36969}, {23261, 35738}, {33703, 37641}, {34508, 36769}, {34783, 36980}, {35229, 36993}, {37333, 40922}, {41106, 41943}

X(42160) = {X(3),X(5321)}-harmonic conjugate of X(42159)
X(42160) = {X(4),X(61)}-harmonic conjugate of X(42162)
X(42160) = {X(6),X(3627)}-harmonic conjugate of X(42161)

X(42161) = GIBERT(3,3,-1) POINT

Barycentrics    Sqrt[3]*a^2*S - a^2*SA + 6*SB*SC : :

X(42161) lies on these lines: {2, 5351}, {3, 5318}, {4, 14}, {5, 36843}, {6, 3627}, {13, 20}, {15, 3529}, {16, 3091}, {17, 376}, {18, 3832}, {30, 5340}, {61, 3146}, {202, 5225}, {381, 5350}, {382, 397}, {395, 3843}, {396, 1657}, {398, 3830}, {546, 18581}, {548, 16644}, {631, 36968}, {3090, 5237}, {3389, 23249}, {3390, 23259}, {3523, 10188}, {3525, 10646}, {3528, 16241}, {3534, 16772}, {3543, 16964}, {3544, 16967}, {3545, 12816}, {3628, 11481}, {3839, 41100}, {3850, 16645}, {3851, 16773}, {3853, 5339}, {3855, 37835}, {5056, 16242}, {5059, 36967}, {5072, 23303}, {5076, 5321}, {5229, 7006}, {5352, 11488}, {6695, 37171}, {8596, 22113}, {8721, 41039}, {10303, 16966}, {10304, 41121}, {11001, 16267}, {11480, 12103}, {11542, 15704}, {14269, 41120}, {14541, 22907}, {14927, 36757}, {15683, 16962}, {15687, 41113}, {16001, 22513}, {16963, 41099}, {17578, 36970}, {23251, 35738}, {33703, 37640}, {34783, 36978}, {35230, 36995}, {37332, 40921}, {41106, 41944}

X(42161) = {X(3),X(5318)}-harmonic conjugate of X(42162)
X(42161) = {X(4),X(62)}-harmonic conjugate of X(42159)
X(42161) = {X(6),X(3627)}-harmonic conjugate of X(42160)

X(42162) = GIBERT(3,3,1) POINT

Barycentrics    Sqrt[3]*a^2*S + a^2*SA + 6*SB*SC : :

X(42162) lies on these lines: {2, 5237}, {3, 5318}, {4, 13}, {5, 5340}, {6, 546}, {14, 3832}, {15, 3146}, {16, 3090}, {17, 20}, {18, 3545}, {30, 36836}, {62, 3091}, {202, 10591}, {203, 5229}, {376, 41121}, {381, 397}, {382, 396}, {395, 3851}, {398, 3843}, {550, 16644}, {621, 22113}, {622, 7938}, {624, 37178}, {630, 11147}, {631, 37832}, {632, 11481}, {635, 37170}, {1327, 15765}, {1328, 18585}, {1657, 16772}, {3364, 23259}, {3365, 23249}, {3389, 31412}, {3391, 9540}, {3392, 13935}, {3412, 17578}, {3522, 16241}, {3523, 36968}, {3525, 5351}, {3529, 5238}, {3543, 12816}, {3544, 11489}, {3627, 11542}, {3628, 36843}, {3839, 41113}, {3845, 5339}, {3855, 37641}, {3857, 11543}, {5055, 16773}, {5056, 41974}, {5067, 16242}, {5068, 37835}, {5071, 41100}, {5072, 11486}, {5076, 11485}, {5079, 23303}, {5225, 7005}, {5349, 14269}, {5365, 41108}, {5459, 37172}, {5869, 41016}, {6560, 35738}, {7006, 10590}, {7684, 33421}, {10303, 10646}, {10645, 17538}, {11001, 41943}, {11303, 22492}, {11480, 15704}, {12155, 32984}, {14538, 31705}, {15022, 16967}, {15682, 16962}, {16268, 41106}, {16626, 16634}, {18436, 36978}, {18586, 41946}, {18587, 41945}, {20415, 22513}, {22114, 22491}, {25154, 40922}, {33703, 36967}, {36252, 41043}

X(42162) = {X(3),X(5318)}-harmonic conjugate of X(42161)
X(42162) = {X(4),X(61)}-harmonic conjugate of X(42160)
X(42162) = {X(6),X(546)}-harmonic conjugate of X(42159)

X(42163) = GIBERT(-3,3,2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA - 6*SB*SC : :

X(42163) lies on these lines: {2, 5339}, {3, 5321}, {4, 395}, {5, 14}, {6, 3091}, {13, 3850}, {15, 3628}, {16, 3627}, {18, 30}, {20, 16645}, {62, 546}, {140, 5352}, {203, 10593}, {302, 32819}, {376, 5365}, {381, 397}, {511, 31706}, {524, 22114}, {531, 630}, {532, 33464}, {547, 41108}, {548, 16242}, {550, 36970}, {590, 35732}, {621, 33412}, {631, 5343}, {632, 5238}, {635, 37351}, {636, 33561}, {1656, 10654}, {2307, 7173}, {3090, 5334}, {3146, 11489}, {3364, 18538}, {3365, 18762}, {3411, 3861}, {3525, 11480}, {3529, 11481}, {3530, 36967}, {3564, 16627}, {3629, 22113}, {3832, 5340}, {3843, 5350}, {3845, 16268}, {3851, 40693}, {3857, 16808}, {5055, 41113}, {5056, 16644}, {5068, 37640}, {5072, 18582}, {5079, 11485}, {5344, 41099}, {5351, 15704}, {5460, 37341}, {5464, 33415}, {5471, 39565}, {5478, 41056}, {5562, 36980}, {6114, 16002}, {6695, 35020}, {6782, 16001}, {7005, 10592}, {8260, 10612}, {8739, 23047}, {10109, 16962}, {10110, 36978}, {10645, 14869}, {10646, 12103}, {10677, 30531}, {11137, 13434}, {11244, 41362}, {11306, 22491}, {11488, 15022}, {11542, 12811}, {11626, 15030}, {11737, 16267}, {11801, 36209}, {12102, 19106}, {12108, 33416}, {12812, 16966}, {14893, 41100}, {15619, 36301}, {15687, 16963}, {15699, 41101}, {15702, 33603}, {16241, 41971}, {16960, 41989}, {20415, 38735}, {22847, 41022}, {22882, 32421}, {22883, 32419}, {23046, 41107}, {31694, 34508}, {36758, 39884}

X(42163) = {X(3),X(5321)}-harmonic conjugate of X(42164)
X(42163) = {X(4),X(22238)}-harmonic conjugate of X(42165)
X(42163) = {X(6),X(3091)}-harmonic conjugate of X(42166)

X(42164) = GIBERT(3,-3,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA - 6*SB*SC : :

X(42164) lies on these lines: {3, 5321}, {4, 396}, {5, 5238}, {6, 3146}, {13, 3853}, {14, 550}, {15, 546}, {16, 15704}, {17, 3845}, {18, 548}, {20, 395}, {30, 62}, {61, 3627}, {140, 36967}, {185, 36980}, {376, 5343}, {381, 16772}, {382, 397}, {547, 12817}, {629, 35304}, {631, 5365}, {632, 10645}, {1657, 40694}, {3090, 11480}, {3091, 23302}, {3522, 16645}, {3529, 5334}, {3530, 37835}, {3543, 5340}, {3628, 5352}, {3830, 5350}, {3832, 16644}, {3858, 37832}, {5059, 37641}, {5073, 10653}, {5076, 11485}, {5237, 11543}, {5893, 11243}, {10616, 41037}, {11481, 17538}, {11542, 12102}, {12101, 16267}, {12811, 16966}, {13598, 36978}, {14540, 22512}, {14869, 16967}, {14893, 16962}, {15681, 41113}, {15686, 16268}, {15687, 41101}, {15688, 41120}, {15690, 41944}, {16242, 33923}, {16963, 19710}, {16965, 41973}, {17578, 37640}, {22844, 36330}, {23046, 41943}, {34200, 41122}, {35403, 41119}, {35404, 41107}

X(42164) = {X(3),X(5321)}-harmonic conjugate of X(42163)
X(42164) = {X(4),X(22236)}-harmonic conjugate of X(42166)
X(42164) = {X(6),X(3146)}-harmonic conjugate of X(42165)

X(42165) = GIBERT(3,3,-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA + 6*SB*SC : :

X(42165) lies on these lines: {3, 5318}, {4, 395}, {5, 5237}, {6, 3146}, {13, 550}, {14, 3853}, {15, 15704}, {16, 546}, {17, 548}, {18, 3845}, {20, 396}, {30, 61}, {62, 3627}, {140, 36968}, {185, 36978}, {376, 5344}, {381, 16773}, {382, 398}, {547, 12816}, {630, 35303}, {631, 5366}, {632, 10646}, {1657, 40693}, {3090, 11481}, {3091, 23303}, {3522, 16644}, {3529, 5335}, {3530, 37832}, {3543, 5339}, {3628, 5351}, {3830, 5349}, {3832, 16645}, {3858, 37835}, {5059, 37640}, {5073, 10654}, {5076, 11486}, {5238, 11542}, {5893, 11244}, {10617, 41036}, {11480, 17538}, {11543, 12102}, {12101, 16268}, {12811, 16967}, {13598, 36980}, {14541, 22513}, {14869, 16966}, {14893, 16963}, {15681, 41112}, {15686, 16267}, {15687, 41100}, {15688, 41119}, {15690, 41943}, {16241, 33923}, {16962, 19710}, {16964, 41974}, {17578, 37641}, {22845, 35752}, {23046, 41944}, {34200, 41121}, {35403, 41120}, {35404, 41108}

X(42165) = {X(3),X(5318)}-harmonic conjugate of X(42166)
X(42165) = {X(4),X(22238)}-harmonic conjugate of X(42163)
X(42165) = {X(6),X(3146)}-harmonic conjugate of X(42164)

X(42166) = GIBERT(3,3,2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA + 6*SB*SC : :

X(42166) lies on these lines: {2, 5340}, {3, 5318}, {4, 396}, {5, 13}, {6, 3091}, {14, 3850}, {15, 3627}, {16, 3628}, {17, 30}, {20, 16644}, {61, 546}, {140, 5351}, {202, 10593}, {303, 32819}, {376, 5366}, {381, 398}, {511, 31705}, {524, 22113}, {530, 629}, {533, 33465}, {547, 41107}, {548, 16241}, {550, 36969}, {615, 35732}, {622, 33413}, {631, 5344}, {632, 5237}, {635, 33560}, {636, 37352}, {1656, 10653}, {3090, 5335}, {3146, 11488}, {3389, 18538}, {3390, 18762}, {3412, 3861}, {3525, 11481}, {3529, 11480}, {3530, 36968}, {3564, 16626}, {3614, 7127}, {3629, 22114}, {3832, 5339}, {3843, 5349}, {3845, 16267}, {3851, 40694}, {3857, 16809}, {5055, 41112}, {5056, 16645}, {5068, 37641}, {5072, 18581}, {5079, 11486}, {5343, 41099}, {5352, 15704}, {5459, 37340}, {5463, 33414}, {5472, 39565}, {5479, 41057}, {5562, 36978}, {6115, 16001}, {6694, 35019}, {6783, 16002}, {7006, 10592}, {8259, 10611}, {8740, 23047}, {10109, 16963}, {10110, 36980}, {10645, 12103}, {10646, 14869}, {10678, 30531}, {11134, 13434}, {11243, 41362}, {11305, 22492}, {11489, 15022}, {11543, 12811}, {11624, 15030}, {11737, 16268}, {11801, 36208}, {12102, 19107}, {12108, 33417}, {12812, 16967}, {14893, 41101}, {15619, 36300}, {15687, 16962}, {15699, 41100}, {15702, 33602}, {16242, 41972}, {16961, 41989}, {20416, 38735}, {22893, 41023}, {22927, 32421}, {22928, 32419}, {23046, 41108}, {31693, 34509}, {36757, 39884}

X(42166) = {X(3),X(5318)}-harmonic conjugate of X(42165)
X(42166) = {X(4),X(22236)}-harmonic conjugate of X(42164)
X(42166) = {X(6),X(3091)}-harmonic conjugate of X(42163)

X(42167) = GIBERT(-1,1,2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42167) lies on these lines: {3, 5321}, {4, 6412}, {6, 42170}, {15, 7584}, {16, 34551}, {397, 6398}, {398, 6200}, {615, 11480}, {1152, 5335}, {2042, 23302}, {5318, 6396}, {5334, 6411}, {5340, 6434}, {6410, 35732}, {10653, 35735}, {16773, 35812}, {16966, 35738}, {19107, 34552}

X(42167) = {X(3),X(5321)}-harmonic conjugate of X(42168)
X(42167) = {X(4),X(6412)}-harmonic conjugate of X(42169)

X(42168) = GIBERT(1,-1,2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42168) lies on these lines: {3, 5321}, {4, 6411}, {{6, 42169}, 15, 7583}, {16, 34552}, {397, 6221}, {398, 6396}, {590, 11480}, {1151, 5335}, {2041, 23302}, {5318, 6200}, {5334, 6412}, {5340, 6433}, {6409, 35740}, {16773, 35813}, {19107, 34551}

X(42168) = {X(3),X(5321)}-harmonic conjugate of X(42167)
X(42168) = {X(4),X(6411)}-harmonic conjugate of X(42170)

X(42169) = GIBERT(1,1,-2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42169) lies on these lines: {3, 5318}, {4, 6412}, {6, 42168}, {15, 34552}, {16, 7584}, {397, 6200}, {398, 6398}, {615, 11481}, {1152, 5334}, {2041, 23303}, {5321, 6396}, {5335, 6411}, {5339, 6434}, {16772, 35812}, {19106, 34551}

X(42169) = {X(3),X(5318)}-harmonic conjugate of X(42170)
X(42169) = {X(4),X(6412)}-harmonic conjugate of X(42167)

X(42170) = GIBERT(1,1,2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42170) lies on these lines: {3, 5318}, {4, 6411}, {6, 42167}, {15, 34551}, {16, 7583}, {397, 6396}, {398, 6221}, {590, 11481}, {1151, 5334}, {2042, 23303}, {5321, 6200}, {5335, 6412}, {5339, 6433}, {6409, 35732}, {10654, 35735}, {11542, 35731}, {16772, 35813}, {16960, 35739}, {16967, 35738}, {19106, 34552}, {33518, 35748}

X(42170) = {X(3),X(5318)}-harmonic conjugate of X(42169)
X(42170) = {X(4),X(6411)}-harmonic conjugate of X(42168)

X(42171) = GIBERT(-1,1,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42171) lies on these lines: {3, 5321}, {4, 5420}, {6, 14813}, {15, 486}, {16, 2044}, {18, 5418}, {372, 5335}, {376, 36470}, {397, 6395}, {398, 6221}, {485, 11489}, {615, 18582}, {2041, 16809}, {2043, 19107}, {2045, 10645}, {2046, 16967}, {3312, 35740}, {3364, 16961}, {3390, 16966}, {5318, 6398}, {5334, 6200}, {5339, 6411}, {5340, 6438}, {5366, 6477}, {6412, 14814}, {8252, 11480}, {10654, 41945}, {11485, 18510}, {11543, 34551}, {16645, 36439}, {22644, 35739}, {31454, 40694}, {36437, 36967}, {36455, 37835}

X(42171) = {X(3),X(5321)}-harmonic conjugate of X(42172)
X(42171) = {X(4),X(6396)}-harmonic conjugate of X(42173)
X(42171) = {X(6),X(14813)}-harmonic conjugate of X(42174)

X(42172) = GIBERT(1,-1,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42172) lies on these lines: {3, 5321}, {4, 5418}, {6, 14814}, {5, 42188}, {15, 485}, {16, 2043}, {18, 5420}, {371, 5335}, {376, 36452}, {397, 6199}, {398, 6398}, {486, 11489}, {590, 18582}, {2042, 16809}, {2044, 19107}, {2045, 16967}, {2046, 10645}, {3365, 16961}, {3389, 16966}, {5318, 6221}, {5334, 6396}, {5339, 6412}, {5340, 6437}, {5350, 9690}, {5366, 6476}, {6411, 14813}, {6449, 35740}, {8253, 11480}, {10654, 41946}, {11485, 18512}, {11543, 34552}, {16645, 36457}, {31487, 40693}, {36437, 37835}, {36455, 36967}

X(42172) = {X(3),X(5321)}-harmonic conjugate of X(42171)
X(42172) = {X(4),X(6200)}-harmonic conjugate of X(42174)
X(42172) = {X(6),X(14814)}-harmonic conjugate of X(42173)

X(42173) = GIBERT(1,1,-SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42173) lies on these lines: {3, 5318}, {4, 5420}, {5, 42189}, {6, 14814}, {15, 2043}, {16, 486}, {17, 5418}, {372, 5334}, {376, 36453}, {397, 6221}, {398, 6395}, {485, 11488}, {615, 18581}, {2042, 16808}, {2044, 19106}, {2045, 16966}, {2046, 10646}, {3365, 16967}, {3389, 16960}, {5321, 6398}, {5335, 6200}, {5339, 6438}, {5340, 6411}, {5365, 6477}, {6412, 14813}, {8252, 11481}, {10653, 41945}, {11486, 18510}, {11542, 34552}, {16644, 36457}, {31454, 40693}, {36437, 37832}, {36455, 36968}

X(42173) = {X(3),X(5318)}-harmonic conjugate of X(42174)
X(42173) = {X(4),X(6396)}-harmonic conjugate of X(42171)
X(42173) = {X(6),X(14814)}-harmonic conjugate of X(42172)

X(42174) = GIBERT(1,1,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42174) lies on these lines: {3, 5318}, {4, 5418}, {5, 42187}, {6, 14813}, {15, 2044}, {16, 485}, {17, 5420}, {371, 5334}, {376, 36469}, {397, 6398}, {398, 6199}, {486, 11488}, {590, 18581}, {623, 35741}, {2041, 16808}, {2043, 19106}, {2045, 10646}, {2046, 16966}, {3364, 16967}, {3390, 16960}, {5321, 6221}, {5335, 6396}, {5339, 6437}, {5340, 6412}, {5349, 9690}, {5365, 6476}, {6411, 14814}, {8253, 11481}, {10653, 41946}, {11486, 18512}, {11542, 34551}, {16644, 36439}, {31487, 40694}, {36437, 36968}, {36455, 37832}

X(42174) = {X(3),X(5318)}-harmonic conjugate of X(42173)
X(42174) = {X(4),X(6200)}-harmonic conjugate of X(42172)
X(42174) = {X(6),X(14813)}-harmonic conjugate of X(42171)

X(42175) = GIBERT(-1,2,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA - 12*SB*SC : :

X(42175) lies on these lines: {3, 16809}, {4, 5420}, {5, 42194}, {6, 42178}, {14, 13846}, {15, 6565}, {16, 35820}, {18, 23251}, {371, 5334}, {372, 5318}, {382, 3392}, {2042, 35821}, {2044, 6564}, {3367, 11480}, {5321, 6200}, {5339, 6221}, {5340, 6395}, {6419, 35740}, {22880, 31706}, {35823, 36454}, {36448, 41107}

X(42175) = {X(3),X(42093)}-harmonic conjugate of X(42176)
X(42175) = {X(4),X(6396)}-harmonic conjugate of X(42177)
X(42175) = {X(6),X(42279)}-harmonic conjugate of X(42178)

X(42176) = GIBERT(1,-2,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA - 12*SB*SC : :

X(42176) lies on these lines: {3, 16809}, {4, 5418}, {14, 13847}, {5, 42192}, {15, 6564}, {16, 35821}, {18, 23261}, {371, 5318}, {372, 5334}, {382, 3391}, {2041, 35820}, {2043, 6565}, {3366, 11480}, {5321, 6396}, {5339, 6398}, {5340, 6199}, {22881, 31706}, {35822, 36436}, {36466, 41107}

X(42176) = {X(3),X(42093)}-harmonic conjugate of X(42175)
X(42176) = {X(4),X(6200)}-harmonic conjugate of X(42178)
X(42176) = {X(6),X(42278)}-harmonic conjugate of X(42177)

X(42177) = GIBERT(1,2,-SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA + 12*SB*SC : :

X(42177) lies on these lines: {3, 16808}, {4, 5420}, {5, 42193}, {13, 13846}, {15, 35820}, {16, 6565}, {17, 23251}, {371, 5335}, {372, 5321}, {382, 3367}, {2041, 35821}, {2043, 6564}, {3392, 11481}, {5318, 6200}, {5339, 6395}, {5340, 6221}, {22925, 31705}, {35823, 36436}, {36466, 41108}

X(42177) = {X(3),X(42094)}-harmonic conjugate of X(42178)
X(42177) = {X(4),X(6396)}-harmonic conjugate of X(42175)
X(42177) = {X(6),X(42278)}-harmonic conjugate of X(42176)

X(42178) = GIBERT(1,2,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA + 12*SB*SC : :

X(42178) lies on these lines: {3, 16808}, {4, 5418}, {5, 42191}, {6, 42175}, {13, 13847}, {15, 35821}, {16, 6564}, {17, 23261}, {371, 5321}, {372, 5335}, {382, 3366}, {2042, 35820}, {2044, 6565}, {3391, 11481}, {5318, 6396}, {5339, 6199}, {5340, 6398}, {19107, 35731}, {22926, 31705}, {35822, 36454}, {36448, 41108}

X(42178) = {X(3),X(42094)}-harmonic conjugate of X(42177)
X(42178) = {X(4),X(6200)}-harmonic conjugate of X(42176)
X(42178) = {X(6),X(42279)}-harmonic conjugate of X(42175)

X(42179) = GIBERT(-1,2*SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 12*Sqrt[3]*SB*SC : :

X(42179) lies on these lines: {3, 42180}, ,{4, 16}, {6, 42182}, {13, 1328}, {61, 23259}, {382, 3392}, {546, 3364}, {2043, 33416}, {2044, 16966}, {3365, 3627}, {3366, 35821}, {3367, 16808}, {3389, 22615}, {3391, 3843}, {5318, 7584}, {10645, 35732}, {11485, 23261}, {18586, 19107}, {22597, 22855}, {36454, 36967}

X(42179) = {X(4),X(16)}-harmonic conjugate of X(42181)

X(42180) = GIBERT(1,-2*SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 12*Sqrt[3]*SB*SC : :

X(42180) lies on these lines: {3, 42179}, {4, 15}, {6, 42181}, {14, 1328}, {62, 23259}, {382, 3367}, {546, 3389}, {2043, 16967}, {2044, 33417}, {3146, 35739}, {3364, 22615}, {3366, 3843}, {3390, 3627}, {3391, 35821}, {3392, 16809}, {5321, 7584}, {11486, 23261}, {18587, 19106}, {22599, 22901}, {35255, 35731}, {36436, 36968}

X(42180) = {X(4),X(15)}-harmonic conjugate of X(42182)

X(42181) = GIBERT(1,2*SQRT(3),-1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 12*Sqrt[3]*SB*SC : :

X(42181) lies on these lines: {3, 42182},, {4, 16}, {6, 42180}, {13, 1327}, {61, 23249}, {382, 3391}, {546, 3365}, {2043, 16966}, {2044, 33416}, {3364, 3627}, {3366, 16808}, {3367, 35820}, {3390, 22644}, {3392, 3843}, {5318, 7583}, {11485, 23251}, {18587, 19107}, {22626, 22855}, {36436, 36967}

X(42181) = {X(4),X(16)}-harmonic conjugate of X(42179)

X(42182) = GIBERT(1,2*SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 12*Sqrt[3]*SB*SC : :

X(42182) lies on these lines: {3, 42181}, {4, 15}, {6, 42179}, {14, 1327}, {62, 23249}, {382, 3366}, {546, 3390}, {2043, 33417}, {2044, 16967}, {3091, 35739}, {3365, 22644}, {3367, 3843}, {3389, 3627}, {3391, 16809}, {3392, 35820}, {5321, 7583}, {10646, 35732}, {11486, 23251}, {18586, 19106}, {22628, 22901}, {36454, 36968}

X(42182) = {X(4),X(15)}-harmonic conjugate of X(42180)

X(42183) = GIBERT(-1,3,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA - 18*SB*SC : :

X(42183) lies on these lines: {3, 42101}, {4, 5420}, {6, 42186}, {16, 22644}, {1327, 33606}, {2042, 19107}, {2044, 16809}, {5318, 6395}, {5321, 6221}, {6199, 35740}, {6200, 35732}, {6411, 14813}, {6560, 36969}, {6561, 36454}, {32787, 36448}

X(42183) = {X(3),X(42101)}-harmonic conjugate of X(42184)
X(42183) = {X(4),X(6396)}-harmonic conjugate of X(42185)
X(42183) = {X(6),X(42281)}-harmonic conjugate of X(42186)

X(42184) = GIBERT(1,-3,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA - 18*SB*SC : :

X(42184) lies on these lines: {3, 42101}, {4, 5418}, {6, 42185}, {16, 22615}, {1328, 33606}, {2041, 19107}, {2043, 16809}, {5318, 6199}, {5321, 6398}, {6412, 14814}, {6445, 35740}, {6560, 36436}, {6561, 36969}, {32788, 36466}

X(42184) = {X(3),X(42101)}-harmonic conjugate of X(42183)
X(42184) = {X(4),X(6200)}-harmonic conjugate of X(42186)
X(42184) = {X(6),X(42280)}-harmonic conjugate of X(42185)

X(42185) = GIBERT(1,3,-SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA + 18*SB*SC : :

X(42185) lies on these lines: {3, 42102}, {4, 5420}, {6, 42184}, {15, 22644}, {1327, 33607}, {2041, 19106}, {2043, 16808}, {5318, 6221}, {5321, 6395}, {6411, 14814}, {6451, 35740}, {6560, 36970}, {6561, 36436}, {32787, 36466}

X(42185) = {X(3),X(42102)}-harmonic conjugate of X(42186)
X(42185) = {X(4),X(6396)}-harmonic conjugate of X(42183)
X(42185) = {X(6),X(42280)}-harmonic conjugate of X(42184)

X(42186) = GIBERT(1,3,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA + 18*SB*SC : :

X(42186) lies on these lines: {3, 42102}, {4, 5418}, {6, 42183}, {15, 22615}, {1328, 33607}, {2042, 19106}, {2044, 16808}, {5318, 6398}, {5321, 6199}, {6221, 35740}, {6396, 35732}, {6412, 14813}, {6560, 36454}, {6561, 36970}, {32788, 36448}

X(42186) = {X(3),X(42102)}-harmonic conjugate of X(42185)
X(42186) = {X(4),X(6200)}-harmonic conjugate of X(42184)
X(42186) = {X(6),X(42281)}-harmonic conjugate of X(42183)

X(42187) = GIBERT(-1,SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42187) lies on these lines: {3, 22615}, {4, 16}, {5, 42174}, {6, 42183}, {13, 14226}, {14, 13703}, {15, 23259}, {20, 3392}, {61, 23273}, {62, 23249}, {486, 5318}, {2041, 35787}, {2042, 35821}, {2044, 6565}, {3071, 11485}, {3091, 3364}, {3146, 3365}, {3366, 6459}, {3391, 3832}, {5321, 6561}, {5335, 7586}, {10653, 36467}, {10654, 36454}, {11480, 14813}, {32488, 33359}

X(42187) = {X(3),X(42283)}-harmonic conjugate of X(42188)
X(42187) = {X(4),X(16)}-harmonic conjugate of X(42189)
X(42187) = {X(6),X(42281)}-harmonic conjugate of X(42190)

X(42188) = GIBERT(1,-SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42188) lies on these lines: {3, 22615}, {4, 15}, {5, 42172}, {6, 42184}, {13, 13705}, {14, 14226}, {16, 23259}, {20, 3367}, {61, 23249}, {62, 23273}, {486, 5321}, {2041, 35821}, {2042, 35787}, {2043, 6565}, {3071, 11486}, {3091, 3389}, {3146, 3390}, {3366, 3832}, {3391, 6459}, {3529, 35739}, {5318, 6561}, {5334, 7586}, {10645, 35732}, {10653, 36436}, {10654, 36449}, {11481, 14814}, {32488, 33361}

X(42188) = {X(3),X(42283)}-harmonic conjugate of X(42187)
X(42188) = {X(4),X(15)}-harmonic conjugate of X(42190)
X(42188) = {X(6),X(42280)}-harmonic conjugate of X(42189)

X(42189) = GIBERT(1,SQRT(3),-1) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42189) lies on these lines: {3, 22644}, {4, 16}, {5, 42173}, {6, 42184}, {13, 14241}, {14, 13823}, {15, 23249}, {20, 3391}, {61, 23267}, {62, 23259}, {485, 5318}, {2041, 35820}, {2042, 35786}, {2043, 6564}, {3070, 11485}, {3091, 3365}, {3146, 3364}, {3367, 6460}, {3389, 31412}, {3392, 3832}, {5321, 6560}, {5335, 7585}, {10646, 35732}, {10653, 36450}, {10654, 36436}, {11480, 14814}, {31414, 40693}, {32489, 33360}

X(42189) = {X(3),X(42284)}-harmonic conjugate of X(42190)
X(42189) = {X(4),X(16)}-harmonic conjugate of X(42187)
X(42189) = {X(6),X(42280)}-harmonic conjugate of X(42188)

X(42190) = GIBERT(1,SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42190) lies on these lines: {3, 22644}, {4, 15}, {5, 42171}, {6, 42183}, {13, 13825}, {14, 14241}, {16, 23249}, {20, 3366}, {61, 23259}, {62, 23267}, {485, 5321}, {2041, 35786}, {2042, 35820}, {2044, 6564}, {3070, 11486}, {3090, 35739}, {3091, 3390}, {3146, 3389}, {3364, 31412}, {3367, 3832}, {3392, 6460}, {5318, 6560}, {5334, 7585}, {10653, 36454}, {10654, 36468}, {11481, 14813}, {31414, 40694}, {32489, 33358}

X(42190) = {X(3),X(42284)}-harmonic conjugate of X(42189)
X(42190) = {X(4),X(15)}-harmonic conjugate of X(42188)
X(42190) = {X(6),X(42281)}-harmonic conjugate of X(42187)

X(42191) = GIBERT(-1,SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42191) lies on these lines: {3, 22615}, {4, 11481}, {5, 42178}, {6, 35732}, {15, 3071}, {395, 36454}, {590, 18581}, {615, 2044}, {1328, 35735}, {2042, 5321}, {3070, 11486}, {3366, 31454}, {6200, 35738}, {6565, 34551}, {8960, 11543}, {11480, 23259}, {15765, 16967}, {18585, 19106}, {22236, 23273}, {22238, 23249}, {36439, 37832}

X(42191) = {X(3),X(42283)}-harmonic conjugate of X(42192)
X(42191) = {X(4),X(11481)}-harmonic conjugate of X(42193)
X(42191) = {X(6),X(35732)}-harmonic conjugate of X(42194)

X(42192) = GIBERT(1,-SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42192) lies on these lines: {3, 22615}, {4, 11480}, {5, 42176}, {16, 3071}, {396, 36436}, {590, 18582}, {615, 2043}, {2041, 5318}, {3070, 11485}, {3391, 31454}, {6565, 34552}, {8960, 11542}, {11481, 23259}, {15765, 19107}, {16966, 18585}, {22236, 23249}, {22238, 23273}, {36457, 37835}

X(42192) = {X(3),X(42283)}-harmonic conjugate of X(42191)
X(42192) = {X(4),X(11480)}-harmonic conjugate of X(42194)
X(42192) = {X(6),X(42282)}-harmonic conjugate of X(42193)

X(42193) = GIBERT(1,SQRT(3),-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 6*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42193) lies on these lines: {3, 22644}, {4, 11481}, {5, 42177}, {15, 3070}, {395, 36436}, {590, 2043}, {615, 18581}, {2041, 5321}, {3071, 11486}, {6564, 34552}, {11480, 23249}, {15765, 19106}, {16967, 18585}, {22236, 23267}, {22238, 23259}, {32785, 35740}, {36457, 37832}

X(42193) = {X(3),X(42284)}-harmonic conjugate of X(42194)
X(42193) = {X(4),X(11481)}-harmonic conjugate of X(42191)
X(42193) = {X(6),X(42282)}-harmonic conjugate of X(42192)

X(42194) = GIBERT(1,SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42194) lies on these lines: {3, 22644}, {4, 11480}, {5, 42175}, {6, 35732}, {16, 3070}, {396, 36454}, {590, 2044}, {615, 18582}, {1327, 35735}, {2042, 5318}, {3071, 11485}, {6396, 35738}, {6564, 34551}, {11481, 23249}, {15765, 16966}, {18585, 19107}, {22236, 23259}, {22238, 23267}, {36439, 37835}

X(42194) = {X(3),X(42284)}-harmonic conjugate of X(42193)
X(42194) = {X(4),X(11480)}-harmonic conjugate of X(42192)
X(42194) = {X(6),X(35732)}-harmonic conjugate of X(42191)

X(42195) = GIBERT(-1,SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42195) lies on these lines: {3, 22615}, {4, 10187}, {6, 14813}, {15, 23273}, {16, 23249}, {486, 16772}, {1588, 34754}, {2042, 6200}, {2044, 6396}, {3364, 32785}, {3366, 8972}, {3390, 13941}, {3392, 18582}, {5318, 5420}, {6411, 15765}, {8252, 36439}, {10645, 23259}, {13785, 35735}, {18586, 23303}, {18762, 34551}, {23267, 34755}, {35814, 40693}

X(42196) = GIBERT(1,-SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42196) lies on these lines: {3, 22615}, {4, 10188}, {6, 14814}, {15, 23249}, {16, 23273}, {486, 16773}, {1588, 34755}, {2041, 6200}, {2043, 6396}, {3365, 13941}, {3367, 18581}, {3389, 32785}, {3391, 8972}, {5321, 5420}, {6411, 18585}, {8252, 36457}, {10646, 23259}, {18587, 23302}, {18762, 34552}, {23267, 34754}, {35814, 40694}

X(42197) = GIBERT(1,SQRT(3),-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 9*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42197) lies on these lines: {3, 22644}, {4, 10187}, {6, 14814}, {15, 23267}, {16, 23259}, {485, 16772}, {1587, 34754}, {2041, 6396}, {2043, 6200}, {3365, 32786}, {3367, 13941}, {3389, 8972}, {3391, 18582}, {5318, 5418}, {6412, 18585}, {8253, 36457}, {10645, 23249}, {18538, 34552}, {18587, 23303}, {23273, 34755}, {35815, 40693}

X(42198) = GIBERT(1,SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S + 9*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42198) lies on these lines: {3, 22644}, {4, 10188}, {6, 14813}, {15, 23259}, {16, 23267}, {485, 16773}, {1587, 34755}, {2042, 6396}, {2044, 6200}, {3364, 8972}, {3366, 18581}, {3390, 32786}, {3392, 13941}, {5321, 5418}, {6412, 15765}, {8253, 36439}, {10646, 23249}, {13665, 35735}, {18538, 34551}, {18586, 23302}, {23273, 34754}, {35815, 40694}

X(42199) = GIBERT(-2,1,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42199) lies on these lines: {3, 5334}, {4, 3591}, {6, 14813}, {15, 615}, {18, 3070}, {372, 5318}, {395, 35822}, {398, 6200}, {2042, 7584}, {2043, 35256}, {2044, 11486}, {2046, 18538}, {3069, 11542}, {3312, 35732}, {3390, 16967}, {3392, 3412}, {5321, 6396}, {5335, 6395}, {5339, 6412}, {6420, 35740}, {6560, 18581}, {9680, 40694}, {10654, 15764}, {13993, 35738}

X(42200) = GIBERT(2,-1,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42200) lies on these lines: {3, 5334}, {4, 3590}, {6, 14814}, {14, 15764}, {15, 590}, {18, 3071}, {371, 5318}, {395, 35823}, {398, 6396}, {2041, 7583}, {2043, 11486}, {2044, 35255}, {2045, 18762}, {3068, 11542}, {3389, 16967}, {3391, 3412}, {5321, 6200}, {5335, 6199}, {5339, 6411}, {6449, 35732}, {6453, 35740}, {6561, 15765}, {9541, 18586}

X(42201) = GIBERT(2,1,-SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42201) lies on these lines: {3, 5335}, {4, 3591}, {6, 14814}, {13, 15764}, {16, 615}, {17, 3070}, {372, 5321}, {396, 35822}, {397, 6200}, {2041, 7584}, {2043, 11485}, {2044, 35256}, {2045, 18538}, {3069, 11543}, {3365, 16966}, {3367, 3411}, {5318, 6396}, {5334, 6395}, {5340, 6412}, {6450, 35732}, {6560, 15765}, {9680, 40693}

X(42202) = GIBERT(2,1,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42202) lies on these lines: {3, 5335}, {4, 3590}, {6, 14813}, {16, 590}, {17, 3071}, {371, 5321}, {396, 35823}, {397, 6396}, {2042, 7583}, {2043, 35255}, {2044, 11485}, {2046, 18762}, {3068, 11543}, {3311, 35732}, {3364, 16966}, {3366, 3411}, {5318, 6200}, {5334, 6199}, {5340, 6411}, {6561, 18582}, {9541, 18587}, {10653, 15764}, {13925, 35738}

X(42203) = GIBERT(-2,2,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA - 12*SB*SC : :

X(42203) lies on these lines: {3, 5321}, {4, 3591}, {398, 6199}, {2044, 11543}, {3311, 35732}, {3312, 5335}, {5318, 6395}, {5334, 6221}, {5339, 6200}, {6417, 35740}, {6452, 14814}, {6565, 19107}, {11485, 13785}, {16809, 35820}

X(42204) = GIBERT(2,-2,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA - 12*SB*SC : :

X(42204) lies on these lines: {3, 5321}, {4, 3590}, {398, 6395}, {2043, 11543}, {3311, 5335}, {5318, 6199}, {5334, 6398}, {5339, 6396}, {6407, 35740}, {6451, 14813}, {6455, 35732}, {6564, 19107}, {11485, 13665}, {16809, 35821}

X(42205) = GIBERT(2,2,-SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA + 12*SB*SC : :

X(42205) lies on these lines: {3, 5318}, {4, 3591}, {397, 6199}, {2043, 11542}, {3312, 5334}, {5321, 6395}, {5335, 6221}, {5340, 6200}, {6452, 14813}, {6456, 35732}, {6565, 19106}, {11486, 13785}, {16808, 35820}

X(42206) = GIBERT(2,2,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA + 12*SB*SC : :

X(42206) lies on these lines: {3, 5318}, {4, 3590}, {397, 6395}, {2044, 11542}, {3311, 5334}, {3312, 35732}, {5321, 6199}, {5335, 6398}, {5340, 6396}, {6451, 14814}, {6564, 19106}, {11486, 13665}, {16808, 35821}, {31412, 35738}

X(42207) = GIBERT(-2,3,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA - 18*SB*SC : :

X(42207) lies on these lines: {4, 3591}, {396, 6565}, {397, 6436}, {1327, 41120}, {2044, 18538}, {3392, 19107}, {3830, 36465}, {5321, 6200}, {5334, 6199}, {5339, 6437}, {5340, 6442}, {5343, 9690}, {6221, 35732}, {6412, 14814}, {8253, 36439}, {11543, 23249}, {15764, 36970}, {16809, 18585}, {18586, 23259}, {19054, 36448}

X(42208) = GIBERT(2,-3,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA - 18*SB*SC : :

X(42208) lies on these lines: {4, 3590}, {396, 6564}, {397, 6435}, {1328, 41120}, {2043, 18762}, {3391, 19107}, {3830, 36446}, {5321, 6396}, {5334, 6395}, {5339, 6438}, {5340, 6441}, {6411, 14813}, {6451, 35732}, {6480, 35740}, {8252, 36457}, {11543, 23259}, {15765, 16809}, {18587, 23249}, {19053, 36466}

X(42209) = GIBERT(2,3,-SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*Sqrt[3]*a^2*SA + 18*SB*SC : :

X(42209) lies on these lines: {4, 3591}, {395, 6565}, {398, 6436}, {1327, 41119}, {2043, 18538}, {3367, 19106}, {3830, 36447}, {5318, 6200}, {5335, 6199}, {5339, 6442}, {5340, 6437}, {5344, 9690}, {6412, 14813}, {6452, 35732}, {8253, 36457}, {11542, 23249}, {15765, 16808}, {18587, 23259}, {19054, 36466}

X(42210) = GIBERT(2,3,SQRT(3)) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*Sqrt[3]*a^2*SA + 18*SB*SC : :

X(42210) lies on these lines: {4, 3590}, {395, 6564}, {398, 6435}, {1328, 41119}, {2044, 18762}, {3366, 19106}, {3830, 36464}, {5318, 6396}, {5335, 6395}, {5339, 6441}, {5340, 6438}, {6200, 35740}, {6398, 35732}, {6411, 14814}, {8252, 36439}, {11542, 23259}, {15764, 36969}, {16808, 18585}, {18586, 23249}, {19053, 36448}

X(42211) = GIBERT(-2,SQRT(3),1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42211) lies on these lines: {3, 18762}, {4, 11409}, {14, 32787}, {15, 3071}, {2044, 11542}, {5318, 6565}, {5321, 35821}, {5334, 18586}, {5335, 7584}, {6561, 15765}, {11481, 14814}, {11485, 23273}, {16242, 36457}, {16644, 36439}, {18539, 37332}

X(42212) = GIBERT(2,-SQRT(3),1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42212) lies on these lines: {3, 18762}, {4, 11408}, {13, 32787}, {16, 3071}, {1328, 15764}, {2043, 11543}, {5318, 35821}, {5321, 6565}, {5334, 7584}, {5335, 18587}, {6561, 18582}, {11480, 14813}, {11486, 23273}, {16241, 36439}, {16645, 36457}, {18539, 37333}

X(42213) = GIBERT(2,SQRT(3),-1) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 3*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42213) lies on these lines: {3, 18538}, {4, 11409}, {14, 32788}, {15, 3070}, {1327, 15764}, {2043, 11542}, {5318, 6564}, {5321, 35820}, {5334, 18587}, {5335, 7583}, {6560, 18581}, {11481, 14813}, {11485, 23267}, {16242, 36439}, {16644, 36457}, {26438, 37332}

X(42214) = GIBERT(2,SQRT(3),1) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 3*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42214) lies on these lines: {3, 18538}, {4, 11408}, {13, 32788}, {16, 3070}, {2044, 11543}, {5318, 35820}, {5321, 6564}, {5334, 7583}, {5335, 18586}, {6560, 15765}, {11480, 14814}, {11486, 23267}, {16241, 36457}, {16645, 36439}, {26438, 37333}

X(42215) = GIBERT(2*SQRT(3),-1,1) POINT

Barycentrics    2*a^2*S + a^2*SA - 2*SB*SC : :

X(42215) lies on these lines: {2, 6221}, {3, 1588}, {4, 1131}, {5, 371}, {6, 30}, {10, 31439}, {15, 34551}, {16, 34552}, {20, 3312}, {35, 19027}, {36, 19029}, {74, 19051}, {140, 486}, {141, 32419}, {143, 12239}, {182, 9687}, {186, 13937}, {230, 9675}, {235, 10880}, {265, 19060}, {355, 1702}, {372, 550}, {376, 6398}, {381, 3068}, {382, 1587}, {388, 31474}, {397, 3364}, {398, 3389}, {403, 13884}, {485, 546}, {487, 11314}, {492, 6390}, {495, 2066}, {496, 2067}, {547, 6437}, {548, 1152}, {549, 615}, {576, 12598}, {631, 6449}, {632, 6453}, {639, 7915}, {952, 35775}, {1124, 18990}, {1132, 3090}, {1270, 26619}, {1327, 6441}, {1328, 5066}, {1335, 15171}, {1351, 39876}, {1353, 35841}, {1478, 19038}, {1479, 18996}, {1483, 35642}, {1504, 7745}, {1595, 11473}, {1596, 5412}, {1656, 9540}, {1657, 6418}, {2043, 11486}, {2044, 11485}, {2460, 38230}, {2550, 31485}, {3070, 3627}, {3091, 8976}, {3092, 6756}, {3093, 13488}, {3102, 32448}, {3146, 6427}, {3298, 15172}, {3299, 7354}, {3301, 6284}, {3316, 5068}, {3317, 9543}, {3522, 6450}, {3523, 6455}, {3524, 6451}, {3525, 6519}, {3526, 6407}, {3528, 6456}, {3529, 6428}, {3530, 5420}, {3534, 6395}, {3543, 23267}, {3545, 8972}, {3579, 13936}, {3583, 19030}, {3585, 19028}, {3594, 12103}, {3628, 5418}, {3629, 32421}, {3818, 36723}, {3830, 18512}, {3832, 13886}, {3843, 23263}, {3845, 6564}, {3850, 13925}, {3851, 13903}, {3853, 6431}, {3857, 35815}, {3858, 8960}, {3861, 6470}, {4299, 18995}, {4302, 19037}, {5010, 13958}, {5054, 6445}, {5055, 32785}, {5058, 5254}, {5073, 6500}, {5076, 23253}, {5092, 13972}, {5204, 13962}, {5217, 13963}, {5305, 6424}, {5326, 31500}, {5334, 18586}, {5335, 18587}, {5409, 15235}, {5411, 18533}, {5413, 37458}, {5414, 9660}, {5591, 26289}, {5787, 19068}, {5870, 36711}, {5886, 9583}, {6033, 19109}, {6202, 36712}, {6214, 37343}, {6215, 8396}, {6250, 14239}, {6290, 13650}, {6321, 19056}, {6396, 8703}, {6410, 33923}, {6411, 12100}, {6412, 34200}, {6420, 15704}, {6429, 9680}, {6430, 41981}, {6433, 11812}, {6435, 33699}, {6438, 15690}, {6439, 11540}, {6446, 15688}, {6452, 10304}, {6468, 10124}, {6476, 41951}, {6480, 11539}, {6496, 15717}, {6497, 21735}, {6501, 17800}, {6502, 9647}, {6699, 13979}, {6823, 10897}, {7280, 18966}, {7388, 7879}, {7485, 9695}, {7525, 9683}, {7687, 13915}, {7715, 35765}, {7728, 19111}, {7741, 18965}, {7951, 13901}, {7968, 34773}, {7969, 22791}, {8276, 13861}, {8725, 19091}, {8983, 9955}, {8991, 20299}, {8994, 20304}, {9582, 13947}, {9585, 34595}, {9602, 37637}, {9616, 26446}, {9646, 10592}, {9655, 31408}, {9661, 10593}, {9676, 40111}, {9682, 12106}, {9690, 15694}, {9821, 19089}, {9956, 13912}, {10272, 10819}, {10283, 35763}, {10386, 35809}, {10733, 19052}, {10738, 19082}, {10742, 19113}, {10749, 19094}, {10895, 13905}, {10896, 13904}, {11292, 12221}, {11313, 12322}, {11542, 18585}, {11543, 15765}, {12121, 19110}, {12163, 19061}, {12240, 13630}, {12257, 36655}, {12293, 19062}, {12515, 19077}, {12699, 18991}, {12702, 19065}, {12918, 19115}, {12971, 36966}, {13624, 13971}, {13748, 36658}, {13759, 26615}, {13763, 33878}, {13881, 22617}, {13883, 18480}, {13902, 18493}, {13910, 19130}, {13911, 18357}, {13975, 31663}, {14216, 19088}, {14830, 19057}, {14880, 18994}, {15484, 31403}, {15686, 41946}, {15687, 35822}, {15699, 32789}, {15760, 18457}, {15800, 19096}, {16232, 37730}, {17578, 23269}, {18481, 18992}, {18525, 19066}, {18534, 19006}, {18581, 34559}, {18582, 34562}, {18583, 19145}, {19004, 41869}, {19055, 38741}, {19059, 20127}, {19081, 38753}, {19087, 20427}, {19108, 38730}, {21737, 26348}, {26288, 26340}, {26441, 36714}, {28174, 35774}, {31419, 31453}, {31859, 35949}, {32810, 32896}, {35771, 35820}, {35777, 37440}, {35788, 38138}, {35789, 38112}, {35814, 41964}, {36490, 36553}, {36492, 36551}, {36512, 36585}, {41949, 41969}, {41953, 41963}

X(42216) = GIBERT(2*SQRT(3),1,-1) POINT

Barycentrics    2*a^2*S - a^2*SA + 2*SB*SC : :

X(42216) lies on these lines: {2, 6398}, {3, 1587}, {4, 1132}, {5, 372}, {6, 30}, {15, 34552}, {16, 34551}, {20, 3311}, {35, 19028}, {36, 19030}, {74, 19052}, {140, 485}, {141, 32421}, {143, 12240}, {182, 13030}, {186, 13884}, {235, 10881}, {265, 19059}, {355, 1703}, {371, 550}, {376, 6221}, {381, 3069}, {382, 1588}, {397, 3365}, {398, 3390}, {403, 13937}, {486, 546}, {488, 11313}, {491, 6390}, {495, 5414}, {496, 6502}, {547, 6438}, {548, 1151}, {549, 590}, {576, 12597}, {631, 6450}, {632, 6454}, {640, 7915}, {952, 35774}, {1124, 15171}, {1131, 3090}, {1160, 21737}, {1271, 26620}, {1327, 5066}, {1328, 6442}, {1335, 18990}, {1351, 39875}, {1353, 35840}, {1478, 19037}, {1479, 18995}, {1483, 35641}, {1505, 7745}, {1595, 11474}, {1596, 5413}, {1656, 13935}, {1657, 6417}, {2043, 11485}, {2044, 11486}, {2362, 37730}, {2459, 38230}, {3071, 3627}, {3091, 13951}, {3092, 13488}, {3093, 6756}, {3103, 32448}, {3146, 6428}, {3295, 31408}, {3297, 15172}, {3299, 6284}, {3301, 7354}, {3316, 10303}, {3317, 5068}, {3522, 6449}, {3523, 6456}, {3524, 6452}, {3525, 6522}, {3526, 6408}, {3528, 6455}, {3529, 6427}, {3530, 5418}, {3534, 6199}, {3543, 23273}, {3545, 13941}, {3579, 13883}, {3583, 19029}, {3585, 19027}, {3592, 12103}, {3628, 5420}, {3629, 32419}, {3818, 36726}, {3830, 18510}, {3832, 13939}, {3843, 23253}, {3845, 6565}, {3850, 13993}, {3851, 13961}, {3853, 6432}, {3857, 35814}, {3858, 35786}, {3861, 6471}, {4294, 31474}, {4299, 18996}, {4302, 19038}, {5010, 13901}, {5013, 31411}, {5024, 31403}, {5054, 6446}, {5055, 32786}, {5062, 5254}, {5073, 6501}, {5076, 23263}, {5092, 13910}, {5204, 13904}, {5217, 13905}, {5305, 6423}, {5334, 18587}, {5335, 18586}, {5408, 15236}, {5410, 18533}, {5412, 37458}, {5590, 26288}, {5787, 19067}, {5871, 36712}, {6033, 19108}, {6200, 8703}, {6201, 36711}, {6214, 8416}, {6215, 37342}, {6251, 14235}, {6289, 13771}, {6321, 19055}, {6409, 33923}, {6411, 34200}, {6412, 12100}, {6419, 15704}, {6429, 41981}, {6430, 16239}, {6434, 11812}, {6436, 33699}, {6437, 15690}, {6440, 11540}, {6445, 15688}, {6451, 10304}, {6469, 10124}, {6477, 41952}, {6481, 11539}, {6496, 21735}, {6497, 15717}, {6500, 17800}, {6699, 13915}, {6823, 10898}, {7280, 18965}, {7389, 7879}, {7687, 13979}, {7715, 35764}, {7728, 19110}, {7741, 18966}, {7951, 13958}, {7968, 22791}, {7969, 34773}, {8277, 13861}, {8725, 19092}, {8960, 15712}, {8982, 36709}, {8983, 13624}, {9690, 15695}, {9708, 31413}, {9821, 19090}, {9955, 13971}, {9956, 13975}, {10272, 10820}, {10283, 35762}, {10386, 35808}, {10733, 19051}, {10738, 19081}, {10742, 19112}, {10749, 19093}, {10895, 13963}, {10896, 13962}, {11291, 12222}, {11314, 12323}, {11542, 15765}, {11543, 18585}, {12121, 19111}, {12163, 19062}, {12239, 13630}, {12256, 36656}, {12293, 19061}, {12515, 19078}, {12699, 18992}, {12702, 19066}, {12918, 19114}, {12965, 36966}, {13639, 26616}, {13644, 33878}, {13749, 36657}, {13881, 22646}, {13912, 31663}, {13936, 18480}, {13959, 18493}, {13969, 20304}, {13972, 19130}, {13973, 18357}, {13980, 20299}, {14216, 19087}, {14830, 19058}, {14880, 18993}, {15686, 41945}, {15687, 35823}, {15699, 32790}, {15760, 18459}, {15800, 19095}, {17578, 23275}, {18481, 18991}, {18525, 19065}, {18534, 19005}, {18581, 34562}, {18582, 34559}, {18583, 19146}, {19003, 41869}, {19056, 38741}, {19060, 20127}, {19082, 38753}, {19088, 20427}, {19109, 38730}, {26289, 26339}, {28174, 35775}, {31439, 31730}, {31859, 35948}, {32811, 32896}, {35770, 35821}, {35776, 37440}, {35788, 38112}, {35789, 38138}, {35815, 41963}, {36490, 36552}, {36491, 36551}, {36512, 36584}, {41950, 41970}, {41954, 41964}

X(42217) = GIBERT(-2, SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 3*Sqrt[3]*SB*SC : :

X(42217) lies on these lines: {3, 18762}, {4, 16}, {6, 35732}, {13, 36447}, {15, 23273}, {62, 23267}, {486, 11488}, {631, 3392}, {1588, 11485}, {2042, 5334}, {2044, 3069}, {3068, 11543}, {3071, 11480}, {3090, 3364}, {3365, 3529}, {3390, 13939}, {3391, 3855}, {6221, 35738}, {9541, 15765}, {11486, 23249}, {13785, 34551}, {14226, 35737}, {14814, 23263}, {35822, 36454}

X(42218) = GIBERT(2, -SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 3*Sqrt[3]*SB*SC : :

X(42218) lies on these lines: {3, 18762}, {4, 15}, {14, 36465}, {16, 23273}, {61, 23267}, {486, 11489}, {631, 3367}, {1588, 11486}, {2041, 5335}, {2043, 3069}, {3068, 11542}, {3071, 11481}, {3090, 3389}, {3365, 13939}, {3366, 3855}, {3390, 3529}, {9541, 18585}, {11480, 35732}, {11485, 23249}, {13785, 34552}, {14813, 23263}, {17538, 35739}, {35822, 36436}

X(42219) = GIBERT(2, SQRT(3),-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 3*Sqrt[3]*SB*SC : :

X(42219) lies on these lines: {3, 18538}, {4, 16}, {13, 36464}, {15, 23267}, {62, 23273}, {485, 11488}, {631, 3391}, {1587, 11485}, {2041, 5334}, {2043, 3068}, {3069, 11543}, {3070, 11480}, {3090, 3365}, {3364, 3529}, {3389, 13886}, {3392, 3855}, {11481, 35732}, {11486, 23259}, {13665, 34552}, {14813, 23253}, {35823, 36436}

X(42220) = GIBERT(2, SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 3*Sqrt[3]*SB*SC : :

X(42220) lies on these lines: {3, 18538}, {4, 15}, {6, 35732}, {14, 36446}, {16, 23267}, {61, 23273}, {485, 11489}, {631, 3366}, {1587, 11486}, {2042, 5335}, {2044, 3068}, {3069, 11542}, {3070, 11481}, {3090, 3390}, {3364, 13886}, {3367, 3855}, {3389, 3529}, {3525, 35739}, {6398, 35738}, {11485, 23259}, {13665, 34551}, {14241, 35737}, {14814, 23253}, {35823, 36454}

X(42221) = GIBERT(-2, SQRT(3),3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42221) lies on these lines: {3, 18762}, {6, 14813}, {13, 615}, {18, 41963}, {2042, 6221}, {2044, 6398}, {2045, 6451}, {3070, 34755}, {3071, 10645}, {3364, 32789}, {3392, 32790}, {5318, 10577}, {5335, 13966}, {6200, 15765}, {6561, 15764}, {10646, 14814}, {11486, 23267}, {11489, 18538}, {11542, 13941}, {18510, 35735}

X(42222) = GIBERT(2, -SQRT(3),3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42222) lies on these lines: {3, 18762}, {6, 14814}, {14, 615}, {17, 41963}, {2041, 6221}, {2043, 6398}, {2046, 6451}, {3070, 34754}, {3071, 10646}, {3367, 32790}, {3389, 32789}, {5321, 10577}, {5334, 13966}, {6200, 18585}, {6565, 15764}, {10645, 14813}, {11485, 23267}, {11488, 18538}, {11543, 13941}

X(42223) = GIBERT(2, SQRT(3),-3) POINT

Barycentrics    2*Sqrt[3]*a^2*S - 9*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42223) lies on these lines: {3, 18538}, {6, 14814}, {13, 590}, {18, 41964}, {2041, 6398}, {2043, 6221}, {2046, 6452}, {3070, 10645}, {3071, 34755}, {3365, 32790}, {3391, 32789}, {5318, 10576}, {5335, 8981}, {6396, 18585}, {6564, 15764}, {8972, 11542}, {10646, 14813}, {11486, 23273}, {11489, 18587}

X(42224) = GIBERT(2, SQRT(3),3) POINT

Barycentrics    2*Sqrt[3]*a^2*S + 9*a^2*SA + 6*Sqrt[3]*SB*SC : :

X(42224) lies on these lines: {3, 18538}, {6, 14813}, {14, 590}, {17, 41964}, {2042, 6398}, {2044, 6221}, {2045, 6452}, {3070, 10646}, {3071, 34754}, {3366, 32789}, {3390, 32790}, {5321, 10576}, {5334, 8981}, {6396, 15765}, {6560, 15764}, {8972, 11543}, {10645, 14814}, {11485, 23273}, {11488, 18586}, {18512, 35735}

X(42225) = GIBERT(2*SQRT(3),-3,3) POINT

Barycentrics    2*a^2*S + 3*a^2*SA - 6*SB*SC : :

X(42225) lies on these lines: {2, 6451}, {3, 18762}, {4, 3590}, {5, 6200}, {6, 30}, {15, 35730}, {20, 6398}, {140, 6411}, {371, 3627}, {372, 15704}, {376, 6452}, {381, 6445}, {382, 6199}, {485, 3853}, {486, 548}, {495, 9660}, {496, 9647}, {546, 1151}, {549, 6565}, {550, 3071}, {590, 3845}, {615, 8703}, {1131, 31487}, {1132, 3528}, {1152, 12103}, {1328, 8252}, {1587, 5073}, {1588, 1657}, {1656, 23263}, {3068, 3830}, {3069, 3534}, {3090, 6455}, {3091, 6449}, {3146, 3311}, {3312, 3529}, {3316, 9543}, {3317, 21734}, {3522, 13951}, {3525, 6496}, {3543, 13665}, {3592, 22644}, {3593, 26615}, {3628, 6409}, {3630, 32419}, {3843, 9540}, {3850, 5418}, {3856, 9680}, {3858, 10576}, {3861, 6468}, {4316, 19029}, {4324, 19027}, {5059, 7582}, {5066, 8253}, {5076, 31412}, {5420, 33923}, {6410, 13993}, {6425, 12102}, {6427, 11541}, {6447, 13886}, {6450, 17538}, {6456, 13939}, {6460, 17800}, {6476, 35812}, {6481, 15686}, {6484, 41991}, {6564, 15687}, {7585, 15682}, {7586, 11001}, {8991, 18383}, {10295, 13937}, {10577, 15712}, {10645, 34551}, {10646, 34552}, {12101, 13846}, {13847, 15690}, {13883, 33697}, {13901, 18513}, {13935, 15696}, {13979, 37853}, {15681, 18510}, {15684, 18512}, {15685, 19053}, {18514, 18965}, {19117, 35820}, {19710, 32788}, {28178, 35774}, {28186, 35775}, {31439, 31673}, {31454, 35786}, {32787, 33699}, {32805, 33457}, {35404, 35822}, {35610, 37705}, {35777, 37814}

X(42226) = GIBERT(2*SQRT(3),3,-3) POINT

Barycentrics    2*a^2*S - 3*a^2*SA + 6*SB*SC : :

X(42226) lies on these lines: {2, 6452}, {3, 18538}, {4, 3591}, {5, 6396}, {6, 30}, {20, 6221}, {140, 6412}, {371, 15704}, {372, 3627}, {376, 6451}, {381, 6446}, {382, 6395}, {485, 548}, {486, 3853}, {546, 1152}, {549, 6564}, {550, 3070}, {590, 8703}, {615, 3845}, {1131, 3528}, {1151, 12103}, {1327, 8253}, {1587, 1657}, {1588, 5073}, {1656, 23253}, {3068, 3534}, {3069, 3830}, {3090, 6456}, {3091, 6450}, {3146, 3312}, {3311, 3529}, {3316, 21734}, {3522, 8976}, {3525, 6497}, {3543, 13785}, {3594, 22615}, {3595, 26616}, {3628, 6410}, {3630, 32421}, {3843, 13935}, {3850, 5420}, {3858, 10577}, {3861, 6469}, {4316, 19030}, {4324, 19028}, {5059, 7581}, {5066, 8252}, {5418, 33923}, {6409, 13925}, {6426, 12102}, {6428, 11541}, {6448, 13939}, {6449, 17538}, {6455, 13886}, {6459, 17800}, {6477, 35813}, {6480, 15686}, {6485, 41991}, {6565, 15687}, {7585, 11001}, {7586, 15682}, {9540, 15696}, {9541, 15681}, {10295, 13884}, {10576, 15712}, {10645, 34552}, {10646, 34551}, {12101, 13847}, {13846, 15690}, {13903, 31414}, {13915, 37853}, {13936, 33697}, {13958, 18513}, {13980, 18383}, {14269, 17851}, {15684, 18510}, {15685, 19054}, {18514, 18966}, {19116, 35821}, {19710, 32787}, {21737, 35247}, {28178, 35775}, {28186, 35774}, {32788, 33699}, {32806, 33456}, {35404, 35823}, {35611, 37705}, {35776, 37814}

X(42227) = GIBERT(-3,1,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - Sqrt[3]*a^2*SA - 2*SB*SC : :

X(42227) lies on these lines: {3, 395}, {4, 372}, {6, 14813}, {14, 2041}, {15, 2045}, {18, 485}, {61, 2042}, {62, 2044}, {397, 3312}, {462, 10133}, {1152, 5339}, {2043, 16964}, {3070, 18581}, {3365, 6561}, {3366, 10194}, {3594, 5340}, {5318, 6395}, {5321, 6398}, {5334, 6396}, {5343, 6454}, {5418, 11489}, {5868, 36714}, {6418, 35740}, {6420, 35732}, {13847, 15765}, {13951, 18582}, {16268, 36455}, {18586, 32788}, {18587, 41946}, {33353, 33439}, {33367, 33443}, {33369, 33437}, {36437, 41101}, {37342, 37824}, {39679, 41034}

X(42228) = GIBERT(3,-1,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + Sqrt[3]*a^2*SA - 2*SB*SC : :

X(42228) lies on these lines: {3, 395}, {4, 371}, {6, 14814}, {14, 2042}, {15, 2046}, {18, 486}, {61, 2041}, {62, 2043}, {397, 3311}, {462, 10132}, {1151, 5339}, {2044, 16964}, {3071, 18581}, {3364, 6560}, {3367, 10195}, {3592, 5340}, {5318, 6199}, {5321, 6221}, {5334, 6200}, {5343, 6453}, {5349, 35740}, {5420, 11489}, {5868, 36709}, {8976, 18582}, {13846, 18585}, {16268, 36437}, {18586, 41945}, {18587, 32787}, {33350, 33438}, {33366, 33436}, {33368, 33442}, {36455, 41101}, {37343, 37824}, {39648, 41034}

X(42229) = GIBERT(3,1,-SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - Sqrt[3]*a^2*SA + 2*SB*SC : :

X(42229) lies on these lines: {3, 396}, {4, 372}, {6, 14814}, {13, 2042}, {16, 2046}, {17, 485}, {61, 2043}, {62, 2041}, {398, 3312}, {463, 10133}, {1152, 5340}, {2044, 16965}, {3070, 18582}, {3390, 6561}, {3391, 10194}, {3594, 5339}, {5318, 6398}, {5321, 6395}, {5335, 6396}, {5344, 6454}, {5418, 11488}, {5869, 36714}, {6450, 35740}, {13847, 18585}, {13951, 18581}, {16267, 36437}, {18586, 41946}, {18587, 32788}, {33351, 33437}, {33367, 33439}, {33369, 33441}, {35739, 41974}, {36455, 41100}, {37342, 37825}, {39679, 41035}

X(42230) = GIBERT(3,1,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + Sqrt[3]*a^2*SA + 2*SB*SC : :

X(42230) lies on these lines: {3, 396}, {4, 371}, {6, 14813}, {13, 2041}, {16, 2045}, {17, 486}, {18, 35730}, {61, 2044}, {62, 2042}, {398, 3311}, {463, 10132}, {1151, 5340}, {2043, 16965}, {3071, 18582}, {3389, 6560}, {3392, 10195}, {3592, 5339}, {5318, 6221}, {5321, 6199}, {5335, 6200}, {5344, 6453}, {5420, 11488}, {5869, 36709}, {6419, 35732}, {8976, 18581}, {13846, 15765}, {16267, 36455}, {18586, 32787}, {18587, 41945}, {33352, 33436}, {33366, 33440}, {33368, 33438}, {36437, 41100}, {37343, 37825}, {39648, 41035}

X(42231) = GIBERT(-3,2,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - Sqrt[3]*a^2*SA - 4*SB*SC : :

X(42231) lies on these lines: {3, 14}, {4, 372}, {15, 10577}, {61, 18586}, {299, 33436}, {371, 398}, {397, 6420}, {485, 22237}, {491, 33451}, {1152, 3367}, {1656, 3366}, {2042, 10654}, {2044, 35822}, {2046, 10576}, {3070, 11543}, {3312, 5340}, {3364, 8960}, {3365, 35821}, {3389, 41973}, {3390, 6564}, {3392, 13785}, {5321, 6396}, {5334, 6200}, {5349, 6454}, {6301, 33369}, {6419, 35732}, {16809, 23251}, {19107, 23261}, {22114, 22874}, {22598, 33353}, {33445, 39388}, {34551, 36470}, {34559, 36453}, {35740, 35771}, {35813, 36970}, {36455, 41120}

X(42232) = GIBERT(3,-2,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + Sqrt[3]*a^2*SA - 4*SB*SC : :

X(42232) lies on these lines: {3, 14}, {4, 371}, {15, 10576}, {61, 18587}, {299, 33437}, {372, 398}, {397, 6419}, {486, 22237}, {492, 33450}, {1151, 3366}, {1656, 3367}, {2041, 10654}, {2043, 35823}, {2045, 10577}, {3071, 11543}, {3311, 5340}, {3364, 35820}, {3389, 6565}, {3390, 41973}, {3391, 13665}, {5321, 6200}, {5334, 6396}, {5349, 6453}, {5365, 35732}, {6305, 33366}, {16809, 23261}, {19107, 23251}, {22114, 22872}, {22627, 33350}, {33444, 39387}, {34552, 36452}, {34562, 36469}, {35812, 36970}, {36437, 41120}

X(42233) = GIBERT(3,2,-SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - Sqrt[3]*a^2*SA + 4*SB*SC : :

X(42233) lies on these lines: {3, 13}, {4, 372}, {16, 10577}, {62, 18587}, {298, 33438}, {371, 397}, {398, 6420}, {485, 22235}, {491, 33449}, {1152, 3392}, {1656, 3391}, {2041, 10653}, {2043, 35822}, {2045, 10576}, {3070, 11542}, {3312, 5339}, {3364, 41974}, {3365, 6564}, {3367, 13785}, {3389, 8960}, {3390, 35821}, {5318, 6396}, {5335, 6200}, {5350, 6454}, {5366, 35732}, {6300, 33367}, {16808, 23251}, {19106, 23261}, {22113, 22919}, {22600, 33351}, {33447, 39388}, {34552, 36453}, {34562, 36470}, {35813, 36969}, {36437, 41119}

X(42234) = GIBERT(3,2,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + Sqrt[3]*a^2*SA + 4*SB*SC : :

X(42234) lies on these lines: {3, 13}, {4, 371}, {16, 10576}, {62, 18586}, {298, 33439}, {372, 397}, {398, 6419}, {486, 22235}, {492, 33448}, {1151, 3391}, {1656, 3392}, {2042, 10653}, {2044, 35823}, {2046, 10577}, {3071, 11542}, {3311, 5339}, {3364, 6565}, {3365, 41974}, {3366, 13665}, {3389, 35820}, {5318, 6200}, {5335, 6396}, {5350, 6453}, {6304, 33368}, {6420, 35732}, {16808, 23261}, {19106, 23251}, {22113, 22917}, {22629, 33352}, {33446, 39387}, {34551, 36469}, {34559, 36452}, {35812, 36969}, {36455, 41119}

X(42235) = GIBERT(-3,2*SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 4*Sqrt[3]*SB*SC : :

X(42235) lies on these lines: {2, 33361}, {3, 3367}, {4, 16}, {5, 3364}, {13, 486}, {14, 371}, {15, 3071}, {30, 3365}, {61, 1588}, {62, 3070}, {372, 16965}, {381, 3391}, {615, 35739}, {621, 33353}, {2042, 3389}, {2043, 16242}, {2045, 33416}, {2046, 16966}, {3105, 41018}, {5238, 23275}, {5318, 18762}, {6200, 16967}, {6777, 33431}, {10645, 23259}, {10646, 14814}, {11486, 23251}, {13749, 41021}, {14233, 41034}, {15765, 37835}, {18585, 36969}, {18587, 35787}, {19054, 36454}, {23273, 34754}, {25559, 35759}, {33436, 40901}

X(42236) = GIBERT(3,-2*SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 4*Sqrt[3]*SB*SC : :

X(42236) lies on these lines: {2, 33359}, {3, 3367}, {4, 15}, {5, 3389}, {13, 371}, {14, 486}, {16, 3071}, {20, 35739}, {30, 3390}, {61, 3070}, {62, 1588}, {372, 16964}, {381, 3366}, {622, 33351}, {2041, 3364}, {2044, 16241}, {2045, 16967}, {2046, 33417}, {5237, 23275}, {5321, 18762}, {5352, 35732}, {6200, 16966}, {6778, 33431}, {10645, 14813}, {10646, 23259}, {11485, 23251}, {13749, 41020}, {14233, 41035}, {15765, 36970}, {18585, 35731}, {18586, 35787}, {19054, 36436}, {23273, 34755}, {33438, 40900}

X(42237) = GIBERT(3,2*SQRT(3),-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 4*Sqrt[3]*SB*SC : :

X(42) lies on these lines: {2, 33358}, {3, 3366}, {4, 16}, {5, 3365}, {13, 485}, {14, 372}, {15, 3070}, {30, 3364}, {61, 1587}, {62, 3071}, {371, 16965}, {381, 3392}, {621, 33350}, {2041, 3390}, {2044, 16242}, {2045, 16966}, {2046, 33416}, {5238, 23269}, {5318, 18538}, {5340, 8960}, {5351, 35732}, {6396, 16967}, {6777, 33430}, {10645, 23249}, {10646, 14813}, {11486, 23261}, {13748, 41021}, {14230, 41034}, {15765, 36969}, {18585, 35739}, {18586, 35786}, {19053, 36436}, {23267, 34754}, {33437, 40901}

X(42238) = GIBERT(3,2*SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 4*Sqrt[3]*SB*SC : :

X(42238) lies on these lines: {2, 33360}, {3, 3366}, {4, 15}, {5, 3390}, {13, 372}, {14, 485}, {16, 3070}, {30, 3389}, {61, 3071}, {62, 1587}, {371, 16964}, {381, 3367}, {622, 33352}, {2042, 3365}, {2043, 16241}, {2045, 33417}, {2046, 16967}, {5237, 23269}, {5321, 18538}, {5339, 8960}, {6396, 16966}, {6778, 33430}, {10645, 14814}, {10646, 23249}, {11485, 23261}, {13748, 41020}, {14230, 41035}, {15765, 37832}, {18585, 36970}, {18587, 35786}, {19053, 36454}, {23267, 34755}, {33439, 40900}

X(42239) = GIBERT(-3,3,2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42239) lies on these lines: {3, 5321}, {4, 615}, {5, 3390}, {6, 35732}, {14, 34551}, {15, 18762}, {30, 3365}, {371, 398}, {372, 5318}, {395, 2044}, {396, 486}, {397, 3312}, {639, 6303}, {1151, 5334}, {2042, 3071}, {3364, 11543}, {3366, 37835}, {3594, 5335}, {5339, 6409}, {5349, 14814}, {5471, 35742}, {5480, 41018}, {6114, 35759}, {10577, 15765}, {11480, 32790}, {12256, 41039}, {15764, 36470}, {16809, 35739}, {18585, 35820}, {22856, 35746}, {32788, 36454}, {34552, 36970}, {35734, 41120}, {35735, 41113}

X(42240) = GIBERT(3,-3,2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42240) lies on these lines: {3, 5321}, {4, 590}, {5, 3389}, {14, 34552}, {15, 18538}, {30, 3364}, {371, 5318}, {372, 398}, {395, 2043}, {396, 485}, {397, 3311}, {640, 6307}, {1152, 5334}, {2041, 3070}, {3365, 11543}, {3367, 37835}, {3592, 5335}, {3845, 35731}, {5339, 6410}, {5349, 14813}, {6409, 35732}, {10576, 18585}, {11480, 32789}, {12257, 41039}, {15765, 35821}, {16809, 35738}, {16964, 35739}, {21736, 41038}, {32787, 36436}, {34551, 36970}

X(42241) = GIBERT(3,3,-2*SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42241) lies on these lines: {3, 5318}, {4, 615}, {5, 3365}, {13, 34552}, {16, 18762}, {30, 3390}, {371, 397}, {372, 5321}, {395, 486}, {396, 2043}, {398, 3312}, {639, 6302}, {1151, 5335}, {2041, 3071}, {3389, 11542}, {3391, 37832}, {3594, 5334}, {5340, 6409}, {5350, 14813}, {6410, 35732}, {10577, 18585}, {11481, 32790}, {12256, 41038}, {15765, 35820}, {16808, 35738}, {19106, 35739}, {32788, 36436}, {34551, 36969}

X(42242) = GIBERT(-3,3,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42242) lies on these lines: {3, 5321}, {4, 372}, {14, 485}, {371, 5334}, {397, 6418}, {398, 3311}, {489, 33353}, {1151, 5339}, {1327, 36445}, {1328, 3392}, {2041, 35739}, {2042, 3389}, {2043, 3367}, {2046, 3366}, {3070, 40694}, {3071, 10654}, {3312, 5318}, {3365, 19107}, {3390, 16809}, {5335, 6420}, {5340, 6432}, {5349, 6450}, {6410, 14814}, {9738, 16002}, {13785, 40693}, {18585, 23251}, {20428, 37342}, {33603, 35737}, {35731, 41108}, {36714, 41038}

X(42243) = GIBERT(3,-3,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + Sqrt[3]*a^2*SA - 6*SB*SC : :

X(42243) lies on these lines: {3, 5321}, {4, 371}, {14, 486}, {372, 5334}, {397, 6417}, {398, 3312}, {490, 33350}, {1152, 5339}, {1327, 3391}, {1328, 36463}, {2041, 3390}, {2044, 3366}, {2045, 3367}, {3070, 10654}, {3071, 40694}, {3311, 5318}, {3364, 19107}, {3389, 16809}, {5335, 6419}, {5340, 6431}, {5349, 6449}, {6200, 35732}, {6221, 35740}, {6409, 14813}, {9739, 16002}, {13665, 40693}, {15765, 23261}, {20428, 37343}, {36709, 41038}

X(42244) = GIBERT(3,3,-SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S - Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42244) lies on these lines: {3, 5318}, {4, 372}, {13, 485}, {371, 5335}, {397, 3311}, {398, 6418}, {489, 33351}, {1151, 5340}, {1327, 36463}, {1328, 3367}, {2041, 3364}, {2044, 3392}, {2045, 3391}, {3070, 40693}, {3071, 10653}, {3312, 5321}, {3365, 16808}, {3390, 19106}, {5334, 6420}, {5339, 6432}, {5350, 6450}, {6396, 35732}, {6410, 14813}, {9738, 16001}, {13785, 40694}, {15765, 23251}, {20429, 37342}, {36714, 41039}

X(42245) = GIBERT(3,3,SQRT(3)) POINT

Barycentrics    Sqrt[3]*a^2*S + Sqrt[3]*a^2*SA + 6*SB*SC : :

X(42245) lies on these lines: {3, 5318}, {4, 371}, {13, 486}, {372, 5335}, {397, 3312}, {398, 6417}, {490, 33352}, {1152, 5340}, {1327, 3366}, {1328, 36445}, {2042, 3365}, {2043, 3391}, {2046, 3392}, {3070, 10653}, {3071, 40693}, {3311, 5321}, {3364, 16808}, {3389, 19106}, {5334, 6419}, {5339, 6431}, {5350, 6449}, {6409, 14814}, {9739, 16001}, {13665, 40694}, {18585, 23261}, {20429, 37343}, {33602, 35737}, {36709, 41039}

X(42246) = GIBERT(-3,SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S - a^2*SA - 2*Sqrt[3]*SB*SC : :

X(42246) lies on these lines: {2, 33367}, {3, 486}, {4, 14}, {15, 23273}, {16, 23259}, {17, 14226}, {61, 1588}, {371, 18581}, {398, 18586}, {488, 22882}, {640, 22598}, {1132, 3367}, {1328, 18587}, {2042, 10654}, {2043, 36452}, {2044, 35823}, {3364, 18582}, {3389, 5334}, {3390, 5335}, {3392, 6459}, {5237, 23275}, {5321, 22615}, {5340, 18585}, {11294, 33440}, {12322, 37178}, {13939, 35739}, {14813, 22236}, {14814, 36843}, {22238, 23261}, {36437, 41113}, {36445, 41112}

X(42247) = GIBERT(3,-SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S + a^2*SA - 2*Sqrt[3]*SB*SC : :

X(42247) lies on these lines: {2, 33369}, {3, 486}, {4, 13}, {15, 23259}, {16, 23273}, {18, 14226}, {62, 1588}, {371, 18582}, {397, 18587}, {488, 22927}, {640, 22600}, {1132, 3392}, {1328, 18586}, {2041, 10653}, {2043, 35823}, {2044, 36469}, {3364, 5335}, {3365, 5334}, {3367, 6459}, {3389, 18581}, {5238, 23275}, {5318, 22615}, {5339, 15765}, {11294, 33442}, {11488, 35730}, {12322, 37177}, {14813, 36836}, {14814, 22238}, {22236, 23261}, {36455, 41112}, {36463, 41113}

X(42248) = GIBERT(3,SQRT(3),-1) POINT

Barycentrics    Sqrt[3]*a^2*S - a^2*SA + 2*Sqrt[3]*SB*SC : :

X(42248) lies on these lines: {2, 33368}, {3, 485}, {4, 14}, {15, 23267}, {16, 23249}, {17, 14241}, {61, 1587}, {372, 18581}, {398, 18587}, {487, 22883}, {639, 22627}, {1131, 3366}, {1327, 18586}, {2041, 10654}, {2043, 35822}, {2044, 36470}, {3365, 18582}, {3389, 5335}, {3390, 5334}, {3391, 6460}, {5237, 23269}, {5321, 22644}, {5340, 15765}, {11293, 33441}, {12323, 37178}, {14540, 21737}, {14813, 36843}, {14814, 22236}, {22238, 23251}, {36455, 41113}, {36463, 41112}

X(42249) = GIBERT(3,SQRT(3),1) POINT

Barycentrics    Sqrt[3]*a^2*S + a^2*SA + 2*Sqrt[3]*SB*SC : :

X(42249) lies on these lines: {2, 33366}, {3, 485}, {4, 13}, {15, 23249}, {16, 23267}, {18, 14241}, {62, 1587}, {372, 18582}, {397, 18586}, {487, 22928}, {639, 22629}, {1131, 3391}, {1327, 18587}, {2042, 10653}, {2043, 36453}, {2044, 35822}, {3364, 5334}, {3365, 5335}, {3366, 6460}, {3390, 18581}, {5238, 23269}, {5318, 22644}, {5339, 18585}, {11293, 33443}, {12323, 37177}, {14541, 21737}, {14813, 22238}, {14814, 36836}, {22236, 23251}, {36437, 41112}, {36445, 41113}

X(42250) = GIBERT(-3,SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA - 2*Sqrt[3]*SB*SC : :

X(42250) lies on these lines: {3, 486}, {4, 395}, {6, 35732}, {18, 15765}, {61, 14813}, {62, 3070}, {371, 11543}, {397, 2044}, {398, 2042}, {1151, 23303}, {1588, 22236}, {2043, 16773}, {3364, 16966}, {5237, 14814}, {5335, 13939}, {6303, 22882}, {11480, 23273}, {11481, 23259}, {16267, 36439}, {16965, 18585}, {18586, 32787}, {23261, 36843}, {31414, 37641}, {32490, 33445}, {33395, 33424}, {34551, 35823}, {34559, 36470}, {35746, 35849}, {36455, 41951}

X(42251) = GIBERT(3,-SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA - 2*Sqrt[3]*SB*SC : :

X(42251) lies on these lines: {3, 486}, {4, 396}, {17, 18585}, {61, 3070}, {62, 14814}, {371, 11542}, {397, 2041}, {398, 2043}, {1151, 23302}, {1588, 22238}, {2044, 16772}, {3389, 16967}, {5238, 14813}, {5334, 13939}, {6302, 22927}, {6565, 35738}, {11480, 23259}, {11481, 23273}, {11488, 35740}, {15765, 16964}, {16268, 36457}, {18587, 32787}, {23261, 35732}, {31414, 37640}, {32490, 33447}, {33392, 33427}, {34552, 35823}, {34562, 36453}, {36437, 41951}

X(42252) = GIBERT(3,SQRT(3),-2) POINT

Barycentrics    Sqrt[3]*a^2*S - 2*a^2*SA + 2*Sqrt[3]*SB*SC : :

X(42252) lies on these lines: {3, 485}, {4, 395}, {18, 18585}, {61, 14814}, {62, 3071}, {372, 11543}, {397, 2043}, {398, 2041}, {1152, 23303}, {1587, 22236}, {2044, 16773}, {3365, 16966}, {5237, 14813}, {5335, 13886}, {6307, 22883}, {6564, 35738}, {11480, 23267}, {11481, 23249}, {15765, 16965}, {16267, 36457}, {18587, 32788}, {23251, 35732}, {31412, 35740}, {32491, 33444}, {33393, 33426}, {34552, 35822}, {34562, 36452}, {36437, 41952}

X(42253) = GIBERT(3,SQRT(3),2) POINT

Barycentrics    Sqrt[3]*a^2*S + 2*a^2*SA + 2*Sqrt[3]*SB*SC : :

X(42253) lies on these lines: {3, 485}, {4, 396}, {6, 35732}, {17, 15765}, {61, 3071}, {62, 14813}, {372, 11542}, {397, 2042}, {398, 2044}, {1152, 23302}, {1587, 22238}, {2043, 16772}, {3365, 35730}, {3390, 16967}, {5238, 14814}, {5334, 13886}, {6306, 22928}, {11480, 23249}, {11481, 23267}, {16268, 36439}, {16964, 18585}, {18586, 32788}, {23251, 36836}, {32491, 33446}, {33394, 33425}, {34551, 35822}, {34559, 36469}, {35746, 35846}, {36455, 41952}

X(42254) = GIBERT(-3,SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA - 2*Sqrt[3]*SB*SC : :

X(42254) lies on these lines: {2, 3364}, {3, 486}, {4, 16}, {6, 14813}, {13, 3317}, {15, 1588}, {20, 3365}, {61, 7582}, {62, 1587}, {371, 2042}, {372, 2044}, {376, 36465}, {395, 485}, {488, 6303}, {638, 33353}, {640, 6301}, {1151, 15765}, {2041, 6565}, {2043, 35821}, {2045, 6200}, {2046, 10577}, {3068, 3366}, {3069, 3390}, {3070, 11486}, {3091, 3391}, {3389, 6459}, {5237, 23263}, {5335, 13941}, {5418, 23303}, {5871, 41021}, {7584, 34551}, {8981, 35738}, {10194, 23302}, {10645, 23273}, {10646, 23259}, {10784, 41020}, {11481, 14814}, {12322, 37173}, {13748, 41034}, {13847, 36439}, {13935, 35739}, {16773, 22615}, {22238, 23251}, {23249, 34755}, {35733, 37640}

X(42255) = GIBERT(3,-SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 2*Sqrt[3]*SB*SC : :

X(42255) lies on these lines: {2, 3367}, {3, 486}, {4, 15}, {6, 14814}, {14, 3317}, {16, 1588}, {20, 3390}, {61, 1587}, {62, 7582}, {371, 2041}, {372, 2043}, {376, 35739}, {396, 485}, {488, 6302}, {638, 33351}, {640, 6300}, {1151, 16644}, {2042, 6565}, {2044, 35821}, {2045, 10577}, {2046, 6200}, {3068, 3391}, {3069, 3365}, {3070, 11485}, {3091, 3366}, {3364, 6459}, {5238, 23263}, {5334, 13941}, {5418, 23302}, {5871, 41020}, {7584, 34552}, {10194, 23303}, {10645, 23259}, {10646, 23273}, {10784, 41021}, {11480, 14813}, {12322, 37172}, {13748, 41035}, {13847, 36457}, {13935, 36967}, {16772, 22615}, {22236, 23251}, {23249, 34754}, {35731, 36445}

X(42256) = GIBERT(3,SQRT(3),-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 2*Sqrt[3]*SB*SC : :

X(42256) lies on these lines: {2, 3365}, {3, 485}, {4, 16}, {6, 14814}, {13, 3316}, {15, 1587}, {20, 3364}, {61, 7581}, {62, 1588}, {371, 2043}, {372, 2041}, {376, 36446}, {395, 486}, {487, 6307}, {637, 33350}, {639, 6305}, {1152, 16645}, {2042, 6564}, {2044, 35820}, {2045, 10576}, {2046, 6396}, {3068, 3389}, {3069, 3367}, {3071, 11486}, {3091, 3392}, {3366, 31412}, {3390, 6460}, {5237, 23253}, {5335, 8972}, {5420, 23303}, {5870, 41021}, {7583, 34552}, {9540, 36968}, {10195, 23302}, {10645, 23267}, {10646, 23249}, {10783, 41020}, {11481, 14813}, {12323, 37173}, {13749, 41034}, {13846, 36457}, {14538, 21737}, {16773, 22644}, {22238, 23261}, {23259, 34755}

X(42257) = GIBERT(3,SQRT(3),3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 2*Sqrt[3]*SB*SC : :

X(42257) lies on these lines: {2, 3366}, {3, 485}, {4, 15}, {6, 14813}, {14, 3316}, {16, 1587}, {20, 3389}, {61, 1588}, {62, 7581}, {371, 2044}, {372, 2042}, {376, 36464}, {396, 486}, {487, 6306}, {631, 35739}, {637, 33352}, {639, 6304}, {1152, 15765}, {2041, 6564}, {2043, 35820}, {2045, 6396}, {2046, 10576}, {3068, 3364}, {3069, 3392}, {3071, 11485}, {3091, 3367}, {3365, 6460}, {3391, 31412}, {5238, 23253}, {5334, 8972}, {5420, 23302}, {5870, 41020}, {6561, 35740}, {7583, 34551}, {9540, 36967}, {10195, 23303}, {10645, 23249}, {10646, 23267}, {10783, 41021}, {11480, 14814}, {12323, 37172}, {13749, 41035}, {13846, 36439}, {13966, 35738}, {14539, 21737}, {16772, 22644}, {22236, 23261}, {23259, 34754}, {35944, 36762}

X(42258) = GIBERT(SQRT(3),-1,2) POINT

Barycentrics    a^2*S + 2*a^2*SA - 2*SB*SC : :

X(42258) lies on these lines: {1, 9647}, {2, 6409}, {3, 486}, {4, 590}, {5, 6200}, {6, 20}, {30, 371}, {140, 6565}, {141, 489}, {156, 9676}, {165, 13973}, {372, 550}, {376, 1152}, {381, 5418}, {382, 485}, {490, 524}, {516, 7969}, {546, 10576}, {548, 6396}, {549, 10577}, {591, 12221}, {631, 6411}, {637, 35949}, {952, 35610}, {1086, 31550}, {1124, 4299}, {1131, 3068}, {1132, 15717}, {1160, 12124}, {1327, 15684}, {1328, 5054}, {1335, 4302}, {1503, 26441}, {1587, 3529}, {1656, 6455}, {1657, 3311}, {1699, 9615}, {1885, 5412}, {1991, 12323}, {2041, 5318}, {2042, 5321}, {2066, 7354}, {2067, 6284}, {2548, 9600}, {2549, 6424}, {2883, 10533}, {3069, 3522}, {3090, 23263}, {3091, 8253}, {3092, 18533}, {3297, 4293}, {3298, 4294}, {3299, 4316}, {3301, 4324}, {3312, 3534}, {3364, 14814}, {3365, 36967}, {3371, 41980}, {3372, 41979}, {3389, 14813}, {3390, 36968}, {3523, 8252}, {3524, 23275}, {3526, 6451}, {3528, 6412}, {3530, 18762}, {3543, 6429}, {3575, 11473}, {3583, 9661}, {3585, 9646}, {3589, 11293}, {3590, 10139}, {3594, 7582}, {3627, 6453}, {3830, 6407}, {3832, 6433}, {3843, 6445}, {3845, 6484}, {3850, 6486}, {3853, 6480}, {4190, 31473}, {4297, 7968}, {5058, 7756}, {5059, 7585}, {5062, 6781}, {5073, 13665}, {5076, 6519}, {5254, 12963}, {5305, 41410}, {5414, 15338}, {5475, 9674}, {5587, 9582}, {5691, 9616}, {5870, 15428}, {5895, 17819}, {6199, 17800}, {6398, 15696}, {6417, 15681}, {6419, 15704}, {6420, 12103}, {6422, 7737}, {6426, 7586}, {6431, 7581}, {6437, 23249}, {6439, 9692}, {6450, 18510}, {6452, 13961}, {6456, 15688}, {6468, 8972}, {6470, 15683}, {6478, 35404}, {6485, 41962}, {6502, 15326}, {6644, 35777}, {6811, 10838}, {7517, 9682}, {7728, 10819}, {7748, 9675}, {7951, 31499}, {8276, 18534}, {8407, 36733}, {8550, 8982}, {8703, 13966}, {8855, 10691}, {8980, 39809}, {8991, 41362}, {8994, 12295}, {8997, 39838}, {8998, 13202}, {9583, 41869}, {9631, 18455}, {9649, 18965}, {9662, 13901}, {9686, 26883}, {9695, 35502}, {9778, 19065}, {9812, 13902}, {9975, 11179}, {10264, 35835}, {10295, 10881}, {10304, 13847}, {10519, 12509}, {10748, 11835}, {10820, 38723}, {10880, 18560}, {11513, 31829}, {11541, 23269}, {11836, 38798}, {12257, 13749}, {12296, 13881}, {12305, 35945}, {12376, 34153}, {12512, 13936}, {12943, 31472}, {12964, 15311}, {12968, 35947}, {12970, 34782}, {13488, 35764}, {13748, 21736}, {13883, 28164}, {13886, 15682}, {13912, 31673}, {13939, 21735}, {13941, 21734}, {13947, 16192}, {13993, 34200}, {14226, 15715}, {14677, 35827}, {15171, 35768}, {15325, 35803}, {15686, 35770}, {15692, 41951}, {15698, 41947}, {15765, 37835}, {17365, 31549}, {17845, 19088}, {18289, 34609}, {18481, 35775}, {18585, 35731}, {18990, 35808}, {19051, 38788}, {19089, 22676}, {19145, 31670}, {22791, 35763}, {23311, 39387}, {28160, 31439}, {28174, 35641}, {28224, 35842}, {32423, 35826}, {32497, 36709}, {32521, 35867}, {33923, 35256}, {34380, 39893}, {34773, 35642}, {35765, 37458}, {36711, 39649}

X(42259) = GIBERT(SQRT(3),1,-2) POINT

Barycentrics    a^2*S - 2*a^2*SA + 2*SB*SC : :

X(42259) lies on these lines: {2, 6410}, {3, 485}, {4, 615}, {5, 6396}, {6, 20}, {30, 372}, {140, 6564}, {141, 490}, {165, 13911}, {371, 550}, {376, 1151}, {381, 5420}, {382, 486}, {489, 524}, {516, 7968}, {546, 10577}, {548, 6200}, {549, 10576}, {591, 12322}, {631, 6412}, {638, 35948}, {952, 35611}, {1086, 31549}, {1124, 4302}, {1131, 15717}, {1132, 3069}, {1161, 12123}, {1327, 5054}, {1328, 15684}, {1335, 4299}, {1503, 8982}, {1588, 3529}, {1656, 6456}, {1657, 3312}, {1885, 5413}, {1991, 12222}, {2041, 5321}, {2042, 5318}, {2066, 15338}, {2067, 15326}, {2549, 6423}, {2883, 10534}, {3068, 3522}, {3090, 23253}, {3091, 8252}, {3093, 18533}, {3297, 4294}, {3298, 4293}, {3299, 4324}, {3301, 4316}, {3311, 3534}, {3364, 36967}, {3365, 14813}, {3385, 41980}, {3386, 41979}, {3389, 36968}, {3390, 14814}, {3523, 8253}, {3524, 23269}, {3526, 6452}, {3528, 6411}, {3530, 18538}, {3543, 6430}, {3575, 11474}, {3589, 11294}, {3591, 10140}, {3592, 7581}, {3627, 6454}, {3830, 6408}, {3832, 6434}, {3843, 6446}, {3845, 6485}, {3850, 6487}, {3853, 6481}, {4297, 7969}, {5010, 9646}, {5058, 6781}, {5059, 7586}, {5062, 7756}, {5073, 13785}, {5076, 6522}, {5217, 31472}, {5254, 12968}, {5305, 41411}, {5414, 7354}, {5691, 13973}, {5871, 15428}, {5895, 17820}, {6199, 9681}, {6221, 15696}, {6284, 6502}, {6395, 17800}, {6418, 15681}, {6419, 12103}, {6420, 15704}, {6421, 7737}, {6425, 7585}, {6432, 7582}, {6438, 23259}, {6445, 31487}, {6449, 18512}, {6451, 9680}, {6455, 15688}, {6469, 13941}, {6471, 15683}, {6479, 35404}, {6484, 41961}, {6644, 35776}, {6813, 10837}, {6872, 31473}, {7280, 9661}, {7728, 10820}, {8277, 18534}, {8400, 36719}, {8550, 26441}, {8703, 8981}, {8854, 10691}, {8960, 33923}, {8972, 21734}, {9600, 31411}, {9778, 19066}, {9812, 13959}, {9974, 11179}, {10264, 35834}, {10295, 10880}, {10304, 13846}, {10519, 12510}, {10748, 11836}, {10819, 38723}, {10881, 18560}, {11514, 31829}, {11541, 23275}, {11835, 38798}, {12256, 13748}, {12295, 13969}, {12297, 13881}, {12305, 21737}, {12306, 35944}, {12375, 34153}, {12512, 13883}, {12963, 35946}, {12964, 34782}, {12970, 15311}, {13202, 13990}, {13488, 35765}, {13886, 21735}, {13893, 16192}, {13925, 34200}, {13936, 28164}, {13939, 15682}, {13967, 39809}, {13975, 31673}, {13980, 41362}, {13989, 39838}, {14241, 15715}, {14677, 35826}, {15171, 35769}, {15325, 35802}, {15515, 31481}, {15686, 35771}, {15692, 41952}, {15698, 41948}, {15765, 37832}, {15815, 31463}, {17365, 31550}, {17845, 19087}, {18290, 34609}, {18481, 35774}, {18585, 35739}, {18990, 35809}, {19052, 38788}, {19090, 22676}, {19146, 31670}, {22791, 35762}, {23312, 39388}, {28174, 35642}, {28224, 35843}, {32423, 35827}, {32494, 36714}, {32521, 35866}, {34380, 39894}, {34773, 35641}, {35764, 37458}, {36712, 39658}

X(42260) = GIBERT(SQRT(3),-1,3) POINT

Barycentrics    a^2*S + 3*a^2*SA - 2*SB*SC : :

X(42260) lies on these lines: {1, 9631}, {2, 12819}, {3, 486}, {4, 5418}, {5, 6409}, {6, 550}, {20, 371}, {30, 485}, {55, 9647}, {56, 9660}, {140, 6411}, {372, 376}, {381, 6455}, {382, 590}, {488, 13712}, {489, 35949}, {491, 7802}, {492, 7782}, {511, 12124}, {546, 8253}, {548, 1152}, {549, 1328}, {631, 6565}, {641, 11147}, {944, 35610}, {962, 35763}, {1124, 15326}, {1131, 9542}, {1132, 15692}, {1335, 15338}, {1370, 18289}, {1504, 6781}, {1588, 3522}, {1614, 9676}, {1656, 6451}, {1657, 3070}, {2043, 3364}, {2044, 3389}, {2045, 33416}, {2046, 33417}, {2066, 4299}, {2067, 4302}, {2549, 12963}, {2777, 10819}, {3068, 3529}, {3069, 3528}, {3090, 35787}, {3102, 35946}, {3146, 6564}, {3311, 3534}, {3312, 15696}, {3317, 15698}, {3523, 10577}, {3526, 6496}, {3530, 8252}, {3543, 6486}, {3592, 12103}, {3627, 35255}, {3851, 32789}, {4293, 35808}, {4294, 35768}, {4297, 35775}, {5059, 6480}, {5073, 6445}, {5218, 35801}, {5286, 41410}, {5691, 9582}, {5731, 35642}, {5878, 10533}, {5925, 17819}, {6194, 35867}, {6361, 35641}, {6407, 13665}, {6408, 15695}, {6410, 7584}, {6412, 13966}, {6418, 15689}, {6419, 6460}, {6425, 7583}, {6426, 19116}, {6432, 15690}, {6433, 18538}, {6438, 41981}, {6447, 18512}, {6450, 15688}, {6452, 41964}, {6454, 7586}, {6456, 18510}, {6484, 31412}, {6497, 13961}, {6519, 13903}, {7288, 35803}, {7387, 9682}, {7388, 7937}, {7745, 9600}, {7747, 9674}, {7756, 9675}, {8276, 39568}, {8854, 34608}, {8972, 23253}, {8983, 28150}, {9543, 15683}, {9615, 41869}, {9646, 12943}, {9649, 19030}, {9661, 12953}, {9662, 19028}, {9677, 34148}, {9691, 15685}, {9693, 31414}, {9778, 35611}, {9862, 35878}, {10147, 12818}, {10194, 15712}, {10299, 23275}, {10304, 13935}, {10483, 31472}, {10820, 38726}, {10880, 35481}, {10881, 35503}, {10895, 31499}, {11001, 35822}, {11265, 34350}, {11473, 18533}, {11825, 35945}, {11835, 23699}, {11836, 38803}, {12082, 35776}, {12148, 12171}, {12244, 12375}, {12248, 35882}, {12253, 35880}, {12257, 22592}, {12383, 35826}, {12962, 19103}, {12964, 20427}, {12968, 19102}, {13172, 35824}, {13199, 35856}, {13200, 35828}, {13847, 34200}, {13911, 28160}, {13912, 28164}, {13939, 19708}, {13973, 31663}, {15681, 32787}, {15684, 41948}, {15686, 19117}, {15720, 32790}, {17928, 35777}, {18587, 35740}, {19145, 29181}, {21734, 35813}, {21735, 23273}, {22484, 41490}, {25406, 35841}, {26295, 36701}, {26441, 40274}, {31730, 35774}, {34781, 35864}, {35400, 41952}, {35731, 36455}

X(42261) = GIBERT(SQRT(3),1,-3) POINT

Barycentrics    a^2*S - 3*a^2*SA + 2*SB*SC : :

X(42261) lies on these lines: {2, 12818}, {3, 485}, {4, 5420}, {5, 6410}, {6, 550}, {20, 372}, {30, 486}, {140, 6412}, {371, 376}, {381, 6456}, {382, 615}, {487, 13835}, {490, 35948}, {491, 7782}, {492, 7802}, {511, 12123}, {546, 8252}, {548, 1151}, {549, 1327}, {631, 6564}, {642, 11147}, {944, 35611}, {962, 35762}, {1038, 9632}, {1124, 15338}, {1131, 15692}, {1335, 15326}, {1370, 18290}, {1505, 6781}, {1587, 3522}, {1656, 6452}, {1657, 3071}, {2041, 35739}, {2043, 3390}, {2044, 3365}, {2045, 33417}, {2046, 33416}, {2549, 12968}, {2777, 10820}, {3068, 3528}, {3069, 3529}, {3090, 35786}, {3103, 35947}, {3146, 6565}, {3311, 9681}, {3312, 3534}, {3316, 15698}, {3523, 10576}, {3524, 31412}, {3526, 6497}, {3530, 8253}, {3543, 6487}, {3594, 12103}, {3627, 35256}, {3851, 32790}, {4293, 35809}, {4294, 35769}, {4297, 35774}, {4299, 5414}, {4302, 6502}, {5010, 31472}, {5059, 6481}, {5073, 6446}, {5218, 35800}, {5267, 31484}, {5286, 41411}, {5731, 35641}, {5878, 10534}, {5925, 17820}, {6194, 35866}, {6361, 35642}, {6407, 15695}, {6408, 13785}, {6409, 7583}, {6411, 8981}, {6417, 15689}, {6419, 9541}, {6420, 6459}, {6425, 19117}, {6426, 7584}, {6431, 15690}, {6434, 18762}, {6437, 41981}, {6448, 18510}, {6449, 15688}, {6451, 41963}, {6453, 7585}, {6455, 18512}, {6485, 33703}, {6496, 13903}, {6522, 13961}, {7288, 35802}, {7389, 7937}, {8277, 39568}, {8589, 31481}, {8855, 34608}, {8960, 21735}, {8982, 40275}, {9540, 10304}, {9647, 19037}, {9660, 18995}, {9683, 35243}, {9778, 35610}, {9862, 35879}, {10148, 12819}, {10195, 15712}, {10299, 23269}, {10819, 38726}, {10880, 35503}, {10881, 35481}, {11001, 35823}, {11266, 34350}, {11474, 18533}, {11824, 35944}, {11835, 38803}, {11836, 23699}, {12082, 35777}, {12147, 12172}, {12244, 12376}, {12248, 35883}, {12253, 35881}, {12256, 22591}, {12383, 35827}, {12963, 19105}, {12969, 19104}, {12970, 20427}, {13172, 35825}, {13199, 35857}, {13200, 35829}, {13846, 34200}, {13886, 19708}, {13911, 31663}, {13941, 23263}, {13971, 28150}, {13973, 28160}, {13975, 28164}, {15681, 32788}, {15683, 35814}, {15684, 41947}, {15686, 19116}, {15720, 32789}, {17928, 35776}, {19146, 29181}, {21734, 35812}, {22485, 41491}, {25406, 35840}, {26294, 36703}, {31463, 37512}, {31730, 35775}, {34781, 35865}, {35400, 41951}

X(42262) = GIBERT(-SQRT(3),2,2) POINT

Barycentrics    a^2*S - 2*a^2*SA - 4*SB*SC : :

X(42262) lies on these lines: {2, 489}, {3, 3367}, {4, 615}, {5, 6}, {11, 3298}, {12, 3297}, {20, 6412}, {30, 5420}, {69, 23312}, {115, 6421}, {140, 6409}, {230, 26468}, {371, 1656}, {372, 381}, {376, 23263}, {382, 6396}, {387, 36680}, {394, 15233}, {403, 3093}, {492, 32488}, {546, 6426}, {547, 8981}, {549, 1328}, {550, 10194}, {590, 1588}, {591, 638}, {599, 640}, {631, 6411}, {639, 3763}, {946, 13973}, {999, 35801}, {1124, 7951}, {1131, 6442}, {1270, 32872}, {1327, 38071}, {1335, 7741}, {1377, 25639}, {1378, 3814}, {1482, 35789}, {1504, 7603}, {1505, 39565}, {1506, 6422}, {1578, 15760}, {1579, 11585}, {1587, 3545}, {1591, 17825}, {1592, 17811}, {1594, 3092}, {1699, 13947}, {1834, 36690}, {1853, 12970}, {2041, 11481}, {2042, 11480}, {2362, 17605}, {2476, 31473}, {3053, 37342}, {3068, 5056}, {3069, 3070}, {3102, 7697}, {3167, 35837}, {3295, 35803}, {3311, 5055}, {3312, 3851}, {3316, 6441}, {3364, 37832}, {3366, 11485}, {3389, 37835}, {3391, 11486}, {3525, 9541}, {3526, 6200}, {3544, 7581}, {3591, 3854}, {3614, 19029}, {3627, 35256}, {3628, 5418}, {3817, 13936}, {3818, 19146}, {3830, 6450}, {3832, 6438}, {3839, 41946}, {3843, 6398}, {3845, 6430}, {3850, 13993}, {3853, 6434}, {3855, 23249}, {3861, 6469}, {5013, 37343}, {5020, 8281}, {5066, 6471}, {5067, 6437}, {5068, 7586}, {5070, 6221}, {5071, 7582}, {5072, 6420}, {5073, 6456}, {5079, 6419}, {5087, 30557}, {5094, 11473}, {5286, 36664}, {5413, 7507}, {5414, 10896}, {5448, 13970}, {5475, 6423}, {5476, 9974}, {5587, 7968}, {5790, 35642}, {5893, 13980}, {5907, 12240}, {5943, 9823}, {6119, 11316}, {6199, 35812}, {6251, 13934}, {6395, 35814}, {6408, 14269}, {6417, 8960}, {6418, 19709}, {6424, 7746}, {6429, 35255}, {6433, 9681}, {6452, 17800}, {6455, 15694}, {6468, 9680}, {6470, 35018}, {6485, 38335}, {6486, 15723}, {6497, 15681}, {6502, 10895}, {6721, 13873}, {6813, 9756}, {7173, 19027}, {7486, 31454}, {7516, 9683}, {7547, 10881}, {7585, 15022}, {7687, 13990}, {7969, 8227}, {7988, 18991}, {7989, 18992}, {8277, 9818}, {8414, 32491}, {8983, 10171}, {9600, 31455}, {9615, 34595}, {9654, 35769}, {9669, 35809}, {9677, 13353}, {9738, 12601}, {9824, 14913}, {9955, 35774}, {9956, 35775}, {9975, 34507}, {10109, 13925}, {10113, 10820}, {10139, 41985}, {10148, 12102}, {10172, 13912}, {10175, 13911}, {10247, 35843}, {10601, 15234}, {10653, 34562}, {10654, 34559}, {10671, 20428}, {10672, 20429}, {11294, 32807}, {11474, 37197}, {11479, 13943}, {11548, 18289}, {12221, 32806}, {12256, 14230}, {12314, 22810}, {12376, 38724}, {12645, 35811}, {12963, 37637}, {13913, 38319}, {13937, 23047}, {13971, 19925}, {13975, 18483}, {14226, 34089}, {15884, 35830}, {16232, 17606}, {16644, 18586}, {16645, 18587}, {18393, 38235}, {18493, 35641}, {18512, 35770}, {18525, 35762}, {19145, 38317}, {21736, 32494}, {23253, 41947}, {23269, 41106}, {31479, 35808}, {32385, 32395}, {32447, 35867}, {32609, 35835}, {35731, 36456}, {35738, 36836}, {35825, 38743}, {35827, 38789}, {35857, 38755}, {35879, 38732}, {36655, 36990}

X(42263) = GIBERT(SQRT(3),-2,2) POINT

Barycentrics    a^2*S + 2*a^2*SA - 4*SB*SC : :

X(42263) lies on these lines: {2, 6411}, {3, 3367}, {4, 590}, {5, 6409}, {6, 30}, {20, 1152}, {115, 36719}, {371, 382}, {372, 1657}, {376, 615}, {381, 6200}, {485, 3627}, {486, 550}, {546, 5418}, {548, 5420}, {631, 23263}, {1124, 10483}, {1132, 41953}, {1327, 33699}, {1328, 8703}, {1478, 9660}, {1479, 9647}, {1539, 10819}, {1579, 12605}, {1587, 6431}, {1588, 3529}, {1656, 35787}, {2043, 11481}, {2044, 11480}, {2066, 12943}, {2067, 12953}, {3068, 3543}, {3070, 3146}, {3092, 6240}, {3093, 18560}, {3297, 7354}, {3298, 6284}, {3311, 5073}, {3312, 17800}, {3524, 32790}, {3534, 6396}, {3545, 32789}, {3830, 6221}, {3839, 32785}, {3843, 6449}, {3845, 6433}, {3851, 6455}, {3853, 6429}, {3861, 9680}, {5055, 6451}, {5059, 6432}, {5070, 6496}, {5076, 6453}, {5413, 37196}, {5475, 9600}, {5895, 12964}, {6144, 32421}, {6199, 15684}, {6395, 15685}, {6398, 15681}, {6407, 35812}, {6408, 35814}, {6421, 7756}, {6422, 7747}, {6424, 7748}, {6426, 7584}, {6434, 15686}, {6438, 11001}, {6439, 12101}, {6441, 15640}, {6445, 14269}, {6447, 35815}, {6452, 15689}, {6456, 35813}, {6468, 15687}, {6469, 19710}, {6471, 19116}, {6480, 38335}, {7526, 9683}, {7530, 9682}, {7581, 11541}, {7582, 41955}, {7583, 22644}, {7586, 15683}, {7969, 41869}, {9582, 18492}, {9655, 35808}, {9668, 35768}, {9674, 39590}, {9676, 10540}, {9677, 37472}, {9690, 35403}, {9739, 12601}, {10304, 32786}, {10620, 35835}, {10880, 35490}, {11473, 12173}, {12103, 13966}, {12239, 13598}, {12257, 14230}, {12375, 38790}, {12819, 14869}, {12902, 35826}, {12970, 17845}, {13473, 13884}, {13749, 26441}, {13911, 31673}, {13935, 17538}, {13939, 41964}, {13951, 15696}, {13973, 31730}, {14233, 21736}, {15484, 36734}, {15682, 23249}, {17578, 31412}, {17579, 31473}, {18525, 35610}, {19107, 35731}, {23302, 36445}, {23303, 36463}, {26617, 32805}, {28146, 35774}, {28160, 35775}, {31439, 33697}, {32419, 40341}, {34780, 35864}, {35824, 38733}, {35878, 38744}, {35882, 38756}, {36718, 41410}

X(42264) = GIBERT(SQRT(3),2,-2) POINT

Barycentrics    a^2*S - 2*a^2*SA + 4*SB*SC : :

X(42264) lies on these lines: {2, 6412}, {3, 3366}, {4, 615}, {5, 6410}, {6, 30}, {20, 1151}, {115, 36733}, {371, 1657}, {372, 382}, {376, 590}, {381, 6396}, {485, 550}, {486, 3627}, {546, 5420}, {548, 5418}, {631, 23253}, {1131, 41954}, {1327, 8703}, {1328, 33699}, {1335, 10483}, {1539, 10820}, {1578, 12605}, {1587, 3529}, {1588, 6432}, {1656, 35786}, {2043, 11480}, {2044, 11481}, {3069, 3543}, {3071, 3146}, {3092, 18560}, {3093, 6240}, {3297, 6284}, {3298, 7354}, {3311, 17800}, {3312, 5073}, {3522, 31412}, {3524, 32789}, {3534, 6200}, {3545, 32790}, {3830, 6398}, {3839, 32786}, {3843, 6450}, {3845, 6434}, {3851, 6456}, {3853, 6430}, {5055, 6452}, {5059, 6431}, {5070, 6497}, {5076, 6454}, {5412, 37196}, {5414, 12943}, {5475, 36719}, {5895, 12970}, {6144, 32419}, {6199, 15685}, {6221, 15681}, {6395, 15684}, {6407, 35815}, {6408, 35813}, {6421, 7747}, {6422, 7756}, {6423, 7748}, {6425, 7583}, {6433, 15686}, {6437, 9541}, {6440, 12101}, {6442, 15640}, {6444, 31403}, {6446, 14269}, {6448, 35814}, {6449, 8960}, {6451, 15689}, {6455, 35812}, {6468, 19710}, {6469, 15687}, {6470, 19117}, {6481, 38335}, {6502, 12953}, {7581, 41956}, {7582, 11541}, {7584, 22615}, {7585, 15683}, {7968, 41869}, {8976, 15696}, {8981, 12103}, {8982, 13748}, {9540, 17538}, {9601, 22646}, {9655, 35809}, {9668, 35769}, {9676, 37477}, {9680, 13925}, {9738, 12602}, {10304, 32785}, {10620, 35834}, {10881, 35490}, {11114, 31473}, {11474, 12173}, {12102, 17852}, {12240, 13598}, {12256, 14233}, {12376, 38790}, {12818, 14869}, {12902, 35827}, {12964, 17845}, {13473, 13937}, {13886, 41963}, {13911, 31730}, {13973, 31673}, {15338, 31472}, {15484, 36718}, {15682, 23259}, {18525, 35611}, {23302, 36463}, {23303, 36445}, {26618, 32806}, {28146, 35775}, {28160, 35774}, {32421, 40341}, {34780, 35865}, {35825, 38733}, {35879, 38744}, {35883, 38756}, {36734, 41411}

X(42265) = GIBERT(SQRT(3),2,2) POINT

Barycentrics    a^2*S + 2*a^2*SA + 4*SB*SC : :

X(42265) lies on these lines: {2, 490}, {3, 3366}, {4, 590}, {5, 6}, {11, 3297}, {12, 3298}, {20, 6411}, {30, 5418}, {69, 23311}, {115, 6422}, {140, 6410}, {230, 26469}, {371, 381}, {372, 1656}, {376, 23253}, {382, 6200}, {387, 36681}, {394, 15234}, {403, 3092}, {491, 32489}, {546, 6425}, {547, 13966}, {549, 1327}, {550, 10195}, {599, 639}, {615, 1587}, {631, 6412}, {637, 1991}, {640, 3763}, {946, 13911}, {999, 35800}, {1124, 7741}, {1132, 6441}, {1271, 32872}, {1328, 38071}, {1329, 31484}, {1335, 7951}, {1377, 3814}, {1378, 25639}, {1478, 9661}, {1479, 9646}, {1482, 35788}, {1504, 39565}, {1505, 7603}, {1506, 6421}, {1578, 11585}, {1579, 15760}, {1588, 3545}, {1591, 17811}, {1592, 17825}, {1594, 3093}, {1699, 13893}, {1834, 36691}, {1853, 12964}, {2041, 11480}, {2042, 11481}, {2066, 10896}, {2067, 10895}, {2362, 17606}, {3053, 37343}, {3068, 3071}, {3069, 5056}, {3103, 7697}, {3167, 35836}, {3295, 35802}, {3311, 3851}, {3312, 5055}, {3317, 6442}, {3365, 37832}, {3367, 11485}, {3390, 37835}, {3392, 11486}, {3525, 23269}, {3526, 6396}, {3544, 7582}, {3590, 3854}, {3614, 19030}, {3627, 35255}, {3628, 5420}, {3817, 13883}, {3818, 19145}, {3830, 6449}, {3832, 6437}, {3839, 41945}, {3843, 6221}, {3845, 6429}, {3850, 13925}, {3853, 6433}, {3855, 23259}, {3861, 6468}, {4193, 31473}, {4302, 31499}, {5013, 37342}, {5020, 8280}, {5066, 6470}, {5067, 6438}, {5068, 7585}, {5070, 6398}, {5071, 7581}, {5072, 6419}, {5073, 6455}, {5079, 6420}, {5087, 30556}, {5094, 11474}, {5254, 31463}, {5286, 36665}, {5412, 7507}, {5448, 13909}, {5475, 6424}, {5476, 9975}, {5587, 7969}, {5790, 35641}, {5893, 8991}, {5907, 12239}, {5943, 9824}, {6118, 11315}, {6199, 35815}, {6250, 13882}, {6395, 35813}, {6407, 14269}, {6417, 19709}, {6423, 7746}, {6430, 35256}, {6434, 16239}, {6451, 17800}, {6456, 15694}, {6471, 35018}, {6484, 38335}, {6487, 15723}, {6496, 15681}, {6721, 13926}, {6811, 9756}, {7173, 19028}, {7486, 31414}, {7526, 9682}, {7530, 9683}, {7547, 10880}, {7586, 15022}, {7687, 8998}, {7748, 9600}, {7968, 8227}, {7988, 18992}, {7989, 18991}, {8276, 9818}, {8406, 32490}, {8909, 9927}, {8983, 19925}, {9583, 18492}, {9654, 35768}, {9669, 35808}, {9675, 39590}, {9676, 37472}, {9677, 10540}, {9739, 12602}, {9823, 14913}, {9955, 35775}, {9956, 35774}, {9974, 34507}, {10109, 13993}, {10113, 10819}, {10140, 41985}, {10147, 12102}, {10171, 13971}, {10172, 13975}, {10175, 13973}, {10247, 35842}, {10589, 31408}, {10601, 15233}, {10653, 34559}, {10654, 34562}, {10667, 20428}, {10668, 20429}, {11473, 37197}, {11479, 13889}, {11548, 18290}, {12222, 32805}, {12257, 14233}, {12313, 22809}, {12375, 38724}, {12645, 35810}, {12968, 37637}, {13884, 23047}, {13912, 18483}, {13977, 38319}, {14230, 21736}, {14241, 34091}, {15883, 35831}, {16232, 17605}, {16644, 18587}, {16645, 18586}, {18395, 38235}, {18493, 35642}, {18510, 35771}, {18525, 35763}, {19146, 38317}, {23263, 41948}, {23275, 41106}, {31474, 35803}, {31479, 35809}, {32384, 32395}, {32447, 35866}, {32609, 35834}, {35738, 36843}, {35824, 38743}, {35826, 38789}, {35856, 38755}, {35878, 38732}, {36656, 36990}

X(42266) = GIBERT(SQRT(3),-2,3) POINT

Barycentrics    a^2*S + 3*a^2*SA - 4*SB*SC : :

X(42266) lies on these lines: {2, 22615}, {3, 3367}, {4, 5418}, {6, 1657}, {20, 372}, {30, 371}, {35, 35801}, {36, 35803}, {74, 35835}, {376, 486}, {378, 9683}, {381, 6409}, {382, 1151}, {485, 3146}, {490, 32419}, {511, 39893}, {515, 35610}, {516, 35641}, {548, 615}, {550, 3071}, {590, 3627}, {639, 35949}, {962, 35810}, {1131, 6478}, {1152, 3534}, {1327, 13886}, {1328, 10304}, {1503, 35864}, {1587, 5059}, {1656, 6411}, {1870, 9631}, {1885, 35764}, {2066, 10483}, {2362, 4333}, {2460, 35831}, {2777, 12375}, {2794, 35828}, {2829, 35882}, {3068, 9681}, {3069, 17538}, {3092, 37196}, {3311, 17800}, {3312, 15681}, {3522, 5420}, {3523, 23263}, {3529, 6419}, {3543, 6484}, {3579, 35789}, {3830, 6449}, {3843, 6455}, {3850, 32789}, {3851, 6451}, {3853, 6486}, {4297, 35762}, {4299, 35769}, {4302, 35809}, {4316, 6502}, {4324, 5414}, {5055, 6496}, {5073, 6221}, {5254, 41410}, {5412, 18560}, {5413, 35471}, {5691, 35788}, {5840, 35856}, {6240, 11473}, {6284, 9647}, {6407, 13846}, {6410, 13785}, {6412, 13951}, {6417, 15685}, {6420, 15704}, {6425, 13665}, {6426, 18510}, {6429, 13903}, {6454, 7584}, {6456, 13847}, {6460, 11001}, {6480, 8981}, {6481, 23273}, {6759, 9676}, {6781, 12968}, {7354, 9660}, {7391, 8280}, {7500, 8854}, {7667, 8855}, {7748, 12963}, {7969, 28146}, {8725, 35869}, {9677, 13352}, {9680, 17578}, {9821, 35867}, {10533, 22802}, {10721, 10819}, {10734, 11835}, {10881, 13619}, {11474, 35481}, {12121, 12376}, {12163, 35837}, {12203, 35766}, {12240, 14855}, {12305, 38738}, {12515, 35853}, {12699, 35763}, {12702, 35843}, {12943, 35800}, {12953, 35802}, {12970, 34785}, {13883, 28172}, {13993, 15690}, {14241, 35409}, {14830, 35699}, {15682, 31412}, {15686, 32788}, {15712, 32790}, {16772, 35740}, {17702, 35826}, {18457, 18565}, {18481, 35642}, {18533, 35765}, {18538, 41948}, {18762, 33923}, {19116, 19710}, {20127, 35827}, {20427, 35865}, {21735, 32786}, {23698, 35824}, {26615, 33364}, {28168, 31439}, {29181, 35840}, {31454, 41954}, {34773, 35811}, {35730, 36836}, {35731, 36967}, {35776, 39568}, {35825, 38741}, {35857, 38753}, {35879, 38730}, {41947, 41949}

X(42267) = GIBERT(SQRT(3),2,-3) POINT

Barycentrics    a^2*S - 3*a^2*SA + 4*SB*SC : :

X(42267) lies on these lines: {2, 22644}, {3, 3366}, {4, 5420}, {6, 1657}, {20, 371}, {30, 372}, {35, 35800}, {36, 35802}, {74, 35834}, {376, 485}, {381, 6410}, {382, 1152}, {486, 3146}, {489, 32421}, {511, 39894}, {515, 35611}, {516, 35642}, {548, 590}, {550, 3070}, {615, 3627}, {640, 35948}, {962, 35811}, {1132, 6479}, {1151, 3534}, {1327, 10304}, {1328, 13939}, {1503, 35865}, {1588, 5059}, {1656, 6412}, {1885, 35765}, {2066, 4324}, {2067, 4316}, {2459, 35830}, {2777, 12376}, {2794, 35829}, {2829, 35883}, {3068, 17538}, {3069, 22615}, {3093, 37196}, {3311, 15681}, {3312, 17800}, {3522, 5418}, {3523, 23253}, {3528, 31412}, {3529, 6420}, {3543, 6485}, {3579, 35788}, {3830, 6450}, {3843, 6456}, {3850, 32790}, {3851, 6452}, {3853, 6487}, {4297, 35763}, {4299, 35768}, {4302, 35808}, {4333, 16232}, {5055, 6497}, {5073, 6398}, {5254, 41411}, {5412, 35471}, {5413, 18560}, {5414, 10483}, {5691, 35789}, {5840, 35857}, {6240, 11474}, {6284, 35769}, {6408, 13847}, {6409, 13665}, {6411, 8976}, {6418, 15685}, {6419, 15704}, {6425, 18512}, {6426, 13785}, {6429, 31487}, {6430, 13961}, {6453, 7583}, {6455, 13846}, {6459, 11001}, {6480, 23267}, {6481, 13966}, {6484, 31454}, {6781, 12963}, {7354, 35809}, {7391, 8281}, {7500, 8855}, {7667, 8854}, {7748, 12968}, {7968, 28146}, {8725, 35868}, {9680, 13886}, {9683, 33524}, {9821, 35866}, {10534, 22802}, {10721, 10820}, {10734, 11836}, {10880, 13619}, {11473, 35481}, {12121, 12375}, {12124, 21737}, {12163, 35836}, {12203, 35767}, {12239, 14855}, {12306, 38738}, {12515, 35852}, {12699, 35762}, {12702, 35842}, {12943, 35801}, {12953, 35803}, {12964, 34785}, {13925, 15690}, {13936, 28172}, {14226, 35409}, {14830, 35698}, {15686, 32787}, {15712, 32789}, {17702, 35827}, {18459, 18565}, {18481, 35641}, {18533, 35764}, {18538, 33923}, {18587, 35739}, {18762, 41947}, {19117, 19710}, {20127, 35826}, {20427, 35864}, {21735, 32785}, {23698, 35825}, {26616, 33365}, {29181, 35841}, {34773, 35810}, {35777, 39568}, {35824, 38741}, {35856, 38753}, {35878, 38730}, {41948, 41950}, {41953, 41970}

X(42268) = GIBERT(-SQRT(3),3,1) POINT

Barycentrics    a^2*S - a^2*SA - 6*SB*SC : :

X(42268) lies on these lines: {2, 12819}, {3, 22615}, {4, 372}, {5, 1151}, {6, 546}, {20, 10577}, {30, 5420}, {32, 13834}, {262, 32470}, {371, 3091}, {381, 485}, {382, 615}, {388, 35803}, {428, 18290}, {497, 35801}, {550, 8252}, {590, 3851}, {632, 6411}, {640, 12322}, {962, 35789}, {1132, 1327}, {1152, 3627}, {1588, 3832}, {1597, 8277}, {1656, 6455}, {1657, 6497}, {2043, 3392}, {2044, 3367}, {3068, 3855}, {3070, 3843}, {3090, 6200}, {3092, 23047}, {3093, 10151}, {3146, 6396}, {3317, 15682}, {3364, 16808}, {3389, 16809}, {3529, 32786}, {3543, 6485}, {3544, 6453}, {3545, 6459}, {3592, 3857}, {3628, 6409}, {3830, 6408}, {3845, 6432}, {3850, 13925}, {3853, 6430}, {3854, 8960}, {3858, 7583}, {3861, 6471}, {5056, 6486}, {5058, 18424}, {5066, 8981}, {5068, 9540}, {5072, 6221}, {5076, 6398}, {5079, 6449}, {5133, 18289}, {5225, 35809}, {5229, 35769}, {5480, 9974}, {5818, 35610}, {6250, 14853}, {6290, 22617}, {6412, 15704}, {6419, 23273}, {6420, 23249}, {6424, 13711}, {6425, 12811}, {6426, 12102}, {6431, 41991}, {6454, 13941}, {6468, 41989}, {6500, 13665}, {6623, 35764}, {6995, 8281}, {7378, 8855}, {7395, 9683}, {7503, 35777}, {7529, 9682}, {7582, 35822}, {7586, 23253}, {9677, 13434}, {9691, 19709}, {9812, 35611}, {10590, 35808}, {10591, 35768}, {10819, 36518}, {10820, 12295}, {11266, 18568}, {12602, 22591}, {13846, 38071}, {13847, 15687}, {13911, 38140}, {13961, 38335}, {13973, 22793}, {14233, 36655}, {14269, 32788}, {15081, 35826}, {17578, 35813}, {18440, 22596}, {18483, 35774}, {19053, 23269}, {19102, 39590}, {19117, 23046}, {19925, 35775}, {22587, 22588}, {22592, 35833}, {23267, 35770}, {32499, 39876}, {33355, 33356}, {33359, 33361}, {33457, 41491}, {35403, 41951}

X(42269) = GIBERT(SQRT(3),3,1) POINT

Barycentrics    a^2*S + a^2*SA + 6*SB*SC : :

X(42269) lies on these lines: {2, 12818}, {3, 22644}, {4, 371}, {5, 1152}, {6, 546}, {20, 10576}, {30, 5418}, {32, 13711}, {34, 9632}, {262, 32471}, {372, 3091}, {381, 486}, {382, 590}, {388, 35802}, {428, 18289}, {497, 35800}, {550, 8253}, {615, 3851}, {632, 6412}, {639, 12323}, {962, 35788}, {1131, 1328}, {1151, 3627}, {1587, 3832}, {1593, 9682}, {1597, 8276}, {1656, 6456}, {1657, 6496}, {2043, 3366}, {2044, 3391}, {3069, 3855}, {3071, 3843}, {3090, 6396}, {3092, 10151}, {3093, 23047}, {3146, 6200}, {3316, 15682}, {3365, 16808}, {3390, 16809}, {3529, 32785}, {3543, 6484}, {3544, 6454}, {3545, 6460}, {3583, 31472}, {3590, 9542}, {3594, 3857}, {3628, 6410}, {3830, 6407}, {3845, 6431}, {3850, 13993}, {3853, 6429}, {3858, 7584}, {3861, 6470}, {5056, 6487}, {5062, 18424}, {5066, 13966}, {5068, 10194}, {5072, 6398}, {5076, 6221}, {5079, 6450}, {5133, 18290}, {5225, 35808}, {5229, 35768}, {5480, 9975}, {5818, 35611}, {6251, 14853}, {6289, 22646}, {6411, 15704}, {6419, 23259}, {6420, 23267}, {6423, 13834}, {6425, 12102}, {6426, 12811}, {6432, 41991}, {6453, 8972}, {6469, 41989}, {6501, 13785}, {6623, 35765}, {6995, 8280}, {7378, 8854}, {7503, 35776}, {7581, 35823}, {7582, 31414}, {7585, 23263}, {7748, 31463}, {9541, 9692}, {9646, 12953}, {9647, 13898}, {9660, 13897}, {9661, 12943}, {9677, 14157}, {9683, 18534}, {9812, 35610}, {10590, 35809}, {10591, 35769}, {10819, 12295}, {10820, 36518}, {11265, 18568}, {12601, 22592}, {13846, 15687}, {13847, 38071}, {13903, 38335}, {13911, 22793}, {13973, 38140}, {14230, 36656}, {14269, 32787}, {15081, 35827}, {18440, 22625}, {18483, 35775}, {19054, 23275}, {19105, 39590}, {19116, 23046}, {19709, 41946}, {19925, 35774}, {22591, 35832}, {22618, 22619}, {23273, 35771}, {31408, 35803}, {32498, 39875}, {33354, 33357}, {33358, 33360}, {33456, 41490}, {35403, 41952}

X(42270) = GIBERT(-SQRT(3),3,2) POINT

Barycentrics    a^2*S - 2*a^2*SA - 6*SB*SC : :

X(42270) lies on these lines: {2, 6409}, {3, 22615}, {4, 615}, {5, 371}, {6, 3091}, {20, 8252}, {30, 10577}, {140, 35821}, {141, 32488}, {372, 546}, {381, 486}, {382, 5420}, {485, 3851}, {495, 35803}, {496, 35801}, {547, 6484}, {591, 12323}, {631, 23263}, {637, 23312}, {1131, 19053}, {1132, 3068}, {1151, 3090}, {1328, 5055}, {1352, 9975}, {1505, 18424}, {1587, 3855}, {1588, 3545}, {1656, 6449}, {1699, 13973}, {1991, 12221}, {2066, 3614}, {2067, 7173}, {3069, 3832}, {3146, 6410}, {3297, 10590}, {3298, 10591}, {3311, 5072}, {3365, 16809}, {3367, 14813}, {3390, 16808}, {3392, 14814}, {3525, 6411}, {3526, 6496}, {3529, 6412}, {3544, 3592}, {3589, 32489}, {3594, 13939}, {3627, 6396}, {3628, 6200}, {3817, 7969}, {3839, 6460}, {3843, 6560}, {3845, 13966}, {3850, 6564}, {3853, 6487}, {3854, 7586}, {3857, 6420}, {3858, 35786}, {3861, 35813}, {5056, 6429}, {5066, 7583}, {5067, 9541}, {5071, 9540}, {5073, 10194}, {5076, 6450}, {5079, 6221}, {5413, 23047}, {6251, 32494}, {6398, 22644}, {6419, 12811}, {6422, 31415}, {6425, 15022}, {6426, 13941}, {6432, 23267}, {6453, 12812}, {6501, 13665}, {6813, 14233}, {6871, 31473}, {7374, 26331}, {7388, 23311}, {7514, 35777}, {7581, 41106}, {7968, 19925}, {7989, 13911}, {8976, 19709}, {8982, 14235}, {9779, 19065}, {10146, 41962}, {10151, 11474}, {10272, 35835}, {10297, 10898}, {10516, 26468}, {10534, 41362}, {10592, 35808}, {10593, 35768}, {10880, 35487}, {11801, 12376}, {12240, 15030}, {12571, 13936}, {12819, 15720}, {13834, 30435}, {13983, 22682}, {15069, 26469}, {15765, 36970}, {18357, 35642}, {18358, 35841}, {18510, 41953}, {18585, 36969}, {19054, 41952}, {19116, 35822}, {22791, 35789}, {23046, 35814}, {23253, 41099}, {23302, 35732}, {32498, 37342}, {35018, 35255}, {35610, 38042}, {35611, 40273}, {35641, 38034}, {35811, 37705}, {35824, 38229}, {35840, 38136}, {35842, 38138}

X(42271) = GIBERT(SQRT(3),-3,2) POINT

Barycentrics    a^2*S + 2*a^2*SA - 6*SB*SC : :

X(42271) lies on these lines: {3, 22615}, {4, 590}, {6, 3146}, {20, 615}, {30, 372}, {140, 35787}, {371, 3627}, {376, 23263}, {381, 6455}, {382, 3070}, {485, 3830}, {486, 1657}, {546, 6200}, {548, 10577}, {550, 6565}, {1132, 13847}, {1152, 3529}, {1328, 13951}, {1587, 15682}, {1588, 6432}, {3068, 17578}, {3069, 5059}, {3090, 6411}, {3091, 6409}, {3364, 19106}, {3389, 19107}, {3522, 8252}, {3534, 5420}, {3543, 6459}, {3583, 9647}, {3585, 9660}, {3592, 23249}, {3594, 11541}, {3832, 8253}, {3843, 5418}, {3845, 6486}, {3853, 6564}, {3861, 35255}, {5072, 6451}, {5073, 6418}, {5076, 6221}, {5079, 6496}, {5893, 10533}, {6396, 15704}, {6408, 13785}, {6412, 17538}, {6425, 31412}, {6429, 8972}, {6431, 23267}, {6437, 13886}, {6453, 12102}, {6460, 6471}, {6479, 35814}, {6485, 13966}, {6492, 41952}, {6500, 15684}, {6811, 14239}, {7968, 28164}, {8976, 9681}, {8981, 15687}, {9542, 10139}, {9543, 41948}, {9646, 18513}, {9661, 18514}, {9974, 31670}, {10145, 35403}, {10248, 13902}, {11001, 13935}, {12084, 35777}, {12103, 18762}, {12819, 15688}, {13936, 28158}, {14230, 26441}, {19117, 35404}, {22728, 32470}, {23311, 35949}, {28178, 35611}, {28186, 35642}, {28212, 35843}, {31295, 31473}, {32497, 36711}, {33699, 35822}, {35763, 40273}, {35771, 35820}

X(42272) = GIBERT(SQRT(3),3,-2) POINT

Barycentrics    a^2*S - 2*a^2*SA + 6*SB*SC : :

X(42272) lies on these lines: {3, 22644}, {4, 615}, {6, 3146}, {20, 590}, {30, 371}, {140, 35786}, {372, 3627}, {376, 23253}, {381, 6456}, {382, 3071}, {485, 1657}, {486, 3830}, {546, 6396}, {548, 10576}, {550, 6564}, {1131, 13846}, {1151, 3529}, {1327, 8976}, {1587, 6431}, {1588, 15682}, {3068, 5059}, {3069, 17578}, {3090, 6412}, {3091, 6410}, {3365, 19106}, {3390, 19107}, {3522, 8253}, {3534, 5418}, {3543, 6460}, {3592, 11541}, {3594, 23259}, {3832, 8252}, {3843, 5420}, {3845, 6487}, {3853, 6565}, {3861, 35256}, {4316, 9661}, {4324, 9646}, {5072, 6452}, {5073, 6417}, {5076, 6398}, {5079, 6497}, {5893, 10534}, {6200, 15704}, {6407, 13665}, {6411, 17538}, {6430, 13941}, {6432, 23273}, {6438, 13939}, {6454, 12102}, {6459, 6470}, {6478, 35815}, {6484, 8981}, {6493, 41951}, {6501, 15684}, {6813, 14235}, {7969, 28164}, {8982, 14233}, {9540, 11001}, {9541, 23269}, {9975, 31670}, {10146, 35403}, {10248, 13959}, {12084, 35776}, {12103, 18538}, {12818, 15688}, {13883, 28158}, {13951, 41949}, {13966, 15687}, {19116, 35404}, {22728, 32471}, {23312, 35948}, {28178, 35610}, {28186, 35641}, {28212, 35842}, {32494, 36712}, {33699, 35823}, {35762, 40273}, {35770, 35821}

X(42273) = GIBERT(SQRT(3),3,2) POINT

Barycentrics    a^2*S + 2*a^2*SA + 6*SB*SC : :

X(42273) lies on these lines: {2, 6410}, {3, 22644}, {4, 590}, {5, 372}, {6, 3091}, {20, 8253}, {30, 10576}, {140, 35820}, {141, 32489}, {371, 546}, {381, 485}, {382, 5418}, {486, 3851}, {495, 35802}, {496, 35800}, {547, 6485}, {591, 12222}, {631, 23253}, {638, 23311}, {1131, 3069}, {1132, 19054}, {1152, 3090}, {1327, 5055}, {1352, 9974}, {1504, 18424}, {1587, 3545}, {1588, 3855}, {1656, 6450}, {1699, 13911}, {1991, 12322}, {3068, 3832}, {3146, 6409}, {3297, 10591}, {3298, 10590}, {3312, 5072}, {3364, 16809}, {3366, 14814}, {3389, 16808}, {3391, 14813}, {3525, 6412}, {3526, 6497}, {3529, 6411}, {3544, 3594}, {3583, 9646}, {3585, 9661}, {3589, 32488}, {3592, 13886}, {3614, 5414}, {3627, 6200}, {3628, 6396}, {3817, 7968}, {3839, 6459}, {3843, 6561}, {3845, 8981}, {3850, 6565}, {3853, 6486}, {3854, 7585}, {3857, 6419}, {3858, 8960}, {3861, 35812}, {5056, 6430}, {5066, 7584}, {5071, 13935}, {5073, 10195}, {5076, 6449}, {5079, 6398}, {5187, 31473}, {5412, 23047}, {6221, 22615}, {6250, 32497}, {6420, 12811}, {6421, 31415}, {6425, 8972}, {6426, 15022}, {6431, 23273}, {6454, 12812}, {6500, 13785}, {6502, 7173}, {6811, 14230}, {7000, 26330}, {7389, 23312}, {7514, 35776}, {7582, 41106}, {7586, 31414}, {7969, 19925}, {7989, 13973}, {8992, 22682}, {9543, 10139}, {9647, 18513}, {9660, 18514}, {9691, 14269}, {9779, 19066}, {10145, 41961}, {10151, 11473}, {10272, 35834}, {10297, 10897}, {10516, 26469}, {10533, 41362}, {10592, 35809}, {10593, 35769}, {10881, 35487}, {10896, 31472}, {11801, 12375}, {12239, 15030}, {12571, 13883}, {12818, 15720}, {13711, 30435}, {13951, 19709}, {14239, 26441}, {15069, 26468}, {15765, 36969}, {18357, 35641}, {18358, 35840}, {18512, 41954}, {18585, 36970}, {19053, 41951}, {19117, 35823}, {22791, 35788}, {23046, 35815}, {23263, 41099}, {23303, 35732}, {32499, 37343}, {35018, 35256}, {35610, 40273}, {35611, 38042}, {35642, 38034}, {35810, 37705}, {35825, 38229}, {35841, 38136}, {35843, 38138}

X(42274) = GIBERT(-SQRT(3),3,3) POINT

Barycentrics    a^2*S - 3*a^2*SA - 6*SB*SC : :

X(42274) lies on these lines: {2, 1328}, {3, 22615}, {4, 5420}, {5, 6}, {20, 35787}, {30, 6412}, {140, 6411}, {187, 37342}, {371, 3090}, {372, 3091}, {381, 615}, {382, 6452}, {492, 41483}, {546, 1152}, {547, 6437}, {574, 37343}, {590, 5055}, {631, 35821}, {632, 6409}, {639, 3619}, {640, 3620}, {642, 12322}, {1124, 3614}, {1132, 7486}, {1151, 3628}, {1327, 5066}, {1335, 7173}, {1587, 5068}, {1588, 5056}, {1656, 3071}, {1659, 8973}, {2041, 3392}, {2042, 3367}, {2043, 16809}, {2044, 16808}, {2045, 16967}, {2046, 16966}, {3055, 9600}, {3068, 5071}, {3069, 3545}, {3070, 3851}, {3085, 35803}, {3086, 35801}, {3297, 10592}, {3298, 10593}, {3311, 5079}, {3312, 5072}, {3316, 35815}, {3317, 3855}, {3366, 34754}, {3391, 34755}, {3523, 23263}, {3526, 6451}, {3530, 12819}, {3544, 6420}, {3592, 12812}, {3594, 12811}, {3627, 6410}, {3631, 23312}, {3817, 35774}, {3832, 6481}, {3843, 6446}, {3845, 6434}, {3850, 6438}, {3854, 23253}, {3857, 6426}, {3858, 6469}, {5020, 9682}, {5067, 6459}, {5070, 6445}, {5076, 6456}, {5603, 35789}, {5818, 35642}, {6119, 11291}, {6248, 32471}, {6251, 21736}, {6419, 15022}, {6430, 41991}, {6431, 13925}, {6435, 7582}, {6436, 7586}, {6440, 23046}, {6442, 11737}, {6468, 15699}, {6622, 35764}, {6997, 18290}, {7392, 8281}, {7393, 9683}, {7509, 35777}, {7539, 18289}, {7603, 31463}, {7687, 10820}, {8276, 11484}, {8277, 11479}, {8940, 22590}, {8981, 10195}, {9582, 19872}, {9632, 9817}, {9690, 15703}, {9780, 35610}, {9955, 13973}, {10109, 13846}, {10175, 35775}, {10588, 35808}, {10589, 35768}, {10590, 35769}, {10591, 35809}, {10595, 35843}, {10819, 12900}, {11272, 32470}, {11314, 23311}, {11480, 35738}, {12376, 15081}, {12571, 13975}, {13653, 23234}, {13665, 19709}, {13880, 32499}, {13886, 35771}, {14494, 33344}, {15066, 15233}, {17530, 31473}, {18510, 32787}, {18586, 23302}, {18587, 23303}, {32419, 32806}, {32813, 41491}, {35841, 40330}, {36450, 41121}, {36468, 41122}, {41950, 41951}

X(42275) = GIBERT(SQRT(3),-3,3) POINT

Barycentrics    a^2*S + 3*a^2*SA - 6*SB*SC : :

X(42275) lies on these lines: {3, 22615}, {4, 5418}, {5, 6411}, {6, 30}, {20, 486}, {34, 9631}, {187, 13834}, {371, 3146}, {372, 3529}, {376, 6565}, {381, 6451}, {382, 485}, {492, 13712}, {546, 6409}, {550, 5420}, {590, 3830}, {615, 1328}, {631, 35787}, {1131, 35815}, {1151, 3627}, {1152, 15704}, {1327, 3068}, {1588, 5059}, {1593, 9683}, {1657, 3071}, {2041, 19106}, {2042, 19107}, {2043, 10646}, {2044, 10645}, {3069, 6481}, {3070, 5073}, {3522, 10577}, {3528, 12819}, {3543, 6480}, {3845, 8253}, {3853, 6433}, {5072, 6496}, {5076, 6449}, {6395, 17800}, {6410, 12103}, {6419, 11541}, {6429, 13925}, {6434, 13966}, {6436, 7582}, {6437, 23251}, {6438, 7584}, {6441, 19117}, {6446, 13785}, {6453, 31412}, {6459, 23267}, {6468, 8981}, {6477, 35814}, {7391, 18289}, {7585, 15640}, {8252, 8703}, {8960, 23253}, {8976, 9690}, {9540, 17578}, {9647, 12953}, {9660, 12943}, {9676, 14157}, {9682, 18534}, {9778, 35789}, {9812, 35763}, {10819, 13202}, {11008, 32421}, {11413, 35777}, {12240, 14641}, {12244, 35835}, {13651, 22646}, {13665, 15684}, {13770, 32492}, {13846, 33699}, {13847, 19710}, {13911, 33697}, {14927, 35841}, {15683, 35823}, {15685, 32788}, {15686, 35256}, {15687, 35255}, {16808, 36455}, {16809, 36437}, {18510, 41946}, {20070, 35843}, {20080, 32419}, {22591, 22809}, {26361, 26615}, {28150, 35774}, {28164, 35775}

X(42276) = GIBERT(SQRT(3),3,-3) POINT

Barycentrics    a^2*S - 3*a^2*SA + 6*SB*SC : :

X(42276) lies on these lines: {3, 22644}, {4, 5420}, {5, 6412}, {6, 30}, {20, 485}, {187, 13711}, {371, 3529}, {372, 3146}, {376, 6564}, {381, 6452}, {382, 486}, {491, 13835}, {546, 6410}, {550, 5418}, {590, 1327}, {615, 3830}, {631, 35786}, {1132, 35814}, {1151, 15704}, {1152, 3627}, {1328, 3069}, {1587, 5059}, {1657, 3070}, {2041, 19107}, {2042, 19106}, {2043, 10645}, {2044, 10646}, {3068, 6480}, {3071, 5073}, {3522, 10576}, {3528, 12818}, {3543, 6481}, {3845, 8252}, {3853, 6434}, {4324, 31472}, {5072, 6497}, {5076, 6450}, {6199, 17800}, {6409, 12103}, {6420, 11541}, {6430, 13993}, {6433, 8981}, {6435, 7581}, {6437, 7583}, {6438, 23261}, {6442, 19116}, {6445, 13665}, {6460, 23273}, {6469, 13966}, {6476, 31414}, {7391, 18290}, {7586, 15640}, {8253, 8703}, {8718, 9677}, {8960, 23269}, {9541, 15683}, {9682, 21312}, {9690, 31454}, {9778, 35788}, {9812, 35762}, {10820, 13202}, {11008, 32419}, {11413, 35776}, {12239, 14641}, {12244, 35834}, {13651, 32495}, {13770, 22617}, {13785, 15684}, {13846, 19710}, {13847, 33699}, {13935, 17578}, {13973, 33697}, {14927, 35840}, {15685, 32787}, {15686, 35255}, {15687, 35256}, {16808, 36437}, {16809, 36455}, {17538, 31412}, {18512, 41945}, {20070, 35842}, {20080, 32421}, {22592, 22810}, {26362, 26616}, {28150, 35775}, {28164, 35774}

X(42277) = GIBERT(SQRT(3),3,3) POINT

Barycentrics    a^2*S + 3*a^2*SA + 6*SB*SC : :

X(42277) lies on these lines: {2, 1327}, {3, 22644}, {4, 5418}, {5, 6}, {20, 35786}, {30, 6411}, {115, 31463}, {140, 6412}, {187, 37343}, {371, 3091}, {372, 3090}, {381, 590}, {382, 6451}, {491, 41484}, {546, 1151}, {547, 6438}, {574, 37342}, {615, 5055}, {631, 35820}, {632, 6410}, {639, 3620}, {640, 3619}, {641, 12323}, {1124, 7173}, {1131, 7486}, {1152, 3628}, {1328, 5066}, {1335, 3614}, {1587, 5056}, {1588, 5068}, {1598, 9683}, {1656, 3070}, {2041, 3366}, {2042, 3391}, {2043, 16808}, {2044, 16809}, {2045, 16966}, {2046, 16967}, {3068, 3545}, {3069, 5071}, {3071, 3851}, {3085, 35802}, {3086, 35800}, {3297, 10593}, {3298, 10592}, {3311, 5072}, {3312, 5079}, {3316, 3855}, {3317, 35814}, {3367, 34754}, {3392, 34755}, {3523, 23253}, {3526, 6452}, {3530, 12818}, {3544, 6419}, {3592, 12811}, {3594, 12812}, {3627, 6409}, {3631, 23311}, {3814, 31484}, {3817, 35775}, {3832, 6480}, {3839, 9541}, {3843, 6445}, {3845, 6433}, {3850, 6437}, {3854, 23263}, {3857, 6425}, {3858, 6468}, {5067, 6460}, {5070, 6446}, {5076, 6455}, {5603, 35788}, {5818, 35641}, {6118, 11292}, {6248, 32470}, {6420, 15022}, {6429, 41991}, {6432, 13993}, {6435, 7585}, {6436, 7581}, {6439, 23046}, {6441, 11737}, {6469, 15699}, {6622, 35765}, {6997, 18289}, {7392, 8280}, {7509, 35776}, {7539, 18290}, {7687, 10819}, {7741, 31472}, {8276, 11479}, {8277, 11484}, {8944, 22621}, {9632, 37697}, {9646, 10896}, {9661, 10895}, {9676, 15033}, {9682, 9818}, {9690, 41963}, {9780, 35611}, {9955, 13911}, {10109, 13847}, {10175, 35774}, {10194, 13966}, {10588, 35809}, {10589, 35769}, {10590, 35768}, {10591, 35808}, {10595, 35842}, {10820, 12900}, {11272, 32471}, {11313, 23312}, {11481, 35738}, {12375, 15081}, {12571, 13912}, {12953, 31499}, {13773, 23234}, {13785, 19709}, {13921, 32498}, {13939, 35770}, {14494, 33345}, {15066, 15234}, {15703, 41946}, {17533, 31473}, {18512, 32788}, {18586, 23303}, {18587, 23302}, {31481, 39565}, {32421, 32805}, {32812, 41490}, {35840, 40330}, {36449, 41122}, {36467, 41121}, {41949, 41952}

X(42278) = GIBERT(0,-2,SQRT(3)) POINT

Barycentrics    3*Sqrt[3]*a^2*SA - 12*SB*SC : :

X(42278) lies on thesse lines: {2, 3}, {6, 42176}, {13, 8960}, {15, 23251}, {16, 23261}, {17, 6564}, {18, 6565}, {371, 5340}, {372, 5339}, {397, 3311}, {398, 3312}, {1151, 3391}, {1152, 3367}, {1587, 42213}, {1588, 42212}, {3070, 11485}, {3071, 11486}, {3364, 42263}, {3365, 42262}, {3389, 42265}, {3390, 42264}, {5318, 6221}, {5321, 6398}, {5334, 6395}, {5335, 6199}, {5349, 6450}, {5350, 6449}, {5365, 6408}, {5366, 6407}, {5611, 12602}, {5615, 12601}, {6200, 42094}, {6289, 33450}, {6290, 33449}, {6396, 42093}, {6411, 42178}, {6412, 42175}, {6445, 42134}, {6446, 42133}, {6451, 42102}, {6452, 42101}, {6455, 35740}, {6456, 42239}, {8976, 42128}, {10576, 16808}, {10577, 16809}, {11480, 42181}, {11481, 42180}, {13951, 42125}, {16628, 22882}, {16629, 22928}, {19106, 42266}, {19107, 42267}, {23249, 42222}, {23253, 42214}, {23259, 42223}, {23263, 42211}, {35820, 42157}, {35821, 42158}, {42115, 42197}, {42116, 42196}, {42167, 42183}, {42170, 42186}, {42247, 42252}, {42248, 42251}

X(42278) = reflection of X(42279) in X(5073)
X(42278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 42279), (5, 3851, 42279), (140, 3843, 42279), (381, 1656, 42279), (382, 1657, 42279), (550, 3830, 42279), (42176, 42177, 6)

X(42279) = GIBERT(0,2,SQRT(3)) POINT

Barycentrics    3*Sqrt[3]*a^2*SA + 12*SB*SC : :

X(42279) lies on thesse lines: {2, 3}, {6, 42175}, {14, 8960}, {15, 23261}, {16, 23251}, {17, 6565}, {18, 6564}, {371, 5339}, {372, 5340}, {397, 3312}, {398, 3311}, {1151, 3366}, {1152, 3392}, {1587, 42214}, {1588, 42211}, {3070, 11486}, {3071, 11485}, {3364, 42265}, {3365, 42264}, {3389, 42263}, {3390, 42262}, {5318, 6398}, {5321, 6221}, {5334, 6199}, {5335, 6395}, {5349, 6449}, {5350, 6450}, {5365, 6407}, {5366, 6408}, {5611, 12601}, {5615, 12602}, {6200, 42093}, {6289, 33448}, {6290, 33451}, {6396, 42094}, {6411, 42176}, {6412, 42177}, {6445, 42133}, {6446, 42134}, {6451, 42101}, {6452, 42102}, {6455, 42240}, {6456, 42241}, {8252, 35739}, {8976, 42125}, {10576, 16809}, {10577, 16808}, {11480, 42179}, {11481, 42182}, {13951, 42128}, {16628, 22883}, {16629, 22927}, {16964, 35731}, {19106, 42267}, {19107, 42266}, {23249, 42221}, {23253, 42213}, {23259, 42224}, {23263, 42212}, {35820, 42158}, {35821, 42157}, {42115, 42195}, {42116, 42198}, {42168, 42184}, {42169, 42185}, {42246, 42253}, {42249, 42250}

X(42279) = reflection of X(42278) in X(5073)
X(42279) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 42278), (5, 3851, 42278), (140, 3843, 42278), (381, 1656, 42278), (382, 1657, 42278), (550, 3830, 42278), (42175, 42178, 6)

X(42280) = GIBERT(0,-3,SQRT(3)) POINT

Barycentrics    3*Sqrt[3]*a^2*SA - 18*SB*SC : :

X(42280) lies on thesse lines: {2, 3}, {6, 42184}, {15, 42181}, {16, 42180}, {61, 3070}, {62, 3071}, {371, 5318}, {372, 5321}, {397, 6419}, {398, 6420}, {485, 42162}, {486, 42159}, {590, 35730}, {1151, 42094}, {1152, 42093}, {3311, 5335}, {3312, 5334}, {3364, 19106}, {3365, 16809}, {3367, 42259}, {3389, 16808}, {3390, 19107}, {3391, 42258}, {3592, 5340}, {3594, 5339}, {5237, 42235}, {5238, 42238}, {5349, 6454}, {5350, 6453}, {5365, 6448}, {5366, 6447}, {6200, 35740}, {6221, 42134}, {6250, 7684}, {6251, 7685}, {6396, 42101}, {6398, 42133}, {6409, 42174}, {6410, 42171}, {6411, 42186}, {6412, 42183}, {6425, 42230}, {6426, 42227}, {6449, 42206}, {6450, 42203}, {6560, 42160}, {6561, 42161}, {6564, 42166}, {6565, 42163}, {10645, 42182}, {10646, 42179}, {11480, 42190}, {11481, 42187}, {11485, 23249}, {11486, 23259}, {18538, 42138}, {18581, 42268}, {18582, 42269}, {18762, 42135}, {22236, 23251}, {22238, 23261}, {22615, 42086}, {22644, 42085}, {31412, 42128}, {34560, 38042}, {35820, 42164}, {35821, 42165}, {36836, 42257}, {36843, 42254}, {42103, 42274}, {42104, 42276}, {42105, 42275}, {42106, 42277}, {42115, 42217}, {42116, 42220}, {42136, 42226}, {42137, 42225}, {42168, 42178}, {42169, 42175}

X(42280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 42281), (5, 546, 42281), (20, 5076, 42281), (381, 3091, 42281), (382, 3146, 42281), (550, 12102, 42281), (42184, 42185, 6)

X(42281) = GIBERT(0,3,SQRT(3)) POINT

Barycentrics    3*Sqrt[3]*a^2*SA + 18*SB*SC : :

X(42281) lies on thesse lines: {2, 3}, {6, 42183}, {15, 42179}, {16, 42182}, {61, 3071}, {62, 3070}, {371, 5321}, {372, 5318}, {397, 6420}, {398, 6419}, {485, 42159}, {486, 42162}, {1151, 42093}, {1152, 42094}, {3311, 5334}, {3312, 5335}, {3364, 16809}, {3365, 19106}, {3366, 42258}, {3389, 19107}, {3390, 16808}, {3392, 42259}, {3592, 5339}, {3594, 5340}, {5237, 42237}, {5238, 42236}, {5349, 6453}, {5350, 6454}, {5365, 6447}, {5366, 6448}, {5479, 35742}, {6200, 42101}, {6221, 42133}, {6250, 7685}, {6251, 7684}, {6396, 42102}, {6398, 42134}, {6409, 42172}, {6410, 42173}, {6411, 42184}, {6412, 42185}, {6425, 42228}, {6426, 42229}, {6449, 42204}, {6450, 42205}, {6560, 42161}, {6561, 42160}, {6564, 42163}, {6565, 42166}, {10645, 42180}, {10646, 42181}, {11480, 42188}, {11481, 42189}, {11485, 23259}, {11486, 23249}, {16964, 35730}, {18538, 42135}, {18581, 42269}, {18582, 42268}, {18762, 42138}, {22236, 23261}, {22238, 23251}, {22615, 42085}, {22644, 42086}, {22795, 35747}, {22797, 35759}, {28178, 34560}, {31412, 42125}, {32789, 35733}, {35731, 36970}, {35820, 42165}, {35821, 42164}, {36836, 42255}, {36843, 42256}, {42103, 42277}, {42104, 42275}, {42105, 42276}, {42106, 42274}, {42115, 42219}, {42116, 42218}, {42136, 42225}, {42137, 42226}, {42167, 42177}, {42170, 42176}

X(42281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 42280), (5, 546, 42280), (20, 5076, 42280), (381, 3091, 42280), (382, 3146, 42280), (550, 12102, 42280), (42183, 42186, 6), (42187, 42190, 6)

X(42282) = GIBERT(0,-SQRT(3),2) POINT

Barycentrics    6*a^2*SA - 6*Sqrt[3]*SB*SC : :

X(42282) lies on thesse lines: {2, 3}, {6, 42192}, {15, 23249}, {16, 23259}, {61, 1587}, {62, 1588}, {371, 5335}, {372, 5334}, {397, 3592}, {398, 3594}, {487, 622}, {488, 621}, {590, 42166}, {615, 42163}, {1151, 5318}, {1152, 5321}, {3070, 22236}, {3071, 22238}, {3311, 42200}, {3312, 42201}, {3364, 42086}, {3365, 18581}, {3367, 13935}, {3389, 18582}, {3390, 42085}, {3391, 9540}, {4301, 36458}, {4857, 36442}, {5237, 23263}, {5238, 23253}, {5270, 36443}, {5339, 6426}, {5340, 6425}, {5343, 6454}, {5344, 6453}, {5351, 42235}, {5352, 42238}, {5365, 42231}, {5366, 42234}, {5870, 33350}, {5871, 33351}, {6200, 42134}, {6221, 42209}, {6396, 42133}, {6398, 42208}, {6409, 35740}, {6410, 42093}, {6411, 42102}, {6412, 42101}, {6419, 42228}, {6420, 42229}, {6449, 42202}, {6450, 42199}, {6451, 42210}, {6452, 42207}, {6455, 42206}, {6456, 42203}, {9541, 42161}, {9671, 36459}, {10645, 42181}, {10646, 42180}, {11480, 42220}, {11481, 42217}, {11485, 23267}, {11486, 23273}, {11488, 31412}, {11542, 13886}, {11543, 13939}, {18435, 34553}, {19107, 35739}, {23251, 36836}, {23261, 36843}, {23302, 42273}, {23303, 42270}, {32785, 42142}, {32786, 42139}, {32789, 42110}, {32790, 42107}, {35742, 36962}, {42087, 42272}, {42088, 42271}, {42115, 42211}, {42116, 42214}, {42164, 42259}, {42165, 42258}, {42179, 42195}, {42182, 42198}

X(42282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3091, 35732), (3, 4, 35732), (5, 3090, 35732), (20, 3146, 35732), (376, 3627, 35732), (381, 3525, 35732), (382, 17538, 35732), (546, 631, 35732), (42192, 42193, 6)

X(42283) = GIBERT(-1,SQRT(3),0) POINT

Barycentrics    -(Sqrt[3]*a^2*S) + 6*Sqrt[3]*SB*SC : :

X(42283) lies on thesse lines: {2, 6411}, {3, 22615}, {4, 6}, {5, 6200}, {15, 42179}, {16, 42180}, {20, 6412}, {30, 615}, {61, 42182}, {62, 42181}, {115, 13807}, {140, 42266}, {187, 6251}, {371, 546}, {372, 3627}, {376, 8252}, {381, 590}, {382, 486}, {485, 3843}, {489, 23312}, {542, 26438}, {550, 10577}, {574, 36709}, {637, 3631}, {638, 3630}, {1132, 6460}, {1151, 3091}, {1152, 2672}, {1327, 18512}, {1328, 3830}, {1384, 13834}, {1656, 6451}, {1657, 5420}, {1991, 33457}, {2041, 42088}, {2042, 42087}, {2043, 23303}, {2044, 23302}, {2045, 42168}, {2046, 42170}, {3054, 6813}, {3055, 6811}, {3068, 3839}, {3069, 3543}, {3090, 6409}, {3297, 5229}, {3298, 5225}, {3311, 42269}, {3312, 5076}, {3367, 19106}, {3392, 19107}, {3529, 6410}, {3545, 6433}, {3592, 31412}, {3620, 12322}, {3818, 18539}, {3832, 6437}, {3845, 6564}, {3850, 6480}, {3851, 5418}, {3853, 7584}, {3855, 6468}, {3856, 35812}, {3857, 6453}, {3858, 8981}, {3859, 6476}, {3861, 7583}, {5024, 36711}, {5066, 35255}, {5072, 6449}, {5073, 6446}, {5079, 6455}, {5200, 31860}, {5412, 10151}, {6420, 12102}, {6426, 13939}, {6434, 13935}, {6435, 14893}, {6436, 19116}, {6439, 41106}, {6441, 7585}, {6454, 13993}, {6469, 13847}, {6477, 35404}, {6481, 13966}, {7388, 34573}, {7526, 35777}, {7741, 9647}, {7951, 9660}, {7968, 31673}, {7969, 18483}, {9600, 31415}, {9675, 18424}, {9681, 9690}, {10645, 14813}, {10646, 14814}, {11008, 12221}, {11294, 23311}, {11381, 12240}, {11473, 23047}, {11480, 35732}, {11481, 42217}, {11485, 42190}, {11486, 42189}, {11801, 35826}, {12323, 20080}, {13665, 14269}, {13846, 41099}, {13911, 18492}, {13973, 41869}, {14234, 14236}, {15171, 35801}, {15492, 31561}, {15684, 41951}, {15687, 35823}, {15765, 16809}, {16808, 18585}, {16814, 31562}, {18323, 18459}, {18357, 35610}, {18510, 38335}, {18581, 42184}, {18582, 42186}, {18586, 42085}, {18587, 42086}, {18990, 35803}, {22236, 42220}, {22238, 42219}, {22591, 22810}, {22596, 32492}, {22625, 32497}, {26617, 32812}, {28174, 35789}, {28186, 35762}, {28224, 35811}, {32494, 32498}, {34754, 42238}, {34755, 42237}, {35641, 40273}, {35738, 42136}, {35740, 42110}, {35763, 38034}, {35841, 39884}, {36657, 41410}, {41950, 41957}, {41952, 41955}, {41959, 41967}, {41961, 41965}, {42103, 42243}, {42104, 42242}, {42105, 42244}, {42106, 42245}, {42107, 42240}, {42108, 42239}, {42109, 42241}, {42111, 42172}, {42112, 42171}, {42113, 42173}, {42114, 42174}, {42115, 42197}, {42116, 42198}, {42143, 42176}, {42144, 42175}, {42145, 42177}, {42146, 42178}

X(42283) = {X(4),X(6)}-harmonic conjugate of X(42284)
X(42283) = {X(42187),X(42188)}-harmonic conjugate of X(3)
X(42283) = {X(42191),X(42192)}-harmonic conjugate of X(3)

X(42284) = GIBERT(1,SQRT(3),0) POINT

Barycentrics    Sqrt[3]*a^2*S + 6*Sqrt[3]*SB*SC : :

X(42284) lies on thesse lines: {2, 6412}, {3, 22644}, {4, 6}, {5, 6396}, {15, 42181}, {16, 42182}, {20, 6411}, {30, 590}, {61, 42180}, {62, 42179}, {115, 13687}, {140, 42267}, {187, 6250}, {371, 3627}, {372, 546}, {376, 8253}, {381, 615}, {382, 485}, {486, 3843}, {490, 23311}, {542, 18539}, {550, 10576}, {574, 36714}, {591, 33456}, {637, 3630}, {638, 3631}, {1131, 6459}, {1151, 2671}, {1152, 3091}, {1327, 3830}, {1328, 18510}, {1384, 13711}, {1656, 6452}, {1657, 5418}, {2041, 42087}, {2042, 42088}, {2043, 23302}, {2044, 23303}, {2045, 42169}, {2046, 42167}, {3054, 6811}, {3055, 6813}, {3068, 3543}, {3069, 3839}, {3090, 6410}, {3297, 5225}, {3298, 5229}, {3311, 5076}, {3312, 42268}, {3366, 19106}, {3391, 19107}, {3529, 6409}, {3545, 6434}, {3620, 12323}, {3818, 26438}, {3832, 6438}, {3845, 6565}, {3850, 6481}, {3851, 5420}, {3853, 7583}, {3855, 6469}, {3856, 35813}, {3857, 6454}, {3858, 13966}, {3859, 6477}, {3861, 7584}, {4324, 31499}, {5024, 36712}, {5066, 35256}, {5072, 6450}, {5073, 6445}, {5079, 6456}, {5200, 41424}, {5210, 21736}, {5413, 10151}, {6419, 12102}, {6425, 13886}, {6433, 9540}, {6435, 19117}, {6436, 14893}, {6440, 41106}, {6441, 31414}, {6442, 7586}, {6453, 13925}, {6468, 9541}, {6476, 35404}, {6480, 8981}, {7389, 34573}, {7526, 35776}, {7968, 18483}, {7969, 31673}, {9661, 10483}, {9690, 15684}, {10645, 14814}, {10646, 14813}, {11008, 12222}, {11293, 23312}, {11381, 12239}, {11474, 23047}, {11480, 42220}, {11481, 35732}, {11485, 42188}, {11486, 42187}, {11801, 35827}, {12322, 20080}, {12953, 31472}, {13785, 14269}, {13847, 41099}, {13911, 41869}, {13973, 18492}, {14238, 14240}, {15171, 35800}, {15492, 31562}, {15687, 35822}, {15765, 16808}, {16809, 18585}, {16814, 31561}, {17851, 41970}, {18323, 18457}, {18357, 35611}, {18512, 38335}, {18581, 42183}, {18582, 42185}, {18586, 42086}, {18587, 42085}, {18990, 35802}, {22236, 42218}, {22238, 42217}, {22592, 22809}, {22596, 32494}, {22625, 32495}, {26618, 32813}, {28174, 35788}, {28186, 35763}, {28224, 35810}, {31415, 36655}, {32497, 32499}, {34754, 42236}, {34755, 42235}, {35642, 40273}, {35738, 42137}, {35740, 42109}, {35762, 38034}, {35840, 39884}, {36658, 41411}, {41949, 41958}, {41951, 41956}, {41960, 41968}, {41962, 41966}, {42103, 42242}, {42104, 42243}, {42105, 42245}, {42106, 42244}, {42107, 42239}, {42108, 42240}, {42110, 42241}, {42111, 42171}, {42112, 42172}, {42113, 42174}, {42114, 42173}, {42115, 42195}, {42116, 42196}, {42143, 42175}, {42144, 42176}, {42145, 42178}, {42146, 42177}

X(42284) = {X(4),X(6)}-harmonic conjugate of X(42283)
X(42284) = {X(42189),X(42190)}-harmonic conjugate of X(3)
X(42284) = {X(42193),X(42194)}-harmonic conjugate of X(3)

leftri

Perpsectors associated with product triangles: X(42285)-X(42286)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, March 23, 2021.

Let T1 = A1B1C1 be a triangle, and let M1 be the matrix whose rows are the normalized barycentrics of A1, B1, C1, respectively.
Let T2 = A2B2C2 be a triangle, and let M2 be the matrix whose rows are the normalized barycentrics of A2, B2, C2, respectively.
Let M be the matrix sum M1 + M2. The triangle sum T1 + T2 is here defined as the triangle whose vertices are given by the rows of M1 + M2. (Note that triangle sum is non-associative and that there is no additive identity, hence no additive inverses.)

In this section "cevian(P)" means the cevian triangle of P, and "anticevian U" means the anticevian triangle of U.

cevian(P)+anticevian(U) is perspective to both cevian(P) and anticevian(U), and the perspector is the P-Ceva conjugate of U, given by

u (-u/p + v/q + w/r) : v (-v/q + w/r + u/p) w (-w/r + u/p + v/q).

If P = X(1), then cevian(P) + anticevian(U) is perspective to ABC for U on a certain cubic through X(1), X(10), and X(1125). If U = X(1), the perspector is X(42285); if U = X(1125), the perspector is X(551).

If P = X(6), then cevian(P) + anticevian(U) is perspective to ABC for U on a certain cubic through X(1), X(141), and X(3589). If U = X(141), the perspector is X(42286).

X(42285) = PERSPECTOR OF THESE TRIANGLES: ABC AND CEVIAN(X(1)) + ANTICEVIAN(X(10))

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

X(42285) lies on these lines: {1,16704}, {2,4674}, {8,31011}, {10,3702}, {19,37168}, {37,519}, {38,39697}, {45,996}, {65,392}, {82,40091}, {225,11105}, {261,40438}, {514,4364}, {517,4698}, {551,17450}, {595,2363}, {596,2292}, {756,4738}, {758,13476}, {759,1621}, {876,30580}, {897,5263}, {984,41683}, {986,6532}, {994,3877}, {1953,3986}, {2214,16685}, {2217,5248}, {2802,3842}, {3230,40747}, {3636,31503}, {3679,31035}, {3828,30818}, {3884,34434}, {3919,25501}, {4370,21822}, {4389,20569}, {4424,24589}, {4425,5620}, {4792,32013}, {6630,24864}, {6690,11734}, {9978,33337}, {14210,39712}, {17057,25378}, {18785,36480}, {18833,40089}, {27268,30116}, {29655,34895}, {31339,39708}

X(42286) = PERSPECTOR OF THESE TRIANGLES: ABC AND CEVIAN(X(6)) + ANTICEVIAN(X(141))

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

X(42286) lies on these lines: {2,17413}, {39,524}, {69,31068}, {141,3266}, {373,468}, {523,4045}, {574,9516}, {597,13410}, {599,31088}, {732,41440}, {2393,6683}, {2854,10007}, {4062,21035}, {5967,8546}, {6329,31506}, {6664,23642}, {7790,40826}, {7919,40429}, {8041,36792}, {8542,15482}, {9019,27375}, {9045,32450}, {15464,17430}, {20582,30749}, {20859,25322}, {25334,33798}, {27376,37778}

leftri

Polarologic and Polelogic centers & T-isogonal-axes: X(42287)-X(42410)

rightri

This preamble and centers X(42287)-X(42410) were contributed by César Eliud Lozada, March 24, 2021.

Let T ' = A1B1C1 and T" = A2B2C2 be two distinct scalene triangles. Let (a'), (b'), (c') be the trilinear polars of the vertices of T ' with respect to T" and let A', B', C' be the trilinear poles of the sidelines of T ' with respect to T". Inversely, let (a"), (b"), (c") be the trilinear polars of the vertices of T" with respect to T ' and A", B", C" the trilinear poles of the sidelines of T" with respect to T '. Then:

  1.  If (a'), (b'), (c') concur in a point P' then (a"), (b"), (c") also concur in a point P".
  2.  If A', B', C' are collinear on a line ℓ' then A", B", C" are also collinear on a line ℓ".
  3.  If T ' and T" satisfy one of the properties (1) or (2), then they satisfy the other property.

T ' and T" have the concurrences in (1) and the collinearities in (2) if the isogonal conjugates of the vertices of T ' with respect to T" are collinear on a line 𝓂'. In this case, the isogonal conjugates of the vertices of T" with respect to T ' are also collinear on a line 𝓂".

Here, points P' and P" are named the polarologic centers of (T ' to T") and (T" to T ') and the tripoles of lines ℓ' and ℓ" are referred as the polelogic centers of (T ' to T") and (T" to T ') . Also, the lines 𝓂' and 𝓂" are introduced as the T-isogonal-axes of (T ' wrt T") and (T" wrt T ').

The appearance of (T, [i, j], [m, n]) in the following list means that triangles ABC and T have polarologic centers X(i), X(j) and polelogic centers X(m), X(n):

(ABC-X3 reflections, [1350, 6], [42287, 69]), (1st anti-circumperp, [20477, 6], [30441, 670]), (anti-Honsberger, [182, 32], [42288, 83]), (anti-tangential-midarc, [42289, 1400], [42290, 7]), (anti-Ursa minor, [42291, 826], [42292, 83]), (anti-Wasat, [1510, 42293], [2963, 8795]), (Aries, [42294, 42295], [42296, 42297]), (9th Brocard, [37174, 2], [42298, 2052]), (circummedial, [183, 6], [42299, 308]), (circumnormal, [3, 6], [275, 95]), (circumorthic, [33971, 6], [42300, 8795]), (1st circumperp, [11495, 6], [42301, 190]), (2nd circumperp, [1001, 6], [42302, 86]), (circumsymmedial, [6, 6], [6, 6]), (circumtangential, [3, 6], [648, 99]), (inner-Conway, [30625, 1], [42303, 668]), (outer-Garcia, [3696, 37], [27475, 75]), (Garcia-reflection, [4106, 650], [42304, 1088]), (Gossard, [42305, 42306], [42307, 42308]), (Honsberger, [42309, 1], [42310, 42311]), (intangents, [42312, 657], [39956, 7]), (Johnson, [39530, 216], [42313, 264]), (3rd mixtilinear, [42314, 6], [42315, 269]), (4th mixtilinear, [42316, 6], [42317, 9]), (5th mixtilinear, [3243, 9], [42318, 7]), (1st Schiffler, [42319, 650], [42320, 42321]), (2nd Schiffler, [42322, 650], [42323, 42324]), (tangential-midarc, [--, 7707], [--, 7]), (2nd tangential-midarc, [--, 10495], [--, 7]), (Ursa-minor, [42325, 10581], [42326, 42311]), (Wasat, [42327, 523], [42328, 86])

The appearance of (T, [i, j], [m, n]) in the following list means that triangles ANTICOMPLEMENTARY and T have polarologic centers X(i), X(j) and polelogic centers X(m), X(n):

(anti-Euler, [42329, 264], [42330, 95]), (3rd anti-Euler, [1510, 42331], [42332, 42333]), (Aquila, [42334, 86], [42335, 1268]), (Bevan antipodal, [42336, 42337], [42338, 42339]), (excentral, [798, 523], [40433, 1268]), (Pelletier, [42340, 42341], [42342, 42343]), (Schroeter, [42344, 690], [42345, 99]), (Soddy, [3669, 3900], [1434, 32008]), (Stammler, [1510, 41298], [40393, 40410]), (tangential, [9494, 826], [42346, 10159]), (X-parabola-tangential, [42347, 33906], [42348, 42349]), (X3-ABC reflections, [42350, 95], [42351, 40410])

The appearance of (T, [i, j], [m, n]) in the following list means that triangles MEDIAL and T have polarologic centers X(i), X(j) and polelogic centers X(m), X(n):

(anti-Aquila, [15569, 3739], [39721, 7]), (Euler, [5480, 141], [42352, 253]), (2nd Euler, [42353, 141], [42354, 42355]), (3rd Euler, [42356, 141], [42357, 190]), (4th Euler, [3826, 141], [42358, 75]), (5th Euler, [3815, 141], [42359, 6]), (excenters-midpoints, [4394, 4885], [42360, 42361]), (extouch, [2321, 142], [646, 4569]), (Feuerbach, [5949, 141], [42362, 42363]), (2nd Hatzipolakis, [42364, --], [--, --]), (incentral, [1100, 3739], [662, 670]), (intouch, [10481, 10], [36838, 668]), (Lemoine, [42365, 20582], [42366, 42367]), (Macbeath, [42368, 140], [42369, 18831]), (orthic, [53, 141], [15352, 670]), (Steiner, [14588, 523], [42370, 892]), (symmedial, [5007, 3934], [827, 42371]), (Yff contact, [32094, 514], [42372, 4555])

The appearance of (T, [i, j], [m, n]) in the following list means that triangles ORTHIC and T have polarologic centers X(i), X(j) and polelogic centers X(m), X(n):

(anti-excenters-reflections, [36990, 6], [42373, 393]), (Euler, [5480, 53], [42374, 6]), (2nd Euler, [42353, 53], [42375, 42376]), (3rd Euler, [42356, 53], [--, --]), (4th Euler, [3826, 53], [--, --]), (5th Euler, [3815, 53], [42377, 34208]), (extouch, [42378, 42379], [42380, 42381]), (Feuerbach, [5949, 53], [--, --]), (2nd Hatzipolakis, [42382, 5101], [42383, 42384]), (incentral, [2646, 42385], [662, 15352]), (intouch, [42386, 42387], [42388, 42389]), (Lemoine, [42390, 42391], [42392, 42393]), (Macbeath, [42394, 428], [42395, 42396]), (Mandart-incircle, [42397, 42385], [--, --]), (medial, [141, 53], [670, 15352]), (Steiner, [42398, 42399], [--, --]), (symmedial, [13366, 42400], [933, 42401]), (Yff contact, [42402, 42403], [--, --])

The appearance of (T, i, j) in the following list means that the tripoles of the T-isogonal-axes of triangles ABC and T are X(i) and X(j):

(ABC-X3 reflections, 2, 2), (1st anti-circumperp, 2, 2), (anti-Honsberger, 3407, 76), (anti-tangential-midarc, 40420, 333), (anti-Ursa minor, 41676, 827), (anti-Wasat, 42404, 42405), (Aries, 42406, 42407), (9th Brocard, --, 6), (circummedial, 2, 2), (circumnormal, 2, 2), (circumorthic, 2, 2), (1st circumperp, 2, 2), (2nd circumperp, 2, 2), (circumsymmedial, 2, 2), (circumtangential, 2, 2), (inner-Conway, 32019, 1), (outer-Garcia, 32017, 81), (Garcia-reflection, 42408, 651), (Gossard, --, --), (Honsberger, 42409, 1), (intangents, 658, 658), (Johnson, 42410, 275), (3rd mixtilinear, 2, 2), (4th mixtilinear, 2, 2), (5th mixtilinear, 8056, 57), (1st Schiffler, --, 651), (2nd Schiffler, --, 651), (tangential-midarc, --, --), (2nd tangential-midarc, --, --), (Ursa minor, --, --), (Wasat, --, 110)

X(42287) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO ABC-X3 REFLECTIONS

Barycentrics    (-a^2+b^2+c^2)*(3*a^4-2*(b^2-c^2)*a^2-(b^2-c^2)*(b^2+3*c^2))*(3*a^4+2*(b^2-c^2)*a^2+(b^2-c^2)*(3*b^2+c^2)) : :
X(42287) = 2*X(6)+X(253) = X(69)-4*X(20208) = 4*X(182)-X(41374) = 2*X(1249)-5*X(3618) = 5*X(3618)-X(31887) = 7*X(3619)-10*X(20200)

The reciprocal polelogic center of these triangles is X(69)

X(42287) lies on the cubics K295, K677 and these lines: {2, 154}, {4, 6330}, {6, 253}, {69, 441}, {95, 3619}, {182, 41374}, {193, 35510}, {264, 1249}, {287, 34156}, {305, 37669}, {438, 35711}, {458, 10002}, {459, 34407}, {525, 2419}, {1441, 5749}, {1494, 1992}, {2373, 37643}, {6225, 26218}, {6337, 40708}, {6340, 12215}, {8797, 20204}, {11427, 18018}, {11433, 13575}, {14853, 15312}, {15594, 31360}, {15740, 33198}, {18019, 37645}, {18918, 37073}, {20563, 28708}, {30786, 36894}, {31886, 36889}, {37188, 42313}

X(42287) = reflection of X(31887) in X(1249)
X(42287) = isotomic conjugate of the polar conjugate of X(3424)
X(42287) = polar conjugate of X(10002)
X(42287) = barycentric product X(69)*X(3424)
X(42287) = barycentric quotient X(i)/X(j) for these (i, j): (3, 1350), (4, 10002), (69, 37668), (1073, 40813), (1503, 1529)
X(42287) = trilinear product X(63)*X(3424)
X(42287) = trilinear quotient X(i)/X(j) for these (i, j): (63, 1350), (92, 10002), (304, 37668)
X(42287) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(3), X(5085)}}
X(42287) = Cevapoint of X(525) and X(12037)
X(42287) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 1350}, {48, 10002}, {204, 40813}, {1973, 37668}
X(42287) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 1350), (4, 10002), (69, 37668), (1073, 40813)

X(42288) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO ANTI-HONSBERGER

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

The reciprocal polelogic center of these triangles is X(83)

X(42288) lies on these lines: {3, 83}, {98, 32085}, {184, 251}, {228, 18098}, {733, 17970}, {878, 18105}, {5171, 26224}, {10547, 39674}, {38834, 40319}

X(42288) = complement of the anticomplementary conjugate of X(32451)
X(42288) = isogonal conjugate of X(14994)
X(42288) = barycentric product X(i)*X(j) for these {i, j}: {6, 42299}, {51, 39283}, {82, 2186}, {83, 263}, {251, 262}
X(42288) = barycentric quotient X(i)/X(j) for these (i, j): (32, 14096), (82, 3403), (83, 20023), (251, 183), (262, 8024), (263, 141)
X(42288) = trilinear product X(i)*X(j) for these {i, j}: {31, 42299}, {82, 263}, {83, 3402}, {251, 2186}, {2179, 39283}
X(42288) = trilinear quotient X(i)/X(j) for these (i, j): (31, 14096), (82, 183), (83, 3403), (262, 1930), (263, 38), (2186, 141)
X(42288) = trilinear pole of the line {3049, 18105}
X(42288) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7786)}} and {{A, B, C, X(3), X(25)}}
X(42288) = X(263)-cross conjugate of-X(42299)
X(42288) = X(i)-isoconjugate-of-X(j) for these {i, j}: {38, 183}, {39, 3403}, {75, 14096}, {182, 1930}
X(42288) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (32, 14096), (82, 3403), (83, 20023), (251, 183)
X(42288) = X(2)-vertex conjugate of-X(83)

X(42289) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO ANTI-TANGENTIAL-MIDARC

Barycentrics    a*(a^2-(b+c)*a-2*b*c)*(b+c)*(a-b+c)*(a+b-c) : :
X(42289) = 5*X(17609)-4*X(40636)

The reciprocal polarologic center of these triangles is X(1400)

X(42289) lies on these lines: {1, 7}, {8, 26125}, {10, 21931}, {12, 3214}, {31, 37543}, {34, 1839}, {37, 65}, {38, 5173}, {42, 226}, {43, 5226}, {45, 41712}, {56, 7225}, {57, 968}, {72, 21039}, {73, 3649}, {181, 28387}, {238, 8543}, {241, 15569}, {256, 17097}, {354, 7004}, {651, 4649}, {674, 1469}, {740, 1441}, {750, 37541}, {756, 41539}, {774, 942}, {916, 7352}, {934, 28842}, {940, 9316}, {941, 5665}, {954, 1253}, {976, 28081}, {984, 7672}, {1001, 1471}, {1064, 39542}, {1066, 6147}, {1100, 1456}, {1125, 17077}, {1193, 3485}, {1201, 30097}, {1214, 1962}, {1245, 28786}, {1254, 3931}, {1386, 34253}, {1402, 39793}, {1411, 32259}, {1423, 3340}, {1427, 37593}, {1450, 15950}, {1457, 17301}, {1463, 4864}, {1736, 30329}, {1738, 21617}, {1757, 29007}, {1818, 5880}, {1834, 21955}, {1836, 14547}, {1854, 2654}, {1892, 2356}, {1953, 3827}, {2310, 5728}, {2318, 3925}, {2340, 2550}, {2647, 17016}, {2650, 12709}, {2772, 5425}, {3293, 3947}, {3487, 37529}, {3660, 17450}, {3682, 12609}, {3685, 41246}, {3751, 8545}, {3812, 25067}, {3869, 24554}, {3870, 30699}, {3883, 17152}, {3886, 4441}, {3911, 30950}, {3993, 4552}, {4038, 17074}, {4365, 6358}, {4658, 34043}, {4848, 21803}, {4854, 6354}, {5249, 25941}, {5263, 10030}, {5311, 8270}, {5435, 26102}, {5714, 37699}, {6051, 37544}, {7073, 10118}, {7237, 15556}, {7613, 30275}, {7677, 16484}, {7986, 15934}, {9364, 37633}, {9791, 17950}, {10460, 40940}, {10474, 39780}, {11246, 22053}, {11375, 17278}, {11518, 18216}, {11526, 16496}, {11529, 33536}, {14942, 21453}, {17017, 34036}, {17080, 17592}, {17257, 19860}, {17609, 40636}, {18421, 30116}, {21454, 29814}, {24341, 41228}, {24789, 28253}, {25453, 28776}, {28082, 28089}, {28774, 29635}, {28780, 29633}, {29640, 37797}, {33128, 37695}, {40934, 41003}, {40958, 41011}

X(42289) = reflection of X(2293) in X(1)
X(42289) = barycentric product X(i)*X(j) for these {i, j}: {10, 5228}, {37, 40719}, {56, 4044}, {57, 3696}, {63, 1893}, {65, 4384}
X(42289) = barycentric quotient X(i)/X(j) for these (i, j): (42, 40779), (56, 42302), (65, 27475), (1001, 333), (1042, 42290), (1400, 1002)
X(42289) = trilinear product X(i)*X(j) for these {i, j}: {3, 1893}, {10, 1471}, {37, 5228}, {42, 40719}, {56, 3696}, {65, 1001}
X(42289) = trilinear quotient X(i)/X(j) for these (i, j): (37, 40779), (57, 42302), (65, 1002), (226, 27475), (1001, 21), (1400, 2279)
X(42289) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1334)}} and {{A, B, C, X(7), X(37)}}
X(42289) = crosssum of X(1) and X(991)
X(42289) = X(1)-Beth conjugate of-X(1400)
X(42289) = X(i)-isoconjugate-of-X(j) for these {i, j}: {9, 42302}, {21, 1002}, {81, 40779}, {284, 27475}
X(42289) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (42, 40779), (56, 42302), (65, 27475), (1001, 333)
X(42289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 7, 1458), (1, 1721, 7675), (1, 3671, 1042), (1, 4295, 4300), (1, 4298, 4322), (1, 4312, 991), (1, 4328, 4327), (1, 5018, 1442), (1, 7274, 4321), (1, 11552, 4337), (1, 12560, 2263), (65, 1284, 1400), (991, 4312, 3000), (1001, 5228, 1471), (3671, 4356, 3668), (4318, 7269, 1)

X(42290) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO ANTI-TANGENTIAL-MIDARC

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

The reciprocal polelogic center of these triangles is X(7)

X(42290) lies on these lines: {6, 1014}, {7, 37}, {25, 1396}, {42, 57}, {56, 1462}, {269, 1400}, {393, 37102}, {479, 1427}, {940, 7411}, {941, 980}, {1119, 1880}, {1423, 19604}, {2054, 2114}, {4334, 5223}, {4344, 37596}, {5228, 42314}, {5435, 16606}, {6610, 28658}, {7268, 16975}, {8693, 15728}, {14624, 31643}, {17245, 39983}, {34253, 37138}, {37650, 39798}, {37681, 39956}

X(42290) = isogonal conjugate of X(37658)
X(42290) = isotomic conjugate of X(28809)
X(42290) = barycentric product X(i)*X(j) for these {i, j}: {7, 1002}, {57, 27475}, {85, 2279}, {226, 42302}, {279, 40779}, {1418, 42310}
X(42290) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3886), (7, 4441), (25, 28044), (56, 1001), (57, 4384), (65, 3696)
X(42290) = trilinear product X(i)*X(j) for these {i, j}: {7, 2279}, {56, 27475}, {57, 1002}, {65, 42302}, {269, 40779}
X(42290) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3886), (7, 4384), (19, 28044), (56, 2280), (57, 1001), (77, 23151)
X(42290) = trilinear pole of the line {512, 3669}
X(42290) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(5308)}} and {{A, B, C, X(2), X(6)}}
X(42290) = X(1469)-cross conjugate of-X(7)
X(42290) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 3886}, {8, 2280}, {9, 1001}, {33, 23151}
X(42290) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 3886), (7, 4441), (25, 28044), (56, 1001)

X(42291) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO ANTI-URSA MINOR

Barycentrics    ((b^4+b^2*c^2+c^4)*a^4-b^4*c^4)*(b^2-c^2) : :
X(42291) = X(9491)-5*X(31279)

The reciprocal polarologic center of these triangles is X(826)

X(42291) lies on these lines: {2, 9494}, {141, 9005}, {512, 625}, {804, 34964}, {881, 7752}, {3934, 8711}, {6292, 9498}, {9491, 31279}, {30217, 39511}

X(42291) = complement of X(9494)
X(42291) = complementary conjugate of the complement of X(42371)
X(42291) = barycentric quotient X(523)/X(42292)
X(42291) = trilinear quotient X(1577)/X(42292)
X(42291) = X(i)-complementary conjugate of-X(j) for these (i, j): (75, 35971), (308, 16592), (561, 15449), (670, 16587)
X(42291) = X(163)-isoconjugate-of-X(42292)
X(42291) = X(523)-reciprocal conjugate of-X(42292)

X(42292) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO ANTI-URSA MINOR

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

The reciprocal polelogic center of these triangles is X(83)

X(42292) lies on these lines: {194, 40382}, {6374, 7786}

X(42292) = barycentric quotient X(523)/X(42291)
X(42292) = trilinear quotient X(1577)/X(42291)
X(42292) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7786)}} and {{A, B, C, X(39), X(308)}}
X(42292) = X(163)-isoconjugate-of-X(42291)
X(42292) = X(523)-reciprocal conjugate of-X(42291)

X(42293) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTI-WASAT TO ABC

Barycentrics    a^4*(-a^2+b^2+c^2)^2*((b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2) : :
Trilinears    cos B sec C csc^2 C - cos C sec B csc^2 B : :
Trilinears    sin^2 A cos A (cos^2 2B - cos^2 2C) : :

The reciprocal polarologic center of these triangles is X(1510)

X(42293) lies on these lines: {2, 42331}, {6, 23286}, {216, 6368}, {230, 231}, {577, 37084}, {3049, 23200}, {15450, 20975}

X(42293) is the intersection of the isogonal conjugate of the polar conjugate of the Lemoine axis (i.e., line X(3049)X(39201)), and the polar conjugate of the isogonal conjugate of the Lemoine axis (i.e., line X(230)X(231)). (Randy Hutson, June 30, 2021)

X(42293) = complement of X(42331)
X(42293) = isogonal conjugate of X(42405)
X(42293) = barycentric product X(i)*X(j) for these {i, j}: {3, 15451}, {5, 39201}, {6, 17434}, {51, 520}, {53, 32320}, {54, 34983}
X(42293) = barycentric quotient X(i)/X(j) for these (i, j): (32, 16813), (51, 6528), (130, 42331), (184, 18831), (216, 6331), (217, 648)
X(42293) = trilinear product X(i)*X(j) for these {i, j}: {31, 17434}, {48, 15451}, {51, 822}, {216, 810}, {217, 656}, {418, 661}
X(42293) = trilinear quotient X(i)/X(j) for these (i, j): (31, 16813), (48, 18831), (51, 823), (158, 42401), (216, 811), (217, 162)
X(42293) = crossdifference of every pair of points on line {X(3), X(95)}
X(42293) = crosspoint of X(i) and X(j) for these (i, j): {2, 1303}, {4, 42401}, {184, 14586}, {647, 39201}
X(42293) = crosssum of X(i) and X(j) for these (i, j): {264, 18314}, {648, 6528}, {850, 40684}
X(42293) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 130), (6, 34980), (647, 15451)
X(42293) = X(i)-complementary conjugate of-X(j) for these (i, j): (31, 130), (1303, 2887)
X(42293) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 16813}, {92, 18831}, {95, 823}, {162, 276}
X(42293) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (32, 16813), (51, 6528), (130, 42331), (184, 18831)
X(42293) = pole wrt polar circle of line X(2)X(276)
X(42293) = perspector of hyperbola {{A,B,C,X(4),X(51)}}
X(42293) = perspector of ABC and orthocevian triangle of X(1303)
X(42293) = X(798)-of-orthic-triangle, if ABC is acute

X(42294) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO ARIES

Barycentrics    (a^16-6*(b^2+c^2)*a^14+4*(4*b^4+b^2*c^2+4*c^4)*a^12-2*(b^2+c^2)*(13*b^4-14*b^2*c^2+13*c^4)*a^10+6*(5*b^8+5*c^8+2*(b^4-5*b^2*c^2+c^4)*b^2*c^2)*a^8-2*(b^4-c^4)*(b^2-c^2)*(13*b^4+14*b^2*c^2+13*c^4)*a^6+4*(b^2-c^2)^2*(4*b^8+4*c^8+3*(b^2+c^2)^2*b^2*c^2)*a^4-6*(b^4-c^4)^3*(b^2-c^2)*a^2+(b^8+c^8+2*(4*b^4+3*b^2*c^2+4*c^4)*b^2*c^2)*(b^2-c^2)^4)*a^2 : :

The reciprocal polarologic center of these triangles is X(42295)

X(42294) lies on this line: {20, 394}

X(42294) = barycentric quotient X(110)/X(42296)
X(42294) = trilinear quotient X(662)/X(42296)
X(42294) = X(661)-isoconjugate-of-X(42296)
X(42294) = X(110)-reciprocal conjugate of-X(42296)

X(42295) = POLAROLOGIC CENTER OF THESE TRIANGLES: ARIES TO ABC

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

The reciprocal polarologic center of these triangles is X(42294)

X(42295) lies on these lines: {2, 6}, {3, 20859}, {22, 1691}, {25, 1501}, {32, 51}, {115, 11550}, {154, 14567}, {182, 1194}, {184, 1196}, {251, 5640}, {419, 3168}, {549, 39524}, {800, 19031}, {1495, 34481}, {1570, 3787}, {1627, 3060}, {1853, 39691}, {1899, 2450}, {1915, 1995}, {1974, 40146}, {2021, 41275}, {2052, 6531}, {2175, 21813}, {2207, 6524}, {2211, 3162}, {2422, 8029}, {2502, 8780}, {3053, 20977}, {3094, 7485}, {3155, 6423}, {3156, 6424}, {3167, 20976}, {3224, 33336}, {3291, 9306}, {3788, 4121}, {3917, 5028}, {4563, 35294}, {5012, 9465}, {5033, 22352}, {5052, 15004}, {6034, 31133}, {6353, 41363}, {6660, 38905}, {6800, 35006}, {7484, 8041}, {7592, 37446}, {8569, 11328}, {8586, 41394}, {8627, 9909}, {8770, 35259}, {10329, 39560}, {11060, 14583}, {11402, 40126}, {13366, 39764}, {13410, 30435}, {14713, 40947}, {16949, 33798}, {16951, 18906}, {18374, 40366}, {20998, 35264}, {21849, 41413}, {23291, 34137}

X(42295) = isogonal conjugate of X(42407)
X(42295) = polar conjugate of the isotomic conjugate of X(40947)
X(42295) = barycentric product X(i)*X(j) for these {i, j}: {3, 41762}, {4, 40947}, {6, 3767}, {19, 2083}, {25, 1899}, {31, 17871}
X(42295) = barycentric quotient X(i)/X(j) for these (i, j): (25, 34405), (110, 42297), (426, 4176), (1632, 670), (1899, 305), (2083, 304)
X(42295) = trilinear product X(i)*X(j) for these {i, j}: {19, 40947}, {25, 2083}, {31, 3767}, {32, 17871}, {48, 41762}, {560, 41760}
X(42295) = trilinear quotient X(i)/X(j) for these (i, j): (19, 34405), (426, 1102), (662, 42297), (1632, 799), (1899, 304), (2083, 69)
X(42295) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3767)}} and {{A, B, C, X(25), X(325)}}
X(42295) = crossdifference of every pair of points on line {X(512), X(6333)}
X(42295) = crosspoint of X(6) and X(2207)
X(42295) = crosssum of X(2) and X(3926)
X(42295) = X(i)-Ceva conjugate of-X(j) for these (i, j): (2, 14713), (6, 39643), (83, 41761), (107, 669)
X(42295) = X(31)-complementary conjugate of-X(14713)
X(42295) = X(i)-isoconjugate-of-X(j) for these {i, j}: {63, 34405}, {661, 42297}
X(42295) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (25, 34405), (110, 42297), (426, 4176), (1632, 670)
X(42295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 1184, 3051), (6, 1611, 394), (6, 1613, 1993), (6, 10601, 20965), (25, 40825, 1501), (394, 1611, 3231), (1196, 1692, 184), (1501, 3124, 25), (1627, 3060, 5017), (1627, 39024, 3060), (1691, 3981, 22), (5359, 5422, 6), (8576, 8577, 3148), (19031, 19034, 800)

X(42296) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO ARIES

Barycentrics    (a^16-2*(3*b^2-2*c^2)*a^14+2*(8*b^4+3*b^2*c^2-10*c^4)*a^12-2*(b^2-c^2)*(13*b^4+23*b^2*c^2+14*c^4)*a^10+2*(b^2-c^2)*(15*b^6+13*c^6+2*(7*b^2+11*c^2)*b^2*c^2)*a^8-2*(b^2-c^2)*(13*b^8+14*c^8+(7*b^4-7*b^2*c^2+5*c^4)*b^2*c^2)*a^6+2*(b^2-c^2)^3*(8*b^6+10*c^6+(25*b^2+21*c^2)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)^2*(3*b^2+c^2)*(b^4+b^2*c^2+2*c^4)*a^2+(b^4+c^4)*(b^2-c^2)^6)*(a^2-c^2)*(a^16+2*(2*b^2-3*c^2)*a^14-2*(10*b^4-3*b^2*c^2-8*c^4)*a^12+2*(b^2-c^2)*(14*b^4+23*b^2*c^2+13*c^4)*a^10-2*(b^2-c^2)*(13*b^6+15*c^6+2*(11*b^2+7*c^2)*b^2*c^2)*a^8+2*(b^2-c^2)*(14*b^8+13*c^8+(5*b^4-7*b^2*c^2+7*c^4)*b^2*c^2)*a^6-2*(b^2-c^2)^3*(10*b^6+8*c^6+(21*b^2+25*c^2)*b^2*c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)^2*(b^2+3*c^2)*(2*b^4+b^2*c^2+c^4)*a^2+(b^4+c^4)*(b^2-c^2)^6)*(a^2-b^2) : :

The reciprocal polelogic center of these triangles is X(42297)

X(42296) lies on these lines: {}

X(42296) = barycentric quotient X(110)/X(42294)
X(42296) = trilinear quotient X(662)/X(42294)
X(42296) = X(661)-isoconjugate-of-X(42294)
X(42296) = X(110)-reciprocal conjugate of-X(42294)

X(42297) = POLELOGIC CENTER OF THESE TRIANGLES: ARIES TO ABC

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

The reciprocal polelogic center of these triangles is X(42296)

X(42297) lies on the circumcircle and these lines: {98, 305}, {111, 42407}, {112, 2396}, {670, 925}, {2715, 4563}, {3563, 34405}, {4609, 22456}, {14659, 37803}

X(42297) = isogonal conjugate of the circumtangential-isogonal conjugate of X(42297)
X(42297) = circumtangential-isogonal conjugate of the isogonal conjugate of X(42297)
X(42297) = barycentric product X(99)*X(42407)
X(42297) = barycentric quotient X(i)/X(j) for these (i, j): (99, 3767), (110, 42295), (648, 41762), (670, 41760), (799, 17871)
X(42297) = trilinear product X(662)*X(42407)
X(42297) = trilinear quotient X(i)/X(j) for these (i, j): (662, 42295), (670, 17871), (799, 3767), (811, 41762)
X(42297) = Collings transform of X(7887)
X(42297) = trilinear pole of the line {6, 6393}
X(42297) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(305), X(2396)}}
X(42297) = Cevapoint of X(523) and X(7887)
X(42297) = X(i)-cross conjugate of-X(j) for these (i, j): (315, 4590), (394, 34537)
X(42297) = X(i)-isoconjugate-of-X(j) for these {i, j}: {661, 42295}, {669, 17871}, {798, 3767}, {810, 41762}
X(42297) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (99, 3767), (110, 42295), (648, 41762), (670, 41760)

X(42298) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 9th BROCARD

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

The reciprocal polelogic center of these triangles is X(2052)

X(42298) lies on these lines: {193, 264}, {2052, 6353}, {14265, 16081}, {40814, 40819}

X(42298) = polar conjugate of X(1351)
X(42298) = barycentric product X(264)*X(7612)
X(42298) = barycentric quotient X(i)/X(j) for these (i, j): (4, 1351), (76, 10008), (264, 1007), (2052, 37174)
X(42298) = trilinear product X(92)*X(7612)
X(42298) = trilinear quotient X(i)/X(j) for these (i, j): (92, 1351), (561, 10008), (1969, 1007)
X(42298) = trilinear pole of the line {3566, 14618}
X(42298) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(193)}} and {{A, B, C, X(76), X(847)}}
X(42298) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 1351}, {560, 10008}, {1007, 9247}
X(42298) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 1351), (76, 10008), (264, 1007)

X(42299) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO CIRCUMMEDIAL

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

The reciprocal polelogic center of these triangles is X(308)

X(42299) lies on the Jerabek hyperbola and these lines: {3, 83}, {6, 32085}, {51, 290}, {54, 9418}, {66, 10550}, {69, 263}, {71, 18082}, {73, 18097}, {74, 32581}, {248, 251}, {695, 7745}, {3527, 39646}, {8795, 34854}, {14970, 36214}, {18092, 34817}, {26714, 39427}, {32451, 42359}, {40425, 41435}

X(42299) = isogonal conjugate of X(14096)
X(42299) = isotomic conjugate of X(14994)
X(42299) = polar conjugate of the complement of X(22240)
X(42299) = barycentric product X(i)*X(j) for these {i, j}: {5, 39283}, {76, 42288}, {83, 262}, {251, 327}, {263, 308}
X(42299) = barycentric quotient X(i)/X(j) for these (i, j): (83, 183), (251, 182), (262, 141), (263, 39), (308, 20023), (327, 8024)
X(42299) = trilinear product X(i)*X(j) for these {i, j}: {75, 42288}, {82, 262}, {83, 2186}, {263, 3112}, {308, 3402}
X(42299) = trilinear quotient X(i)/X(j) for these (i, j): (82, 182), (262, 38), (263, 1964), (308, 3403), (327, 1930)
X(42299) = trilinear pole of the line {647, 4108}
X(42299) = intersection, other than A,B,C, of conic {{A, B, C, X(2), X(11174)}} and Jerabek hyperbola
X(42299) = Cevapoint of X(i) and X(j) for these (i, j): {2, 32451}, {262, 263}
X(42299) = X(263)-cross conjugate of-X(42288)
X(42299) = X(i)-isoconjugate-of-X(j) for these {i, j}: {38, 182}, {183, 1964}, {458, 4020}
X(42299) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (83, 183), (251, 182), (262, 141), (263, 39)

X(42300) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO CIRCUMORTHIC

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

The reciprocal polelogic center of these triangles is X(8795)

X(42300) lies on these lines: {2, 34384}, {6, 95}, {25, 262}, {39, 276}, {54, 1976}, {97, 251}, {111, 4993}, {263, 9792}, {308, 36212}, {327, 2165}, {393, 8795}, {2395, 15412}, {7786, 34386}, {8576, 16037}, {8577, 16032}, {8770, 19188}, {11427, 37872}, {33631, 39286}, {34079, 39277}

X(42300) = polar conjugate of X(39530)
X(42300) = barycentric product X(i)*X(j) for these {i, j}: {54, 327}, {95, 262}, {141, 39283}, {263, 34384}, {275, 42313}
X(42300) = barycentric quotient X(i)/X(j) for these (i, j): (4, 39530), (54, 182), (95, 183), (262, 5), (263, 51), (275, 458)
X(42300) = trilinear product X(i)*X(j) for these {i, j}: {38, 39283}, {95, 2186}, {262, 2167}, {327, 2148}
X(42300) = trilinear quotient X(i)/X(j) for these (i, j): (92, 39530), (262, 1953), (263, 2179), (327, 14213)
X(42300) = trilinear pole of the line {512, 11674}
X(42300) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(4), X(37067)}}
X(42300) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 39530}, {182, 1953}, {183, 2179}
X(42300) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 39530), (54, 182), (95, 183), (262, 5)

X(42301) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 1st CIRCUMPERP

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

The reciprocal polelogic center of these triangles is X(190)

X(42301) lies on this line: {4105, 4626}

X(42301) = barycentric product X(1)*X(42303)
X(42301) = barycentric quotient X(i)/X(j) for these (i, j): (100, 30625), (101, 11495)
X(42301) = trilinear product X(6)*X(42303)
X(42301) = trilinear quotient X(i)/X(j) for these (i, j): (100, 11495), (190, 30625)
X(42301) = trilinear pole of the line {165, 170}
X(42301) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(4626)}} and {{A, B, C, X(2), X(42357)}}
X(42301) = Cevapoint of X(1) and X(4105)
X(42301) = X(i)-isoconjugate-of-X(j) for these {i, j}: {513, 11495}, {649, 30625}
X(42301) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (100, 30625), (101, 11495)

X(42302) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 2nd CIRCUMPERP

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

The reciprocal polelogic center of these triangles is X(86)

X(42302) lies on these lines: {6, 1014}, {9, 86}, {55, 81}, {58, 2195}, {284, 757}, {333, 873}, {673, 1434}, {1019, 1024}, {1174, 1412}, {2160, 7291}, {2258, 2274}, {2319, 2669}, {4663, 7077}, {4833, 23351}, {15569, 40773}, {18166, 34820}, {33635, 40438}, {37657, 39981}, {40439, 42028}

X(42302) = isotomic conjugate of X(4044)
X(42302) = barycentric product X(i)*X(j) for these {i, j}: {81, 27475}, {86, 1002}, {274, 2279}, {333, 42290}, {1019, 32041}, {1434, 40779}
X(42302) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3696), (21, 3886), (34, 1893), (56, 42289), (58, 1001), (81, 4384)
X(42302) = trilinear product X(i)*X(j) for these {i, j}: {21, 42290}, {58, 27475}, {81, 1002}, {86, 2279}, {1014, 40779}, {1019, 37138}
X(42302) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3696), (21, 37658), (57, 42289), (58, 2280), (81, 1001), (86, 4384)
X(42302) = trilinear pole of the line {663, 1019}
X(42302) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(16831)}} and {{A, B, C, X(2), X(749)}}
X(42302) = Cevapoint of X(i) and X(j) for these (i, j): {1, 37657}, {1002, 2279}
X(42302) = crosssum of X(9) and X(25427)
X(42302) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 3696}, {9, 42289}, {10, 2280}, {37, 1001}
X(42302) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 3696), (21, 3886), (34, 1893), (56, 42289)

X(42303) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO INNER-CONWAY

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

The reciprocal polelogic center of these triangles is X(668)

X(42303) lies on this line: {4130, 36838}

X(42303) = isotomic conjugate of the complement of X(4130)
X(42303) = barycentric product X(75)*X(42301)
X(42303) = barycentric quotient X(i)/X(j) for these (i, j): (100, 11495), (190, 30625)
X(42303) = trilinear product X(2)*X(42301)
X(42303) = trilinear quotient X(i)/X(j) for these (i, j): (190, 11495), (668, 30625)
X(42303) = trilinear pole of the line {144, 3059}
X(42303) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(36838)}} and {{A, B, C, X(344), X(2397)}}
X(42303) = X(i)-isoconjugate-of-X(j) for these {i, j}: {649, 11495}, {667, 30625}
X(42303) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (100, 11495), (190, 30625)

X(42304) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO GARCIA-REFLECTION

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

The reciprocal polelogic center of these triangles is X(1088)

X(42304) lies on these lines: {2, 27823}, {7, 3175}, {34, 4248}, {57, 3759}, {65, 145}, {226, 17234}, {1427, 5435}, {3644, 4032}, {9311, 24177}, {19604, 30568}, {31227, 31231}

X(42304) = isogonal conjugate of X(3217)
X(42304) = isotomic conjugate of X(30568)
X(42304) = barycentric product X(i)*X(j) for these {i, j}: {7, 34860}, {57, 40012}, {85, 39956}
X(42304) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3913), (7, 3875), (34, 4186), (56, 3915), (57, 4383), (65, 3214)
X(42304) = trilinear product X(i)*X(j) for these {i, j}: {7, 39956}, {56, 40012}, {57, 34860}
X(42304) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3913), (7, 4383), (56, 16946), (57, 3915), (85, 3875), (226, 3214)
X(42304) = trilinear pole of the line {3667, 4017}
X(42304) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(34791)}} and {{A, B, C, X(2), X(145)}}
X(42304) = X(i)-cross conjugate of-X(j) for these (i, j): (10, 7), (982, 1088)
X(42304) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 3913}, {8, 16946}, {9, 3915}, {41, 3875}
X(42304) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 3913), (7, 3875), (34, 4186), (56, 3915)

X(42305) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO GOSSARD

Barycentrics    (2*a^16-4*(b^2+c^2)*a^14-2*(5*b^4-16*b^2*c^2+5*c^4)*a^12+4*(b^2+c^2)*(8*b^4-17*b^2*c^2+8*c^4)*a^10-(19*b^8+19*c^8+4*(13*b^4-36*b^2*c^2+13*c^4)*b^2*c^2)*a^8-8*(b^4-c^4)*(b^2-c^2)*(2*b^4-11*b^2*c^2+2*c^4)*a^6+4*(b^2-c^2)^2*(5*b^8+5*c^8+(b^4-21*b^2*c^2+c^4)*b^2*c^2)*a^4-4*(b^4-c^4)*(b^2-c^2)^3*(b^4+7*b^2*c^2+c^4)*a^2-(b^2-c^2)^4*(b^8+c^8-4*(b^4+3*b^2*c^2+c^4)*b^2*c^2))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^8-(b^2+c^2)*a^6-(2*b^2-c^2)*(b^2-2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2)*(a^2-b^2+c^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2) : :

The reciprocal polarologic center of these triangles is X(42306)

X(42305) lies on these lines: {402, 32750}, {4240, 16077}

X(42305) = midpoint of X(4240) and X(42308)
X(42305) = reflection of X(42306) in X(402)
X(42305) = X(402)-reciprocal conjugate of-X(42307)

X(42306) = POLAROLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ABC

Barycentrics    (b^2-c^2)^2*(-a^2+b^2+c^2)^2*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)^2*(a^8-(b^2+c^2)*a^6-(2*b^2-c^2)*(b^2-2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(42306) = X(34767)-3*X(42307)

The reciprocal polarologic center of these triangles is X(42305)

X(42306) lies on these lines: {2, 23582}, {402, 32750}, {1650, 14401}, {3163, 11049}, {14920, 39081}, {16177, 38974}, {34767, 42307}

X(42306) = reflection of X(42305) in X(402)
X(42306) = complement of X(42308)
X(42306) = barycentric product X(402)*X(1650)
X(42306) = barycentric quotient X(402)/X(42308)
X(42306) = center of the circumconic {{ A, B, C, X(648), X(15351), X(39062), X(39352) }}
X(42306) = crosspoint of X(i) and X(j) for these (i, j): {2, 1650}, {402, 38240}
X(42306) = X(2)-Ceva conjugate of-X(402)
X(42306) = X(i)-complementary conjugate of-X(j) for these (i, j): (31, 402), (810, 9033), (1495, 23998), (1636, 4369)
X(42306) = X(402)-reciprocal conjugate of-X(42308)

X(42307) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO GOSSARD

Barycentrics    (b^2-c^2)*(-a^2+b^2+c^2)*(a^16+4*(b^2-2*c^2)*a^14-2*(10*b^4-8*b^2*c^2-5*c^4)*a^12+4*(4*b^6+4*c^6+9*(b^2-2*c^2)*b^2*c^2)*a^10+(19*b^8-38*c^8-4*(26*b^4-18*b^2*c^2-13*c^4)*b^2*c^2)*a^8-4*(b^2-c^2)*(8*b^8+4*c^8-(5*b^4+27*b^2*c^2-17*c^4)*b^2*c^2)*a^6+2*(b^2-c^2)^2*(5*b^8+5*c^8+(28*b^4-21*b^2*c^2-26*c^4)*b^2*c^2)*a^4+4*(b^2-c^2)^3*(b^8+2*c^8-(5*b^2-c^2)*(b^2+2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^4*(2*b^8-c^8+2*(2*b^4-3*b^2*c^2-4*c^4)*b^2*c^2))*(a^16-4*(2*b^2-c^2)*a^14+2*(5*b^4+8*b^2*c^2-10*c^4)*a^12+4*(4*b^6+4*c^6-9*(2*b^2-c^2)*b^2*c^2)*a^10-(38*b^8-19*c^8-4*(13*b^4+18*b^2*c^2-26*c^4)*b^2*c^2)*a^8+4*(b^2-c^2)*(4*b^8+8*c^8+(17*b^4-27*b^2*c^2-5*c^4)*b^2*c^2)*a^6+2*(b^2-c^2)^2*(5*b^8+5*c^8-(26*b^4+21*b^2*c^2-28*c^4)*b^2*c^2)*a^4-4*(b^2-c^2)^3*(2*b^8+c^8+(2*b^2+c^2)*(b^2-5*c^2)*b^2*c^2)*a^2+(b^8-2*c^8+2*(4*b^4+3*b^2*c^2-2*c^4)*b^2*c^2)*(b^2-c^2)^4) : :
X(42307) = X(34767)+2*X(42306)

The reciprocal polelogic center of these triangles is X(42308)

X(42307) lies on this line: {34767, 42306}

X(42307) = barycentric quotient X(402)/X(42305)
X(42307) = X(402)-reciprocal conjugate of-X(42305)

X(42308) = POLELOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ABC

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

The reciprocal polelogic center of these triangles is X(42307)

X(42308) lies on these lines: {2, 23582}, {69, 18020}, {287, 16080}, {648, 14401}, {1304, 22456}, {1494, 1651}, {1972, 14919}, {4240, 16077}, {14977, 15459}, {16076, 34582}, {32230, 36889}, {36831, 41208}

X(42308) = reflection of X(4240) in X(42305)
X(42308) = anticomplement of X(42306)
X(42308) = isotomic conjugate of X(1650)
X(42308) = barycentric product X(i)*X(j) for these {i, j}: {99, 15459}, {648, 16077}, {670, 32695}, {1304, 6331}, {1494, 23582}
X(42308) = barycentric quotient X(i)/X(j) for these (i, j): (30, 39008), (74, 3269), (99, 41077), (107, 1637), (110, 1636), (112, 9409)
X(42308) = trilinear product X(i)*X(j) for these {i, j}: {74, 23999}, {162, 16077}, {662, 15459}, {799, 32695}, {811, 1304}, {1494, 24000}
X(42308) = trilinear quotient X(i)/X(j) for these (i, j): (162, 9409), (648, 2631), (662, 1636), (799, 41077), (811, 9033), (823, 1637)
X(42308) = trilinear pole of the line {525, 648}
X(42308) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(3), X(2430)}}
X(42308) = Cevapoint of X(i) and X(j) for these (i, j): {2, 4240}, {30, 648}, {107, 14165}
X(42308) = X(i)-cross conjugate of-X(j) for these (i, j): (5, 39290), (30, 648), (340, 6528), (402, 2)
X(42308) = X(i)-isoconjugate-of-X(j) for these {i, j}: {647, 2631}, {656, 9409}, {661, 1636}, {798, 41077}
X(42308) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (30, 39008), (74, 3269), (99, 41077), (107, 1637)

X(42309) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO HONSBERGER

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

The reciprocal polarologic center of these triangles is X(1)

X(42309) lies on these lines: {1, 7}, {9, 85}, {57, 658}, {142, 348}, {165, 9446}, {169, 1445}, {226, 17093}, {479, 21454}, {518, 9312}, {527, 17079}, {553, 7056}, {664, 3243}, {738, 1434}, {1001, 40719}, {1111, 15299}, {1565, 5805}, {1996, 3911}, {2346, 30494}, {2550, 9436}, {2809, 7672}, {2898, 11019}, {3306, 37780}, {3729, 40704}, {5219, 37757}, {5437, 31627}, {5686, 31994}, {5853, 6604}, {6063, 11679}, {6173, 17078}, {7177, 14377}, {7223, 8581}, {7676, 30502}, {7677, 38859}, {10389, 21453}, {10980, 31526}, {11246, 30623}, {17095, 20195}, {17181, 38150}, {17732, 41857}, {21609, 30568}, {21617, 34847}, {31507, 38250}, {33298, 38200}

X(42309) = midpoint of X(31565) and X(31566)
X(42309) = reflection of X(i) in X(j) for these (i, j): (7, 10481), (30625, 9)
X(42309) = barycentric product X(i)*X(j) for these {i, j}: {7, 40719}, {85, 5228}, {269, 4441}, {279, 4384}, {479, 3886}, {658, 4762}
X(42309) = barycentric quotient X(i)/X(j) for these (i, j): (57, 40779), (269, 1002), (279, 27475), (658, 32041), (738, 42290), (934, 37138)
X(42309) = trilinear product X(i)*X(j) for these {i, j}: {7, 5228}, {57, 40719}, {85, 1471}, {269, 4384}, {279, 1001}, {479, 37658}
X(42309) = trilinear quotient X(i)/X(j) for these (i, j): (7, 40779), (269, 2279), (279, 1002), (479, 42290), (658, 37138), (934, 8693)
X(42309) = homothetic center of Honsberger triangle and polar triangle of Adams circle
X(42309) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(673)}} and {{A, B, C, X(2), X(11038)}}
X(42309) = X(1434)-Beth conjugate of-X(269)
X(42309) = X(i)-isoconjugate-of-X(j) for these {i, j}: {55, 40779}, {200, 2279}, {220, 1002}, {480, 42290}
X(42309) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (57, 40779), (269, 1002), (279, 27475), (658, 32041)
X(42309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 3160, 11038), (7, 3188, 7675), (7, 7176, 4321), (7, 14189, 1)

X(42310) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO HONSBERGER

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

The reciprocal polelogic center of these triangles is X(42311)

X(42310) lies on these lines: {1, 31269}, {37, 31618}, {57, 21453}, {81, 6605}, {105, 2346}, {274, 3693}, {279, 27253}, {985, 40739}, {1219, 27109}, {9445, 13405}, {24600, 25430}, {32041, 34578}

X(42310) = barycentric quotient X(i)/X(j) for these (i, j): (1002, 354), (1170, 5228), (1174, 2280)
X(42310) = trilinear product X(1002)*X(32008)
X(42310) = trilinear quotient X(i)/X(j) for these (i, j): (1002, 1475), (1170, 1471)
X(42310) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(37), X(3693)}}
X(42310) = X(i)-isoconjugate-of-X(j) for these {i, j}: {354, 2280}, {1001, 1475}, {1212, 1471}
X(42310) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1002, 354), (1170, 5228), (1174, 2280)

X(42311) = POLELOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO ABC

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

The reciprocal polelogic center of these triangles is X(42310)

X(42311) lies on the Feuerbach hyperbola and these lines: {1, 1088}, {7, 39789}, {8, 6063}, {9, 85}, {75, 42015}, {279, 27253}, {294, 1170}, {479, 5558}, {943, 40443}, {2346, 14189}, {3254, 4569}, {3296, 7056}, {6601, 6604}, {6606, 34894}, {9442, 10481}, {10390, 23062}, {17682, 33765}

X(42311) = isotomic conjugate of X(3059)
X(42311) = barycentric product X(i)*X(j) for these {i, j}: {7, 31618}, {75, 10509}, {85, 21453}, {1088, 32008}, {1170, 6063}
X(42311) = barycentric quotient X(i)/X(j) for these (i, j): (1, 8012), (7, 1212), (55, 8551), (56, 20229), (57, 2293), (65, 21795)
X(42311) = trilinear product X(i)*X(j) for these {i, j}: {2, 10509}, {7, 21453}, {57, 31618}, {85, 1170}, {273, 40443}, {279, 32008}
X(42311) = trilinear quotient X(i)/X(j) for these (i, j): (2, 8012), (7, 2293), (9, 8551), (57, 20229), (77, 22079), (85, 1212)
X(42311) = trilinear pole of the line {650, 24002}
X(42311) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(65), X(34855)}}
X(42311) = Cevapoint of X(i) and X(j) for these (i, j): {2, 30628}, {7, 1088}, {85, 6604}
X(42311) = X(i)-cross conjugate of-X(j) for these (i, j): (7, 21453), (514, 36838)
X(42311) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 8012}, {9, 20229}, {33, 22079}, {41, 1212}
X(42311) = trilinear product of vertices of Honsberger triangle
X(42311) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 8012), (7, 1212), (55, 8551), (56, 20229)
X(42311) = {X(85), X(32008)}-harmonic conjugate of X(31618)

X(42312) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO INTANGENTS

Barycentrics    a*(-a+b+c)*(a^2+(b+c)*a-2*b*c)*(b-c) : :
X(42312) = 3*X(663)-2*X(3737) = 4*X(3737)-3*X(17418) = X(20293)-3*X(27545) = 2*X(20316)-3*X(26144)

The reciprocal polarologic center of these triangles is X(657)

Let H be the {ABC,intangents}-circumconic (the hyperbola that is the isogonal conjugate of the Soddy line). Then X(42312) is the perspector of H wrt the intangents triangle. (Randy Hutson, June 30, 2021)

X(42312) lies on these lines: {1, 3667}, {33, 7649}, {35, 39225}, {55, 4057}, {200, 7628}, {521, 4895}, {522, 663}, {523, 4724}, {900, 1459}, {1040, 20315}, {1392, 23838}, {1769, 15313}, {1919, 2268}, {2269, 20979}, {2293, 29328}, {2310, 34949}, {2605, 4926}, {3063, 4526}, {3309, 4017}, {3601, 8656}, {3709, 4501}, {3716, 4397}, {3887, 21189}, {3907, 4811}, {4040, 28161}, {4139, 4498}, {4162, 4449}, {4375, 7220}, {4474, 4985}, {4786, 5287}, {4959, 35057}, {4962, 21173}, {7004, 14115}, {7190, 31605}, {7741, 39508}, {8058, 21119}, {17452, 17458}, {20293, 27545}, {20316, 26144}, {21348, 21834}

X(42312) = midpoint of X(4895) and X(6615)
X(42312) = reflection of X(i) in X(j) for these (i, j): (4397, 3716), (4474, 4985), (17418, 663)
X(42312) = reflection of X(43924) in the Nagel line
X(42312) = pole of the trilinear polar of X(1019) with respect to Feuerbach hyperbola
X(42312) = crossdifference of every pair of points on line {X(978), X(1400)}
X(42312) = crosspoint of X(i) and X(j) for these (i, j): {1, 3699}, {21, 31343}
X(42312) = X(i)-Ceva conjugate of-X(j) for these (i, j): (1, 17477), (1019, 650)
X(42312) = X(i)-isoconjugate-of-X(j) for these {i, j}: {65, 8690}, {101, 42304}, {109, 34860}, {651, 39956}
X(42312) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (284, 8690), (513, 42304), (522, 40012), (650, 34860)
X(42312) = barycentric product X(i)*X(j) for these {i, j}: {1, 20317}, {8, 4498}, {9, 4106}, {333, 4139}, {513, 30568}, {514, 3913}
X(42312) = barycentric quotient X(i)/X(j) for these (i, j): (284, 8690), (513, 42304), (522, 40012), (650, 34860), (663, 39956)
X(42312) = trilinear product X(i)*X(j) for these {i, j}: {6, 20317}, {9, 4498}, {21, 4139}, {55, 4106}, {513, 3913}, {514, 3217}
X(42312) = trilinear quotient X(i)/X(j) for these (i, j): (21, 8690), (514, 42304), (650, 39956)

X(42313) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO JOHNSON

Barycentrics    (-a^2+b^2+c^2)*((b^2+2*c^2)*a^2-b^4+b^2*c^2)*((2*b^2+c^2)*a^2+b^2*c^2-c^4) : :
Barycentrics    cos A sec(A - ω) : :
X(42313) = X(69)+2*X(216) = 4*X(141)-X(264) = X(1351)-4*X(10003) = 2*X(1352)+X(42329) = X(1972)+2*X(15595) = X(3164)+5*X(3620) = 7*X(3619)-4*X(14767) = 2*X(39530)-5*X(40330)

The reciprocal polelogic center of these triangles is X(264)

X(42313) lies on these lines: {2, 51}, {3, 287}, {6, 95}, {53, 141}, {69, 216}, {76, 18024}, {99, 30541}, {183, 40802}, {249, 7771}, {253, 3164}, {305, 343}, {394, 1799}, {401, 3098}, {458, 1350}, {599, 1494}, {1351, 10003}, {1352, 39682}, {1441, 3662}, {1503, 35937}, {1972, 15595}, {2351, 37068}, {2373, 15066}, {2710, 6037}, {3094, 40814}, {3314, 12037}, {3618, 36948}, {3619, 8797}, {3763, 40410}, {3818, 40853}, {5447, 28407}, {5562, 37186}, {6330, 11331}, {9289, 22416}, {11178, 40885}, {11257, 20021}, {11821, 32971}, {15644, 37337}, {17811, 40413}, {18018, 34138}, {18092, 34817}, {21356, 36889}, {30786, 37638}, {31884, 35941}, {33190, 39908}, {33257, 35240}, {33533, 40856}, {34507, 40867}, {36952, 41009}, {37174, 39530}, {37188, 42287}

X(42313) = isogonal conjugate of X(10311)
X(42313) = isotomic conjugate of X(458)
X(42313) = polar conjugate of X(33971)
X(42313) = polar conjugate of the anticomplement of X(42353)
X(42313) = barycentric product X(i)*X(j) for these {i, j}: {3, 327}, {69, 262}, {263, 305}, {304, 2186}, {343, 42300}
X(42313) = barycentric quotient X(i)/X(j) for these (i, j): (3, 182), (5, 39530), (69, 183), (184, 34396), (262, 4), (263, 25)
X(42313) = trilinear product X(i)*X(j) for these {i, j}: {48, 327}, {63, 262}, {69, 2186}, {263, 304}, {305, 3402}
X(42313) = trilinear quotient X(i)/X(j) for these (i, j): (48, 34396), (63, 182), (262, 19), (263, 1973), (304, 183), (305, 3403)
X(42313) = trilinear pole of the line {525, 684}
X(42313) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(76)}} and {{A, B, C, X(4), X(14853)}}
X(42313) = Cevapoint of X(6) and X(15577)
X(42313) = crosssum of X(1351) and X(11328)
X(42313) = X(327)-Ceva conjugate of-X(262)
X(42313) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 182}, {92, 34396}, {162, 3288}, {183, 1973}
X(42313) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 182), (5, 39530), (69, 183), (184, 34396)

X(42314) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO 3rd MIXTILINEAR

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

The reciprocal polarologic center of these triangles is X(6)

X(42314) lies on these lines: {1, 1418}, {6, 41}, {37, 4321}, {45, 8581}, {77, 38315}, {109, 1407}, {241, 3242}, {269, 1279}, {388, 17245}, {991, 999}, {1001, 4334}, {1037, 5096}, {1042, 1616}, {1106, 21059}, {1190, 23653}, {1191, 4306}, {1253, 5204}, {1319, 2263}, {2191, 4320}, {2293, 3304}, {2975, 25878}, {3361, 4255}, {3600, 4648}, {4327, 16777}, {4675, 12573}, {5228, 42290}, {5265, 37650}, {6180, 7677}, {6610, 7290}, {7271, 38316}, {7288, 17337}, {9316, 21000}, {20978, 32577}

X(42314) = barycentric product X(i)*X(j) for these {i, j}: {56, 29627}, {57, 3243}, {1407, 10005}
X(42314) = barycentric quotient X(56)/X(42318)
X(42314) = trilinear product X(i)*X(j) for these {i, j}: {56, 3243}, {604, 29627}, {1106, 10005}
X(42314) = trilinear quotient X(i)/X(j) for these (i, j): (57, 42318), (1407, 42315)
X(42314) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(1477)}} and {{A, B, C, X(41), X(3445)}}
X(42314) = X(21)-Beth conjugate of-X(11495)
X(42314) = X(i)-isoconjugate-of-X(j) for these {i, j}: {9, 42318}, {346, 42315}
X(42314) = X(56)-reciprocal conjugate of-X(42318)
X(42314) = X(6)-of-3rd-mixtilinear-triangle
X(42314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (56, 1458, 6), (269, 1420, 1279), (1407, 1617, 3052)

X(42315) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 3rd MIXTILINEAR

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

The reciprocal polelogic center of these triangles is X(269)

X(42315) lies on these lines: {6, 19604}, {7, 1743}, {57, 1279}, {1014, 33628}, {3243, 5228}, {4936, 36807}, {8817, 30813}

X(42315) = barycentric product X(57)*X(42318)
X(42315) = barycentric quotient X(i)/X(j) for these (i, j): (1, 10005), (56, 3243), (57, 29627), (1106, 42314)
X(42315) = trilinear product X(56)*X(42318)
X(42315) = trilinear quotient X(i)/X(j) for these (i, j): (2, 10005), (7, 29627), (57, 3243), (1407, 42314)
X(42315) = trilinear pole of the line {3669, 8643}
X(42315) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1279)}} and {{A, B, C, X(6), X(1743)}}
X(42315) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 10005}, {9, 3243}, {55, 29627}, {346, 42314}
X(42315) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 10005), (56, 3243), (57, 29627), (1106, 42314)

X(42316) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO 4th MIXTILINEAR

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

The reciprocal polarologic center of these triangles is X(6)

X(42316) lies on the cubic K297 and these lines: {1, 5022}, {2, 17747}, {3, 101}, {5, 17732}, {6, 31}, {9, 165}, {19, 7964}, {35, 218}, {36, 34867}, {37, 57}, {39, 1191}, {40, 1212}, {41, 5217}, {45, 1155}, {46, 16601}, {48, 6602}, {56, 1334}, {63, 3693}, {100, 37658}, {105, 6016}, {169, 3579}, {183, 190}, {198, 1615}, {213, 4255}, {219, 15931}, {221, 36074}, {292, 3445}, {346, 14829}, {354, 16777}, {381, 5134}, {386, 31461}, {405, 16549}, {474, 3294}, {517, 34522}, {573, 6244}, {584, 1174}, {595, 9605}, {840, 6017}, {956, 1018}, {958, 3501}, {995, 5024}, {999, 5030}, {1001, 17754}, {1015, 16486}, {1100, 10389}, {1104, 9593}, {1146, 5657}, {1190, 36744}, {1213, 26040}, {1282, 5220}, {1434, 27253}, {1447, 24352}, {1475, 3303}, {1500, 5021}, {1571, 16583}, {1593, 41320}, {1616, 2275}, {1617, 2256}, {1656, 24045}, {1697, 40133}, {1743, 31508}, {1788, 21049}, {1796, 11350}, {1826, 1889}, {2082, 37568}, {2176, 5013}, {2271, 31451}, {2277, 28272}, {2284, 22163}, {2291, 6014}, {2324, 10857}, {2345, 5273}, {2911, 37504}, {2975, 4513}, {3204, 38849}, {3208, 12513}, {3247, 10980}, {3295, 4253}, {3423, 4265}, {3553, 10383}, {3554, 10388}, {3576, 6603}, {3598, 4419}, {3684, 4421}, {3731, 17122}, {3748, 16884}, {3752, 9574}, {3913, 21384}, {3916, 17742}, {4191, 40586}, {4209, 32008}, {4383, 17756}, {4428, 16503}, {4520, 19861}, {4646, 31426}, {4713, 15271}, {4860, 16672}, {4884, 17314}, {5010, 5526}, {5124, 37519}, {5173, 21853}, {5179, 26446}, {5204, 9310}, {5221, 21808}, {5228, 40779}, {5276, 37540}, {5282, 14439}, {5314, 7123}, {5338, 37385}, {5687, 16552}, {5711, 25092}, {5781, 7411}, {5792, 37416}, {5819, 9778}, {7368, 8273}, {7522, 8804}, {7539, 24054}, {8568, 40998}, {9588, 23058}, {10164, 40869}, {10310, 32561}, {11114, 26074}, {12514, 25066}, {14321, 28602}, {15668, 25349}, {15815, 21008}, {15853, 37551}, {16370, 16788}, {16518, 21010}, {17095, 27129}, {17355, 32916}, {17451, 37567}, {17595, 26242}, {17796, 34879}, {21856, 37679}, {22080, 26867}, {27000, 31269}, {28535, 28911}, {30503, 34526}, {31859, 40859}, {40937, 41338}

X(42316) = isogonal conjugate of the isotomic conjugate of X(29616)
X(42316) = barycentric product X(i)*X(j) for these {i, j}: {1, 5223}, {6, 29616}, {220, 10004}
X(42316) = barycentric quotient X(101)/X(32040)
X(42316) = trilinear product X(i)*X(j) for these {i, j}: {6, 5223}, {31, 29616}, {1253, 10004}
X(42316) = trilinear quotient X(i)/X(j) for these (i, j): (55, 42317), (100, 32040), (692, 26716)
X(42316) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(35270)}} and {{A, B, C, X(6), X(103)}}
X(42316) = crossdifference of every pair of points on line {X(514), X(676)}
X(42316) = X(9)-Beth conjugate of-X(57)
X(42316) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 42317}, {513, 32040}, {693, 26716}
X(42316) = X(101)-reciprocal conjugate of X(32040)
X(42316) = X(6)-of-4th-mixtilinear-triangle
X(42316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 41325, 17747), (3, 220, 3207), (3, 3730, 220), (6, 17735, 3052), (6, 21000, 1914), (9, 165, 910), (35, 218, 4258), (39, 14974, 1191), (55, 672, 6), (165, 2951, 15506), (198, 2272, 1615), (213, 31448, 4255), (672, 41423, 55), (3730, 24047, 3), (9574, 16970, 3752)

X(42317) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 4th MIXTILINEAR

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

The reciprocal polelogic center of these triangles is X(9)

X(42317) lies on the Feuerbach hyperbola and these lines: {1, 3207}, {4, 1886}, {6, 3062}, {7, 1419}, {8, 23058}, {9, 41339}, {90, 16572}, {104, 26716}, {218, 38271}, {219, 42015}, {220, 4866}, {1000, 8074}, {1156, 16670}, {1699, 23972}, {2338, 19605}, {2346, 3247}, {2481, 16834}, {3158, 4876}, {3680, 3684}, {4900, 6603}, {16667, 31507}, {17745, 36599}

X(42317) = barycentric product X(650)*X(32040)
X(42317) = barycentric quotient X(i)/X(j) for these (i, j): (9, 29616), (41, 42316), (55, 5223), (57, 10004)
X(42317) = trilinear product X(i)*X(j) for these {i, j}: {522, 26716}, {663, 32040}
X(42317) = trilinear quotient X(i)/X(j) for these (i, j): (7, 10004), (8, 29616), (9, 5223), (55, 42316)
X(42317) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(6), X(1419)}}
X(42317) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 42316}, {55, 10004}, {56, 29616}, {57, 5223}
X(42317) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (9, 29616), (41, 42316), (55, 5223), (57, 10004)

X(42318) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 5th MIXTILINEAR

Barycentrics    (3*a^2-2*(2*b+c)*a+(b-c)*(b-3*c))*(3*a^2-2*(b+2*c)*a+(3*b-c)*(b-c)) : :
X(42318) = X(7)-4*X(4859) = 2*X(9)+X(4373) = 2*X(3161)-5*X(18230)

The reciprocal polelogic center of these triangles is X(7)

X(42318) lies on the circumhyperbola dual of Yff parabola and these lines: {2, 3158}, {7, 1743}, {8, 36807}, {9, 4373}, {75, 3161}, {142, 30712}, {144, 36606}, {335, 15590}, {390, 31183}, {673, 30332}, {903, 6172}, {1088, 5435}, {2550, 31189}, {4384, 10005}, {5222, 27475}, {5226, 21453}, {5838, 17278}, {5936, 16832}, {8056, 16078}, {12630, 29627}, {14189, 36620}, {17132, 36588}, {18228, 42361}, {20157, 42335}, {26007, 31188}, {28626, 29598}, {29628, 39721}, {31191, 40333}, {38186, 39704}

X(42318) = isotomic conjugate of X(29627)
X(42318) = barycentric product X(312)*X(42315)
X(42318) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3243), (8, 10005), (56, 42314)
X(42318) = trilinear product X(8)*X(42315)
X(42318) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3243), (57, 42314), (312, 10005)
X(42318) = trilinear pole of the line {514, 4162}
X(42318) = intersection, other than A,B,C, of conic {{A, B, C, X(1), X(38316)}} and circumhyperbola dual of Yff parabola
X(42318) = Cevapoint of X(i) and X(j) for these (i, j): {2, 24599}, {11, 14330}
X(42318) = X(390)-cross conjugate of-X(7)
X(42318) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 3243}, {9, 42314}, {604, 10005}
X(42318) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 3243), (8, 10005), (56, 42314)

X(42319) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO 1st SCHIFFLER

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

The reciprocal polarologic center of these triangles is X(650)

X(42319) lies on these lines: {523, 4468}, {2788, 4106}, {3912, 29298}, {3935, 8702}

X(42320) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 1st SCHIFFLER

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

The reciprocal polelogic center of these triangles is X(42321)

X(42320) lies on these lines: {}

X(42321) = POLELOGIC CENTER OF THESE TRIANGLES: 1st SCHIFFLER TO ABC

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

The reciprocal polelogic center of these triangles is X(42320)

X(42321) lies on these lines: {}

X(42322) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO 2nd SCHIFFLER

Barycentrics    a*(a^4-(4*b^2-3*b*c+4*c^2)*a^2+(b+c)*(4*b^2-5*b*c+4*c^2)*a-b^4-c^4)*(b-c) : :
X(42322) = 2*X(214)-3*X(30234)

The reciprocal polarologic center of these triangles is X(650)

X(42322) lies on these lines: {11, 4106}, {57, 35355}, {80, 28475}, {100, 4394}, {104, 30199}, {149, 4380}, {214, 30234}, {649, 38325}, {900, 4786}, {1155, 3309}, {3218, 13266}, {3667, 3911}, {3669, 3999}, {4905, 18201}, {9048, 10755}, {14947, 34583}, {17660, 17664}

X(42322) = midpoint of X(i) and X(j) for these {i, j}: {149, 4380}, {649, 38325}
X(42322) = reflection of X(i) in X(j) for these (i, j): (100, 4394), (4106, 11)

X(42323) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO 2nd SCHIFFLER

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

The reciprocal polelogic center of these triangles is X(42324)

X(42323) lies on these lines: {}

X(42324) = POLELOGIC CENTER OF THESE TRIANGLES: 2nd SCHIFFLER TO ABC

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

The reciprocal polelogic center of these triangles is X(42323)

X(42324) lies on these lines: {}

X(42325) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO URSA MINOR

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

The reciprocal polarologic center of these triangles is X(10581)

X(42325) lies on these lines: {7, 23599}, {21, 1019}, {30, 511}, {79, 885}, {442, 4129}, {663, 3960}, {667, 39476}, {764, 4879}, {905, 4794}, {1308, 4564}, {1734, 4724}, {2499, 21212}, {3126, 3647}, {3762, 21302}, {3777, 4775}, {3888, 4752}, {4040, 14838}, {4367, 6161}, {4729, 21385}, {4806, 6701}, {4807, 21677}, {4895, 23738}, {5216, 35637}, {7178, 21201}, {11281, 23814}, {14427, 21390}, {21127, 30295}, {38371, 39772}

X(42325) = barycentric product X(i)*X(j) for these {i, j}: {513, 17263}, {514, 3957}, {650, 32007}, {693, 17745}
X(42325) = barycentric quotient X(513)/X(42326)
X(42325) = trilinear product X(i)*X(j) for these {i, j}: {513, 3957}, {514, 17745}, {649, 17263}, {663, 32007}
X(42325) = trilinear quotient X(514)/X(42326)
X(42325) = crosssum of X(i) and X(j) for these (i, j): {55, 21127}, {513, 3748}, {661, 4068}
X(42325) = X(101)-isoconjugate-of-X(42326)
X(42325) = X(513)-reciprocal conjugate of-X(42326)
X(42325) = X(7)-Waw conjugate of-X(5083)
X(42325) = X(i)-Zayin conjugate of-X(j) for these (i, j): (2, 100), (9, 14722), (220, 101)

X(42326) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO URSA MINOR

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

The reciprocal polelogic center of these triangles is X(42311)

X(42326) lies on these lines: {1, 3826}, {2, 7264}, {10, 1280}, {35, 105}, {57, 24796}, {75, 32019}, {81, 3008}, {142, 17745}, {1002, 18398}, {1170, 5526}, {1212, 34578}, {1219, 19855}, {1224, 31238}, {1255, 26724}, {1698, 39959}, {3673, 42409}, {5222, 25417}, {5308, 27789}, {7178, 37626}, {16706, 32009}, {17095, 34018}, {24789, 25430}, {27304, 38247}, {33129, 40434}

X(42326) = isogonal conjugate of X(17745)
X(42326) = isotomic conjugate of X(17263)
X(42326) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3957), (7, 32007), (513, 42325)
X(42326) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3957), (85, 32007), (514, 42325)
X(42326) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(4), X(38200)}}
X(42326) = X(1174)-cross conjugate of-X(15909)
X(42326) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 3957}, {41, 32007}, {101, 42325}
X(42326) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 3957), (7, 32007), (513, 42325)

X(42327) = POLAROLOGIC CENTER OF THESE TRIANGLES: ABC TO WASAT

Barycentrics    ((b^2+b*c+c^2)*a^2-b^2*c^2)*(b-c) : :
X(42327) = 3*X(2)+X(17217) = X(3768)-7*X(27138) = 3*X(4728)-X(20954) = 3*X(4927)-X(23743) = X(17458)+3*X(27485) = X(18080)+3*X(21606) = X(20906)-3*X(27485)

The reciprocal polarologic center of these triangles is X(523)

X(42327) lies on these lines: {2, 798}, {6, 23149}, {37, 22046}, {75, 21834}, {86, 20981}, {141, 27854}, {512, 17066}, {514, 4408}, {661, 7199}, {693, 27647}, {786, 3250}, {788, 20549}, {802, 6586}, {812, 14838}, {1001, 23400}, {1577, 4481}, {1919, 24601}, {1924, 29458}, {2484, 17215}, {2605, 4107}, {3572, 17234}, {3661, 21055}, {3716, 3834}, {3733, 15668}, {3739, 4132}, {3766, 21123}, {3768, 27106}, {3912, 21099}, {4079, 4374}, {4083, 25127}, {4106, 30094}, {4129, 28840}, {4406, 4502}, {4411, 21206}, {4728, 20954}, {4826, 17159}, {4927, 23743}, {4992, 9400}, {6003, 24220}, {9286, 21238}, {17458, 20906}, {18080, 21606}, {18137, 20953}, {18155, 28372}, {18196, 21763}, {20295, 27159}, {21211, 28217}, {21389, 24354}, {27855, 29985}

X(42327) = midpoint of X(i) and X(j) for these {i, j}: {75, 21834}, {661, 7199}, {798, 17217}, {1577, 4481}, {3250, 3261}, {3766, 21123}, {3835, 21191}, {4079, 4374}, {4406, 4502}, {4826, 17159}, {17458, 20906}
X(42327) = reflection of X(4411) in X(21206)
X(42327) = complement of X(798)
X(42327) = isotomic conjugate of the isogonal conjugate of X(21763)
X(42327) = polar conjugate of the isogonal conjugate of X(22387)
X(42327) = complementary conjugate of X(16592)
X(42327) = barycentric product X(i)*X(j) for these {i, j}: {76, 21763}, {141, 18106}, {264, 22387}, {274, 21836}, {321, 18196}, {514, 25264}
X(42327) = barycentric quotient X(514)/X(42328)
X(42327) = trilinear product X(i)*X(j) for these {i, j}: {10, 18196}, {38, 18106}, {75, 21763}, {86, 21836}, {92, 22387}, {512, 34022}
X(42327) = trilinear quotient X(693)/X(42328)
X(42327) = crossdifference of every pair of points on line {X(2176), X(20990)}
X(42327) = crosspoint of X(2) and X(4602)
X(42327) = crosssum of X(6) and X(1924)
X(42327) = X(i)-complementary conjugate of-X(j) for these (i, j): (1, 16592), (2, 115), (4, 6388), (6, 1084)
X(42327) = X(692)-isoconjugate-of-X(42328)
X(42327) = X(514)-reciprocal conjugate of-X(42328)
X(42327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 17217, 798), (4369, 8062, 8060), (17458, 27485, 20906)

X(42328) = POLELOGIC CENTER OF THESE TRIANGLES: ABC TO WASAT

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

The reciprocal polelogic center of these triangles is X(86)

X(42328) lies on these lines: {75, 6379}, {192, 714}, {257, 4022}, {872, 19565}, {3121, 6385}, {3212, 17497}, {4687, 6376}, {20963, 33296}, {25264, 40433}

X(42328) = isotomic conjugate of X(25264)
X(42328) = barycentric quotient X(i)/X(j) for these (i, j): (274, 34022), (514, 42327), (649, 21763), (661, 21836), (1019, 18196), (1459, 22387)
X(42328) = trilinear quotient X(i)/X(j) for these (i, j): (310, 34022), (513, 21763), (523, 21836), (693, 42327), (905, 22387)
X(42328) = trilinear pole of the line {3835, 6005}
X(42328) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1218)}} and {{A, B, C, X(2), X(749)}}
X(42328) = Cevapoint of X(514) and X(3121)
X(42328) = X(i)-isoconjugate-of-X(j) for these {i, j}: {100, 21763}, {110, 21836}, {692, 42327}
X(42328) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (274, 34022), (514, 42327), (649, 21763), (661, 21836)

X(42329) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ANTI-EULER

Barycentrics    2*(b^2+c^2)*a^10-(6*b^4+5*b^2*c^2+6*c^4)*a^8+2*(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^6-2*(b^6-c^6)*(b^2-c^2)*a^4-(b^2-c^2)^4*b^2*c^2 : :
X(42329) = 3*X(381)-4*X(10003) = 6*X(549)-5*X(40329) = 5*X(631)-4*X(14767) = 2*X(1352)-3*X(42313)

The reciprocal polarologic center of these triangles is X(264)

X(42329) lies on these lines: {2, 26895}, {3, 95}, {4, 216}, {20, 185}, {22, 98}, {30, 30258}, {182, 401}, {184, 8613}, {262, 22240}, {297, 42353}, {324, 26874}, {381, 10003}, {418, 2052}, {436, 26880}, {542, 1972}, {549, 40329}, {577, 41204}, {631, 14767}, {933, 23606}, {1294, 30247}, {1352, 39682}, {1370, 9744}, {1629, 6641}, {1632, 15577}, {3522, 40896}, {3690, 7361}, {3937, 6360}, {6527, 10519}, {6638, 15466}, {6811, 41515}, {6813, 41516}, {8719, 34808}, {9747, 22655}, {11414, 39646}, {12131, 20885}, {14461, 37648}, {14570, 41716}, {15581, 35226}, {17538, 41374}, {18381, 23719}, {18667, 37521}, {22676, 38553}, {22712, 30737}

X(42329) = midpoint of X(20) and X(3164)
X(42329) = reflection of X(i) in X(j) for these (i, j): (4, 216), (264, 3)
X(42329) = anticomplement of X(39530)
X(42329) = polar conjugate of X(42374)
X(42329) = barycentric quotient X(4)/X(42374)
X(42329) = trilinear quotient X(92)/X(42374)
X(42329) = crossdifference of every pair of points on line {X(2451), X(42293)}
X(42329) = X(48)-isoconjugate-of-X(42374)
X(42329) = X(4)-reciprocal conjugate of-X(42374)
X(42329) = pole wrt polar circle of trilinear polar of X(42374) (line X(15451)X(16229))
X(42329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (8982, 26441, 19467), (26907, 42400, 2)

X(42330) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ANTI-EULER

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

The reciprocal polelogic center of these triangles is X(95)

X(42330) lies on these lines: {2, 1629}, {5, 6330}, {69, 36794}, {95, 441}, {182, 41374}, {287, 8550}, {458, 1350}, {577, 11348}, {8797, 17907}, {31360, 41235}, {37665, 42373}

X(42330) = isotomic conjugate of the complement of X(458)
X(42330) = polar conjugate of X(5480)
X(42330) = barycentric product X(264)*X(5481)
X(42330) = barycentric quotient X(4)/X(5480)
X(42330) = trilinear product X(92)*X(5481)
X(42330) = trilinear quotient X(92)/X(5480)
X(42330) = trilinear pole of the line {525, 35474}
X(42330) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(4), X(42373)}}
X(42330) = Cevapoint of X(2) and X(458)
X(42330) = X(48)-isoconjugate-of-X(5480)
X(42330) = X(4)-reciprocal conjugate of-X(5480)

X(42331) = POLAROLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO ANTICOMPLEMENTARY

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

The reciprocal polarologic center of these triangles is X(1510)

X(42331) lies on these lines: {2, 42293}, {69, 15415}, {95, 37084}, {264, 6368}, {325, 523}

X(42331) = anticomplement of X(42293)
X(42331) = isotomic conjugate of X(1303)
X(42331) = anticomplementary conjugate of the anticomplement of X(42405)
X(42331) = barycentric product X(i)*X(j) for these {i, j}: {436, 3267}, {525, 9291}, {1954, 20948}
X(42331) = barycentric quotient X(i)/X(j) for these (i, j): (130, 42293), (436, 112), (850, 9290), (1577, 9251)
X(42331) = trilinear product X(i)*X(j) for these {i, j}: {436, 14208}, {525, 9252}, {656, 9291}, {850, 1954}
X(42331) = trilinear quotient X(i)/X(j) for these (i, j): (436, 32676), (850, 9251)
X(42331) = crosspoint of X(670) and X(42333)
X(42331) = X(823)-anticomplementary conjugate of-X(17035)
X(42331) = X(130)-cross conjugate of-X(2)
X(42331) = X(1576)-isoconjugate-of-X(9251)
X(42331) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (130, 42293), (436, 112), (850, 9290), (1577, 9251)

X(42332) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 3rd ANTI-EULER

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

The reciprocal polelogic center of these triangles is X(42333)

X(42332) lies on these lines: {251, 17005}, {2963, 7769}, {7930, 24861}, {7940, 34816}

X(42332) = isotomic conjugate of the complement of X(7769)
X(42332) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(95), X(18023)}}
X(42332) = Cevapoint of X(2) and X(7769)
X(42332) = X(1510)-cross conjugate of-X(670)

X(42333) = POLELOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO ANTICOMPLEMENTARY

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

The reciprocal polelogic center of these triangles is X(42332)

X(42333) lies on these lines: {69, 18027}, {76, 3964}, {264, 394}, {276, 311}, {1502, 4176}, {5562, 8795}, {11444, 42355}

X(42333) = isotomic conjugate of X(389)
X(42333) = barycentric product X(i)*X(j) for these {i, j}: {76, 40448}, {305, 40402}
X(42333) = barycentric quotient X(i)/X(j) for these (i, j): (95, 19170), (311, 34836), (324, 6750)
X(42333) = trilinear product X(i)*X(j) for these {i, j}: {75, 40448}, {304, 40402}
X(42333) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(8795)}} and {{A, B, C, X(4), X(7395)}}
X(42333) = Cevapoint of X(i) and X(j) for these (i, j): {2, 5562}, {69, 311}
X(42333) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (95, 19170), (311, 34836), (324, 6750)

X(42334) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO AQUILA

Barycentrics    a^3+(b+c)*a^2-(2*b^2+3*b*c+2*c^2)*a-(b+c)^3 : :
X(42334) = 3*X(8)+X(9791) = 2*X(8)+X(24697) = 4*X(1125)-5*X(31248) = 3*X(1654)-X(9791) = 5*X(1698)-4*X(6707) = 5*X(3617)-X(20090) = 3*X(3679)-2*X(4733) = 3*X(3679)-X(24342) = 2*X(9791)-3*X(24697) = 2*X(24325)-3*X(27483) = 2*X(25354)-3*X(31144)

The reciprocal polarologic center of these triangles is X(86)

X(42334) lies on these lines: {1, 1213}, {2, 5625}, {8, 192}, {10, 86}, {81, 8013}, {238, 3686}, {239, 3775}, {291, 4651}, {333, 8298}, {511, 4111}, {519, 25354}, {524, 3416}, {537, 6650}, {594, 1757}, {726, 5564}, {846, 4046}, {1100, 19856}, {1125, 31248}, {1211, 33135}, {1698, 4851}, {2796, 4669}, {2895, 21020}, {3120, 31143}, {3578, 4418}, {3617, 20090}, {3626, 17770}, {3634, 17317}, {3661, 20142}, {3696, 4690}, {3741, 4886}, {3770, 4647}, {3773, 6651}, {3821, 17271}, {3836, 17287}, {3842, 6542}, {3923, 17346}, {3932, 4478}, {3993, 16358}, {4042, 32778}, {4062, 5235}, {4357, 4716}, {4384, 33087}, {4425, 41816}, {4445, 29674}, {4527, 17261}, {4645, 4732}, {4650, 14552}, {4655, 17343}, {4668, 5223}, {4683, 17163}, {4709, 24723}, {4938, 37635}, {4967, 34379}, {5222, 25539}, {5271, 33084}, {5278, 33158}, {5739, 33096}, {9534, 24520}, {9780, 17373}, {10180, 26044}, {16830, 17772}, {17116, 17771}, {17162, 27081}, {17239, 29633}, {17258, 28522}, {17270, 32784}, {17272, 33149}, {17316, 31336}, {17344, 32857}, {17348, 29637}, {17778, 27798}, {20055, 31308}, {24325, 27483}, {29617, 32921}, {31330, 32861}, {32780, 32864}, {32782, 33132}

X(42334) = midpoint of X(8) and X(1654)
X(42334) = reflection of X(i) in X(j) for these (i, j): (1, 1213), (86, 10), (24342, 4733), (24697, 1654)
X(42334) = anticomplement of X(5625)
X(42334) = intersection, other than A,B,C, of conics {{A, B, C, X(79), X(33770)}} and {{A, B, C, X(256), X(40438)}}
X(42334) = X(8)-Beth conjugate of-X(86)
X(42334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 319, 32846), (333, 21085, 33160), (2895, 21020, 33097), (3679, 24342, 4733), (3696, 4690, 33082), (3696, 33082, 24715), (5271, 33084, 33130)

X(42335) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO AQUILA

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

The reciprocal polelogic center of these triangles is X(1268)

X(42335) lies on the circumhyperbola dual of Yff parabola and these lines: {2, 4023}, {6, 28626}, {7, 4364}, {75, 3247}, {86, 4416}, {190, 31336}, {310, 25507}, {335, 29578}, {594, 5308}, {673, 1125}, {1268, 3912}, {3696, 16826}, {3842, 5223}, {5263, 39721}, {5333, 39734}, {5550, 20135}, {14621, 29612}, {17277, 30598}, {17308, 28650}, {20133, 39746}, {20157, 42318}

X(42335) = isotomic conjugate of X(24603)
X(42335) = barycentric quotient X(i)/X(j) for these (i, j): (1, 15569), (10, 4733)
X(42335) = trilinear quotient X(i)/X(j) for these (i, j): (2, 15569), (321, 4733)
X(42335) = intersection, other than A,B,C, of conic {{A, B, C, X(1), X(16831)}} and circumhyperbola dual of Yff parabola
X(42335) = Cevapoint of X(2) and X(16826)
X(42335) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 15569}, {1333, 4733}
X(42335) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 15569), (10, 4733)

X(42336) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO BEVAN ANTIPODAL

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

The reciprocal polarologic center of these triangles is X(42337)

X(42336) lies on these lines: {663, 855}, {664, 9296}

X(42336) = barycentric product X(i)*X(j) for these {i, j}: {57, 6363}, {649, 1122}, {1201, 3669}, {1357, 21362}, {1407, 6615}
X(42336) = barycentric quotient X(i)/X(j) for these (i, j): (604, 8706), (1122, 1978), (1201, 646), (1919, 1261)
X(42336) = trilinear product X(i)*X(j) for these {i, j}: {56, 6363}, {667, 1122}, {1106, 6615}, {1357, 23845}
X(42336) = trilinear quotient X(i)/X(j) for these (i, j): (56, 8706), (667, 1261), (1122, 668), (1201, 3699)
X(42336) = crossdifference of every pair of points on line {X(9), X(30693)}
X(42336) = crosspoint of X(664) and X(42338)
X(42336) = X(i)-isoconjugate-of-X(j) for these {i, j}: {8, 8706}, {644, 32017}, {646, 23617}, {668, 1261}
X(42336) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (604, 8706), (1122, 1978), (1201, 646)

X(42337) = POLAROLOGIC CENTER OF THESE TRIANGLES: BEVAN ANTIPODAL TO ANTICOMPLEMENTARY

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

The reciprocal polarologic center of these triangles is X(42336)

X(42337) lies on these lines: {11, 35065}, {30, 511}, {676, 21186}, {1043, 7253}, {1834, 23757}, {3704, 4086}, {4105, 40500}, {6615, 14284}

X(42337) = isotomic conjugate of X(6613)
X(42337) = barycentric product X(i)*X(j) for these {i, j}: {8, 21120}, {11, 25268}, {312, 6615}, {514, 6736}, {522, 3452}, {650, 20895}
X(42337) = barycentric quotient X(i)/X(j) for these (i, j): (346, 8706), (522, 40420), (650, 1476), (663, 3451), (1122, 4617), (1201, 1461)
X(42337) = trilinear product X(i)*X(j) for these {i, j}: {8, 6615}, {9, 21120}, {341, 6363}, {513, 6736}, {522, 3057}, {650, 3452}
X(42337) = trilinear quotient X(i)/X(j) for these (i, j): (341, 8706), (522, 1476), (650, 3451), (1122, 6614)
X(42337) = crossdifference of every pair of points on line {X(6), X(1604)}
X(42337) = crosspoint of X(i) and X(j) for these (i, j): {2, 6613}, {522, 4397}, {664, 42339}
X(42337) = X(1476)-anticomplementary conjugate of-X(33650)
X(42337) = X(i)-Ceva conjugate of-X(j) for these (i, j): (57, 5514), (312, 1146), (522, 6615)
X(42337) = X(1476)-complementary conjugate of-X(124)
X(42337) = X(i)-isoconjugate-of-X(j) for these {i, j}: {109, 1476}, {651, 3451}, {1106, 8706}, {1261, 6614}
X(42337) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (346, 8706), (522, 40420), (650, 1476), (663, 3451)

X(42338) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO BEVAN ANTIPODAL

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

The reciprocal polelogic center of these triangles is X(42339)

X(42338) lies on these lines: {}

X(42338) = intersection, other than A,B,C, of conics {{A, B, C, X(56), X(57)}} and {{A, B, C, X(81), X(1120)}}

X(42339) = POLELOGIC CENTER OF THESE TRIANGLES: BEVAN ANTIPODAL TO ANTICOMPLEMENTARY

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

The reciprocal polelogic center of these triangles is X(42338)

X(42339) lies on these lines: {2, 34524}, {8, 1997}, {85, 30827}, {312, 20196}, {333, 5316}, {3452, 40420}, {7308, 30608}, {11814, 14942}, {31995, 38255}

X(42339) = isotomic conjugate of X(6692)
X(42339) = barycentric quotient X(1)/X(20323)
X(42339) = trilinear quotient X(2)/X(20323)
X(42339) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(8)}} and {{A, B, C, X(9), X(30827)}}
X(42339) = Cevapoint of X(2) and X(3452)
X(42339) = X(6)-isoconjugate-of-X(20323)
X(42339) = X(1)-reciprocal conjugate of-X(20323)

X(42340) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO PELLETIER

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

The reciprocal polarologic center of these triangles is X(42341)

X(42340) lies on this line: {926, 2170}

X(42341) = POLAROLOGIC CENTER OF THESE TRIANGLES: PELLETIER TO ANTICOMPLEMENTARY

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

The reciprocal polarologic center of these triangles is X(42340)

X(42341) lies on these lines: {2, 30700}, {30, 511}, {38, 3310}, {63, 6139}, {200, 9511}, {210, 1638}, {354, 1639}, {668, 883}, {1015, 17435}, {2488, 4468}, {3681, 4453}, {3873, 30565}, {4014, 21139}, {4025, 4524}, {4147, 21195}, {4367, 21390}, {4449, 20980}, {4919, 21343}, {14430, 30691}, {21051, 40474}, {22086, 32912}

X(42341) = isotomic conjugate of X(14727)
X(42341) = barycentric product X(i)*X(j) for these {i, j}: {513, 40883}, {518, 4885}, {522, 6168}, {672, 20907}, {918, 1376}, {1026, 21139}
X(42341) = barycentric quotient X(i)/X(j) for these (i, j): (518, 30610), (663, 6169), (665, 9309), (918, 32023), (1376, 666), (2254, 9311)
X(42341) = trilinear product X(i)*X(j) for these {i, j}: {518, 4449}, {649, 40883}, {650, 6168}, {665, 3729}, {672, 4885}, {918, 9310}
X(42341) = trilinear quotient X(i)/X(j) for these (i, j): (650, 6169), (665, 9315), (918, 9311), (926, 9439), (1376, 36086), (2254, 9309)
X(42341) = crossdifference of every pair of points on line {X(6), X(1633)}
X(42341) = crosspoint of X(i) and X(j) for these (i, j): {2, 14727}, {518, 883}
X(42341) = crosssum of X(105) and X(884)
X(42341) = X(919)-anticomplementary conjugate of-X(41792)
X(42341) = X(650)-Ceva conjugate of-X(3126)
X(42341) = X(513)-Daleth conjugate of-X(6004)
X(42341) = X(513)-Hirst inverse of-X(3900)
X(42341) = X(i)-isoconjugate-of-X(j) for these {i, j}: {651, 6169}, {666, 9315}, {919, 9311}, {927, 9439}
X(42341) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (518, 30610), (663, 6169), (665, 9309), (918, 32023)

X(42342) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO PELLETIER

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

The reciprocal polelogic center of these triangles is X(42343)

X(42342) lies on these lines: {}

X(42343) = POLELOGIC CENTER OF THESE TRIANGLES: PELLETIER TO ANTICOMPLEMENTARY

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

The reciprocal polelogic center of these triangles is X(42342)

X(42343) lies on these lines: {4554, 31250}, {4885, 30610}, {6667, 14947}, {28743, 32041}

X(42343) = isotomic conjugate of X(31287)
X(42343) = barycentric product X(668)*X(41439)
X(42343) = barycentric quotient X(i)/X(j) for these (i, j): (100, 4421), (190, 25728), (668, 25278)
X(42343) = trilinear product X(190)*X(41439)
X(42343) = trilinear quotient X(i)/X(j) for these (i, j): (190, 4421), (668, 25728)
X(42343) = trilinear pole of the line {518, 1278}
X(42343) = Cevapoint of X(i) and X(j) for these (i, j): {2, 4885}, {650, 5274}
X(42343) = X(i)-isoconjugate-of-X(j) for these {i, j}: {649, 4421}, {667, 25728}
X(42343) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (100, 4421), (190, 25728), (668, 25278)

X(42344) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO SCHROETER

Barycentrics    (b^2-c^2)^4*(2*a^2-b^2-c^2) : :
X(42344) = 2*X(115)+X(1648) = 8*X(115)+X(14444) = X(1641)-4*X(5461) = 4*X(1648)-X(14444) = 2*X(11053)-5*X(14061) = 2*X(14423)+X(14443)

The reciprocal polarologic center of these triangles is X(690)

X(42344) lies on these lines: {115, 125}, {524, 5103}, {1641, 5461}, {11053, 14061}, {11646, 25328}, {14423, 14443}, {23991, 33921}

X(42344) = isotomic conjugate of X(42370)
X(42344) = barycentric product X(i)*X(j) for these {i, j}: {115, 1648}, {338, 21906}, {351, 23105}, {523, 33919}, {690, 8029}
X(42344) = barycentric quotient X(i)/X(j) for these (i, j): (690, 31614), (1648, 4590)
X(42344) = trilinear product X(i)*X(j) for these {i, j}: {661, 33919}, {1109, 21906}, {1648, 2643}
X(42344) = trilinear quotient X(1648)/X(24041)
X(42344) = tripolar centroid of X(8029)
X(42344) = crosspoint of X(i) and X(j) for these (i, j): {671, 42345}, {1648, 33919}
X(42344) = crosssum of X(110) and X(21906)
X(42344) = X(i)-Ceva conjugate of-X(j) for these (i, j): (115, 14443), (1648, 33919)
X(42344) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (690, 31614), (1648, 4590)
X(42344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (115, 6388, 16278), (115, 15359, 39691)

X(42345) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO SCHROETER

Barycentrics    (b^2-c^2)*(a^4-2*b^2*a^2+c^4-2*b^2*c^2+2*b^4)*(a^4-2*c^2*a^2-2*b^2*c^2+2*c^4+b^4) : :
X(42345) = 9*X(2)-4*X(36955) = X(99)-6*X(8029) = X(148)+4*X(12076) = 4*X(620)-9*X(5466) = 9*X(671)-4*X(9293) = 8*X(10279)-3*X(21166)

The reciprocal polelogic center of these triangles is X(99)

X(42345) lies on the cubic K241 and these lines: {2, 36955}, {99, 8029}, {148, 690}, {523, 14061}, {620, 5466}, {671, 9293}, {10279, 21166}, {21089, 21092}

X(42345) = isotomic conjugate of X(14588)
X(42345) = barycentric product X(i)*X(j) for these {i, j}: {523, 40429}, {1648, 14728}
X(42345) = barycentric quotient X(i)/X(j) for these (i, j): (115, 11123), (512, 20976), (514, 17199), (523, 620), (647, 22085), (661, 17467)
X(42345) = trilinear product X(661)*X(40429)
X(42345) = trilinear quotient X(i)/X(j) for these (i, j): (523, 17467), (656, 22085), (661, 20976), (693, 17199), (850, 20903), (1109, 11123)
X(42345) = trilinear pole of the line {1648, 10278}
X(42345) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(14061)}} and {{A, B, C, X(98), X(3448)}}
X(42345) = Cevapoint of X(523) and X(8029)
X(42345) = crossdifference of every pair of points on line {X(20976), X(22085)}
X(42345) = X(524)-cross conjugate of-X(5466)
X(42345) = X(i)-isoconjugate-of-X(j) for these {i, j}: {110, 17467}, {162, 22085}, {163, 620}, {662, 20976}
X(42345) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (115, 11123), (512, 20976), (514, 17199), (523, 620)

X(42346) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO TANGENTIAL

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

The reciprocal polelogic center of these triangles is X(10159)

X(42346) lies on these lines: {2, 10014}, {6, 1078}, {32, 5012}, {83, 3051}, {213, 23475}, {251, 3203}, {1186, 3224}, {1629, 2207}, {1974, 5039}, {2422, 8870}, {3114, 7760}, {5007, 9468}, {7808, 9463}, {12212, 39674}, {14252, 30435}, {18993, 26461}, {18994, 26454}, {39588, 39872}

X(42346) = complement of the anticomplementary conjugate of X(39)
X(42346) = anticomplement of the complementary conjugate of X(6683)
X(42346) = isogonal conjugate of X(3934)
X(42346) = polar conjugate of X(42394)
X(42346) = barycentric product X(i)*X(j) for these {i, j}: {6, 39968}, {32, 31630}, {83, 31613}
X(42346) = barycentric quotient X(i)/X(j) for these (i, j): (1, 20889), (4, 42394), (31, 17445), (32, 20965), (42, 21022), (58, 17176)
X(42346) = trilinear product X(i)*X(j) for these {i, j}: {31, 39968}, {82, 31613}, {560, 31630}, {1923, 31622}
X(42346) = trilinear quotient X(i)/X(j) for these (i, j): (2, 20889), (6, 17445), (31, 20965), (37, 21022), (48, 22062), (81, 17176)
X(42346) = 1st Saragossa point of X(83)
X(42346) = trilinear pole of the line {669, 2513}
X(42346) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7786)}} and {{A, B, C, X(3), X(5039)}}
X(42346) = Cevapoint of X(i) and X(j) for these (i, j): {6, 3051}, {32, 3203}
X(42346) = X(1207)-cross conjugate of-X(83)
X(42346) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 17445}, {6, 20889}, {37, 17176}, {38, 18092}
X(42346) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 20889), (4, 42394), (31, 17445), (32, 20965)
X(42346) = X(592)-vertex conjugate of-X(1173)
X(42346) = {X(83), X(3051)}-harmonic conjugate of X(38854)

X(42347) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO X-PARABOLA-TANGENTIAL

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

The reciprocal polarologic center of these triangles is X(33906)

X(42347) lies on this line: {1648, 8029}

X(42347) = crosspoint of X(892) and X(42348)

X(42348) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO X-PARABOLA-TANGENTIAL

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

The reciprocal polelogic center of these triangles is X(42349)

X(42348) lies on this line: {115, 33799}

X(42349) = POLELOGIC CENTER OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL TO ANTICOMPLEMENTARY

Barycentrics    (3*a^4-2*(b^2+2*c^2)*a^2+2*b^4-2*b^2*c^2+3*c^4)*(3*a^4-2*(2*b^2+c^2)*a^2+3*b^4-2*b^2*c^2+2*c^4) : :
X(42349) = 5*X(99)+16*X(40511) = 16*X(620)+5*X(40429) = X(4590)+20*X(31274)

The reciprocal polelogic center of these triangles is X(42348)

X(42349) lies on these lines: {99, 40511}, {620, 40429}, {3619, 5967}, {4590, 31274}, {18823, 22247}

X(42349) = isotomic conjugate of X(6722)
X(42349) = trilinear pole of the line {690, 14683}
X(42349) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(468)}} and {{A, B, C, X(115), X(31274)}}
X(42349) = Cevapoint of X(2) and X(620)

X(42350) = POLAROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO X3-ABC REFLECTIONS

Barycentrics    a^12-4*(b^2+c^2)*a^10+(4*b^4+5*b^2*c^2+4*c^4)*a^8+3*(b^6+c^6)*a^6-(b^2-c^2)^2*(8*b^4+9*b^2*c^2+8*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(5*b^4-9*b^2*c^2+5*c^4)*a^2-(b^2-c^2)^6 : :
X(42350) = 6*X(547)-5*X(40331) = 5*X(1656)-4*X(6709) = 5*X(3091)-X(40897)

The reciprocal polarologic center of these triangles is X(95)

X(42350) lies on these lines: {3, 233}, {4, 3164}, {5, 95}, {97, 3078}, {547, 40331}, {1351, 3818}, {1656, 6709}, {2967, 37349}, {3091, 40897}, {6321, 31656}, {9792, 32438}, {10003, 40853}, {13352, 18464}, {14941, 30506}, {19210, 35887}, {34836, 35884}

X(42350) = midpoint of X(4) and X(17035)
X(42350) = reflection of X(i) in X(j) for these (i, j): (3, 233), (95, 5)

X(42351) = POLELOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO X3-ABC REFLECTIONS

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

The reciprocal polelogic center of these triangles is X(40410)

X(42351) lies on these lines: {2, 39243}, {95, 27377}, {140, 287}, {264, 36751}, {297, 40410}, {577, 36948}, {1351, 10003}, {8797, 36412}

X(42351) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(3), X(37067)}}

X(42352) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO EULER

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

The reciprocal polelogic center of these triangles is X(253)

X(42352) lies on these lines: {20, 287}, {253, 297}, {13575, 20213}

X(42352) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(20), X(297)}}

X(42353) = POLAROLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 2nd EULER

Barycentrics    (-a^2+b^2+c^2)^2*(3*a^4+(b^2-c^2)^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(42353) = 5*X(631)-3*X(20792) = X(18437)+2*X(34828)

The reciprocal polarologic center of these triangles is X(141)

X(42353) lies on these lines: {2, 26870}, {3, 66}, {4, 20477}, {5, 53}, {30, 36988}, {114, 122}, {131, 14672}, {182, 441}, {297, 42329}, {343, 418}, {465, 5617}, {466, 5613}, {511, 41005}, {577, 3564}, {631, 20792}, {852, 37648}, {1353, 3284}, {1513, 40822}, {1576, 41729}, {1594, 40681}, {2871, 9967}, {2980, 34002}, {3164, 6530}, {5158, 18583}, {5480, 30258}, {5562, 6751}, {6146, 40947}, {6248, 6823}, {6638, 13567}, {6643, 41761}, {6676, 26880}, {6755, 34836}, {6776, 37188}, {8573, 39571}, {8800, 27352}, {10516, 36751}, {10608, 18396}, {10749, 31656}, {10979, 18358}, {11411, 18953}, {11585, 23333}, {13562, 26899}, {15069, 36748}, {17814, 17849}, {18380, 18404}, {19131, 19156}, {19179, 19212}, {21243, 26906}, {25150, 27353}, {26874, 37636}, {34507, 41008}, {36245, 41362}

X(42353) = midpoint of X(i) and X(j) for these {i, j}: {3, 18437}, {4, 20477}, {5562, 6751}
X(42353) = reflection of X(i) in X(j) for these (i, j): (3, 34828), (53, 5), (8800, 27352)
X(42353) = complement of X(33971)
X(42353) = X(1350)-of-orthic-triangle
X(42353) = X(6)-of-2nd-Euler-triangle
X(42353) = QA-P21 (Reflection of QA-P16 in QA-P1) of quadrangle ABCX(4)
X(42353) = complement of the polar conjugate of X(42313)
X(42353) = barycentric product X(i)*X(j) for these {i, j}: {5, 37188}, {343, 6776}, {418, 40822}
X(42353) = barycentric quotient X(i)/X(j) for these (i, j): (216, 40801), (418, 40799)
X(42353) = trilinear product X(1953)*X(37188)
X(42353) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(27354)}} and {{A, B, C, X(5), X(14376)}}
X(42353) = crossdifference of every pair of points on line {X(2485), X(23286)}
X(42353) = X(i)-complementary conjugate of-X(j) for these (i, j): (255, 15819), (263, 24005)
X(42353) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (216, 40801), (418, 40799)
X(42353) = {X(5562), X(8905)}-harmonic conjugate of X(27354)

X(42354) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 2nd EULER

Barycentrics    (a^6-2*b^2*a^4+(b^2-3*c^2)*b^2*a^2+(b^2-c^2)^2*c^2)*(a^6-2*c^2*a^4-(3*b^2-c^2)*c^2*a^2+(b^2-c^2)^2*b^2)/a^2 : :
Barycentrics    sec(2A + ω) : :

The reciprocal polelogic center of these triangles is X(42355)

X(42354) lies on these lines: {6, 311}, {25, 324}, {76, 2987}, {263, 1352}, {264, 8882}, {1976, 14265}, {5012, 5392}, {30535, 40814}

X(42354) = polar conjugate of X(6403)
X(42354) = barycentric quotient X(i)/X(j) for these (i, j): (4, 6403), (5, 41169)
X(42354) = trilinear quotient X(92)/X(6403)
X(42354) = trilinear pole of the line {512, 13449}
X(42354) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(4), X(41231)}}
X(42354) = Cevapoint of X(338) and X(23878)
X(42354) = X(48)-isoconjugate-of-X(6403)
X(42354) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 6403), (5, 41169)

X(42355) = POLELOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO MEDIAL

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

The reciprocal polelogic center of these triangles is X(42354)

X(42355) lies on these lines: {97, 5392}, {264, 34148}, {311, 1975}, {324, 1993}, {5889, 8795}, {11444, 42333}

X(42355) = isotomic conjugate of X(5889)
X(42355) = barycentric product X(76)*X(22261)
X(42355) = trilinear product X(75)*X(22261)
X(42355) = trilinear pole of the line {18314, 30476}
X(42355) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(8795)}} and {{A, B, C, X(3), X(34148)}}
X(42355) = Cevapoint of X(338) and X(520)

X(42356) = POLAROLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 3rd EULER

Barycentrics    (b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :
X(42356) = X(4)+3*X(38037) = X(7)-9*X(9779) = X(9)+3*X(1699) = X(142)-3*X(3817) = X(382)+3*X(38031) = X(390)+7*X(3832) = X(962)+3*X(38057) = X(1001)-3*X(38037) = X(1537)+3*X(38159) = X(2550)-5*X(3091) = X(2951)-9*X(7988) = X(2951)-5*X(20195) = X(3062)+3*X(6173) = X(3062)+15*X(30308) = X(3579)-3*X(38318) = 9*X(7988)-5*X(20195) = 9*X(9779)+X(16112) = 4*X(9955)-X(25557) = X(15254)+2*X(18483) = X(20330)-3*X(38034)

The reciprocal polarologic center of these triangles is X(141).

X(42356) lies on these lines: {2, 7965}, {4, 1001}, {5, 516}, {7, 11}, {9, 1699}, {12, 390}, {20, 7958}, {119, 381}, {142, 1538}, {144, 6067}, {226, 5572}, {235, 1890}, {354, 41857}, {382, 38031}, {480, 3434}, {495, 30331}, {496, 5542}, {497, 8232}, {518, 946}, {527, 3829}, {546, 18242}, {673, 7384}, {908, 3059}, {954, 1479}, {962, 9710}, {971, 9955}, {1012, 38759}, {1329, 2550}, {1445, 1836}, {1484, 2801}, {1537, 38159}, {1721, 17278}, {1742, 17245}, {2346, 3058}, {2951, 7988}, {3062, 3255}, {3243, 11522}, {3485, 5809}, {3543, 38025}, {3545, 35514}, {3614, 7679}, {3627, 38043}, {3652, 5805}, {3739, 21629}, {3812, 21628}, {3847, 5880}, {3854, 11681}, {3855, 38149}, {3925, 9812}, {3988, 20117}, {4026, 36652}, {4187, 38052}, {4197, 10248}, {4301, 24393}, {4312, 7741}, {4326, 5219}, {4335, 17717}, {4336, 5723}, {4343, 5718}, {4423, 10431}, {4999, 6837}, {5068, 40333}, {5072, 38121}, {5087, 10863}, {5220, 5817}, {5222, 21955}, {5223, 24390}, {5432, 7676}, {5528, 15017}, {5584, 6886}, {5691, 38316}, {5698, 6828}, {5728, 12047}, {5732, 8227}, {5759, 6990}, {5779, 5852}, {5806, 12617}, {5845, 24682}, {5850, 24387}, {5853, 12607}, {5886, 31672}, {5927, 15185}, {6147, 20116}, {6284, 6894}, {6601, 11235}, {6668, 6848}, {6690, 19541}, {6691, 6847}, {6839, 10724}, {6849, 11496}, {6866, 10893}, {6870, 10896}, {6896, 10310}, {6900, 11826}, {7354, 7677}, {7377, 16593}, {7675, 11375}, {7982, 38154}, {7989, 9711}, {8236, 15888}, {8255, 14100}, {8545, 15845}, {8728, 38059}, {8732, 10589}, {9355, 17365}, {9441, 17337}, {9581, 12560}, {10157, 40659}, {10171, 37364}, {10177, 27869}, {10442, 10886}, {10478, 35892}, {10593, 30424}, {11038, 37722}, {12245, 16615}, {12612, 21077}, {12699, 38108}, {13271, 34894}, {13405, 15006}, {13464, 15570}, {13729, 15843}, {15171, 18782}, {15842, 17618}, {15911, 18233}, {15950, 30284}, {16160, 31657}, {17243, 28850}, {17348, 28849}, {17527, 38204}, {18222, 30827}, {18393, 18412}, {18480, 32213}, {19512, 24309}, {21153, 41869}, {21258, 34848}, {25524, 37434}, {30329, 39542}, {30379, 31391}, {30628, 31053}, {31162, 38075}, {31394, 36654}, {36971, 41563}, {36990, 38048}, {36991, 38053}

X(42356) = midpoint of X(i) and X(j) for these {i, j}: {4, 1001}, {7, 16112}, {4301, 24393}, {5880, 11372}, {13271, 34894}, {20116, 31871}, {22793, 31658}
X(42356) = reflection of X(i) in X(j) for these (i, j): (3826, 5), (15570, 13464)
X(42356) = complement of X(11495)
X(42356) = X(6)-of-3rd-Euler-triangle
X(42356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 38037, 1001), (7, 7678, 11), (144, 11680, 6067), (1699, 8226, 2886), (1699, 41858, 8226), (2951, 7988, 20195), (3817, 8727, 3816), (4301, 38158, 24393), (7678, 30311, 7), (9779, 10883, 11), (11372, 38150, 5880), (12558, 12571, 5), (14100, 17605, 21617), (14100, 21617, 8255), (30306, 30307, 11), (30309, 30310, 11), (31555, 31556, 5)

X(42357) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 3rd EULER

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

The reciprocal polelogic center of these triangles is X(190)

X(42357) lies on these lines: {}

X(42357) = trilinear pole of the line {2140, 3817}

X(42358) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 4th EULER

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

The reciprocal polelogic center of these triangles is X(75)

X(42358) lies on these lines: {3995, 17018}, {4687, 17786}

X(42358) = isotomic conjugate of the anticomplement of X(4044)
X(42358) = trilinear pole of the line {4129, 6005}
X(42358) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(749)}} and {{A, B, C, X(75), X(3995)}}

X(42359) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 5th EULER

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

The reciprocal polelogic center of these triangles is X(6)

X(42359) lies on these lines: {39, 1975}, {141, 33734}, {308, 31360}, {1843, 9308}, {7754, 27375}, {32451, 42299}

X(42359) = isotomic conjugate of X(32451)
X(42359) = trilinear pole of the line {3005, 30476}
X(42359) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(11174)}} and {{A, B, C, X(4), X(308)}}

X(42360) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO EXCENTERS-MIDPOINTS

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

The reciprocal polelogic center of these triangles is X(42361)

X(42360) lies on these lines: {1999, 4460}, {2403, 5905}, {17315, 18743}

X(42360) = isotomic conjugate of the anticomplement of X(30568)
X(42360) = barycentric quotient X(1)/X(11512)
X(42360) = trilinear quotient X(2)/X(11512)
X(42360) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(34260)}} and {{A, B, C, X(2), X(145)}}
X(42360) = X(6)-isoconjugate-of-X(11512)
X(42360) = X(1)-reciprocal conjugate of-X(11512)

X(42361) = POLELOGIC CENTER OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO MEDIAL

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

The reciprocal polelogic center of these triangles is X(42360)

X(42361) lies on the circumhyperbola dual of Yff parabola and these lines: {2, 24181}, {7, 3174}, {329, 673}, {345, 36807}, {962, 39732}, {1088, 6604}, {1440, 9436}, {9776, 27475}, {11037, 39734}, {18228, 42318}, {24152, 24155}, {24153, 24154}, {26015, 36620}

X(42361) = anticomplement of X(24771)
X(42361) = isogonal conjugate of X(21002)
X(42361) = isotomic conjugate of X(36845)
X(42361) = barycentric quotient X(i)/X(j) for these (i, j): (1, 16572), (3, 22153), (7, 8732), (9, 3174), (10, 21096), (75, 20946)
X(42361) = trilinear quotient X(i)/X(j) for these (i, j): (2, 16572), (8, 3174), (63, 22153), (76, 20946), (85, 8732), (321, 21096)
X(42361) = intersection, other than A,B,C, of circumhyperbola dual of Yff parabola and conic {{A, B, C, X(4), X(39959)}}
X(42361) = Cevapoint of X(i) and X(j) for these (i, j): {2, 20015}, {522, 4904}
X(42361) = X(200)-cross conjugate of-X(2)
X(42361) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 16572}, {19, 22153}, {32, 20946}, {41, 8732}
X(42361) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 16572), (3, 22153), (7, 8732), (9, 3174)

X(42362) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO FEUERBACH

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

The reciprocal polelogic center of these triangles is X(42363)

X(42362) lies on these lines: {}

X(42362) = isotomic conjugate of the anticomplement of X(7265)
X(42362) = trilinear pole of the line {442, 1155}
X(42362) = Cevapoint of X(523) and X(1100)

X(42363) = POLELOGIC CENTER OF THESE TRIANGLES: FEUERBACH TO MEDIAL

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

The reciprocal polelogic center of these triangles is X(42362)

X(42363) lies on these lines: {4436, 16680}, {4623, 17166}, {17159, 21295}, {22280, 22311}

X(42363) = isogonal conjugate of X(16874)
X(42363) = isotomic conjugate of X(17166)
X(42363) = barycentric quotient X(i)/X(j) for these (i, j): (10, 22044), (75, 18154), (514, 23823)
X(42363) = trilinear quotient X(i)/X(j) for these (i, j): (76, 18154), (321, 22044), (693, 23823)
X(42363) = trilinear pole of the line {1211, 3912}
X(42363) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(4623)}} and {{A, B, C, X(7), X(4594)}}
X(42363) = Cevapoint of X(i) and X(j) for these (i, j): {512, 3666}, {513, 17045}, {522, 21233}, {523, 3739}
X(42363) = X(i)-isoconjugate-of-X(j) for these {i, j}: {32, 18154}, {692, 23823}, {1333, 22044}
X(42363) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (10, 22044), (75, 18154), (514, 23823)

X(42364) = POLAROLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 2nd HATZIPOLAKIS

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

X(42364) lies on this line: {1119, 17054}

X(42364) = barycentric product X(1119)*X(14743)
X(42364) = trilinear product X(1435)*X(14743)

X(42365) = POLAROLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO LEMOINE

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

The reciprocal polarologic center of these triangles is X(20582)

X(42365) lies on these lines: {597, 598}, {20583, 35138}

X(42365) = barycentric product X(598)*X(14762)
X(42365) = pole of the trilinear polar of X(42366) with respect to Kiepert hyperbola
X(42365) = X(598)-Ceva conjugate of-X(32069)

X(42366) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO LEMOINE

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

The reciprocal polelogic center of these triangles is X(42367)

X(42366) lies on these lines: {}

X(42366) = Cevapoint of X(i) and X(j) for these (i, j): {523, 42365}, {598, 17436}

X(42367) = POLELOGIC CENTER OF THESE TRIANGLES: LEMOINE TO MEDIAL

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

The reciprocal polelogic center of these triangles is X(42366)

X(42367) lies on the Steiner circumellipse and these lines: {99, 12074}, {141, 671}, {892, 4576}, {3228, 3329}, {4577, 5468}, {7779, 18823}, {9146, 35138}, {14764, 41624}

X(42367) = isogonal conjugate of the Gibert-circumtangential conjugate of X(12074)
X(42367) = isotomic conjugate of X(12073)
X(42367) = barycentric product X(i)*X(j) for these {i, j}: {76, 12074}, {99, 10302}, {670, 39389}
X(42367) = barycentric quotient X(i)/X(j) for these (i, j): (99, 597), (110, 5008), (249, 35357), (599, 17436), (648, 10301), (670, 26235)
X(42367) = trilinear product X(i)*X(j) for these {i, j}: {75, 12074}, {662, 10302}, {799, 39389}
X(42367) = trilinear quotient X(i)/X(j) for these (i, j): (662, 5008), (799, 597), (811, 10301)
X(42367) = trilinear pole of the line {2, 5355}
X(42367) = intersection, other than A,B,C, of Steiner circumellipse and conic {{A, B, C, X(141), X(4576)}}
X(42367) = Cevapoint of X(i) and X(j) for these (i, j): {2, 12073}, {99, 9146}, {523, 20582}, {525, 10300}
X(42367) = X(599)-cross conjugate of-X(4590)
X(42367) = X(i)-isoconjugate-of-X(j) for these {i, j}: {597, 798}, {661, 5008}, {810, 10301}
X(42367) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (99, 597), (110, 5008), (249, 35357), (599, 17436)

X(42368) = POLAROLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO MACBEATH

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

The reciprocal polarologic center of these triangles is X(140)

X(42368) lies on these lines: {5, 264}, {30, 9291}, {140, 276}, {297, 324}, {339, 14978}, {546, 6528}, {3933, 18022}, {5305, 16081}, {8795, 41008}, {40207, 40684}

X(42368) = isotomic conjugate of the isogonal conjugate of X(42400)
X(42368) = polar conjugate of the isogonal conjugate of X(14767)
X(42368) = barycentric product X(i)*X(j) for these {i, j}: {76, 42400}, {264, 14767}, {276, 11197}
X(42368) = trilinear product X(i)*X(j) for these {i, j}: {75, 42400}, {92, 14767}
X(42368) = X(264)-Ceva conjugate of-X(11197)
X(42368) = X(264)-Waw conjugate of-X(40684)
X(42368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (264, 18027, 5), (276, 16089, 140)

X(42369) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO MACBEATH

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

The reciprocal polelogic center of these triangles is X(18831)

X(42369) lies on this line: {6331, 42401}

X(42369) = isotomic conjugate of the isogonal conjugate of X(42401)
X(42369) = barycentric product X(76)*X(42401)
X(42369) = barycentric quotient X(i)/X(j) for these (i, j): (276, 32320), (850, 41219)
X(42369) = trilinear product X(75)*X(42401)
X(42369) = trilinear pole of the line {264, 11197}
X(42369) = Cevapoint of X(i) and X(j) for these (i, j): {264, 17434}, {525, 42368}
X(42369) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (276, 32320), (850, 41219)

X(42370) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO STEINER

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

The reciprocal polelogic center of these triangles is X(892)

X(42370) lies on these lines: {620, 4590}, {4600, 21047}, {9170, 31614}, {9293, 14588}

X(42370) = isotomic conjugate of X(42344)
X(42370) = barycentric product X(892)*X(31614)
X(42370) = barycentric quotient X(i)/X(j) for these (i, j): (99, 33919), (249, 21906), (691, 22260), (892, 8029)
X(42370) = trilinear quotient X(799)/X(33919)
X(42370) = trilinear pole of the line {99, 11123}
X(42370) = Cevapoint of X(i) and X(j) for these (i, j): {99, 1648}, {524, 14588}
X(42370) = X(798)-isoconjugate-of-X(33919)
X(42370) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (99, 33919), (249, 21906), (691, 22260), (892, 8029)

X(42371) = POLELOGIC CENTER OF THESE TRIANGLES: SYMMEDIAL TO MEDIAL

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

The reciprocal polelogic center of these triangles is X(827)

X(42371) lies on the Steiner circumellipse and these lines: {32, 39082}, {39, 9495}, {76, 14970}, {83, 3225}, {99, 689}, {190, 37204}, {308, 3228}, {315, 40359}, {316, 18901}, {626, 14946}, {671, 40016}, {782, 880}, {826, 18828}, {827, 9063}, {1502, 7818}, {3112, 18826}, {3934, 31622}, {4562, 4602}, {4586, 4593}, {4630, 33515}, {6528, 42395}, {18827, 18833}, {41073, 41209}

X(42371) = reflection of X(i) in X(j) for these (i, j): (32, 39082), (39, 39076), (14946, 626)
X(42371) = anticomplement of the complementary conjugate of X(42291)
X(42371) = isogonal conjugate of X(9494)
X(42371) = isotomic conjugate of X(688)
X(42371) = barycentric product X(i)*X(j) for these {i, j}: {3, 42395}, {75, 37204}, {76, 689}, {83, 4609}, {99, 40016}, {308, 670}
X(42371) = barycentric quotient X(i)/X(j) for these (i, j): (75, 2084), (76, 3005), (82, 1924), (83, 669), (99, 3051), (110, 41331)
X(42371) = trilinear product X(i)*X(j) for these {i, j}: {2, 37204}, {48, 42395}, {75, 689}, {76, 4593}, {82, 4609}, {83, 4602}
X(42371) = trilinear quotient X(i)/X(j) for these (i, j): (76, 2084), (82, 9426), (83, 1924), (99, 1923), (308, 798), (561, 3005)
X(42371) = trilinear pole of the line {2, 308}
X(42371) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(9062)}} and {{A, B, C, X(6), X(36881)}}
X(42371) = Cevapoint of X(i) and X(j) for these (i, j): {2, 688}, {512, 3934}, {626, 826}, {670, 4609}
X(42371) = X(i)-cross conjugate of-X(j) for these (i, j): (512, 31622), (670, 689), (688, 2)
X(42371) = X(i)-isoconjugate-of-X(j) for these {i, j}: {32, 2084}, {38, 9426}, {39, 1924}, {512, 1923}
X(42371) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (75, 2084), (76, 3005), (82, 1924), (83, 669)

X(42372) = POLELOGIC CENTER OF THESE TRIANGLES: MEDIAL TO YFF CONTACT

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

The reciprocal polelogic center of these triangles is X(4555)

X(42372) lies on these lines: {1016, 4422}, {4555, 6635}, {4589, 4622}, {4607, 32665}, {4986, 7035}, {6551, 8709}

X(42372) = isotomic conjugate of X(24188)
X(42372) = barycentric product X(190)*X(6635)
X(42372) = barycentric quotient X(i)/X(j) for these (i, j): (101, 8661), (190, 6550), (765, 2087), (901, 21143), (1016, 1647)
X(42372) = trilinear product X(i)*X(j) for these {i, j}: {100, 6635}, {668, 6551}, {1016, 5376}
X(42372) = trilinear quotient X(i)/X(j) for these (i, j): (100, 8661), (668, 6550), (901, 8027)
X(42372) = trilinear pole of the line {190, 6546}
X(42372) = Cevapoint of X(i) and X(j) for these (i, j): {190, 1647}, {519, 32094}
X(42372) = X(i)-isoconjugate-of-X(j) for these {i, j}: {513, 8661}, {667, 6550}, {764, 1960}, {900, 8027}
X(42372) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (101, 8661), (190, 6550), (765, 2087), (901, 21143)

X(42373) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO ANTI-EXCENTERS-REFLECTIONS

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

The reciprocal polelogic center of these triangles is X(393)

X(42373) lies on these lines: {6, 32000}, {263, 12294}, {1976, 15258}, {37665, 42330}

X(42373) = polar conjugate of the anticomplement of X(1350)
X(42373) = barycentric quotient X(25)/X(41266)
X(42373) = trilinear quotient X(19)/X(41266)
X(42373) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(4), X(42330)}}
X(42373) = X(63)-isoconjugate-of-X(41266)
X(42373) = X(25)-reciprocal conjugate of-X(41266)

X(42374) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO EULER

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

The reciprocal polelogic center of these triangles is X(6)

X(42374) lies on these lines: {216, 9308}, {2351, 19189}

X(42374) = polar conjugate of X(42329)
X(42374) = barycentric quotient X(4)/X(42329)
X(42374) = trilinear quotient X(92)/X(42329)
X(42374) = trilinear pole of the line {15451, 16229}
X(42374) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(8795)}} and {{A, B, C, X(4), X(37067)}}
X(42374) = X(48)-isoconjugate-of-X(42329)
X(42374) = X(4)-reciprocal conjugate of-X(42329)

X(42375) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO 2nd EULER

Barycentrics    (S^4-(2*R^2*(6*R^2-5*SW+SC)-SC^2+2*SW^2)*S^2-(2*R^2-SW)*(4*R^2-SW)*(SW*(2*R^2-SW)+SC^2))*(S^4-(2*R^2*(6*R^2-5*SW+SB)-SB^2+2*SW^2)*S^2-(2*R^2-SW)*(4*R^2-SW)*(SW*(2*R^2-SW)+SB^2)) : :

The reciprocal polelogic center of these triangles is X(42376)

X(42375) lies on these lines: {}

X(42376) = POLELOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO ORTHIC

Barycentrics    (S^2-2*(4*R^2-SB-2*SW)*R^2-SW^2)*(S^2-2*(4*R^2-SC-2*SW)*R^2-SW^2)*SB^2*SC^2 : :

The reciprocal polelogic center of these triangles is X(42375)

X(42376) lies on this line: {393, 6193}

X(42376) = Cevapoint of X(139) and X(12077)

X(42377) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO 5th EULER

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

The reciprocal polelogic center of these triangles is X(34208)

X(42377) lies on these lines: {4, 10983}, {297, 34208}, {6353, 6531}

X(42377) = polar conjugate of X(37667)
X(42377) = barycentric quotient X(4)/X(37667)
X(42377) = trilinear quotient X(92)/X(37667)
X(42377) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(93)}} and {{A, B, C, X(6), X(10983)}}
X(42377) = X(48)-isoconjugate-of-X(37667)
X(42377) = X(4)-reciprocal conjugate of-X(37667)

X(42378) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO EXTOUCH

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

The reciprocal polarologic center of these triangles is X(42379)

X(42378) lies on these lines: {8, 210}, {65, 16086}, {72, 13532}, {145, 33118}, {3246, 3621}, {3703, 6737}, {3712, 12437}, {3717, 10950}, {3932, 41575}, {3962, 7270}, {3967, 5086}, {4030, 5837}, {4126, 5795}, {4387, 12625}, {5015, 31165}, {8258, 37539}, {34406, 42379}

X(42378) = pole of the trilinear polar of X(42380) with respect to Feuerbach hyperbola
X(42378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (8, 1265, 1837), (1265, 1837, 4009)

X(42379) = POLAROLOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO ORTHIC

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

The reciprocal polarologic center of these triangles is X(42378)

X(42379) lies on these lines: {4, 65}, {29, 17605}, {53, 1839}, {225, 1852}, {1785, 6284}, {1838, 7354}, {1842, 37376}, {1940, 7541}, {3585, 39529}, {3683, 17555}, {3962, 5081}, {5307, 40271}, {7510, 12047}, {7534, 9579}, {10895, 39585}, {17923, 37605}, {34406, 42378}

X(42379) = pole of the trilinear polar of X(42381) with respect to Feuerbach hyperbola
X(42379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 1118, 1837), (4, 42385, 42387), (1940, 7541, 17606)

X(42380) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO EXTOUCH

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

The reciprocal polelogic center of these triangles is X(42381)

X(42380) lies on these lines: {}

X(42380) = barycentric quotient X(i)/X(j) for these (i, j): (190, 36570), (646, 3772)
X(42380) = trilinear product X(646)*X(40436)
X(42380) = trilinear quotient X(i)/X(j) for these (i, j): (646, 3924), (668, 36570)
X(42380) = Cevapoint of X(650) and X(42378)
X(42380) = X(667)-isoconjugate-of-X(36570)
X(42380) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (190, 36570), (646, 3772)

X(42381) = POLELOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO ORTHIC

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

The reciprocal polelogic center of these triangles is X(42380)

X(42381) lies on these lines: {}

X(42381) = Cevapoint of X(650) and X(42379)

X(42382) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO 2nd HATZIPOLAKIS

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

The reciprocal polarologic center of these triangles is X(5101)

X(42382) lies on these lines: {279, 2355}, {479, 1119}

X(42382) = barycentric product X(278)*X(30623)
X(42382) = trilinear product X(34)*X(30623)

X(42383) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO 2nd HATZIPOLAKIS

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

The reciprocal polelogic center of these triangles is X(42384)

X(42383) lies on these lines: {}

X(42384) = POLELOGIC CENTER OF THESE TRIANGLES: 2nd HATZIPOLAKIS TO ORTHIC

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

The reciprocal polelogic center of these triangles is X(42383)

X(42384) lies on these lines: {646, 653}, {648, 7258}, {1978, 13149}, {16082, 30701}

X(42384) = barycentric quotient X(i)/X(j) for these (i, j): (100, 1473), (107, 4211), (190, 7289), (321, 21107), (644, 7124), (646, 27509)
X(42384) = trilinear product X(i)*X(j) for these {i, j}: {646, 1041}, {1897, 30701}
X(42384) = trilinear quotient X(i)/X(j) for these (i, j): (190, 1473), (313, 21107), (646, 1040), (668, 7289), (823, 4211), (1018, 22363)
X(42384) = trilinear pole of the line {4, 341}
X(42384) = X(i)-isoconjugate-of-X(j) for these {i, j}: {614, 22383}, {649, 1473}, {667, 7289}, {822, 4211}
X(42384) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (100, 1473), (107, 4211), (190, 7289), (321, 21107)

X(42385) = POLAROLOGIC CENTER OF THESE TRIANGLES: INCENTRAL TO ORTHIC

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

The reciprocal polarologic center of these triangles is X(2646)

X(42385) lies on these lines: {1, 7524}, {4, 65}, {11, 133}, {12, 1785}, {29, 243}, {33, 1867}, {53, 1826}, {55, 39585}, {92, 3057}, {210, 318}, {225, 235}, {278, 11376}, {354, 1895}, {407, 42069}, {412, 1940}, {942, 1784}, {1842, 1852}, {1872, 41538}, {2262, 6520}, {2264, 8748}, {2654, 2658}, {3486, 7518}, {5125, 17606}, {6708, 40946}, {7497, 22760}, {7510, 10572}, {7531, 37600}, {7952, 17718}, {10895, 39531}, {16141, 31902}, {17728, 40836}, {22768, 37393}

X(42385) = polar conjugate of the isotomic conjugate of X(6708)
X(42385) = barycentric product X(i)*X(j) for these {i, j}: {4, 6708}, {92, 2654}, {1896, 18592}
X(42385) = trilinear product X(i)*X(j) for these {i, j}: {4, 2654}, {19, 6708}, {158, 40946}, {1896, 2658}
X(42385) = Zosma transform of X(73)
X(42385) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(2654)}} and {{A, B, C, X(65), X(1896)}}
X(42385) = pole of the trilinear polar of X(15352) with respect to Feuerbach hyperbola
X(42385) = crossdifference of every pair of points on line {X(23187), X(36054)}
X(42385) = crosspoint of X(4) and X(1896)
X(42385) = crosssum of X(3) and X(22341)
X(42385) = X(4)-Waw conjugate of-X(40950)
X(42385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 158, 65), (4, 1118, 1836), (4, 1857, 1837), (4, 40149, 1888), (29, 243, 2646), (412, 1940, 1155), (1785, 39574, 12), (42379, 42387, 4)

X(42386) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO INTOUCH

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

The reciprocal polarologic center of these triangles is X(42387)

X(42386) lies on these lines: {7, 354}, {226, 10136}, {658, 17605}, {3474, 10004}, {3706, 7055}, {4114, 10481}, {10360, 10401}, {34398, 42387}

X(42386) = pole of the trilinear polar of X(42388) with respect to Feuerbach hyperbola

X(42387) = POLAROLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO ORTHIC

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

The reciprocal polarologic center of these triangles is X(42386)

X(42387) lies on these lines: {4, 65}, {53, 1886}, {55, 39531}, {225, 10151}, {243, 17605}, {412, 17606}, {3583, 39529}, {6284, 39574}, {7551, 37600}, {12953, 39585}, {34398, 42386}

X(42387) = Zosma transform of X(20277)
X(42387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 1857, 1836), (4, 42385, 42379)

X(42388) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO INTOUCH

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

The reciprocal polelogic center of these triangles is X(42389)

X(42388) lies on these lines: {}

X(42388) = barycentric quotient X(479)/X(2520)
X(42388) = Cevapoint of X(650) and X(42386)
X(42388) = X(479)-reciprocal conjugate of-X(2520)

X(42389) = POLELOGIC CENTER OF THESE TRIANGLES: INTOUCH TO ORTHIC

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

The reciprocal polelogic center of these triangles is X(42388)

X(42389) lies on these lines: {}

X(42389) = Cevapoint of X(650) and X(42387)

X(42390) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO LEMOINE

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

The reciprocal polarologic center of these triangles is X(42391)

X(42390) lies on this line: {597, 598}

X(42390) = pole of the trilinear polar of X(42392) with respect to Kiepert hyperbola

X(42391) = POLAROLOGIC CENTER OF THESE TRIANGLES: LEMOINE TO ORTHIC

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

The reciprocal polarologic center of these triangles is X(42390)

X(42391) lies on these lines: {4, 6}, {3054, 37196}, {3199, 3861}, {3815, 18386}, {3853, 27371}, {18424, 37458}, {37935, 39601}

X(42391) = pole of the trilinear polar of X(42393) with respect to Kiepert hyperbola

X(42392) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO LEMOINE

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

The reciprocal polelogic center of these triangles is X(42393)

X(42392) lies on these lines: {}

X(42392) = Cevapoint of X(523) and X(42390)

X(42393) = POLELOGIC CENTER OF THESE TRIANGLES: LEMOINE TO ORTHIC

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

The reciprocal polelogic center of these triangles is X(42392)

X(42393) lies on these lines: {}

X(42393) = Cevapoint of X(523) and X(42391)

X(42394) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO MACBEATH

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

The reciprocal polarologic center of these triangles is X(428)

X(42394) lies on these lines: {264, 305}, {290, 11245}, {297, 324}, {428, 17984}, {5133, 23962}, {14569, 18027}, {14768, 39931}, {37439, 40822}

X(42394) = polar conjugate of X(42346)
X(42394) = barycentric product X(i)*X(j) for these {i, j}: {92, 20889}, {264, 3934}, {1235, 18092}
X(42394) = barycentric quotient X(i)/X(j) for these (i, j): (4, 42346), (264, 39968), (427, 31613)
X(42394) = trilinear product X(i)*X(j) for these {i, j}: {4, 20889}, {92, 3934}, {264, 17445}, {286, 21022}
X(42394) = trilinear quotient X(92)/X(42346)
X(42394) = intersection, other than A,B,C, of conics {{A, B, C, X(305), X(3934)}} and {{A, B, C, X(325), X(20965)}}
X(42394) = X(48)-isoconjugate-of-X(42346)
X(42394) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 42346), (264, 39968), (427, 31613)
X(42394) = pole wrt polar circle of trilinear polar of X(42346) (line X(669)X(2513))
X(42394) = {X(264), X(18022)}-harmonic conjugate of X(427)

X(42395) = POLELOGIC CENTER OF THESE TRIANGLES: ORTHIC TO MACBEATH

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

The reciprocal polelogic center of these triangles is X(42396)

X(42395) lies on these lines: {689, 22456}, {6528, 42371}

X(42395) = polar conjugate of X(9494)
X(42395) = barycentric product X(i)*X(j) for these {i, j}: {264, 42371}, {689, 18022}
X(42395) = barycentric quotient X(i)/X(j) for these (i, j): (4, 9494), (264, 688), (308, 3049), (648, 41331), (670, 20775), (689, 184)
X(42395) = trilinear product X(i)*X(j) for these {i, j}: {92, 42371}, {264, 37204}, {689, 1969}, {811, 40016}
X(42395) = trilinear quotient X(i)/X(j) for these (i, j): (92, 9494), (689, 9247), (811, 41331)
X(42395) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 9494}, {688, 9247}, {810, 41331}
X(42395) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 9494), (264, 688), (308, 3049), (648, 41331)

X(42396) = POLELOGIC CENTER OF THESE TRIANGLES: MACBEATH TO ORTHIC

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

The reciprocal polelogic center of these triangles is X(42395)

X(42396) lies on the circumconic with center X(1249) (the conic {{A,B,C,X(107),X(648)}}) and these lines: {83, 16080}, {107, 827}, {112, 689}, {251, 324}, {297, 40850}, {427, 9483}, {648, 4577}, {653, 4599}, {685, 4630}, {1799, 6330}, {4580, 15459}, {6336, 31915}, {10301, 17983}, {17409, 18022}, {18020, 35325}, {23962, 36415}

X(42396) = isotomic conjugate of X(2525)
X(42396) = polar conjugate of X(826)
X(42396) = barycentric product X(i)*X(j) for these {i, j}: {4, 4577}, {19, 4593}, {25, 689}, {82, 811}, {83, 648}, {92, 4599}
X(42396) = barycentric quotient X(i)/X(j) for these (i, j): (4, 826), (25, 3005), (28, 2530), (82, 656), (83, 525), (99, 3933)
X(42396) = trilinear product X(i)*X(j) for these {i, j}: {4, 4599}, {19, 4577}, {25, 4593}, {82, 648}, {83, 162}, {92, 827}
X(42396) = trilinear quotient X(i)/X(j) for these (i, j): (19, 3005), (25, 2084), (27, 2530), (28, 21123), (82, 647), (83, 656)
X(42396) = Zosma transform of X(39336)
X(42396) = trilinear pole of the line {4, 83}
X(42396) = intersection, other than A,B,C, of conics {{A, B, C, X(25), X(35325)}} and {{A, B, C, X(107), X(648)}}
X(42396) = Cevapoint of X(i) and X(j) for these (i, j): {83, 4580}, {112, 648}, {428, 2501}, {523, 7745}
X(42396) = X(i)-cross conjugate of-X(j) for these (i, j): (25, 18020), (264, 23582), (827, 4577)
X(42396) = X(i)-isoconjugate-of-X(j) for these {i, j}: {38, 647}, {39, 656}, {48, 826}, {63, 3005}
X(42396) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 826), (25, 3005), (28, 2530), (82, 656)
X(42396) = X(2)-vertex conjugate of-X(4630)

X(42397) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO MANDART-INCIRCLE

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

The reciprocal polarologic center of these triangles is X(42385)

X(42397) lies on these lines: {11, 6022}, {55, 213}, {144, 145}, {211, 766}, {497, 17137}, {1912, 4162}, {2646, 40432}, {3022, 10544}, {3271, 4875}, {3688, 4520}, {3693, 4531}, {5836, 20863}

X(42397) = barycentric product X(55)*X(26562)
X(42397) = trilinear product X(41)*X(26562)
X(42397) = pole of the trilinear polar of X(670) with respect to Feuerbach hyperbola
X(42397) = crosssum of X(1402) and X(6180)
X(42397) = X(670)-Ceva conjugate of-X(650)
X(42397) = {X(3056), X(23497)}-harmonic conjugate of X(3057)

X(42398) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO STEINER

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

The reciprocal polarologic center of these triangles is X(42399)

X(42398) lies on these lines: {99, 3566}, {524, 2076}, {9182, 9218}

X(42399) = POLAROLOGIC CENTER OF THESE TRIANGLES: STEINER TO ORTHIC

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

The reciprocal polarologic center of these triangles is X(42398)

X(42399) lies on these lines: {4, 3566}, {133, 16188}, {512, 39533}, {523, 10151}, {1598, 34952}, {2501, 42403}

X(42399) = polar conjugate of the isotomic conjugate of X(14341)
X(42399) = barycentric product X(4)*X(14341)
X(42399) = trilinear product X(19)*X(14341)

X(42400) = POLAROLOGIC CENTER OF THESE TRIANGLES: SYMMEDIAL TO ORTHIC

Barycentrics    ((b^2+c^2)*a^6-2*(b^4+b^2*c^2+c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^2*b^2*c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Trilinears    (sec A) (2 csc 2A + csc 2B + csc 2C) : :

The reciprocal polarologic center of these triangles is X(13366)

X(42400) lies on these lines: {2, 26895}, {3, 37871}, {4, 51}, {5, 26905}, {25, 9756}, {53, 232}, {115, 138}, {140, 10184}, {184, 33971}, {186, 19192}, {262, 7378}, {264, 3917}, {275, 13366}, {324, 511}, {373, 15466}, {428, 8902}, {436, 1495}, {467, 21243}, {1199, 4994}, {1216, 14978}, {1352, 37192}, {1594, 6750}, {2450, 27371}, {3199, 37988}, {3819, 40684}, {5133, 39569}, {5480, 14569}, {5943, 30506}, {6146, 35717}, {6248, 37174}, {6748, 11245}, {6755, 13567}, {8795, 21638}, {8884, 13367}, {10151, 16311}, {11197, 14767}, {11424, 41365}, {21849, 35360}, {23719, 32767}, {30739, 37873}, {32165, 35887}, {33843, 39906}, {34965, 36412}

X(42400) = isogonal conjugate of the isotomic conjugate of X(42368)
X(42400) = polar conjugate of the isotomic conjugate of X(14767)
X(42400) = barycentric product X(i)*X(j) for these {i, j}: {4, 14767}, {6, 42368}, {275, 11197}
X(42400) = trilinear product X(i)*X(j) for these {i, j}: {19, 14767}, {31, 42368}
X(42400) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(31365)}} and {{A, B, C, X(51), X(8795)}}
X(42400) = pole of the trilinear polar of X(42401) with respect to Jerabek hyperbola
X(42400) = crosspoint of X(4) and X(8795)
X(42400) = crosssum of X(3) and X(418)
X(42400) = X(4)-Waw conjugate of-X(6748)
X(42400) = X(42)-of-orthic-triangle if ABC is acute
X(42400) = pole wrt polar circle of line X(520)X(31296)
X(42400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 42329, 26907), (4, 2052, 51), (4, 13450, 10110), (53, 427, 6747), (275, 41204, 13366), (436, 1629, 1495)

X(42401) = POLELOGIC CENTER OF THESE TRIANGLES: SYMMEDIAL TO ORTHIC

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

The reciprocal polelogic center of these triangles is X(933)

X(42401) lies on these lines: {648, 42405}, {6331, 42369}, {8794, 16081}

X(42401) = isogonal conjugate of the isotomic conjugate of X(42369)
X(42401) = polar conjugate of the complementary conjugate of X(38976)
X(42401) = barycentric product X(i)*X(j) for these {i, j}: {6, 42369}, {276, 15352}
X(42401) = barycentric quotient X(i)/X(j) for these (i, j): (107, 418), (275, 32320), (393, 42293), (523, 41219), (933, 23606), (1093, 15451)
X(42401) = trilinear product X(i)*X(j) for these {i, j}: {31, 42369}, {158, 42405}, {276, 36126}, {811, 8794}, {823, 8795}
X(42401) = trilinear quotient X(i)/X(j) for these (i, j): (158, 42293), (823, 418), (1577, 41219)
X(42401) = trilinear pole of the line {4, 6752}
X(42401) = Cevapoint of X(i) and X(j) for these (i, j): {4, 42293}, {647, 42400}
X(42401) = X(i)-isoconjugate-of-X(j) for these {i, j}: {163, 41219}, {255, 42293}, {418, 822}
X(42401) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (107, 418), (275, 32320), (393, 42293), (523, 41219)

X(42402) = POLAROLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO YFF CONTACT

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

The reciprocal polarologic center of these triangles is X(42403)

X(42402) lies on this line: {6542, 32094}

X(42403) = POLAROLOGIC CENTER OF THESE TRIANGLES: YFF CONTACT TO ORTHIC

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

The reciprocal polarologic center of these triangles is X(42402)

X(42403) lies on these lines: {4, 38360}, {2501, 42399}, {3798, 6994}, {4024, 39532}, {6590, 16228}

X(42404) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ABC WRT ANTI-WASAT

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

The correspondent reciprocal tripole of these triangles is X(42405)

X(42404) lies on these lines: {}

X(42404) = trilinear pole of the line {52, 6751}
X(42404) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(35360)}} and {{A, B, C, X(110), X(14570)}}
X(42404) = Cevapoint of X(5) and X(42293)

X(42405) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ANTI-WASAT WRT ABC

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

The correspondent reciprocal tripole of these triangles is X(42404)

X(42405) lies on the MacBeath circumconic and these lines: {110, 6528}, {275, 287}, {276, 4993}, {648, 42401}, {895, 8795}, {933, 22456}, {2987, 8794}, {4558, 6331}, {35360, 41208}

X(42405) = complement of the anticomplementary conjugate of X(42331)
X(42405) = isogonal conjugate of X(42293)
X(42405) = isotomic conjugate of X(17434)
X(42405) = polar conjugate of X(15451)
X(42405) = barycentric product X(i)*X(j) for these {i, j}: {76, 16813}, {95, 6528}, {99, 8795}, {107, 34384}, {264, 18831}, {275, 6331}
X(42405) = barycentric quotient X(i)/X(j) for these (i, j): (4, 15451), (5, 34983), (54, 39201), (95, 520), (97, 32320), (99, 5562)
X(42405) = trilinear product X(i)*X(j) for these {i, j}: {75, 16813}, {92, 18831}, {95, 823}, {162, 276}, {255, 42401}, {275, 811}
X(42405) = trilinear quotient X(i)/X(j) for these (i, j): (92, 15451), (95, 822), (107, 2179), (162, 217), (275, 810), (276, 656)
X(42405) = orthocorrespondent of X(130)
X(42405) = trilinear pole of the line {3, 95}
X(42405) = intersection, other than A,B,C, of conic {{A, B, C, X(2), X(35360)}} and MacBeath circumconic
X(42405) = Cevapoint of X(i) and X(j) for these (i, j): {264, 18314}, {648, 6528}, {850, 40684}
X(42405) = X(648)-cross conjugate of-X(18831)
X(42405) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 15451}, {51, 822}, {216, 810}, {217, 656}
X(42405) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 15451), (5, 34983), (54, 39201), (95, 520)

X(42406) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ABC WRT ARIES

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

The correspondent reciprocal tripole of these triangles is X(42407)

X(42406) lies on these lines: {2, 311}, {6, 6393}, {114, 41761}, {141, 37067}, {216, 3788}, {230, 34254}, {233, 3734}, {315, 8553}, {325, 1609}, {571, 16925}, {577, 620}, {626, 10979}, {631, 5157}, {1879, 32961}, {1975, 9722}, {3815, 11324}, {7774, 13345}, {7778, 36751}, {7791, 14806}, {7803, 13351}, {7862, 36412}, {11511, 38751}, {14060, 41757}, {17907, 34990}, {36212, 41770}

X(42406) = {X(2), X(40697)}-harmonic conjugate of X(41760)

X(42407) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ARIES WRT ABC

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

The correspondent reciprocal tripole of these triangles is X(42406)

X(42407) lies on these lines: {6, 6393}, {25, 317}, {69, 1976}, {76, 2165}, {95, 41271}, {99, 41761}, {111, 42297}, {251, 7774}, {305, 13854}, {308, 7795}, {393, 3926}, {491, 8577}, {492, 8576}, {577, 32458}, {670, 6389}, {1502, 16081}, {1989, 32833}, {2395, 3267}, {2963, 32832}, {2998, 7836}, {3108, 16989}, {6387, 31639}, {7778, 8770}, {7792, 39951}, {7799, 34288}, {17907, 39645}, {31401, 39968}

X(42407) = isogonal conjugate of X(42295)
X(42407) = isotomic conjugate of X(3767)
X(42407) = polar conjugate of X(41762)
X(42407) = barycentric product X(i)*X(j) for these {i, j}: {69, 34405}, {523, 42297}
X(42407) = barycentric quotient X(i)/X(j) for these (i, j): (3, 40947), (4, 41762), (63, 2083), (69, 1899), (75, 17871), (76, 41760)
X(42407) = trilinear product X(i)*X(j) for these {i, j}: {63, 34405}, {661, 42297}
X(42407) = trilinear quotient X(i)/X(j) for these (i, j): (63, 40947), (69, 2083), (76, 17871), (92, 41762), (304, 1899), (326, 39643)
X(42407) = trilinear pole of the line {512, 6333}
X(42407) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(4), X(7807)}}
X(42407) = Cevapoint of X(2) and X(3926)
X(42407) = X(520)-cross conjugate of-X(670)
X(42407) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 40947}, {25, 2083}, {32, 17871}, {48, 41762}
X(42407) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 40947), (4, 41762), (63, 2083), (69, 1899)

X(42408) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ABC WRT GARCIA-REFLECTION

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

The correspondent reciprocal tripole of these triangles is X(651)

X(42408) lies on these lines: {}

X(42408) = trilinear pole of the line {145, 1837}
X(42408) = Cevapoint of X(650) and X(14923)

X(42409) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ABC WRT HONSBERGER

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

The correspondent reciprocal tripole of these triangles is X(1)

X(42409) lies on these lines: {105, 7676}, {3673, 42326}, {3693, 32019}, {17278, 34018}, {31169, 34578}

X(42410) = TRIPOLE OF THE T-ISOGONAL-AXIS OF THESE TRIANGLES: ABC WRT JOHNSON

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

The correspondent reciprocal tripole of these triangles is X(275)

X(42410) lies on the Kiepert hyperbola and these lines: {4, 5449}, {96, 7542}, {262, 31236}, {275, 3580}, {343, 2986}, {801, 37636}, {5392, 37638}, {5961, 18316}, {13567, 40393}, {26958, 34289}

X(42410) = polar conjugate of X(6240)
X(42410) = barycentric quotient X(i)/X(j) for these (i, j): (3, 12038), (4, 6240)
X(42410) = trilinear quotient X(i)/X(j) for these (i, j): (63, 12038), (92, 6240)
X(42410) = trilinear pole of the line {523, 18403}
X(42410) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(95), X(18817)}}
X(42410) = Cevapoint of X(6) and X(1658)
X(42410) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 12038}, {48, 6240}
X(42410) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 12038), (4, 6240)

X(42411) = X(3)X(39163) ∩ X(4)X(39162)

Barycentrics    Sqrt(-3*S^2+SW^2)*(S^2-3*SB*SC)+2*(S^2-2*SB*SC)*Sqrt(-3*S^2+2*sqrt(-3*S^2+SW^2)*(9*R^2-2*SW)-18*SW*R^2+5*SW^2)+(3*SA-2*SW)*S^2+3*SB*SC*SW : :
Barycentrics    a^2*SA*(y+z)-SB*SC*x : :, where x : y : z = X(40852)
X(42411) = 4*X(3)-3*X(39163) = 2*X(4)-3*X(39162) = X(3146)-3*X(39158) = 5*X(3522)-3*X(39159) = 3*X(39163)-2*X(40852)

Contributed by César Lozada, March 25, 2021.

X(42411) lies on the cubics K004 (Darboux cubic), K187, K852, K855 and these lines: {3, 39163}, {4, 39162}, {20, 3413}, {30, 40851}, {1498, 40993}, {3146, 39158}, {3522, 39159}

X(42411) = reflection of X(i) in X(j) for these (i, j): (40852, 3), (42412, 20)
X(42411) = isogonal conjugate of X(42412)
X(42411) = X(3)-vertex conjugate of-X(40993)
X(42411) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(39162)}} and {{A, B, C, X(4), X(39158)}}
X(42411) = {X(3), X(40852)}-harmonic conjugate of X(39163)

X(42412) = X(3)X(39162) ∩ X(4)X(39163)

Barycentrics    sqrt(-3*S^2+SW^2)*(S^2-3*SB*SC)-2*(S^2-2*SB*SC)*sqrt(-3*S^2+2*sqrt(-3*S^2+SW^2)*(9*R^2-2*SW)-18*SW*R^2+5*SW^2)+(3*SA-2*SW)*S^2+3*SB*SC*SW : :
Barycentrics   a^2*SA*(y+z)-SB*SC*x : :, where x : y : z = X(40851)
X(42412) = 4*X(3)-3*X(39162) = 2*X(4)-3*X(39163) = X(3146)-3*X(39159) = 5*X(3522)-3*X(39158) = 3*X(39162)-2*X(40851)

Contributed by César Lozada, March 25, 2021.

X(42412) lies on the cubics K004 (Darboux cubic), K187, K852, K855 and these lines: {3, 39162}, {4, 39163}, {20, 3413}, {30, 40852}, {1498, 40994}, {3146, 39159}, {3522, 39158}

X(42412) = reflection of X(i) in X(j) for these (i, j): (40851, 3), (42411, 20)
X(42412) = isogonal conjugate of X(42411)
X(42412) = X(3)-vertex conjugate of-X(40994)
X(42412) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(39163)}} and {{A, B, C, X(4), X(32443)}}
X(42412) = {X(3), X(40851)}-harmonic conjugate of X(39162)

leftri

Points on the Cullen cubic: X(42413)-X(42421)

rightri

This preamble and points X(42413)-X(42413) are contributed by Peter Moses, March 25, 2021.

The Cullen cubic is indexed as K369; see K369.

Points (42413)-X(42420) are Gibert points, as in the preamble just before X(42085).

X(42413) = GIBERT(SQRT(3),-5/2,3) POINT

Barycentrics    a^2*S + 3*a^2*SA - 5*SB*SC : :

X(42413) lies on the cubic K369 and these lines: {2, 42271}, {3, 23263}, {4, 5418}, {5, 6496}, {6, 5059}, {20, 1152}, {30, 1587}, {371, 23269}, {372, 11001}, {376, 1328}, {382, 6407}, {485, 15682}, {486, 17538}, {550, 6452}, {590, 17578}, {631, 22615}, {641, 26615}, {1131, 3068}, {1132, 6410}, {1151, 3543}, {1588, 1657}, {1702, 28172}, {3522, 23261}, {3523, 42283}, {3524, 42268}, {3525, 35787}, {3528, 6565}, {3529, 6420}, {3534, 13935}, {3627, 9540}, {3832, 6409}, {3845, 6455}, {3850, 6451}, {3853, 6449}, {3854, 32789}, {5056, 6411}, {5073, 23249}, {5076, 35255}, {6221, 23253}, {6396, 23275}, {6435, 7581}, {6439, 8972}, {6448, 15704}, {6450, 15686}, {6471, 15683}, {6474, 15684}, {6478, 9681}, {6488, 42265}, {6497, 15690}, {6501, 17800}, {7584, 15681}, {7585, 42272}, {8252, 21734}, {9543, 13846}, {9683, 14865}, {10299, 42274}, {10304, 42262}, {10577, 21735}, {11541, 35820}, {12103, 13785}, {12296, 35947}, {12819, 15715}, {13925, 35404}, {15640, 41945}, {15717, 42270}, {18991, 28158}, {19066, 28164}, {23273, 42261}, {29317, 39876}, {35776, 37946}, {42140, 42191}, {42141, 42192}, {42160, 42254}, {42161, 42255}

X(42414) = GIBERT(SQRT(3),5/2,-3) POINT

Barycentrics    a^2*S - 3*a^2*SA + 5*SB*SC : :

X(42414) lies on the cubic K369 and these lines: {2, 42272}, {3, 23253}, {4, 5420}, {5, 6497}, {6, 5059}, {20, 1151}, {30, 1588}, {371, 11001}, {372, 23275}, {376, 1327}, {382, 6408}, {485, 17538}, {486, 15682}, {550, 6451}, {615, 17578}, {631, 22644}, {642, 26616}, {1131, 6409}, {1132, 3069}, {1152, 3543}, {1587, 1657}, {1703, 28172}, {3522, 23251}, {3523, 42284}, {3524, 42269}, {3525, 35786}, {3528, 6564}, {3529, 6419}, {3534, 9540}, {3627, 13935}, {3832, 6410}, {3845, 6456}, {3850, 6452}, {3853, 6450}, {3854, 32790}, {5056, 6412}, {5073, 23259}, {5076, 35256}, {6200, 23269}, {6398, 23263}, {6436, 7582}, {6440, 13941}, {6447, 9541}, {6449, 15686}, {6470, 15683}, {6475, 15684}, {6479, 13939}, {6489, 42262}, {6496, 15690}, {6500, 17800}, {7583, 15681}, {7586, 42271}, {8253, 21734}, {10299, 42277}, {10304, 42265}, {10576, 21735}, {11541, 35821}, {12103, 13665}, {12297, 35946}, {12818, 15715}, {13993, 35404}, {15640, 41946}, {15717, 42273}, {18992, 28158}, {19065, 28164}, {23267, 42260}, {29317, 39875}, {35777, 37946}, {42140, 42193}, {42141, 42194}, {42160, 42256}, {42161, 42257}

X(42415) = GIBERT(20,-7,11) POINT

Barycentrics    (20*a^2*S)/Sqrt[3] + 11*a^2*SA - 14*SB*SC : :

X(42415) lies on the cubic K369 and these lines: {15, 10188}, {546, 18582}, {550, 11486}, {3530, 10645}, {3861, 34754}, {5237, 42122}, {5321, 11737}, {5334, 14869}, {5350, 19107}, {10654, 34200}, {11485, 15687}, {14891, 16961}, {15720, 22237}, {41101, 42136}

X(42416) = GIBERT(20,7,-11) POINT

Barycentrics    (20*a^2*S)/Sqrt[3] - 11*a^2*SA + 14*SB*SC : :

X(42416) lies on the cubic K369 and these lines: {16, 10187}, {546, 18581}, {550, 11485}, {3530, 10646}, {3861, 34755}, {5238, 42123}, {5318, 11737}, {5335, 14869}, {5349, 19106}, {10653, 34200}, {11486, 15687}, {14891, 16960}, {15720, 22235}, {41100, 42137}

X(42417) = GIBERT(9*SQRT(3),-5,8) POINT

Barycentrics    9*a^2*S + 8*a^2*SA - 10*SB*SC : :

X(42417) lies on the cubic K369 and these lines: {2, 489}, {6, 11001}, {30, 6419}, {140, 41951}, {371, 3845}, {372, 15690}, {376, 6426}, {381, 6447}, {486, 15701}, {547, 6453}, {590, 5066}, {615, 6451}, {1327, 3830}, {1328, 6221}, {1384, 19099}, {1588, 19708}, {3070, 6470}, {3312, 3534}, {3316, 23261}, {3522, 6489}, {3533, 10147}, {3543, 3592}, {3545, 6425}, {5054, 9681}, {5055, 41963}, {6200, 11812}, {6396, 8703}, {6408, 15695}, {6409, 15719}, {6420, 15686}, {6429, 23275}, {6441, 15640}, {6476, 6565}, {6482, 41985}, {6500, 15685}, {7584, 15759}, {8396, 13810}, {8960, 14893}, {9541, 13847}, {9680, 15703}, {10109, 42270}, {10577, 11540}, {11917, 13666}, {12100, 13993}, {12101, 35821}, {13663, 33457}, {13786, 26341}, {13846, 41099}, {13951, 15722}, {14269, 41952}, {14891, 35813}, {15697, 19053}, {19710, 42259}, {31487, 35403}, {32789, 41955}, {33699, 35822}, {34200, 41964}, {35737, 36836}, {35812, 38071}, {41100, 42200}, {41101, 42202}

X(42418) = GIBERT(9*SQRT(3),5,-8) POINT

Barycentrics    9*a^2*S - 8*a^2*SA + 10*SB*SC : :

X(42418) lies on the cubic K369 and these lines: {2, 490}, {6, 11001}, {30, 6420}, {140, 41952}, {371, 15690}, {372, 3845}, {376, 6425}, {381, 6448}, {485, 15701}, {547, 6454}, {590, 6452}, {615, 5066}, {1327, 6398}, {1328, 3830}, {1384, 19100}, {1587, 19708}, {3071, 6471}, {3311, 3534}, {3317, 23251}, {3522, 6488}, {3533, 10148}, {3543, 3594}, {3545, 6426}, {5054, 10195}, {5055, 41964}, {5067, 17852}, {6200, 8703}, {6396, 11812}, {6407, 15695}, {6410, 15719}, {6419, 15686}, {6430, 23269}, {6442, 15640}, {6477, 6564}, {6483, 41985}, {6501, 15685}, {7583, 15759}, {8416, 13691}, {8960, 17504}, {8976, 15722}, {10109, 42273}, {10304, 31454}, {10576, 11540}, {11916, 13786}, {12100, 13925}, {12101, 35820}, {13666, 26348}, {13783, 33456}, {13846, 15698}, {13847, 41099}, {14269, 41951}, {14891, 35812}, {15697, 19054}, {15708, 31414}, {17851, 32790}, {19710, 42258}, {33699, 35823}, {34200, 41963}, {35737, 36843}, {35813, 38071}, {41100, 42199}, {41101, 42201}

X(42419) = GIBERT(54,-7,13) POINT

Barycentrics    18*Sqrt[3]*a^2*S + 13*a^2*SA - 14*SB*SC : :

X(42419) lies on the cubic K369 and these lines: {2, 11485}, {6, 19710}, {30, 41974}, {61, 5066}, {398, 10109}, {3412, 14892}, {3830, 5344}, {3845, 40693}, {3860, 11542}, {5238, 12100}, {8703, 22238}, {10646, 15759}, {10654, 33699}, {12101, 12816}, {15690, 41101}, {15713, 22236}, {15716, 37641}, {16268, 41978}, {22235, 41099}, {41107, 42108}, {41112, 42136}, {41113, 42098}, {41990, 42159}

X(42420) = GIBERT(54,7,-13) POINT

Barycentrics    18*Sqrt[3]*a^2*S - 13*a^2*SA + 14*SB*SC : : X(42420) lies on the cubic K369 and these lines: {2, 11486}, {6, 19710}, {30, 41973}, {62, 5066}, {397, 10109}, {3411, 14892}, {3830, 5343}, {3845, 40694}, {3860, 11543}, {5237, 12100}, {8703, 22236}, {10645, 15759}, {10653, 33699}, {12101, 12817}, {15690, 41100}, {15713, 22238}, {15716, 37640}, {16267, 41977}, {22237, 41099}, {41108, 42109}, {41112, 42095}, {41113, 42137}, {41990, 42162}

X(42421) = X(6)X(194)∩X(30)X(182)

Barycentrics    {Sqrt[3]*(1 + 2*Cos[2*w])*Csc[w]*Sec[w], Csc[w]^2, 2*(-1 + Cot[w]^2)} : :
Barycentrics    2*a^6 + 2*a^4*b^2 + a^2*b^4 + 2*a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 : :

X(42421) lies on the cubic K369 and these lines: {5, 39750}, {6, 194}, {30, 182}, {32, 141}, {39, 15870}, {69, 19689}, {81, 19700}, {83, 316}, {187, 10007}, {193, 19692}, {385, 24273}, {511, 32134}, {524, 6661}, {732, 5007}, {736, 7804}, {940, 19670}, {1078, 34573}, {1350, 10788}, {1352, 11842}, {1503, 3398}, {2076, 10346}, {3094, 3972}, {3329, 5116}, {3407, 7792}, {3416, 10789}, {3618, 6655}, {3629, 5039}, {3631, 19702}, {3763, 7793}, {4383, 19669}, {5008, 14994}, {5031, 6680}, {5033, 8357}, {5038, 6329}, {5085, 7470}, {5092, 14881}, {5171, 21167}, {5207, 10583}, {5718, 19683}, {5846, 12194}, {7750, 10345}, {7770, 8177}, {7773, 7948}, {7808, 8364}, {7829, 29012}, {7878, 13331}, {7894, 41747}, {9053, 12195}, {9300, 10352}, {10191, 22352}, {10328, 34482}, {10333, 12206}, {10334, 41624}, {11356, 40825}, {12007, 12177}, {12110, 29181}, {12151, 20583}, {13193, 32242}, {14561, 37243}, {15516, 32135}, {16285, 33786}, {18501, 31670}, {19679, 28369}, {32515, 35426}, {33185, 39603}, {36759, 37341}, {36760, 37340}

X(42421) = 5th-Brocard-to-ABC similarity image of X(141)

leftri

Points on the nine-point circle: X(42422)-X(42426)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, March 26, 2021.

The appearance of {i,j} in the following list means that X(i) and X(j) are a pair of antipodes on the nine-point circle; i.e., each is the reflection of the other in the center, X(5), of the nine-point circle.

{11,119}, {113,125}, {114,115}, {115,114}, {116,118}, {117,124}, {118,116}, {119,11}, {120,5511}, {121,5510}, {122,133}, {123,25640}, {124,117}, {125,113}, {126,5512}, {127,132}, {128,137}, {129,130}, {130,129}, {131,136}, {132,127}, {133,122}, {136,131}, {137,128}, {1312,1313}, {1313,1312}, {1560,14672}, {1566,33331}, {2039,2040}, {2040,2039}, {2679,33330}, {3258,25641}, {3259,31841}, {5099,16188}, {5139,31842}, {5510,121}, {5511,120}, {5512,126}, {5513,25642}, {5520,42422}, {5521,42423}, {5950,5952}, {5952,5950}, {6092,31654}, {10017,39535}, {11569,13994}, {12052,42424}, {12494,13234}, {12624,13249}, {13234,12494}, {13249,12624}, {13994,11569}, {14672,1560}, {16177,18809}, {16188,5099}, {18402,20625}, {18809,16177}, {20625,18402}, {25640,123}, {25641,3258}, {25642,5513}, {25652,42425}, {31654,6092}, {31841,3259}, {31842,5139}, {33330,2679}, {33331,1566}, {33333,35591}, {35591,33333}, {38971,42426}, {39535,10017}, {42422,5520}, {42423,5521}, {42424,12052}, {42425,25652}, {42426,38971}

Theorem (Moses): Suppose that P = p : q : r is a point, and define f(P) = p*((a^2 - b^2 + c^2)*q + (-a^2 - b^2 + c^2)*r)*((-b^2 + c^2)*q*r + p*(c^2*q - b^2*r)) : : . The point f(P) is on the nine-point circle, and f(P) is the center of the rectangular circumhyperbola {{A, B, C, X(4), P}}. The perspector of the hyperbola lies on the orthic axis.

The appearance of {i, {i(1),i(2),...i(n),} {name}} in the following list means that X(i) = f(X(k)) for k = 1,2,...,n, and that the circumhyperbola passing through these points has the indicated name, if there is one:

{11, {1, 4, 7, 8, 9, 21, 79, 80, 84, 90, 104, 177, 256, 294, 314, 885, 941, 943, 981, 983, 987, 989, 1000, 1039, 1041, 1061, 1063, 1156, 1172, 1251, 1320, 1389, 1392, 1476, 1896, 1937, 2298, 2320, 2335, 2344, 2346, 2481, 2648, 2997, 3062, 3065, 3254, 3255, 3296, 3307, 3308, 3427, 3467, 3495, 3551, 3577, 3680, 4180, 4866, 4876, 4900, 5377, 5424, 5551, 5553, 5555, 5556, 5557, 5558, 5559, 5560, 5561, 5665, 6595, 6596, 6597, 6598, 6599, 6601, 7003, 7049, 7091, 7126, 7133, 7149, 7155, 7160, 7161, 7162, 7261, 7284, 7285, 7317, 7319, 7320, 7595, 7707, 8372, 8759, 8809, 9365, 9372, 9442, 10266, 10305, 10307, 10308, 10309, 10390, 10429, 10435, 11279, 11604, 11609, 12641, 12867, 12868, 13143, 13426, 13454, 13602, 13606, 14224, 14496, 14497, 14947, 15173, 15175, 15176, 15179, 15180, 15314, 15315, 15446, 15909, 15910, 15997, 15998, 16005, 16615, 17097, 17098, 17501, 18299, 18490, 19551, 21398, 23836, 23838, 23893, 23959, 24297, 24298, 24300, 24302, 26722, 30479, 30494, 30500, 30513, 31316, 31507, 31509, 32635, 33576, 33653, 33696, 34215, 34216, 34256, 34485, 34894, 34917, 34918, 34919, 35097, 35355, 36121, 36599, 36798, 37518, 38249, 38250, 38251, 38261, 38268, 38270, 38271, 38272, 38274, 38306, 38307, 38308, 39144, 39145, 39768, 40396, 40454, 40565, 40566, 40779, 41527, 42013, 42015, 42017}, {Feuerbach circumhyperbola}}

{113, {4, 110, 14264, 15329, 18881, 39985, 41512}, {}}

{114, {4, 99, 4226, 14265, 34174, 36875, 41173}, {}}

{115, {2, 4, 10, 13, 14, 17, 18, 76, 83, 94, 96, 98, 226, 262, 275, 321, 459, 485, 486, 598, 671, 801, 1029, 1131, 1132, 1139, 1140, 1327, 1328, 1446, 1676, 1677, 1751, 1916, 2009, 2010, 2051, 2052, 2394, 2592, 2593, 2671, 2672, 2986, 2996, 3316, 3317, 3366, 3367, 3370, 3373, 3374, 3381, 3382, 3387, 3388, 3391, 3392, 3397, 3399, 3406, 3407, 3413, 3414, 3424, 3429, 3590, 3591, 3597, 4049, 4052, 4080, 4444, 5392, 5395, 5397, 5401, 5402, 5403, 5404, 5466, 5485, 5487, 5488, 5490, 5491, 5503, 6177, 6178, 6504, 6539, 6568, 6569, 6625, 7578, 7607, 7608, 7612, 8587, 8781, 8796, 8808, 9180, 9221, 9290, 9302, 9381, 10153, 10155, 10159, 10185, 10187, 10188, 10194, 10195, 10290, 10302, 10484, 10511, 11121, 11122, 11140, 11167, 11170, 11172, 11538, 11599, 11602, 11603, 11606, 11608, 11611, 11668, 11669, 12066, 12816, 12817, 12818, 12819, 12820, 12821, 12822, 12823, 13380, 13478, 13576, 13579, 13580, 13581, 13582, 13583, 13584, 13585, 13599, 14223, 14226, 14228, 14229, 14231, 14232, 14234, 14236, 14237, 14238, 14240, 14241, 14243, 14244, 14245, 14458, 14484, 14485, 14488, 14492, 14494, 14534, 14554, 14632, 14633, 16080, 16277, 17503, 17758, 18316, 18366, 18840, 18841, 18842, 18843, 18844, 18845, 21845, 21846, 22235, 22237, 22244, 22245, 24007, 24008, 24624, 27797, 30505, 30588, 31363, 31630, 31943, 32014, 32022, 32130, 32532, 33602, 33603, 33604, 33605, 33606, 33607, 33698, 34087, 34089, 34091, 34258, 34289, 34475, 34899, 35005, 35098, 35353, 36316, 36317, 36907, 37865, 37874, 37892, 38253, 38259, 38309, 39284, 39295, 39641, 39642, 39994, 40012, 40013, 40016, 40017, 40021, 40024, 40030, 40031, 40104, 40105, 40149, 40158, 40159, 40162, 40163, 40167, 40168, 40178, 40393, 40395, 40448, 40515, 40706, 40707, 40718, 40824, 40831, 41194, 41195, 41895, 41899, 42006, 42010, 42011, 42023, 42024, 42035, 42036, 42062, 42063}, {Kiepert circumhyperbola}}

{116, {4, 103, 947, 1002, 1126, 1174, 2141, 3681, 3730, 4184, 7357, 8049, 10623, 13577, 33297, 39961, 39993}, {}}

{117, {4, 109, 7450}, {}}

{118, {4, 101, 4243}, {}}

{119, {4, 100, 3658, 14266, 39991}, {}}

{120, {4, 668, 1292, 4236, 14267}, {}}

{121, {4, 1293, 8050, 39264}, {}}

{122, {4, 20, 253, 1249, 1294, 3346, 3668, 5930, 6188, 8804, 8806, 9307, 10152, 14249, 14615, 14863, 15318, 15319, 16251, 18349, 31361, 33702, 33893, 33897, 35140, 35515, 38808, 39130}, {}}

{123, {4, 961, 998, 1295, 1766, 2995, 3436, 7219, 14257, 16049, 21147, 34263, 39990, 40457, 41364}, {}}

{124, {4, 58, 102, 573, 959, 994, 3417, 3869, 4225, 8048, 9309, 10571, 20028, 34242, 39992}, {}}

{125, {3, 4, 6, 54, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 248, 265, 290, 695, 879, 895, 1173, 1175, 1176, 1177, 1242, 1243, 1244, 1245, 1246, 1439, 1798, 1903, 1942, 1987, 2213, 2435, 2574, 2575, 2992, 2993, 3426, 3431, 3519, 3521, 3527, 3531, 3532, 3657, 4846, 5486, 5504, 5505, 5900, 6145, 6391, 6413, 6414, 6415, 6416, 8044, 8612, 8795, 8811, 8814, 9399, 9513, 10097, 10099, 10100, 10261, 10262, 10293, 10378, 10693, 11138, 11139, 11270, 11559, 11564, 11738, 11744, 12023, 13418, 13452, 13472, 13603, 13622, 13623, 14220, 14374, 14375, 14380, 14457, 14483, 14487, 14490, 14491, 14498, 14528, 14542, 14841, 14843, 14861, 15002, 15077, 15232, 15316, 15317, 15320, 15321, 15328, 15453, 15460, 15461, 15740, 15749, 16000, 16540, 16620, 16623, 16665, 16774, 16835, 16867, 17040, 17505, 17711, 18123, 18124, 18125, 18296, 18363, 18368, 18434, 18532, 18550, 19151, 19222, 20029, 20421, 21400, 22334, 22336, 22466, 26861, 28786, 28787, 28788, 30496, 31366, 31371, 32533, 32585, 32586, 32618, 32619, 33565, 34135, 34136, 34207, 34221, 34222, 34259, 34435, 34436, 34437, 34438, 34439, 34440, 34483, 34567, 34800, 34801, 34802, 34817, 35364, 35373, 35512, 35909, 36214, 36296, 36297, 37142, 38005, 38006, 38257, 38260, 38263, 38264, 38433, 38436, 38439, 38442, 38443, 38445, 38447, 38449, 38534, 38535, 38955, 39372, 39379, 39380, 39381, 39665, 39666, 40048, 40441, 41433, 41435, 41518, 41519, 41897, 41898, 42016, 42021, 42059}, {Jerabek circumhyperbola}}

{126, {4, 670, 1296, 11634, 14263, 14948, 34171, 34574, 36874}, {}}

{127, {4, 22, 251, 315, 1297, 3425, 4456, 4463, 8743, 11605, 11610, 13575, 14495}, {}}

{128, {4, 930}, {}}

{129, {4, 1303, 14586}, {}}

{130, {4, 51, 184, 217, 418, 1298, 5562, 20574, 27372, 31357, 32319}, {}}

{131, {4, 925, 30512}, {}}

{132, {4, 112, 877, 4230, 6528, 35908, 39265}, {}}

{133, {4, 107, 4240}, {}}

{134, {4, 52, 571, 3133, 39110}, {}}

{135, {4, 24, 317, 1299, 5412, 5413, 8745, 8882, 14111, 14518, 34756}, {}}

{136, {4, 93, 225, 254, 264, 393, 847, 1093, 1105, 1179, 1217, 1300, 1826, 6344, 6526, 6531, 8737, 8738, 8741, 8742, 8801, 8884, 14860, 15424, 16263, 17983, 18808, 18846, 18847, 18848, 18849, 18850, 18851, 18852, 18853, 18854, 18855, 24243, 24244, 32085, 34208, 35142, 36611, 36612, 38427, 38428, 40402, 41013, 41515, 41516}, {}}

{137, {4, 5, 53, 311, 327, 1141, 1263, 1487, 2165, 2980, 3459, 3613, 8797, 8800, 10412, 11082, 11087, 11816, 13450, 14225, 15619, 16837, 17500, 17507, 17703, 19712, 19713, 21011, 22261, 22335, 25043, 25148, 27352, 27353, 27356, 27360, 27361, 27364, 32535, 34449, 36300, 36301, 36809, 38305, 38899, 39286, 40449}, {{{A,B,C,X(4),X(5)}}}

{138, {4, 16813, 23232}, {}}

{139, {4, 324, 467, 11547, 14149, 23233, 39114}, {}}

{1312, {4, 1113, 15164, 16071}, {}}

{1313, {4, 1114, 15165, 16070}, {}}

{1560, {4, 648, 4235, 30247}, {}}

{1566, {4, 279, 514, 516, 2724, 6185, 10405, 14377, 14953, 23984, 30807, 34529, 35158}, {}}

{2039, {4, 1380, 3557}, {}}

{2040, {4, 1379, 3558}, {}}

{2679, {4, 32, 237, 263, 511, 512, 2211, 2698, 5360, 9292, 14251, 20022, 27375, 34157, 34214, 34238, 34854, 36892, 37841, 39684, 41520}, {}}

{3258, {4, 30, 477, 523, 1138, 1990, 3260, 3471, 5627, 5641, 6662, 9154, 9214, 11080, 11085, 11815, 13489, 14254, 14387, 14536, 15454, 16104, 19776, 19777, 21765, 32230, 34288, 35906, 36298, 36299, 36889, 36891, 39453, 41522}, {}}

{3259, {4, 56, 513, 517, 859, 945, 953, 957, 1457, 1875, 2183, 10428, 14260, 17101, 17139, 34431, 34434, 38008, 39173}, {}}

{5099, {4, 23, 316, 842, 1383, 8744, 10422, 10561, 10630, 13485, 13574, 14246, 23964, 41896}, {}}

{5139, {4, 25, 683, 1426, 1824, 2207, 2333, 3563, 6524, 8753, 8946, 8948, 14248, 14486, 14593, 15591, 17980, 18384, 31942, 34405, 34428, 36878, 39109, 40801, 41521}, {}}

{5190, {4, 27, 92, 278, 917, 1847, 6336, 8747, 17982, 36124, 36613, 37203, 40444, 40445, 40573, 40574}, {}}

{5509, {4, 31, 4215, 5508, 7139, 26893, 29009}, {}}

{5510, {4, 106, 7419, 14261, 14923, 41439, 41446}, {}}

{5511, {4, 105, 169, 3434, 4228, 14268, 34036, 39732, 40154}, {}}

{5512, {4, 111, 1995, 11185, 14262, 34166, 34241, 38331}, {}}

{5513, {4, 190, 4237}, {}}

{5514, {4, 40, 57, 189, 196, 223, 329, 937, 972, 1817, 2184, 3194, 3345, 8810, 14256, 26735, 34546, 36100, 38009, 40397, 41790}, {}}

{5515, {4, 75, 388, 1010, 1065, 1220, 2345, 4385, 34265, 37842}, {}}

{5516, {4, 145, 519, 3667, 4487}, {}}

{5517, {4, 81, 2994, 5739, 12514, 14258, 27174, 34260}, {}}

{5518, {4, 291, 979, 3223, 3501, 7093, 7224, 7346, 7350, 8817, 8927, 13588, 15323, 32937, 39741, 39976}, {}}

{5519, {4, 218, 518, 3309, 6604, 28914, 34159}, {}}

{5520, {4, 267, 1325, 2687, 4581, 5080, 16548, 29374}, {}}

{5521, {4, 19, 28, 34, 286, 915, 1118, 1119, 5317, 8751, 10977, 14493, 17981, 34406, 34408, 36125, 39267, 41505}, {}}

{5522, {4, 95, 631, 3087, 34285}, {}}

{9151, {4, 13586, 35146, 35511}, {}}

{9193, {4, 8598, 14255, 18823}, {}}

{10017, {4, 280, 515, 522, 2734, 10570, 39457, 40437}, {}}

{11792, {4, 140, 252, 1232, 6748, 11703, 13139, 13381, 13597, 15464, 21012, 22270, 26862, 30102, 36948}, {}}

{13612, {4, 282, 1034, 1490, 3176, 3341, 3347, 5932, 8805, 13614}, {}}

{13613, {4, 1032, 1073, 1498, 3343, 3348, 6617, 8807, 14361, 15324}, {}}

{13994, {4, 6094, 11159, 11568, 13377}, {}}

{13999, {4, 270, 1870, 2190, 5081, 7012, 17515}, {}}

{14672, {4, 2373, 7493, 34165, 41370}, {}}

{15607, {4, 55, 942, 955, 1859, 8021, 14547, 38007, 41509}, {}}

{15608, {4, 59, 7163, 15339}, {}}

{15609, {4, 15, 61, 8446, 11146, 11581, 11600, 13483, 16771, 19778, 34219, 39404, 39407}, {}}

{15610, {4, 16, 62, 8456, 11145, 11582, 11601, 13484, 16770, 19779, 34220, 39405, 39406}, {}}

{15611, {4, 596, 996, 1219, 4373, 4696, 10106, 11115, 17355}, {}}

{15612, {4, 693, 2723, 5179, 14956}, {}}

{16177, {4, 2071, 2693, 10419, 13573, 15262, 15384, 34170, 38937}, {}}

{{16188, {4, 691, 7468, 14221, 18333, 32708, 34175, 35139, 38939}, {}}

{16221, {4, 186, 250, 340, 562, 1825, 1835, 5962, 8739, 8740, 14222, 32710, 38936, 40388}, {}}

{20620, {4, 29, 158, 273, 281, 318, 7040, 8748, 11546, 32706, 36123, 36610, 40446, 40836}, {}}

{20621, {4, 1783, 4238, 18026, 26706}, {}}

{20622, {4, 4241, 26705, 36118, 41321}, {}}

{20625, {4, 1166, 7488, 15620, 18401}, {}}

{31653, {4, 63, 2982, 26872, 40575}, {}}

{31654, {4, 524, 1499, 1992, 6093, 9487, 13608, 15471, 22100, 27088, 34161}, {}}

{31655, {4, 892, 2696, 7472, 34169}, {}}

{31845, {4, 3952, 6011, 13589, 38938}, {}}

{33330, {4, 805, 12833, 13137}, {}}

{33504, {4, 441, 525, 1503, 9476, 14376, 34156, 34403, 36894}, {}}

{35579, {4, 520, 6000, 14379, 36893, 39174}, {}}

{35580, {4, 521, 1433, 6001, 39167, 39175}, {}}

{35581, {4, 526, 5663, 14385, 16169}, {}}

{35582, {4, 542, 690, 1640, 5967, 14357, 36890}, {}}

{35583, {4, 758, 6003, 15556, 27086, 39166}, {}}

{35584, {4, 826, 14378, 29012}, {}}

{35587, {4, 900, 952, 14584, 36944}, {}}

{35588, {4, 924, 1147, 13754}, {}}

{35591, {4, 143, 1154, 1510, 11135, 11136, 15907, 24772, 25044, 27357}, {}}

{35968, {4, 5879, 5897, 11413, 39268}, {}}

{35971, {4, 384, 1031, 2998, 3114, 9230, 14970, 37888, 39953}, {}}

{36471, {4, 2065, 2710, 15388, 37183, 38826, 41363}, {}}

{36472, {4, 249, 14253, 23700, 35296}, {}}

{38957, {4, 78, 18391, 18446}, {}}

{38958, {4, 82, 977, 5015, 11102}, {}}

{38959, {4, 85, 277, 673, 948, 2550, 16054}, {}}

{38960, {4, 86, 966, 3485, 11110, 31359, 40028}, {}}

{38964, {4, 91, 1478, 11103, 18815}, {}}

{38966, {4, 33, 1857, 4183, 7008, 7046, 7079, 34398, 36122, 40169}, {}}

{38967, {4, 37, 405, 1882, 5295, 14549, 41506}, {}}

{38968, {4, 42, 1011, 10449}, {}}

{38970, {4, 297, 6530, 14618, 18027, 23582}, {}}

{38971, {4, 850, 858, 1236, 2697, 5523, 10415, 14364, 21017, 39269, 40421}, {}}

{38974, {4, 401, 15351, 15412, 32545, 34536, 34538, 39682, 40815, 41204}, {}}

The appearance of {i, {i(1),i(2),...i(n),} {name}} in the following list means that X(i) is the perspector of the circumhyperbola passing through the points X(i(k)), and that the hyperbola has indicated name, if there is one:

{230, {4, 99, 4226, 14265, 34174, 36875, 41173}, {}}

{231, {4, 930}, {}}

{232, {4, 112, 877, 4230, 6528, 35908, 39265}, {}}

{468, {4, 648, 4235, 30247}, {}}

{523, {2, 4, 10, 13, 14, 17, 18, 76, 83, 94, 96, 98, 226, 262, 275, 321, 459, 485, 486, 598, 671, 801, 1029, 1131, 1132, 1139, 1140, 1327, 1328, 1446, 1676, 1677, 1751, 1916, 2009, 2010, 2051, 2052, 2394, 2592, 2593, 2671, 2672, 2986, 2996, 3316, 3317, 3366, 3367, 3370, 3373, 3374, 3381, 3382, 3387, 3388, 3391, 3392, 3397, 3399, 3406, 3407, 3413, 3414, 3424, 3429, 3590, 3591, 3597, 4049, 4052, 4080, 4444, 5392, 5395, 5397, 5401, 5402, 5403, 5404, 5466, 5485, 5487, 5488, 5490, 5491, 5503, 6177, 6178, 6504, 6539, 6568, 6569, 6625, 7578, 7607, 7608, 7612, 8587, 8781, 8796, 8808, 9180, 9221, 9290, 9302, 9381, 10153, 10155, 10159, 10185, 10187, 10188, 10194, 10195, 10290, 10302, 10484, 10511, 11121, 11122, 11140, 11167, 11170, 11172, 11538, 11599, 11602, 11603, 11606, 11608, 11611, 11668, 11669, 12066, 12816, 12817, 12818, 12819, 12820, 12821, 12822, 12823, 13380, 13478, 13576, 13579, 13580, 13581, 13582, 13583, 13584, 13585, 13599, 14223, 14226, 14228, 14229, 14231, 14232, 14234, 14236, 14237, 14238, 14240, 14241, 14243, 14244, 14245, 14458, 14484, 14485, 14488, 14492, 14494, 14534, 14554, 14632, 14633, 16080, 16277, 17503, 17758, 18316, 18366, 18840, 18841, 18842, 18843, 18844, 18845, 21845, 21846, 22235, 22237, 22244, 22245, 24007, 24008, 24624, 27797, 30505, 30588, 31363, 31630, 31943, 32014, 32022, 32130, 32532, 33602, 33603, 33604, 33605, 33606, 33607, 33698, 34087, 34089, 34091, 34258, 34289, 34475, 34899, 35005, 35098, 35353, 36316, 36317, 36907, 37865, 37874, 37892, 38253, 38259, 38309, 39284, 39295, 39641, 39642, 39994, 40012, 40013, 40016, 40017, 40021, 40024, 40030, 40031, 40104, 40105, 40149, 40158, 40159, 40162, 40163, 40167, 40168, 40178, 40393, 40395, 40448, 40515, 40706, 40707, 40718, 40824, 40831, 41194, 41195, 41895, 41899, 42006, 42010, 42011, 42023, 42024, 42035, 42036, 42062, 42063}, {Kiepert circumhyperbola}}

{647, {3, 4, 6, 54, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 248, 265, 290, 695, 879, 895, 1173, 1175, 1176, 1177, 1242, 1243, 1244, 1245, 1246, 1439, 1798, 1903, 1942, 1987, 2213, 2435, 2574, 2575, 2992, 2993, 3426, 3431, 3519, 3521, 3527, 3531, 3532, 3657, 4846, 5486, 5504, 5505, 5900, 6145, 6391, 6413, 6414, 6415, 6416, 8044, 8612, 8795, 8811, 8814, 9399, 9513, 10097, 10099, 10100, 10261, 10262, 10293, 10378, 10693, 11138, 11139, 11270, 11559, 11564, 11738, 11744, 12023, 13418, 13452, 13472, 13603, 13622, 13623, 14220, 14374, 14375, 14380, 14457, 14483, 14487, 14490, 14491, 14498, 14528, 14542, 14841, 14843, 14861, 15002, 15077, 15232, 15316, 15317, 15320, 15321, 15328, 15453, 15460, 15461, 15740, 15749, 16000, 16540, 16620, 16623, 16665, 16774, 16835, 16867, 17040, 17505, 17711, 18123, 18124, 18125, 18296, 18363, 18368, 18434, 18532, 18550, 19151, 19222, 20029, 20421, 21400, 22334, 22336, 22466, 26861, 28786, 28787, 28788, 30496, 31366, 31371, 32533, 32585, 32586, 32618, 32619, 33565, 34135, 34136, 34207, 34221, 34222, 34259, 34435, 34436, 34437, 34438, 34439, 34440, 34483, 34567, 34800, 34801, 34802, 34817, 35364, 35373, 35512, 35909, 36214, 36296, 36297, 37142, 38005, 38006, 38257, 38260, 38263, 38264, 38433, 38436, 38439, 38442, 38443, 38445, 38447, 38449, 38534, 38535, 38955, 39372, 39379, 39380, 39381, 39665, 39666, 40048, 40441, 41433, 41435, 41518, 41519, 41897, 41898, 42016, 42021, 42059}, {Jerabek circumhyperbola}}

{650, {1, 4, 7, 8, 9, 21, 79, 80, 84, 90, 104, 177, 256, 294, 314, 885, 941, 943, 981, 983, 987, 989, 1000, 1039, 1041, 1061, 1063, 1156, 1172, 1251, 1320, 1389, 1392, 1476, 1896, 1937, 2298, 2320, 2335, 2344, 2346, 2481, 2648, 2997, 3062, 3065, 3254, 3255, 3296, 3307, 3308, 3427, 3467, 3495, 3551, 3577, 3680, 4180, 4866, 4876, 4900, 5377, 5424, 5551, 5553, 5555, 5556, 5557, 5558, 5559, 5560, 5561, 5665, 6595, 6596, 6597, 6598, 6599, 6601, 7003, 7049, 7091, 7126, 7133, 7149, 7155, 7160, 7161, 7162, 7261, 7284, 7285, 7317, 7319, 7320, 7595, 7707, 8372, 8759, 8809, 9365, 9372, 9442, 10266, 10305, 10307, 10308, 10309, 10390, 10429, 10435, 11279, 11604, 11609, 12641, 12867, 12868, 13143, 13426, 13454, 13602, 13606, 14224, 14496, 14497, 14947, 15173, 15175, 15176, 15179, 15180, 15314, 15315, 15446, 15909, 15910, 15997, 15998, 16005, 16615, 17097, 17098, 17501, 18299, 18490, 19551, 21398, 23836, 23838, 23893, 23959, 24297, 24298, 24300, 24302, 26722, 30479, 30494, 30500, 30513, 31316, 31507, 31509, 32635, 33576, 33653, 33696, 34215, 34216, 34256, 34485, 34894, 34917, 34918, 34919, 35097, 35355, 36121, 36599, 36798, 37518, 38249, 38250, 38251, 38261, 38268, 38270, 38271, 38272, 38274, 38306, 38307, 38308, 39144, 39145, 39768, 40396, 40454, 40565, 40566, 40779, 41527, 42013, 42015, 42017}, {Feuerbach circumhyperbola}}

{676, {4, 279, 514, 516, 2724, 6185, 10405, 14377, 14953, 23984, 30807, 34529, 35158}, {}}

{1637, {4, 30, 477, 523, 1138, 1990, 3260, 3471, 5627, 5641, 6662, 9154, 9214, 11080, 11085, 11815, 13489, 14254, 14387, 14536, 15454, 16104, 19776, 19777, 21765, 32230, 34288, 35906, 36298, 36299, 36889, 36891, 39453, 41522}, {}}

{1886, {4, 4241, 26705, 36118, 41321}, {}}

{2485, {4, 22, 251, 315, 1297, 3425, 4456, 4463, 8743, 11605, 11610, 13575, 14495}, {}}

{2489, {4, 25, 683, 1426, 1824, 2207, 2333, 3563, 6524, 8753, 8946, 8948, 14248, 14486, 14593, 15591, 17980, 18384, 31942, 34405, 34428, 36878, 39109, 40801, 41521}, {}}

{2490, {4, 6553, 17539, 36606, 39697}, {}}

{2491, {4, 32, 237, 263, 511, 512, 2211, 2698, 5360, 9292, 14251, 20022, 27375, 34157, 34214, 34238, 34854, 36892, 37841, 39684, 41520}, {}}

{2492, {4, 23, 316, 842, 1383, 8744, 10422, 10561, 10630, 13485, 13574, 14246, 23964, 41896}, {}}

{2493, {4, 691, 7468, 14221, 18333, 32708, 34175, 35139, 38939}, {}}

{2501, {4, 93, 225, 254, 264, 393, 847, 1093, 1105, 1179, 1217, 1300, 1826, 6344, 6526, 6531, 8737, 8738, 8741, 8742, 8801, 8884, 14860, 15424, 16263, 17983, 18808, 18846, 18847, 18848, 18849, 18850, 18851, 18852, 18853, 18854, 18855, 24243, 24244, 32085, 34208, 35142, 36611, 36612, 38427, 38428, 40402, 41013, 41515, 41516}, {}}

{3003, {4, 110, 14264, 15329, 18881, 39985, 41512}, {}}

{3011, {4, 190, 4237}, {}}

{3012, {4, 658}, {}}

{3018, {4, 476, 7471, 34150}, {}}

{3064, {4, 29, 158, 273, 281, 318, 7040, 8748, 11546, 32706, 36123, 36610, 40446, 40836}, {}}

{3290, {4, 668, 1292, 4236, 14267}, {}}

{3291, {4, 670, 1296, 11634, 14263, 14948, 34171, 34574, 36874}, {}}

{3310, {4, 56, 513, 517, 859, 945, 953, 957, 1457, 1875, 2183, 10428, 14260, 17101, 17139, 34431, 34434, 38008, 39173}, {}}

{3806, {4, 141, 3618, 3867, 8362}, {}}

{4874, {4, 330, 3765, 3923, 6650}, {}}

{5089, {4, 1783, 4238, 18026, 26706}, {}}

{6103, {4, 685, 935, 7473, 16077, 17986, 35907}, {}}

{6129, {4, 40, 57, 189, 196, 223, 329, 937, 972, 1817, 2184, 3194, 3345, 8810, 14256, 26735, 34546, 36100, 38009, 40397, 41790}, {}}

{6130, {4, 401, 15351, 15412, 32545, 34536, 34538, 39682, 40815, 41204}, {}}

{6132, {4, 249, 14253, 23700, 35296}, {}}

{{6586, {4, 103, 947, 1002, 1126, 1174, 2141, 3681, 3730, 4184, 7357, 8049, 10623, 13577, 33297, 39961, 39993}, {}}

{6587, {4, 20, 253, 1249, 1294, 3346, 3668, 5930, 6188, 8804, 8806, 9307, 10152, 14249, 14615, 14863, 15318, 15319, 16251, 18349, 31361, 33702, 33893, 33897, 35140, 35515, 38808, 39130}, {}}

{6588, {4, 961, 998, 1295, 1766, 2995, 3436, 7219, 14257, 16049, 21147, 34263, 39990, 40457, 41364}, {}}

{6589, {4, 58, 102, 573, 959, 994, 3417, 3869, 4225, 8048, 9309, 10571, 20028, 34242, 39992}, {}}

{6590, {4, 75, 388, 1010, 1065, 1220, 2345, 4385, 34265, 37842}, {}}

{6591, {4, 19, 28, 34, 286, 915, 1118, 1119, 5317, 8751, 10977, 14493, 17981, 34406, 34408, 36125, 39267, 41505}, {}}

{6753, {4, 24, 317, 1299, 5412, 5413, 8745, 8882, 14111, 14518, 34756}, {}}

{7649, {4, 27, 92, 278, 917, 1847, 6336, 8747, 17982, 36124, 36613, 37203, 40444, 40445, 40573, 40574}, {}}

{7662, {4, 274, 26643, 39721}, {}}

{8105, {4, 1114, 15165, 16070}, {}}

{8106, {4, 1113, 15164, 16071}, {}}

{8607, {4, 109, 7450}, {}}

{8608, {4, 101, 4243}, {}}

{8609, {4, 100, 3658, 14266, 39991}, {}}

{8610, {4, 1293, 8050, 39264}, {}}

{8755, {4, 7452, 23987, 26704, 36127}, {}}

{8756, {4, 1897, 32704}, {}}

{9125, {4, 524, 1499, 1992, 6093, 9487, 13608, 15471, 22100, 27088, 34161}, {}}

{9189, {4, 8598, 14255, 18823}, {}}

{9209, {4, 376, 1494, 39263, 40138, 40385}, {}}

{10418, {4, 892, 2696, 7472, 34169}, {}}

{11176, {4, 13586, 35146, 35511}, {}}

{{12077, {4, 5, 53, 311, 327, 1141, 1263, 1487, 2165, 2980, 3459, 3613, 8797, 8800, 10412, 11082, 11087, 11816, 13450, 14225, 15619, 16837, 17500, 17507, 17703, 19712, 19713, 21011, 22261, 22335, 25043, 25148, 27352, 27353, 27356, 27360, 27361, 27364, 32535, 34449, 36300, 36301, 36809, 38305, 38899, 39286, 40449}, {{A,B,C,X(4), X(5)}}

{14273, {4, 468, 2501, 5203, 10603, 14052, 18020, 40118}, {}}

{14325, {4, 492, 3068, 18819, 39387}, {}}

{14326, {4, 491, 3069, 18820, 39388}, {}}

{14425, {4, 145, 519, 3667, 4487}, {}}

{14571, {4, 108, 4246, 23706}, {}}

{16040, {4, 1166, 7488, 15620, 18401}, {}}

{16230, {4, 297, 6530, 14618, 18027, 23582}, {}}

{16317, {4, 30256, 35179, 36877}, {}}

{16318, {4, 1289, 2409, 6529, 23977}, {}}

{21348, {4, 291, 979, 3223, 3501, 7093, 7224, 7346, 7350, 8817, 8927, 13588, 15323, 32937, 39741, 39976}, {}}

{33525, {4, 55, 942, 955, 1859, 8021, 14547, 38007, 41509}, {}}

{40134, {4, 3421, 4221, 18816}, {}}

X(42422) = NINE-POINT-CIRCLE ANTIPODE OF X(5520)

Barycentrics    (a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + a^5*c + a^3*b^2*c - 2*a*b^4*c - a^4*c^2 + a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 + a*b^2*c^3 + 2*a^2*c^4 - 2*a*b*c^4 + b^2*c^4 + a*c^5 - c^6)*(a^6*b - a^4*b^3 - a^2*b^5 + b^7 + a^6*c - 2*a^5*b*c + a^3*b^3*c + a*b^5*c - b^6*c + a^2*b^3*c^2 - 3*b^5*c^2 - a^4*c^3 + a^3*b*c^3 + a^2*b^2*c^3 - 2*a*b^3*c^3 + 3*b^4*c^3 + 3*b^3*c^4 - a^2*c^5 + a*b*c^5 - 3*b^2*c^5 - b*c^6 + c^7) : :
X(42422) = X[20] - 3 X[38711], 3 X[381] + X[38588]

X(42422) lies on the nine-point circle and these lines: {2, 2687}, {4, 1290}, {5, 5520}, {11, 30}, {12, 31522}, {20, 38711}, {113, 513}, {114, 7626}, {115, 8609}, {119, 523}, {123, 2072}, {124, 11813}, {125, 517}, {136, 37982}, {381, 14686}, {403, 5146}, {429, 16221}, {431, 16178}, {442, 3258}, {3259, 9955}, {5099, 30444}, {5511, 11799}, {5840, 36167}, {6941, 38514}, {6949, 38570}, {7477, 38952}, {10017, 37565}, {11698, 36909}, {16177, 21530}, {30445, 38971}

X(42422) = midpoint of X(i) and X(j) for these {i,j}: {4, 1290}, {7477, 38952}
X(42422) = reflection of X(5520) in X(5)
X(42422) = reflection of X(119) in the Euler line
X(42422) = complement of X(2687)
X(42422) = orthocentroidal-circle-inverse of X(14686)
X(42422) = complement of the isogonal conjugate of X(2771)
X(42422) = medial-isogonal conjugate of X(2771)
X(42422) = orthic-isogonal conjugate of X(2771)
X(42422) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 2771}, {42, 3013}, {2771, 10}, {37966, 8062}
X(42422) = X(4)-Ceva conjugate of X(2771)
X(42422) = X(100)-of-reflection-of-Euler-triangle in Euler line
X(42422) = X(5520)-of-Johnson-triangle

X(42423) = NINE-POINT-CIRCLE ANTIPODE OF X(5521)

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

X(42423) lies on the nine-point circle and these lines: {2, 915}, {3, 11}, {4, 13397}, {5, 5521}, {115, 18591}, {116, 18589}, {119, 34332}, {123, 11585}, {124, 21616}, {125, 21530}, {135, 429}, {136, 442}, {517, 15608}, {1214, 38964}, {1807, 38957}, {2072, 5520}, {5139, 30444}, {5164, 36472}, {5190, 6881}, {5511, 15760}, {5514, 42018}, {6842, 20620}, {6882, 13999}, {6907, 38966}, {16178, 37982}, {16221, 30447}, {20621, 21664}

X(42423) = midpoint of X(4) and X(13397)
X(42423) = reflection of X(5521) in X(5)
X(42423) = complement of X(915)
X(42423) = complement of the isogonal conjugate of X(912)
X(42423) = medial-isogonal conjugate of X(912)
X(42423) = orthic-isogonal conjugate of X(912)
X(42423) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 912}, {656, 3139}, {912, 10}, {914, 141}, {1737, 5}, {1795, 6713}, {1807, 18254}, {2252, 2}, {3658, 8062}, {8609, 226}, {18838, 1210}
X(42423) = X(4)-Ceva conjugate of X(912)
X(42423) = X(5521)-of-Johnson-triangle

X(42424) = NINE-POINT-CIRCLE ANTIPODE OF X(16221)

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

X(42424) lies on the nine-point circle and these lines: {2, 16167}, {3, 3258}, {4, 10420}, {5, 12052}, {30, 136}, {113, 924}, {115, 3284}, {122, 10257}, {125, 1568}, {128, 34333}, {131, 523}, {133, 36169}, {135, 403}, {137, 13557}, {1560, 36170}, {5099, 15760}, {5139, 11799}, {5448, 35588}, {5520, 37361}, {11585, 16177}, {15241, 21268}, {15544, 36472}, {18531, 38971}

X(42424) = midpoint of X(i) and X(j) for these {i,j}: {4, 10420}, {13557, 18403}
X(42424) = reflection of X(i) in X(j) for these {i,j}: {16221, 5}, {21268, 23323}
X(42424) = reflection of X(131) in the Euler line
X(42424) = complement of X(32710)
X(42424) = complement of the isogonal conjugate of X(17702)
X(42424) = orthoptic-circle-of-Steiner-inellipse-inverse of X(16167)
X(42424) = medial-isogonal conjugate of X(17702)
X(42424) = orthic-isogonal conjugate of X(17702)
X(42424) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 17702}, {656, 3154}, {3018, 226}, {7471, 8062}, {17702, 10}, {36062, 31379}
X(42424) = X(4)-Ceva conjugate of X(17702)
X(42424) = X(16221)-of-Johnson-triangle

X(42425) = NINE-POINT-CIRCLE ANTIPODE OF X(31845)

Barycentrics    (b - c)^2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3)*(-a^4 + a^3*b - a*b^3 + b^4 + a^3*c - a^2*b*c - 2*b^2*c^2 - a*c^3 + c^4) : :
X(42425) = 3 X[381] + X[14663], 5 X[1698] - X[34196], 3 X[1699] + X[21381], 3 X[34311] + X[34789]

X(42425) lies on the nine-point circle and these lines: {2, 6011}, {4, 759}, {5, 25652}, {11, 1365}, {12, 34194}, {30, 38612}, {113, 946}, {114, 14680}, {117, 7683}, {119, 6246}, {120, 37360}, {121, 9956}, {124, 38390}, {125, 867}, {381, 14663}, {1283, 8229}, {1698, 34196}, {1699, 21381}, {2051, 6044}, {6003, 8286}, {6941, 38511}, {10113, 16160}, {10478, 38480}, {10883, 19642}, {15763, 25640}, {20621, 37362}, {34311, 34789}

X(42425) = midpoint of X(4) and X(759)
X(42425) = reflection of X(31845) in X(5)
X(42425) = complement of X(6011)
X(42425) = complement of the isogonal conjugate of X(6003)
X(42425) = polar-circle inverse of X(30250)
X(42425) = medial-isogonal conjugate of X(6003)
X(42425) = orthic-isogonal conjugate of X(6003)
X(42425) = X(31845)-of-Johnson-triangle
X(42425) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 6003}, {6, 7178}, {513, 6734}, {2605, 35193}, {5174, 20316}, {6003, 10}, {8286, 125}, {13739, 8062}, {31603, 2886}, {33116, 3835}, {34772, 513}, {37583, 522}
X(42425) = X(4)-Ceva conjugate of X(6003)

X(42426) = NINE-POINT-CIRCLE ANTIPODE OF X(38971)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10 + a^8*c^2 + a^4*b^4*c^2 - 2*a^2*b^6*c^2 - 2*a^6*c^4 + a^4*b^2*c^4 + b^6*c^4 - 2*a^2*b^2*c^6 + b^4*c^6 + 2*a^2*c^8 - c^10) : :
X(42426) = 3 X[403] - X[5523]

X(42426) lies on the nine-point circle and these lines: {2, 1304}, {4, 842}, {5, 38971}, {23, 14918}, {25, 16221}, {30, 127}, {53, 38970}, {113, 525}, {114, 7630}, {115, 232}, {122, 858}, {125, 468}, {132, 523}, {133, 16229}, {136, 37981}, {147, 250}, {427, 3258}, {647, 1560}, {1554, 2781}, {1843, 2679}, {2453, 37074}, {2715, 36472}, {2967, 39216}, {3818, 16760}, {5094, 14685}, {5139, 10151}, {5159, 35968}, {5512, 37984}, {5520, 37362}, {6103, 17986}, {8705, 12624}, {10152, 39062}, {10632, 15609}, {10633, 15610}, {11799, 14672}, {15451, 18402}, {16188, 18312}, {20389, 41377}, {30716, 36173}, {31842, 36170}, {35583, 41503}, {36183, 38974}, {37937, 40079}

X(42426) = midpoint of X(i) and X(j) for these {i,j}: {4, 935}, {17986, 35907}
X(42426) = reflection of X(38971) in X(5)
X(42426) = reflection of X(132) in the Euler line
X(42426) = complement of X(2697)
X(42426) = complement of the isogonal conjugate of X(2781)
X(42426) = polar-circle-inverse of X(842)
X(42426) = orthoptic-circle-of-Steiner-inellipse-inverse of X(1304)
X(42426) = Moses Radical circle inverse of X(1560)
X(42426) = medial-isogonal conjugate of X(2781)
X(42426) = orthic-isogonal conjugate of X(2781)
X(42426) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 2781}, {31, 6103}, {2781, 10}, {37937, 8062}
X(42426) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 6103}, {4, 2781}
X(42426) = barycentric product X(18312)*X(37937)
X(42426) = barycentric quotient X(i)/X(j) for these {i,j}: {6103, 2697}, {37937, 5649}
X(42426) = {X(34366),X(35907)}-harmonic conjugate of X(6103)
X(42426) = X(112)-of-reflection-of-Euler-triangle-in-Euler-line
X(42426) = X(38971)-of-Johnson-triangle

X(42427) = ISOTOMIC CONJUGATE OF X(39159)

Barycentrics   -(sqrt(-3*S^2+SW^2)-6*SA+SW)*sqrt(2*OH^2*sqrt(-3*S^2+SW^2)-3*S^2-18*SW*R^2+5*SW^2)+(3*SA-SW)*sqrt(-3*S^2+SW^2)+3*SA^2+6*SB*SC-SW^2 : :
Barycentrics    y + z - x : :, where x:y:z = X(40990)

Contributed by César Lozada, March 25, 2021.

X(42422) lies on the cubic K007 (Lucas cubic) and these lines: {2, 40990}, {20, 3413}, {69, 39159}, {6190, 39158}

X(42427) = anticomplement of X(40990)
X(42427) = isotomic conjugate of X(39159)
X(42427) = cyclocevian conjugate of X(42428)
X(42427) = anticomplementary conjugate of the anticomplement of X(40992)
X(42427) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6190)}} and {{A, B, C, X(4), X(32443)}}

X(42428) = ISOTOMIC CONJUGATE OF X(39158)

Barycentrics   (sqrt(-3*S^2+SW^2)-6*SA+SW)*sqrt(2*OH^2*sqrt(-3*S^2+SW^2)-3*S^2-18*SW*R^2+5*SW^2)+(3*SA-SW)*sqrt(-3*S^2+SW^2)+3*SA^2+6*SB*SC-SW^2 : :
Barycentrics    y + z - x : :, where x:y:z = X(40989)

Contributed by César Lozada, March 25, 2021.

X(42428) lies on the cubic K007 (Lucas cubic) and these lines: {2, 40989}, {20, 3413}, {69, 39158}, {6190, 39159}

X(42428) = anticomplement of X(40989)
X(42428) = isotomic conjugate of X(39158)
X(42428) = cyclocevian conjugate of X(42427)
X(42428) = anticomplementary conjugate of the anticomplement of X(40991)
X(42428) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6190)}} and {{A, B, C, X(4), X(39158)}}

leftri

Gibert points on 7th Evans cubic, K1196: X(42429)-X(42436)

rightri

This preamble and points X(42429)-X(42436) are contributed by Peter Moses, March 29, 2021.

See K1196.



X(42429) = GIBERT(3,10,-13) POINT

Barycentrics    Sqrt[3]*a^2*S - 13*a^2*SA + 20*SB*SC : :

X(42429) lies on the cubic K1196 and these lines: {2, 43226}, {6, 42430}, {13, 15681}, {14, 16}, {15, 11001}, {17, 20}, {62, 5059}, {376, 16808}, {382, 16242}, {396, 15704}, {550, 37832}, {617, 22494}, {621, 36386}, {1657, 16965}, {3412, 42165}, {3529, 10653}, {3534, 12816}, {3543, 10646}, {3830, 16967}, {5073, 16645}, {5237, 33703}, {5318, 19710}, {8703, 33417}, {10304, 42105}, {10645, 15686}, {12820, 15688}, {14269, 33416}, {15640, 18581}, {15682, 37835}, {15683, 42086}, {15684, 16809}, {15685, 41101}, {15689, 42094}, {15691, 23302}, {15695, 42098}, {15697, 42092}, {16267, 42090}, {16962, 42127}, {17800, 42154}, {19708, 42106}, {23263, 35739}, {23303, 35404}, {34200, 42102}, {35400, 42115}, {41100, 42131}, {41107, 42087}, {41108, 41972}, {41121, 42137}, {41122, 42104}

X(42430) = GIBERT(-3,10,-13) POINT

Barycentrics    Sqrt[3]*a^2*S + 13*a^2*SA - 20*SB*SC : :

X(42430) lies on the cubic K1196 and these lines: {2, 43227}, {6, 42429}, {13, 15}, {14, 15681}, {16, 11001}, {18, 20}, {61, 5059}, {376, 16809}, {382, 16241}, {395, 15704}, {550, 37835}, {616, 22493}, {622, 36388}, {1657, 16964}, {3411, 42164}, {3529, 10654}, {3534, 12817}, {3543, 10645}, {3830, 16966}, {5073, 16644}, {5238, 33703}, {5321, 19710}, {8703, 33416}, {10304, 42104}, {10646, 15686}, {12821, 15688}, {14269, 33417}, {15640, 18582}, {15682, 37832}, {15683, 42085}, {15684, 16808}, {15685, 41100}, {15689, 42093}, {15691, 23303}, {15695, 42095}, {15697, 42089}, {16268, 42091}, {16963, 42126}, {17800, 42155}, {19708, 42103}, {23302, 35404}, {34200, 42101}, {35400, 42116}, {35931, 36770}, {41101, 42130}, {41107, 41971}, {41108, 42088}, {41121, 42105}, {41122, 42136}

X(42431) = GIBERT(3,4,-3) POINT

Barycentrics    Sqrt[3]*a^2*S - 3*a^2*SA + 8*SB*SC : :

X(42431) lies on the cubic K1196 and these lines: {3, 36969}, {4, 16}, {5, 5351}, {6, 5073}, {13, 20}, {14, 3627}, {15, 1657}, {17, 550}, {30, 61}, {62, 382}, {140, 5350}, {202, 12953}, {203, 10483}, {376, 41121}, {381, 5237}, {395, 3853}, {396, 15704}, {398, 19107}, {530, 633}, {548, 16241}, {549, 12816}, {630, 33623}, {1656, 10646}, {3091, 16242}, {3105, 5869}, {3146, 10653}, {3200, 6759}, {3201, 37495}, {3206, 13352}, {3364, 42263}, {3365, 42264}, {3389, 35820}, {3390, 35821}, {3391, 35740}, {3392, 42241}, {3411, 17578}, {3522, 5366}, {3523, 16966}, {3529, 36967}, {3533, 42114}, {3534, 5352}, {3543, 5365}, {3544, 12820}, {3830, 16268}, {3843, 36843}, {3845, 16773}, {3850, 16967}, {3851, 11481}, {3858, 23303}, {5056, 33416}, {5059, 5335}, {5068, 42089}, {5076, 42153}, {5343, 42104}, {5868, 36992}, {7006, 12943}, {7755, 41408}, {8739, 35490}, {8740, 34797}, {10299, 42142}, {10654, 33703}, {10721, 36209}, {11001, 16962}, {11600, 15444}, {12103, 16772}, {12155, 33192}, {13491, 36978}, {14269, 41944}, {15681, 16267}, {15682, 42160}, {15683, 41112}, {15684, 41108}, {15687, 16963}, {15696, 16644}, {15712, 33417}, {15720, 42098}, {16960, 42090}, {16961, 42101}, {17800, 22236}, {21735, 42092}, {22832, 37463}, {23004, 41021}, {23251, 42206}, {23261, 42205}, {23302, 33923}, {32789, 42210}, {32790, 42209}, {34755, 42093}, {34783, 36979}, {38335, 41122}, {42177, 42266}, {42178, 42267}

X(42431) = {X(6),X(5073)}-harmonic conjugate of X(42432)

X(42432) = GIBERT(-3,4,-3) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA - 8*SB*SC : :

X(42432) lies on the cubic K1196 and these lines: {3, 36970}, {4, 15}, {5, 5352}, {6, 5073}, {13, 3627}, {14, 20}, {16, 1657}, {18, 550}, {30, 62}, {61, 382}, {140, 5349}, {202, 10483}, {203, 12953}, {376, 41122}, {381, 5238}, {395, 15704}, {396, 3853}, {397, 19106}, {531, 634}, {548, 16242}, {549, 12817}, {629, 33625}, {1656, 10645}, {3091, 16241}, {3104, 5868}, {3146, 10654}, {3200, 37495}, {3201, 6759}, {3205, 13352}, {3364, 35820}, {3365, 35821}, {3366, 42240}, {3367, 42239}, {3389, 42263}, {3390, 42264}, {3412, 17578}, {3522, 5365}, {3523, 16967}, {3529, 36968}, {3533, 42111}, {3534, 5351}, {3543, 5366}, {3544, 12821}, {3830, 16267}, {3843, 36836}, {3845, 16772}, {3850, 16966}, {3851, 11480}, {3858, 23302}, {5056, 33417}, {5059, 5334}, {5068, 42092}, {5076, 42156}, {5344, 42105}, {5869, 36994}, {7005, 12943}, {7755, 41409}, {8739, 34797}, {8740, 35490}, {10299, 42139}, {10653, 33703}, {10721, 36208}, {11001, 16963}, {11601, 15445}, {12103, 16773}, {12154, 33192}, {13491, 36980}, {14269, 41943}, {14814, 35739}, {15681, 16268}, {15682, 42161}, {15683, 41113}, {15684, 41107}, {15687, 16962}, {15696, 16645}, {15712, 33416}, {15720, 42095}, {16960, 42102}, {16961, 42091}, {17800, 22238}, {21735, 42089}, {22831, 37464}, {23005, 41020}, {23251, 42204}, {23261, 42203}, {23303, 33923}, {32789, 42208}, {32790, 42207}, {34754, 42094}, {34783, 36981}, {38335, 41121}, {42175, 42266}, {42176, 42267}

X(42432) = {X(6),X(5073)}-harmonic conjugate of X(42431)

X(42433) = GIBERT(3,2,-7) POINT

Barycentrics    Sqrt[3]*a^2*S - 7*a^2*SA + 4*SB*SC : :

X(42433) lies on the cubic K1196 and these lines: {3, 13}, {4, 5351}, {5, 10646}, {6, 15696}, {14, 1657}, {15, 548}, {16, 20}, {18, 30}, {61, 376}, {62, 550}, {140, 36969}, {202, 15338}, {382, 11481}, {395, 15704}, {396, 33923}, {397, 5352}, {398, 12103}, {511, 22843}, {530, 628}, {532, 22845}, {549, 42165}, {624, 33386}, {631, 16966}, {632, 5350}, {635, 5463}, {1250, 4325}, {2041, 42235}, {2042, 42237}, {3366, 42213}, {3367, 42211}, {3389, 42261}, {3390, 42260}, {3412, 11480}, {3522, 5238}, {3523, 10188}, {3524, 42162}, {3526, 16808}, {3528, 10645}, {3529, 36970}, {3530, 5318}, {3534, 22238}, {3627, 37835}, {3832, 42089}, {3843, 16967}, {3853, 23303}, {3855, 42105}, {3861, 42109}, {4330, 19373}, {5059, 42159}, {5067, 42141}, {5070, 42094}, {5073, 16645}, {5319, 41408}, {5335, 21734}, {5339, 15681}, {5344, 15692}, {5474, 5864}, {6777, 38741}, {6779, 41020}, {7006, 15326}, {7486, 42106}, {7765, 19780}, {7860, 30472}, {8739, 35491}, {8740, 35503}, {10304, 41107}, {10654, 17538}, {11001, 16268}, {11296, 36767}, {11305, 33387}, {12816, 15694}, {12820, 35018}, {13630, 36979}, {13966, 35739}, {15683, 33606}, {15686, 41108}, {15688, 36836}, {15689, 41101}, {15705, 41119}, {15712, 42166}, {15717, 18582}, {16111, 36209}, {16239, 42137}, {16267, 34200}, {16772, 16960}, {16961, 42096}, {17578, 42113}, {17800, 19107}, {18581, 33703}, {19708, 41943}, {22531, 41022}, {33414, 33560}, {33417, 42127}, {34755, 42087}, {35751, 35932}, {36208, 38723}, {36447, 42247}, {36464, 42249}, {37641, 41973}, {41971, 42150}

X(42433) = {X(6),X(15696)}-harmonic conjugate of X(42434)

X(42434) = GIBERT(-3,2,-7) POINT

Barycentrics    Sqrt[3]*a^2*S + 7*a^2*SA - 4*SB*SC : :

X(42434) lies on the cubic K1196 and these lines:{3, 14}, {4, 5352}, {5, 10645}, {6, 15696}, {13, 1657}, {15, 20}, {16, 548}, {17, 30}, {61, 550}, {62, 376}, {140, 36970}, {203, 15338}, {382, 11480}, {395, 33923}, {396, 15704}, {397, 12103}, {398, 5351}, {511, 22890}, {531, 627}, {533, 22844}, {549, 42164}, {623, 33387}, {631, 16967}, {632, 5349}, {636, 5464}, {2041, 42238}, {2042, 42236}, {3364, 42261}, {3365, 42260}, {3391, 42214}, {3392, 42212}, {3411, 11481}, {3522, 5237}, {3523, 10187}, {3524, 42159}, {3526, 16809}, {3528, 10646}, {3529, 36969}, {3530, 5321}, {3534, 22236}, {3627, 37832}, {3832, 42092}, {3843, 16966}, {3853, 23302}, {3855, 42104}, {3861, 42108}, {4325, 10638}, {4330, 7051}, {5059, 42162}, {5067, 42140}, {5070, 42093}, {5073, 16644}, {5319, 41409}, {5334, 21734}, {5340, 15681}, {5343, 15692}, {5473, 5865}, {6778, 38741}, {6780, 41021}, {7005, 15326}, {7486, 42103}, {7765, 19781}, {7860, 30471}, {8739, 35503}, {8740, 35491}, {10304, 41108}, {10653, 17538}, {11001, 16267}, {11306, 33386}, {12817, 15694}, {12821, 35018}, {13630, 36981}, {15683, 33607}, {15686, 41107}, {15688, 36843}, {15689, 41100}, {15705, 41120}, {15712, 42163}, {15717, 18581}, {16111, 36208}, {16239, 42136}, {16268, 34200}, {16773, 16961}, {16960, 42097}, {17578, 42112}, {17800, 19106}, {18582, 33703}, {19708, 41944}, {22532, 41023}, {33415, 33561}, {33416, 42126}, {34754, 42088}, {35931, 36329}, {36209, 38723}, {36446, 42248}, {36465, 42246}, {37640, 41974}, {41972, 42151}

X(42434) = {X(6),X(15696)}-harmonic conjugate of X(42433)

X(42435) = GIBERT(21,2,11) POINT

Barycentrics    7*Sqrt[3]*a^2*S + 11*a^2*SA + 4*SB*SC : :

X(42435) lies on the cubic K1196 and these lines: {2, 18}, {6, 31457}, {13, 3627}, {15, 548}, {17, 5072}, {62, 15712}, {396, 3850}, {397, 15686}, {398, 12812}, {1657, 16965}, {3411, 16772}, {3412, 3843}, {5237, 14891}, {5238, 21735}, {11542, 42415}, {12108, 16773}, {12817, 23046}, {12820, 42154}, {14093, 36836}, {14890, 41944}, {14893, 16267}, {15684, 42157}, {15706, 22238}, {17538, 37640}, {19106, 33703}, {32455, 36757}, {38335, 41101}

X(42436) = GIBERT(-21,2,11) POINT

Barycentrics    7*Sqrt[3]*a^2*S - 11*a^2*SA - 4*SB*SC : :

X(42436) lies on the cubic K1196 and these lines: {2, 17}, {6, 31457}, {14, 3627}, {16, 548}, {18, 5072}, {61, 15712}, {395, 3850}, {397, 12812}, {398, 15686}, {1657, 16964}, {3411, 3843}, {3412, 16773}, {5237, 21735}, {5238, 14891}, {11543, 42416}, {12108, 16772}, {12816, 23046}, {12821, 42155}, {14093, 36843}, {14890, 41943}, {14893, 16268}, {15684, 42158}, {15706, 22236}, {17538, 37641}, {19107, 33703}, {32455, 36758}, {38335, 41100}

leftri

Products X(i)*T for selected triangles T: X(42437)-X(42463)

rightri

This preamble and centers X(42437)-(X42463) are contributed by Clark Kimberling and Peter Moses, March 31, 2021.

If X is a normalized triangle center and T a normalized central triangle, then the left-product X*T is a triangle center. Let P = p : q : r be a point. Centers X(42437)-X(42450) are products X(i)*(cevian triangle of P), given by

p (q + r) (p y + q y + p z + r z) : q (p + r) (p x + q x + q z + r z) : (p + q) r (p x + r x + q y + r y).

Centers X(42451)-X(42463) are products X(i)*(anticevian triangle of P), given by p*((p + q - r)*(p - q + r)*x + (p - q - r)*(p + q - r)*y + (p - q - r)*(p - q + r)*z) : :

The appearance of (i,j,k) in the following list means that X(i)*(cevian triangle of X(j) = X(k):

(1,1,2292), (1,2,10), (1,4,65), (1,6,20969), (1,7,1), (1,8,8), (1,20,5930), (2,1,1962), (2,2,2), (2,3,32078), (2,4,51), (2,6,11205), (2,7,354), (2,8,210), (2,20,154), (2,30,3081), (3,2,5), (3,4,4), (3,5,31389), (3,7,946), (3,8,10), (3,20,2883), (4,1,18673), (4,2,3), (4,3,31388), (4,4,185), (4,7,1071), (4,8,72), (4,20,20), (5,2,140), (5,3,3), (5,4,389), (5,7,12005), (5,8,3678), (6,1,4016), (6,2,141), (6,4,6), (6,6,23642), (6,7,3663), (6,8,2321), (7,2,9), (7,7,14100), (7,8,3059), (8,1,2650), (8,2,1), (8,7,65), (8,8,3057), (9,1,2294), (9,2,142), (9,4,2262), (9,7,7), (9,8,9), (10,1,1), (10,2,1125), (10,7,942), (10,8,960), (11,2,3035), (11,7,5083), (11,8,14740), (12,1,2646), (12,2,4999), (13,2,618), (14,2,619), (15,2,623), (15,4,5318), (16,2,624), (16,4,5321), (17,2,629), (18,2,630), (19,1,18674), (19,2,18589), (19,7,41004), (19,8,219), (19,20,8804), (20,2,4), (20,4,11381), (20,7,12688), (20,8,65), (20,20,5895), (21,2,442), (21,7,3649), (21,8,21677), (22,2,427), (22,4,11550), (23,2,858), (24,2,11585), (25,2,1368), (25,4,1899), (26,2,13371), (26,4,18381), (27,2,440), (28,2,21530), (29,2,18641), (29,7,39791), (31,1,4137), (31,2,2887), (31,4,3914), (31,7,3782), (31,8,3703), (32,2,626), (32,4,5254), (32,7,4920), (32,8,4136), (33,2,34822), (33,7,222), (34,2,34823), (35,2,25639), (35,7,12047), (35,8,6734), (36,2,3814), (36,7,30384), (36,8,6735), (37,1,37), (37,2,3739), (37,7,3664), (37,8,3686), (38,2,1215), (38,6,42), (38,8,4030), (39,2,3934), (39,4,7745), (39,6,39), (39,8,4095), (40,2,946), (40,4,12688), (40,7,4), (40,8,1), (41,2,17046), (41,7,3665), (41,8,40997), (42,1,38), (42,2,3741), (42,4,41011), (42,7,3666), (42,8,3706), (43,2,3840), (43,6,23446), (43,7,982), (43,8,312), (44,2,3834), (44,7,4887), (44,8,2325), (45,2,34824), (45,7,4896), (45,8,3707), (46,2,21616), (46,4,1898), (46,7,1479), (46,8,78), (47,2,34825), (48,2,20305), (48,4,1826), (48,7,41007), (49,2,34826), (49,4,1594), (50,2,34827), (51,2,3819), (51,3,3917), (51,4,11245), (52,2,1216), (52,3,5562), (52,4,6146), (53,2,34828), (53,3,577), (54,2,1209), (54,4,3574), (55,2,2886), (55,4,1836), (55,7,226), (55,8,4847), (56,2,1329), (56,4,1837), (56,7,12053), (56,8,6736), (57,2,3452), (57,4,1864), (57,7,497), (57,8,200), (58,2,3454), (58,4,1834), (58,8,3704), (60,2,34829), (61,2,635), (61,4,397), (62,2,636), (62,4,398), (63,2,226), (63,4,1824), (63,7,1836), (63,8,55), (64,2,2883), (64,4,5895), (64,8,5930), (64,20,13155), (65,1,3057), (65,2,960), (65,4,1858), (65,7,950), (65,8,6737), (66,2,206), (67,2,6593), (67,4,40949), (68,2,1147), (68,4,52), (69,2,6), (69,4,1843), (69,6,31390), (69,7,12723), (69,8,40965), (70,2,34116), (71,1,1953), (71,2,34830), (71,4,1839), (71,8,40937), (72,1,65), (72,2,942), (72,4,1829), (72,7,4292), (72,8,950), (73,2,34831), (73,4,40950), (74,2,113), (74,4,13202), (74,8,6739), (75,1,2667), (75,2,37), (75,6,21752), (75,7,21746), (75,8,3688), (76,2,39), (76,4,40951), (76,8,4531), (77,2,20262), (77,4,1827), (78,2,1210), (78,4,1828), (78,7,56), (78,8,1837), (79,2,3647), (79,8,31938), (80,2,214), (80,7,11570), (81,2,1211), (81,4,40952), (81,7,4854), (81,8,4046), (82,2,21249), (83,2,6292), (84,2,6260), (84,4,40953), (84,8,40), (85,2,1212), (85,7,39789), (86,2,1213), (86,4,40954), (86,7,4890), (86,8,4111), (87,2,34832), (87,7,4941), (87,8,4110), (88,2,16594), (88,8,4152), (90,2,41540), (90,7,10052), (92,2,1214), (92,7,39796), (92,20,3198), (93,2,34833), (94,2,34834), (95,2,233), (96,2,34835), (97,2,34836), (98,2,114), (99,2,115), (99,8,40608), (100,1,2611), (100,2,11), (100,4,38389), (100,7,11), (100,8,11), (101,1,3708), (101,2,116), (101,4,1146), (101,7,1565), (101,8,1146), (102,2,117), (103,2,118), (104,2,119), (104,7,1537), (104,8,1145), (105,2,120), (105,8,40609), (106,2,121), (107,2,122), (107,20,122), (108,2,123), (108,8,7358), (109,2,124), (109,4,38357), (109,7,38357), (109,8,2968), (110,2,125), (110,4,125), (110,8,6741), (111,2,126), (112,2,127), (112,4,1562), (112,20,1562),

The appearance of (i,j,k) in the following list means that X(i)*(anticevian triangle of X(j) = X(k):

(1,1,40), (1,2,8), (1,3,3157), (1,6,3556), (1,8,6552), (1,9,1), (1,10,10), (1,11,21132), (2,1,165), (2,2,2), (2,3,3167), (2,6,154), (2,7,32079), (2,9,3158), (2,10,3971), (2,30,34582), (3,1,1158), (3,2,4), (3,3,155), (3,4,6523), (3,5,5), (3,6,3), (3,9,3811), (3,10,21077), (4,1,1490), (4,2,20), (4,3,3), (4,4,3183), (4,5,41481), (4,6,1498), (4,9,40), (4,30,15774), (5,1,6796), (5,2,3), (5,3,1147), (5,5,15912), (5,6,6759), (5,9,8715), (6,1,1766), (6,2,69), (6,3,6), (6,6,159), (6,10,2321), (7,1,2951), (7,2,144), (7,6,3197), (7,7,15913), (7,9,9), (7,10,21084), (8,1,1), (8,2,145), (8,6,221), (8,8,8834), (8,9,2136), (9,1,9), (9,2,7), (9,3,3211), (9,6,18621), (9,7,17113), (9,9,3174), (9,10,3950), (10,1,3), (10,2,1), (10,6,14529), (10,9,3913), (10,10,3159), (11,1,100), (11,2,100), (11,3,36059), (11,9,100), (11,11,15914), (12,1,411), (12,2,2975), (12,9,3871), (13,2,616), (14,2,617), (15,2,621), (15,3,10661), (16,2,622), (16,3,10662), (17,2,627), (18,2,628), (19,2,4329), (19,3,219), (19,10,8804), (20,1,7992), (20,2,3146), (20,3,12164), (20,4,4), (20,6,64), (20,9,11523), (21,1,191), (21,2,2475), (21,10,3178), (22,2,7391), (23,2,5189), (24,2,37444), (25,2,1370), (25,3,394), (25,6,1619), (25,10,21062), (26,2,14790), (26,6,32321), (27,2,3151), (27,3,22139), (28,3,22136), (29,1,2939), (29,2,3152), (31,1,21375), (31,2,6327), (31,3,22130), (31,10,306), (32,2,315), (32,3,23128), (32,10,4153), (33,1,1763), (33,3,222), (33,9,8270), (34,1,16389), (34,3,7078), (36,2,5080), (37,1,573), (37,2,75), (37,6,18611), (37,10,37), (38,2,17165), (38,10,42), (39,2,76), (39,6,15270), (39,10,21067), (40,1,84), (40,2,962), (40,9,6765), (41,1,1759), (41,2,21285), (41,10,21073), (42,1,1764), (42,2,17135), (42,6,23359), (42,10,321), (43,1,20368), (43,2,10453), (43,10,4135), (44,2,320), (44,10,3943), (45,10,4029), (46,2,11415), (46,3,1069), (48,2,21270), (48,6,1631), (48,10,1826), (49,6,2937), (51,2,2979), (51,3,2), (51,6,11206), (52,2,11412), (52,3,68), (52,6,9833), (53,2,20477), (53,3,577), (54,2,2888), (54,3,195), (54,5,15780), (54,6,2917), (55,1,63), (55,2,3434), (55,3,3173), (55,9,3870), (55,10,226), (55,11,40166), (56,2,3436), (56,9,78), (56,10,21075), (57,1,10860), (57,2,329), (57,9,200), (57,10,21060), (58,2,1330), (58,10,21081), (61,2,633), (62,2,634), (63,1,1709), (63,2,5905), (63,6,55), (63,9,2900), (63,10,4028), (64,2,6225), (64,3,1498), (65,1,20), (65,2,3869), (65,3,1), (65,9,8), (65,10,72), (66,2,5596), (66,3,159), (67,2,11061), (67,3,2930), (67,6,15141), (68,2,6193), (68,3,9937), (68,6,155), (69,1,1721), (69,2,193), (69,3,19588), (69,6,6),(71,1,1765), (71,2,17220), (71,5,1953), (71,6,8053), (71,10,22021), (72,1,4), (72,2,3868), (72,6,1), (72,9,3189), (72,10,2901), (73,6,23361), (73,10,41013), (74,2,146), (74,3,399), (74,6,2935), (75,1,1742), (75,2,192), (75,3,23075), (75,6,21767), (75,9,3169), (75,10,21080), (76,1,32462), (76,2,194), (76,3,19597), (76,6,32445), (76,9,32468), (77,2,5942), (77,6,198), (78,1,46), (78,2,12649), (78,6,56), (79,2,3648), (79,9,191), (80,1,6326), (80,2,6224), (80,9,5541), (81,1,2941), (81,2,2895), (81,10,21085), (82,2,21289), (82,10,21083), (83,2,2896), (84,2,6223), (84,6,12335), (84,9,1490), (85,1,170), (85,2,3177), (85,9,3208), (86,1,2938), (86,2,1654), (88,2,30578), (92,1,2947), (92,2,6360), (92,3,20760), (94,2,18301), (95,2,17035), (98,2,147), (99,2,148), (100,1,1768), (100,2,149), (100,10,21093), (100,11,11), (101,2,150), (101,10,21090), (101,11,1146), (102,2,151), (103,2,152), (104,1,2950), (104,2,153), (104,9,6326), (105,2,20344), (106,2,21290), (106,10,21087), (107,2,34186), (108,2,34188), (109,2,33650), (109,11,38357), (110,2,3448), (110,6,10117), (110,10,21098), (111,2,14360), (112,2,13219), (112,3,22146)



X(42437) = X(1)*T, WHERE T = CEVIAN TRIANGLE OF X(1)

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

X(42437) lies on these lines: {1, 41821}, {8, 20090}, {10, 3995}, {442, 20653}, {519, 17589}, {594, 21808}, {1230, 4647}, {2292, 4733}, {2475, 3679}, {3178, 27812}, {3617, 33100}, {3915, 28634}, {4065, 8040}, {4783, 4968}, {5506, 24963}, {6367, 21124}, {27577, 27798}

X(42438) = X(2)*T, WHERE T = CEVIAN TRIANGLE OF X(9)

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

X(42438) lies on these lines: {9, 1174}, {71, 374}, {165, 169}, {198, 1626}, {354, 1212}, {661, 33570}, {1202, 16601}, {2246, 26744}, {2348, 21811}, {3119, 3740}, {5273, 17490}, {5919, 15853}, {8932, 11203}

X(42439) = X(2)*T, WHERE T = CEVIAN TRIANGLE OF X(10)

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

X(42439) lies on these lines: {2, 1051}, {10, 3995}, {210, 21014}, {1046, 9780}, {1213, 1962}, {1698, 2895}, {3120, 27798}, {3828, 23812}, {4977, 6546}, {6367, 8029}, {20966, 21700}, {27790, 28516}

X(42440) = X(3)*T, WHERE T = CEVIAN TRIANGLE OF X(1)

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

] X(42440) lies on these lines: {1, 1437}, {5, 23555}, {12, 17874}, {37, 2333}, {55, 20832}, {65, 21318}, {201, 22300}, {758, 1482}, {774, 1953}, {855, 2292}, {942, 17463}, {1486, 3295}, {1725, 18180}, {1834, 4516}, {1962, 18673}, {2070, 37621}, {2611, 2650}, {3145, 38336}, {3649, 18210}, {3728, 6042}, {3913, 31395}, {4016, 17444}, {4646, 23668}, {4647, 24390}, {10459, 21326}, {11688, 18719}, {15507, 41591}, {16980, 24431}, {18115, 33592}, {18692, 41007}, {20254, 28628}, {23841, 24433}, {28258, 34977}

X(42441) = X(3)*T, WHERE T = CEVIAN TRIANGLE OF X(3)

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^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) : :

Let A'B'C' be the cevian triangle of X(3). Let A" be the circumcenter of AB'C', and define B" and C" cyclically. Then X(42441) = X(4)-of-A"B"C". (Randy Hutson, July 16, 2021)

X(42441) lies on these lines: {2, 1075}, {3, 54}, {4, 3164}, {5, 324}, {52, 418}, {140, 2972}, {155, 6641}, {216, 217}, {233, 31354}, {264, 13599}, {378, 23709}, {381, 35719}, {417, 9730}, {426, 36752}, {631, 12012}, {648, 40448}, {852, 5462}, {1033, 7395}, {1181, 10608}, {1209, 35442}, {1235, 7399}, {1994, 2055}, {3090, 10184}, {3091, 14635}, {3567, 6638}, {3574, 10600}, {5489, 6368}, {5640, 38281}, {5647, 17814}, {5907, 41212}, {6146, 20975}, {6750, 34836}, {7066, 22350}, {7400, 12251}, {10024, 34333}, {11587, 14118}, {12271, 20794}, {13434, 15781}, {13754, 26897}, {14531, 26907}, {14918, 15780}, {15024, 38283}, {17834, 26898}, {31802, 41169}, {40647, 40948}

X(42441) = isogonal conjugate of polar conjugate of X(34836)
X(42441) = crosspoint of X(3) and X(5)
X(42441) = crosssum of X(4) and X(54)

X(42442) = X(3)*T, WHERE T = CEVIAN TRIANGLE OF X(6)

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

X(42442) lies on these lines: {3, 9019}, {6, 2353}, {32, 23635}, {39, 1843}, {159, 9605}, {184, 3456}, {525, 31869}, {732, 1352}, {826, 3574}, {2971, 39590}, {3001, 7819}, {5007, 20975}, {5041, 9407}, {6292, 22424}, {7794, 16893}, {7822, 20819}, {8362, 16789}, {9971, 20960}, {14660, 14886}, {15861, 28343}, {20967, 20969}

X(42443) = X(5)*T, WHERE T = CEVIAN TRIANGLE OF X(1)

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

X(42443) lies on these lines: {1, 859}, {3, 20718}, {37, 2200}, {65, 3724}, {228, 22300}, {500, 513}, {517, 5496}, {692, 943}, {740, 8669}, {758, 1385}, {851, 11553}, {950, 34969}, {1100, 40955}, {1104, 3725}, {1319, 2650}, {1962, 18673}, {2290, 17438}, {2292, 2646}, {2594, 21319}, {3057, 12081}, {3743, 24929}, {4647, 5440}, {5266, 8618}, {15624, 37529}

X(42444) = X(5)*T, WHERE T = CEVIAN TRIANGLE OF X(6)

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

X(42444) lies on these lines: {6, 27375}, {32, 39684}, {39, 3203}, {54, 826}, {182, 732}, {184, 3456}, {211, 20775}, {576, 8546}, {754, 14133}, {3202, 9605}, {5012, 7760}, {5028, 23642}, {5041, 9418}, {7829, 36213}, {9306, 10191}, {13434, 38664}

X(42445) = X(6)*T, WHERE T = CEVIAN TRIANGLE OF X(3)

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*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(42445) lies on these lines: {3, 14533}, {6, 1154}, {25, 5647}, {53, 14978}, {155, 36751}, {184, 10979}, {216, 217}, {233, 3574}, {570, 1216}, {577, 8565}, {1352, 32428}, {2197, 2252}, {3269, 22052}, {5891, 14576}, {7691, 8882}, {11197, 36412}

X(42445) = isogonal conjugate of polar conjugate of X(1209)
X(42445) = isotomic conjugate of polar conjugate of crosspoint of X(5) and X(6)
X(42445) = isotomic conjugate of polar conjugate of crosssum of X(2) and X(54)

X(42446) = X(7)*T, WHERE T = CEVIAN TRIANGLE OF X(1)

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

X(42446) lies on these lines: {1, 1014}, {9, 24394}, {42, 21871}, {55, 199}, {390, 740}, {497, 21020}, {517, 4343}, {756, 21867}, {758, 4326}, {1108, 35270}, {1253, 3747}, {1334, 21039}, {1621, 17868}, {1697, 2292}, {2269, 2310}, {2293, 2650}, {3882, 10868}, {3958, 14100}, {4433, 21033}, {4516, 21811}, {4642, 40934}, {4647, 12575}, {5274, 27798}, {5281, 10180}, {21346, 37555}, {21673, 38930}

X(42447) = X(7)*T, WHERE T = CEVIAN TRIANGLE OF X(4)

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

X(42447) lies on these lines: {6, 3270}, {7, 2808}, {25, 3197}, {33, 11435}, {41, 8554}, {51, 1824}, {55, 584}, {65, 1827}, {184, 18621}, {185, 1902}, {241, 22440}, {390, 9052}, {511, 10394}, {672, 22079}, {674, 14100}, {916, 5728}, {942, 15030}, {950, 14053}, {1253, 20683}, {1334, 2293}, {1409, 2356}, {1442, 14520}, {1837, 40954}, {1843, 1858}, {1863, 5185}, {2171, 2310}, {2389, 3059}, {2772, 30329}, {2807, 18412}, {2810, 40269}, {3022, 4336}, {3057, 9049}, {3100, 4260}, {3271, 40968}, {3779, 4319}, {3781, 7675}, {3917, 10391}, {3990, 14547}, {5338, 11190}, {5650, 17603}, {6000, 15938}, {6284, 15443}, {7680, 15607}, {10382, 26893}, {10393, 22076}, {11436, 40971}, {11446, 37685}, {18725, 26892}, {21933, 42069}, {22277, 41339}

X(42448) = X(8)*T, WHERE T = CEVIAN TRIANGLE OF X(4)

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

X(42448) lies on these lines: {1, 855}, {4, 151}, {8, 29958}, {19, 3330}, {25, 221}, {31, 8615}, {51, 65}, {52, 14988}, {56, 3937}, {64, 11406}, {109, 37259}, {145, 2810}, {184, 3556}, {185, 1829}, {228, 4300}, {244, 17114}, {373, 3812}, {375, 3698}, {392, 11573}, {427, 20306}, {511, 3869}, {513, 7354}, {517, 16980}, {595, 20999}, {674, 3962}, {774, 18210}, {960, 3917}, {1064, 22345}, {1193, 22344}, {1201, 1401}, {1203, 26889}, {1204, 11383}, {1284, 23440}, {1409, 2354}, {1456, 40985}, {1457, 30493}, {1463, 23675}, {1464, 23383}, {1473, 16466}, {1495, 14529}, {1682, 4414}, {1755, 22070}, {1771, 28077}, {1777, 37397}, {1824, 11381}, {1843, 1858}, {1851, 4295}, {1854, 3270}, {1878, 7686}, {1900, 32062}, {2170, 23630}, {2262, 17634}, {2392, 3878}, {2594, 23844}, {2650, 21746}, {2800, 31825}, {2835, 15556}, {2841, 5903}, {2842, 3874}, {3057, 8679}, {3259, 7681}, {3271, 3924}, {3436, 31785}, {3754, 15049}, {3784, 19861}, {3884, 23156}, {3890, 23155}, {3898, 23157}, {4084, 31757}, {4418, 9565}, {4642, 23638}, {5057, 15488}, {5562, 5887}, {5650, 25917}, {5730, 37482}, {6737, 29353}, {7428, 34586}, {7959, 12174}, {10441, 11415}, {10571, 28348}, {11391, 11550}, {12514, 22076}, {12526, 26893}, {12711, 17441}, {13724, 37558}, {13747, 35059}, {13754, 40266}, {15030, 31937}, {17562, 19368}, {18180, 39542}, {19367, 37254}, {20323, 41682}, {23544, 23636}, {26377, 26883}, {26884, 34043}, {33899, 34462}, {37516, 37614}

X(42449) = X(9)*T, WHERE T = CEVIAN TRIANGLE OF X(9)

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

X(42449) lies on these lines: {7, 3177}, {9, 1174}, {77, 2124}, {664, 1223}, {1202, 5572}, {1212, 2293}, {1441, 41006}, {2082, 4326}, {3119, 6666}, {8012, 15185}, {9502, 40937}, {15837, 38375}

X(42450) = X(10)*T, WHERE T = CEVIAN TRIANGLE OF X(4)

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

X(42450) lies on these lines: {1, 8679}, {4, 12930}, {5, 34825}, {6, 1245}, {10, 375}, {25, 14529}, {31, 23843}, {42, 23844}, {44, 10822}, {47, 11334}, {51, 65}, {52, 5887}, {56, 26892}, {72, 674}, {73, 23383}, {124, 7683}, {143, 14988}, {185, 1839}, {221, 17810}, {354, 23154}, {389, 6001}, {511, 960}, {513, 4292}, {515, 34434}, {516, 22300}, {517, 5446}, {518, 29958}, {551, 23156}, {568, 40266}, {581, 3185}, {601, 2933}, {602, 1626}, {758, 31757}, {859, 37836}, {916, 31732}, {946, 31825}, {959, 9309}, {970, 4640}, {997, 37482}, {999, 41682}, {1064, 23361}, {1066, 18613}, {1104, 3271}, {1125, 2392}, {1191, 1469}, {1203, 3220}, {1265, 25304}, {1399, 37259}, {1425, 1456}, {1486, 7078}, {1593, 34935}, {1695, 24708}, {1824, 1898}, {1829, 1852}, {1854, 11436}, {2179, 8608}, {2183, 4300}, {2361, 3145}, {2650, 20961}, {2654, 14055}, {2771, 11557}, {2778, 11807}, {2779, 18483}, {2800, 31760}, {2807, 9856}, {2810, 34791}, {2818, 7686}, {2835, 12432}, {3057, 16980}, {3060, 3869}, {3157, 11365}, {3555, 9026}, {3622, 23155}, {3636, 23157}, {3683, 22076}, {3784, 25524}, {3812, 5943}, {3827, 9969}, {3868, 30438}, {3917, 25917}, {3937, 32636}, {4303, 20470}, {4337, 16453}, {4642, 20962}, {4646, 23638}, {5057, 41723}, {5348, 28077}, {5462, 34339}, {5480, 20306}, {5752, 12514}, {5777, 40635}, {5836, 23841}, {5892, 40296}, {5904, 9049}, {6684, 38472}, {7299, 13733}, {8614, 26884}, {9119, 34146}, {9729, 9943}, {9961, 10574}, {10176, 31737}, {10178, 17704}, {10441, 24703}, {10571, 20122}, {10693, 13417}, {12047, 18180}, {12572, 22299}, {12711, 14557}, {13754, 31937}, {16466, 22654}, {18178, 24210}, {20727, 24511}, {21616, 37536}, {21969, 31165}, {22053, 28270}, {23630, 40133}, {25681, 37521}

X(42450) = crosspoint of X(4) and X(58)
X(42450) = crosssum of X(3) and X(10)
X(42450) = X(178)-of-orthic-triangle if ABC is acute

X(42451) = X(1)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(4)

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

X(42451) lies on these lines: {1, 196}, {4, 3668}, {158, 278}, {347, 1895}, {459, 39130}, {1068, 18678}, {1214, 3346}, {1498, 32714}, {1854, 4295}, {3176, 5930}, {3182, 8802}, {6223, 36118}

X(42452) = X(2)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(7*a^8 - 4*a^6*b^2 - 14*a^4*b^4 + 12*a^2*b^6 - b^8 - 4*a^6*c^2 + 28*a^4*b^2*c^2 - 12*a^2*b^4*c^2 - 12*b^6*c^2 - 14*a^4*c^4 - 12*a^2*b^2*c^4 + 26*b^4*c^4 + 12*a^2*c^6 - 12*b^2*c^6 - c^8) : :

X(42452) lies on these lines: {2, 15312}, {4, 64}, {20, 12090}, {107, 3079}, {631, 3346}, {1249, 10192}, {3090, 33546}, {3424, 7714}, {3523, 20329}, {5667, 33702}, {6353, 16318}, {6618, 11245}

X(42452) = X(2)-of-anticevian-triangle-of-X(4)

X(42453) = X(2)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(5)

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

X(42453) lies on these lines: {3, 1075}, {5, 324}, {30, 568}, {51, 32428}, {143, 13322}, {216, 12012}, {418, 35360}, {523, 10192}, {632, 6662}, {1368, 2967}, {1994, 41202}, {2052, 30258}, {2790, 41580}, {3164, 3168}, {5891, 14640}, {6676, 16318}, {9755, 9909}, {10691, 15312}, {12077, 40588}, {18282, 24385}

X(42454) = X(2)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(11)

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

X(42454) lies on these lines: {11, 15914}, {51, 513}, {55, 885}, {210, 522}, {354, 514}, {523, 1962}, {650, 5432}, {693, 3816}, {900, 33519}, {1621, 17494}, {3058, 11193}, {3703, 4391}, {3752, 23811}, {4124, 21132}, {4995, 11124}, {6284, 11247}, {17728, 35348}

X(42454) = X(650)-Ceva conjugate of X(11)

X(42455) = X(3)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(11)

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

X(42455) lies on these lines: {2, 33528}, {4, 513}, {8, 885}, {10, 522}, {85, 693}, {158, 17924}, {341, 4397}, {442, 1577}, {499, 905}, {514, 946}, {521, 6238}, {667, 2217}, {900, 14304}, {1212, 28143}, {1734, 18395}, {2401, 3086}, {2517, 11024}, {3667, 9948}, {3704, 4086}, {3762, 6366}, {4448, 37009}, {4462, 20220}, {6554, 28132}, {11607, 36802}, {14010, 40213}, {14505, 23100}, {16732, 21134}, {21132, 23615}, {30591, 31936}

X(42455) = Kirikami-Euler image of X(11)

X(42456) = X(4)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(10)

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

X(42456) lies on these lines: {10, 201}, {307, 17864}, {522, 31730}, {648, 15796}, {726, 1766}, {758, 2901}, {1125, 22465}, {1745, 6360}, {3159, 3950}, {3178, 3971}, {3678, 21084}, {3682, 4552}, {18477, 27378}, {20222, 22350}, {22381, 42027}

X(42457) = X(5)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^12 - 7*a^10*b^2 - 2*a^8*b^4 + 18*a^6*b^6 - 17*a^4*b^8 + 5*a^2*b^10 - 7*a^10*c^2 + 16*a^8*b^2*c^2 - 18*a^6*b^4*c^2 + 12*a^4*b^6*c^2 + a^2*b^8*c^2 - 4*b^10*c^2 - 2*a^8*c^4 - 18*a^6*b^2*c^4 + 10*a^4*b^4*c^4 - 6*a^2*b^6*c^4 + 16*b^8*c^4 + 18*a^6*c^6 + 12*a^4*b^2*c^6 - 6*a^2*b^4*c^6 - 24*b^6*c^6 - 17*a^4*c^8 + a^2*b^2*c^8 + 16*b^4*c^8 + 5*a^2*c^10 - 4*b^2*c^10) : :

X(42457) lies on these lines: {3, 14363}, {4, 64}, {107, 1498}, {140, 15274}, {154, 1075}, {1093, 1192}, {1656, 33546}, {1657, 23240}, {3168, 11425}, {3346, 3523}, {3517, 33582}, {5894, 36876}, {8567, 41372}, {10282, 15576}, {10606, 14249}, {14361, 34782}, {15712, 20329}, {17821, 38808}, {35360, 35602}

X(42458) = X(6)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^10 + 3*a^8*b^2 - 14*a^6*b^4 + 14*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 - 14*a^6*c^4 - 14*a^4*b^2*c^4 + 30*a^2*b^4*c^4 - 2*b^6*c^4 + 14*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(42458) lies on these lines: {2, 14091}, {3, 1033}, {4, 41489}, {6, 15311}, {20, 2138}, {112, 36413}, {347, 17903}, {390, 21148}, {393, 800}, {3172, 36965}, {5065, 40138}, {6353, 16318}, {6525, 33581}, {6527, 41678}, {6804, 33546}, {8879, 10565}, {12250, 14642}, {13488, 40065}

X(42458) = polar conjugate of isogonal conjugate of X(1661)
X(42458) = polar conjugate of isotomic conjugate of X(6225)
X(42458) = polar conjugate of cyclocevian conjugate of X(35510)

X(42459) = X(6)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(5)

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

Let E be the inellipse that is the barycentric square of the Euler line, centered at X(23583). Then X(42459) is the intersection of the tangents to E at X(36412) (the barycentric square of X(5)) and X(36413) (the barycentric square of X(20)). (Randy Hutson, July 16, 2021)

X(42459) lies on these lines: {3, 393}, {4, 31363}, {5, 53}, {6, 30}, {20, 1249}, {22, 16318}, {26, 1609}, {140, 36751}, {217, 31802}, {230, 10154}, {232, 1368}, {297, 3164}, {343, 13157}, {376, 33630}, {382, 3087}, {418, 14569}, {427, 22240}, {441, 17907}, {459, 37877}, {548, 36748}, {549, 10979}, {550, 577}, {566, 13371}, {570, 23335}, {800, 5254}, {1033, 11414}, {1074, 40937}, {1108, 23537}, {1172, 37468}, {1350, 15312}, {1657, 38292}, {1658, 8553}, {1865, 8727}, {1907, 26216}, {2052, 26906}, {2070, 41758}, {2165, 13383}, {2207, 12362}, {2794, 34774}, {3146, 40065}, {3163, 15686}, {3284, 15704}, {3627, 5158}, {5059, 5702}, {5112, 41584}, {5179, 40943}, {5286, 39568}, {5304, 34608}, {5305, 7387}, {5359, 34658}, {5721, 40979}, {6527, 20208}, {6530, 42329}, {6747, 26905}, {6823, 27376}, {7735, 9909}, {7736, 34609}, {8703, 18487}, {8745, 12605}, {8963, 15235}, {9308, 41008}, {9605, 34938}, {9607, 13341}, {9722, 15761}, {10226, 15109}, {11063, 12107}, {11563, 16328}, {13155, 20265}, {13322, 41334}, {13406, 18573}, {14091, 40234}, {15355, 30739}, {15466, 20207}, {15594, 30549}, {15681, 33636}, {15699, 36430}, {15912, 41523}, {17849, 23128}, {18591, 37424}, {18643, 37448}, {18685, 36029}, {21309, 34726}, {27377, 40853}, {30435, 31305}, {35007, 40136}, {37201, 41489}, {38920, 41465}, {41480, 41481}

X(42460) = X(7)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(3)

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

X(42460) lies on these lines: {1, 7083}, {3, 77}, {6, 29957}, {55, 15374}, {521, 3126}, {651, 7071}, {912, 6767}, {916, 2293}, {954, 3564}, {1069, 3477}, {1200, 2280}, {3270, 23144}, {3781, 7078}, {5942, 28044}, {13754, 15937}, {17262, 23874}, {18621, 34371}, {20760, 38288}, {22117, 22132}, {23078, 23079}, {36059, 37541}

X(42460) = isogonal conjugate of polar conjugate of X(30694)
X(42460) = isogonal conjugate of X(7)-cross conjugate of X(4)

X(42461) = X(8)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(3)

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

X(42461) lies on these lines: {1, 7083}, {3, 63}, {6, 29958}, {25, 3868}, {38, 1036}, {56, 4641}, {64, 2808}, {65, 1773}, {101, 4306}, {144, 37399}, {145, 28029}, {155, 10680}, {159, 9021}, {218, 1400}, {219, 23620}, {329, 37415}, {394, 23154}, {404, 26866}, {405, 17257}, {517, 39568}, {518, 3556}, {521, 6161}, {651, 1398}, {758, 9798}, {942, 5020}, {956, 3564}, {958, 4643}, {960, 22769}, {999, 1201}, {1046, 1460}, {1069, 3478}, {1147, 16203}, {1245, 6391}, {1423, 5247}, {1425, 23144}, {1482, 12309}, {1593, 12528}, {1597, 40263}, {1598, 24474}, {1633, 3189}, {1722, 28039}, {1724, 21362}, {1858, 16541}, {2178, 21874}, {2286, 20818}, {2292, 3295}, {2800, 9910}, {3191, 19782}, {3193, 11401}, {3218, 37257}, {3219, 37246}, {3220, 11523}, {3487, 25514}, {3732, 7754}, {3869, 8192}, {3874, 11365}, {3876, 7484}, {3901, 8185}, {3913, 4952}, {4185, 5905}, {4186, 12649}, {4223, 11036}, {5044, 16419}, {5777, 11479}, {5904, 8193}, {6147, 7535}, {6642, 24475}, {6743, 24309}, {7011, 23159}, {7078, 7193}, {7289, 37613}, {7393, 31835}, {7532, 20256}, {9909, 37547}, {10529, 28034}, {10822, 34931}, {12109, 17810}, {12164, 22770}, {12635, 22654}, {15934, 28787}, {16049, 20078}, {20007, 37328}, {20013, 35998}, {22120, 22144}, {22148, 23072}, {23168, 38290}, {36059, 41426}, {37541, 38903}

X(42461) = isogonal conjugate of polar conjugate of X(30699)
X(42461) = isogonal conjugate of X(8)-cross conjugate of X(4)

X(42462) = X(9)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(11)

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

X(42462) is the perspector of the circumhyperbola that is the locus of trilinear products of Feuerbach hyperbola antipodes. (Randy Hutson, July 16, 2021)

X(42462) lies on these lines: {7, 514}, {9, 522}, {19, 649}, {37, 650}, {346, 3239}, {393, 7649}, {513, 2262}, {523, 2294}, {654, 4984}, {661, 1901}, {663, 4336}, {665, 30572}, {693, 20905}, {1086, 21133}, {1278, 25259}, {3059, 3900}, {3700, 21033}, {4000, 21202}, {4171, 42337}, {4391, 20895}, {4500, 27417}, {4530, 14393}, {4534, 14442}, {4820, 40137}, {6545, 23760}, {6546, 9318}, {10006, 14476}, {15914, 38375}, {17412, 17418}, {17420, 24121}, {21131, 23775}, {21143, 23764}, {23615, 33573}, {23748, 24002}, {27486, 31346}, {28161, 31325}

X(42462) = isogonal conjugate of X(4619)

X(42463) = X(10)*T, WHERE T = ANTICEVIAN TRIANGLE OF X(3)

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

X(42463) lies on these lines: {1, 1762}, {3, 48}, {6, 27802}, {56, 3173}, {58, 15376}, {60, 28606}, {63, 1437}, {72, 184}, {101, 580}, {110, 3868}, {154, 37547}, {155, 11249}, {182, 5044}, {206, 518}, {255, 20803}, {354, 28787}, {394, 11573}, {405, 26885}, {474, 26889}, {517, 6759}, {525, 24286}, {578, 5777}, {603, 23169}, {692, 3811}, {758, 14529}, {912, 960}, {942, 9306}, {971, 13346}, {975, 5135}, {999, 1201}, {1036, 1069}, {1071, 1092}, {1098, 18042}, {1125, 9028}, {1193, 22130}, {1386, 5045}, {1420, 36059}, {1451, 4245}, {1714, 5137}, {1780, 2352}, {2174, 19763}, {2175, 5266}, {2217, 9928}, {2323, 5752}, {2915, 26893}, {3045, 12532}, {3215, 23067}, {3220, 37482}, {3292, 23154}, {3564, 15985}, {3876, 5012}, {3927, 3955}, {5138, 37594}, {5439, 5651}, {5767, 27410}, {5769, 34831}, {5791, 37527}, {5927, 11424}, {6638, 20764}, {7113, 19762}, {7535, 37543}, {7561, 26942}, {10539, 24474}, {11363, 14054}, {12233, 31832}, {12514, 20986}, {12528, 34148}, {13352, 40263}, {13754, 35203}, {17220, 31900}, {19365, 34048}, {19597, 23075}, {20739, 23620}, {22276, 39582}, {23072, 23089}, {23074, 23172}, {23383, 35327}, {23841, 39523}, {24320, 36742}, {29958, 34986}, {31835, 32046}

X(42463) = isogonal conjugate of polar conjugate of X(3187)
X(42463) = isogonal conjugate of X(10)-cross conjugate of X(4)
X(42463) = isotomic conjugate of polar conjugate of X(5301)

leftri

Perspectors associated with product triangles: X(42464)-X(42471)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, March 28, 2021.

Let T1 = A1B1C1 be a triangle, and let M1 be the matrix whose rows are the normalized barycentrics of A1, B1, C1, respectively. Let T2 = A2B2C2 be a triangle, and let M2 be the matrix whose rows are the normalized barycentrics of A2, B2, C2, respectively. Let M be the matrix product M1*M2. The triangle product T1*T2 is here defined as the triangle whose vertices are given by the rows of M1*M2. The noncommutative operation denoted by *, here named triangle multiplication, is associative.

Abbreviate the cevian triangle of a pont P = p : q : r as cevian(P) and the anticevian triangle of U = u : v : w as anticevian(U). Let

T1 = A'B'C' = anticevian(P), so that A' = -p : q : r
T2 = A''B''C'' = anticevian triangle(U), so that A'' = -u : v : w.

Then T1*T2 is perspective to ABC, and the perspector, f(P,U), is given by

u (p u^2 - q u^2 + r u^2 - 2 r u v - p v^2 + q v^2 + r v^2 + 2 q u w + 2 p v w - p w^2 - q w^2 - r w^2) (p u^2 + q u^2 - r u^2 + 2 r u v - p v^2 - q v^2 - r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 + r w^2)
:
-v (p u^2 - q u^2 + r u^2 - 2 r u v - p v^2 + q v^2 + r v^2 + 2 q u w + 2 p v w - p w^2 - q w^2 - r w^2) (p u^2 + q u^2 + r u^2 - 2 r u v - p v^2 - q v^2 + r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 - r w^2)
:
-w (p u^2 + q u^2 + r u^2 - 2 r u v - p v^2 - q v^2 + r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 - r w^2) (p u^2 + q u^2 - r u^2 + 2 r u v - p v^2 - q v^2 - r v^2 - 2 q u w + 2 p v w - p w^2 + q w^2 + r w^2)

The appearance of (i,j,k) in the following list means that f(X(i),X(j)) = X(k):

(1,1,84)
(1,2,7)
(1,3,1069)
(1,6,7169)
(1,8,6553)
(1,9,200)
(1,10,10)
(1,11,522)
(2,1,3062)
(2,2,2)
(2,3,6391)
(2,6,64)
(2,9,3680)
(2,10,42027)
(3,1,42464)
(3,2,69)
(3,3,15316)
(3,4,3346)
(3,5,5)
(3,6,25)
(3,11,513)
(4,1,3345)
(4,2,253)
(4,3,3)
(4,4,42465)
(5,2,264)
(5,3,68)
(5,5,42466)
(5,6,32319)
(6,1,42467)
(6,2,4)
(6,3,394)
(6,4,42468)
(6,5,13157)
(6,6,34207)
(6,10,226)
(6,11,4391)
(7,1,8917)
(7,2,10405)
(7,6,7152)
(7,9,9)
(8,1,1)
(8,2,4373)
(8,3,42469)
(8,6,2192)
(9,1,57)
(9,2,8)
(9,9,42470)
(9,10,4052)
(9,11,514)
(10,1,4)
(10,2,75)
(10,10,42471)
(10,11,523)
(11,1,513)
(11,2,693)
(11,3,521)
(11,9,3900)
(11,10,522)
(13,2,19776)
(14,2,19777)
(15,2,2992)
(16,2,2993)
(17,2,19712)
(18,2,19713)
(19,2,7219)
(19,3,222)
(20,2,35510)
(20,4,4)
(20,6,154)

The locus of a point X = x : y : z such that f(P,X) lies on this line: at infinity is the bicevian conic of the points X(2) = 1 : 1 : 1 and -p + q + r : p - q + r : p + q - r, given by

(p - q + r)(p + q - r) x^2 + 2 p (p - q - r) y z + (cyclic) = 0.

This conic, denoted by BC(P), has center q + r : r + p : p+ q and perspector

(p - q - r)/(p^2 - q r - r p - p q) : (q - r - p)/(q^2 - r p - p q - q r) : (r - p - q)/(r^2 - p q - q r - r p)

If P lies inside the medial triangle, then BC(P) is an ellipse. If P lies on the nine-point circle, then BC(P) is a right hyperbola.

The appearance of (i, [name]) in the following list means that BC(X(i)) is the named conic:

(2, Steiner inellipse), (3, nine-point circle), (5, bicevian conic of X(2) and X(3), (10, circumellipse of the medial and incentral triangles [see X(34585)], (115, Kiepert circumhyperbola of the medial triangle) (125, Jerabek circumhyperbola of the medial triangle)

The appearance of (i, {n1, n2, ..., nk}) in the next list means that BC(X(i)) is the conic that passes through the points: X(n1), X(n2), ..., X(nk):

(1, {11, 1145, 1146, 2968, 3756, 4904, 6739, 6741, 7358, 8286, 16613, 38992, 39004, 39050, 40608, 40609})
(4, {122, 3184, 13611, 39020, 40616})
(6, {125, 5181, 6388, 7358, 15526, 15595, 17421, 26932, 40618, 40626})
(9, {11, 1086, 8287, 10427, 13609, 16591, 16592, 16593, 16594, 16595, 16596, 16597, 20343, 21623, 26932, 34846, 38989, 39007, 39063, 40615, 40617, 40622, 40629}
(37, {244, 1086, 2968, 4858, 5515, 6377, 16586, 17755, 38995, 39040, 40619, 40624, 40626})
(39, {115, 339, 3124, 5976, 7664, 21208, 36901, 39000, 40619})
(113, {3, 125, 2088, 3134, 39174, 39987})
(114, {3, 115, 868, 17423, 34156, 34810, 38997, 39078})
(116, {3, 118, 354, 2140, 3136, 3789, 5452, 20970, 32664, 39029, 39046, 40586, 40591})
(118, {3, 116, 3138, 14714, 39006})
(119, {3, 11, 3139, 34467, 39175})
(120, {3, 1015, 3140, 3675, 5511, 34160, 39025})
(122, {3, 4, 133, 800, 1249, 3184, 6523, 14363, 15259, 16253, 20208, 23976, 33549, 33580})
(123, {3, 56, 429, 12610, 25640, 36103, 40590})
(124, {3, 65, 117, 478, 3142, 3454, 24220, 34281, 36033, 39037, 39070, 40611, 40613})
(126, {3, 1084, 3143, 5512, 21906, 34158})
(127, {3, 32, 132, 427, 3162, 21248, 22391, 39045, 39071, 39086, 40588, 40959})
(130, {3, 129, 389, 3819, 21243, 34850})
(132, {3, 127, 3150, 35071, 41172})
(136, {3, 131, 216, 6389, 10600, 24245, 24246, 31377, 33553, 34833, 34851, 34853, 35067, 37565, 37864})
(137, {3, 128, 140, 570, 6592, 8562, 15345, 17707, 21975, 23702, 34828, 39171})
(140, {137, 2972, 6592, 8902, 17433, 35442, 39019})
(141, {125, 1084, 3124, 6593, 7668, 8054, 15450, 17413, 36213, 38987, 38988, 38989, 38990, 38991, 38992, 38993, 38994, 38995, 38996, 38997, 38998, 39067, 39068, 39075, 39079, 39080, 40601})
(142, {11, 3119, 6594, 35508, 38991})
(216, {{136, 338, 2972, 14920, 15526, 34834, 36901, 38987, 40624}})
(226, {6506, 26932, 31653, 34591, 35072, 39006})
(230, {441, 868, 35088, 36212, 38987})
(233, {5522, 15526, 35442, 38997, 39081, 40604})
(244, {10, 37, 960, 1125, 3739, 4075, 16597, 18589, 19563, 20529, 21249, 31845, 34587, 34851, 35068, 40607})

Inverse triangles are introduced in the preamble just before X(42005). If T1 and T2 are triangles and T1 is non-degenerate (so that its matrix is invertible), then there exists a unique triangle T such that T1*T = T2, and the solution is T = inverse(T1)*T2. Likewise, there exists a unique triangle T such that T*T1 = T2, and the solution is T = T2*inverse(T1). Following are eight examples:


1. Let P = p : q : r and U = u : v : w. Let T1 = cevian(P) and T2 = cevian(U). The triangle T = A'B'C' such that T1*T = T2 is given by

A' = u (v + w) (2 p u + q u + r u + p v + r v + p w + q w) : v (u + w) (-q u - r u + p v - r v + p w + q w), (u + v) w (-q u - r u + p v + r v + p w - q w)
B' = u (v + w) (q u - r u - p v - r v + p w + q w) : v (u + w) (q u + r u + p v + 2 q v + r v + p w + q w) : (u + v) w (q u + r u - p v - r v - p w + q w)
C' = u (v + w) (-q u + r u + p v + r v - p w - q w) : -v (u + w) (-q u - r u + p v - r v + p w + q w) : (u + v) w (q u + r u + p v + r v + p w + q w + 2 r w)

The locus of a point U = X = x : y : z such that T is perspective to the medial triangle is the cubic given by

(p + r) (q + r) y^2 z + (p + q) (q + r) y z^2 + (cyclic) + 2 (p^2 + q^2 + r^2 + 3(q r + r p + p q)) x y z = 0.

For P = X(148), this cubic is K185. For P = X(150), the cubic passes through X(i) for i = 7, 8, 80, 320, 42482.


2. Let P = p : q : r and U = u : v : w. Let T1 = cevian(P) and T2 = cevian(U). The triangle T = A'B'C' such that T*T1 = T2 is given by

A' = (q + r) (r v + q w) : (p + r) (-r v + q w) : -(p + q) (-r v + q w)
B' = (q + r) (-r u + p w) : (p + r) (r u + p w) : -(p + q) (-r u + p w)
C' = (q + r) (-q u + p v), -(p + r) (-q u + p v), (p + q) (q u + p v)

The locus of a point U = X = x : y : z such that T is perspective to the medial triangle is the cubic given by

p^2 (r (p+r) (q+r) y^2 z - q (p+q) (q+r) y z^2) + (cyclic) = 0.

This cubic passes through X(2) for every point P. For P = X(4), the cubic is K621. The appearance of (i, {n1, n2, ..., nk}) in the next list means that the cubic passes through the points: X(n1), X(n2), ..., X(nk):

(1, {1,2,6,37,81,3293,17147,39949,39964})
(3, {2,3,97,216,577})
(4, [K621],{2,4,6,24,393,847,2052,6515})
(5,{2,5,233,31610,36412})
(6, {2,6,32,39,251,8267})
(7, {1,2,7,279,1088,1445,36845})
(8, {2,8,9,78,312,318,329,346})
(9, {2,9,220,1212,6605})
(10, {2,10,594,1213,6539})
(20, {2,20,1249,6616,18623,27382,36413,37669,41084})
(63, {2,63,394,1214,1812})
(69, {2,3,69,305,1370,3926,20806})
(74, {2,74,323,3003,10419,14264,14910,40353})
(75, {2,10,75,76,310,4043,17135,40004,40005})
(76, {2,76,141,1502,40016})
(99, {2,99,523,4590,14061,14089,31614})
(100, {2,100,650,1252,31615})
(101, {2,101,6586,14085,23990,31616})
(110, {2,110,647,13198,23357})
(190, {2,190,514,1016,6632,14087,27191})
(192, {2,43,192,6376,41840})
(193, {2,193,439,6337,6353,18287})
(239, {2,239,4366,6654,17755})
(385, {2,385,4027,5976,40820})


3. Let P = p : q : r and U = u : v : w. Let T1 = cevian(P) and T2 = anticevian(U). The triangle T = A'B'C' such that T1*T = T2 is given by

A' = u (p u^2 - p u v - q u v + q v^2 - p u w - r u w - q v w - r v w + r w^2)
: v (q u^2 - p u v - q u v + p v^2 + p u w + r u w + q v w + r v w - p w^2 - q w^2 - r w^2)
: w (r u^2 + p u v + q u v - p v^2 - q v^2 - r v^2 - p u w - r u w + q v w + r v w + p w^2)

B' = u (q u^2 - p u v - q u v + p v^2 + p u w + r u w + q v w + r v w - p w^2 - q w^2 - r w^2)
: v (p u^2 - p u v - q u v + q v^2 - p u w - r u w - q v w - r v w + r w^2)
: w (-p u^2 - q u^2 - r u^2 + p u v + q u v + r v^2 + p u w + r u w - q v w - r v w + q w^2)

C' = u (r u^2 + p u v + q u v - p v^2 - q v^2 - r v^2 - p u w - r u w + q v w + r v w + p w^2)
: v (-p u^2 - q u^2 - r u^2 + p u v + q u v + r v^2 + p u w + r u w - q v w - r v w + q w^2)
: w (p u^2 - p u v - q u v + q v^2 - p u w - r u w - q v w - r v w + r w^2)

The triangle T is perspective to ABC for all P and U. The perspector, denoted here by g(P,U), is given by

g(P,U) = u (r u^2 + p u v + q u v - p v^2 - q v^2 - r v^2 - p u w - r u w + q v w + r v w + p w^2) (q u^2 - p u v - q u v + p v^2 + p u w + r u w + q v w + r v w - p w^2 - q w^2 - r w^2)
: v (-p u^2 - q u^2 - r u^2 + p u v + q u v + r v^2 + p u w + r u w - q v w - r v w + q w^2) (q u^2 - p u v - q u v + p v^2 + p u w + r u w + q v w + r v w - p w^2 - q w^2 - r w^2)
: w (r u^2 + p u v + q u v - p v^2 - q v^2 - r v^2 - p u w - r u w + q v w + r v w + p w^2) (-p u^2 - q u^2 - r u^2 + p u v + q u v + r v^2 + p u w + r u w - q v w - r v w + q w^2).

For U = P, this perspector is

(-p^2 + q^2 - r^2) (p^2 + q^2 - r^2) : (p^2 - q^2 - r^2) (p^2 + q^2 - r^2) : (p^2 - q^2 - r^2) (p^2 - q^2 + r^2)

The appearance of (i,j) in the following list means that g(X(i),X(i)) = nX(j):

(1,4), (2,2), (3,68), (4,3346), (5,6662), (6,66), (7,42483), (8,6553), (9,6601), (10,596), (19,42483), (37,13476), (39,27375), (57,34546), (63,6504), (65,42844), (69,6339), (75,2998), (76,42845), (92,34287), (115,36955), (141,6664), (174,189), (188,8), (216, 42846), (365,7357), (366,7), (508,10405), (509,8048), (556,39694), (1577,15412), (2582,2592), (2583,2593), (4179,27807), (4182,42361), (5374,69)


4. Let P = p : q : r and U = u : v : w. Let T1 = cevian(P) and T2 = anticevian(U). The triangle T A'B'C' such that T*T1 = T2 is given by

A' = u (p u - q v + r v + q w - r w) : -v (-q u + r u + p v - p w) : -w (q u - r u - p v + p w)
B' = u (q u - p v + r v - q w) : -v (-p u + r u + q v + p w - r w) : w (-q u + p v - r v + q w)
C' = u (r u - r v - p w + q w) : v (-r u + r v + p w - q w) : w (p u - q u - p v + q v - r w).

The locus of a point U = X = x : y : z for which the anticomplementary triangle is perspective to T is the union of two cubics, given by

Cubic 4.1: p (- p + q + r) y z (y - z) + q (- q + r + p) z x (z - x) + r (- r + p + q) x y (x - y) = 0, and

Cubic 4.2: p x^3 + q y^3 + r z^3 - (r y + q z) y z - (p z + r x) z x - (q x + p y) x y = 0.

The appearance of {i, [name and/or list of points]} in the following list means that for P = X(i), Cubic 4.1 passes through the listed points:

{1, K1077, {1,2,8,9,188,236,3161,7028,8056,24150,24151,24152,24153,24154,24155,24156,24157,24158,39121}
{3, K002, {1,2,3,4,6,9,57,223,282,1073,1249,3341,3342,3343,3344,3349,3350,3351,3352,3356,14481,39162,39163,39164,39165,40989,40990,40991,40992}
{4, {2,4,20,1249,38253}
{5, K612, {2,3,5,6,216,343,2165,34853,40678}
{6, K168, {2,3,6,69,485,486,5374,5408,5409,6337,8770,13388,13389,24245,24246,30556,30557,30558}
{7, {2,7,144,3160,38254}
{8, {2,8,145,3161,38255}
{9, K363, {1,2,7,9,366,1489,3160,7090,13388,13389,14121,19605,40374,41885}
{10, K345,{1,2,9,10,37,226,281,1214,7952,39131}
{11, {2,11,100,650,5375}
{37, {2,10,37,75,6376,16606}
{39, {2,39,76,141,6374}
{44, {2,44,214,320,5239,5240,36668,36669}
{51, {2,51,2979,40588,41378,41379}
{75, {2,75,192,6376,40027,40598}
{76, {2,76,194,6374,32746}
{111, {2,111,14360,15899,38280}
{113, K489,{2,3,6,74,113,403,1989,3003,3580,14993,34834,36896}
{114, {2,98,114,230,36899}
{115, {2,99,115,523,31998}
{116, {2,101,116,6586,39026}
{119, {1,2,9,104,119,1737,2006,8609}
{121, {2,106,121,8610,40595}
{125, {2,110,125,647,36830}
{126, {2,111,126,3291,15899}
{127, {2,112,127,2485,40596}
{132, {2,132,232,1297,5000,5001,41200,41201}
{133, {2,4,30,133,1249,1294,1990,3163,16080}
{140, {2,5,140,216,233,40684}
{141, K836, {2,3,6,39,141,427,5403,5404,14376,40938}
{142,, {1,2,9,142,277,1212,4847}
{178, {2,178,188,236,2090,5430,16015,16016}
{206, K177, {2,3,6,25,32,66,206,1676,1677,3162,19615,41378,41379}
{214, K453, {1,2,9,44,80,88,214,519,3911,4370,19618,40594}
{216, {2,5,216,264,39641,39642}
{220, {2,220,6600,6604,24152,24153}
{223, K965, {2,57,174,189,223,557,558,1659,13388,13389,13390,15495,15891,15892,16662,16663,16664,39122}
{226, {2,63,226,1214,6505,13388,13389}
{230, {2,114,230,325,5976}
{233, {2,17,18,95,140,233}
{236, {2,188,236,1489,7048,41885}
{239, {2,239,6542,6651,41841}
{325, {2,325,385,5976,8290}
{343, {2,343,1993,5408,5409}
{385, {2,385,7779,8290,39091}
{389, {2,5,216,389,577,1147,5562,34836}
{395, {2,299,395,619,30472}
{396, {2,298,396,618,30471}
{402, {2,402,1650,14401,38240,42306}
{442, {1,2,9,21,442,5249,6734,7110,16585,37887,40582,40937}
{478, {2,56,478,509,2362,8048,13388,13389,16232}

The appearance of {i, [list of points]} in the following list means that for P = X(i), Cubic 4.2 passes through the listed points:

{240, {1,2582,2583}
{403, {125,41077}


5. Let P = p : q : r and U = u : v : w. Let T1 = anticevian(P) and T2 = anticevian(U). The triangle T = A'B'C' such that T1*T = T2 is given by

A' = -(q + r) (q r u + p r v + p q w) : (p + r) (q r u + p r v - p q w) : (p + q) (q r u - p r v + p q w)
B' = -2 q r (q + r) (u + w) (q r u + p r v - p q w) : 2 q r (p + r) (u + w) (q r u + p r v + p q w) : -2 q (p + q) r (u + w) (-q r u + p r v + p q w)
C' = -2 q r (q + r) (u + w) (q r u + p r v - p q w): 2 q r (p + r) (u + w) (q r u + p r v + p q w) : -2 q (p + q) r (u + w) (-q r u + p r v + p q w))


6. Let P = p : q : r and U = u : v : w. Let T1 = anticevian(P) and T2 = anticevian(U). The triangle T = A'B'C' such that T*T1 = T2 is given by

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

The locus of a point P = X = x : y : z such that the triangle T is perspective to the medial triangle is the union of two cubics:

u y z (y - z) + v z x (z - x) + w x y (x - y) = 0, and

v w (x - y - z) x^2 + w u (y - z - x) y^2 + u v (z - x - y) z^2 = 0.

The locus of a point P = X = x : y : z such that the triangle T is perspective to the anticomplementary triangle is the Steiner inellipse, given by

x^2 + y^2 + z^2 - 2 y z - 2 z x - 2 x y = 0.

The locust of a point U = X = x : y : z such that T is perspective to the anticomplementary triangle is the following cubic:

p^2 (r (p + q - r) y^2 z - q (p - q + r) y z^2) + q^2 (p (q + r - p) z^2 x - r (q - r + p) z x^2) + r^2 (q (q + p - q) x^2 y - p (r - p + q) x y^2) = 0.


7. Let P = p : q : r and U = u : v : w. Let T1 = anticevian(P) and T2 = cevian(U). The triangle T = A'B'C' such that T1*T = T2 is given by

A' = u (2 p u + p v - q v + r v + p w + q w - r w) : (p + q - r) v (u + w) : (p - q + r) (u + v) w
B' = (p + q - r) u (v + w) : v (-p u + q u + r u + 2 q v + p w + q w - r w) : -(p - q - r) (u + v) w
C' = (p - q + r) u (v + w) : -(p - q - r) v (u + w) : w (-p u + q u + r u + p v - q v + r v + 2 r w)

The triangle T is perspective to ABC for all P and U. The perspector if U = P is given by

g(P) = p (q + r) (p - q + r) (p + q - r) : q (r + p) (q - r + p) (q + r - p) : r (p + q)(r - p + q) (r + p - q).

The appearance of (i,j) in thje following list means that g(X(i)) = X(j):

(1,65), (2,2), (3,5562), (4,64), (5,14978), (6,1843), (7,3062), (8,3680), (9,3059), (10,4647), (11,42547), (13,34296), (14,34295), (20,33893), (37,2667), (39,42548), (57,42549), (65,42550), (66,22262), (67,10417), (69,6391), (75,42027), (76,42551), (99,9293), (100,42552), (115,42553), (141,42554), (188,42017), (190,42555), (216,42556), (668,9267).


8. Let P = p : q : r and U = u : v : w. Let T1 = anticevian(P) and T2 = cevian(U). The triangle T = A'B'C' such that T*T1 = T2 is given by

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

The triangle T is perspective to ABC for all P and U. The perspector is given by

p (- p + q + r) v w : q (p - q + r) w u : r (p + q - r) u v

The locus of a point P = X = x : y : z such that T is perspective to the medial triangle is the conic given by

u (y - z)(x - y - z) + v (z - x)(y - z - x) + w (x - y)(z - x - y) = 0, or, equivalently,

(v - w)(x^2 + y z) + (w - u)(y^2 + z x) + (u - v)(z^2 + x y) = 0.

Clearly, this conic passes through the centroid and the vertices of the medial triangle. The appearance of {i,{n1, n2, ... ,nk} in the following list mean that for U = X(i), the conic passes through the points X(n1), X(n2), ... , X(nk):

{1,{2,9,37,440,1213,3161,4370,5513,6544,6651,15487,16590,16593,17755,21838,24771,27481,31336,36911,38015,39056,39059,40181,40586,40614,40651,41841}}
{3,{2,6,216,233,1196,1249,1560,3162,3163,8105,8106,8968,14091,14401,15595,18311,32750,37891,37895,39034,39078,39081,40179,40582,40583,40601,40937,40938,40939,40940,40941,40942,42306}}
{6,{2,3,39,114,618,619,629,630,641,642,1125,1649,2482,3413,3414,3666,5664,5745,5976,6292,6337,6503,6509,6626,7710,8290,8299,8786,10291,10335,11147,11165,13701,13821,13882,13934,14713,15349,15810,15814,15819,15850,22848,22892,27929,30471,30472,33364,33365,33614,33615,33616,33617,33618,33619,33620,33621,34452,34834,34835,38998,39090,39091,39094,39096,39098,39100,39102,40125,40592,40604,40605,41820,41849}}
{7,{1,2,223,1212,1214,2582,2583,3160,3752,6505,16585,16586,17056,18641,31534,31535,35110,36905,39035,39046,39047,39066,40611,40612,41771}}
{11,{2,650,2238,3008,3290,5375,5452,16588,27942,35113,40869}}
{12,{2,478,5750,17053,34261,39595}}
{13,{2,396,11127,35444,40578,40581,40696,41888}}
{14,{2,395,11126,35443,40579,40580,40695,41887}}
{17,{2,465,10639,11130,23302}}
{18,{2,466,10640,11131,23303}}
{31,{2,16584,17023,19557,32664,33568,40597}}
{32,{2,206,1194,3589,6676,7664,8265,29654,36213,41884}}
{37,{2,10,120,1211,3452,3789,6376,6552,6554,13466,14434,16589,16594,17793,21530,28651,36912,39028,40598,40603,40609}}
{38,{2,661,1575,3912,16587,19584,35123,36906,40585}}
{39,{2,126,141,1368,3739,3741,6338,6374,6389,10472,20339,21246,21248,27854,32746,34021,35073,39080}}
{45,{2,514,519,3936,6547,6631,9460,16610,27751,35121,40587,40594}}
{51,{2,3117,3815,11672,33569,40588}}
{74,{2,3003,5158,9209,36896}}
{85,{2,142,4000,11019,17073,17113,18635,20206,36908,39063,40593}}
{92,{2,226,442,1210,3772,7952,18592,20621,39036,40837}}
{94,{2,623,624,2072,3580,13162,14993,16188,18314}}
{98,{2,230,441,647,8623,22391,23967,36830,36899,36904,38975,41196,41197}}
{99,{2,523,524,3291,5159,8542,9165,15899,23991,31655,31998,35087,39061}}
{196,{2,860,1465,14837,39053}}
{210,{2,2276,6184,29571,33570,40599,40606}}
{216,{2,5,132,3767,6523,6708,13567,20207,34836}}
{253,{2,4,3343,26958,33537,40839}}
{257,{2,4357,16591,24239,28358,39040,41886}}
{269,{2,5437,14986,40183,40194}}
{290,{2,511,23878,24206,39058}}
{292,{2,1966,3836,9470,19564,24325}}
{308,{2,626,3934,39076,39082}}
{311,{2,343,639,640,1209,7746,11585,31842,34853}}
{319,{2,3647,5325,19862,28606,30563}}
{330,{2,75,3840,20258,20343,34832}}
{371,{2,615,5408,10962,24245}}
{372,{2,590,5409,10960,24246}}
{648,{2,30,525,4550,9410,14918,16253,39062,42426}}
{662,{2,14838,16579,35466,39054}}
{664,{2,522,527,10001,15346}}
{694,{2,325,3005,3229,6682,19602,25666,35077,39092}}
{846,{2,239,4988,10026,35085}}
{1225,{2,635,636,21975,31843,37452,37636}}
{1341,{2,13636,39023,39068,40989,40990}}
{1501,{2,6679,7792,19576,40368,40377}}
{1978,{2,3835,3948,20340,20530,21250}}
{2006,{2,908,1577,1737,17057,36909,36914}}
{2415,{2,8,11530,16602,21129,24151,30827}}
{2481,{2,518,3826,4762,33675}}
{3003,{2,113,5309,15760,37648}}
{3108,{2,6665,10691,31128,34573}}
{4554,{2,2886,4885,20335,34852}}

The locus of a point U = X = x : y : z such that T is perspective to the medial triangle is the circumconic given by p y z + q z x + r x y = 0.


The triangle sum, T1 + T2 of two arbitrary triangles T1 and T2 is defined in the preamble just before X(42285). The two distributive laws hold:; i.e., if T is a triangle, then

T*(T1 + T2) = T*T1 + T*T2)
(T1 + T2)*T = (T1 + T*T2)*T

For arbitrary points P and U, the product triangle cevian(P)*anticevian(U) is perspective to anticevian(U)*cevian(P), and the perspector is the P-Ceva conjugate of U.

The product A'B'C' = cevian(P)*cevian(U) is given by
A' = (q u + r u + q v + r w) : r v (u + w) : q w (u + v)
B' = u (q u + r u + q v + r w) : r v (u + w) : q w (u + v)
C' = q u (v + w) : p v (u + w) : w (p u + q v + p w + q w)

In particular, the square of cevian(P) is given by
A' = p (p q + q^2 + p r + r^2) : q r (p + r) : q r (p + q) :
B' = p r (q + r) : q (p^2 + p q + q r + r^2) : p r (p + q) :
C' = p q (q + r) : p q (p + r) : r (p^2 + q^2 + p r + q r)



X(42464) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN(X(3))*ANTICEVIAN(X(1))

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

X(42464) lies on these lines: {40, 3436}, {46, 208}, {198, 1766}, {221, 517}, {318, 1158}, {1800, 2360}, {12704, 22464}

X(42465) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(4))*ANTICEVIAN(X(4))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12 + 6*a^10*b^2 - 29*a^8*b^4 + 36*a^6*b^6 - 9*a^4*b^8 - 10*a^2*b^10 + 5*b^12 - 6*a^10*c^2 + 14*a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 36*a^4*b^6*c^2 + 34*a^2*b^8*c^2 - 10*b^10*c^2 + 15*a^8*c^4 - 20*a^6*b^2*c^4 + 50*a^4*b^4*c^4 - 36*a^2*b^6*c^4 - 9*b^8*c^4 - 20*a^6*c^6 - 20*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 36*b^6*c^6 + 15*a^4*c^8 + 14*a^2*b^2*c^8 - 29*b^4*c^8 - 6*a^2*c^10 + 6*b^2*c^10 + c^12)*(a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + b^12 + 6*a^10*c^2 + 14*a^8*b^2*c^2 - 20*a^6*b^4*c^2 - 20*a^4*b^6*c^2 + 14*a^2*b^8*c^2 + 6*b^10*c^2 - 29*a^8*c^4 + 4*a^6*b^2*c^4 + 50*a^4*b^4*c^4 + 4*a^2*b^6*c^4 - 29*b^8*c^4 + 36*a^6*c^6 - 36*a^4*b^2*c^6 - 36*a^2*b^4*c^6 + 36*b^6*c^6 - 9*a^4*c^8 + 34*a^2*b^2*c^8 - 9*b^4*c^8 - 10*a^2*c^10 - 10*b^2*c^10 + 5*c^12) : :

X(42465) lies on these lines: {20, 3183}, {253, 33546}, {459, 3346}, {1249, 3349}, {1294, 41425}, {3176, 5930}, {8804, 8894}, {10152, 12324}, {15005, 16251}, {15311, 33893}, {28781, 42452}, {33702, 36965}

X(42466) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(5))*ANTICEVIAN(X(5))

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

X(42466) lies on these lines: {3, 15912}, {216, 6663}, {324, 6662}

X(42467) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(6))*ANTICEVIAN(X(1))

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

X(42467) lies on these lines: {1, 40454}, {3, 960}, {19, 13478}, {57, 1848}, {58, 4227}, {63, 573}, {84, 7713}, {103, 40097}, {222, 3666}, {295, 2807}, {312, 1766}, {572, 2339}, {1395, 7004}, {1707, 1768}, {1708, 5928}, {1709, 26118}, {1748, 21370}, {1790, 17185}, {2051, 2285}, {3218, 36850}, {3674, 7177}, {5450, 16579}

X(42468) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(6))*ANTICEVIAN(X(4))

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 + 3*a^8*c^2 + 12*a^6*b^2*c^2 - 30*a^4*b^4*c^2 + 12*a^2*b^6*c^2 + 3*b^8*c^2 - 14*a^6*c^4 + 14*a^4*b^2*c^4 + 14*a^2*b^4*c^4 - 14*b^6*c^4 + 14*a^4*c^6 - 20*a^2*b^2*c^6 + 14*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 - c^10)*(a^10 + 3*a^8*b^2 - 14*a^6*b^4 + 14*a^4*b^6 - 3*a^2*b^8 - b^10 - 3*a^8*c^2 + 12*a^6*b^2*c^2 + 14*a^4*b^4*c^2 - 20*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 30*a^4*b^2*c^4 + 14*a^2*b^4*c^4 + 14*b^6*c^4 + 2*a^4*c^6 + 12*a^2*b^2*c^6 - 14*b^4*c^6 - 3*a^2*c^8 + 3*b^2*c^8 + c^10) : :

X(42468) lies on these lines: {3, 6523}, {394, 14361}, {1093, 3346}, {14919, 20213}, {15466, 42458}

X(42469) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(8))*ANTICEVIAN(X(3))

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

X(42469) lies on these lines: {28, 39696}, {56, 4641}, {1472, 3167}, {7053, 20805}

X(42470) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(9))*ANTICEVIAN(X(9))

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

X(42470) lies on these lines: {1, 6600}, {4, 5853}, {7, 3174}, {8, 24389}, {9, 10388}, {84, 518}, {90, 5223}, {104, 6282}, {200, 6601}, {294, 2324}, {322, 2481}, {516, 10309}, {519, 3427}, {521, 35355}, {527, 10307}, {885, 8058}, {1000, 6666}, {1001, 7160}, {1156, 34784}, {1476, 7672}, {2136, 5665}, {2801, 34256}, {2951, 5856}, {3062, 15733}, {3158, 8730}, {3243, 7091}, {3577, 3880}, {4326, 34919}, {5732, 10305}, {7320, 19860}, {15998, 21627}, {16005, 17768}, {30500, 37569}, {40659, 42015}

X(42471) = PERSPECTOR OF ABC AND THE PRODUCT TRIANGLE ANTICEVIAN((X(10))*ANTICEVIAN(X(10))

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

X(42471) lies on these lines: {1, 3159}, {19, 17915}, {37, 4075}, {65, 2901}, {75, 24046}, {267, 17763}, {321, 596}, {519, 34434}, {726, 13476}, {740, 40504}, {876, 8714}, {994, 14923}, {2214, 39964}, {2218, 22027}, {3175, 6534}, {3249, 4024}, {3634, 25347}, {4066, 42027}, {4674, 17751}, {4737, 31359}, {6532, 31993}, {21081, 21100}, {21208, 27801}, {22045, 39697}, {30942, 39711}

leftri

Gibert points on the cubic K1200: X(42472)-X(42483)

rightri

This preamble and points X(42472)-X(42483) are contributed by Peter Moses, April 2, 2021. See also the preambles just before X(42085), X(42413), and X(42429).

See K1200.



X(42472) = GIBERT (2,7,6) POINT

Barycentrics    a^2*S/Sqrt[3] + 3*a^2*SA + 7*SB*SC) : :

X(42472) lies on the cubic K1200 and these lines: {2, 42088}, {4, 10188}, {5, 5335}, {6, 5068}, {14, 3545}, {15, 3855}, {16, 5071}, {381, 42119}, {546, 42130}, {547, 42127}, {631, 42100}, {1656, 42123}, {3090, 5237}, {3091, 5321}, {3523, 42102}, {3524, 42105}, {3525, 19106}, {3529, 33417}, {3533, 42091}, {3544, 18581}, {3832, 23302}, {3839, 11480}, {3850, 42132}, {3851, 5334}, {3854, 42093}, {3857, 42126}, {3858, 42116}, {5055, 42138}, {5056, 5318}, {5066, 11485}, {5067, 42086}, {5070, 42137}, {5072, 11542}, {5079, 42118}, {5344, 42121}, {5366, 35018}, {7486, 11481}, {10299, 42113}, {10303, 42097}, {10653, 33602}, {11543, 19709}, {12811, 42125}, {12817, 37832}, {15022, 23303}, {15717, 42109}, {16809, 42435}, {16961, 42111}, {16967, 41974}, {19107, 41099}, {37641, 42095}, {38071, 42415}

X(42473) = GIBERT (-2,7,6) POINT

Barycentrics    -a^2*S/Sqrt[3] + 3*a^2*SA + 7*SB*SC) : :

X(42473) lies on the cubic K1200 and these lines: {2, 42087}, {4, 10187}, {5, 5334}, {6, 5068}, {13, 3545}, {15, 5071}, {16, 3855}, {381, 42120}, {546, 42131}, {547, 42126}, {631, 42099}, {1656, 42122}, {3090, 5238}, {3091, 5318}, {3523, 42101}, {3524, 42104}, {3525, 19107}, {3529, 33416}, {3533, 42090}, {3544, 18582}, {3832, 23303}, {3839, 11481}, {3850, 42129}, {3851, 5335}, {3854, 42094}, {3857, 42127}, {3858, 42115}, {5055, 42135}, {5056, 5321}, {5066, 11486}, {5067, 42085}, {5070, 42136}, {5072, 11543}, {5079, 42117}, {5343, 42124}, {5365, 35018}, {7486, 11480}, {10299, 42112}, {10303, 42096}, {10654, 33603}, {11542, 19709}, {12811, 42128}, {12816, 37835}, {15022, 23302}, {15717, 42108}, {16808, 42436}, {16960, 42114}, {16966, 41973}, {19106, 41099}, {37640, 42098}, {38071, 42416}

X(42474) = GIBERT (3,14,16) POINT

Barycentrics    Sqrt[3]*a^2*S + 16*a^2*SA + 28*SB*SC : :

X(42474) lies on the cubic K1200 and these lines: {2, 42088}, {5, 5339}, {6, 5071}, {13, 5055}, {381, 10645}, {395, 5056}, {547, 42114}, {549, 42113}, {1656, 5237}, {3090, 5340}, {3545, 42093}, {3851, 10188}, {3860, 42090}, {5066, 11480}, {5068, 36836}, {5070, 36968}, {5072, 36970}, {5079, 37835}, {7486, 36843}, {10109, 18582}, {11481, 15699}, {11539, 42106}, {11812, 42105}, {12817, 16966}, {14269, 33417}, {15022, 22237}, {15694, 42097}, {15702, 42102}, {15703, 16808}, {15708, 42109}, {15718, 42429}, {22238, 35018}, {37832, 41122}, {38071, 42092}, {41119, 42420}, {41121, 42129}

X(42475) = GIBERT (-3,14,16) POINT

Barycentrics    Sqrt[3]*a^2*S - 16*a^2*SA - 28*SB*SC : :

X(42475) lies on the cubic K1200 and these lines: {2, 42087}, {5, 5340}, {6, 5071}, {14, 5055}, {381, 10646}, {396, 5056}, {547, 42111}, {549, 42112}, {1656, 5238}, {3090, 5339}, {3545, 42094}, {3851, 10187}, {3860, 42091}, {5066, 11481}, {5068, 36843}, {5070, 36967}, {5072, 36969}, {5079, 37832}, {7486, 36836}, {10109, 18581}, {11480, 15699}, {11539, 42103}, {11812, 42104}, {12816, 16967}, {14269, 33416}, {15022, 22235}, {15694, 42096}, {15702, 42101}, {15703, 16809}, {15708, 42108}, {15718, 42430}, {22236, 35018}, {37835, 41121}, {38071, 42089}, {41120, 42419}, {41122, 42132}

X(42476) = GIBERT (11,26,48) POINT

Barycentrics    11*a^2*S/Sqrt[3] + 48*a^2*SA + 52*SB*SC : :

X(42476) lies on the cubic K1200 and these lines: {5, 11480}, {1656, 41978}, {5318, 15702}, {5351, 42128}, {10188, 16967}, {10299, 42094}, {10304, 42109}, {11481, 11540}, {15681, 33417}, {15693, 42098}, {16645, 41985}, {34755, 42132}

X(42477) = GIBERT (-11,26,48) POINT

Barycentrics    11*a^2*S/Sqrt[3] - 48*a^2*SA - 52*SB*SC : :

X(42476) lies on the cubic K1200 and these lines: {5, 11481}, {1656, 41977}, {5321, 15702}, {5352, 42125}, {10187, 16966}, {10299, 42093}, {10304, 42108}, {11480, 11540}, {15681, 33416}, {15693, 42095}, {16644, 41985}, {34754, 42129}

X(42478) = GIBERT (66,13,2) POINT

Barycentrics    11*Sqrt[3]*a^2*S + a^2*SA + 13*SB*SC : :

X(42478) lies on the cubic K1200 and these lines: {5, 37641}, {5059, 42147}, {5335, 38335}, {5351, 10299}, {6435, 36436}, {6436, 36454}, {10304, 11480}, {10653, 17538}, {10654, 15682}, {11486, 11540}, {11488, 15702}, {15681, 42118}, {15690, 42120}, {16967, 33607}, {19708, 41972}, {41122, 42142}, {42119, 42429}

X(42479) = GIBERT (-66,13,2) POINT

Barycentrics    -11*Sqrt[3]*a^2*S + a^2*SA + 13*SB*SC : :

X(42479) lies on the cubic K1200 and these lines: {5, 37640}, {5059, 42148}, {5334, 38335}, {5352, 10299}, {6435, 36454}, {6436, 36436}, {10304, 11481}, {10653, 15682}, {10654, 17538}, {11485, 11540}, {11489, 15702}, {15681, 42117}, {15690, 42119}, {16966, 33606}, {19708, 41971}, {41121, 42139}, {42120, 42430}

X(42480) = GIBERT (117,14,1) POINT

Barycentrics    39*Sqrt[3]*a^2*S + a^2*SA + 28*SB*SC : :

X(42480) lies on the cubic K1200 and these lines: {5, 16268}, {6, 12816}, {61, 15691}, {62, 15709}, {5237, 15692}, {10645, 15759}, {10653, 11001}, {11486, 15701}, {15684, 16965}, {15688, 22236}, {15723, 16963}, {16772, 41983}, {33603, 41112}, {33607, 37641}, {35408, 42431}, {41107, 42137}

X(42481) = GIBERT (-117,14,1) POINT

Barycentrics    39*Sqrt[3]*a^2*S - a^2*SA - 28*SB*SC : :

X(42481) lies on the cubic K1200 and these lines: {5, 16267}, {6, 12816}, {61, 15709}, {62, 15691}, {5238, 15692}, {10646, 15759}, {10654, 11001}, {11485, 15701}, {15684, 16964}, {15688, 22238}, {15723, 16962}, {16773, 41983}, {33602, 41113}, {33606, 37640}, {35408, 42432}, {41108, 42136}

X(42482) = POINT CURSA

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

X(42482) is one of many points in ETC named for a star; see CURSA

X(42482) lies on this line: {80, 320}

X(42483) = ISOGONAL CONJUGATE OF X(1615)

Barycentrics    (a^4 + 4*a^3*b - 10*a^2*b^2 + 4*a*b^3 + b^4 - 4*a^3*c + 4*a^2*b*c + 4*a*b^2*c - 4*b^3*c + 6*a^2*c^2 - 4*a*b*c^2 + 6*b^2*c^2 - 4*a*c^3 - 4*b*c^3 + c^4)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*a^3*c + 4*a^2*b*c - 4*a*b^2*c - 4*b^3*c - 10*a^2*c^2 + 4*a*b*c^2 + 6*b^2*c^2 + 4*a*c^3 - 4*b*c^3 + c^4) : :
X(42483) = X[18230] - 3 X[32079]

X(42483 lies on the cubic K202 and these lines: {2, 17113}, {7, 19605}, {9, 2124}, {144, 200}, {346, 16284}, {527, 36627}, {5431, 16016}, {15891, 16663}, {15892, 16662}, {18230, 32079}, {20059, 41798}

X(42483) = reflection of X(15913) in X(9)
X(42483) = isogonal conjugate of X(1615)
X(42483) = isotomic conjugate of X(30695)
X(42483) = anticomplement of X(17113)
X(42483) = cyclocevian conjugate of X(6601)
X(42483) = isotomic conjugate of the anticomplement of X(279)
X(42483) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2125, 3434}, {8917, 6604}
X(42483) = X(i)-cross conjugate of X(j) for these (i,j): {279, 2}, {3062, 7}, {17426, 3900}
X(42483) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1615}, {6, 2951}, {31, 30695}, {41, 31527}, {55, 2124}, {101, 17427}, {1253, 17113}, {17426, 24013}
X(42483) = cevapoint of X(i) and X(j) for these (i,j): {513, 35508}, {514, 13609}, {2125, 8917}, {3900, 17426}, {6362, 38973}
X(42483) = trilinear pole of line {3900, 7658}
X(42483) = barycentric product X(i)*X(j) for these {i,j}: {75, 8917}, {85, 2125}
X(42483) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2951}, {2, 30695}, {6, 1615}, {7, 31527}, {57, 2124}, {279, 17113}, {513, 17427}, {2125, 9}, {8917, 1}, {35508, 17426}

X(42484) = ISOGONAL CONJUGATE OF X(1619)

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

X(42484) lies on these lines: {2, 15259}, {4, 31367}, {20, 159}, {25, 40185}, {1249, 3162}, {1370, 14615}, {5930, 8900}, {6804, 31362}, {6816, 14249}, {18589, 36103}

X(42484) = reflection of X(33584) in X(31367)
X(42484) = isogonal conjugate of X(1619)
X(42484) = anticomplement of X(15259)
X(42484) = cyclocevian conjugate of X(6339)
X(42484) = isotomic conjugate of the anticomplement of X(2207)
X(42484) = polar conjugate of the isotomic conjugate of X(2139)
X(42484) = X(2139)-anticomplementary conjugate of X(5905)
X(42484) = X(2139)-Ceva conjugate of X(40186)
X(42484) = X(i)-cross conjugate of X(j) for these (i,j): {2207, 2}, {34944, 4}
X(42484) = cevapoint of X(122) and X(512)
X(42484) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1619}, {63, 2138}, {326, 15259}
X(42484) = barycentric product X(i)*X(j) for these {i,j}: {4, 2139}, {253, 40186}
X(42484) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1619}, {25, 2138}, {2139, 69}, {2207, 15259}, {40186, 20}

X(42485) = ISOGONAL CONJUGATE OF X(1610)

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

X(42485) lies on these lines: {2, 15267}, {56, 34279}, {63, 23359}, {65, 19608}, {72, 3588}, {304, 18659}, {306, 22282}, {478, 958}, {960, 40590}, {1214, 12089}, {26702, 41401}

X(42485) = isogonal conjugate of X(1610)
X(42485) = anticomplement of X(15267)
X(42485) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1610}, {6, 23512}, {572, 34267}, {1098, 15267}
X(42485) = cevapoint of X(512) and X(34591)
X(42485) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23512}, {6, 1610}, {34434, 34267}

X(42486) = ISOGONAL CONJUGATE OF X(33786)

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

X(42486) lies on these lines: {32, 8264}, {39, 6374}, {194, 3051}, {315, 14946}, {3186, 19566}, {16985, 40146}, {21080, 32453}, {21814, 22028}

X(42486) = reflection of X(39468) in X(39)
X(42486) = isogonal conjugate of X(33786)
X(42486) = isotomic conjugate of X(8264)
X(42486) = isotomic conjugate of the anticomplement of X(1502)
X(42486) = isotomic conjugate of the complement of X(40907)
X(42486) = X(1502)-cross conjugate of X(2)
X(42486) = cevapoint of X(2) and X(40907)
X(42486) = trilinear pole of line {688, 21262}
X(42486) = X(i)-isoconjugate of X(j) for these (i,j): {1, 33786}, {6, 33782}, {19, 23173}, {31, 8264}, {32, 33788}, {560, 19562}, {4118, 38838}
X(42486) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 33782}, {2, 8264}, {3, 23173}, {6, 33786}, {75, 33788}, {76, 19562}, {38826, 38838}
X(42486) = {X(32),X(40381)}-harmonic conjugate of X(8264)

X(42487) = ISOGONAL CONJUGATE OF X(1629)

Barycentrics    a^2*(a^2 - b^2 - c^2)^2*(a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2)*(a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) : :
X(42487) = X[216] - 3 X[3917], X[264] + 3 X[2979], X[3164] - 9 X[33884], X[6243] - 5 X[40329]

X(42487) lies on these lines: {5, 141}, {185, 31504}, {216, 3289}, {264, 2979}, {3164, 11794}, {6243, 40329}, {6662, 10627}, {10003, 32142}, {10625, 39530}, {11412, 13599}, {15318, 42329}

X(42487) = midpoint of X(10625) and X(39530)
X(42487) = reflection of X(10003) in X(32142)
X(42487) = isogonal conjugate of X(1629)
X(42487) = isogonal conjugate of the polar conjugate of X(36952)
X(42487) = X(15526)-cross conjugate of X(520)
X(42487) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1629}, {19, 36794}, {92, 10312}, {158, 5012}, {393, 18042}, {823, 3050}, {1078, 1096}, {2190, 30506}, {2207, 33764}, {7668, 24000}, {24019, 31296}, {33778, 36417}
X(42487) = barycentric product X(i)*X(j) for these {i,j}: {3, 36952}, {394, 3613}, {520, 11794}, {3926, 27375}, {15526, 27867}
X(42487) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 36794}, {6, 1629}, {184, 10312}, {216, 30506}, {255, 18042}, {326, 33764}, {394, 1078}, {418, 41334}, {520, 31296}, {577, 5012}, {1364, 27010}, {3269, 7668}, {3613, 2052}, {3917, 37125}, {3926, 33769}, {11794, 6528}, {15526, 36901}, {27375, 393}, {27867, 23582}, {28724, 41296}, {34980, 38352}, {36952, 264}, {39201, 3050}

leftri

Gibert points on cubics K1194-K1201: X(42488)-X(42536)

rightri

This preamble and points X(42488)-X(42536) are contributed by Peter Moses, April 5, 2021

For the relevant cubics, see
K1194
K1195
K1197
K1200
K1201

Gibert points are introduced in the preamble just before X(42085)



X(42488) = GIBERT (3,4,7) POINT

Barycentrics    Sqrt[3]*a^2*S + 7*a^2*SA + 8*SB*SC : :

X(42488) lies on the cubic K1194 and these lines: {2, 17}, {3, 36969}, {4, 5352}, {5, 15}, {6, 5070}, {13, 140}, {14, 3090}, {16, 3526}, {18, 396}, {20, 16808}, {61, 1656}, {303, 635}, {381, 5238}, {382, 10645}, {397, 632}, {398, 547}, {546, 36967}, {548, 19106}, {549, 42158}, {569, 3201}, {623, 628}, {624, 11307}, {630, 6669}, {631, 10646}, {1216, 36979}, {2042, 35739}, {2046, 35731}, {3091, 42157}, {3107, 3934}, {3205, 9306}, {3206, 13353}, {3364, 10577}, {3365, 10576}, {3389, 8253}, {3390, 8252}, {3391, 42173}, {3392, 42174}, {3411, 16960}, {3412, 5067}, {3523, 36968}, {3524, 42161}, {3525, 10653}, {3528, 42100}, {3530, 5318}, {3545, 42150}, {3832, 19107}, {3843, 11480}, {3851, 36836}, {3853, 42099}, {3855, 42085}, {3856, 42122}, {3859, 42101}, {3861, 42087}, {5054, 5340}, {5055, 22236}, {5056, 10654}, {5066, 42164}, {5068, 42160}, {5071, 41101}, {5072, 42154}, {5079, 5339}, {5349, 12811}, {5350, 8703}, {5366, 15692}, {5470, 8591}, {6670, 16529}, {6671, 30560}, {7486, 18581}, {7617, 11305}, {7749, 41406}, {7808, 11312}, {8739, 14940}, {9736, 16629}, {9885, 22489}, {10170, 30439}, {10303, 42151}, {10304, 12816}, {10657, 20379}, {11243, 32767}, {11515, 37452}, {11539, 41100}, {11542, 16239}, {11737, 12817}, {15059, 36208}, {15694, 36843}, {15696, 42094}, {15702, 41119}, {15703, 16268}, {15709, 41112}, {15712, 42165}, {15717, 42086}, {15720, 42155}, {17578, 42090}, {20416, 22997}, {21734, 42134}, {22511, 36763}, {24206, 36757}, {32142, 36978}, {33703, 42106}, {34754, 42095}, {35018, 42163}, {42104, 42472}

X(42488) = {X(6),X(5070)}-harmonic conjugate of X(42489)

X(42489) = GIBERT (-3,4,7) POINT

Barycentrics    -(Sqrt[3]*a^2*S) + 7*a^2*SA + 8*SB*SC : :

X(42489) lies on the cubic K1194 and these lines: {2, 18}, {3, 36970}, {4, 5351}, {5, 16}, {6, 5070}, {13, 3090}, {14, 140}, {15, 3526}, {17, 395}, {20, 16809}, {62, 1656}, {302, 636}, {381, 5237}, {382, 10646}, {397, 547}, {398, 632}, {546, 36968}, {548, 19107}, {549, 42157}, {569, 3200}, {623, 11308}, {624, 627}, {629, 6670}, {631, 10645}, {1216, 36981}, {3091, 42158}, {3106, 3934}, {3205, 13353}, {3206, 9306}, {3364, 8253}, {3365, 8252}, {3366, 42171}, {3367, 42172}, {3389, 10577}, {3390, 10576}, {3411, 5067}, {3412, 16961}, {3523, 36967}, {3524, 42160}, {3525, 10654}, {3528, 42099}, {3530, 5321}, {3545, 42151}, {3832, 19106}, {3843, 11481}, {3851, 36843}, {3853, 42100}, {3855, 42086}, {3856, 42123}, {3859, 42102}, {3861, 42088}, {5054, 5339}, {5055, 22238}, {5056, 10653}, {5066, 42165}, {5068, 42161}, {5071, 41100}, {5072, 42155}, {5079, 5340}, {5349, 8703}, {5350, 12811}, {5365, 15692}, {5469, 8591}, {6669, 16530}, {6672, 30559}, {7486, 18582}, {7617, 11306}, {7749, 41407}, {7808, 11311}, {8740, 14940}, {9735, 16628}, {9886, 22490}, {10170, 30440}, {10303, 42150}, {10304, 12817}, {10658, 20379}, {11244, 32767}, {11516, 37452}, {11539, 41101}, {11543, 16239}, {11737, 12816}, {15059, 36209}, {15694, 36836}, {15696, 42093}, {15702, 41120}, {15703, 16267}, {15709, 41113}, {15712, 42164}, {15717, 42085}, {15720, 42154}, {17578, 42091}, {20415, 22998}, {21734, 42133}, {24206, 36758}, {32142, 36980}, {33703, 42103}, {34755, 42098}, {35018, 42166}, {36251, 36766}, {36770, 37177}, {42105, 42473}

X(42489) = {X(6),X(5070)}-harmonic conjugate of X(42488)

X(42490) = GIBERT (3,2,8) POINT

Barycentrics    Sqrt[3]*a^2*S + 8*a^2*SA + 4*SB*SC : :

X(42490) lies on the cubic K1194 and these lines: {2, 5339}, {3, 13}, {5, 11480}, {6, 631}, {15, 3526}, {18, 15694}, {20, 23302}, {61, 5054}, {62, 15720}, {140, 16645}, {381, 5352}, {382, 10645}, {395, 10303}, {396, 3523}, {397, 3524}, {398, 3525}, {547, 42160}, {548, 18582}, {549, 22238}, {599, 628}, {627, 40341}, {630, 11297}, {632, 10654}, {635, 13083}, {1656, 5238}, {1657, 37832}, {2041, 8253}, {2042, 8252}, {3091, 42474}, {3412, 11486}, {3522, 42166}, {3528, 5318}, {3530, 11481}, {3628, 42150}, {3642, 11309}, {3763, 11307}, {3832, 42087}, {3843, 16966}, {3851, 36967}, {3853, 42090}, {3856, 42104}, {3859, 42144}, {3861, 42114}, {5055, 12817}, {5056, 42164}, {5067, 5321}, {5070, 16964}, {5071, 5349}, {5072, 42432}, {5079, 36970}, {5237, 15693}, {5344, 19708}, {5350, 17538}, {6671, 11311}, {6694, 11302}, {6778, 38634}, {7486, 42119}, {8703, 42162}, {10304, 42165}, {10653, 15712}, {11134, 37515}, {11488, 15717}, {12100, 42151}, {14093, 41121}, {14869, 42149}, {14891, 41112}, {15028, 36980}, {15688, 42431}, {15696, 42097}, {15700, 16267}, {15701, 16962}, {15705, 22235}, {15706, 41107}, {15715, 33604}, {15718, 41100}, {15723, 41108}, {16239, 18581}, {16242, 42435}, {16808, 17800}, {17578, 42110}, {21734, 42088}, {31467, 41407}, {33923, 42161}, {37835, 41971}, {42169, 42219}, {42170, 42217}

X(42490) = {X(6),X(631)}-harmonic conjugate of X(42491)

X(42491) = GIBERT (-3,2,8) POINT

Barycentrics    -(Sqrt[3]*a^2*S) + 8*a^2*SA + 4*SB*SC : :

X(42491) lies on the cubic K1194 and these lines: {2, 5340}, {3, 14}, {5, 11481}, {6, 631}, {16, 3526}, {17, 15694}, {20, 23303}, {61, 15720}, {62, 5054}, {140, 16644}, {381, 5351}, {382, 10646}, {395, 3523}, {396, 10303}, {397, 3525}, {398, 3524}, {547, 42161}, {548, 18581}, {549, 22236}, {599, 627}, {628, 40341}, {629, 11298}, {632, 10653}, {636, 13084}, {1656, 5237}, {1657, 37835}, {2041, 8252}, {2042, 8253}, {3091, 42475}, {3411, 11485}, {3522, 42163}, {3528, 5321}, {3530, 11480}, {3628, 42151}, {3643, 11310}, {3763, 11308}, {3832, 42088}, {3843, 16967}, {3851, 36968}, {3853, 42091}, {3856, 42105}, {3859, 42145}, {3861, 42111}, {5055, 12816}, {5056, 42165}, {5067, 5318}, {5070, 16965}, {5071, 5350}, {5072, 42431}, {5079, 36969}, {5238, 15693}, {5343, 19708}, {5349, 17538}, {6672, 11312}, {6695, 11301}, {6777, 38634}, {7486, 42120}, {8703, 42159}, {10304, 42164}, {10654, 15712}, {11137, 37515}, {11489, 15717}, {12100, 42150}, {14093, 41122}, {14869, 42152}, {14891, 41113}, {15028, 36978}, {15688, 42432}, {15696, 42096}, {15700, 16268}, {15701, 16963}, {15705, 22237}, {15706, 41108}, {15715, 33605}, {15718, 41101}, {15723, 41107}, {16239, 18582}, {16241, 42436}, {16809, 17800}, {17578, 42107}, {21734, 42087}, {31467, 41406}, {33923, 42160}, {37832, 41972}, {42167, 42220}, {42168, 42218}

X(42491) = {X(6),X(631)}-harmonic conjugate of X(42490)

X(42492) = GIBERT (4,9,17) POINT

Barycentrics    (4*a^2*S)/Sqrt[3] + 17*a^2*SA + 18*SB*SC : :

X(42492) lies on the cubic K1194 and these lines: {5, 11480}, {14, 15699}, {16, 632}, {18, 10188}, {30, 42472}, {140, 5344}, {547, 42139}, {549, 16966}, {550, 33417}, {3530, 42141}, {3628, 11485}, {3857, 10645}, {3858, 42108}, {5335, 10124}, {8703, 42105}, {10109, 42119}, {11539, 18582}, {11737, 42130}, {11812, 42127}, {12108, 42142}, {12812, 42116}, {14869, 42146}, {15704, 42114}, {15711, 42100}, {15712, 42098}, {15713, 42123}, {16239, 42132}, {17504, 42094}, {23046, 42090}, {37640, 41985}

X(42493) = GIBERT (-4,9,17) POINT

Barycentrics    (-4*a^2*S)/Sqrt[3] + 17*a^2*SA + 18*SB*SC : :

X(42493) lies on the cubic K1194 and these lines: {5, 11481}, {13, 15699}, {15, 632}, {17, 10187}, {30, 42473}, {140, 5343}, {547, 42142}, {549, 16967}, {550, 33416}, {3530, 42140}, {3628, 11486}, {3857, 10646}, {3858, 42109}, {5334, 10124}, {8703, 42104}, {10109, 42120}, {11539, 18581}, {11737, 42131}, {11812, 42126}, {12108, 42139}, {12812, 42115}, {14869, 42143}, {15704, 42111}, {15711, 42099}, {15712, 42095}, {15713, 42122}, {16239, 42129}, {17504, 42093}, {23046, 42091}, {37641, 41985}

X(42494) = GIBERT (6,7,6) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 7*SB*SC : :

X(42494) lies on the cubic K1194 and these lines: {2, 5340}, {3, 5366}, {4, 15}, {5, 37641}, {6, 5068}, {13, 3090}, {18, 42114}, {20, 5350}, {61, 3855}, {62, 5071}, {140, 5344}, {381, 5343}, {395, 15022}, {396, 3832}, {397, 5056}, {398, 3091}, {550, 42132}, {622, 31275}, {631, 37832}, {1656, 5335}, {1657, 42138}, {3146, 16644}, {3522, 23302}, {3523, 5318}, {3524, 42161}, {3525, 16965}, {3528, 36969}, {3533, 16966}, {3543, 16772}, {3544, 40694}, {3545, 40693}, {3839, 22236}, {3850, 5334}, {3851, 11542}, {3854, 5339}, {3858, 5365}, {5059, 42094}, {5067, 10653}, {5073, 42124}, {5238, 15682}, {5351, 15709}, {5862, 22113}, {7486, 22238}, {10299, 42086}, {10303, 42155}, {10654, 42435}, {11486, 35018}, {15712, 42127}, {15717, 42165}, {16241, 17538}, {16267, 41106}, {16653, 22531}, {16964, 41099}, {17578, 36836}, {21735, 42092}, {41974, 42089}

X(42495) = GIBERT (-6,7,6) POINT

Barycentrics    -Sqrt[3]*a^2*S + 3*a^2*SA + 7*SB*SC : :

X(42495) lies on the cubic K1194 and these lines: {2, 5339}, {3, 5365}, {4, 16}, {5, 37640}, {6, 5068}, {14, 3090}, {17, 42111}, {20, 5349}, {61, 5071}, {62, 3855}, {140, 5343}, {381, 5344}, {395, 3832}, {396, 15022}, {397, 3091}, {398, 5056}, {550, 42129}, {621, 31275}, {631, 37835}, {1656, 5334}, {1657, 42135}, {3146, 16645}, {3522, 23303}, {3523, 5321}, {3524, 42160}, {3525, 16964}, {3528, 36970}, {3533, 16967}, {3543, 16773}, {3544, 40693}, {3545, 40694}, {3839, 22238}, {3850, 5335}, {3851, 11543}, {3854, 5340}, {3858, 5366}, {5059, 42093}, {5067, 10654}, {5073, 42121}, {5237, 15682}, {5352, 15709}, {5863, 22114}, {7486, 22236}, {10299, 42085}, {10303, 42154}, {10653, 42436}, {11485, 35018}, {15712, 42126}, {15717, 42164}, {16242, 17538}, {16268, 41106}, {16652, 22532}, {16965, 41099}, {17578, 36843}, {21735, 42089}, {41973, 42092}

X(42496) = GIBERT (12,5,7) POINT

Barycentrics    4*Sqrt[3]*a^2*S + 7*a^2*SA + 10*SB*SC : :

X(42496) lies on the cubic K1194 and these lines: {5, 37640}, {6, 547}, {13, 15}, {14, 11737}, {16, 11812}, {17, 395}, {61, 3850}, {62, 16239}, {140, 16644}, {382, 22235}, {397, 3530}, {398, 12811}, {524, 6669}, {531, 35019}, {546, 10654}, {548, 42152}, {549, 11488}, {3412, 3861}, {3845, 11485}, {3853, 22236}, {3859, 5339}, {3860, 5321}, {5066, 18582}, {5334, 38071}, {5335, 8703}, {5340, 12103}, {5343, 41991}, {5352, 41981}, {6329, 6670}, {6783, 22566}, {9540, 34552}, {10109, 11543}, {10124, 23302}, {10611, 31710}, {10653, 12100}, {11480, 15690}, {11481, 41983}, {11486, 11539}, {11540, 16242}, {12101, 41119}, {12102, 42147}, {12812, 40694}, {12816, 42108}, {13886, 18587}, {13935, 34551}, {13939, 18586}, {14891, 41943}, {14892, 42098}, {14893, 42117}, {15640, 33604}, {15686, 42116}, {15687, 42128}, {15691, 42086}, {15699, 37641}, {15759, 41107}, {16772, 33923}, {16963, 41984}, {17504, 42416}, {18581, 42474}, {19107, 33607}, {19710, 42127}, {20253, 41621}, {22495, 36770}, {23046, 42142}, {33699, 42119}, {34200, 42118}, {35018, 37835}, {35404, 42134}, {36763, 41745}, {41101, 42136}, {41108, 42110}, {41987, 42093}

X(42496) = {X(6),X(547)}-harmonic conjugate of X(42497)

X(42497) = GIBERT (-12,5,7) POINT

Barycentrics    -4*Sqrt[3]*a^2*S + 7*a^2*SA + 10*SB*SC : :

X(42497) lies on the cubic K1194 and these lines: {5, 37641}, {6, 547}, {13, 11737}, {14, 16}, {15, 11812}, {18, 396}, {61, 16239}, {62, 3850}, {140, 16645}, {382, 22237}, {397, 12811}, {398, 3530}, {524, 6670}, {530, 35020}, {546, 10653}, {548, 42149}, {549, 11489}, {3411, 3861}, {3845, 11486}, {3853, 22238}, {3859, 5340}, {3860, 5318}, {5066, 18581}, {5334, 8703}, {5335, 38071}, {5339, 12103}, {5344, 41991}, {5351, 41981}, {6329, 6669}, {6782, 22566}, {9540, 34551}, {10109, 11542}, {10124, 23303}, {10612, 31709}, {10654, 12100}, {11480, 41983}, {11481, 15690}, {11485, 11539}, {11540, 16241}, {12101, 41120}, {12102, 42148}, {12812, 40693}, {12817, 42109}, {13886, 18586}, {13935, 34552}, {13939, 18587}, {14891, 41944}, {14892, 42095}, {14893, 42118}, {15640, 33605}, {15686, 42115}, {15687, 42125}, {15691, 42085}, {15699, 37640}, {15759, 41108}, {16773, 33923}, {16962, 41984}, {17504, 42415}, {18582, 42475}, {19106, 33606}, {19710, 42126}, {20252, 41620}, {23046, 42139}, {33699, 42120}, {34200, 42117}, {35018, 37832}, {35404, 42133}, {41100, 42137}, {41107, 42107}, {41987, 42094}

X(42497) = {X(6),X(547)}-harmonic conjugate of X(42496)

X(42498) = GIBERT (1,10,23) POINT

Barycentrics    (a^2*S)/Sqrt[3] + 23*a^2*SA + 20*SB*SC : :

X(42498) lies on the cubic K1195 and these lines: {2, 10645}, {6, 15723}, {13, 10124}, {15, 16239}, {16, 3533}, {18, 10187}, {62, 632}, {140, 19106}, {547, 42430}, {3525, 42086}, {3526, 11481}, {3628, 42099}, {5070, 42096}, {5352, 41992}, {10188, 11542}, {10646, 11539}, {11267, 12043}, {11540, 36969}, {15694, 16808}, {15709, 42114}, {15713, 42102}, {15721, 42113}, {16961, 42435}, {16962, 23303}, {16967, 36836}, {41984, 42143}

X(42499) = GIBERT (-1,10,23) POINT

Barycentrics    -((a^2*S)/Sqrt[3]) + 23*a^2*SA + 20*SB*SC : :

X(42499) lies on the cubic K1195 and these lines: {2, 10646}, {6, 15723}, {14, 10124}, {15, 3533}, {16, 16239}, {17, 10188}, {61, 632}, {140, 19107}, {547, 42429}, {3525, 42085}, {3526, 11480}, {3628, 42100}, {5070, 42097}, {5351, 41992}, {10187, 11543}, {10645, 11539}, {11268, 12043}, {11540, 36970}, {15694, 16809}, {15709, 42111}, {15713, 42101}, {15721, 42112}, {16960, 42436}, {16963, 23302}, {16966, 36843}, {41984, 42146}

X(42500) = GIBERT (3,5,16) POINT

Barycentrics    Sqrt[3]*a^2*S + 16*a^2*SA + 10*SB*SC : :

X(42500) lies on the cubic K1195 and these lines: {2, 5321}, {3, 5350}, {4, 42474}, {6, 15702}, {13, 549}, {14, 10124}, {15, 11539}, {16, 11812}, {17, 12108}, {30, 33417}, {61, 140}, {376, 42102}, {381, 42112}, {396, 5054}, {397, 631}, {398, 3525}, {547, 10645}, {616, 33475}, {619, 31274}, {632, 37835}, {3523, 42155}, {3524, 5318}, {3526, 10654}, {3530, 36968}, {3533, 36836}, {3534, 42110}, {3628, 36970}, {3845, 42430}, {5055, 42087}, {5066, 42108}, {5067, 5349}, {5070, 42164}, {5071, 42101}, {5238, 16239}, {5335, 33604}, {6669, 35303}, {8703, 16966}, {10109, 19107}, {10303, 16773}, {10304, 42098}, {10653, 15701}, {11481, 15708}, {11488, 15721}, {11540, 11543}, {12100, 37832}, {12821, 35018}, {14890, 16963}, {14891, 42146}, {15681, 42114}, {15688, 42109}, {15689, 42106}, {15693, 18582}, {15694, 23303}, {15695, 42105}, {15699, 36967}, {15700, 42086}, {15703, 42085}, {15705, 42142}, {15706, 42091}, {15707, 42132}, {15709, 16645}, {15711, 42138}, {15712, 42165}, {15713, 16242}, {15715, 42134}, {15718, 42128}, {15720, 42148}, {15722, 41119}, {15723, 42116}, {15759, 42100}, {16808, 34200}, {16962, 42121}, {17504, 36969}, {19708, 42094}, {19709, 42090}, {30471, 37688}, {38071, 42099}, {41121, 42123}, {42133, 42475}

X(42500) = {X(6),X(15702)}-harmonic conjugate of X(42501)

X(42501) = GIBERT (-3,5,16) POINT

Barycentrics    -(Sqrt[3]*a^2*S) + 16*a^2*SA + 10*SB*SC : :

X(42501) lies on the cubic K1195 and these lines: {2, 5318}, {3, 5349}, {4, 42475}, {6, 15702}, {13, 10124}, {14, 549}, {15, 11812}, {16, 11539}, {18, 12108}, {30, 33416}, {62, 140}, {376, 42101}, {381, 42113}, {395, 5054}, {397, 3525}, {398, 631}, {547, 10646}, {617, 33474}, {618, 31274}, {632, 37832}, {3523, 42154}, {3524, 5321}, {3526, 10653}, {3530, 36967}, {3533, 36843}, {3534, 42107}, {3628, 36969}, {3845, 42429}, {5055, 42088}, {5066, 42109}, {5067, 5350}, {5070, 42165}, {5071, 42102}, {5237, 16239}, {5334, 33605}, {6670, 35304}, {8703, 16967}, {10109, 19106}, {10303, 16772}, {10304, 42095}, {10654, 15701}, {11480, 15708}, {11489, 15721}, {11540, 11542}, {12100, 37835}, {12820, 35018}, {14890, 16962}, {14891, 42143}, {15681, 42111}, {15688, 42108}, {15689, 42103}, {15693, 18581}, {15694, 23302}, {15695, 42104}, {15699, 36968}, {15700, 42085}, {15703, 42086}, {15705, 42139}, {15706, 42090}, {15707, 42129}, {15709, 16644}, {15711, 42135}, {15712, 42164}, {15713, 16241}, {15715, 42133}, {15718, 42125}, {15720, 42147}, {15722, 41120}, {15723, 42115}, {15759, 42099}, {16809, 34200}, {16963, 42124}, {17504, 36970}, {19708, 42093}, {19709, 42091}, {30472, 37688}, {38071, 42100}, {41122, 42122}, {42134, 42474}

X(42501) = {X(6),X(15702)}-harmonic conjugate of X(42500)

X(42502) = GIBERT (27,17,16) POINT

Barycentrics    9*Sqrt[3]*a^2*S + 16*a^2*SA + 34*SB*SC : :

X(42502) lies on the cubic K1195 and these lines: {2, 397}, {13, 8703}, {14, 5066}, {17, 12100}, {61, 3860}, {396, 3830}, {398, 41106}, {549, 41974}, {3412, 14893}, {3534, 16772}, {3628, 42420}, {3845, 16267}, {5318, 11001}, {5321, 41099}, {5335, 33604}, {5340, 19708}, {5350, 15682}, {10188, 11539}, {11488, 15697}, {11812, 23302}, {12101, 42147}, {15685, 42152}, {15690, 41943}, {15693, 41112}, {15698, 16644}, {15701, 42148}, {15722, 42151}, {15759, 16965}, {16962, 33699}, {19709, 40693}, {41101, 42136}, {41113, 42110}, {41973, 41987}

X(42503) = GIBERT (-27,17,16) POINT

Barycentrics    -9*Sqrt[3]*a^2*S + 16*a^2*SA + 34*SB*SC : :

X(42503) lies on the cubic K1195 and these lines: {2, 398}, {13, 5066}, {14, 8703}, {18, 12100}, {62, 3860}, {395, 3830}, {397, 41106}, {549, 41973}, {3411, 14893}, {3534, 16773}, {3628, 42419}, {3845, 16268}, {5318, 41099}, {5321, 11001}, {5334, 33605}, {5339, 19708}, {5349, 15682}, {10187, 11539}, {10612, 36769}, {11489, 15697}, {11812, 23303}, {12101, 42148}, {15685, 42149}, {15690, 41944}, {15693, 41113}, {15698, 16645}, {15701, 42147}, {15722, 42150}, {15759, 16964}, {16963, 33699}, {19709, 40694}, {41100, 42137}, {41112, 42107}, {41974, 41987}

X(42504) = GIBERT (27,10,65) POINT

Barycentrics    9*Sqrt[3]*a^2*S + 65*a^2*SA + 20*SB*SC : :

X(42504) lies on the cubic K1195 and these lines: {2, 5238}, {6, 15693}, {13, 8703}, {15, 11812}, {17, 15695}, {18, 15713}, {61, 15719}, {62, 19711}, {381, 10188}, {549, 3411}, {3412, 15706}, {3830, 16241}, {3845, 5352}, {5066, 36967}, {5237, 12100}, {5349, 10109}, {5350, 19710}, {10645, 11001}, {10654, 33605}, {12108, 42419}, {12817, 42107}, {15640, 37832}, {15697, 41121}, {15698, 41100}, {15711, 16772}, {15722, 16963}, {15759, 16267}, {16966, 41099}, {19708, 41943}, {22165, 36388}, {33416, 33606}

X(42505) = GIBERT (-27,10,65) POINT

Barycentrics    -9*Sqrt[3]*a^2*S + 65*a^2*SA + 20*SB*SC : :

X(42505) lies on the cubic K1195 and these lines: {2, 5237}, {6, 15693}, {14, 8703}, {16, 11812}, {17, 15713}, {18, 15695}, {61, 19711}, {62, 15719}, {381, 10187}, {549, 3412}, {3411, 15706}, {3830, 16242}, {3845, 5351}, {5066, 36968}, {5238, 12100}, {5349, 19710}, {5350, 10109}, {10646, 11001}, {10653, 33604}, {12108, 42420}, {12816, 42110}, {15640, 37835}, {15697, 41122}, {15698, 41101}, {15711, 16773}, {15722, 16962}, {15759, 16268}, {16967, 41099}, {19708, 41944}, {22165, 36386}, {33417, 33607}

X(42506) = GIBERT (27,10,11) POINT

Barycentrics    9*Sqrt[3]*a^2*S + 11*a^2*SA + 20*SB*SC : :

X(42506) lies on the cubic K1195 and these lines: {2, 17}, {6, 25565}, {13, 3830}, {14, 5066}, {15, 11001}, {16, 11812}, {30, 3412}, {61, 3845}, {396, 8703}, {397, 12100}, {546, 42419}, {3107, 11055}, {3411, 15703}, {3534, 16962}, {3860, 42166}, {5237, 15719}, {5238, 15690}, {5335, 15697}, {5340, 15685}, {5351, 19711}, {5469, 10611}, {5470, 14136}, {5858, 22489}, {10109, 16268}, {10304, 41974}, {10653, 15698}, {11481, 15693}, {11540, 42420}, {12821, 33604}, {14893, 41973}, {15640, 36969}, {15682, 42157}, {15695, 42158}, {15711, 42148}, {15759, 16772}, {16529, 36329}, {16808, 37640}, {16961, 37832}, {16966, 42480}, {18582, 41122}, {19708, 42152}, {19709, 42156}, {19710, 42434}, {22510, 36382}, {22571, 35693}, {22602, 36391}, {22631, 36390}, {22688, 36384}, {22846, 33627}, {25151, 36387}, {25157, 36393}, {25158, 36395}, {25159, 36396}, {25160, 36397}, {25217, 36367}, {33605, 42473}, {33606, 42095}, {35384, 42130}, {35730, 35735}, {36763, 36767}, {41020, 41028}, {41971, 42108}

X(42507) = GIBERT (-27,10,11) POINT

Barycentrics    -9*Sqrt[3]*a^2*S + 11*a^2*SA + 20*SB*SC : :

X(42507) lies on the cubic K1195 and these lines: {2, 18}, {6, 25565}, {13, 5066}, {14, 3830}, {15, 11812}, {16, 11001}, {30, 3411}, {62, 3845}, {395, 8703}, {398, 12100}, {546, 42420}, {3106, 11055}, {3412, 15703}, {3534, 16963}, {3860, 42163}, {5237, 15690}, {5238, 15719}, {5334, 15697}, {5339, 15685}, {5352, 19711}, {5469, 14137}, {5470, 10612}, {5859, 22490}, {10109, 16267}, {10304, 41973}, {10654, 15698}, {11480, 15693}, {11540, 42419}, {12820, 33605}, {14893, 41974}, {15640, 36970}, {15682, 42158}, {15695, 42157}, {15711, 42147}, {15759, 16773}, {16530, 35751}, {16809, 37641}, {16960, 37835}, {16967, 42481}, {18581, 41121}, {19708, 42149}, {19709, 42153}, {19710, 42433}, {22511, 36383}, {22572, 35697}, {22604, 36394}, {22633, 36392}, {22690, 36385}, {22891, 33626}, {25161, 36389}, {25167, 36398}, {25168, 36399}, {25169, 36400}, {25170, 36401}, {25214, 36369}, {33604, 42472}, {33607, 42098}, {35384, 42131}, {36402, 41127}, {36403, 41103}, {41021, 41029}, {41972, 42109}

X(42508) = GIBERT (27,10,-16) POINT

Barycentrics    9*Sqrt[3]*a^2*S - 16*a^2*SA + 20*SB*SC : :

X(42508) lies on the cubic K1195 and these lines: {2, 5340}, {6, 11001}, {14, 3830}, {17, 15722}, {61, 3534}, {397, 19708}, {3411, 35403}, {3845, 22238}, {3860, 42161}, {5054, 41974}, {5055, 10187}, {5066, 16645}, {5318, 42475}, {5335, 33604}, {5339, 15682}, {8703, 10653}, {10646, 15693}, {11481, 11812}, {12100, 42151}, {12816, 42095}, {15534, 36331}, {15640, 42108}, {15685, 42158}, {15690, 22236}, {15697, 42120}, {15701, 42156}, {15704, 42419}, {15716, 16267}, {15759, 40693}, {16965, 19709}, {35384, 42126}, {36392, 36394}, {41099, 42094}, {41108, 42097}, {41122, 42127}, {42474, 42477}

X(42509) = GIBERT (-27,10,-16) POINT

Barycentrics    -9*Sqrt[3]*a^2*S - 16*a^2*SA + 20*SB*SC : :

X(42509) lies on the cubic K1195 and these lines: {2, 5339}, {6, 11001}, {13, 3830}, {18, 15722}, {62, 3534}, {398, 19708}, {3412, 35403}, {3845, 22236}, {3860, 42160}, {5054, 41973}, {5055, 10188}, {5066, 16644}, {5321, 42474}, {5334, 33605}, {5340, 15682}, {8703, 10654}, {10645, 15693}, {11480, 11812}, {12100, 42150}, {12817, 42098}, {15534, 35750}, {15640, 42109}, {15685, 42157}, {15690, 22238}, {15697, 42119}, {15701, 42153}, {15704, 42420}, {15716, 16268}, {15759, 40694}, {16964, 19709}, {35384, 42127}, {36390, 36391}, {41099, 42093}, {41107, 42096}, {41121, 42126}, {42475, 42476}

X(42510) = GIBERT (9,1,-7) POINT

Barycentrics    3*Sqrt[3]*a^2*S - 7*a^2*SA + 2*SB*SC : :

X(42510) lies on the cubic K1197 and these lines: {2, 13}, {4, 16963}, {6, 8703}, {14, 15682}, {15, 19708}, {17, 15702}, {18, 3839}, {20, 41973}, {30, 5339}, {61, 10304}, {62, 376}, {202, 10385}, {381, 5350}, {395, 3830}, {396, 15693}, {397, 5054}, {398, 15681}, {532, 37173}, {547, 5340}, {549, 36843}, {631, 16267}, {3090, 10187}, {3107, 36322}, {3146, 3411}, {3412, 10299}, {3523, 41943}, {3524, 5237}, {3534, 10654}, {3543, 12817}, {3545, 16965}, {3845, 18581}, {3860, 42094}, {5055, 16773}, {5066, 16645}, {5071, 41977}, {5318, 19709}, {5351, 15692}, {5352, 15710}, {5464, 5863}, {5859, 34511}, {6200, 36449}, {6396, 36468}, {9115, 36363}, {10109, 42121}, {10646, 15698}, {11001, 34755}, {11296, 33459}, {11480, 15759}, {11481, 12100}, {11489, 36969}, {11539, 42156}, {11542, 15713}, {11543, 33699}, {11812, 16644}, {12101, 42105}, {12816, 37835}, {14269, 42165}, {15640, 36970}, {15683, 16964}, {15685, 42088}, {15687, 42153}, {15689, 42147}, {15690, 42090}, {15697, 36967}, {15700, 16772}, {15701, 42092}, {15703, 42166}, {15709, 33607}, {15711, 42420}, {15719, 16241}, {19710, 42123}, {22236, 34200}, {22907, 36386}, {22998, 36344}, {36436, 42246}, {36448, 42191}, {36454, 42248}, {36466, 42193}, {36962, 41031}, {38335, 42163}, {42140, 42429}, {42416, 42475}, {42432, 42436}

X(42510) = {X(6),X(8703)}-harmonic conjugate of X(42511)

X(42511) = GIBERT (-9,1,-7) POINT

Barycentrics    -3*Sqrt[3]*a^2*S - 7*a^2*SA + 2*SB*SC : :

X(42511) lies on the cubic K1197 and these lines: {2, 14}, {4, 16962}, {6, 8703}, {13, 15682}, {16, 19708}, {17, 3839}, {18, 15702}, {20, 41974}, {30, 5340}, {61, 376}, {62, 10304}, {203, 10385}, {381, 5349}, {395, 15693}, {396, 3830}, {397, 15681}, {398, 5054}, {533, 37172}, {547, 5339}, {549, 36836}, {631, 16268}, {3090, 10188}, {3106, 36323}, {3146, 3412}, {3411, 10299}, {3523, 41944}, {3524, 5238}, {3534, 10653}, {3543, 12816}, {3545, 16964}, {3845, 18582}, {3860, 42093}, {5055, 16772}, {5066, 16644}, {5071, 41978}, {5321, 19709}, {5351, 15710}, {5352, 15692}, {5463, 5862}, {5858, 34511}, {6200, 36467}, {6396, 36450}, {6782, 36767}, {9117, 36362}, {10109, 42124}, {10645, 15698}, {11001, 34754}, {11295, 33458}, {11480, 12100}, {11481, 15759}, {11488, 36970}, {11539, 42153}, {11542, 33699}, {11543, 15713}, {11812, 16645}, {12101, 42104}, {12817, 37832}, {14269, 42164}, {15640, 36969}, {15683, 16965}, {15685, 42087}, {15687, 42156}, {15689, 42148}, {15690, 42091}, {15697, 36968}, {15700, 16773}, {15701, 42089}, {15703, 42163}, {15709, 33606}, {15711, 42419}, {15719, 16242}, {19710, 42122}, {22238, 34200}, {22861, 36388}, {22997, 36319}, {35752, 36772}, {36436, 42249}, {36448, 42194}, {36454, 42247}, {36466, 42192}, {36961, 41030}, {38335, 42166}, {42141, 42430}, {42415, 42474}, {42431, 42435}

X(42511) = {X(6),X(8703)}-harmonic conjugate of X(42510)

X(42512) = GIBERT (15,11,19) POINT

Barycentrics    5*Sqrt[3]*a^2*S + 19*a^2*SA + 22*SB*SC : :

X(42512) lies on the cubic K1197 and these lines: {2, 16960}, {13, 15692}, {14, 5071}, {15, 41099}, {17, 631}, {30, 11480}, {396, 1656}, {632, 40693}, {3091, 5365}, {3146, 12820}, {3522, 42161}, {3843, 42150}, {3858, 42154}, {3859, 42160}, {5076, 16772}, {5318, 15695}, {5339, 41989}, {10654, 19709}, {11542, 15713}, {12812, 42475}, {14093, 42086}, {15693, 41112}, {15694, 23302}, {15696, 42162}, {15697, 41121}, {15711, 42155}, {15712, 42156}, {16241, 19708}, {16267, 42089}, {16961, 37640}, {17538, 36969}, {17578, 36967}, {35434, 42116}, {41943, 42085}

X(42513) = GIBERT (-15,11,19) POINT

Barycentrics    -5*Sqrt[3]*a^2*S + 19*a^2*SA + 22*SB*SC : :

X(42513) lies on the cubic K1197 and these lines: {2, 16961}, {13, 5071}, {14, 15692}, {16, 41099}, {18, 631}, {30, 11481}, {395, 1656}, {632, 40694}, {3091, 5366}, {3146, 12821}, {3522, 42160}, {3843, 42151}, {3858, 42155}, {3859, 42161}, {5076, 16773}, {5321, 15695}, {5340, 41989}, {10653, 19709}, {11543, 15713}, {12812, 42474}, {14093, 42085}, {15693, 41113}, {15694, 23303}, {15696, 42159}, {15697, 41122}, {15711, 42154}, {15712, 42153}, {16242, 19708}, {16268, 42092}, {16960, 37641}, {17538, 36970}, {17578, 36968}, {35434, 42115}, {41944, 42086}

X(42514) = GIBERT (18,65,-50) POINT

Barycentrics    3*Sqrt[3]*a^2*S - 25*a^2*SA + 65*SB*SC : :

X(42514) lies on the cubic K1197 and these lines: {6, 15640}, {14, 15682}, {30, 5344}, {3543, 36843}, {3830, 42143}, {5343, 15684}, {5350, 15683}, {10645, 11001}, {15685, 42124}, {15719, 42429}, {35384, 42117}, {41099, 42113}, {41107, 42141}

X(42515) = GIBERT (-18,65,-50) POINT

Barycentrics    -3*Sqrt[3]*a^2*S - 25*a^2*SA + 65*SB*SC : :

X(42515) lies on the cubic K1197 and these lines: {6, 15640}, {13, 15682}, {30, 5343}, {3543, 36836}, {3830, 42146}, {5344, 15684}, {5349, 15683}, {10646, 11001}, {15685, 42121}, {15719, 42430}, {35384, 42118}, {41099, 42112}, {41108, 42140}

X(42516) = GIBERT (30,1,14) POINT

Barycentrics    5*Sqrt[3]*a^2*S + 7*a^2*SA + SB*SC : :

X(42516) lies on the cubic K1197 and these lines: {6, 9542}, {14, 5071}, {15, 19708}, {30, 5335}, {61, 631}, {376, 34754}, {396, 3091}, {3522, 22236}, {3859, 5343}, {5334, 19709}, {5350, 17578}, {10187, 40694}, {10653, 17538}, {10654, 12817}, {11486, 15711}, {11489, 15694}, {15697, 42120}, {15714, 42116}, {15715, 34755}, {35403, 42117}, {35434, 42134}, {37832, 42435}, {41101, 42140}, {41113, 42473}, {41971, 42112}, {42150, 42429}

X(42517) = GIBERT (-30,1,14) POINT

Barycentrics    -5*Sqrt[3]*a^2*S + 7*a^2*SA + SB*SC : :

X(42517) lies on the cubic K1197 and these lines: {6, 9542}, {13, 5071}, {16, 19708}, {30, 5334}, {62, 631}, {376, 34755}, {395, 3091}, {3522, 22238}, {3859, 5344}, {5335, 19709}, {5349, 17578}, {10188, 40693}, {10653, 12816}, {10654, 17538}, {11485, 15711}, {11488, 15694}, {15697, 42119}, {15714, 42115}, {15715, 34754}, {35403, 42118}, {35434, 42133}, {37835, 42436}, {41100, 42141}, {41112, 42472}, {41972, 42113}, {42151, 42430}

X(42518) = GIBERT (45,26,34) POINT

Barycentrics    15*Sqrt[3]*a^2*S + 34*a^2*SA + 52*SB*SC : :

X(42518) lies on the cubic K1197 and these lines: {3, 33607}, {13, 15695}, {14, 16960}, {17, 15694}, {30, 36836}, {396, 41099}, {1656, 16267}, {3843, 41101}, {5071, 42153}, {5340, 14093}, {10646, 15693}, {11488, 15697}, {11542, 15713}, {15696, 41943}, {15711, 41112}, {16962, 35403}, {19708, 33604}, {41121, 41971}

X(42519) = GIBERT (-45,26,34) POINT

Barycentrics    -15*Sqrt[3]*a^2*S + 34*a^2*SA + 52*SB*SC : :

X(42519) lies on the cubic K1197 and these lines: {3, 33606}, {13, 16961}, {14, 15695}, {18, 15694}, {30, 36843}, {395, 41099}, {1656, 16268}, {3843, 41100}, {5071, 42156}, {5339, 14093}, {10645, 15693}, {11489, 15697}, {11543, 15713}, {15696, 41944}, {15711, 41113}, {16963, 35403}, {19708, 33605}, {41122, 41972}

X(42520) = GIBERT (45,2,13) POINT

Barycentrics    15*Sqrt[3]*a^2*S + 13*a^2*SA + 4*SB*SC : :

X(42520) lies on the cubic K1197 and these lines: {2, 16961}, {6, 15693}, {14, 16960}, {15, 19708}, {16, 15711}, {17, 5071}, {18, 42481}, {30, 61}, {62, 15692}, {381, 33607}, {631, 16963}, {632, 41943}, {1656, 3412}, {1992, 36386}, {3091, 16267}, {3843, 12817}, {5238, 15714}, {5321, 42419}, {8703, 34754}, {10654, 12816}, {11485, 15695}, {14093, 22236}, {15694, 16962}, {15697, 36968}, {15698, 34755}, {15713, 16241}, {16808, 37640}, {16964, 35403}, {19107, 41112}, {33416, 35381}, {41121, 42110}

X(42521) = GIBERT (-45,2,13) POINT

Barycentrics    -15*Sqrt[3]*a^2*S + 13*a^2*SA + 4*SB*SC : :

X(42521) lies on the cubic K1197 and these lines: {2, 16960}, {6, 15693}, {13, 16961}, {15, 15711}, {16, 19708}, {17, 42480}, {18, 5071}, {30, 62}, {61, 15692}, {381, 33606}, {631, 16962}, {632, 41944}, {1656, 3411}, {1992, 36388}, {3091, 16268}, {3843, 12816}, {5237, 15714}, {5318, 42420}, {8703, 34755}, {10653, 12817}, {11486, 15695}, {14093, 22238}, {15694, 16963}, {15697, 36967}, {15698, 34754}, {15713, 16242}, {16809, 37641}, {16965, 35403}, {19106, 41113}, {33417, 35381}, {41122, 42107}

X(42522) = GIBERT (8 Sqrt[3],1,6) POINT

Barycentrics    8*a^2*S + 6*a^2*SA + 2*SB*SC : :

X(42522) lies on the cubic K1201 and these lines: {2, 3311}, {3, 9542}, {4, 6199}, {6, 3523}, {20, 371}, {147, 13653}, {372, 15692}, {376, 9543}, {485, 3839}, {549, 6500}, {576, 26516}, {590, 6470}, {597, 33365}, {631, 6417}, {638, 13639}, {1131, 6561}, {1132, 35815}, {1151, 10304}, {1504, 5304}, {1588, 5056}, {3068, 3071}, {3069, 6431}, {3070, 42413}, {3090, 13903}, {3146, 7583}, {3299, 5265}, {3301, 5281}, {3312, 15717}, {3316, 13785}, {3522, 6221}, {3524, 6418}, {3525, 19116}, {3528, 6407}, {3529, 18512}, {3530, 6501}, {3543, 6459}, {3545, 13925}, {3590, 42277}, {3629, 33364}, {3832, 13886}, {3854, 18538}, {5058, 37665}, {5059, 23267}, {5067, 18510}, {5068, 8976}, {5261, 13905}, {5274, 13904}, {5418, 13941}, {5420, 6419}, {6395, 10299}, {6408, 15698}, {6409, 9692}, {6425, 6460}, {6427, 35255}, {6441, 32786}, {6445, 21735}, {6447, 42216}, {6449, 21734}, {6450, 15705}, {6474, 15688}, {6995, 10880}, {8703, 9691}, {8960, 23259}, {9680, 35770}, {9690, 33923}, {9695, 16661}, {10145, 14093}, {13665, 17578}, {13935, 15708}, {13961, 15702}, {14986, 19038}, {15640, 35822}, {15697, 42259}, {15703, 34089}, {15721, 19053}, {19145, 33748}, {20070, 31439}, {26521, 39561}, {31412, 41955}, {31414, 42263}, {32789, 41947}, {32805, 32898}

X(42522) = {X(6),X(3523)}-harmonic conjugate of X(42523)

X(42523) = GIBERT (-8 Sqrt[3],1,6) POINT

Barycentrics    -8*a^2*S + 6*a^2*SA + 2*SB*SC : :

X(42523) lies on the cubic K1201 and these lines: {2, 3312}, {3, 9543}, {4, 6395}, {6, 3523}, {20, 372}, {147, 13773}, {371, 15692}, {376, 19116}, {486, 3839}, {549, 6501}, {576, 26521}, {597, 33364}, {615, 6471}, {631, 6418}, {637, 13759}, {1131, 35814}, {1132, 6560}, {1152, 10304}, {1505, 5304}, {1587, 5056}, {3068, 6432}, {3069, 3070}, {3071, 42414}, {3090, 13961}, {3146, 7584}, {3299, 5281}, {3301, 5265}, {3311, 9542}, {3317, 13665}, {3522, 6398}, {3524, 6417}, {3525, 19117}, {3528, 6408}, {3529, 18510}, {3530, 6500}, {3543, 6460}, {3545, 13993}, {3591, 42274}, {3629, 33365}, {3832, 13939}, {3854, 18762}, {5059, 23273}, {5062, 37665}, {5067, 18512}, {5068, 13951}, {5261, 13963}, {5274, 13962}, {5418, 6420}, {5420, 8972}, {6199, 10299}, {6407, 15698}, {6426, 6459}, {6428, 35256}, {6442, 32785}, {6446, 21735}, {6448, 42215}, {6449, 15705}, {6450, 21734}, {6454, 9541}, {6475, 15688}, {6485, 9681}, {6995, 10881}, {9540, 15708}, {10146, 14093}, {13785, 17578}, {13847, 31412}, {13903, 15702}, {14986, 19037}, {15640, 35823}, {15697, 42258}, {15703, 34091}, {15721, 19054}, {17554, 31473}, {19146, 33748}, {26516, 39561}, {31414, 41954}, {32790, 41948}, {32806, 32898}

X(42523) = {X(6),X(3523)}-harmonic conjugate of X(42522)

X(42524) = GIBERT (9 Sqrt[3],2,-23) POINT

Barycentrics    9*a^2*S - 23*a^2*SA + 4*SB*SC : :

X(42524) lies on the cubic K1201 and these lines: {2, 1327}, {30, 35813}, {371, 19708}, {372, 8703}, {376, 6454}, {382, 6489}, {397, 15764}, {549, 42418}, {615, 33699}, {1152, 3534}, {1657, 10148}, {3070, 15713}, {3524, 8960}, {3594, 14093}, {3830, 6450}, {3845, 6487}, {5066, 42259}, {5420, 41099}, {6200, 15759}, {6398, 15695}, {6410, 15693}, {6412, 15716}, {6419, 34200}, {6420, 10304}, {6426, 15688}, {6434, 6565}, {6440, 18510}, {6446, 13847}, {6452, 13846}, {6456, 15701}, {6459, 6479}, {6460, 15719}, {6477, 42215}, {6481, 15690}, {6485, 11001}, {6522, 15689}, {7583, 12100}, {9543, 35771}, {12101, 35256}, {13935, 15640}, {14869, 41952}, {15300, 35825}, {15533, 39894}, {15682, 42261}, {15696, 17852}, {15698, 35815}, {15700, 35812}, {15711, 32787}, {19709, 35820}, {19711, 42216}, {34091, 41106}, {41950, 41958}, {41962, 42225}

X(42525) = GIBERT (-9 Sqrt[3],2,-23) POINT

Barycentrics    -9*a^2*S - 23*a^2*SA + 4*SB*SC : :

X(42525) lies on the cubic K1201 and these lines: {2, 1328}, {30, 35812}, {371, 8703}, {372, 19708}, {376, 6453}, {382, 6488}, {398, 15764}, {549, 42417}, {590, 33699}, {1151, 3534}, {1657, 10147}, {3071, 15713}, {3524, 9681}, {3543, 9680}, {3592, 14093}, {3830, 6449}, {3845, 6486}, {4677, 9582}, {5066, 42258}, {5418, 41099}, {6221, 15695}, {6396, 15759}, {6409, 15693}, {6411, 15716}, {6419, 10304}, {6420, 34200}, {6425, 15688}, {6433, 6564}, {6439, 18512}, {6445, 13846}, {6451, 13847}, {6455, 15701}, {6459, 15719}, {6460, 6478}, {6476, 42216}, {6480, 15690}, {6484, 11001}, {6519, 15689}, {7584, 12100}, {8960, 15681}, {9540, 15640}, {10141, 31487}, {12101, 35255}, {14869, 41951}, {15300, 35824}, {15533, 39893}, {15682, 42260}, {15686, 31454}, {15698, 35814}, {15700, 35813}, {15711, 32788}, {19709, 35821}, {19711, 42215}, {34089, 41106}, {41949, 41957}, {41961, 42226}

X(42526) = GIBERT (18 Sqrt[3],22,35) POINT

Barycentrics    18*a^2*S + 35*a^2*SA + 44*SB*SC : :

X(42526) lies on the cubic K1201 and these lines: {2, 3312}, {3, 41952}, {381, 6425}, {485, 15701}, {590, 3830}, {1587, 11540}, {3071, 19709}, {3311, 10109}, {3534, 23251}, {3590, 15702}, {5055, 31487}, {5066, 23259}, {5418, 15695}, {6221, 41099}, {6412, 13665}, {6430, 35822}, {6447, 38071}, {6449, 33699}, {6519, 14893}, {8703, 23249}, {8981, 41106}, {9540, 12101}, {11001, 18538}, {11812, 32785}, {13785, 13846}, {15640, 35255}, {15690, 31412}, {32806, 32892}, {35403, 41963}, {42277, 42417}

X(42527) = GIBERT (-18 Sqrt[3],22,35) POINT

Barycentrics    -18*a^2*S + 35*a^2*SA + 44*SB*SC : :

X(42527) lies on the cubic K1201 and these lines: {2, 3311}, {3, 41951}, {381, 6426}, {486, 15701}, {615, 3830}, {1588, 11540}, {3070, 19709}, {3312, 10109}, {3534, 23261}, {3591, 15702}, {5055, 41952}, {5066, 23249}, {5420, 15695}, {6398, 41099}, {6411, 13785}, {6429, 35823}, {6448, 38071}, {6450, 33699}, {6522, 14893}, {8703, 23259}, {9680, 15694}, {11001, 18762}, {11812, 32786}, {12101, 13935}, {13665, 13847}, {13966, 41106}, {15640, 35256}, {32805, 32892}, {35403, 41964}, {42274, 42418}

X(42528) = GIBERT (3,2,-11) POINT

Barycentrics    Sqrt[3]*a^2*S - 11*a^2*SA + 4*SB*SC : :

X(42528) lies on the cubic K1201 and these lines: {2, 12820}, {3, 13}, {6, 15688}, {14, 3534}, {15, 8703}, {16, 376}, {18, 20}, {30, 10646}, {61, 3522}, {62, 548}, {299, 7782}, {381, 33416}, {395, 550}, {396, 34200}, {549, 16966}, {617, 33622}, {631, 42431}, {3106, 22676}, {3411, 15696}, {3524, 37832}, {3528, 5238}, {3543, 42089}, {3642, 5463}, {3830, 16967}, {3839, 42113}, {5054, 16808}, {5055, 42097}, {5066, 42109}, {5071, 42105}, {5318, 12100}, {5321, 15686}, {5334, 15697}, {5335, 41943}, {5350, 14869}, {5352, 33923}, {5464, 35751}, {6777, 38749}, {7584, 35739}, {7739, 41408}, {10124, 42110}, {10299, 42162}, {10304, 10645}, {11001, 12817}, {11057, 30472}, {11303, 36770}, {11480, 14093}, {11486, 15695}, {11488, 15710}, {11539, 42145}, {11542, 15759}, {11543, 15691}, {11812, 42137}, {12103, 16773}, {12816, 15701}, {13083, 35752}, {15681, 16645}, {15684, 42095}, {15685, 42093}, {15689, 16963}, {15690, 41108}, {15692, 18582}, {15693, 33417}, {15694, 42094}, {15698, 33602}, {15699, 42102}, {15700, 42127}, {15702, 42141}, {15706, 42128}, {15708, 42134}, {15709, 42114}, {15712, 42165}, {15714, 42124}, {15716, 42132}, {15717, 42161}, {15719, 42142}, {16267, 42118}, {16268, 42085}, {16772, 41974}, {16960, 19708}, {17504, 23302}, {17538, 42149}, {19710, 41122}, {21734, 42152}, {21735, 40693}, {22238, 42434}, {22845, 40899}, {33699, 42107}, {35733, 42237}, {36209, 37853}, {36436, 42179}, {36438, 42178}, {36454, 42181}, {36456, 42177}, {41120, 42140}, {41983, 42146}

X(42528) = {X(6),X(15688)}-harmonic conjugate of X(42529)

X(42529) = GIBERT (-3,2,-11) POINT

Barycentrics    -(Sqrt[3]*a^2*S) - 11*a^2*SA + 4*SB*SC : :

X(42529) lies on the cubic K1201 and these lines: {2, 12821}, {3, 14}, {6, 15688}, {13, 3534}, {15, 376}, {16, 8703}, {17, 20}, {30, 10645}, {61, 548}, {62, 3522}, {298, 7782}, {381, 33417}, {395, 34200}, {396, 550}, {549, 16967}, {616, 33624}, {631, 42432}, {3107, 22676}, {3412, 15696}, {3524, 37835}, {3528, 5237}, {3543, 42092}, {3643, 5464}, {3830, 16966}, {3839, 42112}, {5054, 16809}, {5055, 42096}, {5066, 42108}, {5071, 42104}, {5318, 15686}, {5321, 12100}, {5334, 41944}, {5335, 15697}, {5349, 14869}, {5351, 33923}, {5463, 36329}, {6778, 38749}, {7739, 41409}, {10124, 42107}, {10299, 42159}, {10304, 10646}, {11001, 12816}, {11057, 30471}, {11481, 14093}, {11485, 15695}, {11489, 15710}, {11539, 42144}, {11542, 15691}, {11543, 15759}, {11812, 42136}, {12103, 16772}, {12817, 15701}, {13084, 36330}, {15681, 16644}, {15684, 42098}, {15685, 42094}, {15689, 16962}, {15690, 41107}, {15692, 18581}, {15693, 33416}, {15694, 42093}, {15698, 33603}, {15699, 42101}, {15700, 42126}, {15702, 42140}, {15706, 42125}, {15708, 42133}, {15709, 42111}, {15712, 42164}, {15714, 42121}, {15716, 42129}, {15717, 42160}, {15719, 42139}, {16267, 42086}, {16268, 42117}, {16773, 41973}, {16961, 19708}, {17504, 23303}, {17538, 42152}, {19710, 41121}, {21734, 42149}, {21735, 40694}, {22236, 42433}, {22844, 40898}, {33699, 42110}, {36208, 37853}, {36436, 42182}, {36438, 42175}, {36454, 42180}, {36456, 42176}, {41119, 42141}, {41983, 42143}

X(42529) = {X(6),X(15688)}-harmonic conjugate of X(42528)

X(42530) = GIBERT (19,14,23) POINT

Barycentrics    (19*a^2*S)/Sqrt[3] + 23*a^2*SA + 28*SB*SC : :

X(42530) lies on the cubic K1201 and these lines: {13, 15706}, {14, 42132}, {15, 3839}, {16, 17}, {3412, 18581}, {3544, 11488}, {5076, 16808}, {15681, 16644}, {16241, 19708}, {16645, 16960}, {16964, 42098}, {16967, 37640}, {18582, 33703}, {33699, 42087}, {37832, 42135}, {42100, 42162}

X(42531) = GIBERT (-19,14,23) POINT

Barycentrics    (-19*a^2*S)/Sqrt[3] + 23*a^2*SA + 28*SB*SC : :

X(42531) lies on the cubic K1201 and these lines: {13, 42129}, {14, 15706}, {15, 18}, {16, 3839}, {3411, 18582}, {3544, 11489}, {5076, 16809}, {15681, 16645}, {16242, 19708}, {16644, 16961}, {16965, 42095}, {16966, 37641}, {18581, 33703}, {33699, 42088}, {37835, 42138}, {42099, 42159}

X(42532) = GIBERT (27,2,13) POINT

Barycentrics    9*Sqrt[3]*a^2*S + 13*a^2*SA + 4*SB*SC : :

X(42532) lies on the cubic K1201 and these lines: {2, 18}, {6, 15693}, {13, 3830}, {14, 42132}, {15, 8703}, {16, 15698}, {17, 19709}, {62, 12100}, {381, 3412}, {396, 5066}, {397, 19710}, {398, 10109}, {3411, 5054}, {3534, 22236}, {3839, 41973}, {3845, 16267}, {5237, 15711}, {5238, 19708}, {5335, 42430}, {5352, 15759}, {5469, 36330}, {8584, 36757}, {10653, 15697}, {10654, 12817}, {11001, 34754}, {11055, 32465}, {11295, 36366}, {11488, 41120}, {11812, 16242}, {12820, 42104}, {15534, 36386}, {15640, 19106}, {15682, 40693}, {15685, 16965}, {15686, 41974}, {15695, 42433}, {15701, 16963}, {15713, 16772}, {15716, 22238}, {16966, 41122}, {18581, 33605}, {33602, 42140}, {33699, 42147}, {34200, 42420}, {36969, 42144}, {36970, 41119}, {37832, 41113}

X(42533) = GIBERT (-27,2,13) POINT

Barycentrics    -9*Sqrt[3]*a^2*S + 13*a^2*SA + 4*SB*SC : :

X(42533) lies on the cubic K1201 and these lines: {2, 17}, {6, 15693}, {13, 42129}, {14, 3830}, {15, 15698}, {16, 8703}, {18, 19709}, {61, 12100}, {381, 3411}, {395, 5066}, {397, 10109}, {398, 19710}, {3412, 5054}, {3534, 22238}, {3839, 41974}, {3845, 16268}, {5237, 19708}, {5238, 15711}, {5334, 42429}, {5351, 15759}, {5470, 35752}, {8584, 36758}, {10653, 12816}, {10654, 15697}, {11001, 34755}, {11055, 32466}, {11296, 36368}, {11489, 41119}, {11812, 16241}, {12821, 42105}, {15534, 36388}, {15640, 19107}, {15682, 40694}, {15685, 16964}, {15686, 41973}, {15695, 42434}, {15701, 16962}, {15713, 16773}, {15716, 22236}, {16967, 41121}, {18582, 33604}, {33603, 42141}, {33699, 42148}, {34200, 42419}, {36969, 41120}, {36970, 42145}, {37835, 41112}

X(42534) = X(5)X(182)∩X(6)X(76)

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

X(42534) lies on K1200 and these lines: {2, 1501}, {3, 10007}, {5, 182}, {6, 76}, {32, 141}, {39, 4048}, {67, 13193}, {69, 7787}, {98, 10516}, {110, 32242}, {115, 19120}, {287, 39685}, {384, 3094}, {511, 7804}, {518, 10791}, {524, 5039}, {597, 5034}, {599, 12150}, {698, 3734}, {1078, 3763}, {1180, 10328}, {1350, 12110}, {1352, 3398}, {1469, 10797}, {2023, 5026}, {2076, 3972}, {2456, 14561}, {2458, 5103}, {2916, 33717}, {3056, 10798}, {3114, 14603}, {3242, 12195}, {3329, 10334}, {3416, 12194}, {3618, 5038}, {3619, 7793}, {3934, 8177}, {3981, 16950}, {4027, 7875}, {4045, 29012}, {5007, 14994}, {5012, 39668}, {5017, 10350}, {5033, 8361}, {5041, 41622}, {5050, 12177}, {5085, 13860}, {5092, 6683}, {5116, 7786}, {5182, 7884}, {5480, 10358}, {5846, 10800}, {5969, 11286}, {6393, 6661}, {6680, 10104}, {6776, 10359}, {7606, 14762}, {7772, 32449}, {7789, 13356}, {7794, 15870}, {7815, 33185}, {7839, 41747}, {7887, 7943}, {7893, 12206}, {7915, 39603}, {8041, 16949}, {8722, 21167}, {10346, 19689}, {10353, 11606}, {10519, 10788}, {10790, 37485}, {10799, 12589}, {11261, 35375}, {11338, 39080}, {12151, 19570}, {12192, 14982}, {12202, 41735}, {12203, 36990}, {12251, 35427}, {12588, 12835}, {14001, 34870}, {14036, 39652}, {14370, 31360}, {15069, 39872}, {15482, 17508}, {15514, 22486}, {16932, 20859}, {18501, 33878}, {18502, 31670}, {18841, 39874}, {19130, 40279}, {24733, 41259}, {32135, 39515}, {41443, 42286}

X(42534) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 7770, 24256}, {6, 24273, 76}, {69, 7787, 12212}, {141, 42421, 32}, {182, 3818, 14880}, {182, 7808, 3589}, {597, 13196, 5034}, {1676, 1677, 7834}, {3329, 12215, 13331}, {3618, 39141, 5038}, {7878, 32451, 6}, {10358, 13355, 5480}, {24206, 39750, 10104}

X(42535) = X(5)X(32)∩X(6)X(98)

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

X(42535) lies on K1200 and these lines: {2, 1501}, {4, 34870}, {5, 32}, {6, 98}, {39, 14880}, {83, 7887}, {115, 40250}, {182, 3815}, {183, 5017}, {187, 15819}, {385, 13330}, {598, 7610}, {1007, 39141}, {1078, 3053}, {1656, 38905}, {2076, 22712}, {2080, 7737}, {2548, 3398}, {3054, 41412}, {3055, 5033}, {3094, 5999}, {3934, 39603}, {4027, 7777}, {5013, 12203}, {5034, 9300}, {5038, 7736}, {5039, 5306}, {5104, 6194}, {5182, 11184}, {5254, 13356}, {6036, 39750}, {6055, 7753}, {7612, 11170}, {7735, 12212}, {7747, 40279}, {7752, 10349}, {7773, 10350}, {7787, 32961}, {7793, 16924}, {7808, 8361}, {7815, 7819}, {7901, 10345}, {7912, 10333}, {7925, 10334}, {7934, 10347}, {8177, 35432}, {8667, 22486}, {8787, 11163}, {9596, 10799}, {9599, 12835}, {9752, 10788}, {9770, 12151}, {10335, 14931}, {10359, 31404}, {10631, 14537}, {11361, 39652}, {11842, 15484}, {12054, 31401}, {23055, 33005}, {31489, 39560}, {39685, 40814}

X(42535) = crosssum of X(1664) and X(1665)
X(42535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 13860, 2023}, {32, 5475, 10796}

X(42536) = X(5)X(524)∩X(6)X(598)

Barycentrics    5*a^6 - 15*a^4*b^2 + 9*a^2*b^4 + 2*b^6 - 15*a^4*c^2 - 18*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(42536) = 2 X[1153] - 3 X[7606], 3 X[5476] - X[8176], X[8182] + 3 X[20423], X[11165] - 9 X[14848], 9 X[14853] - X[23334]

X(42536) lies on K1200 and these lines: {2, 8586}, {5, 524}, {6, 598}, {262, 7610}, {511, 1153}, {542, 40277}, {543, 10796}, {574, 597}, {575, 32479}, {2080, 8182}, {5026, 11164}, {5038, 7738}, {5475, 8584}, {5485, 11170}, {5969, 11165}, {7608, 42011}, {7619, 25555}, {7777, 10487}, {7786, 22486}, {7899, 21358}, {8859, 13330}, {9855, 10485}, {10166, 13192}, {14853, 23334}, {15019, 42008}

X(42536) = midpoint of X(i) and X(j) for these {i,j}: {576, 7617}, {8584, 20112}
X(42536) = reflection of X(7619) in X(25555)
X(42536) = {X(6),X(11317)}-harmonic conjugate of X(8787)

leftri

Gibert points on the Brocard-Kiepert quartic, Q073: X(42537)-X(42546)

rightri

This preamble and points X(42537)-X(42546) are contributed by Peter Moses, April 6, 2021

Gibert points are introduced in the preamble just before X(42085); for the Brocard-Kiepert quartic, see Q073.



X(42537) = GIBERT (-6 SQRT(3),25,-22) POINT

Barycentrics    -3*a^2*S - 11*a^2*SA + 25*SB*SC : :

X(42537) lies on the curce Q073 and these lines: {30, 1588}, {376, 42268}, {1132, 13847}, {1151, 3543}, {1327, 3068}, {1328, 6396}, {3146, 31414}, {3316, 42266}, {3534, 32786}, {3830, 35255}, {3845, 6451}, {5059, 6426}, {6419, 33703}, {6441, 15640}, {6479, 35821}, {6497, 15686}, {6500, 35400}, {9541, 33699}, {9543, 41952}, {10194, 17538}, {15685, 23259}, {15697, 42283}, {15710, 35787}, {35409, 35820}, {36449, 42141}, {36467, 42140}

X(42538) = GIBERT (6 SQRT(3),25,-22) POINT

Barycentrics    3*a^2*S - 11*a^2*SA + 25*SB*SC : :

X(42538) lies on the curce Q073 and these lines: {30, 1587}, {376, 42269}, {1131, 13846}, {1152, 3543}, {1327, 6200}, {1328, 3069}, {3146, 32788}, {3317, 42267}, {3529, 8960}, {3534, 32785}, {3830, 35256}, {3845, 6452}, {5059, 6425}, {6420, 33703}, {6442, 15640}, {6478, 35820}, {6496, 15686}, {6501, 35400}, {10195, 17538}, {15681, 31412}, {15685, 23249}, {15697, 42284}, {15710, 35786}, {35409, 35821}, {36450, 42140}, {36468, 42141}

X(42539) = GIBERT (-16 SQRT(3),25,2) POINT

Barycentrics    -8*a^2*S + a^2*SA + 25*SB*SC : :

X(42539) lies on the curce Q073 and these lines: {2, 1328}, {4, 6501}, {5, 6474}, {30, 17851}, {615, 42413}, {1131, 3071}, {1132, 1152}, {3068, 3854}, {3091, 13903}, {3146, 6448}, {3311, 3832}, {3317, 3522}, {3543, 6395}, {5056, 6407}, {6420, 23263}, {6471, 7586}, {9542, 15022}, {9543, 42270}, {15705, 42225}

X(42540) = GIBERT (16 SQRT(3),25,2) POINT

Barycentrics    8*a^2*S + a^2*SA + 25*SB*SC : :

X(42540) lies on the curce Q073 and these lines: {2, 1327}, {4, 6500}, {5, 6475}, {590, 42414}, {1131, 1151}, {1132, 3070}, {3069, 3854}, {3091, 13961}, {3146, 6447}, {3312, 3832}, {3316, 3522}, {3543, 6199}, {5056, 6408}, {6419, 23253}, {6470, 7585}, {9542, 14241}, {15705, 42226}, {21734, 31412}

X(42541) = GIBERT (-64 SQRT(3),25,62) POINT

Barycentrics    -32*a^2*S + 31*a^2*SA + 25*SB*SC : :

X(42541) lies on the curce Q073 and these lines: {20, 6408}, {3069, 3839}, {3312, 3317}, {5059, 6475}, {5420, 6419}, {6199, 15708}, {6451, 15692}, {6459, 41964}, {9542, 32788}, {13785, 15640}

X(42542) = GIBERT (64 SQRT(3),25,62) POINT

Barycentrics    32*a^2*S + 31*a^2*SA + 25*SB*SC : :

X(42542) lies on the curce Q073 and these lines: {20, 6407}, {3068, 3839}, {3311, 3316}, {5059, 6474}, {5418, 6420}, {6395, 15708}, {6452, 15692}, {6460, 41963}, {6498, 31487}, {13665, 15640}

X(42543) = GIBERT (-9,98,-155) POINT

Barycentrics    -3*Sqrt[3]*a^2*S - 155*a^2*SA + 196*SB*SC : :

X(42543) lies on the curce Q073 and these lines: {1657, 3412}, {3411, 15704}, {3534, 33416}, {5238, 5366}, {10646, 12817}, {10654, 11001}, {11543, 42430}, {12816, 15685}, {15681, 42153}

X(42544) = GIBERT (9,98,-155) POINT

Barycentrics    3*Sqrt[3]*a^2*S - 155*a^2*SA + 196*SB*SC : :

X(42544) lies on the curce Q073 and these lines: {1657, 3411}, {3412, 15704}, {3534, 33417}, {5237, 5365}, {10645, 12816}, {10653, 11001}, {11542, 42429}, {12817, 15685}, {15681, 42156}

X(42545) = GIBERT (-45,98,-79) POINT

Barycentrics    -15*Sqrt[3]*a^2*S - 79*a^2*SA + 196*SB*SC : :

X(42545) lies on the curce Q073 and these lines: {13, 382}, {546, 10188}, {550, 12817}, {3411, 42088}, {3529, 16268}, {3530, 42099}, {3855, 10645}, {19106, 42415}

X(42546) = GIBERT (45,98,-79) POINT

Barycentrics    15*Sqrt[3]*a^2*S - 79*a^2*SA + 196*SB*SC : :

X(42546) lies on the curve Q073 and these lines: {14, 382}, {546, 10187}, {550, 12816}, {3412, 42087}, {3529, 16267}, {3530, 42100}, {3855, 10646}, {19107, 42416}

X(42547) = X(11)X(15914)∩X(100)X(693)

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

X(42547) lies on these lines: {11, 15914}, {100, 693}, {513, 34789}, {514, 21635}, {2804, 23770}, {3035, 11124}, {3738, 4010}, {4939, 42455}, {6366, 30592}, {11934, 13274}, {35100, 38752}

X(42548) = X(6)X(76)∩X(32)X(39684)

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

X(42548) lies on these lines: {6, 76}, {32, 39684}, {39, 3051}, {99, 42346}, {194, 40382}, {217, 5052}, {385, 1207}, {511, 14133}, {688, 23099}, {1500, 2309}, {1613, 7786}, {2086, 5368}, {3118, 27374}, {3231, 6683}, {3934, 17176}, {5007, 9427}, {5041, 32748}, {6292, 14822}, {6461, 12992}, {7772, 18899}, {7839, 38382}, {9019, 13330}, {9419, 42442}, {9865, 34482}, {10339, 32476}, {13331, 16285}, {14778, 32450}, {17475, 25264}

X(42548) = isogonal conjugate of X(31622)
X(42548) = trilinear product X(19)*X(23210)

X(42549) = X(69)X(189)∩X(84)X(517)

Barycentrics    a*(a*b + b^2 + a*c - 2*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)*(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(42549) lies on these lines: {57, 1422}, {65, 1413}, {69, 189}, {84, 517}, {354, 2208}, {1364, 17832}, {1433, 18732}, {2097, 7129}, {2188, 17441}, {2192, 3827}, {5908, 15239}, {8808, 14554}

X(42549) = crosspoint of X(57) and X(34546)
X(42549) = crosssum of X(9) and X(1604)

X(42550) = X(1)X(1437)∩X(65)X(15267)

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

X(42550) lies on these lines: {1, 1437}, {65, 15267}, {72, 1089}, {314, 2995}, {758, 10441}, {960, 19608}, {1409, 2171}, {2292, 22345}, {7015, 10570}, {18697, 41600}, {20653, 22076}

X(42550) = isogonal conjugate of X(40452)
X(42550) = crosspoint of X(65) and X(42485)
X(42550) = crosssum of X(21) and X(1610)

X(42551) = X(2)X(17042)∩X(6)X(194)

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

X(42551) lies on these lines: {2, 17042}, {6, 194}, {39, 4074}, {76, 3981}, {141, 14820}, {525, 882}, {538, 21849}, {695, 9230}, {706, 40951}, {732, 1843}, {755, 3222}, {2353, 3504}, {3094, 31360}, {3934, 41440}, {20081, 20977}, {31506, 41622}

X(42551) = isogonal conjugate of X(38834)
X(42551) = X(i)-isoconjugate-of X(j) for these {i,j}: {82, 1613}, {251, 1740}
X(42551) = crosspoint of X(76) and X(42486)
X(42551) = crosssum of X(32) and X(33786)
X(42551) = trilinear product X(i)*X(j) for these {i,j}: {38, 2998}, {39, 18832}, {141, 3223}, {561, 19606}
X(42551) = trilinear quotient X(i)/X(j) for these (i,j): (1930, 194), (2998, 82), (3223, 251), (3665, 1424), (18832, 83), (19606, 560)

X(42552) = X(11)X(11193)∩X(149)X(693)

Barycentrics    a*(a - b - c)*(b - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + a*b*c - b^2*c + a*c^2 + b*c^2 - c^3)*(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(42552) lies on these lines: {11, 11193}, {149, 693}, {513, 22321}, {654, 14418}, {900, 1830}, {1768, 3309}, {2254, 3722}, {3446, 15313}, {3738, 14740}, {3887, 5083}, {8760, 12747}, {11247, 37718}, {20095, 30613}

X(42552) = isogonal conjugate of X(40577)
X(42552) = crosssum of X(i) and X(j) for these {i,j}: {513, 38863}, {5540, 11193}
X(42552) = crossdifference of every pair of points on line X(1421)X(5540)
X(42552) = trilinear product X(i)*X(j) for these {i,j}: {109, 34896}, {522, 3446}, {663, 8047}

X(42553) = X(99)X(523)∩X(115)X(8029)

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

X(42553) lies on these lines: {99, 523}, {115, 8029}, {512, 39846}, {620, 11123}, {671, 9293}, {1649, 31274}, {2482, 36955}, {6722, 8371}, {8151, 15561}, {10278, 14061}, {10279, 38224}, {12042, 16220}, {14444, 33919}, {32204, 38750}

X(42554) = X(6)X(76)∩X(69)X(1369)

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

X(42554) lies on these lines: {6, 76}, {69, 1369}, {141, 6665}, {305, 3763}, {339, 11574}, {524, 31390}, {594, 20911}, {826, 22260}, {1235, 3867}, {1799, 2916}, {3266, 34573}, {3313, 14994}, {3589, 11205}, {3619, 9464}, {3631, 36792}, {3926, 13351}, {3933, 42442}, {4509, 23885}, {8788, 26190}, {10191, 40022}, {17949, 40043}, {20898, 21038}, {22289, 22308}, {33907, 35522}, {41256, 41916}

X(42554) = isotomic conjugate of isogonal conjugate of X(6292)

X(42555) = X(190)X(6634)∩X(514)X(4440)

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

X(42555) lies on these lines: {190, 6634}, {514, 4440}, {522, 21100}, {812, 14759}, {900, 3035}, {1086, 21204}, {2786, 3762}, {2796, 5592}, {3667, 9282}, {9262, 18645}, {17197, 21211}, {24821, 32212}, {32094, 42372}

X(42555) = isogonal conjugate of X(41405)
X(42555) = isotomic conjugate of X(6631)

X(42556) = X(3)X(95)∩X(51)X(216)

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

X(42556) lies on these lines: {3, 95}, {51, 216}, {185, 20775}, {511, 42441}, {1033, 26865}, {1075, 26876}, {3164, 26874}, {3611, 23197}, {6638, 12012}, {10263, 30258}, {11197, 14767}, {20975, 31364}, {23209, 40947}, {41212, 42445}

leftri

Gibert points on the KHO quartic Q168: X(42557)-X(42561)

rightri

This preamble and points X(42557)-X(42560) are contributed by Peter Moses, April 7, 2021. See also the preambles just before X(42085), X(42413), and X(42429).

This section gives four new points on the following KHO quartic (using KHO coordinates (x,y,z), introduced at KHO curves.

2*x^4 + 30*x^2*y^2 - 108*y^4 - 27*x^2*y*z + 216*y^3*z - 171*y^2*z^2 + 81*y*z^3 - 18*z^4 = 0.

This quartic passes through X(i) for these i: 2,5,17,18,371,372,1131,1132,3068,3069,3070,3071,8976,13951,31412,39641,39642, 42557, 42558, 42559, 42560.



X(42557) = GIBERT (-5 SQRT(3),8,13) POINT

Barycentrics    5*a^2*S - 13*a^2*SA - 16*SB*SC : :

X(42557) lies on the quartic K168 and these lines: {4, 6487}, {5, 372}, {6, 15703}, {140, 6484}, {376, 6565}, {381, 6481}, {486, 3525}, {1132, 3523}, {1656, 35770}, {1657, 42262}, {3068, 3317}, {3069, 6436}, {3071, 12108}, {3090, 35814}, {3091, 12818}, {3146, 5420}, {3830, 6396}, {3839, 42274}, {3843, 6485}, {3854, 13935}, {5054, 6200}, {5070, 35771}, {6411, 15722}, {6419, 8253}, {6434, 14269}, {6438, 19709}, {6477, 6560}, {6480, 15694}, {7586, 10576}, {8960, 13993}, {10124, 32790}, {12100, 18762}, {12102, 42267}, {13846, 42527}, {13847, 18512}, {13939, 35812}, {13941, 35822}, {14226, 42525}, {14241, 42277}, {14893, 35256}, {15705, 23259}, {19710, 42283}, {21734, 42266}, {33923, 35821}, {35762, 37712}

X(42558) = GIBERT (5 SQRT(3),8,13) POINT

Barycentrics    5*a^2*S + 13*a^2*SA + 16*SB*SC : :

X(42558) lies on the quartic K168 and these lines: {4, 6486}, {5, 371}, {6, 15703}, {140, 6485}, {376, 6564}, {381, 6480}, {485, 3525}, {1131, 3523}, {1656, 35771}, {1657, 42265}, {3068, 6435}, {3069, 3316}, {3070, 12108}, {3090, 35815}, {3091, 12819}, {3146, 5418}, {3830, 6200}, {3839, 9541}, {3843, 6484}, {3854, 9540}, {5054, 6396}, {5070, 35770}, {6412, 15722}, {6420, 8252}, {6433, 14269}, {6437, 19709}, {6476, 6561}, {6481, 15694}, {7585, 10577}, {8972, 35823}, {10124, 32789}, {12100, 18538}, {12102, 42266}, {13846, 18510}, {13847, 42526}, {13886, 35813}, {14226, 42274}, {14241, 42524}, {14893, 35255}, {15705, 23249}, {19710, 42284}, {21734, 42267}, {33923, 35820}, {35763, 37712}

X(42559) = GIBERT (-3/SQRT(2),-1/2,1) POINT

Barycentrics    -Sqrt[3/2]*a^2*S + a^2*SA - SB*SC : :

X(42559) lies on the quartic K168 and these lines: {6, 20}, {17, 14782}, {18, 14783}, {10653, 14784}, {10654, 14785}

X(42560) = GIBERT (3/SQRT(2),-1/2,1) POINT

Barycentrics    Sqrt[3/2]*a^2*S + a^2*SA - SB*SC : :

X(42560) lies on the quartic K168 and these lines:: {6, 20}, {17, 14783}, {18, 14782}, {10653, 14785}, {10654, 14784}

X(42561) = GIBERT (-2 SQRT(3),3,2) POINT

Barycentrics    -a^2*S + a^2*SA + 3*SB*SC : :
X(42561) = 2 X[6450] - 3 X[13935], X[6450] - 3 X[13951], 2 X[6450] + 3 X[23263], 2 X[13951] + X[23263]

X(31412) = the associated Gibert point, (2 SQRT(3),3,2)

X(42561) lies on the curve Q168 and these lines: {2, 489}, {3, 18762}, {4, 372}, {5, 1588}, {6, 3091}, {20, 615}, {30, 6450}, {69, 32488}, {114, 13653}, {140, 6455}, {371, 3090}, {376, 1328}, {378, 8277}, {381, 1587}, {382, 6408}, {485, 3545}, {487, 26362}, {488, 7620}, {491, 12221}, {515, 13959}, {516, 13947}, {546, 3312}, {550, 6497}, {590, 5056}, {626, 637}, {631, 6561}, {632, 6449}, {638, 5860}, {639, 7375}, {641, 26619}, {946, 19065}, {962, 13973}, {1056, 35801}, {1058, 35803}, {1124, 10590}, {1131, 3854}, {1152, 2672}, {1271, 23312}, {1335, 10591}, {1377, 31418}, {1504, 31415}, {1593, 13943}, {1656, 9540}, {1657, 35256}, {1699, 13936}, {1702, 10175}, {1703, 18483}, {2066, 10588}, {2067, 10589}, {3070, 3832}, {3093, 6623}, {3128, 13052}, {3297, 5261}, {3298, 5274}, {3364, 18582}, {3365, 42159}, {3367, 42254}, {3389, 18581}, {3390, 42162}, {3392, 42255}, {3522, 42263}, {3523, 8252}, {3524, 42260}, {3525, 6200}, {3528, 42266}, {3529, 6396}, {3533, 6486}, {3535, 8979}, {3543, 6430}, {3544, 6419}, {3564, 26469}, {3583, 13963}, {3585, 13962}, {3591, 5059}, {3592, 8972}, {3594, 42284}, {3614, 19038}, {3618, 32489}, {3627, 6398}, {3628, 6221}, {3634, 9616}, {3817, 18991}, {3818, 39875}, {3830, 13961}, {3839, 6471}, {3843, 23253}, {3850, 13665}, {3851, 6500}, {3855, 6564}, {3857, 6428}, {5055, 8981}, {5066, 19117}, {5067, 5418}, {5068, 7585}, {5070, 9691}, {5071, 10576}, {5072, 6417}, {5076, 42226}, {5079, 6199}, {5177, 31473}, {5225, 5414}, {5229, 6502}, {5411, 23047}, {5412, 6622}, {5448, 19061}, {5475, 26456}, {5587, 19066}, {5590, 7388}, {5640, 12239}, {5691, 13971}, {5818, 35775}, {5870, 36655}, {5893, 19087}, {5895, 13980}, {6202, 36656}, {6253, 13953}, {6256, 13964}, {6284, 13954}, {6352, 31562}, {6353, 8281}, {6409, 10303}, {6420, 23267}, {6422, 31404}, {6425, 32789}, {6426, 42272}, {6427, 12811}, {6448, 12102}, {6451, 14869}, {6452, 12103}, {6454, 42276}, {6456, 15704}, {6474, 15703}, {6485, 33703}, {6496, 12108}, {6499, 38071}, {7000, 13748}, {7173, 18996}, {7354, 13955}, {7486, 8253}, {7512, 35777}, {7687, 19110}, {7714, 18290}, {7728, 13979}, {7988, 8983}, {7989, 13883}, {8164, 35808}, {8227, 13902}, {8855, 8889}, {8964, 8990}, {9615, 19862}, {9974, 14853}, {10194, 10299}, {10515, 36664}, {10592, 31474}, {10721, 13969}, {10722, 13967}, {10723, 13989}, {10724, 13991}, {10728, 13977}, {10733, 13990}, {10735, 13992}, {10895, 19029}, {10896, 19027}, {11292, 26361}, {11294, 32805}, {11479, 19005}, {11488, 35732}, {11489, 42282}, {12111, 12240}, {12173, 13937}, {12245, 35789}, {12296, 13934}, {12509, 22617}, {12943, 18966}, {12953, 13958}, {12970, 32064}, {13846, 42522}, {13972, 36990}, {13975, 41869}, {13976, 34789}, {13981, 36962}, {13982, 36961}, {14561, 39876}, {14651, 33431}, {14784, 41976}, {14785, 41975}, {15681, 42537}, {15682, 42267}, {15690, 42527}, {17538, 42275}, {17578, 42264}, {17820, 41362}, {18539, 39864}, {18945, 19356}, {18992, 19925}, {19056, 23514}, {19060, 23515}, {19078, 38161}, {19082, 23513}, {19088, 23332}, {19109, 36519}, {19111, 36518}, {21734, 42539}, {22553, 22587}, {23269, 35786}, {26462, 31411}, {33345, 33371}, {35733, 42235}, {35738, 42203}, {35739, 42100}, {35822, 41106}, {41956, 42523}, {42119, 42239}, {42120, 42241}, {42125, 42280}, {42128, 42281}, {42135, 42209}, {42138, 42207}

X(42561) = midpoint of X(13935) and X(23263)
X(42561) = reflection of X(13935) in X(13951)
X(42561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 489, 33365}, {2, 1132, 3071}, {2, 3071, 6459}, {4, 486, 3069}, {4, 3069, 6460}, {4, 13939, 372}, {5, 1588, 3068}, {5, 13785, 1588}, {6, 3091, 31412}, {6, 42270, 3091}, {371, 3090, 32785}, {371, 42274, 3090}, {372, 486, 13939}, {372, 6565, 42268}, {372, 13939, 3069}, {372, 42268, 4}, {376, 3317, 5420}, {376, 35821, 42413}, {381, 7584, 1587}, {485, 7582, 19054}, {485, 35823, 7582}, {486, 6565, 4}, {486, 32498, 12256}, {486, 42268, 372}, {546, 3312, 23249}, {615, 23261, 20}, {615, 42271, 6410}, {631, 23275, 6561}, {1132, 42262, 6459}, {1152, 42283, 3146}, {1328, 3317, 42413}, {1328, 5420, 35821}, {1587, 7584, 19053}, {1656, 42215, 9540}, {3071, 42262, 2}, {3090, 23273, 371}, {3146, 13941, 1152}, {3544, 13886, 42277}, {3545, 7582, 485}, {3545, 14226, 35823}, {3545, 35823, 19054}, {3627, 13993, 6398}, {3832, 7586, 3070}, {3843, 42216, 23253}, {3850, 19116, 13665}, {3851, 18510, 7583}, {3855, 7581, 6564}, {5068, 7585, 42265}, {5072, 6417, 18538}, {5420, 35821, 376}, {6396, 22615, 3529}, {6409, 32790, 10303}, {6410, 23261, 42271}, {6410, 42271, 20}, {6419, 42277, 13886}, {6420, 42269, 23267}, {6560, 35787, 4}, {6561, 10577, 631}, {6564, 7581, 31414}, {8252, 42258, 3523}, {10895, 19029, 31408}, {11294, 32805, 33364}, {18762, 23259, 32786}, {23269, 41099, 35786}, {23273, 42274, 32785}, {42217, 42218, 32786}, {42242, 42244, 6565}

leftri

Gibert points: X(42562)-X(42593)

rightri

This preamble and points X(42562)-X(42593) are contributed by Peter Moses, April 7, 2021. See also the preambles just before X(42085), X(42413), and X(42429).

Points X(42562)-X(42575) lie on the cubic K458. Points X(42576)-X(42579) lie on the cubic K369.



X(42562) = GIBERT (12-7*SQRT(3), 5-3*SQRT(3), 3-2*SQRT(3)) POINT

Barycentrics    a^2*((-7 + 4*Sqrt[3])*S + (3 - 2*Sqrt[3])*SA) + (10 - 6*Sqrt[3])*SB*SC : :

X(42562) lies on the cubic K458 and these lines: {2, 371}, {5, 3389}, {6, 42565}, {13, 3070}, {14, 34559}, {15, 3071}, {16, 17}, {61, 3392}, {62, 590}, {302, 33351}, {303, 641}, {372, 2042}, {396, 3390}, {615, 3364}, {2044, 42266}, {2045, 10576}, {2046, 6200}, {2460, 6671}, {3102, 33392}, {3103, 22691}, {3367, 42251}, {3391, 37832}, {5318, 42226}, {5335, 6396}, {5420, 11488}, {6449, 36456}, {6564, 42244}, {6565, 42159}, {8253, 42156}, {8981, 23303}, {10195, 22235}, {10645, 42170}, {10646, 42201}, {11303, 33352}, {13993, 41943}, {14814, 16966}, {16809, 42212}, {16960, 42202}, {16965, 35740}, {16967, 18762}, {18586, 23261}, {19106, 42194}, {22615, 36454}, {23259, 42175}, {35732, 35821}, {35770, 36468}, {35814, 36449}, {35820, 36437}, {36439, 42258}, {36463, 42269}, {42113, 42178}, {42130, 42279}, {42150, 42187}, {42152, 42254}, {42177, 42277}, {42179, 42432}, {42219, 42494}, {42228, 42274}, {42245, 42261}

X(42562) = {X(2),X(5418)}-harmonic conjugate of X(42564)
X(42562) = {X(16),X(17)}-harmonic conjugate of X(42563)
X(42562) = {X(371),X(10577)}-harmonic conjugate of X(42564)

X(42563) = GIBERT (12+7*SQRT(3), 5+3*SQRT(3), 3+2*SQRT(3)) POINT

Barycentrics    a^2*((7 + 4*Sqrt[3])*S + (3 + 2*Sqrt[3])*SA) + (10 + 6*Sqrt[3])*SB*SC : :

X(42563) lies on the cubic K458 and these lines: {2, 372}, {5, 3390}, {6, 42564}, {13, 3071}, {14, 34562}, {15, 3070}, {16, 17}, {61, 3391}, {62, 615}, {302, 33352}, {303, 642}, {371, 2041}, {396, 3389}, {549, 35731}, {590, 3365}, {2043, 42267}, {2045, 6396}, {2046, 10577}, {2459, 6671}, {3102, 22691}, {3103, 33394}, {3366, 42253}, {3392, 35740}, {5318, 42225}, {5335, 6200}, {5418, 11488}, {6450, 36438}, {6564, 42159}, {6565, 42245}, {8252, 42156}, {8960, 42152}, {10194, 22235}, {10645, 42169}, {10646, 42202}, {11303, 33351}, {13925, 41943}, {13966, 23303}, {14813, 16966}, {15765, 35739}, {16809, 42214}, {16960, 42201}, {16965, 42241}, {16967, 18538}, {18587, 23251}, {19106, 42192}, {22644, 36436}, {23249, 42176}, {35771, 36449}, {35815, 36468}, {35820, 42282}, {35821, 36455}, {36445, 42268}, {36457, 42259}, {42113, 42177}, {42130, 42278}, {42150, 42189}, {42178, 42274}, {42181, 42432}, {42217, 42494}, {42227, 42277}, {42244, 42260}

X(42563) = {X(2),X(5420)}-harmonic conjugate of X(42565)
X(42563) = {X(16),X(17)}-harmonic conjugate of X(42562)
X(42563) = {X(372),X(10576)}-harmonic conjugate of X(42565)

X(42564) = GIBERT (-12-7*SQRT(3), 5+3*SQRT(3), 3+2*SQRT(3)) POINT

Barycentrics    a^2 ((-7-4 Sqrt[3]) S+(3+2 Sqrt[3]) SA)+(10+6 Sqrt[3]) SB SC : :

X(42564) lies on the cubic K458 and these lines: {2, 371}, {5, 3364}, {6, 42563}, {13, 34562}, {14, 3070}, {15, 18}, {16, 3071}, {61, 590}, {62, 3367}, {302, 641}, {303, 33353}, {372, 2041}, {395, 3365}, {615, 3389}, {2043, 42266}, {2045, 6200}, {2046, 10576}, {2460, 6672}, {3102, 33395}, {3103, 22692}, {3366, 37835}, {3392, 42250}, {5321, 42226}, {5334, 6396}, {5420, 11489}, {6449, 36438}, {6564, 42242}, {6565, 42162}, {8253, 42153}, {8981, 23302}, {10195, 22237}, {10645, 42199}, {10646, 42168}, {11304, 33350}, {13993, 41944}, {14813, 16967}, {16808, 42211}, {16961, 42200}, {16964, 35739}, {16966, 18762}, {18587, 23261}, {19107, 42193}, {22615, 36436}, {23259, 42177}, {35770, 36450}, {35814, 36467}, {35820, 36455}, {35821, 42282}, {36445, 42269}, {36457, 42258}, {42112, 42176}, {42131, 42278}, {42149, 42255}, {42151, 42188}, {42175, 42277}, {42180, 42431}, {42220, 42495}, {42230, 42274}, {42243, 42261}

X(42564) = {X(2),X(5418)}-harmonic conjugate of X(42562)
X(42564) = {X(15),X(18)}-harmonic conjugate of X(42565)
X(42564) = {X(371),X(10577)}-harmonic conjugate of X(42562)

X(42565) = GIBERT (-12+7*SQRT(3), 5-3*SQRT(3), 3-2*SQRT(3)) POINT

Barycentrics    a^2*((7 - 4*Sqrt[3])*S + (3 - 2*Sqrt[3])*SA) + (10 - 6*Sqrt[3])*SB*SC : :

X(42565) lies on the cubic K458 and these lines: {2, 372}, {5, 3365}, {6, 42562}, {13, 34559}, {14, 3071}, {15, 18}, {16, 3070}, {61, 615}, {62, 3366}, {302, 642}, {303, 33350}, {371, 2042}, {395, 3364}, {590, 3390}, {2044, 42267}, {2045, 10577}, {2046, 6396}, {2459, 6672}, {3102, 22692}, {3103, 33393}, {3367, 37835}, {3391, 42252}, {5321, 42225}, {5334, 6200}, {5418, 11489}, {6450, 36456}, {6564, 42162}, {6565, 42243}, {8252, 42153}, {8960, 42149}, {10194, 22237}, {10645, 42200}, {10646, 42167}, {11304, 33353}, {13925, 41944}, {13966, 23302}, {14814, 16967}, {16242, 35739}, {16808, 42213}, {16961, 42199}, {16964, 42239}, {16966, 18538}, {18586, 23251}, {19107, 42191}, {22644, 36454}, {23249, 42178}, {25189, 35757}, {32787, 35731}, {35732, 35820}, {35771, 36467}, {35815, 36450}, {35821, 36437}, {36439, 42259}, {36463, 42268}, {42112, 42175}, {42131, 42279}, {42151, 42190}, {42176, 42274}, {42182, 42431}, {42218, 42495}, {42229, 42277}, {42242, 42260}

X(42565) = {X(2),X(5420)}-harmonic conjugate of X(42563)
X(42565) = {X(15),X(18)}-harmonic conjugate of X(42564)
X(42565) = {X(372),X(10576)}-harmonic conjugate of X(42563)

X(42566) = GIBERT (5*SQRT(3), 9, 24) POINT

Barycentrics    a^2*(5*S + 24*SA) + 18*SB*SC : :

X(42566) lies on the cubic K458 and these lines: {2, 6437}, {5, 6200}, {6, 3525}, {376, 8253}, {590, 5054}, {615, 10124}, {631, 6434}, {1587, 6440}, {1657, 42277}, {3068, 6442}, {3070, 3523}, {3071, 6468}, {3090, 6433}, {3146, 6411}, {3628, 6480}, {5418, 6199}, {6396, 12108}, {6436, 32787}, {6438, 10303}, {6445, 42270}, {6469, 13886}, {6476, 35255}, {6481, 14869}, {6486, 12812}, {6490, 42262}, {6560, 15718}, {9690, 15703}, {10576, 33923}, {12100, 18538}, {12103, 42273}, {13665, 15722}, {18762, 41963}, {21734, 42265}, {23267, 42418}, {31454, 32786}

X(42567) = GIBERT (-5*SQRT(3), 9, 24) POINT

Barycentrics    a^2*(5*S - 24*SA) - 18*SB*SC : :

X(42567) lies on the cubic K458 and these lines: {2, 6438}, {5, 6396}, {6, 3525}, {376, 8252}, {590, 10124}, {615, 5054}, {631, 6433}, {1588, 6439}, {1657, 42274}, {3069, 6441}, {3070, 6469}, {3071, 3523}, {3090, 6434}, {3146, 6412}, {3628, 6481}, {5420, 6395}, {6200, 12108}, {6435, 32788}, {6437, 10303}, {6446, 42273}, {6468, 13939}, {6477, 35256}, {6480, 14869}, {6487, 12812}, {6491, 42265}, {6561, 15718}, {10577, 33923}, {12100, 18762}, {12103, 42270}, {13785, 15722}, {15703, 41946}, {18538, 41964}, {21734, 42262}, {23273, 42417}

X(42568) = GIBERT (5*SQRT(3), 2, 12) POINT

Barycentrics    a^2*(5*S + 12*SA) + 4*SB*SC : :

X(42568) lies on the cubic K458 and these lines: {2, 6429}, {3, 35815}, {4, 6433}, {5, 1151}, {6, 3523}, {30, 12818}, {140, 6437}, {371, 5054}, {376, 3070}, {381, 6484}, {382, 6486}, {485, 12103}, {486, 10124}, {549, 6431}, {590, 3146}, {1152, 12100}, {1588, 3317}, {1656, 6480}, {1657, 6200}, {3068, 21734}, {3071, 6468}, {3090, 10141}, {3312, 15718}, {3526, 42557}, {3530, 6432}, {3592, 12108}, {3830, 6449}, {3839, 42258}, {3854, 32785}, {3860, 22615}, {5067, 10139}, {5079, 6482}, {6221, 10577}, {6407, 15703}, {6410, 7581}, {6411, 8981}, {6417, 15722}, {6430, 15717}, {6434, 10299}, {6438, 15712}, {6439, 32789}, {6445, 10576}, {6451, 8960}, {6453, 42262}, {6455, 35812}, {6459, 41955}, {6487, 15706}, {6488, 12102}, {6490, 18762}, {6496, 35822}, {6519, 6565}, {8972, 42414}, {9541, 10147}, {9615, 37712}, {9692, 42561}, {10195, 42225}, {15533, 33364}, {15693, 35770}, {15701, 35814}, {15705, 32787}, {23253, 41950}, {23275, 41945}, {26516, 31884}, {35404, 42269}, {42263, 42558}

X(42569) = GIBERT (-5*SQRT(3), 2, 12) POINT

Barycentrics    a^2*(5*S - 12*SA) - 4*SB*SC : :

X(42569) lies on the cubic K458 and these lines: {2, 6430}, {3, 35814}, {4, 6434}, {5, 1152}, {6, 3523}, {30, 12819}, {140, 6438}, {372, 5054}, {376, 3071}, {381, 6485}, {382, 6487}, {485, 10124}, {486, 12103}, {549, 6432}, {615, 3146}, {1151, 12100}, {1587, 3316}, {1656, 6481}, {1657, 6396}, {3069, 21734}, {3070, 6469}, {3090, 10142}, {3311, 15718}, {3526, 42558}, {3530, 6431}, {3594, 8981}, {3830, 6450}, {3839, 42259}, {3854, 32786}, {3860, 22644}, {5067, 10140}, {5079, 6483}, {6398, 10576}, {6408, 15703}, {6409, 7582}, {6412, 13966}, {6418, 15722}, {6429, 15717}, {6433, 10299}, {6437, 15712}, {6440, 32790}, {6446, 10577}, {6454, 42265}, {6456, 35813}, {6460, 17852}, {6486, 15706}, {6489, 12102}, {6491, 18538}, {6497, 35823}, {6522, 6564}, {10194, 42226}, {13941, 42413}, {15533, 33365}, {15693, 35771}, {15701, 35815}, {15705, 32788}, {23263, 41949}, {23269, 41946}, {26521, 31884}, {35404, 42268}, {42264, 42557}

X(42570) = GIBERT (10*SQRT(3), 11, 6) POINT

Barycentrics    a^2*(5*S + 3*SA) + 11*SB*SC : :

X(42570) lies on the cubic K458 and these lines: {2, 6430}, {5, 1587}, {6, 3854}, {376, 485}, {590, 21734}, {631, 42558}, {1131, 3068}, {1657, 6445}, {3070, 3523}, {3071, 3839}, {3311, 14893}, {3525, 6460}, {3830, 6459}, {3860, 19117}, {5420, 23267}, {6410, 41952}, {6472, 9541}, {6476, 8960}, {6564, 7582}, {7583, 23263}, {8972, 42414}, {8976, 33923}, {9540, 12103}, {12100, 42526}, {13886, 42260}, {13925, 19710}, {13935, 15703}, {15682, 35815}, {15705, 42259}, {19053, 41951}, {35771, 41099}, {35822, 41106}, {41954, 42265}, {41965, 42264}

X(42571) = GIBERT (-10*SQRT(3), 11, 6) POINT

Barycentrics    a^2*(5*S - 3*SA) - 11*SB*SC : :

X(42571) lies on the cubic K458 and these lines: {2, 6429}, {5, 1588}, {6, 3854}, {376, 486}, {615, 21734}, {631, 42557}, {1132, 3069}, {1657, 6446}, {3070, 3839}, {3071, 3523}, {3312, 14893}, {3525, 6459}, {3830, 6460}, {3860, 19116}, {5418, 23273}, {6409, 41951}, {6565, 7581}, {7584, 23253}, {9540, 15703}, {9541, 12108}, {9690, 18762}, {12100, 42527}, {12103, 13935}, {13939, 42261}, {13941, 42413}, {13951, 33923}, {13993, 19710}, {15682, 35814}, {15705, 42258}, {19054, 41952}, {31412, 35823}, {35770, 41099}, {41953, 42262}, {41966, 42263}

X(42572) = GIBERT (15*SQRT(3), 13, 8) POINT

Barycentrics    a^2*(15*S + 8*SA) + 26*SB*SC : :

X(42572) lies on the cubic K458 and these lines: {2, 6438}, {5, 6420}, {6, 14226}, {30, 35815}, {371, 35404}, {376, 3070}, {485, 5054}, {590, 12100}, {1327, 3830}, {1657, 6519}, {3068, 42538}, {3071, 3839}, {3146, 41945}, {3523, 31414}, {3525, 17852}, {3860, 6564}, {5215, 13701}, {6280, 13932}, {6437, 15640}, {6472, 42272}, {6481, 11540}, {7583, 14893}, {8960, 12103}, {8976, 15718}, {10138, 15694}, {11737, 35770}, {13663, 13678}, {15703, 41952}, {17851, 32789}, {19710, 42525}, {23046, 35771}, {23267, 42418}, {41961, 42276}, {42216, 42558}, {42265, 42523}

X(42573) = GIBERT (-15*SQRT(3), 13, 8) POINT

Barycentrics    a^2*(15*S - 8*SA) - 26*SB*SC : :

X(42573) lies on the cubic K458 and these lines: {2, 6437}, {5, 6419}, {6, 14226}, {30, 35814}, {372, 35404}, {376, 3071}, {486, 5054}, {615, 12100}, {1328, 3830}, {1657, 6522}, {3069, 42537}, {3070, 3839}, {3146, 41946}, {3860, 6565}, {5215, 13821}, {6279, 13850}, {6438, 15640}, {6473, 42271}, {6480, 11540}, {7584, 14893}, {10137, 15694}, {11737, 35771}, {13783, 13798}, {13951, 15718}, {15703, 41951}, {19710, 42524}, {23046, 35770}, {23273, 42417}, {41962, 42275}, {42215, 42557}, {42262, 42522}

X(42574) = GIBERT (26*SQRT(3), 9, -30) POINT

Barycentrics    a^2*(13*S - 15*SA) + 9*SB*SC : :

X(42574) lies on the cubic K458 and these lines: {2, 6440}, {5, 6398}, {20, 6442}, {3068, 6412}, {3071, 42414}, {3090, 6477}, {3528, 6476}, {3544, 6479}, {3590, 6434}, {6199, 15688}, {6200, 7581}, {6396, 13886}, {6436, 6459}, {6468, 19054}, {6481, 31412}, {6561, 11001}, {13941, 41947}, {15684, 23259}, {15701, 42216}, {15709, 23267}, {15723, 18538}, {18510, 42537}

X(42575) = GIBERT (-26*SQRT(3), 9, -30) POINT

Barycentrics    a^2*(13*S + 15*SA) - 9*SB*SC : :

X(42575) lies on the cubic K458 and these lines: {2, 6439}, {5, 6221}, {20, 6441}, {3069, 6411}, {3070, 42413}, {3090, 6476}, {3528, 6477}, {3544, 6478}, {3591, 6433}, {6200, 13939}, {6395, 9541}, {6396, 7582}, {6435, 6460}, {6469, 19053}, {6480, 42561}, {6560, 11001}, {8972, 41948}, {9542, 42417}, {9543, 32790}, {9681, 35814}, {15684, 23249}, {15701, 42215}, {15709, 23273}, {15723, 18762}, {18512, 42538}, {31414, 42275}

X(42576) = GIBERT (9*SQRT(3), 26, -20) POINT

Barycentrics    a^2*(9*S - 20*SA) + 52*SB*SC : :

X(42576) lies on the cubic K369 and these lines: {2, 42272}, {4, 10148}, {6, 15640}, {20, 41952}, {30, 3592}, {1152, 12101}, {3068, 41959}, {3071, 6471}, {3534, 23251}, {3594, 35404}, {3830, 6398}, {3845, 5420}, {3860, 42261}, {6407, 15685}, {6410, 41106}, {6433, 11001}, {6435, 42263}, {6483, 38335}, {8252, 41099}, {8253, 8703}, {12100, 22644}, {14269, 42524}, {15681, 35812}, {15695, 42265}, {15698, 42284}, {15711, 42269}, {19709, 42267}, {23261, 33699}, {32787, 42538}

X(42577) = GIBERT (-9*SQRT(3), 26, -20) POINT

Barycentrics    a^2*(9*S + 20*SA) - 52*SB*SC : :

X(42577) lies on the cubic K369 and these lines: {2, 42271}, {4, 10147}, {6, 15640}, {20, 41951}, {30, 3594}, {1151, 12101}, {3069, 41960}, {3070, 6470}, {3534, 23261}, {3592, 35404}, {3830, 6221}, {3845, 5418}, {3860, 42260}, {6408, 15685}, {6409, 41106}, {6434, 11001}, {6436, 42264}, {6482, 38335}, {8252, 8703}, {8253, 41099}, {12100, 22615}, {14269, 42525}, {15681, 35813}, {15695, 42262}, {15698, 42283}, {15711, 42268}, {19709, 42266}, {23251, 33699}, {32788, 42537}

X(42578) = GIBERT (11*SQRT(3), 10, 12) POINT

Barycentrics    a^2*(11*S + 12*SA) + 20*SB*SC : :

X(42578) lies on the cubic K369 and these lines: {2, 41948}, {6, 5056}, {30, 485}, {372, 15723}, {590, 15717}, {1132, 32787}, {1587, 3316}, {1656, 6436}, {3070, 21735}, {3071, 3855}, {3312, 5070}, {3529, 41954}, {3590, 32789}, {3592, 41991}, {5055, 6499}, {5072, 6419}, {6199, 8960}, {6396, 8976}, {6408, 35822}, {6409, 23269}, {6410, 15719}, {6429, 42413}, {6438, 10195}, {6440, 32785}, {6441, 18538}, {6447, 6564}, {6451, 42267}, {6459, 41952}, {6468, 23253}, {6480, 35405}, {6489, 42216}, {6497, 15718}, {6500, 42262}, {8972, 42414}, {10303, 41970}, {11541, 41961}, {42272, 42540}, {42273, 42522}

X(42579) = GIBERT (-11*SQRT(3), 10, 12) POINT

Barycentrics    a^2*(11*S - 12*SA) - 20*SB*SC : :

X(42579) lies on the cubic K369 and these lines: {2, 41947}, {6, 5056}, {30, 486}, {371, 15723}, {615, 15717}, {1131, 32788}, {1588, 3317}, {1656, 6435}, {3070, 3855}, {3071, 21735}, {3311, 5070}, {3529, 41953}, {3591, 32790}, {3594, 41991}, {5055, 6498}, {5072, 6420}, {6200, 13951}, {6395, 23251}, {6407, 35823}, {6409, 15719}, {6410, 23275}, {6430, 42414}, {6437, 10194}, {6439, 32786}, {6442, 18762}, {6448, 6565}, {6452, 42266}, {6460, 41951}, {6469, 23263}, {6481, 35405}, {6488, 42215}, {6496, 15718}, {6501, 42265}, {10303, 41969}, {11541, 41962}, {13941, 42413}, {17851, 42267}, {42270, 42523}, {42271, 42539}

X(42580) = GIBERT (-3, 6, 7) POINT

Barycentrics    a^2*(Sqrt[3]*S - 7*SA) - 12*SB*SC : :

X(42580) lies on these lines: {2, 5238}, {3, 16809}, {4, 5351}, {5, 13}, {6, 5079}, {14, 1656}, {15, 3628}, {16, 3091}, {17, 5055}, {61, 3090}, {140, 36967}, {202, 3614}, {303, 22845}, {381, 36843}, {382, 42528}, {396, 35018}, {398, 547}, {546, 5237}, {548, 42430}, {549, 12817}, {619, 20378}, {621, 630}, {623, 7944}, {629, 11304}, {631, 36970}, {632, 5321}, {636, 21359}, {1657, 10187}, {3146, 42089}, {3180, 33464}, {3200, 13434}, {3364, 42191}, {3365, 42193}, {3389, 42274}, {3390, 42277}, {3525, 10645}, {3526, 42157}, {3529, 42103}, {3530, 5349}, {3544, 11489}, {3545, 41100}, {3627, 10646}, {3830, 42491}, {3832, 42431}, {3843, 36968}, {3850, 16773}, {3851, 16645}, {3855, 42151}, {3857, 42121}, {5054, 42434}, {5056, 37832}, {5066, 41944}, {5067, 10654}, {5068, 10653}, {5070, 5339}, {5071, 16268}, {5072, 16808}, {5076, 42100}, {5318, 12811}, {5340, 16963}, {5350, 38071}, {5366, 42510}, {5418, 42235}, {5420, 42237}, {5460, 33414}, {5469, 8724}, {5611, 22849}, {6670, 11289}, {6777, 20416}, {7006, 7173}, {7486, 42152}, {9761, 22844}, {10095, 36979}, {10109, 16267}, {10303, 42085}, {10576, 35730}, {10612, 16529}, {10658, 15027}, {11267, 12010}, {11272, 32465}, {11305, 22490}, {11444, 36981}, {11543, 12812}, {12103, 42101}, {12108, 42087}, {14869, 42135}, {15022, 18582}, {15686, 42543}, {15699, 16772}, {15703, 41101}, {15720, 42529}, {16961, 42098}, {18874, 36978}, {22891, 33422}, {23515, 36209}, {31694, 36767}, {31706, 39555}, {33417, 36836}, {33561, 36770}, {33923, 42501}, {34755, 42110}, {35739, 42226}, {41120, 41943}, {41991, 42102}, {42086, 42473}, {42144, 42493}, {42156, 42475}

X(42580) = {X(6),X(5079)}-harmonic conjugate of X(42581)

X(42581) = GIBERT (3, 6, 7) POINT

Barycentrics    a^2*(Sqrt[3]*S + 7*SA) + 12*SB*SC : :

X(42581) lies on these lines: {2, 5237}, {3, 16808}, {4, 5352}, {5, 14}, {6, 5079}, {13, 1656}, {15, 3091}, {16, 3628}, {18, 5055}, {62, 3090}, {140, 36968}, {203, 3614}, {302, 22844}, {381, 36836}, {382, 42529}, {395, 35018}, {397, 547}, {546, 5238}, {548, 42429}, {549, 12816}, {618, 20377}, {622, 629}, {624, 7944}, {630, 11303}, {631, 36969}, {632, 5318}, {635, 21360}, {1657, 10188}, {3146, 42092}, {3181, 33465}, {3201, 13434}, {3364, 42274}, {3365, 42277}, {3389, 42192}, {3390, 42194}, {3525, 10646}, {3526, 42158}, {3529, 42106}, {3530, 5350}, {3544, 11488}, {3545, 41101}, {3627, 10645}, {3830, 42490}, {3832, 42432}, {3843, 36967}, {3850, 16772}, {3851, 16644}, {3855, 42150}, {3857, 42124}, {5054, 42433}, {5056, 37835}, {5066, 41943}, {5067, 10653}, {5068, 10654}, {5070, 5340}, {5071, 16267}, {5072, 16809}, {5076, 42099}, {5321, 12811}, {5339, 16962}, {5349, 38071}, {5365, 42511}, {5418, 42236}, {5420, 42238}, {5459, 33415}, {5470, 8724}, {5615, 22895}, {6565, 35730}, {6669, 11290}, {6778, 20415}, {7005, 7173}, {7486, 42149}, {9763, 22845}, {10095, 36981}, {10109, 16268}, {10303, 42086}, {10611, 16530}, {10657, 15027}, {11268, 12010}, {11272, 32466}, {11306, 22489}, {11444, 36979}, {11542, 12812}, {12103, 42102}, {12108, 42088}, {14869, 42138}, {15022, 18581}, {15686, 42544}, {15699, 16773}, {15703, 41100}, {15720, 42528}, {16001, 36766}, {16960, 42095}, {18874, 36980}, {22846, 33423}, {23515, 36208}, {31705, 39554}, {33416, 36843}, {33923, 42500}, {34754, 42107}, {41119, 41944}, {41991, 42101}, {42085, 42472}, {42145, 42492}, {42153, 42474}

X(42581) = {X(6),X(5079)}-harmonic conjugate of X(42580)

X(42582) = GIBERT (SQRT(3), 3, 4) POINT

Barycentrics    a^2*(S + 4*SA) + 6*SB*SC : :

X(42582) lies on these lines: {2, 490}, {3, 22644}, {4, 6409}, {5, 371}, {6, 3090}, {140, 6564}, {372, 3628}, {373, 12240}, {381, 5418}, {382, 6496}, {485, 615}, {486, 5055}, {491, 23311}, {546, 6200}, {547, 7583}, {549, 35820}, {550, 35786}, {631, 23251}, {632, 6396}, {640, 7867}, {1151, 3091}, {1327, 15694}, {1587, 5067}, {1588, 3316}, {2045, 42241}, {2046, 42239}, {2066, 7173}, {2067, 3614}, {3054, 12968}, {3068, 5056}, {3069, 7486}, {3146, 6411}, {3297, 10589}, {3298, 10588}, {3311, 5079}, {3317, 41950}, {3364, 42191}, {3365, 16966}, {3366, 18585}, {3389, 42192}, {3390, 16967}, {3391, 15765}, {3523, 42264}, {3524, 23253}, {3525, 6410}, {3526, 6456}, {3530, 42267}, {3544, 6425}, {3545, 6429}, {3583, 31499}, {3592, 8972}, {3594, 32786}, {3832, 42263}, {3843, 42260}, {3845, 42266}, {3850, 6484}, {3851, 6407}, {3855, 9541}, {3857, 42225}, {5054, 42261}, {5066, 35787}, {5068, 6459}, {5070, 5420}, {5072, 6221}, {5076, 6451}, {5901, 35788}, {6412, 10303}, {6419, 12812}, {6424, 31415}, {6426, 23267}, {6432, 13941}, {6453, 12811}, {6455, 42275}, {6478, 11737}, {6485, 41992}, {6486, 41991}, {6487, 16239}, {6501, 13951}, {6931, 31473}, {7516, 35776}, {7581, 13847}, {7584, 8960}, {7741, 9646}, {7852, 32490}, {7951, 9661}, {7969, 10175}, {7988, 13893}, {8227, 13911}, {8407, 13882}, {8854, 11548}, {9542, 41965}, {9605, 13711}, {9674, 18424}, {9975, 14561}, {10109, 35823}, {10146, 41970}, {10171, 13883}, {10283, 35842}, {10592, 35768}, {10593, 35808}, {11291, 32459}, {12222, 32812}, {12818, 15700}, {12964, 23332}, {13881, 31463}, {13966, 15699}, {14869, 42226}, {15325, 35800}, {15692, 42414}, {15703, 41952}, {18357, 35763}, {19709, 42417}, {23302, 42240}, {23303, 35740}, {32497, 37342}, {35610, 38034}, {35641, 38042}, {35732, 42095}, {35739, 42214}, {35765, 37942}, {35827, 40685}, {35878, 38229}, {41948, 41949}, {41953, 42522}, {41954, 41964}, {42098, 42282}, {42150, 42243}, {42151, 42245}

X(42582) = {X(6),X(3090)}-harmonic conjugate of X(42583)

X(42583) = GIBERT (-SQRT(3), 3, 4) POINT

Barycentrics    a^2*(S - 4*SA) - 6*SB*SC : :

X(42583) lies on these lines: {2, 489}, {3, 22615}, {4, 6410}, {5, 372}, {6, 3090}, {140, 6565}, {371, 3628}, {373, 12239}, {381, 5420}, {382, 6497}, {485, 5055}, {486, 590}, {492, 23312}, {546, 6396}, {547, 7584}, {549, 35821}, {550, 35787}, {631, 23261}, {632, 6200}, {639, 7867}, {1152, 3091}, {1328, 15694}, {1587, 3317}, {1588, 5067}, {2045, 42240}, {2046, 35740}, {3054, 12963}, {3068, 7486}, {3069, 5056}, {3146, 6412}, {3297, 10588}, {3298, 10589}, {3312, 5079}, {3316, 41949}, {3364, 16966}, {3365, 42193}, {3367, 15765}, {3389, 16967}, {3390, 42194}, {3392, 18585}, {3523, 42263}, {3524, 23263}, {3525, 6409}, {3526, 6455}, {3530, 42266}, {3533, 9541}, {3544, 6426}, {3545, 6430}, {3592, 32785}, {3594, 13941}, {3614, 6502}, {3832, 42264}, {3843, 42261}, {3845, 42267}, {3850, 6485}, {3851, 6408}, {3857, 42226}, {5054, 42260}, {5066, 35786}, {5068, 6460}, {5070, 5418}, {5072, 6398}, {5076, 6452}, {5414, 7173}, {5901, 35789}, {6411, 10303}, {6420, 12812}, {6423, 31415}, {6425, 23273}, {6431, 8972}, {6454, 12811}, {6456, 42276}, {6479, 11737}, {6484, 41992}, {6486, 16239}, {6487, 41991}, {6500, 8976}, {6933, 31473}, {7516, 35777}, {7582, 13846}, {7583, 35018}, {7852, 32491}, {7968, 10175}, {7988, 13947}, {8227, 13973}, {8400, 13934}, {8855, 11548}, {8960, 19116}, {8981, 15699}, {9605, 13834}, {9616, 19872}, {9974, 14561}, {10109, 35822}, {10145, 41969}, {10171, 13936}, {10283, 35843}, {10592, 35769}, {10593, 35809}, {11292, 32459}, {12221, 32813}, {12819, 15700}, {12970, 23332}, {13955, 31472}, {14869, 42225}, {15325, 35801}, {15692, 42413}, {15703, 41951}, {18357, 35762}, {19709, 42418}, {23302, 42239}, {23303, 42241}, {31414, 41954}, {32488, 32807}, {32494, 37343}, {35611, 38034}, {35642, 38042}, {35732, 42098}, {35739, 42281}, {35764, 37942}, {35826, 40685}, {35879, 38229}, {41947, 41950}, {41953, 41963}, {42095, 42282}, {42150, 42242}, {42151, 42244}

X(42583) = {X(6),X(3090)}-harmonic conjugate of X(42582)

X(42584) = GIBERT (2, 3, -5) POINT

Barycentrics    a^2*(2*Sqrt[3]*S - 15*SA) + 18*SB*SC : :

X(42584) lies on these lines: {3, 42134}, {5, 42091}, {6, 15704}, {13, 15690}, {14, 16}, {15, 12103}, {17, 41981}, {20, 11485}, {140, 19106}, {376, 42124}, {382, 42121}, {396, 15691}, {546, 10646}, {547, 42528}, {548, 5318}, {549, 42094}, {550, 5340}, {632, 42106}, {1657, 42117}, {3146, 42115}, {3522, 42128}, {3528, 42132}, {3529, 11486}, {3530, 16808}, {3534, 5335}, {3543, 42129}, {3627, 11481}, {3628, 42102}, {3845, 42089}, {3853, 23303}, {3861, 16967}, {5059, 42126}, {5066, 33416}, {5073, 11489}, {5237, 42101}, {5334, 17800}, {5350, 33417}, {5351, 12102}, {7667, 37776}, {8703, 18582}, {8972, 42213}, {10645, 42165}, {10653, 19710}, {11001, 42130}, {11488, 15696}, {12100, 16966}, {12101, 16242}, {12816, 42500}, {13941, 42211}, {14869, 42114}, {14892, 42501}, {15681, 42119}, {15686, 42090}, {15687, 42095}, {15694, 42472}, {15712, 42098}, {15759, 37832}, {16645, 35404}, {17538, 42116}, {22238, 42112}, {23302, 33923}, {33703, 42125}, {34200, 36969}, {34755, 42164}, {35738, 42193}, {36843, 42104}, {41991, 42493}, {42087, 42158}, {42096, 42151}, {42099, 42148}, {42415, 42420}, {42496, 42529}

X(42585) = GIBERT (-2, 3, -5) POINT

Barycentrics    a^2*(2*Sqrt[3]*S + 15*SA) - 18*SB*SC : :

X(42585) lies on these lines: {3, 42133}, {5, 42090}, {6, 15704}, {13, 15}, {14, 15690}, {16, 12103}, {18, 41981}, {20, 11486}, {140, 19107}, {376, 42121}, {382, 42124}, {395, 15691}, {546, 10645}, {547, 42529}, {548, 5321}, {549, 42093}, {550, 5339}, {632, 42103}, {1657, 42118}, {3146, 42116}, {3522, 42125}, {3528, 42129}, {3529, 11485}, {3530, 16809}, {3534, 5334}, {3543, 42132}, {3627, 11480}, {3628, 42101}, {3845, 42092}, {3853, 23302}, {3861, 16966}, {5059, 42127}, {5066, 33417}, {5073, 11488}, {5238, 42102}, {5335, 17800}, {5349, 33416}, {5352, 12102}, {7667, 37775}, {8703, 18581}, {8972, 42214}, {10646, 42164}, {10654, 19710}, {11001, 42131}, {11489, 15696}, {12100, 16967}, {12101, 16241}, {12817, 42501}, {13941, 42212}, {14869, 42111}, {14892, 42500}, {15681, 42120}, {15686, 42091}, {15687, 42098}, {15694, 42473}, {15712, 42095}, {15759, 37835}, {16644, 35404}, {17538, 42115}, {22236, 42113}, {23303, 33923}, {33703, 42128}, {34200, 36970}, {34754, 42165}, {35738, 42192}, {36836, 42105}, {41991, 42492}, {42088, 42157}, {42097, 42150}, {42100, 42147}, {42416, 42419}, {42497, 42528}

X(42586) = GIBERT (9, 14, -20) POINT

Barycentrics    a^2*(3*Sqrt[3]*S - 20*SA) + 28*SB*SC : :

X(42586) lies on these lines: {3, 10188}, {6, 15683}, {14, 35400}, {30, 5339}, {376, 5318}, {381, 10646}, {382, 41944}, {547, 42091}, {549, 42094}, {3534, 5238}, {3543, 16645}, {3843, 10187}, {3845, 42491}, {3851, 42546}, {5350, 15698}, {10124, 42474}, {11480, 15691}, {11481, 15687}, {11485, 15681}, {11737, 42105}, {12103, 41112}, {14093, 36969}, {14269, 42433}, {15682, 36843}, {15684, 36968}, {15685, 42158}, {15686, 42086}, {15688, 42431}, {15689, 42156}, {15690, 42161}, {15694, 19106}, {15700, 42098}, {15703, 42528}, {15714, 42137}, {15718, 16808}, {17800, 41100}, {19710, 22236}, {22235, 42165}, {34200, 42145}, {35403, 42095}, {35404, 42113}, {35408, 42497}, {41101, 42543}, {41943, 42127}, {42096, 42429}, {42109, 42473}, {42131, 42154}, {42419, 42509}

X(42587) = GIBERT (-9, 14, -20) POINT

Barycentrics    a^2*(3*Sqrt[3]*S + 20*SA) - 28*SB*SC : :

X(42587) lies on these lines: {3, 10187}, {6, 15683}, {13, 35400}, {30, 5340}, {376, 5321}, {381, 10645}, {382, 41943}, {547, 42090}, {549, 42093}, {3534, 5237}, {3543, 16644}, {3843, 10188}, {3845, 42490}, {3851, 42545}, {5349, 15698}, {10124, 42475}, {11480, 15687}, {11481, 15691}, {11486, 15681}, {11737, 42104}, {12103, 41113}, {14093, 36970}, {14269, 42434}, {15682, 36836}, {15684, 36967}, {15685, 42157}, {15686, 42085}, {15688, 42432}, {15689, 42153}, {15690, 42160}, {15694, 19107}, {15700, 42095}, {15703, 42529}, {15714, 42136}, {15718, 16809}, {17800, 41101}, {19710, 22238}, {22237, 42164}, {34200, 42144}, {35403, 42098}, {35404, 42112}, {35408, 42496}, {41100, 42544}, {41944, 42126}, {42097, 42430}, {42108, 42472}, {42130, 42155}, {42420, 42508}

X(42588) = GIBERT (18,13,-10) POINT

Barycentrics    a^2*(3*Sqrt[3]*S - 5*SA) + 13*SB*SC : :

X(42588) lies on these lines: {2, 5318}, {4, 3411}, {6, 15640}, {13, 19708}, {16, 41106}, {17, 15710}, {376, 5352}, {395, 42508}, {396, 15697}, {397, 15683}, {538, 5863}, {549, 5344}, {3524, 41121}, {3528, 33604}, {3534, 5335}, {3543, 5339}, {3545, 12816}, {3830, 37641}, {3839, 42148}, {3845, 42127}, {3855, 41944}, {5054, 42494}, {5055, 5366}, {5066, 42134}, {5071, 42151}, {5334, 33699}, {5340, 10304}, {6221, 36446}, {6398, 36465}, {8703, 11488}, {10653, 15682}, {11001, 34754}, {11486, 12101}, {11489, 36969}, {11542, 15695}, {11812, 42128}, {12817, 42105}, {14226, 36447}, {14241, 36464}, {15690, 42131}, {15698, 36968}, {15701, 42123}, {15702, 42162}, {15708, 42166}, {15715, 42433}, {15719, 18582}, {16242, 42472}, {16962, 17538}, {16963, 42495}, {19106, 41113}, {19711, 42132}, {22238, 42519}, {22513, 35749}, {23006, 36344}, {33603, 40694}, {33610, 37172}, {33703, 41974}, {35409, 42432}, {36445, 42233}, {36463, 42234}, {42087, 42516}

X(42589) = GIBERT (-18,13,-10) POINT

Barycentrics    a^2*(3*Sqrt[3]*S + 5*SA) - 13*SB*SC : :

X(42589) lies on these lines: {2, 5321}, {4, 3412}, {6, 15640}, {14, 19708}, {15, 41106}, {18, 15710}, {376, 5351}, {395, 15697}, {396, 42509}, {398, 15683}, {538, 5862}, {549, 5343}, {3524, 41122}, {3528, 33605}, {3534, 5334}, {3543, 5340}, {3545, 12817}, {3830, 37640}, {3839, 42147}, {3845, 42126}, {3855, 41943}, {5054, 42495}, {5055, 5365}, {5066, 42133}, {5071, 42150}, {5335, 33699}, {5339, 10304}, {6221, 36464}, {6398, 36447}, {8703, 11489}, {10654, 15682}, {11001, 34755}, {11485, 12101}, {11488, 36970}, {11543, 15695}, {11812, 42125}, {12816, 42104}, {14226, 36465}, {14241, 36446}, {15690, 42130}, {15698, 36967}, {15701, 42122}, {15702, 42159}, {15708, 42163}, {15715, 42434}, {15719, 18581}, {16241, 42473}, {16962, 42494}, {16963, 17538}, {19107, 41112}, {19711, 42129}, {22236, 42518}, {22512, 36327}, {23013, 36319}, {33602, 40693}, {33611, 37173}, {33703, 41973}, {35409, 42431}, {36445, 42232}, {36463, 42231}, {42088, 42517}

X(42590) = GIBERT (6,9,17) POINT

Barycentrics    a^2*(2*Sqrt[3]*S + 17*SA) + 18*SB*SC : :

X(42590) lies on these lines: {3, 42134}, {5, 36836}, {6, 42492}, {13, 140}, {15, 12812}, {17, 16239}, {61, 3628}, {395, 10187}, {397, 10124}, {546, 5352}, {547, 41108}, {549, 42162}, {630, 33477}, {632, 11542}, {3090, 42124}, {3091, 42122}, {3412, 42497}, {3525, 42132}, {3530, 37832}, {3544, 42116}, {3627, 42092}, {3845, 42490}, {3850, 16241}, {3857, 11480}, {3859, 36967}, {5067, 22237}, {5079, 42117}, {5238, 12811}, {5344, 15701}, {5349, 14892}, {10109, 42147}, {10303, 42118}, {10645, 12102}, {11539, 42156}, {11737, 42157}, {11812, 16965}, {12108, 33417}, {14869, 18582}, {14891, 42431}, {15022, 42135}, {15699, 42152}, {15704, 42098}, {15713, 42151}, {16772, 35018}, {16962, 42503}, {22236, 42143}, {33413, 35303}, {40694, 42519}, {41983, 42433}

X(42591) = GIBERT (-6,9,17) POINT

Barycentrics    a^2*(2*Sqrt[3]*S - 17*SA) - 18*SB*SC : :

X(42591) lies on these lines: {3, 42133}, {5, 36843}, {6, 42492}, {14, 140}, {16, 12812}, {18, 16239}, {62, 3628}, {396, 10188}, {398, 10124}, {546, 5351}, {547, 41107}, {549, 42159}, {629, 33476}, {632, 11543}, {3090, 42121}, {3091, 42123}, {3411, 42496}, {3525, 42129}, {3530, 37835}, {3544, 42115}, {3627, 42089}, {3845, 42491}, {3850, 16242}, {3857, 11481}, {3859, 36968}, {5067, 22235}, {5079, 42118}, {5237, 12811}, {5343, 15701}, {5350, 14892}, {10109, 42148}, {10303, 42117}, {10646, 12102}, {11539, 42153}, {11737, 42158}, {11812, 16964}, {12108, 33416}, {14869, 18581}, {14891, 42432}, {15022, 42138}, {15699, 42149}, {15704, 42095}, {15713, 42150}, {16773, 35018}, {16963, 42502}, {22238, 42146}, {33412, 35304}, {40693, 42518}, {41983, 42434}

X(42592) = GIBERT (9,12,25) POINT

Barycentrics    a^2*(3*Sqrt[3]*S + 25*SA) + 24*SB*SC : :

X(42592) lies on these lines: {2, 3412}, {3, 36969}, {4, 42515}, {5, 10188}, {13, 14869}, {15, 3090}, {17, 3525}, {61, 42129}, {62, 632}, {140, 41100}, {395, 41992}, {398, 3628}, {546, 5352}, {630, 7828}, {631, 41119}, {1656, 41108}, {3091, 16241}, {3146, 42092}, {3526, 16267}, {3544, 42157}, {3627, 42430}, {3857, 36967}, {5054, 41974}, {5067, 41122}, {5070, 41943}, {5072, 5238}, {5076, 10645}, {5237, 5335}, {5351, 42132}, {5365, 15022}, {7486, 41101}, {10646, 12108}, {12812, 16964}, {12816, 21735}, {15704, 42102}, {15720, 41121}, {16808, 17538}, {23046, 42504}, {41989, 42164}, {42152, 42516}, {42433, 42500}

X(42593) = GIBERT (-9,12,25) POINT

Barycentrics    a^2*(3*Sqrt[3]*S - 25*SA) - 24*SB*SC : :

X(42593) lies on these lines: {2, 3411}, {3, 36970}, {4, 42514}, {5, 10187}, {14, 14869}, {16, 3090}, {18, 3525}, {61, 632}, {62, 42132}, {140, 41101}, {396, 41992}, {397, 3628}, {546, 5351}, {629, 7828}, {631, 41120}, {1656, 41107}, {3091, 16242}, {3146, 42089}, {3526, 16268}, {3544, 42158}, {3627, 42429}, {3857, 36968}, {5054, 41973}, {5067, 41121}, {5070, 41944}, {5072, 5237}, {5076, 10646}, {5238, 5334}, {5352, 42129}, {5366, 15022}, {7486, 41100}, {10645, 12108}, {12812, 16965}, {12817, 21735}, {15704, 42101}, {15720, 41122}, {16809, 17538}, {23046, 42505}, {41989, 42165}, {42149, 42517}, {42434, 42501}

leftri

Gibert points on the cubic K1204: X(42594)-X(42609)

rightri

This preamble and points X(42594)-X(42609) are contributed by Peter Moses, April 13, 2021. See also the preambles just before X(42085), X(42413), and X(42429). See K1204.



X(42594) = GIBERT (3,23,52) POINT

Barycentrics    a^2*(Sqrt[3]*S + 52*SA) + 46*SB*SC : :

X(42594) lies on the cubic K1204 and these lines: {2, 5321}, {13, 10124}, {15, 41984}, {17, 632}, {18, 16239}, {140, 42528}, {549, 42429}, {631, 42474}, {3525, 5366}, {3526, 42151}, {3533, 42491}, {5054, 42106}, {5318, 11539}, {10109, 42430}, {14890, 19106}, {15694, 42086}, {15702, 42586}, {15703, 42101}, {15713, 42110}, {15721, 42102}, {15723, 23302}, {16967, 41971}, {23303, 41943}, {33416, 42496}, {33417, 42493}, {36967, 41985}, {41107, 42492}

X(42595) = GIBERT (-3,23,52) POINT

Barycentrics    a^2*(Sqrt[3]*S - 52*SA) - 46*SB*SC : :

X(42595) lies on the cubic K1204 and these lines: {2, 5318}, {14, 10124}, {16, 41984}, {17, 16239}, {18, 632}, {140, 42529}, {549, 42430}, {631, 42475}, {3525, 5365}, {3526, 42150}, {3533, 42490}, {5054, 42103}, {5321, 11539}, {10109, 42429}, {14890, 19107}, {15694, 42085}, {15702, 42587}, {15703, 42102}, {15713, 42107}, {15721, 42101}, {15723, 23303}, {16966, 41972}, {23302, 41944}, {33416, 42492}, {33417, 42497}, {36968, 41985}, {41108, 42493}

X(42596) = GIBERT (3,10,25) POINT

Barycentrics    a^2*(Sqrt[3]*S + 25*SA) + 20*SB*SC : :

X(42596) lies on the cubic K1204 and these lines: {2, 5238}, {3, 42429}, {5, 42099}, {6, 3411}, {13, 140}, {15, 16239}, {17, 15694}, {18, 632}, {61, 3533}, {62, 11539}, {398, 33606}, {549, 5350}, {630, 36770}, {631, 16966}, {1656, 42434}, {3525, 16242}, {3530, 19106}, {3545, 42515}, {3628, 5349}, {3850, 42430}, {5054, 42158}, {5055, 42587}, {5067, 10645}, {5070, 16809}, {5079, 42529}, {5344, 10303}, {5351, 15702}, {5352, 42500}, {10124, 16962}, {11300, 33414}, {11485, 42499}, {11540, 16267}, {12108, 36969}, {12815, 36782}, {15684, 42543}, {15699, 42432}, {15709, 41100}, {15713, 42166}, {15720, 42528}, {15721, 42161}, {15723, 22236}, {16241, 42153}, {16773, 16960}, {16967, 42490}, {35381, 42533}, {36843, 42505}, {42092, 42489}, {42124, 42435}

X(42596) = {X(6),X(3526)}-harmonic conjugate of X(42597)

X(42597) = GIBERT (-3,10,25) POINT

Barycentrics    a^2*(Sqrt[3]*S - 25*SA) - 20*SB*SC : :

X(42597) lies on the cubic K1204 and these lines: {2, 5237}, {3, 42430}, {5, 42100}, {6, 3411}, {14, 140}, {16, 16239}, {17, 632}, {18, 15694}, {61, 11539}, {62, 3533}, {397, 33607}, {549, 5349}, {631, 16967}, {1656, 42433}, {3525, 16241}, {3530, 19107}, {3545, 42514}, {3628, 5350}, {3850, 42429}, {5054, 42157}, {5055, 42586}, {5067, 10646}, {5070, 16808}, {5079, 42528}, {5343, 10303}, {5351, 42501}, {5352, 15702}, {10124, 16963}, {11299, 33415}, {11310, 36770}, {11486, 42498}, {11540, 16268}, {12108, 36970}, {15684, 42544}, {15699, 42431}, {15709, 41101}, {15713, 42163}, {15720, 42529}, {15721, 42160}, {15723, 22238}, {16242, 42156}, {16772, 16961}, {16966, 42491}, {35381, 42532}, {36836, 42504}, {42089, 42488}, {42121, 42436}

X(42597) = {X(6),X(3526)}-harmonic conjugate of X(42596)

X(42598) = GIBERT (3,3,4) POINT

Barycentrics    a^2*(Sqrt[3]*S + 4*SA) + 6*SB*SC : :

X(42598) lies on the cubic K1204 and these lines: {2, 397}, {3, 5318}, {4, 16644}, {5, 14}, {6, 3090}, {13, 140}, {15, 546}, {16, 632}, {18, 547}, {20, 5350}, {30, 5352}, {62, 3628}, {203, 10592}, {302, 22113}, {376, 42490}, {381, 5349}, {395, 1656}, {548, 36969}, {549, 16965}, {550, 16241}, {590, 42251}, {615, 42253}, {618, 6673}, {624, 7889}, {630, 2482}, {631, 5340}, {633, 33458}, {636, 6669}, {1151, 42218}, {1152, 42220}, {1216, 36978}, {2045, 42252}, {2046, 42250}, {2307, 3614}, {3091, 5321}, {3146, 11480}, {3364, 18762}, {3365, 18538}, {3367, 35730}, {3523, 42155}, {3524, 5344}, {3525, 5335}, {3526, 10653}, {3528, 5366}, {3529, 42094}, {3530, 42158}, {3533, 42491}, {3544, 5334}, {3545, 5339}, {3627, 5238}, {3832, 42154}, {3843, 42150}, {3845, 41943}, {3850, 16964}, {3851, 10654}, {3853, 36967}, {3857, 42117}, {3858, 36970}, {3861, 42432}, {5054, 41119}, {5055, 40694}, {5056, 37640}, {5066, 16962}, {5067, 16645}, {5070, 42149}, {5072, 11485}, {5076, 42106}, {5079, 18581}, {5351, 14869}, {5365, 41106}, {5478, 22892}, {6036, 6115}, {6694, 37351}, {6783, 20416}, {7005, 10593}, {7486, 37641}, {7789, 37178}, {8703, 42431}, {10109, 42503}, {10124, 41100}, {10170, 11624}, {10187, 42436}, {10303, 11481}, {10616, 20429}, {10645, 15704}, {10646, 12108}, {11303, 33413}, {11539, 41107}, {11543, 12812}, {11737, 41108}, {12100, 42433}, {12102, 42122}, {12103, 19106}, {12811, 16809}, {12816, 15686}, {13350, 31705}, {14136, 22847}, {14892, 42435}, {15022, 42095}, {15694, 41112}, {15699, 42489}, {15712, 36968}, {15765, 36469}, {16239, 16242}, {17538, 42134}, {18358, 36757}, {18585, 36453}, {21402, 33517}, {22114, 37786}, {33414, 40334}, {33425, 33447}, {33427, 33446}, {34754, 42135}, {35018, 37835}, {35256, 35738}, {35732, 42262}, {35786, 42280}, {35787, 42281}, {38071, 41101}, {41991, 42530}, {41997, 42001}, {42192, 42273}, {42194, 42270}, {42265, 42282}

X(42598) = {X(6),X(3090)}-harmonic conjugate of X(42599)

X(42599) = GIBERT (-3,3,4) POINT

Barycentrics    a^2*(Sqrt[3]*S - 4*SA) - 6*SB*SC : :

X(42599) lies on the cubic K1204 and these lines: {2, 398}, {3, 5321}, {4, 16645}, {5, 13}, {6, 3090}, {14, 140}, {15, 632}, {16, 546}, {17, 547}, {20, 5349}, {30, 5351}, {61, 3628}, {202, 10592}, {303, 22114}, {376, 42491}, {381, 5350}, {396, 1656}, {548, 36970}, {549, 16964}, {550, 16242}, {590, 42250}, {615, 42252}, {619, 6674}, {623, 7889}, {629, 2482}, {631, 5339}, {634, 33459}, {635, 6670}, {1151, 42217}, {1152, 42219}, {1216, 36980}, {2045, 42251}, {2046, 42253}, {3091, 5318}, {3146, 11481}, {3389, 18762}, {3390, 18538}, {3523, 42154}, {3524, 5343}, {3525, 5334}, {3526, 10654}, {3528, 5365}, {3529, 42093}, {3530, 42157}, {3533, 42490}, {3544, 5335}, {3545, 5340}, {3627, 5237}, {3832, 42155}, {3843, 42151}, {3845, 41944}, {3850, 16965}, {3851, 10653}, {3853, 36968}, {3857, 42118}, {3858, 36969}, {3861, 42431}, {5054, 41120}, {5055, 40693}, {5056, 37641}, {5066, 16963}, {5067, 16644}, {5070, 42152}, {5072, 11486}, {5076, 42103}, {5079, 18582}, {5352, 14869}, {5366, 41106}, {5479, 22848}, {6036, 6114}, {6695, 37352}, {6782, 20415}, {7006, 10593}, {7127, 7173}, {7486, 37640}, {7789, 37177}, {8703, 42432}, {10109, 42502}, {10124, 41101}, {10170, 11626}, {10188, 42435}, {10303, 11480}, {10617, 20428}, {10645, 12108}, {10646, 15704}, {11304, 33412}, {11539, 41108}, {11542, 12812}, {11737, 41107}, {12100, 42434}, {12102, 42123}, {12103, 19107}, {12811, 16808}, {12817, 15686}, {13349, 31706}, {14137, 22893}, {14892, 42436}, {15022, 42098}, {15694, 41113}, {15699, 42488}, {15712, 36967}, {15765, 36470}, {16239, 16241}, {17538, 42133}, {18358, 36758}, {18585, 36452}, {21401, 33518}, {22113, 37785}, {31694, 36769}, {33415, 40335}, {33424, 33444}, {33426, 33445}, {34755, 42138}, {35018, 37832}, {35255, 35738}, {35732, 42265}, {35786, 42281}, {35787, 42280}, {38071, 41100}, {41991, 42531}, {41998, 42002}, {42191, 42273}, {42193, 42270}, {42262, 42282}

X(42599) = {X(6),X(3090)}-harmonic conjugate of X(42598)

X(42600) = GIBERT (SQRT(3),7,17) POINT

Barycentrics    a^2*(S + 17*SA) + 14*SB*SC : :

X(42600) lies on the cubic K1204 and these lines: {2, 1328}, {6, 10124}, {140, 6410}, {485, 3525}, {486, 41949}, {547, 42275}, {549, 42276}, {590, 3312}, {631, 42267}, {632, 3592}, {1656, 42271}, {3069, 3533}, {3628, 42263}, {5054, 42277}, {5067, 22615}, {5070, 42260}, {6396, 15709}, {6411, 15699}, {6412, 15713}, {6429, 9680}, {6438, 8253}, {6440, 11540}, {6446, 13665}, {6487, 10576}, {6519, 41953}, {6564, 15702}, {9540, 10194}, {10147, 42262}, {10303, 42261}, {11812, 42264}, {15701, 42284}, {15720, 22644}, {15723, 18510}, {41950, 41960}

X(42601) = GIBERT (-SQRT(3),7,17) POINT

Barycentrics    a^2*(S - 17*SA) - 14*SB*SC : :

X(42601) lies on the cubic K1204 and these lines: {2, 1327}, {6, 10124}, {140, 6409}, {485, 41950}, {486, 3525}, {547, 42276}, {549, 42275}, {615, 3311}, {631, 42266}, {632, 3594}, {1656, 42272}, {3068, 3533}, {3628, 42264}, {5054, 42274}, {5067, 22644}, {5070, 42261}, {6200, 15709}, {6411, 15713}, {6412, 15699}, {6430, 16239}, {6437, 8252}, {6439, 11540}, {6445, 13785}, {6486, 9681}, {6522, 41954}, {6565, 15702}, {9542, 35823}, {10148, 42265}, {10195, 13935}, {10303, 42260}, {11812, 42263}, {13665, 17851}, {15701, 42283}, {15720, 22615}, {15723, 18512}, {41949, 41959}

X(42602) = GIBERT (3*SQRT(3),5,7) POINT

Barycentrics    a^2*(3*S + 7*SA) + 10*SB*SC : :

X(42602) lies on the cubic K1204 and these lines: {2, 372}, {3, 1327}, {4, 9680}, {5, 3592}, {6, 547}, {17, 36468}, {18, 36450}, {30, 5418}, {371, 1328}, {376, 6564}, {381, 590}, {486, 5055}, {546, 9681}, {549, 6412}, {615, 15703}, {639, 32811}, {1131, 15708}, {1151, 3845}, {1152, 11539}, {1504, 19099}, {1506, 19105}, {1656, 6428}, {3068, 5071}, {3070, 5054}, {3071, 19709}, {3090, 8960}, {3091, 35812}, {3241, 35788}, {3524, 31412}, {3526, 6522}, {3533, 6454}, {3534, 22644}, {3543, 6200}, {3582, 31472}, {3590, 7486}, {3828, 35774}, {3830, 42260}, {3832, 6453}, {3839, 9540}, {3843, 41963}, {3850, 6425}, {3851, 31454}, {5056, 6419}, {5062, 22541}, {5066, 8981}, {5067, 6420}, {5309, 13711}, {6281, 22616}, {6396, 15702}, {6398, 15723}, {6410, 11812}, {6411, 15686}, {6421, 19100}, {6426, 16239}, {6432, 42578}, {6446, 13665}, {6449, 38335}, {6450, 42418}, {6459, 41106}, {6460, 14241}, {6469, 10124}, {6486, 42413}, {6565, 8972}, {7583, 13847}, {8703, 23251}, {9646, 11238}, {9661, 11237}, {9974, 22165}, {10109, 13925}, {10304, 35820}, {10385, 35802}, {10577, 13886}, {11050, 35790}, {11737, 42215}, {13893, 38021}, {13902, 38074}, {13903, 42270}, {13951, 41948}, {13973, 38083}, {14269, 42258}, {14891, 42226}, {14893, 42263}, {15681, 42284}, {15682, 35786}, {15687, 35255}, {15689, 42272}, {15692, 23249}, {15693, 42259}, {15719, 23269}, {15765, 42151}, {16267, 36467}, {16268, 36449}, {18510, 41951}, {18512, 32790}, {18581, 36439}, {18582, 36457}, {18585, 42150}, {19708, 42267}, {21356, 35840}, {23261, 38071}, {31145, 35810}, {34200, 42264}, {34551, 42491}, {34552, 42490}, {34559, 42149}, {34562, 42152}, {34627, 35763}, {35382, 41953}, {35815, 42561}, {35821, 41099}, {35878, 41135}, {36436, 42142}, {36437, 36968}, {36438, 42132}, {36445, 42243}, {36446, 42230}, {36454, 42139}, {36455, 36967}, {36456, 42129}, {36463, 42245}, {36464, 42228}, {41950, 41968}

X(42602) = {X(6),X(547)}-harmonic conjugate of X(42603)

X(42603) = GIBERT (-3*SQRT(3),5,7) POINT

Barycentrics    a^2*(3*S - 7*SA) - 10*SB*SC : :

X(42603) lies on the cubic K1204 and these lines: {2, 371}, {3, 1328}, {5, 3594}, {6, 547}, {17, 36449}, {18, 36467}, {30, 5420}, {99, 32807}, {140, 9681}, {372, 1327}, {376, 6565}, {381, 615}, {485, 5055}, {546, 17852}, {549, 6411}, {590, 15703}, {632, 9680}, {640, 32810}, {1132, 15708}, {1151, 11539}, {1152, 3845}, {1505, 19100}, {1506, 19102}, {1656, 6427}, {3069, 5071}, {3070, 19709}, {3071, 5054}, {3090, 19053}, {3091, 35813}, {3241, 35789}, {3524, 42260}, {3526, 6519}, {3533, 6453}, {3534, 22615}, {3543, 6396}, {3591, 7486}, {3828, 35775}, {3830, 42261}, {3832, 6454}, {3839, 13935}, {3843, 41964}, {3850, 6426}, {5056, 6420}, {5058, 19101}, {5066, 13966}, {5067, 6419}, {5309, 13834}, {6200, 15702}, {6221, 15723}, {6278, 22645}, {6409, 11812}, {6412, 15686}, {6422, 19099}, {6425, 16239}, {6431, 42579}, {6445, 13785}, {6449, 42417}, {6450, 38335}, {6459, 14226}, {6460, 41106}, {6468, 10124}, {6487, 42414}, {6564, 13941}, {7584, 13846}, {8703, 23261}, {8976, 41947}, {9541, 15721}, {9975, 22165}, {10109, 13993}, {10304, 35821}, {10385, 35803}, {10576, 13939}, {11050, 35791}, {11737, 42216}, {13911, 38083}, {13947, 38021}, {13959, 38074}, {13961, 42273}, {14269, 42259}, {14891, 42225}, {14893, 42264}, {15681, 42283}, {15682, 35787}, {15687, 35256}, {15689, 42271}, {15692, 23259}, {15693, 42258}, {15719, 23275}, {15765, 42150}, {16267, 36450}, {16268, 36468}, {18510, 32789}, {18512, 41952}, {18581, 36457}, {18582, 36439}, {18585, 42151}, {19708, 42266}, {21356, 35841}, {23251, 38071}, {31145, 35811}, {31412, 35814}, {34200, 42263}, {34551, 42490}, {34552, 42491}, {34559, 42152}, {34562, 42149}, {34627, 35762}, {35382, 41954}, {35820, 41099}, {35879, 41135}, {36436, 42139}, {36437, 36967}, {36438, 42129}, {36445, 42244}, {36447, 42227}, {36454, 42142}, {36455, 36968}, {36456, 42132}, {36463, 42242}, {36465, 42229}, {41949, 41967}

X(42603) = {X(6),X(547)}-harmonic conjugate of X(42602)

X(42604) = GIBERT (16*SQRT(3),25,26) POINT

Barycentrics    a^2*(8*S + 13*SA) + 25*SB*SC : :

X(42604) lies on the cubic K1204 and these lines: {2, 6398}, {3, 42540}, {4, 9691}, {371, 3832}, {485, 15022}, {590, 3146}, {1131, 6410}, {3316, 5059}, {3522, 35820}, {3543, 6445}, {3590, 42273}, {3854, 42215}, {5056, 6418}, {5068, 7585}, {6199, 42539}, {6564, 15683}, {8972, 41945}, {10146, 16239}, {15705, 23249}, {15717, 31412}, {32786, 41952}, {35731, 42189}, {41970, 42570}

X(42605) = GIBERT (-16*SQRT(3),25,26) POINT

Barycentrics    a^2*(8*S - 13*SA) - 25*SB*SC : :

X(42605) lies on the cubic K1204 and these lines: {2, 6221}, {3, 42539}, {372, 3832}, {486, 15022}, {615, 3146}, {1132, 6409}, {3317, 5059}, {3522, 35821}, {3543, 6446}, {3591, 42270}, {3854, 42216}, {5056, 6417}, {5068, 7586}, {6395, 42540}, {6565, 15683}, {10145, 16239}, {13941, 41946}, {15705, 23259}, {15717, 42561}, {31414, 42579}, {32785, 41951}, {41969, 42571}

X(42606) = GIBERT (27*SQRT(3),35,52) POINT

Barycentrics    a^2*(27*S + 52*SA) + 70*SB*SC : :

X(42606) lies on the cubic K1204 and these lines: {2, 3590}, {371, 5066}, {590, 3830}, {3070, 15693}, {3316, 6409}, {3543, 6488}, {3845, 6453}, {3860, 35812}, {6396, 11812}, {6411, 15697}, {6437, 42417}, {6522, 10195}, {8703, 42267}, {8976, 41947}, {10148, 15702}, {12100, 41952}, {12101, 41963}, {13846, 23273}, {15709, 17852}, {31454, 41106}, {32785, 41950}, {35822, 41948}, {41099, 41945}, {41967, 42266}, {42572, 42574}

X(42607) = GIBERT (-27*SQRT(3),35,52) POINT

Barycentrics    a^2*(27*S - 52*SA) - 70*SB*SC : :

X(42607) lies on the cubic K1204 and these lines: {2, 3591}, {372, 5066}, {615, 3830}, {3071, 15693}, {3317, 6410}, {3543, 6489}, {3845, 6454}, {3860, 35813}, {6200, 11812}, {6412, 15697}, {6438, 42418}, {6519, 10194}, {8703, 42266}, {10147, 15702}, {12100, 41951}, {12101, 41964}, {13847, 23267}, {13951, 41948}, {32786, 41949}, {35823, 41947}, {41099, 41946}, {41968, 42267}, {42573, 42575}

X(42608) = GIBERT (27*SQRT(3),35,25) POINT

Barycentrics    a^2*(27*S + 25*SA) + 70*SB*SC : :

X(42608) lies on the cubic K1204 and these lines: {2, 6454}, {6, 5066}, {30, 10147}, {485, 3830}, {1132, 35771}, {1327, 6200}, {3534, 41952}, {3592, 3845}, {5418, 8703}, {6410, 11812}, {6446, 42418}, {6453, 15682}, {6489, 11539}, {6560, 15693}, {9681, 33699}, {10124, 10148}, {10195, 15719}, {13939, 35822}, {14241, 42277}, {15640, 23253}, {15698, 23269}, {23275, 31412}, {35255, 42576}, {42274, 42572}

X(42609) = GIBERT (-27*SQRT(3),35,25) POINT

Barycentrics    a^2*(27*S - 25*SA) - 70*SB*SC : :

X(42609) lies on the cubic K1204 and these lines: {2, 6453}, {6, 5066}, {30, 10148}, {486, 3830}, {1131, 35770}, {1328, 6396}, {3534, 41951}, {3594, 3845}, {5420, 8703}, {6409, 11812}, {6445, 42417}, {6454, 15682}, {6488, 11539}, {6561, 15693}, {9681, 11540}, {10124, 10147}, {10194, 15719}, {13886, 35823}, {14226, 42274}, {15640, 23263}, {15698, 23275}, {23269, 35786}, {35256, 42577}, {42277, 42573}

X(42610) = GIBERT (3,8,14) POINT

Barycentrics    a^2*(Sqrt[3]*S + 14*SA) + 16*SB*SC : :

X(42610) lies on the cubic K1205 and these lines: {2, 397}, {5, 11480}, {6, 5070}, {14, 1656}, {20, 42472}, {61, 15703}, {140, 42161}, {382, 33417}, {546, 42474}, {547, 5339}, {548, 42114}, {631, 42088}, {632, 5340}, {3090, 5343}, {3146, 42500}, {3412, 42129}, {3525, 42155}, {3526, 11481}, {3528, 42110}, {3530, 42094}, {3533, 42166}, {3628, 16644}, {3855, 42096}, {3856, 42090}, {5054, 42433}, {5055, 36836}, {5056, 42154}, {5067, 23302}, {5079, 16241}, {5352, 19709}, {6669, 11310}, {6673, 11312}, {7486, 16772}, {10109, 42160}, {10124, 42151}, {11539, 42162}, {12812, 42150}, {15699, 42152}, {15701, 42431}, {15702, 42165}, {15717, 42097}, {16239, 18582}, {21734, 42102}, {36843, 37832}, {41983, 42586}, {41984, 42510}, {42159, 42475}

X(42611) = GIBERT (-3,8,14) POINT

Barycentrics    a^2*(Sqrt[3]*S - 14*SA) - 16*SB*SC : :

X(42611) lies on the cubic K1205 and these lines: {2, 398}, {5, 11481}, {6, 5070}, {13, 1656}, {20, 42473}, {62, 15703}, {140, 42160}, {382, 33416}, {546, 42475}, {547, 5340}, {548, 42111}, {631, 42087}, {632, 5339}, {3090, 5344}, {3146, 42501}, {3411, 42132}, {3525, 42154}, {3526, 11480}, {3528, 42107}, {3530, 42093}, {3533, 42163}, {3628, 16645}, {3855, 42097}, {3856, 42091}, {5054, 42434}, {5055, 36843}, {5056, 42155}, {5067, 23303}, {5079, 16242}, {5351, 19709}, {6670, 11309}, {6674, 11311}, {7486, 16773}, {10109, 42161}, {10124, 42150}, {11539, 42159}, {12812, 42151}, {15699, 42149}, {15701, 42432}, {15702, 42164}, {15717, 42096}, {16239, 18581}, {16644, 42590}, {21734, 42101}, {33415, 36770}, {36836, 37835}, {41983, 42587}, {41984, 42511}, {42162, 42474}

X(42612) = GIBERT (45,12,-1) POINT

Barycentrics    a^2*(15*Sqrt[3]*S - SA) + 24*SB*SC : :

X(42612) lies on the cubic K1205 and these lines: {13, 3544}, {14, 397}, {16, 14869}, {61, 3529}, {62, 5079}, {382, 41107}, {3412, 34200}, {3526, 42518}, {3530, 41943}, {3851, 16268}, {5068, 42521}, {5238, 42123}, {5344, 12820}, {5350, 12821}, {10299, 41100}, {11542, 42499}, {12103, 34754}, {15681, 41974}, {15688, 42508}, {30472, 33465}, {35018, 41121}, {38071, 42503}

X(42613) = GIBERT (-45,12,-1) POINT

Barycentrics    a^2*(15*Sqrt[3]*S + SA) - 24*SB*SC : :

X(42613) lies on the cubic K1205 and these lines: {13, 398}, {14, 3544}, {15, 14869}, {61, 5079}, {62, 3529}, {382, 41108}, {3411, 34200}, {3526, 42519}, {3530, 41944}, {3851, 16267}, {5068, 42520}, {5237, 42122}, {5343, 12821}, {5349, 12820}, {10299, 41101}, {11543, 42498}, {12103, 34755}, {15681, 41973}, {15688, 42509}, {30471, 33464}, {35018, 41122}, {38071, 42502}

leftri

Central angle points: X(42614)-X(42621)

rightri

This preamble and points X(42614)-X(42621) are contributed by Clark Kimberling and Peter Moses, April 15, 2021.

Suppose the P is a point in the plane of a triangle ABC. Let A' = angle BPC, B' = angle CPA, C' = angle APB, and assume that A' = t*A + u*(B + C) for some real numbers t and u. Since B + C = π - A, the angle A' is given by

A' = A'(t) = t*(3A - π)/2 + π - A, and B' and C' are determined cyclically, and the corresponding point P = P(t) is given by

P(t)= 1/(cot(A) + cot(A - t*(3*A - π)/2)) : : ,

so that P(t) is a major center, here named the t-central angle point. Let L denote the locus of P(t) as t traverses the real number line, and note that P(2 - t) = isogonal conjugate of P(t), so that if a point X is on L, then the isogonal conjugate of X is also on L. The appearance of (t,i) in the following list means P = X(i) is the point for which A'(t) = angle BPC, B'(t) = angle CPA, C'(t) = angle APB. The list is presented in isogonal conjugate pairs:

(-2,5964); (4,5963)
(-1,41622); (3,42621)
(-2/3,42619; (8/3,42620)
(0,4); (2,3)
(1/3,42615); (5/3,42616)
(1/2,42618); (3/2,42614)
(2/3,13); (4/3,15)
(1,1); (1,1)



X(42614) = (3/2)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A - 3*(3*A - Pi)/4]) : :

X(42614) is the point P for which the central angles are given by
BPC = (5A+ π)/4, CPA = (5B + π)/4, APB = (5C + π)/4.

X(42614) lies on these lines: {3, 10231}, {36, 266}, {56, 1130}, {10215, 12523}

X(42614) = isogonal conjugate of X(42617)
X(42614) = circumcircle-inverse of X(10231)

X(42615) = (1/3)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A + (-3*A + Pi)/6]) : :

X(42615) is the point P for which the central angles are given by
BPC = (-3A+ π)/6, CPA = (-3B + π)/6, APB = (-3C + π)/6.

X(42615) lies on this line: {7951, 10651}

X(42615) = isogonal conjugate of X(42616)

X(42616) = (5/3)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A - 5*(3*A - Pi)/6]) : :

X(42616) is the point P for which the central angles are given by
BPC = 3A/2 + π/6, CPA = 3B/2 + π/6, APB = 3C/2 + π/6.

X(42616) lies on these lines: {3, 39151}, {35, 19551}, {36, 33655}, {993, 7026}, {1324, 19305}, {2153, 6104}, {2161, 32613}, {5010, 11752}, {6187, 39150}, {8715, 36931}

X(42616) = isogonal conjugate of X(42615)
X(42616) = circumcircle-inverse of X(39151)

X(42617) = (1/2)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A + (-3*A + Pi)/4]) : :

X(42617) is the point P for which the central angles are given by
BPC = -A/4 + 3π/4, CPA = -B/4 + 3π/4, APB = -C/4 + 3π/4.

X(42617) lies on these lines: {188, 3814}, {8133, 30370}

X(42617) = isogonal conjugate of X(42614)

X(42618) = (-2/3)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A + (3*A - Pi)/3]) : :

X(42618) is the point P for which the central angles are given by
BPC = -2A + 4π/3, CPA = -2B + 4π/3, APB = -2C + 4π/3.

X(42618) lies on this line: {20429, 36981}

X(42618) = isogonal conjugate of X(42619)

X(42619) = (8/3)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A - 4*(3*A - Pi)/3]) : :

X(42619) is the point P for which the central angles are given by
BPC = 3A - π/3, CPA = 3B - π/3, APB = 3C - π/3.

X(42619) lies on these lines: {3, 11601}, {13, 36249}, {18, 38943}, {2070, 11586}, {6104, 11082}, {7502, 40105}, {8173, 34008}

X(42619) = isogonal conjugate of X(42618)
X(42619) = circumcircle-inverse of X(11601)

X(42620) = (3)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A - 3*(3*A - Pi)/2]) : :

X(42620) is the point P for which the central angles are given by
BPC = (7A - π)/2, CPA = (7B - π)/2, APB = (7C - π)/2.

X(42620) lies on this line: {3, 33599}

X(42620) = isogonal conjugate of X(42621)
X(42620) = circumcircle-inverse of X(33599)

X(42621) = (-1)-CENTRAL ANGLE POINT

Barycentrics    1/(Cot[A] + Cot[A +(3*A - Pi)/2]) : :

X(42621) is the point P for which the central angles are given by
BPC = (-5A + 3π)/2, CPA = (-5B + 3π)/2, APB = (-5C - 3π)/2.

X(42621) lies on this line: (pending)

X(42621) = isogonal conjugate of X(42620)

X(42622) = DAO-LOZADA TANGENTIAL PERSPECTOR OF X(1)

Barycentrics    a*sin(A/2)*(1+sin(A/2)) : :

This introduction and centers X(46622)-X(42624) were contributed by César E. Lozada, April 16, 2021.

Let ABC be an acute triangle with circumcircle Ω and P a finite point not on Ω. Denote as {{a'}}, {{b'}}, {{c'}} the circles {{B, C, P}}, {{C, A, P}}, {{A, B, P}}, respectively. Also, denote {{a"}} the circle internally tangent to Ω and externally tangent to {{b'}} and {{c'}} and let A" be its center. Define {{b"}}, {{c"}}, B", C" cyclically. Let Ta, Tb, Tc be the touchpoints of Ω and {{a"}}, {{b"}} and {{c"}}, respectively. Then the lines ATa, BTb, CTc concur at a point Z'(P). (Dao Thanh Oai, April 10, 2021)

If P = u : v : w (exact trilinears), then Z'(P) = S*u*ρ(A)+a*R*(u^2-v*w*cos(A)+w*u*cos(B)+u*v*cos(C)) : : , where ρ(A) is the radius of {{a'}}. Z'(P) is named here the Dao-Lozada tangential perspector of P.

The appearance of (i,j) in the following list means that Z'(X(i))=X(j): (1, 42622), (3, 55), (4, 25), (13, 42623), (14, 42624), (15, 6)

Dao Thanh Oai also conjectured that triangles ABC and A"B"C" are perspective with perspector Z"(P). Although this conjecture has not been proved yet, the appearance of (i,j) in the following list means that Z"(X(i))=X(j): (1, 1130), (3, 35), (4, 4), (15, 61).

X(42622) lies on these lines: {1, 20183}, {3, 1130}, {55, 259}, {56, 266}, {57, 13444}, {188, 7587}, {999, 10231}

X(42622) = isogonal conjugate of X(7048)
X(42622) = anticomplement of the complementary conjugate of X(236)
X(42622) = X(266)-Ceva conjugate of-X(6)
X(42622) = barycentric product X(i)*X(j) for these {i, j}: {1, 173}, {6, 7057}, {55, 18886}, {236, 266}, {259, 2089}
X(42622) = barycentric quotient X(i)/X(j) for these (i, j): (31, 258), (56, 21456), (173, 75)
X(42622) = trilinear product X(i)*X(j) for these {i, j}: {6, 173}, {31, 7057}, {41, 18886}
X(42622) = trilinear quotient X(i)/X(j) for these (i, j): (6, 258), (57, 21456), (173, 2), (236, 556), (259, 7028), (266, 1488)
X(42622) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(259)}} and {{A, B, C, X(173), X(18888)}}
X(42622) = pole of the trilinear polar of X(266) with respect to circumcircle
X(42622) = crosssum of X(i) and X(j) for these (i, j): {1, 20183}, {188, 7028}
X(42622) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 258}, {9, 21456}, {174, 7028}, {188, 1488}
X(42622) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 258), (56, 21456), (173, 75)

X(42623) = DAO-LOZADA TANGENTIAL PERSPECTOR OF X(13)

Barycentrics    (SB+SC)*(SQRT(3)*SB+S)*(SQRT(3)*SC+S)*(SA-SQRT(3)*S-2*b*c) : :

X(42623) lies on these lines: {6, 2151}, {36, 3179}, {55, 199}, {56, 2306}, {396, 10648}, {999, 39153}, {2153, 11142}, {21310, 39151}

X(42623) = barycentric product X(i)*X(j) for these {i, j}: {1, 3179}, {13, 202}, {79, 5357}, {1251, 37773}
X(42623) = barycentric quotient X(202)/X(298)
X(42623) = trilinear product X(i)*X(j) for these {i, j}: {6, 3179}, {36, 11072}, {202, 2153}, {1251, 19373}
X(42623) = trilinear quotient X(i)/X(j) for these (i, j): (1251, 7026), (2151, 7006), (2153, 14358)
X(42623) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(5240)}} and {{A, B, C, X(36), X(56)}}
X(42623) = pole of the trilinear polar of X(2153) with respect to circumcircle
X(42623) = X(2153)-Ceva conjugate of-X(6)
X(42623) = X(i)-isoconjugate-of-X(j) for these {i, j}: {319, 11073}, {1082, 7026}
X(42623) = X(202)-reciprocal conjugate of-X(298)

X(42624) = DAO-LOZADA TANGENTIAL PERSPECTOR OF X(14)

Barycentrics    (SB+SC)*(SQRT(3)*SB-S)*(SQRT(3)*SC-S)*(SA+SQRT(3)*S+2*b*c) : :

X(42624) lies on these lines: {3, 39150}, {35, 7150}, {45, 55}, {56, 2306}, {559, 7202}, {759, 14359}, {958, 7043}, {2154, 11141}, {2174, 10638}, {3913, 36930}, {7005, 17104}

X(42624) = barycentric product X(i)*X(j) for these {i, j}: {1, 7150}, {14, 7005}, {16, 14359}, {80, 5357}, {559, 7126}
X(42624) = barycentric quotient X(31)/X(41225)
X(42624) = trilinear product X(i)*X(j) for these {i, j}: {6, 7150}, {35, 11072}, {2152, 14359}, {2154, 7005}, {2161, 5357}
X(42624) = trilinear quotient X(i)/X(j) for these (i, j): (6, 41225), (2152, 203), (2174, 5353)
X(42624) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(2154)}} and {{A, B, C, X(16), X(3285)}}
X(42624) = X(2152)-cross conjugate of-X(6)
X(42624) = X(i)-isoconjugate-of-X(j) for these {i, j}: {320, 11073}, {554, 5239}
X(42624) = X(31)-reciprocal conjugate of-X(41225)

leftri

Gibert points on the cubic K1191 (Evans 5th cubic): X(42625)-X(42648)

rightri

This preamble and points X(42625)-X(42648) are contributed by Peter Moses, April 16, 2021. See K1191 and the preambles just before X(42085), X(42413), and X(42429).



X(42625) = GIBERT (3,2,-8) POINT

Barycentrics    a^2*(Sqrt[3]*S - 8*SA) + 4*SB*SC : :

X(42625) lies on the cubic K1191 and these lines: {2, 42088}, {3, 13}, {4, 42491}, {6, 376}, {14, 15681}, {15, 15688}, {16, 3534}, {18, 17800}, {20, 395}, {30, 11481}, {62, 15696}, {381, 10646}, {382, 5351}, {396, 10304}, {397, 3528}, {398, 17538}, {547, 42145}, {548, 22236}, {549, 42086}, {550, 10654}, {599, 616}, {617, 40341}, {618, 11296}, {1657, 5237}, {3522, 36836}, {3523, 42165}, {3524, 5318}, {3525, 5350}, {3526, 42431}, {3529, 16773}, {3530, 42161}, {3543, 23303}, {3763, 11300}, {3830, 16242}, {3839, 42109}, {3845, 42089}, {3850, 42611}, {5054, 36969}, {5055, 19106}, {5059, 42163}, {5066, 42105}, {5071, 42102}, {5076, 42489}, {5321, 11001}, {5335, 19708}, {5859, 36326}, {6411, 36457}, {6412, 36439}, {8252, 36455}, {8253, 36437}, {8703, 10653}, {10124, 42114}, {10299, 42598}, {10645, 14093}, {11134, 37480}, {11485, 15695}, {11486, 15689}, {11489, 15683}, {11539, 42137}, {11543, 19710}, {11812, 42138}, {12100, 18582}, {12103, 40694}, {13961, 35739}, {14269, 16967}, {14869, 42610}, {14893, 42111}, {15640, 42139}, {15684, 16809}, {15685, 19107}, {15686, 42085}, {15687, 42113}, {15690, 42090}, {15691, 42117}, {15692, 23302}, {15693, 37832}, {15694, 16808}, {15697, 42119}, {15699, 42106}, {15700, 42128}, {15701, 16966}, {15702, 42134}, {15704, 42149}, {15706, 42132}, {15707, 33417}, {15708, 42142}, {15709, 42594}, {15711, 41119}, {15712, 42162}, {15716, 41121}, {15717, 42166}, {15759, 41112}, {16268, 42126}, {16772, 21735}, {16963, 42099}, {17504, 42092}, {19709, 33416}, {20791, 36978}, {33699, 42103}, {33703, 42599}, {33923, 40693}, {34200, 42118}, {35404, 42143}, {37641, 42087}, {41120, 42136}, {41944, 42125}, {42108, 42515}, {42144, 42497}, {42169, 42218}, {42170, 42220}

X(42625) = {X(6),X(376)}-harmonic conjugate of X(42626)

X(42626) = GIBERT (3,-2,8) POINT

Barycentrics    a^2*(Sqrt[3]*S + 8*SA) - 4*SB*SC : :

X(42626) lies on the cubic K1191 and these lines: {2, 42087}, {3, 14}, {4, 42490}, {6, 376}, {13, 15681}, {15, 3534}, {16, 15688}, {17, 17800}, {20, 396}, {30, 11480}, {61, 15696}, {381, 10645}, {382, 5352}, {395, 10304}, {397, 17538}, {398, 3528}, {547, 42144}, {548, 22238}, {549, 42085}, {550, 10653}, {599, 617}, {616, 40341}, {619, 11295}, {1657, 5238}, {3522, 36843}, {3523, 42164}, {3524, 5321}, {3525, 5349}, {3526, 42432}, {3529, 16772}, {3530, 42160}, {3543, 23302}, {3763, 11299}, {3830, 16241}, {3839, 42108}, {3845, 42092}, {3850, 42610}, {5054, 36970}, {5055, 19107}, {5059, 42166}, {5066, 42104}, {5071, 42101}, {5076, 42488}, {5318, 11001}, {5334, 19708}, {5858, 36324}, {6411, 36439}, {6412, 36457}, {8252, 36437}, {8253, 36455}, {8703, 10654}, {10124, 42111}, {10299, 42599}, {10646, 14093}, {11137, 37480}, {11485, 15689}, {11486, 15695}, {11488, 15683}, {11539, 42136}, {11542, 19710}, {11812, 42135}, {12100, 18581}, {12103, 40693}, {14269, 16966}, {14869, 42611}, {14893, 42114}, {15640, 42142}, {15684, 16808}, {15685, 19106}, {15686, 42086}, {15687, 42112}, {15690, 42091}, {15691, 42118}, {15692, 23303}, {15693, 37835}, {15694, 16809}, {15697, 42120}, {15699, 42103}, {15700, 42125}, {15701, 16967}, {15702, 42133}, {15704, 42152}, {15706, 42129}, {15707, 33416}, {15708, 42139}, {15709, 42595}, {15711, 41120}, {15712, 42159}, {15716, 41122}, {15717, 42163}, {15759, 41113}, {16267, 42127}, {16773, 21735}, {16962, 42100}, {17504, 42089}, {19709, 33417}, {20791, 36980}, {33699, 42106}, {33703, 42598}, {33923, 40694}, {34200, 42117}, {35404, 42146}, {37640, 42088}, {41119, 42137}, {41943, 42128}, {42109, 42514}, {42145, 42496}, {42167, 42217}, {42168, 42219}

X(42626) = {X(6),X(376)}-harmonic conjugate of X(42625)

X(42627) = GIBERT (4,3,5) POINT

Barycentrics    a^2*(4*Sqrt[3]*S + 15*SA) + 18*SB*SC : :

X(42627) lies on the cubic K1191 and these lines: {2, 42492}, {5, 5334}, {6, 3628}, {13, 12100}, {15, 546}, {16, 17}, {30, 11480}, {61, 12812}, {62, 42590}, {396, 547}, {548, 5318}, {549, 5335}, {550, 5366}, {590, 34562}, {615, 34559}, {632, 11486}, {3411, 16960}, {3530, 42092}, {3627, 42116}, {3845, 42119}, {3850, 42098}, {3853, 16772}, {3856, 42093}, {3857, 42133}, {3858, 42126}, {3859, 42147}, {3860, 42154}, {3861, 42085}, {5066, 5321}, {5238, 42102}, {5352, 42109}, {6676, 37776}, {8703, 42127}, {10109, 41120}, {10124, 42089}, {10641, 16198}, {10645, 12103}, {10653, 11812}, {10654, 11737}, {11481, 12108}, {12102, 36836}, {12811, 22236}, {14869, 42115}, {14891, 42155}, {14892, 16962}, {14893, 19107}, {15687, 42130}, {15690, 41121}, {15691, 36969}, {15699, 37640}, {15704, 42134}, {15712, 42120}, {15720, 22235}, {15759, 41119}, {16239, 40693}, {16241, 34200}, {16267, 33416}, {18538, 42193}, {18581, 35018}, {18762, 42191}, {33923, 42086}, {34754, 42107}, {35738, 42224}, {38071, 42415}, {41107, 42500}, {41113, 42474}, {41989, 42159}, {42091, 42490}, {42145, 42162}

X(42627) = {X(6),X(3628)}-harmonic conjugate of X(42628)

X(42628) = GIBERT (-4,3,5) POINT

Barycentrics    a^2*(4*Sqrt[3]*S - 15*SA) - 18*SB*SC : :

X(42628) lies on the cubic K1191 and these lines: {2, 42493}, {5, 5335}, {6, 3628}, {14, 12100}, {15, 18}, {16, 546}, {30, 11481}, {61, 42591}, {62, 12812}, {395, 547}, {548, 5321}, {549, 5334}, {550, 5365}, {590, 34559}, {615, 34562}, {632, 11485}, {3412, 16961}, {3530, 42089}, {3627, 42115}, {3845, 42120}, {3850, 42095}, {3853, 16773}, {3856, 42094}, {3857, 42134}, {3858, 42127}, {3859, 42148}, {3860, 42155}, {3861, 42086}, {5066, 5318}, {5237, 42101}, {5351, 42108}, {6676, 37775}, {8703, 42126}, {10109, 41119}, {10124, 42092}, {10642, 16198}, {10646, 12103}, {10653, 11737}, {10654, 11812}, {11480, 12108}, {12102, 36843}, {12811, 22238}, {14869, 42116}, {14891, 42154}, {14892, 16963}, {14893, 19106}, {15687, 42131}, {15690, 41122}, {15691, 36970}, {15699, 37641}, {15704, 42133}, {15712, 42119}, {15720, 22237}, {15759, 41120}, {16239, 40694}, {16242, 34200}, {16268, 33417}, {18538, 42194}, {18582, 35018}, {18762, 42192}, {33923, 42085}, {34755, 42110}, {35738, 42221}, {38071, 42416}, {41108, 42501}, {41112, 42475}, {41989, 42162}, {42090, 42491}, {42144, 42159}

X(42628) = {X(6),X(3628)}-harmonic conjugate of X(42627)

X(42629) = GIBERT (5,6,-3) POINT

Barycentrics    a^2*(5*Sqrt[3]*S - 9*SA) + 36*SB*SC : :

X(42629) lies on the cubic K1191 and these lines: {2, 10646}, {4, 16961}, {6, 382}, {13, 15681}, {14, 12821}, {15, 3529}, {16, 546}, {17, 550}, {30, 34754}, {61, 42109}, {62, 42105}, {376, 42512}, {381, 12820}, {1657, 16960}, {3364, 42185}, {3365, 42186}, {3412, 42087}, {3528, 18582}, {3530, 33417}, {3544, 5237}, {3830, 42521}, {3851, 16967}, {3855, 42111}, {5079, 11481}, {5334, 41974}, {5335, 42112}, {5340, 42099}, {5344, 42090}, {5350, 35018}, {5351, 42110}, {5366, 42092}, {10299, 42091}, {11488, 42529}, {11542, 42434}, {11543, 42416}, {14269, 16809}, {14869, 42138}, {15640, 42520}, {15688, 16241}, {15707, 42528}, {15720, 16966}, {16644, 42504}, {17504, 37832}, {19710, 33607}, {22844, 33959}, {23302, 34200}, {23303, 38071}, {36967, 42506}, {36992, 39874}, {37640, 42514}, {41100, 42125}, {41101, 42096}, {41107, 42117}, {42098, 42433}, {42108, 42612}, {42160, 42613}, {42166, 42584}, {42192, 42209}, {42194, 42210}, {42283, 42564}, {42284, 42565}, {42533, 42588}

X(42629) = {X(6),X(382)}-harmonic conjugate of X(42630)

X(42630) = GIBERT (5,-6,3) POINT

Barycentrics    a^2*(5*Sqrt[3]*S + 9*SA) - 36*SB*SC : :

X(42630) lies on the cubic K1191 and these lines: {2, 10645}, {4, 16960}, {6, 382}, {13, 12820}, {14, 15681}, {15, 546}, {16, 3529}, {18, 550}, {30, 34755}, {61, 42104}, {62, 42108}, {376, 42513}, {381, 12821}, {1657, 16961}, {3389, 42183}, {3390, 42184}, {3411, 42088}, {3528, 18581}, {3530, 33416}, {3544, 5238}, {3830, 42520}, {3851, 16966}, {3855, 42114}, {5079, 11480}, {5334, 42113}, {5335, 41973}, {5339, 42100}, {5343, 42091}, {5349, 35018}, {5352, 42107}, {5365, 42089}, {6200, 35733}, {10299, 42090}, {11489, 42528}, {11542, 42415}, {11543, 42433}, {14269, 16808}, {14869, 42135}, {15640, 42521}, {15688, 16242}, {15707, 42529}, {15720, 16967}, {16645, 42505}, {17504, 37835}, {19710, 33606}, {22845, 33960}, {23302, 38071}, {23303, 34200}, {35730, 42186}, {36968, 42507}, {36994, 39874}, {37641, 42515}, {41100, 42097}, {41101, 42128}, {41108, 42118}, {42095, 42434}, {42109, 42613}, {42161, 42612}, {42163, 42585}, {42191, 42207}, {42193, 42208}, {42283, 42562}, {42284, 42563}, {42532, 42589}

X(42630) = {X(6),X(382)}-harmonic conjugate of X(42629)

X(42631) = GIBERT (9,4,-19) POINT

Barycentrics    a^2*(3*Sqrt[3]*S - 19*SA) + 8*SB*SC : :

X(42631) lies on the cubic K1191 and these lines: {2, 10646}, {3, 16267}, {6, 15695}, {13, 12100}, {14, 11001}, {15, 8703}, {16, 3534}, {17, 15692}, {18, 30}, {20, 16963}, {61, 15688}, {62, 376}, {381, 5351}, {395, 19710}, {396, 15759}, {398, 15691}, {530, 30471}, {531, 33611}, {549, 42158}, {550, 42613}, {3411, 12103}, {3412, 21735}, {3522, 42520}, {3524, 16965}, {3830, 11481}, {3845, 16242}, {5055, 42431}, {5066, 19106}, {5238, 14093}, {5318, 11812}, {5340, 15700}, {5352, 10304}, {5862, 36329}, {6778, 35750}, {6779, 36383}, {7811, 35932}, {9885, 35752}, {10109, 33416}, {10653, 19708}, {10654, 15697}, {11539, 42165}, {11540, 42594}, {11543, 42430}, {12101, 23303}, {14269, 42489}, {14893, 42580}, {15640, 18581}, {15681, 16268}, {15682, 42100}, {15683, 42149}, {15685, 36970}, {15686, 16964}, {15689, 22238}, {15690, 34755}, {15693, 41121}, {15698, 16241}, {15701, 37832}, {15702, 42161}, {15706, 42156}, {15708, 42162}, {15709, 42581}, {15710, 42152}, {15711, 42118}, {15713, 16966}, {15714, 16772}, {15716, 16644}, {15719, 18582}, {15764, 42259}, {16809, 33699}, {16960, 42504}, {16962, 34200}, {16967, 41099}, {19107, 41120}, {19711, 23302}, {19780, 39593}, {25235, 36318}, {33603, 42544}, {35404, 42599}, {36994, 41027}, {38335, 42586}, {40694, 42589}, {41106, 42089}, {41983, 42598}, {42099, 42507}, {42137, 42501}, {42481, 42521}, {42490, 42518}, {42511, 42529}

X(42631) = {X(6),X(15695)}-harmonic conjugate of X(42632)

X(42632) = GIBERT (9,-4,19) POINT

Barycentrics    a^2*(3*Sqrt[3]*S + 19*SA) - 8*SB*SC : :

X(42632) lies on the cubic K1191 and these lines: {2, 10645}, {3, 16268}, {6, 15695}, {13, 11001}, {14, 12100}, {15, 3534}, {16, 8703}, {17, 30}, {18, 15692}, {20, 16962}, {61, 376}, {62, 15688}, {381, 5352}, {395, 15759}, {396, 19710}, {397, 15691}, {530, 33610}, {531, 30472}, {549, 42157}, {550, 42612}, {3411, 21735}, {3412, 12103}, {3522, 42521}, {3524, 16964}, {3830, 11480}, {3845, 16241}, {5055, 42432}, {5066, 19107}, {5237, 14093}, {5321, 11812}, {5339, 15700}, {5351, 10304}, {5863, 35751}, {6777, 36331}, {6780, 36382}, {7811, 35931}, {9886, 36330}, {10109, 33417}, {10653, 15697}, {10654, 19708}, {11539, 42164}, {11540, 42595}, {11542, 42429}, {12101, 23302}, {14269, 42488}, {14893, 42581}, {15640, 18582}, {15681, 16267}, {15682, 42099}, {15683, 42152}, {15685, 36969}, {15686, 16965}, {15689, 22236}, {15690, 34754}, {15693, 41122}, {15698, 16242}, {15701, 37835}, {15702, 42160}, {15706, 42153}, {15708, 42159}, {15709, 42580}, {15710, 42149}, {15711, 42117}, {15713, 16967}, {15714, 16773}, {15716, 16645}, {15719, 18581}, {15764, 42258}, {16808, 33699}, {16961, 42505}, {16963, 34200}, {16966, 41099}, {19106, 41119}, {19711, 23303}, {19781, 39593}, {25236, 36320}, {33602, 42543}, {35404, 42598}, {36992, 41026}, {38335, 42587}, {40693, 42588}, {41106, 42092}, {41983, 42599}, {42100, 42506}, {42136, 42500}, {42480, 42520}, {42491, 42519}, {42510, 42528}

X(42632) = {X(6),X(15695)}-harmonic conjugate of X(42631)

X(42633) = GIBERT (12,1,5) POINT

Barycentrics    a^2*(4*Sqrt[3]*S + 5*SA) + 2*SB*SC : :

X(42633) lies on the cubic K1191 and these lines: {2, 42493}, {5, 14}, {6, 549}, {13, 12820}, {15, 8703}, {16, 17504}, {18, 42592}, {30, 5335}, {62, 15712}, {140, 37641}, {193, 11301}, {381, 42496}, {395, 11539}, {397, 15704}, {546, 5365}, {547, 11488}, {550, 10653}, {597, 619}, {618, 3629}, {632, 16645}, {3180, 37340}, {3627, 40693}, {3830, 33602}, {3845, 10654}, {3856, 5343}, {3857, 5339}, {3858, 42156}, {3859, 42494}, {3860, 42142}, {5066, 5334}, {5318, 33699}, {5321, 16267}, {6199, 36455}, {6395, 36437}, {6670, 33475}, {6671, 33459}, {10109, 42132}, {10124, 11489}, {10645, 15714}, {11307, 40898}, {11481, 15711}, {11486, 12100}, {11543, 15699}, {11737, 42125}, {12101, 42126}, {12817, 42502}, {13084, 20583}, {14869, 16242}, {14891, 42115}, {14892, 42139}, {14893, 42128}, {15686, 34754}, {15690, 42120}, {15713, 16241}, {16960, 41108}, {18581, 42512}, {18582, 38071}, {19116, 34551}, {19117, 34552}, {19710, 42122}, {23303, 41943}, {33606, 37835}, {34200, 42116}, {35404, 42085}, {36969, 42147}, {36970, 42138}, {41107, 42087}, {41112, 42137}, {41113, 42098}, {41944, 42500}, {42129, 42492}, {42143, 42475}, {42148, 42529}

X(42633) = {X(6),X(549)}-harmonic conjugate of X(42634)

X(42634) = GIBERT (-12,1,5) POINT

Barycentrics    a^2*(4*Sqrt[3]*S - 5*SA) - 2*SB*SC : :

X(42634) lies on the cubic K1191 and these lines: {2, 42492}, {5, 13}, {6, 549}, {14, 12821}, {15, 17504}, {16, 8703}, {17, 42593}, {30, 5334}, {61, 15712}, {140, 37640}, {193, 11302}, {381, 42497}, {396, 11539}, {398, 15704}, {546, 5366}, {547, 11489}, {550, 10654}, {597, 618}, {619, 3629}, {632, 16644}, {3181, 37341}, {3627, 40694}, {3830, 33603}, {3845, 10653}, {3856, 5344}, {3857, 5340}, {3858, 42153}, {3859, 42495}, {3860, 42139}, {5066, 5335}, {5318, 16268}, {5321, 33699}, {6199, 36437}, {6395, 36455}, {6669, 33474}, {6672, 33458}, {10109, 42129}, {10124, 11488}, {10646, 15714}, {11308, 40899}, {11480, 15711}, {11485, 12100}, {11542, 15699}, {11737, 42128}, {12101, 42127}, {12816, 42503}, {13083, 20583}, {14869, 16241}, {14891, 42116}, {14892, 42142}, {14893, 42125}, {15686, 34755}, {15690, 42119}, {15713, 16242}, {16961, 41107}, {18581, 38071}, {18582, 42513}, {19116, 34552}, {19117, 34551}, {19710, 42123}, {23302, 41944}, {33607, 37832}, {34200, 42115}, {35404, 42086}, {36969, 42135}, {36970, 42148}, {41108, 42088}, {41112, 42095}, {41113, 42136}, {41943, 42501}, {42132, 42493}, {42146, 42474}, {42147, 42528}

X(42634) = {X(6),X(549)}-harmonic conjugate of X(42633)

X(42635) = GIBERT (45,4,23) POINT

Barycentrics    a^2*(15*Sqrt[3]*S + 23*SA) + 8*SB*SC : :

X(42635) lies on the cubic K1191 and these lines: {2, 18}, {6, 15707}, {13, 42140}, {15, 15688}, {16, 17504}, {17, 11737}, {30, 34754}, {62, 15700}, {381, 42518}, {396, 38071}, {546, 3412}, {549, 42520}, {550, 42612}, {3523, 42521}, {3524, 42517}, {3528, 41100}, {3529, 42511}, {3545, 42516}, {3839, 16960}, {3851, 41108}, {5067, 33606}, {5238, 34200}, {5318, 41971}, {5351, 15715}, {10645, 15710}, {11485, 14269}, {12821, 42496}, {13903, 36469}, {13961, 36453}, {14893, 33607}, {15681, 22236}, {15687, 41101}, {15706, 34755}, {16773, 41978}, {37640, 42090}, {37832, 42125}, {42147, 42506}

X(42636) = GIBERT (-45,4,23) POINT

Barycentrics    a^2*(15*Sqrt[3]*S - 23*SA) - 8*SB*SC : :

X(42636) lies on the cubic K1191 and these lines: {2, 17}, {6, 15707}, {14, 42141}, {15, 17504}, {16, 15688}, {18, 11737}, {30, 34755}, {61, 15700}, {381, 42519}, {395, 38071}, {546, 3411}, {549, 42521}, {550, 42613}, {3523, 42520}, {3524, 42516}, {3528, 41101}, {3529, 42510}, {3545, 42517}, {3839, 16961}, {3851, 41107}, {5067, 33607}, {5237, 34200}, {5321, 41972}, {5352, 15715}, {10646, 15710}, {11486, 14269}, {12820, 42497}, {13903, 36452}, {13961, 36470}, {14893, 33606}, {15681, 22238}, {15687, 41100}, {15706, 34754}, {16772, 41977}, {37641, 42091}, {37835, 42128}, {42148, 42507}

X(42637) = GIBERT (2 SQRT(3),1,-6) POINT

Barycentrics    a^2*(S - 3*SA) + SB*SC : :

X(42637) lies on the cubic K1191 and these lines: {2, 6410}, {3, 1587}, {4, 5420}, {5, 6456}, {6, 3522}, {14, 35739}, {20, 1152}, {30, 6450}, {140, 6452}, {165, 19066}, {371, 3528}, {372, 376}, {382, 35256}, {485, 3524}, {486, 3529}, {488, 35948}, {489, 5860}, {490, 5591}, {511, 26295}, {516, 13959}, {548, 3312}, {549, 6497}, {550, 1588}, {590, 15717}, {615, 3146}, {631, 6560}, {638, 2482}, {1131, 8253}, {1132, 13847}, {1151, 10304}, {1204, 18924}, {1327, 15709}, {1656, 23253}, {1657, 6446}, {2045, 42220}, {2046, 42219}, {3070, 3523}, {3090, 35820}, {3091, 42264}, {3093, 37460}, {3098, 39875}, {3311, 8703}, {3317, 15682}, {3525, 6564}, {3530, 13665}, {3533, 42277}, {3534, 6408}, {3543, 42262}, {3545, 22644}, {3832, 8252}, {3839, 42583}, {4297, 19065}, {4316, 13963}, {4324, 13962}, {5056, 42284}, {5059, 6434}, {5067, 42269}, {5068, 32790}, {5071, 35786}, {5073, 18762}, {5217, 31408}, {5418, 10299}, {5590, 11293}, {5894, 17820}, {6036, 12975}, {6200, 7581}, {6221, 33923}, {6225, 10534}, {6409, 7585}, {6418, 15688}, {6426, 7586}, {6430, 32788}, {6432, 41945}, {6438, 42523}, {6448, 19116}, {6454, 6561}, {6455, 19117}, {6481, 23273}, {6485, 11001}, {6487, 6565}, {6522, 12103}, {7389, 33364}, {7738, 12968}, {7822, 11291}, {7889, 11292}, {7967, 35611}, {7968, 9778}, {7987, 13902}, {8277, 12082}, {8976, 15712}, {9681, 35770}, {9733, 35944}, {9862, 9986}, {10303, 42265}, {10820, 12244}, {11418, 30552}, {11495, 19013}, {11836, 14654}, {12124, 36701}, {12172, 26376}, {12239, 20791}, {12257, 40275}, {12323, 26362}, {12512, 18992}, {12963, 26463}, {12969, 26457}, {13785, 15704}, {13846, 15705}, {13883, 16192}, {13925, 17504}, {13947, 28164}, {13961, 15681}, {13980, 17845}, {14813, 42127}, {14814, 42126}, {15515, 31411}, {15696, 42215}, {15698, 35822}, {15720, 18538}, {15815, 31403}, {17576, 31473}, {17578, 42270}, {19055, 38736}, {19059, 38726}, {19108, 38747}, {19110, 37853}, {19112, 38759}, {19145, 33750}, {22615, 35813}, {23275, 42275}, {42217, 42584}, {42218, 42585}

X(42637) = {X(6),X(3522)}-harmonic conjugate of X(42638)

X(42638) = GIBERT (2 SQRT(3),-1,6) POINT

Barycentrics    a^2*(S + 3*SA) - SB*SC : :

X(42638) lies on the cubic K1191 and these lines: {2, 6409}, {3, 1588}, {4, 5418}, {5, 6455}, {6, 3522}, {20, 1151}, {30, 6449}, {140, 6451}, {165, 19065}, {186, 9683}, {371, 376}, {372, 3528}, {382, 35255}, {485, 3529}, {486, 3524}, {487, 35949}, {489, 5590}, {490, 5861}, {511, 26294}, {515, 9582}, {516, 9615}, {548, 3311}, {549, 6496}, {550, 1587}, {590, 3146}, {615, 15717}, {631, 6561}, {637, 2482}, {1131, 13846}, {1132, 8252}, {1152, 10304}, {1204, 18923}, {1328, 15709}, {1656, 23263}, {1657, 6445}, {2045, 42217}, {2046, 42218}, {3071, 3523}, {3085, 9647}, {3086, 9660}, {3090, 35821}, {3091, 42263}, {3092, 37460}, {3098, 39876}, {3312, 8703}, {3316, 15682}, {3525, 6565}, {3530, 13785}, {3533, 42274}, {3534, 6407}, {3543, 42265}, {3545, 22615}, {3832, 8253}, {3839, 42582}, {4297, 9616}, {4316, 13905}, {4324, 13904}, {5056, 42283}, {5059, 6433}, {5067, 42268}, {5068, 32789}, {5071, 35787}, {5073, 18538}, {5420, 10299}, {5591, 11294}, {5894, 17819}, {6036, 12974}, {6225, 10533}, {6396, 7582}, {6398, 33923}, {6410, 7586}, {6417, 15688}, {6425, 7585}, {6429, 32787}, {6431, 41946}, {6437, 42522}, {6447, 19117}, {6453, 6560}, {6456, 19116}, {6480, 23267}, {6484, 11001}, {6486, 6564}, {6519, 12103}, {6781, 31411}, {7388, 33365}, {7737, 9674}, {7738, 12963}, {7822, 11292}, {7889, 11291}, {7967, 35610}, {7969, 9778}, {7987, 13959}, {8276, 12082}, {8960, 23269}, {8991, 17845}, {9583, 31730}, {9649, 18996}, {9662, 19038}, {9682, 12088}, {9690, 42226}, {9691, 18512}, {9694, 33524}, {9732, 35945}, {9862, 9987}, {10303, 42262}, {10590, 31499}, {10819, 12244}, {11417, 30552}, {11495, 19014}, {11835, 14654}, {12123, 36703}, {12171, 26375}, {12240, 20791}, {12256, 40274}, {12322, 26361}, {12512, 18991}, {12962, 26462}, {12968, 26456}, {13665, 15704}, {13847, 15705}, {13893, 28164}, {13897, 31500}, {13903, 15681}, {13936, 16192}, {13951, 15712}, {13993, 17504}, {14813, 42126}, {14814, 42127}, {15326, 31408}, {15696, 42216}, {15698, 35823}, {15720, 18762}, {17578, 42273}, {19056, 38736}, {19060, 38726}, {19109, 38747}, {19111, 37853}, {19113, 38759}, {19146, 33750}, {22644, 35812}, {31473, 37267}, {42219, 42584}, {42220, 42585}

X(42638) = {X(6),X(3522)}-harmonic conjugate of X(42637)

X(42639) = GIBERT (12 SQRT(3),13,17) POINT

Barycentrics    a^2*(12*S + 17*SA) + 26*SB*SC : :

X(42639) lies on the cubic K1191 and these lines: {2, 6395}, {3, 3590}, {5, 6419}, {6, 42518}, {30, 6449}, {140, 6522}, {371, 23046}, {372, 41948}, {381, 13925}, {485, 549}, {547, 13951}, {590, 8703}, {1131, 15688}, {1151, 35404}, {1327, 19710}, {1328, 38071}, {1587, 10124}, {1588, 14892}, {3068, 5066}, {3070, 17504}, {3311, 11737}, {3316, 5054}, {3543, 10137}, {3830, 8972}, {3839, 13903}, {3845, 6437}, {5055, 13886}, {6199, 41106}, {6221, 12101}, {6396, 42572}, {6398, 11540}, {6445, 15640}, {6473, 15694}, {6560, 15711}, {6564, 33699}, {7583, 13847}, {8981, 15687}, {10109, 19054}, {11539, 35822}, {11812, 32785}, {12100, 13665}, {12108, 31414}, {12811, 31487}, {13897, 15170}, {14093, 23269}, {14869, 41946}, {15686, 41952}, {15690, 23249}, {15701, 23267}, {15713, 42216}, {32789, 42606}, {41947, 42582}, {41950, 41955}, {42276, 42608}

X(42640) = GIBERT (-12 SQRT(3),13,17) POINT

Barycentrics    a^2*(12*S - 17*SA) - 26*SB*SC : :

X(42640) lies on the cubic K1191 and these lines: {2, 6199}, {3, 3591}, {5, 6420}, {6, 42518}, {30, 6450}, {140, 6519}, {371, 41947}, {372, 23046}, {381, 13993}, {486, 549}, {547, 8976}, {615, 8703}, {1132, 15688}, {1152, 35404}, {1327, 38071}, {1328, 19710}, {1587, 14892}, {1588, 10124}, {3069, 5066}, {3071, 17504}, {3312, 11737}, {3317, 5054}, {3543, 10138}, {3627, 17852}, {3830, 13941}, {3839, 13961}, {3845, 6438}, {5055, 13939}, {6200, 42573}, {6221, 11540}, {6395, 41106}, {6398, 12101}, {6446, 15640}, {6472, 15694}, {6561, 15711}, {6565, 33699}, {7584, 13846}, {10109, 19053}, {11539, 35823}, {11812, 32786}, {12100, 13785}, {13954, 15170}, {13966, 15687}, {14093, 23275}, {14869, 41945}, {15686, 41951}, {15690, 23259}, {15701, 23273}, {15713, 42215}, {32790, 42607}, {41948, 42583}, {41949, 41956}, {42275, 42609}

X(42641) = GIBERT (15 SQRT(3),26,-8) POINT

Barycentrics    a^2*(15*S - 8*SA) + 52*SB*SC : :

X(42641) lies on the cubic K1191 and these lines: {2, 6410}, {4, 42573}, {30, 6437}, {382, 6419}, {546, 13847}, {1151, 42576}, {1152, 11737}, {1327, 34200}, {3311, 42577}, {3529, 31454}, {3851, 6522}, {3855, 41946}, {6395, 6565}, {6429, 42572}, {6430, 41106}, {6431, 35404}, {6432, 12101}, {6438, 23046}, {6449, 13846}, {6460, 41947}, {6560, 38071}, {6564, 15707}, {8253, 17504}, {10137, 15685}, {15686, 42568}, {15687, 19116}, {15688, 42264}, {15700, 42265}, {19709, 42569}, {23249, 41954}, {41952, 42414}

X(42642) = GIBERT (15 SQRT(3),-26,8) POINT

Barycentrics    a^2*(15*S + 8*SA) - 52*SB*SC : :

X(42642) lies on the cubic K1191 and these lines: {2, 6409}, {4, 42572}, {30, 6438}, {382, 6420}, {546, 13846}, {1151, 11737}, {1152, 42577}, {1328, 34200}, {3312, 42576}, {3529, 17852}, {3851, 6519}, {3855, 41945}, {6199, 6564}, {6429, 41106}, {6430, 42573}, {6431, 12101}, {6432, 35404}, {6437, 23046}, {6450, 13847}, {6459, 41948}, {6561, 38071}, {6565, 15707}, {8252, 17504}, {10138, 15685}, {15686, 42569}, {15687, 19117}, {15688, 42263}, {15700, 42262}, {19709, 42568}, {23259, 41953}, {41951, 42413}

X(42643) = GIBERT (20 SQRT(3),3,21) POINT

Barycentrics    a^2*(20*S + 21*SA) + 6*SB*SC : :

X(42643) lies on the cubic K1191 and these lines: {2, 6199}, {6, 3530}, {30, 6437}, {140, 42567}, {371, 546}, {382, 1131}, {548, 6480}, {550, 1587}, {3068, 15687}, {3311, 14869}, {3528, 6445}, {3592, 13993}, {3851, 8972}, {3853, 35815}, {3855, 13903}, {3856, 12819}, {5079, 23273}, {6200, 34200}, {6395, 10299}, {6398, 17504}, {6425, 42226}, {6431, 12108}, {6433, 33923}, {6434, 14891}, {6435, 41963}, {6441, 13966}, {6447, 23249}, {6468, 42216}, {6476, 41958}, {7585, 9690}, {8981, 10195}, {9542, 15710}, {10300, 18457}, {11737, 42215}, {15681, 23267}, {18762, 31454}, {23259, 38071}

X(42644) = GIBERT (-20 SQRT(3),3,21) POINT

Barycentrics    a^2*(20*S - 21*SA) - 6*SB*SC : :

X(42644) lies on the cubic K1191 and these lines: {2, 6395}, {6, 3530}, {30, 6438}, {140, 42566}, {372, 546}, {382, 1132}, {548, 6481}, {550, 1588}, {3069, 15687}, {3312, 14869}, {3528, 6446}, {3594, 13925}, {3851, 13941}, {3853, 35814}, {3855, 13961}, {3856, 12818}, {5079, 23267}, {6199, 10299}, {6221, 17504}, {6396, 34200}, {6426, 42225}, {6432, 12108}, {6433, 14891}, {6434, 33923}, {6436, 41964}, {6442, 8981}, {6448, 23259}, {6469, 42215}, {6477, 41957}, {7586, 15688}, {10194, 13966}, {10300, 18459}, {11737, 42216}, {15681, 23273}, {23249, 38071}

X(42645) = GIBERT (SQRT(3/2),1,0) POINT

Barycentrics    Sqrt[2]*a^2*S + 4*SB*SC : :

X(42645) lies on the cubic K1191 and these lines: {3, 3373}, {4, 6}, {5, 41975}, {550, 41976}, {1151, 14785}, {1152, 14784}, {3371, 35821}, {3372, 6564}, {3385, 35820}, {3386, 6565}, {3627, 41979}, {14782, 42265}, {14783, 42262}

X(42646) = GIBERT (-SQRT(3/2),1,0) POINT

Barycentrics    Sqrt[2]*a^2*S - 4*SB*SC : :

X(42646) lies on the cubic K1191 and these lines: {3, 3374}, {4, 6}, {5, 41976}, {550, 41975}, {1151, 14784}, {1152, 14785}, {3371, 6564}, {3372, 35821}, {3385, 6565}, {3386, 35820}, {3627, 41980}, {14782, 42262}, {14783, 42265}

X(42647) = GIBERT (2 SQRT(6),3,3) POINT

Barycentrics    a^2*(2*Sqrt[2]*S + 3*SA) + 6*SB*SC : :

X(42647) lies on the cubic K1191 and these lines: {5, 6}, {549, 41980}, {550, 41976}, {3373, 6396}, {3387, 6200}, {3845, 41979}, {8972, 14785}, {13941, 14784}, {14782, 23267}, {14783, 23273}

X(42648) = GIBERT (-2 SQRT(6),3,3) POINT

Barycentrics    a^2*(2*Sqrt[2]*S - 3*SA) - 6*SB*SC : :

X(42648) lies on the cubic K1191 and these lines: {5, 6}, {549, 41979}, {550, 41975}, {3374, 6396}, {3388, 6200}, {3845, 41980}, {8972, 14784}, {13941, 14785}, {14782, 23273}, {14783, 23267}

X(42649) = X(187)X(237)∩X(654)X(4041)

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

X(42649) lies on these lines: {187, 237}, {654, 4041}, {2355, 18344}, {4063, 21132}

X(42649) = center of circle {{X(15), X(16), X(35), X(484), X(3483), X(14102)}}
X(42649) = X(664)-isoconjugate of X(3467)
X(42649) = crosspoint of X(109) and X(2160)
X(42649) = crosssum of X(522) and X(3219)
X(42649) = crossdifference of every pair of points on line {2, 24148}
X(42649) = barycentric product X(i)*X(j) for these {i,j}: {522, 21773}, {649, 27529}, {650, 3336}, {663, 17483}, {3064, 23070}, {3737, 21863}, {3738, 11069}
X(42649) = barycentric quotient X(i)/X(j) for these {i,j}: {3063, 3467}, {3336, 4554}, {11069, 35174}, {17483, 4572}, {21773, 664}, {27529, 1978}

X(42650) = X(187)X(237)∩X(3574)X(32478)

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

X(42650) lies on these lines: {187, 237}, {3574, 32478}

X(42650) = center of circle {{X(15), X(16), X(54), X(1157), X(3482), X(18335)}}
X(42650) = crosspoint of X(i) and X(j) for these (i,j): {51, 32737}, {53, 933}
X(42650) = crosssum of X(i) and X(j) for these (i,j): {95, 41298}, {97, 6368}
X(42650) = crossdifference of every pair of points on line {2, 34520}
X(42650) = barycentric product X(195)*X(12077)

X(42651) = X(110)X(933)∩X(184)X(2081)

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

X(42651) lies on these lines: {110, 933}, {184, 2081}, {187, 237}, {520, 14696}, {3124, 15450}, {6130, 32193}

X(42651) = center of circle {{X(15), X(16), X(125), X(184), X(2081), X(13558)}}
X(42651) = Parry-circle-inverse of X(15451)
X(42651) = barycentric product X(647)*X(41203)
X(42651) = barycentric quotient X(41203)/X(6331)
X(42651) = {X(5638),X(5639)}-harmonic conjugate of X(15451)

X(42652) = X(187)X(237)∩X(385)X(804)

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

X(42652) lies on these lines: {187, 237}, {385, 804}, {691, 9150}, {805, 881}, {2086, 2679}, {5970, 14898}, {22329, 25423}

X(42652) = center of circle {{X(15), X(16), X(385), X(805), X(5970), X(32531)}}
X(42652) =X(35146)-isoconjugate of X(37134)
X(42652) =crosspoint of X(805) and X(5970)
X(42652) =crosssum of X(804) and X(5969)
X(42652) =crossdifference of every pair of points on line {2, 18829}
X(42652) =barycentric product X(i)*X(j) for these {i,j}: {804, 5106}, {1691, 11182}, {2086, 14607}, {5027, 5969}
X(42652) =barycentric quotient X(i)/X(j) for these {i,j}: {5027, 35146}, {5106, 18829}, {11182, 18896}
X(42652) ={X(669),X(3231)}-harmonic conjugate of X(351)

X(42653) = X(187)X(237)∩X(520)X(34975)

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

X(42653) lies on these lines: {187, 237}, {520, 34975}, {523, 21179}, {650, 6367}, {690, 905}, {804, 21260}, {810, 3725}, {1962, 4041}, {3309, 11615}, {3709, 9279}, {3743, 14838}, {4155, 6586}, {4983, 21828}, {6370, 20517}, {8574, 21837}, {9147, 21301}, {9148, 31251}, {9402, 17990}, {11176, 31288}, {21192, 31947}, {24782, 25686}, {25084, 25473}

X(42653) = center of circle {{X(15), X(16), X(501), X(3743), X(5127), X(14838), X(14873), X(39149)}}
X(42653) = X(4705)-Ceva conjugate of X(512)
X(42653) = X(i)-isoconjugate of X(j) for these (i,j): {99, 267}, {190, 40143}, {662, 1029}, {799, 3444}, {4610, 21353}
X(42653) = crosspoint of X(37) and X(110)
X(42653) = crosssum of X(i) and X(j) for these (i,j): {81, 523}, {86, 4467}
X(42653) = crossdifference of every pair of points on line {2, 1029}
X(42653) = barycentric product X(i)*X(j) for these {i,j}: {37, 31947}, {42, 21192}, {191, 661}, {451, 647}, {501, 4024}, {512, 2895}, {513, 21873}, {523, 1030}, {649, 21081}, {798, 20932}, {2501, 22136}, {3700, 8614}, {3709, 41808}, {4556, 21723}, {4705, 40592}
X(42653) = barycentric quotient X(i)/X(j) for these {i,j}: {191, 799}, {451, 6331}, {501, 4610}, {512, 1029}, {667, 40143}, {669, 3444}, {798, 267}, {1030, 99}, {2895, 670}, {4079, 502}, {8614, 4573}, {20932, 4602}, {21081, 1978}, {21192, 310}, {21873, 668}, {22136, 4563}, {31947, 274}, {40592, 4623}

X(42654) = X(110)X(250)∩X(187)X(237)

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

X(42654) lies on these lines: {110, 250}, {187, 237}, {247, 3258}, {523, 15448}, {879, 35260}, {1503, 22264}, {2433, 26864}, {2485, 20998}, {9155, 13480}, {13394, 40550}, {14165, 16229}, {35259, 41167}

X(42654) = center of circle {{X(15), X(16), X(647), X(1495), X(14685), X(16319), X(35901)}}
X(42654) = Parry-circle-inverse of X(9409)
X(42654) = crossdifference of every pair of points on line {2, 30227}
X(42654) = {X(5638),X(5639)}-harmonic conjugate of X(9409)

X(42655) = X(110)X(898)∩X(187)X(237)

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

X(42655) lies on these lines: {110, 898}, {187, 237}, {238, 4367}, {875, 2176}, {1083, 24286}

X(42655) = center of circle {{X(15), X(16), X(667), X(1083), X(3230), X(11650), X(11651), X(11652)}}
X(42655) = Parry-circle-inverse of X(890)
X(42655) = crosspoint of X(11651) and X(11652)
X(42655) = crossdifference of every pair of points on line {2, 24289}
X(42655) = {X(5638),X(5639)}-harmonic conjugate of X(890)

X(42656) = X(187)X(237)∩X(526)X(14157)

Barycentrics    a^2*(b - c)*(b + c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 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(42656) lies on these lines: {187, 237}, {526, 14157}

X(42656) = center of circle {{X(15), X(16), X(1138), X(2132), X(6794), X(12112), X(14354)}}
X(42656) = X(36034)-isoconjugate of X(40705)
X(42656) = crosspoint of X(i) and X(j) for these (i,j): {1304, 1990}, {1495, 14560}
X(42656) = crosssum of X(i) and X(j) for these (i,j): {1494, 3268}, {9033, 14919}
X(42656) = barycentric product X(i)*X(j) for these {i,j}: {399, 1637}, {1272, 14398}, {1495, 14566}, {19303, 36035}
X(42656) = barycentric quotient X(i)/X(j) for these {i,j}: {1637, 40705}, {14398, 1138}

X(42657) = X(40)X(30574)∩X(187)X(237)

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

X(42657) lies on these lines: {40, 30574}, {187, 237}, {652, 4814}, {654, 4895}, {692, 36072}, {2254, 2775}, {2774, 3746}, {4041, 9404}, {4088, 12514}, {5250, 14432}, {21132, 21385}

X(42657) = center of circle {{X(15), X(16), X(3065), X(3464), X(5540), X(6126)}}
X(42657) = X(34921)-Ceva conjugate of X(6)
X(42657) = X(i)-isoconjugate of X(j) for these (i,j): {75, 34921}, {109, 40716}, {651, 21739}, {664, 3065}, {4554, 19302}, {4585, 26743}, {14147, 17078}
X(42657) = crosspoint of X(i) and X(j) for these (i,j): {6, 34921}, {109, 2161}
X(42657) = crosssum of X(i) and X(j) for these (i,j): {57, 30572}, {514, 3582}, {522, 3218}, {4707, 11263}
X(42657) = crossdifference of every pair of points on line {2, 21739}
X(42657) = barycentric product X(i)*X(j) for these {i,j}: {37, 35055}, {484, 650}, {522, 19297}, {663, 17484}, {3063, 17791}, {3064, 23071}, {3737, 21864}, {11076, 35057}
X(42657) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34921}, {484, 4554}, {650, 40716}, {663, 21739}, {3063, 3065}, {17484, 4572}, {19297, 664}, {35055, 274}

leftri

Points on the Lemoine axis: X(42658)-X(42671)

rightri

This preamble and points X(42658-X(42671) are contributed by Peter Moses, April 19, 2021.

Suppose that P' = p' : q' : r' is a point on a line p x + q y + r z = 0 and that u x + v y + w z = 0 is a line, L. Then the point P'' = (p/u)*p' : (q/v)*q' + (r/w)*r' (p/u)*p' : (q/v)*q' + (r/w)*r' lies on L For example, if P' is on the Euler line and L is the Lemoine axis, X(187)X(237), then P'' is on L. Points X(42658)-X(42671) are obtained in this manner, where, in the same order, P' = X(i) for i = 20, 23, 378, 429, 447, 460, 469, 858, 860, 1113, 1114, 1981, 2074, 2409.



X(42658) = CROSSSUM OF X(4) AND X(525)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 - b^2 - c^2)*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(42658) = X[647] + 2 X[9409], 3 X[647] - 2 X[15451], 3 X[8644] - 4 X[34952], 3 X[9409] + X[15451], X[15451] - 3 X[39201]

X(42658) lies on these lines: {3, 2435}, {112, 32649}, {184, 2430}, {187, 237}, {520, 11589}, {523, 37931}, {878, 40319}, {2485, 2881}, {2524, 39469}, {3198, 14308}, {6130, 16229}, {7669, 34190}, {8057, 15427}, {9517, 39228}, {38608, 39071}

X(42658) = midpoint of X(9409) and X(39201)
X(42658) = reflection of X(i) in X(j) for these {i,j}: {647, 39201}, {16229, 6130}
X(42658) = isogonal conjugate of the anticomplement of X(39020)
X(42658) = isogonal conjugate of the isotomic conjugate of X(8057)
X(42658) = isogonal conjugate of the polar conjugate of X(6587)
X(42658) = anticomplement of complementary conjugate of X(39020)
X(42658) = pole wrt polar circle of line X(253)X(264)
X(42658) = X(20402)-complementary conjugate of X(34846)
X(42658) = X(i)-Ceva conjugate of X(j) for these (i,j): {20, 1562}, {112, 3172}, {520, 647}, {1301, 6}, {3532, 3269}, {22089, 2524}, {32713, 184}, {34285, 125}
X(42658) = X(i)-isoconjugate of X(j) for these (i,j): {64, 811}, {75, 1301}, {107, 19611}, {108, 5931}, {162, 253}, {459, 662}, {648, 2184}, {799, 41489}, {823, 1073}, {2155, 6331}, {4592, 6526}, {6528, 19614}, {8809, 36797}, {14208, 15384}, {15394, 36126}, {16096, 36092}, {23052, 35571}, {24019, 34403}, {32676, 41530}
X(42658) = crosspoint of X(i) and X(j) for these (i,j): {3, 112}, {6, 1301}, {110, 41894}, {6525, 32713}, {6587, 8057}
X(42658) = crosssum of X(i) and X(j) for these (i,j): {2, 8057}, {4, 525}, {25, 2451}, {27, 7253}, {523, 26958}, {3265, 15394}
X(42658) = crossdifference of every pair of points on line {2, 253}
X(42658) = barycentric product X(i)*X(j) for these {i,j}: {3, 6587}, {6, 8057}, {20, 647}, {25, 20580}, {48, 17898}, {71, 21172}, {73, 14331}, {74, 14345}, {110, 1562}, {112, 122}, {154, 525}, {204, 24018}, {222, 14308}, {512, 37669}, {520, 1249}, {521, 30456}, {523, 15905}, {610, 656}, {652, 5930}, {810, 18750}, {822, 1895}, {905, 3198}, {1301, 39020}, {1394, 8611}, {1459, 8804}, {1559, 2430}, {1636, 10152}, {2501, 35602}, {3049, 14615}, {3172, 3265}, {6368, 33629}, {9033, 15291}, {14249, 32320}, {15466, 39201}, {17434, 38808}, {20975, 36841}, {23286, 42459}
X(42658) = barycentric quotient X(i)/X(j) for these {i,j}: {20, 6331}, {32, 1301}, {122, 3267}, {154, 648}, {204, 823}, {512, 459}, {520, 34403}, {525, 41530}, {610, 811}, {647, 253}, {652, 5931}, {669, 41489}, {810, 2184}, {822, 19611}, {1249, 6528}, {1562, 850}, {2489, 6526}, {2972, 14638}, {3049, 64}, {3172, 107}, {3198, 6335}, {6525, 15352}, {6587, 264}, {8057, 76}, {14308, 7017}, {14345, 3260}, {15291, 16077}, {15451, 13157}, {15905, 99}, {17898, 1969}, {20580, 305}, {30456, 18026}, {32320, 15394}, {33629, 18831}, {35602, 4563}, {37669, 670}, {38808, 42405}, {39201, 1073}, {40933, 13149}, {42293, 8798}

X(42659) = CROSSSUM OF X(2) AND X(9517)

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

X(42659) lies on these lines: {2, 25644}, {3, 14417}, {22, 2799}, {23, 9979}, {25, 1637}, {184, 39469}, {187, 237}, {523, 37969}, {684, 22085}, {690, 3455}, {878, 14582}, {1576, 2491}, {3268, 6636}, {4108, 39214}, {7669, 10117}, {8428, 14273}, {9131, 11616}, {9517, 16165}, {9529, 12082}, {20975, 23216}

X(42659) = isogonal conjugate of the isotomic conjugate of X(9517)
X(42659) = isogonal conjugate of the polar conjugate of X(2492)
X(42659) = X(i)-Ceva conjugate of X(j) for these (i,j): {935, 6}, {9076, 125}, {10097, 3049}, {32729, 184}, {34437, 3269}
X(42659) = X(i)-isoconjugate of X(j) for these (i,j): {67, 811}, {75, 935}, {92, 17708}, {162, 18019}, {799, 8791}, {823, 34897}, {2157, 6331}, {37221, 41676}
X(42659) = crosspoint of X(i) and X(j) for these (i,j): {6, 935}, {112, 1177}, {1576, 14908}, {2492, 9517}, {4630, 9076}
X(42659) = crosssum of X(i) and X(j) for these (i,j): {2, 9517}, {525, 858}, {935, 17708}, {9019, 23285}
X(42659) = crossdifference of every pair of points on line {2, 339}
X(42659) = barycentric product X(i)*X(j) for these {i,j}: {3, 2492}, {6, 9517}, {23, 647}, {184, 9979}, {248, 33752}, {316, 3049}, {512, 22151}, {520, 8744}, {523, 10317}, {525, 18374}, {669, 37804}, {810, 16568}, {2200, 21205}, {2433, 16165}, {2435, 28343}, {3292, 10561}, {6593, 10097}, {10510, 30491}, {14908, 18311}, {37765, 39201}
X(42659) = barycentric quotient X(i)/X(j) for these {i,j}: {23, 6331}, {32, 935}, {184, 17708}, {647, 18019}, {669, 8791}, {2492, 264}, {3049, 67}, {8744, 6528}, {9517, 76}, {9979, 18022}, {10317, 99}, {18374, 648}, {22151, 670}, {30491, 10512}, {37804, 4609}, {39201, 34897}
X(42659) = {X(669),X(34952)}-harmonic conjugate of X(8644)

X(42660) = CROSSSUM OF X(2) AND X(8675)

Barycentrics    a^4*(b - c)*(b + c)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) : :
X(42660) = 2 X[669] - 3 X[34952], X[669] - 3 X[39201], 4 X[14270] - 3 X[34952], 2 X[14270] - 3 X[39201], 3 X[17414] - 2 X[18117]

X(42660) lies on these lines: {3, 523}, {32, 3049}, {39, 2451}, {187, 237}, {826, 22089}, {3050, 3053}, {3566, 35463}, {3800, 39228}, {8029, 37457}, {8675, 10564}, {8722, 38354}, {9605, 39520}, {12054, 39513}, {12073, 39477}, {26316, 39495}, {32231, 34291}, {34347, 40799}

X(42660) = midpoint of X(3005) and X(9409)
X(42660) = reflection of X(i) in X(j) for these {i,j}: {669, 14270}, {21731, 647}, {34291, 32231}, {34952, 39201}
X(42660) = circumcircle-inverse of X(6795)
X(42660) = isogonal conjugate of the isotomic conjugate of X(8675)
X(42660) = X(1302)-Ceva conjugate of X(6)
X(42660) = X(i)-isoconjugate of X(j) for these (i,j): {75, 1302}, {76, 36149}, {561, 32738}, {662, 34289}, {799, 34288}, {811, 4846}, {3260, 36083}
X(42660) = crosspoint of X(i) and X(j) for these (i,j): {6, 1302}, {10419, 32681}
X(42660) = crosssum of X(i) and X(j) for these (i,j): {2, 8675}, {512, 5309}, {523, 37648}, {525, 15760}
X(42660) = crossdifference of every pair of points on line {2, 3003}
X(42660) = bicentric difference of trilinear product P(9)*P(86) and trilinear product U(9)*U(86)
X(42660) = barycentric product X(i)*X(j) for these {i,j}: {6, 8675}, {32, 30474}, {378, 647}, {512, 15066}, {523, 5063}, {669, 32833}, {2433, 10564}, {2623, 5891}, {3569, 11653}
X(42660) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1302}, {378, 6331}, {512, 34289}, {560, 36149}, {669, 34288}, {1501, 32738}, {3049, 4846}, {5063, 99}, {8675, 76}, {15066, 670}, {30474, 1502}, {32833, 4609}
X(42660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {669, 14270, 34952}, {669, 39201, 14270}

X(42661) = CROSSSUM OF X(110) AND X(4612)

Barycentrics    a^2*(b - c)*(b + c)^2*(a*b + b^2 + a*c + c^2) : :
X(42661) = 3 X[351] - 2 X[667], 3 X[9147] - X[31291], 3 X[9148] - 4 X[21260]

X(42661) lies on these lines: {37, 18002}, {187, 237}, {523, 4391}, {690, 2530}, {804, 21301}, {830, 3743}, {2512, 4843}, {3801, 6370}, {4024, 4705}, {4041, 4155}, {4079, 17411}, {4983, 8034}, {9147, 31291}, {9148, 21260}, {17994, 18344}

X(42661) = reflection of X(8639) in X(647)
X(42661) = X(i)-Ceva conjugate of X(j) for these (i,j): {37, 3124}, {1402, 20975}, {34434, 35506}
X(42661) = X(i)-isoconjugate of X(j) for these (i,j): {99, 2363}, {163, 40827}, {261, 36098}, {662, 14534}, {757, 8707}, {799, 1169}, {811, 1798}, {873, 32736}, {1509, 36147}, {2185, 6648}, {2298, 4610}, {4556, 30710}, {4581, 24041}, {4636, 31643}
X(42661) = crosspoint of X(i) and X(j) for these (i,j): {512, 4705}, {8687, 18772}
X(42661) = crosssum of X(i) and X(j) for these (i,j): {110, 4612}, {523, 6703}, {1798, 15420}, {3910, 4999}, {4581, 14534}
X(42661) = crossdifference of every pair of points on line {2, 261}
X(42661) = barycentric product X(i)*X(j) for these {i,j}: {42, 21124}, {181, 3910}, {429, 647}, {512, 1211}, {513, 21810}, {523, 2092}, {594, 6371}, {649, 20653}, {661, 2292}, {669, 1228}, {798, 18697}, {872, 4509}, {1193, 4024}, {1500, 3004}, {1577, 3725}, {2171, 17420}, {2300, 4036}, {2354, 4064}, {2501, 22076}, {2643, 3882}, {3005, 27067}, {3666, 4705}, {3704, 7180}, {3709, 41003}, {4017, 21033}, {4079, 4357}, {7178, 40966}, {14394, 38882}
X(42661) = barycentric quotient X(i)/X(j) for these {i,j}: {181, 6648}, {429, 6331}, {512, 14534}, {523, 40827}, {669, 1169}, {798, 2363}, {872, 36147}, {960, 4631}, {1193, 4610}, {1211, 670}, {1228, 4609}, {1500, 8707}, {2092, 99}, {2292, 799}, {3049, 1798}, {3124, 4581}, {3666, 4623}, {3725, 662}, {3882, 24037}, {3910, 18021}, {4024, 1240}, {4079, 1220}, {4705, 30710}, {6371, 1509}, {7109, 32736}, {18697, 4602}, {20653, 1978}, {20967, 4612}, {20975, 15420}, {21033, 7257}, {21124, 310}, {21810, 668}, {22076, 4563}, {27067, 689}, {40966, 645}

X(42662) = CROSSSUM OF X(1) AND X(4707)

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

X(42662) lies on these lines: {1, 2785}, {58, 2774}, {74, 106}, {101, 112}, {187, 237}, {214, 40613}, {248, 38865}, {976, 4088}, {1010, 24353}, {1015, 3269}, {1017, 9408}, {1459, 3960}, {2254, 8578}, {3430, 9527}, {3924, 30574}, {4040, 29118}, {4449, 29094}, {4724, 29029}, {6184, 9475}, {9412, 21781}, {9862, 41190}

X(42662) = isogonal conjugate of X(35169)
X(42662) = isogonal conjugate of the anticomplement of X(35122)
X(42662) = X(4242)-Ceva conjugate of X(2183)
X(42662) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35169}, {100, 16099}, {162, 40715}
X(42662) = crosspoint of X(1897) and X(2161)
X(42662) = crosssum of X(i) and X(j) for these (i,j): {1, 4707}, {514, 30117}, {1459, 3218}
X(42662) = crossdifference of every pair of points on line {2, 1762}
X(42662) = barycentric product X(i)*X(j) for these {i,j}: {101, 867}, {447, 647}, {649, 16086}
X(42662) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35169}, {447, 6331}, {647, 40715}, {649, 16099}, {867, 3261}, {16086, 1978}
X(42662) = {X(39665),X(39666)}-harmonic conjugate of X(649)

X(42663) = CROSSSUM OF X(99) AND X(2396)

Barycentrics    a^2*(b - c)*(b + c)*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(42663) = 3 X[351] - 2 X[3569], 3 X[351] - 4 X[5027], 9 X[351] - 8 X[5113], 5 X[351] - 4 X[9208], 3 X[3569] - 4 X[5113], X[3569] - 3 X[9135], 5 X[3569] - 6 X[9208], 3 X[5027] - 2 X[5113], 2 X[5027] - 3 X[9135], 5 X[5027] - 3 X[9208], 4 X[5113] - 9 X[9135], 10 X[5113] - 9 X[9208], 3 X[5652] - 2 X[24284], 2 X[6333] - 3 X[11123], 5 X[9135] - 2 X[9208], 3 X[9148] - 4 X[24284], 3 X[14398] - 2 X[22260]

X(42663) lies on these lines: {6, 2872}, {69, 6131}, {110, 3565}, {187, 237}, {399, 2780}, {523, 32220}, {542, 32121}, {690, 24981}, {804, 25046}, {1510, 21006}, {1976, 17994}, {2422, 14601}, {2444, 17993}, {2514, 3050}, {2971, 3124}, {3566, 6562}, {3800, 14316}, {5652, 6033}, {6088, 10765}, {6132, 35364}, {6333, 11123}, {9517, 11641}, {13306, 31299}, {14398, 22260}, {14610, 39905}, {18105, 20188}, {21905, 39689}

X(42663) = reflection of X(i) in X(j) for these {i,j}: {69, 6131}, {351, 9135}, {2514, 3050}, {3005, 3288}, {3569, 5027}, {9148, 5652}, {35364, 6132}, {39905, 14610}
X(42663) = Parry-circle-inverse of X(8651)
X(42663) = X(i)-Ceva conjugate of X(j) for these (i,j): {511, 2086}, {1976, 3124}, {10425, 6}, {39644, 20975}
X(42663) = X(i)-isoconjugate of X(j) for these (i,j): {69, 36105}, {75, 10425}, {99, 8773}, {304, 32697}, {662, 8781}, {670, 36051}, {799, 2987}, {4592, 35142}, {4602, 32654}, {24037, 35364}
X(42663) = crosspoint of X(i) and X(j) for these (i,j): {6, 10425}, {99, 6531}, {512, 2422}, {2207, 32696}
X(42663) = crosssum of X(i) and X(j) for these (i,j): {99, 2396}, {512, 36212}, {3926, 6333}
X(42663) = crossdifference of every pair of points on line {2, 2987}
X(42663) = barycentric product X(i)*X(j) for these {i,j}: {114, 2422}, {230, 512}, {460, 647}, {523, 1692}, {661, 8772}, {798, 1733}, {882, 12829}, {2489, 3564}, {2491, 14265}, {3124, 4226}, {5477, 9178}, {6041, 34174}, {6132, 14384}, {14398, 36875}, {32696, 41181}
X(42663) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 10425}, {230, 670}, {460, 6331}, {512, 8781}, {669, 2987}, {798, 8773}, {1084, 35364}, {1692, 99}, {1733, 4602}, {1924, 36051}, {1973, 36105}, {1974, 32697}, {2422, 40428}, {2489, 35142}, {4226, 34537}, {8772, 799}, {9426, 32654}, {12829, 880}, {14398, 36891}
X(42663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3569, 5027, 351}, {3569, 9135, 5027}, {5191, 21731, 351}, {5638, 5639, 8651}

X(42664) = CROSSSUM OF X(2) AND X(23879)

Barycentrics    a^2*(b - c)*(b + c)*(a*b + b^2 + a*c + b*c + c^2) : :
X(42664) = 7 X[27138] - 5 X[31072], 4 X[30476] - 5 X[30835]

X(42664) lies on these lines: {187, 237}, {523, 661}, {525, 16892}, {650, 4132}, {850, 3835}, {1499, 38329}, {2611, 20974}, {2786, 27469}, {3050, 7252}, {4129, 24083}, {4139, 4893}, {4467, 4481}, {4502, 4526}, {4750, 28372}, {4785, 36900}, {4826, 21828}, {4979, 21123}, {14349, 23879}, {20295, 31296}, {21051, 21715}, {21196, 27647}, {21721, 31946}, {23878, 31147}, {27138, 31072}, {30476, 30835}

X(42664) = midpoint of X(i) and X(j) for these {i,j}: {3005, 8663}, {20295, 31296}
X(42664) = reflection of X(i) in X(j) for these {i,j}: {649, 647}, {850, 3835}, {3804, 8653}
X(42664) = isogonal conjugate of the isotomic conjugate of X(23879)
X(42664) = X(i)-Ceva conjugate of X(j) for these (i,j): {835, 20966}, {34819, 20982}, {39967, 3124}
X(42664) = X(i)-isoconjugate of X(j) for these (i,j): {58, 37218}, {81, 835}, {99, 2214}
X(42664) = crosspoint of X(i) and X(j) for these (i,j): {834, 14349}, {835, 40394}, {4033, 31359}
X(42664) = crosssum of X(i) and X(j) for these (i,j): {2, 23879}, {523, 17398}, {525, 7536}
X(42664) = crossdifference of every pair of points on line {2, 58}
X(42664) = barycentric product X(i)*X(j) for these {i,j}: {6, 23879}, {10, 834}, {37, 14349}, {58, 23282}, {313, 8637}, {386, 523}, {469, 647}, {512, 5224}, {661, 28606}, {798, 33935}, {3122, 33948}, {3709, 33949}, {3876, 4017}
X(42664) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 37218}, {42, 835}, {386, 99}, {469, 6331}, {798, 2214}, {834, 86}, {3876, 7257}, {5224, 670}, {8637, 58}, {14349, 274}, {23282, 313}, {23879, 76}, {28606, 799}, {33935, 4602}
X(42664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 21834, 4024}, {21051, 21719, 21720}, {21051, 23948, 21726}

X(42665) = CROSSSUM OF X(4) AND X(9979)

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

X(42665) lies on these lines: {25, 2881}, {122, 125}, {184, 1636}, {187, 237}, {526, 15106}, {686, 39469}, {1184, 2508}, {1576, 2445}, {1637, 17994}, {3258, 38368}, {3292, 13303}, {5489, 14424}, {5505, 14380}, {9517, 14908}, {14416, 42442}

X(42665) = X(i)-Ceva conjugate of X(j) for these (i,j): {67, 3269}, {2373, 38356}, {10423, 6}, {14908, 20975}, {32709, 10602}, {40347, 3124}
X(42665) = X(i)-isoconjugate of X(j) for these (i,j): {2, 36095}, {75, 10423}, {112, 37220}, {162, 2373}, {811, 1177}, {823, 18876}
X(42665) = crosspoint of X(i) and X(j) for these (i,j): {6, 10423}, {112, 895}, {523, 10097}
X(42665) = crosssum of X(i) and X(j) for these (i,j): {4, 9979}, {110, 4235}, {468, 525}, {1974, 14273}, {2393, 2485}
X(42665) = crossdifference of every pair of points on line {2, 112}
X(42665) = barycentric product X(i)*X(j) for these {i,j}: {71, 21109}, {520, 5523}, {523, 14961}, {525, 2393}, {647, 858}, {656, 18669}, {810, 20884}, {1236, 3049}, {1459, 21017}, {3265, 14580}, {5181, 10097}, {19510, 30491}, {34158, 35522}
X(42665) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 36095}, {32, 10423}, {647, 2373}, {656, 37220}, {858, 6331}, {2393, 648}, {3049, 1177}, {5523, 6528}, {14580, 107}, {14961, 99}, {18669, 811}, {34158, 691}, {39201, 18876}, {39469, 36823}
X(42665) = {X(17414),X(39201)}-harmonic conjugate of X(647)

X(42666) = CROSSSUM OF X(2) AND X(6370)

Barycentrics    a^2*(b - c)*(b + c)^2*(a^2 - b^2 + b*c - c^2) : :
X(42666) = 3 X[351] - 2 X[1960], 3 X[1962] - X[4895], X[2650] - 3 X[14413]

X(42666) lies on these lines: {10, 18003}, {187, 237}, {291, 18009}, {523, 10015}, {526, 6126}, {690, 2254}, {758, 3960}, {804, 24462}, {1769, 6089}, {1962, 4895}, {2491, 40986}, {2492, 22108}, {2642, 2643}, {2650, 14413}, {3743, 3887}, {4041, 4838}, {4142, 4151}, {4707, 4736}, {21756, 23648}, {21784, 23861}

X(42666) = midpoint of X(2254) and X(2292)
X(42666) = isogonal conjugate of the isotomic conjugate of X(6370)
X(42666) = X(i)-Ceva conjugate of X(j) for these (i,j): {759, 20982}, {1464, 2088}, {2433, 3709}, {36069, 6}
X(42666) = X(i)-isoconjugate of X(j) for these (i,j): {2, 37140}, {60, 35174}, {75, 36069}, {76, 32671}, {99, 759}, {110, 14616}, {261, 2222}, {593, 36804}, {655, 2185}, {662, 24624}, {799, 34079}, {850, 9274}, {1414, 6740}, {1577, 9273}, {2006, 4612}, {2161, 4610}, {2341, 4573}, {2617, 39277}, {4556, 18359}, {4623, 6187}, {4636, 18815}, {14838, 39295}, {17104, 35139}, {32678, 34016}, {32680, 40214}, {36066, 36815}
X(42666) = crosspoint of X(6) and X(36069)
X(42666) = crosssum of X(i) and X(j) for these (i,j): {2, 6370}, {523, 35466}, {758, 14838}, {3738, 16579}
X(42666) = crossdifference of every pair of points on line {2, 662}
X(42666) = barycentric product X(i)*X(j) for these {i,j}: {1, 2610}, {6, 6370}, {10, 21828}, {12, 654}, {36, 4024}, {42, 4707}, {181, 3904}, {320, 4079}, {512, 3936}, {513, 4053}, {523, 2245}, {526, 8818}, {647, 860}, {661, 758}, {756, 3960}, {798, 35550}, {1089, 21758}, {1109, 1983}, {1464, 3700}, {1500, 4453}, {1577, 3724}, {1835, 8611}, {2088, 6742}, {2171, 3738}, {2433, 6739}, {2624, 6757}, {2643, 4585}, {3218, 4705}, {3708, 4242}, {3709, 41804}, {4036, 7113}, {4041, 18593}, {6358, 8648}
X(42666) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 37140}, {32, 36069}, {36, 4610}, {181, 655}, {512, 24624}, {526, 34016}, {560, 32671}, {654, 261}, {661, 14616}, {669, 34079}, {756, 36804}, {758, 799}, {798, 759}, {860, 6331}, {1464, 4573}, {1576, 9273}, {1983, 24041}, {2088, 4467}, {2171, 35174}, {2245, 99}, {2361, 4612}, {2610, 75}, {2623, 39277}, {3218, 4623}, {3709, 6740}, {3724, 662}, {3904, 18021}, {3936, 670}, {3960, 873}, {4024, 20566}, {4053, 668}, {4079, 80}, {4511, 4631}, {4585, 24037}, {4705, 18359}, {4707, 310}, {6370, 76}, {8648, 2185}, {8818, 35139}, {14270, 40214}, {18593, 4625}, {21758, 757}, {21828, 86}, {35550, 4602}
X(42666) = {X(3724),X(8648)}-harmonic conjugate of X(14270)

X(42667) = CROSSSUM OF X(2) AND X(2575)

Barycentrics    a^2*(b^2 - c^2)*SA*(a^2*(1 - J)*SA - 2*SB*SC) : :

X(42667) lies on these lines: {3, 2575}, {25, 8106}, {98, 1113}, {187, 237}, {228, 2579}, {1114, 35278}, {1344, 9756}, {1799, 22340}, {1995, 9174}, {15167, 20975}

X(42667) = reflection of X(42668) in X(14270)
X(42667) = isogonal conjugate of X(15165)
X(42667) = isogonal conjugate of the anticomplement of X(15167)
X(42667) = isogonal conjugate of the isotomic conjugate of X(2575)
X(42667) = isogonal conjugate of the polar conjugate of X(8106)
X(42667) = circumcircle-inverse of X(13414)
X(42667) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 15167}, {1114, 6}
X(42667) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15165}, {2, 2581}, {69, 2587}, {75, 1114}, {76, 2577}, {92, 8116}, {99, 2588}, {162, 22339}, {264, 1823}, {648, 2582}, {662, 2592}, {799, 8105}, {811, 2574}, {1577, 39299}, {2578, 6331}, {2584, 6528}, {24041, 39240}
X(42667) = crosspoint of X(i) and X(j) for these (i,j): {6, 1114}, {110, 41941}, {112, 15461}, {2575, 8106}
X(42667) = crosssum of X(i) and X(j) for these (i,j): {2, 2575}, {4, 2593}, {69, 22340}, {525, 1313}, {1114, 8116}, {2592, 39240}
X(42667) = crossdifference of every pair of points on line {2, 2592}
X(42667) = barycentric product X(i)*X(j) for these {i,j}: {1, 2579}, {3, 8106}, {6, 2575}, {19, 2585}, {31, 2583}, {32, 22340}, {48, 2589}, {184, 2593}, {512, 8115}, {647, 1113}, {656, 2576}, {661, 1822}, {810, 2580}, {822, 2586}, {1114, 15167}, {3049, 15164}, {20975, 39298}, {23110, 41942}, {32661, 39241}
X(42667) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 15165}, {31, 2581}, {32, 1114}, {184, 8116}, {512, 2592}, {560, 2577}, {647, 22339}, {669, 8105}, {798, 2588}, {810, 2582}, {1113, 6331}, {1576, 39299}, {1822, 799}, {1973, 2587}, {2575, 76}, {2576, 811}, {2579, 75}, {2583, 561}, {2585, 304}, {2589, 1969}, {2593, 18022}, {3049, 2574}, {3124, 39240}, {8106, 264}, {8115, 670}, {9247, 1823}, {15167, 22340}, {22340, 1502}
X(42667) = {X(237),X(5191)}-harmonic conjugate of X(42668)

X(42668) = CROSSSUM OF X(2) AND X(2574)

Barycentrics    a^2*(b^2 - c^2)*SA*(a^2*(1 + J)*SA - 2*SB*SC) : :

X(42668) lies on these lines: {3, 2574}, {25, 8105}, {98, 1114}, {187, 237}, {228, 2578}, {1113, 35278}, {1345, 9756}, {1799, 22339}, {1995, 9173}, {15166, 20975}

X(42668) = reflection of X(42667) in X(14270)
X(42668) = circumcircle-inverse of X(13415)
X(42668) = isogonal conjugate of X(15164)
X(42668) = isogonal conjugate of the anticomplement of X(15166)
X(42668) = isogonal conjugate of the isotomic conjugate of X(2574)
X(42668) = isogonal conjugate of the polar conjugate of X(8105)
X(42668) = circumcircle-inverse of X(13415)
X(42668) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 15166}, {1113, 6}
X(42668) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15164}, {2, 2580}, {69, 2586}, {75, 1113}, {76, 2576}, {92, 8115}, {99, 2589}, {162, 22340}, {264, 1822}, {648, 2583}, {662, 2593}, {799, 8106}, {811, 2575}, {1577, 39298}, {2579, 6331}, {2585, 6528}, {24041, 39241}
X(42668) = crosspoint of X(i) and X(j) for these (i,j): {6, 1113}, {110, 41942}, {112, 15460}, {2574, 8105}
X(42668) = crosssum of X(i) and X(j) for these (i,j): {2, 2574}, {4, 2592}, {69, 22339}, {525, 1312}, {1113, 8115}, {2593, 39241}
X(42668) = crossdifference of every pair of points on line {2, 2593}
X(42668) = {X(237),X(5191)}-harmonic conjugate of X(42667)
X(42668) = barycentric product X(i)*X(j) for these {i,j}: {1, 2578}, {3, 8105}, {6, 2574}, {19, 2584}, {31, 2582}, {32, 22339}, {48, 2588}, {184, 2592}, {512, 8116}, {647, 1114}, {656, 2577}, {661, 1823}, {810, 2581}, {822, 2587}, {1113, 15166}, {3049, 15165}, {20975, 39299}, {23109, 41941}, {32661, 39240}
X(42668) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 15164}, {31, 2580}, {32, 1113}, {184, 8115}, {512, 2593}, {560, 2576}, {647, 22340}, {669, 8106}, {798, 2589}, {810, 2583}, {1114, 6331}, {1576, 39298}, {1823, 799}, {1973, 2586}, {2574, 76}, {2577, 811}, {2578, 75}, {2582, 561}, {2584, 304}, {2588, 1969}, {2592, 18022}, {3049, 2575}, {3124, 39241}, {8105, 264}, {8116, 670}, {9247, 1822}, {15166, 22339}, {22339, 1502}

X(42669) = CROSSSUM OF X(2) AND X(8680)

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

X(42669) lies on these lines: {1, 19}, {10, 22061}, {41, 2650}, {56, 40978}, {65, 2200}, {73, 2333}, {101, 758}, {187, 237}, {213, 1042}, {572, 25081}, {604, 40977}, {607, 2658}, {859, 1755}, {993, 22099}, {1460, 3725}, {1464, 39690}, {1944, 5088}, {1945, 17963}, {1951, 26884}, {2176, 2178}, {2179, 23383}, {2292, 9310}, {2654, 40975}, {4020, 23361}, {4245, 24511}, {4390, 21020}, {4456, 20727}, {6603, 20718}, {8235, 12520}, {9247, 14529}, {12081, 17439}, {14963, 22098}, {18047, 35544}, {20963, 21748}

X(42669) = isogonal conjugate of X(35145)
X(42669) = isogonal conjugate of the anticomplement of X(35075)
X(42669) = isogonal conjugate of the isotomic conjugate of X(8680)
X(42669) = X(2249)-Ceva conjugate of X(6)
X(42669) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35145}, {2, 37142}, {21, 1952}, {29, 40843}, {75, 2249}, {296, 31623}, {314, 1945}, {333, 1937}, {521, 41207}, {522, 41206}, {2713, 17899}
X(42669) = crosspoint of X(i) and X(j) for these (i,j): {6, 2249}, {1951, 2202}
X(42669) = crosssum of X(i) and X(j) for these (i,j): {2, 8680}, {333, 40882}, {1952, 40843}, {9391, 16573}
X(42669) = crossdifference of every pair of points on line {2, 656}
X(42669) = barycentric product X(i)*X(j) for these {i,j}: {1, 851}, {6, 8680}, {10, 26884}, {42, 5088}, {65, 1936}, {72, 1430}, {73, 243}, {162, 9391}, {226, 1951}, {647, 1981}, {656, 23353}, {798, 15418}, {1042, 7360}, {1214, 2202}, {1400, 1944}, {1409, 1948}, {1425, 15146}, {1880, 6518}, {2249, 35075}
X(42669) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35145}, {31, 37142}, {32, 2249}, {851, 75}, {1400, 1952}, {1402, 1937}, {1409, 40843}, {1415, 41206}, {1430, 286}, {1936, 314}, {1944, 28660}, {1951, 333}, {1981, 6331}, {2202, 31623}, {5088, 310}, {8680, 76}, {9391, 14208}, {15418, 4602}, {23353, 811}, {26884, 86}, {32674, 41207}
X(42669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {65, 2200, 23621}, {73, 2333, 23619}, {3724, 5202, 3747}

X(42670) = CROSSSUM OF X(2) AND X(8674)

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

X(42670) lies on these lines: {3, 2775}, {21, 2787}, {35, 9508}, {55, 4730}, {163, 692}, {187, 237}, {405, 14431}, {659, 21201}, {993, 4922}, {2530, 22160}, {3271, 20975}, {4010, 5248}, {4455, 7669}, {4705, 21789}, {6050, 11934}, {8674, 16164}, {16865, 30709}, {34858, 40352}

X(42670) = midpoint of X(21) and X(16158)
X(42670) = isogonal conjugate of X(35156)
X(42670) = isogonal conjugate of the anticomplement of X(35090)
X(42670) = isogonal conjugate of the isotomic conjugate of X(8674)
X(42670) = X(i)-Ceva conjugate of X(j) for these (i,j): {1290, 6}, {1983, 2251}
X(42670) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35156}, {75, 1290}, {99, 5620}, {190, 21907}, {664, 11604}
X(42670) = crosspoint of X(i) and X(j) for these (i,j): {6, 1290}, {692, 6187}
X(42670) = crosssum of X(i) and X(j) for these (i,j): {2, 8674}, {320, 693}, {513, 33129}
X(42670) = crossdifference of every pair of points on line {2, 16732}
X(42670) = barycentric product X(i)*X(j) for these {i,j}: {6, 8674}, {512, 37783}, {513, 17796}, {523, 19622}, {647, 2074}, {650, 5172}, {661, 5127}, {667, 32849}, {1290, 35090}, {1946, 37799}, {2433, 16164}
X(42670) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35156}, {32, 1290}, {667, 21907}, {798, 5620}, {2074, 6331}, {3063, 11604}, {5127, 799}, {5172, 4554}, {8674, 76}, {17796, 668}, {19622, 99}, {32849, 6386}, {37783, 670}
X(42670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {667, 8641, 4775}, {1960, 8648, 667}, {8636, 8637, 667}, {8645, 8648, 1960}

X(42671) = CROSSSUM OF X(2) AND X(1508)

Barycentrics    a^2*(2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6) : :
X(42671) = X[401] - 3 X[35278], 2 X[441] - 3 X[35282]

X(42671) lies on the cubics K784 and I785 and these lines: {3, 64}, {6, 33582}, {22, 5188}, {23, 9157}, {25, 32}, {31, 8615}, {39, 184}, {50, 19596}, {51, 5007}, {69, 15594}, {98, 419}, {110, 2710}, {115, 460}, {157, 206}, {159, 577}, {160, 22052}, {161, 10316}, {187, 237}, {232, 248}, {297, 2794}, {394, 30270}, {401, 35278}, {418, 23208}, {420, 9862}, {441, 1503}, {468, 10991}, {511, 3506}, {571, 20987}, {574, 26864}, {800, 1974}, {1384, 41424}, {1576, 2393}, {1660, 22401}, {1661, 3053}, {1692, 1976}, {1843, 14575}, {1915, 13357}, {2080, 14673}, {2187, 37586}, {2409, 34156}, {2908, 42447}, {3003, 7669}, {3095, 34986}, {3292, 37916}, {3313, 33801}, {3398, 5943}, {3424, 11348}, {3455, 40352}, {3767, 6620}, {3785, 10565}, {3796, 37479}, {4224, 5337}, {4512, 8235}, {5008, 34417}, {5012, 37335}, {5041, 13366}, {5065, 19459}, {5117, 9873}, {5158, 19153}, {5595, 26953}, {5596, 6389}, {5938, 14961}, {6292, 7499}, {6688, 21513}, {6750, 6756}, {6800, 21163}, {7494, 7800}, {7712, 34099}, {7772, 11402}, {7795, 14826}, {7816, 35926}, {7889, 37439}, {8573, 33578}, {8721, 11206}, {8779, 9475}, {8854, 39648}, {8855, 39679}, {8968, 36709}, {9407, 20975}, {9605, 17809}, {9924, 15905}, {10547, 10551}, {11328, 13335}, {11430, 32444}, {11550, 14003}, {11672, 39072}, {14096, 22352}, {14913, 37893}, {15513, 41275}, {15585, 41008}, {16318, 23976}, {17810, 30435}, {18437, 34776}, {18475, 35934}, {19761, 37052}, {20897, 35007}, {21309, 31860}, {21458, 30737}, {21637, 23635}, {21639, 34569}, {23606, 34750}, {26881, 37184}, {26882, 37114}, {32621, 33871}, {34565, 34571}, {34774, 41005}, {35225, 41580}, {35259, 37344}, {35265, 35296}, {37457, 37512}

X(42671) = reflection of X(3284) in X(1576)
X(42671) = isogonal conjugate of X(35140)
X(42671) = isogonal conjugate of the anticomplement of X(23976)
X(42671) = isogonal conjugate of the isotomic conjugate of X(1503)
X(42671) = isogonal conjugate of the polar conjugate of X(16318)
X(42671) = polar conjugate of the isotomic conjugate of X(8779)
X(42671) = circumcircle-inverse of X(107)-of-1st-Brocard-triangle
X(42671) = X(34129)-complementary conjugate of X(20305)
X(42671) = X(i)-Ceva conjugate of X(j) for these (i,j): {232, 1692}, {248, 32}, {1297, 6}, {1503, 8779}, {21458, 1503}, {32696, 512}, {34156, 23976}
X(42671) = crosspoint of X(i) and X(j) for these (i,j): {6, 1297}, {25, 1976}, {248, 34156}, {685, 23590}, {1503, 16318}, {15384, 32687}, {34135, 41200}, {34136, 41201}
X(42671) = crosssum of X(i) and X(j) for these (i,j): {2, 1503}, {8, 857}, {69, 325}, {122, 39473}, {160, 3289}, {297, 39265}, {5002, 41199}, {5003, 41198}, {6330, 14944}
X(42671) = crossdifference of every pair of points on line {2, 2419}
X(42671) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35140}, {63, 6330}, {69, 8767}, {75, 1297}, {162, 2419}, {336, 39265}, {799, 34212}, {811, 2435}, {1959, 9476}, {3265, 36092}, {3267, 36046}, {14944, 19611}, {15407, 40703}
X(42671) = barycentric product X(i)*X(j) for these {i,j}: {1, 2312}, {3, 16318}, {4, 8779}, {6, 1503}, {19, 8766}, {25, 441}, {32, 30737}, {39, 21458}, {67, 28343}, {74, 6793}, {98, 9475}, {111, 35282}, {132, 248}, {232, 34156}, {512, 34211}, {520, 23977}, {525, 2445}, {647, 2409}, {822, 24024}, {1297, 23976}, {1976, 15595}, {2435, 15639}, {3172, 16096}, {6103, 40080}, {32713, 39473}
X(42671) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35140}, {25, 6330}, {32, 1297}, {441, 305}, {647, 2419}, {669, 34212}, {1503, 76}, {1973, 8767}, {1976, 9476}, {2211, 39265}, {2312, 75}, {2409, 6331}, {2445, 648}, {3049, 2435}, {3172, 14944}, {6793, 3260}, {8766, 304}, {8779, 69}, {9475, 325}, {14600, 15407}, {16318, 264}, {21458, 308}, {23976, 30737}, {23977, 6528}, {28343, 316}, {30737, 1502}, {34211, 670}, {35282, 3266}
X(42671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {51, 34396, 5007}, {110, 37183, 36212}, {157, 206, 216}, {184, 3148, 39}, {1495, 5191, 187}, {1974, 40947, 800}, {7669, 18374, 3003}, {11206, 37188, 8721}, {36212, 37183, 18860}

X(42672) = X(16)X(627)∩X(17)X(76)

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

X(42672) lies on the Kiepert circumhyperbola of the Brocard triangle and these lines: {2, 5469}, {3, 623}, {5, 33383}, {16, 627}, {17, 76}, {141, 22737}, {182, 3642}, {302, 39554}, {532, 599}, {636, 16629}, {1352, 22687}, {3314, 22508}, {3618, 22683}, {3643, 24206}, {3734, 42673}, {5965, 22685}, {6673, 11311}, {9743, 16652}, {11300, 13084}, {11307, 16967}, {18582, 20377}, {22911, 37341}, {33225, 42489}, {34508, 36759}

X(42672) = midpoint of X(627) and X(22907)
X(42672) = reflection of X(22737) in X(141)
X(42672) = X(17)-of-Brocard-triangle
X(42672) = 1st-Brocard-isogonal conjugate of X(42674)
X(42672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14144, 36782}, {3107, 11132, 35689}

X(42673) = X(15)X(628)∩X(18)X(76)

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

X(42673) lies on the Kiepert circumhyperbola of the Brocard triangle and these lines: {2, 5470}, {3, 624}, {5, 33382}, {15, 628}, {18, 76}, {141, 22736}, {182, 3643}, {303, 39555}, {533, 599}, {635, 16628}, {1352, 22689}, {3314, 22506}, {3618, 22685}, {3642, 24206}, {3734, 42672}, {5965, 22683}, {6674, 11312}, {9743, 16653}, {11299, 13083}, {11308, 16966}, {18581, 20378}, {22866, 37340}, {33225, 42488}, {34509, 36760}

X(42673) = midpoint of X(628) and X(22861)
X(42673) = reflection of X(22736) in X(141)
X(42673) = X(18)-of-Brocard-triangle
X(42673) = 1st-Brocard-isogonal conjugate of X(42675)
X(42673) = {X(3106),X(11133)}-harmonic conjugate of X(35688)

X(42674) = X(2)X(5470)∩X(4)X(623)

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

X(42674) lies on these lines: {2, 5470}, {3, 33421}, {4, 623}, {6, 22527}, {76, 36759}, {182, 2782}, {384, 36760}, {622, 636}, {629, 16967}, {1078, 30559}, {1352, 3642}, {3552, 30560}, {3643, 24206}, {3934, 13349}, {6295, 25157}, {7697, 33389}, {7816, 13350}, {10614, 31712}, {11185, 36252}, {18582, 37178}, {22683, 39561}, {22708, 24273}, {22737, 22907}

X(42674) = X(61)-of-Brocard-triangle
X(42674) = 1st-Brocard-isogonal conjugate of X(42672)
X(42674) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {182, 3734, 42675}, {3734, 22687, 22689}, {25157, 35917, 6295}

X(42675) = X(2)X(5469)∩X(4)X(624)

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

X(42675) lies on these lines: {2, 5469}, {3, 33420}, {4, 624}, {6, 22526}, {76, 36760}, {182, 2782}, {384, 36759}, {621, 635}, {630, 16966}, {1078, 30560}, {1352, 3643}, {3552, 30559}, {3642, 24206}, {3934, 13350}, {6582, 25167}, {7697, 33388}, {7816, 13349}, {10613, 31711}, {11185, 36251}, {18581, 37177}, {22685, 39561}, {22707, 24273}, {22736, 22861}

X(42675) = X(62)-of-Brocard-triangle
X(42675) = 1st-Brocard-isogonal conjugate of X(42673)
X(42675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {182, 3734, 42674}, {3734, 22689, 22687}, {25167, 35918, 6582}

leftri

Centers on cubic K588: X(42676)-X(42681)

rightri

This preamble and centers X(42676)-X(42681) were contributed by César Eliud Lozada, April 19, 2021.

The three internal bisectors AI, BI, CI of ABC are rotated about each corresponding vertex of ABC of a same angle θ, all outwardly or all inwardly. The six rotated bisectors define a triangle NaNbNc which is perspective to ABC at a point P. The locus of P is K588. (Reference: Bernard Gibert, CTC K588)

For a given angle θ, the perspector P(θ), here denoted by K588(θ), has barycentrics coordinates:

  P(θ) = a*sin(A/2 - θ)/sin(A/2 + θ) : :

or, equivalently,

  P(t) = a*((a+b+c)*(-a+b+c)*t-S*(1-t^2))/((a+b+c)*(-a+b+c)*t+S*(1-t^2)) : :, where t = tan(θ/2)

Some perspectors K588(θ) are shown in the following table:

θ -π/2 = -90° -5π/12 = -75° -π/3 = -60° -π/4 = -45° -π/6 = -30° -π/12 = -15° 0 π/12 = 15° π/6 = 30° π/4 = 45° π/3 = 60° 5π/12 = 75° π/2 = 90°
K588(θ) X(1) X(42676) X(42677) X(6212) X(39150) X(42678) X(1) X(42679) X(39151) X(6213) X(42680) X(42681) X(1)

Note: P(θ) and P(-θ) are isogonal conjugates.

X(42676) = K588(-5 π/12 = -75°)

Barycentrics    a*(-2*S+(2+sqrt(3))*(a+b+c)*(a+b-c))*(-2*S+(2+sqrt(3))*(a+b+c)*(a-b+c))*(2*S+(2+sqrt(3))*(-a+b+c)*(a+b+c)) : :
Barycentrics    a*sin(A/2 - 5*Pi/12)/sin(A/2 + 5*Pi/12) : :

X(42676) lies on the cubic K588 and these lines: {1, 3389}, {10, 17}, {58, 7968}, {3302, 33655}

X(42676) = isogonal conjugate of X(42681)
X(42676) = trilinear product X(i)*X(j) for these {i, j}: {17, 3299}, {62, 3302}
X(42676) = trilinear quotient X(i)/X(j) for these (i, j): (17, 3300), (62, 3301), (303, 32792)
X(42676) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3367)}} and {{A, B, C, X(10), X(42678)}}
X(42676) = X(i)-isoconjugate-of-X(j) for these {i, j}: {18, 3301}, {61, 3300}

X(42677) = K588(π/3 = -60°)

Barycentrics    a*(2*S+(-a+b+c)*(a+b+c)*sqrt(3))*(-2*S+(a+b+c)*(a-b+c)*sqrt(3))*(-2*S+(a+b+c)*(a+b-c)*sqrt(3)) : :
Barycentrics    a*sin(A/2 - Pi/3)/sin(A/2 + Pi/3) : :

X(42677) lies on the cubics K261a, K588, K1146 and these lines: {1, 15}, {3, 42623}, {10, 13}, {14, 79}, {16, 3647}, {35, 6104}, {37, 2160}, {58, 11072}, {62, 6191}, {100, 2381}, {202, 7059}, {532, 3578}, {553, 1081}, {559, 7005}, {846, 2952}, {3383, 19551}, {5237, 11789}, {5240, 5267}, {11142, 42616}, {37830, 40693}

X(42677) = isogonal conjugate of X(42680)
X(42677) = barycentric product X(i)*X(j) for these {i, j}: {299, 11072}, {2306, 40714}
X(42677) = barycentric quotient X(i)/X(j) for these (i, j): (2152, 5353), (2174, 7006), (2306, 554)
X(42677) = trilinear product X(i)*X(j) for these {i, j}: {13, 5357}, {79, 7005}, {559, 1251}, {1081, 10638}
X(42677) = trilinear quotient X(i)/X(j) for these (i, j): (16, 5353), (35, 7006), (559, 1082), (1081, 554), (1251, 33653), (2153, 11073)
X(42677) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(14)}} and {{A, B, C, X(10), X(39150)}}
X(42677) = X(i)-isoconjugate-of-X(j) for these {i, j}: {14, 5353}, {79, 7006}, {554, 1250}, {1082, 33653}
X(42677) = X(2152)-reciprocal conjugate of-X(5353)
X(42677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 2306, 39153), (1, 3179, 2306), (37, 3579, 42680), (1251, 2306, 1), (1251, 3179, 39153)

X(42678) = K588(-π/12 = -15°)

Barycentrics    a*(2*S-(sqrt(3)-2)*(-a+b+c)*(a+b+c))*(2*S+(a+b+c)*(a-b+c)*(sqrt(3)-2))*(2*S+(a+b+c)*(a+b-c)*(sqrt(3)-2)) : :
Barycentrics    a*sin(A/2 - Pi/12)/sin(A/2 + Pi/12) : :

X(42678) lies on the cubic K588 and these lines: {1, 3364}, {10, 18}, {58, 7968}, {3302, 7052}

X(42678) = isogonal conjugate of X(42679)
X(42678) = trilinear product X(i)*X(j) for these {i, j}: {18, 3299}, {61, 3302}
X(42678) = trilinear quotient X(i)/X(j) for these (i, j): (18, 3300), (61, 3301), (302, 32792)
X(42678) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3392)}} and {{A, B, C, X(10), X(42676)}}
X(42678) = X(i)-isoconjugate-of-X(j) for these {i, j}: {17, 3301}, {62, 3300}

X(42679) = K588(π/12 = 15°)

Barycentrics    a*(-2*S+(a+b+c)*(a+b-c)*(sqrt(3)-2))*(2*S+(sqrt(3)-2)*(-a+b+c)*(a+b+c))*(-2*S+(a+b+c)*(a-b+c)*(sqrt(3)-2)) : :
Barycentrics    a*sin(A/2+Pi/12)/sin(A/2-Pi/12) : :

X(42679) lies on the cubic K588 and these lines: {1, 3390}, {10, 17}, {58, 7969}, {3300, 33655}

X(42679) = isogonal conjugate of X(42678)
X(42679) = trilinear product X(i)*X(j) for these {i, j}: {17, 3301}, {62, 3300}
X(42679) = trilinear quotient X(i)/X(j) for these (i, j): (17, 3302), (62, 3299), (303, 32791)
X(42679) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3366)}} and {{A, B, C, X(10), X(42681)}}
X(42679) = X(i)-isoconjugate-of-X(j) for these {i, j}: {18, 3299}, {61, 3302}

X(42680) = K588(π/3 = 60°)

Barycentrics    a*(2*S-sqrt(3)*(-a+b+c)*(a+b+c))*(2*S+sqrt(3)*(a+b+c)*(a-b+c))*(2*S+sqrt(3)*(a+b+c)*(a+b-c)) : :
Barycentrics    a*sin(A/2 + Pi/3)/sin(A/2 -Pi/3) : :

X(42680) lies on the cubics K261b, K588, K1146 and these lines: {1, 16}, {10, 14}, {13, 79}, {15, 3647}, {35, 6105}, {37, 2160}, {58, 11073}, {61, 6192}, {100, 2380}, {203, 7060}, {533, 3578}, {553, 554}, {846, 2953}, {1082, 7006}, {3376, 7126}, {5238, 11752}, {5239, 5267}, {21311, 42624}, {37833, 40694}

X(42680) = isogonal conjugate of X(42677)
X(42680) = barycentric product X(298)*X(11073)
X(42680) = barycentric quotient X(i)/X(j) for these (i, j): (2151, 5357), (2174, 7005), (2307, 559)
X(42680) = trilinear product X(i)*X(j) for these {i, j}: {14, 5353}, {79, 7006}, {554, 1250}, {1082, 33653}
X(42680) = trilinear quotient X(i)/X(j) for these (i, j): (15, 5357), (35, 7005), (554, 1081), (1082, 559), (1250, 10638), (2154, 11072)
X(42680) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(13)}} and {{A, B, C, X(10), X(39151)}}
X(42680) = X(i)-isoconjugate-of-X(j) for these {i, j}: {13, 5357}, {79, 7005}, {559, 1251}, {1081, 10638}
X(42680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 33654, 39152), (1, 41225, 33654), (14, 2154, 39150), (37, 3579, 42677), (33653, 33654, 1), (33653, 41225, 39152)

X(42681) = K588(5 π/12 = 75°)

Barycentrics    a*(2*S+(2+sqrt(3))*(a+b+c)*(a+b-c))*(2*S+(2+sqrt(3))*(a+b+c)*(a-b+c))*(2*S-(2+sqrt(3))*(-a+b+c)*(a+b+c)) : :
Barycentrics    a*sin(A/2+5*Pi/12)/sin(A/2-5*Pi/12) : :

X(42681) lies on the cubic K588 and these lines: {1, 3365}, {10, 18}, {58, 7969}, {3300, 7052}

X(42681) = isogonal conjugate of X(42676)
X(42681) = trilinear product X(i)*X(j) for these {i, j}: {18, 3301}, {61, 3300}
X(42681) = trilinear quotient X(i)/X(j) for these (i, j): (18, 3302), (61, 3299), (302, 32791)
X(42681) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3391)}} and {{A, B, C, X(10), X(42679)}}
X(42681) = X(i)-isoconjugate-of-X(j) for these {i, j}: {17, 3299}, {62, 3302}

leftri

Gibert (i,j,k) points on cubics K1206a and K1206b: X(42682)-X(42695)

rightri

This preamble and points X(42682)-X(42695) are contributed by Peter Moses, April 19, 2021

See
K1206

Gibert points are introduced in the preamble just before X(42085)



X(42682) = GIBERT (5,-7,2) POINT

Barycentrics    (5*a^2*S)/Sqrt[3] + 2*a^2*SA - 14*SB*SC : :

X(42682) lies on the cubic K1206a and these lines:{5, 42630}, {6, 17578}, {14, 16}, {15, 3858}, {398, 42104}, {631, 42087}, {632, 16809}, {1656, 5349}, {3091, 23302}, {3522, 23303}, {3534, 42513}, {3627, 42613}, {3843, 18582}, {3859, 16772}, {3860, 12821}, {3861, 34754}, {5071, 11480}, {5076, 5318}, {5238, 41989}, {5334, 42165}, {5339, 42109}, {5343, 42097}, {5365, 11481}, {10646, 42531}, {10654, 35403}, {11485, 41119}, {12101, 42520}, {12812, 42122}, {12817, 15713}, {14093, 42089}, {15692, 42139}, {15693, 42090}, {15694, 42130}, {15695, 42129}, {15696, 18581}, {15712, 33416}, {15714, 37835}, {16773, 42112}, {16960, 42117}, {16964, 42102}, {17538, 42096}, {19709, 42103}, {35407, 42131}, {35434, 42105}, {41099, 42110}, {42099, 42599}, {42146, 42530}, {42514, 42517}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 17578, 42683}, {5321, 19107, 42088}, {5321, 42108, 395}, {5349, 42085, 42107}, {19107, 42088, 42108}, {36970, 42136, 5321}, {42093, 42164, 23302}, {42101, 42126, 42147}

X(42683) = GIBERT (5,7,-2) POINT

Barycentrics    (5*a^2*S)/Sqrt[3] - 2*a^2*SA + 14*SB*SC : :

X(42683) lies on the cubic K1206b and these lines:{5, 42629}, {6, 17578}, {13, 15}, {16, 3858}, {397, 42105}, {631, 42088}, {632, 16808}, {1656, 5350}, {3091, 23303}, {3522, 23302}, {3534, 42512}, {3627, 42612}, {3843, 18581}, {3859, 16773}, {3860, 12820}, {3861, 34755}, {5071, 11481}, {5076, 5321}, {5237, 41989}, {5335, 42164}, {5340, 42108}, {5344, 42096}, {5366, 11480}, {10645, 42530}, {10653, 35403}, {11486, 41120}, {12101, 42521}, {12812, 42123}, {12816, 15713}, {14093, 42092}, {15692, 42142}, {15693, 42091}, {15694, 42131}, {15695, 42132}, {15696, 18582}, {15712, 33417}, {15714, 37832}, {16772, 42113}, {16961, 42118}, {16965, 42101}, {17538, 42097}, {19709, 42106}, {35407, 42130}, {35434, 42104}, {41099, 42107}, {42100, 42598}, {42143, 42531}, {42515, 42516}

X(42683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 17578, 42682}, {5318, 19106, 42087}, {5318, 42109, 396}, {5350, 42086, 42110}, {19106, 42087, 42109}, {36969, 42137, 5318}, {42094, 42165, 23303}, {42102, 42127, 42148}

X(42684) = GIBERT (5,-1,10) POINT

Barycentrics    (5*a^2*S)/Sqrt[3] + 10*a^2*SA - 2*SB*SC : :

X(42684) lies on the cubic K1206b and these lines:{4, 11480}, {6, 10304}, {14, 549}, {15, 548}, {395, 15698}, {396, 3534}, {485, 42168}, {486, 42167}, {547, 42630}, {3526, 5321}, {3628, 5352}, {3856, 16966}, {3857, 19107}, {5055, 42085}, {5066, 36967}, {5072, 42092}, {5238, 5318}, {5349, 33417}, {7486, 42107}, {8703, 34754}, {10303, 42119}, {10646, 15759}, {10654, 15706}, {11001, 42518}, {11488, 15683}, {11489, 15717}, {11540, 42143}, {12103, 16960}, {14890, 37835}, {15022, 42093}, {15640, 16644}, {15684, 42102}, {15709, 42154}, {16241, 23046}, {16772, 17800}, {16808, 33699}, {19710, 33607}, {34200, 34755}, {36836, 42088}, {37640, 42508}, {42099, 42598}, {42111, 42500}, {42118, 42529}, {42124, 42434}, {42140, 42490}

X(42684) = {X(6),X(10304)}-harmonic conjugate of X(42685)
X(42684) = {X(16772),X(42090)}-harmonic conjugate of X(42109)

X(42685) = GIBERT (5,1,-10) POINT

Barycentrics    (5*a^2*S)/Sqrt[3] - 10*a^2*SA + 2*SB*SC : :

X(42685) lies on the cubic K1206a and these lines:{4, 11481}, {6, 10304}, {13, 549}, {16, 548}, {395, 3534}, {396, 15698}, {485, 42170}, {486, 42169}, {547, 42629}, {3526, 5318}, {3628, 5351}, {3856, 16967}, {3857, 19106}, {5055, 42086}, {5066, 36968}, {5072, 42089}, {5237, 5321}, {5350, 33416}, {7486, 42110}, {8703, 34755}, {10303, 42120}, {10645, 15759}, {10653, 15706}, {11001, 42519}, {11488, 15717}, {11489, 15683}, {11540, 42146}, {12103, 16961}, {14890, 37832}, {15022, 42094}, {15640, 16645}, {15684, 42101}, {15709, 42155}, {16242, 23046}, {16773, 17800}, {16809, 33699}, {19710, 33606}, {34200, 34754}, {36843, 42087}, {37641, 42509}, {42100, 42599}, {42114, 42501}, {42117, 42528}, {42121, 42433}, {42141, 42491}

X(42685) = {X(16773),X(42091)}-harmonic conjugate of X(42108)

X(42686) = GIBERT (-5,1,10) POINT

Barycentrics    (-5*a^2*S)/Sqrt[3] + 10*a^2*SA + 2*SB*SC : :

X(42686) lies on the cubic K1206b and these lines:{4, 11481}, {5, 42629}, {6, 15717}, {13, 11540}, {16, 396}, {395, 10304}, {548, 10646}, {3526, 18582}, {3530, 34755}, {3534, 5321}, {3628, 5237}, {3856, 19106}, {3857, 16967}, {5054, 42512}, {5055, 42089}, {5066, 16242}, {5072, 42086}, {5335, 15709}, {5349, 42628}, {5351, 15704}, {6560, 42167}, {6561, 42168}, {7486, 42120}, {8703, 16961}, {10303, 23302}, {10653, 42502}, {11480, 15698}, {11485, 15706}, {11489, 42164}, {12100, 42521}, {12108, 16960}, {15022, 42165}, {15640, 42139}, {15683, 16645}, {15684, 42129}, {15685, 42513}, {15713, 33607}, {16966, 42501}, {17800, 18581}, {23046, 36968}, {33699, 42100}, {37832, 42505}, {37835, 42584}, {41100, 42627}, {41121, 42492}, {42091, 42599}, {42118, 42499}, {42136, 42528}, {42143, 42433}, {42145, 42489}, {42500, 42510}

X(42685) = {X(6),X(10304)}-harmonic conjugate of X(42684)
X(42686) = {X(10646),X(16773)}-harmonic conjugate of X(42087)

X(42687) = GIBERT (5,1,10) POINT

Barycentrics    (5*a^2*S)/Sqrt[3] + 10*a^2*SA + 2*SB*SC : :

X(42687) lies on the cubic K1206a and these lines:{4, 11480}, {5, 42630}, {6, 15717}, {14, 11540}, {15, 395}, {396, 10304}, {548, 10645}, {3526, 18581}, {3530, 34754}, {3534, 5318}, {3628, 5238}, {3856, 19107}, {3857, 16966}, {5054, 42513}, {5055, 42092}, {5066, 16241}, {5072, 42085}, {5334, 15709}, {5350, 42627}, {5352, 15704}, {6560, 42169}, {6561, 42170}, {7486, 42119}, {8703, 16960}, {10303, 23303}, {10654, 42503}, {11481, 15698}, {11486, 15706}, {11488, 42165}, {12100, 42520}, {12108, 16961}, {15022, 42164}, {15640, 42142}, {15683, 16644}, {15684, 42132}, {15685, 42512}, {15713, 33606}, {16967, 42500}, {17800, 18582}, {23046, 36967}, {33699, 42099}, {37832, 42585}, {37835, 42504}, {41101, 42628}, {41122, 42493}, {42090, 42598}, {42117, 42498}, {42137, 42529}, {42144, 42488}, {42146, 42434}, {42501, 42511}

X(42687) = {X(10645),X(16772)}-harmonic conjugate of X(42088)

X(42688) = GIBERT (10,-8,5) POINT

Barycentrics    (10*a^2*S)/Sqrt[3] + 5*a^2*SA - 16*SB*SC : :

X(42688) lies on the cubic K1206a and these lines:{4, 11408}, {6, 15684}, {381, 42512}, {395, 3534}, {548, 11489}, {549, 42125}, {3526, 5321}, {3628, 42119}, {3830, 42520}, {3857, 42132}, {5055, 16241}, {5066, 42133}, {5072, 23302}, {5334, 15704}, {7486, 42135}, {10303, 42122}, {11486, 16964}, {11488, 23046}, {14269, 34754}, {15640, 42118}, {15683, 42144}, {15685, 34755}, {15695, 33606}, {15706, 23303}, {15709, 42143}, {15717, 42129}, {42086, 42164}, {42095, 42498}

X(42689) = GIBERT (10,8,-5) POINT

Barycentrics    (10*a^2*S)/Sqrt[3] - 5*a^2*SA + 16*SB*SC : :

X(42689) lies on the cubic K1206b and these lines:{4, 11409}, {6, 15684}, {381, 42513}, {396, 3534}, {548, 11488}, {549, 42128}, {3526, 5318}, {3628, 42120}, {3830, 42521}, {3857, 42129}, {5055, 16242}, {5066, 42134}, {5072, 23303}, {5335, 15704}, {7486, 42138}, {10303, 42123}, {11485, 16965}, {11489, 23046}, {14269, 34755}, {15640, 42117}, {15683, 42145}, {15685, 34754}, {15695, 33607}, {15706, 23302}, {15709, 42146}, {15717, 42132}, {42085, 42165}, {42098, 42499}

X(42690) = GIBERT (-10,8,5) POINT

Barycentrics    (-10*a^2*S)/Sqrt[3] + 5*a^2*SA + 16*SB*SC : :

X(42690) lies on the cubic K1206a and these lines:{4, 11409}, {14, 5055}, {16, 15684}, {548, 42126}, {549, 42119}, {632, 42415}, {3526, 18581}, {3534, 5321}, {3628, 5334}, {3830, 12821}, {3857, 42128}, {5066, 42139}, {5072, 18582}, {5335, 23046}, {5339, 33416}, {5343, 42628}, {7486, 42143}, {10303, 42117}, {10304, 42130}, {11480, 41122}, {11489, 15704}, {15022, 42132}, {15640, 42123}, {15683, 42136}, {15706, 42089}, {17800, 19107}, {33602, 42138}, {42088, 42159}, {42141, 42497}

X(42691) = GIBERT (10,8,5) POINT

Barycentrics    (10*a^2*S)/Sqrt[3] + 5*a^2*SA + 16*SB*SC : :

X(42691) lies on the cubic K1206b and these lines:{4, 11408}, {13, 5055}, {15, 15684}, {548, 42127}, {549, 42120}, {632, 42416}, {3526, 18582}, {3534, 5318}, {3628, 5335}, {3830, 12820}, {3857, 42125}, {5066, 42142}, {5072, 18581}, {5334, 23046}, {5340, 33417}, {5344, 42627}, {7486, 42146}, {10303, 42118}, {10304, 42131}, {11481, 41121}, {11488, 15704}, {15022, 42129}, {15640, 42122}, {15683, 42137}, {15706, 42092}, {17800, 19106}, {33603, 42135}, {42087, 42162}, {42140, 42496}

X(42692) = GIBERT (-5,7,2) POINT

Barycentrics    (-5*a^2*S)/Sqrt[3] + 2*a^2*SA + 14*SB*SC : :

X(42692) lies on the cubic K1206a and these lines:{5, 15}, {6, 3839}, {14, 14893}, {376, 23303}, {395, 3830}, {396, 41106}, {398, 42103}, {549, 42630}, {1657, 5349}, {3146, 11489}, {3523, 42087}, {3525, 42095}, {3853, 16961}, {3854, 5339}, {3857, 16960}, {3860, 16808}, {5054, 42085}, {5066, 34754}, {5318, 42159}, {5334, 42166}, {5343, 42098}, {5365, 11480}, {10124, 10645}, {10646, 12817}, {11543, 12102}, {12100, 36970}, {12101, 42629}, {12103, 19107}, {12108, 16967}, {15687, 34755}, {15703, 42111}, {16773, 42104}, {21734, 42140}, {33923, 42136}, {35408, 42429}, {36836, 42473}, {37835, 42144}, {41973, 42627}, {41989, 42415}, {42109, 42153}, {42121, 42531}, {42489, 42585}, {42499, 42632}

X(42692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5321, 16809, 23302}, {5321, 42107, 42147}, {5349, 18581, 42108}, {16809, 23302, 42107}, {42093, 42163, 42088}, {42101, 42125, 395}

X(42693) = GIBERT (5,7,2) POINT

Barycentrics    (5*a^2*S)/Sqrt[3] + 2*a^2*SA + 14*SB*SC : :

X(42693) lies on the cubic K1206b and these lines:{5, 16}, {6, 3839}, {13, 14893}, {376, 23302}, {395, 41106}, {396, 3830}, {397, 42106}, {549, 42629}, {1657, 5350}, {3146, 11488}, {3523, 42088}, {3525, 42098}, {3853, 16960}, {3854, 5340}, {3857, 16961}, {3860, 16809}, {5054, 42086}, {5066, 34755}, {5321, 42162}, {5335, 42163}, {5344, 42095}, {5366, 11481}, {10124, 10646}, {10645, 12816}, {11542, 12102}, {12100, 36969}, {12101, 42630}, {12103, 19106}, {12108, 16966}, {15687, 34754}, {15703, 42114}, {16772, 42105}, {21734, 42141}, {33923, 42137}, {35408, 42430}, {36843, 42472}, {37832, 42145}, {41974, 42628}, {41989, 42416}, {42108, 42156}, {42124, 42530}, {42488, 42584}, {42498, 42631}

X(42693) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5318, 16808, 23303}, {5318, 42110, 42148}, {5350, 18582, 42109}, {16808, 23303, 42110}, {42094, 42166, 42087}, {42102, 42128, 396}

X(42694) = GIBERT (-15,32,5) POINT

Barycentrics    -5*Sqrt[3]*a^2*S + 5*a^2*SA + 64*SB*SC : :

X(24694) lies on the cubic K1206a and these lines:{4, 14}, {5, 42630}, {18, 33699}, {548, 16809}, {3146, 12821}, {3526, 42093}, {3534, 42489}, {3628, 42432}, {3839, 42520}, {3856, 16964}, {3857, 42147}, {5055, 5352}, {5066, 5349}, {5072, 36970}, {5237, 15684}, {5365, 16267}, {10646, 17800}, {12817, 23046}, {15687, 33606}, {15704, 37835}, {15717, 19107}, {41988, 42521}, {42096, 42477}, {42103, 42488}, {42107, 42498}, {42121, 42433}, {42495, 42631}

X(42695) = GIBERT (15,32,5) POINT

Barycentrics    5*Sqrt[3]*a^2*S + 5*a^2*SA + 64*SB*SC : :

X(42695) lies on the cubic K1206b and these lines:{4, 13}, {5, 42629}, {17, 33699}, {548, 16808}, {3146, 12820}, {3526, 42094}, {3534, 42488}, {3628, 42431}, {3839, 42521}, {3856, 16965}, {3857, 42148}, {5055, 5351}, {5066, 5350}, {5072, 36969}, {5238, 15684}, {5366, 16268}, {10645, 17800}, {12816, 23046}, {15687, 33607}, {15704, 37832}, {15717, 19106}, {41988, 42520}, {42097, 42476}, {42106, 42489}, {42110, 42499}, {42124, 42434}, {42494, 42632}

X(42696) = ISOTOMIC CONJUGATE OF X(3296)

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

X(42696) lies on these lines: {1, 4464}, {2, 594}, {4, 33941}, {6, 4399}, {7, 8}, {9, 4431}, {10, 3875}, {37, 27480}, {80, 4986}, {81, 19825}, {86, 145}, {141, 17119}, {142, 4060}, {144, 17346}, {183, 7172}, {190, 391}, {192, 966}, {193, 4363}, {239, 2345}, {264, 7046}, {269, 25719}, {280, 20477}, {307, 36889}, {313, 4441}, {314, 1000}, {321, 14555}, {326, 3872}, {329, 4886}, {332, 37728}, {344, 2321}, {345, 5271}, {346, 17277}, {347, 33298}, {350, 3974}, {491, 32793}, {492, 32794}, {497, 33931}, {519, 10436}, {524, 7231}, {536, 17257}, {599, 4478}, {894, 1992}, {956, 1444}, {1007, 3705}, {1058, 33939}, {1086, 3620}, {1211, 30699}, {1213, 17318}, {1219, 7268}, {1265, 3596}, {1266, 4668}, {1267, 32806}, {1278, 1654}, {1387, 4561}, {2324, 28827}, {2551, 33938}, {2968, 40680}, {3008, 4058}, {3161, 17335}, {3187, 19822}, {3226, 26076}, {3241, 17394}, {3247, 24603}, {3305, 42032}, {3434, 17163}, {3436, 21273}, {3578, 20078}, {3589, 4405}, {3598, 37671}, {3616, 4460}, {3617, 3672}, {3619, 3661}, {3621, 3945}, {3625, 3664}, {3626, 3663}, {3632, 3879}, {3644, 17256}, {3662, 21356}, {3679, 4357}, {3681, 21867}, {3686, 3729}, {3687, 5219}, {3706, 30758}, {3707, 25728}, {3739, 17299}, {3758, 7229}, {3759, 5749}, {3763, 4395}, {3790, 38057}, {3886, 30331}, {3911, 11679}, {3912, 4007}, {3943, 17259}, {3946, 17308}, {3996, 16992}, {4034, 4416}, {4043, 28809}, {4046, 37703}, {4072, 25072}, {4329, 5178}, {4346, 17273}, {4359, 18141}, {4373, 39710}, {4389, 4452}, {4393, 28604}, {4398, 17271}, {4402, 16706}, {4420, 7269}, {4440, 4821}, {4454, 17347}, {4470, 17379}, {4472, 16884}, {4643, 4686}, {4644, 11008}, {4648, 4699}, {4664, 5296}, {4669, 17274}, {4675, 4739}, {4688, 4851}, {4690, 4726}, {4727, 31238}, {4740, 6646}, {4748, 17247}, {4751, 5308}, {4764, 17258}, {4772, 17300}, {4798, 4910}, {4816, 4888}, {4852, 17303}, {4869, 17295}, {4873, 25101}, {4916, 17391}, {4980, 5905}, {5015, 32006}, {5082, 5195}, {5084, 33932}, {5222, 17289}, {5257, 17133}, {5295, 5722}, {5391, 32805}, {5554, 24547}, {5736, 20013}, {5739, 17484}, {5750, 16834}, {7081, 34229}, {7222, 17364}, {7228, 40341}, {8025, 20046}, {9780, 17322}, {10327, 26234}, {10444, 11362}, {10446, 12245}, {14213, 26872}, {14552, 32939}, {14828, 20015}, {15668, 17388}, {16685, 27640}, {16815, 17242}, {16816, 17280}, {16833, 17353}, {16969, 28252}, {17014, 17381}, {17135, 30962}, {17142, 25291}, {17184, 19819}, {17228, 37756}, {17229, 17278}, {17234, 29616}, {17239, 17301}, {17240, 29627}, {17245, 17309}, {17246, 17251}, {17262, 17330}, {17264, 18230}, {17268, 29628}, {17269, 17337}, {17281, 17348}, {17282, 29594}, {17296, 24199}, {17302, 29593}, {17310, 27147}, {17311, 34824}, {17319, 28635}, {17327, 17395}, {17332, 20073}, {17350, 37654}, {17352, 24599}, {17354, 37681}, {17358, 29590}, {17365, 20080}, {17373, 26806}, {17378, 31145}, {17383, 29591}, {17396, 29610}, {18228, 42034}, {18697, 33937}, {19789, 32782}, {19804, 34255}, {19826, 33146}, {20012, 37632}, {20050, 41847}, {20879, 26871}, {20947, 26105}, {21020, 33088}, {21085, 33144}, {21271, 21279}, {21868, 28358}, {23897, 27556}, {24048, 24058}, {24963, 31012}, {26038, 30963}, {26042, 26801}, {26048, 26107}, {26774, 27192}, {27484, 41325}, {28329, 28639}, {28641, 29580}, {29016, 36706}, {30828, 33077}, {31130, 33090}, {32791, 32813}, {32792, 32812}, {32797, 32811}, {32798, 32810}, {32800, 32807}, {35550, 36500}, {38000, 42049}

X(42696) = isotomic conjugate of X(3296)
X(42696) = anticomplement of the isogonal conjugate of X(25417)
X(42696) = isotomic conjugate of the isogonal conjugate of X(3295)
X(42696) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {58, 30562}, {8652, 514}, {25417, 8}, {28625, 1654}, {30590, 21291}, {30598, 69}, {32042, 20295}, {34819, 192}, {37211, 513}, {42030, 3436}
X(42696) = X(3697)-cross conjugate of X(3305)
X(42696) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3296}, {1973, 30679}
X(42696) = cevapoint of X(8) and X(41915)
X(42696) = crosspoint of X(4998) and X(32042)
X(42696) = crosssum of X(3271) and X(4834)
X(42696) = barycentric product X(i)*X(j) for these {i,j}: {7, 42032}, {75, 3305}, {76, 3295}, {274, 3697}, {312, 7190}, {4917, 40014}
X(42696) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3296}, {69, 30679}, {3295, 6}, {3305, 1}, {3697, 37}, {4917, 1743}, {7190, 57}, {42032, 8}
X(42696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 8, 319}, {7, 319, 69}, {8, 75, 69}, {8, 31995, 32099}, {8, 32087, 75}, {10, 3875, 17321}, {69, 75, 42697}, {75, 319, 7}, {75, 320, 31995}, {75, 5564, 8}, {75, 17360, 7321}, {75, 20930, 20880}, {142, 4060, 17294}, {239, 2345, 3618}, {320, 32099, 69}, {391, 4461, 190}, {594, 4361, 2}, {594, 17366, 17293}, {894, 5839, 1992}, {894, 29617, 5839}, {1086, 4445, 3620}, {1278, 1654, 4419}, {2321, 4384, 344}, {2345, 4371, 239}, {3008, 4058, 17286}, {3616, 4460, 17393}, {3616, 5936, 28653}, {3617, 3672, 5224}, {3621, 3945, 17377}, {3626, 3663, 17270}, {3632, 25590, 3879}, {3661, 4000, 3619}, {3661, 17117, 4000}, {3679, 17151, 4357}, {3739, 17299, 17316}, {4034, 4659, 4416}, {4361, 17293, 17366}, {4363, 17362, 193}, {4389, 32025, 5232}, {4399, 4665, 6}, {4402, 29611, 16706}, {4452, 4678, 5232}, {4452, 5232, 4389}, {4460, 5936, 3616}, {4478, 7263, 599}, {4644, 17363, 11008}, {4678, 5232, 32025}, {4690, 4726, 17276}, {4699, 6542, 4648}, {4739, 17372, 4675}, {4751, 17315, 5308}, {4772, 20055, 17300}, {4821, 17343, 4440}, {4852, 17303, 26626}, {4886, 42029, 329}, {5224, 17160, 3672}, {5564, 32087, 69}, {7321, 17360, 21296}, {15668, 17388, 29585}, {16816, 17280, 37650}, {17116, 17363, 4644}, {17229, 17278, 29579}, {17245, 17309, 29583}, {17281, 17348, 26685}, {17293, 17366, 2}, {17360, 21296, 69}, {17393, 28653, 3616}, {31995, 32099, 320}, {34255, 41915, 19804}, {36928, 36929, 388}

X(42697) = ISOTOMIC CONJUGATE OF X(1000)

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

X(42697) lies on this lines: {1, 1266}, {2, 45}, {4, 33940}, {6, 4395}, {7, 8}, {9, 4480}, {10, 4887}, {38, 24445}, {63, 9816}, {79, 15434}, {81, 19789}, {86, 3445}, {141, 17118}, {142, 344}, {144, 17277}, {145, 17160}, {183, 3598}, {192, 4648}, {193, 4361}, {239, 1992}, {264, 1119}, {274, 957}, {307, 8797}, {312, 9776}, {314, 3296}, {321, 18141}, {326, 7190}, {329, 19804}, {332, 16137}, {333, 9965}, {341, 11024}, {345, 5249}, {346, 17234}, {347, 20477}, {348, 22464}, {350, 30947}, {374, 41772}, {376, 39552}, {391, 17347}, {443, 1265}, {491, 32794}, {492, 32793}, {497, 4459}, {519, 4896}, {524, 4405}, {527, 3707}, {536, 4675}, {537, 24693}, {553, 11679}, {573, 29382}, {594, 3620}, {599, 4665}, {870, 9432}, {894, 3618}, {940, 30699}, {948, 40862}, {966, 4699}, {982, 24463}, {988, 1125}, {1000, 20569}, {1001, 24280}, {1007, 7179}, {1043, 11036}, {1056, 20924}, {1213, 17255}, {1227, 17165}, {1267, 32805}, {1268, 36606}, {1278, 17300}, {1376, 21320}, {1443, 4861}, {1444, 37227}, {1447, 34229}, {1633, 26241}, {1654, 4772}, {1698, 4357}, {1958, 18162}, {1964, 25570}, {1996, 5231}, {1997, 5437}, {2321, 17298}, {2345, 3619}, {2481, 34919}, {2551, 33944}, {2796, 24331}, {3161, 17263}, {3187, 19819}, {3210, 5712}, {3241, 36588}, {3244, 3664}, {3306, 4054}, {3421, 33934}, {3434, 17140}, {3474, 3757}, {3475, 32932}, {3589, 7231}, {3616, 17320}, {3617, 17271}, {3623, 3945}, {3633, 3879}, {3644, 17317}, {3661, 21356}, {3685, 38053}, {3687, 4654}, {3717, 38052}, {3739, 17257}, {3753, 20925}, {3758, 5222}, {3759, 4402}, {3763, 7227}, {3826, 27549}, {3834, 17281}, {3872, 17079}, {3883, 4312}, {3886, 5542}, {3912, 4659}, {3943, 17313}, {3980, 33144}, {4009, 30758}, {4029, 28301}, {4307, 32922}, {4310, 5263}, {4359, 5905}, {4371, 17363}, {4399, 40341}, {4418, 29638}, {4429, 7613}, {4431, 17296}, {4441, 29824}, {4461, 4869}, {4488, 17336}, {4569, 6063}, {4618, 36944}, {4643, 4688}, {4664, 5308}, {4667, 4982}, {4670, 17301}, {4673, 11037}, {4676, 16020}, {4679, 30796}, {4686, 4727}, {4691, 17270}, {4726, 17299}, {4738, 32097}, {4739, 17275}, {4740, 6542}, {4741, 24699}, {4747, 17014}, {4748, 17254}, {4751, 5296}, {4764, 17315}, {4795, 16666}, {4798, 41311}, {4821, 17375}, {4859, 17353}, {4886, 41915}, {4899, 38200}, {4902, 4967}, {4911, 32006}, {4947, 24456}, {5057, 26234}, {5224, 32089}, {5232, 17273}, {5278, 20078}, {5391, 8243}, {5554, 17895}, {5695, 25557}, {5698, 16823}, {5739, 17483}, {5744, 30608}, {5749, 16706}, {5750, 17304}, {5839, 11008}, {5902, 20894}, {5936, 39707}, {5949, 27704}, {6172, 17335}, {6356, 40680}, {6666, 25728}, {6687, 17278}, {6745, 40719}, {7172, 37671}, {7229, 17289}, {7240, 18194}, {7359, 27509}, {7736, 33891}, {8817, 21436}, {8822, 37113}, {9780, 17250}, {10444, 31730}, {10446, 29309}, {10527, 41804}, {14213, 26871}, {14829, 21454}, {15668, 17246}, {16064, 24822}, {16670, 41140}, {16709, 17183}, {16815, 17333}, {16816, 20072}, {16825, 24695}, {17132, 29571}, {17133, 29605}, {17146, 21283}, {17164, 35550}, {17169, 30939}, {17170, 39731}, {17184, 19822}, {17227, 29611}, {17235, 17303}, {17236, 28604}, {17245, 17262}, {17249, 28653}, {17259, 17334}, {17261, 27147}, {17264, 29627}, {17265, 17340}, {17279, 31243}, {17282, 17355}, {17286, 21255}, {17297, 29616}, {17318, 17392}, {17322, 25581}, {17323, 17398}, {17344, 28634}, {17349, 31300}, {17350, 37650}, {17362, 20080}, {17377, 20014}, {17740, 27757}, {17776, 27186}, {17868, 18161}, {17889, 29861}, {17891, 18207}, {18135, 39995}, {19276, 39544}, {19785, 29833}, {19825, 32782}, {20054, 32093}, {20195, 25101}, {20533, 36494}, {20568, 21290}, {20879, 26872}, {20905, 20927}, {20917, 40875}, {21061, 29747}, {24165, 26098}, {24248, 24325}, {24334, 36538}, {24357, 41842}, {24589, 31018}, {24616, 35596}, {24715, 31178}, {24723, 39581}, {24789, 26065}, {24833, 36474}, {26040, 32937}, {26042, 26149}, {26229, 37762}, {26245, 37540}, {26842, 28605}, {28827, 40880}, {28968, 37800}, {29069, 36698}, {30035, 34830}, {30589, 37635}, {30598, 39709}, {30712, 39710}, {32025, 33800}, {32791, 32812}, {32792, 32813}, {32797, 32810}, {32798, 32811}, {32799, 32807}, {34255, 42029}, {35171, 35175}

X(42697) = isogonal conjugate of X(34446)
X(42697) = isotomic conjugate of X(1000)
X(42697) = isotomic conjugate of the anticomplement of X(40587)
X(42697) = isotomic conjugate of the isogonal conjugate of X(999)
X(42697) = complement of X(20073)
X(42697) = anticomplement of X(45)
X(42697) = anticomplement of the isogonal conjugate of X(89)
X(42697) = anticomplement of the isotomic conjugate of X(20569)
X(42697) = polar conjugate of the isogonal conjugate of X(22129)
X(42697) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2, 21291}, {6, 17488}, {58, 30564}, {89, 8}, {649, 39364}, {2163, 2}, {2320, 329}, {2364, 144}, {4588, 514}, {4597, 20295}, {4604, 513}, {5385, 3952}, {5549, 4468}, {20569, 6327}, {28607, 192}, {28658, 1654}, {30588, 1330}, {30608, 3436}, {34073, 17494}, {39428, 17310}, {39704, 69}, {40833, 21282}
X(42697) = X(i)-Ceva conjugate of X(j) for these (i,j): {20569, 2}, {20925, 28808}
X(42697) = X(i)-cross conjugate of X(j) for these (i,j): {3306, 17079}, {3753, 3306}, {3872, 28808}, {4054, 20925}, {40587, 2}
X(42697) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34446}, {31, 1000}, {604, 36916}, {1973, 30680}
X(42697) = cevapoint of X(i) and X(j) for these (i,j): {999, 22129}, {3241, 5744}, {3306, 3872}, {3753, 4054}
X(42697) = crosspoint of X(4597) and X(4998)
X(42697) = crosssum of X(3271) and X(4775)
X(42697) = crossdifference of every pair of points on line {1960, 3063}
X(42697) = barycentric product X(i)*X(j) for these {i,j}: {1, 20925}, {7, 28808}, {8, 17079}, {75, 3306}, {76, 999}, {85, 3872}, {86, 4054}, {190, 21183}, {264, 22129}, {274, 3753}, {1231, 17519}, {3261, 35281}, {20569, 40587}, {30608, 36595}, {36919, 40833}
X(42697) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1000}, {6, 34446}, {8, 36916}, {69, 30680}, {999, 6}, {3306, 1}, {3672, 14556}, {3753, 37}, {3872, 9}, {4054, 10}, {4997, 36596}, {17079, 7}, {17519, 1172}, {20925, 75}, {21183, 514}, {22129, 3}, {28808, 8}, {35281, 101}, {36595, 5219}, {36914, 36920}, {36919, 4908}, {40587, 45}
X(42697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4346, 4389}, {2, 4440, 4419}, {2, 4454, 190}, {2, 20073, 45}, {7, 8, 320}, {7, 75, 69}, {7, 31598, 1122}, {7, 31995, 75}, {7, 32087, 21296}, {8, 320, 69}, {10, 4887, 17274}, {45, 31139, 34824}, {45, 31244, 31285}, {45, 34824, 2}, {69, 75, 42696}, {75, 85, 3262}, {75, 319, 32087}, {75, 320, 8}, {75, 7321, 7}, {75, 17361, 5564}, {75, 20930, 20895}, {86, 4398, 3672}, {142, 3729, 344}, {190, 29451, 29505}, {192, 26806, 4648}, {239, 4644, 1992}, {319, 21296, 69}, {391, 20059, 17347}, {594, 7232, 3620}, {894, 4000, 3618}, {903, 4389, 4346}, {1086, 4363, 2}, {1086, 17369, 17290}, {1278, 17300, 17314}, {2345, 3662, 3619}, {3306, 4054, 28808}, {3662, 17116, 2345}, {3663, 10436, 17321}, {3672, 4373, 4398}, {3739, 17276, 17257}, {3758, 37756, 5222}, {3834, 17281, 29579}, {3943, 17313, 29583}, {3945, 4452, 4360}, {4000, 7222, 894}, {4359, 5905, 14555}, {4361, 17365, 193}, {4363, 17290, 17369}, {4461, 4869, 17233}, {4472, 17325, 2}, {4488, 18230, 17336}, {4659, 6173, 3912}, {4665, 7238, 599}, {4670, 17301, 26626}, {4699, 6646, 966}, {4726, 17376, 17299}, {4739, 17345, 17275}, {4751, 17258, 5296}, {4862, 25590, 4357}, {4888, 17151, 3879}, {5222, 35578, 3758}, {5564, 17361, 32099}, {5839, 17364, 11008}, {7228, 7263, 6}, {7321, 31995, 69}, {16816, 20072, 37654}, {17117, 17364, 5839}, {17160, 17378, 145}, {17254, 29576, 4748}, {17278, 17351, 26685}, {17290, 17369, 2}, {17318, 17392, 29585}, {17320, 41847, 3616}, {17354, 27191, 2}, {17361, 32099, 69}, {17740, 31019, 30828}, {20331, 28363, 27638}, {20331, 30958, 2}, {21296, 32087, 319}, {24594, 30566, 2}, {24715, 31178, 36479}, {24894, 25659, 2}, {26039, 26104, 2}, {26769, 26817, 2}, {26976, 27107, 2}, {27037, 27160, 2}, {27266, 27316, 2}, {31244, 31285, 2}, {31285, 34824, 31244}

leftri

Points on the line X(2)X(37): X(42698)-X(42724)

rightri

This preamble and points X(42698-X(42724) are contributed by Peter Moses, April 21, 2021

Suppose that P' = p' : q' : r' is a point on a line p x + q y + r z = 0 and that u x + v y + w z = 0 is a line, L. Then the point P'' = (p/u)*p' : (q/v)*q' + (r/w)*r' (p/u)*p' : (q/v)*q' + (r/w)*r' lies on L For example, if P' is on the Euler line and L is the line X(2)X(37), then P'' is on L. Points X(42698)-X(42715) are obtained in this manner from the Euler line, where, in the same order, P' = X(i) for i = 5, 20, 24, 186, 237, 297, 378, 404, 405, 406, 407, 447, 451, 458, 461, 468, 469, 475. Points X(42716)-X(42724) are obtained from points P' on the line at infinity, with indices in this order: 30, 511, 515, 516, 518, 524, 528, 674, 1499.



X(42698) = X(2)X(37)∩X(68)X(72)

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

X(42698) lies on these lines: {2, 37}, {5, 14213}, {68, 72}, {306, 18588}, {343, 18695}, {7283, 17515}, {18027, 27801}, {18855, 41013}, {23120, 23130}

X(42698) = crosspoint of X(i) and X(j) for these (i,j): {18695, 28706}, {20336, 27801}
X(42698) = X(i)-isoconjugate of X(j) for these (i,j): {28, 2148}, {54, 1474}, {58, 8882}, {275, 2206}, {608, 35196}, {649, 933}, {1333, 2190}, {1919, 18831}, {2167, 2203}, {2169, 5317}, {6591, 36134}, {7649, 14586}, {8747, 14533}
X(42698) = barycentric product X(i)*X(j) for these {i,j}: {5, 20336}, {10, 18695}, {37, 28706}, {72, 311}, {216, 27801}, {304, 21011}, {305, 21807}, {306, 14213}, {321, 343}, {324, 3998}, {668, 6368}, {906, 15415}, {1332, 18314}, {1953, 40071}, {2618, 4561}, {6386, 15451}, {16697, 28654}
X(42698) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 28}, {10, 2190}, {37, 8882}, {51, 2203}, {53, 5317}, {71, 2148}, {72, 54}, {78, 35196}, {100, 933}, {216, 1333}, {306, 2167}, {311, 286}, {313, 40440}, {321, 275}, {343, 81}, {668, 18831}, {906, 14586}, {1331, 36134}, {1332, 18315}, {1953, 1474}, {2618, 7649}, {3682, 2169}, {3990, 14533}, {3998, 97}, {4064, 2616}, {5562, 1437}, {6332, 39177}, {6335, 16813}, {6368, 513}, {7069, 2299}, {12077, 6591}, {14213, 27}, {14391, 14399}, {15451, 667}, {16697, 593}, {17434, 22383}, {18314, 17924}, {18695, 86}, {20336, 95}, {21011, 19}, {21807, 25}, {27801, 276}, {28706, 274}, {30493, 1408}, {35307, 32674}, {35442, 18210}, {41013, 8884}
X(42698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {321, 20336, 3998}

X(42699) = X(2)X(37)∩X(20)X(3198)

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

X(42699) lies on these lines: {2, 37}, {20, 3198}, {72, 15740}, {100, 34168}, {306, 1231}, {440, 20235}, {1105, 4219}, {1259, 23661}, {4183, 7283}

X(42699) = X(i)-isoconjugate of X(j) for these (i,j): {27, 33581}, {28, 2155}, {58, 41489}, {64, 1474}, {459, 2206}, {649, 1301}, {2184, 2203}, {2204, 8809}, {5317, 19614}, {8747, 14642}
X(42699) = barycentric product X(i)*X(j) for these {i,j}: {20, 20336}, {72, 14615}, {304, 8804}, {305, 3198}, {306, 18750}, {321, 37669}, {610, 40071}, {668, 8057}, {1231, 27382}, {1562, 4601}, {3710, 33673}, {3718, 5930}, {3998, 15466}, {4561, 17898}, {6335, 20580}, {6386, 42658}, {15905, 27801}
X(42699) = barycentric quotient X(i)/X(j) for these {i,j}: {20, 28}, {37, 41489}, {71, 2155}, {72, 64}, {100, 1301}, {122, 18210}, {154, 2203}, {228, 33581}, {306, 2184}, {307, 8809}, {321, 459}, {610, 1474}, {1249, 5317}, {1562, 3125}, {1895, 8747}, {3198, 25}, {3682, 19614}, {3694, 30457}, {3718, 5931}, {3990, 14642}, {3998, 1073}, {5379, 15384}, {5930, 34}, {6587, 6591}, {7070, 2299}, {8057, 513}, {8804, 19}, {14308, 18344}, {14345, 14399}, {14615, 286}, {15905, 1333}, {17898, 7649}, {18623, 1396}, {18750, 27}, {20336, 253}, {20580, 905}, {27382, 1172}, {30456, 608}, {35602, 1437}, {36908, 1435}, {37669, 81}, {40933, 1398}, {41013, 6526}, {41086, 7151}, {42658, 667}
X(42699) = {X(321),X(3998)}-harmonic conjugate of X(20336)

X(42700) = X(2)X(37)∩X(54)X(72)

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

X(42700) lies on these lines: {2, 37}, {24, 1748}, {54, 72}, {63, 21078}, {100, 1824}, {190, 31631}, {228, 3425}, {254, 5552}, {306, 16577}, {908, 22001}, {1214, 3936}, {1708, 22021}, {1737, 2901}, {1812, 3219}, {1867, 11681}, {3187, 8609}, {3262, 17479}, {3694, 3969}, {4416, 16585}, {4552, 40149}, {4567, 18879}, {5016, 37528}, {5278, 40937}, {5294, 25078}, {6745, 22027}, {7283, 11103}, {11517, 26377}, {18607, 32859}, {20227, 26747}, {20305, 27725}, {20769, 21367}, {22000, 22003}, {22002, 22020}, {25083, 32933}, {25252, 27287}

X(42700) = crosspoint of X(4567) and X(6335)
X(42700) = crosssum of X(3125) and X(22383)
X(42700) = X(i)-isoconjugate of X(j) for these (i,j): {27, 2351}, {28, 1820}, {58, 2165}, {68, 1474}, {91, 1333}, {513, 36145}, {514, 32734}, {649, 925}, {1790, 14593}, {2168, 18180}, {2206, 5392}, {17167, 41271}, {21102, 32692}
X(42700) = barycentric product X(i)*X(j) for these {i,j}: {24, 20336}, {37, 7763}, {47, 313}, {72, 317}, {100, 6563}, {306, 1748}, {321, 1993}, {571, 27801}, {668, 924}, {3998, 11547}, {4570, 17881}, {6386, 34952}, {9723, 41013}, {18605, 28654}, {27808, 34948}
X(42700) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 91}, {24, 28}, {37, 2165}, {47, 58}, {52, 18180}, {71, 1820}, {72, 68}, {100, 925}, {101, 36145}, {228, 2351}, {313, 20571}, {317, 286}, {321, 5392}, {571, 1333}, {692, 32734}, {924, 513}, {1147, 1437}, {1748, 27}, {1824, 14593}, {1993, 81}, {6335, 30450}, {6563, 693}, {6753, 6591}, {7763, 274}, {8745, 5317}, {9723, 1444}, {14397, 14399}, {17881, 21207}, {18605, 593}, {20336, 20563}, {30451, 22383}, {34948, 3733}, {34952, 667}, {41013, 847}
X(42700) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 3998, 321}

X(42701) = X(2)X(37)∩X(100)X(842)

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

X(42701) lies on these lines: {2, 37}, {63, 24048}, {72, 3431}, {100, 842}, {249, 1931}, {908, 22003}, {1468, 22836}, {2174, 3219}, {3218, 4053}, {3578, 16585}, {3936, 18593}, {3969, 16577}, {4062, 16598}, {4467, 7265}, {7283, 13746}, {16553, 21376}, {25080, 41809}, {25083, 39767}

X(42701) = crosssum of X(3125) and X(14399)
X(42701) = trilinear pole of line {526, 32679}
X(42701) = crossdifference of every pair of points on line {667, 6186}
X(42701) = X(i)-isoconjugate of X(j) for these (i,j): {58, 1989}, {79, 34079}, {86, 11060}, {94, 2206}, {265, 1474}, {476, 649}, {513, 32678}, {514, 14560}, {667, 32680}, {759, 2160}, {1333, 2166}, {1790, 18384}, {1919, 35139}, {3122, 39295}, {4556, 15475}, {6186, 24624}, {6591, 36061}, {7649, 32662}, {11075, 14158}, {18653, 40355}, {22383, 36129}
X(42701) = barycentric product X(i)*X(j) for these {i,j}: {35, 35550}, {37, 7799}, {50, 27801}, {72, 340}, {100, 3268}, {186, 20336}, {190, 32679}, {313, 6149}, {319, 758}, {320, 3678}, {321, 323}, {526, 668}, {1978, 2624}, {2088, 4601}, {2245, 33939}, {3218, 3969}, {3219, 3936}, {3998, 14165}, {4036, 10411}, {4053, 34016}, {4420, 41804}, {4511, 40999}, {4585, 7265}, {6335, 8552}, {6386, 14270}, {16577, 32851}, {18593, 42033}
X(42701) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 2166}, {35, 759}, {37, 1989}, {50, 1333}, {72, 265}, {100, 476}, {101, 32678}, {186, 28}, {190, 32680}, {213, 11060}, {319, 14616}, {321, 94}, {323, 81}, {340, 286}, {526, 513}, {668, 35139}, {692, 14560}, {758, 79}, {906, 32662}, {1154, 18180}, {1331, 36061}, {1824, 18384}, {1897, 36129}, {2088, 3125}, {2174, 34079}, {2245, 2160}, {2594, 1411}, {2624, 649}, {3219, 24624}, {3268, 693}, {3678, 80}, {3724, 6186}, {3936, 30690}, {3969, 18359}, {4036, 10412}, {4053, 8818}, {4420, 6740}, {4511, 3615}, {4567, 39295}, {4705, 15475}, {6126, 14158}, {6149, 58}, {7206, 15065}, {7799, 274}, {8552, 905}, {9126, 30234}, {14270, 667}, {16186, 18210}, {16577, 2006}, {20336, 328}, {22115, 1437}, {27801, 20573}, {32679, 514}, {34397, 2203}, {34834, 18609}, {35550, 20565}, {39149, 30602}, {39495, 4164}, {40999, 18815}, {41013, 6344}

X(42702) = X(2)X(37)∩X(647)X(656)

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

X(42702) lies on these lines: {2, 37}, {48, 184}, {213, 40799}, {232, 240}, {237, 1755}, {647, 656}, {828, 22057}, {1101, 19622}, {2198, 41526}, {4053, 8607}

X(42702) = isotomic conjugate of the polar conjugate of X(5360)
X(42702) = crosspoint of X(295) and X(1214)
X(42702) = crosssum of X(i) and X(j) for these (i,j): {242, 1172}, {6531, 36120}
X(42702) = crossdifference of every pair of points on line {28, 667}
X(42702) = X(i)-isoconjugate of X(j) for these (i,j): {27, 98}, {28, 1821}, {58, 16081}, {81, 36120}, {86, 6531}, {286, 1910}, {287, 8747}, {290, 1474}, {336, 5317}, {514, 685}, {649, 22456}, {693, 36104}, {2966, 7649}, {3122, 41174}, {3261, 32696}, {4025, 20031}, {6591, 36036}, {17924, 36084}
X(42702) = barycentric product X(i)*X(j) for these {i,j}: {37, 36212}, {69, 5360}, {71, 1959}, {72, 511}, {100, 684}, {213, 6393}, {228, 325}, {232, 3998}, {237, 20336}, {240, 3682}, {297, 3990}, {306, 1755}, {321, 3289}, {668, 39469}, {692, 6333}, {906, 2799}, {1332, 3569}, {3949, 17209}, {4055, 40703}, {4064, 23997}, {4567, 41172}, {9417, 40071}
X(42702) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 16081}, {42, 36120}, {71, 1821}, {72, 290}, {100, 22456}, {213, 6531}, {228, 98}, {237, 28}, {511, 286}, {684, 693}, {692, 685}, {906, 2966}, {1331, 36036}, {1755, 27}, {2200, 1910}, {2211, 5317}, {2491, 6591}, {3289, 81}, {3569, 17924}, {3682, 336}, {3990, 287}, {4055, 293}, {4567, 41174}, {5360, 4}, {6333, 40495}, {6393, 6385}, {9417, 1474}, {9418, 2203}, {20336, 18024}, {32656, 36084}, {32739, 36104}, {36212, 274}, {39469, 513}, {41172, 16732}

X(42703) = X(2)X(37)∩X(100)X(2857)

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

X(42703) lies on these lines: {2, 37}, {100, 2857}, {297, 40703}, {327, 27801}, {422, 4601}, {668, 5641}, {850, 4036}, {1330, 21595}, {1824, 34405}, {2901, 3905}, {3454, 21421}, {4053, 35551}, {5360, 20022}, {13485, 20553}, {17864, 40071}, {18022, 40363}, {20932, 30660}

X(42703) = X(i)-isoconjugate of X(j) for these (i,j): {27, 14600}, {58, 1976}, {86, 14601}, {98, 2206}, {248, 1474}, {293, 2203}, {649, 2715}, {667, 36084}, {1333, 1910}, {1459, 32696}, {1919, 2966}, {1980, 36036}, {2422, 4556}, {15628, 16947}, {22383, 36104}
X(42703) = barycentric product X(i)*X(j) for these {i,j}: {240, 40071}, {297, 20336}, {306, 40703}, {313, 1959}, {321, 325}, {511, 27801}, {668, 2799}, {868, 4601}, {1502, 5360}, {2396, 4036}, {3569, 6386}, {6333, 6335}, {6393, 41013}
X(42703) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 1910}, {37, 1976}, {72, 248}, {100, 2715}, {190, 36084}, {213, 14601}, {228, 14600}, {232, 2203}, {240, 1474}, {297, 28}, {306, 293}, {313, 1821}, {321, 98}, {325, 81}, {511, 1333}, {668, 2966}, {684, 22383}, {868, 3125}, {1755, 2206}, {1783, 32696}, {1897, 36104}, {1959, 58}, {1978, 36036}, {2491, 1980}, {2799, 513}, {3569, 667}, {3701, 15628}, {3998, 17974}, {4036, 2395}, {4463, 11610}, {4705, 2422}, {5360, 32}, {6333, 905}, {6335, 685}, {6393, 1444}, {6530, 5317}, {15523, 3404}, {16230, 6591}, {16591, 1428}, {17209, 849}, {20336, 287}, {21833, 15630}, {27801, 290}, {36212, 1437}, {40071, 336}, {40703, 27}, {41013, 6531}

X(42704) = X(2)X(37)∩X(72)X(74)

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

X(42704) lies on these lines: {2, 37}, {72, 74}, {306, 18593}, {518, 11322}, {1150, 25083}, {1214, 3969}, {3306, 3970}, {3681, 37400}, {3694, 3936}, {3811, 5264}, {4527, 16598}, {5295, 33089}, {7283, 17584}, {24048, 24611}, {27801, 40832}

X(42704) = crossdifference of every pair of points on line {667, 14399}
X(42704) = X(i)-isoconjugate of X(j) for these (i,j): {58, 34288}, {513, 36149}, {514, 32738}, {649, 1302}, {1474, 4846}, {2206, 34289}, {11125, 32681}, {14399, 36083}
X(42704) = barycentric product X(i)*X(j) for these {i,j}: {37, 32833}, {100, 30474}, {321, 15066}, {378, 20336}, {668, 8675}, {5063, 27801}, {6386, 42660}
X(42704) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 34288}, {72, 4846}, {100, 1302}, {101, 36149}, {321, 34289}, {378, 28}, {692, 32738}, {5063, 1333}, {5891, 18180}, {8675, 513}, {15066, 81}, {30474, 693}, {32833, 274}, {42660, 667}

X(42705) = X(2)X(37)∩X(72)X(3784)

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

X(42705) lies on these lines: {2, 37}, {72, 3784}, {190, 1817}, {306, 4466}, {404, 32939}, {1265, 37180}, {1812, 4561}, {3264, 18662}, {4552, 30713}, {7283, 37168}, {18141, 22021}

X(42705) = barycentric product X(i)*X(j) for these {i,j}: {306, 32939}, {404, 20336}, {4563, 21721}
X(42705) = barycentric quotient X(i)/X(j) for these {i,j}: {404, 28}, {21721, 2501}, {32939, 27}
X(42705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3998, 20336, 345}

X(42706) = X(2)X(37)∩X(72)X(306)

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

X(42706) lies on these lines: {2, 37}, {27, 7283}, {72, 306}, {304, 18607}, {405, 5271}, {1104, 3187}, {1259, 11679}, {1999, 16050}, {2901, 40940}, {3151, 7270}, {3159, 20106}, {3198, 10327}, {3694, 40161}, {3933, 18651}, {4463, 32862}, {14021, 34255}, {21062, 22020}, {27801, 40447}

X(42706) = isotomic conjugate of the polar conjugate of X(5295)
X(42706) = X(i)-isoconjugate of X(j) for these (i,j): {28, 2215}, {649, 36077}
X(42706) = barycentric product X(i)*X(j) for these {i,j}: {69, 5295}, {306, 5271}, {405, 20336}, {1264, 1882}
X(42706) = barycentric quotient X(i)/X(j) for these {i,j}: {71, 2215}, {100, 36077}, {405, 28}, {1882, 1118}, {3694, 2335}, {4574, 36080}, {5271, 27}, {5295, 4}, {5320, 2203}, {23882, 17925}, {37543, 1396}, {39585, 8747}
X(42706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {321, 17776, 37}, {345, 20336, 3998}, {440, 3695, 306}

X(42707) = X(2)X(37)∩X(25)X(32929)

Barycentrics    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) : :

X(42707) lies on these lines: {2, 37}, {25, 32929}, {304, 5905}, {306, 21062}, {405, 3702}, {429, 3695}, {1230, 1441}, {1824, 10327}, {3685, 37325}, {4082, 22027}, {4239, 32932}, {7283, 16049}, {17016, 41813}, {17742, 22001}, {20932, 33066}, {26689, 27643}

X(42707) = barycentric product X(i)*X(j) for these {i,j}: {313, 12514}, {321, 5739}, {406, 20336}, {27174, 28654}, {27801, 36744}
X(42707) = barycentric quotient X(i)/X(j) for these {i,j}: {406, 28}, {5739, 81}, {12514, 58}, {27174, 593}, {36744, 1333}

X(42708) = X(2)X(37)∩X(10)X(41501)

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

X(42708) lies on these lines: {2, 37}, {10, 41501}, {72, 1478}, {92, 17275}, {125, 21692}, {226, 4053}, {329, 21873}, {338, 21690}, {407, 21677}, {594, 6354}, {1109, 8013}, {1211, 16732}, {1834, 4647}, {1848, 22007}, {2901, 30143}, {4036, 23930}, {4046, 17874}, {4415, 21810}, {4665, 20237}, {5743, 20236}, {5745, 22003}, {7211, 14973}, {17056, 18698}, {21020, 40967}, {21029, 39245}, {21096, 24044}, {24048, 28609}, {30690, 31143}

X(42708) = X(18698)-Ceva conjugate of X(21674)
X(42708) = crosspoint of X(321) and X(6358)
X(42708) = crosssum of X(1333) and X(2150)
X(42708) = X(i)-isoconjugate of X(j) for these (i,j): {1333, 40430}, {2150, 17097}, {2189, 40442}
X(42708) = barycentric product X(i)*X(j) for these {i,j}: {10, 18698}, {75, 21674}, {313, 2650}, {321, 17056}, {349, 21811}, {407, 20336}, {1089, 3664}, {1441, 21677}, {1577, 22003}, {2646, 34388}, {4033, 23755}, {4036, 17136}, {5745, 6358}
X(42708) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 40430}, {12, 17097}, {201, 40442}, {407, 28}, {2646, 60}, {2650, 58}, {3664, 757}, {5745, 2185}, {6737, 1098}, {17056, 81}, {18698, 86}, {21674, 1}, {21677, 21}, {21748, 2150}, {21811, 284}, {22003, 662}, {23755, 1019}, {30604, 4833}, {40950, 270}
X(42708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {321, 27569, 42034}

X(42709) = X(2)X(37)∩X(190)X(21368)

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

X(42709) lies on these lines: {2, 37}, {190, 21368}, {304, 40075}, {306, 2064}, {341, 5086}, {668, 14206}, {693, 15416}, {976, 4385}, {1008, 32931}, {1089, 5293}, {3145, 7283}, {3729, 36572}, {3936, 17789}, {3952, 14956}, {4033, 20920}, {4647, 36499}, {4673, 36500}, {7035, 31905}, {7081, 36559}, {11679, 36504}, {17788, 33077}, {18750, 21286}, {21600, 35516}, {27538, 37193}, {32932, 36497}, {33136, 36568}

X(42709) = X(i)-isoconjugate of X(j) for these (i,j): {32, 16099}, {1919, 35169}
X(42709) = barycentric product X(i)*X(j) for these {i,j}: {75, 16086}, {447, 20336}, {867, 7035}, {6386, 42662}
X(42709) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 16099}, {447, 28}, {668, 35169}, {867, 244}, {16086, 1}, {20336, 40715}, {42662, 667}
X(42709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {306, 2064, 20929}, {321, 4358, 37759}, {321, 32779, 75}

X(42710) = X(2)X(37)∩X(304)X(32859)

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

X(42710) lies on these lines: {2, 37}, {304, 32859}, {1230, 35550}, {1325, 7283}, {2895, 20932}, {3695, 30447}, {3701, 6757}, {3702, 5259}, {3948, 20896}, {4053, 32858}, {4115, 17781}, {4601, 31614}, {14210, 42045}, {18697, 41809}, {20988, 32929}, {21810, 32782}

X(42710) = isotomic conjugate of X(40143)
X(42710) = isotomic conjugate of the isogonal conjugate of X(21873)
X(42710) = X(i)-Ceva conjugate of X(j) for these (i,j): {20932, 21081}, {28654, 321}
X(42710) = X(2895)-cross conjugate of X(321)
X(42710) = crosspoint of X(4601) and X(27808)
X(42710) = X(i)-isoconjugate of X(j) for these (i,j): {31, 40143}, {58, 3444}, {267, 1333}, {849, 21353}, {1029, 2206}
X(42710) = barycentric product X(i)*X(j) for these {i,j}: {10, 20932}, {75, 21081}, {76, 21873}, {191, 313}, {321, 2895}, {451, 20336}, {1030, 27801}, {3701, 41808}, {4033, 21192}, {6386, 42653}, {21723, 24037}, {27808, 31947}, {28654, 40592}
X(42710) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40143}, {10, 267}, {37, 3444}, {191, 58}, {321, 1029}, {451, 28}, {501, 849}, {594, 21353}, {1030, 1333}, {1089, 502}, {2895, 81}, {6757, 30602}, {8614, 1408}, {20932, 86}, {21081, 1}, {21192, 1019}, {21723, 2643}, {21873, 6}, {22136, 1437}, {31947, 3733}, {40592, 593}, {41808, 1014}, {42653, 667}
X(42710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {321, 27705, 4359}

X(42711) = X(2)X(37)∩X(72)X(290)

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

X(42711) lies on these lines: {2, 37}, {65, 1237}, {72, 290}, {183, 3403}, {210, 35544}, {313, 3967}, {319, 30660}, {518, 4485}, {1212, 18050}, {1824, 40717}, {3914, 21238}, {16583, 21412}, {16605, 21435}, {20723, 22278}, {21403, 40071}

X(42711) = crosspoint of X(3403) and X(20023)
X(42711) = X(i)-isoconjugate of X(j) for these (i,j): {58, 263}, {81, 3402}, {262, 2206}, {649, 26714}, {1333, 2186}, {17187, 42288}
X(42711) = barycentric product X(i)*X(j) for these {i,j}: {10, 3403}, {37, 20023}, {182, 27801}, {183, 321}, {458, 20336}, {668, 23878}, {3288, 6386}
X(42711) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 2186}, {37, 263}, {42, 3402}, {100, 26714}, {182, 1333}, {183, 81}, {321, 262}, {458, 28}, {3288, 667}, {3403, 86}, {6784, 3121}, {10311, 2203}, {14994, 16696}, {18098, 42288}, {20023, 274}, {20336, 42313}, {23878, 513}, {27801, 327}, {33971, 5317}

X(42712) = X(2)X(37)∩X(9)X(3702)

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

X(42712) lies on these lines: {2, 37}, {9, 3702}, {198, 32929}, {306, 21068}, {391, 4673}, {1089, 3950}, {1108, 26770}, {1441, 4044}, {1449, 4742}, {1696, 5695}, {2171, 21071}, {2321, 3701}, {2324, 27410}, {3247, 4968}, {3294, 3692}, {3686, 3902}, {3986, 4647}, {3992, 4058}, {4007, 4723}, {4066, 4098}, {4072, 4125}, {4082, 21039}, {4696, 17314}, {5179, 21070}, {5279, 37422}, {7101, 41013}, {7283, 37402}, {17751, 21871}

X(42712) = X(4673)-Ceva conjugate of X(4061)
X(42712) = X(i)-isoconjugate of X(j) for these (i,j): {649, 5545}, {1408, 25430}, {1412, 2334}, {5936, 16947}, {7203, 34074}
X(42712) = barycentric product X(i)*X(j) for these {i,j}: {10, 4673}, {75, 4061}, {312, 5257}, {313, 4512}, {318, 4101}, {321, 391}, {341, 3671}, {461, 20336}, {646, 4841}, {668, 4843}, {1449, 30713}, {1577, 30728}, {2321, 19804}, {3596, 37593}, {3616, 3701}, {3699, 4815}, {3710, 5342}, {3952, 4811}, {4033, 4765}, {4047, 7017}, {4258, 27801}, {4801, 30730}, {6386, 8653}
X(42712) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 5545}, {210, 2334}, {391, 81}, {461, 28}, {644, 4627}, {646, 4633}, {1449, 1412}, {2321, 25430}, {3616, 1014}, {3671, 269}, {3699, 4614}, {3701, 5936}, {4033, 4624}, {4047, 222}, {4061, 1}, {4069, 8694}, {4082, 4866}, {4101, 77}, {4258, 1333}, {4512, 58}, {4515, 34820}, {4673, 86}, {4765, 1019}, {4771, 1429}, {4778, 7203}, {4801, 17096}, {4811, 7192}, {4815, 3676}, {4819, 1319}, {4827, 7252}, {4829, 1284}, {4841, 3669}, {4843, 513}, {5257, 57}, {8653, 667}, {14625, 1462}, {19804, 1434}, {30713, 40023}, {30728, 662}, {30730, 4606}, {37593, 56}
X(42712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {321, 22016, 22040}, {4043, 20336, 321}, {22016, 27569, 321}

X(42713) = X(2)X(37)∩X(72)X(5486)

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

X(42713) lies on these lines: {2, 37}, {69, 21873}, {72, 5486}, {100, 2770}, {141, 21810}, {213, 9516}, {228, 40102}, {304, 4643}, {468, 3712}, {523, 1577}, {524, 14210}, {527, 4115}, {668, 18823}, {740, 39688}, {744, 4368}, {908, 22007}, {1213, 18697}, {1279, 4742}, {1654, 20932}, {1930, 4364}, {2325, 22003}, {3230, 9022}, {3663, 24067}, {3701, 4377}, {3718, 4851}, {3834, 24076}, {3912, 4053}, {3936, 16581}, {3948, 16732}, {3954, 42286}, {3985, 8680}, {4026, 4714}, {4035, 18589}, {4039, 21254}, {4062, 16597}, {4078, 4125}, {4118, 22196}, {4363, 33942}, {4436, 20857}, {4553, 5360}, {4567, 4590}, {4647, 17514}, {4708, 20911}, {4971, 4986}, {5277, 24335}, {5695, 19309}, {6542, 26147}, {7267, 16702}, {7283, 11116}, {17251, 33936}, {17296, 24048}, {17318, 33937}, {17444, 30059}, {21076, 27559}, {22017, 24092}, {24058, 24199}, {27801, 40826}

X(42713) = isotomic conjugate of the isogonal conjugate of X(21839)
X(42713) = X(14210)-Ceva conjugate of X(4062)
X(42713) = crosspoint of X(3266) and X(14210)
X(42713) = crosssum of X(923) and X(32740)
X(42713) = crossdifference of every pair of points on line {667, 1333}
X(42713) = X(i)-isoconjugate of X(j) for these (i,j): {27, 14908}, {28, 36060}, {58, 111}, {81, 923}, {86, 32740}, {284, 7316}, {310, 19626}, {513, 36142}, {514, 32729}, {649, 691}, {667, 36085}, {671, 2206}, {892, 1919}, {895, 1474}, {897, 1333}, {1412, 5547}, {1437, 36128}, {1790, 8753}, {4556, 9178}, {4786, 32648}, {6629, 41936}, {30234, 36045}
X(42713) = barycentric product X(i)*X(j) for these {i,j}: {10, 14210}, {37, 3266}, {75, 4062}, {76, 21839}, {100, 35522}, {187, 27801}, {313, 896}, {321, 524}, {351, 6386}, {468, 20336}, {594, 16741}, {668, 690}, {1089, 6629}, {1441, 3712}, {1648, 4601}, {1978, 2642}, {3701, 7181}, {3998, 37778}, {4024, 24039}, {4033, 4750}, {4036, 5468}, {4647, 31013}, {6335, 14417}, {6390, 41013}, {14419, 27808}, {16702, 28654}
X(42713) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 897}, {37, 111}, {42, 923}, {65, 7316}, {71, 36060}, {72, 895}, {100, 691}, {101, 36142}, {126, 16756}, {187, 1333}, {190, 36085}, {210, 5547}, {213, 32740}, {228, 14908}, {321, 671}, {351, 667}, {468, 28}, {524, 81}, {668, 892}, {690, 513}, {692, 32729}, {896, 58}, {922, 2206}, {1648, 3125}, {1649, 14419}, {1824, 8753}, {1826, 36128}, {2205, 19626}, {2482, 16702}, {2642, 649}, {3266, 274}, {3292, 1437}, {3712, 21}, {3908, 32583}, {3952, 5380}, {4024, 23894}, {4036, 5466}, {4062, 1}, {4553, 36827}, {4705, 9178}, {4750, 1019}, {4933, 4653}, {4938, 4658}, {5380, 34574}, {6390, 1444}, {6629, 757}, {7181, 1014}, {7813, 16696}, {9125, 30234}, {11183, 4164}, {14210, 86}, {14273, 6591}, {14417, 905}, {14419, 3733}, {14424, 2530}, {14432, 3737}, {16597, 7292}, {16702, 593}, {16741, 1509}, {20336, 30786}, {21814, 41272}, {21839, 6}, {21906, 3121}, {22105, 18108}, {23106, 16733}, {23889, 4556}, {24038, 6629}, {24039, 4610}, {27801, 18023}, {30595, 4840}, {30605, 4833}, {31013, 40438}, {35522, 693}, {36792, 16741}, {41013, 17983}, {41586, 18180}
X(42713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3948, 35550, 16732}, {27565, 27697, 37}, {27569, 27705, 75}, {27586, 27727, 192}

X(42714) = X(2)X(37)∩X(72)X(319)

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

X(42714) lies on these lines: {2, 37}, {69, 21594}, {72, 319}, {86, 33939}, {313, 1089}, {314, 33775}, {594, 26601}, {1269, 1930}, {1386, 3702}, {1909, 20932}, {1978, 37842}, {2321, 4150}, {2478, 42696}, {2901, 4360}, {3159, 4357}, {3759, 41249}, {3969, 27052}, {4066, 6541}, {5224, 33935}, {5295, 5564}, {5695, 23868}, {7148, 23498}, {17233, 22021}, {18133, 20911}, {20234, 20337}, {21070, 22012}, {22018, 24044}, {25354, 42031}

X(42714) = X(1333)-isoconjugate of X(2214)
X(42714) = barycentric product X(i)*X(j) for these {i,j}: {10, 33935}, {313, 28606}, {321, 5224}, {349, 3876}, {386, 27801}, {469, 20336}, {668, 23879}, {799, 23282}, {1577, 33948}, {3701, 33949}, {6386, 42664}, {14349, 27808}
X(42714) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 2214}, {386, 1333}, {469, 28}, {3876, 284}, {4033, 835}, {5224, 81}, {14349, 3733}, {23282, 661}, {23879, 513}, {27808, 37218}, {28606, 58}, {33935, 86}, {33948, 662}, {33949, 1014}, {42664, 667}
X(42714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 312, 18147}, {75, 20947, 28653}, {321, 20336, 75}, {321, 27569, 37}, {1089, 18697, 313}

X(42715) = X(2)X(37)∩X(72)X(10327)

Barycentrics    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) : :

X(42715) lies on these lines: {2, 37}, {72, 10327}, {305, 27801}, {306, 4006}, {427, 3006}, {442, 1230}, {614, 2901}, {1441, 28654}, {1759, 15487}, {2049, 4968}, {3187, 16502}, {3702, 37060}, {3718, 5905}, {4228, 7283}, {5256, 33937}, {5287, 33942}, {6735, 20238}, {30713, 35550}

X(42715) = barycentric product X(i)*X(j) for these {i,j}: {475, 20336}, {27801, 36743}
X(42715) = barycentric quotient X(i)/X(j) for these {i,j}: {475, 28}, {36743, 1333}

X(42716) = X(2)X(37)∩X(100)X(1302)

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

X(42716) lies on these lines: {2, 37}, {100, 1302}, {190, 15455}, {645, 648}, {2517, 3573}, {3260, 7359}, {4240, 24001}, {4391, 4585}, {4567, 30528}, {4601, 6035}, {6386, 9211}, {15413, 15418}, {18752, 39767}

X(42716) = X(14399)-cross conjugate of X(30)
X(42716) = cevapoint of X(30) and X(14399)
X(42716) = trilinear pole of line {30, 14206}
X(42716) = X(i)-isoconjugate of X(j) for these (i,j): {58, 2433}, {74, 649}, {513, 2159}, {514, 40352}, {667, 2349}, {1459, 8749}, {1474, 14380}, {1494, 1919}, {1980, 33805}, {2206, 2394}, {3120, 32640}, {3125, 36034}, {4025, 40354}, {4466, 32715}, {6591, 35200}, {7649, 18877}, {11125, 40353}, {18210, 36131}, {22383, 36119}
X(42716) = barycentric product X(i)*X(j) for these {i,j}: {30, 668}, {100, 3260}, {190, 14206}, {306, 24001}, {321, 2407}, {646, 6357}, {1495, 6386}, {1637, 4601}, {1784, 4561}, {1978, 2173}, {2420, 27801}, {4033, 18653}, {4240, 20336}, {4554, 7359}, {4567, 41079}, {4600, 36035}, {6335, 11064}, {7035, 11125}, {14399, 31625}
X(42716) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 513}, {37, 2433}, {72, 14380}, {100, 74}, {101, 2159}, {190, 2349}, {321, 2394}, {644, 15627}, {668, 1494}, {692, 40352}, {906, 18877}, {1099, 11125}, {1331, 35200}, {1332, 14919}, {1495, 667}, {1637, 3125}, {1783, 8749}, {1784, 7649}, {1897, 36119}, {1978, 33805}, {1990, 6591}, {2173, 649}, {2407, 81}, {2420, 1333}, {3163, 14399}, {3260, 693}, {3284, 22383}, {4036, 12079}, {4240, 28}, {4570, 36034}, {5379, 1304}, {5380, 9139}, {5642, 14419}, {6335, 16080}, {6357, 3669}, {7359, 650}, {9033, 18210}, {9406, 1919}, {9407, 1980}, {11064, 905}, {11125, 244}, {14206, 514}, {14395, 7117}, {14398, 3121}, {14399, 1015}, {14400, 2170}, {18653, 1019}, {20336, 34767}, {23347, 2203}, {24001, 27}, {35266, 30234}, {36035, 3120}, {41013, 18808}, {41079, 16732}
X(42716) = {X({}),X(1)}-harmonic conjugate of X({}[[1]][[3]])

X(42717) = X(2)X(37)∩X(100)X(110)

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

X(42717) lies on these lines: {2, 37}, {39, 36230}, {69, 24499}, {72, 9513}, {100, 110}, {190, 27805}, {650, 3570}, {4567, 5649}, {6331, 6335}, {7283, 37041}, {18047, 21859}, {18061, 34460}

X(42717) = crosspoint of X(4554) and X(4589)
X(42717) = crosssum of X(i) and X(j) for these (i,j): {513, 4164}, {3063, 4455}
X(42717) = trilinear pole of line {511, 1959}
X(42717) = crossdifference of every pair of points on line {667, 3125}
X(42717) = X(i)-isoconjugate of X(j) for these (i,j): {27, 878}, {58, 2395}, {86, 2422}, {98, 649}, {248, 7649}, {290, 1919}, {293, 6591}, {513, 1910}, {514, 1976}, {667, 1821}, {879, 1474}, {1459, 6531}, {2715, 3120}, {2966, 3122}, {3121, 36036}, {3125, 36084}, {3261, 14601}, {3404, 18108}, {4107, 34238}, {4466, 32696}, {4610, 15630}, {18210, 36104}, {22383, 36120}
X(42717) = barycentric product X(i)*X(j) for these {i,j}: {37, 2396}, {72, 877}, {100, 325}, {190, 1959}, {237, 6386}, {240, 4561}, {297, 1332}, {313, 23997}, {321, 2421}, {511, 668}, {670, 5360}, {1331, 40703}, {1755, 1978}, {1783, 6393}, {2799, 4567}, {3405, 4568}, {3569, 4601}, {4033, 17209}, {4230, 20336}, {4553, 20022}, {5379, 6333}, {6335, 36212}, {14966, 27801}
X(42717) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 2395}, {72, 879}, {100, 98}, {101, 1910}, {190, 1821}, {213, 2422}, {228, 878}, {232, 6591}, {237, 667}, {240, 7649}, {297, 17924}, {325, 693}, {511, 513}, {644, 15628}, {668, 290}, {684, 18210}, {692, 1976}, {877, 286}, {906, 248}, {1331, 293}, {1332, 287}, {1755, 649}, {1783, 6531}, {1897, 36120}, {1959, 514}, {2396, 274}, {2421, 81}, {2491, 3121}, {2799, 16732}, {3289, 22383}, {3405, 10566}, {3569, 3125}, {4230, 28}, {4553, 20021}, {4561, 336}, {4567, 2966}, {4570, 36084}, {4600, 36036}, {5360, 512}, {5379, 685}, {5380, 9154}, {5976, 14296}, {6335, 16081}, {6386, 18024}, {6393, 15413}, {9155, 14419}, {9417, 1919}, {9418, 1980}, {14966, 1333}, {16591, 7212}, {17209, 1019}, {23997, 58}, {36212, 905}, {36213, 4164}

X(42718) = X(2)X(37)∩X(190)X(653)

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

X(42718) lies on these lines: {2, 37}, {100, 9056}, {190, 653}, {646, 4561}, {651, 29000}, {1332, 4033}, {2398, 4397}, {4585, 13136}, {21362, 29733}

X(42718) = X(14304)-cross conjugate of X(35516)
X(42718) = X(i)-isoconjugate of X(j) for these (i,j): {11, 32643}, {56, 2432}, {102, 649}, {513, 32677}, {667, 36100}, {1397, 2399}, {1919, 34393}, {2170, 36040}, {6591, 36055}, {7004, 32667}, {7117, 36067}, {22383, 36121}
X(42718) = barycentric product X(i)*X(j) for these {i,j}: {100, 35516}, {312, 2406}, {345, 24035}, {515, 668}, {646, 34050}, {1978, 2182}, {2425, 28659}, {3718, 23987}, {4998, 14304}, {7452, 20336}
X(42718) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 2432}, {59, 36040}, {100, 102}, {101, 32677}, {190, 36100}, {312, 2399}, {515, 513}, {644, 15629}, {668, 34393}, {1331, 36055}, {1897, 36121}, {2149, 32643}, {2182, 649}, {2406, 57}, {2425, 604}, {4397, 15633}, {7012, 36067}, {7115, 32667}, {7452, 28}, {8755, 6591}, {14304, 11}, {23987, 34}, {24035, 278}, {34050, 3669}, {35516, 693}, {39471, 7004}

X(42719) = X(2)X(37)∩X(190)X(658)

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

X(42719) lies on these lines: {2, 37}, {100, 9057}, {190, 658}, {220, 18738}, {651, 28999}, {668, 30728}, {811, 7259}, {1023, 1577}, {1897, 3699}, {3730, 29477}, {3882, 29421}, {4562, 14727}, {4585, 31633}, {17742, 21579}, {35517, 40869}

X(42719) = cevapoint of X(676) and X(17747)
X(42719) = trilinear pole of line {516, 30807}
X(42719) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2424}, {32, 2400}, {103, 649}, {244, 36039}, {513, 911}, {667, 36101}, {677, 1015}, {1086, 32642}, {1919, 18025}, {2310, 32668}, {3937, 40116}, {6591, 36056}, {7649, 32657}, {14936, 24016}, {15634, 32739}, {20974, 35184}, {22084, 32701}, {22383, 36122}
X(42719) = barycentric product X(i)*X(j) for these {i,j}: {75, 2398}, {100, 35517}, {190, 30807}, {304, 41321}, {341, 23973}, {346, 24015}, {516, 668}, {561, 2426}, {676, 7035}, {799, 17747}, {910, 1978}, {4033, 14953}, {4241, 20336}, {4554, 40869}, {4572, 41339}, {6335, 26006}, {9502, 36803}
X(42719) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2424}, {75, 2400}, {100, 103}, {101, 911}, {190, 36101}, {516, 513}, {644, 2338}, {666, 9503}, {668, 18025}, {676, 244}, {693, 15634}, {765, 677}, {906, 32657}, {910, 649}, {1110, 32642}, {1252, 36039}, {1262, 32668}, {1331, 36056}, {1332, 1815}, {1886, 6591}, {1897, 36122}, {2398, 1}, {2426, 31}, {3234, 910}, {4241, 28}, {7045, 24016}, {9502, 665}, {14953, 1019}, {17747, 661}, {23973, 269}, {24014, 676}, {24015, 279}, {26006, 905}, {28345, 22108}, {28346, 9508}, {30807, 514}, {35517, 693}, {39470, 3942}, {40869, 650}, {41321, 19}, {41339, 663}

X(42720) = X(2)X(37)∩X(100)X(190)

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

X(42720) lies on these lines: {2, 37}, {8, 14947}, {39, 28598}, {100, 190}, {514, 4169}, {644, 4561}, {646, 1978}, {662, 10330}, {664, 6558}, {666, 39272}, {668, 30730}, {672, 27919}, {874, 4583}, {876, 4562}, {883, 1025}, {919, 35574}, {1016, 30731}, {1018, 4568}, {1023, 32094}, {1500, 25263}, {1644, 4937}, {1909, 25244}, {2321, 24318}, {3177, 25278}, {3252, 3930}, {3257, 5387}, {3616, 19895}, {3681, 36294}, {3695, 37165}, {3729, 9318}, {3938, 7032}, {4363, 31063}, {4518, 13576}, {4595, 21272}, {4606, 37215}, {4986, 24036}, {6376, 25237}, {6632, 31615}, {6634, 31628}, {7080, 32034}, {7283, 37009}, {9055, 20331}, {14439, 17755}, {16549, 17141}, {16705, 28594}, {16720, 26759}, {17136, 18047}, {17152, 33299}, {18055, 20244}, {20345, 39350}, {20955, 26757}, {21431, 27071}, {21604, 26794}, {22003, 22033}, {24330, 31052}, {24407, 27912}, {25268, 30610}, {26770, 33938}, {27025, 33944}, {27096, 33930}, {27109, 33937}, {28742, 33933}, {30790, 30857}

X(42720) = isogonal conjugate of X(43929)
X(42720) = anticomplement of X(27918)
X(42720) = isotomic conjugate of the isogonal conjugate of X(2284)
X(42720) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {660, 150}, {692, 39362}, {765, 20345}, {813, 149}, {1016, 20554}, {1110, 33888}, {1252, 17794}, {4562, 21293}, {5378, 69}, {23990, 30667}, {34067, 4440}, {36081, 25049}
X(42720) = X(i)-Ceva conjugate of X(j) for these (i,j): {874, 23354}, {1016, 4437}, {4583, 3952}, {35574, 100}, {36803, 668}
X(42720) = X(i)-cross conjugate of X(j) for these (i,j): {665, 518}, {918, 3912}, {1026, 883}, {2254, 30941}, {4437, 1016}, {4925, 9436}
X(42720) = cevapoint of X(i) and X(j) for these (i,j): {518, 665}, {812, 3008}, {918, 3912}, {2254, 3930}
X(42720) = crosspoint of X(i) and X(j) for these (i,j): {190, 4562}, {668, 36803}
X(42720) = crosssum of X(649) and X(8632)
X(42720) = trilinear pole of line {518, 3717}
X(42720) = crossdifference of every pair of points on line {667, 1015}
X(42720) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1027}, {56, 1024}, {57, 884}, {105, 649}, {244, 919}, {513, 1438}, {604, 885}, {608, 23696}, {650, 1416}, {663, 1462}, {666, 3248}, {667, 673}, {875, 6654}, {985, 29956}, {1015, 36086}, {1086, 32666}, {1106, 28132}, {1459, 8751}, {1474, 10099}, {1919, 2481}, {1980, 18031}, {2170, 32735}, {2191, 2440}, {2195, 3669}, {2254, 41934}, {3271, 36146}, {3733, 18785}, {5377, 21143}, {6591, 36057}, {7649, 32658}, {22383, 36124}, {23349, 36816}
X(42720) = barycentric product X(i)*X(j) for these {i,j}: {8, 883}, {75, 1026}, {76, 2284}, {99, 3932}, {100, 3263}, {120, 35574}, {190, 3912}, {241, 646}, {312, 1025}, {341, 41353}, {344, 2414}, {518, 668}, {644, 40704}, {664, 3717}, {665, 31625}, {666, 4437}, {670, 20683}, {672, 1978}, {799, 3930}, {874, 22116}, {918, 1016}, {1018, 18157}, {1861, 4561}, {2223, 6386}, {2254, 7035}, {2283, 3596}, {2340, 4572}, {3252, 27853}, {3286, 27808}, {3570, 40217}, {3693, 4554}, {3699, 9436}, {3952, 30941}, {4033, 18206}, {4088, 4600}, {4238, 20336}, {4562, 17755}, {4583, 8299}, {4601, 24290}, {4602, 39258}, {4966, 6540}, {6184, 36803}, {6335, 25083}, {10029, 30720}, {30610, 40883}, {36801, 39775}
X(42720) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1027}, {8, 885}, {9, 1024}, {55, 884}, {59, 32735}, {72, 10099}, {78, 23696}, {100, 105}, {101, 1438}, {109, 1416}, {120, 23770}, {190, 673}, {218, 2440}, {241, 3669}, {344, 2402}, {346, 28132}, {518, 513}, {644, 294}, {646, 36796}, {651, 1462}, {665, 1015}, {666, 6185}, {668, 2481}, {672, 649}, {765, 36086}, {883, 7}, {906, 32658}, {918, 1086}, {919, 41934}, {926, 3271}, {1016, 666}, {1018, 18785}, {1025, 57}, {1026, 1}, {1110, 32666}, {1252, 919}, {1331, 36057}, {1332, 1814}, {1642, 1643}, {1783, 8751}, {1818, 1459}, {1861, 7649}, {1897, 36124}, {1978, 18031}, {2223, 667}, {2254, 244}, {2276, 29956}, {2283, 56}, {2284, 6}, {2340, 663}, {2414, 277}, {3126, 3675}, {3252, 3572}, {3263, 693}, {3286, 3733}, {3570, 6654}, {3675, 764}, {3693, 650}, {3699, 14942}, {3717, 522}, {3912, 514}, {3930, 661}, {3932, 523}, {3939, 2195}, {3952, 13576}, {4076, 36802}, {4088, 3120}, {4238, 28}, {4437, 918}, {4447, 4367}, {4554, 34018}, {4561, 31637}, {4564, 36146}, {4578, 28071}, {4684, 4778}, {4712, 2254}, {4899, 3667}, {4925, 3756}, {4966, 4977}, {4998, 927}, {5089, 6591}, {6184, 665}, {6558, 6559}, {8299, 659}, {9436, 3676}, {9454, 1919}, {9455, 1980}, {14439, 1635}, {15149, 17925}, {16593, 6084}, {17755, 812}, {18157, 7199}, {18206, 1019}, {20662, 8659}, {20683, 512}, {20752, 22383}, {20776, 23225}, {20778, 22384}, {22116, 876}, {23102, 3126}, {23225, 22096}, {23829, 17205}, {23891, 36816}, {24290, 3125}, {25083, 905}, {27919, 4375}, {30941, 7192}, {31625, 36803}, {34230, 23345}, {35293, 14413}, {36801, 33676}, {39258, 798}, {40217, 4444}, {40704, 24002}, {40730, 875}, {40883, 4885}, {41353, 269}, {42341, 4014}
X(42720) = trilinear product X(i)*X(j) for these {i,j}: {2, 1026}, {8, 1025}, {9, 883}, {75, 2284}, {99, 3930}, {100, 3912}, {101, 3263}, {190, 518}, {241, 3699}, {306, 4238}, {312, 2283}, {346, 41353}, {644, 9436}, {646, 1458}, {651, 3717}, {660, 17755}, {662, 3932}, {664, 3693}, {665, 7035}, {666, 4712}, {668, 672}, {670, 39258}, {765, 918}, {799, 20683}, {874, 3252}, {1016, 2254}, {1018, 30941}, {1332, 1861}, {1818, 6335}, {1897, 25083}, {1978, 2223}, {2340, 4554}, {2397, 36819}, {2414, 3870}, {3286, 4033}, {3570, 22116}, {3573, 40217}, {3675, 6632}, {3939, 40704}, {3952, 18206}, {4088, 4567}, {4437, 36086}, {4447, 27805}, {4555, 14439}, {4557, 18157}, {4561, 5089}, {4562, 8299}, {4571, 5236}, {4600, 24290}, {4606, 4684}, {4899, 27834}, {4925, 5382}, {4966, 37212}, {6386, 9454}, {6558, 34855}, {17464, 35574}, {24004, 34230}, {27853, 40730}, {34253, 36801}, {36803, 42079}
X(42720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 192, 24403}, {190, 3807, 3952}, {3693, 40883, 3263}, {4595, 33946, 21272}, {21272, 25272, 33946}

X(42721) = X(2)X(37)∩X(99)X(100)

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

X(42721) lies on these lines: {2, 37}, {99, 100}, {190, 35181}, {693, 874}, {889, 35147}, {1290, 35574}, {3266, 3712}, {4062, 16741}, {4601, 9170}, {4933, 14210}, {5380, 39296}, {5468, 24039}, {7283, 37014}, {14608, 21839}, {16703, 33160}

X(42721) = X(i)-cross conjugate of X(j) for these (i,j): {4750, 16741}, {14419, 524}
X(42721) = cevapoint of X(i) and X(j) for these (i,j): {524, 14419}, {4062, 4750}
X(42721) = trilinear pole of line {524, 14210}
X(42721) = crossdifference of every pair of points on line {667, 3121}
X(42721) = X(i)-isoconjugate of X(j) for these (i,j): {58, 9178}, {111, 649}, {513, 923}, {514, 32740}, {663, 7316}, {667, 897}, {671, 1919}, {691, 3122}, {1333, 23894}, {1459, 8753}, {1474, 10097}, {2206, 5466}, {3120, 32729}, {3121, 36085}, {3125, 36142}, {3248, 5380}, {3261, 19626}, {4750, 41936}, {6591, 36060}, {7649, 14908}, {10566, 41272}, {22383, 36128}
X(42721) = barycentric product X(i)*X(j) for these {i,j}: {10, 24039}, {100, 3266}, {187, 6386}, {190, 14210}, {313, 23889}, {321, 5468}, {524, 668}, {646, 7181}, {670, 21839}, {690, 4601}, {799, 4062}, {896, 1978}, {3712, 4554}, {3952, 16741}, {4033, 6629}, {4235, 20336}, {4567, 35522}, {4583, 4760}, {4750, 7035}, {5380, 36792}, {5467, 27801}, {6335, 6390}, {14419, 31625}, {16702, 27808}
X(42721) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 23894}, {37, 9178}, {72, 10097}, {100, 111}, {101, 923}, {187, 667}, {190, 897}, {321, 5466}, {351, 3121}, {468, 6591}, {524, 513}, {644, 5547}, {651, 7316}, {668, 671}, {690, 3125}, {692, 32740}, {896, 649}, {906, 14908}, {922, 1919}, {1016, 5380}, {1331, 36060}, {1332, 895}, {1783, 8753}, {1897, 36128}, {2482, 14419}, {2642, 3122}, {3266, 693}, {3292, 22383}, {3712, 650}, {3793, 3803}, {3908, 42007}, {4062, 661}, {4235, 28}, {4567, 691}, {4570, 36142}, {4600, 36085}, {4601, 892}, {4750, 244}, {4760, 659}, {4831, 4790}, {4933, 4893}, {4938, 4813}, {5026, 4164}, {5380, 10630}, {5467, 1333}, {5468, 81}, {5642, 14399}, {6335, 17983}, {6386, 18023}, {6390, 905}, {6629, 1019}, {7181, 3669}, {7267, 4367}, {7813, 2530}, {14210, 514}, {14417, 18210}, {14419, 1015}, {14432, 2170}, {14567, 1980}, {16702, 3733}, {16741, 7192}, {20336, 14977}, {21839, 512}, {23889, 58}, {24038, 4750}, {24039, 86}, {27088, 30234}, {35522, 16732}

X(42722) = X(2)X(37)∩X(190)X(693)

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

X(42722) lies on these lines: {2, 37}, {190, 693}, {644, 666}, {646, 7035}, {1577, 32094}, {3699, 4397}, {3762, 6633}, {4462, 6631}, {4554, 4582}, {4801, 32028}, {4978, 32106}, {21362, 29738}

X(42722) = X(1643)-cross conjugate of X(528)
X(42722) = cevapoint of X(528) and X(1643)
X(42722) = X(i)-isoconjugate of X(j) for these (i,j): {649, 840}, {667, 37131}, {1919, 18821}
X(42722) = barycentric product X(i)*X(j) for these {i,j}: {528, 668}, {646, 5723}, {1642, 36803}, {1643, 31625}, {1978, 2246}
X(42722) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 840}, {190, 37131}, {528, 513}, {668, 18821}, {1642, 665}, {1643, 1015}, {2246, 649}, {5723, 3669}, {14190, 23345}, {17780, 14191}, {35113, 1643}
X(42722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {350, 17264, 4358}

X(42723) = X(2)X(37)∩X(100)X(101)

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

X(42723) lies on these lines: {2, 37}, {100, 101}, {650, 17780}, {1150, 36258}, {3909, 7239}, {4427, 35310}, {7283, 37040}, {16975, 34362}, {21580, 27134}, {32933, 34361}

X(42723) = crossdifference of every pair of points on line {244, 667}
X(42723) = X(i)-isoconjugate of X(j) for these (i,j): {244, 36087}, {513, 2224}, {649, 675}, {667, 37130}, {1086, 32682}, {15397, 29240}
X(42723) = barycentric product X(i)*X(j) for these {i,j}: {100, 3006}, {668, 674}, {765, 23887}, {1978, 2225}, {4033, 14964}, {4249, 20336}, {6386, 8618}
X(42723) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 675}, {101, 2224}, {190, 37130}, {674, 513}, {1110, 32682}, {1252, 36087}, {2225, 649}, {3006, 693}, {4249, 28}, {8618, 667}, {14964, 1019}, {23887, 1111}

X(42724) = X(2)X(37)∩X(100)X(9084)

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

X(42724) lies on these lines: {2, 37}, {100, 9084}, {190, 1434}, {213, 1332}, {1001, 4742}, {3159, 24171}, {3262, 30830}, {3896, 39688}, {3986, 18697}, {3992, 4078}, {6335, 17983}, {9227, 22028}, {21078, 22047}

X(42724) = trilinear pole of line {1499, 14207}
X(42724) = X(i)-isoconjugate of X(j) for these (i,j): {58, 21448}, {86, 39238}, {649, 1296}, {667, 37216}, {1919, 35179}, {2206, 5485}, {4750, 32648}, {14419, 36045}
X(42724) = barycentric product X(i)*X(j) for these {i,j}: {37, 11059}, {190, 14207}, {313, 36277}, {321, 1992}, {668, 1499}, {1384, 27801}, {4033, 4786}, {4232, 20336}, {4601, 6791}, {6386, 8644}, {27808, 30234}
X(42724) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 21448}, {100, 1296}, {190, 37216}, {213, 39238}, {321, 5485}, {668, 35179}, {1384, 1333}, {1499, 513}, {1992, 81}, {4232, 28}, {4786, 1019}, {6791, 3125}, {8644, 667}, {9125, 14419}, {11059, 274}, {14207, 514}, {27088, 16702}, {30234, 3733}, {36277, 58}

leftri

Gibert points on cubic K1207: X(42725)-X(42730)

rightri

This preamble and points X(42725)-X(42730) are contributed by Peter Moses, April 21, 2021

See
K1207

Gibert points are introduced in the preamble just before X(42085)



X(42725) = GIBERT (6 SQRT(6),5,4) POINT

Barycentrics    3*Sqrt[2]*a^2*S + 2*a^2*SA + 5*SB*SC : :

X(42725) lies on the cubic K1207 and these lines: {2, 41975}, {6, 3545}, {5055, 42647}, {10304, 41980}, {14785, 31487}, {15682, 42645}, {15692, 41976}

X(42726) = GIBERT (-6 SQRT(6),5,4) POINT

Barycentrics    -3*Sqrt[2]*a^2*S + 2*a^2*SA + 5*SB*SC : :

X(42726) lies on the cubic K1207 and these lines: {2, 41976}, {6, 3545}, {5055, 42648}, {10304, 41979}, {14784, 31487}, {15682, 42646}, {15692, 41975}

X(42727) = GIBERT (3 SQRT(3/2),5,1) POINT

Barycentrics    (3*a^2*S)/Sqrt[2] + a^2*SA + 10*SB*SC : :

X(42727) lies on the cubic K1207 and these lines: {6, 1327}, {30, 41976}, {381, 42645}, {3830, 12822}, {5066, 41980}, {11737, 41975}, {12101, 41979}, {14269, 42646}, {33699, 42647}

X(42728) = GIBERT (-3 SQRT(3/2),5,1) POINT

Barycentrics    (-3*a^2*S)/Sqrt[2] + a^2*SA + 10*SB*SC : :

X(42728) lies on the cubic K1207 and these lines: {6, 1327}, {30, 41975}, {381, 42646}, {3830, 12823}, {5066, 41979}, {11737, 41976}, {12101, 41980}, {14269, 42645}, {33699, 42648}

X(42729) = GIBERT (2 SQRT(2/3),1,-2) POINT

Barycentrics    (Sqrt[2]*a^2*S)/3 - a^2*SA + SB*SC : :

X(42729) lies on the cubic K1207 and these lines: {3, 42647}, {4, 41975}, {6, 20}, {376, 41980}, {382, 42648}, {3523, 42645}, {11001, 41979}, {14784, 42226}, {14785, 42225}, {21735, 41976}

X(42730) = GIBERT (2 SQRT(2/3),-1,2) POINT

Barycentrics    (Sqrt[2]*a^2*S)/3 + a^2*SA - SB*SC : :

X(42730) lies on the cubic K1207 and these lines: {3, 42648}, {4, 41976}, {6, 20}, {376, 41979}, {382, 42647}, {3523, 42646}, {11001, 41980}, {14784, 42225}, {14785, 42226}, {21735, 41975}

leftri

Centers of circles that pass through X(13) and X(14): X(42731)-X(42738)

rightri

This preamble and points X(42731)-X(42738) are contributed by Clark Kimberling and Peter Moses, April 22, 2021

See X(13) for a list of circles that pass through X(13) and X(14), and see X(15) for circles through X(15) and X(16).



X(42731) = CENTER OF CIRCLE {{{13,14,112}}

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

X(42731) lies on these lines: {3, 24978}, {115, 125}, {140, 41078}, {186, 523}, {2797, 9979}, {2799, 38748}, {2848, 9409}, {5489, 6130}, {14656, 39857}

X(42731) = tripolar centroid of X(275)
X(42731) = crossdifference of every pair of points on line {110, 216}

X(42732) = CENTER OF CIRCLE {{{13,14,2079}}

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

X(42732) lies on these lines: {115, 125}, {523, 31667}, {6140, 10279}

X(42732) = tripolar centroid of X(13585)

X(42733) = CENTER OF CIRCLE {{{4,13,14}}

Barycentrics    (b^2 - c^2)*(-2*a^8 + 2*a^6*b^2 + 3*a^4*b^4 - 4*a^2*b^6 + b^8 + 2*a^6*c^2 - 8*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 2*b^6*c^2 + 3*a^4*c^4 + 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(42733) lies on these lines: {3, 14566}, {4, 523}, {5, 5664}, {30, 18556}, {115, 125}, {378, 39201}, {381, 525}, {512, 32062}, {520, 15030}, {647, 33843}, {879, 1499}, {1316, 1649}, {1995, 30474}, {2777, 30497}, {2797, 3268}, {2799, 9880}, {3265, 32815}, {3906, 22682}, {5094, 9209}, {5478, 23870}, {5479, 23871}, {6249, 33752}, {6644, 22089}, {9007, 11180}, {14223, 14639}, {18809, 42426}

X(42733) = reflection of X(381) in X(39491)
X(42733) = pole wrt orthocentroidal circle of van Aubel line
X(42733) = tripolar centroid of X(16080)
X(42733) = X(16075)-isoconjugate of X(36034)
X(42733) = crossdifference of every pair of points on line {110, 3284}
X(42733) = barycentric product X(i)*X(j) for these {i,j}: {1637, 16076}, {1651, 2394}
X(42733) = barycentric quotient X(i)/X(j) for these {i,j}: {1637, 16075}, {1651, 2407}, {2433, 41433}
X(42733) = {X(868),X(1640)}-harmonic conjugate of X(8371)

X(42734) = CENTER OF CIRCLE {{{13,14,616}}

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

X(42734) lies on these lines: {115, 125}, {298, 523}, {618, 1649}, {5466, 11121}, {8029, 23870}, {9205, 11123}, {14082, 32553}, {14424, 23871}

X(42734) = reflection of X(42735) in X(9148)
X(42734) = tripolar centroid of X(40706)
X(42734) = {X(9200),X(11182)}-harmonic conjugate of X(8371)

X(42735) = CENTER OF CIRCLE {{{13,14,617}}

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

X(42735) lies on these lines: {115, 125}, {299, 523}, {619, 1649}, {5466, 11122}, {8029, 23871}, {9204, 11123}, {14081, 32552}, {14424, 23870}

X(42735) = reflection of X(42734) in X(9148)
X(42735) = tripolar centroid of X(40707)
X(42735) = {X(9201),X(11182)}-harmonic conjugate of X(8371)

X(42736) = CENTER OF CIRCLE {{{13,14,125}}

Barycentrics    (b^2 - c^2)*(-2*a^10 + 3*a^8*b^2 + 5*a^6*b^4 - 10*a^4*b^6 + 3*a^2*b^8 + b^10 + 3*a^8*c^2 - 16*a^6*b^2*c^2 + 11*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 6*b^8*c^2 + 5*a^6*c^4 + 11*a^4*b^2*c^4 - 22*a^2*b^4*c^4 + 5*b^6*c^4 - 10*a^4*c^6 + 8*a^2*b^2*c^6 + 5*b^4*c^6 + 3*a^2*c^8 - 6*b^2*c^8 + c^10) : :

Let A', B', C' be the orthic axis intercepts of lines BC, CA, AB, resp. Let LA, LB, LC be the reflections of the orthic axis in BC, CA, AB, resp. Let AB, AC be the orthogonal projections of A' on LB, LC, resp. Define BC, BA, CA, CB cyclically. The circumcircles of A'ABAC, B'BCBA, C'CACB are coaxial, and X(42736) is the centroid of their centers. (See Hyacinthos #21468, Antreas Hatzipolakis, Jan 30, 2013) (Randy Hutson, May 31, 2021)

X(42736) lies on these lines: {2, 9033}, {98, 9189}, {115, 125}, {523, 22264}, {526, 10189}, {1499, 37984}, {2848, 24930}, {3268, 15059}, {5466, 16080}, {6130, 9003}, {9140, 14697}, {12099, 39469}, {15475, 34310}, {16315, 33921}

X(42737) = CENTER OF CIRCLE {{{13,14,110}}

Barycentrics    (b^2 - c^2)*(-2*a^10 + 6*a^8*b^2 - 7*a^6*b^4 + 5*a^4*b^6 - 3*a^2*b^8 + b^10 + 6*a^8*c^2 - 10*a^6*b^2*c^2 + 5*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 - 7*a^6*c^4 + 5*a^4*b^2*c^4 - 10*a^2*b^4*c^4 + 2*b^6*c^4 + 5*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(42737) lies on these lines: {115, 125}, {323, 523}, {526, 8029}, {804, 14698}, {1499, 7574}, {1649, 6132}, {5466, 13582}, {5642, 13290}

X(42737) = Hutson-Parry-circle-inverse of X(1116)
X(42737) = crossdifference of every pair of points on line {110, 15544}
X(42737) = {X(13636),X(13722)}-harmonic conjugate of X(1116)

X(42738) = CENTER OF CIRCLE {{{13,14,98}}

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

X(42738) lies on these lines: {98, 523}, {115, 125}, {525, 11632}, {2394, 14651}, {2782, 5664}, {2793, 16230}, {2799, 6055}, {5489, 11623}, {9180, 16080}, {12042, 18556}, {14566, 38224}

X(42738) = Dao-Moses-Telv-circle-inverse of X(1640)
X(42738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {125, 1637, 8371}, {8371, 14420, 13291}

X(42739) = CENTER OF CIRCLE {{{13,14,74}}

Barycentrics    (b^2 - c^2)*(-2*a^14 + 4*a^12*b^2 + 9*a^10*b^4 - 35*a^8*b^6 + 40*a^6*b^8 - 18*a^4*b^10 + a^2*b^12 + b^14 + 4*a^12*c^2 - 30*a^10*b^2*c^2 + 39*a^8*b^4*c^2 + a^6*b^6*c^2 - 24*a^4*b^8*c^2 + 15*a^2*b^10*c^2 - 5*b^12*c^2 + 9*a^10*c^4 + 39*a^8*b^2*c^4 - 84*a^6*b^4*c^4 + 42*a^4*b^6*c^4 - 15*a^2*b^8*c^4 + 9*b^10*c^4 - 35*a^8*c^6 + a^6*b^2*c^6 + 42*a^4*b^4*c^6 - 2*a^2*b^6*c^6 - 5*b^8*c^6 + 40*a^6*c^8 - 24*a^4*b^2*c^8 - 15*a^2*b^4*c^8 - 5*b^6*c^8 - 18*a^4*c^10 + 15*a^2*b^2*c^10 + 9*b^4*c^10 + a^2*c^12 - 5*b^2*c^12 + c^14) : :

X(42739) lies on these lines: {115, 125}, {523, 1138}, {5664, 10264}

leftri

Dao-Lester and Dao-Parry circles: X(42740)-X(42747)

rightri

This preamble and centers X(42740)-X(42747) were contributed by César Eliud Lozada, April 22, 2021.

These constructions are based on two problems in the paper "Generalizations of some famous classical Euclidean geometry theorems", by Dao Thanh Oai & als., published in International Journal of Computer Discovered Mathematics, Vol. 1, No. 3, 2016, pp. 13-20.

Problem 1 (A generalization of the Lester circle associated with the Neuberg cubic). Let ABC be a triangle and P a point on the Neuberg cubic of ABC. Let Pa be the reflection of P in the line BC, and define Pb and Pc cyclically. It is known that lines APa, BPb, CPc concur at a point Q(P). Then P, Q(P) and the two Fermat points of ABC lie on a circle. (Remark: If P=X(3) then Q(P)=X(5) and the given circle is the Lester circle of ABC. See Lester circle in WolframMathworld).

The described circle is named here the Dao-Lester circle of P. Its center O(P) lies on the line X(115)X(125). For P = x : y : z (barycentrics) on the Neuberg cubic of ABC, O(P) has coordinates:

  O(P) = (2*(S^2-3*SA^2)*x^2+4*(S^2-3*SB*SC)*y*z-(S^2-3*SB^2)*y^2-(S^2-3*SC^2)*z^2-2*(S^2-3*SA*SB)*x*y-2*(S^2-3*SA*SC)*x*z)*(SB-SC) : :

The appearance of (i, j) in the following list means that O(X(i))=X(j):
(1, 42740), (3, 1116), (4, 42733), (15, 9201), (16, 9200), (30, 690), (74, 42739), (399, 690), (484, 30574), (616, 42734), (617, 42735), (1157, 42731), (2132, 42733), (3464, 30574), (5623, 9200), (5624, 9201), (5667, 42731), (5668, 14446), (5669, 14447), (5670, 42739), (5671, 1116), (5672, 4120), (5673, 4120), (5674, 42735), (5675, 42734), (5677, 42740), (8172, 14446), (8173, 14447)

Problem 2 (A generalization of the Parry circle associated with two isogonal conjugate points). Let ℍ be a rectangular circum-hyperbola of ABC and ℓ be the line isogonal conjugate of ℍ. The tangent line to the ℍ at X(4) meets ℓ at point K. The line through K and the center of ℍ meets ℍ at P1, P2. Let P1*, P2*, K* be the isogonal conjugates of P1, P2 and K, respectively. Let K' be the inverse point of K* with respect to the circumcircle of ABC. Then the five points P1*, P2*, K*, K' and X(110) lie on a circle. Furthermore K lie on the Jerabek hyperbola. (Remark: It can be proved that if ℍ is the Kiepert hyperbola of ABC, then the given circle is the Parry circle of ABC).

The last circle is named here the Dao-Parry circle of ℍ. Its center O(ℍ) lies on the line X(110)X(351). If P = x : y : z is any point on ℍ, other than A, B, C, X(4), then:

  O(ℍ) = (SB+SC)*(SA-SB)*(SA-SC)*(-(y-z)*SA*x-(x+z)*SB*y+(x+y)*SC*z)*(x*((y-z)*S^2-4*SA*(SB*y-SC*z))+(x+z)*SB^2*y-(x+y)*SC^2*z) : :

O(ℍ)=X(42741), X(351), X(526) for ℍ = Feuerbach, Kiepert, Jerabek circum-hyperbola, respectively. In general, since ℍ is a rectangular circum-hyperbola of ABC, its center ℍo lies on the nine-point-circle of ABC. The appearance of (i, j) in the following list means that if ℍo = X(i) then O(ℍ) = X(j):
(11, 42741), (113, 42742), (114, 42743), (115, 351), (116, 42744), (118, 42745), (119, 42746), (120, 42747), (125, 526), (3258, 526), (5099, 351), (5520, 42741), (16188, 42743), (25641, 42742), (42422, 42746)

X(42740) = CENTER OF THE DAO-LESTER CIRCLE OF X(1)

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

X(42740) lies on these lines: {115, 125}, {523, 3649}, {4926, 6129}, {11705, 35051}, {11706, 35052}

X(42741) = CENTER OF THE DAO-PARRY CIRCLE OF THE FEUERBACH CIRCUM-HYPERBOLA

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

X(42741) lies on these lines: {21, 900}, {28, 39534}, {36, 238}, {110, 351}, {404, 24920}, {665, 1333}, {759, 6089}, {1283, 9508}, {1635, 19302}, {2070, 39210}, {2605, 42653}, {4225, 28284}, {4228, 26275}, {5047, 24959}, {5260, 21714}, {7202, 18181}, {7419, 28396}, {8648, 34442}, {8674, 16164}, {11115, 26078}, {16047, 28779}, {17588, 26144}, {22586, 35053}

X(42741) = isogonal conjugate of the anticomplement of X(38982)
X(42741) = barycentric product X(i)*X(j) for these {i, j}: {81, 8674}, {274, 42670}, {513, 37783}, {514, 5127}, {693, 19622}, {905, 2074}
X(42741) = barycentric quotient X(i)/X(j) for these (i, j): (81, 35156), (649, 5620), (1333, 1290)
X(42741) = trilinear product X(i)*X(j) for these {i, j}: {58, 8674}, {86, 42670}, {513, 5127}, {514, 19622}, {649, 37783}, {1019, 17796}
X(42741) = trilinear quotient X(i)/X(j) for these (i, j): (58, 1290), (86, 35156), (513, 5620), (1019, 21907)
X(42741) = intersection, other than A,B,C, of conics {{A, B, C, X(36), X(759)}} and {{A, B, C, X(110), X(513)}}
X(42741) = pole of the trilinear polar of X(37140) with respect to circumcircle
X(42741) = crossdifference of every pair of points on line {X(37), X(115)}
X(42741) = crosspoint of X(i) and X(j) for these (i, j): {58, 36069}, {110, 759}
X(42741) = crosssum of X(i) and X(j) for these (i, j): {10, 6370}, {523, 758}
X(42741) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 1290}, {42, 35156}, {100, 5620}, {1018, 21907}
X(42741) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (81, 35156), (649, 5620), (1333, 1290)
X(42741) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (110, 15329, 42746), (1634, 5467, 42747), (3733, 21789, 4833)

X(42742) = CENTER OF THE DAO-PARRY CIRCLE OF THE CIRCUM-HYPERBOLA WITH CENTER X(113)

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

X(42742) lies on these lines: {30, 113}, {110, 351}, {399, 14933}, {3134, 5972}, {5663, 39987}, {7471, 16171}

X(42742) = midpoint of X(i) and X(j) for these {i, j}: {110, 15329}, {399, 14933}
X(42742) = reflection of X(3134) in X(5972)
X(42742) = barycentric product X(1511)*X(2410)
X(42742) = barycentric quotient X(i)/X(j) for these (i, j): (1511, 2411), (1553, 41079)
X(42742) = trilinear product X(1553)*X(36034)
X(42742) = trilinear quotient X(i)/X(j) for these (i, j): (250, 36117), (1553, 36035)
X(42742) = intersection, other than A,B,C, of conics {{A, B, C, X(30), X(110)}} and {{A, B, C, X(113), X(2437)}}
X(42742) = crossdifference of every pair of points on line {X(115), X(2433)}
X(42742) = X(125)-isoconjugate-of-X(36117)
X(42742) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1511, 2411), (1553, 41079)
X(42742) = intersection of Simson line of X(110) and tangent to circumcircle at X(110)

X(42743) = CENTER OF THE DAO-PARRY CIRCLE OF THE CIRCUM-HYPERBOLA WITH CENTER X(114)

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

X(42743) lies on these lines: {110, 351}, {237, 511}, {325, 3233}, {542, 5191}, {804, 4226}, {3289, 39689}, {3569, 14966}, {4230, 17994}, {7468, 20403}, {14999, 36885}, {20976, 23584}

X(42743) = midpoint of X(1634) and X(5467)
X(42743) = barycentric product X(i)*X(j) for these {i, j}: {511, 14999}, {542, 2421}
X(42743) = barycentric quotient X(i)/X(j) for these (i, j): (237, 14998), (511, 14223)
X(42743) = trilinear product X(i)*X(j) for these {i, j}: {542, 23997}, {1755, 14999}
X(42743) = trilinear quotient X(i)/X(j) for these (i, j): (1755, 14998), (1959, 14223)
X(42743) = intersection, other than A,B,C, of conics {{A, B, C, X(110), X(511)}} and {{A, B, C, X(237), X(1576)}}
X(42743) = crossdifference of every pair of points on line {X(115), X(2395)}
X(42743) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (237, 14998), (511, 14223)
X(42743) = {X(110), X(15329)}-harmonic conjugate of X(351)

X(42744) = CENTER OF THE DAO-PARRY CIRCLE OF THE CIRCUM-HYPERBOLA WITH CENTER X(116)

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

X(42744) lies on these lines: {110, 351}, {239, 514}, {926, 4184}, {4466, 17198}

X(42744) = barycentric product X(i)*X(j) for these {i, j}: {86, 2774}, {2073, 4025}
X(42744) = barycentric quotient X(i)/X(j) for these (i, j): (58, 2690), (1459, 38535), (2073, 1897)
X(42744) = trilinear product X(i)*X(j) for these {i, j}: {81, 2774}, {905, 2073}
X(42744) = trilinear quotient X(i)/X(j) for these (i, j): (81, 2690), (905, 38535), (2073, 1783)
X(42744) = crossdifference of every pair of points on line {X(42), X(115)}
X(42744) = X(37)-isoconjugate-of-X(2690)
X(42744) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (58, 2690), (1459, 38535)
X(42744) = {X(110), X(15329)}-harmonic conjugate of X(42745)

X(42745) = CENTER OF THE DAO-PARRY CIRCLE OF THE CIRCUM-HYPERBOLA WITH CENTER X(118)

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

X(42745) lies on these lines: {110, 351}, {516, 14953}, {926, 4243}

X(42745) = {X(110), X(15329)}-harmonic conjugate of X(42744)

X(42746) = CENTER OF THE DAO-PARRY CIRCLE OF THE CIRCUM-HYPERBOLA WITH CENTER X(119)

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

X(42746) lies on these lines: {110, 351}, {517, 859}, {900, 3658}, {4246, 39534}

X(42746) = {X(110), X(15329)}-harmonic conjugate of X(42741)

X(42747) = CENTER OF THE DAO-PARRY CIRCLE OF THE CIRCUM-HYPERBOLA WITH CENTER X(120)

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

X(42747) lies on these lines: {110, 351}, {518, 2223}, {4236, 6084}

X(42747) = trilinear product X(1818)*X(7476)
X(42747) = {X(1634), X(5467)}-harmonic conjugate of X(42741)

X(42748) = ANTIPODE OF X(1) IN THE DAO-LESTER CIRCLE OF X(1)

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

Dao-Lester circles are defined in the preamble just before X(42740).

X(42748) lies on these lines: {1,42740}, {79, 523}, {690, 13178}, {4926, 15079}

X(42748) = reflection of X(1) in X(42740)

X(42749) = ANTIPODE OF X(79) IN THE DAO-LESTER CIRCLE OF X(1)

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

X(42749) lies on these lines: {1, 523}, {79,42740}, {1699, 28473}, {5692, 35057}

X(42749) = reflection of X(79) in X(42740)

leftri

Points on the Sherman line: X(42750)-X(42772)

rightri

This preamble and points X(42750-X(42772) are contributed by Peter Moses, April 23, 2021

Suppose that P' = p' : q' : r' is a point on a line p x + q y + r z = 0 and that u x + v y + w z = 0 is a line, L. Then the point P'' = (p/u)*p' : (q/v)*q' + (r/w)*r' (p/u)*p' : (q/v)*q' + (r/w)*r' lies on L For example, if P' is on the line at infinity and L is the Sherman line, X(3259)X(3326), then the point P'*X(10015) is on L. Points X(42750)-X(42772) are obtained in this manner from the line at infinity, where, in the same order, P' = X(i) for i = 30, 511, 512, 513, 514, 515, 516, 517, 518, 523, 524, 525, 527, 528, 536, 537, 726, 740, 758, 912, 918, 926, 971.



X(42750) = X(113)X(133)∩X(3259)X(3326)

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

X(42750) lies on these lines: {113, 133}, {513, 942}, {520, 24474}, {522, 18483}, {523, 946}, {1354, 11125}, {1845, 8677}, {2804, 12611}, {3259, 3326}, {6087, 11719}, {7978, 14224}

X(42750) = crossdifference of every pair of points on line {15627, 17796}
X(42750) = X(i)-isoconjugate of X(j) for these (i,j): {74, 36037}, {1309, 35200}, {2159, 13136}, {2349, 32641}, {15627, 37136}, {36034, 38955}
X(42750) = barycentric product X(i)*X(j) for these {i,j}: {30, 10015}, {859, 41079}, {908, 11125}, {1637, 17139}, {1769, 14206}, {2173, 36038}, {2804, 6357}, {3260, 3310}, {3262, 14399}, {11064, 39534}, {14400, 22464}
X(42750) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 13136}, {1495, 32641}, {1637, 38955}, {1769, 2349}, {1990, 1309}, {2173, 36037}, {3310, 74}, {8677, 14919}, {10015, 1494}, {11125, 34234}, {14395, 1809}, {14399, 104}, {14581, 14776}, {23220, 18877}, {36038, 33805}, {39534, 16080}

X(42751) = X(114)X(132)∩X(3259)X(3326)

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

X(42751) lies on these lines: {114, 132}, {512, 1064}, {513, 3666}, {3259, 3326}, {3310, 8677}, {15451, 39212}, {23224, 40956}

X(42751) = crossdifference of every pair of points on line {248, 5291}
X(42751) = X(i)-isoconjugate of X(j) for these (i,j): {98, 36037}, {293, 1309}, {336, 14776}, {1821, 32641}, {1910, 13136}, {2250, 2966}, {15628, 37136}, {36084, 38955}
X(42751) = barycentric product X(i)*X(j) for these {i,j}: {297, 8677}, {325, 3310}, {511, 10015}, {859, 2799}, {1755, 36038}, {1769, 1959}, {3569, 17139}, {36212, 39534}
X(42751) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 1309}, {237, 32641}, {511, 13136}, {859, 2966}, {1755, 36037}, {1769, 1821}, {2211, 14776}, {3310, 98}, {3569, 38955}, {8677, 287}, {10015, 290}, {23220, 248}, {39534, 16081}

X(42752) = X(98)X(17981)∩X(3259)X(3326)

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

X(42752) lies on these lines: {98, 17981}, {115, 2971}, {1356, 3122}, {1402, 40934}, {3259, 3326}, {7983, 11609}

X(42752) = X(859)-Ceva conjugate of X(3310)
X(42752) = crosspoint of X(i) and X(j) for these (i,j): {859, 3310}, {1769, 21801}
X(42752) = crosssum of X(i) and X(j) for these (i,j): {110, 16704}, {13136, 38955}
X(42752) = crossdifference of every pair of points on line {645, 4558}
X(42752) = X(i)-isoconjugate of X(j) for these (i,j): {99, 36037}, {104, 4600}, {645, 37136}, {662, 13136}, {799, 32641}, {909, 4601}, {1309, 4592}, {1812, 39294}, {2250, 4590}, {2720, 7257}, {4567, 34234}, {4570, 18816}, {24041, 38955}
X(42752) = barycentric product X(i)*X(j) for these {i,j}: {115, 859}, {244, 21801}, {512, 10015}, {517, 3125}, {523, 3310}, {647, 39534}, {661, 1769}, {798, 36038}, {908, 3122}, {1015, 17757}, {1400, 35015}, {1457, 21044}, {1465, 4516}, {1880, 35014}, {2183, 3120}, {2397, 8034}, {2501, 8677}, {2680, 17946}, {2804, 7180}, {3121, 3262}, {3124, 17139}, {4079, 23788}, {14571, 18210}, {14618, 23220}
X(42752) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 13136}, {517, 4601}, {669, 32641}, {798, 36037}, {859, 4590}, {1457, 4620}, {1769, 799}, {2183, 4600}, {2489, 1309}, {2680, 17790}, {3121, 104}, {3122, 34234}, {3124, 38955}, {3125, 18816}, {3310, 99}, {4516, 36795}, {8034, 2401}, {8677, 4563}, {10015, 670}, {17139, 34537}, {17757, 31625}, {21801, 7035}, {23220, 4558}, {35015, 28660}, {36038, 4602}, {39534, 6331}

X(42753) = X(25)X(34)∩X(3259)X(3326)

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

X(42753) lies on the cubic K583 and these lines: {2, 19884}, {11, 2969}, {25, 34}, {43, 3987}, {63, 36280}, {104, 36125}, {244, 1357}, {517, 14260}, {661, 17435}, {764, 1647}, {1393, 42448}, {1421, 20999}, {1465, 23981}, {1482, 3870}, {1577, 3141}, {1772, 2841}, {3159, 22000}, {3259, 3326}, {3817, 21318}, {5400, 22321}, {5687, 12876}, {6127, 16110}, {7004, 38389}, {9955, 18115}, {11681, 29641}, {14008, 18175}, {14249, 37372}, {14455, 26098}, {15507, 16586}, {15635, 23345}, {15906, 34586}, {17605, 21807}, {20060, 29840}, {21339, 36197}, {24590, 36279}

X(42753) = X(i)-Ceva conjugate of X(j) for these (i,j): {517, 1769}, {1465, 3310}, {3262, 10015}, {23345, 764}, {36125, 513}
X(42753) = crosspoint of X(i) and X(j) for these (i,j): {88, 693}, {269, 1022}, {513, 1411}, {517, 1769}, {908, 23788}, {1875, 39534}, {3262, 10015}, {14260, 23345}
X(42753) = crosssum of X(i) and X(j) for these (i,j): {44, 692}, {100, 4511}, {104, 36037}, {200, 1023}, {17780, 36944}, {32641, 34858}
X(42753) = crossdifference of every pair of points on line {644, 906}
X(42753) = X(i)-isoconjugate of X(j) for these (i,j): {100, 36037}, {101, 13136}, {104, 765}, {190, 32641}, {219, 39294}, {644, 37136}, {646, 32669}, {909, 1016}, {1110, 18816}, {1252, 34234}, {1309, 1331}, {1795, 15742}, {1809, 7012}, {2149, 36795}, {2250, 4567}, {2342, 4998}, {2423, 6632}, {2720, 3699}, {4561, 14776}, {4570, 38955}, {4571, 36110}, {5377, 36819}, {7035, 34858}, {9268, 36944}
X(42753) = barycentric product X(i)*X(j) for these {i,j}: {11, 1465}, {57, 35015}, {88, 3259}, {244, 908}, {278, 35014}, {513, 10015}, {514, 1769}, {517, 1086}, {649, 36038}, {661, 23788}, {693, 3310}, {764, 2397}, {859, 16732}, {905, 39534}, {1015, 3262}, {1022, 23757}, {1111, 2183}, {1427, 14010}, {1457, 4858}, {1565, 14571}, {1785, 3942}, {1875, 26932}, {2170, 22464}, {2804, 3669}, {3125, 17139}, {3326, 34051}, {8677, 17924}, {15635, 26611}, {16082, 35012}, {16726, 17757}, {17205, 21801}, {21132, 24029}, {23981, 40166}
X(42753) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 36795}, {34, 39294}, {244, 34234}, {513, 13136}, {517, 1016}, {649, 36037}, {667, 32641}, {764, 2401}, {859, 4567}, {908, 7035}, {1015, 104}, {1086, 18816}, {1357, 34051}, {1457, 4564}, {1465, 4998}, {1769, 190}, {1977, 34858}, {2087, 36944}, {2183, 765}, {2804, 646}, {2969, 16082}, {3122, 2250}, {3125, 38955}, {3248, 909}, {3259, 4358}, {3262, 31625}, {3310, 100}, {6591, 1309}, {7117, 1809}, {8027, 2423}, {8677, 1332}, {10015, 668}, {14260, 5376}, {14571, 15742}, {17139, 4601}, {22096, 14578}, {23220, 906}, {23757, 24004}, {23788, 799}, {23981, 31615}, {35014, 345}, {35015, 312}, {36038, 1978}, {39534, 6335}

X(42754) = X(19)X(57)∩X(3259)X(3326)

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

X(42754) lies on these lines: {19, 57}, {116, 2973}, {192, 31053}, {344, 30852}, {903, 4638}, {908, 2397}, {1022, 2401}, {1086, 1358}, {2183, 22464}, {2265, 5723}, {2280, 17301}, {2405, 5236}, {2969, 7004}, {3254, 10695}, {3259, 3326}, {3675, 7336}, {4957, 21044}, {6336, 34234}, {6545, 23760}, {16200, 41570}, {16560, 37771}, {16732, 23773}, {30858, 36910}

X(42754) = X(34179)-anticomplementary conjugate of X(30578)
X(42754) = X(i)-Ceva conjugate of X(j) for these (i,j): {908, 10015}, {1022, 6545}, {4089, 1647}, {6336, 514}, {22464, 1769}
X(42754) = X(i)-isoconjugate of X(j) for these (i,j): {100, 32641}, {101, 36037}, {104, 1252}, {212, 39294}, {644, 2720}, {692, 13136}, {765, 909}, {906, 1309}, {1016, 34858}, {1110, 34234}, {1332, 14776}, {1809, 7115}, {2250, 4570}, {2342, 4564}, {3699, 32669}, {3939, 37136}, {4571, 32702}, {4587, 36110}, {6065, 34051}, {14578, 15742}, {18816, 23990}
X(42754) = crosspoint of X(i) and X(j) for these (i,j): {279, 6548}, {514, 2006}, {903, 3261}, {908, 10015}
X(42754) = crosssum of X(i) and X(j) for these (i,j): {101, 2323}, {220, 23344}, {902, 32739}, {909, 32641}
X(42754) = crossdifference of every pair of points on line {1110, 3939}
X(42754) = barycentric product X(i)*X(j) for these {i,j}: {7, 35015}, {11, 22464}, {244, 3262}, {273, 35014}, {513, 36038}, {514, 10015}, {517, 1111}, {523, 23788}, {693, 1769}, {859, 21207}, {903, 3259}, {908, 1086}, {1145, 6549}, {1358, 6735}, {1457, 34387}, {1465, 4858}, {1565, 1785}, {1875, 17880}, {2183, 23989}, {2397, 6545}, {2427, 23100}, {2804, 3676}, {2973, 22350}, {3120, 17139}, {3261, 3310}, {3668, 14010}, {4025, 39534}, {6548, 23757}, {16727, 21801}, {17205, 17757}, {24029, 40166}
X(42754) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 104}, {278, 39294}, {513, 36037}, {514, 13136}, {517, 765}, {649, 32641}, {859, 4570}, {908, 1016}, {1015, 909}, {1086, 34234}, {1111, 18816}, {1457, 59}, {1465, 4564}, {1647, 36944}, {1769, 100}, {1785, 15742}, {1875, 7012}, {2183, 1252}, {2397, 6632}, {2804, 3699}, {2969, 36123}, {3120, 38955}, {3125, 2250}, {3248, 34858}, {3259, 519}, {3262, 7035}, {3271, 2342}, {3310, 101}, {3326, 6735}, {3669, 37136}, {3675, 36819}, {3937, 1795}, {4858, 36795}, {6545, 2401}, {6735, 4076}, {7004, 1809}, {7649, 1309}, {8677, 1331}, {10015, 190}, {14010, 1043}, {14260, 9268}, {17139, 4600}, {21143, 2423}, {22464, 4998}, {23220, 32656}, {23757, 17780}, {23788, 99}, {24029, 31615}, {35012, 22350}, {35014, 78}, {35015, 8}, {36038, 668}, {39534, 1897}

X(42755) = X(4)X(522)∩X(3259)X(3326)

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

X(42755) lies on these lines: {4, 522}, {19, 652}, {46, 21186}, {65, 513}, {117, 14304}, {208, 7649}, {407, 656}, {514, 3577}, {661, 1901}, {1359, 6087}, {1420, 21172}, {1537, 2804}, {1769, 1846}, {1845, 2849}, {3259, 3326}, {7982, 8058}, {8677, 14299}, {23726, 23760}, {37259, 39199}

X(42755) = orthic-isogonal conjugate of X(35015)
X(42755) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 35015}, {522, 1769}, {653, 8755}, {36127, 1457}
X(42755) = crosspoint of X(i) and X(j) for these (i,j): {4, 23987}, {522, 14304}, {653, 22464}
X(42755) = crosssum of X(i) and X(j) for these (i,j): {109, 36040}, {652, 2342}
X(42755) = crossdifference of every pair of points on line {2323, 15629}
X(42755) = X(i)-isoconjugate of X(j) for these (i,j): {102, 36037}, {1309, 36055}, {1809, 36067}, {6081, 15501}, {13136, 32677}, {15629, 37136}, {32641, 36100}, {32643, 36795}
X(42755) = barycentric product X(i)*X(j) for these {i,j}: {515, 10015}, {653, 10017}, {1465, 14304}, {2182, 36038}, {2406, 35015}, {2804, 34050}, {3310, 35516}, {24035, 35014}
X(42755) = barycentric quotient X(i)/X(j) for these {i,j}: {515, 13136}, {1455, 37136}, {1769, 36100}, {2182, 36037}, {3310, 102}, {8755, 1309}, {10015, 34393}, {10017, 6332}, {14304, 36795}, {23987, 39294}, {35015, 2399}

X(42756) = X(354)X(513)∩X(3259)X(3326)

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

X(42756) lies on these lines: {118, 20622}, {196, 7649}, {354, 513}, {514, 5603}, {522, 1699}, {523, 23615}, {676, 1360}, {1427, 6129}, {1459, 34036}, {2093, 14837}, {2849, 11125}, {3259, 3326}, {4064, 22000}, {8058, 25568}, {9521, 14414}, {20988, 39199}

X(42756) = crosssum of X(36819) and X(37628)
X(42756) = crossdifference of every pair of points on line {1795, 2338}
X(42756) = X(i)-isoconjugate of X(j) for these (i,j): {103, 36037}, {104, 677}, {911, 13136}, {1309, 36056}, {2338, 37136}, {18816, 32642}, {32641, 36101}, {34234, 36039}
X(42756) = barycentric product X(i)*X(j) for these {i,j}: {516, 10015}, {676, 908}, {910, 36038}, {1769, 30807}, {1785, 39470}, {3310, 35517}, {17747, 23788}, {26006, 39534}
X(42756) = barycentric quotient X(i)/X(j) for these {i,j}: {516, 13136}, {676, 34234}, {910, 36037}, {1456, 37136}, {1769, 36101}, {1886, 1309}, {2183, 677}, {3310, 103}, {8677, 1815}, {10015, 18025}, {23220, 32657}

X(42757) = X(1)X(513)∩X(3259)X(3326)

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

X(42757) lies on on the cubic K583 and these lines: {1, 513}, {56, 6129}, {119, 2804}, {517, 37629}, {521, 1482}, {522, 946}, {523, 42440}, {900, 1537}, {1361, 1769}, {1459, 23842}, {1486, 8641}, {2849, 11700}, {3259, 3326}, {3577, 3900}, {3738, 25485}, {4017, 39791}, {4077, 41003}, {4397, 11681}, {4777, 27471}, {5903, 21189}, {6073, 23101}, {6089, 21731}, {23706, 23981}, {38505, 38514}, {39200, 42670}

X(42757) = X(i)-Ceva conjugate of X(j) for these (i,j): {693, 10015}, {934, 1465}, {15632, 24028}, {21664, 3326}
X(42757) = X(35012)-cross conjugate of X(1361)
X(42757) = crosspoint of X(i) and X(j) for these (i,j): {693, 10015}, {934, 1465}, {15632, 24028}, {23981, 39173}
X(42757) = crosssum of X(i) and X(j) for these (i,j): {522, 10265}, {692, 32641}
X(42757) = crossdifference of every pair of points on line {44, 14578}
X(42757) = X(i)-isoconjugate of X(j) for these (i,j): {104, 36037}, {190, 41933}, {909, 13136}, {1309, 1795}, {1809, 36110}, {32641, 34234}, {32669, 36795}
X(42757) = barycentric product X(i)*X(j) for these {i,j}: {513, 26611}, {514, 24028}, {517, 10015}, {651, 3326}, {693, 23980}, {905, 21664}, {908, 1769}, {1086, 15632}, {1361, 4391}, {1465, 2804}, {2183, 36038}, {2401, 23101}, {3261, 42078}, {3262, 3310}, {6335, 35012}, {13149, 41215}, {15413, 42072}, {21801, 23788}, {24029, 35015}
X(42757) = barycentric quotient X(i)/X(j) for these {i,j}: {517, 13136}, {667, 41933}, {1361, 651}, {1457, 37136}, {1769, 34234}, {2183, 36037}, {2804, 36795}, {3310, 104}, {3326, 4391}, {10015, 18816}, {14571, 1309}, {15632, 1016}, {21664, 6335}, {23101, 2397}, {23220, 14578}, {23706, 39294}, {23980, 100}, {24028, 190}, {26611, 668}, {35012, 905}, {39534, 16082}, {41220, 36054}, {42072, 1783}, {42078, 101}

X(42758) = X(55)X(513)∩X(3259)X(3326)

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

X(42758) lies on these lines: {12, 42455}, {43, 21189}, {55, 513}, {120, 20621}, {226, 522}, {514, 31397}, {521, 3870}, {523, 17874}, {614, 6129}, {650, 40131}, {656, 40952}, {693, 7179}, {764, 21105}, {900, 12831}, {905, 999}, {926, 1362}, {1056, 2401}, {1470, 22091}, {1491, 29639}, {1734, 5902}, {1769, 3310}, {1946, 8069}, {2099, 3900}, {2530, 4449}, {2804, 36038}, {3126, 3675}, {3259, 3326}, {3309, 18446}, {3667, 41561}, {3738, 41553}, {4397, 29641}, {4705, 21118}, {4885, 30742}, {15632, 23981}, {34230, 36819}

X(42758) = X(34230)-Ceva conjugate of X(3675)
X(42758) = crossdifference of every pair of points on line {104, 294}
X(42758) = X(i)-isoconjugate of X(j) for these (i,j): {104, 36086}, {105, 36037}, {294, 37136}, {666, 909}, {673, 32641}, {919, 34234}, {927, 2342}, {1309, 36057}, {1438, 13136}, {2720, 14942}, {14776, 31637}, {18816, 32666}, {32669, 36796}
X(42758) = barycentric product X(i)*X(j) for these {i,j}: {241, 2804}, {517, 918}, {518, 10015}, {665, 3262}, {672, 36038}, {908, 2254}, {1025, 35015}, {1769, 3912}, {2397, 3675}, {3263, 3310}, {3930, 23788}, {17139, 24290}, {21801, 23829}, {25083, 39534}
X(42758) = barycentric quotient X(i)/X(j) for these {i,j}: {517, 666}, {518, 13136}, {665, 104}, {672, 36037}, {918, 18816}, {1457, 36146}, {1458, 37136}, {1465, 927}, {1769, 673}, {2183, 36086}, {2223, 32641}, {2254, 34234}, {2427, 5377}, {2804, 36796}, {3262, 36803}, {3310, 105}, {3675, 2401}, {5089, 1309}, {8677, 1814}, {10015, 2481}, {22464, 34085}, {23220, 32658}, {23225, 14578}, {24029, 39293}, {24290, 38955}, {36038, 18031}

X(42759) = X(65)X(225)∩X(3259)X(3326)

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

X(42759) lies on these lines: {65, 225}, {125, 136}, {522, 867}, {1365, 2611}, {1647, 7336}, {2969, 3270}, {3259, 3326}, {5563, 33147}, {7984, 11604}, {10222, 41571}

X(42759) = X(17139)-Ceva conjugate of X(10015)
X(42759) = crosspoint of X(i) and X(j) for these (i,j): {65, 3657}, {850, 4080}, {3668, 4049}, {10015, 17139}
X(42759) = crosssum of X(i) and X(j) for these (i,j): {21, 3658}, {1576, 3285}
X(42759) = crossdifference of every pair of points on line {5546, 23090}
X(42759) = barycentric product of Kiepert hyperbola intercepts of the Sherman line
X(42759) = X(i)-isoconjugate of X(j) for these (i,j): {104, 4570}, {110, 36037}, {163, 13136}, {249, 2250}, {643, 2720}, {645, 32669}, {662, 32641}, {909, 4567}, {1101, 38955}, {1309, 4575}, {1795, 5379}, {2193, 39294}, {4592, 14776}, {4600, 34858}, {5546, 37136}, {18315, 35321}
X(42759) = barycentric product X(i)*X(j) for these {i,j}: {115, 17139}, {226, 35015}, {338, 859}, {517, 16732}, {523, 10015}, {525, 39534}, {661, 36038}, {850, 3310}, {908, 3120}, {1086, 17757}, {1111, 21801}, {1577, 1769}, {1785, 4466}, {2183, 21207}, {2677, 21907}, {2804, 7178}, {3125, 3262}, {3259, 4080}, {4024, 23788}, {4049, 23757}, {6354, 14010}, {6549, 21942}, {8677, 14618}, {21044, 22464}, {35014, 40149}
X(42759) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 38955}, {225, 39294}, {512, 32641}, {517, 4567}, {523, 13136}, {661, 36037}, {859, 249}, {908, 4600}, {1769, 662}, {2183, 4570}, {2489, 14776}, {2501, 1309}, {2643, 2250}, {2677, 32849}, {2680, 5291}, {2804, 645}, {3120, 34234}, {3121, 34858}, {3122, 909}, {3125, 104}, {3259, 16704}, {3262, 4601}, {3310, 110}, {4017, 37136}, {7180, 2720}, {8034, 2423}, {8677, 4558}, {10015, 99}, {14010, 7058}, {14571, 5379}, {16732, 18816}, {17139, 4590}, {17757, 1016}, {21801, 765}, {22464, 4620}, {23220, 32661}, {23788, 4610}, {35014, 1812}, {35015, 333}, {36038, 799}, {39534, 648}
X(42759) = {X(2969),X(38357)}-harmonic conjugate of X(38389)

X(42760) = X(126)X(1560)∩X(3259)X(3326)

Barycentrics    (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) : :

X(42760) lies on these lines: {126, 1560}, {523, 29639}, {1366, 4750}, {3259, 3326}, {3310, 10015}, {7658, 40940}, {17069, 37520}

X(42760) = crossdifference of every pair of points on line {5547, 14908}
X(42760) = X(i)-isoconjugate of X(j) for these (i,j): {111, 36037}, {691, 2250}, {897, 32641}, {909, 5380}, {923, 13136}, {1309, 36060}, {5547, 37136}, {36142, 38955}
X(42760) = barycentric product X(i)*X(j) for these {i,j}: {524, 10015}, {690, 17139}, {859, 35522}, {896, 36038}, {908, 4750}, {1769, 14210}, {2804, 7181}, {3262, 14419}, {3266, 3310}, {4062, 23788}, {6390, 39534}, {14432, 22464}
X(42760) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 32641}, {468, 1309}, {517, 5380}, {524, 13136}, {690, 38955}, {859, 691}, {896, 36037}, {1769, 897}, {2642, 2250}, {3310, 111}, {4750, 34234}, {8677, 895}, {10015, 671}, {14419, 104}, {17139, 892}, {23220, 14908}, {30605, 36921}, {39534, 17983}

X(42761) = X(37)X(226)∩X(3259)X(3326)

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

X(42761) lies on these lines: {3, 36735}, {37, 226}, {115, 127}, {1086, 16596}, {1367, 4466}, {3259, 3326}, {14919, 21907}, {16595, 40622}, {17073, 17276}, {20278, 21659}

X(42761) = X(i)-complementary conjugate of X(j) for these (i,j): {647, 31845}, {759, 30476}, {1807, 3835}, {2161, 20316}, {3049, 35069}, {22383, 214}, {24624, 21259}, {32671, 5972}, {34079, 8062}
X(42761) = X(i)-Ceva conjugate of X(j) for these (i,j): {4080, 525}, {17139, 8677}, {41804, 9033}
X(42761) = crosssum of X(14591) and X(41502)
X(42761) = crossdifference of every pair of points on line {1576, 14776}
X(42761) = X(i)-isoconjugate of X(j) for these (i,j): {112, 36037}, {162, 32641}, {163, 1309}, {250, 2250}, {643, 32702}, {662, 14776}, {909, 5379}, {933, 35321}, {2194, 39294}, {5546, 36110}, {13136, 32676}, {32669, 36797}
X(42761) = barycentric product X(i)*X(j) for these {i,j}: {125, 17139}, {307, 35015}, {339, 859}, {525, 10015}, {656, 36038}, {850, 8677}, {908, 4466}, {1441, 35014}, {1565, 17757}, {1769, 14208}, {1785, 17216}, {2804, 17094}, {3262, 18210}, {3265, 39534}, {3267, 3310}, {4064, 23788}, {6356, 14010}, {21207, 22350}
X(42761) = barycentric quotient X(i)/X(j) for these {i,j}: {125, 38955}, {226, 39294}, {512, 14776}, {517, 5379}, {523, 1309}, {525, 13136}, {647, 32641}, {656, 36037}, {859, 250}, {1769, 162}, {2804, 36797}, {3120, 36123}, {3259, 37168}, {3310, 112}, {3708, 2250}, {4017, 36110}, {4466, 34234}, {7180, 32702}, {8677, 110}, {10015, 648}, {16732, 16082}, {17139, 18020}, {17757, 15742}, {18210, 104}, {22350, 4570}, {23220, 1576}, {35012, 859}, {35014, 21}, {35015, 29}, {36038, 811}, {39534, 107}

X(42762) = X(57)X(652)∩X(3259)X(3326)

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

X(42762) lies on these lines: {57, 652}, {513, 17603}, {514, 5219}, {661, 6545}, {1638, 3321}, {3259, 3326}, {3835, 23726}, {4025, 31164}, {4120, 23737}, {4369, 23728}, {4885, 23727}, {6332, 30828}, {14475, 23730}, {17605, 23615}, {25259, 31053}, {33573, 40629}

X(42762) = crossdifference of every pair of points on line {2342, 4845}
X(42762) = X(i)-isoconjugate of X(j) for these (i,j): {1156, 32641}, {2291, 36037}, {2342, 37139}, {2720, 41798}, {4845, 37136}, {13136, 34068}, {32728, 36795}
X(42762) = barycentric product X(i)*X(j) for these {i,j}: {527, 10015}, {908, 1638}, {1155, 36038}, {1323, 2804}, {1769, 30806}, {3262, 14413}, {6366, 22464}, {17139, 30574}, {23757, 36887}
X(42762) = barycentric quotient X(i)/X(j) for these {i,j}: {527, 13136}, {1055, 32641}, {1155, 36037}, {1457, 14733}, {1465, 37139}, {1638, 34234}, {1769, 1156}, {3310, 2291}, {6139, 2342}, {6610, 37136}, {10015, 1121}, {14413, 104}, {14414, 1809}, {22464, 35157}, {23710, 1309}, {30573, 36944}, {30574, 38955}

X(42763) = X(513)X(11934)∩X(3259)X(3326)

Barycentrics    (b - c)*(-(a^2*b) + b^3 - a^2*c + 2*a*b*c - b^2*c - b*c^2 + c^3)*(-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(42763) lies on these lines: {513, 11934}, {517, 10015}, {676, 1155}, {908, 2804}, {3259, 3326}, {3328, 23766}, {6550, 23745}

X(42763) = crossdifference of every pair of points on line {218, 32641}
X(42763) = X(i)-isoconjugate of X(j) for these (i,j): {840, 36037}, {32641, 37131}
X(42763) = barycentric product X(i)*X(j) for these {i,j}: {528, 10015}, {1643, 3262}, {2246, 36038}, {2804, 5723}
X(42763) = barycentric quotient X(i)/X(j) for these {i,j}: {528, 13136}, {1643, 104}, {1769, 37131}, {2246, 36037}, {3310, 840}, {10015, 18821}

X(42764) = X(3120)X(3126)∩X(3259)X(3326)

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

X(42764) lies on these lines: {693, 33151}, {3120, 3126}, {3259, 3326}, {4415, 40166}, {4526, 4728}, {4885, 17595}, {7658, 24177}, {10015, 26611}

X(42764) = crossdifference of every pair of points on line {32641, 32718}
X(42764) = X(i)-isoconjugate of X(j) for these (i,j): {104, 34075}, {739, 36037}, {898, 909}, {4607, 34858}, {32641, 37129}, {32669, 36798}, {32718, 34234}
X(42764) = barycentric product X(i)*X(j) for these {i,j}: {536, 10015}, {891, 3262}, {899, 36038}, {908, 4728}, {1769, 6381}, {3310, 35543}, {3994, 23788}, {14430, 22464}, {14431, 17139}
X(42764) = barycentric quotient X(i)/X(j) for these {i,j}: {517, 898}, {536, 13136}, {890, 34858}, {891, 104}, {899, 36037}, {908, 4607}, {1646, 2423}, {1769, 37129}, {2183, 34075}, {2397, 5381}, {2804, 36798}, {3230, 32641}, {3262, 889}, {3310, 739}, {3768, 909}, {4728, 34234}, {10015, 3227}, {14431, 38955}, {23757, 36872}, {28603, 36921}, {30583, 36944}, {36038, 31002}

X(42765) = X(908)X(1769)∩X(3259)X(3326)

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

X(42765) lies on these lines: {908, 1769}, {2254, 32856}, {3259, 3326}, {3262, 36038}, {4768, 21241}, {10015, 17757}, {18201, 25380}

X(42765) = X(2382)-isoconjugate of X(36037)
X(42765) = barycentric product X(i)*X(j) for these {i,j}: {537, 10015}, {908, 36848}, {20331, 36038}
X(42765) = barycentric quotient X(i)/X(j) for these {i,j}: {537, 13136}, {3310, 2382}, {10015, 18822}, {20331, 36037}, {36848, 34234}

X(42766) = X(513)X(20359)∩X(3259)X(3326)

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

X(42766) lies on these lines: {513, 20359}, {522, 3944}, {1459, 33144}, {3259, 3326}, {3837, 20366}, {20316, 27538}

X(42766) = X(36814)-Ceva conjugate of X(21140)
X(42766) = X(i)-isoconjugate of X(j) for these (i,j): {727, 36037}, {2720, 8851}, {8709, 34858}, {13136, 34077}, {20332, 32641}, {32669, 36799}
X(42766) = barycentric product X(i)*X(j) for these {i,j}: {517, 20908}, {726, 10015}, {908, 3837}, {1575, 36038}, {2397, 21140}, {3310, 35538}, {17139, 21053}
X(42766) = barycentric quotient X(i)/X(j) for these {i,j}: {726, 13136}, {908, 8709}, {1463, 37136}, {1575, 36037}, {1769, 20332}, {2804, 36799}, {3009, 32641}, {3310, 727}, {3837, 34234}, {6373, 909}, {10015, 3226}, {20908, 18816}, {21053, 38955}, {21140, 2401}, {22092, 1795}, {36038, 32020}

X(42767) = X(513)X(21334)∩X(3259)X(3326)

Barycentrics    (b - c)*(b + 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(42767) lies on these lines: {513, 21334}, {522, 24210}, {523, 4415}, {656, 3914}, {804, 3027}, {1577, 4424}, {1769, 23788}, {3259, 3326}, {3971, 4086}

X(42767) = crossdifference of every pair of points on line {2311, 32641}
X(42767) = X(i)-isoconjugate of X(j) for these (i,j): {741, 36037}, {909, 4584}, {1808, 36110}, {2311, 37136}, {4589, 34858}, {13136, 18268}, {32641, 37128}, {32669, 36800}
X(42767) = barycentric product X(i)*X(j) for these {i,j}: {740, 10015}, {812, 17757}, {908, 4010}, {1577, 15507}, {1769, 3948}, {1785, 24459}, {2238, 36038}, {2804, 16609}, {3262, 21832}, {3310, 35544}, {3766, 21801}, {4037, 23788}, {6735, 7212}
X(42767) = barycentric quotient X(i)/X(j) for these {i,j}: {517, 4584}, {740, 13136}, {908, 4589}, {1284, 37136}, {1769, 37128}, {2238, 36037}, {2804, 36800}, {3262, 4639}, {3310, 741}, {3747, 32641}, {4010, 34234}, {4155, 2250}, {4455, 909}, {10015, 18827}, {15507, 662}, {17757, 4562}, {21801, 660}, {21832, 104}, {36038, 40017}

X(42768) = X(12)X(523)∩X(3259)X(3326)

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

X(42768) lies on these lines: {1, 35050}, {12, 523}, {37, 661}, {65, 656}, {73, 3657}, {513, 2646}, {522, 12047}, {526, 3028}, {652, 2260}, {1104, 6129}, {1769, 14299}, {3259, 3326}, {3737, 37816}, {6003, 21740}, {11009, 35057}, {14310, 36035}

X(42768) = X(i)-Ceva conjugate of X(j) for these (i,j): {1020, 2245}, {1577, 2610}
X(42768) = crosspoint of X(i) and X(j) for these (i,j): {1, 4242}, {758, 4551}, {1577, 36038}
X(42768) = crosssum of X(759) and X(3737)
X(42768) = crossdifference of every pair of points on line {2250, 2341}
X(42768) = X(i)-isoconjugate of X(j) for these (i,j): {110, 40437}, {759, 36037}, {1793, 36110}, {2250, 37140}, {2341, 37136}, {2720, 6740}, {13136, 34079}, {24624, 32641}, {36069, 38955}
X(42768) = barycentric product X(i)*X(j) for these {i,j}: {517, 4707}, {523, 16586}, {525, 1845}, {758, 10015}, {1577, 34586}, {1769, 3936}, {2245, 36038}, {2610, 17139}, {2804, 18593}, {3262, 21828}, {3310, 35550}, {3960, 17757}, {4053, 23788}, {4453, 21801}
X(42768) = barycentric quotient X(i)/X(j) for these {i,j}: {661, 40437}, {758, 13136}, {859, 37140}, {1464, 37136}, {1769, 24624}, {1845, 648}, {2245, 36037}, {2610, 38955}, {3310, 759}, {3724, 32641}, {4707, 18816}, {10015, 14616}, {16586, 99}, {17757, 36804}, {21828, 104}, {34586, 662}, {42666, 2250}

X(42769) = X(3)X(513)∩X(3259)X(3326)

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

X(42769) lies on these lines: {3, 513}, {73, 1459}, {119, 34332}, {521, 37700}, {522, 12608}, {661, 18591}, {1643, 5158}, {1745, 23800}, {3259, 3326}, {3835, 18589}, {6261, 15313}, {17102, 24457}, {20315, 34823}, {21530, 31946}, {41172, 41179}

X(42769) = complement of X(43933)
X(42769) = X(i)-complementary conjugate of X(j) for these (i,j): {101, 26011}, {255, 35014}, {765, 8677}, {906, 3911}, {1331, 517}, {1795, 33646}, {2397, 20305}, {2427, 226}, {4575, 15325}, {22350, 11}, {23981, 1210}, {24029, 16608}, {32656, 8609}
X(42769) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 35014}, {513, 8677}, {13397, 517}
X(42769) = crosssum of X(i) and X(j) for these (i,j): {100, 6099}, {1783, 14776}
X(42769) = crossdifference of every pair of points on line {8609, 15500}
X(42769) = X(i)-isoconjugate of X(j) for these (i,j): {104, 36106}, {913, 13136}, {915, 36037}, {1309, 36052}, {1897, 15381}, {6099, 36123}, {32641, 37203}, {32698, 34234}
X(42769) = barycentric product X(i)*X(j) for these {i,j}: {119, 905}, {912, 10015}, {914, 1769}, {2252, 36038}
X(42769) = barycentric quotient X(i)/X(j) for these {i,j}: {119, 6335}, {912, 13136}, {1769, 37203}, {2183, 36106}, {2252, 36037}, {3310, 915}, {8609, 1309}, {8677, 2990}, {22383, 15381}, {23220, 32655}

X(42770) = X(650)X(1086)∩X(3259)X(3326)

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

X(42770) lies on these lines: {241, 5089}, {650, 1086}, {908, 1465}, {1566, 3323}, {3259, 3326}, {3328, 21105}, {4077, 4858}, {4468, 26932}, {5532, 21118}, {16732, 23749}

X(42770) = X(i)-isoconjugate of X(j) for these (i,j): {909, 5377}, {919, 36037}, {13136, 32666}, {32641, 36086}, {32669, 36802}
X(42770) = barycentric product X(i)*X(j) for these {i,j}: {918, 10015}, {2254, 36038}, {3262, 3675}, {4088, 23788}, {9436, 35015}
X(42770) = barycentric quotient X(i)/X(j) for these {i,j}: {517, 5377}, {665, 32641}, {918, 13136}, {1769, 36086}, {2254, 36037}, {2804, 36802}, {3310, 919}, {3675, 104}, {5236, 39294}, {10015, 666}, {22464, 39293}, {35015, 14942}

X(42771) = X(11)X(1491)∩X(3259)X(3326)

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

X(42771) lies on these lines: {11, 1491}, {55, 24488}, {661, 2310}, {764, 3328}, {2183, 41215}, {2223, 2356}, {3022, 4775}, {3259, 3326}, {3271, 8641}, {4705, 5532}, {15615, 35505}

X(42771) = crossdifference of every pair of points on line {666, 2401}
X(42771) = X(i)-isoconjugate of X(j) for these (i,j): {104, 39293}, {666, 37136}, {927, 36037}, {1814, 39294}, {13136, 36146}, {32641, 34085}, {32669, 36803}
X(42771) = barycentric product X(i)*X(j) for these {i,j}: {517, 17435}, {665, 2804}, {672, 35015}, {926, 10015}, {5089, 35014}
X(42771) = barycentric quotient X(i)/X(j) for these {i,j}: {926, 13136}, {1769, 34085}, {2183, 39293}, {2356, 39294}, {2804, 36803}, {3310, 927}, {8638, 32641}, {17435, 18816}, {35015, 18031}

X(42772) = X(57)X(513)∩X(3259)X(3326)

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

X(42772) lies on these lines: {25, 23224}, {57, 513}, {514, 7682}, {521, 19541}, {3259, 3326}, {15762, 39212}

X(42772) = crossdifference of every pair of points on line {6603, 15501}
X(42772) = X(972)-isoconjugate of X(36037)
X(42772) = barycentric product X(i)*X(j) for these {i,j}: {971, 10015}, {2272, 36038}
X(42772) = barycentric quotient X(i)/X(j) for these {i,j}: {971, 13136}, {2272, 36037}, {3310, 972}

leftri

Gibert (i,j,k) points on the cubic K1208: X(42773)-X(42784)

rightri

This preamble and points X(42773)-X(424784) are contributed by Peter Moses, April 24, 2021.

Gibert points are introduced in the preamble just before X(42085). See K1208.



X(42773) = GIBERT (3,2,12) POINT

Barycentrics    Sqrt[3]*a^2*S + 12*a^2*SA + 4*SB*SC : :

X(42773) lies on the cubic K1208 and these lines: {2, 42164}, {3, 13}, {5, 42626}, {6, 3523}, {15, 15720}, {18, 42116}, {62, 15693}, {140, 5339}, {376, 5350}, {395, 42479}, {396, 15717}, {397, 10299}, {398, 631}, {546, 42474}, {549, 22236}, {550, 42092}, {1656, 10645}, {1657, 42098}, {3090, 42500}, {3522, 23302}, {3524, 16772}, {3525, 5343}, {3526, 5352}, {3528, 5366}, {3530, 22238}, {3533, 5321}, {3534, 42488}, {3545, 42587}, {3839, 42515}, {3843, 42529}, {3850, 42090}, {3851, 33417}, {3854, 42108}, {5054, 5238}, {5055, 42434}, {5056, 42087}, {5070, 36967}, {5073, 16966}, {5318, 21735}, {5351, 15700}, {8703, 42586}, {10303, 42147}, {10304, 42166}, {10654, 14869}, {11134, 13347}, {11481, 15712}, {11489, 42687}, {11539, 42159}, {12100, 40693}, {12108, 40694}, {15692, 42148}, {15694, 16964}, {15696, 37832}, {15701, 42507}, {15702, 42599}, {15703, 42596}, {15706, 41943}, {15713, 42509}, {15714, 41119}, {15716, 16267}, {15718, 16962}, {15722, 16963}, {16239, 42160}, {17800, 42581}, {18582, 33923}, {21734, 42165}, {22237, 23303}, {34200, 42161}, {41120, 42591}, {41973, 42129}, {41981, 42138}, {42095, 42157}, {42124, 42151}, {42132, 42431}, {42139, 42684}, {42436, 42532}

X(42773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16241, 42156}, {3, 42156, 42625}, {3, 42490, 16644}, {6, 3523, 42774}, {140, 11480, 5339}, {546, 42610, 42474}, {549, 22236, 42491}, {631, 36836, 16645}, {3524, 16772, 36843}, {3526, 5352, 42154}, {5054, 5238, 42153}, {5340, 42625, 42158}, {11539, 42159, 42611}, {15712, 42152, 11481}, {16241, 42625, 16644}, {42156, 42158, 5340}

X(42774) = GIBERT (-3,2,12) POINT

Barycentrics    Sqrt[3]*a^2*S - 12*a^2*SA - 4*SB*SC : :

X(42774) lies on the cubic K1208 and these lines: {2, 42165}, {3, 14}, {5, 42625}, {6, 3523}, {16, 15720}, {17, 42115}, {61, 15693}, {140, 5340}, {376, 5349}, {395, 15717}, {396, 42478}, {397, 631}, {398, 10299}, {546, 42475}, {549, 22238}, {550, 42089}, {1656, 10646}, {1657, 42095}, {3090, 42501}, {3522, 23303}, {3524, 16773}, {3525, 5344}, {3526, 5351}, {3528, 5365}, {3530, 22236}, {3533, 5318}, {3534, 42489}, {3545, 42586}, {3839, 42514}, {3843, 42528}, {3850, 42091}, {3851, 33416}, {3854, 42109}, {5054, 5237}, {5055, 42433}, {5056, 42088}, {5070, 36968}, {5073, 16967}, {5321, 21735}, {5352, 15700}, {8703, 42587}, {10303, 42148}, {10304, 42163}, {10653, 14869}, {11137, 13347}, {11480, 15712}, {11488, 42686}, {11539, 42162}, {12100, 40694}, {12108, 40693}, {15692, 42147}, {15694, 16965}, {15696, 37835}, {15701, 42506}, {15702, 42598}, {15703, 42597}, {15706, 41944}, {15713, 42508}, {15714, 41120}, {15716, 16268}, {15718, 16963}, {15722, 16962}, {16239, 42161}, {17800, 42580}, {18581, 33923}, {21734, 42164}, {22235, 23302}, {34200, 42160}, {41119, 42590}, {41974, 42132}, {41981, 42135}, {42098, 42158}, {42121, 42150}, {42129, 42432}, {42142, 42685}, {42435, 42533}

X(42774) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16242, 42153}, {3, 42153, 42626}, {3, 42491, 16645}, {6, 3523, 42773}, {140, 11481, 5340}, {546, 42611, 42475}, {549, 22238, 42490}, {631, 36843, 16644}, {3524, 16773, 36836}, {3526, 5351, 42155}, {5054, 5237, 42156}, {5339, 42626, 42157}, {11539, 42162, 42610}, {15712, 42149, 11480}, {16242, 42626, 16645}, {42153, 42157, 5339}

X(42775) = GIBERT (6,11,6) POINT

Barycentrics    Sqrt[3]*a^2*S + 3*a^2*SA + 11*SB*SC : :

X(42774) lies on the cubic K1208 and these lines: {2, 42165}, {4, 15}, {5, 5344}, {6, 3854}, {13, 3855}, {18, 3545}, {61, 41099}, {140, 42131}, {397, 3091}, {546, 5365}, {547, 42588}, {631, 42431}, {1656, 5366}, {1657, 42146}, {3090, 16242}, {3146, 42626}, {3522, 42094}, {3523, 5350}, {3524, 42581}, {3525, 36969}, {3528, 42546}, {3529, 12820}, {3533, 42086}, {3543, 42598}, {3544, 10653}, {3832, 5339}, {3839, 5349}, {3850, 42128}, {3851, 5335}, {3858, 5334}, {5056, 5318}, {5059, 23302}, {5067, 42161}, {5068, 5340}, {5071, 16965}, {5343, 11542}, {7486, 42155}, {10299, 16966}, {12816, 15702}, {15022, 42148}, {15683, 42490}, {15688, 42590}, {15708, 42610}, {15709, 42433}, {15720, 42137}, {16644, 17578}, {17538, 42488}, {19106, 21735}, {22237, 42107}, {35018, 42493}, {40694, 41106}, {41121, 42160}, {42114, 42158}, {42429, 42592}

X(42775) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 17, 42119}, {4, 42142, 42494}, {4, 42494, 11488}, {17, 42106, 4}, {397, 3091, 42495}, {397, 42495, 37641}, {1656, 5366, 42120}, {1656, 42138, 5366}, {3523, 5350, 42141}, {3832, 22235, 5339}, {3832, 42166, 37640}, {5068, 5340, 11489}, {5339, 22235, 37640}, {5339, 42166, 22235}, {5340, 42110, 5068}, {5350, 42098, 3523}, {18582, 42140, 11488}

X(42776) = GIBERT (-6,11,6) POINT

Barycentrics    -Sqrt[3]*a^2*S + 3*a^2*SA + 11*SB*SC : :

X(42776) lies on the cubic K1208 and these lines: {2, 42164}, {4, 16}, {5, 5343}, {6, 3854}, {14, 3855}, {17, 3545}, {62, 41099}, {140, 42130}, {398, 3091}, {546, 5366}, {547, 42589}, {631, 42432}, {1656, 5365}, {1657, 42143}, {3090, 16241}, {3146, 42625}, {3522, 42093}, {3523, 5349}, {3524, 42580}, {3525, 36970}, {3528, 42545}, {3529, 12821}, {3533, 42085}, {3543, 42599}, {3544, 10654}, {3832, 5340}, {3839, 5350}, {3850, 42125}, {3851, 5334}, {3858, 5335}, {5056, 5321}, {5059, 23303}, {5067, 42160}, {5068, 5339}, {5071, 16964}, {5344, 11543}, {7486, 42154}, {10299, 16967}, {12817, 15702}, {15022, 42147}, {15683, 42491}, {15688, 42591}, {15708, 42611}, {15709, 42434}, {15720, 42136}, {16645, 17578}, {17538, 42489}, {19107, 21735}, {22235, 42110}, {35018, 42492}, {40693, 41106}, {41122, 42161}, {42111, 42157}, {42430, 42593}

X(42776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 18, 42120}, {4, 42139, 42495}, {4, 42495, 11489}, {18, 42103, 4}, {398, 3091, 42494}, {398, 42494, 37640}, {1656, 5365, 42119}, {1656, 42135, 5365}, {3523, 5349, 42140}, {3832, 22237, 5340}, {3832, 42163, 37641}, {5068, 5339, 11488}, {5339, 42107, 5068}, {5340, 22237, 37641}, {5340, 42163, 22237}, {5349, 42095, 3523}, {18581, 42141, 11489}

X(42777) = GIBERT (15,7,8) POINT

Barycentrics    5*Sqrt[3]*a^2*S + 8*a^2*SA + 14*SB*SC : :

X(42777) lies on the cubic K1208 and these lines: {2, 42517}, {6, 5071}, {13, 15}, {16, 15713}, {17, 632}, {61, 3858}, {381, 42692}, {395, 1656}, {397, 631}, {398, 3091}, {546, 42694}, {3412, 42164}, {3522, 16772}, {3843, 5349}, {3845, 12821}, {5076, 42147}, {5321, 41099}, {5335, 19708}, {5340, 17538}, {5965, 22847}, {10124, 34755}, {10653, 15693}, {11480, 15697}, {11485, 41119}, {11488, 15692}, {11812, 42686}, {12812, 37835}, {15686, 42684}, {15687, 34754}, {15688, 42687}, {15694, 23302}, {15695, 41112}, {15696, 42152}, {15700, 42685}, {15711, 41107}, {15712, 16241}, {15714, 41943}, {16242, 42627}, {16808, 42633}, {16961, 37832}, {16966, 42521}, {17131, 37352}, {17578, 22235}, {18582, 19709}, {23303, 42634}, {31693, 35019}, {35403, 42128}, {35434, 42085}, {38335, 42691}, {41101, 42138}, {41121, 42110}, {41985, 42636}, {42095, 42503}, {42101, 42516}, {42108, 42511}

X(42777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5071, 42778}, {13, 16267, 42496}, {13, 42496, 396}, {396, 42087, 16962}, {11542, 16267, 396}, {11542, 42496, 13}, {15694, 42512, 23302}

X(42778) = GIBERT (-15,7,8) POINT

Barycentrics    5*Sqrt[3]*a^2*S - 8*a^2*SA - 14*SB*SC : :

X(42778) lies on the cubic K1208 and these lines: {2, 42516}, {6, 5071}, {14, 16}, {15, 15713}, {18, 632}, {62, 3858}, {381, 42693}, {396, 1656}, {397, 3091}, {398, 631}, {546, 42695}, {3411, 42165}, {3522, 16773}, {3843, 5350}, {3845, 12820}, {5076, 42148}, {5318, 41099}, {5334, 19708}, {5339, 17538}, {5965, 22893}, {10124, 34754}, {10654, 15693}, {11481, 15697}, {11486, 41120}, {11489, 15692}, {11812, 42687}, {12812, 37832}, {15686, 42685}, {15687, 34755}, {15688, 42686}, {15694, 23303}, {15695, 41113}, {15696, 42149}, {15700, 42684}, {15711, 41108}, {15712, 16242}, {15714, 41944}, {16241, 42628}, {16809, 42634}, {16960, 37835}, {16967, 42520}, {17131, 37351}, {17578, 22237}, {18581, 19709}, {23302, 42633}, {31694, 35020}, {35403, 42125}, {35434, 42086}, {38335, 42690}, {41100, 42135}, {41122, 42107}, {41985, 42635}, {42098, 42502}, {42102, 42517}, {42109, 42510}

X(42778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5071, 42777}, {14, 16268, 42497}, {14, 42497, 395}, {395, 42088, 16963}, {11543, 16268, 395}, {11543, 42497, 14}, {15694, 42513, 23303}

X(42779) = GIBERT (15,4,3) POINT

Barycentrics    5*Sqrt[3]*a^2*S + 3*a^2*SA + 8*SB*SC : :

X(42779) lies on the cubic K1208 and these lines: {2, 17}, {3, 42612}, {6, 3851}, {13, 398}, {14, 3855}, {15, 397}, {16, 15720}, {18, 11542}, {61, 382}, {140, 16960}, {396, 3530}, {1657, 34754}, {3107, 32450}, {3411, 42598}, {3412, 3528}, {3529, 16965}, {3543, 42520}, {3544, 40694}, {5056, 16961}, {5071, 33607}, {5079, 37835}, {5238, 15688}, {5318, 42630}, {5335, 42112}, {5343, 42162}, {5344, 19107}, {5350, 15687}, {5351, 15700}, {5366, 10654}, {10188, 42627}, {10299, 10646}, {11485, 42431}, {11737, 41122}, {14269, 41108}, {14869, 42500}, {14892, 33606}, {15681, 22236}, {15699, 42521}, {15707, 36843}, {16242, 41977}, {16772, 17504}, {16773, 42496}, {16962, 34200}, {18581, 22235}, {22114, 33560}, {22846, 33464}, {33923, 42416}, {35739, 42230}, {37641, 42581}, {38071, 42166}, {41101, 42161}, {41121, 42153}, {42137, 42415}, {42165, 42633}, {42593, 42610}

X(42779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 42636, 41944}, {6, 3851, 42780}, {15, 397, 41974}, {61, 5340, 42432}, {62, 16267, 42488}, {62, 40693, 16267}, {62, 42488, 41944}, {3412, 10653, 5352}, {5340, 42432, 36969}, {41107, 42635, 15681}

X(42780) = GIBERT (-15,4,3) POINT

Barycentrics    5*Sqrt[3]*a^2*S - 3*a^2*SA - 8*SB*SC : :

X(42780) lies on the cubic K1208 and these lines: {2, 18}, {3, 42613}, {6, 3851}, {13, 3855}, {14, 397}, {15, 15720}, {16, 398}, {17, 11543}, {62, 382}, {140, 16961}, {395, 3530}, {1657, 34755}, {3106, 32450}, {3411, 3528}, {3412, 42599}, {3529, 16964}, {3543, 42521}, {3544, 40693}, {5056, 16960}, {5071, 33606}, {5079, 37832}, {5237, 15688}, {5321, 42629}, {5334, 42113}, {5343, 19106}, {5344, 42159}, {5349, 15687}, {5352, 15700}, {5365, 10653}, {10187, 42628}, {10299, 10645}, {11486, 42432}, {11737, 41121}, {14269, 41107}, {14869, 42501}, {14892, 33607}, {15681, 22238}, {15699, 42520}, {15707, 36836}, {16241, 41978}, {16772, 42497}, {16773, 17504}, {16963, 34200}, {18582, 22237}, {22113, 33561}, {22891, 33465}, {33923, 42415}, {37640, 42580}, {38071, 42163}, {41100, 42160}, {41122, 42156}, {42136, 42416}, {42164, 42634}, {42592, 42611}

X(42780) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 42635, 41943}, {6, 3851, 42779},{16, 398, 41973}, {61, 16268, 42489}, {61, 40694, 16268}, {61, 42489, 41943}, {62, 5339, 42431}, {3411, 10654, 5351}, {5339, 42431, 36970}, {41108, 42636, 15681}

X(42781) = GIBERT (25,9,8) POINT

Barycentrics    25*a^2*S/Sqrt[3] + 8*a^2*SA + 18*SB*SC : :

X(42781) lies on the cubic K1208 and these lines: {2, 42517}, {6, 3544}, {14, 38071}, {16, 14869}, {18, 11542}, {382, 5318}, {396, 34200}, {397, 10299}, {550, 42684}, {3530, 16960}, {5335, 42626}, {5349, 42138}, {16267, 42501}, {16773, 42627}, {33602, 42094}, {42124, 42506}, {42691, 42692}

X(42782) = GIBERT (-25,9,8) POINT

Barycentrics    25*a^2*S/Sqrt[3] - 8*a^2*SA - 18*SB*SC : :

X(42782) lies on the cubic K1208 and these lines: {2, 42516}, {6, 3544}, {13, 38071}, {15, 14869}, {17, 11543}, {382, 5321}, {395, 34200}, {398, 10299}, {550, 42685}, {3530, 16961}, {5334, 42625}, {5350, 42135}, {16268, 42500}, {16772, 42628}, {33603, 42093}, {42121, 42507}, {42690, 42693}

X(42783) = GIBERT (SQRT(6),1,1) POINT

Barycentrics    Sqrt[2]*a^2*S + a^2*SA + 2*SB*SC : :

X(42783) lies on the cubic K1208 and these lines: {2, 41975}, {3, 41976}, {4, 41979}, {5, 6}, {30, 42645}, {371, 3387}, {372, 3373}, {546, 42646}, {590, 3372}, {615, 3386}, {1587, 14782}, {1588, 14783}, {3068, 14785}, {3069, 14784}, {3070, 3385}, {3071, 3371}, {3374, 6419}, {3388, 6420}

X(42783) = crosspoint of X(3373) and X(3387)
X(42783) = crosssum of X(3371) and X(3385)
X(42783) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11542, 11543, 42647}, {41976, 41980, 3}

X(42784) = GIBERT (-SQRT(6),1,1) POINT

Barycentrics    Sqrt[2]*a^2*S - a^2*SA - 2*SB*SC : :

X(42784) lies on the cubic K1208 and these lines: {2, 41976}, {3, 41975}, {4, 41980}, {5, 6}, {30, 42646}, {371, 3388}, {372, 3374}, {546, 42645}, {590, 3371}, {615, 3385}, {1587, 14783}, {1588, 14782}, {3068, 14784}, {3069, 14785}, {3070, 3386}, {3071, 3372}, {3373, 6419}, {3387, 6420}

X(42784) = crosspoint of X(3374) and X(3388)
X(42784) = crosssum of X(3372) and X(3386)
X(42784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11542, 11543, 42648}, {41975, 41979, 3}

X(42785) = X(5)X(3631)∩X(6)X(13)

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

X(42785) lies on the cubic K1209 and these lines: {5, 3631}, {6, 13}, {20, 5092}, {140, 3098}, {141, 10109}, {182, 3627}, {193, 25561}, {262, 8782}, {511, 3090}, {546, 39561}, {576, 12811}, {597, 12101}, {3091, 5097}, {3524, 31670}, {3545, 7926}, {3589, 8703}, {3618, 15682}, {3620, 20423}, {3629, 5066}, {3630, 11178}, {3763, 25565}, {3845, 6329}, {3851, 5965}, {3858, 12007}, {3861, 18583}, {5068, 14853}, {5070, 33878}, {5072, 5102}, {5073, 12017}, {5640, 13417}, {5943, 12058}, {7706, 34584}, {7875, 14492}, {10168, 15689}, {11541, 20190}, {15699, 21850}, {15701, 19924}, {18376, 41593}, {18394, 39874}, {19709, 40341}, {22234, 39884}, {22819, 23259}, {22820, 23249}, {32455, 41990}, {34754, 37332}, {34755, 37333}

X(42785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5476, 19130, 3818}

X(42786) = X(5)X(3098)∩X(6)X(17)

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

X(42786) lies on the cubic K1209 and these lines: {2, 1495}, {5, 3098}, {6, 17}, {141, 547}, {156, 182}, {511, 3090}, {542, 15703}, {575, 40330}, {576, 3631}, {632, 17508}, {1350, 5079}, {1352, 5067}, {1514, 7399}, {3054, 41412}, {3091, 14810}, {3523, 29323}, {3526, 29012}, {3589, 11178}, {3620, 7486}, {3630, 18583}, {3763, 5055}, {3850, 21167}, {3851, 29317}, {5070, 10516}, {5071, 31670}, {5072, 31884}, {5097, 20080}, {5169, 5888}, {5480, 35018}, {6644, 32600}, {6683, 32429}, {6688, 37643}, {6723, 32305}, {7388, 22819}, {7389, 22820}, {7405, 11438}, {7571, 15066}, {7822, 35002}, {9301, 15821}, {9873, 16896}, {9993, 16988}, {10168, 18440}, {10182, 34775}, {10564, 14787}, {11430, 14786}, {12055, 31455}, {12584, 23515}, {12900, 19140}, {15088, 32273}, {15720, 33751}, {16187, 37454}, {18376, 35228}, {18553, 39874}, {20582, 21850}, {21358, 25565}, {31415, 41413}, {32455, 38079}, {34417, 37990}, {37353, 41462}

X(42786) = crossdifference of every pair of points on line {1510, 9210}
X(42786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 34573, 3098}, {1656, 24206, 38317}, {3763, 5055, 19130}, {10576, 10577, 7755}, {16966, 16967, 7746}, {24206, 38317, 34507}

X(42787) = X(3)X(147)∩X(5)X(3098)

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

X(42787) lies on the cubic K1209 and these lines: {3, 147}, {5, 3098}, {30, 3096}, {32, 549}, {140, 262}, {376, 18503}, {548, 9873}, {550, 9996}, {626, 22566}, {631, 9301}, {2023, 7749}, {2076, 2548}, {3094, 5305}, {3530, 26316}, {3627, 10356}, {3628, 9993}, {5054, 10583}, {5092, 7882}, {5901, 12497}, {6704, 14881}, {7789, 7865}, {7811, 7871}, {7905, 12054}, {9857, 34773}, {9983, 32516}, {9984, 10272}, {10877, 15325}, {22803, 24206}, {37450, 40252}

X(42788) = X(5)X(99)∩X(30)X(1506)

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

X(42788) lies on the cubic K1209 and these lines: {5, 99}, {30, 1506}, {83, 549}, {140, 143}, {547, 7789}, {550, 7608}, {3628, 7874}, {5038, 31406}, {5054, 10583}, {5055, 7891}, {5116, 31401}, {7777, 32151}, {9698, 12042}, {16239, 39784}, {17005, 37243}, {18501, 33274}, {25561, 32190}, {31489, 40279}, {33024, 38733}

X(42789) = 1ST MONTESDEOCA-EULER POINT

Barycentrics    a*(b*c*SB*SC + Sqrt[2]*a*SA*Sqrt[SA*SB*SC]) : :

The mapping x(3) + t x(4) 8 t SA SB SC X(3) + a^2 b^2 c^2 X(4) is an involution of the Euler line. The mapping includes {X(3), X(4)} and {X(1113), X(1114)} as involutory pairs. The fixed points of the mapping are

(a (b c SBSC + Sqrt[2] a SA Sqrt[SA SB SC]) : : and (a (b c SB SC - Sqrt[2] a SA Sqrt[SA SB SC]) : : ,

These points are real if and only if the reference triangle ABC is acute. (Angel Montesdeoca, April 26, 2021.) The points are here named the 1st Montesdeoca-Euler point and 2nd Montesdeoca-Euler point, respectively.

X(42789) lies on the curves K114 and Q023 and this line: {2, 3}

For a figure showing X(42789) and X(42790) on the curves K114 and Q023, see Euler line, points, and curves. (Angel Montesdeoca, April 27, 2021)j

X(42789) = reflection of X(42790) in X(186)
X(42789) = circumcircle-inverse of X(42790)
X(42789) = X(74)-Ceva conjugate of X(42790)
X(42789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 378, X(42790)}, {3, 4, X(42790)}, {5, 3520, X(42790)}, {20, 24, X(42790)}, {21, 7414, X(42790)}, {22, 18533, X(42790)}, {23, 10295, X(42790)}, {25, 376, X(42790)}, {26, 35471, X(42790)}, {27, 7430, X(42790)}, {28, 3651, X(42790)}, {29, 7421, X(42790)}, {140, 14865, X(42790)}, {237, 35474, X(42790)}, {381, 35473, X(42790)}, {382, 21844, X(42790)}, {384, 35476, X(42790)}, {403, 2071, X(42790)}, {411, 37117, X(42790)}, {412, 37115, X(42790)}, {427, 35921, X(42790)}, {468, 7464, X(42790)}, {470, 35469, X(42790)}, {471, 35470, X(42790)}, {548, 34484, X(42790)}, {549, 13596, X(42790)}, {550, 3518, X(42790)}, {631, 1593, X(42790)}, {1006, 4219, X(42790)}, {1012, 37441, X(42790)}, {1113, 1114, X(42790)}, {1325, 37979, X(42790)}, {1594, 14118, X(42790)}, {1597, 3524, X(42790)}, {1598, 3528, X(42790)}, {1656, 35475, X(42790)}, {1658, 34797, X(42790)}, {1995, 35485, X(42790)}, {2070, 13619, X(42790)}, {2073, 36026, X(42790)}, {2074, 36001, X(42790)}, {3088, 7509, X(42790)}, {3090, 3516, X(42790)}, {3091, 35477, X(42790)}, {3146, 32534, X(42790)}, {3147, 12085, X(42790)}, {3153, 37970, X(42790)}, {3515, 3529, X(42790)}, {3517, 17538, X(42790)}, {3522, 10594, X(42790)}, {3523, 35502, X(42790)}, {3541, 7503, X(42790)}, {3542, 11413, X(42790)}, {3543, 35472, X(42790)}, {3545, 11410, X(42790)}, {3575, 7512, X(42790)}, {3627, 17506, X(42790)}, {3628, 35478, X(42790)}, {3843, 23040, X(42790)}, {4185, 6876, X(42790)}, {4220, 4227, X(42790)}, {4221, 4231, X(42790)}, {4222, 37403, X(42790)}, {4230, 7422, X(42790)}, {4235, 7418, X(42790)}, {4238, 7425, X(42790)}, {4241, 7440, X(42790)}, {4242, 14127, X(42790)}, {4246, 7429, X(42790)}, {4249, 7433, X(42790)}, {4250, 7442, X(42790)}, {5000, 40894, X(42790)}, {5001, 40895, X(42790)}, {5059, 35479, X(42790)}, {5198, 21735, X(42790)}, {5899, 35489, X(42790)}, {6143, 14130, X(42790)}, {6240, 7488, X(42790)}, {6353, 21312, X(42790)}, {6636, 7576, X(42790)}, {6644, 35481, X(42790)}, {6875, 37194, X(42790)}, {6905, 37305, X(42790)}, {6906, 7412, X(42790)}, {6998, 7431, X(42790)}, {7411, 36009, X(42790)}, {7413, 7436, X(42790)}, {7441, 7463, X(42790)}, {7452, 7454, X(42790)}, {7456, 7461, X(42790)}, {7470, 27369, X(42790)}, {7480, 36164, X(42790)}, {7482, 36166, X(42790)}, {7487, 10323, X(42790)}, {7496, 35484, X(42790)}, {7501, 7580, X(42790)}, {7502, 18559, X(42790)}, {7505, 12084, X(42790)}, {7513, 37120, X(42790)}, {7517, 35503, X(42790)}, {7526, 37119, X(42790)}, {7527, 37118, X(42790)}, {7531, 37195, X(42790)}, {7556, 37196, X(42790)}, {7577, 18570, X(42790)}, {10018, 12086, X(42790)}, {10151, 37948, X(42790)}, {10298, 35480, X(42790)}, {10299, 11403, X(42790)}, {11250, 16868, X(42790)}, {11284, 35483, X(42790)}, {12082, 37460, X(42790)}, {14709, 14710, X(42790)}, {15559, 37126, X(42790)}, {15702, 35501, X(42790)}, {15750, 33703, X(42790)}, {16042, 35492, X(42790)}, {16049, 31384, X(42790)}, {16386, 37951, X(42790)}, {17562, 37426, X(42790)}, {18535, 19708, X(42790)}, {18560, 22467, X(42790)}, {18859, 37943, X(42790)}, {21669, 37289, X(42790)}, {26863, 33923, X(42790)}, {30267, 30733, X(42790)}, {34864, 35482, X(42790)}, {37114, 37200, X(42790)}, {37925, 37931, X(42790)}, {37934, 37946, X(42790)}, {37959, 37961, X(42790)}

X(42790) = 2ND MONTESDEOCA-EULER POINT

Barycentrics    a*(b*c*SB*SC - Sqrt[2]*a*SA*Sqrt[SA*SB*SC]) : :

X(42790) lies on the curves K114 and Q023 and this line: {2, 3}

X(42790) = reflection of X(42789) in X(186)
X(42790) = circumcircle-inverse of X(42789)
X(42790) = X(74)-Ceva conjugate of X(42789)
X(42790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 378, X(42789)}, {3, 4, X(42789)}, {5, 3520, X(42789)}, {20, 24, X(42789)}, {21, 7414, X(42789)}, {22, 18533, X(42789)}, {23, 10295, X(42789)}, {25, 376, X(42789)}, {26, 35471, X(42789)}, {27, 7430, X(42789)}, {28, 3651, X(42789)}, {29, 7421, X(42789)}, {140, 14865, X(42789)}, {237, 35474, X(42789)}, {381, 35473, X(42789)}, {382, 21844, X(42789)}, {384, 35476, X(42789)}, {403, 2071, X(42789)}, {411, 37117, X(42789)}, {412, 37115, X(42789)}, {427, 35921, X(42789)}, {468, 7464, X(42789)}, {470, 35469, X(42789)}, {471, 35470, X(42789)}, {548, 34484, X(42789)}, {549, 13596, X(42789)}, {550, 3518, X(42789)}, {631, 1593, X(42789)}, {1006, 4219, X(42789)}, {1012, 37441, X(42789)}, {1113, 1114, X(42789)}, {1325, 37979, X(42789)}, {1594, 14118, X(42789)}, {1597, 3524, X(42789)}, {1598, 3528, X(42789)}, {1656, 35475, X(42789)}, {1658, 34797, X(42789)}, {1995, 35485, X(42789)}, {2070, 13619, X(42789)}, {2073, 36026, X(42789)}, {2074, 36001, X(42789)}, {3088, 7509, X(42789)}, {3090, 3516, X(42789)}, {3091, 35477, X(42789)}, {3146, 32534, X(42789)}, {3147, 12085, X(42789)}, {3153, 37970, X(42789)}, {3515, 3529, X(42789)}, {3517, 17538, X(42789)}, {3522, 10594, X(42789)}, {3523, 35502, X(42789)}, {3541, 7503, X(42789)}, {3542, 11413, X(42789)}, {3543, 35472, X(42789)}, {3545, 11410, X(42789)}, {3575, 7512, X(42789)}, {3627, 17506, X(42789)}, {3628, 35478, X(42789)}, {3843, 23040, X(42789)}, {4185, 6876, X(42789)}, {4220, 4227, X(42789)}, {4221, 4231, X(42789)}, {4222, 37403, X(42789)}, {4230, 7422, X(42789)}, {4235, 7418, X(42789)}, {4238, 7425, X(42789)}, {4241, 7440, X(42789)}, {4242, 14127, X(42789)}, {4246, 7429, X(42789)}, {4249, 7433, X(42789)}, {4250, 7442, X(42789)}, {5000, 40894, X(42789)}, {5001, 40895, X(42789)}, {5059, 35479, X(42789)}, {5198, 21735, X(42789)}, {5899, 35489, X(42789)}, {6143, 14130, X(42789)}, {6240, 7488, X(42789)}, {6353, 21312, X(42789)}, {6636, 7576, X(42789)}, {6644, 35481, X(42789)}, {6875, 37194, X(42789)}, {6905, 37305, X(42789)}, {6906, 7412, X(42789)}, {6998, 7431, X(42789)}, {7411, 36009, X(42789)}, {7413, 7436, X(42789)}, {7441, 7463, X(42789)}, {7452, 7454, X(42789)}, {7456, 7461, X(42789)}, {7470, 27369, X(42789)}, {7480, 36164, X(42789)}, {7482, 36166, X(42789)}, {7487, 10323, X(42789)}, {7496, 35484, X(42789)}, {7501, 7580, X(42789)}, {7502, 18559, X(42789)}, {7505, 12084, X(42789)}, {7513, 37120, X(42789)}, {7517, 35503, X(42789)}, {7526, 37119, X(42789)}, {7527, 37118, X(42789)}, {7531, 37195, X(42789)}, {7556, 37196, X(42789)}, {7577, 18570, X(42789)}, {10018, 12086, X(42789)}, {10151, 37948, X(42789)}, {10298, 35480, X(42789)}, {10299, 11403, X(42789)}, {11250, 16868, X(42789)}, {11284, 35483, X(42789)}, {12082, 37460, X(42789)}, {14709, 14710, X(42789)}, {15559, 37126, X(42789)}, {15702, 35501, X(42789)}, {15750, 33703, X(42789)}, {16042, 35492, X(42789)}, {16049, 31384, X(42789)}, {16386, 37951, X(42789)}, {17562, 37426, X(42789)}, {18535, 19708, X(42789)}, {18560, 22467, X(42789)}, {18859, 37943, X(42789)}, {21669, 37289, X(42789)}, {26863, 33923, X(42789)}, {30267, 30733, X(42789)}, {34864, 35482, X(42789)}, {37114, 37200, X(42789)}, {37925, 37931, X(42789)}, {37934, 37946, X(42789)}, {37959, 37961, X(42789)}

X(42791) = GIBERT (9,-1,16) POINT

Barycentrics    3*Sqrt[3]*a^2*S + 16*a^2*SA - 2*SB*SC : :

X(42791) lies on these lines: {2, 5321}, {3, 42511}, {6, 19708}, {13, 19710}, {14, 11812}, {15, 8703}, {16, 15759}, {17, 30}, {18, 549}, {61, 34200}, {376, 397}, {395, 10645}, {396, 3534}, {398, 3524}, {524, 33622}, {547, 42157}, {550, 16962}, {3525, 33603}, {3530, 16268}, {3543, 42598}, {3830, 42087}, {3845, 23302}, {3860, 19107}, {5054, 41120}, {5055, 42164}, {5066, 16241}, {5071, 5349}, {5237, 15714}, {5318, 11001}, {5334, 33605}, {5339, 15702}, {5344, 33604}, {6411, 36467}, {6412, 36450}, {7801, 35304}, {10109, 36970}, {10304, 22236}, {10653, 15695}, {10654, 15693}, {11485, 42510}, {11539, 16964}, {12101, 37832}, {12108, 41973}, {12817, 42107}, {12820, 42099}, {14891, 16963}, {15681, 42152}, {15682, 16644}, {15683, 42156}, {15685, 41119}, {15686, 16267}, {15688, 42148}, {15689, 40693}, {15690, 41107}, {15691, 16965}, {15692, 16773}, {15694, 42163}, {15697, 42155}, {15700, 40694}, {15701, 23303}, {15703, 42160}, {15706, 42149}, {15708, 42153}, {15709, 42773}, {15710, 36843}, {15712, 41944}, {15713, 41122}, {15719, 16645}, {15722, 42089}, {15764, 35823}, {16242, 19711}, {16961, 41971}, {16967, 42595}, {17578, 42515}, {19709, 42085}, {23046, 42488}, {33458, 35931}, {33459, 33627}, {33607, 42429}, {33699, 42124}, {36448, 42198}, {36466, 42196}, {37835, 42493}, {38071, 42432}, {41106, 42101}, {42100, 42496}, {42121, 42507}, {42138, 42430}

X(42791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 19708, 42792}, {15, 42528, 42633}, {15, 42631, 42532}, {10645, 41101, 12100}, {12100, 41101, 395}, {15686, 16267, 42165}, {15690, 41107, 42088}, {15701, 41113, 23303}, {15713, 42117, 41122}, {41107, 42529, 15690}, {41119, 42090, 15685}

X(42792) = GIBERT (9,1,-16) POINT

Barycentrics    3*Sqrt[3]*a^2*S - 16*a^2*SA + 2*SB*SC : :

X(42792) lies on these lines: {2, 5318}, {3, 42510}, {6, 19708}, {13, 11812}, {14, 19710}, {15, 15759}, {16, 8703}, {17, 549}, {18, 30}, {62, 34200}, {376, 398}, {395, 3534}, {396, 10646}, {397, 3524}, {524, 33624}, {547, 42158}, {550, 16963}, {3525, 33602}, {3530, 16267}, {3543, 42599}, {3830, 42088}, {3845, 23303}, {3860, 19106}, {5054, 41119}, {5055, 42165}, {5066, 16242}, {5071, 5350}, {5238, 15714}, {5321, 11001}, {5335, 33604}, {5340, 15702}, {5343, 33605}, {6411, 36449}, {6412, 36468}, {7801, 35303}, {10109, 36969}, {10304, 22238}, {10653, 15693}, {10654, 15695}, {11486, 42511}, {11539, 16965}, {12101, 37835}, {12108, 41974}, {12816, 42110}, {12821, 42100}, {14891, 16962}, {15681, 42149}, {15682, 16645}, {15683, 42153}, {15685, 41120}, {15686, 16268}, {15688, 42147}, {15689, 40694}, {15690, 41108}, {15691, 16964}, {15692, 16772}, {15694, 42166}, {15697, 42154}, {15700, 40693}, {15701, 23302}, {15703, 42161}, {15706, 42152}, {15708, 42156}, {15709, 42774}, {15710, 36836}, {15712, 41943}, {15713, 41121}, {15719, 16644}, {15722, 42092}, {15764, 35822}, {16241, 19711}, {16960, 41972}, {16966, 42594}, {17578, 42514}, {19709, 42086}, {22513, 36767}, {23046, 42489}, {33458, 33626}, {33459, 35932}, {33606, 42430}, {33699, 42121}, {36448, 42195}, {36466, 42197}, {37832, 42492}, {38071, 42431}, {41106, 42102}, {42099, 42497}, {42124, 42506}, {42135, 42429}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 19708, 42791}, {16, 42529, 42634}, {16, 42632, 42533}, {10646, 41100, 12100}, {12100, 41100, 396}, {15686, 16268, 42164}, {15690, 41108, 42087}, {15701, 41112, 23302}, {15713, 42118, 41121}, {41108, 42528, 15690}, {41120, 42091, 15685}

X(42793) = GIBERT (-9,1,18) POINT

Barycentrics    3*Sqrt[3]*a^2*S - 18*a^2*SA - 2*SB*SC : :

X(42793) lies on these lines: {3, 42511}, {4, 11481}, {13, 140}, {14, 550}, {16, 15712}, {18, 42136}, {395, 3522}, {396, 3523}, {397, 15720}, {398, 10646}, {549, 42506}, {631, 42518}, {1656, 42161}, {1657, 42163}, {3411, 34200}, {3533, 42166}, {3850, 16242}, {3853, 42631}, {3858, 36968}, {5054, 42502}, {5056, 42165}, {5059, 16645}, {5073, 42599}, {5340, 42476}, {5349, 42121}, {5350, 42089}, {8703, 41973}, {10187, 42431}, {10299, 22238}, {10304, 42509}, {12103, 41944}, {12108, 41100}, {15691, 42503}, {15699, 42505}, {16964, 41981}, {16965, 42501}, {17538, 42519}, {21735, 42147}, {22235, 23302}, {33416, 42693}, {35018, 42110}, {36768, 37341}, {38071, 42593}, {42087, 42149}, {42128, 42151}

X(42793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 41974, 42598}, {11481, 42686, 23303}

X(42794) = GIBERT (9,1,18) POINT

Barycentrics    3*Sqrt[3]*a^2*S + 18*a^2*SA + 2*SB*SC : :

X(42794) lies on these lines: {3, 42510}, {4, 11480}, {13, 550}, {14, 140}, {15, 15712}, {17, 42137}, {395, 3523}, {396, 3522}, {397, 10645}, {398, 15720}, {549, 42507}, {631, 42519}, {1656, 42160}, {1657, 42166}, {3412, 34200}, {3533, 42163}, {3850, 16241}, {3853, 42632}, {3858, 36967}, {5054, 42503}, {5056, 42164}, {5059, 16644}, {5073, 42598}, {5339, 42477}, {5349, 42092}, {5350, 42124}, {8703, 41974}, {10188, 42432}, {10299, 22236}, {10304, 42508}, {12103, 41943}, {12108, 41101}, {15691, 42502}, {15699, 42504}, {16964, 42500}, {16965, 41981}, {17538, 42518}, {21735, 42148}, {22237, 23303}, {33417, 42692}, {35018, 42107}, {38071, 42592}, {42088, 42152}, {42125, 42150}

X(42794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 41973, 42599}, {11480, 42687, 23302}

X(42795) = GIBERT (15,-2,35) POINT

Barycentrics    5*Sqrt[3]*a^2*S + 35*a^2*SA - 4*SB*SC : :

X(42795) lies on these lines: {4, 5352}, {5, 12821}, {13, 3534}, {14, 549}, {15, 10304}, {16, 15759}, {17, 15704}, {30, 42687}, {395, 41971}, {397, 548}, {550, 42777}, {618, 36329}, {3090, 42694}, {3523, 42513}, {3526, 42157}, {3628, 36970}, {5055, 33417}, {5066, 19107}, {5072, 42610}, {5321, 11540}, {5343, 10303}, {5474, 38224}, {7850, 30471}, {