| 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) |
Line X(1099)X(1784) is the tangent to the inellipse that is the trilinear square of the Euler line, at X(1099) (the trilinear square of X(30)).
X(4240) is the unique point on the Euler line whose trilinear polar is parallel to the Euler line. X(24001) is the trilinear product X(2)*X(4240).
X(24001) lies on these lines: {19, 27}, {662, 811}, {24000, 24024}
X(24001) = trilinear pole of line X(1099)X(1784)
X(24001) = trilinear product X(i)*X(j) for these {i,j}: {2, 4240}, {4, 2407}, {30, 648}, {99, 1990}, {107, 11064}, {112, 3260}, {162, 14206}, {264, 2420}, {662, 1784}, {670, 14581}, {811, 2173}
Line X(11)X(1111) is the tangent to the inellipse that is the trilinear square of the Gergonne line, at X(1111) (the trilinear square of X(514)). This inellipse has center X(11019) and Brianchon point (perspector) X(1088).
Let P1 and P2 be the two points on the Gergonne line whose trilinear polars are parallel to the Gergonne line. P1 and P2 lie on the circumconic centered at X(1086) (hyperbola {{A, B, C, X(2), X(7)}}), and circle {{X(2), X(109), X(675)}}. The midpoint of P1 and P2 is X(1638). X(24002) is the trilinear product P1*P2.
X(24002) lies on these lines: {7, 513}, {85, 20949}, {109, 2860}, {226, 4776}, {273, 2400}, {279, 2401}, {514, 7216}, {522, 693}, {651, 666}, {655, 658}, {764, 7185}, {918, 3261}, {929, 934}, {1014, 17212}, {1090, 1111}, {1441, 20504}, {1445, 21390}, {1446, 23100}, {2402, 4000}, {2517, 4411}, {3323, 7336}, {3663, 23798}, {3667, 23819}, {3669, 4560}, {4130, 4885}, {4378, 7176}, {4397, 20907}, {4462, 10029}, {4486, 21206}, {4554, 4582}, {4617, 13149}, {4905, 19594}, {5228, 21007}, {7658, 14282}, {20520, 21189}
X(24002) = reflection of X(4130) in X(4885)
X(24002) = isotomic conjugate of X(644)
X(24002) = trilinear pole of line X(11)X(1111)
X(24002) = crossdifference of every pair of points on line X(41)X(1253)
X(24002) = trilinear product X(i)*X(j) for these {i,j}: {2, 3676}, {7, 514}, {10, 17096}, {11, 658}, {34, 15413}, {56, 3261}, {57, 693}, {59, 23100}, {65, 7199}, {75, 3669}, {77, 17924}, {85, 513}, {226, 7192}, {244, 4554}, {273, 905}, {278, 4025}, {279, 522}, {307, 17925}, {314, 7216}, {331, 1459}, {479, 8834}, {552, 4024}, {553, 4608}, {650, 1088}, {651, 1111}, {653, 1565}, {664, 1086}, {738, 4397}, {934, 4858}, {961, 4509}, {1014, 1577}, {2401, 22464}, {20615, 20949}
The inellipse that is the trilinear square of the Nagel line passes through X(1), X(75), X(341), X(872), X(1089) and X(4738). The Brianchon point (perspector) of this inellipse is X(7035). The vertex conjugate of the foci of this inellipse is X(765),
X(24003) lies on these lines: {1, 1120}, {2, 38}, {10, 11}, {120, 124}, {900, 3035}, {3740, 3741} et al
X(24003) = isotomic conjugate of X(32016)
X(24003) = midpoint of X(1) and X(4738)
X(24003) = complement of X(244)
X(24003) = perspector of mid- and side-triangles of Gemini triangles 15 and 16
Line X(519)X(3992) is the tangent to the inellipse that is the trilinear square of the Nagel line, at X(4738) (the trilinear square of X(519)).
X(17780) is the unique point on the Nagel line whose trilinear polar is parallel to the Nagel line. X(24004) is the trilinear product X(2)*X(17780).
X(24004) lies on these lines: {2, 37}, {100, 9059}, {190, 646}, {662, 7258}, {666, 5387}, {889, 3572}, {1016, 4585}, {2325, 3264} et al
X(24004) = isotomic conjugate of X(1022)
X(24004) = trilinear pole of line X(519)X(3992)
X(24004) = crossdifference of every pair of points on line X(667)X(3248)
X(24004) = trilinear product X(i)*X(j) for these {i,j}: {2, 17780}, {44, 668}, {75, 1023}, {86, 4169}, {99, 3943}, {100, 4358}, {101, 3264}, {145, 2415}, {190, 519}, {646, 1319}, {651, 4723}, {662, 3992}, {664, 2325}, {765, 3762}, {900, 1016}, {902, 1978}, {1145, 13136}, {1275, 4528}, {1317, 4582}, {1635, 7035}, {1639, 4998}, {1647, 6632}, {1897, 3977}, {2251, 6386}, {3257, 4738}, {3689, 4554}, {3699, 3911}, {3952, 16704}, {4120, 4600}, {4370, 4555}, {4432, 4562}, {4439, 4586}, {4561, 8756}, {4564, 4768}, {4584, 4783}, {4597, 4908}, {4601, 4730}, {4606, 4742}, {4633, 4819}, {4969, 6540}, {5440, 6335}, {6079, 16594}
X(24004) = barycentric product X(i)*X(j) for these {i,j}: {44, 1978}, {75, 17780}, {76, 1023}, {99, 3992}, {100, 3264}, {101, 9456}, {190, 4358}, {274, 4169}, {519, 668}, {646, 3911}, {664, 4723}, {765, 900}, {789, 4439}, {799, 3943}, {902, 6386}, {1016, 3762}, {2325, 4554}, {2415, 18743}, {3689, 4572}, {3977, 6335}, {4033, 16704}, {4120, 4601}, {4432, 4583}, {4555, 4738}, {4589, 4783}, {4768, 4998}, {4975, 6540}, {6079, 20900}
The inellipse that is the trilinear square of the orthic axis passes through X(1109) and X(2310). The Brianchon point (perspector) of this inellipse is X(158). The vertex conjugate of the foci of this inellipse is X(1096).
X(24005) lies on these lines: {2, 326}, {6, 1210}, {9, 1737}, {10, 37}, {11, 2262}, {198, 1837}, {226, 7363} et al
X(24005) = complement of X(326)
Line X(1109)X(3708) is the tangent to the inellipse that is the trilinear square of the orthic axis, at X(1109) (the trilinear square of X(523)).
X(24007) and X(24008) are the two points on the orthic axis whose trilinear polars are parallel to the orthic axis. X(24006) is the trilinear product X(24007)*X(24008).
X(24006) is the perspector of the anticevian triangle of X(19) and tangential triangle, wrt the excentral triangle, of the bianticevian conic of X(1) and X(4). This conic is a rectangular hyperbola passing through X(1), X(4), X(19), and the vertices of their anticevian triangles. It has center X(107) and is the excentral isogonal conjugate of line X(40)X(2939), the anticomplementary conjugate of line X(20)X(1330), and the anticomplementary isotomic conjugate of line X(1654)X(3164).
X(24006) lies on these lines: {4, 6003}, {19, 798}, {92, 14207}, {108, 2689}, {136, 15608}, {225, 4017}, {240, 522}, {424, 2501}, {661, 3064}, {810, 2616}, {823, 24000}, {1824, 4132}, {4064, 4086} et al
X(24006) = isogonal conjugate of X(4575)
X(24006) = isotomic conjugate of X(4592)
X(24006) = pole wrt polar circle of trilinear polar of X(662) (line X(1)X(21))
X(24006) = polar conjugate of X(662)
X(24006) = trilinear pole of line X(1109)X(3708)
X(24006) = crossdifference of every pair of points on line X(48)X(255)
X(24006) = trilinear product X(i)*X(j) for these {i,j}: {2, 2501}, {4, 523}, {6, 14618}, {10, 7649}, {19, 1577}, {25, 850}, {27, 4024}, {28, 4036}, {30, 18808}, {34, 4086}, {37, 17924}, {76, 2489}, {92, 661}, {98, 16230}, {99, 8754}, {107, 250}, {110, 2970}, {112, 338}, {162, 1109}, {225, 522}, {226, 3064}, {264, 512}, {393, 525}, {520, 1093}, {526, 6344}, {671, 14273}, {798, 1969}, {811, 2643}, {1096, 14208}, {4064, 8747}, {24007, 24008}
X(24006) = barycentric product X(4)*X(1577)
X(24007) is one of two points on the orthic axis (X(24008) is the other) whose trilinear polar is parallel to the orthic axis.
The trilinear polar of X(24007) meets the line at infinity at X(523).
X(24007) lies on the Kiepert hyperbola, the Dao-Moses-Telv circle, circle {{X(2), X(98), X(112)}} and these lines: {230, 231}, {671, 2480}
X(24007) = reflection of X(24008) in X(1637)
X(24007) = antipode of X(24008) in the Dao-Moses-Telv circle
X(24007) = pole wrt polar circle of trilinear polar of X(2479) (line X(2)X(24008))
X(24007) = polar conjugate of X(2479)
X(24007) = X(107)-Ceva conjugate of X(24008)
X(24007) = X(125)-cross conjugate of X(24008)
X(24007) = X(25)-vertex conjugate of X(24008)
X(24007) = PU(4)-harmonic conjugate of X(24008)
X(24007) = {X(i),X(j)}-harmonic conjugate of X(24008) for these {i,j}: {468, 6103}, {647, 6130}, {2501, 16230}, {3018, 11657}
X(24008) is one of two points on the orthic axis (X(24007) is the other) whose trilinear polar is parallel to the orthic axis.
The trilinear polar of X(24008) meets the line at infinity at X(523).
X(24008) lies on the Kiepert hyperbola, the Dao-Moses-Telv circle, circle {{X(2), X(98), X(112)}} and these lines: {230, 231}, {671, 2479}
X(24008) = reflection of X(24007) in X(1637)
X(24008) = antipode of X(24007) in the Dao-Moses-Telv circle
X(24008) = pole wrt polar circle of trilinear polar of X(2480) (line X(2)X(24007))
X(24008) = polar conjugate of X(2480)
X(24008) = X(107)-Ceva conjugate of X(24007)
X(24008) = X(125)-cross conjugate of X(24007)
X(24008) = X(25)-vertex conjugate of X(24007)
X(24008) = PU(4)-harmonic conjugate of X(24007)
X(24008) = {X(i),X(j)}-harmonic conjugate of X(24007) for these {i,j}: {468, 6103}, {647, 6130}, {2501, 16230}, {3018, 11657}
The inellipse that is the trilinear square of the Soddy line passes through X(1), X(1088), X(1097) and X(24014). This inellipse is tangent to the Soddy line at X(1).
X(24009) lies on these lines: {1, 85}, {2, 24010}, {4904, 11019}
X(24009) = midpoint of X(1) and X(24014)
X(24009) = complement of X(24010)
X(24010) lies on the inellipse centered at X(10) and these lines: {2, 24009}, {8, 3177}, {10, 24014}, {200, 644}, {244, 2968}, {756, 7046}, {1146, 2310}, {3022, 3119}, {3679, 24028}, {4642, 23667}, {23529, 24034}
X(24010) = reflection of X(24014) in X(10)
X(24010) = isogonal conjugate of X(24013)
X(24010) = isotomic conjugate of X(24011)
X(24010) = anticomplement of X(24009)
X(24010) = trilinear square of X(3900)
X(24010) = antipode of X(24014) in inellipse centered at X(10)
X(24010) = crossdifference of every pair of points on line X(1461)X(4617)
X(24010) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24013}, {31, 24011}, {934, 934}
As a point P moves on the Soddy line, the locus of the trilinear pole of the tangent at P to hyperbola {{A, B, C, X(1), P}} is the trilinear polar of X(24011) (line X(658)X(3732)).
X(24011) lies on these lines: {657, 658}, {1275, 10025}, {4635, 15419}
X(24011) = isogonal conjugate of X(24012)
X(24011) = isotomic conjugate of X(24010)
X(24011) = trilinear square of X(658)
X(24011) = trilinear pole of line X(658)X(3732)
X(24011) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24012}, {31, 24010}, {657, 657}
X(24012) lies on these lines: {1, 658}, {244, 14714}, {872, 7071}, {1097, 7038}, {3022, 14936} et al
X(24012) = isogonal conjugate of X(24011)
X(24012) = trilinear square of X(657)
X(24012) = crossdifference of every pair of points on line X(658)X(3732)
X(24012) = trilinear pole, wrt incentral triangle, of Soddy line
X(24012) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24011}, {658, 658}
Lies on the inellipse centered at X(10) and these lines: {658, 14837}, {1262, 6610}, {1323, 7045}, {3669, 4617}, {4637, 7254}, {7128, 7177} et al
X(24013) = isogonal conjugate of X(24010)
X(24013) = trilinear square of X(934)
X(24013) = trilinear pole of line X(1461)X(4617)
X(24013) = vertex conjugate of foci of inellipse centered at X(24009) (the trilinear square of the Soddy line)
X(24014) lies on the inellipse centered at X(10), the inellipse centered at X(24009), and these lines: {1, 85}, {2, 24026}, {10, 24010}, {149, 10580}, {200, 1897}, {244, 3914}, {1088, 4312}, {1089, 7952}, {1109, 1962}, {6533, 17102} et al
X(24014) = reflection of X(1) in X(24009)
X(24014) = reflection of X(24010) in X(10)
X(24014) = antipode of X(1) in inellipse centered at X(24009)
X(24014) = antipode of X(24010) in inellipse centered at X(10)
X(24014) = trilinear square of X(516)
Line X(516)X(24014) is the tangent to the inellipse that is the trilinear square of the Soddy line, at X(24014) (the trilinear square of X(516)).
Let P1 and P2 be as at X(23973). X(24015) is the trilinear product P1*P2.
X(24015) lies on these lines: {2, 85}, {658, 3732}, {664, 1897}, {934, 9057}, {1275, 4585} et al
X(24015) = trilinear product X(516)*X(658)
X(24015) = trilinear pole of line X(516)X(24014)
X(24015) = crossdifference of every pair of points on line X(8641)X(24012)
Let A', B', C' be the intersections of the Soddy line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(24016).
X(24016) lies on the circumcircle and these lines: {7, 2723}, {77, 2739}, {100, 677}, {101, 1262}, {103, 1458}, {105, 1456}, {107, 7192}, {108, 4617}, {109, 7339}, {112, 7254}, {269, 2717}, {279, 2724}, {658, 9057}, {663, 6614}, {934, 24012}, {1042, 2700} et al
X(24016) = trilinear pole of line X(6)X(911)
X(24016) = Ψ(X(6), X(911))
X(24016) = trilinear product X(103)*X(934) (circumcircle-X(1) antipodes)
X(24017) lies on this line: {2, 24020}
X(24017) = complement of X(24020)
The trilinear polar of X(24018) passes through X(2632).
X(24018) lies on these lines: {69, 15411}, {100, 2727}, {514, 661}, {521, 656}, {525, 8611}, {822, 4131}, {1019, 6003} et al
X(24018) = isogonal conjugate of X(24019)
X(24018) = isotomic conjugate of X(823)
X(24018) = X(6)-isoconjugate of X(107)
X(24018) = trilinear product X(i)*X(j) for these {i,j}: {2, 520}, {6, 3265}, {63, 656}
X(24018) = crossdifference of every pair of points on line X(19)X(31)
X(24018) = polar conjugate of X(36126)
X(24018) = pole wrt polar circle of trilinear polar of X(36126) (line X(19)X(158))
X(24019) is the trilinear product of the circumcircle intercepts of the van Aubel line.
X(24019) lies on these lines: {19, 2159}, {27, 21621}, {28, 911}, {101, 107}, {108, 112}, {158, 1910}, {162, 163}, {648, 3732}, {662, 811}, {692, 1783}, {909, 1474}, {923, 1096}, {1172,1905} et al
X(24019) = isogonal conjugate of X(24018)
X(24019) = trilinear pole of line X(19)X(31)
X(24019) = polar conjugate of X(14208)
X(24019) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24018}, {2, 520}, {6, 3265}, {48, 14208}
X(24019) = trilinear product X(i)*X(j) for these {i,j}: {4, 112}, {6, 107}, {19, 162}
X(24019) = barycentric product of circumcircle-intercepts of line X(1)X(29)
X(24020) lies on these lines: {2, 24017}, {304, 1956}, {326, 4592}
X(24020) = isogonal conjugate of X(24022)
X(24020) = isotomic conjugate of X(24021)
X(24020) = anticomplement of X(24017)
X(24021) lies on these lines: {1784, 1955}
X(24021) = isotomic conjugate of X(24020)
X(24021) = trilinear square of X(107)
X(24022) lies on these lines: {1784, 1955}
X(24022) = isogonal conjugate of X(24020)
X(24022) = trilinear square of X(24019)
X(24022) = vertex conjugate of foci of inellipse centered at X(24017) (the trilinear square of the van Aubel line)
X(24023) lies on the inellipse centered at X(10), the inellipse centered at X(24017), and these lines: {31, 92}, {1111, 4292} {11031, 24031}
X(24023) = trilinear square of X(1503)
Line X(2312)X(24023) is the tangent to the inellipse that is the trilinear square of the van Aubel line, at X(24023) (the trilinear square of X(1503)).
Let P1 and P2 be as at X(23977). X(24024) is the trilinear product P1*P2.
X(24024) lies on these lines: {1, 29}, {24000, 24001}
X(24024) = trilinear product X(107)*X(1503)
X(24024) = trilinear pole of line X(2312)X(24023)
The inellipse that is the trilinear square of line X(1)X(3) passes through X(1), X(255), X(269), X(1079), X(1103), X(1106), X(1253), X(1254) and X(24028). The Brianchon point (perspector) of this inellipse is X(7045).
X(24025) lies on these lines: {1, 88}, {2, 24026}, {3, 2817}, {10, 2968}, {227, 4297}, {498, 15065}, {516, 1465}, {651, 1768}, {676, 2804}, {758, 1735}, {774, 3216}, {899, 1736}, {1062,6796} et al
X(24025) = midpoint of X(1) and X(24028)
X(24025) = complement of X(24026)
X(24026) lies on the inellipse centered at X(10) and these lines: {1, 318}, {2, 24025}, {8, 80}, {11, 123}, {92, 1699}, {124, 20620}, {321, 4712}, {522, 7004}, {1099, 1125}, {1111, 3120}, {4647, 4736} et al
X(24026) = reflection of X(24028) in X(10)
X(24026) = isogonal conjugate of X(24027)
X(24026) = isotomic conjugate of X(7045)
X(24026) = anticomplement of X(24025)
X(24026) = trilinear square of X(522)
X(24026) = polar conjugate of isogonal conjugate of X(34591)
X(24026) = antipode of X(24028) in inellipse centered at X(10)
X(24026) = pole wrt polar circle of trilinear polar of X(7128) (line X(108)X(109))
X(24026) = polar conjugate of X(7128)
X(24026) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24027}, {31, 7045}, {48, 7128}, {109, 109}
X(24027) lies on these lines: {1, 7012}, {36, 59}, {109, 1459}, {1106, 16944}, {1275, 5384}, {1415, 3063}, {1758, 4570} et al
X(24027) = isogonal conjugate of X(24026)
X(24027) = trilinear square of X(109)
X(24027) = vertex conjugate of foci of inellipse centered at X(24025) (the trilinear square of line X(1)X(3))
X(24028) lies on the inellipse centered at X(10), the inellipse centered at X(24025), and these lines: {1, 88}, {8, 24031}, {10, 24026}, {31, 998}, {38, 12647}, {40, 109}, {46, 1106}, {55, 1411}, {65, 1066}, {80, 2310}, {201, 5690}, {269, 2093}, {347, 4566}, {517, 1457}, {519, 1735}, {656, 6739}, {756, 15065}, {774, 10573}, {952, 7004}, {1074, 1111}, {1103, 7991}, {1109, 2292}, {1110, 2717}, {1145, 1769}, {1210, 3987}, {1253, 1718}, {1254, 5903} et al
X(24028) = reflection of X(1) in X(24025)
X(24028) = reflection of X(24026) in X(10)
X(24028) = antipode of X(1) in inellipse centered at X(24025)
X(24028) = antipode of X(24026) in inellipse centered at X(10)
X(24028) = trilinear square of X(517)
Line X(517)X(1457) is the tangent to the inellipse that is the trilinear square of line X(1)X(3), at X(24028) (the trilinear square of X(517)).
Let P1 and P2 be as at X(23981). X(24029) is the trilinear product P1*P2.
X(24029) lies on the circumconic centered at X(23980) and these lines: {2, 7}, {101, 651}, {108, 1331}, {109, 9058}, {190, 653}, {513, 2283}, {901, 2222}, {1360, 6068} et al
X(24029) = isotomic conjugate of polar conjugate of X(23706)
X(24029) = trilinear pole of line X(517)X(1457)
X(24029) = crossdifference of every pair of points on line X(663)X(2310)
X(24029) = X(19)-isoconjugate of X(37628)
X(24029) = trilinear product X(i)*X(j) for these {i,j}: {7, 2427}, {101, 22464}, {109, 908}, {190, 1457}, {517, 651}, {664, 2183}, {2222, 16586}
The inellipse that is the trilinear square of line X(1)X(4) passes through X(1), X(158), X(7138) and X(24034).
X(24030) lies on these lines: {1, 318}, {2, 24030}, {10, 7358}, {11, 118}, {1359, 4297}, {1745, 1895} et al
X(24030) = midpoint of X(1) and X(24034)
X(24030) = complement of X(24031)
X(24031) lies on the inellipse centered at X(10) and these lines: {1, 271}, {2, 24030}, {10, 24034}, {36, 7111}, {63, 100}, {78, 255}, {84, 1295}, {1099, 6734}, {2155, 7097} et al
X(24031) = reflection of X(24034) in X(10)
X(24031) = isogonal conjugate of X(24033)
X(24031) = isotomic conjugate of X(24032)
X(24031) = anticomplement of X(24030)
X(24031) = antipode of X(24034) in inellipse centered at X(10)
X(24031) = trilinear square of X(521)
X(24031) = X(i)-isoconjugate of X(j) for these (i,j): {1, 24033}, {31, 24032}, {108, 108}
As a point P moves on line X(1)X(4), the locus of the trilinear pole of the tangent at P to hyperbola {{A, B, C, X(1), P}} is line X(653)X(1020), the trilinear polar of X(24032).
X(24032) lies on these lines: {652, 653}, {693, 13149}, {7045, 15146} et al
X(24032) = isogonal conjugate of X(2638)
X(24032) = isotomic conjugate of X(24031)
X(24032) = trilinear square of X(653)
X(24032) = trilinear pole of line X(653)X(1020)
X(24032) = polar conjugate of X(34591)
X(24032) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2638}, {31, 24031}, {652, 652}
X(24033) lies on these lines: {108, 6129}, {516, 1785}, {910, 7115}, {1456, 1875} et al
X(24033) = isogonal conjugate of X(24031)
X(24033) = trilinear square of X(108)
X(24033) = vertex conjugate of foci of inellipse centered at X(24030) (the trilinear square of line X(1)X(4))
X(24034) lies on the inellipse centered at X(10), the inellipse centered at X(24030), and these lines: {1, 318}, {7, 80}, {10, 24031}, {158, 5691}, {244, 1210}, {1109, 2650} et al
X(24034) = reflection of X(1) in X(24030)
X(24034) = reflection of X(24031) in X(10)
X(24034) = antipode of X(1) in inellipse centered at X(24030)
X(24034) = antipode of X(24031) in inellipse centered at X(10)
X(24034) = trilinear square of X(515)
Line X(515)X(24034) is the tangent to the inellipse that is the trilinear square of line X(1)X(4), at X(24030) (the trilinear square of X(515)).
Let P1 and P2 be as at X(23987). X(24035) is the trilinear product P1*P2.
X(24035) lies on these lines: {2, 92}, {108, 9056}, {653, 1020}, {664, 6335}
X(24035) = trilinear product X(515)*X(653)
X(24035) = trilinear pole of line X(515)X(24034)
X(24035) = crossdifference of every pair of points on line X(1946)X(2638)
The inellipse that is the trilinear square of line X(1)X(6) passes through X(1), X(31), X(200), X(678), X(756) and X(4712). The Brianchon point (perspector) of this inellipse is X(765). The vertex conjugate of the foci of this inellipse is X(1110).
X(24036) lies on these lines: {1, 644}, {2, 1111}, {6, 4574}, {9, 48}, {10, 1146}, {37, 537}, {45, 9259}, {100, 5540}, {106, 2297}, {292, 5283}, {519, 3693}, {620, 14838}, {665, 1639}, {672, 758}, {918, 3960}, {1001, 11716}, {1017, 11224}, {1018, 2170}, {1083, 11712}, {1108, 3950} et al
X(24036) = midpoint of X(1) and X(4712)
X(24036) = complement of X(1111)
The trilinear square of line X(2)X(6) is the inellipse centered at X(21254). This inellipse passes through X(31), X(75), X(757), X(873), X(4094) and X(24038). The vertex conjugate of the foci of this inellipse is X(24041).
X(24037) lies on these lines: {99, 4367}, {512, 9425}, {662, 1924}, {799, 1577}, {1016, 1509}, {1580, 14210} et al
X(24037) = isotomic conjugate of X(2643)
X(24037) = trilinear square of X(99)
X(24037) = trilinear pole of line X(662)X(799)
X(24038) lies on the inellipse centered at X(10), the inellipse centered at X(21254), and these lines: {31, 4592}, {63, 2157}, {75, 799}, {244, 4357}, {350, 1227}, {1111, 4359}, {1366, 7067}, {2642, 4750}, {3218, 17755}, {6629, 14210} et al
X(24038) = trilinear square of X(524)
Line X(6629)X(14210) is the tangent to the inellipse that is the trilinear square of line X(2)X(6), at X(24038) (the trilinear square of X(524)).
X(5468) is the unique point on line X(2)X(6) whose trilinear polar is parallel to line X(2)X(6). X(24039) is the trilinear product X(2)*X(5468).
X(24039) lies on these lines: {1, 75}, {99, 8691}, {662, 799}
X(24039) = isotomic conjugate of X(23894)
X(24039) = trilinear pole of line X(6629)X(14210)
X(24039) = trilinear product X(i)*X(j) for these {i,j}: {2, 5468}, {76, 5467}, {99, 524}, {110, 3266}, {187, 670}, {662, 14210}, {690, 4590}, {799, 896}, {892, 2482}, {1992, 2418}, {4601, 14419}
The Brianchon point (perspector) of this inellipse is X(1109). The vertex conjugate of the foci of this inellipse is X(2643).
X(24040) lies on these lines: {2, 24041}, {115, 3700}, {2615, 7180}, {4092, 12072}, {4369, 8287}
X(24040) = complement of X(24041)
X(24040) = complementary conjugate of X(21254)
X(24041) is the vertex conjugate of the foci of the inellipse that is the trilinear square of line X(2)X(6). This inellipse is centered at X(21254) and has Brianchon point (perspector) X(24037).
X(24041) lies on these lines: {2, 24040}, {99, 4467}, {249, 1931}, {261, 4998}, {415, 4620}, {593, 1252}, {661, 662}, {799, 4575}, {896, 1101}, {1580, 14210} et al
X(24041) = isogonal conjugate of X(2643)
X(24041) = isotomic conjugate of X(1109)
X(24041) = anticomplement of X(24040)
X(24041) = trilinear square of X(662)
X(24041) = trilinear pole of line X(163)X(662)
X(24041) = complement of anticomplementary conjugate of X(21295)
X(24041) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2643}, {31, 1109}, {661, 661}
See Antreas Hatzipolakis and Peter Moses Hyacinthos 28370.
X(24042) lies on these lines: {1,4}, {3,20107}, {30,6713}, {382,10893}, {517,6246}, {519,10738}, {1319,16174}, {2077,10724}, {3627,7681}, {3667,21179}, {3814,5840}, {3830,22753}, {3843,11496}, {3845,7680}, {4299,10598}, {4302,6968}, {4324,6949}, {4881,8227}, {5046,6684}, {5176,14217}, {5450,10896}, {5806,16125}, {6681,23513}, {6796,12953}, {6923,10165}, {6929,10175}, {6965,10172}, {7704,21842}, {7743,11715}, {7956,15687}, {8666,11928}, {10265,12764}, {10525,11362}
X(24042) = midpoint of X(i) and X(j) for these {i,j}: {4, 3583}, {2077, 10724}, {5176, 14217}
X(24042) = reflection of X(i) in X(j) for these {i,j}: {1319, 16174}, {1519, 18483}, {11715, 7743}
X(24042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5225, 6256), (4, 5603, 18513)
See Antreas Hatzipolakis and Peter Moses Hyacinthos 28370.
X(24043) lies on these lines: {4,11538}, {30,137}
X(24044) lies on these lines: {2, 24049}, {8, 4115}, {10, 4037}, {37, 1574}, {321, 1930}, {514, 17762}, {594, 4075}, {1089, 4103}, {1921, 4043}, {2321, 4053}, {3625, 21839}, {3626, 21879}, {3969, 24056}, {4058, 21810}, {4060, 21873}, {4135, 22206}, {4647, 21921}, {5295, 18250}, {17233, 24050}, {21883, 25092}, {24045, 24064}, {24047, 24052}, {24069, 24071}, {24077, 24080}
X(24045) lies on these lines: {2, 1796}, {3, 5134}, {4, 101}, {5, 3730}, {6, 9665}, {9, 6990}, {11, 4253}, {41, 3583}, {85, 5074}, {115, 2176}, {116, 17753}, {169, 1699}, {190, 7752}, {218, 10896}, {220, 381}, {312, 4153}, {382, 3207}, {388, 9327}, {430, 3190}, {469, 22000}, {499, 5030}, {573, 15908}, {595, 3767}, {626, 4713}, {672, 7741}, {847, 1826}, {908, 21073}, {910, 22793}, {946, 5179}, {995, 5254}, {1018, 11681}, {1055, 10483}, {1146, 22791}, {1212, 9955}, {1334, 7951}, {1479, 4251}, {1724, 17737}, {1759, 5057}, {1802, 18406}, {2140, 17671}, {2280, 4857}, {2476, 3294}, {3061, 11813}, {3501, 3814}, {3585, 9310}, {3760, 4766}, {3825, 17754}, {3835, 23100}, {3944, 16600}, {4056, 9318}, {4193, 16549}, {4209, 6710}, {4256, 9598}, {4258, 9668}, {4262, 6284}, {4417, 21070}, {5011, 12699}, {5046, 16788}, {5087, 25066}, {5692, 21029}, {5903, 21044}, {6603, 18480}, {6604, 10708}, {7746, 17735}, {7748, 21008}, {7755, 21793}, {8818, 16777}, {9259, 9651}, {9664, 18755}, {9956, 21872}, {11680, 16552}, {13881, 14974}, {16601, 17605}, {17233, 24057}, {17451, 18393}, {17717, 25092}, {21384, 24387}, {24044, 24064}, {24049, 24071}, {24068, 24070}
X(24046) lies on these lines: {1, 88}, {2, 3670}, {3, 17054}, {5, 1086}, {6, 5708}, {8, 1739}, {10, 982}, {28, 34}, {31, 3336}, {36, 3924}, {38, 1698}, {39, 20271}, {40, 5573}, {42, 18398}, {43, 3874}, {45, 16853}, {46, 595}, {56, 1324}, {65, 995}, {72, 16610}, {75, 24166}, {85, 17205}, {145, 3987}, {191, 748}, {227, 3660}, {386, 942}, {405, 17595}, {484, 3915}, {496, 3756}, {497, 12534}, {519, 3976}, {573, 20227}, {581, 10202}, {596, 4385}, {602, 5535}, {726, 20923}, {758, 978}, {899, 5904}, {936, 1257}, {975, 5437}, {984, 3634}, {986, 1125}, {991, 9940}, {997, 11512}, {998, 4320}, {999, 15955}, {1015, 3959}, {1064, 15016}, {1068, 4000}, {1089, 17155}, {1104, 4257}, {1149, 5697}, {1193, 5902}, {1201, 5903}, {1210, 1785}, {1254, 3361}, {1279, 3579}, {1465, 4306}, {1468, 3337}, {1471, 15932}, {1616, 12702}, {1722, 18193}, {1724, 3218}, {1735, 3086}, {1736, 5704}, {1737, 23536}, {1738, 10916}, {1837, 6788}, {2140, 24185}, {2275, 3125}, {2292, 3624}, {2650, 5313}, {2901, 3210}, {3120, 7741}, {3159, 18743}, {3214, 17449}, {3216, 3868}, {3242, 9709}, {3290, 3730}, {3293, 3873}, {3454, 3662}, {3555, 3999}, {3616, 4424}, {3632, 4695}, {3663, 9843}, {3666, 5439}, {3673, 21208}, {3678, 16569}, {3735, 16604}, {3742, 3931}, {3772, 24910}, {3778, 3841}, {3782, 4187}, {3821, 17065}, {3825, 3944}, {3831, 24165}, {3833, 17591}, {3836, 4446}, {3878, 21214}, {3970, 17756}, {4253, 16583}, {4255, 15934}, {4283, 5791}, {4347, 9364}, {4359, 10479}, {4392, 9780}, {4414, 5259}, {4415, 17527}, {4419, 17559}, {4646, 5045}, {4657, 24923}, {4658, 5256}, {4859, 5705}, {5011, 16502}, {5030, 16968}, {5044, 16602}, {5121, 21616}, {5221, 16466}, {5247, 18201}, {5248, 17596}, {5264, 7191}, {5272, 12514}, {5400, 12528}, {5687, 17597}, {5709, 13329}, {6682, 19858}, {6997, 15434}, {7004, 9581}, {7204, 10521}, {7226, 19877}, {7759, 24699}, {8572, 10246}, {9534, 24620}, {10039, 23675}, {10306, 15287}, {10449, 17490}, {10571, 18838}, {11227, 15852}, {11263, 14815}, {12609, 24239}, {14964, 16716}, {15048, 21049}, {15839, 16200}, {15854, 25405}, {16600, 17754}, {16611, 21384}, {16700, 18180}, {16706, 20083}, {16817, 24627}, {16818, 24629}, {16823, 24464}, {16975, 21951}, {17734, 24914}, {17861, 24173}, {20320, 24218}, {21077, 24231}, {24789, 24880}
X(24047) lies on these lines: {1, 5030}, {2, 1796}, {3, 101}, {5, 5134}, {9, 2173}, {21, 16549}, {35, 672}, {36, 1334}, {39, 595}, {41, 5010}, {54, 71}, {55, 4253}, {56, 9327}, {58, 2276}, {100, 16552}, {106, 16969}, {140, 17747}, {165, 169}, {171, 25092}, {190, 1078}, {213, 4256}, {218, 4262}, {404, 3294}, {484, 17451}, {573, 10310}, {574, 2176}, {596, 9108}, {644, 5303}, {902, 5299}, {993, 3501}, {995, 5013}, {1018, 2975}, {1155, 16601}, {1191, 5024}, {1212, 3579}, {1385, 21872}, {1475, 3746}, {1571, 16968}, {1724, 17756}, {1759, 25082}, {1766, 7549}, {2141, 4184}, {2170, 11010}, {2329, 5267}, {3052, 9605}, {3208, 8666}, {3218, 3970}, {3295, 5022}, {3336, 21808}, {3496, 24036}, {3683, 25068}, {3693, 3916}, {3930, 6763}, {4189, 16788}, {4284, 5301}, {4420, 17746}, {4520, 17614}, {4640, 25066}, {4652, 17742}, {4653, 17750}, {4713, 7815}, {4877, 17303}, {5074, 17095}, {5179, 6684}, {5248, 17754}, {5254, 17734}, {5445, 21044}, {6603, 13624}, {7280, 9310}, {7573, 22000}, {7772, 21793}, {8671, 23851}, {8715, 21384}, {14439, 17744}, {14829, 21070}, {14964, 17524}, {16600, 17596}, {16784, 23649}, {17682, 17729}, {24044, 24052}
X(24048) lies on these lines: {1, 6}, {2, 24050}, {144, 22003}, {344, 22047}, {346, 4115}, {594, 8818}, {758, 21033}, {1018, 3949}, {1400, 4067}, {1761, 16553}, {1766, 3984}, {2171, 3678}, {3169, 3899}, {4006, 21871}, {4058, 21060}, {4361, 24076}, {5295, 18492}, {5949, 17757}, {17314, 24049}, {21075, 21675}
X(24248) = X(1)-of-obverse-triangle-of-X(69)
X(24049) lies on these lines: {1, 4037}, {2, 24044}, {37, 1698}, {145, 4115}, {321, 16831}, {1018, 4099}, {2901, 3294}, {3175, 3970}, {3178, 3947}, {3632, 21879}, {3633, 21839}, {3661, 3995}, {3679, 21816}, {17314, 24048}, {24045, 24071}, {24077, 24092}
X(24050) lies on these lines: {2, 24048}, {7, 22003}, {37, 527}, {72, 12563}, {75, 22047}, {142, 4053}, {226, 21069}, {344, 4115}, {514, 18714}, {3674, 3970}, {4043, 20924}, {6666, 21873}, {8818, 24086}, {17233, 24044}, {17243, 24067}, {17732, 24051}, {21879, 25072}, {22008, 22011}, {24077, 24081}, {24185, 24530}
X(24051) lies on these lines: {1, 4115}, {2, 24044}, {10, 37}, {519, 21816}, {1125, 4037}, {1509, 24074}, {1959, 3970}, {3244, 21879}, {3635, 21839}, {3995, 16826}, {4664, 22003}, {16777, 24067}, {17732, 24050}, {21820, 25092}
X(24052) lies on these lines: {2, 24064}, {37, 1054}, {171, 21883}, {1018, 4551}, {3699, 4115}, {17122, 21820}, {24044, 24047}
X(24053) lies on these lines: {1, 6}, {17732, 24064}, {24044, 24047}
X(24054) lies on these lines: {2, 1796}, {22, 5134}, {101, 7391}, {220, 5064}, {427, 17747}, {3730, 5133}, {4713, 21248}, {8750, 13854}
X(24055) lies on these lines: {2, 1796}, {23, 5134}, {101, 5189}, {858, 17747}, {3730, 5169}, {24085, 24088}
X(24056) lies on these lines: {312, 22038}, {3969, 24044}, {4153, 4177}, {17233, 24060}, {21070, 22010}, {22008, 22009}, {24061, 24071}
X(24057) lies on these lines: {190, 7917}, {4150, 4174}, {17233, 24045}, {17732, 24074}, {24070, 24080}
X(24058) lies on these lines: {2, 24067}, {37, 4472}, {75, 24224}, {226, 306}, {594, 24076}, {4043, 18157}, {4053, 4431}, {4115, 17277}, {4967, 21810}, {17233, 24044}, {17243, 24081}
X(24059) lies on these lines: {2, 24092}, {350, 22012}, {873, 4568}, {1621, 4115}, {3969, 24044}, {4359, 22011}, {21817, 22035}
X(24060) lies on these lines: {2, 24044}, {306, 1230}, {1920, 4043}, {3741, 4037}, {3969, 24071}, {4115, 17135}, {17233, 24056}, {20500, 22032}
X(24061) lies on these lines: {2, 24044}, {3687, 21070}, {3840, 4037}, {4115, 10453}, {24056, 24071}
X(24062) lies on these lines: {37, 3631}, {3912, 4053}, {3970, 22008}, {17233, 24044}, {17243, 24090}, {17297, 22003}, {24076, 24081}, {24083, 24084}
X(24063) lies on these lines: {2, 24069}, {37, 25269}, {75, 4053}, {304, 3970}, {4687, 21816}, {4751, 21810}, {17233, 24044}, {17243, 24077}
X(24064) lies on these lines: {2, 24052}, {37, 17889}, {4043, 22009}, {4417, 22038}, {17233, 24056}, {17717, 21883}, {17732, 24053}, {21090, 21923}, {22015, 22019}, {24044, 24045}
X(24065) lies on these lines: {10, 20970}, {306, 1230}, {519, 16886}, {3695, 4115}, {3936, 22011}, {3969, 24044}, {4035, 22017}, {4071, 21081}, {7230, 24070}, {17233, 24045}, {20500, 21092}, {20501, 22008}, {21066, 21076}
X(24066) lies on these lines: {2, 24048}, {226, 4053}, {312, 22047}, {1848, 22000}, {1870, 3191}, {3969, 24044}, {4115, 17776}, {5905, 22003}, {21072, 21077}, {24080, 24081}
X(24067) lies on these lines: {2, 24058}, {6, 4115}, {37, 39}, {190, 757}, {321, 4967}, {519, 21873}, {594, 4075}, {2171, 7276}, {2321, 3971}, {3175, 17133}, {3686, 21879}, {3949, 4099}, {3950, 4053}, {3995, 17011}, {4065, 4272}, {5105, 19582}, {5257, 21816}, {8818, 24070}, {16777, 24051}, {17233, 24081}, {17242, 22047}, {17243, 24050}, {17262, 22003}, {17314, 24048}, {21101, 21820}
X(24068) lies on these lines: {1, 3159}, {2, 596}, {5, 4884}, {10, 75}, {38, 1089}, {145, 14774}, {190, 595}, {192, 4065}, {194, 4568}, {518, 2901}, {519, 3869}, {522, 3913}, {537, 3874}, {714, 3743}, {1046, 24821}, {1107, 22036}, {1125, 3971}, {1126, 4360}, {1330, 21289}, {1698, 17155}, {1724, 3891}, {2176, 4115}, {2276, 21067}, {2292, 4692}, {3157, 23874}, {3175, 3555}, {3216, 3952}, {3293, 17147}, {3295, 17262}, {3454, 3703}, {3634, 24165}, {3670, 3701}, {3741, 4066}, {3831, 4125}, {3953, 4358}, {3987, 4723}, {3992, 24443}, {4013, 11681}, {4424, 4696}, {5267, 8669}, {5283, 22011}, {6693, 17602}, {6763, 17763}, {7226, 10479}, {9053, 15171}, {9369, 15955}, {16777, 24051}, {17114, 24816}, {17314, 17732}, {18135, 21208}, {18140, 24166}, {21101, 25092}, {24045, 24070}
X(24069) lies on these lines: {2, 24063}, {244, 19862}, {321, 3452}, {3701, 21081}, {3969, 24086}, {4053, 4358}, {24044, 24071}
X(24070) lies on these lines: {2, 24074}, {10, 115}, {190, 14061}, {226, 1358}, {1018, 4013}, {4120, 21090}, {4370, 5257}, {7230, 24065}, {8818, 24067}, {19862, 24956}, {21089, 21092}, {24045, 24068}, {24057, 24080}, {24072, 24076}
X(24071) lies on these lines: {2, 24052}, {37, 3120}, {321, 20500}, {3936, 22038}, {3969, 24060}, {4728, 22031}, {21090, 21092}, {24044, 24069}, {24045, 24049}, {24056, 24061}, {24073, 24086}, {24076, 24078}
X(24072) lies on these lines: {4568, 18035}, {21091, 21092}, {24070, 24076}
X(24073) lies on these lines: {306, 20500}, {1826, 2970}, {21091, 22032}, {24071, 24086}
X(24074) lies on these lines: {2, 24070}, {10, 148}, {99, 101}, {543, 21711}, {551, 894}, {1509, 24051}, {2891, 7283}, {3219, 5540}, {4037, 6629}, {6651, 17205}, {6758, 25265}, {17732, 24057}, {24187, 24956}
X(24075) lies on these lines: {2, 24044}, {8, 4037}, {37, 19877}, {321, 5308}, {346, 24883}, {3621, 4115}, {17230, 21070}, {20053, 21839}
X(24076) lies on these lines: {2, 24063}, {37, 86}, {321, 4033}, {536, 4053}, {545, 22003}, {594, 24058}, {918, 22031}, {3739, 21810}, {3943, 22047}, {4115, 4422}, {4361, 24048}, {4755, 21816}, {6370, 12078}, {17233, 24077}, {17243, 24050}, {17348, 21873}, {24062, 24081}, {24070, 24072}, {24071, 24078}
X(24077) lies on these lines: {2, 24058}, {37, 24598}, {321, 3596}, {3159, 19858}, {3247, 3995}, {4037, 17242}, {4115, 17349}, {4135, 21829}, {17233, 24076}, {17243, 24063}, {22016, 22036}, {24044, 24080}, {24049, 24092}, {24050, 24081}
X(24078) lies on these lines: {824, 4079}, {24071, 24076}
X(24079) lies on these lines: {17233, 24045}, {24082, 24083}
X(24080) lies on these lines: {2, 596}, {190, 7751}, {192, 21067}, {712, 20943}, {3661, 4066}, {4568, 20081}, {6376, 22036}, {7230, 17233}, {18832, 22039}, {24044, 24077}, {24057, 24070}, {24066, 24081}
X(24081) lies on these lines: {2, 24044}, {37, 25358}, {3912, 4037}, {3969, 4075}, {3995, 17292}, {4115, 6542}, {17233, 24067}, {17243, 24058}, {17776, 24880}, {24050, 24077}, {24062, 24076}, {24066, 24080}
X(24082) lies on these lines: {24079, 24083}
X(24083) lies on these lines: {10, 8663}, {661, 22043}, {3261, 3835}, {4502, 23803}, {16892, 22037}, {24062, 24084}, {24079, 24082}
X(24084) lies on these lines: {2, 24089}, {1577, 8611}, {3676, 7265}, {3835, 22044}, {24062, 24083}
X(24085) lies on these lines: {693, 4079}, {24055, 24088}, {24062, 24083}, {24071, 24076}
X(24086) lies on these lines: {10, 528}, {142, 24224}, {3912, 18073}, {3969, 24069}, {6666, 21675}, {8287, 22003}, {8818, 24050}, {21090, 21091}, {24070, 24072}, {24071, 24073}
X(24087) lies on these lines: {24079, 24082}
X(24088) lies on these lines: {850, 4079}, {24055, 24085}, {24079, 24082}
X(24089) lies on these lines: {2, 24084}, {37, 4976}, {522, 649}, {4151, 4170}, {4468, 7265}, {4765, 22042}, {21196, 22043}
X(24090) lies on these lines: {2, 24048}, {10, 20360}, {37, 524}, {321, 4052}, {594, 24058}, {3912, 21810}, {4053, 4357}, {4115, 17280}, {6646, 22003}, {17233, 24067}, {17243, 24062}, {17353, 21873}, {21879, 25101}, {24044, 24077}
X(24091) lies on these lines: {3969, 24044}, {4053, 4071}, {22038, 22047}, {24062, 24083}
X(24092) lies on these lines: {2, 24059}, {75, 2140}, {1001, 4115}, {16777, 24051}, {17233, 24044}, {24049, 24077}
X(24093) lies on these lines: {2, 24097}, {8, 900}, {10, 6550}, {191, 21385}, {513, 5836}, {514, 1125}, {523, 2292}, {659, 2975}, {676, 3445}, {891, 3878}, {1089, 3762}, {1220, 4581}, {1329, 3837}, {2826, 5690}, {3766, 20955}, {4448, 21105}, {5330, 21343}, {6366, 10912}, {10015, 24174}, {19947, 21198}, {24095, 24098}, {24101, 24104}, {24106, 24127}, {24111, 24112}, {24118, 24121}, {24122, 24130}, {24123, 24126}
X(24094) lies on these lines: {1, 522}, {1265, 14429}, {24098, 24099}, {24117, 24119}
X(24095) lies on these lines: {2, 24117}, {514, 894}, {523, 17246}, {4120, 17233}, {4361, 4750}, {16574, 21832}, {16892, 21133}, {20505, 21119}, {24093, 24098}, {24100, 24116}, {24101, 24107}, {24102, 24105}, {24106, 24112}, {24108, 24122}, {24118, 24133}, {24123, 24124}, {24129, 24130}
X(24096) lies on these lines: {2, 24098}, {37, 650}, {219, 4435}, {346, 1639}, {900, 16885}, {3287, 23617}, {24097, 24116}, {24112, 24121}
X(24097) lies on these lines: {1, 6550}, {2, 24093}, {10, 514}, {56, 659}, {145, 900}, {513, 3057}, {523, 1222}, {891, 5903}, {1482, 2826}, {2804, 15347}, {3336, 21385}, {3837, 11681}, {3904, 19582}, {4977, 5484}, {14421, 21201}, {18220, 23770}, {24096, 24116}, {24130, 24142}
X(24098) lies on these lines: {2, 24096}, {192, 918}, {344, 1639}, {513, 11997}, {523, 3728}, {650, 25067}, {812, 4032}, {900, 17365}, {1638, 4000}, {2804, 20507}, {21104, 23757}, {21133, 24457}, {24093, 24095}, {24094, 24099}, {24118, 24129}, {24130, 24133}
X(24099) lies on these lines: {1, 900}, {2, 24093}, {12, 3837}, {513, 12579}, {514, 3634}, {659, 5253}, {764, 4977}, {891, 3754}, {1022, 1224}, {1125, 6550}, {2826, 5901}, {4448, 23764}, {4943, 6161}, {18004, 21222}, {24094, 24098}, {24118, 24119}
X(24100) lies on these lines: {1459, 2309}, {24094, 24098}, {24095, 24116}, {24103, 24111}, {24115, 24118}
X(24101) lies on these lines: {16892, 20511}, {24093, 24104}, {24095, 24107}, {24102, 24106}, {24108, 24126}
X(24102) lies on these lines: {24095, 24105}, {24101, 24106}, {24125, 24132}
X(24103) lies on these lines: {2, 24118}, {141, 4728}, {812, 1654}, {1635, 17277}, {3762, 3963}, {4977, 21135}, {24093, 24095}, {24100, 24111}, {24121, 24129}
X(24104) lies on these lines: {2, 24142}, {764, 24169}, {1621, 4448}, {17494, 24381}, {24093, 24101}
X(24105) lies on these lines: {2, 24119}, {24095, 24102}
X(24106) lies on these lines: {24093, 24127}, {24095, 24112}, {24101, 24102}
X(24107) lies on these lines: {2, 24093}, {514, 6685}, {900, 17135}, {3741, 6550}, {24095, 24101}, {24120, 24126}
X(24108) lies on these lines: {2, 24093}, {514, 6686}, {900, 10453}, {3840, 6550}, {24095, 24122}, {24101, 24126}
X(24109) lies on these lines: {2, 24124}, {514, 4480}, {523, 2254}, {24093, 24095}, {24129, 24133}, {24134, 24135}
X(24110) lies on these lines: {2, 24123}, {45, 6546}, {4389, 6545}, {14475, 17290}, {16892, 24457}, {24093, 24095}
X(24111) lies on these lines: {2, 24126}, {55, 21105}, {514, 17596}, {982, 21132}, {16892, 20505}, {21118, 23761}, {24093, 24112}, {24095, 24101}, {24100, 24103}
X(24112) lies on these lines: {4642, 17420}, {24093, 24111}, {24095, 24106}, {24096, 24121}, {24126, 24128}
X(24113) lies on these lines: {2, 24096}, {345, 1639}, {650, 824}, {918, 17490}, {20505, 20508}, {21120, 21129}, {24093, 24111}
X(24114) lies on these lines: {21121, 21134}, {24093, 24101}, {24095, 24102}
X(24115) lies on these lines: {2, 24096}, {650, 25091}, {918, 3210}, {1639, 17776}, {24093, 24101}, {24100, 24118}, {24132, 24133}
X(24116) lies on these lines: {10459, 17418}, {24093, 24111}, {24095, 24100}, {24096, 24097}
X(24117) lies on these lines: {2, 24095}, {514, 3662}, {523, 17340}, {579, 21832}, {661, 17452}, {4120, 17314}, {24094, 24119}, {24096, 24097}
X(24118) lies on these lines: {2, 24103}, {6, 1635}, {513, 16814}, {514, 24199}, {649, 2269}, {661, 1213}, {665, 21796}, {812, 17300}, {3572, 4581}, {4728, 17234}, {14284, 21834}, {21106, 21127}, {24093, 24121}, {24095, 24133}, {24096, 24097}, {24098, 24129}, {24099, 24119}, {24100, 24115}
X(24119) lies on these lines: {2, 24105}, {661, 23897}, {2176, 14407}, {4079, 21879}, {14825, 17192}, {24094, 24117}, {24099, 24118}
X(24120) lies on these lines: {2, 24095}, {63, 21832}, {3969, 4120}, {4988, 6546}, {21124, 21129}, {24093, 24101}, {24107, 24126}, {24123, 24138}
X(24121) lies on these lines: {2, 24095}, {9, 21832}, {594, 4120}, {4988, 6544}, {24093, 24118}, {24096, 24112}, {24103, 24129}
X(24122) lies on these lines: {24093, 24130}, {24095, 24108}
X(24123) lies on these lines: {2, 24110}, {88, 6546}, {4750, 4763}, {6545, 24183}, {10015, 21115}, {14442, 16594}, {24093, 24126}, {24095, 24124}, {24120, 24138}
X(24124) lies on these lines: {2, 24109}, {16892, 21129}, {24095, 24123}
X(24125) lies on these lines: {3125, 14442}, {24102, 24132}, {24127, 24129}
X(24126) lies on these lines: {2, 24111}, {100, 21105}, {244, 21132}, {514, 1054}, {659, 22379}, {2254, 2804}, {3120, 6545}, {24093, 24123}, {24101, 24108}, {24107, 24120}, {24112, 24128}, {24129, 24131}
X(24127) lies on these lines: {514, 9259}, {7200, 21132}, {21133, 21138}, {24093, 24106}, {24125, 24129}
X(24128) lies on these lines: {1, 514}, {2, 24093}, {8, 6550}, {513, 14923}, {900, 3621}, {2826, 12245}, {12513, 13266}, {24112, 24126}
X(24129) lies on these lines: {2, 24110}, {190, 6546}, {513, 21889}, {514, 4440}, {649, 16560}, {812, 3904}, {900, 4088}, {1086, 6545}, {2786, 21385}, {4422, 6544}, {4473, 10196}, {7202, 21143}, {24095, 24130}, {24098, 24118}, {24103, 24121}, {24109, 24133}, {24125, 24127}, {24126, 24131}
X(24130) lies on these lines: {2, 24103}, {69, 812}, {513, 17351}, {514, 3663}, {1635, 17349}, {3596, 3762}, {4444, 21143}, {4728, 17232}, {24093, 24122}, {24095, 24129}, {24097, 24142}, {24098, 24133}
X(24131) lies on these lines: {11, 244}, {519, 4645}, {903, 1644}, {1054, 4440}, {1320, 24715}, {6547, 6550}, {21087, 24821}, {24126, 24129}
X(24132) lies on these lines: {2, 24105}, {514, 3959}, {24093, 24122}, {24102, 24125}, {24115, 24133}
X(24133) lies on these lines: {2, 24093}, {241, 514}, {900, 6542}, {3912, 6550}, {24095, 24118}, {24098, 24130}, {24109, 24129}, {24115, 24132}
X(24134) lies on these lines: {1015, 21196}, {1086, 3004}, {16892, 21138}, {24109, 24135}
X(24135) lies on these lines: {2, 24140}, {2170, 4369}, {21104, 21139}, {24109, 24134}
X(24136) lies on these lines: {244, 23772}, {1647, 4088}, {21140, 24193}, {24109, 24134}, {24126, 24129}
X(24137) lies on these lines: {3121, 24900}, {24109, 24134}
X(24138) lies on these lines: {1086, 21131}, {10015, 20509}, {24120, 24123}, {24125, 24127}
X(24139) lies on these lines: {3801, 21142}
X(24140) lies on these lines: {2, 24135}, {115, 661}, {2223, 21805}, {2325, 3930}, {3119, 4521}, {5375, 17439}
X(24141) lies on these lines: {2, 24096}, {75, 918}, {523, 25124}, {650, 25099}, {659, 8424}, {900, 17332}, {1639, 17280}, {17069, 21202}, {24093, 24122}, {24095, 24118}, {24103, 24121}
X(24142) lies on these lines: {2, 24104}, {764, 3821}, {1001, 4448}, {24093, 24095}, {24097, 24130}, {24099, 24118}
X(24143) lies on these lines: {4513, 14427}, {24096, 24097}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.
X(24144) lies on the cubic K466 and these lines: {3,252}, {54,1263}, {12026,19268}, {14072,21975}
X(24144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (930, 1141, 252), (1263, 6343, 10285)
X(24144) = circumcircle-inverse of X(252)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.
X(24145) lies on these lines: {2,2006}, {81,1086}, {278,23958}, {651,1994}, {3960,4560}, {5718,17395}, {5723,17484}, {17013,19785}, {17732,24046}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.
X(24146) lies on these lines: {2,2006}, {343,17483}, {3448,21028}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.
X(24147) lies on these lines: {140,1493}, {195,252}, {1157,19553}, {5965,10615}, {7159,14102}
X(24147) = midpoint of X(1157) and X(19553)
X(24147) = {X(195),X(252)}-harmonic conjugate of X(13856)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.
X(24148) lies on these lines: {239,514}, {323,4858}, {1994,14213}, {3187,17035}, {3219,18662}, {17479,19742}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28378.
X(24149) lies on these lines: {2,7}, {37,16698}, {81,18359}, {323,6358}, {1994,14213}, {17168,22002}
Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(4098) and X(24150) are equal.
X(24150) lies on the cubic K1077 and these lines: {1, 3161}, {2, 8051}, {9, 2137}
X(24150) = X(8051)-Ceva conjugate of X(6553)
X(24150) = X(i)-cross conjugate of X(j) for these (i,j): {9, 3161}, {6555, 145}
X(24150) = X(i)-isoconjugate of X(j) for these (i,j): {1616, 8056}, {3445, 23511}, {5382, 17071}, {8055, 16945}
X(24150) = barycentric product X(i)X(j) for these {i,j}: {145, 6553}, {3161, 8051}
X(24150) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 4452}, {1743, 23511}, {2137, 19604}, {3052, 1616}, {3158, 2136}, {3161, 8055}, {4849, 21896}, {6553, 4373}, {6555, 6552}, {20818, 23089}
X(24151) lies on the cubic K1077 and these lines: {2, 8051}, {8, 3452}, {9, 8056}, {3038, 5573}, {4373, 18228}, {15601, 17958}
X(24151) = complement of X(8051)
X(24151) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 21627}, {31, 8056}, {41, 346}, {1616, 142}, {2136, 141}, {4452, 17046}, {6552, 21244}, {8055, 2887}, {21896, 17052}, {23511, 2886}
X(24151) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8056}, {19604, 3680}
X(24151) = crosspoint of X(2) and X(8055)
X(24151) = X(i)-isoconjugate of X(j) for these (i,j): {1743, 2137}, {3052, 8051}
X(24151) = barycentric product X(i)X(j) for these {i,j}: {2136, 4373}, {3680, 4452}, {6552, 19604}, {6557, 23511}, {8055, 8056}
X(24151) = barycentric quotient X(i)/X(j) for these {i,j}: {1616, 1420}, {2136, 145}, {3445, 2137}, {3680, 6553}, {8055, 18743}, {8056, 8051}, {21896, 4848}, {23511, 5435}
X(24152) lies on the cubic K1077 and this line: {9,55}
X(24152) = {X(9),X(200)}-harmonic conjugate of X(24153)
X(24152) = {X(55),X(3693)}-harmonic conjugate of X(24153)
X(24153) lies on the cubic K1077 and this line: {9,55}
X(24153) = {X(9),X(200)}-harmonic conjugate of X(24152)
X(24153) = {X(55),X(3693)}-harmonic conjugate of X(24152)
X(24154) lies on the conic {{A,B,C,X(2),X(7)}} and the cubic K1077, and on this line: {1,142}
X(24154) = complement X(24314)
X(24154) = {X(1),X(142)}-harmonic conjugate of X(24155)
X(24155) lies on the conic {{A,B,C,X(2),X(7)}} and the cubic K1077, and on this line: {1,142}
X(24155) = complement X(24313)
X(24155) = {X(1),X(142)}-harmonic conjugate of X(24154)
X(24156) lies on the cubic K1077, and on this line: {277,5745}
X(24157) lies on the cubic K1077, and on this line: {277,5745}
X(24158) lies on these lines: {1, 236}, {2, 1488}, {188, 3161}
X(24158) = X(9)-cross conjugate of X(236)
X(24158) = X(289)-isoconjugate of X(8078)
X(24158) = barycentric product X(7057)*X(12644)
X(24158) = barycentric quotient X(12644)/X(7048)
X(24159) lies on these lines: {1, 224}, {2, 3670}, {3, 1086}, {5, 17054}, {6, 6147}, {7, 58}, {8, 24222}, {38, 19854}, {46, 3011}, {72, 24789}, {88, 17566}, {142, 975}, {221, 15253}, {226, 19372}, {244, 499}, {278, 4306}, {344, 3159}, {348, 17205}, {386, 2140}, {387, 11036}, {405, 3782}, {498, 24443}, {595, 4295}, {614, 12047}, {936, 4859}, {942, 3772}, {946, 990}, {982, 24161}, {986, 10198}, {988, 1125}, {995, 3485}, {997, 24178}, {1056, 15955}, {1279, 12699}, {1330, 19851}, {1387, 3445}, {1453, 4654}, {1478, 3924}, {1479, 3120}, {1616, 22791}, {1698, 24223}, {1714, 3868}, {1722, 21077}, {1724, 5905}, {1738, 3811}, {1739, 5552}, {1788, 17734}, {1834, 15934}, {1930, 4441}, {2345, 22011}, {3616, 4653}, {3649, 16466}, {3682, 24779}, {3752, 11374}, {3767, 20271}, {3953, 10527}, {3987, 10528}, {4252, 24470}, {4255, 5719}, {4256, 5703}, {4310, 19843}, {4346, 17558}, {4363, 17698}, {4364, 16844}, {4389, 11110}, {4415, 11108}, {4419, 16845}, {4642, 10056}, {4694, 10529}, {4962, 21172}, {5044, 17278}, {5230, 5902}, {5248, 24248}, {5266, 5880}, {5272, 21616}, {5439, 17720}, {5573, 8227}, {5687, 17724}, {5711, 17061}, {5758, 13329}, {6700, 24175}, {7040, 7649}, {7483, 17595}, {9579, 16485}, {9895, 24476}, {10200, 17063}, {10916, 17064}, {11269, 18398}, {13411, 24177}, {14023, 24699}, {14450, 17127}, {17597, 24390}, {17719, 24174}, {17871, 23555}, {19755, 20256}, {23581, 24204}, {24166, 24187}
X(24160) lies on these lines: {1, 2476}, {2, 3670}, {10, 3699}, {35, 3120}, {37, 24937}, {58, 226}, {65, 17734}, {72, 24880}, {73, 2006}, {106, 23675}, {140, 1086}, {171, 11263}, {386, 3772}, {495, 15955}, {595, 3011}, {894, 6693}, {995, 11375}, {1125, 13161}, {1201, 5443}, {1279, 9955}, {1616, 18493}, {1656, 17054}, {1834, 5719}, {3487, 5292}, {3584, 4642}, {3782, 7483}, {3811, 17064}, {3838, 5266}, {3841, 5293}, {3915, 18393}, {3924, 7951}, {3944, 5248}, {3970, 17737}, {4256, 13411}, {4415, 6675}, {4861, 24222}, {5687, 17783}, {5904, 24892}, {6788, 17606}, {7746, 20271}, {7780, 24699}, {16706, 20108}, {17095, 17205}, {17724, 24390}, {17749, 24789}, {19862, 24171}, {19878, 25072}, {24170, 24187}
X(24161) lies on these lines: {1, 442}, {2, 986}, {3, 17889}, {10, 3699}, {21, 3120}, {35, 24715}, {43, 11374}, {58, 11263}, {140, 1054}, {171, 12609}, {226, 5247}, {238, 1780}, {405, 3944}, {498, 24440}, {499, 1725}, {758, 24880}, {846, 6675}, {896, 14450}, {905, 16744}, {946, 7413}, {978, 11375}, {982, 24159}, {984, 19854}, {988, 23681}, {1010, 1125}, {1046, 3649}, {1086, 4999}, {1104, 3838}, {1330, 4892}, {1699, 13442}, {1717, 23708}, {1722, 5219}, {1738, 13411}, {1962, 24936}, {2476, 3924}, {2650, 24883}, {3011, 5255}, {3523, 7613}, {3584, 3987}, {3624, 4657}, {3754, 17734}, {3782, 24953}, {3821, 19270}, {3846, 16817}, {3868, 24892}, {3925, 5293}, {3976, 10527}, {4425, 11110}, {5015, 21241}, {5272, 8227}, {5497, 24387}, {5886, 15973}, {5988, 24181}, {6536, 17557}, {6857, 24248}, {7379, 16020}, {7483, 17596}, {8143, 11230}, {8616, 12699}, {13740, 25385}, {16887, 24187}, {17123, 21616}, {17593, 19862}, {17733, 18134}, {17737, 21808}, {19269, 24788}, {21264, 24774}, {23536, 24541}
X(24161) = perspector of Gemini triangle 7 and cross-triangle of Gemini triangles 3 and 7
X(24162) lies on these lines: {7, 614}, {10, 24173}, {56, 2218}, {75, 24178}, {142, 3666}, {226, 20227}, {244, 307}, {982, 4357}, {988, 1125}, {1086, 18589}, {1441, 23675}, {1444, 17205}, {4656, 5750}, {17282, 20106}, {17863, 23536}, {21174, 23808}, {23663, 23677}
X(24163) lies on these lines: {2, 3670}, {25, 17054}, {31, 57}, {38, 17306}, {191, 5272}, {427, 1086}, {612, 24443}, {1194, 20271}, {1739, 10327}, {3924, 5322}, {3987, 20020}, {4694, 19993}, {17871, 21210}
X(24164) lies on these lines: {2, 3670}, {238, 244}, {614, 3337}, {858, 1086}, {1995, 17054}, {2228, 3006}, {3920, 4642}, {5297, 24443}, {9465, 20271}
X(24165) lies on these lines: {1, 3210}, {2, 726}, {10, 38}, {21, 8720}, {42, 17140}, {43, 17490}, {57, 4362}, {63, 16825}, {75, 982}, {76, 20340}, {86, 17600}, {142, 24180}, {190, 17123}, {210, 537}, {244, 321}, {310, 4609}, {312, 4871}, {354, 740}, {404, 8669}, {514, 8027}, {518, 4685}, {519, 3873}, {522, 21204}, {536, 3742}, {551, 1962}, {553, 5847}, {614, 3923}, {714, 4688}, {750, 3891}, {752, 11246}, {756, 24589}, {846, 16823}, {899, 4090}, {968, 24331}, {984, 19804}, {1015, 21345}, {1054, 7081}, {1086, 2887}, {1125, 24598}, {1201, 17164}, {1215, 3752}, {1266, 24210}, {1269, 20598}, {1386, 4697}, {1575, 21101}, {2321, 3726}, {2886, 7263}, {3058, 17764}, {3159, 6532}, {3244, 3896}, {3338, 17733}, {3452, 21093}, {3634, 24068}, {3663, 4425}, {3666, 24325}, {3687, 24231}, {3696, 21342}, {3703, 3836}, {3705, 7897}, {3706, 3999}, {3720, 3993}, {3729, 4011}, {3739, 21080}, {3757, 17596}, {3771, 17740}, {3782, 3846}, {3819, 14839}, {3826, 4884}, {3831, 24046}, {3835, 21197}, {3953, 4647}, {3966, 4655}, {3967, 16602}, {4003, 6682}, {4025, 20518}, {4028, 5542}, {4038, 4360}, {4135, 4358}, {4385, 24174}, {4418, 7191}, {4438, 24789}, {4453, 20525}, {4514, 24715}, {4641, 4974}, {4651, 17154}, {4659, 5573}, {4671, 9335}, {4699, 17157}, {4703, 17276}, {4706, 4946}, {4709, 17135}, {4847, 20882}, {4850, 6685}, {4865, 5880}, {5263, 17598}, {6327, 24692}, {6377, 16606}, {7186, 25048}, {8167, 17262}, {10009, 17149}, {10980, 17151}, {11031, 20880}, {11269, 19789}, {11679, 18193}, {13407, 17748}, {13476, 22316}, {14829, 18201}, {16569, 24620}, {16604, 17459}, {16703, 17205}, {17146, 20011}, {17447, 23944}, {17448, 22184}, {18143, 25140}, {18144, 25121}, {18169, 21352}, {18837, 20888}, {19805, 23537}, {20459, 21369}, {20892, 23677}, {20911, 24215}, {21085, 24235}, {21416, 24221}, {21814, 22011}, {22232, 22343}, {24239, 25385}
X(24166) lies on these lines: {2, 22011}, {75, 24046}, {76, 596}, {244, 1930}, {330, 514}, {626, 1086}, {712, 16604}, {982, 16739}, {1574, 9055}, {3216, 17141}, {3760, 17155}, {3953, 20911}, {4025, 21201}, {4441, 24226}, {4446, 24180}, {6534, 18146}, {16600, 24631}, {18140, 24068}, {18837, 20888}, {19805, 24177}, {20530, 22036}, {20893, 24221}, {21443, 24207}, {24159, 24187}, {24185, 24190}
X(24167) lies on these lines: {1, 1392}, {10, 982}, {38, 3634}, {57, 4347}, {58, 17513}, {191, 7292}, {244, 1125}, {519, 3953}, {551, 986}, {596, 3831}, {726, 18137}, {942, 4719}, {978, 4067}, {993, 17054}, {995, 4084}, {1104, 4973}, {1393, 4298}, {1698, 4392}, {1739, 3626}, {3159, 4871}, {3244, 3976}, {3293, 17449}, {3315, 3746}, {3336, 7191}, {3337, 4351}, {3579, 4906}, {3624, 9335}, {3625, 24440}, {3635, 4642}, {3636, 4424}, {3678, 16610}, {3741, 24176}, {3742, 3743}, {3752, 3874}, {3782, 3825}, {3881, 3999}, {3892, 4646}, {4003, 5439}, {4066, 17155}, {4134, 17749}, {4695, 4701}, {4850, 18398}, {4868, 5045}, {5248, 17595}, {5573, 12514}, {8715, 17597}, {10916, 24177}, {11263, 24239}, {16887, 24189}, {17050, 24185}, {17063, 19862}, {21208, 24214}
X(24168) lies on these lines: {1, 9335}, {10, 982}, {36, 88}, {38, 3828}, {57, 16428}, {244, 519}, {484, 7292}, {551, 17063}, {726, 24196}, {756, 3634}, {758, 16610}, {978, 4084}, {986, 19862}, {995, 3919}, {997, 8056}, {1086, 3814}, {1125, 4424}, {1149, 4674}, {1698, 7226}, {2292, 19878}, {3244, 24440}, {3625, 3976}, {3626, 3953}, {3635, 3987}, {3636, 4642}, {3666, 3833}, {3742, 4868}, {3752, 5883}, {3831, 24176}, {3840, 4717}, {3874, 4849}, {4067, 17749}, {4125, 17155}, {4134, 16569}, {4392, 19875}, {5121, 11813}, {10176, 16602}, {17164, 19847}, {20335, 24185}, {20517, 21186}
X(24169) lies on these lines: {2, 846}, {6, 4987}, {10, 38}, {35, 404}, {43, 3662}, {82, 171}, {141, 21085}, {142, 16056}, {244, 4972}, {354, 4085}, {516, 19649}, {614, 4660}, {756, 24988}, {764, 24104}, {899, 17184}, {908, 6686}, {946, 19514}, {982, 4429}, {1051, 20090}, {1086, 1215}, {1155, 6679}, {1376, 17290}, {1738, 3741}, {1961, 17302}, {2292, 17674}, {2887, 3752}, {3125, 23914}, {3263, 3663}, {3336, 8258}, {3589, 4697}, {3666, 3836}, {3740, 17235}, {3782, 21093}, {3791, 17366}, {3816, 25377}, {3831, 19810}, {3840, 3914}, {3846, 16610}, {3912, 4970}, {3925, 6682}, {4000, 4362}, {4028, 21255}, {4383, 4655}, {4415, 24003}, {4434, 17061}, {4438, 17595}, {4640, 17356}, {4672, 11246}, {4682, 17382}, {4734, 17232}, {4871, 24210}, {5047, 12579}, {5249, 6685}, {5268, 17304}, {6377, 18905}, {6541, 17147}, {7191, 17766}, {7262, 17352}, {10180, 17245}, {11263, 20108}, {13741, 24851}, {15523, 17495}, {16062, 24174}, {16707, 17205}, {17023, 24610}, {17122, 19786}, {17123, 24723}, {17234, 17592}, {17282, 17594}, {21241, 24239}, {21242, 21949}, {21817, 22011}, {24170, 24211}, {24318, 25345}, {25349, 25357}
X(24169) = complement of X(32930)
X(24170) lies on these lines: {2, 2140}, {10, 274}, {39, 20255}, {75, 24046}, {116, 6656}, {141, 1574}, {171, 17200}, {335, 21067}, {384, 17729}, {514, 7187}, {538, 21025}, {995, 21281}, {1086, 3934}, {1258, 3997}, {1575, 21240}, {1739, 20911}, {1930, 24443}, {3125, 16720}, {3159, 20947}, {3216, 17137}, {3263, 3670}, {3634, 4357}, {3721, 4568}, {3831, 20888}, {4642, 14210}, {4651, 17208}, {5247, 6629}, {5280, 24602}, {6381, 24214}, {7770, 14377}, {8362, 21258}, {9436, 10521}, {16589, 25349}, {17210, 19874}, {17759, 21070}, {19810, 24177}, {22008, 24530}, {22011, 24185}, {24160, 24187}, {24169, 24211}
X(24171) lies on these lines: {1, 6904}, {3, 12592}, {4, 5573}, {10, 982}, {36, 1612}, {58, 1414}, {244, 1210}, {386, 5542}, {387, 10980}, {443, 3677}, {515, 17054}, {553, 16466}, {614, 4292}, {936, 4310}, {946, 1086}, {975, 4353}, {978, 24231}, {988, 1125}, {995, 3671}, {1738, 3976}, {1739, 6736}, {3086, 23681}, {3333, 4000}, {3361, 3668}, {3624, 4656}, {3646, 4419}, {3660, 5930}, {3752, 21620}, {3755, 5045}, {3924, 4311}, {3953, 4847}, {4859, 19843}, {5265, 22464}, {5272, 12572}, {6700, 11512}, {7174, 17582}, {9843, 13161}, {10445, 20227}, {10461, 16752}, {10595, 15839}, {11019, 19790}, {11496, 15287}, {16610, 21075}, {16825, 24215}, {17749, 21060}, {19862, 24160}, {23675, 24443}
X(24172) lies on these lines: {1, 1447}, {10, 4986}, {11, 4920}, {37, 17048}, {75, 3831}, {85, 3976}, {142, 20861}, {244, 20880}, {307, 1210}, {726, 3760}, {982, 3673}, {1086, 17046}, {1111, 3953}, {1193, 20247}, {1334, 24403}, {1930, 3840}, {3125, 20257}, {3662, 7938}, {3664, 18398}, {3934, 21101}, {3970, 24786}, {3999, 4059}, {16887, 23824}, {17050, 20271}, {17197, 18167}, {17448, 21138}, {18833, 21443}, {18837, 20888}, {20435, 21330}, {21084, 24575}
X(24173) lies on these lines: {10, 24162}, {75, 3831}, {142, 24195}, {240, 1210}, {244, 1441}, {273, 1860}, {322, 3976}, {1086, 20305}, {1149, 21271}, {1393, 3668}, {3663, 21208}, {3778, 23677}, {3840, 18697}, {4858, 20259}, {10436, 18906}, {16609, 20227}, {17861, 24046}, {17863, 24443}
X(24174) lies on these lines: {1, 474}, {2, 986}, {3, 1054}, {5, 17889}, {8, 244}, {10, 982}, {34, 9364}, {36, 20842}, {38, 9780}, {40, 5272}, {43, 942}, {46, 238}, {55, 16422}, {57, 1722}, {58, 11116}, {65, 978}, {72, 16569}, {75, 3831}, {88, 2975}, {106, 22837}, {142, 5530}, {145, 4695}, {171, 16478}, {277, 291}, {386, 5883}, {404, 3924}, {405, 17596}, {442, 20338}, {499, 1772}, {517, 21214}, {518, 6048}, {614, 5255}, {726, 24182}, {750, 5262}, {756, 19877}, {758, 17749}, {774, 5704}, {846, 11108}, {899, 3868}, {946, 5121}, {960, 16602}, {984, 1698}, {995, 3754}, {1015, 4051}, {1046, 4383}, {1086, 1329}, {1149, 14923}, {1210, 1738}, {1247, 16575}, {1254, 5435}, {1278, 22220}, {1393, 1788}, {1401, 23841}, {1466, 2647}, {1479, 24715}, {1575, 20271}, {1714, 24779}, {1724, 3336}, {1735, 17064}, {2275, 21951}, {2478, 24851}, {2944, 19517}, {3061, 3125}, {3091, 7613}, {3120, 4193}, {3178, 17234}, {3214, 3873}, {3216, 5902}, {3290, 3501}, {3293, 18398}, {3339, 23511}, {3579, 8616}, {3616, 4642}, {3624, 4424}, {3632, 4694}, {3679, 3953}, {3701, 17155}, {3731, 25086}, {3756, 3813}, {3826, 4446}, {3833, 17592}, {3915, 7292}, {3923, 13741}, {3944, 4187}, {3959, 16604}, {3961, 9709}, {3980, 13740}, {4000, 17048}, {4208, 11031}, {4253, 16611}, {4385, 24165}, {4386, 16787}, {4413, 5293}, {4414, 5047}, {4418, 5192}, {4420, 9350}, {4662, 21342}, {4674, 5697}, {4904, 20255}, {4926, 15079}, {5055, 5492}, {5084, 24248}, {5248, 17601}, {5266, 16498}, {5400, 15071}, {5529, 12635}, {6682, 19853}, {6735, 23675}, {8582, 13161}, {8715, 17715}, {9843, 24210}, {10015, 24093}, {10479, 25122}, {11533, 16863}, {12514, 17123}, {12609, 17717}, {12782, 24181}, {14837, 19950}, {16062, 24169}, {16583, 17754}, {16605, 21384}, {16736, 18178}, {16969, 21888}, {17337, 18253}, {17606, 24430}, {17697, 24850}, {17719, 24159}, {17748, 18134}, {17756, 21808}, {18180, 18792}, {21075, 24231}, {21173, 23345}, {23536, 24982}, {24311, 24342}, {24369, 24653}, {24372, 24923}, {24777, 24785}, {24789, 24914}
X(24175) lies on these lines: {1, 11024}, {2, 2415}, {7, 23511}, {10, 982}, {43, 5542}, {57, 169}, {142, 2092}, {165, 16020}, {226, 16610}, {244, 4847}, {321, 24183}, {329, 4887}, {333, 17205}, {516, 5272}, {519, 18141}, {551, 3750}, {553, 4383}, {978, 3671}, {1054, 10164}, {1074, 1210}, {1086, 3452}, {1125, 11512}, {1266, 18743}, {1699, 7613}, {1722, 4298}, {1738, 11019}, {1743, 21454}, {2550, 5573}, {2999, 3664}, {3210, 3950}, {3306, 24779}, {3622, 7963}, {3666, 24181}, {3668, 5435}, {3687, 21255}, {3742, 3755}, {3756, 21949}, {3772, 6692}, {3782, 5316}, {3817, 5121}, {3834, 4035}, {3873, 4924}, {3911, 24789}, {3912, 17490}, {3971, 24200}, {4000, 5437}, {4021, 17022}, {4031, 4641}, {4082, 17155}, {4301, 21214}, {4310, 8580}, {4353, 5268}, {4359, 24219}, {4384, 24215}, {4862, 18228}, {5325, 17337}, {5745, 17278}, {6555, 15590}, {6700, 24159}, {6736, 23675}, {7658, 23801}, {8582, 23536}, {9843, 19802}, {14837, 21198}, {16569, 21060}, {16736, 17197}, {16832, 24214}, {17282, 20106}, {17593, 19862}, {21342, 24393}
X(24175) = complement of X(30568)
X(24176) lies on these lines: {1, 17495}, {2, 3159}, {5, 7263}, {10, 38}, {75, 24046}, {244, 4647}, {386, 17490}, {519, 942}, {537, 4015}, {540, 24470}, {726, 3634}, {1070, 10916}, {1086, 3454}, {1125, 3666}, {1574, 21067}, {1575, 22011}, {1698, 17155}, {2292, 6533}, {2901, 5439}, {3293, 17140}, {3626, 24235}, {3720, 4065}, {3741, 24167}, {3752, 20108}, {3828, 6534}, {3831, 24168}, {3836, 24180}, {4000, 20083}, {4066, 20891}, {4142, 8714}, {4257, 19851}, {4361, 5708}, {9843, 17863}, {16825, 20367}, {16853, 17262}, {17205, 20911}, {17591, 19858}, {17749, 24620}, {20888, 21208}, {21187, 21203}, {21240, 24185}
X(24177) lies on these lines: {1, 6904}, {2, 2415}, {6, 553}, {7, 2999}, {10, 38}, {19, 57}, {42, 5542}, {43, 24231}, {63, 3008}, {81, 17205}, {142, 3666}, {200, 4310}, {226, 1086}, {239, 24215}, {244, 3914}, {306, 17495}, {312, 1266}, {329, 4862}, {345, 17282}, {354, 3755}, {376, 16485}, {496, 18257}, {497, 5573}, {516, 614}, {527, 4383}, {528, 4906}, {551, 19336}, {612, 4353}, {726, 4082}, {899, 21060}, {940, 3946}, {950, 17054}, {968, 1125}, {980, 17050}, {982, 1738}, {986, 24178}, {1122, 14557}, {1149, 4342}, {1193, 3671}, {1201, 4301}, {1210, 1785}, {1462, 23601}, {1463, 23638}, {1722, 12527}, {1743, 9965}, {1817, 17189}, {2221, 14377}, {2550, 3677}, {2968, 20309}, {3011, 10164}, {3120, 3817}, {3210, 3912}, {3264, 18739}, {3306, 19785}, {3452, 3782}, {3474, 7290}, {3662, 3687}, {3664, 5256}, {3672, 17022}, {3710, 17674}, {3720, 4356}, {3772, 3911}, {3823, 4884}, {3875, 18141}, {3915, 5493}, {3924, 4297}, {3925, 4003}, {3944, 5121}, {3950, 17147}, {4021, 5287}, {4031, 17366}, {4114, 17365}, {4198, 4292}, {4346, 18228}, {4349, 17017}, {4357, 19804}, {4384, 24214}, {4392, 25006}, {4398, 18743}, {4415, 5316}, {4419, 7308}, {4430, 4924}, {4512, 16020}, {4642, 23675}, {4850, 5249}, {4858, 20205}, {4887, 5905}, {4896, 17012}, {5222, 10481}, {5272, 24248}, {5313, 11551}, {5435, 22464}, {5712, 6173}, {5743, 17235}, {5745, 17067}, {5853, 17597}, {6692, 17720}, {6703, 17382}, {6736, 24440}, {7058, 24378}, {7191, 20097}, {7963, 24558}, {8582, 13161}, {10521, 24590}, {10916, 24167}, {13411, 24159}, {14552, 16833}, {14555, 17274}, {16700, 17197}, {16752, 17185}, {17020, 17483}, {17063, 24210}, {17740, 20106}, {17889, 24239}, {19805, 24166}, {19810, 24170}, {20367, 24790}, {24181, 25080}
X(24178) lies on these lines: {1, 142}, {2, 988}, {4, 5272}, {5, 5121}, {8, 23675}, {10, 982}, {12, 16610}, {20, 16020}, {28, 36}, {43, 21620}, {56, 24789}, {72, 24231}, {75, 24162}, {171, 12436}, {225, 7288}, {226, 978}, {238, 4292}, {244, 6734}, {274, 4357}, {377, 614}, {388, 1722}, {404, 3011}, {442, 24239}, {475, 499}, {553, 1046}, {631, 1072}, {905, 19947}, {946, 17889}, {958, 17278}, {960, 1086}, {962, 7613}, {986, 24177}, {995, 12609}, {997, 24159}, {1010, 1125}, {1054, 6684}, {1074, 3086}, {1191, 5880}, {1193, 5249}, {1210, 17063}, {1329, 16602}, {1698, 8056}, {1714, 3338}, {1739, 10039}, {1743, 4355}, {1834, 3742}, {2292, 24564}, {2475, 7292}, {3008, 4298}, {3216, 13407}, {3306, 5230}, {3315, 5178}, {3612, 23604}, {3616, 3914}, {3710, 17155}, {3752, 5530}, {3813, 21949}, {3912, 17670}, {3976, 4847}, {4201, 16823}, {4340, 16475}, {4383, 10404}, {4696, 24988}, {4719, 17056}, {4904, 21240}, {4968, 17674}, {5253, 17518}, {5267, 11102}, {5268, 17582}, {5290, 23511}, {5302, 17337}, {5345, 17562}, {5794, 17054}, {6700, 17719}, {7963, 25055}, {8583, 23681}, {10624, 24715}, {10916, 24216}, {11321, 17023}, {12572, 17123}, {13728, 24774}, {16458, 19758}, {16569, 21075}, {16828, 24778}, {17749, 21077}, {20880, 24548}, {21189, 23808}, {24443, 24987}
X(24179) lies on these lines: {1, 1441}, {7, 90}, {36, 18655}, {46, 17220}, {57, 24220}, {69, 10916}, {75, 997}, {77, 1111}, {86, 3673}, {273, 4341}, {307, 499}, {614, 17188}, {988, 1125}, {1086, 17073}, {1447, 10446}, {1479, 18650}, {3664, 11019}, {3729, 25078}, {3875, 22836}, {5231, 17272}, {7289, 17197}, {8583, 24547}, {10826, 21270}, {13374, 24471}, {16831, 25065}, {18698, 19861}, {20880, 24540}, {24207, 24227}, {24559, 25252}
X(24180) lies on these lines: {37, 39}, {38, 3778}, {142, 24165}, {522, 21194}, {732, 17049}, {742, 13476}, {894, 7191}, {993, 16684}, {1086, 21249}, {3836, 24176}, {3874, 5847}, {3963, 21210}, {4446, 24166}, {6646, 17154}, {10436, 17155}, {21193, 23808}, {21208, 21238}
X(24181) lies on these lines: {1, 142}, {2, 24771}, {7, 16572}, {10, 4904}, {170, 17889}, {218, 226}, {220, 17278}, {354, 15493}, {946, 16020}, {1086, 1212}, {1125, 20269}, {1210, 21258}, {3663, 16601}, {3666, 24175}, {4869, 6764}, {5222, 5249}, {5882, 9317}, {5988, 24161}, {9957, 20328}, {12053, 17761}, {12610, 13442}, {12782, 24174}, {16699, 17205}, {17023, 24781}, {17158, 17234}, {17306, 19855}, {17760, 24199}, {19862, 24784}, {20335, 21075}, {24177, 25080}
X(24182) lies on these lines: {10, 3662}, {75, 3840}, {726, 24174}, {1278, 4871}, {3741, 4772}, {4066, 20891}, {4135, 20923}, {5836, 24325}, {7263, 21257}, {10009, 24215}, {17351, 24742}, {20892, 23677}
X(24183) lies on these lines: {2, 45}, {10, 244}, {75, 24184}, {100, 474}, {142, 14554}, {145, 3315}, {169, 3306}, {321, 24175}, {551, 678}, {1054, 3624}, {1266, 4358}, {1500, 4850}, {1647, 25351}, {2254, 23808}, {3008, 24593}, {3636, 3722}, {3762, 4453}, {3834, 3936}, {3943, 17495}, {3994, 24200}, {4033, 19804}, {4359, 24195}, {4442, 4871}, {4450, 5272}, {4706, 9507}, {4728, 25381}, {6545, 24123}, {9324, 25055}, {16729, 17205}, {17230, 20255}
X(24184) lies on these lines: {2, 1266}, {9, 20092}, {10, 4392}, {75, 24183}, {145, 11024}, {169, 3218}, {1022, 17494}, {4025, 21198}, {4389, 24589}, {7263, 16594}, {14475, 25381}, {17244, 17495}, {17250, 19804}, {23809, 24457}
X(24185) lies on these lines: {2, 4115}, {115, 116}, {244, 17761}, {335, 4103}, {514, 3125}, {537, 3739}, {596, 20255}, {620, 1125}, {942, 2809}, {2140, 24046}, {3017, 4000}, {4359, 4986}, {4904, 6741}, {14377, 17054}, {17050, 24167}, {17179, 17497}, {17310, 17495}, {17758, 24443}, {20335, 24168}, {21192, 24226}, {21240, 24176}, {22011, 24170}, {24050, 24530}, {24166, 24190}
X(24186) lies on these lines: {37, 24938}, {125, 1086}, {244, 1109}, {3035, 3666}, {3120, 24224}, {3670, 24982}, {3752, 24885}, {3937, 21138}, {4025, 23824}, {4357, 20903}, {4359, 24988}, {5205, 17495}, {21208, 24226}, {21209, 23811}
X(24187) lies on these lines: {2, 4115}, {99, 1125}, {350, 20893}, {620, 1086}, {903, 17322}, {11263, 17200}, {16887, 24161}, {24074, 24956}, {24159, 24166}, {24160, 24170}
X(24188) lies on these lines: {2, 24416}, {11, 244}, {75, 24233}, {106, 14028}, {190, 11814}, {335, 19961}, {519, 1738}, {903, 24228}, {6550, 7336}, {6788, 25436}, {21112, 21140}, {24222, 25351}
X(24189) lies on these lines: {75, 24046}, {244, 14210}, {625, 1086}, {6381, 20340}, {6629, 18201}, {16887, 24167}, {24200, 24240}
X(24190) lies on these lines: {2, 2140}, {7, 17499}, {10, 3662}, {75, 21240}, {76, 1086}, {86, 2241}, {116, 5025}, {150, 17680}, {330, 17205}, {384, 14377}, {1266, 21071}, {1278, 21070}, {1500, 17234}, {3210, 3912}, {3496, 10436}, {3501, 17282}, {3552, 17729}, {3834, 20691}, {3836, 12782}, {3959, 20924}, {4000, 17034}, {4389, 16589}, {6656, 21258}, {7263, 21024}, {7786, 25350}, {9941, 24325}, {16706, 17750}, {17030, 17050}, {17033, 24790}, {24166, 24185}
X(24191) lies on these lines: {244, 693}, {350, 1266}, {1086, 3835}, {3663, 25377}, {3912, 17495}, {4025, 21208}, {6381, 20340}, {6549, 7192}
X(24192) lies on these lines: {2, 24198}, {11, 4106}, {244, 4025}, {350, 1266}, {1086, 4885}, {1111, 3676}, {2170, 11068}, {3026, 14027}, {3911, 20367}, {4887, 9436}, {4998, 24203}, {17197, 18197}, {17205, 18155}, {20651, 24196}, {21208, 23799}, {23791, 23823}, {24209, 24236}, {24216, 24232}
X(24193) lies on these lines: {2, 24419}, {11, 244}, {334, 24233}, {350, 1266}, {2228, 3006}, {21140, 24136}
X(24194) lies on these lines: {244, 4467}, {320, 17731}, {350, 1266}, {693, 17205}, {908, 3008}, {1086, 4369}, {3120, 20295}, {5209, 24234}, {17197, 23803}, {17761, 23554}, {23799, 23824}
X(24195) lies on these lines: {115, 116}, {142, 24173}, {244, 1109}, {514, 16726}, {522, 3122}, {942, 2810}, {1125, 21254}, {3125, 17197}, {3666, 6692}, {3739, 9055}, {4000, 24884}, {4359, 24183}, {16732, 17205}, {21200, 23810}
X(24196) lies on these lines: {244, 514}, {726, 24168}, {1086, 21260}, {6381, 20340}, {8714, 21208}, {20517, 21210}, {20651, 24192}
X(24197) lies on these lines: {244, 661}, {693, 3122}, {1086, 23301}, {2228, 3006}, {6381, 20340}, {18891, 24238}
X(24198) lies on these lines: {2, 24192}, {11, 5519}, {244, 7658}, {650, 1086}, {899, 17067}, {908, 3008}, {1111, 4468}, {1936, 3011}, {3835, 17761}, {16751, 17205}
X(24199) lies on these lines: {1, 4780}, {2, 2415}, {6, 4795}, {7, 391}, {8, 5542}, {9, 4480}, {10, 3662}, {37, 1266}, {69, 4034}, {75, 142}, {86, 3946}, {141, 4688}, {190, 6666}, {192, 4098}, {193, 4888}, {226, 5233}, {239, 3664}, {274, 1432}, {319, 4545}, {320, 3686}, {333, 553}, {344, 4659}, {514, 24118}, {516, 16823}, {519, 17117}, {527, 7321}, {536, 17245}, {545, 16814}, {551, 17396}, {594, 3834}, {894, 3008}, {903, 17258}, {908, 24589}, {966, 17274}, {1086, 1213}, {1100, 4395}, {1125, 3685}, {1278, 3950}, {1699, 9801}, {1738, 4085}, {1781, 3218}, {2325, 17263}, {2345, 17282}, {3244, 17391}, {3264, 18143}, {3616, 4356}, {3620, 3679}, {3625, 17373}, {3626, 17287}, {3634, 17326}, {3644, 4029}, {3661, 4772}, {3672, 16831}, {3687, 3936}, {3696, 4684}, {3707, 17347}, {3717, 3826}, {3742, 4890}, {3759, 4667}, {3817, 9950}, {3828, 21087}, {3875, 4648}, {3879, 4361}, {3883, 5880}, {3943, 4726}, {3945, 4402}, {4000, 10436}, {4021, 16826}, {4044, 20923}, {4058, 17230}, {4060, 17295}, {4292, 16817}, {4298, 16824}, {4346, 5296}, {4353, 16830}, {4363, 17278}, {4389, 4751}, {4398, 4687}, {4399, 17374}, {4440, 17260}, {4452, 5308}, {4454, 18230}, {4464, 17390}, {4472, 17384}, {4654, 14555}, {4665, 17231}, {4670, 17366}, {4686, 17243}, {4698, 17246}, {4740, 17242}, {4851, 17119}, {4852, 17392}, {4862, 16832}, {4869, 17294}, {4887, 6646}, {4896, 16816}, {4898, 17151}, {5121, 25385}, {5287, 19789}, {5564, 17297}, {5750, 16706}, {6533, 12047}, {7227, 17357}, {7232, 17275}, {7238, 17344}, {9436, 24631}, {9776, 11679}, {10455, 16752}, {10521, 17739}, {15668, 17301}, {16610, 21796}, {16709, 17197}, {16738, 17205}, {16819, 24214}, {17077, 22464}, {17118, 17279}, {17133, 17315}, {17140, 25006}, {17160, 17317}, {17164, 24564}, {17207, 17474}, {17259, 17276}, {17265, 17281}, {17290, 17303}, {17299, 17313}, {17330, 17345}, {17337, 17351}, {17348, 17365}, {17356, 17369}, {17362, 17376}, {17382, 17398}, {17449, 21027}, {17760, 24181}, {17889, 24230}, {20234, 20893}, {20236, 20880}, {20881, 25001}, {20888, 20891}, {20892, 20913}, {20907, 21195}, {21242, 24216}, {23677, 24207}, {23682, 24212}
X(24199) = complement of X(17261)
X(24199) = anticomplement of X(25072)
X(24200) lies on these lines: {10, 38}, {244, 4442}, {350, 1266}, {354, 4743}, {897, 18201}, {1086, 4892}, {2796, 7292}, {3616, 3980}, {3971, 24175}, {3994, 24183}, {4062, 17495}, {4831, 4974}, {5212, 24231}, {16610, 21093}, {16741, 17205}, {24189, 24240}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28380.
X(24201) lies on these lines: {1,953}, {57,901}, {59,1421}, {65,13756}, {226,3259}, {354,3025}, {513,5083}, {517,1387}, {1319,18593}, {2835,15635}, {5570,12016}, {11028,18839}, {14115,18240}
X(24201) = midpoint of X(65) and X(13756)
X(24201) = reflection of X(14115) in X(18240)
X(24202) lies on these lines: {1, 1441}, {7, 5563}, {36, 17220}, {86, 99}, {322, 22837}, {1086, 17043}, {1111, 1442}, {1201, 17189}, {1429, 24220}, {2893, 24387}, {3582, 5740}, {3673, 17394}, {4360, 24209}, {4909, 14828}, {5738, 10072}, {7741, 21270}, {16826, 25065}, {17134, 21842}, {17197, 18162}, {17859, 17877}, {21625, 24213}, {24212, 24227}, {24559, 25078}
X(24203) lies on these lines: {1, 85}, {2, 644}, {7, 104}, {11, 150}, {56, 17753}, {75, 997}, {81, 1625}, {86, 99}, {101, 673}, {190, 18061}, {404, 20244}, {517, 1447}, {658, 18240}, {938, 5543}, {946, 4911}, {1086, 9259}, {1319, 5088}, {1320, 21272}, {1429, 6996}, {1434, 5563}, {1482, 3212}, {1633, 23392}, {1807, 7269}, {2170, 3732}, {2329, 17681}, {3086, 6604}, {3315, 4453}, {3674, 13464}, {3699, 4986}, {3811, 17158}, {3879, 24213}, {4059, 20323}, {4360, 17861}, {4845, 11019}, {4858, 21602}, {4919, 21232}, {4998, 24192}, {5736, 15956}, {5886, 7179}, {5901, 17084}, {7176, 24928}, {7247, 12047}, {7272, 18393}, {8286, 23674}, {9310, 17682}, {9317, 17439}, {9327, 17758}, {9446, 17626}, {10186, 24248}, {11376, 17181}, {17160, 24209}, {17197, 18723}, {17719, 24228}, {24559, 25083}
X(24204) lies on these lines: {1, 1441}, {4, 1111}, {277, 2345}, {321, 17284}, {349, 3673}, {1086, 18636}, {16502, 16732}, {17197, 18727}, {17877, 18693}, {18698, 19836}, {23581, 24159}
X(24205) lies on these lines: {1, 1441}, {1086, 18637}, {1111, 3583}, {16732, 16784}, {17197, 18728}, {17877, 18694}, {18208, 24233}
X(24206) lies on these lines: {2, 98}, {3, 2916}, {4, 3096}, {5, 141}, {6, 17}, {30, 14810}, {66, 6759}, {67, 14643}, {69, 576}, {76, 3399}, {115, 3094}, {126, 12494}, {140, 1503}, {159, 7393}, {193, 7486}, {262, 3314}, {265, 12584}, {315, 10358}, {323, 7570}, {325, 14994}, {343, 5943}, {373, 3580}, {381, 1350}, {389, 7405}, {394, 7539}, {403, 12294}, {427, 3819}, {498, 12589}, {499, 12588}, {517, 3844}, {518, 6583}, {524, 547}, {549, 11645}, {550, 21167}, {575, 3564}, {578, 14786}, {597, 1353}, {599, 1351}, {620, 4048}, {621, 16002}, {622, 16001}, {631, 17508}, {632, 20190}, {732, 7764}, {754, 10796}, {858, 5650}, {1078, 5207}, {1386, 11230}, {1469, 7951}, {1495, 7495}, {1513, 15819}, {1533, 16261}, {1568, 7699}, {1594, 1843}, {1595, 13348}, {1691, 7749}, {1853, 16419}, {1974, 7505}, {1993, 7571}, {2030, 3054}, {2072, 9967}, {2076, 7747}, {2080, 7810}, {2393, 20300}, {2548, 5039}, {2777, 4550}, {2782, 4045}, {2783, 3821}, {2792, 24295}, {2794, 9996}, {2854, 16511}, {2896, 12110}, {3056, 7741}, {3091, 7938}, {3095, 7794}, {3242, 5790}, {3292, 14389}, {3313, 5576}, {3398, 7889}, {3416, 5886}, {3524, 14927}, {3525, 25406}, {3526, 5085}, {3533, 23294}, {3549, 19126}, {3574, 11444}, {3618, 5067}, {3620, 5056}, {3629, 22330}, {3734, 23698}, {3826, 21252}, {3836, 17047}, {3841, 25144}, {3917, 5133}, {3923, 24250}, {4260, 6881}, {4265, 7489}, {4549, 18420}, {5017, 5475}, {5050, 5070}, {5052, 7603}, {5071, 20423}, {5079, 11477}, {5116, 11646}, {5138, 6861}, {5157, 10539}, {5169, 7998}, {5171, 7800}, {5181, 13162}, {5422, 11225}, {5448, 14128}, {5510, 20551}, {5562, 14788}, {5663, 6698}, {5846, 5901}, {5891, 18388}, {5907, 7399}, {6144, 11482}, {6194, 9993}, {6247, 17704}, {6248, 6656}, {6289, 11314}, {6290, 11313}, {6403, 7577}, {6639, 19131}, {6643, 15435}, {6676, 15448}, {6680, 10104}, {6688, 13567}, {6803, 18931}, {6829, 10477}, {6998, 17307}, {7380, 17234}, {7395, 18396}, {7403, 15644}, {7404, 13346}, {7485, 11550}, {7504, 15988}, {7514, 15577}, {7533, 15107}, {7569, 20806}, {7605, 15019}, {7619, 9830}, {7692, 21736}, {7753, 12212}, {7765, 13108}, {7769, 12215}, {7780, 20576}, {7795, 9737}, {7811, 10788}, {7819, 13335}, {7831, 11676}, {7844, 20398}, {7853, 15980}, {7867, 13355}, {7868, 13860}, {7876, 11257}, {7919, 14651}, {7934, 22677}, {8252, 19146}, {8253, 19145}, {8362, 13334}, {8370, 13449}, {8542, 12596}, {9127, 10162}, {9669, 10387}, {9698, 13331}, {9753, 16990}, {9821, 15821}, {9863, 16895}, {10193, 15578}, {10203, 14940}, {10255, 18438}, {10347, 12176}, {11171, 14981}, {11173, 18584}, {11291, 12975}, {11292, 12974}, {11303, 20429}, {11304, 20428}, {11459, 14789}, {11548, 23292}, {11574, 11585}, {11695, 12359}, {13154, 15581}, {13347, 14216}, {13352, 14787}, {13399, 20791}, {13561, 16239}, {13862, 16986}, {14848, 15533}, {14913, 20303}, {14982, 15061}, {14984, 15088}, {15026, 21230}, {15060, 15101}, {17283, 21554}, {18906, 23514}
X(24206) = midpoint of X(182) and X(1352)
X(24206) = complement of X(182)
X(24206) = X(5)-of-1st-Brocard-triangle
X(24206) = X(141)-of-McCay-triangle
X(24206) = centroid of PU(5)PU(11)
X(24207) lies on these lines: {75, 21210}, {244, 20236}, {514, 18194}, {982, 17861}, {1086, 17047}, {3551, 3667}, {3670, 23689}, {3953, 23690}, {17065, 21208}, {17197, 18168}, {18208, 24220}, {21185, 24802}, {21443, 24166}, {23677, 24199}, {24179, 24227}, {24224, 24230}
X(24208) lies on these lines: {7, 3585}, {10, 75}, {273, 10481}, {516, 23689}, {1099, 1111}, {1441, 4021}, {3244, 20930}, {3672, 10056}, {3946, 16732}, {4353, 23690}, {4464, 17791}, {4698, 25065}, {4887, 23521}, {12649, 21296}, {20171, 21255}, {24214, 24219}
X(24209) lies on these lines: {2, 25076}, {7, 5270}, {10, 75}, {44, 4957}, {78, 17151}, {85, 4896}, {91, 3668}, {498, 3672}, {516, 1733}, {519, 3262}, {527, 16732}, {850, 4025}, {1111, 4887}, {1441, 3664}, {1737, 18815}, {3008, 4858}, {3739, 25065}, {3875, 22836}, {3950, 20171}, {4021, 13411}, {4353, 23689}, {4360, 24202}, {4452, 5552}, {4847, 17871}, {5294, 20886}, {6737, 20895}, {7264, 23521}, {17132, 20881}, {17160, 24203}, {17355, 20236}, {18698, 24993}, {24192, 24236}, {25001, 25072}
X(24210) lies on these lines: {1, 4}, {2, 968}, {5, 3931}, {6, 24703}, {8, 4104}, {10, 312}, {11, 114}, {36, 4221}, {37, 2886}, {42, 908}, {43, 3452}, {55, 17720}, {57, 12717}, {63, 11269}, {65, 15488}, {81, 5057}, {142, 4335}, {149, 3920}, {171, 516}, {181, 517}, {192, 3705}, {238, 2328}, {256, 314}, {284, 2201}, {329, 3751}, {354, 3782}, {381, 5725}, {386, 21616}, {390, 3749}, {442, 6051}, {496, 20256}, {511, 21334}, {518, 4415}, {519, 4514}, {612, 3434}, {614, 19785}, {693, 23793}, {740, 3687}, {756, 25006}, {846, 5745}, {940, 1836}, {960, 1834}, {982, 3663}, {984, 4656}, {986, 1210}, {988, 3086}, {1001, 3772}, {1010, 1125}, {1045, 21246}, {1054, 6692}, {1076, 4295}, {1086, 3742}, {1211, 3706}, {1266, 24165}, {1279, 17061}, {1329, 4646}, {1362, 12915}, {1402, 4192}, {1500, 20544}, {1621, 3011}, {1707, 5698}, {1722, 5084}, {1725, 3670}, {1737, 4424}, {1756, 22097}, {1839, 2303}, {1961, 20539}, {1999, 4388}, {2292, 6734}, {2310, 11031}, {2550, 5268}, {2887, 3912}, {3006, 3995}, {3008, 17123}, {3058, 3744}, {3120, 3720}, {3175, 3703}, {3616, 23536}, {3622, 23675}, {3664, 4038}, {3672, 5274}, {3673, 6063}, {3679, 3974}, {3696, 5743}, {3702, 5051}, {3717, 3971}, {3727, 23903}, {3736, 17182}, {3748, 17724}, {3750, 13405}, {3752, 3816}, {3814, 4868}, {3817, 4356}, {3821, 3840}, {3826, 21949}, {3838, 15569}, {3883, 4362}, {3896, 5741}, {3911, 17596}, {3961, 5853}, {4000, 5272}, {4001, 4683}, {4011, 17353}, {4021, 17600}, {4028, 4417}, {4123, 22836}, {4138, 18134}, {4292, 24851}, {4307, 9812}, {4310, 10580}, {4343, 21617}, {4353, 17598}, {4358, 4972}, {4359, 4442}, {4383, 4679}, {4416, 4703}, {4419, 24477}, {4423, 16849}, {4429, 18743}, {4432, 6679}, {4642, 24982}, {4689, 5432}, {4743, 5212}, {4780, 5233}, {4862, 10980}, {4871, 24169}, {4891, 4966}, {5046, 17016}, {5080, 17015}, {5230, 5250}, {5247, 12572}, {5255, 10624}, {5266, 15171}, {5269, 9580}, {5292, 12514}, {5316, 16569}, {5710, 12701}, {5711, 12699}, {5718, 17605}, {5739, 17156}, {6358, 23690}, {6381, 18057}, {6686, 11814}, {7174, 24392}, {8167, 17278}, {8582, 24440}, {9284, 21345}, {9711, 21896}, {9843, 24174}, {10164, 17601}, {10448, 24541}, {10458, 17167}, {10582, 23681}, {11238, 17599}, {11375, 19765}, {11599, 14534}, {14008, 25060}, {14009, 25058}, {14829, 24723}, {15280, 25098}, {15507, 20967}, {16826, 23682}, {17063, 24177}, {17122, 24715}, {17197, 18169}, {17595, 17728}, {17715, 17725}, {17723, 20182}, {18905, 20363}, {19815, 20106}, {21935, 24987}, {23812, 23823}, {24386, 25353}
X(24210) = complement of X(32932)
X(24211) lies on these lines: {1, 147}, {10, 1930}, {37, 17046}, {39, 24318}, {65, 4920}, {75, 18835}, {116, 16600}, {171, 4911}, {307, 1210}, {626, 4071}, {742, 4109}, {894, 7797}, {976, 21285}, {1086, 17048}, {3664, 13407}, {3767, 24333}, {3831, 4357}, {3944, 17753}, {4056, 5264}, {4434, 7767}, {4847, 23667}, {4865, 7776}, {4872, 5255}, {5209, 16887}, {5283, 25353}, {7278, 24222}, {9436, 13161}, {16788, 20267}, {17790, 25364}, {20255, 25345}, {20880, 21935}, {24169, 24170}
X(24212) lies on these lines: {1, 9551}, {38, 20236}, {239, 17153}, {256, 2481}, {514, 18207}, {982, 17861}, {1086, 17049}, {3662, 21278}, {3670, 23690}, {3953, 23689}, {4021, 17600}, {4353, 13161}, {4858, 17446}, {6385, 16887}, {7184, 24237}, {7321, 23823}, {16706, 18082}, {16732, 18183}, {17197, 18170}, {23682, 24199}, {24202, 24227}
X(24213) lies on these lines: {7, 1210}, {10, 75}, {11, 1122}, {269, 1256}, {946, 24471}, {1086, 24005}, {1111, 3668}, {1423, 10445}, {1737, 4862}, {3123, 23663}, {3664, 11019}, {3772, 20205}, {3879, 24203}, {3986, 25065}, {4000, 20262}, {4328, 18391}, {4373, 25005}, {4452, 6735}, {4847, 17272}, {6736, 17151}, {7318, 15866}, {9843, 10436}, {10521, 21628}, {21625, 24202}
X(24214) lies on these lines: {1, 7}, {10, 3761}, {38, 20880}, {39, 20335}, {76, 3831}, {85, 986}, {142, 5283}, {194, 3662}, {213, 527}, {226, 980}, {239, 4001}, {274, 4357}, {310, 3741}, {519, 17137}, {538, 21240}, {726, 1930}, {982, 3673}, {1086, 1107}, {1111, 3670}, {1125, 4368}, {1193, 18600}, {1266, 17143}, {1975, 24549}, {2176, 17276}, {2887, 3933}, {3008, 16552}, {3665, 3782}, {3666, 4059}, {3720, 17169}, {3727, 7200}, {3760, 3840}, {3912, 25264}, {3944, 17181}, {3946, 20963}, {3953, 7264}, {4000, 21384}, {4384, 24177}, {4398, 17144}, {4656, 16831}, {4872, 24851}, {5254, 17046}, {6381, 24170}, {6646, 16827}, {6706, 25073}, {7195, 22097}, {7754, 24586}, {9436, 13161}, {9534, 17272}, {10521, 20367}, {16819, 24199}, {16832, 24175}, {16975, 20257}, {17197, 18171}, {17758, 25092}, {20018, 21296}, {21208, 24167}, {24208, 24219}
X(24215) lies on these lines: {1, 7}, {2, 23649}, {10, 274}, {42, 18600}, {57, 20606}, {76, 3840}, {85, 982}, {142, 1107}, {171, 1434}, {194, 3912}, {239, 24177}, {304, 726}, {330, 3662}, {335, 7187}, {519, 21281}, {527, 2176}, {538, 21071}, {1086, 17448}, {1111, 3953}, {1201, 20347}, {1231, 24218}, {1266, 17144}, {1401, 20358}, {1565, 4920}, {2200, 18723}, {2275, 20335}, {2664, 21060}, {3008, 21384}, {3673, 3976}, {3687, 24621}, {3721, 7200}, {3761, 3831}, {3771, 3926}, {3950, 25264}, {4364, 25130}, {4384, 24175}, {4416, 16827}, {4419, 24654}, {4424, 7278}, {4656, 16826}, {4685, 16711}, {4694, 7264}, {4713, 24652}, {4847, 23682}, {4871, 18135}, {10009, 24182}, {13161, 24241}, {16284, 24440}, {16720, 21101}, {16779, 17691}, {16825, 24171}, {16969, 17276}, {16975, 17050}, {17095, 17719}, {17197, 18172}, {17449, 20247}, {17866, 24219}, {18157, 21080}, {20036, 21296}, {20911, 24165}, {24656, 25349}, {25102, 25350}
X(24216) lies on these lines: {1, 631}, {2, 16496}, {11, 3999}, {37, 17051}, {106, 519}, {244, 1738}, {354, 5718}, {497, 18193}, {516, 18201}, {517, 6018}, {518, 3756}, {527, 24398}, {614, 24597}, {908, 1647}, {982, 3663}, {1054, 5853}, {1210, 3976}, {1357, 15310}, {1447, 3945}, {1699, 4902}, {1737, 4694}, {2191, 5272}, {3011, 3315}, {3035, 4864}, {3664, 17722}, {3676, 4905}, {3705, 17232}, {3717, 4871}, {3741, 4967}, {3742, 17245}, {3749, 5435}, {3816, 21342}, {3953, 13161}, {3961, 6692}, {4038, 4909}, {4847, 17063}, {4860, 17721}, {4888, 10980}, {4899, 24003}, {5045, 5530}, {5048, 22102}, {5211, 5847}, {5537, 22942}, {5542, 17717}, {10164, 17715}, {10580, 17594}, {10916, 24178}, {17197, 18173}, {17597, 17728}, {17889, 24386}, {21214, 24391}, {21242, 24199}, {24192, 24232}
X(24217) lies on these lines: {1, 5}, {2, 2177}, {43, 3816}, {149, 750}, {171, 497}, {238, 11269}, {354, 3944}, {519, 5233}, {940, 11238}, {982, 3663}, {1086, 17051}, {1647, 4850}, {1699, 4888}, {1757, 4679}, {1834, 21214}, {2886, 17245}, {2887, 17232}, {3058, 3550}, {3306, 24715}, {3338, 24851}, {3434, 17122}, {3622, 21935}, {3632, 4023}, {3679, 5241}, {3705, 17242}, {3720, 11680}, {3741, 5224}, {3742, 17889}, {3755, 5121}, {3829, 17056}, {3846, 10453}, {3848, 21949}, {3911, 17601}, {3914, 17063}, {3945, 4038}, {4080, 17146}, {4256, 10199}, {4424, 24223}, {4425, 17249}, {4429, 4871}, {4693, 17740}, {4854, 17591}, {4883, 17605}, {4890, 24230}, {4902, 10980}, {5268, 24392}, {5284, 24892}, {6048, 17527}, {7777, 17319}, {10582, 17064}, {16592, 23903}, {17197, 18174}, {17234, 21241}, {17592, 24239}, {17596, 17728}, {18201, 24248}
X(24218) lies on these lines: {57, 7009}, {75, 982}, {225, 1210}, {226, 24235}, {244, 17862}, {345, 726}, {614, 20223}, {1231, 24215}, {1441, 11031}, {1891, 4292}, {3971, 17719}, {4310, 17155}, {11019, 24225}, {17197, 18175}, {20320, 24046}, {23661, 24443}
X(24219) lies on these lines: {1, 21271}, {10, 24162}, {75, 24046}, {244, 18698}, {307, 1210}, {511, 942}, {982, 17861}, {1086, 17052}, {1227, 4357}, {1441, 3953}, {3159, 18147}, {3666, 3911}, {3739, 9055}, {4022, 21210}, {4256, 4360}, {4359, 24175}, {4887, 14755}, {17151, 17495}, {17197, 18179}, {17866, 24215}, {24208, 24214}
X(24220) lies on these lines: {1, 9551}, {2, 573}, {3, 142}, {4, 991}, {5, 141}, {6, 17197}, {7, 1020}, {10, 3781}, {11, 21746}, {30, 6176}, {57, 24179}, {63, 22000}, {69, 5816}, {81, 1746}, {86, 572}, {116, 117}, {140, 6707}, {222, 226}, {355, 4851}, {381, 17313}, {514, 18161}, {517, 3739}, {908, 1150}, {952, 17390}, {980, 3663}, {995, 4000}, {1086, 17053}, {1429, 24202}, {1482, 4361}, {1656, 17327}, {1699, 1742}, {1766, 10436}, {1777, 12047}, {1953, 4858}, {1959, 20236}, {2876, 17059}, {3091, 4869}, {3136, 3917}, {3452, 5737}, {3589, 19512}, {3616, 10470}, {3667, 21191}, {3729, 22019}, {3741, 20788}, {3814, 21244}, {3817, 3840}, {3834, 5482}, {3878, 21233}, {3912, 21069}, {3946, 13464}, {4224, 17188}, {4301, 17050}, {4399, 5844}, {4402, 5734}, {4445, 5790}, {4852, 10222}, {4859, 11522}, {5249, 19645}, {5587, 17296}, {5733, 6776}, {5736, 21617}, {5786, 21620}, {5788, 21077}, {5792, 20818}, {5799, 8728}, {5806, 6706}, {5901, 17045}, {6210, 8227}, {6831, 18635}, {7146, 17861}, {7377, 17234}, {7381, 18647}, {7384, 17300}, {7406, 18648}, {7522, 17811}, {7749, 20666}, {9956, 17239}, {10444, 16831}, {10476, 19858}, {10888, 17022}, {11376, 20270}, {11679, 22020}, {12053, 19765}, {13329, 21554}, {15488, 15973}, {16435, 19701}, {16552, 16713}, {16574, 17139}, {17077, 17220}, {17262, 22031}, {17355, 20258}, {17792, 20544}, {18208, 24207}, {19549, 20108}, {20245, 21061}, {20274, 21210}, {20883, 21429}
X(24221) lies on these lines: {75, 3840}, {85, 982}, {1086, 20528}, {3663, 3817}, {3752, 3774}, {3912, 17490}, {20893, 24166}, {21416, 24165}
X(24222) lies on these lines: {1, 5}, {2, 106}, {8, 24159}, {10, 244}, {36, 19335}, {109, 388}, {141, 3679}, {150, 3945}, {519, 3936}, {535, 902}, {542, 6126}, {595, 20060}, {668, 5224}, {982, 24223}, {1056, 24618}, {1074, 1111}, {1086, 1145}, {1149, 3814}, {1320, 24864}, {1647, 6702}, {1737, 4694}, {1739, 6735}, {1769, 3762}, {1772, 3670}, {2093, 4902}, {2789, 5988}, {2802, 3120}, {2948, 24345}, {3216, 12607}, {3244, 21935}, {3315, 6788}, {3944, 6018}, {3963, 4692}, {3976, 18395}, {3987, 10915}, {4013, 25031}, {4389, 20568}, {4737, 18040}, {4861, 24160}, {4967, 4986}, {5255, 5270}, {5541, 24416}, {5790, 17597}, {7208, 24318}, {7278, 24211}, {10246, 17783}, {11236, 16483}, {13259, 14430}, {16486, 17556}, {17249, 18159}, {17460, 21630}, {24188, 25351}
X(24223) lies on these lines: {1, 631}, {10, 4392}, {80, 17595}, {982, 24222}, {1086, 17057}, {1647, 17461}, {1698, 24159}, {1734, 4443}, {1737, 3663}, {1739, 4446}, {3245, 17721}, {3585, 15434}, {3670, 18395}, {4414, 6788}, {4424, 24217}, {4859, 19875}, {4967, 10468}, {5718, 5902}, {10260, 14793}, {20516, 21199}
X(24224) lies on these lines: {2, 22003}, {11, 1365}, {75, 24058}, {115, 116}, {142, 24086}, {514, 7202}, {522, 2486}, {942, 2801}, {1111, 24237}, {3120, 24186}, {3667, 4459}, {3739, 4422}, {4359, 20881}, {4605, 18815}, {4858, 17761}, {5954, 7336}, {7146, 17861}, {17863, 21617}, {18208, 24229}, {24207, 24230}
X(24225) lies on these lines: {1, 4552}, {11, 1365}, {244, 17888}, {514, 3271}, {522, 1086}, {596, 3976}, {942, 2783}, {1111, 2310}, {2293, 24799}, {2643, 4151}, {3122, 20525}, {3667, 4014}, {3675, 4459}, {4081, 4904}, {4516, 17761}, {8299, 22003}, {8714, 23823}, {11019, 24218}, {17197, 17463}, {17205, 17877}, {17885, 24802}, {21185, 24232}, {23798, 24236}
X(24226) lies on these lines: {244, 17877}, {1111, 7004}, {3666, 13006}, {4025, 17205}, {4441, 24166}, {21192, 24185}, {21208, 24186}
X(24227) lies on these lines: {1, 24224}, {86, 23823}, {1086, 11725}, {3120, 18645}, {24179, 24207}, {24202, 24212}
X(24228) lies on these lines: {11, 1111}, {903, 24188}, {1647, 4453}, {4887, 9436}, {17719, 24203}
X(24229) lies on these lines: {982, 17861}, {4424, 23689}, {4694, 23690}, {18208, 24224}, {24232, 24236}
X(24230) lies on these lines: {1, 9551}, {1086, 17065}, {1111, 20567}, {1210, 24231}, {3551, 23821}, {3663, 3817}, {4890, 24217}, {16732, 18168}, {17197, 18194}, {17861, 18208}, {17889, 24199}, {24207, 24224}
X(24231) lies on these lines: {1, 7}, {2, 18193}, {8, 7613}, {10, 3662}, {11, 3999}, {38, 5249}, {43, 24177}, {44, 5852}, {57, 6211}, {72, 24178}, {142, 984}, {144, 16020}, {145, 4780}, {171, 553}, {226, 262}, {238, 527}, {240, 5236}, {244, 908}, {291, 20335}, {306, 17155}, {320, 5847}, {329, 5272}, {335, 726}, {354, 3782}, {511, 1463}, {518, 1086}, {519, 4645}, {528, 4864}, {536, 4966}, {537, 3717}, {611, 5228}, {613, 6180}, {614, 5905}, {740, 1266}, {748, 17781}, {894, 1125}, {942, 13161}, {946, 3976}, {978, 24171}, {986, 21620}, {988, 3487}, {1001, 17276}, {1054, 6745}, {1111, 23690}, {1155, 17724}, {1210, 24230}, {1278, 4133}, {1279, 17768}, {1386, 17365}, {1707, 9965}, {1737, 24232}, {1757, 3008}, {1785, 1876}, {1836, 17597}, {2285, 3338}, {2298, 5557}, {2550, 16496}, {2886, 21342}, {3006, 17154}, {3011, 3218}, {3120, 17449}, {3210, 4028}, {3242, 5880}, {3315, 5057}, {3416, 7232}, {3452, 17063}, {3474, 3749}, {3475, 17594}, {3616, 17247}, {3624, 3986}, {3632, 15590}, {3634, 17291}, {3667, 4017}, {3670, 5530}, {3677, 4654}, {3685, 4440}, {3687, 24165}, {3696, 7263}, {3705, 4138}, {3742, 4415}, {3744, 11246}, {3751, 4000}, {3756, 5087}, {3775, 4967}, {3790, 17232}, {3834, 3932}, {3868, 23536}, {3869, 23675}, {3873, 3914}, {3874, 23537}, {3883, 4655}, {3911, 17719}, {3944, 11019}, {3946, 4649}, {3953, 12047}, {3982, 17598}, {4003, 5718}, {4014, 15310}, {4026, 17235}, {4031, 17725}, {4078, 17234}, {4104, 19804}, {4114, 17716}, {4357, 24325}, {4416, 16825}, {4432, 17767}, {4644, 16475}, {4663, 17366}, {4733, 4739}, {4847, 17889}, {4854, 4883}, {4859, 5223}, {4860, 17720}, {4871, 21093}, {4890, 5045}, {4941, 12053}, {4974, 17771}, {4989, 16477}, {5212, 24200}, {5220, 17278}, {5263, 7321}, {5266, 24470}, {5268, 9776}, {5279, 6763}, {5293, 12436}, {5772, 19875}, {5843, 15251}, {5846, 7238}, {5853, 24715}, {5988, 9436}, {6173, 7174}, {6646, 16823}, {7191, 17483}, {7290, 24695}, {7292, 17484}, {13405, 17596}, {15254, 17334}, {15481, 17337}, {15569, 17246}, {16569, 21060}, {17064, 24477}, {17140, 17184}, {17156, 19789}, {17205, 18792}, {17368, 19862}, {17595, 17718}, {17766, 24692}, {19868, 24342}, {20348, 21214}, {20470, 24405}, {21075, 24174}, {21077, 24046}
X(24232) lies on these lines: {244, 4391}, {1086, 17072}, {1111, 4905}, {1737, 24231}, {4147, 6547}, {6381, 20340}, {6549, 23789}, {20517, 21208}, {21185, 24225}, {24192, 24216}, {24229, 24236}
X(24233) lies on these lines: {11, 1111}, {75, 24188}, {334, 24193}, {1647, 3766}, {18208, 24205}, {24192, 24216}
X(24234) lies on these lines: {11, 6002}, {36, 516}, {667, 17197}, {4905, 23821}, {5209, 24194}, {8714, 17205}, {24192, 24216}
X(24235) lies on these lines: {11, 1365}, {226, 24218}, {244, 1109}, {514, 18191}, {522, 3120}, {542, 553}, {942, 952}, {1086, 2968}, {1111, 7004}, {1125, 16598}, {1357, 2789}, {1772, 3670}, {3626, 24176}, {3666, 4353}, {3935, 17495}, {4025, 16727}, {4359, 25006}, {4904, 6741}, {17197, 18210}, {21085, 24165}
X(24236) lies on these lines: {244, 20907}, {1111, 3667}, {17761, 23886}, {21178, 21210}, {23798, 24225}, {24192, 24209}, {24229, 24232}
X(24237) lies on these lines: {2, 21362}, {7, 1020}, {10, 2810}, {11, 1357}, {57, 2051}, {116, 5514}, {142, 3589}, {226, 4896}, {244, 13246}, {514, 3942}, {522, 17463}, {545, 22031}, {651, 24618}, {812, 1015}, {908, 24593}, {946, 20418}, {1111, 24224}, {1125, 15507}, {1358, 3026}, {1407, 13478}, {2140, 6173}, {2310, 3667}, {2347, 24778}, {3675, 4459}, {4129, 8287}, {4675, 17750}, {4817, 15634}, {7184, 24212}, {10436, 10469}, {10478, 21454}, {17355, 19593}, {20258, 21255}
X(24238) lies on these lines: {1111, 4170}, {18208, 24205}, {18891, 24197}, {24229, 24232}
X(24239) lies on these lines: {1, 2}, {4, 988}, {5, 13161}, {7, 18193}, {9, 7736}, {11, 114}, {22, 14793}, {25, 8071}, {35, 19649}, {36, 4220}, {37, 3815}, {38, 908}, {44, 9300}, {55, 16434}, {56, 19544}, {57, 6210}, {140, 5266}, {142, 17063}, {171, 3911}, {183, 3879}, {226, 262}, {230, 1100}, {238, 5745}, {240, 1848}, {244, 5249}, {325, 3846}, {354, 5718}, {427, 1785}, {442, 24178}, {443, 11512}, {496, 3931}, {497, 17594}, {516, 17596}, {553, 18201}, {611, 4383}, {613, 940}, {946, 986}, {950, 7413}, {984, 3452}, {999, 5725}, {1007, 17321}, {1072, 6830}, {1086, 3838}, {1104, 4999}, {1279, 6690}, {1319, 5724}, {1368, 17102}, {1447, 3664}, {1449, 7735}, {1699, 24248}, {1707, 5744}, {1733, 4359}, {1738, 2886}, {1834, 4719}, {1836, 17595}, {2476, 23536}, {3055, 3723}, {3058, 4689}, {3290, 21796}, {3361, 7390}, {3523, 4339}, {3550, 10164}, {3598, 4888}, {3662, 4138}, {3663, 3817}, {3670, 8229}, {3674, 3865}, {3677, 5219}, {3742, 3756}, {3744, 5432}, {3745, 17726}, {3749, 5218}, {3751, 24477}, {3755, 24386}, {3782, 4003}, {3813, 4646}, {3826, 16602}, {3848, 17245}, {3914, 4850}, {3925, 16610}, {3928, 24695}, {3953, 13407}, {3967, 4884}, {3976, 21620}, {4000, 17064}, {4008, 19804}, {4044, 10079}, {4054, 17155}, {4078, 18743}, {4090, 4899}, {4104, 5233}, {4298, 7385}, {4307, 5435}, {4310, 5226}, {4353, 10171}, {4388, 24627}, {4415, 5087}, {4416, 7774}, {4438, 11174}, {4518, 11814}, {4643, 9766}, {4648, 24752}, {4657, 7778}, {4734, 4780}, {4851, 15271}, {5133, 8070}, {5217, 21487}, {5255, 6684}, {5304, 16667}, {5306, 16666}, {5710, 24914}, {5716, 7288}, {5717, 6998}, {5847, 14829}, {6184, 25075}, {6636, 14792}, {6692, 17122}, {7322, 20196}, {7392, 10629}, {7484, 8069}, {7952, 8889}, {9592, 24247}, {11263, 24167}, {12609, 24046}, {15283, 25098}, {15491, 17243}, {17592, 24217}, {17597, 17718}, {17598, 17719}, {17599, 17720}, {17889, 24177}, {18483, 24851}, {21241, 24169}, {21795, 25074}, {24165, 25385}
X(24240) lies on these lines: {1, 7616}, {307, 1210}, {3582, 15903}, {3831, 4967}, {17048, 17245}, {24189, 24200}, {24192, 24216}
X(24241) lies on these lines: {1, 147}, {7, 3944}, {10, 304}, {37, 25353}, {171, 4872}, {222, 226}, {942, 4920}, {982, 3663}, {1125, 24549}, {2276, 24318}, {3314, 3912}, {3840, 4357}, {3997, 5074}, {4021, 17599}, {4799, 4987}, {4860, 4887}, {5275, 24694}, {5276, 24712}, {6063, 17861}, {10436, 17064}, {13161, 24215}, {14828, 17719}, {16783, 20267}
X(24242) lies on the cubic K1079 and these lines: {1,289}, {57,2089}, {164,10215}, {173,266}, {188,258}, {505,10231}, {7588,10233}, {8094,10234}
X(24242) = isogonal conjugate of X(8078)
X(24242) = X(9)-cross conjugate of X(258)
X(24242) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8078}, {56, 5430}, {266, 12646}
X(24242) = cevapoint of X(1) and X(18291)
X(24242) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 8078}, {236, 173}
X(24242) = barycentric product X(174)*X(12644)
X(24242) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8078}, {9, 5430}, {259, 12646}, {12644, 556}
X(24243) lies on the curves Q124, K070a, K170, and on these lines: {4,487}, {136,13430}, {193,13439}, {254,488}, {264,3128}, {393,494}, {427,1007}, {648,19041}, {1300,1307}, {1322,12221}, {1826,13386}, {8884,8982}, {13441,14593}
X(24243) = isogonal conjugate of X(10133)
X(24243) = isotomic conjugate of X(487)
X(24243) = polar conjugate of X(3069)
X(24243) = X(19217)-anticomplementary conjugate of X(638)
X(24243) = X(i)-cross conjugate of X(j) for these (i,j): {486, 2}, {494, 5491}
X(24243) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10133}, {6, 19216}, {31, 487}, {48, 3069}, {63, 6424}, {163, 17432}, {1973, 8223}
X(24243) = cevapoint of X(i) and X(j) for these (i,j): {2, 12221}, {4, 3536}, {494, 8946}
X(24243) = trilinear pole of line {2501, 14333}
X(24243) = barycentric product X(i)*X(j) for these {i,j}: {4, 5491}, {75, 19217}, {76, 8946}, {264, 494}, {1307, 14618}
X(24243) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 19216}, {2, 487}, {4, 3069}, {6, 10133}, {25, 6424}, {69, 8223}, {494, 3}, {523, 17432}, {1307, 4558}, {5491, 69}, {6406, 19033}, {6421, 19447}, {8946, 6}, {19217, 1}
X(24244) lies on the curves Q124, K070b, K170, and on these lines: {4,488}, {136,13441}, {193,13428}, {254,487}, {264,3127}, {393,493}, {427,1007}, {648,19042}, {1300,1306}, {1321,12222}, {1588,8950}, {1826,13387}, {13430,14593}
X(24244) = isogonal conjugate of X(10132)
X(24244) = isotomic conjugate of X(488)
X(24244) = polar conjugate of X(3068)
X(24244) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {8950, 6360}, {19218, 637}
X(24244) = X(i)-cross conjugate of X(j) for these (i,j): {485, 2}, {493, 5490}
X(24244) = cevapoint of X(i) and X(j) for these (i,j): {2, 12222}, {4, 3535}, {493, 8948}
X(24244) = trilinear pole of line {2501, 14334}
X(24244) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10132}, {6, 19215}, {31, 488}, {48, 3068}, {63, 6423}, {163, 17431}, {255, 5200}, {1973, 8222}
X(24244) = barycentric product X(i)*X(j) for these {i,j}: {4, 5490}, {75, 19218}, {76, 8948}, {264, 493}, {1306, 14618}
X(24244) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 19215}, {2, 488}, {4, 3068}, {6, 10132}, {25, 6423}, {69, 8222}, {393, 5200}, {493, 3}, {523, 17431}, {1306, 4558}, {5490, 69}, {6291, 19032}, {6422, 19446}, {8948, 6}, {8950, 8911}, {19218, 1}
X(24245) lies on the cubic K168 and these lines: {3,486}, {6,8940}, {20,1322}, {69,1590}, {131,6560}, {570,18289}, {631,13429}, {1579,10666}, {1609,18290}, {2165,6676}, {5418,8961}, {7586,21464}, {8576,11513}, {10963,11291}
X(24245) = complement of X(24243)
X(24245) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 486}, {487, 2887}, {3069, 20305}, {6424, 226}, {10133, 10}, {17432, 21253}, {19216, 141}
X(24245) = X(2)-Ceva conjugate of X(486)
X(24245) = X(372)-isoconjugate of X(19217)
X(24245) = crosspoint of X(2) and X(487)
X(24245) = crosssum of X(6) and X(8946)
X(24245) = barycentric product X(i)*X(j) for these {i,j}: {486, 487}, {3069, 11091}
X(24245) = barycentric quotient X(i)/X(j) for these {i,j}: {487, 491}, {3069, 1586}, {6414, 494}, {6424, 5412}, {8576, 8946}, {10133, 372}, {11091, 5491}
X(24246) lies on the cubic K168 and these lines: {3,485}, {6,8944}, {20,1321}, {69,1589}, {131,6561}, {570,18290}, {631,13440}, {1609,18289}, {2165,6676}, {7585,21463}, {8577,11514}, {10961,11292}
X(24246) = complement of X(24244)
X(24246) = X(24246) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 485}, {488, 2887}, {3068, 20305}, {6423, 226}, {10132, 10}, {17431, 21253}, {19215, 141}
X(24246) = X(24246) = X(2)-Ceva conjugate of X(485)
X(24246) = X(24246) = X(i)-isoconjugate of X(j) for these (i,j): {92, 8950}, {371, 19218}
X(24246) = X(24246) = crosspoint of X(2) and X(488)
X(24246) = X(24246) = crosssum of X(6) and X(8948)
X(24246) = X(24246) = barycentric product X(i)*X(j) for these {i,j}: {485, 488}, {3068, 11090}
X(24246) = X(24246) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 8950}, {488, 492}, {3068, 1585}, {6413, 493}, {6423, 5413}, {8577, 8948}, {10132, 371}, {11090, 5490}
X(24247) lies on these lines: {1, 5286}, {2, 9317}, {4, 3061}, {8, 672}, {9, 515}, {20, 3496}, {169, 17647}, {257, 7791}, {348, 17046}, {355, 25066}, {377, 17451}, {944, 2329}, {986, 7738}, {997, 5179}, {1018, 12647}, {1146, 1376}, {1212, 5794}, {1475, 12649}, {1573, 6184}, {1759, 4299}, {2170, 3434}, {2266, 3189}, {2267, 2345}, {2549, 2795}, {2784, 3923}, {3488, 16503}, {3509, 4293}, {3693, 5252}, {3721, 9597}, {3727, 9598}, {3785, 17739}, {3878, 17732}, {4051, 5082}, {4165, 17740}, {4317, 17736}, {4513, 10944}, {5013, 21965}, {5289, 17747}, {5438, 23058}, {5816, 25078}, {6506, 15842}, {9592, 24239}, {10527, 21029}, {10573, 16549}, {11185, 18061}, {17754, 18391}, {24265, 24275}
X(24247) = anticomplement of X(24249)
X(24247) = X(7)-of-1st-Brocard-triangle
X(24247) = 1st-Brocard-isogonal conjugate of X(24264)
X(24247) = 1st-Brocard-isotomic conjugate of X(24248)
Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B' and C' cyclically. A'B'C' is also the anticomplementary triangle of the 1st Brocard triangle. Also, X(24248) = X(1)-of-A'B'C'. (Randy Hutson, October 15, 2018)
X(24248) lies on these lines: {1, 7}, {2, 846}, {3, 1284}, {4, 240}, {6, 17768}, {8, 726}, {9, 1738}, {10, 2996}, {11, 17595}, {30, 7986}, {31, 19785}, {35, 24309}, {37, 5880}, {38, 3434}, {40, 1423}, {42, 5905}, {43, 329}, {45, 3826}, {46, 1400}, {55, 3782}, {57, 12717}, {63, 1711}, {65, 17635}, {69, 740}, {72, 18252}, {75, 23690}, {79, 941}, {86, 11104}, {141, 5695}, {144, 1757}, {145, 17766}, {149, 4392}, {171, 3474}, {182, 5091}, {190, 4429}, {191, 1714}, {192, 4645}, {193, 17770}, {194, 3865}, {226, 17594}, {228, 14455}, {238, 4000}, {291, 24458}, {344, 3836}, {345, 2887}, {354, 12722}, {377, 2292}, {387, 1046}, {497, 982}, {499, 17077}, {517, 1469}, {518, 17276}, {519, 11160}, {527, 3751}, {528, 3242}, {536, 3416}, {573, 1756}, {611, 5762}, {896, 24597}, {938, 9801}, {940, 4854}, {942, 12723}, {946, 988}, {966, 24697}, {968, 5249}, {984, 2550}, {1001, 1086}, {1058, 3976}, {1125, 17304}, {1150, 4442}, {1155, 17720}, {1193, 11415}, {1210, 21629}, {1266, 3883}, {1352, 2783}, {1376, 4415}, {1386, 17301}, {1403, 4192}, {1463, 3057}, {1478, 4424}, {1479, 3670}, {1633, 7295}, {1698, 5296}, {1699, 24239}, {1720, 2961}, {1722, 12572}, {1733, 17861}, {1737, 12618}, {1836, 3666}, {1899, 18202}, {1957, 17903}, {2385, 7289}, {2478, 24443}, {2549, 2795}, {2551, 24440}, {2782, 19637}, {2784, 5921}, {2792, 6776}, {3052, 17061}, {3056, 15310}, {3058, 17597}, {3061, 7738}, {3094, 24289}, {3210, 4388}, {3218, 11269}, {3240, 17484}, {3339, 7996}, {3436, 4642}, {3475, 3750}, {3496, 5286}, {3550, 9778}, {3618, 4672}, {3649, 19765}, {3662, 3685}, {3673, 4008}, {3677, 9580}, {3679, 10005}, {3683, 24789}, {3696, 4643}, {3721, 9598}, {3727, 9597}, {3736, 17139}, {3752, 24703}, {3772, 4640}, {3784, 21334}, {3842, 24693}, {3844, 17281}, {3868, 12530}, {3875, 5847}, {3886, 17274}, {3891, 4450}, {3924, 6872}, {3932, 17262}, {3938, 20075}, {3946, 16475}, {3961, 17784}, {3993, 17316}, {4001, 17156}, {4003, 17721}, {4026, 4363}, {4085, 17767}, {4133, 17294}, {4137, 25254}, {4259, 20718}, {4389, 5263}, {4440, 24349}, {4443, 24316}, {4512, 23681}, {4644, 4649}, {4656, 5268}, {4675, 15569}, {4676, 16706}, {4679, 16610}, {4683, 5739}, {4689, 17718}, {4703, 14555}, {4716, 5839}, {4733, 17251}, {4743, 17771}, {4850, 5057}, {4966, 7232}, {4995, 17783}, {5084, 24174}, {5132, 16382}, {5218, 17601}, {5220, 17334}, {5222, 16468}, {5248, 24159}, {5250, 23536}, {5255, 6361}, {5272, 24177}, {5292, 16566}, {5530, 9612}, {5712, 17592}, {5724, 12943}, {5745, 17064}, {5759, 9441}, {5800, 8680}, {5853, 16496}, {6007, 10477}, {6327, 17147}, {6392, 17739}, {6857, 24161}, {6986, 8238}, {7390, 8245}, {8424, 13723}, {9812, 17591}, {10186, 24203}, {10385, 17715}, {10431, 11031}, {11019, 18193}, {11677, 17446}, {12047, 12610}, {12514, 23537}, {12579, 13736}, {13097, 16056}, {14450, 19767}, {15254, 17278}, {15485, 16020}, {16569, 18228}, {16600, 17732}, {16830, 17247}, {16910, 25248}, {17018, 17483}, {17150, 20064}, {17164, 17676}, {17183, 18792}, {17593, 17717}, {17950, 18391}, {18201, 24217}, {20017, 20290}, {24282, 24293}, {24347, 25372}, {24424, 24445}
X(24248) = X(8)-of-1st-Brocard-triangle
X(24248) = reflection of X(24282) in X(24293)
X(24248) = complement of X(24280)
X(24248) = anticomplement of X(3923)
X(24248) = 1st-Brocard-isogonal conjugate of X(24265)
X(24248) = 1st-Brocard-isotomic conjugate of X(24247)
X(24249) lies on these lines: {1, 20335}, {2, 9317}, {10, 7795}, {55, 21232}, {56, 17048}, {76, 16822}, {142, 515}, {355, 17062}, {730, 16825}, {999, 6647}, {1111, 16788}, {1352, 2784}, {1837, 17046}, {2329, 3673}, {2795, 3734}, {3061, 17681}, {3212, 3496}, {3663, 9620}, {3735, 24281}, {3754, 14377}, {3816, 17044}, {3980, 4112}, {4011, 24255}, {5088, 17754}, {11343, 16609}, {17050, 19860}, {17619, 24784}, {17683, 21921}, {17738, 24282}, {24263, 24279}, {24264, 24283}
X(24249) = complement of X(24247)
X(24249) = 1st-Brocard-isogonal conjugate of X(24266)
X(24249) = 1st-Brocard-isotomic conjugate of X(24274)
X(24249) = X(9)-of-1st-Brocard-triangle
X(24250) lies on these lines: {1, 19890}, {2, 1083}, {5, 10}, {9, 2957}, {11, 14839}, {36, 1009}, {75, 7336}, {141, 513}, {542, 5150}, {760, 908}, {1211, 3140}, {1352, 24265}, {3923, 24206}, {3944, 19950}, {4011, 21243}, {4092, 18151}, {4422, 21252}, {5057, 17292}, {11679, 19987}, {17047, 17279}, {24254, 25385}
X(24250) = X(11)-of-1st-Brocard-triangle
X(24250) = {X(3923),X(24206)}-harmonic conjugate of X(24251)
X(24251) lies on these lines: {2, 24265}, {141, 20718}, {542, 24253}, {1352, 24264}, {3821, 24269}, {3923, 24206}, {3980, 21243}, {5880, 17047}, {19950, 24274}
X(24251) = X(12)-of-1st-Brocard-triangle
X(24251) = {X(3923),X(24206)}-harmonic conjugate of X(24250)
X(24252) lies on these lines: {1231, 1973}, {1352, 2784}, {3734, 24269}, {3735, 24268}, {3923, 24263}, {3980, 9306}, {9317, 14826}
X(24252) = X(19)-of-1st-Brocard-triangle
X(24252) = 1st-Brocard-isogonal conjugate of X(24268)
X(24253) lies on these lines: {10, 7193}, {182, 2783}, {184, 3980}, {542, 24251}, {692, 24325}, {726, 2330}, {740, 5135}, {1376, 23095}, {1580, 4279}, {2175, 16825}, {3821, 5091}, {4418, 5012}, {16549, 19554}, {19561, 24295}, {24259, 24267}
X(24253) = X(35)-of-1st-Brocard-triangle
X(24253) = {X(182),X(3923)}-harmonic conjugate of X(5150)
X(24254) lies on these lines: {2, 3125}, {6, 8682}, {10, 626}, {32, 16822}, {37, 712}, {65, 21240}, {141, 758}, {213, 20911}, {274, 2106}, {304, 17750}, {517, 3739}, {742, 3997}, {766, 17792}, {894, 20924}, {1015, 24631}, {1018, 24326}, {1111, 24330}, {1215, 21232}, {1572, 4384}, {1573, 17755}, {1930, 2295}, {2284, 4363}, {2795, 3734}, {3727, 16818}, {3754, 20255}, {3980, 9306}, {4372, 5264}, {4376, 16788}, {4418, 9317}, {5091, 5108}, {7816, 24850}, {9620, 10436}, {10800, 16825}, {14210, 24512}, {14839, 24325}, {16549, 16720}, {16705, 25248}, {17034, 17762}, {17357, 21331}, {17489, 21802}, {17499, 20955}, {24250, 25385}, {24256, 24295}, {24275, 24281}
X(24254) = X(37)-of-1st-Brocard-triangle
X(24254) = 1st-Brocard-isogonal conjugate of X(24271)
X(24254) = complement of X(3735)
X(24255) lies on these lines: {1, 17789}, {2, 19933}, {6, 744}, {321, 730}, {1045, 17788}, {2278, 4381}, {2309, 20234}, {2795, 24259}, {3285, 4836}, {3923, 4112}, {3980, 24266}, {4011, 24249}, {4703, 5692}, {4812, 7032}, {6327, 9857}, {14210, 18138}, {16732, 24327}, {20432, 21352}
X(24255) = X(38)-of-1st-Brocard-triangle
X(24255) = 1st-Brocard-isogonal conjugate of X(24272)
X(24256) lies on these lines: {2, 694}, {5, 141}, {6, 76}, {13, 25167}, {14, 25157}, {32, 8177}, {39, 698}, {51, 8891}, {69, 7785}, {99, 5116}, {182, 2782}, {183, 5017}, {194, 3618}, {262, 7778}, {384, 1691}, {385, 12212}, {518, 12263}, {524, 5052}, {538, 597}, {575, 13196}, {591, 22723}, {599, 7809}, {611, 10079}, {613, 10063}, {726, 3946}, {730, 1386}, {736, 7804}, {742, 21443}, {1078, 2076}, {1180, 10191}, {1235, 2211}, {1350, 13860}, {1352, 6033}, {1428, 18982}, {1501, 16932}, {1503, 6248}, {1915, 16950}, {1974, 12143}, {1991, 22722}, {2021, 8369}, {2330, 13077}, {3095, 7795}, {3098, 7815}, {3104, 23024}, {3105, 23018}, {3329, 9865}, {3763, 7887}, {3788, 11272}, {5007, 15870}, {5012, 10328}, {5034, 17130}, {5038, 12215}, {5039, 7751}, {5050, 13108}, {5085, 11257}, {5092, 7816}, {5108, 7698}, {5207, 16044}, {5939, 22498}, {6309, 9605}, {6656, 10292}, {7664, 10160}, {7783, 12055}, {7787, 9983}, {7800, 9821}, {7808, 8149}, {8024, 20965}, {8267, 11205}, {9230, 14603}, {9516, 11060}, {9902, 16475}, {10130, 15107}, {10329, 16276}, {12251, 14853}, {13511, 19121}, {16964, 23019}, {16965, 23025}, {17353, 17760}, {24254, 24295}, {25324, 25325}
X(24256) = midpoint of X(6) and X(76)
X(24256) = midpoint of PU(181)
X(24256) = complement of X(3094)
X(24256) = anticomplement of X(10007)
X(24256) = X(39)-of-1st-Brocard-triangle
X(24256) = 1st-Brocard-isogonal conjugate of X(24273)
X(24257) lies on these lines: {3, 740}, {40, 3875}, {182, 2783}, {192, 6211}, {239, 6210}, {355, 4085}, {511, 24728}, {515, 990}, {912, 4523}, {946, 3946}, {1352, 2784}, {1429, 4008}, {1999, 20368}, {2187, 17862}, {2321, 6684}, {2791, 24283}, {2792, 6776}, {2796, 11179}, {3564, 4655}, {3576, 3886}, {4133, 10164}, {4297, 4780}, {4672, 5050}, {4743, 18481}, {5085, 5695}, {5278, 11203}, {5853, 5882}, {5988, 9744}, {7413, 17592}, {7609, 17349}, {7613, 9317}, {8235, 9534}, {8550, 17768}, {14912, 24695}
X(24257) = X(40)-of-1st-Brocard-triangle
X(24258) lies on these lines: {1083, 3923}, {1233, 9447}, {3734, 24263}, {4048, 4112}, {9306, 24259}
X(24258) = X(41)-of-1st-Brocard-triangle
X(24258) = 1st-Brocard-isogonal conjugate of X(24274)
X(24259) lies on these lines: {2, 846}, {31, 17031}, {42, 726}, {43, 3729}, {76, 4039}, {171, 350}, {190, 24463}, {321, 2239}, {384, 1580}, {516, 1764}, {672, 21369}, {986, 1008}, {1009, 24850}, {1086, 24425}, {1215, 24326}, {1269, 1918}, {1376, 4713}, {1707, 17026}, {2223, 12263}, {2245, 24688}, {2795, 24255}, {3734, 4112}, {3747, 20913}, {3769, 4479}, {3961, 17794}, {4286, 25347}, {4362, 4441}, {4436, 24327}, {4640, 21264}, {4697, 24512}, {5091, 24271}, {5108, 24279}, {9306, 24258}, {10453, 20101}, {17135, 17766}, {24253, 24267}
X(24259) = X(42)-of-1st-Brocard-triangle
X(24259) = 1st-Brocard-isogonal conjugate of X(24275)
X(24260) lies on these lines: {2, 846}, {43, 726}, {194, 17795}, {239, 9902}, {516, 3840}, {962, 18788}, {1376, 17793}, {2239, 4362}, {3434, 3741}, {3663, 6685}, {3729, 16569}, {3734, 24276}, {3771, 12610}, {4112, 5091}, {4192, 24728}, {10453, 17766}, {16557, 17754}, {17290, 24425}, {24264, 24294}
X(24260) = X(43)-of-1st-Brocard-triangle
X(24260) = 1st-Brocard-isogonal conjugate of X(24276)
X(24261) lies on these lines: {2, 24278}, {2787, 4107}, {2795, 3734}, {16822, 17130}, {17738, 24281}
X(24261) = X(44)-of-1st-Brocard-triangle
X(24261) = 1st-Brocard-isogonal conjugate of X(24277)
X(24262) lies on these lines: {2, 24277}, {10, 7869}, {2795, 3734}, {7751, 16822}
X(24262) = X(45)-of-1st-Brocard-triangle
X(24262) = 1st-Brocard-isogonal conjugate of X(24278)
X(24263) lies on these lines: {349, 9247}, {3734, 24258}, {3735, 24288}, {3923, 24252}, {4112, 4172}, {24249, 24279}
X(24263) = X(48)-of-1st-Brocard-triangle
X(24264) lies on these lines: {2, 1083}, {6, 6007}, {31, 10800}, {32, 1580}, {38, 9997}, {55, 14839}, {63, 760}, {75, 2175}, {99, 24282}, {101, 1376}, {171, 2242}, {182, 2783}, {184, 4418}, {517, 993}, {573, 8424}, {692, 4363}, {740, 5138}, {846, 3735}, {1352, 24251}, {1486, 17049}, {1572, 1707}, {1754, 18235}, {1836, 24630}, {2278, 4436}, {2329, 9441}, {2330, 3729}, {2344, 24578}, {2795, 24268}, {3550, 11364}, {3556, 9565}, {3734, 4112}, {3875, 19133}, {3980, 9306}, {4039, 7751}, {4858, 24329}, {5114, 24696}, {5135, 5695}, {9620, 17594}, {13323, 24850}, {16792, 17301}, {17792, 24309}, {24249, 24283}, {24260, 24294}
X(24264) = X(55)-of-1st-Brocard-triangle
X(24264) = 1st-Brocard-isogonal conjugate of X(24247)
X(24265) lies on these lines: {2, 24251}, {182, 2783}, {312, 1397}, {613, 14839}, {1352, 24250}, {1428, 3729}, {2175, 4676}, {3501, 12194}, {3734, 24258}, {4011, 9306}, {4672, 5138}, {5091, 24280}, {13478, 20545}, {24247, 24275}, {24276, 24294}
X(24265) = X(56)-of-1st-Brocard-triangle
X(24265) = 1st-Brocard-isogonal conjugate of X(24248)
X(24266) lies on these lines: {2, 9317}, {182, 2783}, {239, 3501}, {515, 997}, {604, 1229}, {1083, 4011}, {2795, 24283}, {3008, 16968}, {3061, 6996}, {3980, 24255}, {4858, 24334}, {7803, 17023}, {16609, 21477}
X(24266) = X(57)-of-1st-Brocard-triangle
X(24266) = 1st-Brocard-isogonal conjugate of X(24249)
X(24267) lies on these lines: {6, 726}, {182, 2782}, {384, 4279}, {515, 12618}, {1230, 2206}, {3770, 5009}, {3923, 4112}, {4713, 23095}, {24253, 24259}, {24275, 24294}
X(24267) = X(58)-of-1st-Brocard-triangle
X(24267) = 1st-Brocard-isogonal conjugate of X(3821)
X(24268) lies on these lines: {1, 4}, {2, 9317}, {3, 16609}, {6, 8680}, {9, 25252}, {57, 5088}, {63, 194}, {92, 2202}, {213, 22001}, {284, 286}, {321, 4390}, {326, 21246}, {355, 16603}, {379, 17451}, {527, 1992}, {572, 17861}, {604, 17863}, {730, 4362}, {870, 1438}, {940, 7223}, {966, 24435}, {980, 13478}, {993, 4124}, {1400, 17134}, {1429, 3673}, {1431, 24728}, {1441, 2268}, {2278, 16732}, {2792, 6776}, {2795, 24264}, {3735, 24252}, {3923, 4112}, {3980, 24293}, {4011, 24294}, {4384, 5745}, {4393, 5905}, {5091, 24283}, {5244, 7354}, {5283, 25080}, {6996, 7146}, {7384, 17084}, {7709, 17596}, {9028, 16973}, {9741, 16833}, {17738, 18906}, {17797, 21299}
X(24268) = X(63)-of-1st-Brocard-triangle
X(24268) = 1st-Brocard-isogonal conjugate of X(24252)
X(24269) lies on these lines: {182, 2783}, {740, 4260}, {1009, 1733}, {1284, 20236}, {1352, 2791}, {2549, 2795}, {2886, 16579}, {3734, 24252}, {3821, 24251}, {14453, 17647}
X(24269) = X(65)-of-1st-Brocard-triangle
X(24270) lies on these lines: {30, 18382}, {182, 2794}, {1899, 2549}, {2386, 7761}, {2790, 9996}, {3734, 21243}, {7514, 23333}
X(24270) = X(66)-of-1st-Brocard-triangle
X(24270) = 1st-Brocard-isogonal conjugate of X(184)
X(24271) lies on these lines: {2, 99}, {6, 536}, {30, 1211}, {76, 5337}, {81, 538}, {172, 4044}, {187, 16046}, {321, 5291}, {535, 11355}, {754, 2895}, {980, 1975}, {1083, 1316}, {1230, 17587}, {2795, 4418}, {3175, 16785}, {3923, 4112}, {3934, 21495}, {3948, 5277}, {4383, 11286}, {4721, 20769}, {5091, 24259}, {5278, 11352}, {5283, 19281}, {5739, 7737}, {7815, 21537}, {7816, 21511}, {11354, 19701}, {14033, 14555}, {15271, 16431}, {16609, 24850}, {16732, 24335}, {21024, 24632}, {24277, 24288}
X(24271) = X(81)-of-1st-Brocard-triangle
X(24271) = 1st-Brocard-isogonal conjugate of X(24254)
X(24272) lies on these lines: {10, 20994}, {3735, 4112}, {3821, 24273}, {3923, 4172}, {4418, 9317}
X(24272) = X(82)-of-1st-Brocard-triangle
X(24272) = 1st-Brocard-isogonal conjugate of X(24255)
X(24273) lies on these lines: {2, 4048}, {3, 2916}, {6, 76}, {69, 20088}, {75, 17741}, {99, 10007}, {141, 384}, {182, 7697}, {183, 3407}, {511, 13111}, {518, 9903}, {597, 19570}, {599, 754}, {623, 23019}, {624, 23025}, {698, 17128}, {782, 17997}, {1003, 21358}, {1350, 22677}, {1503, 12252}, {1691, 3934}, {1915, 8891}, {3053, 6308}, {3094, 3734}, {3114, 10010}, {3314, 9866}, {3416, 17766}, {3552, 3619}, {3589, 7797}, {3821, 24272}, {4074, 4563}, {5031, 7832}, {5085, 14880}, {5103, 16044}, {5149, 11646}, {6683, 12055}, {6704, 7834}, {7716, 12144}, {7804, 12212}, {7808, 13331}, {7935, 18500}, {8369, 13086}, {8627, 10130}, {9743, 9751}, {10387, 13078}, {11324, 21001}, {12156, 15533}, {13330, 18548}, {13586, 20582}, {14621, 20917}
X(24273) = X(83)-of-1st-Brocard-triangle
X(24273) = 1st-Brocard-isogonal conjugate of X(24256)
X(24273) = 1st-Brocard-isotomic conjugate of X(4045)
X(24274) lies on these lines: {971, 1212}, {984, 6184}, {2549, 2795}, {2550, 17435}, {3125, 7613}, {3501, 9941}, {3721, 4862}, {19950, 24251}
X(24274) = X(85)-of-1st-Brocard-triangle
X(24274) = 1st-Brocard-isogonal conjugate of X(24258)
X(24274) = 1st-Brocard-isotomic conjugate of X(24249)
X(24275) lies on these lines: {1, 4037}, {2, 99}, {6, 519}, {10, 17735}, {30, 1213}, {32, 4195}, {39, 13740}, {58, 21024}, {76, 17688}, {86, 538}, {187, 4234}, {754, 1654}, {964, 5283}, {966, 7737}, {1010, 16589}, {1043, 20970}, {1220, 1500}, {1573, 5263}, {1724, 12194}, {2322, 14581}, {2702, 2758}, {2795, 24342}, {3125, 4418}, {3735, 3923}, {3934, 16061}, {4201, 7756}, {5224, 7761}, {5254, 17698}, {5275, 16394}, {5277, 11115}, {5793, 14974}, {7748, 16062}, {7798, 17379}, {7804, 17277}, {7816, 16060}, {7848, 17271}, {7865, 17238}, {8818, 24935}, {9346, 10453}, {9597, 19836}, {9598, 19784}, {9903, 16552}, {11286, 17259}, {11287, 17327}, {15048, 17398}, {16975, 24552}, {17330, 18907}, {23897, 24880}, {24247, 24265}, {24254, 24281}, {24267, 24294}
X(24275) = X(86)-of-1st-Brocard-triangle
X(24275) = 1st-Brocard-isogonal conjugate of X(24259)
X(24275) = 1st-Brocard-isotomic conjugate of X(3821)
X(24276) lies on these lines: {10, 3500}, {3734, 24260}, {3923, 18906}, {24265, 24294}
X(24276) = X(87)-of-1st-Brocard-triangle
X(24276) = 1st-Brocard-isogonal conjugate of X(24260)
X(24277) lies on these lines: {2, 24262}, {952, 16594}, {3734, 24278}, {3923, 5091}, {24271, 24288}
X(24277) = X(88)-of-1st-Brocard-triangle
X(24277) = 1st-Brocard-isogonal conjugate of X(24261)
X(24278) lies on these lines: {2, 24261}, {3734, 24277}
X(24278) = X(89)-of-1st-Brocard-triangle
X(24278) = 1st-Brocard-isogonal conjugate of X(24262)
X(24279) lies on the Brocard circle and these lines: {2, 5991}, {3, 142}, {6, 514}, {86, 4237}, {238, 19950}, {384, 24502}, {726, 17976}, {894, 5088}, {1023, 3729}, {1083, 3923}, {5011, 20179}, {5108, 24259}, {5179, 17353}, {24249, 24263}
X(24279) = X(101)-of-1st-Brocard-triangle
X(24279) = 1st-Brocard-isogonal conjugate of X(2786)
X(24279) = inverse-in-circumcircle of X(8618)
X(24279) = isogonal conjugate of antitomic conjugate of X(101)
X(24280) lies on these lines: {1, 7240}, {2, 846}, {7, 3685}, {8, 144}, {63, 12717}, {69, 5695}, {72, 12530}, {75, 5698}, {78, 1721}, {145, 726}, {190, 2550}, {192, 4307}, {193, 740}, {312, 3474}, {344, 5880}, {345, 1836}, {346, 4645}, {390, 4454}, {518, 12718}, {527, 3886}, {545, 3242}, {960, 17635}, {962, 20348}, {984, 20073}, {990, 4511}, {1266, 7290}, {2345, 24723}, {2783, 6776}, {2792, 5921}, {2795, 24282}, {3241, 17132}, {3332, 25252}, {3522, 24728}, {3616, 3663}, {3617, 4660}, {3620, 4655}, {3621, 17766}, {3705, 9812}, {3868, 12723}, {3872, 12652}, {3873, 12722}, {3876, 18252}, {3877, 12721}, {3883, 4659}, {3912, 4312}, {4000, 4676}, {4018, 21848}, {4190, 25253}, {4295, 7283}, {4310, 4440}, {4385, 6361}, {4387, 11246}, {4419, 5263}, {4442, 24597}, {4779, 11038}, {5057, 17740}, {5091, 24265}, {5550, 17304}, {6734, 21629}, {6872, 17164}, {6904, 19582}, {7081, 9778}, {9780, 17355}, {9965, 10453}, {11358, 13097}, {11415, 20245}, {15310, 25304}, {17127, 19789}, {17135, 20078}, {17165, 20075}, {17770, 20080}, {17776, 20292}
X(24280) = anticomplement of X(24248)
X(24280) = X(145)-of-1st-Brocard-triangle
X(24281) lies on these lines: 1, 16377}, {2, 24262}, {6, 514}, {8, 7794}, {10, 19886}, {32, 3212}, {101, 21138}, {192, 1016}, {239, 980}, {519, 599}, {664, 1015}, {903, 24807}, {952, 1086}, {1506, 17084}, {3125, 9317}, {3699, 13466}, {3734, 24282}, {3735, 24249}, {3752, 5662}, {3912, 17720}, {4000, 6547}, {4237, 5170}, {4360, 6631}, {4393, 4555}, {4482, 9055}, {5091, 24286}, {6633, 17318}, {9259, 21208}, {17119, 24873}, {17395, 24864}, {17738, 24261}, {24254, 24275}
X(24281) = X(190)-of-1st-Brocard-triangle
X(24281) = 1st-Brocard-isogonal conjugate of X(4107)
X(24281) = 1st-Brocard-isotomic conjugate of X(2786)
X(24282) lies on these lines: {2, 3125}, {7, 20924}, {8, 315}, {40, 7350}, {65, 304}, {69, 758}, {75, 517}, {76, 3212}, {85, 12709}, {99, 24264}, {145, 17141}, {192, 712}, {193, 8682}, {213, 21216}, {222, 664}, {239, 1572}, {668, 6382}, {766, 25304}, {874, 24351}, {883, 4454}, {894, 9620}, {942, 18156}, {1376, 4561}, {1655, 25270}, {1930, 5903}, {2795, 24280}, {3496, 16822}, {3501, 17760}, {3734, 24281}, {3869, 20911}, {3905, 5255}, {3923, 18906}, {4165, 4766}, {4647, 25307}, {4713, 21138}, {5902, 14210}, {7763, 17084}, {10449, 17762}, {14839, 24349}, {17165, 21272}, {17279, 21331}, {17738, 24249}, {18135, 25253}, {18140, 19582}, {24248, 24293}
X(24282) = reflection of X(24248) in X(24293)
X(24282) = X(192)-of-1st-Brocard-triangle
X(24283) lies on these lines: {2, 846}, {43, 10025}, {57, 497}, {165, 1447}, {200, 726}, {982, 14942}, {986, 13727}, {1376, 24352}, {1403, 7580}, {1707, 24600}, {2791, 24257}, {2795, 24266}, {3663, 13405}, {3673, 9441}, {3729, 8580}, {3870, 4970}, {4660, 4847}, {5091, 24268}, {14828, 17592}, {24249, 24264}
X(24283) = X(200)-of-1st-Brocard-triangle
X(24284) lies on these lines: {2, 3569}, {6, 9035}, {113, 126}, {125, 127}, {141, 526}, {441, 525}, {512, 625}, {620, 690}, {684, 879}, {804, 4107}, {850, 3288}, {888, 6131}, {1499, 16235}, {1640, 3268}, {2492, 3589}, {2799, 14316}, {3049, 3267}, {3050, 23285}, {3618, 14398}, {3800, 9134}, {5108, 18332}, {5652, 6033}, {22159, 23148}
X(24284) = X(647)-of-1st-Brocard-triangle
X(24285) lies on these lines: {2, 24290}, {141, 8674}, {512, 4369}, {525, 14838}, {620, 690}, {812, 8659}, {826, 21196}, {2487, 3566}, {2787, 4107}, {3063, 15413}, {3309, 4885}, {3835, 6004}, {8062, 17066}, {14316, 21209}, {17023, 23829}
X(24285) = complement of X(24290)
X(24285) = X(650)-of-1st-Brocard-triangle
X(24286) lies on these lines: {1, 512}, {99, 110}, {513, 1386}, {659, 8297}, {663, 1201}, {818, 15413}, {826, 4560}, {840, 14665}, {891, 4435}, {1027, 6372}, {1385, 3309}, {2775, 11699}, {2786, 4164}, {2787, 4107}, {3566, 14529}, {4025, 8633}, {4378, 16971}, {4750, 5040}, {5029, 14419}, {5091, 24281}, {7927, 17166}, {8034, 17017}
X(24286) = X(659)-of-1st-Brocard-triangle
X(24287) lies on these lines: {86, 24506}, {141, 900}, {665, 4486}, {798, 18160}, {802, 15413}, {812, 17738}, {816, 21007}, {876, 4010}, {1019, 1577}, {2254, 21261}, {2530, 3835}, {2642, 5224}, {2786, 23596}, {2787, 4107}, {3733, 8060}, {3766, 20913}, {3798, 17072}, {4444, 23829}, {4705, 21196}, {4750, 21053}, {6707, 24959}, {17069, 21051}
X(24287) = X(661)-of-1st-Brocard-triangle
X(24287) = 1st-Brocard-isogonal conjugate of X(24288)
X(24288) lies on these lines: {6, 514}, {1316, 5091}, {1724, 19950}, {3735, 24263}, {24271, 24277}
X(24288) = X(662)-of-1st-Brocard-triangle
X(24288) = 1st-Brocard-isogonal conjugate of X(24287)
X(24289) lies on these lines: {6, 513}, {9, 1575}, {37, 24338}, {192, 4562}, {292, 3123}, {350, 3662}, {536, 599}, {646, 20532}, {730, 4660}, {1045, 18795}, {1086, 2481}, {3094, 24248}, {3888, 17475}, {3959, 17435}, {4876, 4947}, {6646, 17759}, {9025, 16973}, {11650, 14419}, {17282, 20530}, {17303, 25382}
X(24289) = X(668)-of-1st-Brocard-triangle
X(24289) = 1st-Brocard-isotomic conjugate of X(2787)
X(24290) lies on these lines: {2, 24285}, {6, 8674}, {115, 125}, {119, 20623}, {512, 661}, {513, 21389}, {514, 23780}, {521, 20980}, {525, 1577}, {647, 21837}, {649, 6004}, {650, 1734}, {656, 3709}, {665, 1642}, {826, 4024}, {832, 2484}, {918, 4437}, {926, 20455}, {1018, 4551}, {1635, 8659}, {1643, 3887}, {1769, 4526}, {2509, 3063}, {2530, 3250}, {2642, 14407}, {2775, 5540}, {2786, 23596}, {2821, 3030}, {3239, 17072}, {3251, 14438}, {3287, 6003}, {3566, 14321}, {3800, 4841}, {3906, 4931}, {3912, 23829}, {4017, 4171}, {4079, 8061}, {4086, 21958}, {4130, 7655}, {4486, 21261}, {4750, 5098}, {5029, 14419}, {6161, 8632}, {7180, 8611}, {9508, 9509}, {10097, 10693}, {10099, 18785}, {13277, 20331}, {21348, 23800}, {21797, 21828}
X(24290) = anticomplement of X(24285)
X(24290) = X(693)-of-1st-Brocard-triangle
X(24290) = 1st-Brocard-isotomic conjugate of X(5091)
X(24291) lies on these lines: {1, 76}, {2, 9317}, {32, 17739}, {34, 9312}, {85, 2647}, {99, 17596}, {141, 25434}, {257, 384}, {286, 447}, {515, 4357}, {535, 17274}, {668, 1965}, {712, 3729}, {846, 4112}, {986, 1975}, {996, 3875}, {1757, 10791}, {3061, 7770}, {3212, 4195}, {3585, 17211}, {3734, 3735}, {3905, 4385}, {3923, 18906}, {3944, 11185}, {5086, 24995}, {5293, 6376}, {7789, 21965}, {10436, 17861}, {17451, 17686}, {24254, 24275}, {24293, 24342}
X(24291) = X(894)-of-1st-Brocard-triangle
X(24292) lies on these lines: {2, 11711}, {537, 3751}, {2787, 4107}, {3923, 4112}, {16732, 24350}
X(24292) = X(896)-of-1st-Brocard-triangle
X(24293) lies on these lines: {1, 20924}, {2, 19933}, {10, 12933}, {75, 730}, {758, 4259}, {760, 4660}, {894, 10791}, {991, 24325}, {2795, 3734}, {3094, 3735}, {3673, 12263}, {3980, 24268}, {4112, 4418}, {4645, 9857}, {24248, 24282}, {24291, 24342}
X(24293) = midpoint of X(24248) and X(24282)
X(24293) = X(984)-of-1st-Brocard-triangle
X(24294) lies on these lines: {1, 83}, {2, 4112}, {10, 8301}, {99, 2108}, {101, 17793}, {238, 730}, {560, 18044}, {668, 8300}, {993, 19263}, {1083, 4368}, {1125, 9259}, {1281, 19950}, {2329, 12263}, {2787, 4164}, {2795, 3734}, {3821, 24272}, {4011, 24268}, {4154, 17738}, {5291, 17031}, {12194, 17752}, {12782, 17743}, {24260, 24264}, {24265, 24276}, {24267, 24275}
X(24294) = X(238)-of-1st-Brocard-triangle
X(24295) lies on these lines: {1, 3790}, {2, 846}, {5, 516}, {10, 82}, {37, 39}, {44, 3775}, {141, 4672}, {182, 2784}, {267, 5506}, {319, 16477}, {519, 597}, {551, 17264}, {594, 4974}, {631, 24728}, {740, 3589}, {752, 3844}, {984, 17354}, {1213, 20546}, {1215, 17724}, {1698, 4660}, {2345, 16825}, {2792, 24206}, {3619, 24695}, {3624, 3729}, {3661, 16468}, {3663, 17322}, {3740, 12722}, {3741, 5294}, {3763, 4655}, {3797, 3993}, {3836, 17357}, {3842, 4422}, {3912, 20132}, {4026, 4432}, {4407, 15481}, {4527, 4852}, {4676, 17371}, {5745, 20545}, {6535, 17150}, {10022, 12040}, {12579, 13728}, {15310, 25144}, {16475, 17286}, {17123, 19808}, {17229, 17772}, {17235, 17767}, {17260, 19856}, {17307, 24697}, {18139, 23812}, {19561, 24253}, {24254, 24256}, {24267, 24275}, {24317, 25372}, {24340, 25364}
X(24295) = complement of X(3821)
X(24295) = X(1125)-of-1st-Brocard-triangle
X(24296) lies on these lines: {2, 99}, {30, 5718}, {36, 17720}, {182, 5091}, {538, 1150}, {980, 6996}, {2276, 24630}, {2795, 4414}, {3936, 7761}, {4653, 11355}, {5114, 17139}, {7739, 24597}, {7798, 16704}, {7804, 11352}, {17595, 24618}
X(24296) = X(1150)-of-1st-Brocard-triangle
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.
X(24297) lies on the Feuerbach hyperbola and these lines: {1, 6946}, {4, 12762}, {7, 952}, {8, 4767}, {9, 2802}, {11, 1000}, {21, 10914}, {100, 2320}, {104, 1155}, {517, 1156}, {519, 3254}, {1317, 18490}, {1320, 3935}, {1656, 7320}, {2800, 3062}, {2826, 23836}, {2829, 10307}, {3427, 12247}, {3577, 10698}, {3625, 6598}, {3887, 4792}, {4900, 12653}, {5424, 10087}, {5557, 10106}, {5559, 12053}, {5561, 9897}, {5854, 6601}, {6594, 9623}, {7319, 8148}, {7972, 14563}, {10039, 13606}, {11604, 12531}, {12641, 21630}, {12738, 17097}, {13602, 16173}
X(24297) = midpoint of X(4900) and X(12653)
X(24297) = reflection of X(7972) in X(14563)
X(24297) = isogonal conjugate of X(5126)
X(24297) = antigonal conjugate of X(1000)
X(24297) = antipode of X(1000) in the Feuerbach hyperbola
X(24297) = trilinear pole of the line {45, 650}
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.
X(24298) lies on the Feuerbach hyperbola and these lines: {1, 6901}, {9, 21090}, {11, 943}, {21, 149}, {104, 5842}, {388, 15173}, {497, 15175}, {952, 17097}, {1156, 5762}, {1479, 3467}, {2320, 3434}, {2346, 6881}, {4302, 15446}, {5083, 5557}, {6596, 21630}, {7319, 10526}
X(24298) = antigonal conjugate of X(943)
X(24298) = isotomic conjugate of the anticomplement of X(17796)
X(24298) = antipode of X(943) in the Feuerbach hyperbola
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.
X(24299) lies on these lines: {1, 3}, {21, 912}, {78, 6883}, {140, 5440}, {225, 7510}, {226, 7491}, {284, 8609}, {355, 6861}, {382, 5715}, {443, 10806}, {515, 6841}, {631, 12649}, {938, 6954}, {944, 6824}, {950, 6842}, {952, 6675}, {971, 13743}, {1006, 14054}, {1100, 5755}, {1104, 5396}, {1125, 6881}, {1621, 21740}, {2320, 5768}, {3358, 12687}, {3487, 6868}, {3488, 6825}, {3526, 5705}, {3560, 18446}, {3616, 6826}, {3655, 5787}, {3683, 5694}, {3868, 6875}, {3897, 6857}, {3916, 7508}, {4313, 6850}, {4330, 16155}, {5178, 6989}, {5248, 5887}, {5259, 6326}, {5267, 12005}, {5436, 5720}, {5439, 6924}, {5603, 6869}, {5703, 6827}, {5722, 6863}, {5731, 6851}, {5735, 15696}, {5745, 13607}, {5761, 6987}, {5777, 7489}, {5780, 16857}, {5842, 11281}, {5901, 20420}, {6847, 10587}, {6882, 13411}, {6906, 13369}, {6928, 11374}, {6951, 11015}, {7497, 11363}, {8728, 10943}, {9942, 10179}, {10165, 10916}, {10526, 17718}, {11520, 21165}
X(24299) = midpoint of X(1) and X(10902)
X(24299) = X(5576)-of-2nd circumperp triangle
X(24299) = X(7568)-of-hexyl triangle
X(24299) = X(10902)-of-anti-Aquila triangle
X(24299) = X(14130)-of-Ascella triangle
X(24299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3576, 11249), (1, 7987, 12704), (1, 16208, 7982), (3, 1385, 13151), (1385, 15178, 1319), (3587, 7987, 3), (3601, 18443, 3), (6906, 18444, 13369), (10246, 16202, 1)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.
X(24300) lies on the Feuerbach hyperbola and this line: {11, 1061}
X(24300) = antigonal conjugate of X(1061)
X(24300) = antipode of X(1061) in the Feuerbach hyperbola
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.
X(24301) lies on these lines: {1, 3}, {5, 11363}, {10, 7542}, {26, 1829}, {184, 912}, {355, 3549}, {515, 15760}, {518, 19131}, {944, 3547}, {946, 12605}, {952, 6676}, {1125, 11585}, {1386, 9967}, {1902, 7526}, {2072, 11230}, {3616, 6643}, {3817, 10297}, {5886, 18531}, {5901, 12362}, {6146, 12259}, {6639, 9956}, {6923, 11393}, {6928, 11392}, {7404, 7718}, {7494, 7967}, {7713, 9714}, {7723, 11699}, {7968, 10897}, {7969, 10898}, {9619, 23115}, {9715, 11396}, {9928, 19357}, {9955, 18404}, {10024, 18480}, {10165, 10257}, {10575, 12262}, {11720, 12358}, {12266, 12363}, {18563, 22793}, {18670, 22054}
X(24301) = midpoint of X(1) and X(15177)
X(24301) = X(15177)-of-anti-Aquila triangle
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28386.
X(24302) lies on the Feuerbach hyperbola and these lines: {4, 7972}, {8, 6702}, {79, 14217}, {80, 5048}, {84, 13253}, {952, 5560}, {1317, 5561}, {1392, 2802}, {1476, 11009}, {2098, 15446}, {2099, 15180}, {3065, 12737}, {5424, 5919}, {5557, 11011}, {7284, 16200}, {10308, 10698}, {12331, 21398}, {12740, 13143}
X(24302) = trilinear pole of the line {650, 15492}
See César Lozada, Hyacinthos 28387.
X(24303) lies on these lines: {6, 16459}, {15, 1495}, {22, 2923}
X(24303) = isogonal conjugate of the cyclocevian conjugate of X(19778)
X(24303) = TCC-perspector of X(15)
See César Lozada, Hyacinthos 28387.
X(24304) lies on these lines: {3, 30486}, {6, 16460}, {16, 1495}, {22, 2924}, {6000, 36756}, {15919, 24303
X(24304) = isogonal conjugate of the cyclocevian conjugate of X(19779)
X(24304) = TCC-perspector of X(16)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28391.
X(24305) lies on these lines: {5,16764}, {30,11671}, {137,24147}, {195,10126}, {1157,14072}, {5965,6592}, {14143,22051}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28391.
X(24306) lies on these lines: {2,3}, {137,24147}, {1263,19552}, {3448,11584}, {5663,24043}, {8254,20414}
X(24306) = midpoint of X(1263) and X(19552)
X(24306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5, 20030), (5, 10205, 140), (3850, 10289, 5)
Perspectors involving obverse triangles: X(24307)-X(24309)
Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. The obverse triangle of P is here introduced as the triangle A'B'C', where A' = p : r : q, B' = r : q : p, C' = q : p : r. If P is a triangle center, then the obverse triangle of P is a central triangle. (Clark Kimberling, October 3, 2018)
The obverse triangle of X(1) is the central triangle having A-vertex a : c : b = AX(75)∩GAX(9), where GAGBGC = medial. Following is a list of central triangles perspective to the obverse triangle of X(1), provided by Peter Moses, October 3, 2018:
ABC (TCCT 6.1) at X(75)
medial (TCCT 6.2) at X(9)
anticomplementary (TCCT 6.3) at X(1654)
tangential (TCCT 6.5) at X(8424)
excentral (TCCT 6.7) at X(9)
second circumperp (TCCT 6.22) at X(993)
Fuhrmann (TCCT 8.25) at X(10)
first Sharygin (see ETC X(8229)) at X(8424)
second Sharygin (see ETC X(8229)) at X(4363)
Aquilla (see ETC X(5586) / T(1,2) TCCT (6.40) at X(3679)
inner Garcia (see ETC X(5587)) at X(1)
outer Garcia (see ETC X(5587)) at X(3679)
second extouch (see ETC X(5927)) at X(9)
tangential of excentral at X(9)
Garcia reflection, see X(15995) at X(9)
second Zaniah, see X(18214) at X(9)
intangents, at X(24307)
extangents, at X(24308)
first circumperp, at X(24309)
X(24307) lies on these lines: {1,1463}, {9,3063}, {55,23857}, {3938,4336}
X(24308) lies on these lines: {2,71}, {40,15973}, {55,1740}, {65,984}, {75,21231}, {1762,15985}, {6254,9840}, {15830,20258}
X(24309) lies on these lines: {1,7225}, {3,142}, {9,1633}, {35,24248}, {40,990}, {55,3663}, {87,572}, {100,3729}, {105,4859}, {106,1292}, {165,846}, {182,15310}, {515,996}, {517,3098}, {527,12329}, {573,9441}, {726,8715}, {993,4660}, {1060,11719}, {1155,12723}, {1293,9082}, {1376,17355}, {1473,4847}, {1621,17304}, {1699,19649}, {1738,7295}, {2182,17668}, {2550,3220}, {2726,2736}, {2796,13173}, {2809,7289}, {2823,3359}, {3008,7083}, {3056,5091}, {3295,4353}, {3434,7293}, {3474,5285}, {3817,16434}, {3923,19548}, {4186,8582}, {4219,10860}, {4292,8193}, {4297,9305}, {4298,12410}, {4421,17132}, {4640,18252}, {5853,22769}, {6011,6015}, {6210,13329}, {6684,12618}, {7411,10434}, {7580,10445}, {7688,19262}, {7987,12652}, {8666,17766}, {10164,19544}, {10267,12517}, {12530,16566}, {16686,17278}, {17792,24264}
Perspectors involving N-obverse triangles: X(24310)-X(24312)
Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. The N-obverse triangle of P is here introduced as the triangle A'B'C', where A' = - p : r : q, B' = r : - q : p, C' = q : p : - r. If P is a triangle center, then the obverse triangle of P is a central triangle. (Clark Kimberling, October 3, 2018)
The N-obverse triangle of X(1) is the central triangle having A-vertex - a : c : b = AX(75)∩GAX(1), where GAGBGC = medial. Following is a list of central triangles perspective to the N-obverse triangle of X(1), provided by Peter Moses, October 3, 2018:
ABC (TCCT 6.1) at X(75)
medial (TCCT 6.2) at X(1)
tangential (TCCT 6.5) at X(8301)
second Sharygin (see ETC X(8229)) at X(8301)
outer Garcia (see ETC X(5587)) at X(1)
first Zaniah, see X(18214) at X(1)
extangents at X(24310)
first Sharygin at X(24311)
fifth extouch at X(24312)
X(24310) lies on these lines: {1,3}, {2,71}, {7,22097}, {9,1730}, {12,9548}, {19,27}, {37,19730}, {43,209}, {48,1817}, {200,22275}, {219,11347}, {226,573}, {238,19728}, {255,1396}, {307,1848}, {312,4019}, {329,2183}, {345,21281}, {380,18163}, {440,16608}, {469,20305}, {518,3198}, {916,2947}, {970,7066}, {1001,19727}, {1423,3782}, {1435,7013}, {1465,22134}, {1708,1766}, {1726,16548}, {1745,5752}, {1763,3684}, {1790,2302}, {1802,11349}, {1812,1958}, {1836,6210}, {1869,6734}, {1943,5088}, {2245,3772}, {2264,4641}, {2269,5712}, {2550,6817}, {2997,19645}, {3101,3164}, {3185,20718}, {3197,15509}, {3306,7573}, {3509,7075}, {3687,8896}, {3752,21866}, {3781,16056}, {3868,18673}, {3955,11428}, {4192,11435}, {4897,18197}, {5219,21363}, {5312,8274}, {5745,15830}, {6197,7554}, {7193,10536}, {7330,7534}, {7513,11471}, {9536,18661}, {10436,17185}, {11679,16574}, {16520,21001}, {16609,20606}, {17135,17784}, {17221,23958}, {18134,22370}, {18161,18607}, {18615,20875}
X(24311) lies on these lines: {6,1281}, {45,1213}, {256,4363}, {381,2796}, {536,8245}, {740,3913}, {846,17259}, {4361,4471}, {4425,17327}
X(24312) lies on these lines: {8,1038}, {57,9363}, {388,4000}, {5226,5262}
X(24313) lies on the cubics K1078 and K1082, and on these lines: {2,24153}, {8,9}, {4373,24154}
X(24313) = reflection of X(4373) in X(24154)
X(24313) = anticomplement X(24155)
X(24313) = isotomic conjugate of the anticomplement X(24157)
X(24313) = X(24153)-anticomplementary conjugate of X(3436)
X(24313) = X(24157)-cross conjugate of X(2)
X(24313) = barycentric quotient X(24157)/X(24155)
X(24313) = {X(8),X(9)}-harmonic conjugate of X(24314)
X(24314) lies on the cubics K1078 and K1082, and on these lines: {2,24152}, {8,9}, {4373,24155}
X(24314) = anticomplement X(24154)
X(24314) = reflection of X(4373) in X(24155)
X(24314) = isotomic of the anticomplement X(24156)
X(24314) = X(24152)-anticomplementary conjugate of X(3436)
X(24314) = X(24156)-cross conjugate of X(2)
X(24314) = barycentric quotient X(24156)/X(24154)
X(24314) = {X(8),X(9)}-harmonic conjugate of X(24313)
X(24315) lies on these lines: {2, 1762}, {3, 8680}, {4, 24683}, {5, 24682}, {6, 16609}, {9, 15669}, {10, 9028}, {48, 1441}, {55, 24424}, {69, 21076}, {75, 18162}, {85, 7119}, {86, 423}, {219, 21231}, {284, 17861}, {379, 2173}, {527, 6684}, {534, 946}, {742, 8177}, {1215, 1376}, {1281, 24351}, {1788, 4644}, {1826, 18650}, {2178, 4032}, {2278, 16732}, {2294, 5736}, {2304, 17866}, {2391, 4758}, {2640, 17889}, {3812, 4670}, {3821, 24337}, {3826, 4472}, {3923, 4381}, {4364, 6690}, {4414, 4419}, {4443, 24345}, {5025, 24726}, {5133, 24686}, {5169, 24687}, {7289, 10436}, {8299, 24329}, {17134, 22054}, {18690, 20883}, {21011, 21270}, {22769, 24325}, {25342, 25369}
X(24315) = reflection of X(24316) in X(24317)
X(24315) = complement of X(24316)
X(24315) = anticomplement of X(24317)
X(24316) lies on these lines: {1, 9028}, {2, 1762}, {3, 24683}, {4, 8680}, {7, 2260}, {9, 18589}, {19, 307}, {38, 497}, {40, 534}, {63, 1848}, {69, 1959}, {71, 4329}, {226, 8557}, {281, 20305}, {524, 12635}, {527, 946}, {545, 11235}, {631, 24684}, {960, 4643}, {1001, 4364}, {1370, 24686}, {1468, 1935}, {1839, 18655}, {1944, 17181}, {2173, 24580}, {2183, 5813}, {2294, 5738}, {2886, 4363}, {2893, 25252}, {4357, 7289}, {4381, 24348}, {4443, 24248}, {4454, 24400}, {5327, 24695}, {7791, 24726}, {12589, 17447}, {16063, 24687}, {17043, 20818}, {17221, 20074}, {17272, 18725}, {17321, 18162}, {21061, 22006}, {24318, 24334}, {24319, 24333}, {24336, 24694}
X(24316) = reflection of X(24315) in X(24317)
X(24316) = anticomplement of X(24315)
X(24317) lies on these lines: {2, 1762}, {3, 24682}, {5, 8680}, {11, 24424}, {140, 24684}, {226, 19720}, {534, 6684}, {631, 24683}, {1125, 9028}, {2173, 24581}, {2294, 5740}, {3816, 4364}, {3821, 25346}, {4363, 24318}, {4419, 10589}, {4443, 5988}, {5224, 18161}, {5845, 25358}, {7485, 24686}, {7496, 24687}, {7824, 24726}, {8167, 24328}, {8609, 16603}, {17052, 25078}, {17322, 18162}, {21012, 21271}, {24295, 25372}, {24327, 24348}
X(24317) = midpoint of X(24315) and X(24316)
X(24317) = complement of X(24315)
X(24317) = anticomplement of X(25341)
X(24317) = {X(24318),X(24319)}-harmonic conjugate of X(4363)
X(24318) lies on these lines: {1, 19887}, {2, 7}, {10, 514}, {39, 24211}, {100, 24712}, {116, 24036}, {214, 544}, {291, 5988}, {325, 4071}, {524, 4434}, {545, 25367}, {1083, 1125}, {1376, 24694}, {1565, 21232}, {1647, 24403}, {2246, 24582}, {2276, 24241}, {2329, 4564}, {2786, 13277}, {2886, 25355}, {3035, 5845}, {3257, 17256}, {3501, 17181}, {3570, 4416}, {4292, 16377}, {4363, 24317}, {4364, 25377}, {4419, 24398}, {4425, 25349}, {4438, 25343}, {4465, 11814}, {4472, 25359}, {4865, 9766}, {6647, 7181}, {7208, 24222}, {17046, 25066}, {21013, 21272}, {21029, 25244}, {21616, 24455}, {22279, 22325}, {24169, 25345}, {24316, 24334}, {24323, 25341}, {24348, 25382}
X(24318) = {X(4363),X(24317)}-harmonic conjugate of X(24319)
X(24319) lies on these lines: {2, 24334}, {11, 25371}, {527, 6763}, {545, 25366}, {946, 24705}, {958, 24702}, {2486, 2886}, {3035, 25358}, {3944, 4419}, {4363, 24317}, {4472, 25367}, {4708, 5836}, {4999, 24700}, {5988, 24463}, {6856, 7961}, {21014, 21273}, {24316, 24333}, {24324, 25341}, {24328, 24694}, {24341, 24456}
X(24319) = {X(4363),X(24317)}-harmonic conjugate of X(24318)
X(24320) lies on these lines: {1, 7083}, {2, 1473}, {3, 9}, {6, 7193}, {7, 4223}, {10, 24309}, {21, 17257}, {22, 3219}, {25, 63}, {28, 8822}, {45, 4265}, {55, 846}, {56, 87}, {57, 5020}, {72, 13730}, {75, 242}, {101, 991}, {105, 4310}, {154, 3955}, {169, 990}, {170, 5584}, {191, 8185}, {197, 4640}, {218, 4260}, {219, 511}, {220, 1350}, {222, 9306}, {228, 20835}, {240, 6059}, {255, 2212}, {329, 4224}, {405, 4357}, {474, 17353}, {518, 1486}, {611, 2175}, {613, 3271}, {894, 19310}, {920, 10037}, {956, 3883}, {958, 1503}, {960, 20876}, {999, 7290}, {1001, 4364}, {1158, 9910}, {1259, 3145}, {1351, 2323}, {1370, 21015}, {1460, 1707}, {1466, 23085}, {1597, 18540}, {1598, 5709}, {1633, 2550}, {1716, 5247}, {1736, 16551}, {1957, 7337}, {1995, 3218}, {2003, 3167}, {2178, 3286}, {2195, 2810}, {2267, 22390}, {2911, 4259}, {3173, 20122}, {3211, 5751}, {3242, 16686}, {3295, 7174}, {3303, 7301}, {3305, 7293}, {3306, 11284}, {3428, 6210}, {3452, 16434}, {3487, 17560}, {3662, 16048}, {3717, 5687}, {3718, 7283}, {3784, 17811}, {3786, 16876}, {3923, 24336}, {3927, 20831}, {3929, 5285}, {3937, 5651}, {4186, 6734}, {4192, 15509}, {4220, 5273}, {4228, 5905}, {4254, 16517}, {4363, 24332}, {5017, 16514}, {5096, 16885}, {5220, 12329}, {5250, 8192}, {5257, 16849}, {5329, 7262}, {5745, 19544}, {5750, 16852}, {6090, 22128}, {6211, 10310}, {6511, 23181}, {6642, 24467}, {6646, 17522}, {6677, 20266}, {7009, 18750}, {7082, 10832}, {7308, 16419}, {8817, 15344}, {9798, 12514}, {10436, 19309}, {10477, 23151}, {11108, 17306}, {11344, 22345}, {11350, 22060}, {14520, 22153}, {15494, 16678}, {15650, 20833}, {17260, 19314}, {18228, 19649}, {18589, 24701}, {24325, 24333}
X(24320) = anticomplement of X(25365)
X(24321) lies on these lines: {2, 1762}, {22, 8680}, {427, 24686}, {1370, 24683}, {2173, 24584}, {5133, 24682}, {7485, 24684}, {17492, 21016}
X(24321) = anticomplement of X(25343)
X(24322) lies on these lines: {2, 1762}, {23, 8680}, {858, 24687}, {1281, 2786}, {2173, 24585}, {5169, 24682}, {7496, 24684}, {16063, 24683}, {21017, 21274}
X(24322) = anticomplement of X(25344)
X(24323) lies on these lines: {2, 25366}, {545, 4995}, {1215, 1376}, {3454, 4643}, {4364, 9318}, {4400, 9022}, {4670, 5439}, {21018, 21276}, {24318, 25341}, {24330, 24335}, {24345, 24463}
X(24324) lies on these lines: {2, 2161}, {291, 24345}, {536, 5440}, {545, 6174}, {812, 14288}, {1215, 1376}, {3262, 7113}, {3739, 24618}, {3753, 4670}, {3925, 4472}, {4364, 5432}, {8299, 24346}, {21019, 21277}, {24319, 25341}, {24350, 25382}
X(24325) lies on these lines: {1, 75}, {2, 38}, {6, 4974}, {7, 4655}, {8, 4699}, {10, 141}, {11, 25385}, {31, 4697}, {37, 39}, {42, 4359}, {43, 19804}, {55, 3980}, {56, 4032}, {81, 3791}, {85, 4334}, {105, 24339}, {145, 4772}, {171, 3757}, {192, 3616}, {226, 3846}, {229, 409}, {238, 894}, {239, 4649}, {240, 17913}, {321, 3720}, {354, 3741}, {511, 17049}, {519, 3696}, {536, 551}, {594, 4966}, {689, 741}, {692, 24253}, {730, 21443}, {742, 1386}, {750, 4434}, {752, 3883}, {758, 18805}, {872, 3216}, {899, 24589}, {940, 4362}, {946, 21629}, {960, 20258}, {964, 4812}, {976, 16454}, {985, 16998}, {991, 24293}, {993, 8680}, {1001, 3923}, {1086, 3821}, {1089, 18137}, {1220, 7194}, {1278, 3622}, {1441, 1458}, {1469, 16609}, {1574, 21897}, {1575, 3774}, {1582, 5009}, {1621, 4418}, {1698, 4751}, {1738, 4085}, {1757, 17277}, {1909, 1921}, {1962, 17147}, {1965, 8033}, {1999, 4038}, {2235, 23660}, {2550, 24693}, {2805, 21630}, {2886, 20256}, {2887, 5249}, {3210, 17592}, {3244, 4709}, {3264, 4710}, {3286, 16684}, {3416, 4675}, {3485, 7201}, {3624, 4687}, {3636, 4686}, {3664, 5847}, {3666, 24165}, {3679, 17297}, {3685, 16484}, {3706, 4883}, {3716, 4809}, {3740, 4090}, {3742, 3840}, {3751, 4384}, {3752, 6685}, {3758, 16468}, {3761, 21615}, {3773, 3912}, {3782, 4425}, {3790, 17244}, {3797, 16826}, {3831, 5439}, {3878, 20718}, {3879, 17772}, {3891, 5311}, {3932, 17245}, {3953, 4022}, {3979, 3996}, {4011, 4423}, {4078, 4439}, {4357, 24231}, {4364, 25354}, {4365, 4980}, {4368, 24330}, {4385, 20923}, {4407, 24603}, {4416, 17771}, {4429, 25351}, {4431, 4527}, {4440, 9791}, {4458, 24353}, {4472, 9055}, {4523, 17050}, {4535, 17233}, {4645, 17153}, {4660, 5880}, {4663, 17348}, {4664, 25055}, {4676, 15485}, {4681, 15808}, {4683, 17483}, {4684, 4967}, {4698, 19862}, {4703, 5905}, {4716, 17117}, {4722, 19742}, {4755, 19883}, {4968, 20891}, {4991, 16666}, {5211, 17722}, {5220, 17259}, {5223, 16832}, {5247, 16817}, {5248, 24850}, {5695, 17118}, {5698, 7222}, {5749, 16020}, {5836, 24182}, {5852, 17332}, {5886, 20430}, {5949, 20546}, {6211, 21554}, {6532, 20108}, {6541, 17243}, {6646, 24697}, {6686, 16602}, {6703, 17061}, {7081, 17122}, {7321, 24723}, {7336, 19890}, {8025, 17150}, {8299, 24326}, {9284, 16592}, {9505, 24505}, {9507, 25368}, {9941, 24190}, {10180, 17155}, {10459, 20892}, {10479, 18398}, {10582, 20173}, {10980, 18229}, {11997, 12053}, {14839, 24254}, {15254, 17351}, {15523, 18139}, {16477, 17120}, {17017, 19684}, {17031, 24512}, {17135, 21020}, {17365, 17770}, {17599, 19701}, {18201, 24627}, {19623, 24345}, {19856, 25120}, {21197, 24675}, {21926, 24390}, {22769, 24315}, {24320, 24333}, {24377, 24440}, {25345, 25373}
X(24325) = complement of X(984)
X(24325) = anticomplement of X(3842)
X(24325) = isotomic conjugate of isogonal conjugate of X(20985)
X(24325) = polar conjugate of isogonal conjugate of X(22099)
X(24326) lies on these lines: {1, 16720}, {2, 37}, {38, 9055}, {42, 742}, {55, 4363}, {76, 21021}, {141, 3930}, {244, 25350}, {518, 24690}, {524, 4030}, {538, 4692}, {712, 4424}, {756, 3123}, {894, 1914}, {902, 4797}, {984, 4493}, {1018, 24254}, {1045, 3961}, {1215, 24259}, {1403, 7211}, {1500, 1930}, {1573, 4986}, {1621, 24358}, {2242, 7267}, {3571, 24448}, {3663, 21101}, {3681, 4643}, {3703, 4665}, {3727, 17760}, {3744, 4670}, {3758, 5332}, {3873, 24691}, {3934, 7264}, {3938, 9022}, {3970, 21240}, {4396, 7081}, {4644, 20101}, {4659, 17594}, {4660, 4799}, {4894, 7759}, {4972, 25345}, {6664, 21035}, {7187, 25303}, {7235, 12837}, {7272, 7761}, {8299, 24325}, {17318, 17599}, {18059, 21217}, {20255, 21808}, {20331, 24631}, {20691, 20911}, {24309, 24333}, {24431, 24454}
X(24327) lies on these lines: {1, 4494}, {2, 3122}, {6, 4039}, {10, 674}, {37, 700}, {39, 714}, {76, 24696}, {313, 2309}, {536, 12263}, {696, 17760}, {730, 4377}, {1213, 25120}, {1215, 17369}, {1964, 3963}, {2234, 20913}, {2278, 4112}, {3264, 21352}, {3821, 25374}, {3923, 4381}, {3934, 6007}, {4363, 24405}, {4364, 17793}, {4422, 6690}, {4436, 24259}, {4472, 9055}, {5988, 25346}, {8299, 24425}, {9025, 25102}, {16732, 24255}, {17337, 25106}, {17398, 25124}, {18048, 18209}, {21022, 21278}, {21238, 21746}, {24317, 24348}, {24340, 24345}
X(24327) = complement of X(4443)
X(24327) = anticomplement of X(25347)
X(24328) lies on these lines: {3, 527}, {6, 1423}, {7, 198}, {37, 7289}, {45, 16560}, {48, 6180}, {55, 1633}, {56, 4644}, {100, 4454}, {222, 1020}, {226, 15509}, {513, 14205}, {524, 9840}, {536, 3913}, {545, 4421}, {954, 3220}, {958, 4643}, {999, 4667}, {1001, 4364}, {1215, 1376}, {1282, 24341}, {1473, 21319}, {1696, 4648}, {1732, 4641}, {2097, 7146}, {2178, 17365}, {2182, 8545}, {2183, 5228}, {2262, 7190}, {2270, 4328}, {2347, 7225}, {2391, 6767}, {3247, 18725}, {3553, 24471}, {3663, 4254}, {3929, 21483}, {4413, 4470}, {4428, 20834}, {4471, 24405}, {4654, 11347}, {4659, 5687}, {4715, 11194}, {4747, 5253}, {5085, 23693}, {5745, 23089}, {6600, 24309}, {7175, 37519}, {8167, 24317}, {8715, 17132}, {11343, 17274}, {11495, 15624}, {16367, 17333}, {16672, 18735}, {16777, 18161}, {23085, 24700}, {24319, 24694}, {24352, 24424}
X(24329) lies on these lines: {2, 25348}, {9, 20544}, {744, 4372}, {2175, 20236}, {2330, 20927}, {3923, 15507}, {4363, 24346}, {4376, 4381}, {4858, 24264}, {5698, 7379}, {8299, 24315}, {17046, 24689}, {19133, 20171}, {21023, 21280}, {24428, 24456}
X(24329) = anticomplement of X(25348)
X(24330) lies on these lines: {1, 538}, {2, 45}, {6, 4441}, {10, 4721}, {38, 25368}, {42, 536}, {43, 4659}, {44, 24592}, {75, 2238}, {76, 2295}, {141, 20347}, {171, 4396}, {213, 20888}, {291, 24448}, {321, 742}, {350, 894}, {524, 17135}, {527, 3741}, {672, 17351}, {756, 25384}, {870, 17475}, {1111, 24254}, {1215, 24259}, {1836, 4643}, {1909, 10027}, {2276, 3729}, {3120, 25345}, {3720, 4670}, {3734, 16788}, {3758, 4479}, {3760, 17750}, {3780, 17143}, {3923, 4376}, {3934, 16549}, {4368, 24325}, {4400, 5255}, {4418, 9318}, {4644, 10453}, {4651, 4665}, {4664, 17032}, {4672, 17031}, {4686, 21904}, {4971, 20011}, {5263, 17794}, {5264, 7751}, {6685, 17132}, {9055, 17165}, {15985, 17220}, {17002, 21793}, {17018, 17318}, {17137, 21024}, {17140, 24403}, {17355, 20335}, {17790, 18152}, {20292, 24699}, {24323, 24335}, {25353, 25385}
X(24331) lies on these lines: {1, 2}, {36, 23407}, {45, 537}, {75, 4693}, {106, 789}, {142, 4660}, {238, 3758}, {244, 24464}, {274, 4653}, {392, 20358}, {528, 24693}, {740, 17119}, {752, 4675}, {894, 15485}, {968, 24165}, {984, 24841}, {993, 24333}, {1001, 3923}, {1150, 17450}, {1215, 4423}, {1621, 3980}, {2177, 24589}, {3242, 3842}, {3246, 4670}, {3750, 19804}, {3976, 11110}, {3986, 16517}, {4011, 5284}, {4085, 17278}, {4090, 7308}, {4378, 4448}, {4655, 7238}, {4688, 4702}, {5257, 16973}, {6173, 24692}, {6706, 25130}, {7176, 13462}, {10180, 17599}, {11365, 20834}, {14621, 16801}, {15569, 25384}, {21027, 21283}, {24349, 24821}
X(24332) lies on these lines: {75, 7193}, {92, 3955}, {182, 4858}, {184, 14213}, {242, 894}, {984, 2607}, {1001, 24346}, {1733, 2175}, {1944, 3781}, {1985, 21368}, {3060, 24149}, {3410, 24146}, {3923, 15507}, {4363, 24320}, {4459, 7295}, {6358, 9306}, {11422, 24148}, {11442, 21028}
X(24333) lies on these lines: {2, 7}, {41, 20880}, {69, 4071}, {75, 3684}, {85, 2329}, {87, 2191}, {169, 17050}, {524, 4865}, {742, 4362}, {846, 4419}, {870, 1438}, {993, 24331}, {1001, 24352}, {1111, 16788}, {1215, 1376}, {1281, 9451}, {1282, 2550}, {1478, 9903}, {1486, 8424}, {2082, 20257}, {2246, 24596}, {2886, 5845}, {2887, 4643}, {3035, 25355}, {3158, 4659}, {3496, 17753}, {3729, 21101}, {3742, 4670}, {3767, 24211}, {3923, 4376}, {4011, 4713}, {4364, 25363}, {4434, 8667}, {4444, 8774}, {4454, 5281}, {4644, 24477}, {4754, 9840}, {6180, 22163}, {6647, 7223}, {7032, 24403}, {7201, 11683}, {7264, 16783}, {11680, 24712}, {17739, 21281}, {21029, 21285}, {24309, 24326}, {24316, 24319}, {24320, 24325}
X(24333) = complement of X(25353)
X(24334) lies on these lines: {2, 24319}, {19, 20258}, {40, 24705}, {46, 527}, {55, 25371}, {75, 1429}, {322, 7175}, {524, 8256}, {604, 20895}, {958, 24700}, {1215, 1376}, {1281, 24451}, {1329, 24702}, {1766, 21246}, {1944, 3501}, {2268, 24993}, {2550, 4470}, {2886, 4472}, {3035, 4364}, {3507, 16571}, {3923, 15507}, {4381, 25382}, {4419, 17596}, {4454, 9318}, {4659, 5438}, {4670, 5836}, {4858, 24266}, {5783, 21233}, {7961, 17567}, {17452, 24540}, {21030, 21286}, {24316, 24318}, {24343, 24358}
X(24335) lies on these lines: {2, 25370}, {55, 4363}, {75, 5301}, {141, 1761}, {191, 4643}, {524, 1046}, {536, 5266}, {846, 4364}, {1333, 18697}, {1781, 17279}, {2108, 24340}, {3454, 24704}, {3694, 17351}, {3923, 4381}, {3931, 4670}, {3980, 25363}, {4026, 4472}, {4419, 4427}, {5255, 9022}, {5695, 19761}, {8680, 24850}, {16732, 24271}, {17587, 20896}, {17762, 19623}, {20654, 21287}, {24323, 24330}, {24346, 24425}
X(24336) lies on these lines: {2, 25371}, {3, 24700}, {4, 24702}, {9, 6996}, {11, 25369}, {69, 4032}, {72, 527}, {75, 1423}, {536, 5836}, {573, 21233}, {742, 17792}, {894, 20769}, {960, 12545}, {984, 2550}, {1215, 1376}, {1706, 4385}, {2269, 24547}, {2486, 2886}, {3035, 4472}, {3718, 3729}, {3923, 24320}, {4454, 6555}, {4643, 5794}, {4665, 8256}, {5082, 7961}, {8299, 25375}, {20245, 21033}, {24316, 24694}
X(24337) lies on these lines: {2, 25372}, {206, 24706}, {3821, 24315}, {3923, 25343}, {21034, 21288}, {24424, 24444}
X(24338) lies on these lines: {1, 513}, {2, 19945}, {6, 4941}, {37, 24289}, {43, 23343}, {45, 1575}, {75, 24517}, {190, 3123}, {192, 23354}, {244, 903}, {256, 17246}, {291, 545}, {350, 4389}, {536, 984}, {751, 18822}, {894, 24503}, {900, 24427}, {986, 24433}, {1015, 24722}, {1054, 16561}, {1086, 4947}, {1655, 9295}, {1769, 24409}, {2254, 24408}, {3122, 4440}, {3242, 9025}, {3248, 4499}, {3571, 9318}, {3821, 17790}, {4363, 24456}, {4364, 24450}, {4419, 4443}, {4562, 4664}, {4713, 24458}, {6646, 24437}, {7321, 22172}, {9296, 18140}, {16495, 24482}, {17273, 22167}, {17290, 20530}, {17334, 24575}, {24346, 24428}, {24396, 24398}, {24403, 24413}, {24404, 24414}
X(24339) lies on these lines: {2, 25373}, {105, 24325}, {141, 17744}, {251, 20898}, {984, 4376}, {1369, 21037}, {4363, 7295}, {4364, 24340}
X(24339) = anticomplement of X(25373)
X(24340) lies on these lines: {2, 4381}, {38, 190}, {58, 4672}, {83, 744}, {141, 17799}, {1078, 4836}, {1281, 25347}, {2108, 24335}, {2161, 2886}, {2244, 24614}, {3096, 4837}, {3923, 4283}, {4364, 24339}, {6292, 24707}, {21038, 21289}, {24295, 25364}, {24327, 24345}
X(24340) = anticomplement of X(25374)
X(24341) lies on these lines: {1, 5696}, {2, 2310}, {9, 1721}, {37, 4335}, {38, 4346}, {45, 846}, {75, 4073}, {144, 21039}, {170, 1212}, {171, 1711}, {198, 8245}, {377, 4331}, {774, 4208}, {982, 1086}, {984, 2550}, {1282, 24328}, {1458, 10861}, {1734, 17057}, {1742, 17668}, {2293, 24554}, {2345, 12725}, {3000, 24635}, {3779, 7201}, {3925, 24430}, {4051, 9025}, {4363, 24411}, {4454, 4712}, {4518, 24451}, {5296, 10868}, {17868, 25304}, {24319, 24456}
X(24341) = anticomplement of X(25375)
X(24342) lies on these lines: {1, 75}, {2, 846}, {3, 10892}, {8, 20090}, {9, 46}, {10, 894}, {40, 15973}, {81, 21020}, {165, 7413}, {190, 3842}, {238, 3739}, {239, 4991}, {320, 3775}, {321, 1961}, {333, 4697}, {516, 2938}, {524, 3416}, {726, 16830}, {756, 4756}, {758, 3786}, {870, 10009}, {873, 18059}, {896, 5235}, {966, 24695}, {984, 4363}, {986, 2049}, {996, 7312}, {1001, 4436}, {1051, 19717}, {1100, 4716}, {1125, 3685}, {1215, 3699}, {1386, 4688}, {1441, 5018}, {1699, 2941}, {1738, 5750}, {1762, 3925}, {1770, 19857}, {1909, 4710}, {1962, 5333}, {2292, 14005}, {2550, 4470}, {2640, 4429}, {2663, 3293}, {2795, 24275}, {2887, 19808}, {2895, 8013}, {2959, 4660}, {3122, 24923}, {3336, 16574}, {3624, 4657}, {3634, 17260}, {3696, 4649}, {3706, 4038}, {3836, 17289}, {3932, 7227}, {3993, 16826}, {4026, 4472}, {4205, 24851}, {4357, 19856}, {4365, 17019}, {4384, 14621}, {4527, 17315}, {4650, 5737}, {4655, 5224}, {4672, 17277}, {4676, 4751}, {4693, 15569}, {4699, 16825}, {4967, 5847}, {5564, 17772}, {5695, 15668}, {5902, 10477}, {5999, 9746}, {6998, 8245}, {7262, 19732}, {8025, 17163}, {8424, 19329}, {9780, 17350}, {11110, 24850}, {11263, 24931}, {11533, 17164}, {12047, 21246}, {16477, 17348}, {17258, 17767}, {17301, 25055}, {17592, 19701}, {17778, 21085}, {17790, 25382}, {18140, 24731}, {19868, 24231}, {19879, 20258}, {20880, 23689}, {23690, 24993}, {24174, 24311}, {24291, 24293}
X(24342) = complement of X(9791)
X(24342) = anticomplement of X(25354)
X(24343) lies on these lines: {2, 25376}, {9, 25107}, {894, 4110}, {2162, 6382}, {3242, 4363}, {3739, 24758}, {3923, 4713}, {10009, 24502}, {24334, 24358}
X(24344) lies on these lines: {2, 4432}, {89, 519}, {100, 4363}, {239, 17126}, {750, 4693}, {2163, 4793}, {3240, 3758}, {3679, 24616}, {3923, 9458}, {3952, 24451}, {4392, 24464}, {4427, 9330}, {5561, 21251}, {21042, 21291}
X(24345) lies on these lines: {1, 523}, {2, 24348}, {6, 24365}, {99, 24714}, {115, 24711}, {230, 8557}, {291, 24324}, {524, 3416}, {545, 13174}, {645, 21254}, {894, 9421}, {1046, 12607}, {1086, 2640}, {1109, 14616}, {1281, 24350}, {2161, 17719}, {2948, 24222}, {2959, 17768}, {3035, 16575}, {3110, 8672}, {3571, 9318}, {4381, 24351}, {4427, 17487}, {4436, 24820}, {4443, 24315}, {4590, 17103}, {5539, 24722}, {10026, 17369}, {13610, 17365}, {14985, 17724}, {19623, 24325}, {21043, 21221}, {24323, 24463}, {24327, 24340}, {24405, 24416}
X(24346) lies on these lines: {1, 523}, {2, 25379}, {6, 4124}, {11, 2161}, {75, 3573}, {92, 23353}, {110, 14616}, {116, 24713}, {242, 8679}, {281, 12586}, {674, 1944}, {692, 4858}, {894, 24482}, {900, 24411}, {993, 8680}, {1001, 24332}, {1146, 5848}, {3826, 5805}, {3877, 5263}, {3923, 25382}, {4363, 24329}, {4448, 9318}, {4459, 16686}, {4579, 18151}, {4679, 17369}, {8299, 24324}, {21045, 21293}, {24335, 24425}, {24338, 24428}, {24347, 24442}, {24401, 24455}, {24406, 24416}
X(24347) lies on these lines: {1, 23879}, {116, 16562}, {163, 17886}, {525, 3109}, {544, 2948}, {894, 24504}, {2247, 24619}, {2607, 9318}, {5845, 14985}, {21046, 21294}, {21381, 24712}, {24248, 25372}, {24346, 24442}
X(24348) lies on these lines: {2, 24345}, {10, 523}, {99, 24711}, {524, 551}, {545, 11599}, {620, 24714}, {903, 3120}, {2615, 16598}, {3923, 25362}, {4381, 24316}, {4443, 25346}, {4590, 6626}, {4934, 21254}, {17793, 25367}, {21047, 21295}, {24317, 24327}, {24318, 25382}, {25347, 25364}
X(24349) lies on these lines: {1, 87}, {2, 38}, {4, 4812}, {7, 8}, {9, 16823}, {10, 3662}, {37, 2275}, {42, 3210}, {43, 17490}, {57, 7081}, {63, 3757}, {78, 4327}, {79, 4894}, {81, 3891}, {142, 3717}, {145, 740}, {182, 4579}, {190, 1001}, {210, 19804}, {226, 3705}, {238, 17350}, {239, 3751}, {274, 3786}, {312, 354}, {318, 1876}, {321, 3873}, {341, 3812}, {345, 3475}, {346, 11038}, {386, 596}, {390, 4454}, {404, 18048}, {519, 4740}, {527, 3883}, {536, 3241}, {668, 10009}, {698, 4754}, {742, 4644}, {869, 24621}, {870, 18906}, {871, 20023}, {899, 24620}, {938, 20171}, {942, 4385}, {962, 9801}, {985, 7766}, {1002, 4441}, {1086, 4429}, {1089, 18398}, {1125, 17368}, {1219, 1432}, {1266, 3755}, {1279, 4676}, {1281, 9451}, {1362, 6063}, {1376, 8850}, {1386, 3758}, {1698, 17291}, {1757, 16825}, {1836, 4514}, {1892, 5081}, {2263, 3872}, {2292, 25124}, {2321, 4684}, {2805, 9802}, {2975, 13733}, {3061, 20706}, {3175, 4883}, {3240, 17495}, {3242, 4363}, {3243, 3886}, {3306, 5205}, {3600, 4032}, {3617, 4772}, {3621, 4821}, {3622, 4704}, {3623, 4788}, {3632, 4709}, {3644, 20057}, {3681, 4359}, {3699, 4413}, {3701, 20923}, {3702, 3889}, {3703, 18134}, {3739, 5772}, {3742, 3967}, {3753, 4737}, {3759, 4663}, {3761, 21443}, {3773, 17230}, {3774, 17756}, {3790, 3912}, {3797, 17316}, {3813, 21927}, {3844, 17227}, {3869, 20348}, {3874, 10449}, {3932, 17234}, {3938, 4418}, {3961, 3980}, {3974, 18141}, {3976, 21330}, {3994, 17450}, {4026, 4389}, {4030, 11246}, {4078, 17244}, {4090, 16569}, {4253, 22011}, {4260, 20913}, {4301, 9950}, {4318, 4861}, {4353, 17023}, {4384, 5223}, {4388, 5905}, {4393, 4649}, {4416, 5850}, {4419, 9791}, {4430, 17135}, {4440, 24248}, {4470, 25384}, {4651, 4661}, {4664, 15569}, {4671, 17146}, {4678, 4732}, {4686, 20050}, {4687, 5550}, {4692, 5902}, {4696, 20892}, {4697, 17716}, {4702, 15570}, {4706, 21870}, {4726, 20053}, {4751, 19877}, {4767, 24594}, {4884, 17056}, {4891, 22034}, {4901, 6173}, {4966, 17233}, {5014, 20292}, {5045, 22016}, {5220, 17277}, {5247, 19851}, {5603, 20430}, {5846, 17365}, {5847, 17364}, {5852, 17347}, {5904, 9534}, {6194, 8924}, {6327, 17483}, {6685, 17591}, {7172, 21454}, {7174, 10436}, {9369, 19860}, {9785, 11997}, {10580, 20173}, {11688, 23853}, {14839, 24282}, {15254, 17336}, {15481, 17335}, {16475, 17120}, {16484, 25269}, {16496, 17116}, {16606, 20284}, {16710, 18792}, {17018, 17147}, {17126, 20045}, {17169, 18157}, {17276, 24723}, {17754, 19587}, {18144, 22279}, {18194, 23579}, {18412, 20236}, {19586, 20917}, {20909, 25301}, {21020, 25294}, {21197, 24749}, {24331, 24821}
X(24349) = complement of X(31302)
X(24349) = anticomplement of X(984)
X(24349) = isotomic conjugate of isogonal conjugate of X(21010)
X(24349) = {X(7),X(8)}-harmonic conjugate of X(4645)
X(24349) = polar conjugate of isogonal conjugate of X(22163)
X(24350) lies on these lines: {625, 24716}, {659, 24353}, {922, 20912}, {1281, 24345}, {2795, 24714}, {3923, 4381}, {16732, 24292}, {21048, 21298}, {24324, 25382}
X(24351) lies on these lines: {2, 3122}, {6, 7155}, {8, 674}, {55, 190}, {75, 20358}, {76, 6007}, {192, 700}, {194, 714}, {313, 21299}, {335, 24445}, {523, 7985}, {874, 24282}, {1043, 5327}, {1278, 17142}, {1281, 24315}, {1654, 25291}, {1918, 17350}, {2234, 24621}, {3242, 4363}, {3596, 21746}, {3779, 17787}, {4381, 24345}, {4419, 17794}, {9025, 24524}, {17349, 25277}, {17379, 25295}, {17755, 25375}, {17786, 17792}, {17793, 24456}, {19582, 19765}
X(24351) = anticomplement of X(4443)
X(24352) lies on these lines: {2, 45}, {6, 10025}, {7, 17747}, {44, 24600}, {55, 9318}, {200, 536}, {218, 7264}, {220, 3673}, {497, 5845}, {527, 11019}, {742, 20173}, {982, 24398}, {1001, 24333}, {1376, 24283}, {2391, 4342}, {3242, 14942}, {3693, 17262}, {3870, 17318}, {3967, 4659}, {4421, 24685}, {4643, 4847}, {4644, 10580}, {4670, 10582}, {4971, 20015}, {6604, 21049}, {9436, 17276}, {11235, 24694}, {11238, 24712}, {14548, 17365}, {14828, 16777}, {17132, 20103}, {17251, 25006}, {17597, 24403}, {17790, 18153}, {24328, 24424}
X(24352) = anticomplement of X(25355)
X(24353) lies on these lines: {10, 2774}, {659, 24350}, {810, 17899}, {2786, 3716}, {3700, 8062}, {4458, 24325}, {21050, 21300}
X(24354) lies on these lines: {2, 25356}, {75, 8632}, {190, 14408}, {514, 21112}, {659, 24350}, {786, 3737}, {812, 14288}, {816, 4086}, {894, 3768}, {900, 4375}, {1919, 20906}, {3758, 23650}, {4363, 4491}, {4472, 24721}, {4508, 21606}, {4509, 8060}, {9318, 24416}, {17303, 21261}, {20949, 20981}, {21055, 21304}, {21260, 24698}
X(24355) lies on these lines: {1, 17046}, {2, 21687}, {3, 4361}, {3187, 7146}, {4372, 24367}, {24432, 24435}
X(24356) lies on these lines: {659, 24350}, {1281, 2786}, {8631, 23807}, {21056, 21305}
X(24357) lies on these lines: {1, 742}, {2, 37}, {45, 17755}, {85, 24656}, {335, 4389}, {354, 24691}, {518, 4643}, {537, 24441}, {740, 17318}, {982, 25349}, {984, 4026}, {1001, 3923}, {1215, 4713}, {1279, 4670}, {1486, 8424}, {1621, 4376}, {2667, 3938}, {3616, 16720}, {3873, 24690}, {3932, 4665}, {4078, 25352}, {4377, 21615}, {4419, 9791}, {4429, 25357}, {4660, 24699}, {4709, 17133}, {4797, 8616}, {12618, 20430}, {17063, 25350}, {18135, 21021}, {20331, 24629}
X(24357) = anticomplement of X(25384)
X(24358) lies on these lines: {1, 9055}, {2, 4376}, {6, 17755}, {9, 141}, {21, 16720}, {37, 86}, {63, 24691}, {75, 4366}, {100, 4760}, {171, 4797}, {238, 742}, {304, 4426}, {344, 4473}, {385, 20947}, {513, 9470}, {524, 1757}, {536, 1279}, {545, 24416}, {673, 3739}, {812, 14288}, {846, 25349}, {1001, 3923}, {1086, 4657}, {1447, 20530}, {1486, 24822}, {1621, 24326}, {1914, 3263}, {1966, 17790}, {2082, 24735}, {2243, 24602}, {2345, 20533}, {3219, 24690}, {3496, 20255}, {3512, 15985}, {3701, 4400}, {3758, 20132}, {3836, 24699}, {4011, 4713}, {4026, 4472}, {4078, 4667}, {4154, 21254}, {4358, 4396}, {4364, 24339}, {4370, 4795}, {4437, 4851}, {4440, 7222}, {4465, 9318}, {4553, 7077}, {4567, 4590}, {4708, 17260}, {4715, 17297}, {4872, 20541}, {5263, 25384}, {5291, 14210}, {8424, 24700}, {11814, 25342}, {17596, 25350}, {17739, 21025}, {18589, 20258}, {20483, 20553}, {24003, 24685}, {24334, 24343}
X(24358) = anticomplement of X(25357)
See X(24310).
X(24359) lies on these lines: {1, 900}, {1086, 17718}, {4448, 9318}
X(24360) lies on these lines: {2, 23898}, {24362, 24368}
X(24361) lies on these lines: {2, 23898}
X(24362) lies on these lines: {2, 23901}, {4039, 24435}, {4362, 4412}, {24360, 24368}, {24365, 24369}, {24372, 24378}, {24376, 24377}
X(24363) lies on these lines: {2, 23902}, {7, 8}, {4039, 17314}, {4371, 24435}, {6857, 24640}, {20539, 21271}, {21085, 21675}, {24364, 24372}
X(24364) lies on these lines: {1, 142}, {2, 23897}, {194, 966}, {390, 16527}, {3212, 9278}, {3618, 16913}, {4419, 21879}, {7735, 9509}, {7738, 17691}, {9607, 17277}, {24363, 24372}
X(24365) lies on these lines: {2, 23902}, {6, 24345}, {8, 3826}, {474, 24640}, {1001, 7235}, {1376, 24435}, {4039, 4361}, {4733, 11281}, {8424, 16609}, {24362, 24369}, {24372, 24376}, {24377, 24378}
X(24366) lies on these lines: {2, 23897}, {6, 17680}, {8, 141}, {1213, 1655}, {3589, 17686}, {4085, 24656}, {4657, 24439}, {6707, 14005}, {7864, 17352}, {16926, 17398}, {17045, 24435}, {24372, 24373}
X(24367) lies on these lines: {2, 23909}, {55, 4361}, {4372, 24355}
X(24368) lies on these lines: {2, 23910}, {24360, 24362}
X(24369) lies on these lines: {2, 23913}, {1376, 4361}, {2640, 17277}, {24174, 24653}, {24362, 24365}, {24370, 24384}, {24374, 24376}
X(24370) lies on these lines: {2, 23915}, {24360, 24362}, {24369, 24384}
X(24371) lies on these lines: {2, 23902}, {3, 24640}, {8, 3475}, {55, 7235}, {56, 24432}, {999, 24641}, {17718, 21085}
X(24372) lies on these lines: {2, 23913}, {6, 2640}, {10, 75}, {982, 4361}, {1045, 3125}, {1376, 1961}, {1740, 3959}, {2277, 17891}, {2643, 24530}, {17063, 17380}, {17872, 24478}, {24174, 24923}, {24362, 24378}, {24363, 24364}, {24365, 24376}, {24366, 24373}, {24443, 24575}
X(24373) lies on these lines: {2, 23915}, {8, 4446}, {24366, 24372}
X(24374) lies on these lines: {1, 75}, {2, 23901}, {261, 11711}, {7170, 24575}, {24369, 24376}, {24383, 24384}
X(24375) lies on these lines: {2, 23897}, {3622, 4000}
X(24376) lies on these lines: {2, 23936}, {1054, 2606}, {24362, 24377}, {24365, 24372}, {24369, 24374}
X(24377) lies on these lines: {2, 23913}, {1376, 4360}, {2640, 17349}, {2669, 3212}, {24325, 24440}, {24362, 24376}, {24365, 24378}
X(24378) lies on these lines: {2, 23897}, {75, 24439}, {86, 4000}, {99, 3008}, {239, 9278}, {333, 16722}, {645, 1266}, {662, 1429}, {673, 2669}, {1434, 16711}, {4360, 24435}, {4384, 6626}, {4395, 19623}, {4417, 17680}, {4560, 7178}, {5222, 17103}, {5550, 14005}, {7058, 24177}, {14829, 24610}, {24362, 24372}, {24365, 24377}
X(24379) lies on these lines: {2, 23949}, {17061, 17069}, {24380, 24381}
X(24380) lies on these lines: {2, 23950}, {24379, 24381}
X(24381) lies on these lines: {2, 23951}, {523, 661}, {693, 18059}, {2517, 24674}, {2533, 3907}, {17494, 24104}, {21085, 21196}, {23948, 23954}, {24379, 24380}
X(24382) lies on these lines: {2, 23949}, {1734, 21212}, {2254, 3004}, {4041, 4369}
X(24383) lies on these lines: {2, 23909}, {740, 1001}, {24362, 24365}, {24374, 24384}
X(24384) lies on these lines: {2, 23897}, {10, 620}, {141, 404}, {1211, 19308}, {2134, 6626}, {2482, 6537}, {4000, 5550}, {5224, 7891}, {5743, 21511}, {7267, 20653}, {10026, 17103}, {17292, 24636}, {17694, 21024}, {24369, 24370}, {24374, 24383}, {24435, 24953}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28392.
X(24385) lies on these lines: {2,16762}, {3,16766}, {5,195}, {30,1141}, {137,24147}, {140,15345}, {252,10126}, {523,10096}, {546,23337}, {1154,12026}, {3850,15307}, {6592,10615}, {7604,13469}, {11016,16239}, {13856,23280}, {14051,20414}
X(24385) = midpoint of X(i) and X(j) for these {i,j}: {5, 19553}, {137, 24147}, {1157, 1263}
X(24385) = reflection of X(6592) in X(10614)
X(24385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195, 3459, 5), (16762, 16764, 2), (16766, 16768, 3)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24386) lies on these lines: {1,6856}, {2,3158}, {8,18220}, {9,5274}, {10,496}, {11,210}, {142,2886}, {200,10589}, {226,3873}, {312,17774}, {321,4939}, {497,4512}, {516,11235}, {518,3817}, {519,5055}, {522,21204}, {527,1699}, {528,10164}, {758,946}, {908,4661}, {950,10527}, {1058,5705}, {1125,12437}, {1210,3753}, {1329,4711}, {1420,5175}, {1706,5704}, {2136,9780}, {2321,3705}, {2550,6692}, {2887,17059}, {2900,4666}, {3090,6765}, {3091,6762}, {3189,3624}, {3243,5226}, {3434,3911}, {3616,12625}, {3617,3680}, {3626,10912}, {3634,3913}, {3687,4923}, {3717,4903}, {3755,24239}, {3756,21949}, {3794,17197}, {3825,3956}, {3838,5542}, {3847,4662}, {3877,5837}, {3892,21620}, {3894,12047}, {3921,4187}, {3928,9812}, {3939,17123}, {4031,20292}, {4035,10453}, {4134,21616}, {4314,4999}, {4525,11813}, {4669,5854}, {4731,8582}, {4859,21267}, {4863,6745}, {4906,17070}, {5056,6764}, {5087,21060}, {5325,11238}, {5573,17067}, {5744,9580}, {5784,17626}, {5795,9581}, {5818,12629}, {6557,10005}, {6600,8167}, {6601,6666}, {6736,17606}, {6737,11376}, {7681,9842}, {7741,21075}, {8666,18519}, {9669,12572}, {10106,10529}, {10896,12527}, {12513,19925}, {12632,19877}, {17889,24216}, {18483,18540}
X(24386) = midpoint of X(3928) and X(9812)
X(24386) = reflection of X(3817) in X(3829)
X(24386) = complement of X(3158)
X(24386) = X(i)-complementary conjugate of X(j) for these (i,j): {7, 2885}, {56, 3161}, {269, 12640}, {1293, 4521}, {3445, 9}, {3676, 5510}, {4373, 1329}, {8056, 3452}, {10029, 20540}, {16079, 8}, {16945, 37}, {19604, 10}
X(24386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 3813, 21627), (10, 21627, 12640), (11, 4847, 3452), (497, 5231, 5745), (2886, 11019, 142), (3756, 21949, 24175), (6734, 12053, 5837)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24387) lies on these lines: {1,2476}, {2,3746}, {3,11235}, {4,535}, {5,519}, {8,3814}, {10,11}, {12,3244}, {21,4857}, {35,149}, {72,11813}, {78,23708}, {80,4861}, {115,17448}, {116,20257}, {140,528}, {145,7951}, {214,5533}, {226,3881}, {354,11263}, {355,22837}, {377,10072}, {381,12513}, {382,11194}, {404,3582}, {405,11238}, {442,551}, {474,10199}, {495,3635}, {496,1125}, {497,5248}, {499,3434}, {501,19642}, {515,10943}, {516,6705}, {518,9955}, {529,546}, {536,7764}, {758,946}, {956,10896}, {958,9669}, {960,7743}, {993,1479}, {1058,10198}, {1210,3754}, {1212,21090}, {1329,3626}, {1385,1484}, {1478,10529}, {1506,20691}, {1656,3913}, {1770,4973}, {2475,5563}, {2550,10200}, {2801,12608}, {2893,24202}, {2975,3583}, {3035,20107}, {3058,7483}, {3086,6904}, {3120,3953}, {3241,5141}, {3303,10197}, {3304,17532}, {3337,20292}, {3419,11376}, {3452,4015}, {3526,4421}, {3555,17605}, {3584,7504}, {3625,7173}, {3632,11681}, {3634,3816}, {3678,4847}, {3679,4193}, {3680,6975}, {3811,8227}, {3817,18908}, {3820,3847}, {3826,19878}, {3828,9710}, {3838,5045}, {3851,11236}, {3868,18393}, {3872,10826}, {3874,12047}, {3878,6734}, {3880,9956}, {3889,10129}, {3892,13407}, {3925,19862}, {3968,8582}, {4085,20108}, {4297,15908}, {4301,6831}, {4309,6910}, {4324,5303}, {4330,17549}, {4426,9665}, {4479,7796}, {4658,14009}, {4669,17533}, {4745,9711}, {4972,19864}, {4999,15171}, {5046,5258}, {5057,6763}, {5080,5288}, {5082,10589}, {5129,5274}, {5223,7678}, {5231,9614}, {5267,6284}, {5270,17577}, {5299,17737}, {5450,10525}, {5497,24161}, {5537,6972}, {5603,6873}, {5690,13463}, {5777,22835}, {5790,10912}, {5794,11373}, {5881,6941}, {5882,6842}, {5886,22836}, {6154,7294}, {6690,15172}, {6765,7988}, {6828,11522}, {6829,9624}, {6830,7982}, {6871,11240}, {6881,12437}, {6882,11362}, {6933,10056}, {6943,7991}, {6990,11523}, {7486,12632}, {7681,19925}, {7752,17144}, {7956,12571}, {7989,12629}, {9670,16370}, {10175,10915}, {10948,17647}, {11019,12446}, {11260,18480}, {11928,22758}, {12635,18493}, {14872,21635}, {14923,18395}, {15888,17530}, {16174,20117}, {16829,17669}, {17046,17761}, {17529,19883}, {21073,24036}, {21384,24045}
X(24387) = midpoint of X(i) and X(j) for these {i,j}: {4, 8666}, {5, 3813}, {355, 22837}, {946, 10916}, {5450, 10525}, {5690, 13463}, {10915, 21627}, {11260, 18480}
X(24387) = complement of X(8715)
X(24387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 7741, 3814), (10, 11, 3825), (10, 12053, 3884), (10, 21630, 3057), (21, 10707, 4857), (496, 2886, 1125), (1125, 2886, 3841), (1479, 10527, 993), (3813, 3829, 5), (4847, 21616, 3678), (5231, 9614, 12514), (9710, 17527, 3828), (10175, 21627, 10915), (10914, 17606, 10)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24388) lies on these lines: {1,475}, {10,1001}, {141,15733}, {142,17059}, {281,497}, {515,1597}, {518,12618}, {522,3663}, {527,21629}, {740,10916}, {946,1871}, {1210,3755}, {2321,3693}, {2345,6601}, {3174,17284}, {3886,6734}, {3939,17353}, {3946,11019}, {4523,21616}, {4858,23529}, {4953,17246}, {5572,16608}, {5856,17351}, {10915,17765}, {18216,18634}
X(24388) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 5452}, {3433, 9}, {13577, 1329}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24389) lies on these lines: {2,3174}, {9,497}, {10,1001}, {11,3059}, {142,2886}, {226,15185}, {390,6734}, {480,4863}, {516,1158}, {518,946}, {527,11235}, {528,10265}, {908,7678}, {960,4342}, {1210,2550}, {1445,3434}, {3243,3485}, {3826,9843}, {4134,18254}, {4326,5231}, {4662,18255}, {4686,4953}, {5223,9614}, {5249,11025}, {5542,11263}, {5745,8730}, {6067,14100}, {7674,18230}, {7675,10527}, {11680,21617}, {12609,20116}, {12731,21631}, {17059,21255}
X(24389) = midpoint of X(9) and X(6601)
X(24389) = reflection of X(6600) in X(6666)
X(24389) = complement of X(3174)
X(24389) = X(4578)-Ceva conjugate of X(522)
X(24389) = crosssum of X(56) and X(21059)
X(24389) = barycentric product X(8)*X(24181)
X(24389) = barycentric quotient X(24181)/X(7)
X(24389) = {X(2886),X(5572)}-harmonic conjugate of X(142)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24390) lies on these lines: {1,442}, {2,496}, {3,3434}, {4,956}, {5,8}, {9,9614}, {10,11}, {12,519}, {21,149}, {30,2975}, {35,528}, {39,21956}, {40,5231}, {55,7483}, {56,11112}, {63,3650}, {65,10916}, {72,946}, {75,17181}, {78,5886}, {92,15763}, {100,140}, {145,495}, {200,8227}, {210,21616}, {226,3555}, {325,17143}, {354,12609}, {355,1532}, {377,999}, {381,3436}, {388,17532}, {390,6857}, {404,15325}, {405,497}, {443,14986}, {474,2550}, {498,3913}, {499,1376}, {515,15908}, {516,3916}, {517,6734}, {518,12047}, {529,3585}, {546,5080}, {548,5303}, {551,3841}, {594,17444}, {631,17784}, {726,21927}, {867,17164}, {908,9955}, {942,20612}, {944,5175}, {952,4861}, {954,6601}, {958,1479}, {962,5789}, {993,6284}, {997,11376}, {1001,11517}, {1056,5177}, {1086,3953}, {1125,3925}, {1191,1714}, {1210,3753}, {1212,21073}, {1260,6832}, {1278,7906}, {1319,10949}, {1329,3679}, {1385,10609}, {1478,12513}, {1519,5777}, {1537,5887}, {1538,9947}, {1565,20880}, {1621,6675}, {1656,5552}, {1697,5705}, {1698,3816}, {1737,5836}, {1738,17055}, {1883,4968}, {2170,21029}, {2475,18990}, {2478,9669}, {2551,10591}, {2646,10959}, {2932,6940}, {2968,23528}, {3006,3695}, {3035,5533}, {3058,5248}, {3090,7080}, {3091,3421}, {3136,17135}, {3244,3822}, {3303,10198}, {3338,5880}, {3340,15844}, {3452,3697}, {3582,6691}, {3583,5258}, {3584,6668}, {3614,3625}, {3616,8728}, {3617,3820}, {3621,5141}, {3622,4197}, {3624,3826}, {3626,3814}, {3632,7951}, {3649,3874}, {3656,11682}, {3678,11813}, {3746,6690}, {3811,4863}, {3817,21075}, {3824,5049}, {3838,13407}, {3847,9711}, {3869,6841}, {3870,11374}, {3873,6147}, {3878,21677}, {3880,10039}, {3881,11263}, {3893,10915}, {3927,11415}, {3933,4441}, {3976,17889}, {4002,8582}, {4026,19863}, {4294,16370}, {4299,11194}, {4353,21955}, {4413,10200}, {4423,17590}, {4511,5178}, {4662,5087}, {4678,5154}, {4853,5587}, {4857,5251}, {4865,17733}, {4875,5179}, {4882,7988}, {4904,17046}, {4915,7989}, {5044,7743}, {5045,5249}, {5084,5274}, {5100,7081}, {5173,14054}, {5176,18357}, {5180,11684}, {5219,6765}, {5226,6764}, {5250,5791}, {5254,16975}, {5260,10707}, {5267,15338}, {5291,7745}, {5292,5710}, {5305,17737}, {5432,8715}, {5439,11019}, {5450,11826}, {5484,17677}, {5499,6224}, {5559,12653}, {5657,6922}, {5686,7678}, {5690,6882}, {5697,13463}, {5711,11269}, {5722,19860}, {5724,15955}, {5726,11519}, {5744,6361}, {5745,10624}, {5815,9779}, {5833,10384}, {5842,11012}, {5853,13411}, {5854,8068}, {5855,11009}, {5881,18242}, {5904,18393}, {6174,7294}, {6244,6890}, {6653,7824}, {6705,17613}, {6735,9956}, {6736,10175}, {6737,13464}, {6743,7958}, {6762,9612}, {6763,17768}, {6829,10595}, {6830,12245}, {6833,10306}, {6862,10679}, {6871,9654}, {6872,9668}, {6889,10806}, {6910,20075}, {6913,10531}, {6917,10680}, {6929,10522}, {6933,10528}, {6937,7967}, {6980,10942}, {6984,10597}, {6991,20007}, {7288,16371}, {7354,8666}, {7373,11240}, {7680,7982}, {7767,20553}, {8256,18395}, {9578,12629}, {9581,9623}, {9655,20076}, {9780,17527}, {9874,18228}, {10058,13271}, {10523,10912}, {10525,22758}, {10572,12690}, {10585,11239}, {10944,22837}, {11256,12749}, {11373,19861}, {11544,17483}, {11698,12531}, {12514,12701}, {12527,18483}, {12608,13257}, {12670,21631}, {14019,16020}, {14740,16174}, {15733,16193}, {15868,22767}, {15950,22836}, {16552,17747}, {16842,19855}, {17514,19858}, {17549,20066}, {17577,20060}, {17597,24159}, {17605,21077}, {17724,24160}
X(24390) = complement of X(3871)
X(24390) = midpoint of X(i) and X(j) for these {i,j}: {3585, 5288}, {4861, 5086}
X(24390) = reflection of X(i) in X(j) for these {i,j}: {35, 4999}, {15338, 5267}, {20612, 942}
X(24390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2886, 442), (2, 5082, 5687), (5, 8, 17757), (8, 5603, 5730), (8, 11680, 5), (10, 11, 4187), (10, 10914, 1145), (10, 12053, 392), (10, 21630, 3884), (21, 149, 15171), (145, 2476, 495), (377, 10529, 999), (497, 19843, 405), (499, 1376, 13747), (946, 4847, 72), (958, 1479, 11113), (958, 11235, 1479), (1125, 3925, 17529), (1329, 3829, 7741), (1329, 7741, 17533), (1698, 3816, 17575), (2550, 3086, 474), (2551, 10591, 17556), (2886, 3813, 1), (3006, 3702, 3695), (3244, 3822, 15888), (3434, 10527, 3), (3617, 4193, 3820), (3626, 3814, 21031), (3632, 7951, 12607), (3679, 3829, 17533), (3679, 7741, 1329), (3816, 9710, 1698), (3820, 10593, 4193), (4863, 11375, 3811), (6675, 15172, 1621), (6980, 12645, 10942), (7173, 21031, 3814), (9669, 9708, 2478), (12608, 14872, 13257), (17046, 20257, 4904)
X(24390) = complement of the isogonal conjugate of X(20615)
X(24390) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 4075}, {596, 1329}, {20615, 10}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24391) lies on these lines: {1,5745}, {2,3984}, {3,519}, {4,527}, {8,57}, {9,938}, {10,141}, {11,3962}, {20,3928}, {40,5768}, {56,6737}, {63,950}, {65,4847}, {72,1210}, {78,3911}, {144,10392}, {145,3601}, {165,3189}, {200,1467}, {210,8582}, {226,2476}, {329,9581}, {341,4899}, {354,21677}, {355,2095}, {377,553}, {387,3946}, {405,5325}, {443,3679}, {452,3929}, {497,12526}, {515,5709}, {516,5787}, {517,6245}, {528,5493}, {529,20420}, {551,6675}, {579,2321}, {610,5839}, {758,946}, {908,5154}, {912,6260}, {936,6692}, {958,6738}, {960,11019}, {986,3755}, {1001,6744}, {1040,15954}, {1125,5791}, {1329,21060}, {1376,6743}, {1385,5771}, {1737,5904}, {1770,4880}, {1834,3663}, {1837,12527}, {1998,10393}, {2093,5082}, {2136,6764}, {2323,3562}, {2478,3951}, {2550,3339}, {2551,5223}, {2886,3671}, {2900,10884}, {3008,17054}, {3243,5775}, {3244,5855}, {3419,4292}, {3485,5231}, {3487,5705}, {3522,12536}, {3617,9776}, {3626,5708}, {3632,15803}, {3730,21096}, {3811,6684}, {3813,4301}, {3825,4127}, {3869,12053}, {3876,5316}, {3894,13407}, {3901,12047}, {3916,4304}, {3927,5722}, {3940,6700}, {4001,5016}, {4067,21616}, {4208,6173}, {4298,5794}, {4313,20008}, {4314,4640}, {4654,5177}, {4667,5717}, {4685,16056}, {4711,10855}, {4855,20013}, {5044,9843}, {5046,17781}, {5128,17784}, {5175,9579}, {5220,18250}, {5273,5436}, {5435,5438}, {5657,6765}, {5690,9940}, {5770,6705}, {5777,7682}, {5795,18391}, {5805,5850}, {5843,22792}, {5854,13226}, {6282,12245}, {6763,10572}, {6769,14647}, {6824,13464}, {6847,7982}, {9943,15733}, {10165,22836}, {10175,21077}, {10202,10915}, {10477,20258}, {10529,11682}, {10578,18231}, {11529,19843}, {12832,14740}, {14054,18389}, {14986,15829}, {15933,17558}, {17449,23675}, {21214,24216}
X(24391) = midpoint of X(i) and X(j) for these {i,j}: {8, 6762}, {20, 12625}, {2136, 6764}, {12245, 12629}
X(24391) = reflection of X(i) in X(j) for these {i,j}: {946, 10916}, {3244, 11260}, {3811, 6684}, {4301, 3813}, {5882, 8666}, {12437, 3}, {12635, 1125}, {12640, 11362}
X(24391) = complement of X(11523)
X(24391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 942, 142), (10, 3874, 21620), (63, 12649, 950), (72, 1210, 3452), (145, 5744, 3601), (1737, 5904, 21075), (3868, 6734, 226), (3927, 5722, 12572), (3928, 12625, 20), (5175, 9965, 9579), (5435, 20007, 5438), (5777, 7682, 9842), (5791, 15934, 1125), (6744, 18249, 1001)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24392) lies on these lines: {1,442}, {2,3158}, {4,6762}, {5,6765}, {8,3452}, {9,497}, {10,1058}, {11,200}, {40,6899}, {55,5231}, {57,3434}, {63,149}, {72,9614}, {120,5272}, {142,10580}, {145,5226}, {165,528}, {210,11238}, {226,3243}, {312,4901}, {354,6173}, {355,7956}, {390,5745}, {392,3679}, {496,936}, {516,3928}, {518,1699}, {519,3545}, {521,11193}, {522,6545}, {527,9812}, {674,10439}, {748,3939}, {946,11523}, {956,3586}, {1024,2319}, {1125,3189}, {1146,5574}, {1210,1706}, {1329,4882}, {1420,10529}, {1697,6734}, {1698,3913}, {1738,5573}, {1768,13271}, {1837,4853}, {2481,20935}, {2550,5437}, {2898,9436}, {2999,17721}, {3036,4900}, {3058,4512}, {3086,5438}, {3091,6764}, {3174,3925}, {3295,5705}, {3340,12649}, {3555,9612}, {3601,10527}, {3616,12437}, {3617,12541}, {3622,12536}, {3632,5087}, {3633,10827}, {3677,3914}, {3681,10707}, {3703,4873}, {3705,3886}, {3706,4007}, {3756,8056}, {3811,8227}, {3816,8580}, {3829,7988}, {3870,5219}, {3872,5727}, {3873,4654}, {3875,7179}, {3911,17784}, {3940,7743}, {3944,16496}, {3966,4034}, {4323,20008}, {4326,6067}, {4373,16078}, {4423,6600}, {4514,11679}, {4677,5854}, {4859,21949}, {4862,21342}, {5049,17528}, {5123,8168}, {5175,10106}, {5178,19861}, {5225,12527}, {5261,9797}, {5268,24217}, {5269,11269}, {5436,19843}, {5691,12513}, {5722,9623}, {5748,20015}, {5784,12915}, {5791,15172}, {5837,9785}, {5855,11224}, {5880,10980}, {6666,7674}, {6745,10589}, {7082,13274}, {7174,24210}, {7989,12607}, {8055,10005}, {9624,22836}, {9708,18527}, {9780,12632}, {10453,17296}, {10591,21075}, {11522,12635}, {11531,13463}, {12526,12701}, {16572,21073}, {17284,19589}, {17597,23681}, {21740,22837}
X(24392) = reflection of X(i) in X(j) for these {i,j}: {1699, 11235}, {3158, 2}
X(24392) = X(8)-waw conjugate of X(9)
X(24392) = X(6558)-Ceva conjugate of X(522)
X(24392) = crosspoint of X(8) and X(4373)
X(24392) = crosssum of X(i) and X(j) for these (i,j): {56, 3052}, {604, 21059}
X(24392) = barycentric product X(i)*X(j) for these {i,j}: {8, 4859}, {333, 21949}
X(24392) = barycentric quotient X(i)/X(j) for these {i,j}: {4859, 7}, {21949, 226}
X(24392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 5274, 3452), (8, 6557, 6555), (8, 12053, 15829), (8, 21627, 3680), (11, 4863, 200), (63, 149, 9580), (497, 4847, 9), (1210, 5082, 1706), (2550, 11019, 5437), (3617, 12541, 12640), (3870, 11680, 5219), (3925, 10582, 20195), (8056, 21267, 3756)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24393) lies on these lines: {1,4878}, {2,3243}, {7,3617}, {8,9}, {10,141}, {11,210}, {55,5325}, {75,4899}, {144,4678}, {145,18230}, {200,5218}, {226,3681}, {333,7256}, {355,382}, {480,6737}, {515,3358}, {519,1001}, {527,1478}, {528,4669}, {551,15570}, {944,21153}, {946,3678}, {952,6594}, {958,6600}, {960,4342}, {971,5690}, {984,3755}, {1210,3697}, {1445,10106}, {1788,4321}, {2346,5260}, {2551,4866}, {2886,21060}, {3008,3242}, {3036,5856}, {3057,7064}, {3059,4111}, {3158,5273}, {3174,4882}, {3189,5234}, {3219,20095}, {3244,4974}, {3340,8232}, {3621,8236}, {3625,6541}, {3671,9710}, {3703,4061}, {3706,4082}, {3711,6745}, {3715,4863}, {3740,11019}, {3751,4667}, {3786,17197}, {3876,12053}, {3913,18249}, {3929,17784}, {3946,7174}, {3983,8582}, {3984,11526}, {4015,10916}, {4133,4439}, {4310,17067}, {4314,5302}, {4344,16670}, {4345,15829}, {4349,4663}, {4661,5249}, {4668,5698}, {4690,5845}, {4691,5850}, {4710,4738}, {4711,15733}, {4864,17337}, {4915,8275}, {4929,16833}, {5252,12573}, {5534,5770}, {5657,5732}, {5790,5805}, {5806,12599}, {5817,12245}, {5904,11551}, {6067,21031}, {6692,8580}, {6734,7705}, {7951,21075}, {8256,15587}, {9053,17348}, {9623,11041}, {9668,12572}, {9780,11038}, {9948,11495}, {9956,20330}, {10039,18412}, {10164,13226}, {10389,20015}, {10573,15298}, {10624,15650}, {10950,15837}, {12447,12513}, {12527,12943}, {12564,12855}, {12647,15299}, {18357,18482}, {21342,24175}
X(24393) = midpoint of X(i) and X(j) for these {i,j}: {8, 9}, {2550, 5223}, {7674, 12625}
X(24393) = reflection of X(i) in X(j) for these {i,j}: {1, 6666}, {142, 10}, {5542, 3826}, {12437, 6600}, {18482, 18357}, {20330, 9956}
X(24393) = complement of X(3243)
X(24393) = X(8)-beth conjugate of X(142)
X(24393) = crosssum of X(56) and X(1471)
X(24393) = barycentric product X(4924)*X(6557)
X(24393) = barycentric quotient X(4924)/X(5435)
X(24393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 2321, 4923), (8, 3717, 2321), (8, 5686, 9), (8, 5837, 12640), (8, 10005, 4901), (10, 5542, 3826), (10, 21255, 3823), (210, 4847, 3452), (958, 6743, 12437), (1697, 5809, 15006), (3679, 5223, 2550), (3703, 4113, 4061), (3706, 4126, 4082), (3826, 5542, 142), (9780, 11038, 20195)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28395.
X(24394) lies on these lines: {1,17959}, {30,511}, {37,4097}, {210,2321}, {612,1962}, {2136,2292}, {2294,3174}, {2643,19589}, {2650,3680}, {3169,3728}, {3678,4527}, {3686,11997}, {3742,3946}, {3743,3913}, {3747,3939}, {3753,3755}, {3754,4743}, {3773,3956}, {3786,3877}, {3794,4483}, {3873,3875}, {3919,4780}, {3943,21889}, {3950,21867}, {3968,4085}, {4015,4535}, {4068,6600}, {4133,4134}, {4433,4516}, {4512,4877}, {6601,18698}, {10158,10180}, {12541,17164}, {17163,18697}, {21871,22312}
X(24394) = barycentric product X(i)*X(j) for these {i,j}: {8, 16611}, {9, 4442}, {2321, 7292}, {3701, 16784}, {7017, 23230}
X(24394) = barycentric quotient X(i)/X(j) for these {i,j}: {2832, 17096}, {4442, 85}, {6019, 16611}, {7292, 1434}, {16611, 7}, {16784, 1014}, {23230, 222}
X(24395) lies on these lines: {1, 522}, {988, 1125}, {1331, 17888}, {2550, 10039}, {2607, 9318}, {4011, 25377}, {6604, 24695}, {24397, 24416}
X(24396) lies on these lines: {1, 905}, {2, 24449}, {6, 43}, {2254, 9318}, {2426, 3550}, {3571, 24410}, {4859, 17063}, {4925, 24420}, {24338, 24398}, {24397, 24412}
X(24397) lies on these lines: {1, 513}, {8, 19945}, {45, 16604}, {87, 1001}, {244, 4997}, {528, 4947}, {978, 23343}, {982, 24433}, {984, 1698}, {1909, 4389}, {3038, 17071}, {3123, 24841}, {3242, 10912}, {3616, 24485}, {24395, 24416}, {24396, 24412}, {24420, 24427}
X(24398) lies on these lines: {1, 514}, {2, 2415}, {87, 2191}, {527, 24216}, {982, 24352}, {3570, 3875}, {3961, 23669}, {4384, 19965}, {4419, 24318}, {4454, 25377}, {4919, 21138}, {24311, 25368}, {24338, 24396}, {24399, 24400}, {24409, 24422}, {24413, 24419}, {24420, 24423}
X(24399) lies on these lines: {1, 513}, {10, 75}, {244, 4080}, {537, 3123}, {982, 1647}, {1125, 24487}, {1739, 25025}, {3216, 23343}, {3670, 24433}, {3976, 23869}, {4444, 24403}, {4947, 24715}, {17063, 25377}, {24398, 24400}, {24413, 24414}
X(24400) lies on these lines: {1, 525}, {19, 27}, {1025, 3729}, {4454, 24316}, {24398, 24399}, {24403, 24409}
X(24401) lies on these lines: {1, 824}, {3, 4363}, {812, 24410}, {894, 24484}, {2607, 9318}, {17354, 17671}, {24346, 24455}, {24442, 24453}
X(24402) lies on these lines: {1, 522}, {2, 846}, {109, 14628}, {740, 4585}
X(24403) lies on these lines: {1, 514}, {2, 37}, {524, 17145}, {527, 17449}, {1018, 21208}, {1107, 25261}, {1334, 24172}, {1642, 5222}, {1647, 24318}, {2176, 20247}, {2275, 25237}, {3570, 4360}, {3571, 24419}, {3722, 24685}, {3726, 20347}, {3952, 4465}, {4392, 4419}, {4396, 20045}, {4444, 24399}, {4643, 20042}, {4713, 17165}, {5276, 6654}, {6545, 23810}, {7032, 24333}, {9259, 17136}, {9309, 12915}, {16604, 25244}, {17140, 24330}, {17318, 17780}, {17597, 24352}, {20244, 20271}, {20248, 21785}, {21138, 21272}, {21282, 24699}, {24338, 24413}, {24400, 24409}, {24404, 24429}
X(24404) lies on these lines: {1, 812}, {7, 24484}, {39, 1086}, {190, 16549}, {1655, 4440}, {2607, 9318}, {3821, 25373}, {6646, 24505}, {17483, 24504}, {24338, 24414}, {24403, 24429}
X(24405) lies on these lines: {1, 513}, {2, 23343}, {7, 2283}, {37, 3675}, {42, 19945}, {45, 672}, {100, 903}, {527, 551}, {545, 8299}, {674, 1469}, {894, 24487}, {942, 24433}, {984, 5902}, {1086, 4557}, {1644, 4413}, {3110, 4840}, {3720, 24494}, {4068, 17246}, {4363, 24327}, {4389, 20347}, {4436, 4440}, {4447, 7238}, {4448, 9318}, {4471, 24328}, {4862, 15624}, {5091, 23344}, {6646, 16684}, {8053, 17276}, {16494, 24482}, {16679, 17365}, {16687, 17483}, {17290, 20335}, {17724, 23845}, {20470, 24231}, {24345, 24416}
X(24406) lies on these lines: {1, 513}, {43, 19945}, {192, 21140}, {545, 2108}, {903, 1054}, {984, 3753}, {4912, 24441}, {4947, 21320}, {16569, 23343}, {24346, 24416}, {24409, 24428}
X(24407) lies on these lines: {1, 514}, {2, 45}, {524, 20042}, {527, 1647}, {536, 17780}, {678, 24685}, {3570, 17160}, {4393, 18822}, {4659, 9458}, {4971, 20058}, {24419, 24423}
X(24408) lies on these lines: {1, 514}, {2, 1266}, {536, 9458}, {545, 24709}, {676, 24409}, {1647, 4419}, {2254, 24338}, {17132, 25377}, {24441, 25378}
X(24409) lies on these lines: {1, 522}, {551, 3923}, {676, 24408}, {894, 24485}, {1769, 24338}, {2191, 2414}, {3756, 4370}, {4000, 24980}, {4448, 9318}, {4904, 5856}, {11125, 24427}, {24398, 24422}, {24400, 24403}, {24406, 24428}
X(24410) lies on these lines: {1, 522}, {4, 9}, {34, 2405}, {109, 24026}, {318, 7012}, {348, 22464}, {812, 24401}, {867, 2886}, {993, 14127}, {1376, 9458}, {1457, 10538}, {1818, 3685}, {1935, 23528}, {2284, 4513}, {2792, 21293}, {3035, 16594}, {3073, 17869}, {3100, 24307}, {3571, 24396}, {4318, 4861}, {5263, 9414}, {5438, 6789}, {24416, 24418}
X(24411) lies on these lines: {1, 522}, {85, 10001}, {527, 1478}, {900, 24346}, {990, 997}, {1026, 3729}, {1146, 5851}, {1275, 9312}, {1376, 23343}, {1768, 2250}, {1996, 5231}, {2263, 3872}, {3035, 4370}, {4363, 24341}, {4858, 9355}, {18816, 24026}, {24338, 24396}
X(24412) lies on these lines: {1, 8678}, {3571, 9318}, {4561, 23686}, {24396, 24397}
X(24413) lies on these lines: {1, 512}, {38, 75}, {87, 24492}, {982, 4493}, {1739, 20006}, {2276, 18795}, {3670, 19950}, {4392, 4443}, {9318, 24423}, {19951, 24443}, {24338, 24403}, {24396, 24397}, {24398, 24419}, {24399, 24414}
X(24414) lies on these lines: {1, 788}, {10, 4443}, {3122, 19974}, {24338, 24404}, {24399, 24413}
X(24415) lies on these lines: {1, 525}, {81, 4360}, {918, 9318}
X(24416) lies on these lines: {1, 900}, {2, 24188}, {106, 4432}, {190, 244}, {545, 24358}, {1054, 1086}, {1411, 24816}, {1769, 24338}, {1772, 24845}, {4427, 4440}, {5541, 24222}, {9318, 24354}, {9457, 24841}, {24345, 24405}, {24346, 24406}, {24395, 24397}, {24410, 24418}, {24428, 24461}
X(24417) lies on these lines: {31, 9811}, {812, 1769}, {824, 21189}, {918, 4364}, {1638, 3782}, {3960, 9259}
X(24418) lies on these lines: {1, 513}, {145, 19945}, {984, 3624}, {3242, 17792}, {3622, 24507}, {3976, 24433}, {4389, 25303}, {17290, 25102}, {21214, 23343}, {24410, 24416}
X(24419) lies on these lines: {1, 659}, {2, 24193}, {190, 244}, {291, 4465}, {2254, 24338}, {3248, 3315}, {3571, 24403}, {4809, 24427}, {9318, 24420}, {16676, 20331}, {24398, 24413}, {24407, 24423}, {24421, 24460}
X(24420) lies on these lines: {1, 667}, {37, 982}, {4925, 24396}, {9318, 24419}, {16495, 17449}, {17597, 18170}, {24338, 24403}, {24397, 24427}, {24398, 24423}
X(24421) lies on these lines: {1, 667}, {982, 1107}, {24338, 24404}, {24419, 24460}
X(24422) lies on these lines: {1, 513}, {200, 19945}, {23343, 23511}, {24398, 24409}
X(24423) lies on these lines: {1, 513}, {45, 25427}, {239, 19945}, {292, 24722}, {870, 4389}, {984, 4363}, {2664, 23343}, {9318, 24413}, {16826, 24510}, {24398, 24420}, {24407, 24419}
X(24424) lies on these lines: {1, 5327}, {11, 24317}, {35, 24684}, {38, 497}, {55, 24315}, {71, 17863}, {527, 3874}, {534, 10624}, {536, 960}, {596, 17132}, {950, 9028}, {1001, 3923}, {1479, 24682}, {1953, 25255}, {2294, 17220}, {2486, 2886}, {3663, 18589}, {3782, 25361}, {4019, 18147}, {4032, 16777}, {4294, 24683}, {4454, 17140}, {5432, 25341}, {5572, 13476}, {6508, 18662}, {7289, 10889}, {11235, 24441}, {11238, 25362}, {12635, 17318}, {17861, 21231}, {20531, 25348}, {24248, 24445}, {24328, 24352}, {24337, 24444}
X(24425) lies on these lines: {1, 714}, {38, 190}, {45, 4011}, {75, 3747}, {670, 1965}, {744, 1930}, {1001, 3923}, {1086, 24259}, {1191, 5695}, {1918, 20891}, {4056, 4837}, {4364, 4368}, {4376, 4381}, {5263, 24450}, {8299, 24327}, {13476, 17351}, {17153, 17280}, {17290, 24260}, {18137, 20964}, {23383, 24850}, {24335, 24346}
X(24426) lies on these lines: {1, 667}, {2, 24448}, {6, 982}, {38, 3570}, {3571, 24403}, {9318, 24413}, {24338, 24396}
X(24427) lies on these lines: {1, 514}, {2, 726}, {900, 24338}, {1647, 4392}, {3807, 17793}, {4809, 24419}, {9458, 17495}, {11125, 24409}, {17449, 17484}, {24397, 24420}, {24399, 24413}
X(24428) lies on these lines: {1, 514}, {2, 846}, {740, 3570}, {1083, 19950}, {1647, 3218}, {2607, 3571}, {3724, 13589}, {6548, 17960}, {6654, 8300}, {17194, 18646}, {24329, 24456}, {24338, 24346}, {24406, 24409}, {24416, 24461}
X(24429) lies on these lines: {1, 513}, {8, 192}, {10, 24004}, {38, 20042}, {986, 6788}, {1001, 18792}, {1125, 19945}, {2607, 3571}, {2796, 3122}, {3123, 4432}, {3293, 23343}, {3821, 18046}, {3909, 14752}, {4389, 16887}, {4424, 24433}, {24403, 24404}
X(24430) lies on these lines: {1, 90}, {2, 7004}, {3, 3460}, {9, 1040}, {11, 982}, {20, 201}, {22, 21368}, {31, 1776}, {33, 63}, {37, 10391}, {38, 497}, {55, 846}, {57, 1736}, {73, 12528}, {84, 1038}, {210, 9371}, {212, 3100}, {238, 7082}, {240, 1857}, {244, 10589}, {255, 6198}, {388, 774}, {390, 7226}, {522, 17860}, {651, 20277}, {756, 5218}, {916, 20122}, {920, 3072}, {958, 1854}, {971, 1214}, {986, 1837}, {990, 1708}, {991, 16577}, {1001, 24434}, {1062, 3074}, {1254, 5229}, {1376, 24433}, {1393, 3091}, {1465, 5927}, {1473, 16560}, {1478, 1725}, {1709, 8270}, {1710, 8185}, {1726, 3220}, {1735, 5587}, {1777, 7701}, {1807, 6914}, {1859, 24310}, {1864, 3666}, {1870, 18477}, {2201, 16567}, {2292, 3486}, {2293, 3989}, {2361, 7262}, {2635, 17080}, {2654, 3868}, {3056, 9017}, {3075, 24467}, {3468, 8757}, {3661, 22418}, {3670, 9581}, {3703, 4073}, {3731, 10383}, {3876, 22072}, {3925, 24341}, {3929, 7070}, {3955, 10535}, {3976, 11376}, {4003, 17604}, {4376, 9414}, {4392, 5274}, {4424, 5727}, {4459, 4493}, {4650, 5348}, {5220, 7074}, {5285, 21375}, {5745, 16870}, {5777, 17102}, {5784, 25091}, {6001, 24806}, {7008, 23052}, {7046, 24031}, {7193, 11429}, {7299, 9630}, {7416, 23067}, {10394, 14547}, {10395, 23537}, {11220, 22053}, {11502, 17596}, {13243, 17074}, {15064, 24025}, {16452, 22347}, {17606, 24174}, {18161, 21318}, {20305, 22069}, {22410, 23585}
X(24431) lies on these lines: {1, 195}, {2, 24433}, {11, 38}, {37, 20752}, {55, 846}, {63, 5348}, {201, 7354}, {343, 21920}, {354, 1736}, {495, 1725}, {497, 7226}, {756, 5432}, {774, 15888}, {1393, 3614}, {1776, 3920}, {2167, 2602}, {2292, 10950}, {2310, 3058}, {2361, 3219}, {2801, 16577}, {3074, 9630}, {3670, 17606}, {3989, 14547}, {4392, 10589}, {5492, 5903}, {8679, 21318}, {9414, 24448}, {11031, 17602}, {24326, 24454}
X(24432) lies on these lines: {1, 2}, {3, 7235}, {56, 24371}, {58, 1733}, {740, 1792}, {23922, 24953}, {24355, 24435}
X(24433) lies on these lines: {1, 6}, {2, 24431}, {8, 2397}, {10, 522}, {21, 4570}, {38, 1647}, {65, 24029}, {212, 9639}, {244, 6667}, {528, 2310}, {756, 6690}, {774, 12607}, {942, 24405}, {982, 24397}, {986, 24338}, {1376, 24430}, {1725, 17757}, {1735, 5123}, {2801, 16578}, {2810, 17463}, {2836, 21362}, {2886, 7069}, {3035, 7004}, {3036, 24028}, {3580, 21920}, {3670, 24399}, {3754, 5492}, {3976, 24418}, {4026, 25024}, {4424, 24429}, {4429, 25005}, {4640, 23703}, {4694, 24871}, {5794, 18340}, {6211, 20872}, {8758, 17615}, {17256, 24490}, {19945, 24443}, {20962, 21326}, {20989, 21368}
X(24434) lies on these lines: {1, 6597}, {2, 24431}, {38, 2886}, {756, 3035}, {984, 1376}, {1001, 24430}, {1040, 15296}, {1736, 3742}, {1818, 3989}, {2550, 7226}, {2801, 16579}, {3816, 7069}, {3878, 5492}, {6690, 7004}, {9959, 23846}, {11031, 17061}
X(24435) lies on these lines: {1, 142}, {2, 21675}, {9, 22003}, {48, 75}, {56, 4361}, {63, 18653}, {100, 21231}, {141, 9317}, {144, 18661}, {239, 2260}, {284, 18698}, {354, 4852}, {966, 24268}, {1376, 24365}, {1474, 18697}, {1761, 14953}, {2173, 11683}, {2194, 4418}, {2287, 8680}, {2294, 16054}, {2360, 4647}, {2646, 3739}, {2893, 4466}, {3187, 17490}, {3419, 6739}, {3958, 8822}, {4039, 24362}, {4360, 24378}, {4371, 24363}, {4483, 14210}, {5086, 20305}, {5271, 6350}, {6741, 23902}, {16713, 17136}, {17043, 24390}, {17045, 24366}, {17052, 24780}, {17119, 37519}, {18162, 20880}, {24355, 24432}, {24384, 24953}
X(24436) lies on these lines: {1, 2836}, {3, 7609}, {9, 1030}, {21, 24697}, {22, 7262}, {23, 896}, {25, 4650}, {35, 4557}, {36, 238}, {55, 846}, {56, 7312}, {58, 3122}, {999, 5429}, {1046, 20831}, {1473, 17063}, {1633, 24715}, {1757, 20872}, {2112, 24484}, {2607, 4459}, {2792, 7427}, {3751, 20468}, {3929, 7298}, {4068, 23861}, {4655, 17522}, {5150, 24482}, {5284, 6536}, {5358, 24851}, {6186, 13486}, {7193, 8540}, {7280, 15601}, {7293, 17123}
X(24437) lies on these lines: {1, 6}, {2, 21699}, {8, 4446}, {10, 24520}, {38, 3571}, {69, 4443}, {75, 670}, {86, 3728}, {141, 24575}, {190, 22167}, {239, 4022}, {256, 524}, {291, 594}, {314, 714}, {319, 3778}, {668, 21238}, {700, 17157}, {846, 4068}, {982, 4361}, {1030, 8298}, {1045, 16696}, {1654, 3122}, {2228, 17287}, {2664, 22271}, {3123, 17273}, {3670, 4716}, {3764, 17363}, {3961, 20990}, {4392, 17162}, {4445, 4484}, {4735, 17372}, {4941, 17255}, {6542, 21035}, {6646, 24338}, {7170, 9457}, {12329, 16877}, {12782, 17299}, {16556, 20474}, {16738, 25295}, {17065, 17275}, {17135, 17148}, {17253, 24456}, {17256, 22172}, {17277, 21330}, {17289, 20456}, {17362, 24478}, {17365, 24463}
X(24438) lies on these lines: {1, 474}, {8, 4446}, {2292, 24450}, {2664, 22299}, {8256, 24643}
X(24439) lies on these lines: {1, 142}, {75, 24378}, {982, 4361}, {4657, 24366}, {5271, 17148}
X(24440) lies on these lines: {1, 474}, {2, 4642}, {4, 9365}, {8, 982}, {9, 16605}, {10, 75}, {12, 17889}, {21, 17601}, {36, 2933}, {38, 3617}, {40, 238}, {43, 65}, {46, 4650}, {56, 1054}, {57, 9363}, {72, 6048}, {100, 3924}, {145, 244}, {279, 291}, {386, 3754}, {484, 1724}, {498, 24161}, {517, 978}, {518, 21896}, {519, 3976}, {750, 17016}, {899, 3869}, {958, 17596}, {960, 16569}, {988, 9623}, {1046, 2640}, {1086, 12607}, {1104, 3550}, {1149, 3885}, {1201, 14923}, {1220, 3980}, {1329, 3944}, {1574, 3735}, {1575, 3061}, {1697, 5272}, {1698, 4424}, {1707, 5128}, {1772, 10573}, {2176, 21888}, {2275, 4051}, {2276, 21951}, {2292, 4903}, {2550, 24478}, {2551, 24248}, {2650, 3240}, {2810, 17114}, {2899, 8421}, {3057, 16610}, {3086, 24028}, {3120, 11681}, {3169, 20227}, {3208, 3290}, {3214, 3868}, {3216, 4674}, {3244, 24168}, {3293, 5902}, {3339, 3751}, {3501, 16583}, {3623, 9335}, {3625, 24167}, {3632, 3953}, {3633, 4694}, {3666, 3698}, {3670, 3679}, {3730, 16611}, {3755, 17065}, {3782, 21031}, {3840, 4673}, {3871, 17715}, {3878, 17749}, {3914, 24982}, {4073, 17872}, {4392, 4678}, {4414, 5260}, {4415, 9711}, {4696, 17155}, {4714, 10479}, {4850, 10459}, {4859, 6706}, {4882, 16496}, {5121, 12053}, {5143, 13738}, {5250, 17123}, {5261, 7613}, {5262, 17716}, {5264, 16478}, {5293, 9709}, {5529, 5730}, {5552, 17719}, {6735, 23536}, {6736, 24177}, {6762, 18193}, {7991, 23511}, {8582, 24210}, {9364, 21147}, {16284, 24215}, {17145, 20047}, {17451, 17756}, {18178, 18792}, {19582, 24003}, {20271, 20691}, {21935, 24991}, {24325, 24377}
X(24441) lies on these lines: {1, 4715}, {2, 45}, {6, 17247}, {9, 17323}, {37, 7232}, {144, 17045}, {192, 4445}, {320, 16672}, {514, 23352}, {519, 4643}, {524, 3241}, {527, 551}, {536, 984}, {537, 24357}, {597, 6172}, {599, 742}, {3247, 17345}, {3644, 17252}, {3662, 16675}, {3663, 17259}, {3672, 17332}, {3729, 17327}, {3731, 17235}, {3763, 17249}, {3821, 3828}, {3834, 16676}, {4357, 17262}, {4361, 17246}, {4384, 16590}, {4393, 17488}, {4428, 20834}, {4484, 24456}, {4659, 4708}, {4665, 4748}, {4670, 25055}, {4677, 4690}, {4681, 17272}, {4698, 4862}, {4704, 17273}, {4718, 17270}, {4755, 6173}, {4762, 24457}, {4798, 19883}, {4908, 17237}, {4912, 24406}, {5296, 7263}, {5308, 7238}, {5851, 10186}, {6144, 17393}, {6646, 16777}, {6707, 7222}, {11235, 24424}, {15533, 17389}, {15668, 17276}, {16673, 17376}, {16674, 17300}, {16677, 17234}, {16814, 17304}, {16884, 17347}, {16885, 17302}, {17118, 17248}, {17119, 17256}, {17236, 17267}, {17264, 21358}, {17307, 25269}, {17319, 17329}, {17321, 17334}, {17324, 17336}, {17790, 18146}, {24408, 25378}
X(24442) lies on these lines: {1, 826}, {894, 24488}, {3185, 4418}, {24346, 24347}, {24401, 24453}
X(24443) lies on these lines: {1, 88}, {2, 986}, {3, 3924}, {5, 3120}, {6, 1406}, {8, 982}, {10, 38}, {12, 1086}, {21, 17596}, {31, 46}, {34, 9316}, {35, 4218}, {39, 3125}, {40, 614}, {42, 942}, {43, 3868}, {55, 17054}, {56, 2933}, {57, 961}, {58, 1325}, {63, 1722}, {65, 1193}, {72, 899}, {73, 18838}, {75, 1237}, {141, 20653}, {145, 3976}, {171, 5262}, {192, 22220}, {201, 3772}, {227, 1458}, {239, 24610}, {240, 5125}, {321, 3831}, {354, 4646}, {377, 11031}, {386, 2650}, {405, 4414}, {442, 20966}, {484, 595}, {496, 1647}, {498, 24159}, {517, 1201}, {518, 3214}, {519, 3953}, {580, 5535}, {581, 15016}, {612, 24163}, {672, 16583}, {726, 3701}, {740, 21330}, {748, 12514}, {756, 1698}, {758, 3216}, {774, 1210}, {846, 5047}, {896, 1724}, {902, 3579}, {958, 17595}, {960, 16610}, {964, 3980}, {975, 17124}, {976, 1376}, {978, 3869}, {984, 4699}, {988, 19860}, {995, 5903}, {1042, 1465}, {1104, 1155}, {1107, 21951}, {1125, 4424}, {1149, 3057}, {1329, 3782}, {1334, 3290}, {1388, 8572}, {1401, 16980}, {1403, 13738}, {1451, 1454}, {1574, 3954}, {1575, 3721}, {1697, 5573}, {1706, 3677}, {1737, 21935}, {1738, 3778}, {1788, 4000}, {1837, 7004}, {1838, 2181}, {1930, 24170}, {1959, 24598}, {2140, 24786}, {2170, 2275}, {2171, 2277}, {2269, 20227}, {2276, 20271}, {2294, 4261}, {2310, 9581}, {2476, 17889}, {2478, 24248}, {2999, 3339}, {3008, 24633}, {3011, 6684}, {3178, 18139}, {3218, 5247}, {3244, 4694}, {3293, 3874}, {3555, 17449}, {3616, 17063}, {3617, 4392}, {3622, 9335}, {3660, 4322}, {3663, 8582}, {3666, 3812}, {3681, 6048}, {3691, 16605}, {3696, 4022}, {3698, 4003}, {3702, 3840}, {3720, 3931}, {3724, 16453}, {3726, 20691}, {3727, 16604}, {3743, 3833}, {3753, 10459}, {3755, 21346}, {3764, 5880}, {3821, 5051}, {3826, 21035}, {3876, 16569}, {3877, 21214}, {3913, 17597}, {3923, 5192}, {3936, 17748}, {3938, 5687}, {3944, 4193}, {3992, 24068}, {4284, 16547}, {4296, 9364}, {4310, 7080}, {4346, 8165}, {4385, 17155}, {4418, 13740}, {4429, 4446}, {4645, 24478}, {5011, 5299}, {5046, 24851}, {5128, 7290}, {5177, 7613}, {5249, 5530}, {5250, 5272}, {5254, 21044}, {5255, 7191}, {5264, 17469}, {5283, 21921}, {5297, 24164}, {5396, 5885}, {5445, 17734}, {5563, 15955}, {5692, 17749}, {5711, 17017}, {5904, 21805}, {6187, 23850}, {6650, 16044}, {7069, 17606}, {7264, 21208}, {8056, 8583}, {8728, 21674}, {9310, 9620}, {9317, 9413}, {10479, 21020}, {11319, 24850}, {11512, 19861}, {11533, 17535}, {12526, 23511}, {12579, 14020}, {12702, 16483}, {13161, 24982}, {15048, 21950}, {16478, 17126}, {16549, 16600}, {16552, 16611}, {16700, 18178}, {16720, 25350}, {16824, 24627}, {16825, 24464}, {17053, 17452}, {17114, 23154}, {17145, 20051}, {17187, 18180}, {17495, 17751}, {17541, 17738}, {17750, 21840}, {17758, 24185}, {17863, 24173}, {19945, 24433}, {19951, 24413}, {20284, 20707}, {20360, 24530}, {20366, 25141}, {21342, 21896}, {21725, 23929}, {23661, 24218}, {23675, 24171}, {23928, 24923}, {24178, 24987}, {24372, 24575}
X(24443) = complement of X(25253)
X(24443) = anticomplement of X(25079)
X(24444) lies on these lines: {24337, 24424}
X(24445) lies on these lines: {1, 15076}, {7, 17446}, {55, 24822}, {69, 17872}, {75, 23682}, {335, 24351}, {982, 1086}, {984, 4363}, {986, 5880}, {3959, 9025}, {4073, 4446}, {4459, 4493}, {7184, 18161}, {17063, 17290}, {18252, 20227}, {20930, 23688}, {24248, 24424}
X(24446) lies on these lines: {1, 442}, {2, 21692}, {199, 8301}, {1376, 24365}, {2160, 2319}, {2606, 9451}, {3187, 3873}, {4362, 4412}
X(24447) lies on these lines: {1, 512}, {38, 1755}, {55, 16378}, {86, 1621}, {659, 9318}, {894, 24492}, {7225, 9316}, {20347, 20475}, {24346, 24347}
X(24448) lies on these lines: {2, 24426}, {38, 190}, {291, 24330}, {984, 4376}, {3571, 24326}, {4363, 4493}, {9414, 24431}
X(24449) lies on these lines: {1, 2389}, {2, 24396}, {982, 1086}, {8255, 17056}, {15726, 21856}
X(24450) lies on these lines: {2, 3122}, {10, 17790}, {37, 1045}, {45, 846}, {86, 3728}, {190, 756}, {256, 1213}, {274, 714}, {291, 4472}, {984, 4363}, {1046, 5220}, {1654, 21699}, {1740, 17038}, {2292, 24438}, {2663, 22271}, {3571, 9458}, {4364, 24338}, {4427, 9330}, {4492, 24456}, {5263, 24425}, {5283, 24696}, {5988, 25359}, {6007, 16589}, {17398, 24575}, {18140, 24688}
X(24451) lies on these lines: {2, 3123}, {8, 9025}, {75, 7155}, {190, 1376}, {192, 1740}, {1281, 24334}, {3242, 4363}, {3952, 24344}, {4110, 25311}, {4414, 9458}, {4443, 25382}, {4454, 17794}, {4518, 24341}, {6382, 18830}, {17790, 24717}
X(24452) lies on these lines: {2, 4432}, {9, 484}, {75, 519}, {545, 984}, {1654, 7229}, {3679, 4363}, {3828, 17354}, {4859, 15668}
X(24453) lies on these lines: {1, 23877}, {894, 24498}, {3716, 9318}, {24401, 24442}
X(24454) lies on these lines: {543, 4736}, {620, 7266}, {984, 3571}, {2786, 13277}, {4376, 9414}, {4971, 23928}, {24326, 24431}
X(24455) lies on these lines: {1, 514}, {2, 2140}, {101, 2481}, {517, 25368}, {894, 24491}, {993, 24331}, {3570, 17143}, {4251, 6654}, {6684, 25342}, {17761, 18785}, {19974, 24514}, {21616, 24318}, {24346, 24401}
X(24456) lies on these lines: {1, 4503}, {2, 3123}, {37, 19584}, {75, 4941}, {244, 25378}, {256, 17321}, {291, 4419}, {982, 3122}, {984, 4026}, {1086, 3816}, {2228, 4664}, {3662, 22172}, {3663, 17065}, {3672, 24478}, {3764, 17320}, {3778, 17247}, {4022, 17249}, {4110, 25121}, {4363, 24338}, {4446, 17246}, {4484, 24441}, {4492, 24450}, {4699, 22174}, {5988, 25369}, {17236, 21330}, {17238, 22167}, {17253, 24437}, {17257, 24575}, {17333, 20456}, {17396, 23659}, {17793, 24351}, {19945, 25377}, {24319, 24341}, {24329, 24428}
X(24457) lies on these lines: {1, 513}, {10, 522}, {11, 244}, {44, 1643}, {45, 650}, {521, 10912}, {523, 2292}, {661, 14437}, {693, 4389}, {946, 3667}, {984, 4777}, {1283, 4491}, {1491, 14434}, {1854, 15313}, {2787, 14286}, {2804, 21112}, {2827, 11715}, {3242, 9001}, {3309, 7986}, {3445, 6129}, {3649, 4017}, {3663, 23810}, {3669, 15306}, {3900, 11525}, {4762, 24441}, {4885, 17290}, {4926, 15079}, {4977, 21105}, {5542, 6006}, {6163, 9268}, {8641, 23855}, {14205, 15726}, {16892, 24110}, {21133, 24098}, {23809, 24184}
X(24457) = intouch-to-excentral similarity image of X(5)
X(24458) lies on these lines: 2, 3123}, {75, 20598}, {192, 2227}, {291, 24248}, {350, 4941}, {984, 4493}, {3571, 4376}, {3662, 23462}, {4443, 25349}, {4713, 24338}, {7148, 20081}
X(24459) lies on these lines: {525, 656}, {647, 14208}, {659, 812}, {905, 15411}, {1565, 3942}, {1577, 3709}, {3265, 21107}, {4086, 23878}, {4391, 21225}, {4643, 9034}, {6563, 21117}, {7212, 16591}, {7650, 23882}
X(24460) lies on these lines: {1, 514}, {2, 1334}, {41, 6654}, {894, 24509}, {2481, 9317}, {3570, 17144}, {24419, 24421}
X(24461) lies on these lines: {1, 4777}, {894, 16494}, {900, 14191}, {1001, 3923}, {4448, 9318}, {24416, 24428}
X(24462) lies on these lines: {1, 926}, {10, 20525}, {38, 4453}, {43, 3310}, {171, 654}, {240, 522}, {659, 17990}, {876, 6373}, {918, 984}, {982, 1638}, {2254, 2786}, {2488, 25084}, {2530, 3805}, {3123, 21208}, {3550, 6139}, {3709, 4040}, {4443, 14315}, {4502, 6005}, {9511, 17596}, {21010, 23188}
X(24463) lies on these lines: {1, 6007}, {2, 3123}, {38, 4440}, {75, 256}, {190, 24259}, {291, 4363}, {560, 8296}, {894, 20456}, {984, 2550}, {987, 19844}, {1582, 8424}, {2277, 16571}, {3728, 6646}, {3778, 17116}, {3961, 21320}, {4022, 7321}, {4364, 24338}, {4389, 4947}, {4446, 17118}, {4471, 8297}, {4659, 12782}, {4941, 17321}, {5988, 24319}, {17117, 23659}, {17300, 22167}, {17305, 19945}, {17365, 24437}, {17790, 24688}, {24323, 24345}, {25347, 25382}
X(24464) lies on these lines: {1, 3}, {10, 3765}, {38, 3980}, {213, 1759}, {244, 24331}, {292, 3735}, {752, 3764}, {758, 869}, {984, 4363}, {1107, 3753}, {1739, 4384}, {2664, 5692}, {3778, 4660}, {4283, 4429}, {4392, 24344}, {4443, 24715}, {4674, 16975}, {4703, 21936}, {5283, 16549}, {6327, 20966}, {16823, 24046}, {16825, 24443}
See Antreas Hatzipolakis, César Lozada, and Ercole Suppa, Hyacinthos 28398 and Hyacinthos 28399.
X(24465) lies on these lines: {5, 7702}, {7, 100}, {11, 57}, {30, 18838}, {56, 1387}, {65, 952}, {80, 3339}, {109, 1086}, {149, 21454}, {214, 3671}, {226, 3035}, {388, 1145}, {516, 3660}, {528, 553}, {938, 10724}, {942, 5840}, {1155, 5762}, {1317, 3340}, {1320, 3600}, {1466, 10090}, {1470, 11729}, {1537, 4295}, {1617, 3474}, {2099, 11046}, {2802, 4298}, {2829, 4292}, {2834, 3937}, {3036, 4848}, {3218, 5857}, {3333, 14217}, {3336, 8068}, {3337, 5533}, {3361, 16173}, {3911, 5087}, {4355, 5541}, {4440, 14594}, {4654, 6174}, {4860, 13274}, {5221, 12019}, {5708, 10738}, {5854, 10106}, {6147, 10044}, {7972, 18421}, {9945, 12739}, {10404, 10956}, {10742, 18541}, {11529, 12119}, {14151, 20095}, {15803, 21154}, {17579, 18419}, {17724, 23703}
X(24465) = midpoint of X(4292) and X(12736)
X(24465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5221, 13273, 12832), (12832, 13273, 12019)
See Ercole Suppa, Hyacinthos 28400.
X(24466) lies on these lines: {2,10724}, {3,11}, {4,3035}, {20,100}, {30,119}, {40,550}, {55,6948}, {80,165}, {104,376}, {120,7427}, {140,22938}, {149,3522}, {214,516}, {411,17100}, {515,1145}, {517,1317}, {548,11012}, {631,6667}, {944,5854}, {971,6068}, {1001,6955}, {1155,12743}, {1320,5731}, {1350,5848}, {1376,6938}, {1387,3576}, {1484,8703}, {1657,10742}, {1753,1862}, {2771,16111}, {2800,10609}, {2802,4297}, {2803,3184}, {2806,14689}, {2886,6950}, {2932,7580}, {2950,10860}, {2951,6282}, {3036,5657}, {3058,10269}, {3146,20400}, {3529,10728}, {3534,6244}, {3925,6914}, {4188,7681}, {4190,11496}, {4299,10087}, {4304,12736}, {4314,18240}, {4316,5537}, {4333,5812}, {4413,6930}, {4421,12115}, {4880,5844}, {4996,5842}, {5010,6907}, {5204,13274}, {5217,6850}, {5326,6980}, {5432,6923}, {5434,10679}, {5533,7280}, {5538,15228}, {5732,5856}, {5759,5851}, {6200,13913}, {6224,9778}, {6246,6684}, {6361,10698}, {6690,6951}, {6702,10164}, {6827,12764}, {6891,12953}, {6921,10893}, {6925,12761}, {6934,12775}, {6961,10896}, {7354,10956}, {7680,17579}, {7957,17660}, {7972,7991}, {7982,12735}, {7987,16173}, {8227,17563}, {8674,16163}, {8735,22055}, {9541,19082}, {9803,9963}, {10074,22770}, {10165,16174}, {10167,15528}, {10265,12690}, {10304,10707}, {10711,11001}, {10767,15035}, {10768,21166}, {10778,15055}, {11698,15704}, {11729,12699}, {11849,15888}, {12116,13271}, {12248,17538}, {12773,15696}, {13268,16190}, {14740,14872}
X(24466) = midpoint of X(i) and X(j) for these {i,j}: {20,100}, {40,12119}, {104,13199}, {1657,10742}, {3529,10728}, {4316,5537}, {5538,15228}, {6361,10698}, {7957,17660}, {7972,7991}, {9803,9963}, {10711,11001}, {11698,15704}
X(24466) = reflection of X(i) in X(j) for these {i,j}: {4,3035}, {11,3}, {149,20418}, {1537,214}, {6154,10993}, {6246,6684}, {6326,9945}, {7982,12735}, {10738,6713}, {12690,10265}, {12699,11729}, {14217,1387}, {14872,14740}, {22938,140}
X(24466) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,11,21154}, {3,10525,5433}, {3,10738,6713}, {3,11826,15908}, {20,10310,11827}, {376,13199,104}, {1155,12743,12832}, {3576,14217,1387}, {5010,6907,21155}, {6713,10738,11}
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.
X(24467) lies on these lines: {1, 1399}, {3, 63}, {4, 3218}, {5, 57}, {7, 6824}, {8, 6948}, {9, 140}, {11, 90}, {12, 17700}, {26, 3220}, {30, 84}, {36, 5693}, {38, 601}, {40, 550}, {46, 355}, {56, 920}, {65, 22758}, {104, 3869}, {144, 6926}, {155, 222}, {165, 5534}, {191, 3576}, {226, 6862}, {255, 1062}, {329, 6891}, {392, 16203}, {405, 10202}, {411, 13243}, {484, 5881}, {499, 7082}, {513, 8279}, {517, 1158}, {518, 11248}, {527, 6705}, {546, 18540}, {548, 3587}, {549, 3929}, {602, 896}, {603, 1060}, {631, 3219}, {632, 7308}, {758, 5450}, {908, 6958}, {938, 6930}, {942, 3560}, {960, 10269}, {971, 6985}, {982, 3073}, {993, 5884}, {997, 5694}, {999, 12709}, {1001, 13373}, {1147, 7193}, {1155, 11499}, {1181, 22129}, {1210, 6929}, {1216, 3784}, {1364, 6238}, {1385, 5289}, {1407, 17814}, {1445, 5779}, {1454, 1478}, {1465, 8757}, {1482, 4018}, {1483, 1697}, {1484, 9614}, {1565, 7183}, {1656, 3306}, {1699, 7701}, {1708, 5777}, {1709, 10943}, {1737, 18961}, {1776, 3086}, {1858, 8071}, {2003, 12161}, {2077, 5904}, {2096, 6850}, {2323, 16266}, {2771, 6261}, {2800, 8666}, {2801, 6796}, {3072, 4650}, {3091, 23958}, {3157, 17102}, {3305, 3526}, {3333, 5901}, {3336, 5587}, {3337, 8227}, {3338, 3649}, {3358, 5762}, {3359, 5690}, {3487, 6892}, {3555, 10679}, {3564, 7289}, {3628, 5437}, {3651, 11220}, {3781, 5447}, {3811, 12341}, {3868, 6906}, {3870, 11849}, {3876, 6940}, {3911, 6959}, {3937, 5562}, {4292, 6917}, {4293, 7098}, {4640, 10267}, {4995, 7162}, {5223, 10270}, {5248, 12005}, {5249, 6861}, {5250, 10246}, {5251, 15016}, {5273, 6989}, {5435, 5811}, {5442, 5660}, {5535, 5691}, {5704, 6973}, {5708, 6913}, {5714, 6859}, {5720, 6924}, {5744, 6825}, {5761, 6935}, {5768, 6868}, {5780, 16417}, {5789, 18541}, {5844, 6762}, {5851, 18243}, {5905, 6833}, {6001, 11249}, {6326, 7280}, {6734, 6923}, {6846, 21454}, {6847, 9965}, {6869, 9799}, {6875, 18444}, {6883, 9940}, {6887, 9776}, {6888, 17483}, {6890, 20078}, {6905, 12528}, {6922, 13226}, {6938, 12649}, {6972, 17484}, {7686, 18761}, {10200, 15297}, {10461, 15952}, {10525, 10916}, {10526, 12616}, {10680, 12672}, {10785, 11415}, {10942, 21031}, {11012, 15071}, {12532, 18861}, {12705, 22791}, {13465, 19861}, {15888, 17699}, {16574, 19543}, {18732, 22659}, {19549, 21371}
X(24467) = midpoint of X(i) and X(j) for these {i,j}: {84, 5709}, {6869, 9799}
X(24467) = reflection of X(i) in X(j) for these (i,j): (10525, 10916), (10526, 12616), (12699, 10943)
X(24467) = X(156)-of-excentral triangle
X(24467) = X(13561)-of-6th-mixtilinear-triangle
X(24467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7171, 550), (40, 10085, 18481), (57, 7330, 5), (84, 3928, 5709), (90, 17437, 11), (255, 7004, 1062), (1155, 14872, 11499), (1709, 12704, 12699), (1768, 6763, 40), (3587, 9841, 548), (4640, 12675, 10267), (4652, 18446, 3), (5435, 5811, 6944), (5720, 15803, 6924), (10884, 21165, 3)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.
X(24468) lies on these lines: {1, 3}, {9, 6990}, {10, 6900}, {30, 6763}, {48, 16553}, {63, 2894}, {191, 12699}, {920, 9580}, {946, 6884}, {1732, 1766}, {3218, 20066}, {3219, 18483}, {3651, 3874}, {3868, 16132}, {4654, 14526}, {5127, 5358}, {5250, 15674}, {5587, 6894}, {5762, 15908}, {5904, 6985}, {6284, 16113}, {7098, 10624}, {7289, 9047}, {7411, 12005}, {10021, 16139}, {11499, 15104}, {12248, 12625}, {16127, 20078}
X(24468) = reflection of X(7982) in X(11014)
X(24468) = X(35)-of-tangential-of-excentral-triangle
X(24468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 5709, 5535), (40, 6766, 12703), (40, 12704, 3576), (484, 12702, 40), (2095, 5584, 15016), (11009, 11010, 1697)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.
X(24469) lies on these lines: {1, 22504}, {9, 114}, {40, 99}, {46, 9860}, {57, 98}, {63, 147}, {84, 2794}, {165, 12178}, {542, 3928}, {1697, 7970}, {1709, 12182}, {2782, 5709}, {3218, 5984}, {3220, 9861}, {3333, 11710}, {3929, 6054}, {5437, 6036}, {6033, 7330}, {6769, 13173}, {7061, 7350}, {12514, 21636}, {18540, 22505}
X(24469) = reflection of X(i) in X(j) for these (i,j): (1, 22504), (6769, 13173)
X(24469) = X(22504)-of-Aquila-triangle
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.
X(24470) lies on these lines: {1, 550}, {3, 7}, {4, 5708}, {5, 57}, {11, 79}, {12, 3336}, {20, 15934}, {30, 553}, {36, 3649}, {40, 4355}, {46, 495}, {56, 5901}, {58, 1086}, {63, 8728}, {65, 952}, {78, 17563}, {84, 5805}, {140, 226}, {144, 17582}, {273, 7546}, {329, 16408}, {354, 1770}, {355, 3339}, {376, 11036}, {382, 938}, {386, 17365}, {388, 5690}, {390, 3296}, {404, 17483}, {442, 3218}, {443, 3927}, {474, 5905}, {484, 15888}, {496, 1836}, {516, 5045}, {517, 4298}, {527, 5044}, {528, 3881}, {529, 3754}, {540, 24176}, {545, 3159}, {546, 1210}, {548, 4114}, {549, 4654}, {596, 5846}, {632, 5219}, {944, 1159}, {946, 20418}, {962, 7373}, {975, 17276}, {990, 8144}, {999, 4295}, {1056, 12702}, {1125, 17235}, {1155, 13407}, {1385, 3671}, {1387, 5563}, {1399, 15253}, {1407, 5707}, {1420, 10283}, {1434, 1565}, {1466, 6924}, {1478, 5221}, {1479, 4860}, {1482, 3600}, {1483, 3340}, {1595, 1892}, {1656, 5435}, {1657, 3488}, {1788, 9654}, {1876, 6756}, {2094, 17528}, {2095, 6850}, {2099, 4317}, {2476, 23958}, {2646, 11551}, {3073, 15251}, {3075, 15252}, {3219, 17529}, {3295, 3474}, {3306, 17527}, {3333, 4312}, {3361, 5886}, {3526, 5226}, {3530, 3982}, {3534, 4313}, {3579, 21620}, {3585, 12019}, {3601, 8703}, {3616, 17571}, {3627, 5722}, {3628, 3911}, {3648, 5284}, {3662, 17698}, {3678, 5852}, {3824, 5745}, {3845, 9581}, {3851, 5704}, {3868, 11112}, {3916, 5249}, {3925, 6763}, {3928, 5791}, {3937, 18180}, {3940, 6904}, {3947, 11231}, {4252, 24159}, {4303, 5453}, {4304, 12103}, {4308, 10247}, {4315, 23339}, {4316, 10543}, {4325, 5425}, {4338, 12701}, {4757, 5855}, {4880, 21677}, {4973, 4999}, {5049, 10624}, {5253, 14450}, {5260, 9782}, {5266, 24231}, {5267, 11281}, {5298, 5443}, {5326, 5442}, {5434, 5903}, {5586, 11529}, {5735, 9841}, {5777, 5843}, {5779, 6864}, {5789, 6843}, {5840, 6583}, {5842, 12005}, {5844, 10106}, {5857, 12609}, {5902, 7354}, {6284, 18398}, {6361, 6767}, {6831, 13226}, {6887, 8732}, {6894, 13243}, {6915, 13257}, {7091, 12700}, {7682, 22792}, {9655, 18391}, {9657, 10573}, {9776, 11108}, {10593, 17728}, {11009, 12735}, {11019, 22793}, {11041, 18526}, {11496, 20330}, {11518, 15704}, {11544, 12047}, {11827, 15016}, {13374, 18260}, {14054, 17616}, {15170, 17609}, {15681, 15933}, {16056, 22458}, {16415, 20805}, {16863, 18228}, {17484, 17531}, {17580, 20059}
X(24470) = midpoint of X(i) and X(j) for these {i,j}: {65, 18990}, {942, 4292}, {1770, 15171}
X(24470) = reflection of X(i) in X(j) for these (i,j): (5044, 12436), (12433, 942), (15172, 5045)
X(24470) = X(143)-of-intouch-triangle
X(24470) = X(5901)-of-2nd-anti-circumperp-tangential-triangle
X(24470) = X(6101)-of-inverse-in-incircle-triangle
X(24470) = X(10263)-of-incircle-circles-triangle
X(24470) = excentral-to-intouch similarity image of X(5)
X(24470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7, 6147), (3, 6147, 5719), (4, 21454, 5708), (46, 10404, 495), (79, 3337, 11), (354, 1770, 15171), (443, 9965, 3927), (553, 4292, 942), (999, 4295, 22791), (1836, 3338, 496), (4654, 15803, 11374), (5435, 5714, 1656), (5557, 15228, 1), (5708, 18541, 4), (11374, 15803, 549)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.
See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.
X(24471) lies on these lines: {1, 1350}, {6, 57}, {7, 8}, {12, 3844}, {37, 1423}, {46, 611}, {56, 77}, {60, 757}, {72, 17272}, {86, 1431}, {141, 226}, {181, 9436}, {193, 17490}, {210, 5232}, {241, 1400}, {273, 1875}, {279, 959}, {307, 3665}, {354, 3056}, {511, 942}, {513, 21202}, {517, 3663}, {524, 553}, {599, 4654}, {613, 3338}, {674, 5173}, {742, 4032}, {760, 18252}, {946, 24213}, {960, 3674}, {1100, 1429}, {1108, 18161}, {1155, 2330}, {1210, 5480}, {1284, 15569}, {1319, 1442}, {1351, 5708}, {1357, 1366}, {1358, 2836}, {1401, 9025}, {1432, 3863}, {1466, 7013}, {1470, 1804}, {1503, 4292}, {1834, 5929}, {1836, 12589}, {1837, 21279}, {1843, 1876}, {1887, 7282}, {2099, 7190}, {2257, 18725}, {2260, 3942}, {2262, 4000}, {2264, 7291}, {2269, 3666}, {2285, 6180}, {3057, 3672}, {3218, 15988}, {3242, 3340}, {3339, 3751}, {3361, 16475}, {3435, 7053}, {3487, 10519}, {3589, 3911}, {3618, 5435}, {3619, 5226}, {3629, 4031}, {3630, 4114}, {3631, 3982}, {3668, 3827}, {3673, 10446}, {3676, 9002}, {3739, 15985}, {3740, 5224}, {3763, 5219}, {3812, 10436}, {3875, 3880}, {4021, 9957}, {4260, 10481}, {4267, 4719}, {4269, 16696}, {4298, 5847}, {4341, 22769}, {4359, 15983}, {4419, 21871}, {4452, 14923}, {4662, 17270}, {4663, 5221}, {4731, 5936}, {4862, 5903}, {4888, 5902}, {4909, 5049}, {5083, 9024}, {5085, 15803}, {5092, 5122}, {5572, 21746}, {5846, 10106}, {6385, 18033}, {7023, 7177}, {7197, 14256}, {7269, 11011}, {7274, 18421}, {7686, 17861}, {8679, 20617}, {9004, 22277}, {9021, 15556}, {9037, 18838}, {9612, 10516}, {10360, 18935}, {10391, 18650}, {10456, 10477}, {10520, 12915}, {10914, 17151}, {12722, 15310}, {12723, 15726}, {13374, 24179}, {16603, 17239}, {17084, 17322}, {17189, 18180}, {17276, 21853}, {17705, 22464}, {18440, 18541}, {21239, 24005}
X(24471) = midpoint of X(i) and X(j) for these {i,j}: {65, 1469}, {17276, 21853}
X(24471) = X(53)-of-intouch triangle
X(24471) = X(1386)-of-2nd anti-circumperp-tangential triangle
X(24471) = X(20477)-of-inverse-in-incircle triangle
X(24471) = intouch-isogonal conjugate of X(4854)
X(24471) = excentral-to-intouch similarity image of X(6)
X(24471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2097, 7289), (7, 3212, 75), (7, 5933, 6604), (65, 1122, 7), (1423, 7146, 37)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28401.
X(24472) lies on these lines: {1, 99}, {7, 147}, {46, 10053}, {56, 11710}, {57, 98}, {65, 1355}, {114, 226}, {115, 1210}, {148, 938}, {291, 23996}, {354, 3027}, {388, 9864}, {542, 553}, {620, 13411}, {942, 2782}, {950, 23698}, {1281, 11031}, {1836, 12185}, {1837, 13182}, {1876, 12131}, {2783, 5083}, {2784, 4298}, {2785, 12016}, {2786, 11028}, {2787, 12736}, {2794, 4292}, {2795, 10122}, {3218, 5985}, {3338, 10069}, {3339, 9860}, {3340, 7970}, {3488, 13172}, {3586, 10723}, {3601, 21166}, {3671, 21636}, {3911, 6036}, {4654, 6054}, {4697, 16598}, {5708, 12188}, {5722, 6321}, {5984, 21454}, {5988, 9436}, {8591, 15933}, {9579, 10722}, {9581, 14639}, {10072, 12258}, {10404, 12184}, {11019, 11599}, {11374, 15561}, {11518, 23235}, {13178, 18391}, {13188, 15934}
X(24472) = midpoint of X(65) and X(3023)
X(24472) = incircle-inverse of X(741)
X(24472) = center of the circle {{X(65), X(1356), X(3023)}}
X(24472) = X(129)-of-intouch-triangle
X(24472) = X(1298)-of-inverse-in-incircle-triangle
X(24472) = X(11710)-of-2nd anti-circumperp-tangential-triangle
X(24472) = {X(1), X(10089)}-harmonic conjugate of X(11711)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.
X(24473) lies on these lines: {1,3052}, {2,72}, {3,11520}, {7,3419}, {8,4004}, {9,17542}, {30,1071}, {46,4421}, {56,12559}, {57,5440}, {63,15934}, {65,519}, {78,5708}, {210,3828}, {226,17530}, {354,392}, {376,517}, {381,912}, {405,3929}, {474,11523}, {518,599}, {527,5728}, {528,11570}, {537,21080}, {549,10202}, {597,9021}, {956,11529}, {960,3901}, {971,3543}, {995,3999}, {997,4860}, {999,4930}, {1125,3962}, {1159,3872}, {1210,17533}, {1385,16139}, {1698,4533}, {1739,4849}, {1858,11238}, {2093,3243}, {2095,18446}, {2771,10706}, {2796,12723}, {2802,4744}, {2886,11551}, {3057,3881}, {3058,12711}, {3218,17549}, {3219,16861}, {3244,4757}, {3306,3940}, {3333,5730}, {3338,12635}, {3339,5687}, {3488,9965}, {3524,9940}, {3534,13369}, {3545,5777}, {3601,19704}, {3626,3922}, {3632,10107}, {3634,4005}, {3649,10916}, {3653,13373}, {3654,11239}, {3656,11240}, {3681,3921}, {3697,3812}, {3698,4745}, {3740,4539}, {3742,5692}, {3754,4669}, {3811,5221}, {3829,12047}, {3833,4134}, {3839,5806}, {3869,5045}, {3877,5049}, {3878,17609}, {3889,9957}, {3892,5919}, {3899,10179}, {3902,17145}, {3927,16857}, {3951,11108}, {3984,16408}, {4067,19883}, {4127,19862}, {4234,5208}, {4654,14054}, {4677,5836}, {4737,20892}, {4906,5315}, {4980,5295}, {5057,18527}, {5122,23958}, {5563,16126}, {5570,10072}, {5693,13374}, {5722,5905}, {5844,10273}, {5887,6583}, {6147,6734}, {6875,15178}, {6906,10222}, {7373,11682}, {7672,18419}, {7682,13257}, {9841,11531}, {10056,13750}, {10122,17525}, {10156,15721}, {10246,21165}, {11111,15933}, {11220,15683}, {11227,15692}, {12005,14110}, {12109,21849}, {12527,17706}, {14450,22793}, {15650,19536}, {15803,19705}, {16465,17579}, {17619,21077}, {19251,22345}, {21969,23154}
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.
Let A'B'C' be the orthic triangle. Let LA be the reflection of line B'C' in the internal angle bisector of A, and define LB and LC cyclically. Let A" = LB∩LC and define B" and C" cyclically. X(24474) = X(3)-of-A"B"C". (Randy Hutson, November 30, 2018)
X(24474) lies on these lines: {1,3}, {2,5761}, {4,912}, {5,72}, {7,6850}, {8,6826}, {10,6881}, {19,3211}, {20,13369}, {28,110}, {30,1071}, {34,3157}, {37,5755}, {52,1866}, {63,3560}, {68,5130}, {78,6911}, {119,21077}, {140,5439}, {142,3754}, {155,1829}, {210,9956}, {225,1830}, {226,6842}, {227,5399}, {283,18180}, {329,5804}, {355,518}, {381,5715}, {382,971}, {389,12109}, {392,5771}, {429,12259}, {443,10597}, {500,15852}, {511,13408}, {515,3874}, {516,5884}, {546,5927}, {550,10167}, {579,8609}, {758,946}, {916,12162}, {938,5758}, {944,3873}, {950,7491}, {952,3555}, {960,5791}, {962,5768}, {1046,3073}, {1064,2650}, {1066,1254}, {1104,5398}, {1210,6882}, {1320,12776}, {1389,4861}, {1393,22350}, {1439,22464}, {1479,1858}, {1512,10942}, {1656,5044}, {1699,3901}, {1770,5840}, {1828,5446}, {1836,10525}, {1837,10526}, {1870,3562}, {1871,15762}, {1891,12134}, {1898,3583}, {1902,11472}, {1998,19541}, {2262,2323}, {2771,7728}, {2778,12262}, {2800,4084}, {2802,9946}, {2808,13474}, {2836,9970}, {3090,3876}, {3218,6906}, {3219,6920}, {3419,6917}, {3487,6825}, {3488,6868}, {3529,11220}, {3577,6762}, {3649,15908}, {3651,18444}, {3678,10175}, {3681,5818}, {3753,5690}, {3811,11499}, {3812,10198}, {3817,4067}, {3827,19149}, {3851,10157}, {3869,5603}, {3877,6857}, {3878,5745}, {3881,5882}, {3889,7967}, {3892,13607}, {3894,5691}, {3916,6914}, {3927,6913}, {3940,6918}, {3962,5694}, {4018,8727}, {4185,9928}, {4219,15062}, {4297,12005}, {4311,5083}, {4325,12119}, {4463,5797}, {4848,12736}, {4857,16155}, {5231,18493}, {5295,20237}, {5435,6961}, {5440,6924}, {5480,9021}, {5497,6011}, {5534,18518}, {5587,5904}, {5657,6989}, {5692,8227}, {5703,6954}, {5704,6978}, {5713,5752}, {5714,6982}, {5720,11523}, {5722,5812}, {5728,5762}, {5734,5744}, {5748,6981}, {5763,6922}, {5770,6847}, {5787,5878}, {5841,10572}, {5844,10914}, {5883,6684}, {6147,6907}, {6261,12559}, {6288,14872}, {6797,19914}, {6863,11374}, {6885,14923}, {6908,11036}, {6985,11520}, {7970,13190}, {7978,13218}, {7983,12190}, {7984,12382}, {8261,16139}, {9581,18397}, {9856,18544}, {10156,15720}, {10449,20928}, {10531,11415}, {10625,11573}, {10693,12261}, {10698,13279}, {10705,13119}, {11246,11826}, {11571,12750}, {12635,22753}, {12650,12687}, {12675,18481}, {12695,13743}, {12711,15171}, {13099,13314}, {13226,17654}, {13729,17484}, {17563,17612}, {17625,18990}, {17661,22799}, {17857,18491}, {18239,22792}, {18254,23513}, {18357,18908}
X(24474) = reflection of X(3) in X(942)
X(24474) = anticomplement of X(31837)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.
X(24475) lies on these lines: {1,1399}, {3,3218}, {4,17483}, {5,226}, {7,6917}, {10,5885}, {30,1071}, {40,3894}, {52,23154}, {57,6924}, {65,952}, {72,140}, {145,6948}, {182,9021}, {354,5887}, {355,5270}, {381,12528}, {495,13750}, {496,1858}, {517,550}, {518,5690}, {548,10167}, {549,9940}, {632,5044}, {758,1385}, {916,5876}, {938,6929}, {946,1484}, {960,13373}, {971,3627}, {1125,5694}, {1482,3873}, {1657,11220}, {1735,5399}, {1807,3075}, {1870,23070}, {2095,6985}, {2167,19210}, {2800,3881}, {2801,18480}, {3157,12161}, {3337,6326}, {3339,5534}, {3487,5770}, {3526,3876}, {3555,5844}, {3560,15934}, {3562,18455}, {3576,3901}, {3628,5439}, {3635,10284}, {3670,5396}, {3678,11231}, {3845,5806}, {3850,5927}, {3869,10246}, {3877,17571}, {3878,15178}, {3889,10247}, {3916,7508}, {3927,6883}, {4067,10165}, {4325,5903}, {4430,12245}, {4880,10902}, {5045,10283}, {5083,19907}, {5208,15952}, {5221,11499}, {5446,12109}, {5536,16132}, {5563,6265}, {5693,5886}, {5708,6911}, {5728,5843}, {5730,16203}, {5883,9956}, {5904,15016}, {5905,6928}, {6001,22791}, {6101,11573}, {6824,11036}, {6868,9965}, {6885,21454}, {6902,17484}, {6907,14054}, {6923,12649}, {6936,20078}, {6942,23958}, {6955,20013}, {7171,7982}, {7330,11518}, {10269,12635}, {10273,10914}, {10942,18838}, {11009,11571}, {11227,15712}, {11230,20117}, {11362,13145}, {11567,11715}, {12009,19862}, {12558,22798}, {12699,15071}, {12711,15172}, {13243,21669}, {14872,18357}, {15096,18406}, {15931,16139}, {21740,22765}
X(24475) = anticomplement of X(31835)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28407.
X(24476) lies on these lines: {1,159}, {6,169}, {7,8}, {38,22097}, {46,12329}, {72,141}, {81,105}, {182,10202}, {210,3844}, {511,13408}, {517,990}, {611,13750}, {613,5570}, {912,1352}, {1071,1503}, {1172,18178}, {1284,17447}, {1738,21867}, {1824,3782}, {1843,23154}, {1858,12589}, {1876,6180}, {2002,18838}, {2352,18607}, {2771,14982}, {2809,3755}, {2879,23770}, {3057,3100}, {3313,11573}, {3555,5846}, {3589,5439}, {3619,3876}, {3666,17441}, {3744,20999}, {3751,5902}, {3763,5044}, {3821,4523}, {3873,19993}, {3874,5847}, {4267,18183}, {4463,17184}, {5085,9940}, {5728,5845}, {5777,10516}, {5848,11570}, {5903,16496}, {9028,18389}, {9053,10914}, {9895,24159}, {11220,14927}, {12723,17768}, {16475,18398}
Centers associated with the obverse and N-obverse triangles of X(1): X(24477) - X(24481))
These centers were contributed by Randy Hutson, October 5, 2018. Obverse and N-obverse triangles are introduced in preambles just before X(24307) and X(24310).
X(24477) lies on these lines: {1, 5745}, {2, 210}, {3, 3189}, {4, 10916}, {7, 2886}, {8, 56}, {9, 11019}, {10, 1056}, {11, 329}, {38, 11269}, {46, 5082}, {48, 3684}, {55, 5744}, {57, 2550}, {63, 497}, {69, 1447}, {72, 3086}, {75, 7196}, {78, 7288}, {142, 10980}, {144, 5274}, {145, 2646}, {165, 5853}, {200, 3911}, {226, 5231} et al
X(24478) lies on these lines: {1,2092}, {6, 291}, {9, 12782}, {38, 1654}, {43, 3779}, {69, 982}, {71, 238}, {75, 256}, {239, 3778}, {319, 4022}, {350, 21257}, {614, 22370}, {894, 23659}, {966, 984}, {978, 3781}, {1045, 21746} et al
X(24479) lies on these lines: {39, 8868}, {239, 1916}, {291, 1580}, {511, 1757}, {672, 1931}, {1959, 3930} et al
X(24479) = isogonal conjugate of X(19557)
X(24479) = isotomic conjugate of X(18037)
X(24479) = cevapoint of circumcircle intercepts of line PU(6) (line X(37)X(513))
X(24480) lies on these lines: {101, 3923}, {292, 4493}, {870, 1438}, {1001, 9259}
X(24481) lies on these lines: {274, 284}, {893, 7242}, {1376, 18755} : :
Collineation mappings involving trilinear obverse triangles: X(24482)-X(24518)
Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. Trilinears for P are p/a : q/b : r/c. The trilinear obverse triangle of P is here introduced as the triangle A'B'C', where A' = p/a : r/c : q/b, B' = r/c : q/b : p/a, C' = q/b : p/a : r/c (trilinears). The same triangle is given by barycentrics A' = b c p : b^2 r : c^2 q, B' = a^2 r : c a q : p c^2 p, C' = a^2 q : b^2 p : a b r. For example, the trilinear obverse triangle of X(2) has first barycentric b c : b^2 : c^2. (Clark Kimberling, October 5, 2018)
X(24482) lies on these lines: {2, 513}, {6, 3573}, {8, 24004}, {9, 1026}, {44, 751}, {45, 660}, {55, 765}, {75, 4124}, {190, 3271}, {238, 993}, {244, 24722}, {320, 4679}, {344, 9309}, {518, 1992}, {644, 10755}, {668, 25333}, {846, 9282}, {894, 24346}, {1001, 3257}, {1052, 17596}, {1083, 4585}, {1086, 4499}, {1621, 6163}, {2876, 25316}, {3056, 17336}, {3097, 3764}, {3122, 9359}, {3768, 24493}, {3792, 4759}, {3799, 4370}, {3888, 4422}, {4459, 18151}, {4473, 4553}, {4579, 16686}, {4965, 20881}, {5150, 24436}, {6373, 24508}, {9025, 17264}, {16494, 24405}, {16495, 24338}, {17777, 18191}, {24483, 24490}, {24484, 24486}, {24488, 24492}, {24502, 24505}, {24509, 24515}
X(24483) lies on these lines: {2, 656}, {24482, 24490}, {24484, 24489}, {24485, 24518}
X(24484) lies on these lines: {2, 661}, {3, 6}, {7, 24404}, {100, 20974}, {171, 23636}, {291, 9471}, {513, 5701}, {537, 4876}, {672, 3792}, {894, 24401}, {984, 2113}, {1655, 17300}, {2110, 20670}, {2112, 24436}, {2170, 24715}, {2284, 3730}, {2810, 6184}, {3061, 4655}, {3684, 20729}, {3937, 23988}, {3939, 10756}, {8298, 20465}, {14825, 21143}, {17234, 17671}, {17265, 17675}, {23466, 24508}, {24482, 24486}, {24483, 24489}, {24488, 24494}, {24502, 24510}
X(24485) lies on these lines: {2, 513}, {145, 24004}, {894, 24409}, {3616, 24397}, {16495, 25382}, {24483, 24518}
X(24486) lies on these lines: {2, 650}, {37, 2275}, {1001, 22116}, {1015, 24841}, {6376, 17263}, {12782, 16484}, {16706, 20269}, {24482, 24484}, {24502, 24508}, {24509, 24510}
X(24487) lies on these lines: {1, 24004}, {2, 513}, {75, 25382}, {87, 18040}, {551, 726}, {751, 2228}, {894, 24405}, {1125, 24399}, {3248, 4033}, {24502, 24503}
X(24488) lies on these lines: {2, 1491}, {42, 2183}, {101, 23646}, {659, 24499}, {876, 14947}, {894, 24442}, {2503, 20975}, {17463, 24712}, {24482, 24492}, {24484, 24494}
X(24489) lies on these lines: {2, 8060}, {6, 20839}, {21339, 24713}, {24483, 24484}
X(24490) lies on these lines: {1, 3122}, {17256, 24433}, {24482, 24483}
X(24491) lies on these lines: {1, 6}, {2, 649}, {672, 4759}, {748, 21369}, {874, 17277}, {875, 24494}, {894, 24455}, {1018, 4432}, {2112, 5150}, {4090, 23415}, {4676, 16549}, {9360, 16569}, {17335, 23891}, {24482, 24484}, {24493, 24517}, {24505, 24508}
X(24492) lies on these lines: {2, 659}, {38, 3248}, {87, 24413}, {190, 1621}, {846, 9359}, {894, 24447}, {17601, 20331}, {24482, 24488}
X(24493) lies on these lines: {2, 798}, {2245, 24512}, {3768, 24482}, {16482, 20979}, {24483, 24484}, {24491, 24517}
X(24494) lies on these lines: {2, 513}, {6, 31}, {875, 24491}, {993, 1083}, {2810, 8299}, {3240, 16467}, {3251, 9320}, {3271, 20331}, {3720, 24405}, {3873, 4664}, {9016, 14439}, {17135, 24004}, {24484, 24488}
X(24495) lies on these lines: {2, 513}, {7, 24801}, {354, 4664}, {749, 1100}, {4595, 23456}, {10453, 24004}, {20331, 25048}, {24498, 24516}
X(24496) lies on these lines: {1, 39}, {2, 514}, {672, 1023}, {1575, 2087}, {1698, 19934}, {3573, 5030}, {4089, 20335}, {24482, 24484}, {24508, 24510}
X(24497) lies on these lines: {2, 812}, {45, 1015}, {1001, 4752}, {8671, 23855}, {24482, 24484}
X(24498) lies on these lines: {2, 2254}, {55, 2316}, {894, 24453}, {982, 4419}, {24482, 24483}, {24484, 24488}, {24495, 24516}
X(24499) lies on these lines: {2, 650}, {57, 9319}, {81, 3658}, {100, 1814}, {222, 7045}, {241, 2275}, {659, 24488}, {664, 7117}, {673, 14936}, {1027, 14947}, {1155, 3240}, {1376, 9503}, {1447, 8608}, {3732, 13006}, {6376, 18751}, {7493, 8758}, {9358, 17074}, {9441, 12782}, {20096, 21859}, {24482, 24483}
X(24500) lies on these lines: {662, 20975}, {895, 5546}, {2611, 24711}, {3448, 24713}, {4552, 25051}, {4565, 14060}, {17463, 21221}, {24482, 24488}, {24483, 24484}, {24505, 24516}
X(24501) lies on these lines: {2, 650}, {55, 22116}, {2275, 3666}, {2991, 4571}, {3750, 12782}, {24482, 24488}
X(24502) lies on these lines: {1, 87}, {2, 649}, {6, 874}, {101, 24815}, {384, 24279}, {889, 4363}, {1977, 1978}, {2108, 19579}, {3249, 21211}, {3758, 24004}, {4116, 24696}, {4568, 24825}, {4672, 17499}, {6382, 23538}, {8054, 21224}, {10009, 24343}, {18046, 24731}, {24482, 24505}, {24484, 24510}, {24486, 24508}, {24487, 24503}
X(24503) lies on these lines: {2, 798}, {894, 24338}, {6386, 21762}, {24487, 24502}
X(24504) lies on these lines: {2, 661}, {39, 1931}, {99, 20982}, {100, 20670}, {148, 2170}, {329, 1655}, {335, 21220}, {662, 2503}, {756, 9510}, {894, 24347}, {896, 3240}, {1005, 2651}, {2185, 2644}, {2608, 3219}, {2641, 4641}, {3930, 20536}, {4366, 20974}, {6625, 17451}, {17483, 24404}, {18139, 18140}, {21221, 24712}, {24482, 24488}
X(24505) lies on these lines: {2, 661}, {8, 192}, {39, 2669}, {141, 18140}, {148, 24715}, {244, 21220}, {645, 10754}, {799, 3124}, {894, 2607}, {3952, 25047}, {6646, 24404}, {9505, 24325}, {24482, 24502}, {24491, 24508}, {24500, 24516}
X(24506) lies on these lines: {2, 2642}, {39, 4526}, {86, 24287}, {798, 8062}, {900, 17398}, {992, 4435}, {1213, 24959}, {3716, 3768}, {4800, 24512}, {7253, 8061}, {8060, 17217}, {8632, 14288}, {8674, 21053}, {17362, 21714}
X(24507) lies on these lines: {2, 513}, {3621, 24004}, {3622, 24418}
X(24508) lies on these lines: {2, 812}, {190, 1015}, {291, 4432}, {668, 4422}, {3227, 4370}, {4473, 9263}, {6373, 24482}, {23466, 24484}, {24486, 24502}, {24491, 24505}, {24496, 24510}
X(24509) lies on these lines: {1, 4704}, {2, 649}, {874, 17349}, {894, 24460}, {23466, 24484}, {24482, 24515}, {24486, 24510}
X(24510) lies on these lines: {2, 513}, {335, 24722}, {2276, 3758}, {4664, 4795}, {6542, 24004}, {16826, 24423}, {24484, 24502}, {24486, 24509}, {24496, 24508}
X(24511) lies on these lines: {2, 1755}, {6, 3725}, {9, 1215}, {10, 2179}, {39, 14815}, {71, 3683}, {169, 2083}, {354, 20785}, {375, 8608}, {579, 7262}, {1125, 4020}, {2260, 4641}, {2276, 3764}, {4672, 17754}, {5275, 20665}, {10176, 14964}, {14963, 15049}, {20470, 22099}, {20610, 24631}, {22061, 23383}
X(24512) lies on these lines: {1, 39}, {2, 6}, {10, 3780}, {32, 16783}, {37, 38}, {42, 1100}, {43, 1051}, {55, 16693}, {58, 16850}, {65, 3727}, {75, 17027}, {76, 4754}, {83, 1509}, {171, 1914}, {213, 1125}, {244, 21840}, {274, 17034}, {284, 16056}, {350, 894}, {384, 17103}, {404, 18755}, {405, 5021}, {474, 2271}, {519, 16971}, {551, 3230}, {572, 4192}, {579, 8731}, {594, 17135}, {612, 16973}, {614, 16972}, {750, 2280}, {851, 2278}, {899, 16666}, {942, 3721}, {999, 19263}, {1002, 1390}, {1030, 4210}, {1107, 1475}, {1172, 4212}, {1193, 16604}, {1468, 4426}, {1574, 3293}, {1621, 17735}, {1931, 19237}, {1985, 3330}, {2176, 3616}, {2229, 5109}, {2235, 4670}, {2239, 21764}, {2241, 5264}, {2242, 16788}, {2245, 24493}, {2269, 20359}, {2345, 10453}, {2549, 11355}, {3122, 24513}, {3125, 5883}, {3216, 20970}, {3290, 3742}, {3550, 10987}, {3571, 9506}, {3622, 16969}, {3664, 20335}, {3678, 25089}, {3684, 17122}, {3693, 4883}, {3735, 5902}, {3739, 24592}, {3741, 5750}, {3751, 3789}, {3758, 4465}, {3770, 18152}, {3783, 4649}, {3812, 21951}, {3833, 16611}, {3874, 3954}, {4161, 23493}, {4184, 5124}, {4199, 5019}, {4204, 5115}, {4251, 5277}, {4253, 5283}, {4254, 16059}, {4283, 4476}, {4357, 24690}, {4360, 17759}, {4363, 4441}, {4368, 4672}, {4435, 14438}, {4503, 4667}, {4559, 15950}, {4651, 17362}, {4653, 5030}, {4657, 24691}, {4675, 24712}, {4697, 24259}, {4800, 24506}, {5013, 19765}, {5110, 13588}, {5120, 16058}, {5253, 21008}, {5254, 23903}, {5332, 16779}, {5439, 16583}, {5710, 16781}, {5711, 16502}, {5819, 20995}, {6625, 16044}, {6626, 16927}, {6685, 20228}, {7109, 16685}, {10176, 21839}, {10436, 17026}, {10459, 17448}, {10582, 16970}, {14210, 24254}, {14968, 18907}, {16514, 16826}, {16517, 17022}, {16521, 17021}, {16522, 17012}, {16523, 17316}, {16525, 17032}, {16552, 16589}, {16569, 16667}, {16777, 17597}, {16782, 17023}, {16818, 21240}, {16884, 17018}, {17031, 24325}, {17045, 25349}, {17126, 21793}, {17141, 25263}, {17365, 20347}, {17499, 18140}, {17752, 25303}, {20464, 23532}
X(24512) = {X(2),X(6)}-harmonic conjugate of X(2238)
X(24513) lies on these lines: {2, 256}, {6, 23398}, {42, 51}, {291, 4672}, {672, 3778}, {1009, 14815}, {1193, 23414}, {1201, 23473}, {2085, 16600}, {2276, 3764}, {3122, 24512}, {3720, 22172}, {4022, 24690}, {17135, 22167}, {17759, 24478}, {21753, 23444}, {22343, 23632}
X(24514) lies on these lines: {1, 1655}, {2, 7}, {6, 350}, {31, 385}, {37, 17032}, {41, 384}, {42, 192}, {43, 3729}, {44, 17028}, {72, 1008}, {75, 2238}, {76, 213}, {171, 16997}, {190, 2276}, {193, 10453}, {194, 1193}, {218, 7770}, {238, 16998}, {239, 4441}, {257, 3869}, {316, 4805}, {330, 1201}, {386, 25264}, {536, 21904}, {748, 17000}, {750, 16999}, {870, 16514}, {1575, 17351}, {1654, 4388}, {1743, 17026}, {1909, 2176}, {1958, 16956}, {2106, 8033}, {2251, 3972}, {2280, 4366}, {2295, 6376}, {2887, 16991}, {3195, 9308}, {3314, 4766}, {3570, 4386}, {3720, 17379}, {3741, 4416}, {3758, 4465}, {3759, 4479}, {3760, 17034}, {3780, 17144}, {3783, 3923}, {3789, 5263}, {3963, 7109}, {3997, 6381}, {4110, 25287}, {4431, 4685}, {4672, 17793}, {4676, 8299}, {5025, 24995}, {5276, 14621}, {6180, 7196}, {6327, 17007}, {6645, 9310}, {6679, 17003}, {7754, 16466}, {7766, 21764}, {12263, 16476}, {16468, 17031}, {16552, 17030}, {16670, 17029}, {16969, 25303}, {17001, 17126}, {17002, 17127}, {17018, 17319}, {17135, 17363}, {17334, 25349}, {17347, 24690}, {17349, 24592}, {17743, 18031}, {17750, 18140}, {17751, 20109}, {19974, 24455}
X(24515) lies on these lines: {2, 650}, {39, 874}, {75, 2275}, {5263, 22116}, {6376, 17279}, {17280, 21226}, {24482, 24509}, {24484, 24502}, {24491, 24505}
X(24516) lies on these lines: {1, 3122}, {2, 1491}, {190, 22116}, {320, 3675}, {2245, 3573}, {14947, 23838}, {24482, 24484}, {24495, 24498}, {24500, 24505}
X(24517) lies on these lines: {2, 513}, {10, 24004}, {37, 23354}, {75, 24338}, {740, 3679}, {751, 17250}, {16709, 22174}, {20006, 25036}, {24491, 24493}, {24500, 24505}
X(24518) lies on these lines: {2, 661}, {24483, 24485}
Collineation mappings involving trilinear N-obverse triangles: X(24519)-X(24536)
Let P = p : q : r (barycentrics) be a point in the plane of a triangle ABC. Trilinears for P are p/a : q/b : r/c. The trilinear N-obverse triangle of P is here introduced as the triangle A'B'C', where A' = - p/a : r/c : q/b, B' = r/c : - q/b : p/a, C' = q/b : p/a : - r/c (trilinears). The same triangle is given by barycentrics A' = - b c p : b^2 r : c^2 q, B' = a^2 r : - c a q : p c^2 p, C' = a^2 q : b^2 p : - a b r. For example, the trilinear obverse triangle of X(2) has first barycentric - b c : b^2 : c^2. (Clark Kimberling, October 5, 2018)
X(24519) lies on these lines: {2, 17448}, {1386, 3779}, {1654, 16744}, {2275, 3759}, {3616, 24944}, {18194, 24520}, {24880, 24962}
X(24520) lies on these lines: {2, 3728}, {3, 6}, {10, 24437}, {43, 24530}, {244, 24919}, {17038, 24944}, {18194, 24519}, {22167, 24958}, {24525, 24528}, {24526, 24527}
X(24521) lies on these lines: {2, 3056}, {24522, 24529}
X(24522) lies on these lines: {2, 17448}, {194, 1992}, {330, 5839}, {3189, 3779}, {3227, 24897}, {3780, 9263}, {16722, 20090}, {24521, 24529}
X(24523) lies on these lines: {2, 3056}, {1100, 2275}, {1246, 6384}, {1964, 24478}, {3769, 3779}, {3794, 15985}, {16574, 18788}, {18194, 24519}
X(24524) lies on these lines: {1, 668}, {2, 17448}, {6, 17743}, {7, 8}, {41, 18047}, {42, 17149}, {43, 6384}, {76, 519}, {99, 8715}, {145, 350}, {183, 12513}, {190, 3208}, {192, 21219}, {193, 17787}, {194, 20691}, {239, 20917}, {274, 3679}, {304, 4737}, {312, 3765}, {313, 17377}, {321, 20055}, {325, 12607}, {330, 1575}, {335, 3959}, {341, 18156}, {386, 18148}, {529, 7750}, {535, 7802}, {536, 20081}, {740, 9902}, {799, 2177}, {869, 19567}, {894, 4110}, {1078, 8666}, {1334, 17336}, {1655, 4664}, {1965, 3870}, {1966, 3751}, {1975, 3913}, {2170, 18055}, {2176, 10027}, {2275, 9263}, {2276, 21226}, {2295, 3758}, {3187, 19803}, {3241, 18135}, {3244, 6381}, {3264, 17378}, {3501, 4595}, {3570, 9310}, {3596, 3879}, {3621, 4441}, {3625, 20888}, {3632, 3761}, {3633, 3760}, {3673, 18159}, {3681, 18059}, {3691, 17335}, {3729, 4050}, {3759, 3780}, {3769, 4039}, {3770, 17299}, {3873, 20352}, {3948, 17389}, {3963, 17363}, {3975, 17316}, {4251, 4482}, {4377, 4725}, {4385, 17762}, {4386, 6645}, {4474, 20954}, {4505, 4672}, {4651, 22293}, {4738, 7278}, {4774, 7199}, {4851, 20923}, {4852, 18144}, {4865, 21590}, {4876, 9311}, {7257, 17103}, {7773, 11236}, {8026, 20537}, {9025, 24351}, {10987, 17692}, {14584, 20566}, {16549, 23891}, {17373, 20891}, {17375, 20892}, {17386, 18137}, {17393, 18133}, {19581, 19586}, {20060, 20553}, {21223, 21877}, {24661, 25120}, {25274, 25290}, {25276, 25281}, {25277, 25279}, {25282, 25294}, {25292, 25295}
X(24525) lies on these lines: {2, 24657}, {42, 2260}, {24520, 24528}
X(24526) lies on these lines: {2, 24658}, {24520, 24527}
X(24527) lies on these lines: {2, 24660}, {24520, 24526}
X(24528) lies on these lines: {2, 17448}, {6, 31}, {8, 16606}, {43, 2275}, {200, 292}, {291, 2319}, {312, 21893}, {518, 20284}, {893, 3751}, {1015, 16569}, {1575, 5839}, {2162, 23579}, {2277, 21904}, {3121, 3681}, {3240, 23632}, {3961, 23543}, {4651, 25293}, {4661, 8620}, {6384, 9263}, {6685, 16975}, {20012, 21877}, {20271, 22173}, {24520, 24525}
X(24529) lies on these lines: {1, 4531}, {517, 3500}, {3212, 20460}, {3759, 6647}, {24521, 24522}
X(24530) lies on these lines: {2, 37}, {6, 662}, {39, 2669}, {43, 24520}, {81, 4272}, {82, 8301}, {86, 2092}, {190, 21796}, {256, 2234}, {291, 872}, {319, 21857}, {386, 2274}, {749, 1469}, {869, 4446}, {897, 5968}, {980, 5224}, {986, 6042}, {1045, 3122}, {1054, 13610}, {1333, 19308}, {1654, 16696}, {1698, 21730}, {1740, 3764}, {1757, 3216}, {1964, 24478}, {2275, 3759}, {2352, 19345}, {2643, 24372}, {2664, 21035}, {2895, 18601}, {3125, 18714}, {3783, 4022}, {3959, 18041}, {4277, 17379}, {4360, 17053}, {5069, 17349}, {6542, 21858}, {6703, 25059}, {7235, 24883}, {8610, 17319}, {16700, 17778}, {16726, 20090}, {17256, 21892}, {17315, 20691}, {18042, 21008}, {20360, 24443}, {22008, 24170}, {24050, 24185}, {24880, 25065}, {24890, 24899}
X(24531) lies on these lines: {2, 25138}
X(24532) lies on these lines: {650, 1459}, {5256, 23654}, {23655, 25142}
X(24533) lies on these lines: {2, 669}, {43, 18197}, {44, 513}, {512, 24782}, {667, 17072}, {876, 7180}, {890, 20295}, {1499, 19543}, {2517, 4874}, {2533, 3907}, {3004, 25300}, {3835, 8640}, {4083, 22090}, {4467, 17989}, {5518, 22386}, {8630, 24601}, {8637, 21260}, {8651, 25084}, {16695, 21051}, {18093, 18106}, {19308, 21005}, {21191, 24674}, {21351, 22322}, {24755, 25128}
X(24533) = complement of X(26148)
X(24534) lies on these lines: {42, 23658}, {661, 3907}, {669, 2451}, {8640, 23466}, {22199, 23574}
X(24535) lies on these lines: {2, 23655}, {7234, 17033}
X(24536) lies on these lines: {2, 3056}, {2275, 3759}
Collineation mappings involving Gemini triangle 1: X(24537)-X(24571)
Following is a list of central triangles, by barycentric coordinates of A-vertex. The full names are Gemini triangle 1, Gemini triangle 2, Gemini triangle 3, etc. (Gemini is a constellation of stars - and the Latin noun for twin.) These triangles are cited in the sequel. (Clark Kimberling, October 6, 2018)
Gemini 1 a : b + c : b + c
Gemini 2 - a : b + c : b + c
Gemini 3 a : a + b : a + c
Gemini 4 - a : a + b : a + c
Gemini 5 a : a - b : a - c
Gemini 6 a : b - a : c - a
Gemini 7 a : a - c : a - b
Gemini 8 a : c - a : b - a (Garcia reflection triangle)
Gemini 9 a : b + c - a : b + c - a
Gemini 10 - a : b + c - a : b + c - a
Gemini 11 a : a + b + c : a + b + c
Gemini 12 - a : a + b + c : a + b + c
Gemini 13 b + c : a : a
Gemini 14 b + c : 2a : 2a
Gemini 15 b + c : b : c (Gergonne line extraversion triangle; see X(10180). See note below.)
Gemini 16 b + c : c : b (See note below.)
Gemini 17 b + c : b - c : c - b (See note below.)
Gemini 18 b + c : c - b : b - c (See note below.)
Gemini 19 b + c : a + b : a + c (See note below.)
Gemini 20 2b + 2c : a : a (See note below.)
Gemini 21 a + b + c : a : a
Gemini 22 a + b + c : - a : - a
Gemini 23 a + b + c : b + c : b + c
Gemini 24 a + b + c : - b - c : - b - c
Gemini 25 a + b + c : a + b : a + c
Gemini 26 a + b + c : a + c : a + b
Gemini 27 a - b - c : a : a
Gemini 28 a - b - c : b + c : b + c (See note below.)
Gemini 29 a : b - c : c - b (See note below.)
Gemini 30 a : c - b : b - c (Inner Conway triangle; see note below.)
Gemini 31 b c : a^2 : a^2
Gemini 32 - b c : a^2 : a^2
Gemini 33 a^2 : b c : b c
Gemini 34 - a^2 : b c : b c
Gemini 35 cos A : 1 : 1
Gemini 36 - cos A : 1 : 1
Gemini 37 sec A : 1 : 1
Gemini 38 - sec A : 1 : 1
Gemini 39 -a + b + c : a + b + c : a + b + c
Gemini 40 a + b + c : - a + b + c : - a + b + c
Notes: (Randy Hutson, beginning November 9, 2018)
Gemini triangle 15 = complement of cevian triangle of X(75)
Gemini triangle 16 = complement of incentral triangle
Gemini triangle 17 = anticomplement of cevian triangle of X(75)
Gemini triangle 17 = anticomplement of anticomplement of Gemini triangle 15
Gemini triangle 18 = anticomplement of incentral triangle
Gemini triangle 19 = medial triangle of obverse triangle of X(1)
Gemini triangle 19 = obverse triangle of X(10)
Gemini triangle 20 = complement of Gemini triangle 28
Gemini triangle 23 = complement of Gemini triangle 39
Gemini triangle 28 = anticomplement of Gemini triangle 20
Gemini triangle 29 = anticomplement of extouch triangle
Gemini triangle 29 = anticomplement of anticomplement of 1st Zaniah triangle
Gemini triangle 30 = anticomplement of intouch triangle
Gemini triangle 30 = anticomplement of anticomplement of 2nd Zaniah triangle
Gemini triangle 33 = unary cofactor triangle of 2nd Sharygin triangle
Gemini triangle 34 = unary cofactor triangle of 1st Sharygin triangle
Gemini triangle 39 = anticomplement of Gemini triangle 23
For more Gemini triangles, see the preambles just before X(26153) and X(27378).
If T is a central triangle A'B'C' with A' of the form f(a,b,c) : g(a,b,c) : g(a,b,c), then the (A,B,C,X(2); A',B',C',X(2)) collineation image of the Euler line is the Euler line. Examples include Gemini triangles 1,2,9,10,11,12,13,14,21,22,23,24,27,28, and 31-40.
Let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 1, as in centers X(24537)-X(24571). Then
m(X) = a (a + b - c)(a - b + c) x + (b + c - a) (a + b - c) (a + c) y + (b + c - a) (a - b + c) (a + b) z : : ,
and m(X) is on the Euler line if and only if X is on the Euler line.
X(24537) lies on these lines: {1, 17860}, {2, 3}, {8, 394}, {10, 255}, {77, 1441}, {92, 4296}, {189, 1220}, {1074, 1125}, {1437, 5767}, {1944, 3869}, {2199, 20262}, {5930, 18652}, {19861, 24552}, {24540, 24546}
X(24538) lies on these lines: {2, 3}, {4294, 23541}, {4296, 6350}, {8270, 19860}, {10436, 22464}, {11427, 19767}, {19765, 23292}, {24553, 24555}
X(24539) lies on these lines: {2, 3}, {5248, 23541}, {10436, 24999}, {19861, 24542}
X(24540) lies on these lines: {1, 20895}, {2, 6}, {3, 17183}, {7, 5253}, {56, 20245}, {105, 24669}, {319, 25005}, {326, 17863}, {404, 10446}, {604, 21246}, {1329, 21286}, {1388, 18654}, {3672, 24558}, {3879, 24982}, {4855, 10889}, {5289, 21273}, {7190, 10436}, {9310, 20258}, {10455, 16454}, {14868, 19752}, {17182, 19645}, {17201, 19520}, {17272, 24998}, {17321, 24999}, {17452, 24334}, {20880, 24179}, {24537, 24546}, {24545, 24550}, {24554, 24559}
X(24541) lies on these lines: {1, 2}, {21, 946}, {36, 12609}, {37, 5830}, {40, 6910}, {56, 5249}, {63, 3485}, {65, 4999}, {140, 3753}, {142, 5253}, {149, 4314}, {214, 3841}, {225, 11109}, {226, 2975}, {283, 11110}, {354, 11281}, {377, 3576}, {392, 5771}, {404, 10165}, {405, 5812}, {442, 1385}, {443, 12116}, {452, 5715}, {474, 10267}, {515, 2476}, {516, 4189}, {517, 7483}, {908, 958}, {944, 6856}, {950, 11680}, {956, 11374}, {960, 15950}, {993, 12047}, {1001, 5832}, {1004, 8273}, {1072, 13740}, {1512, 6863}, {1519, 3560}, {1621, 12053}, {1699, 6872}, {1770, 5267}, {1953, 22447}, {2323, 5257}, {2475, 4297}, {2478, 8227}, {2550, 4855}, {2646, 2886}, {3035, 3698}, {3057, 6690}, {3072, 19270}, {3193, 17557}, {3218, 3671}, {3306, 7288}, {3359, 6977}, {3434, 3601}, {3436, 5219}, {3452, 5260}, {3523, 10268}, {3555, 5719}, {3628, 17619}, {3812, 5433}, {3817, 5046}, {3824, 5126}, {3838, 7354}, {3869, 5745}, {3877, 13464}, {3925, 10959}, {3947, 20060}, {4187, 11230}, {4190, 7987}, {4295, 4652}, {4423, 10966}, {4512, 11522}, {4857, 5426}, {4870, 17781}, {4972, 24985}, {5045, 14054}, {5084, 10532}, {5141, 19925}, {5154, 10171}, {5177, 5731}, {5178, 12437}, {5187, 7988}, {5204, 5880}, {5250, 5603}, {5251, 5443}, {5258, 21077}, {5265, 9776}, {5290, 20076}, {5303, 20292}, {5432, 5836}, {5439, 15325}, {5536, 15674}, {5587, 6933}, {5691, 6871}, {5730, 5791}, {5795, 11681}, {5985, 21636}, {6598, 24386}, {6974, 12705}, {7504, 10175}, {7705, 10172}, {8609, 17398}, {8666, 13407}, {8728, 10943}, {9578, 10585}, {9624, 12704}, {9812, 17576}, {9955, 11113}, {10404, 11194}, {10436, 22464}, {10448, 24210}, {10597, 17559}, {10680, 11108}, {10785, 18443}, {10806, 17582}, {11112, 13624}, {11114, 18483}, {11260, 15888}, {11344, 22753}, {11520, 24477}, {12001, 16853}, {12512, 17548}, {12513, 17718}, {12699, 16370}, {13408, 13745}, {15670, 22937}, {16137, 24473}, {16202, 16408}, {16295, 20470}, {16418, 18493}, {17202, 17588}, {17353, 25024}, {17528, 18544}, {17529, 22935}, {17530, 18480}, {17532, 18481}, {23536, 24161}, {24390, 24929}, {24554, 24555}
X(24542) lies on these lines: {2, 11}, {10, 3722}, {31, 18139}, {35, 17674}, {88, 5550}, {238, 3936}, {244, 1125}, {405, 20999}, {678, 3634}, {748, 3771}, {902, 3836}, {1015, 24956}, {1054, 3624}, {1086, 4427}, {1279, 3006}, {1283, 5051}, {1478, 11346}, {1623, 16865}, {2246, 5257}, {3011, 4358}, {3120, 4432}, {3240, 17352}, {3271, 3909}, {3315, 3616}, {3683, 17184}, {3685, 4442}, {3712, 17495}, {3720, 6679}, {3891, 17776}, {3932, 20045}, {3952, 4422}, {3995, 17061}, {4062, 4974}, {4202, 5248}, {4302, 17679}, {4450, 8616}, {4453, 14432}, {4473, 4756}, {4552, 15253}, {4689, 17356}, {4712, 5294}, {4760, 25357}, {4966, 16704}, {5192, 10198}, {5297, 17263}, {5750, 14439}, {7664, 16597}, {9347, 17244}, {11731, 24871}, {15338, 17690}, {16020, 17740}, {16342, 19836}, {16842, 23858}, {17126, 17234}, {17127, 18134}, {17398, 20331}, {17766, 21026}, {19861, 24539}
X(24543) lies on these lines: {2, 3}, {3100, 24552}
X(24544) lies on these lines: {2,3}
X(24545) lies on these lines: {1, 20237}, {2, 31}, {3, 1519}, {6, 24996}, {394, 24997}, {908, 1460}, {1010, 19861}, {2049, 24987}, {5711, 10601}, {5783, 25006}, {24540, 24550}, {24546, 24549}
X(24546) lies on these lines: {1, 21420}, {2, 32}, {24537, 24540}, {24545, 24549}, {24555, 24571}
X(24547) lies on these lines: {2, 37}, {7, 3869}, {9, 24612}, {10, 20895}, {65, 21273}, {142, 2171}, {322, 5232}, {392, 17183}, {960, 20245}, {1108, 16713}, {1266, 24548}, {1400, 21233}, {1441, 4357}, {2269, 24336}, {2292, 3663}, {3100, 5263}, {3262, 5224}, {3702, 10447}, {3718, 4696}, {3766, 25020}, {3875, 19860}, {3877, 10446}, {3925, 21333}, {4858, 5257}, {4967, 24982}, {5250, 10444}, {5252, 21286}, {5831, 20270}, {7190, 10436}, {8583, 24179}, {17869, 19858}, {19859, 20320}, {20235, 25017}, {24552, 24566}, {25000, 25023}
X(24548) lies on these lines: {2, 39}, {142, 17451}, {978, 21405}, {1266, 24547}, {20880, 24178}, {24537, 24540}
X(24549) lies on these lines: {1, 75}, {2, 41}, {7, 4195}, {10, 16992}, {31, 17137}, {32, 21240}, {56, 9436}, {63, 16050}, {69, 5247}, {85, 2647}, {141, 4426}, {142, 11321}, {183, 3831}, {218, 17353}, {239, 16787}, {315, 2887}, {384, 3662}, {405, 4357}, {441, 18639}, {742, 16974}, {894, 17688}, {940, 3912}, {976, 3263}, {993, 16887}, {1125, 24241}, {1220, 14828}, {1944, 3061}, {1973, 15149}, {1975, 24214}, {3494, 6384}, {3721, 4376}, {3915, 17152}, {3924, 20911}, {4056, 17211}, {4384, 16780}, {4386, 20255}, {4950, 16886}, {5249, 19281}, {5255, 21281}, {7770, 20335}, {9259, 24652}, {10448, 16705}, {11319, 20347}, {11320, 17184}, {13735, 17274}, {14829, 17284}, {16061, 17754}, {16783, 16818}, {16912, 17248}, {16914, 17236}, {17023, 19701}, {17050, 20172}, {17282, 17682}, {24545, 24546}, {24550, 24566}
X(24550) lies on these lines: {1, 2}, {141, 24991}, {392, 8731}, {1064, 18465}, {1385, 4199}, {4322, 5484}, {16056, 17614}, {22053, 24723}, {24540, 24545}, {24549, 24566}
X(24551) lies on these lines: {1, 2}, {63, 1284}, {497, 24669}, {2171, 17754}, {3794, 6210}, {3925, 24655}, {5250, 8731}, {17194, 17202}, {20258, 20557}, {21334, 22370}, {24552, 24563}
X(24552) lies on these lines: {1, 321}, {2, 11}, {6, 17135}, {8, 13740}, {9, 4981}, {10, 748}, {21, 15654}, {31, 1150}, {36, 16393}, {38, 3923}, {56, 11115}, {75, 7191}, {76, 18064}, {81, 10453}, {86, 4441}, {141, 6327}, {145, 1220}, {190, 7226}, {238, 5278}, {244, 3980}, {312, 3920}, {333, 17127}, {392, 1824}, {595, 10479}, {599, 20290}, {612, 4358}, {614, 4359}, {740, 17017}, {750, 3840}, {756, 4011}, {870, 18059}, {894, 3873}, {956, 11354}, {958, 11319}, {960, 4463}, {982, 4418}, {999, 16394}, {1008, 4388}, {1010, 3616}, {1125, 3914}, {1215, 3938}, {1386, 3187}, {1479, 5051}, {1836, 17184}, {2176, 18091}, {2177, 6685}, {2887, 24943}, {2975, 4195}, {3100, 24543}, {3112, 6382}, {3219, 4676}, {3240, 3996}, {3242, 17165}, {3589, 21283}, {3662, 20292}, {3763, 21282}, {3886, 3896}, {3974, 20020}, {3995, 4387}, {4042, 19742}, {4202, 19836}, {4361, 17163}, {4362, 17469}, {4363, 17140}, {4376, 25368}, {4383, 4651}, {4414, 6682}, {4514, 17289}, {4649, 19738}, {4666, 10436}, {4670, 4883}, {4673, 17016}, {4693, 17600}, {4847, 5294}, {4865, 15523}, {4871, 17124}, {5047, 19853}, {5224, 20553}, {5248, 16342}, {5251, 11346}, {5259, 19858}, {5260, 17697}, {5271, 7290}, {5272, 24589}, {5297, 18743}, {5695, 17147}, {5710, 17751}, {6284, 17676}, {6679, 21242}, {6686, 9350}, {7292, 19804}, {7770, 20556}, {9580, 17306}, {9780, 13741}, {9812, 13727}, {11105, 11390}, {11108, 19874}, {11322, 16678}, {11393, 24989}, {13728, 15171}, {14829, 17126}, {15569, 19791}, {16405, 23853}, {16475, 17156}, {16738, 20992}, {16825, 21020}, {16975, 24275}, {17155, 17598}, {17293, 18082}, {17353, 25006}, {17398, 21956}, {17526, 19843}, {17698, 24390}, {17716, 17763}, {17871, 20886}, {17884, 20236}, {17887, 21407}, {19861, 24537}, {24540, 24545}, {24547, 24566}, {24551, 24563}
X(24553) lies on these lines: {1, 23529}, {2, 6}, {7, 17073}, {48, 21279}, {281, 1442}, {377, 1800}, {1014, 5746}, {1125, 15299}, {1958, 3434}, {3616, 24554}, {10431, 17188}, {10446, 24580}, {17183, 24609}, {24538, 24555}
X(24554) lies on these lines: {1, 5785}, {2, 37}, {7, 24635}, {9, 7190}, {21, 990}, {63, 4328}, {81, 2257}, {86, 16743}, {142, 22464}, {144, 1212}, {219, 7269}, {672, 7201}, {991, 10861}, {1001, 3100}, {1108, 3945}, {1423, 17451}, {1621, 4319}, {2293, 24341}, {2975, 4327}, {3616, 24553}, {3668, 5249}, {3673, 25255}, {3681, 21039}, {3729, 25082}, {3755, 24987}, {4648, 8609}, {4666, 18216}, {4859, 25065}, {5333, 24567}, {6173, 17092}, {7174, 19860}, {7229, 25066}, {11220, 17194}, {12530, 23407}, {14923, 17868}, {15988, 16973}, {16578, 20195}, {16579, 17080}, {16589, 25004}, {17019, 17811}, {17248, 25000}, {19861, 24557}, {20880, 25252}, {23681, 25080}, {24540, 24559}, {24541, 24555}
X(24555) lies on these lines: {2, 39}, {7, 22070}, {1424, 23636}, {3177, 16588}, {24538, 24553}, {24541, 24554}, {24546, 24571}
X(24556) lies on these lines: {2, 6}, {21, 10309}, {27, 17182}, {329, 1014}, {1010, 19861}, {1043, 17862}, {1408, 24954}, {1412, 3452}, {1434, 5905}, {1817, 17183}, {4658, 8582}, {14007, 24987}, {16054, 17167}, {16700, 25091}
X(24557) lies on these lines: {2, 6}, {9, 1014}, {21, 3062}, {144, 1434}, {274, 25001}, {314, 20905}, {651, 5257}, {1412, 7308}, {1621, 1958}, {2297, 3305}, {3193, 17551}, {3562, 19859}, {5308, 5783}, {15601, 16948}, {16054, 17183}, {16696, 25067}, {16887, 25019}, {17260, 23617}, {19861, 24554}, {24563, 24564}
X(24558) lies on these lines: {1, 2}, {21, 10305}, {56, 9965}, {144, 7677}, {153, 2478}, {214, 4294}, {329, 1420}, {392, 9940}, {404, 10306}, {443, 5901}, {452, 1385}, {474, 10595}, {631, 10698}, {908, 4308}, {944, 6919}, {1387, 5082}, {1388, 2551}, {1466, 5253}, {1482, 17567}, {3306, 4323}, {3434, 18220}, {3522, 4881}, {3523, 3877}, {3576, 17576}, {3672, 24540}, {3869, 5265}, {3890, 5281}, {3897, 5129}, {4187, 7967}, {4188, 10310}, {4189, 22775}, {4855, 9785}, {5046, 12667}, {5084, 10246}, {5177, 5886}, {5289, 7288}, {5328, 6049}, {5330, 6921}, {5435, 11682}, {5603, 6904}, {5731, 6260}, {5744, 15829}, {5748, 10106}, {6850, 11729}, {7963, 24177}, {8148, 17564}, {10283, 16408}, {12245, 13747}, {13462, 20214}
X(24559) lies on these lines: {1, 2}, {297, 17923}, {1447, 1959}, {3685, 24563}, {4881, 14953}, {5249, 17084}, {16054, 17614}, {24179, 25252}, {24202, 25078}, {24203, 25083}, {24540, 24554}
X(24560) lies on these lines: {2, 661}, {394, 23092}, {441, 525}, {693, 17215}, {4077, 10436}, {4529, 25258}, {8062, 18155}
X(24561) lies on these lines: {2, 667}, {1010, 3309}, {1220, 20317}, {3737, 24563}, {4162, 5263}
X(24562) lies on these lines: {2, 650}, {394, 22383}, {405, 8760}, {441, 525}, {652, 4131}, {1001, 11934}, {3126, 8641}, {3309, 8642}, {4130, 25259}, {4190, 8142}, {4765, 14838}, {6129, 16757}, {7253, 16751}, {14077, 19860}
X(24563) lies on these lines: {2, 31}, {81, 24997}, {142, 8543}, {516, 21495}, {2323, 5750}, {2975, 20258}, {3685, 24559}, {3737, 24561}, {5847, 25007}, {7190, 10436}, {15310, 16377}, {16830, 25024}, {24551, 24552}, {24557, 24564}
X(24564) lies on these lines: {1, 2}, {21, 18653}, {142, 3869}, {355, 16842}, {388, 3305}, {392, 8728}, {404, 7688}, {405, 18481}, {442, 9955}, {443, 3587}, {515, 5047}, {517, 17529}, {553, 11684}, {944, 17552}, {946, 4197}, {950, 5284}, {960, 3649}, {984, 23675}, {1213, 4875}, {1266, 24547}, {1512, 1656}, {1837, 8167}, {2292, 24178}, {2478, 3646}, {3057, 3826}, {3177, 17248}, {3218, 18249}, {3219, 4298}, {3340, 20195}, {3436, 7308}, {3475, 3984}, {3522, 21628}, {3615, 14005}, {3648, 4292}, {3740, 15888}, {3742, 21677}, {3876, 21620}, {4054, 19582}, {4190, 4512}, {4205, 6739}, {4297, 16865}, {4355, 20078}, {4423, 5794}, {4513, 17303}, {4881, 15674}, {5234, 20076}, {5253, 5745}, {5259, 5441}, {5260, 6666}, {5270, 5506}, {5302, 5434}, {5316, 11681}, {5484, 17260}, {5586, 12526}, {5790, 16855}, {5919, 9710}, {6175, 18483}, {6675, 13151}, {6684, 17531}, {6701, 12047}, {7354, 15254}, {7987, 12617}, {9956, 17575}, {10164, 17572}, {10176, 13407}, {10395, 13384}, {10404, 17781}, {11025, 24393}, {11108, 18525}, {12536, 24389}, {13624, 15670}, {17164, 24199}, {18250, 20060}, {24557, 24563}
X(24565) lies on these lines: {1, 280}, {2, 3}, {86, 6527}, {189, 1394}, {253, 3945}, {285, 1498}, {347, 10436}, {388, 1359}, {1295, 11496}, {1804, 2975}, {3100, 19861}, {3616, 10538}, {4296, 19860}
X(24566) lies on these lines: {2, 3}, {24547, 24552}, {24549, 24550}
X(24567) lies on these lines: {2, 3}, {1778, 13567}, {5333, 24554}, {10436, 20223}
X(24568) lies on these lines: {2,3}
X(24569) lies on these lines: {2,3}
X(24570) lies on these lines: {2, 3}, {348, 3668}, {1724, 18928}, {1935, 18915}, {1944, 3485}, {3616, 24553}, {19766, 23292}
X(24571) lies on these lines: {2, 3}, {24546, 24555}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28414.
X(24572) lies on these lines: {3,12278}, {5,1986}, {54,15059}, {68,6640}, {125,12038}, {1352,6639}, {2072,5449}, {4549,18404}, {5889,10255}, {8548,11898}, {10024,12162}, {10254,15058}, {10721,15062}, {11585,12606}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28418.
X(24573) lies on these lines: {4,11671}, {5,252}, {137,195}, {546,6288}, {5501,21230}, {10285,14140}, {14072,20120}, {16336,20414}
X(24573) = reflection of X(252) in X(5)
X(24573) = (X(i),X(j))-harmonic conjugate of X(k) for these {i,j,k}: {15307,19552,5}
Centers associated with the trilinear obverse and trilinear N-obverse triangles of X(2): X(24574) - X(24579))
These centers were contributed by Randy Hutson, October 6, 2018. Obverse and N-obverse triangles are introduced in preambles just before X(24482) and X(24519).
X(24574) lies on these lines: {2, 742}, {190, 2276}, {291, 751}, {3060, 9018}, {3809, 4562}
X(24575) is the perspector of the trilinear N-obverse triangle of X(2) and (cross-triangle of trilinear obverse triangle of X(2) and trilinear N-obverse triangle of X(2)).
X(24575) lies on these lines: {1, 3688}, {2, 3728}, {6, 256}, {39, 1045}, {43, 2277}, {75, 291}, {171, 2260}, {239, 3778}, {319, 2228}, {387, 986}, {740, 4283}, {894, 20456}, {982, 4000}, {983, 12329}, {1582, 17798} et al
X(24576) lies on these lines: {239, 1967}, {291, 3978}, {1580, 1922} et al
X(24576) = isogonal conjugate of X(19580)
X(24576) = isotomic conjugate of X(19581)
X(24577) lies on these lines: {43, 798}, {661, 2276}, {1022, 9460} et al
X(24578) is the perspector of the trilinear N-obverse triangle of X(2) and the unary cofactor triangle of the trilinear obverse triangle of X(2).
X(24578) lies on these lines: {1, 39}, {6, 1045}, {9, 75}, {40, 21384}, {57, 7075}, {99, 2311}, {165, 2319}, {191, 2795}, {194, 3212}, {239, 672}, {659, 8659}, {726, 3508}, {869, 17756}, {970, 1490}, {986, 3125}, {1050, 2176}, {1107, 5836} et al
X(24579) lies on these lines: {1, 3511}, {6, 3503}, {9, 1966}, {40, 511}, {57, 893}, {1697, 8844} et al
Collineation mappings involving Gemini triangle 2: X(24580)-X(24638)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 2, as in centers X(24580)-X(24638). Then
m(X) = a x - (a + c) y - (a + b) z : : , and m(X) is on the Euler line if and only if X is on the Euler line.
Fixed points: m(X(2)) = X(2), m(X(239))= X(239), m(X(649))= X(649)
Cycles: m(X(57)) = X(63), m(X(63)) = X(57), and m(X(88)) = X(190), m(X(190)) = X(88)
(Clark Kimberling, October 7, 2018)
X(24580) lies on these lines: {2, 3}, {69, 662}, {241, 2275}, {281, 17134}, {284, 5738}, {307, 610}, {347, 653}, {348, 7291}, {673, 2164}, {910, 5813}, {911, 5773}, {1631, 11677}, {2173, 24316}, {3002, 24597}, {3101, 6349}, {3306, 5011}, {3912, 4855}, {4329, 17073}, {4466, 24683}, {5256, 8555}, {5739, 24632}, {5740, 5802}, {10446, 24553}, {18634, 18650}, {20110, 20818}, {24582, 24612}, {24583, 24596}, {24587, 24636}
X(24581) lies on these lines: {2, 3}, {216, 23585}, {284, 5740}, {572, 25000}, {1055, 16603}, {2173, 24317}, {4359, 17864}, {4384, 24582}, {4466, 24684}, {5222, 24737}, {5741, 24632}, {7291, 17095}, {17043, 21271}, {19804, 20926}, {20305, 22054}, {20913, 24636}
X(24582) lies on these lines: {2, 11}, {498, 17683}, {662, 24619}, {908, 3234}, {1015, 24918}, {1018, 6710}, {1146, 17136}, {1958, 25000}, {2246, 24318}, {3119, 5745}, {3936, 24602}, {4209, 11681}, {4359, 20901}, {4384, 24581}, {4437, 17780}, {4604, 17346}, {4750, 4763}, {5741, 24586}, {17044, 21272}, {19804, 20940}, {24580, 24612}
X(24583) lies on these lines: {2, 12}, {594, 18654}, {1959, 17023}, {3869, 17397}, {4384, 24581}, {5303, 6999}, {17045, 21273}, {17327, 21286}, {17398, 20245}, {24580, 24596}
X(24584) lies on these lines: {2, 3}, {2173, 24321}, {3088, 3547}, {4466, 24686}, {4872, 16706}, {17023, 18651}, {17492, 18636}, {18750, 24612}, {19804, 24596}
X(24585) lies on these lines: {2, 3}, {2173, 24322}, {4466, 24687}, {6084, 24623}, {18637, 21274}
X(24586) lies on these lines: {1, 16060}, {2, 31}, {6, 24691}, {8, 4209}, {10, 11321}, {32, 21240}, {41, 17137}, {55, 3912}, {57, 85}, {63, 15487}, {65, 16822}, {69, 3684}, {75, 3509}, {81, 17208}, {141, 4386}, {142, 16992}, {183, 20335}, {239, 982}, {304, 3496}, {312, 17738}, {315, 17046}, {385, 3662}, {660, 3681}, {673, 2319}, {940, 16502}, {1150, 24592}, {1395, 15149}, {1460, 16412}, {1759, 1930}, {2243, 4376}, {2329, 21281}, {3263, 5282}, {3416, 8301}, {3500, 19806}, {3550, 17284}, {3677, 16834}, {3687, 11347}, {3721, 3905}, {3741, 20172}, {3750, 17316}, {3759, 17731}, {3831, 7770}, {3996, 17294}, {4357, 5275}, {4393, 17598}, {4416, 6180}, {4426, 20255}, {4850, 24613}, {5269, 16831}, {5329, 11329}, {5741, 24582}, {6996, 20368}, {7754, 24214}, {9310, 17152}, {14377, 20888}, {14828, 17298}, {16816, 18201}, {16826, 17716}, {16833, 18193}, {16910, 21935}, {16993, 17248}, {16995, 17236}, {17211, 20267}, {17277, 17754}, {17279, 17735}, {19591, 21371}, {19732, 24603}, {24587, 24591}, {24595, 24637}
X(24587) lies on these lines: {2, 32}, {10, 19271}, {19, 21442}, {35, 5300}, {63, 20432}, {75, 5341}, {239, 3959}, {379, 1150}, {980, 21997}, {1760, 20234}, {2175, 17138}, {2244, 4381}, {3765, 24630}, {4118, 4412}, {4202, 20083}, {4645, 5138}, {5737, 19281}, {17047, 21275}, {24580, 24636}, {24586, 24591}, {24598, 24610}, {24617, 24621}
X(24588) lies on these lines: {2, 35}, {32, 24789}, {63, 169}, {81, 24790}, {86, 142}, {239, 3874}, {2140, 20769}, {3647, 16815}, {3912, 17682}, {4251, 5249}, {5299, 16752}, {16831, 17683}, {19808, 24603}, {24592, 24632}
X(24589) lies on these lines: {2, 37}, {8, 4002}, {10, 244}, {57, 5278}, {81, 17121}, {86, 17012}, {88, 274}, {100, 16823}, {142, 3936}, {210, 17140}, {239, 16971}, {354, 4651}, {404, 16817}, {443, 5016}, {518, 17146}, {519, 17450}, {551, 3902}, {693, 14475}, {748, 3980}, {750, 16825}, {756, 24165}, {899, 24325}, {908, 24199}, {982, 4981}, {1086, 5241}, {1104, 19284}, {1150, 3306}, {1441, 3911}, {1465, 17077}, {1647, 21027}, {1698, 4968}, {2087, 22192}, {2177, 24331}, {2999, 19684}, {3006, 3826}, {3121, 16604}, {3218, 17277}, {3624, 3702}, {3626, 4935}, {3634, 3701}, {3687, 18139}, {3706, 3848}, {3707, 4031}, {3720, 3896}, {3740, 17165}, {3742, 17135}, {3758, 14997}, {3759, 14996}, {3762, 25038}, {3828, 4692}, {3840, 21020}, {3891, 5268}, {3923, 17125}, {3977, 6666}, {4054, 5316}, {4360, 17021}, {4362, 17124}, {4385, 19877}, {4389, 24184}, {4418, 17123}, {4427, 15254}, {4647, 19862}, {4682, 17150}, {4696, 9780}, {4706, 15569}, {4723, 19875}, {4742, 25055}, {4883, 20011}, {4975, 19883}, {5249, 5741}, {5253, 16824}, {5263, 7292}, {5271, 5437}, {5272, 24552}, {5739, 9776}, {5743, 17184}, {6377, 20899}, {6381, 20913}, {7081, 9342}, {7283, 17536}, {7321, 17484}, {8041, 21954}, {14210, 17023}, {16704, 16709}, {16738, 16753}, {17013, 17394}, {17259, 17595}, {17367, 20911}, {21418, 23521}, {24596, 24605}
X(24589) = complement of X(31035)
X(24590) lies on these lines: {1, 11349}, {2, 40}, {7, 2270}, {9, 7229}, {19, 273}, {46, 3008}, {57, 279}, {63, 169}, {77, 2262}, {198, 7190}, {239, 4209}, {910, 5228}, {942, 5256}, {1697, 5308}, {1817, 8726}, {2183, 8545}, {2999, 3339}, {3218, 24599}, {3295, 5287}, {3305, 3730}, {3306, 5011}, {3333, 17014}, {3359, 7397}, {3753, 21514}, {3895, 17316}, {5813, 9436}, {8257, 16548}, {10444, 20905}, {10521, 24177}, {11343, 19860}, {12514, 16832}, {16412, 19861}, {17092, 18725}, {21370, 24600}
X(24591) lies on these lines: {2, 41}, {3, 3419}, {10, 19314}, {63, 6996}, {75, 16560}, {78, 21554}, {239, 4051}, {908, 23151}, {1150, 3765}, {1210, 19310}, {1726, 20882}, {1958, 17077}, {3306, 16054}, {3434, 9441}, {3911, 11329}, {4384, 24618}, {5745, 16367}, {9843, 19316}, {14829, 20917}, {16551, 20236}, {17206, 20925}, {19545, 22449}, {24586, 24587}, {24592, 24605}
X(24592) lies on these lines: {1, 2}, {9, 4441}, {31, 20172}, {44, 24330}, {63, 20605}, {71, 20174}, {75, 672}, {76, 3691}, {274, 1475}, {310, 333}, {321, 17755}, {350, 17277}, {748, 20154}, {940, 1206}, {1086, 24690}, {1150, 24586}, {1334, 17143}, {1468, 11321}, {2238, 17348}, {2276, 4361}, {2280, 16992}, {2308, 14621}, {2350, 4359}, {2886, 4766}, {3112, 18087}, {3219, 17738}, {3686, 20335}, {3696, 8299}, {3739, 24512}, {3797, 4365}, {3883, 13576}, {3975, 18152}, {4395, 25349}, {4416, 20347}, {4423, 20156}, {4479, 17335}, {5247, 17686}, {14829, 24602}, {16552, 20888}, {16748, 18206}, {16752, 17208}, {16887, 24790}, {16994, 20180}, {17050, 17137}, {17117, 17759}, {17149, 18739}, {17152, 20257}, {17349, 24514}, {19804, 24629}, {20179, 21764}, {24588, 24632}, {24591, 24605}
X(24593) lies on these lines: {2, 44}, {57, 321}, {88, 239}, {190, 3218}, {312, 23958}, {484, 4742}, {518, 17780}, {649, 693}, {752, 1647}, {896, 4759}, {899, 4753}, {908, 24237}, {1150, 3306}, {1155, 4702}, {1997, 20078}, {3008, 24183}, {3230, 16826}, {3262, 4359}, {3336, 3702}, {3337, 4968}, {3689, 17145}, {3891, 18193}, {3911, 3936}, {3999, 20045}, {4001, 6692}, {4031, 4054}, {4393, 4850}, {4434, 17449}, {4450, 11019}, {4706, 17162}, {4722, 6686}, {4767, 5205}, {4981, 17122}, {5087, 17491}, {5278, 5437}, {5372, 17791}, {5744, 14154}, {6327, 17728}, {9352, 10453}, {16602, 19742}, {16610, 16704}, {16729, 18206}, {17318, 17595}, {17763, 18201}
X(24594) lies on these lines: {2, 45}, {1001, 4781}, {1150, 3306}, {1647, 24693}, {3218, 17335}, {3711, 17146}, {3891, 17124}, {4358, 4659}, {4359, 5437}, {4393, 24620}, {4413, 17780}, {4670, 16610}, {4759, 17125}, {4767, 24349}, {4850, 16826}, {5741, 9776}, {17020, 19738}, {17051, 21283}, {17277, 24616}, {17318, 17495}
X(24595) lies on these lines: {2, 48}, {63, 1746}, {75, 21368}, {92, 21367}, {212, 20242}, {355, 20999}, {1150, 3765}, {1726, 14213}, {1985, 7193}, {3306, 24618}, {4384, 24606}, {4850, 24624}, {5303, 15486}, {11064, 21239}, {14206, 16551}, {20879, 21375}, {24586, 24637}
X(24596) lies on these lines: {1, 17683}, {2, 11}, {8, 17682}, {31, 3008}, {41, 17050}, {63, 169}, {81, 277}, {142, 2280}, {220, 20244}, {239, 3873}, {333, 16748}, {1150, 24586}, {1914, 17278}, {2246, 24333}, {2975, 4209}, {3306, 24600}, {3739, 4376}, {4000, 5276}, {4512, 16832}, {5260, 17691}, {6996, 9778}, {7195, 21454}, {9310, 20257}, {9780, 17681}, {16454, 16818}, {17026, 24602}, {19804, 24584}, {21258, 21285}, {24580, 24583}, {24589, 24605}
X(24597) lies on these lines: {2, 6}, {4, 162}, {8, 5266}, {20, 5721}, {21, 387}, {22, 5324}, {31, 3434}, {44, 17720}, {57, 15474}, {58, 377}, {63, 1723}, {77, 3911}, {88, 279}, {238, 11269}, {239, 17740}, {283, 1724}, {345, 3187}, {386, 6910}, {464, 2193}, {497, 2361}, {580, 6836}, {582, 6899}, {614, 24216}, {631, 5396}, {896, 24248}, {908, 1743}, {1104, 12649}, {1191, 10529}, {1269, 19818}, {1453, 6734}, {1465, 5435}, {1707, 3914}, {1748, 17903}, {1754, 10431}, {1760, 3218}, {1788, 4296}, {1824, 21848}, {1834, 6872}, {1865, 6994}, {1999, 5336}, {2082, 24611}, {2308, 24892}, {2550, 17126}, {3002, 24580}, {3008, 3306}, {3011, 3751}, {3015, 24608}, {3052, 20075}, {3086, 3562}, {3120, 24695}, {3193, 16471}, {3216, 6921}, {3240, 5218}, {3332, 10883}, {3436, 5230}, {3617, 5724}, {3718, 4358}, {3752, 18607}, {3769, 10327}, {3772, 4641}, {3782, 20078}, {3791, 4438}, {3875, 3977}, {4031, 17067}, {4190, 4252}, {4197, 4340}, {4259, 18191}, {4339, 5178}, {4389, 19823}, {4398, 19824}, {4442, 24280}, {4663, 17718}, {4850, 5222}, {4888, 5249}, {4902, 23681}, {4967, 5271}, {5219, 16670}, {5220, 17602}, {5231, 16469}, {5256, 5745}, {5269, 25006}, {5294, 11679}, {5706, 6837}, {5707, 6832}, {5725, 9780}, {5729, 15252}, {6776, 8229}, {6857, 19767}, {7191, 24477}, {7739, 24296}, {10449, 17526}, {10527, 16466}, {11240, 16483}, {14206, 17861}, {16342, 19766}, {16477, 17717}, {16729, 18135}, {17249, 19786}, {17250, 19832}, {17366, 17595}, {17367, 24627}, {18750, 19788}
X(24597) = anticomplement of X(30811)
X(24598) lies on these lines: {2, 39}, {3, 1244}, {6, 662}, {32, 19308}, {37, 24077}, {43, 20456}, {63, 978}, {81, 386}, {86, 4261}, {87, 23659}, {99, 11320}, {192, 17053}, {239, 2275}, {291, 869}, {379, 24617}, {894, 2277}, {1015, 4393}, {1054, 17795}, {1125, 24165}, {1150, 24614}, {1575, 3661}, {1724, 1931}, {1740, 3778}, {1959, 24443}, {1964, 4446}, {2092, 17379}, {2234, 4443}, {2276, 16826}, {2309, 17065}, {3002, 24580}, {3008, 24635}, {3009, 12782}, {3095, 19522}, {3122, 24696}, {3216, 18206}, {3596, 17148}, {3666, 16604}, {3752, 17367}, {3766, 22092}, {4253, 11349}, {4359, 17030}, {4664, 8610}, {4850, 17023}, {5013, 16367}, {5069, 17277}, {5222, 24625}, {5224, 16696}, {7032, 20862}, {7976, 20352}, {9605, 16412}, {11337, 19762}, {12263, 17155}, {16700, 18134}, {16726, 17378}, {16728, 17352}, {16887, 18601}, {17027, 20868}, {17316, 17756}, {17331, 21892}, {17350, 21796}, {17363, 21857}, {17377, 21858}, {17389, 20691}, {19804, 21608}, {24587, 24610}
X(24599) lies on these lines: {1, 2}, {6, 4747}, {9, 4402}, {44, 4454}, {144, 673}, {333, 18600}, {346, 4361}, {379, 9965}, {391, 4000}, {597, 4470}, {966, 17325}, {1266, 6172}, {1278, 17755}, {3161, 17151}, {3177, 17490}, {3210, 25237}, {3218, 24590}, {3672, 17277}, {3752, 4875}, {3759, 3945}, {3875, 18230}, {3946, 5296}, {3973, 4488}, {4307, 4974}, {4371, 17279}, {4395, 4419}, {4405, 17269}, {4461, 17117}, {4700, 6173}, {4748, 17382}, {4859, 21296}, {4869, 5839}, {4936, 8055}, {5232, 16706}, {5435, 9312}, {5437, 17474}, {6996, 20070}, {7195, 21454}, {7222, 16669}, {7613, 24692}, {12245, 19512}, {16054, 16704}, {17314, 17337}, {20078, 24630}
X(24599) = anticomplement of X(29627)
X(24600) lies on these lines: {1, 2}, {44, 24352}, {57, 658}, {142, 14548}, {728, 17158}, {1449, 14828}, {1453, 13727}, {1707, 24283}, {1743, 10025}, {2402, 3676}, {3306, 24596}, {3333, 17682}, {3361, 4209}, {3598, 5838}, {3673, 16572}, {3693, 3875}, {3975, 18153}, {4000, 9436}, {4395, 25355}, {5435, 14189}, {6173, 24694}, {7290, 14942}, {9776, 15493}, {15299, 24014}, {16750, 18206}, {17755, 20173}, {21370, 24590}
X(24601) lies on these lines: {2, 667}, {27, 18344}, {239, 4083}, {514, 4508}, {649, 3766}, {673, 6008}, {1019, 4391}, {2483, 3261}, {2484, 4374}, {2515, 4411}, {3063, 17217}, {3250, 4107}, {3309, 6996}, {3835, 8632}, {4040, 4776}, {4063, 4384}, {4408, 10566}, {4462, 24612}, {4782, 16815}, {8630, 24533}, {8646, 25299}, {17367, 24719}, {20906, 21389}, {20981, 21191}
X(24602) lies on these lines: {2, 31}, {100, 2725}, {172, 20255}, {239, 244}, {320, 3570}, {333, 873}, {673, 4607}, {799, 3975}, {1019, 4391}, {1086, 4396}, {1150, 3306}, {2177, 17316}, {2243, 24358}, {3218, 17755}, {3263, 3509}, {3662, 16997}, {3691, 17206}, {3760, 14377}, {3831, 17686}, {3936, 24582}, {4358, 17738}, {4372, 20271}, {4495, 4817}, {5011, 14210}, {5235, 6629}, {5277, 21240}, {5280, 24170}, {9310, 21281}, {10448, 22267}, {14829, 24592}, {16784, 17023}, {17026, 24596}, {17680, 21935}, {24612, 24615}
X(24603) lies on these lines: {1, 2}, {6, 4798}, {37, 4431}, {44, 4472}, {75, 4044}, {76, 19804}, {86, 3686}, {142, 4751}, {165, 7406}, {192, 3986}, {274, 3975}, {333, 1509}, {346, 5936}, {379, 5179}, {516, 7384}, {527, 17256}, {594, 4698}, {673, 1268}, {908, 24633}, {950, 16053}, {958, 16412}, {966, 4416}, {993, 11329}, {1086, 1213}, {1100, 6707}, {1211, 21240}, {1266, 4364}, {1654, 3664}, {2321, 4687}, {2345, 25101}, {3305, 3730}, {3501, 7308}, {3619, 20195}, {3663, 4699}, {3666, 16589}, {3674, 16609}, {3691, 18206}, {3707, 3758}, {3723, 4399}, {3729, 5296}, {3740, 20683}, {3797, 25354}, {3842, 4439}, {3879, 15668}, {3883, 20156}, {3925, 20544}, {3943, 4755}, {3946, 17322}, {3948, 4359}, {4021, 17117}, {4022, 21699}, {4058, 17242}, {4060, 17315}, {4363, 4480}, {4383, 17750}, {4395, 25358}, {4407, 24325}, {4413, 21477}, {4473, 17260}, {4648, 17270}, {4667, 17346}, {4670, 17330}, {4675, 17251}, {4690, 17392}, {4700, 4758}, {4733, 15569}, {4739, 17246}, {4741, 4896}, {4748, 17274}, {4772, 17247}, {4887, 17254}, {5232, 17298}, {5235, 6629}, {5249, 17746}, {5251, 21511}, {5260, 11349}, {5267, 19308}, {5717, 14007}, {5743, 20255}, {5745, 6626}, {5750, 17277}, {5814, 16456}, {6376, 18739}, {6381, 20913}, {6537, 20337}, {6684, 6996}, {6999, 19925}, {7227, 16814}, {7377, 10175}, {8185, 11340}, {10022, 16590}, {11353, 20172}, {16752, 17210}, {17067, 17305}, {17238, 21255}, {17239, 17245}, {17259, 17303}, {17278, 17327}, {17280, 25072}, {17337, 17385}, {17348, 17398}, {19732, 24586}, {19808, 24588}
X(24603) = complement of X(16826)
X(24604) lies on these lines: {2, 3}, {7, 610}, {19, 347}, {57, 279}, {198, 8232}, {284, 3945}, {673, 1436}, {910, 948}, {942, 17014}, {1448, 2999}, {1730, 4253}, {2182, 10402}, {2270, 7013}, {3008, 15803}, {3188, 5435}, {3601, 5308}, {3668, 18594}, {4258, 5712}, {4384, 5088}, {5745, 23058}, {9776, 17023}, {14552, 24632}
X(24604) = anticomplement of X(30809)
X(24605) lies on these lines: {2, 3}, {305, 799}, {607, 18659}, {3162, 17903}, {4384, 21370}, {5089, 17134}, {14189, 17080}, {18596, 20235}, {24589, 24596}, {24591, 24592}
X(24606) lies on these lines: {2, 3}, {7, 2303}, {63, 1930}, {81, 17170}, {190, 5279}, {272, 673}, {284, 18650}, {333, 7291}, {1172, 4329}, {1474, 18589}, {3188, 17080}, {4280, 16580}, {4384, 24595}
X(24607) lies on these lines: {2, 3}, {81, 18652}, {284, 18634}, {307, 651}, {333, 348}, {347, 2322}, {1172, 17073}, {1440, 16713}, {2897, 18643}, {4384, 17080}, {5235, 24635}, {7177, 18206}
X(24608) lies on these lines: {2, 3}, {3015, 24597}, {7291, 17079}
X(24609) lies on these lines: {2, 3}, {32, 5712}, {57, 348}, {69, 284}, {333, 3926}, {387, 19758}, {579, 3618}, {610, 4357}, {1119, 17086}, {3053, 17056}, {3601, 3912}, {3785, 18134}, {3933, 14552}, {4384, 5745}, {4648, 5337}, {4850, 5222}, {5273, 25083}, {5737, 7789}, {5746, 8822}, {12437, 17294}, {16834, 24391}, {17183, 24553}, {17316, 24929}
X(24610) lies on these lines: {2, 3}, {239, 24443}, {1054, 4384}, {1150, 24621}, {1958, 3662}, {4466, 24726}, {7187, 7291}, {14829, 24378}, {17023, 24169}, {17797, 17872}, {20913, 24614}, {24587, 24598}
X(24611) lies on these lines: {1, 11337}, {2, 19}, {3, 1829}, {9, 21368}, {21, 7713}, {22, 1040}, {30, 5155}, {33, 20243}, {34, 16049}, {40, 78}, {46, 386}, {57, 77}, {63, 573}, {92, 23512}, {165, 5314}, {193, 3218}, {224, 3430}, {608, 1465}, {908, 1766}, {1062, 2915}, {1150, 24633}, {1214, 11350}, {1452, 4225}, {1726, 21363}, {1753, 6838}, {1824, 19544}, {2082, 24597}, {2172, 16586}, {3149, 8251}, {3306, 5011}, {4228, 5338}, {4384, 24595}, {5057, 12717}, {5119, 17461}, {5219, 16548}, {5744, 7291}, {6197, 6988}, {10444, 18690}, {18651, 20266}, {21361, 21375}, {24589, 24596}
X(24612) lies on these lines: {2, 12}, {6, 20245}, {8, 6996}, {9, 24547}, {63, 169}, {141, 21286}, {144, 673}, {218, 329}, {239, 3869}, {527, 1405}, {604, 21246}, {908, 16788}, {1150, 3765}, {1229, 5227}, {1766, 20895}, {2285, 24993}, {2995, 20891}, {3008, 12527}, {3421, 7397}, {3434, 7406}, {3452, 9310}, {3661, 5176}, {3687, 5016}, {4361, 21273}, {4462, 24601}, {4696, 11679}, {5080, 7377}, {5228, 20347}, {5235, 5744}, {5773, 23151}, {7384, 11680}, {11682, 16834}, {11684, 16816}, {12526, 16833}, {12531, 20055}, {16777, 18654}, {16832, 17683}, {18750, 24584}, {19798, 20921}, {24580, 24582}, {24602, 24615}
X(24613) lies on these lines: {2, 82}, {306, 4097}, {4384, 24637}, {4850, 24586}, {16555, 20898}, {16784, 17023}
X(24614) lies on these lines: {2, 32}, {10, 36}, {141, 662}, {239, 3752}, {1150, 24598}, {2244, 24340}, {3661, 18047}, {3662, 18048}, {16784, 17023}, {16830, 19270}, {17055, 21289}, {20913, 24610}
X(24615) lies on these lines: {2, 87}, {63, 20370}, {1150, 17026}, {2319, 17149}, {4384, 24620}, {4598, 6376}, {5209, 24632}, {6382, 16557}, {24602, 24612}
X(24616) lies on these lines: {2, 44}, {8, 4781}, {63, 4659}, {190, 1150}, {3218, 4384}, {3219, 5372}, {3240, 4753}, {3679, 24344}, {4393, 16704}, {4689, 20048}, {4767, 5220}, {4793, 16558}, {5278, 23958}, {14996, 16826}, {14997, 24627}, {17057, 21291}, {17277, 24594}
X(24617) lies on these lines: {2, 99}, {36, 759}, {81, 17205}, {86, 17382}, {88, 4049}, {105, 4236}, {110, 5091}, {162, 14119}, {239, 9278}, {244, 19642}, {274, 21997}, {333, 16759}, {379, 24598}, {448, 24781}, {523, 897}, {645, 4440}, {662, 1086}, {673, 24625}, {757, 17366}, {1414, 5723}, {1931, 3008}, {1963, 3946}, {2226, 4615}, {3015, 24597}, {5196, 7292}, {6185, 14953}, {6626, 16815}, {17058, 21221}, {17103, 17367}, {20913, 24610}, {24587, 24621}
X(24618) lies on these lines: {1, 21554}, {2, 101}, {11, 5091}, {36, 80}, {57, 1111}, {88, 4049}, {294, 3008}, {572, 17077}, {651, 24237}, {668, 14829}, {673, 812}, {911, 1375}, {1056, 24222}, {1156, 3667}, {1565, 5723}, {2398, 10695}, {2481, 6996}, {3306, 24595}, {3739, 24324}, {4384, 24591}, {4657, 25367}, {4858, 16560}, {4986, 11679}, {5435, 13478}, {5662, 11998}, {9708, 19261}, {16832, 24685}, {17059, 21293}, {17181, 17367}, {17595, 24296}
X(24619) lies on these lines: {2, 98}, {81, 17198}, {116, 163}, {150, 5546}, {333, 16703}, {662, 24582}, {673, 24624}, {850, 1821}, {2247, 24347}, {2249, 2864}, {4237, 6740}, {16562, 17886}, {21253, 24917}
X(24620) lies on these lines: {2, 37}, {8, 244}, {88, 330}, {145, 3812}, {194, 16815}, {236, 16018}, {239, 3306}, {354, 20012}, {404, 19851}, {899, 24349}, {1054, 16825}, {1086, 5233}, {1122, 21454}, {1266, 5316}, {1999, 5437}, {2550, 5211}, {3177, 5744}, {3218, 17349}, {3241, 17450}, {3617, 17480}, {3622, 4719}, {3687, 21255}, {3711, 24841}, {3720, 4734}, {3753, 20037}, {3891, 9342}, {3977, 17338}, {4384, 24615}, {4389, 5241}, {4393, 24594}, {5212, 5542}, {5222, 17497}, {8056, 11679}, {9335, 17135}, {9534, 24046}, {9776, 17778}, {10453, 17063}, {16569, 24165}, {16704, 16710}, {16753, 17178}, {16817, 19278}, {17012, 17379}, {17028, 19565}, {17230, 20255}, {17277, 17595}, {17367, 21216}, {17722, 24693}, {17749, 24176}, {19742, 23958}, {19998, 22295}
X(24620) = anticomplement of X(30829)
X(24621) lies on these lines: {1, 3210}, {2, 39}, {57, 239}, {63, 16827}, {69, 18827}, {81, 16915}, {192, 3247}, {347, 3164}, {385, 11329}, {869, 24349}, {988, 16823}, {1107, 19804}, {1150, 24610}, {1575, 20917}, {1730, 4209}, {2234, 24351}, {3009, 17155}, {3122, 24717}, {3687, 24215}, {3705, 23682}, {4189, 23407}, {4384, 24615}, {4393, 14996}, {4699, 17148}, {5361, 16816}, {6904, 20018}, {7520, 19851}, {7754, 16412}, {7783, 16367}, {7793, 19308}, {9902, 20340}, {12251, 19522}, {16349, 16994}, {16574, 17349}, {16710, 17379}, {16831, 25264}, {16998, 21511}, {17065, 21299}, {17114, 20036}, {17316, 17759}, {17394, 20170}, {21218, 24635}, {24587, 24617}
X(24621) = anticomplement of X(30830)
X(24622) lies on these lines: {2, 647}, {75, 3700}, {512, 25299}, {523, 18154}, {649, 3766}, {650, 3261}, {661, 4374}, {1021, 4384}, {1577, 17899}, {1639, 21611}, {3239, 20907}, {3661, 21719}, {3757, 4477}, {4025, 4391}, {4086, 4458}, {4359, 4467}, {4394, 4408}, {4522, 23684}, {4841, 7199}, {4976, 20954}, {6590, 20906}, {7180, 17066}, {9404, 17277}, {17069, 18155}, {20891, 21437}, {20909, 21433}, {20923, 21610}
X(24622) = complement of X(25258)
X(24622) = anticomplement of X(25084)
X(24623) lies on these lines: {2, 659}, {88, 673}, {239, 891}, {335, 875}, {514, 4508}, {649, 693}, {890, 4366}, {900, 4784}, {918, 7192}, {1960, 16826}, {2483, 7199}, {2786, 4932}, {2826, 6996}, {4374, 21389}, {4375, 4728}, {4384, 21385}, {4393, 21343}, {4724, 4776}, {6084, 17494}, {16729, 17755}, {18197, 21391}, {24593, 24628}
X(24624) lies on the Kiepert hyperbola and these lines: {2, 662}, {4, 162}, {5, 60}, {10, 21}, {11, 110}, {27, 653}, {58, 3585}, {76, 799}, {81, 226}, {83, 4599}, {86, 4604}, {88, 4049}, {98, 8229}, {125, 24916}, {149, 643}, {190, 321}, {229, 1210}, {239, 11611}, {265, 1175}, {267, 2166}, {655, 16548}, {658, 1434}, {660, 17763}, {673, 24619}, {823, 2052}, {897, 5466}, {1090, 2625}, {1098, 5046}, {1109, 21381}, {1155, 5196}, {1171, 1989}, {1325, 1737}, {1411, 5331}, {1465, 18609}, {1492, 14009}, {1746, 2051}, {1837, 11101}, {1931, 11608}, {2194, 14008}, {2349, 2394}, {2580, 2593}, {2581, 2592}, {2606, 2608}, {2617, 5400}, {2651, 5057}, {3109, 12019}, {3257, 4080}, {3448, 8286}, {3583, 5127}, {3615, 7741}, {3911, 18653}, {4052, 4921}, {4359, 8052}, {4467, 14223}, {4606, 5325}, {4850, 24595}, {5297, 7474}, {5333, 18645}, {5741, 7058}, {5972, 24955}, {7951, 9275}, {10826, 13746}, {13576, 14956}, {14204, 15343}, {14628, 18163}, {19810, 20566}
X(24624) = isogonal conjugate of X(2245)
X(24624) = isotomic conjugate of X(3936)
X(24624) = polar conjugate of X(860)
X(24624) = trilinear product of circumcircle intercepts of line X(2)X(1577)
X(24624) = trilinear pole of line X(1)X(523) (the perspectrix of Gemini triangles 10 and 27)
X(24625) lies on these lines: {1, 3799}, {2, 668}, {88, 1022}, {214, 5313}, {239, 292}, {673, 24617}, {2229, 17029}, {2275, 17367}, {3123, 4499}, {3248, 3888}, {3779, 7189}, {4383, 9259}, {4585, 9456}, {5069, 17380}, {5222, 24598}, {6184, 17014}, {8632, 20332}, {8649, 14997}, {9460, 16610}, {16604, 16815}, {17292, 17448}, {17795, 20985}, {19945, 24722}
X(24626) lies on these lines: {2, 669}, {649, 3766}, {3004, 6084}, {4384, 18197}
X(24627) lies on these lines: {1, 19278}, {2, 7}, {8, 988}, {10, 1054}, {11, 24723}, {21, 22344}, {38, 7081}, {75, 17595}, {81, 20228}, {88, 274}, {171, 6682}, {191, 19864}, {239, 980}, {244, 16823}, {312, 17262}, {320, 5718}, {333, 3752}, {405, 23085}, {740, 17593}, {750, 16830}, {752, 17722}, {846, 3840}, {940, 17394}, {942, 19270}, {982, 3757}, {984, 5205}, {1155, 5263}, {1401, 9564}, {1999, 3666}, {2329, 23622}, {3187, 5372}, {3210, 11679}, {3230, 16826}, {3336, 19863}, {3661, 17740}, {3685, 4414}, {3741, 17596}, {3769, 17599}, {3883, 5211}, {3916, 13740}, {3936, 17288}, {3977, 17280}, {4054, 4440}, {4195, 4652}, {4201, 6734}, {4203, 22060}, {4358, 17261}, {4362, 17591}, {4384, 24615}, {4388, 24239}, {4389, 17720}, {4643, 5233}, {4655, 17717}, {4734, 17156}, {5241, 17256}, {5271, 17148}, {5439, 11110}, {5708, 19273}, {5737, 19804}, {6292, 17292}, {10453, 17594}, {14997, 24616}, {15934, 19279}, {16610, 17277}, {16704, 17012}, {16817, 24046}, {16824, 24443}, {17020, 19742}, {17117, 17495}, {17367, 24597}, {18201, 24325}, {19243, 23169}, {19260, 23206}, {21242, 24715}
X(24628) lies on these lines: {1, 21937}, {2, 896}, {57, 85}, {171, 3747}, {190, 3509}, {239, 18201}, {291, 4753}, {320, 24685}, {325, 4987}, {335, 4434}, {649, 693}, {982, 4393}, {1155, 3712}, {3218, 17755}, {4771, 17731}, {5221, 16822}, {6629, 16611}, {17316, 17601}, {17335, 17754}
X(24629) lies on these lines: {2, 38}, {239, 750}, {1150, 3306}, {4359, 17026}, {4850, 17023}, {5222, 17497}, {5253, 16822}, {16818, 24046}, {17316, 17450}, {19804, 24592}, {20331, 24357}
X(24630) lies on these lines: {2, 36}, {27, 3687}, {63, 169}, {83, 226}, {101, 908}, {239, 758}, {515, 3912}, {527, 666}, {1019, 4391}, {1111, 3218}, {1150, 3761}, {1441, 16566}, {1836, 24264}, {2242, 17720}, {2276, 24296}, {2323, 17139}, {3262, 16548}, {3573, 5057}, {3765, 24587}, {5222, 5905}, {5279, 20236}, {5745, 6626}, {5773, 20347}, {5792, 23151}, {7397, 12115}, {8680, 17755}, {11364, 17719}, {17209, 24632}, {17798, 20544}, {20078, 24599}
X(24631) lies on these lines: {1, 16061}, {2, 38}, {10, 17670}, {39, 712}, {56, 16822}, {57, 85}, {75, 17026}, {76, 17048}, {171, 239}, {354, 3703}, {740, 17027}, {894, 3329}, {985, 4974}, {1015, 24254}, {1086, 25345}, {1475, 20911}, {2350, 4359}, {3290, 17353}, {3509, 17277}, {3670, 16818}, {3677, 16831}, {3687, 4260}, {3736, 5256}, {3752, 16606}, {3759, 4771}, {3980, 20172}, {4393, 17716}, {4697, 14621}, {4734, 17014}, {5222, 17490}, {5269, 16834}, {9436, 24199}, {10180, 17397}, {16600, 24166}, {16815, 18201}, {16826, 17598}, {16832, 18193}, {17369, 25368}, {20331, 24326}, {20610, 24511}
X(24632) lies on these lines: {2, 58}, {8, 14953}, {21, 3912}, {27, 318}, {35, 306}, {57, 85}, {63, 1930}, {69, 284}, {81, 5299}, {86, 17227}, {141, 1333}, {239, 3670}, {286, 20236}, {307, 651}, {314, 17738}, {334, 2311}, {344, 4877}, {379, 1150}, {524, 4273}, {599, 3285}, {1010, 17308}, {1043, 17294}, {1211, 5277}, {1761, 18697}, {1778, 17353}, {1817, 3687}, {2303, 4357}, {3286, 21477}, {3662, 17189}, {3729, 8822}, {4253, 5278}, {4267, 11343}, {4269, 16574}, {4276, 21511}, {4278, 21495}, {4281, 16060}, {4388, 17188}, {4653, 17316}, {5021, 19732}, {5209, 24615}, {5222, 16704}, {5235, 6629}, {5308, 17588}, {5333, 17200}, {5739, 24580}, {5741, 24581}, {10479, 19281}, {11110, 16831}, {14552, 24604}, {14829, 18148}, {16047, 17266}, {16050, 17284}, {17052, 21287}, {17209, 24630}, {21024, 24271}, {21245, 24890}, {24588, 24592}
X(24633) lies on these lines: {2, 65}, {9, 24993}, {37, 21273}, {63, 169}, {71, 1229}, {144, 4699}, {239, 2975}, {673, 11683}, {908, 24603}, {1150, 24611}, {1400, 21233}, {1441, 16574}, {1959, 17023}, {2262, 16713}, {3008, 24443}, {3177, 17490}, {3666, 20247}, {3687, 9568}, {3739, 20245}, {4850, 5222}, {4852, 18654}, {5057, 7384}, {5086, 6999}, {5745, 17451}, {11682, 16831}, {11684, 16815}, {12526, 16832}, {17275, 21286}, {17755, 20892}, {20895, 21061}, {21371, 25001}
X(24634) lies on these lines: {2, 66}, {315, 4599}, {17068, 21288}
X(24635) lies on these lines: {1, 21}, {2, 85}, {3, 7291}, {7, 24554}, {8, 25083}, {9, 77}, {37, 144}, {57, 17451}, {71, 18161}, {75, 16713}, {92, 15149}, {142, 17092}, {169, 11349}, {219, 1442}, {220, 394}, {227, 18231}, {277, 15474}, {294, 21511}, {321, 25242}, {326, 2287}, {329, 5308}, {379, 5088}, {572, 16551}, {579, 18726}, {672, 7146}, {756, 11678}, {857, 17181}, {958, 4296}, {966, 5942}, {1005, 2000}, {1108, 3672}, {1214, 3160}, {1444, 1760}, {1743, 25065}, {2267, 16560}, {2310, 24708}, {2340, 3681}, {2346, 8271}, {3000, 24341}, {3008, 24598}, {3101, 5584}, {3218, 5228}, {3661, 6184}, {3666, 17014}, {3912, 25082}, {3928, 16579}, {3929, 16577}, {3998, 14552}, {4419, 8609}, {4511, 23151}, {4517, 23612}, {4558, 7054}, {4850, 5222}, {5235, 24607}, {5249, 10481}, {5745, 17080}, {10025, 16826}, {13726, 18732}, {14021, 17170}, {14953, 16699}, {15853, 25091}, {17147, 17158}, {17272, 25078}, {17284, 24036}, {17447, 20992}, {20173, 25237}, {21218, 24621}
X(24636) lies on these lines: {2, 99}, {4359, 17886}, {17292, 24384}, {17397, 20538}, {19804, 20951}, {20913, 24581}, {24580, 24587}
X(24637) lies on these lines: {2, 560}, {4384, 24613}, {20627, 21366}, {24586, 24595}
X(24638) lies on these lines: {2, 187}, {379, 1150}, {649, 3766}, {2245, 17277}, {3285, 15668}, {3734, 11351}, {4257, 17679}, {8626, 21241}, {16568, 20912}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28423.
X(24639) lies on this line: {1830,7649}
X(24640) lies on these lines:
X(24641) lies on these lines:
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28423.
X(24642) lies on this line: {8758,22465}
X(24643) lies on these lines:
X(24644) lies on the cubic K1083 and these lines: {1,971}, {2,165}, {4,5726}, {7,11522}, {9,5836}, {88,24637}, {144,4301}, {390,5691}, {518,11224}, {946,3361}, {954,1750}, {962,5234}, {990,16487}, {1001,2951}, {1012,13462}, {1156,11526}, {1709,10980}, {2550,7989}, {2800,10398}, {3243,16112}, {3339,12705}, {3656,5843}, {3679,5817}, {3731,12652}, {3869,4853}, {4321,9814}, {4326,8543}, {5175,6736}, {5572,15071}, {5759,9589}, {5779,7982}, {5805,15803}, {5833,6837}, {5880,16209}, {7965,9580}, {7995,12560}, {9588,18230}, {9819,15298}, {9955,10270}, {10268,22793}, {10679,18529}, {12672,18412}, {12699,15909}
X(24644) = reflection of X(3679) in X(5817)
X(24644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11372, 3062), (1001, 2951, 7987)
X(24645) lies on the cubic K1083 and these lines: {7,1699}, {63,5303}, {88,24637}, {145,4297}, {165,518}, {515,3339}, {1071,3361}, {1768,14151}, {3873,11224}, {5223,10164}, {5603,7992}, {5658,10398}, {5691,21454}, {5745,11407}, {6667,11219}, {7967,9819}, {7989,9776}, {7995,12563}, {10582,13243}, {10860,11531}
X(24646) lies on the cubics K040 and K1083, and on and these lines: {1,3308}, {2,11}, {3,23477}, {9,2590}, {516,1381}, {518,2446}, {1382,3911}, {3295,23517}
X(24646) = X(i)-Hirst inverse of X(j) for these (i,j): {1, 3308}, {9, 3307}
X(24646) = X(9)-line conjugate of X(2590)
X(24646) = crossdifference of every pair of points on line {665, 2590}
X(24647) lies on the cubics K040 and K1083, and on and these lines: {1,3307}, {2,11}, {3,23517}, {9,2591}, {516,1382}, {518,2447}, {1381,3911}, {3295,23477}
X(24647) = X(i)-Hirst inverse of X(j) for these (i,j): {1, 3307}, {9, 3308}
X(24647) = X(9)-line conjugate of X(2591)
X(24647) = crossdifference of every pair of points on line {665, 2591}
X(24648) lies on the cubic K1083 and these lines: {7,3308}, {57,100}, {145,3307}, {518,2447}, {1381,3218}
X(24649) lies on the cubic K1083 and these lines: {7,3307}, {57,100}, {145,3308}, {518,2446}, {1382,3218}
X(24650) lies on the cubics K289 and K907, and on these lines: {4,2575}, {6,15461}, {24,15460}, {25,110}, {51,13415}, {193,2574}, {511,1113}, {1312,3580}, {1345,5640}, {1347,19130}, {2104,2393}
X(24651) lies on the cubics K289 and K907, and on these lines: {4,2574}, {6,15460}, {24,15461}, {25,110}, {51,13414}, {193,2575}, {511,1114}, {1313,3580}, {1344,5640}, {1346,19130}, {2105,2393}
Collineation mappings involving Gemini triangle 3: X(24652)-X(24679)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 3, as in centers X(24652)-X(24679). Then
m(X) = a (a b + a c - b c) x + (a^2 b - a^2 c + a b ^2 + b^2 c) y + (a^2 c - a^2 b + a c ^2 + c^2 b) z : :
(Clark Kimberling, October 8, 2018)
X(24652) lies on these lines: {1, 20255}, {2, 17448}, {8, 24770}, {141, 21214}, {1125, 16846}, {1193, 4851}, {1279, 3616}, {2176, 24691}, {2275, 17279}, {3208, 25350}, {4713, 24215}, {9259, 24549}, {24653, 24655}, {24657, 24670}
X(24653) lies on these lines: {1, 3836}, {2, 3728}, {1125, 19518}, {3741, 17278}, {3757, 16986}, {4657, 20335}, {24174, 24369}, {24652, 24655}, {24658, 24660}, {24661, 24663}, {24671, 24672}
X(24654) lies on these lines: {1, 142}, {2, 17448}, {7, 16969}, {330, 344}, {2176, 4644}, {3009, 3475}, {4419, 24215}, {4851, 20036}, {4916, 20018}, {5819, 16524}, {5839, 16827}, {24661, 24669}, {24662, 24677}, {24672, 24679}, {25136, 25295}
X(24655) lies on these lines: {2, 3056}, {1284, 5880}, {1376, 15668}, {1818, 5794}, {2550, 3616}, {2886, 17245}, {3035, 17398}, {3925, 24551}, {4039, 4851}, {4645, 11375}, {4648, 4682}, {4660, 5886}, {4871, 17265}, {5432, 22370}, {17300, 17718}, {24652, 24653}, {24661, 24671}, {24663, 24672}, {24735, 25106}
X(24656) lies on these lines: {1, 21264}, {2, 17448}, {8, 3739}, {37, 1655}, {85, 24357}, {86, 17752}, {274, 20691}, {551, 3934}, {1100, 17033}, {1334, 4754}, {2295, 4670}, {3208, 4363}, {3616, 20530}, {3780, 17348}, {3788, 10197}, {4039, 4682}, {4085, 24366}, {4664, 20081}, {4675, 21281}, {10987, 16919}, {16827, 21904}, {17137, 17376}, {17792, 25124}, {24215, 25349}, {24661, 24662}
X(24657) lies on these lines: {2, 24525}, {2352, 4657}, {24652, 24670}
X(24658) lies on these lines: {2, 24526}, {24653, 24660}
X(24659) lies on these lines: {2, 18194}, {1125, 3883}, {1964, 3840}, {3759, 6686}, {3971, 16696}, {4434, 4682}, {6685, 17394}, {17448, 25106}, {24652, 24653}, {24660, 24668}, {24671, 24678}
X(24660) lies on these lines: {2, 24527}, {17792, 24739}, {24653, 24658}, {24659, 24668}
X(24661) lies on these lines: {1, 75}, {2, 18194}, {37, 87}, {43, 1100}, {171, 21769}, {330, 21080}, {1125, 14823}, {1385, 1742}, {1386, 21214}, {1449, 2664}, {3009, 17379}, {3248, 4687}, {3510, 20530}, {3723, 24696}, {3731, 9359}, {3759, 16569}, {3941, 8616}, {4419, 7240}, {4991, 17749}, {7032, 16826}, {7184, 17321}, {9902, 18147}, {15668, 18170}, {16525, 17754}, {16679, 16690}, {16710, 17155}, {16831, 20140}, {17260, 23524}, {17349, 23532}, {17398, 20532}, {24524, 25120}, {24653, 24663}, {24654, 24669}, {24655, 24671}, {24656, 24662}
X(24662) lies on these lines: {1, 1575}, {2, 24527}, {24654, 24677}, {24656, 24661}
X(24663) lies on these lines: {2, 17448}, {86, 1429}, {388, 1319}, {4417, 21214}, {4648, 16706}, {16705, 17078}, {24653, 24661}, {24655, 24672}
X(24664) lies on these lines: {2, 25103}, {24653, 24658}
X(24665) lies on these lines: {2, 24748}, {21191, 24667}
X(24666) lies on these lines: {2, 23655}, {238, 23568}, {649, 2666}, {650, 4449}, {659, 24768}, {663, 4369}, {676, 20508}, {1459, 4874}, {7662, 17418}, {14296, 17215}, {24747, 25142}
X(24666) = anticomplement of X(25143)
X(24667) lies on these lines: {2, 24525}, {3, 142}, {21191, 24665}, {24652, 24653}
X(24668) lies on these lines: {2, 17448}, {1125, 21240}, {3624, 16476}, {4648, 5550}, {24659, 24660}
X(24669) lies on these lines: {1, 20258}, {2, 3056}, {7, 21}, {71, 17754}, {105, 24540}, {497, 24551}, {1125, 6210}, {1740, 4000}, {2345, 4876}, {3475, 17379}, {3816, 17234}, {4039, 5839}, {4648, 20335}, {6690, 17381}, {24575, 24737}, {24654, 24661}
X(24670) lies on these lines: {2, 24679}, {1964, 3720}, {24652, 24657}
X(24671) lies on these lines: {2, 25118}, {3837, 17719}, {24653, 24672}, {24655, 24661}, {24659, 24678}, {24673, 24676}
X(24672) lies on these lines: {2, 18194}, {1279, 3616}, {4682, 17394}, {24653, 24671}, {24654, 24679}, {24655, 24663}
X(24673) lies on these lines: {2, 24754}, {17392, 25356}, {21191, 25423}, {24671, 24676}
X(24674) lies on these lines: {2, 20983}, {661, 4874}, {1459, 3837}, {1980, 19684}, {2517, 24381}, {3835, 4367}, {4010, 6129}, {8640, 23803}, {19701, 21005}, {21191, 24533}, {23301, 23655}, {24749, 25126}
X(24675) lies on these lines: {2, 23655}, {521, 25380}, {522, 6589}, {788, 6685}, {810, 17072}, {3221, 25142}, {3835, 7234}, {3907, 4885}, {21191, 24533}, {21197, 24325}
X(24676) lies on these lines: {2, 23656}, {3837, 4107}, {21191, 24533}, {24671, 24673}
X(24677) lies on these lines: {2, 25312}, {24654, 24662}
X(24678) lies on these lines: {2, 3056}, {69, 3769}, {86, 171}, {319, 4039}, {1279, 3616}, {1284, 24723}, {1964, 16706}, {2887, 17234}, {3662, 21010}, {3745, 17300}, {4871, 17283}, {7081, 15985}, {10436, 18788}, {17720, 21299}, {22279, 22281}, {24653, 24661}, {24659, 24671}
X(24679) lies on these lines: {1, 3696}, {2, 24670}, {1966, 16777}, {3403, 17318}, {17394, 17752}, {18194, 25102}, {24652, 24653}, {24654, 24672}, {24656, 24661}
See Angel Montesdeoca, HG130918.
X(24680) lies on this line: {1709,4915}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28430.
X(24681) lies on these lines: {502,8702}, {7984,11009}
Collineation mappings involving Gemini triangle 4: X(24682)-X(24726)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 4, as in centers X(24682)-X(24726). Then m(X) = a x - (a + b) y - (a + c) y : : . (Clark Kimberling, October 8, 2018)
X(24682) lies on these lines: {2, 2173}, {3, 24317}, {4, 8680}, {5, 24315}, {10, 534}, {19, 20305}, {25, 25343}, {30, 25362}, {307, 1839}, {379, 4466}, {405, 25359}, {442, 25363}, {469, 1762}, {524, 17733}, {527, 10916}, {946, 9028}, {1479, 24424}, {1656, 25341}, {1953, 21270}, {1995, 25344}, {3741, 4643}, {3838, 4670}, {4329, 21231}, {4419, 5225}, {4644, 11269}, {4655, 4837}, {4672, 24706}, {4708, 15254}, {5020, 25360}, {5133, 24321}, {5169, 24322}, {7377, 16560}, {7522, 25361}, {7770, 25364}, {17438, 20074}, {24696, 24711}
X(24683) lies on these lines: {1, 534}, {2, 2173}, {3, 24316}, {4, 24315}, {19, 18650}, {20, 8680}, {40, 9028}, {42, 3000}, {48, 4329}, {69, 1761}, {144, 3949}, {443, 25363}, {464, 1762}, {527, 3811}, {610, 18589}, {631, 24317}, {976, 4419}, {1370, 24321}, {1953, 20061}, {1973, 17170}, {2267, 5813}, {3090, 25341}, {3524, 25362}, {4294, 24424}, {4466, 24580}, {4640, 4643}, {4670, 5880}, {4837, 24714}, {5845, 11495}, {6353, 25360}, {6857, 25359}, {7490, 25361}, {7493, 25344}, {7494, 25343}, {14001, 25364}, {16063, 24322}, {17251, 18253}, {20074, 21271}, {24685, 24702}, {24694, 24700}, {24695, 24696}
X(24684) lies on these lines: {2, 2173}, {3, 8680}, {5, 25341}, {35, 24424}, {48, 21231}, {140, 24317}, {474, 25363}, {524, 17748}, {534, 1125}, {536, 8669}, {631, 24316}, {1441, 22054}, {1762, 7573}, {2278, 16609}, {4466, 24581}, {4643, 24685}, {4670, 6685}, {4672, 4836}, {5054, 25362}, {6676, 25360}, {6684, 9028}, {7483, 25359}, {7485, 24321}, {7495, 25344}, {7496, 24322}, {7499, 25343}, {7536, 25361}, {7807, 25364}, {17438, 21271}, {18650, 20305}, {24688, 24714}
X(24685) lies on these lines: {2, 2246}, {10, 544}, {11, 25342}, {100, 9318}, {101, 21232}, {214, 514}, {239, 385}, {320, 24628}, {527, 1155}, {659, 812}, {678, 24407}, {740, 4396}, {742, 4434}, {910, 20335}, {1055, 6647}, {1211, 5745}, {1215, 1376}, {3035, 5845}, {3570, 17755}, {3722, 24403}, {3758, 17754}, {3912, 4070}, {4109, 7767}, {4251, 17048}, {4362, 8667}, {4419, 17601}, {4421, 24352}, {4643, 24684}, {4721, 24850}, {4763, 22108}, {4892, 24699}, {5432, 25353}, {9451, 24841}, {16831, 18785}, {16832, 24618}, {17136, 21139}, {17439, 21272}, {24003, 24358}, {24683, 24702}, {25358, 25359}
X(24686) lies on these lines: {2, 2173}, {22, 25343}, {25, 25344}, {427, 24321}, {1370, 24316}, {1995, 25360}, {4466, 24584}, {5133, 24315}, {7391, 8680}, {7485, 24317}, {7571, 25341}, {16545, 16607}, {17442, 17492}
X(24687) lies on these lines: {2, 2173}, {23, 25344}, {858, 24322}, {4466, 24585}, {5169, 24315}, {5189, 8680}, {6636, 25343}, {7496, 24317}, {7570, 25341}, {13595, 25360}, {16063, 24316}, {16546, 21234}, {18669, 21274}
X(24688) lies on these lines: {2, 2234}, {10, 536}, {39, 25347}, {75, 3123}, {76, 714}, {141, 2486}, {321, 2228}, {674, 12263}, {700, 21443}, {726, 4735}, {1269, 3778}, {2245, 24259}, {3122, 20913}, {3728, 18133}, {3739, 21257}, {3741, 17237}, {3770, 24575}, {3834, 3838}, {3934, 6007}, {4479, 17250}, {4655, 4837}, {4688, 20340}, {4749, 17031}, {6385, 24732}, {17229, 25140}, {17445, 21278}, {17790, 24463}, {18040, 22167}, {18140, 24450}, {18143, 21330}, {21024, 25141}, {24684, 24714}, {24707, 24711}
X(24689) lies on these lines: {41, 25348}, {744, 4950}, {4643, 24713}, {4655, 24702}, {4799, 4837}, {16551, 17047}, {17046, 24329}, {17447, 21280}
X(24690) lies on these lines: {2, 44}, {10, 4754}, {38, 742}, {42, 524}, {63, 4376}, {69, 2276}, {72, 16720}, {141, 672}, {172, 17206}, {213, 16887}, {319, 17759}, {350, 6646}, {518, 24326}, {527, 3741}, {536, 17135}, {896, 4797}, {899, 25350}, {1086, 24592}, {1107, 17137}, {1575, 17344}, {2238, 4416}, {3219, 24358}, {3691, 20255}, {3720, 4364}, {3840, 4465}, {3873, 24357}, {4022, 24513}, {4357, 24512}, {4389, 17027}, {4396, 14829}, {4419, 10453}, {4424, 8682}, {4441, 17276}, {4640, 4760}, {4651, 4690}, {4655, 4799}, {4683, 24712}, {4725, 20011}, {4981, 25384}, {7196, 17950}, {16552, 21240}, {17026, 17274}, {17032, 17378}, {17152, 17448}, {17184, 25345}, {17272, 17754}, {17345, 20347}, {17347, 24514}
X(24691) lies on these lines: {2, 44}, {6, 24586}, {7, 21264}, {43, 524}, {63, 24358}, {69, 1575}, {141, 17754}, {193, 21904}, {291, 3416}, {350, 17276}, {354, 24357}, {527, 3840}, {536, 10453}, {672, 17279}, {742, 982}, {1086, 17026}, {2176, 24652}, {2275, 17137}, {2276, 4851}, {3218, 4376}, {3662, 25345}, {3741, 4363}, {3868, 16720}, {3873, 24326}, {4253, 21240}, {4426, 17206}, {4440, 4479}, {4650, 4797}, {4657, 24512}, {4667, 6685}, {4725, 20012}, {4799, 24712}, {7232, 20335}, {16887, 17750}, {17027, 17301}, {17149, 17790}, {17299, 17759}, {17345, 20530}, {17448, 21281}, {20255, 21384}, {24694, 24699}
X(24692) lies on these lines: {2, 4759}, {7, 4660}, {10, 527}, {44, 25351}, {69, 4709}, {79, 3831}, {320, 519}, {528, 7238}, {545, 4439}, {551, 4675}, {726, 4440}, {740, 17374}, {752, 1086}, {899, 17491}, {908, 23831}, {1125, 17305}, {1155, 4892}, {1738, 17770}, {1836, 3840}, {2796, 3912}, {2887, 11246}, {3218, 21241}, {3474, 3771}, {3634, 24697}, {3679, 4741}, {3741, 20292}, {3821, 14621}, {3828, 17256}, {3834, 4432}, {3836, 4422}, {3923, 4312}, {3932, 17767}, {3993, 17316}, {4084, 4259}, {4085, 17365}, {4090, 5905}, {4693, 17297}, {4715, 4753}, {4732, 17344}, {4784, 4785}, {4871, 5057}, {4966, 17764}, {6173, 24331}, {6327, 24165}, {7613, 24599}, {15310, 24827}, {17449, 20042}, {17766, 24231}
X(24693) lies on these lines: {1, 4743}, {2, 4432}, {10, 527}, {45, 2796}, {69, 4732}, {142, 20181}, {320, 3679}, {519, 4675}, {528, 24331}, {551, 17067}, {740, 17316}, {752, 4384}, {984, 4440}, {1150, 21027}, {1647, 24594}, {1698, 17354}, {1738, 17023}, {1739, 3764}, {2550, 24325}, {2640, 4429}, {3306, 21242}, {3696, 17374}, {3739, 4660}, {3754, 4259}, {3821, 17325}, {3826, 3923}, {3836, 17284}, {3842, 24248}, {3925, 3980}, {4085, 10436}, {4307, 4974}, {4359, 4865}, {4407, 17274}, {4413, 25385}, {4439, 4659}, {4644, 4753}, {4663, 4796}, {4693, 17244}, {4703, 20292}, {4709, 4851}, {9780, 24697}, {17256, 19875}, {17450, 21283}, {17722, 24620}
X(24694) lies on these lines: {2, 2246}, {9, 4872}, {10, 24701}, {55, 25353}, {169, 17046}, {239, 7837}, {320, 17026}, {524, 4362}, {527, 1836}, {742, 4865}, {984, 20539}, {1376, 24318}, {2082, 17062}, {2886, 5845}, {3684, 7179}, {3729, 4119}, {3741, 4643}, {4056, 16552}, {4109, 7776}, {4376, 4438}, {4655, 4799}, {4911, 21384}, {5275, 24241}, {6173, 24600}, {9318, 11680}, {11235, 24352}, {17451, 21285}, {24316, 24336}, {24319, 24328}, {24683, 24700}, {24691, 24699}
X(24695) lies on these lines: {1, 527}, {2, 896}, {4, 1046}, {6, 17768}, {7, 238}, {8, 752}, {31, 5905}, {38, 20078}, {43, 2635}, {44, 5880}, {46, 2183}, {56, 15507}, {57, 5121}, {58, 17139}, {69, 3923}, {79, 1714}, {144, 984}, {145, 537}, {171, 329}, {193, 740}, {226, 1707}, {320, 4676}, {387, 24851}, {513, 4259}, {516, 3751}, {524, 5695}, {553, 5272}, {579, 1756}, {583, 24744}, {611, 5857}, {612, 17781}, {651, 4331}, {846, 5712}, {966, 24342}, {982, 9965}, {1001, 17365}, {1125, 4896}, {1281, 7774}, {1386, 17276}, {1468, 11415}, {1738, 1743}, {1742, 5759}, {1757, 2550}, {1836, 4641}, {1992, 2796}, {2094, 18201}, {2308, 19785}, {2650, 6872}, {2795, 7737}, {2835, 5903}, {3120, 24597}, {3242, 5852}, {3416, 17351}, {3475, 8616}, {3618, 3821}, {3619, 24295}, {3648, 19767}, {3663, 16475}, {3685, 17364}, {3729, 5847}, {3758, 24723}, {3779, 15310}, {3826, 16885}, {3928, 24239}, {3979, 10385}, {3980, 14555}, {4000, 16468}, {4011, 18141}, {4295, 5247}, {4310, 20059}, {4353, 16491}, {4383, 11246}, {4418, 5739}, {4523, 12530}, {4645, 17350}, {4675, 15254}, {4748, 19856}, {4862, 16469}, {5057, 11269}, {5222, 16477}, {5263, 17347}, {5327, 24316}, {5744, 17717}, {5745, 16570}, {5850, 16496}, {5988, 7735}, {5992, 7766}, {6173, 15601}, {6604, 24395}, {6650, 20158}, {7290, 24231}, {9776, 17123}, {9791, 17379}, {10186, 11364}, {14912, 24257}, {16476, 17753}, {16830, 17333}, {17063, 21454}, {17122, 18228}, {17126, 17484}, {17127, 17483}, {17165, 20064}, {17716, 20214}, {24683, 24696}, {24728, 25406}
X(24696) lies on these lines: {1, 536}, {2, 2234}, {6, 1045}, {37, 1740}, {39, 4443}, {42, 3758}, {43, 44}, {45, 2664}, {75, 2309}, {76, 24327}, {87, 1100}, {190, 869}, {192, 1964}, {194, 714}, {256, 4261}, {386, 4672}, {513, 3795}, {674, 12782}, {740, 5145}, {872, 17350}, {899, 17335}, {978, 15254}, {995, 4432}, {1193, 4676}, {1278, 17445}, {1580, 2278}, {1613, 21883}, {2235, 2276}, {2274, 3685}, {2667, 17379}, {3009, 4664}, {3097, 4735}, {3122, 24598}, {3240, 4781}, {3248, 4393}, {3720, 4479}, {3723, 24661}, {3736, 3923}, {3739, 16571}, {3759, 22343}, {3783, 4643}, {3875, 18170}, {4116, 24502}, {4360, 7032}, {4377, 9902}, {4446, 21746}, {4493, 21814}, {4644, 24722}, {4655, 24707}, {4836, 24714}, {4837, 24726}, {4851, 7184}, {4852, 18194}, {5069, 24575}, {5110, 8424}, {5114, 24264}, {5283, 24450}, {7786, 25347}, {9020, 9941}, {9025, 20691}, {17443, 17891}, {21238, 21299}, {24682, 24711}, {24683, 24695}
X(24697) lies on these lines: {1, 524}, {2, 896}, {5, 7609}, {8, 192}, {9, 46}, {10, 190}, {21, 24436}, {58, 86}, {81, 6536}, {319, 3993}, {333, 4425}, {726, 17258}, {752, 16830}, {846, 1211}, {960, 3792}, {966, 24248}, {1001, 17253}, {1043, 12579}, {1046, 4205}, {1054, 5241}, {1423, 10404}, {1503, 8245}, {1757, 4026}, {1962, 2895}, {2792, 6998}, {3120, 5235}, {3244, 3883}, {3616, 4741}, {3622, 5625}, {3624, 4675}, {3633, 7174}, {3634, 24692}, {3647, 24931}, {3685, 3775}, {3686, 4716}, {3717, 4691}, {3773, 17261}, {3821, 17277}, {3836, 17260}, {3842, 4645}, {3844, 16814}, {3923, 5224}, {3944, 5737}, {4001, 4038}, {4259, 5692}, {4389, 16825}, {4416, 4649}, {4505, 25280}, {4657, 16468}, {4676, 17250}, {4708, 19856}, {4748, 5698}, {4886, 4970}, {4974, 17302}, {5057, 14009}, {5248, 7301}, {5302, 19879}, {5327, 25359}, {5333, 8040}, {5695, 17251}, {5739, 17592}, {5743, 17596}, {6210, 12699}, {6646, 24325}, {6650, 17755}, {6651, 17292}, {6707, 17306}, {9534, 24717}, {9780, 24693}, {10180, 17778}, {15254, 17237}, {15569, 17344}, {16477, 17023}, {16823, 17254}, {17116, 17767}, {17307, 24295}, {17319, 17772}, {17367, 20142}, {17889, 19732}, {22174, 24923}
X(24698) lies on these lines: {513, 4486}, {667, 25356}, {1919, 17397}, {4675, 21191}, {4785, 24699}, {6542, 17458}, {17256, 20979}, {17308, 21262}, {21260, 24354}, {24716, 24718}
X(24699) lies on these lines: {2, 2243}, {10, 527}, {79, 4721}, {171, 25345}, {238, 25357}, {239, 320}, {315, 20271}, {536, 24715}, {742, 4645}, {1836, 4713}, {3120, 4396}, {3509, 20541}, {3662, 20179}, {3726, 20553}, {3836, 24358}, {4364, 16830}, {4465, 5057}, {4660, 24357}, {4670, 24707}, {4675, 17023}, {4708, 19856}, {4785, 24698}, {4805, 5902}, {4892, 24685}, {4911, 20255}, {5277, 17211}, {7759, 24046}, {7780, 24160}, {14023, 24159}, {17256, 17369}, {20292, 24330}, {21282, 24403}, {24691, 24694}
X(24700) lies on these lines: {2, 24702}, {3, 24336}, {21, 25371}, {527, 3916}, {572, 21233}, {958, 24334}, {1444, 4032}, {2182, 5745}, {4640, 4670}, {4643, 24684}, {4644, 4650}, {4798, 24703}, {4999, 24319}, {5433, 25369}, {8424, 24358}, {17440, 21273}, {23085, 24328}, {24683, 24694}
X(24701) lies on these lines: {1, 30}, {4, 85}, {10, 24694}, {19, 25365}, {25, 18651}, {77, 1892}, {304, 1330}, {1368, 1763}, {1370, 17441}, {1930, 5814}, {4655, 24705}, {4670, 5880}, {5777, 5928}, {6998, 17181}, {16502, 23537}, {18589, 24320}, {18596, 21530}, {19544, 21621}
X(24702) lies on these lines: {2, 24700}, {4, 24336}, {9, 7377}, {56, 25369}, {527, 1210}, {958, 24319}, {1329, 24334}, {1766, 21244}, {2478, 25371}, {3452, 5782}, {3741, 4643}, {4640, 4708}, {4655, 24689}, {4670, 5087}, {4748, 5698}, {5816, 21233}, {17452, 21286}, {24683, 24685}
X(24703) lies on these lines: {1, 529}, {2, 1155}, {3, 12608}, {4, 960}, {5, 12514}, {6, 24210}, {7, 3742}, {8, 3967}, {9, 1699}, {10, 381}, {11, 63}, {12, 5250}, {20, 12679}, {21, 11375}, {30, 997}, {31, 17720}, {38, 17721}, {40, 1329}, {46, 4187}, {55, 908}, {57, 3816}, {65, 2478}, {72, 1479}, {78, 6284}, {79, 3624}, {80, 3899}, {142, 8167}, {144, 5274}, {149, 3681}, {165, 3035}, {190, 3705}, {191, 7741}, {200, 528}, {210, 3434}, {214, 16128}, {226, 1001}, {238, 3772}, {283, 7299}, {306, 4387}, {312, 3416}, {321, 3966}, {329, 497}, {354, 5905}, {355, 3878}, {382, 17647}, {392, 1478}, {404, 24954}, {405, 12047}, {430, 1211}, {452, 3485}, {474, 1770}, {499, 3916}, {513, 3784}, {515, 5289}, {516, 1376}, {517, 6929}, {527, 11019}, {614, 3782}, {748, 3120}, {758, 5722}, {846, 17717}, {946, 958}, {950, 12635}, {962, 2551}, {965, 1839}, {968, 5718}, {978, 24851}, {982, 17276}, {993, 5886}, {1056, 10179}, {1086, 5272}, {1125, 16418}, {1158, 6922}, {1191, 13161}, {1519, 3428}, {1621, 17718}, {1633, 19649}, {1697, 12607}, {1698, 17530}, {1706, 9589}, {1708, 14022}, {1709, 15842}, {1737, 17556}, {1788, 6919}, {1837, 3869}, {1848, 11323}, {2550, 3740}, {2646, 6872}, {2887, 4011}, {2975, 11376}, {3036, 14217}, {3057, 3436}, {3058, 3870}, {3185, 4192}, {3218, 17728}, {3219, 11680}, {3244, 4930}, {3286, 17182}, {3295, 21077}, {3305, 3925}, {3306, 11246}, {3419, 3583}, {3421, 3880}, {3612, 16154}, {3616, 4870}, {3647, 16159}, {3650, 16153}, {3685, 4417}, {3687, 5695}, {3689, 20075}, {3696, 14555}, {3702, 10371}, {3706, 5739}, {3715, 25006}, {3720, 24725}, {3730, 20544}, {3741, 4643}, {3752, 24248}, {3756, 18193}, {3771, 4432}, {3811, 15171}, {3812, 4295}, {3813, 9614}, {3814, 6980}, {3817, 5745}, {3826, 7308}, {3829, 3929}, {3840, 4655}, {3846, 3923}, {3848, 9776}, {3873, 17484}, {3877, 5080}, {3890, 20060}, {3913, 10624}, {3914, 4383}, {3927, 9669}, {3940, 9668}, {3952, 5014}, {3962, 12649}, {3971, 4865}, {4009, 10327}, {4193, 24914}, {4297, 6259}, {4299, 17614}, {4301, 5795}, {4302, 5440}, {4307, 4682}, {4310, 4906}, {4312, 5437}, {4338, 17575}, {4358, 6327}, {4421, 6745}, {4423, 5249}, {4425, 4657}, {4428, 13405}, {4465, 4799}, {4511, 11114}, {4512, 5219}, {4517, 20556}, {4641, 11269}, {4645, 18743}, {4652, 5433}, {4654, 10582}, {4662, 5082}, {4675, 24708}, {4798, 24700}, {4847, 5220}, {4853, 13463}, {4854, 5256}, {4855, 15338}, {4857, 5904}, {4862, 5573}, {4952, 17765}, {4957, 17890}, {4973, 10199}, {4999, 8227}, {5016, 25253}, {5044, 22793}, {5068, 18231}, {5119, 17757}, {5123, 5657}, {5187, 17606}, {5218, 5748}, {5223, 24392}, {5248, 11374}, {5251, 18393}, {5273, 5832}, {5302, 19843}, {5328, 9778}, {5436, 11281}, {5537, 15813}, {5603, 6976}, {5691, 15829}, {5705, 18253}, {5709, 7681}, {5712, 15569}, {5720, 5842}, {5727, 5855}, {5730, 10572}, {5731, 12678}, {5744, 10589}, {5762, 7956}, {5784, 10431}, {5837, 19925}, {5853, 21060}, {5887, 6928}, {5909, 11254}, {6001, 6827}, {6210, 20545}, {6691, 15803}, {6734, 10896}, {6836, 12688}, {6837, 15823}, {6865, 9943}, {6893, 7686}, {7270, 19582}, {7290, 17061}, {7354, 19861}, {7991, 8256}, {8165, 20070}, {8583, 9579}, {8616, 17719}, {8666, 11373}, {9581, 12526}, {9809, 11220}, {10401, 17183}, {10738, 18254}, {10895, 24987}, {10947, 17615}, {10950, 11682}, {10980, 17051}, {11108, 12609}, {11238, 17781}, {11362, 12700}, {12053, 12513}, {12116, 14872}, {12433, 12559}, {12571, 18249}, {13271, 14740}, {15320, 17259}, {15601, 17070}, {16152, 17525}, {16569, 24715}, {17123, 17278}, {17139, 18165}, {17732, 25066}, {20076, 20323}
X(24704) lies on these lines: {58, 25370}, {79, 4363}, {536, 24851}, {1761, 21245}, {1836, 4643}, {3454, 24335}, {4016, 21287}, {4644, 17491}, {4655, 4837}, {4708, 19856}
X(24705) lies on these lines: {1, 527}, {9, 10446}, {40, 24334}, {65, 25371}, {71, 20258}, {226, 17139}, {314, 4044}, {573, 21246}, {946, 24319}, {960, 12545}, {966, 10456}, {1400, 17183}, {1764, 2183}, {1944, 3496}, {1999, 17781}, {2269, 20245}, {2300, 3663}, {3686, 10447}, {3741, 4643}, {4640, 4670}, {4655, 24701}, {4708, 5087}, {5257, 10455}, {5745, 10478}, {5795, 12435}, {5819, 10442}, {7175, 8822}, {10441, 12572}, {10465, 15829}, {10480, 12527}, {17276, 21769}, {21061, 22031}
X(24706) lies on these lines: {66, 25372}, {206, 24337}, {4672, 24682}, {16544, 21247}, {17453, 21288}
X(24707) lies on these lines: {2, 2244}, {42, 320}, {83, 25374}, {744, 2896}, {1078, 25346}, {3096, 4381}, {4640, 17237}, {4655, 24696}, {4670, 24699}, {6292, 24340}, {16556, 21249}, {17457, 21289}, {24688, 24711}
X(24708) lies on these lines: {1, 527}, {2, 3000}, {6, 4335}, {9, 1742}, {40, 13598}, {43, 44}, {85, 25375}, {87, 8424}, {144, 2293}, {165, 2183}, {170, 1212}, {405, 1044}, {978, 15601}, {1001, 4334}, {1042, 11106}, {1423, 20992}, {1633, 2267}, {2269, 9309}, {2310, 24635}, {2340, 6172}, {2951, 5819}, {3208, 9025}, {3576, 15507}, {3672, 20978}, {3870, 20072}, {4499, 23407}, {4675, 24703}, {6007, 21384}, {8572, 21214}, {17026, 24717}, {17139, 17194}
X(24709) lies on these lines: {2, 4432}, {10, 10707}, {11, 4422}, {88, 2796}, {100, 11814}, {121, 5541}, {149, 24003}, {190, 1647}, {244, 4440}, {519, 4767}, {528, 9458}, {545, 24408}, {551, 4945}, {3315, 21093}, {3716, 4728}, {3840, 4683}, {3952, 20042}, {3994, 5211}, {4465, 4643}, {4655, 24710}, {4800, 14286}, {4871, 5057}, {17460, 21290}
X(24710) lies on these lines: {2, 4759}, {89, 25378}, {4643, 5057}, {4655, 24709}, {16558, 21251}, {17461, 21291}
X(24711) lies on these lines: {2, 24714}, {99, 24348}, {115, 24345}, {523, 13178}, {2611, 24500}, {2640, 8287}, {2643, 21221}, {4837, 24717}, {24682, 24696}, {24688, 24707}, {24712, 24722}, {24713, 24715}
X(24712) lies on these lines: {1, 544}, {2, 2246}, {4, 7960}, {10, 10712}, {11, 5845}, {38, 497}, {41, 17181}, {80, 514}, {81, 226}, {100, 24318}, {116, 5540}, {150, 2170}, {210, 4553}, {291, 812}, {294, 3008}, {319, 3807}, {524, 17763}, {527, 1156}, {672, 4872}, {1475, 4911}, {1565, 9317}, {1621, 25353}, {1699, 9355}, {1959, 3930}, {2112, 17397}, {2280, 7179}, {2503, 8287}, {3061, 21285}, {3732, 21044}, {3925, 4472}, {4056, 4253}, {4423, 25367}, {4465, 4643}, {4644, 11269}, {4670, 25383}, {4675, 24512}, {4683, 24690}, {4712, 20539}, {4799, 24691}, {5074, 5526}, {5276, 24241}, {5308, 9502}, {7247, 17474}, {11238, 24352}, {11680, 24333}, {14439, 20533}, {17308, 20540}, {17439, 20096}, {17463, 24488}, {21221, 24504}, {21381, 24347}, {24711, 24722}
X(24713) lies on these lines: {101, 25379}, {116, 24346}, {692, 4466}, {1565, 5848}, {3448, 24500}, {4643, 24689}, {16560, 21252}, {17463, 21293}, {21339, 24489}, {24711, 24715}
X(24714) lies on these lines: {2, 24711}, {99, 24345}, {523, 11711}, {620, 24348}, {662, 21254}, {2795, 24350}, {4836, 24696}, {4837, 24683}, {17467, 21295}, {24684, 24688}
X(24715) lies on these lines: {1, 528}, {2, 4432}, {4, 9365}, {8, 537}, {10, 190}, {11, 1054}, {30, 8481}, {35, 24161}, {42, 20292}, {43, 1836}, {46, 16560}, {55, 17889}, {65, 24836}, {72, 24835}, {75, 4660}, {79, 3293}, {80, 900}, {88, 1647}, {100, 3120}, {106, 14028}, {142, 16484}, {148, 24505}, {149, 244}, {165, 17064}, {171, 3914}, {238, 516}, {239, 752}, {291, 812}, {319, 4709}, {320, 519}, {334, 874}, {335, 740}, {390, 7613}, {484, 2161}, {497, 17063}, {515, 24813}, {517, 3792}, {536, 24699}, {545, 3679}, {668, 4783}, {846, 3925}, {851, 5143}, {894, 4085}, {899, 5057}, {956, 24826}, {978, 12699}, {982, 3434}, {984, 2550}, {1320, 24131}, {1376, 3944}, {1479, 24174}, {1581, 19637}, {1633, 24436}, {1644, 4945}, {1654, 4732}, {1698, 4422}, {1737, 24846}, {1739, 3583}, {1757, 17768}, {1770, 5247}, {1837, 24840}, {1929, 20531}, {1961, 4854}, {2006, 23703}, {2170, 24484}, {2475, 4642}, {2607, 2640}, {2786, 13178}, {2802, 24864}, {2810, 4014}, {2836, 21889}, {2886, 17596}, {3057, 24837}, {3210, 4865}, {3240, 24725}, {3306, 24217}, {3416, 9055}, {3474, 4650}, {3509, 21956}, {3550, 3772}, {3585, 3987}, {3632, 9041}, {3685, 3836}, {3696, 4690}, {3699, 21093}, {3722, 20095}, {3749, 23681}, {3750, 5249}, {3751, 4312}, {3755, 4649}, {3782, 3961}, {3791, 20101}, {3821, 5263}, {3834, 4702}, {3842, 9791}, {3879, 4780}, {3912, 4693}, {3923, 4429}, {4026, 4472}, {4051, 9597}, {4080, 17780}, {4259, 5903}, {4307, 17014}, {4366, 17397}, {4370, 19875}, {4407, 17254}, {4409, 4668}, {4418, 4972}, {4442, 17763}, {4443, 24464}, {4473, 9780}, {4514, 24165}, {4518, 5992}, {4640, 21949}, {4651, 4683}, {4695, 5080}, {4716, 5847}, {4753, 20072}, {4850, 17722}, {4862, 16496}, {4947, 24399}, {5014, 17155}, {5090, 24814}, {5134, 16611}, {5235, 21027}, {5252, 24816}, {5255, 23537}, {5272, 9580}, {5308, 20533}, {5541, 24222}, {5587, 24828}, {5657, 24817}, {5687, 24820}, {5688, 24832}, {5689, 24831}, {5695, 17269}, {5790, 24844}, {5853, 24231}, {6154, 17724}, {6174, 9324}, {6187, 13589}, {8193, 24822}, {8197, 24823}, {8204, 24824}, {8214, 24838}, {8215, 24839}, {8616, 24789}, {9614, 11512}, {9802, 17460}, {9857, 24825}, {10039, 24845}, {10624, 24178}, {10791, 24815}, {10914, 24834}, {10915, 24847}, {10916, 24848}, {11900, 24830}, {12701, 21214}, {13883, 24819}, {13893, 24842}, {13936, 24818}, {13947, 24843}, {15485, 17278}, {16173, 24871}, {16569, 24703}, {17122, 24210}, {17308, 17738}, {17484, 21805}, {17491, 19998}, {17495, 21282}, {17715, 20075}, {17716, 19785}, {17777, 24003}, {18193, 24392}, {21098, 21944}, {21242, 24627}, {24711, 24713}
X(24716) lies on these lines: {625, 24350}, {4655, 4837}, {16568, 21256}, {17472, 21298}, {24698, 24718}
X(24717) lies on these lines: {2, 2234}, {8, 536}, {75, 21299}, {76, 6007}, {194, 4443}, {320, 1836}, {714, 20081}, {1278, 21278}, {3122, 24621}, {3210, 3764}, {4277, 4734}, {4655, 10449}, {4740, 20352}, {4741, 17135}, {4837, 24711}, {9025, 17144}, {9534, 24697}, {16571, 21257}, {17026, 24708}, {17155, 23633}, {17299, 25311}, {17790, 24451}, {22016, 25279}
X(24718) lies on these lines: {10, 4913}, {513, 23301}, {650, 20316}, {656, 18155}, {1577, 10479}, {3716, 6003}, {4874, 9013}, {6002, 17072}, {7662, 25128}, {17478, 21300}, {23783, 23874}, {24698, 24716}
X(24719) lies on these lines: {2, 4782}, {8, 4083}, {79, 4905}, {320, 350}, {514, 4122}, {522, 4810}, {523, 4382}, {649, 3837}, {659, 3835}, {663, 4992}, {667, 1125}, {812, 1491}, {1019, 23815}, {1698, 4063}, {3309, 12699}, {3667, 4458}, {3669, 10404}, {3700, 4813}, {3777, 6002}, {4170, 6004}, {4380, 9508}, {4448, 4679}, {4498, 21051}, {4724, 4806}, {4728, 4874}, {4762, 4824}, {4784, 4785}, {5880, 6008}, {17367, 24601}, {17458, 23656}, {24698, 24716}
X(24720) lies on these lines: {2, 4724}, {10, 514}, {145, 4449}, {512, 23815}, {513, 3716}, {522, 693}, {523, 3776}, {650, 25380}, {661, 4521}, {663, 3616}, {918, 4522}, {1125, 4794}, {1577, 4905}, {1734, 4978}, {2523, 16612}, {2526, 25128}, {3624, 4040}, {3667, 4010}, {3669, 3907}, {3810, 7178}, {4041, 4801}, {4106, 7659}, {4142, 21188}, {4389, 4406}, {4394, 4830}, {4462, 21052}, {4474, 21222}, {4728, 6006}, {4762, 4913}, {4784, 4785}, {4791, 23796}, {4823, 8714}, {4977, 20316}, {6372, 21260}, {8672, 23301}, {8689, 24924}, {16737, 18299}, {20521, 21189}
X(24720) = complement of X(4724)
X(24721) lies on these lines: {291, 812}, {513, 4486}, {514, 4774}, {659, 25381}, {3768, 17256}, {3776, 4382}, {3837, 4375}, {4369, 4728}, {4458, 4810}, {4472, 24354}, {4491, 25356}, {4784, 4785}, {6542, 21303}, {8632, 17397}, {17308, 21261}
X(24722) lies on these lines: {1, 545}, {37, 7240}, {87, 17276}, {244, 24482}, {291, 513}, {292, 24423}, {335, 24510}, {668, 25382}, {1015, 24338}, {1045, 7277}, {1086, 9359}, {2234, 20072}, {3122, 4499}, {3248, 4440}, {3783, 4715}, {4398, 23524}, {4644, 24696}, {5539, 24345}, {7184, 17351}, {7321, 22343}, {19945, 24625}, {24711, 24712}
X(24723) lies on these lines: {1, 320}, {2, 1155}, {8, 536}, {9, 4429}, {10, 190}, {11, 24627}, {21, 1633}, {31, 19786}, {37, 4645}, {38, 4514}, {42, 4683}, {43, 4703}, {65, 17950}, {75, 23690}, {85, 4331}, {141, 3685}, {145, 4741}, {171, 4425}, {192, 3416}, {238, 3821}, {257, 19637}, {274, 19643}, {319, 740}, {333, 3914}, {516, 4357}, {518, 6646}, {528, 17254}, {744, 2292}, {846, 2887}, {894, 4026}, {960, 4201}, {968, 18134}, {984, 4660}, {986, 3764}, {1001, 3662}, {1010, 1770}, {1086, 16823}, {1125, 17305}, {1279, 17235}, {1284, 24678}, {1330, 3931}, {1386, 17302}, {1456, 17086}, {1621, 17184}, {1654, 3696}, {1698, 17354}, {1738, 17277}, {1757, 4085}, {1999, 4854}, {2245, 5051}, {2345, 24280}, {2550, 17257}, {2895, 3896}, {3057, 5484}, {3210, 3966}, {3219, 4972}, {3242, 17255}, {3286, 17202}, {3616, 4675}, {3617, 20073}, {3661, 5695}, {3663, 3883}, {3666, 4388}, {3745, 20101}, {3751, 17347}, {3755, 4416}, {3757, 3782}, {3758, 24695}, {3775, 17764}, {3790, 17262}, {3792, 3878}, {3823, 16814}, {3826, 17260}, {3836, 17263}, {3844, 17280}, {3846, 17596}, {3869, 4259}, {3879, 4356}, {3886, 17272}, {3920, 4450}, {3923, 17289}, {3932, 17261}, {3993, 17315}, {4003, 5211}, {4295, 13725}, {4307, 17321}, {4312, 10436}, {4337, 18465}, {4360, 5847}, {4364, 16830}, {4415, 7081}, {4417, 17594}, {4418, 19808}, {4488, 5772}, {4649, 17770}, {4663, 20072}, {4749, 5262}, {4966, 17288}, {5014, 7226}, {5220, 17333}, {5846, 17246}, {5992, 20716}, {6284, 24726}, {7290, 17304}, {7321, 24325}, {11110, 12609}, {11375, 19278}, {12047, 19270}, {12514, 16062}, {13735, 19869}, {14829, 24210}, {15569, 17300}, {16475, 17380}, {17123, 24169}, {17276, 24349}, {22053, 24550}
X(24724) lies on these lines: {1, 3849}, {79, 4754}, {524, 4442}, {896, 25383}, {1836, 4643}, {4440, 7779}, {4465, 5057}, {4760, 4892}, {4784, 4785}, {9055, 21282}, {16702, 17204}
X(24725) lies on these lines: {1, 535}, {2, 896}, {4, 2650}, {6, 3120}, {7, 244}, {11, 17365}, {31, 226}, {38, 5905}, {42, 1836}, {43, 20292}, {79, 386}, {81, 3944}, {142, 17125}, {329, 756}, {516, 2177}, {614, 4654}, {748, 5249}, {750, 908}, {899, 5880}, {902, 17718}, {982, 17483}, {984, 17484}, {986, 14450}, {1046, 2476}, {1150, 17770}, {1201, 10404}, {1215, 6327}, {1468, 12047}, {1647, 4860}, {1724, 11263}, {1962, 5712}, {2308, 3772}, {2550, 21805}, {3218, 17717}, {3240, 24715}, {3452, 17124}, {3649, 3924}, {3664, 9345}, {3720, 24703}, {3782, 17017}, {3838, 4641}, {3915, 13407}, {3923, 3936}, {3980, 5741}, {4011, 18139}, {4054, 5847}, {4062, 5695}, {4138, 5294}, {4295, 4642}, {4300, 5812}, {4392, 17722}, {4414, 5718}, {4415, 5311}, {4417, 4418}, {4425, 19684}, {4519, 17374}, {4644, 11269}, {4675, 4679}, {4865, 17165}, {5226, 9340}, {5230, 5714}, {5739, 21020}, {5902, 6788}, {6536, 19701}, {6690, 17775}, {9355, 10883}, {9612, 21935}, {16272, 20277}, {17126, 17719}, {17276, 17723}, {17300, 17777}, {17449, 17721}, {17771, 21242}, {19767, 24851}, {20430, 21326}
X(24725) = {X(33096),X(33097)}-harmonic conjugate of X(2)
X(24725) lies on these lines: {2, 2173}, {190, 3661}, {320, 1999}, {384, 25364}, {4466, 24610}, {4655, 10449}, {4837, 24696}, {5025, 24315}, {6284, 24723}, {6655, 8680}, {7791, 24316}, {7824, 24317}, {24688, 24707}
Let U = obverse triangle of X(1), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by
b^3 x^3-c^3 x^3+a b^2 x^2 y-b c^2 x^2 y-a^2 b x y^2+a c^2 x y^2-a^3 y^3+c^3 y^3+b^2 c x^2 z-a c^2 x^2 z-a^2 c y^2 z+b c^2 y^2 z-a b^2 x z^2+a^2 c x z^2+a^2 b y z^2-b^2 c y z^2+a^3 z^3-b^3 z^3 = 0.
This cubic passes through X(i) for i = 1,2,9,239,516,3496,3509,13174,16557,17738, 24727,24728. (Peter Moses, October 8, 2018)
X(24727) lies on these lines: {1,6}, {39,1045}, {57,6063}, {63,17027}, {239,19565}, {672,3685}, {740,24578}, {894,1475}, {2082,17033}, {2111,17738}, {2344,5138}, {2350,4418}, {3218,17029}, {3305,17032}, {3306,17028}, {3496,16574}, {3501,3886}, {3509,17031}, {3691,17260}, {3923,4253}, {3928,16557}, {4063,4762}, {4876,9052}, {5364,10453}, {10436,17030}
See X(24727).
X(24728) lies on the cubic introduced at X(24727) and on these lines: {1,7}, {2,8245}, {3,3923}, {4,3821}, {40,726}, {69,2784}, {103,789}, {165,3729}, {192,18788}, {376,2796}, {511,24257}, {515,4660}, {572,2344}, {573,24578}, {631,24295}, {740,1350}, {944,17766}, {971,18252}, {1403,7580}, {1431,24268}, {1503,4655}, {1699,17304}, {1766,3509}, {1768,16566}, {2783,3098}, {2938,6194}, {3522,24280}, {3831,12618}, {3840,21629}, {3980,4220}, {4011,19649}, {4192,24260}, {4672,5085}, {5918,17635}, {6210,16825}, {6685,10445}, {6776,17770}, {7155,17738}, {7411,11688}, {8669,12512}, {8931,16822}, {9740,17132}, {9841,12717}, {10164,17355}, {10167,12723}, {11220,12530}, {12725,19645}, {24309,24326}
Let U = obverse triangle of X(6), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by
b^6 x^3-c^6 x^3+a^2 b^4 x^2 y-b^2 c^4 x^2 y-a^4 b^2 x y^2+a^2 c^4 x y^2-a^6 y^3+c^6 y^3+b^4 c^2 x^2 z-a^2 c^4 x^2 z-a^4 c^2 y^2 z+b^2 c^4 y^2 z-a^2 b^4 x z^2+a^4 c^2 x z^2+a^4 b^2 y z^2-b^4 c^2 y z^2+a^6 z^3-b^6 z^3 = 0.
This cubic passes through X(i) for i = 2,3,6,385,1503,3511,5989,6660,8301,8424,15588, 24729,24730. (Peter Moses, October 8, 2018)
X(24729) lies on these lines: {3,6}, {25,8840}, {237,12215}, {385,19566}, {732,3511}, {3499,8265}, {7824,22062}, {8424,23851}, {15588,20854}, {21006,23878}
See X(24729).
X(24730) lies on the cubic introduced at X(24729) and on these lines: {4,6}, {154,1975}, {159,698}, {206,3492}, {1853,7851}, {2782,6759}, {3511,15270}, {3556,8301}, {6000,14880}, {7789,10192}
Let U = obverse triangle of X(75), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by
a^3 b^3 x^3-a^3 c^3 x^3+a^3 b^2 c x^2 y-a^2 b c^3 x^2 y-a^2 b^3 c x y^2+a b^2 c^3 x y^2-a^3 b^3 y^3+b^3 c^3 y^3+a^2 b^3 c x^2 z-a^3 b c^2 x^2 z-a^3 b^2 c y^2 z+a b^3 c^2 y^2 z-a b^3 c^2 x z^2+a^2 b c^3 x z^2+a^3 b c^2 y z^2-a b^2 c^3 y z^2+a^3 c^3 z^3-b^3 c^3 z^3 = 0.
This cubic passes through X(i) for i = 2,75,350,6376,19567,20935,24731,24732. (Peter Moses, October 8, 2018)
X(24731) lies on these lines: {10,75}, {141,17762}, {350,4645}, {940,2162}, {1909,17302}, {1966,17738}, {2533,17159}, {3416,17144}, {3502,3905}, {3836,20947}, {4785,4823}, {18046,24502}, {18140,24342}
See X(24731).
X(24732) lies on the cubic introduced at X(24731) and on these lines: {6,19579}, {37,2998}, {69,350}, {75,700}, {141,3863}, {256,9230}, {789,1582}, {1921,21257}, {3978,24575}, {4485,5224}, {6374,18277}, {6384,21264}, {6385,24688}, {9229,17788}, {17234,18149}
Let U = obverse triangle of X(76), and let V(X) = N-obverse triangle of X. The locus of X such that V(X) is perspective to U is the cubic given by
a^6 b^6 x^3-a^6 c^6 x^3+a^6 b^4 c^2 x^2 y-a^4 b^2 c^6 x^2 y-a^4 b^6 c^2 x y^2+a^2 b^4 c^6 x y^2-a^6 b^6 y^3+b^6 c^6 y^3+a^4 b^6 c^2 x^2 z-a^6 b^2 c^4 x^2 z-a^6 b^4 c^2 y^2 z+a^2 b^6 c^4 y^2 z-a^2 b^6 c^4 x z^2+a^4 b^2 c^6 x z^2+a^6 b^2 c^4 y z^2-a^2 b^4 c^6 y z^2+a^6 c^6 z^3-b^6 c^6 z^3 = 0.
This cubic passes through X(i) for i = 2,76,3978,6374,19573,24733,24734. (Peter Moses, October 8, 2018)
X(24733) lies on these lines: {76,141}, {3978,5207}, {7797,9230}, {8920,14880}
See X(24733).
X(24734) lies on the cubic introduced at X(24733) and on these lines: {32,8790}, {39,6374}, {76,14820}, {315,1899}
Collineation mappings involving Gemini triangle 5: X(24735)-X(24758)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 5, as in centers X(24735)-X(24758). Then
m(X) = a(a b + a c - b c) x + (a^2 c - a^2 b + a b^2 + b^2 c - 2 a b c) y + (a^2 b - a^2 c + a c^2 + b c^2 - 2 a b c) z : :
(Clark Kimberling, October 9, 2018)
X(24735) lies on these lines: {2, 17448}, {8, 1279}, {44, 21281}, {85, 1418}, {1655, 17301}, {1909, 17278}, {2082, 24358}, {3208, 4422}, {3691, 4643}, {3765, 24789}, {3772, 3975}, {3780, 4851}, {3959, 17755}, {4713, 20257}, {4754, 24796}, {17026, 21025}, {17277, 17743}, {17352, 17752}, {18194, 24746}, {20255, 21384}, {24655, 25106}, {24736, 24738}
X(24736) lies on these lines: {2, 3728}, {1125, 3242}, {3840, 4438}, {18194, 24757}, {24735, 24738}, {24741, 24743}
X(24737) lies on these lines: {2, 17448}, {348, 2275}, {1475, 4644}, {3242, 3589}, {5222, 24581}, {6714, 16020}, {16713, 16744}, {18194, 24744}, {24575, 24669}, {24753, 24758}
X(24738) lies on these lines: {2, 3056}, {6714, 17337}, {24735, 24736}
X(24739) lies on these lines: {2, 17448}, {274, 4602}, {551, 6683}, {1575, 17144}, {2275, 18135}, {4465, 23649}, {4657, 17095}, {4698, 5550}, {7834, 10199}, {17235, 24798}, {17792, 24660}, {18194, 24745}
X(24740) lies on these lines: {2, 24525}
X(24741) lies on these lines: {2, 24526}, {24736, 24743}
X(24742) lies on these lines: {2, 18194}, {10, 1279}, {3008, 21257}, {3759, 4871}, {3840, 17348}, {17351, 24182}, {17352, 20340}, {24735, 24736}, {24743, 24751}, {24746, 24757}, {25117, 25279}
X(24743) lies on these lines: {2, 24527}, {24736, 24741}, {24742, 24751}
X(24744) lies on these lines: {2, 3728}, {583, 24695}, {18194, 24737}
X(24745) lies on these lines: {2, 24527}, {43, 16969}, {1575, 14823}, {3216, 3795}, {18194, 24739}
X(24746) lies on these lines: {2, 3728}, {9, 46}, {16569, 17279}, {18194, 24735}, {21020, 24958}, {24742, 24757}, {24750, 24751}
X(24747) lies on these lines: {2, 669}, {659, 4885}, {24666, 25142}, {24755, 24756}
X(24748) lies on these lines: {2, 24665}
X(24749) lies on these lines: {2, 23655}, {43, 3835}, {44, 513}, {171, 23568}, {4449, 4885}, {8640, 14426}, {21051, 22090}, {21197, 24349}, {24674, 25126}
X(24750) lies on these lines: {2, 24525}, {24735, 24736}, {24746, 24751}
X(24751) lies on these lines: {2, 17448}, {17385, 19877}, {24742, 24743}, {24746, 24750}
X(24752) lies on these lines: {2, 3056}, {4, 9}, {8, 1284}, {69, 1447}, {329, 20487}, {345, 8844}, {497, 22370}, {978, 1818}, {1376, 4203}, {1788, 4645}, {2886, 5224}, {3035, 17352}, {4000, 24478}, {4073, 16609}, {4648, 24239}, {10446, 20544}, {18194, 24737}
X(24753) lies on these lines: {2, 18194}, {8, 1279}, {24737, 24758}
X(24754) lies on these lines: {2, 24673}, {14408, 16816}, {16815, 23650}
X(24755) lies on these lines: {2, 20983}, {1150, 1980}, {1491, 4369}, {3741, 8640}, {4147, 4367}, {4397, 9508}, {4874, 17420}, {10453, 23506}, {24533, 25128}, {24747, 24756}
X(24756) lies on these lines: {2, 23655}, {513, 3716}, {17063, 21197}, {24747, 24755}
X(24757) lies on these lines: {2, 3056}, {8, 1279}, {43, 17352}, {256, 16706}, {3741, 17123}, {18194, 24736}, {24742, 24746}
X(24758) lies on these lines: {2, 24670}, {3242, 16823}, {3739, 24343}, {17348, 21214}, {18194, 24739}, {24735, 24736}, {24737, 24753}
Collineation mappings involving Gemini triangle 6: X(24759)-X(24770)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 6, as in centers X(24759)-X(24770). Then
m(X) = a(a b + a c - 3 b c) x + (a^2 b - 3 a^2 c - a b^2 - b^2 c + 4 a b c) y + (a^2 c - 3 a^2 b - a c^2 -b c^2 + 4 a b c) z : :
(Clark Kimberling, October 9, 2018)
X(24759) lies on these lines: {2, 24761}, {8, 20530}, {145, 3834}, {536, 9311}, {24760, 24765}
X(24760) lies on these lines: {4851, 4865}, {24759, 24765}
X(24761) lies on these lines: {1, 142}, {2, 24759}, {145, 20530}, {1997, 4393}
X(24762) lies on these lines: {1, 20530}, {2, 24759}, {1201, 4852}, {3622, 4000}, {24766, 24767}
X(24763) lies on these lines: {4865, 4950}
X(24764) lies on these lines: (none)
X(24765) lies on these lines: {2, 24766}, {519, 17232}, {4852, 4906}, {24759, 24760}
X(24766) lies on these lines: {1, 87}, {2, 24765}, {43, 4852}, {3723, 4906}, {17336, 18194}, {24762, 24767}
X(24767) lies on these lines: {24762, 24766}
X(24768) lies on these lines: {659, 24666}, {4369, 4879}
X(24769) lies on these lines: {42, 4382}, {4369, 4449}, {4380, 23655}
X(24770) lies on these lines: {2, 24759}, {8, 24652}, {10, 20530}, {3617, 4000}, {17050, 25102}, {17235, 25280}
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28432.
X(24771) lies on these lines: {2, 24181}, {9, 497}, {37, 3677}, {40, 15487}, {57, 16593}, {190, 1088}, {200, 220}, {440, 17742}, {518, 15490}, {644, 3870}, {1040, 2324}, {2325, 15286}, {3161, 10580}, {4370, 8557}, {4853, 15853}, {16572, 21096}, {17755, 20173}
X(24771) = complement of the isogonal conjugate of X(21002)
X(24771) = barycentric product X(i)*X(j) for these {i,j}: {8, 3174}, {220, 20946}, {341, 21002}, {346, 16572} , {728, 8732}
X(24771) = trilinear product X(i)*X(j) for these {i,j}: {9, 3174}, {200, 16572}, {346, 21002}, {480, 8732}
X(24771) = medial-isotomic conjugate of X(200)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28437.
X(24772) lies on these lines: {4,14980}, {1154,14072}, {5663,15907}
Collineation mappings involving Gemini triangle 7: X(24773)-X(24795)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 7, as in centers X(24773)-X(24795). Then
m(X) = a^2 (a - b - c) x + b ( -a b + b^2 + a c - 2 b c + c^2) y + c (-a c + c^2 + a b - 2 b c + b^2) z : :
(Clark Kimberling, October 9, 2018)
X(24773) lies on these lines: {2, 17861}, {307, 3008}, {326, 17282}, {3772, 20268}, {4056, 16580}, {4859, 10436}, {4904, 17366}, {16551, 16888}, {17278, 20269}, {17384, 24774}, {24777, 24788}
X(24774) lies on these lines: {1, 6706}, {2, 277}, {10, 4904}, {37, 7264}, {65, 2140}, {72, 20335}, {116, 17606}, {142, 442}, {354, 17758}, {496, 20328}, {650, 23100}, {905, 16604}, {984, 1698}, {1111, 1212}, {1155, 14377}, {1385, 9317}, {1447, 17682}, {1737, 21258}, {2348, 24805}, {3057, 17761}, {3085, 4000}, {3663, 25073}, {3752, 24786}, {3753, 17050}, {4059, 4253}, {4657, 24779}, {10914, 20257}, {13728, 24178}, {16593, 21073}, {17384, 24773}, {21264, 24161}, {22011, 24787}
X(24775) lies on these lines: {2, 21436}, {244, 7658}, {905, 7208}, {3756, 4904}, {4885, 4939}, {20269, 24786}
X(24776) lies on these lines: {2, 20890}, {20269, 24787}
X(24777) lies on these lines: {2, 21414}, {24174, 24785}, {24773, 24788}
X(24778) lies on these lines: {2, 2415}, {2347, 24237}, {3008, 17077}, {4904, 17245}, {11512, 19863}, {16706, 24780}, {16828, 24178}, {17278, 20269}, {17384, 24784}, {20905, 25065}, {21173, 22343}, {25076, 25243}
X(24779) lies on these lines: {1, 142}, {2, 17861}, {63, 15474}, {75, 24781}, {219, 1086}, {278, 3911}, {281, 17885}, {387, 5883}, {499, 17073}, {885, 21189}, {948, 1743}, {1214, 24789}, {1445, 1723}, {1479, 18589}, {1714, 24174}, {1737, 18634}, {3306, 24175}, {3332, 7613}, {3682, 24159}, {3739, 19854}, {4021, 19785}, {4361, 4904}, {4657, 24774}, {6737, 21255}, {10573, 16608}, {16752, 17189}
X(24780) lies on these lines: {1, 16608}, {2, 17861}, {35, 18589}, {57, 1744}, {80, 20305}, {86, 4612}, {142, 10090}, {256, 21189}, {284, 4466}, {499, 1733}, {905, 16696}, {1030, 16581}, {1375, 1781}, {2294, 24884}, {3624, 4657}, {3739, 24784}, {4904, 17045}, {7110, 16732}, {16706, 24778}, {17052, 24435}, {17322, 24781}
X(24781) lies on these lines: {2, 277}, {75, 24779}, {92, 15474}, {226, 17682}, {239, 4904}, {379, 4911}, {443, 5440}, {448, 24617}, {475, 1895}, {666, 3008}, {693, 905}, {1010, 1125}, {1086, 1944}, {1375, 1447}, {3811, 18134}, {4000, 20227}, {4859, 10436}, {5249, 16054}, {17023, 24181}, {17322, 24780}
X(24782) lies on these lines: {1, 4477}, {2, 647}, {43, 4524}, {241, 514}, {512, 24533}, {659, 6363}, {693, 3310}, {1021, 2999}, {1491, 21189}, {3239, 25098}, {3666, 3700}, {3752, 17069}, {4383, 9404}, {4467, 4850}, {4885, 6589}, {4897, 21894}, {21828, 24924}, {24793, 24794}
X(24782) = complement of isotomic conjugate of X(37137)
X(24783) lies on these lines: {905, 24794}
X(24784) lies on these lines: {2, 277}, {116, 2646}, {142, 13747}, {905, 1107}, {1125, 4904}, {3739, 24780}, {3752, 20267}, {5440, 17046}, {9317, 9956}, {10039, 17044}, {14377, 17605}, {17062, 17614}, {17384, 24778}, {17619, 24249}, {19862, 24181}
X(24785) lies on these lines: {2, 23581}, {24174, 24777}
X(24786) lies on these lines: {1, 17048}, {2, 1930}, {39, 1111}, {41, 2224}, {244, 17758}, {325, 17192}, {350, 4099}, {498, 4000}, {1015, 7278}, {1089, 3934}, {1447, 5280}, {1739, 17050}, {2140, 24443}, {2275, 7208}, {2276, 7264}, {2548, 4056}, {3294, 25073}, {3670, 20335}, {3752, 24774}, {3970, 24172}, {3987, 20257}, {4372, 7815}, {4376, 7808}, {4642, 17761}, {4647, 21264}, {4680, 7800}, {6292, 16886}, {6683, 16720}, {7272, 9596}, {20269, 24775}, {21208, 21808}
X(24787) lies on these lines: {2, 21404}, {3925, 4904}, {20269, 24776}, {22011, 24774}
X(24788) lies on these lines: {2, 25090}, {905, 16716}, {3739, 24780}, {3772, 17068}, {19269, 24161}, {20267, 20268}, {20269, 24776}, {24773, 24777}
X(24789) lies on these lines: {1, 3925}, {2, 37}, {3, 2218}, {6, 5249}, {7, 4641}, {9, 3782}, {11, 1040}, {12, 1722}, {31, 5880}, {32, 24588}, {43, 17718}, {44, 5905}, {55, 1738}, {56, 24178}, {57, 1723}, {63, 1086}, {72, 24159}, {81, 4675}, {141, 5271}, {142, 940}, {200, 17724}, {218, 226}, {223, 5723}, {238, 1836}, {239, 18134}, {241, 277}, {244, 24892}, {273, 18679}, {306, 4361}, {333, 3662}, {377, 1104}, {405, 23537}, {497, 16020}, {499, 17102}, {612, 3826}, {614, 2886}, {650, 24793}, {748, 3120}, {942, 1714}, {958, 23536}, {975, 17529}, {978, 11375}, {980, 24790}, {988, 24953}, {990, 8226}, {1001, 3914}, {1125, 16458}, {1211, 4384}, {1214, 24779}, {1279, 3434}, {1376, 3011}, {1445, 6354}, {1698, 7322}, {1707, 11246}, {1717, 7741}, {1721, 7965}, {1743, 4654}, {1754, 5805}, {1999, 17234}, {2006, 8056}, {2178, 16438}, {2275, 16699}, {2352, 16056}, {2550, 3744}, {2887, 3966}, {2999, 5718}, {3187, 4851}, {3216, 11374}, {3219, 17276}, {3242, 25006}, {3305, 4415}, {3474, 7613}, {3654, 4674}, {3670, 5791}, {3683, 24248}, {3742, 11269}, {3757, 4429}, {3759, 17778}, {3765, 24735}, {3784, 18191}, {3812, 5230}, {3816, 17070}, {3823, 10327}, {3836, 4362}, {3911, 24175}, {3924, 5794}, {3929, 4862}, {3944, 4679}, {3946, 20182}, {3980, 6679}, {3999, 24477}, {4001, 7232}, {4197, 5262}, {4208, 5716}, {4357, 19732}, {4363, 5294}, {4416, 19723}, {4423, 16849}, {4438, 24165}, {4643, 5278}, {4656, 6666}, {4847, 17597}, {4886, 16816}, {4966, 17156}, {5137, 9306}, {5219, 23511}, {5222, 5712}, {5231, 5573}, {5247, 10404}, {5256, 17056}, {5268, 17602}, {5287, 17245}, {5292, 5439}, {5433, 11512}, {5721, 18443}, {5737, 17290}, {5739, 17348}, {5745, 17067}, {6734, 17054}, {6745, 17783}, {7270, 19851}, {7292, 11680}, {7299, 7702}, {7365, 8732}, {7522, 19728}, {8270, 15253}, {8616, 24715}, {10528, 21896}, {11376, 21214}, {11679, 17282}, {12513, 23675}, {12609, 16466}, {15474, 18607}, {16062, 16817}, {16569, 17719}, {16580, 19724}, {16736, 18603}, {16885, 17781}, {17022, 20195}, {17023, 19701}, {17063, 17728}, {17073, 20268}, {17092, 18625}, {17127, 20292}, {17299, 20483}, {17325, 19744}, {17749, 24160}, {20269, 24776}, {20470, 23339}, {21020, 24943}, {24046, 24880}, {24174, 24914}, {25048, 25308}
X(24790) lies on these lines: {1, 142}, {2, 3760}, {31, 14377}, {42, 17758}, {81, 24588}, {116, 21935}, {213, 1086}, {274, 16706}, {609, 4209}, {673, 5299}, {980, 24789}, {1111, 16583}, {1193, 2140}, {1201, 17761}, {1475, 17205}, {3008, 16552}, {3216, 20335}, {3290, 7264}, {3294, 3663}, {3739, 16828}, {3752, 24774}, {3772, 20267}, {3987, 21232}, {5280, 17682}, {5283, 17278}, {7795, 19846}, {16600, 20880}, {16752, 17023}, {16831, 19785}, {16887, 24592}, {17033, 24190}, {17326, 24919}, {17366, 20963}, {17489, 20893}, {19864, 21264}, {20367, 24177}
X(24791) lies on these lines: {2, 25095}
X(24792) lies on these lines: {905, 4762}, {1769, 17067}
X(24793) lies on these lines: {2, 21438}, {650, 24789}, {693, 905}, {3772, 4885}, {5249, 22383}, {5272, 15283}, {15280, 17064}, {23733, 23806}, {24782, 24794}
X(24794) lies on these lines: {905, 24783}, {24782, 24793}
X(24795) lies on these lines: {2, 25272}, {141, 3679}, {277, 291}, {333, 17205}, {3008, 3732}, {4000, 21232}, {17278, 21138}, {21264, 24161}
Collineation mappings involving the Garcia reflection triangle (Gemini triangle 8): X(24796)-X(24805)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 8, as in centers X(24796)-X(24805). Then
m(X) = a (a^2 - a b + 2 b^2 - a c - 4 c b + 2c^2) x - (2a^b + a b^2 - b^3 + 2 a^2 c + 3 a b c + 2 b^2 c - 4 a c^2 - 3 b c^2 + 2 c^3) y - (2a^c + a c^2 - c^3 + 2 a^2 b + 3 a b c + 2 b c^2 - 4 a b^2 - 3 b^2 c + 2 b^3) z : :
(Clark Kimberling, October 9, 2018)
X(24796) lies on these lines: {1, 1358}, {2, 24797}, {7, 8}, {56, 10481}, {277, 2348}, {279, 1319}, {354, 14268}, {553, 19723}, {1086, 2082}, {1111, 1837}, {1323, 1388}, {1334, 17276}, {1420, 21314}, {1467, 7271}, {1565, 11376}, {1697, 4862}, {3303, 3663}, {3361, 20121}, {3665, 11375}, {4373, 12632}, {4675, 21808}, {4754, 24735}, {4888, 11518}, {4902, 7991}, {5221, 10521}, {9436, 24914}, {12701, 17170}
X(24797) lies on these lines: {2, 24796}, {7, 21}, {8, 1358}, {9, 17107}, {85, 25280}, {279, 3476}, {388, 10481}, {1788, 7195}, {4862, 12053}, {4902, 11522}, {5290, 20121}, {6604, 7185}, {10106, 21314}
X(24798) lies on these lines: {2, 24796}, {7, 5550}, {10, 1358}, {12, 10481}, {65, 760}, {279, 5252}, {307, 1122}, {348, 1319}, {1071, 1367}, {1111, 17606}, {1323, 10944}, {1565, 3057}, {2348, 20269}, {3669, 17448}, {4059, 7179}, {6604, 11011}, {7195, 24914}, {9578, 21314}, {17235, 24739}
X(24799) lies on these lines: {1, 87}, {2, 24802}, {1111, 14100}, {2293, 24225}, {7671, 17861}
X(24800) lies on these lines: {2, 24796}, {7, 350}, {42, 1358}, {85, 25287}
X(24801) lies on these lines: {2, 24796}, {7, 24495}, {43, 1358}, {65, 1088}, {1836, 4106}
X(24802) lies on these lines: {1, 6}, {2, 24799}, {982, 21267}, {2310, 4862}, {4888, 21346}, {7671, 25065}, {17885, 24225}, {21185, 24207}
X(24803) lies on these lines: {2, 24796}, {7, 37}, {57, 20602}, {63, 17107}, {239, 1358}, {552, 553}, {3669, 4380}, {5228, 7185}
X(24804) lies on these lines: (none)
X(24805) lies on these lines: {2, 24796}, {7, 10588}, {85, 1319}, {1111, 2646}, {1125, 1358}, {1434, 1447}, {2348, 24774}, {3598, 10404}, {3649, 10521}, {3674, 4870}, {3748, 7264}, {5183, 17753}, {5433, 10481}, {6706, 9318}, {7195, 11375}
Centers related to obverse triangles: X(24806)-X(24854)
This preamble and centers X(24806)-X(24854) were contributed by César Eliud Lozada, October 10, 2018.
(ABC, 75), (anti-tangential-midarc, 24806), (anticomplementary, 1654), (Aquila, 3679), (1st circumperp, 24309), (2nd circumperp, 993), (excentral, 9), (extangents, 24308), (2nd extouch, 9), (Fuhrmann, 10), (inner-Garcia, 1), (outer-Garcia, 3679), (Garcia-reflection, 9), (intangents, 24307), (medial, 9), (1st Sharygin, 8424), (2nd Sharygin, 4363), (tangential, 8424), (2nd Zaniah, 9), (N-obverse of X(1), 75), (trilinear obverse of X(2), 894),
(anti-Artzt, 3679, 24807), (Artzt, 3679, 24808), (1st Parry, 1, 24809), (2nd Parry, 1, 24810), (1st tri-squares, 3679, 24811), (2nd tri-squares, 3679, 24812)
(ABC, 1, 190), (ABC-X3 reflections, 1, 24813), (anti-Aquila, 1, 4432), (anti-Ara, 1, 24814), (5th anti-Brocard, 1, 24815), (2nd anti-circumperp-tangential, 1, 24816), (anti-Euler, 1, 24817), (anti-inner-Grebe, 1, 24818), (anti-outer-Grebe, 1, 24819), (anti-Mandart-incircle, 1, 24820), (anticomplementary, 1, 4440), (Aquila, 1, 24821), (Ara, 1, 24822), (1st Auriga, 1, 24823), (2nd Auriga, 1, 24824), (5th Brocard, 1, 24825), (2nd circumperp tangential, 1, 24826), (Ehrmann-mid, 1, 24827), (Euler, 1, 24828), (outer-Garcia, 1, 24715), (Gossard, 1, 24830), (inner-Grebe, 1, 24831), (outer-Grebe, 1, 24832), (Johnson, 1, 24833), (inner-Johnson, 1, 24834), (outer-Johnson, 1, 24835), (1st Johnson-Yff, 1, 24836), (2nd Johnson-Yff, 1, 24837), (Lucas homothetic, 1, 24838), (Lucas(-1) homothetic, 1, 24839), (Mandart-incircle, 1, 24840), (medial, 1, 1086), (5th mixtilinear, 1, 24841), (3rd tri-squares-central, 1, 24842), (4th tri-squares-central, 1, 24843), (X3-ABC reflections, 1, 24844), (Yff contact, 10, 190), (inner-Yff, 1, 24845), (outer-Yff, 1, 24846), (inner-Yff tangents, 1, 24847), (outer-Yff tangents, 1, 24848)
(ABC, 75), (anti-tangential-midarc, 24849), (extangents, 24310), (5th extouch, 24312), (outer-Garcia, 1), (medial, 1), (1st Sharygin, 24311), (2nd Sharygin, 8301), (tangential, 8301), (1st Zaniah, 1), (obverse of X(1), 75), (trilinear N-obverse of X(2), 239)
(excenters-midpoints, 1, 24850), (Garcia-reflection, 1, 24851), (2nd Schiffler, 1, 24852)
(ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (anticomplementary, 192), (2nd Brocard, 6), (circumsymmedial, 6), (2nd Ehrmann, 6), (excentral, 1045), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (incentral, 192), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd mixtilinear, 24853), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6), (obverse of X(1), 894), (trilinear N-obverse of X(2), 6)
(1st Ehrmann,6,24854)
(ABC, 6), (anti-Conway, 6), (2nd anti-Conway, 6), (anti-inner-Grebe, 6), (anti-outer-Grebe, 6), (anti-Honsberger, 6), (2nd Brocard, 6), (circumsymmedial, 6), (2nd Ehrmann, 6), (9th Fermat-Dao, 6), (10th Fermat-Dao, 6), (13th Fermat-Dao, 6), (14th Fermat-Dao, 6), (inner-Grebe, 6), (outer-Grebe, 6), (incentral, 2), (1st Kenmotu diagonals, 6), (2nd Kenmotu diagonals, 6), (2nd orthosymmedial, 6), (symmedial, 6), (tangential, 6), (inner tri-equilateral, 6), (outer tri-equilateral, 6), (N-obverse of X(1), 239), trilinear obverse of X(2), 6)
(1st Ehrmann,6,24829)
X(24806) lies on these lines: {1,3}, {2,1457}, {8,73}, {9,4559}, {10,10571}, {34,92}, {75,77}, {87,1411}, {109,993}, {201,3869}, {221,958}, {222,956}, {223,9623}, {227,5836}, {269,7223}, {348,21281}, {355,1745}, {386,4848}, {388,1042}, {603,2975}, {944,4303}, {962,2654}, {995,3911}, {1044,7354}, {1064,18391}, {1193,1788}, {1201,7288}, {1406,22759}, {1442,17015}, {1450,5435}, {1458,3476}, {1464,5252}, {1465,3753}, {1943,16821}, {2263,8680}, {3486,4300}, {3679,4551}, {4296,6360}, {4306,10106}, {4334,5434}, {4386,17966}, {4392,18419}, {5433,21214}, {5657,22350}, {5731,22053}, {6001,24430}, {6708,19372}, {15656,21767}, {21147,23555}
X(24806) = X(23853)-of-2nd anti-circumperp-tangential triangle
X(24806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65, 1319, 1403), (221, 958, 1935), (1042, 10459, 388)
The reciprocal orthologic center of these triangles is X(3679)
X(24807) lies on these lines: {2,952}, {519,4482}, {903,24281}, {2784,11161}, {2789,8593}, {13637,24811}, {13757,24812}
X(24807) = reflection of X(903) in X(24281)
The reciprocal orthologic center of these triangles is X(3679).
Let P be a point on the Steiner circumellipse. Let A' be the orthocenter of BCP, and define B' and C' cyclically. Let Q be the centroid of A'B'C'. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(190), Q = X(24808). See also X(98) and X(6054). (Randy Hutson, October 15, 2018)
X(24808) lies on these lines: {2,952}, {4,10743}, {10,98}, {100,7427}, {355,21554}, {515,13635}, {517,3799}, {953,3006}, {2789,6054}, {2826,10711}, {3090,15251}, {5690,7385}, {5818,7380}, {9956,16830}, {13638,24811}, {13758,24812}
The reciprocal orthologic center of these triangles is X(1)
X(24809) lies on these lines: {2,2784}, {10,5168}, {110,1010}, {351,24810}, {952,9978}, {2789,9485}, {2826,13250}
X(24809) = reflection of X(24810) in X(351)
X(24809) = X(190)-of-1st Parry triangle
X(24809) = X(24813)-of-2nd Parry triangle
The reciprocal orthologic center of these triangles is X(1)
X(24810) lies on these lines: {10,5029}, {351,24809}, {952,9980}, {2784,9147}, {2789,5466}, {2826,13251}
X(24810) = reflection of X(24809) in X(351)
X(24810) = X(190)-of-2nd Parry triangle
X(24810) = X(24813)-of-1st Parry triangle
The reciprocal orthologic center of these triangles is X(3679)
X(24811) lies on these lines: {2,24812}, {952,3068}, {2784,13653}, {2789,13640}, {13637,24807}, {13638,24808}
The reciprocal orthologic center of these triangles is X(3679)
X(24812) lies on these lines: {2,24811}, {952,3069}, {2784,13773}, {2789,13760}, {13757,24807}, {13758,24808}
The reciprocal parallelogic center of these triangles is X(1)
X(24813) lies on these lines: {2,24828}, {3,190}, {4,1086}, {20,4440}, {30,903}, {35,24845}, {36,24846}, {40,537}, {55,24816}, {56,24840}, {98,103}, {104,900}, {165,24821}, {182,24815}, {371,24819}, {372,24818}, {376,545}, {382,24827}, {515,24715}, {517,24841}, {528,944}, {631,4422}, {673,971}, {726,19589}, {1350,9055}, {1593,24814}, {1897,3937}, {2796,4297}, {3098,24825}, {3428,24826}, {3523,4473}, {3524,4370}, {3576,4432}, {4409,17538}, {4437,10519}, {4997,19515}, {5845,6776}, {6284,24837}, {7354,24836}, {9041,12245}, {9540,24842}, {10304,17487}, {10305,18283}, {10310,24820}, {11248,24847}, {11249,24848}, {11414,24822}, {11822,24823}, {11823,24824}, {11824,24831}, {11825,24832}, {11827,24835}, {11828,24838}, {11829,24839}, {13935,24843}, {16593,21151}
X(24813) = midpoint of X(20) and X(4440)
X(24813) = reflection of X(i) in X(j) for these (i,j): (4, 1086), (190, 3), (382, 24827)
X(24813) = anticomplement of X(24828)
X(24813) = X(190)-of-ABC-X3 reflections triangle
X(24813) = X(1086)-of-anti-Euler triangle
X(24813) = X(24816)-of-anti-Mandart-incircle triangle
X(24813) = X(24840)-of-2nd circumperp tangential triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24814) lies on these lines: {4,4440}, {25,190}, {33,24840}, {34,24816}, {235,24828}, {427,1086}, {428,545}, {468,4422}, {528,12135}, {537,1829}, {900,1830}, {903,5064}, {1593,24813}, {1598,24844}, {1843,9055}, {2786,5185}, {2796,12132}, {4432,11363}, {4473,6353}, {5090,24715}, {5411,24818}, {7487,24817}, {7713,24821}, {7714,17487}, {11380,24815}, {11383,24820}, {11384,24823}, {11385,24824}, {11386,24825}, {11388,24831}, {11389,24832}, {11390,24834}, {11391,24835}, {11392,24836}, {11393,24837}, {11394,24838}, {11395,24839}, {11396,24841}, {11398,24845}, {11399,24846}, {11400,24847}, {11401,24848}, {11832,24830}, {13884,24842}, {13937,24843}, {22479,24826}
X(24814) = X(190)-of-anti-Ara triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24815) lies on these lines: {32,190}, {83,1086}, {98,24828}, {101,24502}, {182,24813}, {528,12195}, {537,12194}, {545,12150}, {900,13194}, {1078,4422}, {2786,4027}, {2796,12191}, {4432,11364}, {4440,7787}, {4473,7793}, {9055,12212}, {10788,24817}, {10789,24821}, {10790,24822}, {10791,24715}, {10792,24831}, {10793,24832}, {10794,24834}, {10795,24835}, {10796,24833}, {10797,24836}, {10798,24837}, {10799,24840}, {10800,24841}, {10801,24845}, {10802,24846}, {10803,24847}, {10804,24848}, {11380,24814}, {11490,24820}, {11837,24823}, {11838,24824}, {11839,24830}, {11840,24838}, {11841,24839}, {11842,24844}, {12835,24816}, {13885,24842}, {13938,24843}, {18502,24827}, {18993,24818}, {18994,24819}, {22520,24826}
X(24815) = X(190)-of-5th anti-Brocard triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24816) lies on these lines: {1,24840}, {3,24845}, {4,24837}, {11,24828}, {12,1086}, {34,24814}, {56,190}, {57,24821}, {65,537}, {388,4440}, {528,3885}, {545,5434}, {726,1463}, {900,1317}, {903,11237}, {999,24844}, {1227,3264}, {1319,4432}, {1362,2786}, {1411,24416}, {1469,9055}, {1478,24833}, {2099,24841}, {2796,10106}, {3585,24827}, {3837,20366}, {4293,24817}, {4370,5298}, {4422,5433}, {4473,7288}, {5252,24715}, {11509,24820}, {12835,24815}, {17114,24068}, {18954,24822}, {18955,24823}, {18956,24824}, {18957,24825}, {18958,24830}, {18959,24831}, {18960,24832}, {18961,24834}, {18962,24835}, {18963,24838}, {18964,24839}, {18965,24842}, {18966,24843}, {18967,24848}, {18995,24818}, {18996,24819}
X(24816) = reflection of X(24840) in X(1)
X(24816) = X(190)-of-2nd anti-circumperp-tangential triangle
X(24816) = X(24813)-of-Mandart-incircle triangle
X(24816) = X(24840)-of-5th mixtilinear triangle
X(24816) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 4440, 24836), (999, 24844, 24846)
The reciprocal parallelogic center of these triangles is X(1)
X(24817) lies on these lines: {2,24833}, {3,4440}, {4,190}, {5,4473}, {24,24822}, {30,17487}, {40,2796}, {104,24826}, {376,545}, {388,24845}, {497,24846}, {515,24821}, {528,11827}, {537,944}, {631,1086}, {673,21168}, {900,13199}, {903,3524}, {2786,13172}, {3085,24836}, {3086,24837}, {3090,4422}, {3091,24827}, {3545,4370}, {4293,24816}, {4294,24840}, {4409,21735}, {4432,5603}, {5657,24715}, {5762,20533}, {6776,9055}, {7487,24814}, {7581,24819}, {7582,24818}, {7967,24841}, {9862,24825}, {10783,24831}, {10784,24832}, {10785,24834}, {10786,24835}, {10788,24815}, {10805,24847}, {10806,24848}, {11491,24820}, {11843,24823}, {11844,24824}, {11845,24830}, {11846,24838}, {11847,24839}, {13886,24842}, {13939,24843}
X(24817) = reflection of X(i) in X(j) for these (i,j): (4, 190), (4440, 3)
X(24817) = anticomplement of X(24833)
X(24817) = X(190)-of-anti-Euler triangle
X(24817) = X(4440)-of-ABC-X3 reflections triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24818) lies on these lines: {2,24843}, {6,190}, {372,24813}, {528,19065}, {537,18992}, {545,19053}, {900,19112}, {1086,3069}, {1587,24828}, {2786,19108}, {2796,19057}, {3068,4422}, {3299,24845}, {3301,24846}, {4370,19054}, {4432,18991}, {4440,7586}, {4473,7585}, {5411,24814}, {6418,24844}, {7582,24817}, {7584,24833}, {7968,24841}, {13785,24827}, {13936,24715}, {18993,24815}, {18995,24816}, {18999,24820}, {19003,24821}, {19005,24822}, {19007,24823}, {19009,24824}, {19011,24825}, {19013,24826}, {19017,24830}, {19023,24834}, {19025,24835}, {19027,24836}, {19029,24837}, {19031,24838}, {19033,24839}, {19037,24840}, {19047,24847}, {19049,24848}
X(24818) = X(190)-of-anti-inner-Grebe triangle
X(24818) = {X(4473), X(7585)}-harmonic conjugate of X(24842)
The reciprocal parallelogic center of these triangles is X(1)
X(24819) lies on these lines: {2,24842}, {6,190}, {371,24813}, {528,19066}, {537,18991}, {545,19054}, {900,19113}, {1086,3068}, {1588,24828}, {2786,19109}, {2796,19058}, {3069,4422}, {3299,24846}, {3301,24845}, {4370,19053}, {4432,18992}, {4440,7585}, {4473,7586}, {6417,24844}, {7581,24817}, {7583,24833}, {7969,24841}, {13665,24827}, {13883,24715}, {18994,24815}, {18996,24816}, {19000,24820}, {19004,24821}, {19006,24822}, {19008,24823}, {19010,24824}, {19012,24825}, {19014,24826}, {19018,24830}, {19024,24834}, {19026,24835}, {19028,24836}, {19030,24837}, {19032,24838}, {19034,24839}, {19038,24840}, {19048,24847}, {19050,24848}
X(24819) = X(190)-of-anti-outer-Grebe triangle
X(24819) = {X(4473), X(7586)}-harmonic conjugate of X(24843)
The reciprocal parallelogic center of these triangles is X(1)
X(24820) lies on these lines: {3,537}, {4,528}, {6,983}, {35,24821}, {55,190}, {56,24841}, {100,4440}, {105,4578}, {197,24822}, {335,11329}, {480,673}, {518,6211}, {545,4421}, {900,13205}, {1001,4422}, {1083,4069}, {1086,1376}, {1621,4473}, {2786,13173}, {2796,8715}, {3295,4432}, {3303,17697}, {3939,14839}, {4370,4428}, {4436,24345}, {4557,8301}, {5687,24715}, {7202,16504}, {8177,9055}, {8865,18755}, {9025,19589}, {9041,12513}, {10310,24813}, {11383,24814}, {11490,24815}, {11491,24817}, {11492,24823}, {11493,24824}, {11494,24825}, {11496,24828}, {11497,24831}, {11498,24832}, {11499,24833}, {11500,24835}, {11501,24836}, {11502,24837}, {11503,24838}, {11504,24839}, {11507,24845}, {11508,24846}, {11509,24816}, {11510,24848}, {11848,24830}, {11849,24844}, {13887,24842}, {13940,24843}, {16686,23343}, {18491,24827}, {18999,24818}, {19000,24819}
X(24820) = reflection of X(i) in X(j) for these (i,j): (24826, 3), (24834, 1086)
X(24820) = X(190)-of-anti-Mandart-incircle triangle
X(24820) = X(24841)-of-2nd circumperp tangential triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24821) lies on these lines: {1,190}, {8,2796}, {10,4440}, {35,24820}, {36,24826}, {57,24816}, {165,24813}, {239,726}, {244,4756}, {291,4465}, {320,4439}, {335,16831}, {515,24817}, {517,24844}, {518,4693}, {519,4480}, {528,3632}, {545,3679}, {846,17165}, {900,5541}, {903,17250}, {984,4363}, {1046,24068}, {1054,3952}, {1086,1698}, {1125,4473}, {1282,2786}, {1697,24840}, {1699,24828}, {2108,17794}, {3099,24825}, {3624,4422}, {3633,9041}, {3751,9055}, {3932,7238}, {4009,18201}, {4370,16676}, {4753,17160}, {5220,17119}, {5587,24833}, {5588,24832}, {5589,24831}, {5850,20533}, {6763,16560}, {7713,24814}, {8185,24822}, {8186,24823}, {8187,24824}, {8188,24838}, {8189,24839}, {9324,17780}, {9457,13541}, {9578,24836}, {9581,24837}, {10789,24815}, {10826,24834}, {10827,24835}, {11852,24830}, {13888,24842}, {13942,24843}, {16832,17755}, {18492,24827}, {19003,24818}, {19004,24819}, {20375,21391}, {21087,24131}, {24331,24349}
X(24821) = reflection of X(i) in X(j) for these (i,j): (1, 190), (320, 4439), (4440, 10), (17160, 4753), (24841, 4432)
X(24821) = X(190)-of-Aquila triangle
X(24821) = X(4440)-of-outer-Garcia triangle
X(24821) = {X(4432), X(24841)}-harmonic conjugate of X(1)
The reciprocal parallelogic center of these triangles is X(1)
X(24822) lies on these lines: {3,1086}, {22,4440}, {24,24817}, {25,190}, {55,24445}, {159,9055}, {197,24820}, {528,12410}, {537,9798}, {545,9909}, {900,13222}, {1486,24358}, {1598,24828}, {1995,4473}, {2786,13175}, {2796,9876}, {4363,20834}, {4422,5020}, {4432,11365}, {4446,7083}, {5594,24832}, {5595,24831}, {7517,24844}, {8185,24821}, {8190,24823}, {8191,24824}, {8192,24841}, {8193,24715}, {8194,24838}, {8195,24839}, {9818,24827}, {10037,24845}, {10046,24846}, {10790,24815}, {10828,24825}, {10829,24834}, {10830,24835}, {10831,24836}, {10832,24837}, {10833,24840}, {10834,24847}, {10835,24848}, {11414,24813}, {11853,24830}, {13889,24842}, {13943,24843}, {18954,24816}, {19005,24818}, {19006,24819}, {22654,24826}
X(24822) = X(190)-of-Ara triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24823) lies on these lines: {55,537}, {190,5597}, {528,12454}, {545,11207}, {900,13228}, {1086,5599}, {2786,13176}, {2796,12345}, {4432,11366}, {4440,5601}, {5598,24841}, {8186,24821}, {8190,24822}, {8196,24828}, {8197,24715}, {8198,24831}, {8199,24832}, {8200,24833}, {8201,24838}, {8202,24839}, {9041,12455}, {9055,12452}, {11384,24814}, {11492,24820}, {11493,24826}, {11822,24813}, {11837,24815}, {11843,24817}, {11861,24825}, {11865,24834}, {11867,24835}, {11869,24836}, {11871,24837}, {11873,24840}, {11875,24844}, {11877,24845}, {11879,24846}, {11881,24847}, {11883,24848}, {13890,24842}, {13944,24843}, {18495,24827}, {18955,24816}, {19007,24818}, {19008,24819}
X(24823) = reflection of X(24824) in X(55)
X(24823) = X(190)-of-1st Auriga triangle
X(24823) = X(24841)-of-2nd Auriga triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24824) lies on these lines: {55,537}, {190,5598}, {528,12455}, {545,11208}, {900,13230}, {1086,5600}, {2786,13177}, {2796,12346}, {4432,11367}, {4440,5602}, {5597,24841}, {8187,24821}, {8191,24822}, {8203,24828}, {8204,24715}, {8205,24831}, {8206,24832}, {8207,24833}, {8208,24838}, {8209,24839}, {9041,12454}, {9055,12453}, {11385,24814}, {11492,24826}, {11493,24820}, {11823,24813}, {11838,24815}, {11844,24817}, {11862,24825}, {11866,24834}, {11868,24835}, {11870,24836}, {11872,24837}, {11874,24840}, {11876,24844}, {11878,24845}, {11880,24846}, {11882,24847}, {11884,24848}, {13891,24842}, {13945,24843}, {18497,24827}, {18956,24816}, {19009,24818}, {19010,24819}
X(24824) = reflection of X(24823) in X(55)
X(24824) = X(190)-of-2nd Auriga triangle
X(24824) = X(24841)-of-1st Auriga triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24825) lies on these lines: {32,190}, {528,12495}, {537,9941}, {545,7811}, {900,13235}, {903,7865}, {1086,3096}, {2786,8782}, {2796,9878}, {2896,4440}, {3094,9055}, {3098,24813}, {3099,24821}, {4422,7846}, {4432,11368}, {4473,10583}, {4568,24502}, {9301,24844}, {9857,24715}, {9862,24817}, {9993,24828}, {9994,24831}, {9995,24832}, {9996,24833}, {9997,24841}, {10038,24845}, {10047,24846}, {10828,24822}, {10871,24834}, {10872,24835}, {10873,24836}, {10874,24837}, {10875,24838}, {10876,24839}, {10877,24840}, {10878,24847}, {10879,24848}, {11386,24814}, {11494,24820}, {11861,24823}, {11862,24824}, {11885,24830}, {13892,24842}, {13946,24843}, {18500,24827}, {18957,24816}, {19011,24818}, {19012,24819}, {22744,24826}
X(24825) = X(190)-of-5th Brocard triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24826) lies on these lines: {3,537}, {20,528}, {36,24821}, {55,24841}, {56,190}, {104,24817}, {335,16367}, {518,9441}, {545,11194}, {659,918}, {900,22560}, {956,24715}, {958,1086}, {999,4432}, {1001,4310}, {2786,22514}, {2796,8666}, {2975,4440}, {3428,24813}, {3913,9041}, {4473,5253}, {9055,22769}, {10966,24840}, {11492,24824}, {11493,24823}, {12114,24834}, {16412,17755}, {18761,24827}, {19013,24818}, {19014,24819}, {22479,24814}, {22520,24815}, {22654,24822}, {22744,24825}, {22753,24828}, {22755,24830}, {22756,24831}, {22757,24832}, {22758,24833}, {22759,24836}, {22760,24837}, {22761,24838}, {22762,24839}, {22763,24842}, {22764,24843}, {22765,24844}, {22766,24845}, {22767,24846}, {22768,24847}
X(24826) = reflection of X(i) in X(j) for these (i,j): (24820, 3), (24835, 1086)
X(24826) = X(190)-of-2nd circumperp tangential triangle
X(24826) = X(24841)-of-anti-Mandart-incircle triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24827) lies on these lines: {4,4440}, {5,4422}, {30,1086}, {190,381}, {382,24813}, {528,22791}, {545,3845}, {546,24828}, {900,22938}, {903,3830}, {978,12699}, {1478,24837}, {1479,24836}, {2786,22515}, {2796,18483}, {3091,24817}, {3545,4473}, {3583,24840}, {3585,24816}, {3818,9055}, {3843,24844}, {4370,5066}, {4409,14893}, {4432,9955}, {4437,18358}, {5845,21850}, {9818,24822}, {10895,24845}, {10896,24846}, {13785,24818}, {18491,24820}, {18492,24821}, {18495,24823}, {18497,24824}, {18500,24825}, {18502,24815}, {18507,24830}, {18509,24831}, {18511,24832}, {18516,24834}, {18517,24835}, {18520,24838}, {18522,24839}, {18525,24841}, {18542,24847}, {18544,24848}, {18761,24826}, {18762,24843}
X(24827) = midpoint of X(i) and X(j) for these {i,j}: {4, 24833}, {382, 24813}, {903, 3830}, {12699, 24715}, {18507, 24830}, {18525, 24841}
X(24827) = reflection of X(i) in X(j) for these (i,j): (4370, 5066), (4432, 9955), (4437, 18358), (24828, 546)
X(24827) = X(190)-of-Ehrmann-mid triangle
X(24827) = X(24833)-of-Euler triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24828) lies on these lines: {2,24813}, {3,4422}, {4,190}, {5,1086}, {11,24816}, {12,24840}, {20,4473}, {30,4370}, {98,24815}, {114,118}, {119,900}, {235,24814}, {355,528}, {371,24842}, {372,24843}, {381,545}, {511,4437}, {515,4432}, {537,946}, {546,24827}, {673,5817}, {903,3545}, {971,16593}, {1331,1862}, {1352,5779}, {1478,24846}, {1479,24845}, {1482,9041}, {1587,24818}, {1588,24819}, {1598,24822}, {1699,24821}, {2796,9880}, {3091,4440}, {3839,17487}, {3850,4409}, {3932,15310}, {5480,9055}, {5587,24715}, {5603,24841}, {6201,24832}, {6202,24831}, {7330,16560}, {8196,24823}, {8203,24824}, {8212,24838}, {8213,24839}, {9993,24825}, {10531,24847}, {10532,24848}, {10893,24834}, {10894,24835}, {10895,24836}, {10896,24837}, {11496,24820}, {11897,24830}, {16561,18540}, {16594,19515}, {22753,24826}
X(24828) = midpoint of X(i) and X(j) for these {i,j}: {4, 190}, {24833, 24844}
X(24828) = reflection of X(i) in X(j) for these (i,j): (3, 4422), (1086, 5), (24827, 546)
X(24828) = complement of X(24813)
X(24828) = X(190)-of-Euler triangle
X(24828) = X(1086)-of-Johnson triangle
X(24828) = X(4422)-of-X3-ABC reflections triangle
X(24828) = X(24833)-of-Ehrmann-mid triangle
X(24828) = {X(381), X(24844)}-harmonic conjugate of X(24833)
The reciprocal orthologic center of these triangles is X(6)
X(24829) lies on the line {3,3923}
The reciprocal parallelogic center of these triangles is X(1)
X(24830) lies on these lines: {30,903}, {190,402}, {528,12626}, {537,12438}, {545,1651}, {900,13268}, {1086,1650}, {2786,13179}, {2796,12347}, {4240,4440}, {4422,15183}, {4432,11831}, {9055,12583}, {11832,24814}, {11839,24815}, {11845,24817}, {11848,24820}, {11852,24821}, {11853,24822}, {11885,24825}, {11897,24828}, {11900,24715}, {11901,24831}, {11902,24832}, {11903,24834}, {11904,24835}, {11905,24836}, {11906,24837}, {11907,24838}, {11908,24839}, {11909,24840}, {11910,24841}, {11911,24844}, {11912,24845}, {11913,24846}, {11914,24847}, {11915,24848}, {13894,24842}, {13948,24843}, {18507,24827}, {18958,24816}, {19017,24818}, {19018,24819}, {22755,24826}
X(24830) = midpoint of X(4240) and X(4440)
X(24830) = reflection of X(i) in X(j) for these (i,j): (190, 402), (1650, 1086), (18507, 24827)
X(24830) = X(190)-of-Gossard triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24831) lies on these lines: {6,190}, {528,12627}, {537,3641}, {545,5861}, {900,13269}, {1086,5591}, {1271,4440}, {2786,6319}, {2796,9882}, {4432,11370}, {5589,24821}, {5595,24822}, {5605,24841}, {5689,24715}, {6202,24828}, {6215,24833}, {8198,24823}, {8205,24824}, {8216,24838}, {8217,24839}, {8974,24842}, {9994,24825}, {10040,24845}, {10048,24846}, {10783,24817}, {10792,24815}, {10919,24834}, {10921,24835}, {10923,24836}, {10925,24837}, {10927,24840}, {10929,24847}, {10931,24848}, {11388,24814}, {11497,24820}, {11824,24813}, {11901,24830}, {11916,24844}, {13949,24843}, {18509,24827}, {18959,24816}, {22756,24826}
X(24831) = reflection of X(24832) in X(190)
X(24831) = X(190)-of-inner-Grebe triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24832) lies on these lines: {6,190}, {528,12628}, {537,3640}, {545,5860}, {900,13270}, {1086,5590}, {1270,4440}, {2786,6320}, {2796,9883}, {4432,11371}, {5588,24821}, {5594,24822}, {5604,24841}, {5688,24715}, {6201,24828}, {6214,24833}, {8199,24823}, {8206,24824}, {8218,24838}, {8219,24839}, {8975,24842}, {9995,24825}, {10041,24845}, {10049,24846}, {10784,24817}, {10793,24815}, {10920,24834}, {10922,24835}, {10924,24836}, {10926,24837}, {10928,24840}, {10930,24847}, {10932,24848}, {11389,24814}, {11498,24820}, {11825,24813}, {11902,24830}, {11917,24844}, {13950,24843}, {18511,24827}, {18960,24816}, {22757,24826}
X(24832) = reflection of X(24831) in X(190)
X(24832) = X(190)-of-outer-Grebe triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24833) lies on these lines: {1,24836}, {2,24817}, {3,1086}, {4,4440}, {5,190}, {12,24845}, {30,903}, {88,19515}, {355,537}, {381,545}, {517,3792}, {528,1482}, {673,5762}, {900,10738}, {946,2796}, {952,24841}, {1351,5845}, {1352,9055}, {1478,24816}, {1479,24840}, {1656,4422}, {2786,6321}, {2969,22148}, {3090,4473}, {3545,17487}, {3843,4409}, {4370,5055}, {4432,5886}, {5587,24821}, {5805,20430}, {6214,24832}, {6215,24831}, {7583,24819}, {7584,24818}, {8200,24823}, {8207,24824}, {8220,24838}, {8221,24839}, {8976,24842}, {9041,12645}, {9996,24825}, {10796,24815}, {10942,24847}, {10943,24848}, {11499,24820}, {13951,24843}, {15310,20358}, {22758,24826}
X(24833) = midpoint of X(i) and X(j) for these {i,j}: {4, 4440}, {24834, 24835}
X(24833) = reflection of X(i) in X(j) for these (i,j): (3, 1086), (4, 24827), (190, 5), (24844, 24828)
X(24833) = complement of X(24817)
X(24833) = X(190)-of-Johnson triangle
X(24833) = X(1086)-of-X3-ABC reflections triangle
X(24833) = X(4440)-of-Euler triangle
X(24833) = X(24827)-of-anti-Euler triangle
X(24833) = X(24844)-of-Ehrmann-mid triangle
X(24833) = X(24845)-of-outer-Johnson triangle
X(24833) = X(24846)-of-inner-Johnson triangle
X(24833) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 24844, 24828), (24836, 24837, 1)
The reciprocal parallelogic center of these triangles is X(1)
X(24834) lies on these lines: {11,190}, {12,24847}, {355,537}, {528,944}, {545,11235}, {900,13271}, {1086,1376}, {2786,13180}, {2796,12348}, {3434,4440}, {4432,11373}, {4473,10584}, {9055,12586}, {10523,24845}, {10785,24817}, {10794,24815}, {10826,24821}, {10829,24822}, {10871,24825}, {10893,24828}, {10914,24715}, {10919,24831}, {10920,24832}, {10944,24836}, {10945,24838}, {10946,24839}, {10947,24840}, {10948,24846}, {10949,24848}, {11390,24814}, {11865,24823}, {11866,24824}, {11903,24830}, {11928,24844}, {12114,24826}, {13895,24842}, {13952,24843}, {18516,24827}, {18961,24816}, {19023,24818}, {19024,24819}
X(24834) = reflection of X(i) in X(j) for these (i,j): (24820, 1086), (24835, 24833)
X(24834) = X(190)-of-inner-Johnson triangle
X(24834) = X(24847)-of-outer-Johnson triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24835) lies on these lines: {11,24848}, {12,190}, {72,24715}, {355,537}, {528,962}, {545,11236}, {900,13272}, {958,1086}, {2786,13181}, {2796,12349}, {3436,4440}, {4432,11374}, {4473,10585}, {9055,12587}, {10523,24846}, {10786,24817}, {10795,24815}, {10827,24821}, {10830,24822}, {10872,24825}, {10894,24828}, {10921,24831}, {10922,24832}, {10950,24837}, {10951,24838}, {10952,24839}, {10953,24840}, {10954,24845}, {10955,24847}, {11391,24814}, {11500,24820}, {11827,24813}, {11867,24823}, {11868,24824}, {11904,24830}, {11929,24844}, {13896,24842}, {13953,24843}, {18517,24827}, {18962,24816}, {19025,24818}, {19026,24819}
X(24835) = reflection of X(i) in X(j) for these (i,j): (24826, 1086), (24834, 24833)
X(24835) = X(190)-of-outer-Johnson triangle
X(24835) = X(24848)-of-inner-Johnson triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24836) lies on these lines: {1,24833}, {4,24840}, {5,24846}, {7,528}, {12,190}, {55,3782}, {56,1086}, {65,24715}, {109,1365}, {226,2796}, {388,4440}, {495,24845}, {517,1463}, {537,5252}, {545,11237}, {900,13273}, {1284,5172}, {1400,1989}, {1479,24827}, {2786,13182}, {2807,4014}, {3085,24817}, {4432,11375}, {4473,10588}, {6690,9791}, {7354,24813}, {9055,12588}, {9578,24821}, {9654,24844}, {10797,24815}, {10831,24822}, {10873,24825}, {10895,24828}, {10923,24831}, {10924,24832}, {10944,24834}, {10956,24847}, {10957,24848}, {11392,24814}, {11501,24820}, {11869,24823}, {11870,24824}, {11905,24830}, {11930,24838}, {11931,24839}, {13897,24842}, {13954,24843}, {19027,24818}, {19028,24819}, {22759,24826}
X(24836) = reflection of X(24845) in X(495)
X(24836) = X(190)-of-1st Johnson-Yff triangle
X(24836) = X(24833)-of-inner-Yff triangle
X(24836) = X(24837)-of-inner-Yff tangents triangle
X(24836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24833, 24837), (388, 4440, 24816)
The reciprocal parallelogic center of these triangles is X(1)
X(24837) lies on these lines: {1,24833}, {4,24816}, {5,24845}, {11,190}, {55,1086}, {496,24846}, {497,4440}, {528,2098}, {537,1837}, {545,11238}, {900,13274}, {903,3058}, {1478,24827}, {2786,13183}, {2796,12053}, {3057,24715}, {3086,24817}, {3254,4516}, {3939,7336}, {4432,11376}, {4473,10589}, {6284,24813}, {9055,12589}, {9581,24821}, {9669,24844}, {10798,24815}, {10832,24822}, {10874,24825}, {10896,24828}, {10925,24831}, {10926,24832}, {10950,24835}, {10958,24847}, {10959,24848}, {11393,24814}, {11502,24820}, {11871,24823}, {11872,24824}, {11906,24830}, {11932,24838}, {11933,24839}, {13898,24842}, {13955,24843}, {19029,24818}, {19030,24819}, {22760,24826}
X(24837) = reflection of X(24846) in X(496)
X(24837) = X(190)-of-2nd Johnson-Yff triangle
X(24837) = X(24833)-of-outer-Yff triangle
X(24837) = X(24836)-of-outer-Yff tangents triangle
X(24837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24833, 24836), (497, 4440, 24840)
The reciprocal parallelogic center of these triangles is X(1)
X(24838) lies on these lines: {190,493}, {528,12636}, {537,12440}, {545,12152}, {900,13275}, {1086,8222}, {2786,13184}, {2796,12352}, {4432,11377}, {4440,6462}, {6461,24839}, {8188,24821}, {8194,24822}, {8201,24823}, {8208,24824}, {8210,24841}, {8212,24828}, {8214,24715}, {8216,24831}, {8218,24832}, {8220,24833}, {9055,12590}, {10875,24825}, {10945,24834}, {10951,24835}, {11394,24814}, {11503,24820}, {11828,24813}, {11840,24815}, {11846,24817}, {11907,24830}, {11930,24836}, {11932,24837}, {11947,24840}, {11949,24844}, {11951,24845}, {11953,24846}, {11955,24847}, {11957,24848}, {13899,24842}, {13956,24843}, {18520,24827}, {18963,24816}, {19031,24818}, {19032,24819}, {22761,24826}
X(24838) = X(190)-of-Lucas homothetic triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24839) lies on these lines: {190,494}, {528,12637}, {537,12441}, {545,12153}, {900,13276}, {1086,8223}, {2786,13185}, {2796,12353}, {4432,11378}, {4440,6463}, {6461,24838}, {8189,24821}, {8195,24822}, {8202,24823}, {8209,24824}, {8211,24841}, {8213,24828}, {8215,24715}, {8217,24831}, {8219,24832}, {8221,24833}, {9055,12591}, {10876,24825}, {10946,24834}, {10952,24835}, {11395,24814}, {11504,24820}, {11829,24813}, {11841,24815}, {11847,24817}, {11908,24830}, {11931,24836}, {11933,24837}, {11948,24840}, {11950,24844}, {11952,24845}, {11954,24846}, {11956,24847}, {11958,24848}, {13900,24842}, {13957,24843}, {18522,24827}, {18964,24816}, {19033,24818}, {19034,24819}, {22762,24826}
X(24839) = X(190)-of-Lucas(-1) homothetic triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24840) lies on these lines: {1,24816}, {3,24846}, {4,24836}, {11,244}, {12,24828}, {33,24814}, {55,190}, {56,24813}, {87,24307}, {497,4440}, {513,23772}, {522,3271}, {528,10394}, {537,3057}, {545,3058}, {903,11238}, {950,2796}, {1479,24833}, {1697,24821}, {1837,24715}, {2098,24841}, {2646,4432}, {2786,3022}, {3056,9055}, {3295,24844}, {3583,24827}, {3667,4014}, {4294,24817}, {4370,4995}, {4422,5432}, {4473,5218}, {10385,17487}, {10799,24815}, {10833,24822}, {10877,24825}, {10927,24831}, {10928,24832}, {10947,24834}, {10953,24835}, {10965,24847}, {10966,24826}, {11873,24823}, {11874,24824}, {11909,24830}, {11947,24838}, {11948,24839}, {13901,24842}, {13958,24843}, {19037,24818}, {19038,24819}
X(24840) = reflection of X(i) in X(j) for these (i,j): (4014, 24225), (24816, 1)
X(24840) = X(190)-of-Mandart-incircle triangle
X(24840) = X(24813)-of-2nd anti-circumperp-tangential triangle
X(24840) = X(24816)-of-5th mixtilinear triangle
X(24840) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 4440, 24837), (2310, 4459, 11), (3295, 24844, 24845)
The reciprocal parallelogic center of these triangles is X(1)
X(24841) lies on these lines: {1,190}, {2,4126}, {8,599}, {55,24826}, {56,24820}, {88,17780}, {100,17154}, {145,528}, {239,335}, {244,3699}, {320,519}, {517,24813}, {545,3241}, {726,4693}, {900,1120}, {918,1280}, {952,24833}, {984,24331}, {1015,24486}, {1317,20098}, {1621,20068}, {1647,4997}, {1830,1897}, {2098,24840}, {2099,24816}, {2786,7983}, {2796,3244}, {3123,24397}, {3242,4363}, {3315,3952}, {3616,4422}, {3622,4473}, {3679,17227}, {3685,4864}, {3891,4430}, {3996,17155}, {3999,5205}, {4080,10707}, {4310,4429}, {4366,20145}, {4370,16672}, {4465,17794}, {4552,14151}, {4645,7238}, {5083,14594}, {5597,24824}, {5598,24823}, {5603,24828}, {5604,24832}, {5605,24831}, {6559,17435}, {7081,21342}, {7174,16831}, {7967,24817}, {7968,24818}, {7969,24819}, {8192,24822}, {8210,24838}, {8211,24839}, {9457,24416}, {9997,24825}, {10247,24844}, {10800,24815}, {10944,24834}, {10950,24835}, {11038,16593}, {11396,24814}, {11910,24830}, {12635,17480}, {13902,24842}, {13959,24843}, {18525,24827}
X(24841) = midpoint of X(145) and X(4440)
X(24841) = reflection of X(i) in X(j) for these (i,j): (8, 1086), (190, 1), (3685, 4864), (18525, 24827), (24821, 4432)
X(24841) = X(190)-of-5th mixtilinear triangle
X(24841) = X(24820)-of-2nd anti-circumperp-tangential triangle
X(24841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 24821, 4432), (3242, 24349, 5263), (4080, 20042, 10707)
The reciprocal parallelogic center of these triangles is X(1)
X(24842) lies on these lines: {2,24819}, {6,344}, {190,3068}, {371,24828}, {528,13911}, {537,8983}, {545,13846}, {590,1086}, {900,13922}, {2161,7347}, {2786,8997}, {2796,13908}, {4432,13883}, {4440,8972}, {4473,7585}, {8974,24831}, {8975,24832}, {8976,24833}, {9055,13910}, {9540,24813}, {13884,24814}, {13885,24815}, {13886,24817}, {13887,24820}, {13888,24821}, {13889,24822}, {13890,24823}, {13891,24824}, {13892,24825}, {13893,24715}, {13894,24830}, {13895,24834}, {13896,24835}, {13897,24836}, {13899,24838}, {13900,24839}, {13901,24840}, {13902,24841}, {13903,24844}, {13904,24845}, {13905,24846}, {13906,24847}, {13907,24848}, {18538,24827}, {18965,24816}, {22763,24826}
X(24842) = X(190)-of-3rd tri-squares-central triangle
X(24842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 4422, 24843), (4473, 7585, 24818)
The reciprocal parallelogic center of these triangles is X(1)
X(24843) lies on these lines: {2,24818}, {6,344}, {190,3069}, {372,24828}, {528,13973}, {537,13971}, {545,13847}, {615,1086}, {900,13991}, {2161,7348}, {2786,13989}, {2796,13968}, {4432,13936}, {4440,13941}, {4473,7586}, {9055,13972}, {13935,24813}, {13937,24814}, {13938,24815}, {13939,24817}, {13940,24820}, {13942,24821}, {13943,24822}, {13944,24823}, {13945,24824}, {13946,24825}, {13947,24715}, {13948,24830}, {13949,24831}, {13950,24832}, {13951,24833}, {13952,24834}, {13953,24835}, {13954,24836}, {13955,24837}, {13956,24838}, {13957,24839}, {13958,24840}, {13959,24841}, {13961,24844}, {13962,24845}, {13963,24846}, {13964,24847}, {13965,24848}, {18762,24827}, {18966,24816}, {22764,24826}
X(24843) = X(190)-of-4th tri-squares-central triangle
X(24843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 4422, 24842), (4473, 7586, 24819)
The reciprocal parallelogic center of these triangles is X(1)
X(24844) lies on these lines: {3,190}, {5,4440}, {30,17487}, {140,4473}, {355,2796}, {381,545}, {517,24821}, {528,12645}, {537,1482}, {900,12331}, {903,5055}, {999,24816}, {1086,1656}, {1351,9055}, {1598,24814}, {2786,13188}, {3295,24840}, {3526,4422}, {3843,24827}, {4370,5054}, {4409,5072}, {4432,10246}, {5790,24715}, {5843,20533}, {5845,11898}, {6417,24819}, {6418,24818}, {7517,24822}, {9301,24825}, {9654,24836}, {9669,24837}, {10247,24841}, {11842,24815}, {11849,24820}, {11875,24823}, {11876,24824}, {11911,24830}, {11916,24831}, {11917,24832}, {11928,24834}, {11929,24835}, {11949,24838}, {11950,24839}, {12000,24847}, {12001,24848}, {13903,24842}, {13961,24843}, {22765,24826}
X(24844) = reflection of X(i) in X(j) for these (i,j): (3, 190), (4440, 5), (24833, 24828)
X(24844) = X(190)-of-X3-ABC reflections triangle
X(24844) = X(4440)-of-Johnson triangle
X(24844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24816, 24846, 999), (24828, 24833, 381), (24840, 24845, 3295)
The reciprocal parallelogic center of these triangles is X(1)
X(24845) lies on these lines: {1,190}, {3,24816}, {5,24837}, {12,24833}, {35,24813}, {388,24817}, {495,24836}, {498,1086}, {499,4422}, {528,12647}, {545,10056}, {611,9055}, {900,10087}, {903,3584}, {1479,24828}, {1772,24416}, {2786,10086}, {2796,10054}, {3085,4440}, {3086,4473}, {3295,24840}, {3299,24818}, {3301,24819}, {4370,10072}, {10037,24822}, {10038,24825}, {10039,24715}, {10040,24831}, {10041,24832}, {10523,24834}, {10801,24815}, {10895,24827}, {10954,24835}, {11398,24814}, {11507,24820}, {11877,24823}, {11878,24824}, {11912,24830}, {11951,24838}, {11952,24839}, {13904,24842}, {13962,24843}, {22766,24826}
X(24845) = midpoint of X(190) and X(24847)
X(24845) = reflection of X(24836) in X(495)
X(24845) = X(190)-of-inner-Yff triangle
X(24845) = X(24833)-of-1st Johnson-Yff triangle
X(24845) = X(24846)-of-inner-Yff tangents triangle
X(24845) = X(24847)-of-outer-Yff triangle
X(24845) = {X(3295), X(24844)}-harmonic conjugate of X(24840)
The reciprocal parallelogic center of these triangles is X(1)
X(24846) lies on these lines: {1,190}, {3,24840}, {5,24836}, {36,24813}, {496,24837}, {497,24817}, {498,4422}, {499,1086}, {528,5729}, {545,10072}, {613,9055}, {673,1733}, {900,10090}, {903,3582}, {920,16560}, {999,24816}, {1210,2796}, {1478,24828}, {1737,24715}, {2786,10089}, {3085,4473}, {3086,4440}, {3299,24819}, {3301,24818}, {4370,10056}, {10046,24822}, {10047,24825}, {10048,24831}, {10049,24832}, {10523,24835}, {10802,24815}, {10896,24827}, {10948,24834}, {11399,24814}, {11508,24820}, {11879,24823}, {11880,24824}, {11913,24830}, {11953,24838}, {11954,24839}, {13905,24842}, {13963,24843}, {22767,24826}
X(24846) = midpoint of X(190) and X(24848)
X(24846) = reflection of X(24837) in X(496)
X(24846) = X(190)-of-outer-Yff triangle
X(24846) = X(24833)-of-2nd Johnson-Yff triangle
X(24846) = X(24845)-of-outer-Yff tangents triangle
X(24846) = X(24848)-of-inner-Yff triangle
X(24846) = {X(999), X(24844)}-harmonic conjugate of X(24816)
The reciprocal parallelogic center of these triangles is X(1)
X(24847) lies on these lines: {1,190}, {12,24834}, {528,12648}, {545,11239}, {900,13278}, {1086,5552}, {2786,13189}, {2796,12356}, {4440,10528}, {4473,10586}, {9055,12594}, {10531,24828}, {10803,24815}, {10805,24817}, {10834,24822}, {10878,24825}, {10915,24715}, {10929,24831}, {10930,24832}, {10942,24833}, {10955,24835}, {10956,24836}, {10958,24837}, {10965,24840}, {11248,24813}, {11400,24814}, {11509,24816}, {11881,24823}, {11882,24824}, {11914,24830}, {11955,24838}, {11956,24839}, {12000,24844}, {13906,24842}, {13964,24843}, {18542,24827}, {19047,24818}, {19048,24819}, {22768,24826}
X(24847) = reflection of X(190) in X(24845)
X(24847) = X(190)-of-inner-Yff tangents triangle
X(24847) = X(24834)-of-1st Johnson-Yff triangle
The reciprocal parallelogic center of these triangles is X(1)
X(24848) lies on these lines: {1,190}, {11,24835}, {528,12649}, {545,11240}, {900,13279}, {1086,10527}, {2786,13190}, {2796,12357}, {4440,10529}, {4473,10587}, {9055,12595}, {10532,24828}, {10804,24815}, {10806,24817}, {10835,24822}, {10879,24825}, {10916,24715}, {10931,24831}, {10932,24832}, {10943,24833}, {10949,24834}, {10957,24836}, {10959,24837}, {10966,24826}, {11249,24813}, {11401,24814}, {11510,24820}, {11883,24823}, {11884,24824}, {11915,24830}, {11957,24838}, {11958,24839}, {12001,24844}, {13907,24842}, {13965,24843}, {18544,24827}, {18967,24816}, {19049,24818}, {19050,24819}
X(24848) = reflection of X(190) in X(24846)
X(24848) = X(190)-of-outer-Yff tangents triangle
X(24848) = X(24835)-of-2nd Johnson-Yff triangle
X(24849) lies on these lines: {1,15973}, {8,73}, {43,65}, {56,1740}, {978,1403}, {986,5887}, {1193,3485}, {1457,20036}, {7355,24310}
The reciprocal orthologic center of these triangles is X(1)
X(24850) lies on these lines: {1,4234}, {2,24851}, {3,3923}, {8,896}, {10,30}, {21,4418}, {35,1215}, {58,740}, {79,4892}, {100,24852}, {171,7283}, {190,5293}, {386,4672}, {405,3980}, {474,4011}, {540,21081}, {726,5266}, {846,1010}, {902,4968}, {964,4414}, {978,2234}, {986,4195}, {993,15952}, {1003,16822}, {1009,24259}, {1043,1046}, {1045,4281}, {1054,13741}, {1086,1125}, {1089,4434}, {1155,3831}, {1580,11104}, {1755,3501}, {1770,2887}, {1834,8258}, {2292,4427}, {3178,3712}, {3210,16478}, {3550,4385}, {3634,16052}, {3648,4683}, {3741,3916}, {3821,17698}, {3985,5277}, {4205,12579}, {4252,5695}, {4267,4436}, {4650,10449}, {4754,4760}, {5248,24325}, {6651,16917}, {6679,23537}, {7262,9534}, {7295,19845}, {7816,24254}, {8680,24335}, {11110,24342}, {11319,24443}, {12512,17355}, {12567,19259}, {13323,24264}, {13740,17596}, {16061,17738}, {16609,24271}, {17164,17539}, {17697,24174}, {23383,24425}
X(24850) = midpoint of X(i) and X(j) for these {i,j}: {100, 24852}, {1043, 1046}
X(24850) = reflection of X(1834) in X(8258)
X(24850) = complement of X(24851)
X(24850) = {X(4252), X(5695)}-harmonic conjugate of X(17733)
The reciprocal orthologic center of these triangles is X(1)
X(24851) lies on these lines: {1,30}, {2,24850}, {3,3944}, {4,240}, {5,17596}, {8,4683}, {10,190}, {11,24852}, {21,3120}, {35,17719}, {36,15952}, {40,1756}, {145,17491}, {148,257}, {171,1770}, {238,23537}, {405,17889}, {442,846}, {497,3976}, {498,17601}, {513,18178}, {516,5255}, {522,21134}, {535,15955}, {726,5015}, {740,1330}, {896,3648}, {982,1479}, {987,19645}, {988,1699}, {1010,4425}, {1046,1834}, {1054,4187}, {1125,4234}, {1193,5057}, {1698,16052}, {1714,7262}, {1738,12572}, {2292,2475}, {2476,4414}, {2478,24174}, {2549,3061}, {2650,14450}, {2782,3865}, {2887,7283}, {3338,24217}, {3496,5254}, {3583,3670}, {3585,4424}, {3663,4911}, {3735,7748}, {3750,13407}, {3821,13740}, {3914,5247}, {3923,16062}, {3924,11114}, {3953,4857}, {3980,17533}, {4205,24342}, {4292,24210}, {4309,17715}, {4385,4660}, {4415,5293}, {4418,5051}, {4642,5080}, {4650,5292}, {4653,11263}, {4655,10449}, {4703,9534}, {4872,24214}, {4887,6744}, {5046,24443}, {5129,7613}, {5134,16600}, {5302,21949}, {5358,24436}, {5988,6998}, {6175,21674}, {6536,14005}, {6650,17685}, {6651,17673}, {6656,17738}, {7847,18061}, {8040,17551}, {9612,17594}, {9670,17597}, {10896,17595}, {11110,12579}, {13741,24169}, {16478,19785}, {18483,24239}
X(24851) = reflection of X(i) in X(j) for these (i,j): (1046, 1834), (5255, 13161), (24852, 11)
X(24851) = anticomplement of X(24850)
X(24851) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 24248, 986), (21, 3120, 24161), (3782, 6284, 1)
The reciprocal orthologic center of these triangles is X(1)
X(24852) lies on these lines: {11,24851}, {30,80}, {100,24850}, {214,4234}, {1222,2802}, {2796,21630}, {4418,12746}, {6702,17677}
X(24852) = reflection of X(i) in X(j) for these (i,j): (100, 24850), (24851, 11)
X(24853) lies on the line {220,2276}
The reciprocal orthologic center of these triangles is X(6)
X(24854) lies on the line {3,8301}
See Angel Montesdeoca, HG140918.
X(24855) lies on these lines: {2,6}, {5,8585}, {30,10418}, {111,858}, {113,9193}, {115,16317}, {126,6390}, {187,468}, {427,18424}, {625,6719}, {690,9209}, {1503,2502}, {1513,1514}, {2030,5972}, {3291,5159}, {5107,6388}, {5210,7493}, {5915,10564}, {6791,13857}, {9759,14982}
See Angel Montesdeoca, HG190918.
X(24856) lies on these lines: {7,10405}, {281,1886}
See Alexandr Skutin and Angel Montesdeoca, HG220918.
X(24857) lies on these lines: {81,16489}, {88,551}
See Alexandr Skutin and Angel Montesdeoca, HG220918.
X(24858) lies on these lines: {1,4370}, {2,4738}, {57,1317}, {81,16490}, {88,519}, {89,3241}, {106,8028}, {551,4767}, {900,1022}, {4677,8056}
X(24858) = reflection of X(4767) in X(551)
X(24858) = isogonal conjugate of X(16489)
X(24858) = anticomplement of X(36912)
See Antreas Hatzipolakis and Angel Montesdeoca, HG280918.
X(24859) lies on this line: {5,11014}
See Antreas Hatzipolakis and Angel Montesdeoca, HG280918.
X(24860) lies on these lines: {10095,10620}, {12006,16835}
See Angel Montesdeoca, HG290918.
X(24861) lies on these lines: {251,15271}, {263,3763}
See Angel Montesdeoca, HG031018.
X(24862) lies on these lines: {4, 16813}, {115, 130}, {125, 137}, {184, 11082}, {338, 23290}, {1879, 6752}, {3574, 5095}, {7687, 18402}, {13851, 16337}, {15451, 20975}
X(24862) = trilinear pole, wrt orthic triangle, of line X(4)X(54)
X(24862) = polar conjugate of isotomic conjugate of X(39019)
See Angel Montesdeoca, HG041018.
X(24863) lies on this line: {37, 109}
Collineation mappings involving Gemini triangle 9: X(24864)-X(24879)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 9, as in centers X(24864)-X(24879). Then
m(X) = (a^3 - a^2 b - 2 a b^2 - a^2 c + 5 a b c - 2 a c^2) x + (- 2 a^3 + a^2 b + 2 a b^2 - b^3 + 3 a^2 c - 7 a b c + 2 b^2 c + 3 a c^2 + b c^2 - 2 c^3) y + (- 2 a^3 + a^2 c + 2 a c^2 - c^3 + 3 a^2 b - 7 a b c + 2 b c^2 + 3 a b^2 + b^2 c - 2 b^3) z : : ,
and m(X) is on the Euler line if and only if X is on the Euler line.
(Clark Kimberling, October 10, 2018)
X(24864) lies on these lines: {1, 2}, {11, 10700}, {80, 17460}, {106, 1145}, {514, 1000}, {952, 24828}, {996, 1016}, {1086, 4792}, {1168, 25026}, {1317, 24867}, {1320, 24222}, {2802, 24715}, {3057, 18340}, {3120, 12653}, {3476, 23703}, {3722, 7972}, {4346, 6549}, {4389, 4555}, {4675, 24868}, {5252, 24865}, {6631, 17305}, {10713, 16594}, {13541, 21630}, {17119, 25035}, {17395, 24281}, {17724, 25416}
X(24865) lies on these lines: {2, 3}, {1387, 24869}, {5252, 24864}, {11376, 18340}, {23708, 24871}
X(24866) lies on these lines: {2, 3}, {24870, 24871}
X(24867) lies on these lines: {2, 3}, {1317, 24864}
X(24868) lies on these lines: {2, 6}, {1086, 1320}, {4675, 24864}, {17374, 25034}
X(24869) lies on these lines: {1, 2}, {1387, 24865}, {17056, 25040}, {18220, 18340}, {24874, 24878}
X(24870) lies on these lines: {2, 7}, {106, 1086}, {4675, 24864}, {14475, 23838}, {24866, 24871}
X(24871) lies on these lines: {1, 2}, {1319, 25026}, {1387, 4674}, {4694, 24433}, {11731, 24542}, {16173, 24715}, {17395, 24873}, {23708, 24865}, {24866, 24870}
X(24872) lies on this line: {2, 3}
X(24873) lies on these lines: {2, 37}, {10, 25029}, {1086, 1145}, {4675, 24864}, {14475, 23757}, {17119, 24281}, {17237, 25030}, {17395, 24871}, {25031, 25035}
X(24874) lies on these lines: {2, 37}, {16589, 25037}, {24869, 24878}
X(24875) lies on this line: {2, 3}
X(24876) lies on this line: {2, 3}
X(24877) lies on this line: {2, 3}
X(24878) lies on these lines: {2, 3}, {24869, 24874}
X(24879) lies on this line: {2, 3}
Collineation mappings involving Gemini triangle 10: X(24880)-X(24925)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 10, as in centers X(24880)-X(24925). Then
m(X) = a (a^2 + a b + a c + b c) x + (-a^2 b + b^2 - a^2 c - a b c - a c^2 - b c^2) y + (-a^2 c + c^2 - a^2 b - a b c - a b^2 - b^2 c) z : : ,
and m(X) is on the Euler line if and only if X is on the Euler line.
(Clark Kimberling, October 10, 2018)
X(24880) lies on these lines: {1, 2}, {3, 759}, {5, 580}, {6, 24937}, {58, 442}, {72, 24160}, {79, 896}, {171, 3841}, {191, 3120}, {201, 2006}, {333, 3454}, {377, 4257}, {595, 2886}, {748, 7741}, {758, 24161}, {851, 4278}, {991, 6889}, {1010, 6693}, {1046, 11263}, {1054, 1247}, {1324, 16453}, {1724, 2476}, {1754, 6828}, {1834, 4653}, {3019, 3090}, {3294, 17737}, {3579, 21949}, {3647, 24851}, {3678, 17719}, {3772, 5791}, {3822, 5247}, {3825, 17123}, {4220, 5358}, {4252, 17528}, {4256, 7483}, {4658, 17056}, {5251, 21935}, {5400, 6960}, {5432, 24934}, {5433, 24885}, {5445, 24916}, {5713, 6858}, {5745, 23537}, {5956, 11231}, {6175, 16948}, {6211, 24468}, {6831, 13329}, {7413, 7683}, {7828, 17277}, {7887, 20154}, {9581, 24933}, {12514, 17064}, {17070, 18253}, {17275, 24935}, {17278, 24884}, {17337, 17527}, {17776, 24081}, {23897, 24275}, {24046, 24789}, {24519, 24962}, {24530, 25065}, {24881, 24910}, {24888, 24904}, {24903, 24909}, {24919, 24922}
X(24881) lies on these lines: {2, 3}, {1718, 3216}, {24880, 24910}, {24883, 24916}, {24889, 24890}
X(24882) lies on these lines: {2, 3}, {386, 20277}, {1698, 21686}, {2939, 4466}, {3216, 3468}, {24889, 24918}, {24895, 24897}
X(24883) lies on these lines: {1, 2}, {4, 162}, {6, 2476}, {21, 1834}, {23, 5358}, {30, 16948}, {58, 2475}, {81, 442}, {149, 595}, {213, 17737}, {267, 3336}, {333, 5051}, {346, 24075}, {377, 2363}, {411, 5721}, {579, 16548}, {580, 6840}, {582, 6903}, {896, 3648}, {897, 9214}, {940, 4197}, {1046, 3120}, {1150, 16062}, {1330, 16704}, {1449, 24937}, {1479, 17127}, {1724, 5046}, {1751, 7557}, {1754, 6895}, {2006, 15556}, {2650, 24161}, {2895, 3454}, {3019, 5068}, {3218, 23537}, {3332, 6870}, {3649, 17070}, {3769, 5300}, {3772, 3868}, {3876, 17720}, {4193, 4383}, {4202, 14829}, {4205, 5235}, {4252, 17579}, {4418, 8258}, {4653, 15674}, {4854, 18253}, {4921, 16052}, {5080, 5247}, {5154, 14997}, {5178, 5266}, {5315, 24387}, {5396, 6853}, {5706, 6828}, {5707, 6829}, {5722, 24933}, {6703, 14005}, {6941, 7592}, {7235, 24530}, {11680, 16466}, {15325, 24925}, {17337, 17534}, {17362, 24935}, {17366, 24890}, {17669, 20142}, {24881, 24916}, {24903, 24913}, {24919, 24923}
X(24883) = anticomplement of X(25645)
X(24884) lies on these lines: {2, 7}, {6, 17058}, {140, 3019}, {284, 1375}, {1781, 4466}, {2160, 16581}, {2294, 24780}, {3017, 17366}, {3763, 16408}, {4000, 24195}, {15803, 24933}, {16054, 17052}, {17278, 24880}, {24899, 24919}
X(24885) lies on these lines: {2, 11}, {140, 24902}, {1054, 2607}, {3752, 24186}, {5433, 24880}, {6739, 13747}, {14310, 21180}, {15059, 19629}, {15368, 18191}
X(24886) lies on this line: {2, 3}
X(24887) lies on this line: {2, 3}
X(24888) lies on these lines: {2, 31}, {17728, 24893}, {24880, 24904}, {24890, 24892}, {24896, 24916}
X(24889) lies on these lines: {2, 32}, {24881, 24890}, {24882, 24918}, {24897, 24908}
X(24890) lies on these lines: {2, 6}, {1333, 17052}, {2893, 3285}, {4261, 25065}, {4675, 24937}, {17278, 24880}, {17366, 24883}, {20195, 24902}, {21245, 24632}, {24530, 24899}, {24881, 24889}, {24888, 24892}
X(24891) lies on these lines: {2, 39}, {17278, 24902}, {24881, 24889}, {24918, 24925}
X(24892) lies on these lines: {1, 2}, {9, 17737}, {11, 212}, {31, 2886}, {38, 3772}, {44, 17605}, {63, 3120}, {149, 8616}, {238, 11680}, {244, 24789}, {321, 4438}, {345, 4365}, {442, 1468}, {672, 16567}, {750, 3925}, {756, 17720}, {851, 2217}, {896, 1836}, {902, 3434}, {958, 21935}, {1150, 2887}, {1155, 21949}, {1834, 10448}, {2177, 6690}, {2292, 5791}, {2308, 24597}, {2476, 5247}, {2611, 20254}, {2933, 4191}, {3035, 9350}, {3218, 17889}, {3219, 3944}, {3681, 17719}, {3691, 3767}, {3722, 4863}, {3782, 17070}, {3816, 17125}, {3826, 17124}, {3838, 4641}, {3846, 5278}, {3868, 24161}, {3914, 4414}, {3915, 24390}, {4001, 4138}, {4650, 20292}, {4799, 25383}, {5057, 7262}, {5127, 14008}, {5284, 24217}, {5904, 24160}, {6327, 21241}, {6679, 21242}, {16968, 21029}, {17278, 17728}, {17449, 24477}, {24888, 24890}
X(24893) lies on these lines: {2, 44}, {656, 24921}, {17278, 24880}, {17728, 24888}
X(24894) lies on these lines: {2, 45}, {17245, 17531}, {17278, 24880}, {17282, 24935}
X(24895) lies on these lines: {2, 6}, {5292, 25065}, {7235, 24530}, {24882, 24897}
X(24896) lies on these lines: {1, 2}, {17728, 24923}, {17737, 21061}, {24888, 24916}
X(24897) lies on these lines: {1, 24530}, {2, 39}, {1654, 18171}, {2223, 19345}, {3227, 24522}, {17053, 17143}, {24882, 24895}, {24889, 24908}
X(24898) lies on these lines: {1, 2}, {896, 20084}, {1834, 15674}, {2475, 16948}, {6933, 14997}, {11684, 17070}
X(24899) lies on these lines: {1, 2}, {5741, 7828}, {21124, 24921}, {24530, 24890}, {24884, 24919}
X(24900) lies on these lines: {2, 650}, {43, 21727}, {647, 4789}, {661, 18197}, {3121, 24137}, {3666, 17161}, {4369, 16751}, {4850, 21196}, {14077, 19767}, {14838, 16754}
X(24901) lies on these lines: {2, 31}, {17278, 24880}
X(24902) lies on these lines: {1, 2}, {140, 24885}, {896, 6701}, {1743, 24937}, {5127, 7504}, {5400, 6853}, {17278, 24891}, {20195, 24890}, {23897, 24956}, {24904, 24910}
X(24903) lies on these lines: {2, 3}, {24880, 24909}, {24883, 24913}
X(24904) lies on these lines: {2, 3}, {3615, 5972}, {7741, 24955}, {24880, 24888}, {24902, 24910}
X(24905) lies on this line: {2, 3}
X(24906) lies on this line: {2, 3}
X(24907) lies on these lines: {2, 3}, {7235, 24530}
X(24908) lies on these lines: {2, 3}, {24889, 24897}
X(24909) lies on these lines: {2, 7}, {1014, 1213}, {1170, 24934}, {5433, 24936}, {7235, 24530}, {24880, 24903}
X(24910) lies on these lines: {2, 36}, {3772, 24046}, {24880, 24881}, {24902, 24904}
X(24911) lies on these lines: {2, 38}, {3017, 4695}, {3214, 5956}, {3216, 5253}, {6703, 20966}, {24880, 24888}
X(24912) lies on these lines: {2, 7}, {56, 24931}, {227, 5956}, {1211, 1412}, {5433, 24934}, {24880, 24881}
X(24913) lies on these lines: {1, 24907}, {2, 7}, {1412, 3580}, {24880, 24888}, {24883, 24903}
X(24914) lies on these lines: {1, 140}, {2, 65}, {3, 1737}, {4, 1155}, {5, 46}, {7, 10588}, {8, 1319}, {10, 56}, {11, 40}, {12, 57}, {30, 10826}, {35, 5722}, {36, 355}, {43, 2594}, {55, 1210}, {63, 1329}, {72, 18838}, {73, 899}, {78, 3035}, {79, 19919}, {80, 5442}, {100, 20118}, {109, 7299}, {119, 24467}, {165, 6284}, {201, 3772}, {226, 3634}, {333, 1408}, {354, 3085}, {381, 1770}, {388, 5435}, {392, 10200}, {404, 5794}, {405, 11509}, {406, 1887}, {427, 1452}, {442, 1454}, {475, 1875}, {484, 7741}, {495, 3338}, {496, 5119}, {497, 5704}, {498, 942}, {499, 517}, {515, 5204}, {516, 10896}, {518, 5552}, {519, 1388}, {549, 3612}, {550, 12019}, {553, 3947}, {580, 5348}, {590, 2362}, {604, 17275}, {615, 16232}, {631, 2646}, {750, 1451}, {920, 6842}, {936, 21677}, {938, 5218}, {940, 5530}, {950, 5217}, {958, 1470}, {962, 5183}, {978, 24806}, {986, 17720}, {993, 19524}, {997, 13747}, {999, 10039}, {1038, 1722}, {1125, 2099}, {1145, 20586}, {1158, 1532}, {1213, 2285}, {1214, 1714}, {1284, 17279}, {1317, 3632}, {1376, 6734}, {1385, 10573}, {1393, 17278}, {1399, 1724}, {1400, 17303}, {1402, 5955}, {1403, 3831}, {1415, 4426}, {1420, 3679}, {1428, 3416}, {1445, 3826}, {1450, 10459}, {1464, 16569}, {1466, 5745}, {1467, 8580}, {1469, 17077}, {1478, 9956}, {1479, 3579}, {1512, 12114}, {1617, 9709}, {1656, 12047}, {1697, 9588}, {1699, 5128}, {1702, 19029}, {1703, 19030}, {1706, 5231}, {1728, 6907}, {1768, 6259}, {1771, 2361}, {1776, 6932}, {1858, 6825}, {1864, 6908}, {1882, 14018}, {1898, 6838}, {1905, 3541}, {1935, 9364}, {1940, 5125}, {2067, 13973}, {2093, 8227}, {2098, 11362}, {2245, 3142}, {2263, 17337}, {2475, 9352}, {2476, 5880}, {2478, 4640}, {2551, 5744}, {2975, 25005}, {3011, 17054}, {3057, 3086}, {3090, 4295}, {3091, 3474}, {3149, 12616}, {3212, 17095}, {3218, 11681}, {3256, 5259}, {3303, 11019}, {3305, 16140}, {3333, 15888}, {3336, 7951}, {3339, 3649}, {3340, 3624}, {3359, 15908}, {3361, 5434}, {3419, 5172}, {3436, 5123}, {3476, 3617}, {3486, 3523}, {3524, 4305}, {3576, 10950}, {3582, 3654}, {3583, 15079}, {3584, 18398}, {3586, 15338}, {3614, 9612}, {3616, 11011}, {3627, 4333}, {3653, 24926}, {3680, 13996}, {3683, 5084}, {3697, 17625}, {3698, 19843}, {3714, 17740}, {3752, 5230}, {3814, 7702}, {3816, 5250}, {3828, 4298}, {3833, 12432}, {3838, 6933}, {3844, 12588}, {3872, 8256}, {3873, 13751}, {3880, 10529}, {3884, 10199}, {3916, 17619}, {3925, 5705}, {4187, 4679}, {4188, 5086}, {4193, 24703}, {4292, 10175}, {4293, 5818}, {4299, 5122}, {4309, 18527}, {4423, 9843}, {4511, 17566}, {4646, 11269}, {4654, 19876}, {4860, 21620}, {4863, 5687}, {4999, 19860}, {5045, 10056}, {5048, 12245}, {5057, 5154}, {5061, 5814}, {5080, 7705}, {5087, 6931}, {5131, 10483}, {5141, 20292}, {5193, 5288}, {5224, 10401}, {5225, 9778}, {5235, 5323}, {5248, 14882}, {5255, 17721}, {5316, 18249}, {5326, 11529}, {5439, 10198}, {5535, 5812}, {5587, 7354}, {5691, 15326}, {5698, 6919}, {5708, 13407}, {5710, 24239}, {5711, 17723}, {5727, 7987}, {5770, 14872}, {5836, 10527}, {5883, 15556}, {5886, 5903}, {5887, 6959}, {5901, 25415}, {5902, 11374}, {5919, 14986}, {6001, 6834}, {6175, 18977}, {6361, 10591}, {6502, 13911}, {6691, 19861}, {6702, 13273}, {6735, 12513}, {6745, 24391}, {6796, 10265}, {6833, 7686}, {6848, 12688}, {6883, 11507}, {6891, 14110}, {6964, 17634}, {7080, 24477}, {7181, 9312}, {7195, 24798}, {7235, 24530}, {7580, 10395}, {7742, 11499}, {7989, 9579}, {8162, 21625}, {8543, 17534}, {8581, 8732}, {9436, 24796}, {9708, 22759}, {9947, 11575}, {9957, 10072}, {10085, 13226}, {10090, 12619}, {10179, 10586}, {10366, 11347}, {10371, 14829}, {10571, 17749}, {10592, 24470}, {10624, 11238}, {10786, 12675}, {10827, 18990}, {12647, 24928}, {12678, 18242}, {12831, 20400}, {12943, 19925}, {13161, 17595}, {13373, 15867}, {13883, 18995}, {13893, 19028}, {13912, 19038}, {13936, 18996}, {13947, 19027}, {13975, 19037}, {15016, 18397}, {15228, 18514}, {16371, 17647}, {17078, 17090}, {17124, 21674}, {17734, 24046}, {18525, 21578}, {18965, 18991}, {18966, 18992}, {22791, 23708}, {24174, 24789}, {24836, 25351}, {24880, 24881}
X(24915) lies on these lines: {2, 85}, {7, 24944}, {1434, 16589}, {7235, 24530}
X(24916) lies on these lines: {1, 24904}, {2, 11}, {81, 22156}, {110, 8286}, {125, 24624}, {2607, 3120}, {5445, 24880}, {8287, 15059}, {24881, 24883}, {24888, 24896}
X(24917) lies on these lines: {2, 101}, {21253, 24619}
X(24918) lies on these lines: {2, 99}, {1015, 24582}, {3125, 15903}, {6710, 20331}, {6714, 7483}, {7874, 17674}, {24882, 24889}, {24891, 24925}
X(24919) lies on these lines: {2, 37}, {244, 24520}, {1654, 16752}, {17326, 24790}, {24880, 24922}, {24883, 24923}, {24884, 24899}
X(24920) lies on these lines: {2, 900}, {1698, 21714}, {2642, 17398}, {13277, 24938}
X(24921) lies on these lines: {2, 649}, {656, 24893}, {4728, 24961}, {21124, 24899}
X(24922) lies on these lines: {2, 7}, {17052, 19308}, {17276, 24957}, {24530, 24890}, {24880, 24919}
X(24923) lies on these lines: {1, 24530}, {2, 38}, {6, 20746}, {10, 24437}, {1500, 4261}, {1698, 21699}, {3122, 24342}, {3670, 17322}, {4022, 19856}, {4272, 4658}, {4657, 24046}, {5009, 19329}, {5045, 5956}, {5333, 20966}, {9050, 17749}, {17278, 24880}, {17728, 24896}, {22174, 24697}, {22220, 24958}, {23928, 24443}, {24174, 24372}, {24883, 24919}
X(24924) lies on these lines: {2, 661}, {649, 4106}, {650, 4379}, {656, 24893}, {693, 1635}, {1125, 4761}, {1638, 16892}, {1698, 4160}, {2254, 4874}, {2487, 3700}, {2490, 6546}, {2527, 23729}, {3676, 21115}, {3763, 9013}, {3835, 4979}, {3911, 4077}, {4024, 17069}, {4120, 4897}, {4191, 23864}, {4367, 21052}, {4378, 14430}, {4382, 4394}, {4383, 18199}, {4406, 14408}, {4467, 4931}, {4763, 17494}, {4776, 4932}, {4789, 4838}, {4804, 9508}, {4928, 20295}, {6590, 7658}, {6608, 15584}, {8689, 24720}, {11068, 21183}, {11123, 12072}, {14425, 21116}, {14838, 16754}, {21828, 24782}
X(24925) lies on these lines: {2, 3}, {5433, 24880}, {15325, 24883}, {24891, 24918}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28442.
X(24926) lies on these lines: {1,3}, {10,5444}, {79,21578}, {80,1125}, {214,3918}, {498,7967}, {515,5443}, {551,10572}, {944,7951}, {950,16173}, {1387,10543}, {1476,5424}, {1479,3622}, {1483,5432}, {2320,5248}, {2975,4067}, {3583,5901}, {3584,10944}, {3585,15950}, {3616,3822}, {3636,5441}, {3655,11375}, {3678,4511}, {3723,5356}, {3754,4881}, {3897,5251}, {3901,11194}, {4188,21398}, {4302,10595}, {4324,22791}, {4848,5442}, {5288,22836}, {5445,10165}, {5494,6126}, {5557,12563}, {5727,15079}, {5731,10483}, {5836,15015}, {6284,10283}, {7294,11545}, {7972,10039}, {9897,9956}, {11715,12691}, {18393,18481}, {18493,18514}
X(24926) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,11009}, {1,56,5425}, {1,484,11011}, {1,1385,36}, {1,1420,18398}, {1,2646,3746}, {1,3576,5903}, {1,3612,5697}, {1,5010,1482}, {1,7280,2099}, {1,11010,10222}, {1,21842,5563}, {3,11009,3245}, {1385,11567,3}, {1420,18398,5563}, {2646,15178,1}, {3612,5697,35}, {3746,14803,35}, {10039,13607,7972}, {11011,13624,484}, {18398,21842,1420}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28442.
X(24927) lies on these lines: {1,3}, {119,1125}, {140,6735}, {214,13607}, {355,10200}, {551,12608}, {631,12648}, {944,6944}, {952,17614}, {1483,5440}, {1519,5901}, {2320,5553}, {3616,6893}, {3897,5084}, {4308,6827}, {4311,7491}, {5554,7967}, {5731,10531}, {5777,12773}, {5836,12737}, {5886,6256}, {6265,12675}, {6713,10039}, {6848,10586}, {6882,10106}, {6923,11373}, {9956,12751}, {10165,10915}, {10942,17527}
X(24927) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3359,1482}, {1,3576,11248}, {1,5193,942}, {1,7987,12703}, {1,16203,10202}, {1,16209,7982}, {1385,15178,2646}, {1388,22768,1}, {10246,16203,1}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28442.
X(24928) lies on these lines: {1,3}, {4,4308}, {5,10106}, {7,10595}, {8,17567}, {10,6691}, {11,18480}, {12,11230}, {21,15179}, {30,4311}, {37,5053}, {58,18211}, {72,17624}, {104,1476}, {145,5440}, {210,5288}, {214,3244}, {219,1732}, {226,5901}, {355,3086}, {376,9785}, {381,9613}, {382,9614}, {388,5886}, {392,2975}, {404,10914}, {405,11035}, {452,3487}, {474,3872}, {495,1125}, {496,515}, {497,18481}, {499,5252}, {519,8256}, {529,551}, {550,10624}, {938,6049}, {944,5722}, {946,1387}, {952,1210}, {956,5044}, {958,12128}, {960,8666}, {997,12513}, {998,3445}, {1056,3436}, {1058,5731}, {1064,4322}, {1066,1201}, {1100,4266}, {1108,1731}, {1191,3157}, {1193,5399}, {1317,22935}, {1386,8679}, {1412,15952}, {1475,17439}, {1478,9955}, {1519,22792}, {1538,6256}, {1621,10569}, {1656,9578}, {1657,9580}, {1699,9655}, {1706,16417}, {1737,10944}, {1743,22147}, {1776,5887}, {1828,1870}, {1836,4317}, {1837,10072}, {1872,15500}, {2257,20818}, {2320,3296}, {2649,9444}, {2771,10074}, {2841,11700}, {3028,11699}, {3035,10915}, {3073,9363}, {3218,5330}, {3419,10529}, {3485,6930}, {3486,3655}, {3555,4511}, {3582,17606}, {3585,12764}, {3600,5603}, {3623,11041}, {3633,3689}, {3636,12577}, {3656,4295}, {3680,17573}, {3752,15854}, {3753,4861}, {3813,17647}, {3871,4881}, {3877,3916}, {3884,4640}, {3885,4188}, {3898,5267}, {3911,5690}, {3927,15829}, {3940,6762}, {4002,17531}, {4253,6603}, {4292,22791}, {4293,12699}, {4297,15171}, {4298,13464}, {4299,12701}, {4304,15172}, {4314,15170}, {4848,5844}, {4853,9709}, {5030,21872}, {5083,19907}, {5176,17619}, {5248,10179}, {5265,5657}, {5270,17605}, {5284,14150}, {5290,9624}, {5427,22937}, {5433,10039}, {5434,12047}, {5435,12245}, {5438,12629}, {5533,18976}, {5542,5625}, {5550,8164}, {5691,9669}, {5727,18526}, {5836,22837}, {5882,11019}, {6147,10283}, {6284,21578}, {6377,9434}, {6735,13747}, {6738,13607}, {6797,12737}, {6921,12648}, {6948,12700}, {6970,18391}, {7320,10299}, {7354,22793}, {8227,9654}, {8583,9708}, {9259,16583}, {9581,18525}, {9612,18493}, {9623,16408}, {9856,12114}, {10090,20586}, {10572,18527}, {10573,17728}, {10593,19925}, {11036,17576}, {11037,11111}, {11194,12514}, {12573,20330}, {12616,20418}, {12650,19541}, {13407,15950}, {14100,16132}, {15180,15446}, {17100,17652}, {17644,18908}
X(24928) = midpoint of X(i) and X(j) for these {i,j}: {1,56}, {46,2098}, {4299,12701}, {4311,12053}, {10074,12740}, {10090,20586}
X(24928) = reflection of X(i) in X(j) for these {i,j}: {10,6691}, {1329,1125}, {9957,20789}
X(24928) = complement of isogonal conjugate of X15617
X(24928) = complement of complement of X(36977)
X(24928) = X(24)-of-incircle-circles-triangle
X(24928) = inner-Johnson-to-ABC similarity image of X(18480)
X(24928) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,9957}, {1,35,5919}, {1,36,3057}, {1,46,2098}, {1,57,1482}, {1,65,10222}, {1,999,942}, {1,1319,1385}, {1, 1385, 24929}, {1,1388,15178}, {1,1420,3}, {1,3304,5045}, {1,3337,11009}, {1,3338,2099}, {1,3339,16200}, {1,3340,10247}, {1,3361,7982}, {1,3576,3295}, {1,3601,6767}, {1,3612,3303}, {1,5563,65}, {1,5902,11011}, {1,5903,5048}, {1,7373,5049}, {1,13462,40}, {1,15803,7962}, {1,21842,2646}, {3,1420,5126}, {4,11373,7743}, {36,3057,3579}, {56,2098,46}, {499,5252,9956}, {938,6049,7967}, {944,14986,5722}, {946,4315,18990}, {956,19861,5044}, {1056,3616,11374}, {1125,5795,17527}, {1319,2646,21842}, {1319,20323,1}, {1387,18990,946}, {1388,3304,1}, {1478,11376,9955}, {2646,21842,1385}, {3086,3476,355}, {3337,11009,65}, {4861,5253,3753}, {5045,9940,16218}, {5045,15178,1}, {5048,5903,11278}, {5126,9957,3}, {5433,10039,11231}, {5563,11009,3337}, {5708,10247,3340}, {6583,11567,10222}, {6583,15178,11567}, {7373,10246,1}, {7962,15803,12702}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28477.
X(24929) lies on these lines: {1,3}, {2,3419}, {4,4313}, {5,950}, {7,376}, {8,5791}, {9,3940}, {10,6675}, {11,6881}, {12,6841}, {20,3487}, {21,72}, {28,1255}, {30,226}, {33,7497}, {37,101}, {41,16601}, {42,8731}, {63,16370}, {73,500}, {74,934}, {78,405}, {79,4324}, {80,3584}, {100,3753}, {104,2346}, {105,20219}, {140,1210}, {142,214}, {145,3897}, {169,4258}, {191,3962}, {200,9708}, {210,5251}, {212,5398}, {228,859}, {329,11111}, {355,3085}, {377,3824}, {378,1876}, {381,3586}, {382,9612}, {386,1104}, {388,4305}, {390,5603}, {392,1621}, {404,5439}, {443,1058}, {474,4855}, {495,515}, {496,1125}, {497,5886}, {498,1837}, {516,8255}, {518,993}, {519,5745}, {548,4114}, {549,3911}, {550,3982}, {553,8703}, {579,1100}, {582,1451}, {610,3247}, {631,938}, {674,1386}, {758,4640}, {855,21319}, {894,4234}, {908,11113}, {910,4262}, {912,6914}, {936,2900}, {944,5787}, {946,4314}, {951,7100}, {954,971}, {956,3870}, {958,3811}, {960,5248}, {975,7535}, {976,10448}, {991,6610}, {995,1279}, {997,1001}, {1000,2320}, {1006,5728}, {1026,16443}, {1043,5295}, {1056,5731}, {1064,2293}, {1071,6906}, {1212,4251}, {1400,14636}, {1426,7414}, {1447,13634}, {1464,4337}, {1478,17718}, {1479,9955}, {1656,9581}, {1657,9579}, {1699,9668}, {1737,5432}, {1766,21848}, {1770,3649}, {1785,7510}, {1817,17019}, {1829,14017}, {1836,4302}, {1858,5694}, {1864,7489}, {1870,4219}, {1892,18533}, {1895,7531}, {2003,23071}, {2256,3211}, {2269,5755}, {2271,16968}, {2329,3991}, {2475,11015}, {2649,9367}, {2650,9340}, {2687,14733}, {2771,10058}, {2807,11700}, {2808,15730}, {2975,3555}, {2999,16485}, {3024,11699}, {3058,15950}, {3086,6989}, {3146,5714}, {3149,5806}, {3158,9623}, {3189,19843}, {3218,17549}, {3244,5855}, {3306,16371}, {3475,4293}, {3476,3655}, {3485,4294}, {3522,11036}, {3524,5435}, {3525,5704}, {3534,4654}, {3560,5777}, {3582,5444}, {3583,17605}, {3585,5441}, {3622,6904}, {3636,12436}, {3647,4067}, {3656,10385}, {3671,16137}, {3678,5302}, {3679,3689}, {3683,5692}, {3693,16788}, {3697,4420}, {3720,16056}, {3752,4256}, {3820,6745}, {3868,3916}, {3871,10914}, {3874,5267}, {3876,16865}, {3892,15570}, {3920,4224}, {3927,11523}, {3984,15650}, {3996,16821}, {4134,15481}, {4276,18165}, {4297,18990}, {4299,10404}, {4309,12701}, {4428,5289}, {4649,5429}, {4652,11520}, {4857,5443}, {4870,18393}, {4999,10916}, {5088,14828}, {5110,20227}, {5175,6856}, {5218,18391}, {5249,11112}, {5250,5730}, {5252,10056}, {5256,21483}, {5262,7523}, {5281,5657}, {5287,11347}, {5290,9655}, {5325,15673}, {5341,16777}, {5396,14547}, {5428,10122}, {5434,21578}, {5437,16417}, {5438,16408}, {5450,12675}, {5453,13754}, {5529,17123}, {5687,19860}, {5691,9654}, {5705,12625}, {5720,6913}, {5727,5790}, {5761,5812}, {5768,6935}, {5771,5844}, {5794,10198}, {5804,6927}, {5836,8715}, {5882,6245}, {5887,12711}, {5901,12053}, {5927,6912}, {6175,9963}, {6261,9856}, {6284,12047}, {6326,14100}, {6684,6738}, {6692,17564}, {6700,17527}, {6705,13607}, {6734,7483}, {6762,7160}, {6796,7686}, {6909,10167}, {6910,12649}, {6920,9844}, {7098,16139}, {7308,16857}, {7354,13407}, {7428,22345}, {7508,17010}, {7520,9538}, {8068,12743}, {8227,9669}, {9578,18525}, {9605,16780}, {9614,18493}, {9619,16781}, {9785,10595}, {9946,15558}, {9947,17857}, {10039,10950}, {10165,11019}, {10175,12019}, {10176,15254}, {10304,21454}, {10360,18931}, {10386,10624}, {10436,19276}, {10592,19925}, {11246,11551}, {11281,12609}, {11552,15228}, {11712,15746}, {11715,12735}, {12109,15489}, {12114,12260}, {12512,12563}, {12514,12635}, {12575,13464}, {12672,21740}, {12690,17530}, {13726,19767}, {15008,15299}, {15677,17484}, {15837,18412}, {16054,16826}, {16457,19859}, {16583,18755}, {16843,19753}, {17525,17781}, {17558,20007}, {17576,20214}, {17637,22936}, {18482,21617}, {19520,19861}
X(24929) = complement of X3419
X(24929) = midpoint of X(i) and X/j) for these {i,j}: {1,55}, {72,16465}, {226,4304}, {954,7675}, {956,3870}, {1012,18446}, {1836,4302}, {2099,5119}, {10058,12739}
X(24929) = reflection of X(i) in X(j) for these {i,j}: {10,6690}, {226,5719}, {495,13405}, {942,11018}, {2886,1125}, {5173,5045}, {18407,9955}
X(24929) = X(22)-of-incircle-circles-triangle
X(24929) = outer-Johnson-to-ABC similarity image of X(18480)
X(24929) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,942}, {1,35,65}, {1,36,354}, {1,56,5045}, {1,57,15934}, {1,165,11529}, {1,484,5425}, {1,999,5049}, {1, 1385, 24928}, {1,1420,7373}, {1,1697,1482}, {1,2646,1385}, {1,3057,10222}, {1,3295,9957}, {1,3576,999}, {1,3601,3}, {1,3612,56}, {1,3746,3057}, {1,5010,5902}, {1,5119,2099}, {1,5563,17609}, {1,5697,11011}, {1,7280,18398}, {1,7962,10247}, {1,7987,3333}, {1,9819,16200}, {1,10383,18443}, {1,10389,6767}, {1,13384,10246}, {1,14793,5570}, {1,14800,13751}, {1,15803,11518}, {1,21842,20323}, {1,22766,16193}, {1,22768,13373}, {2,3488,5722}, {3,57,5122}, {3,1482,5709}, {3,5708,15803}, {3,10246,18443}, {3,10247,2095}, {3,15934,57}, {3,18443,11227}, {4,5703,11374}, {8,6857,5791}, {12,10543,10572}, {12,10572,18480}, {35,65,3579}, {35,5425,484}, {55,2099,5119}, {55,5172,35}, {56,3612,13624}, {57,15934,942}, {78,405,5044}, {140,12433,1210}, {388,4305,18481}, {484,5425,65}, {497,5886,7743}, {498,1837,9956}, {550,6147,4292}, {936,5436,11108}, {942,5122,57}, {944,6847,5787}, {946,4314,15171}, {950,13411,5}, {999,3576,5126}, {1058,3616,11373}, {1319,3748,1}, {1479,11375,9955}, {1621,4511,392}, {1737,5432,11231}, {2646,3748,1319}, {3085,3486,355}, {3485,4294,12699}, {3576,6282,3}, {3586,5219,381}, {3601,13384,10383}, {3649,15338,1770}, {3868,4189,3916}, {3940,16418,9}, {4297,21620,18990}, {4313,5703,4}, {4420,5260,3697}, {5010,5902,1155}, {5045,13624,56}, {5049,5126,999}, {5119,13462,3428}, {5248,22836,960}, {5697,11011,11278}, {5708,11518,942}, {5720,6913,10157}, {5731,10578,1056}, {5761,6868,5812}, {5901,15172,12053}, {6261,11496,9856}, {6284,12047,22793}, {6767,10246,1}, {6767,10383,942}, {6909,18444,10167}, {6935,7967,5768}, {10165,11019,15325}, {10386,22791,10624}, {10389,13384,1}, {11230,18527,11}, {11518,15803,5708}, {14547,22350,5396}
See Antreas Hatzipolakis, Angel Montesdeoca, and Peter Moses, Hyacinthos 28446 and Hyacinthos 28449.
X(24930) lies on cubic K818 and these lines: {4,74}, {122,6723}, {402,5972}, {10745,23515}, {12295,23240}, {16111,22337}, {16163,23239}
X(24930) = midpoint of X(i) and X(j) for these {i,j}: {107, 125}, {12295, 23240}, {16111, 22337}
X(24930) = reflection of X(i) in X(j) for these {i,j}: {122, 6723}, {5972, 6716}}
X(24930) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 14847, 107)
Collineation mappings involving Gemini triangle 11: X(24931)-X(24962)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 11, as in centers X(24931)-X(24962). Then
m(X) = a (a^2 + a b + a c + b c) x + (a^2 b + 2 a b^2 + b^3 + a^2 c + 3 a b c + 2 b^2 c + a c^2 + b c^2) y + (a^2 c + 2 a c^2 + c^3 + a^2 b + 3 a b c + 2 b c^2 + a b^2 + b^2 c) z : : ,
and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, October 11, 2018)
X(24931) lies on these lines: {1, 2}, {56, 24912}, {58, 1211}, {261, 1078}, {284, 1213}, {333, 6693}, {405, 759}, {1010, 3454}, {1324, 16287}, {3647, 24697}, {3752, 5956}, {3763, 16408}, {4053, 11374}, {4205, 4653}, {4256, 13728}, {4658, 6703}, {5743, 17698}, {7308, 24933}, {7832, 17307}, {11263, 24342}, {17327, 19273}, {19933, 19936}, {24932, 24954}, {24934, 24938}, {24941, 24950}, {24949, 24952}
X(24932) lies on these lines: {1, 21670}, {2, 3}, {24931, 24954}, {24935, 24942}, {24936, 24955}
X(24933) lies on these lines: {1, 21671}, {2, 3}, {943, 21015}, {3624, 24937}, {5506, 18598}, {5722, 24883}, {5747, 25063}, {7308, 24931}, {9581, 24880}, {11374, 24936}, {15803, 24884}, {24942, 24956}
X(24934) lies on these lines: {1, 21672}, {2, 3}, {1170, 24909}, {5432, 24880}, {5433, 24912}, {15325, 24936}, {24931, 24938}
X(24935) lies on these lines: {1, 20654}, {2, 6}, {1125, 21076}, {1333, 3454}, {3061, 7110}, {3943, 24946}, {4053, 11374}, {8818, 24275}, {17275, 24880}, {17282, 24894}, {17362, 24883}, {24932, 24942}, {24941, 24943}, {24944, 24947}, {24957, 24958}
X(24936) lies on these lines: {1, 2}, {3, 229}, {21, 17056}, {58, 15674}, {81, 6675}, {581, 6884}, {846, 14450}, {1211, 17557}, {1330, 17588}, {1962, 24161}, {2475, 4653}, {3247, 24937}, {3936, 11110}, {4197, 19765}, {4648, 6910}, {5047, 5718}, {5224, 19334}, {5433, 24909}, {9782, 17596}, {11374, 24933}, {11375, 17080}, {13408, 21161}, {15325, 24934}, {15670, 16948}, {16342, 18134}, {17245, 17531}, {18139, 19270}, {24932, 24955}, {24944, 24952}, {24945, 24962}
X(24937) lies on these lines: {1, 21675}, {2, 7}, {5, 572}, {6, 24880}, {37, 24160}, {86, 17052}, {284, 442}, {594, 5719}, {1100, 3017}, {1449, 24883}, {1743, 24902}, {1761, 11263}, {1765, 6852}, {1901, 4877}, {3247, 24936}, {3624, 24933}, {4053, 11374}, {4675, 24890}, {5044, 21873}, {5816, 6858}, {7110, 21921}, {8818, 11108}, {24944, 24957}, {24947, 24958}
X(24938) lies on these lines: {1, 21676}, {2, 11}, {37, 24186}, {4010, 24959}, {13277, 24920}, {24931, 24934}
X(24939) lies on these lines: {1, 21679}, {2, 3}
X(24940) lies on these lines: {1, 21680}, {2, 3}
X(24941) lies on these lines: {1, 20655}, {2, 31}, {2206, 21245}, {24931, 24950}, {24935, 24943}
X(24942) lies on these lines: {1, 21681}, {2, 32}, {24932, 24935}, {24933, 24956}, {24945, 24951}
X(24943) lies on these lines: {1, 2}, {31, 141}, {55, 3763}, {69, 2308}, {210, 17357}, {748, 1211}, {756, 17279}, {902, 3619}, {968, 17306}, {1011, 1626}, {1150, 6679}, {1213, 2280}, {1468, 17698}, {1962, 4657}, {2187, 7499}, {2223, 7822}, {2887, 24552}, {2895, 16468}, {3052, 21358}, {3416, 17469}, {3620, 21747}, {3662, 4418}, {3683, 17237}, {3744, 3844}, {3745, 17231}, {3773, 3891}, {3775, 5278}, {3923, 17184}, {3930, 17303}, {3989, 17776}, {4046, 17366}, {4365, 19785}, {4676, 4683}, {5743, 17125}, {6535, 17286}, {6703, 9345}, {21020, 24789}, {24935, 24941}
X(24944) lies on these lines: {1, 21699}, {2, 37}, {7, 24915}, {86, 16589}, {1268, 1500}, {1655, 16709}, {3616, 24519}, {3624, 24945}, {17038, 24520}, {24935, 24947}, {24936, 24952}, {24937, 24957}
X(24945) lies on these lines: {1, 21700}, {2, 39}, {1045, 3216}, {3624, 24944}, {24936, 24962}, {24942, 24951}
X(24946) lies on these lines: {1, 2}, {1211, 15674}, {3454, 15680}, {3943, 24935}, {6740, 6919}, {20084, 24850}
X(24947) lies on these lines: {1, 2}, {1931, 24956}, {24935, 24944}, {24937, 24958}
X(24948) lies on these lines: {1, 21727}, {2, 650}, {37, 17161}, {661, 1019}, {1491, 4057}, {3709, 4467}, {4010, 8043}, {4789, 25084}
X(24949) lies on these lines: {2, 3}, {24931, 24952}
X(24950) lies on these lines: {2, 3}, {3624, 24955}, {24931, 24941}
X(24951) lies on these lines: {2, 3}, {24942, 24945}
X(24952) lies on these lines: {1, 21673}, {2, 7}, {24931, 24949}, {24936, 24944}
X(24953) lies on these lines: {1, 5791}, {2, 12}, {3, 3925}, {5, 5251}, {8, 6690}, {9, 11375}, {10, 2646}, {11, 405}, {21, 2886}, {36, 8728}, {48, 1213}, {55, 5082}, {63, 3649}, {65, 5745}, {72, 354}, {100, 9710}, {120, 19314}, {140, 355}, {149, 15676}, {210, 13411}, {377, 15326}, {404, 3826}, {442, 993}, {443, 5204}, {452, 10896}, {468, 5130}, {495, 5258}, {496, 5259}, {497, 2894}, {498, 9708}, {499, 11108}, {590, 19026}, {615, 19025}, {631, 4413}, {632, 10942}, {908, 5302}, {946, 3683}, {956, 10198}, {960, 15950}, {988, 24789}, {1001, 1259}, {1212, 22070}, {1376, 6910}, {1377, 13958}, {1378, 13901}, {1453, 17723}, {1468, 17056}, {1470, 19520}, {1479, 16418}, {1621, 3813}, {1656, 10526}, {1834, 10448}, {1837, 5705}, {2260, 17398}, {2352, 19858}, {2478, 7173}, {2550, 5217}, {2829, 6937}, {3035, 6224}, {3058, 5248}, {3086, 4423}, {3090, 10894}, {3333, 3624}, {3419, 10543}, {3428, 6824}, {3485, 5273}, {3525, 10786}, {3526, 6713}, {3560, 15908}, {3616, 12635}, {3634, 13747}, {3698, 6684}, {3763, 12587}, {3782, 24161}, {3816, 5047}, {3820, 10954}, {3829, 16858}, {3841, 5267}, {3868, 11281}, {3869, 18253}, {3901, 16137}, {3916, 11246}, {3983, 6745}, {4026, 16342}, {4299, 17528}, {4302, 17571}, {4429, 19278}, {4512, 12701}, {4652, 5880}, {4679, 5812}, {4699, 7891}, {4870, 5325}, {4972, 16347}, {4995, 5687}, {5044, 16193}, {5054, 18518}, {5067, 10599}, {5070, 11929}, {5084, 10953}, {5094, 11391}, {5129, 10589}, {5177, 12943}, {5219, 5234}, {5220, 5550}, {5221, 5744}, {5225, 11106}, {5231, 5436}, {5247, 5718}, {5316, 17590}, {5326, 10955}, {5584, 6847}, {5719, 5904}, {5837, 11011}, {5842, 6875}, {5886, 12704}, {6147, 6763}, {6174, 9709}, {6832, 7958}, {6837, 7965}, {6852, 7680}, {6853, 18242}, {6856, 10895}, {6861, 11249}, {6878, 10785}, {6889, 12114}, {6892, 10310}, {6914, 11826}, {6920, 7681}, {6976, 10893}, {7484, 10830}, {7515, 23207}, {7561, 19857}, {7786, 12933}, {7807, 16819}, {7808, 10795}, {7914, 10872}, {9669, 16866}, {10021, 16139}, {10200, 16842}, {10523, 17527}, {10944, 24987}, {11111, 12953}, {11238, 17561}, {11499, 21155}, {11680, 16865}, {11904, 15184}, {12572, 17605}, {12699, 16617}, {13181, 14061}, {13214, 15059}, {15338, 16370}, {16478, 17726}, {17057, 18357}, {17560, 20988}, {17566, 19877}, {17698, 19863}, {19273, 19784}, {23922, 24432}, {24384, 24435}, {24931, 24934}
X(24954) lies on these lines: {1, 3820}, {2, 65}, {3, 4679}, {9, 5433}, {10, 2098}, {11, 936}, {12, 8583}, {55, 6700}, {56, 3452}, {63, 6691}, {72, 10200}, {78, 3816}, {210, 3086}, {377, 5087}, {388, 5328}, {404, 24703}, {405, 22768}, {443, 17605}, {474, 1836}, {496, 4863}, {499, 5044}, {631, 3683}, {908, 10404}, {946, 4413}, {956, 1125}, {975, 17723}, {978, 17720}, {997, 1837}, {1058, 3689}, {1155, 17567}, {1319, 2551}, {1329, 5252}, {1376, 12701}, {1388, 5795}, {1408, 24556}, {1698, 3628}, {1770, 16417}, {2099, 8582}, {2646, 5084}, {3035, 5250}, {3303, 6745}, {3305, 4999}, {3333, 3624}, {3419, 3825}, {3421, 20323}, {3476, 8165}, {3616, 11260}, {3649, 5437}, {3678, 10199}, {3679, 11373}, {3698, 5603}, {3740, 10527}, {3782, 11512}, {3872, 9711}, {3925, 6769}, {4005, 24477}, {4193, 5794}, {4423, 13411}, {4640, 6921}, {4662, 10529}, {4860, 15650}, {5057, 17572}, {5204, 12572}, {5221, 6692}, {5233, 10371}, {5289, 24982}, {5293, 17721}, {5438, 6284}, {5880, 17531}, {5919, 7080}, {6001, 6967}, {6910, 15254}, {6926, 12688}, {6944, 14110}, {6983, 7686}, {7288, 18228}, {10179, 10528}, {11230, 19854}, {12047, 16408}, {12053, 20103}, {12514, 13747}, {12609, 16862}, {15481, 18230}, {17303, 21801}, {17556, 17647}, {24931, 24932}, {24936, 24944}
X(24955) lies on these lines: {1, 21054}, {2, 11}, {110, 8287}, {1125, 21098}, {1290, 5954}, {3624, 24950}, {4092, 6742}, {5972, 24624}, {7741, 24904}, {8286, 15059}, {18004, 24957}, {24932, 24936}
X(24956) lies on these lines: {1, 21711}, {2, 99}, {39, 16614}, {1015, 24542}, {1125, 4115}, {1213, 15670}, {1931, 24947}, {3731, 4370}, {5283, 22398}, {5475, 11346}, {14148, 16711}, {16430, 21004}, {19862, 24070}, {23897, 24902}, {24074, 24187}, {24933, 24942}
X(24957) lies on these lines: {1, 21043}, {2, 45}, {115, 662}, {645, 14061}, {1125, 21089}, {1698, 21047}, {4092, 11725}, {6543, 17398}, {13178, 17467}, {17276, 24922}, {18004, 24955}, {24935, 24958}, {24937, 24944}
X(24958) lies on these lines: {1, 21713}, {2, 37}, {3923, 20369}, {7760, 17379}, {21020, 24746}, {22167, 24520}, {22220, 24923}, {24935, 24957}, {24937, 24947}
X(24959) lies on these lines: {1, 21714}, {2, 900}, {1213, 24506}, {4010, 24938}, {6707, 24287}, {8062, 9013}, {14321, 24961}, {18004, 24955}
X(24960) lies on these lines: {1, 21719}, {2, 647}, {523, 24961}, {649, 21260}, {1021, 1698}, {1213, 9404}, {2523, 4391}, {2605, 21721}, {3700, 17303}, {20316, 22383}
X(24961) lies on these lines: {1, 21721}, {2, 650}, {523, 24960}, {4024, 17303}, {4728, 24921}, {14321, 24959}, {18147, 21438}
X(24962) lies on these lines: {1, 21725}, {2, 668}, {99, 16592}, {14061, 16613}, {24519, 24880}, {24936, 24945}
X(24963) lies on these lines: {1, 2}, {17275, 24967}
X(24964) lies on this line: {2, 3}
X(24965) lies on this line: {2, 3}
X(24966) lies on this line: {2, 3}
X(24967) lies on these lines: {2, 6}, {4969, 24968}, {17275, 24963}
X(24968) lies on these lines: {1, 2}, {1051, 9782}, {4969, 24967}
X(24969) lies on these lines: {1, 2}
X(24970) lies on this line: {2, 3}
X(24971) lies on this line: {2, 3}
X(24972) lies on this line: {2, 3}
4th intersections of inconics and the circumcircles of their polar triangles: X(24973)-X(24980)
Let E be an inconic (other than the incircle) with Brianchon point (perspector) P. Let A'B'C' be the polar triangle of E, which is the cevian triangle of P. Let O* be the circumcircle of A'B'C' (i.e., the cevian circle of P). E and O* intersect in 4 points: A', B', C' and a 4th intersection which is a triangle center. The appearance of (E,i,O*,j) in the following list means that inconic E with perspector X(i) intersects circle O* (other than at A', B', C') at X(j):
(Brocard inellipse, 6, symmedial circle, 24973)
(incentral inellipse, 1, incentral circle, 23063)
(Kiepert parabola, 99, 2nd Steiner circle, 24974)
(Lemoine inellipse, 598, 3rd Lemoine circle, 20383)
(MacBeath inconic, 264, MacBeath circle, 24977)
(Mandart inellipse, 8, Mandart circle, 11)
(orthic inconic, 4, nine-point circle, 125)
(Steiner inellipse, 2, nine-point circle, 115)
(Yff parabola, 190, cevian circle of X(190), 24979)
Contributed by Randy Hutson, October 11, 2018.
X(24973) lies on the Brocard inellipse, the symmedial circle, and these lines: {511, 9482}, {323, 5104}, {805, 8041}, {2679, 3005}
X(24974) lies on the Kiepert parabola, the 2nd Steiner circle, and these lines: {476, 8029}, {523, 14683}, {669, 11641}, {868, 1649}, {11123, 14731}
X(24975) lies on these lines: {2, 6}, {3, 18121}, {30, 14356}, {114, 6593}, {140, 18114}, {297, 14590}, {620, 2492}, {868, 5467}, {1576, 6033} et al
X(24975) = complement of isotomic conjugate of isogonal conjugate of X(2088)
X(24975) = complement of isotomic conjugate of X(39295)
X(24975) = complement of complement of X(2407)
X(24976) lies on these lines: {99, 12074}, {351, 523}, {620, 2793}, {1499, 11616}, {6088, 9125}
X(24977) lies on the MacBeath inconic, the MacBeath circle, and these lines: {3, 1291}, {23, 2967}, {2972, 3258}, {12091, 18403}, {13409, 14731}
X(24978) lies on these lines: {83, 14223}, {297, 525}, {523, 5926}, {526, 12236}, {620, 2492}, {648, 16813}, {1263, 10264}, {1637, 6334}, {2394, 18366}, {6132, 8151}, {9517, 16230}, {10214, 11557}, {15412, 20577}
X(24979) lies on the Yff parabola, the cevian circle of X(190), and these lines: {514, 20096}, {927, 6545}, {1566, 3259}, {3239, 17777}
X(24980) lies on these lines: {1, 2}, {226, 5091}, {597, 8255}, {908, 3573}, {1083, 3452}, {3035, 6366}, {3310, 23988}, {3675, 3911}, {15252, 24003}
X(24980) = complement of crossdifference of X(109) and X(649)
X(24980) = complement of isotomic conjugate of X(39293)
X(24980) = complement of complement of X(2398)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28452.
X(24981) lies on these lines: {2,98}, {4,6053}, {23,5965}, {51,1353}, {74,3528}, {113,137}, {115,20976}, {155,382}, {159,2930}, {265,3851}, {373,8550}, {539,10540}, {541,11820}, {550,5562}, {1112,1843}, {1495,3564}, {1503,3292}, {1511,3530}, {2502,6388}, {2777,3529}, {2931,9908}, {2948,3632}, {3124,5477}, {3167,11550}, {3233,6070}, {3544,14644}, {3631,5181}, {3636,11720}, {3855,7687}, {4576,14928}, {5079,14643}, {5191,14981}, {5448,18430}, {5655,12902}, {5907,10619}, {6030,15108}, {6154,8674}, {6329,6593}, {6699,15720}, {6791,20998}, {7706,18445}, {7984,20057}, {9155,10991}, {10116,18350}, {10264,14869}, {10299,12317}, {10620,15688}, {11008,11061}, {11225,13595}, {11422,19130}, {11441,21659}, {11693,22250}, {11737,11801}, {11800,12824}, {12295,15687}, {12310,20850}, {12412,22109}, {13366,18583}, {13417,14984}, {13431,14449}, {13605,15808}, {14389,18553}, {15036,15715}, {15039,20397}, {15040,15700}, {15303,20583}
X(24981) = midpoint of X(i) and X(j) for these {i,j}: {12121,12308}, {12383,14094}
X(24981) = reflection of X(i) in X(j) for these {i,j}: {4,6053}, {113,5609}, {125,110}, {265,16534}, {3448,5972}, {6070,3233}, {10990,16163}, {12317,20417}, {15063,399}, {16003,1511}
X(24981) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {110,125,5642}, {110,3448,5972}, {3448,5972,125}, {6723,9140,125}, {9143,14683,110}, {12317,15035,20417}
Collineation mappings involving Gemini triangle 13: X(24982)-X(25024)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 13, as in centers X(24982)-X(25024). Then
m(X) = (-a^2 b + b^3 - a^2 c - b^2 c - b c^2 + c^3) x + b(a^2 - b^2 - 2 a c + c ^2) y + c(a^2 - c^2 -2a b + b^2) z : : ,
and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, October 12, 2018)
X(24982) lies on these lines: {1, 2}, {3, 1512}, {4, 3359}, {5, 1519}, {9, 5555}, {11, 5836}, {12, 3812}, {20, 10270}, {21, 2077}, {40, 2478}, {57, 3436}, {63, 1788}, {65, 908}, {72, 3820}, {80, 14803}, {84, 377}, {100, 950}, {119, 125}, {142, 7679}, {165, 6872}, {210, 9711}, {226, 11681}, {329, 8165}, {354, 12607}, {355, 474}, {388, 3306}, {392, 5690}, {404, 515}, {405, 11248}, {443, 5818}, {495, 5439}, {496, 10914}, {516, 5046}, {517, 4187}, {518, 21031}, {942, 17757}, {944, 17567}, {946, 4193}, {952, 17614}, {958, 1470}, {962, 6919}, {1058, 3895}, {1145, 9957}, {1266, 23521}, {1319, 6691}, {1376, 1837}, {1385, 13747}, {1476, 5176}, {1699, 5187}, {1706, 3434}, {1738, 21935}, {1739, 23537}, {1834, 25091}, {1861, 4429}, {1877, 5294}, {2093, 11415}, {2345, 24005}, {2475, 19925}, {2476, 7705}, {2646, 3035}, {2886, 3698}, {2975, 3911}, {3036, 20118}, {3057, 3816}, {3218, 12527}, {3219, 18250}, {3339, 5905}, {3361, 20076}, {3419, 9709}, {3452, 3869}, {3486, 4855}, {3576, 6921}, {3579, 11113}, {3614, 3838}, {3628, 11729}, {3670, 24186}, {3681, 24391}, {3693, 21049}, {3701, 17862}, {3717, 20905}, {3740, 21677}, {3742, 15888}, {3754, 3814}, {3817, 5154}, {3826, 5784}, {3832, 21628}, {3868, 21075}, {3877, 11362}, {3879, 24540}, {3897, 10165}, {3914, 24440}, {3918, 6702}, {3922, 17605}, {3925, 10958}, {4026, 25099}, {4188, 4297}, {4189, 10164}, {4190, 5691}, {4197, 15016}, {4292, 5080}, {4298, 20060}, {4357, 24986}, {4413, 5794}, {4423, 10965}, {4512, 9588}, {4642, 24210}, {4731, 9710}, {4967, 24547}, {4972, 24983}, {5051, 24991}, {5084, 5250}, {5087, 10107}, {5177, 11024}, {5179, 16549}, {5251, 5445}, {5260, 5745}, {5261, 9776}, {5289, 24954}, {5316, 5837}, {5437, 9578}, {5438, 5727}, {5687, 5722}, {5711, 10601}, {5749, 20262}, {5782, 17303}, {5790, 16203}, {5791, 19520}, {5828, 11037}, {5880, 10895}, {5883, 13407}, {5902, 21077}, {5903, 21616}, {6174, 10543}, {6245, 6904}, {6871, 7989}, {6920, 12775}, {6931, 8227}, {6957, 12705}, {6992, 10268}, {7483, 11231}, {7504, 10172}, {7951, 12609}, {8728, 10202}, {9955, 17533}, {10265, 12751}, {10404, 11236}, {10596, 17559}, {10679, 11108}, {10786, 18443}, {10805, 17582}, {11112, 18480}, {12000, 16853}, {12053, 14923}, {12512, 15680}, {12513, 17728}, {12699, 17556}, {12709, 18236}, {13161, 24443}, {13373, 17529}, {13906, 19065}, {13911, 19047}, {13964, 19066}, {13973, 19048}, {14011, 17167}, {15829, 20196}, {16371, 18481}, {16417, 18525}, {17063, 23675}, {17528, 18542}, {18239, 18242}, {23536, 24174}, {24985, 24988}, {25001, 25004}
X(24983) lies on these lines: {2, 3}, {10, 23528}, {1210, 23542}, {1737, 23518}, {1785, 23661}, {4972, 24982}, {19861, 21147}, {24986, 24992}
X(24984) lies on these lines: {1, 23518}, {2, 3}, {8, 343}, {10, 1074}, {1441, 4357}, {3100, 5174}, {4296, 17923}, {4972, 19860}, {5797, 18180}, {5906, 7078}, {17862, 23537}, {25000, 25002}
X(24985) lies on these lines: {2, 3}, {1125, 23541}, {4972, 24541}, {13411, 23542}, {17282, 19861}, {24982, 24988}
X(24986) lies on these lines: {2, 6}, {7, 11681}, {10, 20895}, {56, 21286}, {75, 25005}, {1229, 24005}, {1329, 20245}, {1400, 21244}, {4187, 17183}, {4193, 10446}, {4357, 24982}, {4972, 24991}, {5554, 17321}, {17270, 19861}, {17452, 25369}, {24983, 24992}, {25001, 25007}, {25018, 25021}
X(24987) lies on these lines: {1, 2}, {4, 5250}, {5, 392}, {9, 3436}, {12, 908}, {20, 10268}, {21, 515}, {35, 17647}, {40, 377}, {55, 5794}, {63, 388}, {65, 5249}, {72, 495}, {80, 5259}, {92, 225}, {140, 17614}, {142, 4848}, {149, 12575}, {165, 4190}, {191, 5270}, {210, 12607}, {226, 3869}, {283, 1010}, {322, 5224}, {329, 5261}, {355, 405}, {390, 5175}, {394, 5711}, {404, 6684}, {442, 517}, {443, 5657}, {474, 11249}, {516, 2475}, {518, 8261}, {527, 11684}, {529, 18253}, {535, 3647}, {536, 4918}, {758, 13407}, {944, 6857}, {946, 2476}, {950, 1621}, {952, 6675}, {956, 5791}, {958, 5252}, {962, 5177}, {966, 8557}, {982, 23675}, {986, 23536}, {1001, 1837}, {1004, 5584}, {1072, 16062}, {1104, 5724}, {1108, 5742}, {1145, 6797}, {1213, 8609}, {1220, 5294}, {1224, 2990}, {1266, 24999}, {1319, 4999}, {1385, 7483}, {1441, 4357}, {1478, 12514}, {1519, 6842}, {1616, 17721}, {1697, 3434}, {1699, 6871}, {1738, 4642}, {1758, 5484}, {1788, 3306}, {1861, 5174}, {2049, 24545}, {2078, 5176}, {2292, 13161}, {2322, 8755}, {2323, 5750}, {2345, 3692}, {2346, 5178}, {2478, 5587}, {2550, 22370}, {2551, 3305}, {2646, 6690}, {2784, 5985}, {2886, 3057}, {2975, 5745}, {3146, 21628}, {3193, 14005}, {3218, 4298}, {3219, 12527}, {3290, 21965}, {3294, 5179}, {3295, 3419}, {3359, 6897}, {3452, 11681}, {3475, 11520}, {3485, 11682}, {3576, 6910}, {3577, 6933}, {3579, 11112}, {3600, 5744}, {3614, 5087}, {3698, 3826}, {3710, 4385}, {3717, 3963}, {3740, 21031}, {3753, 5690}, {3755, 24554}, {3813, 5919}, {3816, 17606}, {3817, 5141}, {3822, 3878}, {3868, 21620}, {3873, 24391}, {3876, 21075}, {3883, 5016}, {3890, 11680}, {3895, 5082}, {3897, 5882}, {3898, 24387}, {3911, 5253}, {3916, 18990}, {3925, 5836}, {3932, 25099}, {4018, 6147}, {4084, 11551}, {4187, 9956}, {4188, 10164}, {4189, 4297}, {4193, 10175}, {4197, 11362}, {4293, 4652}, {4331, 17257}, {4413, 10966}, {4424, 23537}, {4512, 5691}, {4520, 17747}, {4640, 7354}, {4855, 5218}, {4967, 24993}, {5044, 17757}, {5046, 19925}, {5051, 23541}, {5080, 12572}, {5084, 5818}, {5123, 10957}, {5187, 7989}, {5219, 10585}, {5229, 5698}, {5248, 10572}, {5251, 14798}, {5267, 21578}, {5289, 11375}, {5290, 5905}, {5296, 20262}, {5330, 13464}, {5436, 5727}, {5603, 6856}, {5687, 19520}, {5692, 21077}, {5707, 16458}, {5720, 10786}, {5725, 16466}, {5730, 11374}, {5731, 6245}, {5775, 11037}, {5790, 11108}, {5847, 15988}, {5855, 11281}, {5903, 12609}, {6907, 12672}, {6925, 12705}, {7330, 12115}, {7680, 14110}, {7741, 17057}, {7951, 21616}, {8236, 24389}, {8608, 16699}, {9612, 11415}, {9943, 17616}, {9948, 11220}, {9955, 17530}, {9957, 24390}, {10389, 12625}, {10597, 17582}, {10680, 16408}, {10806, 17559}, {10895, 24703}, {10943, 17527}, {10944, 24953}, {11113, 18480}, {11231, 13747}, {11237, 17781}, {11344, 11500}, {12001, 16863}, {12635, 17718}, {12699, 17532}, {12702, 17528}, {13907, 19065}, {13911, 19049}, {13965, 19066}, {13973, 19050}, {14007, 24556}, {16370, 18481}, {16418, 18525}, {17072, 19948}, {17525, 22798}, {17577, 18483}, {21935, 24210}, {24178, 24443}, {25255, 25260}
X(24988) lies on these lines: {2, 11}, {8, 3315}, {10, 244}, {43, 18139}, {88, 9780}, {121, 1054}, {321, 17888}, {404, 23850}, {474, 20999}, {678, 19862}, {748, 4450}, {756, 24169}, {899, 3836}, {1086, 3952}, {1125, 3722}, {1213, 20331}, {1623, 17572}, {1738, 4358}, {2246, 5750}, {3006, 3823}, {3120, 24003}, {3240, 17234}, {3634, 5051}, {3740, 17184}, {3771, 9350}, {3890, 6018}, {3932, 17495}, {4126, 20068}, {4359, 24186}, {4389, 9330}, {4422, 4427}, {4440, 4756}, {4453, 14430}, {4660, 17125}, {4696, 24178}, {4768, 21180}, {4966, 19998}, {5014, 5272}, {5257, 14439}, {5297, 16706}, {5741, 16569}, {6187, 19846}, {9324, 19872}, {9347, 17367}, {9458, 17719}, {16062, 19877}, {16862, 23858}, {17126, 17352}, {17724, 17780}, {24982, 24985}
X(24989) lies on these lines: {2, 3}, {321, 23541}, {1861, 4972}, {1876, 17184}, {3192, 5741}, {3920, 5081}, {5014, 23050}, {11393, 24552}
X(24990) lies on these lines: {2, 3}, {4442, 23541}
X(24991) lies on these lines: {2, 31}, {10, 20237}, {141, 24550}, {4429, 21912}, {4972, 24986}, {4981, 24998}, {5051, 24982}, {21935, 24440}, {23541, 24997}, {24992, 24995}
X(24992) lies on these lines: {2, 32}, {10, 21420}, {24983, 24986}, {24991, 24995}, {25002, 25018}
X(24993) lies on these lines: {1, 20895}, {2, 37}, {7, 3436}, {9, 24633}, {38, 23677}, {65, 20245}, {69, 5554}, {77, 1441}, {86, 3262}, {322, 3945}, {517, 17183}, {960, 21273}, {1266, 24994}, {1319, 18654}, {2171, 21246}, {2268, 24334}, {2269, 25371}, {2285, 24612}, {3596, 4696}, {3663, 8582}, {3875, 19861}, {4357, 24982}, {4467, 25022}, {4858, 5750}, {4967, 24987}, {4972, 25012}, {4981, 24996}, {5224, 25005}, {8583, 17151}, {10401, 21286}, {17451, 20258}, {17861, 20880}, {18698, 24209}, {23690, 24342}, {25004, 25023}, {25083, 25255}
X(24994) lies on these lines: {2, 39}, {1266, 24993}, {3831, 21405}, {24983, 24986}
X(24995) lies on these lines: {2, 41}, {8, 3905}, {9, 17550}, {10, 1930}, {31, 315}, {32, 4805}, {69, 5230}, {76, 21935}, {115, 4721}, {213, 626}, {218, 7866}, {325, 1193}, {672, 6656}, {742, 16886}, {758, 17211}, {2251, 6680}, {2295, 20541}, {2887, 17137}, {3314, 17033}, {3662, 16906}, {3721, 25345}, {4136, 17489}, {4165, 21216}, {4167, 17497}, {4197, 10436}, {4357, 5051}, {4429, 10030}, {4950, 16974}, {5025, 24514}, {5086, 24291}, {5255, 20553}, {7762, 21764}, {7776, 16466}, {7933, 17350}, {24991, 24992}, {24996, 25012}
X(24996) lies on these lines: {1, 2}, {6, 24545}, {181, 908}, {355, 11358}, {515, 13588}, {1512, 4192}, {2478, 9548}, {4972, 24986}, {4981, 24993}, {5087, 10406}, {23541, 25003}, {24995, 25012}
X(24997) lies on these lines: {1, 2}, {11, 25135}, {81, 24563}, {394, 24545}, {497, 22370}, {515, 4203}, {518, 20487}, {908, 20545}, {1423, 20557}, {1512, 19540}, {3925, 25144}, {3989, 25245}, {4972, 25010}, {4981, 25024}, {23541, 24991}
X(24998) lies on these lines: {2, 44}, {693, 17420}, {3262, 5224}, {4357, 24982}, {4389, 25005}, {4981, 24991}, {17272, 24540}
X(24999) lies on these lines: {2, 45}, {7, 2975}, {1266, 24987}, {2292, 3663}, {4357, 24982}, {10436, 24539}, {10915, 20895}, {17183, 19531}, {17250, 25005}, {17274, 19860}, {17321, 24540}
X(25000) lies on these lines: {2, 6}, {10, 774}, {75, 23978}, {198, 21270}, {307, 20262}, {311, 18740}, {348, 5942}, {379, 5816}, {442, 5796}, {572, 24581}, {573, 857}, {594, 25243}, {1958, 24582}, {2183, 20305}, {3207, 20074}, {4271, 8287}, {4357, 20905}, {5729, 9780}, {5753, 8728}, {17239, 25067}, {17248, 24554}, {17863, 24005}, {24547, 25023}, {24984, 25002}
X(25001) lies on these lines: {2, 37}, {8, 5728}, {9, 1441}, {10, 774}, {45, 17895}, {85, 144}, {274, 24557}, {318, 9780}, {322, 391}, {338, 1213}, {377, 9962}, {379, 1766}, {672, 4032}, {1713, 5278}, {2183, 21231}, {2262, 21271}, {2293, 25375}, {3262, 17277}, {3294, 17866}, {3553, 5736}, {3710, 4968}, {3717, 3963}, {3729, 20880}, {3731, 17861}, {3886, 19860}, {4384, 20895}, {4858, 6666}, {5819, 20061}, {6605, 10025}, {11997, 13576}, {16601, 25255}, {16676, 23521}, {16732, 16814}, {17220, 21871}, {17355, 18698}, {20236, 25101}, {20881, 24199}, {21371, 24633}, {24209, 25072}, {24982, 25004}, {24986, 25007}, {25082, 25252}
X(25002) lies on these lines: {2, 39}, {349, 1212}, {3177, 6063}, {3717, 3963}, {5179, 17866}, {17760, 21404}, {20905, 21418}, {24984, 25000}, {24992, 25018}
X(25003) lies on these lines: {2, 6}, {321, 24005}, {442, 5811}, {3948, 17862}, {3995, 21049}, {4972, 21912}, {5051, 24982}, {23541, 24996}, {25013, 25019}
X(25004) lies on these lines: {2, 6}, {442, 5817}, {3948, 20905}, {5051, 8582}, {16589, 24554}, {24982, 25001}, {24993, 25023}, {25010, 25011}
X(25005) lies on these lines: {1, 2}, {5, 1537}, {11, 8256}, {20, 1512}, {40, 5046}, {46, 5080}, {56, 5176}, {57, 20060}, {65, 5123}, {75, 24986}, {80, 17100}, {100, 1837}, {104, 355}, {140, 3897}, {149, 9581}, {165, 15680}, {319, 24540}, {322, 5740}, {346, 24005}, {377, 2096}, {392, 10284}, {405, 11849}, {443, 5770}, {474, 5790}, {496, 1145}, {515, 4188}, {517, 4193}, {908, 4848}, {944, 4881}, {946, 5154}, {952, 13747}, {962, 5187}, {984, 25010}, {993, 5445}, {1158, 2475}, {1311, 8685}, {1320, 11373}, {1329, 3869}, {1376, 5086}, {1385, 17566}, {1788, 3218}, {2476, 3753}, {2478, 5657}, {2550, 15297}, {2551, 3219}, {2975, 24914}, {3035, 10950}, {3036, 6691}, {3208, 21013}, {3306, 9578}, {3339, 17483}, {3501, 21044}, {3579, 11114}, {3586, 20066}, {3614, 10129}, {3681, 21031}, {3754, 7951}, {3814, 5903}, {3816, 3890}, {3820, 3876}, {3825, 5697}, {3838, 3922}, {3841, 17057}, {3868, 17757}, {3871, 5722}, {3873, 12607}, {3877, 4187}, {4147, 21105}, {4189, 6684}, {4190, 12616}, {4197, 5885}, {4373, 24213}, {4389, 24998}, {4429, 24433}, {4661, 24391}, {4855, 5727}, {5141, 10175}, {5175, 10395}, {5177, 5811}, {5178, 8668}, {5221, 11236}, {5224, 24993}, {5252, 5253}, {5422, 5711}, {5435, 20076}, {5603, 6931}, {5836, 11680}, {5881, 10265}, {5884, 12665}, {6681, 21842}, {6702, 7741}, {6945, 12672}, {7354, 9352}, {7704, 23513}, {9335, 23675}, {9711, 21677}, {9803, 17857}, {9947, 17616}, {10107, 17605}, {10164, 17548}, {10786, 18444}, {10895, 20292}, {11112, 18357}, {12531, 20118}, {12666, 18242}, {12702, 17556}, {13587, 18481}, {14011, 17174}, {15079, 24387}, {15679, 22798}, {15803, 20067}, {16371, 18525}, {17072, 21132}, {17250, 24999}, {17533, 22791}, {17579, 18480}, {21935, 24440}
X(25006) lies on these lines: {1, 2}, {5, 3697}, {9, 3434}, {11, 3740}, {12, 4662}, {38, 1738}, {40, 10431}, {63, 2550}, {65, 9710}, {75, 1233}, {92, 1861}, {100, 5745}, {142, 3873}, {210, 908}, {226, 3681}, {319, 14828}, {321, 3717}, {329, 5686}, {354, 3826}, {355, 7580}, {497, 3305}, {515, 7411}, {516, 3219}, {517, 3690}, {518, 3925}, {527, 20292}, {528, 3683}, {594, 3693}, {756, 24210}, {946, 3876}, {950, 5178}, {952, 13151}, {958, 20835}, {966, 3692}, {984, 3914}, {1001, 4863}, {1004, 5794}, {1005, 5086}, {1109, 4712}, {1211, 8286}, {1329, 3983}, {1441, 4967}, {1512, 5790}, {1621, 5853}, {1654, 10025}, {1697, 10395}, {1734, 23811}, {1836, 5220}, {2346, 5284}, {2475, 12527}, {2476, 21075}, {2911, 17275}, {3058, 15254}, {3059, 8255}, {3242, 24789}, {3306, 24477}, {3416, 4042}, {3419, 9708}, {3452, 11680}, {3485, 3984}, {3555, 8728}, {3587, 5657}, {3663, 7226}, {3678, 12047}, {3686, 4071}, {3689, 6690}, {3691, 20556}, {3696, 3703}, {3706, 3932}, {3715, 24703}, {3782, 21949}, {3786, 17167}, {3814, 3956}, {3820, 3921}, {3841, 13407}, {3883, 5014}, {3886, 17776}, {3890, 21627}, {3951, 4295}, {3967, 4126}, {4001, 4645}, {4082, 4671}, {4090, 25385}, {4197, 21620}, {4312, 20078}, {4314, 16865}, {4357, 4972}, {4359, 24235}, {4392, 24177}, {4416, 6327}, {4430, 5542}, {4431, 17163}, {4512, 20075}, {4514, 17277}, {4679, 11235}, {4684, 18139}, {4733, 6741}, {4849, 5718}, {4866, 6871}, {4883, 17245}, {4899, 17165}, {5044, 7743}, {5045, 17529}, {5046, 18250}, {5082, 5250}, {5173, 21617}, {5174, 14004}, {5177, 5815}, {5223, 5905}, {5234, 6872}, {5258, 17647}, {5263, 5294}, {5269, 24597}, {5273, 17784}, {5302, 6284}, {5316, 24386}, {5534, 6889}, {5687, 5791}, {5690, 8727}, {5783, 24545}, {5836, 21677}, {5837, 14923}, {5850, 17483}, {5904, 12609}, {6692, 9342}, {6769, 6837}, {6907, 18908}, {7160, 12620}, {7174, 19785}, {7308, 24392}, {7674, 18230}, {7991, 12617}, {9711, 17606}, {10584, 20196}, {10883, 11362}, {10914, 14022}, {11108, 18530}, {12699, 15650}, {15587, 17616}, {17072, 21183}, {17140, 24199}, {17251, 24352}, {17278, 17597}, {17286, 24388}, {17353, 24552}, {20070, 21628}, {21283, 25101}
X(25007) lies on these lines: {1, 2}, {297, 1861}, {344, 24005}, {355, 21477}, {515, 21495}, {908, 16609}, {1213, 25099}, {1512, 6996}, {1738, 25010}, {4297, 21537}, {5249, 16603}, {5750, 15988}, {5790, 21526}, {5847, 24563}, {6684, 21511}, {10164, 21508}, {10521, 17483}, {16431, 18481}, {18525, 21539}, {24986, 25001}
X(25008) lies on these lines: {2, 661}, {656, 18155}, {693, 17420}, {4025, 4391}, {4077, 4357}, {4131, 17215}, {4374, 25022}, {14208, 23788}
X(25009) lies on these lines: {2, 650}, {443, 8760}, {656, 4811}, {850, 6587}, {1577, 4765}, {2550, 11934}, {3239, 17896}, {4025, 4391}, {4077, 4521}, {4380, 23819}, {4397, 7649}, {4462, 7178}, {4468, 24002}, {8142, 17576}, {17862, 21438}
X(25010) lies on these lines: {2, 31}, {10, 20633}, {984, 25005}, {1423, 11681}, {1738, 25007}, {1756, 3814}, {4193, 6210}, {4357, 24982}, {4972, 24997}, {17306, 19860}, {25004, 25011}
X(25011) lies on these lines: {1, 2}, {140, 1512}, {142, 11681}, {354, 9711}, {355, 16862}, {377, 18492}, {442, 22798}, {474, 18481}, {496, 4002}, {515, 17531}, {517, 17575}, {908, 3649}, {1266, 24993}, {1329, 5249}, {1738, 22174}, {1788, 3305}, {2551, 3306}, {2975, 6692}, {3218, 18250}, {3359, 6898}, {3436, 5437}, {3648, 12572}, {3698, 3816}, {3742, 21031}, {3753, 17527}, {3813, 4731}, {3814, 6701}, {3820, 5439}, {3826, 14100}, {3833, 13407}, {3848, 15888}, {3869, 5316}, {3911, 5260}, {4187, 9955}, {4197, 10175}, {4297, 17572}, {5047, 6684}, {5068, 21628}, {5080, 12436}, {5084, 6361}, {5221, 17781}, {5250, 17559}, {5251, 5442}, {5253, 5795}, {5440, 15174}, {5586, 5905}, {5750, 7110}, {6666, 8543}, {6919, 11024}, {8165, 9776}, {8728, 17612}, {9842, 9961}, {9956, 17529}, {10164, 16865}, {16408, 18525}, {25004, 25010}
X(25012) lies on these lines: {2, 3}, {3914, 17862}, {4972, 24993}, {24995, 24996}
X(25013) lies on these lines: {2, 3}, {25003, 25019}
X(25014) lies on this line: {2, 3}
X(25015) lies on these lines: {2, 3}, {10, 307}, {78, 3936}, {387, 5738}, {936, 3454}, {938, 1834}, {948, 1211}, {1074, 1210}, {1213, 6554}, {1257, 16086}, {1330, 2287}, {1869, 18589}, {4359, 6734}, {4429, 11024}, {4658, 15936}, {5249, 23542}, {5703, 17056}, {8680, 21671}, {12609, 23518}, {17052, 18634}, {17863, 23537}, {18673, 25361}
X(25016) lies on these lines: {2, 3}, {3679, 23541}
X(25017) lies on these lines: {2, 3}, {8, 13567}, {10, 774}, {1446, 4357}, {3695, 25243}, {5296, 6554}, {19860, 23541}, {20235, 24547}, {20905, 23537}
X(25018) lies on these lines: {2, 3}, {24986, 25021}, {24992, 25002}
X(25019) lies on these lines: {2, 7}, {10, 774}, {141, 25067}, {198, 18650}, {306, 3965}, {857, 10445}, {1125, 1496}, {1441, 20262}, {1730, 21062}, {1861, 4429}, {2183, 18589}, {2270, 4329}, {2293, 21914}, {2321, 25243}, {2654, 19860}, {3663, 20905}, {4001, 17811}, {5051, 8582}, {5256, 18928}, {5942, 9312}, {16887, 24557}, {17220, 21068}, {25003, 25013}
X(25020) lies on these lines: {2, 900}, {522, 21052}, {1769, 23541}, {2815, 3877}, {3766, 24547}
X(25021) lies on these lines: {2, 39}, {1738, 21935}, {24986, 25018}
X(25022) lies on these lines: {2, 647}, {4374, 25008}, {4467, 24993}, {14837, 17896}
X(25023) lies on these lines: {2, 7}, {75, 24005}, {1738, 21935}, {3755, 5554}, {3821, 8582}, {24547, 25000}, {24986, 25001}, {24993, 25004}
X(25024) lies on these lines: {2, 38}, {9, 1405}, {10, 1733}, {21, 6211}, {201, 1220}, {256, 4642}, {518, 25099}, {740, 25245}, {1240, 3718}, {1756, 3754}, {1757, 15988}, {2292, 17261}, {3508, 21808}, {3617, 4073}, {3717, 3963}, {4026, 24433}, {4357, 24982}, {4981, 24997}, {6187, 7295}, {7081, 11031}, {7174, 15839}, {16830, 24563}, {17353, 24541}
Collineation mappings involving Gemini triangle 14: X(25025)-X(25041)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 13, as in centers X(25025)-X(25041). Then
m(X) = : : (a^2 b + a b^2 + 2 b^3 - a^2 c + 2 a b c - 3 b^2 c + a c^2 - 3 b c^2 + 2 c^3) x + (4 a^2 b + 2 a b^2 - 2 b^3 - 10 a b c + 2 b^2 c + 4 b c^2) y + (4 a^2 c - 10 a b c + 4 b^2 c + 2 a c^2 + 2 b c^2 - 2 c^3) z : :
and m(X) is on the Euler line if and only if X is on the Euler line.
(Clark Kimberling, October 12, 2018)
X(25025) lies on these lines: {1, 2}, {88, 10713}, {121, 4674}, {1739, 24399}, {17237, 25029}, {25036, 25037}
X(25026) lies on these lines: {2, 3}, {1145, 25030}, {1168, 24864}, {1319, 24871}
X(25027) lies on this line: {2, 3}
X(25028) lies on this line: {2, 3}
X(25029) lies on these lines: {2, 6}, {10, 24873}, {17237, 25025}
X(25030) lies on these lines: {1, 2}, {1145, 25026}, {17237, 24873}
X(25031) lies on these lines: {1, 2}, {514, 4364}, {2802, 25351}, {4013, 24222}, {4389, 6549}, {6667, 11731}, {21630, 25040}, {24873, 25035}
X(25032) lies on this line: {2, 3}
X(25033) lies on this line: {2, 3}
X(25034) lies on these lines: {2, 37}, {121, 1086}, {17237, 25025}, {17374, 24868}
X(25035) lies on these lines: {2, 45}, {17119, 24864}, {17237, 25025}, {24873, 25031}
X(25036) lies on these lines: {2, 37}, {190, 20568}, {4033, 4738}, {20006, 24517}, {25025, 25037}
X(25037) lies on these lines: {2, 6}, {16589, 24874}, {25025, 25036}
X(25038) lies on these lines: {2, 650}, {3762, 24589}
X(25039) lies on this line: {2, 3}
X(25040) lies on these lines: {2, 3}, {17056, 24869}, {21630, 25031}
X(25041) lies on this line: {2, 3}
See Peter Moses, Hyacinthos 28457.
X(25042) lies on these lines: {3,54}, {140,1141}, {252,549}, {546,7604}, {4994,18559}, {12026,19268}, {13367,20574}
X(25042) = isogonal conjugate of X(22335)
X(25042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 54, 1157), (3, 16762, 12307), (10610, 18016, 3)
See Peter Moses, Hyacinthos 28457.
X(25043) lies on the conic {{A,B,C,X(4),X(5)}}, the cubics K028, K115, K526, and on these lines: {2,3459}, {3,252}, {4,93}, {17,11082}, {18,11087}, {26,1601}, {30,15619}, {343,565}, {566,2165}, {631,18016}, {1263,21230}, {1487,1656}, {1658,21394}, {2937,11816}, {2962,18395}, {2980,17714}, {5449,6368}, {5965,20414}, {10095,17500}, {12044,23181}, {14072,19268}, {14247,14265}, {20424,22335}
X(25043) = isogonal conjugate of X(25044)
X(25043) = anticomplement X(15345)
X(25043) = cevapoint of X(3519) and X(21975)
X(25043) = crosspoint of X(11140) and X(20572)
X(25043) = X(i)-cross conjugate of X(j) for these (i,j): {233, 324}, {1209, 5}, {6368, 930}
X(25043) = antigonal conjugate of X(38899)
X(25043) = X(i)-isoconjugate of X(j) for these (i,j): {49, 2190}, {54, 2964}, {1994, 2148}, {2167, 2965}, {2169, 3518}
X(25043) = barycentric product X(i)*X(j) for these {i,j}: {5, 11140}, {93, 343}, {216, 20572}, {311, 2963}, {324, 3519}, {930, 18314}, {2962, 14213}
X(25043) = Kirikami-Euler image of X(5)
X(25043) = pedal antipodal perspector of X(54)
X(25043) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 1994}, {51, 2965}, {53, 3518}, {93, 275}, {216, 49}, {233, 1493}, {311, 7769}, {930, 18315}, {1487, 288}, {1953, 2964}, {2962, 2167}, {2963, 54}, {3519, 97}, {10412, 2413}, {11140, 95}, {12077, 1510}, {20572, 276}
X(25043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (93, 11140, 3519), (252, 930, 3), (3519, 18370, 4)
See Peter Moses, Hyacinthos 28457.
X(25044) lies on the cubic K009 and these lines: {2,252}, {3,54}, {4,137}, {5,23338}, {24,3432}, {32,14586}, {49,15787}, {96,7578}, {140,24305}, {184,20574}, {186,15620}, {288,13472}, {1147,15958}, {2120,3567}, {3090,7604}, {3484,6241}, {3574,8154}, {8254,14143}, {8884,18559}, {10285,24385}, {11671,24144}, {11815,15454}
X(25044) = isogonal conjugate of X(25043)
X(25044) = X(i)-Ceva conjugate of X(j) for these (i,j): {288, 14533}, {933, 1510}, {1166, 54}
X(25044) = isogonal conjugate of the anticomplement X(15345)
X(25044) = crosssum of X(3519) and X(21975)
X(25044) = X(i)-isoconjugate of X(j) for these (i,j): {5, 2962}, {252, 1087}, {564, 12044}, {930, 2618}, {1953, 11140}, {2963, 14213}
X(25044) = barycentric product X(i)*X(j) for these {i,j}: {49, 275}, {54, 1994}, {95, 2965}, {97, 3518}, {288, 1493}, {1510, 18315}, {2167, 2964}, {15721, 16306}
X(25044) = barycentric quotient X(i)/X(j) for these {i,j}: {49, 343}, {54, 11140}, {275, 20572}, {1510, 18314}, {1994, 311}, {2148, 2962}, {2964, 14213}, {2965, 5}, {3518, 324}, {8882, 93}, {14533, 3519}, {14586, 930}
X(25044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 1157, 3), (1493, 6150, 15345), (6150, 15345, 3)
Points associated with a family of ellipses: X(25045)-X(25050)
Let E be the circumellipse with center X(141), given by
a^2 (b^2 + c^2) y z + b^2 (c^2 + a^2) x z + c^2 (a^2 + b^2) x y = 0.
This ellipse is the isogonal conjugate of the anticomplement of the de Longchamps line, and it passes through X(i) for i = 67, 110, 660, 670, 694, 1634, 4553, 4576, 8050, 20021.
For arbitrary fixed nonzero constant k, let E(k) be the ellipse obtained from E by dilation from X(141) with ratio k; then E(k) is given by
(b^2 + c^2) ((a^2 + b^2) (a^2 + c^2) k^2 - b^2 c^2) x^2 + 2 (a^2 b^2 c^2 + (b^2 + c^2) (c^2 + a^2) (a^2 + b^2) k^2) y z + (cyclic) = 0.
If you have GeoGebra, you can view Lionheart-Moses Ellipses.
The major axis of every E(k) passes through X(i) for i = 6, 1344, 2574, 8105, 9173, 13414, and the minor axis, through X(i) for i = 6, 1345, 2575, 8106, 9174, 13415, 14899. The two axes are parallel to the Simson lines of X(1113) and X(1114).
These ellipses are associated with a problem in navigation posed by William Lionheart (see the link at X(6)), in connection with the fact that if a point P moves on E(k), the sum of squared distances from P to the sidelines BC, CA, AB stays constant. The value of the constant is
(a^2 b^2 c^2 + (b^2 + c^2) (c^2 + a^2) (a^2 + b^2) k^2) / (4 (a^2 + b^2 + c^2) R^2).
Suppose that P = p : q : r is a point on the circumcircle. Then the point
LM(P,k) = (b^2 + c^2) k ((2 a^2 + b^2 + c^2) p + (-a^2 - c^2) q + (-a^2 - b^2) r) - a^2 ((b^2 + c^2) p + (c^2 + a^2) q + (a^2 + b^2) r) : :
is on the ellipse E(k). The point LM(P,k) is here named the Lionheart-Moses k-image of P. Centers X(25045)-X(25050) are examples of such images, for k = 1, as are X(3448), X(11061), X(25332), and X(25333); see also X(25314)-X(25336). (Peter Moses, October 10, 2018)
The elliipse E(1) is the X(82)-anticomplementary conjugate of the infinity line; specifically, if a point (b - c)(a + t) is on the infinity line, then its image on E(1) is the point
- a^2 (b^2 + c^2) / ((a - c) (b + t) - (a - b) (c + t))
+ b^2 (c^2 + a^2) / ((b - a) (c + t) - (b - c) (a + t))
+ c^2 (a^2 + b^2) / ((c - b) (a + t) - (c - a) (b + t)) : :
X(25045) lies on these lines: {4, 94}, {23, 14611}, {1302, 14919}, {2986, 10752}, {6194, 7493}, {9003, 11061}
X(25045) = {30, 21289}, {82, 30}, {251, 18668}, {1495, 21217}, {2173, 2896}, {4599, 3268}, {14206, 1369}
X(25046) lies on these lines: {2, 98}, {6, 17500}, {22, 10330}, {69, 160}, {193, 8264}, {237, 3564}, {325, 6038}, {526, 11061}, {1503, 14957}, {2871, 14570}, {2987, 10753}, {3060, 8267}, {3618, 7668}, {14958, 15588}, {16684, 18612}, {23360, 23371}
X(25046) = reflection of X(i) in X(j) for these {i,j}: {69, 1634}, {25051, 6}, {25317, 25314}
X(25046) = anticomplement X(20021)
X(25046) = X(20022)-Ceva conjugate of X(2)
X(25046) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 511}, {237, 21217}, {511, 21289}, {1755, 2896}, {1959, 1369}, {3112, 14957}, {3405, 69}, {4593, 14295}, {20022, 6327}
X(25046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25051, 25317), (25051, 25314, 6)
X(25047) =
3 X[2] - 4 X[3124], 2 X[6] - 3 X[25315], X[25052] - 3 X[25315], X[25052] - 4 X[25322], 3 X[25315] - 4 X[25322], 3 X[25052] - 4 X[25325], 9 X[25315] - 4 X[25325], 3 X[6] - 2 X[25325], 3 X[25322] - X[25325], 2 X[25322] + X[25334], X[25052] + 2 X[25334], 3 X[25315] + 2 X[25334], 2 X[25325] + 3 X[25334]
X(25047) lies on these lines: {2, 694}, {6, 10330}, {69, 6664}, {94, 2996}, {110, 10754}, {111, 4563}, {148, 690}, {193, 2854}, {194, 11002}, {698, 20977}, {3060, 8267}, {3952, 24505}, {5468, 20998}, {7783, 15018}, {17154, 21220}
X(25047) = X(25047) = midpoint of X(6) and X(25334)
X(25047) = reflection of X(i) in X(j) for these {i,j}: {6, 25322}, {4576, 3124}, {25052, 6}
X(25047) = anticomplement X(4576)
X(25047) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 512}, {83, 17217}, {251, 7192}, {308, 21305}, {512, 21289}, {661, 1369}, {669, 21217}, {798, 2896}, {827, 21295}, {4593, 670}, {4599, 4576}, {4630, 6758}, {10566, 17137}, {18070, 315}, {18082, 21301}, {18098, 20295}, {18105, 8}, {18108, 17135}
X(25047) = X(6)-line conjugate of X(14778)
X(25047) = crossdifference of every pair of points on line {5027, 14778}
X(25047) = barycentric product X(1)*X(18060)
X(25047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25322, 25315), (3124, 4576, 2), (25052, 25315, 6), (25322, 25334, 25052)
X(25048) lies on these lines: {1, 3122}, {2, 4553}, {6, 82}, {8, 4033}, {75, 3056}, {86, 17049}, {105, 1332}, {149, 3448}, {190, 3271}, {239, 674}, {256, 17445}, {291, 3248}, {320, 9025}, {335, 9016}, {350, 20863}, {513, 4440}, {519, 21100}, {545, 4499}, {651, 10755}, {662, 8301}, {903, 4014}, {1086, 3888}, {1964, 24478}, {2836, 3868}, {3060, 3891}, {3270, 14942}, {3573, 16686}, {3688, 17277}, {3759, 3779}, {3770, 17142}, {3778, 18170}, {3799, 4422}, {3885, 7671}, {4124, 18151}, {4360, 21746}, {4443, 4787}, {4446, 7032}, {4473, 16482}, {4517, 17335}, {4922, 9263}, {4969, 9054}, {6007, 17160}, {7186, 24165}, {13476, 20090}, {16507, 22323}, {16706, 17792}, {17121, 22277}, {17790, 20352}, {18042, 23868}, {20331, 24495}
X(25048) = anticomplement X(4553)
X(25048) = reflection of X(i) in X(j) for these {i,j}: {190, 3271}, {320, 20358}, {3888, 1086}, {25050, 6}
X(25048) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 513}, {83, 20295}, {251, 514}, {308, 21304}, {513, 21289}, {514, 1369}, {649, 2896}, {667, 21217}, {827, 4427}, {1176, 20294}, {3112, 21301}, {4628, 190}, {10566, 69}, {18070, 21287}, {18105, 1654}, {18108, 8}
X(25048) = barycentric product X(1)*X(18061)
X(25048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25050, 25316), (190, 3271, 24482), (1964, 24478, 24530)
X(25049) lies on these lines: {7, 3240}, {69, 8050}, {75, 3909}, {149, 20295}, {150, 2774}, {675, 1331}, {2810, 23989}, {4440, 21220}, {6650, 8049}, {11061, 17220}, {17029, 17484}
X(25049) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 514}, {83, 513}, {251, 17494}, {308, 21301}, {513, 2896}, {514, 21289}, {649, 21217}, {693, 1369}, {3112, 20295}, {4599, 4427}, {6591, 8878}, {10566, 8}, {18070, 1330}, {18105, 1655}, {18108, 2}, {18833, 21304}
X(25050) = 3 X[3799] - 2 X[4437], 2 X[6] - 3 X[25316], X[25048] - 3 X[25316], X[25048] - 4 X[25323], 3 X[25316] - 4 X[25323]
X(25050) lies on these lines: {6, 82}, {69, 4553}, {100, 1814}, {190, 2876}, {319, 20552}, {518, 2113}, {2110, 22116}, {2836, 2895}, {2991, 10760}, {3573, 20468}, {3799, 4437}, {3827, 17789}, {3888, 5845}, {8674, 11061}, {10327, 18138}, {14839, 20455}, {20248, 21278} : :
X(25050) = reflection of X(i) in X(j) for these {i,j}: {6, 25323}, {69, 4553}, {25048, 6}
X(25050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25323, 25316), (25048, 25316, 6)
X(25050) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 518}, {83, 20347}, {251, 239}, {518, 21289}, {672, 2896}, {1176, 3100}, {2223, 21217}, {2356, 8878}, {3112, 20556}, {3912, 1369}, {4628, 918}
Barycentrics a^6*b^2 - a^2*b^6 + a^6*c^2 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 + 2*a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - b^2*c^6 : :
X(25051) = 3 X[2] - 4 X[7668], 4 X[6] - 3 X[25314], 2 X[25046] - 3 X[25314], 2 X[6] - 3 X[25317], X[25046] - 3 X[25317], 3 X[25314] - 8 X[25324], X[25046] - 4 X[25324], 3 X[25317] - 4 X[25324]
X(25051) lies on these lines: {2, 1634}, {4, 542}, {6, 17500}, {69, 290}, {98, 4558}, {99, 14060}, {148, 804}, {311, 6467}, {338, 2854}, {524, 14957}, {526, 3448}, {1232, 11574}, {2407, 9512}, {2782, 14570}, {3260, 8681}, {4226, 7669}, {4552, 24500}, {6128, 6997}, {6148, 7386}, {12042, 22085}
X(25051) = reflection of X(i) in X(j) for these {i,j}: {6, 25324}, {69, 20021}, {1634, 7668}, {14570, 20975}, {25046, 6}, {25314, 25317}
X(25051) = anticomplement X(1634)
X(25051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25046, 25314), (6, 25324, 25317), (1634, 7668, 2), (9512, 22143, 2407), (12188, 22143, 9512), (25046, 25317, 6)
X(25051) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 523}, {83, 7192}, {251, 4560}, {308, 17217}, {512, 21217}, {523, 21289}, {661, 2896}, {827, 6758}, {1577, 1369}, {3112, 512}, {4577, 21295}, {4580, 4329}, {4593, 4576}, {4599, 99}, {10566, 75}, {18070, 69}, {18082, 513}, {18097, 693}, {18098, 514}, {18105, 192}, {18107, 17149}, {18108, 1}
X(25052) lies on these lines: {2, 17413}, {6, 10330}, {67, 69}, {99, 895}, {194, 1992}, {690, 11061}, {2393, 3266}, {2407, 5989}, {2930, 5468}, {3124, 3618}, {5095, 14928}, {7664, 15118}, {11059, 11188}
X(25052) = reflection of X(i) in X(j) for these {i,j}: {6, 25325}, {69, 4576}, {25047, 6}, {25334, 25322}
X(25052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25047, 25315), (6, 25334, 25322), (25322, 25334, 25047)
X(25052) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 524}, {83, 17491}, {187, 21217}, {251, 17497}, {308, 21298}, {524, 21289}, {896, 2896}, {3112, 316}, {4599, 690}, {14210, 1369}, {22105, 21221}
X(25053) lies on these lines: {4, 23962}, {22, 10330}, {69, 110}, {287, 6515}, {339, 1112}, {1370, 2882}, {3060, 18018}, {3448, 9517}
X(25053) = anticomplement of X(35325)
X(25053) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 525}, {83, 7253}, {251, 17498}, {308, 21300}, {525, 21289}, {647, 21217}, {656, 2896}, {661, 8878}, {798, 10340}, {1176, 4560}, {1799, 7192}, {3112, 850}, {4580, 8}, {4599, 110}, {10566, 3868}, {14208, 1369}, {18070, 4}, {18082, 4391}, {18097, 521}, {18098, 25259}, {18105, 21216}, {18108, 3187}
X(25054) = 3 X[2] - 4 X[1084], X[670] - 3 X[3228], 2 X[1084] - 3 X[3228], 2 X[6] - 3 X[25318], 4 X[6] - 3 X[25319], 3 X[25319] - 8 X[25326], 3 X[25318] - 4 X[25326], 9 X[25319] - 8 X[25327], 9 X[25318] - 4 X[25327], 3 X[6] - 2 X[25327], 3 X[25326] - X[25327], 4 X[25327] - 3 X[25332], 3 X[25319] - 2 X[25332], 3 X[25318] - X[25332], 4 X[25326] - X[25332]
X(25054) lies on these lines: {2, 670}, {6, 19585}, {69, 694}, {148, 804}, {193, 8264}, {194, 1992}, {385, 9512}, {511, 19566}, {1370, 16098}, {1655, 4664}, {1916, 20975}, {2882, 20065}, {3978, 6379}, {4440, 21220}, {4922, 9263}, {6339, 6527}, {10340, 11061}, {10754, 11596}, {14360, 14948}
X(25054) = reflection of X(i) in X(j) for these {i,j}: {2, 3228}, {6, 25326}, {69, 694}, {670, 1084}, {1370, 16098}, {14360, 14948}, {25319, 25318}, {25332, 6}
X(25054) = anticomplement X(670)
X(25054) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2, 21305}, {6, 17217}, {25, 21300}, {31, 512}, {32, 7192}, {37, 21304}, {42, 21301}, {82, 688}, {163, 4576}, {213, 20295}, {512, 6327}, {513, 17138}, {523, 21275}, {560, 523}, {604, 4374}, {649, 17137}, {661, 315}, {662, 670}, {667, 17135}, {669, 8}, {688, 21289}, {798, 69}, {810, 1370}, {1084, 21221}, {1333, 17159}, {1402, 21302}, {1501, 4560}, {1576, 21295}, {1910, 14295}, {1918, 513}, {1919, 75}, {1924, 2}, {1927, 804}, {1973, 850}, {1974, 7253}, {1980, 1}, {2084, 1369}, {2205, 514}, {2206, 17166}, {2489, 21270}, {3049, 4329}, {3063, 20245}, {3113, 9006}, {3121, 150}, {3122, 21293}, {3124, 21294}, {3709, 21286}, {4017, 21280}, {4079, 21287}, {4117, 148}, {4455, 20554}, {5029, 20560}, {7180, 21285}, {9178, 21298}, {9236, 3005}, {9247, 6563}, {9426, 192}, {9427, 21220}, {9494, 21217}, {14574, 6758}, {18105, 21278}, {18267, 876}, {22383, 18659}
X(25054) = X(512)-Ceva conjugate of X(2)
X(25054) = X(9428)-cross conjugate of X(2)
X(25054) = X(2)-Hirst inverse of X(1084)
X(25054) = cevapoint of X(9431) and X(23180)
X(25054) = polar conjugate of isogonal conjugate of X(23180)
X(25054) = barycentric product X(i)*X(j) for these {i,j}: {76, 9431}, {264, 23180}, {512, 9428}
X(25054) = barycentric quotient X(i)/X(j) for these {i,j}: {9428, 670}, {9431, 6}, {23180, 3}, {23868, 15593}
X(25054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25326, 25318), (6, 25332, 25319), (670, 1084, 2), (670, 3228, 1084), (25318, 25332, 6)
X(25055) lies on these lines: {1, 2}, {3, 9589}, {5, 3655}, {11, 13384}, {21, 25056}, {30, 1699}, {35, 4428}, {36, 1001}, {40, 549}, {56, 4355}, {57, 5298}, {86, 2163}, {99, 12258}, {115, 9875}, {140, 3654}, {165, 3524}, {191, 3338}, {210, 5049}, {214, 9963}, {226, 13462}, {238, 4715}, {274, 4479}, {354, 3894}, {355, 547}, {376, 946}, {377, 4857}, {381, 1385}, {392, 3742}, {405, 5563}, {442, 3829}, {452, 4317}, {474, 3746}, {484, 3306}, {515, 3545}, {516, 10304}, {517, 5054}, {518, 4539}, {524, 16475}, {528, 15015}, {540, 5429}, {553, 3361}, {597, 3751}, {599, 1386}, {620, 9881}, {631, 7991}, {671, 11711}, {726, 9334}, {758, 15671}, {903, 4432}, {940, 5315}, {944, 5071}, {952, 15699}, {956, 8167}, {960, 3901}, {962, 15692}, {984, 4694}, {993, 5284}, {999, 4423}, {1022, 4448}, {1054, 11731} et al
X(25055) = midpoint of X(1) and X(19875)
X(25055) = midpoint of X(2) and X(38314)
X(25055) = reflection of X(i) in X(j) for these (i,j): (1, 38314), (2, 19883), (19875, 2)
X(25055) = {X(1),X(2)}-harmonic conjugate of X(3679)
X(25055) = homothetic center of anti-Aquila triangle and (cross-triangle of Aquila and anti-Aquila triangles)
X(25056) lies on these lines: {2, 319}, {21, 25055}
X(25057) lies on these lines: {2, 44}, {100, 993}
X(25058) lies on these lines: {1, 21}, {2, 314}, {37, 333}, {42, 3786}, {86, 3666}, {274, 3210}, {940, 2305}, {960, 5331}, {1010, 3931}, {1402, 4184} et al
X(25058) = {X(2),X(25060)}-harmonic conjugate of X(25059)
X(25059) lies on these lines: {2, 314}, {27, 1880}, {42, 5208}, {43, 3786}, {57, 77}, {65, 5331}, {86, 3752}, {239, 257}, {940, 11329}, {986, 4281}, {1043, 4646}, {1402, 4225}, {1403, 4267}, {2258, 3869} et al
X(25059) = {X(2),X(25060)}-harmonic conjugate of X(25058)
X(25060) lies on these lines: {1, 994}, {2, 314}, {21, 3931}, {37, 5235}, {58, 4414}, {81, 593}, {86, 4850}, {187, 980}, {286, 3672}, {333, 3187}, {940, 1030}, {2292, 4281} et al
X(25060) = {X(25058),X(25059)}-harmonic conjugate of X(2)
Collineation mappings involving Gemini triangle 15: X(25061)-X(25101)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 15 (the Gergonne line extraversion triangle; see X(10180)), as in centers X(25061)-X(25100). Then
m(X) = a [(b + c)(a - b - c) x + (b^2 - a b - b c) y + (c^2 - a c - b c) z]) : :
(Clark Kimberling, October 13, 2018)
X(25061) lies on these lines: {2, 21404}, {37, 17061}, {756, 23988}, {846, 6184}, {5943, 20684}, {16601, 25070}
X(25062) lies on these lines: {1, 1802}, {2, 23581}, {6, 10122}, {37, 1210}, {169, 1040}, {212, 1729}, {1212, 17102}, {1698, 17916}, {3008, 3666}, {3074, 8558}, {4000, 7264}, {5089, 6684}, {12005, 20752}, {25065, 25071}, {25066, 25096}
X(25063) lies on these lines: {1, 1783}, {2, 23581}, {9, 255}, {37, 216}, {169, 9817}, {226, 241}, {284, 3469}, {946, 5089}, {3075, 8558}, {3731, 7523}, {5747, 24933}, {12059, 22164}, {16600, 17720}, {17911, 24026}, {25067, 25068}
X(25064) lies on these lines: {2, 23581}, {3624, 17916}, {16601, 25069}
X(25065) lies on these lines: {1, 1170}, {2, 17861}, {37, 142}, {41, 16551}, {214, 17457}, {241, 3664}, {256, 14947}, {284, 16560}, {573, 7146}, {758, 16574}, {975, 988}, {986, 3743}, {1418, 4896}, {1743, 24635}, {1781, 11349}, {2171, 20367}, {2183, 18726}, {2809, 17447}, {3306, 16673}, {3666, 3911}, {3673, 4687}, {3739, 24209}, {3752, 16579}, {3912, 4019}, {3946, 8609}, {3986, 24213}, {4261, 24890}, {4266, 18161}, {4424, 12736}, {4698, 24208}, {4859, 24554}, {4888, 17092}, {5292, 24895}, {5662, 16696}, {5718, 18593}, {6586, 21182}, {7671, 24802}, {7832, 17289}, {16566, 21511}, {16601, 25067}, {16826, 24202}, {16831, 24179}, {17046, 21069}, {17047, 25348}, {17353, 24036}, {17355, 25083}, {20881, 25245}, {20905, 24778}, {24530, 24880}, {25062, 25071}, {25070, 25074}, {25097, 25101}
X(25066) lies on these lines: {1, 728}, {2, 277}, {3, 9}, {6, 3694}, {10, 1146}, {11, 21073}, {32, 44}, {36, 17744}, {37, 39}, {41, 5440}, {45, 5013}, {56, 17742}, {57, 21526}, {63, 21477}, {65, 16549}, {72, 672}, {78, 218}, {169, 1376}, {187, 15492}, {200, 16572}, {210, 2223}, {220, 997}, {241, 17284}, {344, 3926}, {346, 14986}, {354, 3970}, {355, 24247}, {392, 1334}, {475, 5089}, {500, 15984}, {517, 3061}, {518, 4253}, {519, 4515}, {574, 16814}, {579, 16298}, {894, 16061}, {939, 2297}, {942, 17754}, {960, 3730}, {988, 3731}, {1018, 3057}, {1100, 7772}, {1108, 2321}, {1155, 1759}, {1210, 8568}, {1214, 14376}, {1222, 6558}, {1329, 5179}, {1384, 3973}, {1385, 2329}, {1418, 21255}, {1427, 20106}, {1475, 3555}, {1574, 16605}, {1575, 16583}, {1723, 3713}, {1743, 3965}, {2049, 5283}, {2082, 5687}, {2170, 10914}, {2276, 3931}, {2345, 5831}, {2646, 16788}, {3053, 16885}, {3208, 9957}, {3218, 21540}, {3219, 21495}, {3305, 11343}, {3306, 21519}, {3496, 3579}, {3679, 4875}, {3691, 3697}, {3721, 20331}, {3744, 5299}, {3752, 16600}, {3753, 17451}, {3825, 21090}, {3846, 21856}, {3878, 21872}, {3912, 3933}, {3916, 5282}, {3929, 21539}, {3950, 21625}, {3998, 5294}, {4005, 17746}, {4357, 8362}, {4416, 7767}, {4422, 7789}, {4640, 24047}, {4641, 5337}, {4643, 7800}, {4851, 7758}, {4885, 23100}, {5007, 16669}, {5041, 16666}, {5087, 24045}, {5120, 5227}, {5437, 21542}, {5439, 21808}, {5525, 5563}, {5703, 5749}, {6292, 17237}, {6390, 25100}, {7123, 22131}, {7229, 24554}, {7308, 21514}, {7794, 17231}, {7795, 17279}, {7819, 17353}, {7822, 17357}, {7854, 17344}, {7855, 17374}, {8609, 17281}, {9310, 17614}, {11019, 21096}, {12447, 15853}, {13006, 17102}, {13161, 15048}, {16043, 17257}, {16060, 17260}, {16675, 22332}, {16728, 16887}, {17046, 24318}, {17619, 21044}, {17732, 24703}, {17747, 21616}, {18758, 20372}, {21258, 25355}, {25062, 25096}, {25087, 25091}
X(25066) = complement of X(3673)
X(25066) = {X(37),X(39)}-harmonic conjugate of X(37592)
X(25067) lies on these lines: {1, 480}, {2, 37}, {9, 241}, {57, 1696}, {141, 25019}, {144, 1418}, {200, 18216}, {219, 8257}, {220, 1445}, {374, 18161}, {474, 990}, {517, 19250}, {518, 21346}, {650, 24098}, {1212, 3160}, {1214, 7308}, {1376, 4319}, {1427, 18228}, {1465, 5316}, {1738, 21955}, {2257, 4383}, {2310, 15587}, {2324, 5228}, {2340, 5572}, {3242, 3445}, {3452, 3668}, {3740, 21039}, {3755, 8582}, {3912, 3965}, {4328, 5437}, {4413, 9371}, {4641, 17811}, {4875, 17277}, {5129, 15852}, {5723, 6666}, {6172, 17092}, {7174, 8583}, {8609, 17337}, {16601, 25065}, {16696, 24557}, {17239, 25000}, {17355, 25097}, {25063, 25068}, {25083, 25101}
X(25068) lies on these lines: {1, 4515}, {2, 277}, {9, 474}, {10, 4875}, {37, 3624}, {44, 5277}, {72, 17754}, {169, 4413}, {210, 4253}, {218, 936}, {392, 3501}, {612, 9605}, {672, 5044}, {960, 16549}, {1125, 3693}, {1212, 1698}, {2082, 9709}, {2276, 6051}, {2329, 17614}, {2345, 3086}, {3061, 3753}, {3305, 16412}, {3616, 3991}, {3634, 24036}, {3683, 24047}, {3697, 21384}, {3731, 11512}, {3740, 16552}, {3742, 3970}, {3816, 21073}, {3930, 5045}, {3931, 17756}, {4390, 24928}, {4679, 17732}, {4936, 8583}, {5750, 5830}, {8580, 16572}, {16600, 16610}, {16728, 17210}, {17303, 19854}, {17355, 25100}, {25063, 25067}, {25085, 25093}, {25088, 25096}, {25089, 25092}
X(25069) lies on these lines: {1, 3119}, {2, 21436}, {37, 24025}, {1054, 3731}, {1639, 17724}, {5400, 9502}, {11019, 20310}, {11814, 24036}, {16600, 17720}, {16601, 25064}, {17435, 18240}
X(25070) lies on these lines: {1, 14827}, {2, 20890}, {37, 2886}, {9502, 11031}, {16601, 25061}, {25065, 25074}, {25075, 25096}
X(25071) lies on these lines: {2, 21414}, {37, 17046}, {16599, 25104}, {25062, 25065}
X(25071) = complement of X(21414)
X(25072) lies on these lines: {1, 4924}, {2, 2415}, {6, 4909}, {9, 3664}, {10, 344}, {37, 3008}, {45, 142}, {144, 4896}, {346, 16832}, {516, 7385}, {519, 16484}, {527, 16814}, {551, 3618}, {749, 984}, {1654, 3912}, {1721, 10164}, {1723, 3305}, {1736, 13411}, {1738, 3634}, {1743, 5308}, {2321, 17259}, {2325, 3739}, {3589, 4755}, {3624, 4310}, {3626, 17233}, {3635, 3759}, {3686, 17243}, {3707, 4851}, {3828, 17342}, {3842, 19868}, {3875, 4098}, {3879, 17335}, {3945, 3973}, {3950, 4384}, {4000, 16676}, {4029, 4361}, {4357, 17263}, {4416, 17244}, {4419, 20195}, {4422, 4698}, {4431, 16815}, {4667, 16885}, {4700, 17390}, {4856, 17349}, {4888, 6172}, {4907, 5218}, {4967, 17264}, {5222, 16673}, {5257, 17279}, {5296, 17284}, {5435, 7274}, {6687, 17045}, {7961, 20196}, {8232, 10481}, {15492, 17392}, {16601, 25065}, {16675, 17278}, {16677, 17301}, {17023, 17338}, {17050, 22019}, {17067, 17246}, {17234, 17329}, {17257, 21255}, {17280, 24603}, {19878, 24160}, {21879, 24050}, {24209, 25001}, {25073, 25086}, {25081, 25097}
X(25072) = complement of X(24199)
X(25073) lies on these lines: {2, 986}, {37, 17048}, {846, 17681}, {1107, 21232}, {3294, 24786}, {3663, 24774}, {3934, 3985}, {6706, 24214}, {17596, 17682}, {17672, 21674}, {17686, 24850}, {25062, 25065}, {25072, 25086}, {25093, 25095}
X(25074) lies on these lines: {2, 277}, {37, 982}, {43, 1212}, {672, 8731}, {1107, 3961}, {3693, 3741}, {3991, 10453}, {3999, 14746}, {4212, 5089}, {4685, 4875}, {5268, 5283}, {6685, 24036}, {17717, 21856}, {21795, 24239}, {25065, 25070}, {25080, 25096}
X(25075) lies on these lines: {2, 277}, {37, 3848}, {672, 11227}, {1212, 16569}, {3693, 3840}, {5121, 21795}, {6184, 24239}, {6686, 24036}, {25070, 25096}
X(25076) lies on these lines: {2, 24209}, {37, 17067}, {3008, 8609}, {7658, 25084}, {16601, 25065}, {24778, 25243}, {25083, 25097}
X(25077) lies on these lines: {2, 20881}, {6666, 8609}, {16601, 25065}
X(25078) lies on these lines: {1, 1257}, {2, 17861}, {6, 22836}, {9, 48}, {10, 5721}, {36, 5279}, {37, 39}, {71, 3878}, {78, 1723}, {241, 21255}, {346, 3086}, {404, 1781}, {519, 1108}, {573, 3061}, {579, 758}, {583, 4053}, {672, 21078}, {1018, 17452}, {1212, 12447}, {1214, 20106}, {1826, 3814}, {2171, 16549}, {2257, 3811}, {2260, 3874}, {2264, 5440}, {2277, 16600}, {2294, 5883}, {2321, 8609}, {3454, 18591}, {3663, 25083}, {3693, 3950}, {3729, 24179}, {3731, 7963}, {3986, 16601}, {3991, 6744}, {4016, 4286}, {4149, 21059}, {4261, 20083}, {5227, 8666}, {5816, 24247}, {8804, 21616}, {10582, 16673}, {14439, 21809}, {16552, 21033}, {16566, 21495}, {16578, 17073}, {17052, 24317}, {17272, 24635}, {24202, 24559}
X(25078) = complement of X(17861)
X(25079) lies on these lines: {1, 341}, {2, 986}, {3, 4011}, {9, 4020}, {10, 11}, {35, 4432}, {39, 3985}, {72, 3840}, {140, 9959}, {312, 978}, {386, 25123}, {404, 24850}, {474, 3923}, {499, 4438}, {537, 3953}, {595, 4434}, {740, 3216}, {899, 3702}, {942, 4871}, {960, 3831}, {975, 25496}, {982, 25492}, {1043, 5529}, {1125, 1215}, {1149, 4696}, {1193, 4358}, {1201, 3701}, {1437, 5150}, {2887, 21616}, {3109, 6789}, {3120, 17674}, {3230, 4095}, {3293, 4975}, {3555, 4090}, {3624, 4687}, {3670, 19847}, {3729, 11512}, {3741, 5044}, {3743, 20108}, {3836, 12047}, {3842, 19863}, {3980, 16408}, {4003, 19862}, {4383, 17733}, {4385, 21214}, {4418, 17531}, {4422, 4999}, {4465, 16720}, {4673, 6048}, {5205, 5255}, {6051, 6685}, {6651, 7824}, {6682, 19864}, {6700, 9371}, {8728, 25385}, {9955, 21241}, {12567, 16302}, {17048, 20530}, {17355, 25068}, {17777, 24851}, {17793, 21336}, {21232, 25103}, {24945, 25124}, {25062, 25095}, {25063, 25078}, {25065, 25093}
X(25079) = complement of X(24443)
X(25080) lies on these lines: {1, 21}, {2, 17861}, {10, 3998}, {35, 3101}, {37, 226}, {57, 2294}, {284, 1762}, {323, 5483}, {345, 18697}, {347, 18625}, {515, 13442}, {740, 4847}, {912, 5453}, {984, 3190}, {1376, 11221}, {1419, 18675}, {1723, 5256}, {1726, 2268}, {1730, 17451}, {1754, 4137}, {1781, 1817}, {2256, 3173}, {2335, 15314}, {2801, 3989}, {2954, 13329}, {3601, 18673}, {3664, 18607}, {3666, 3946}, {3724, 5322}, {3841, 23604}, {3929, 3958}, {4413, 10158}, {4647, 19843}, {5173, 20718}, {5283, 24268}, {5294, 24036}, {10180, 16598}, {14838, 21209}, {16601, 25061}, {16603, 21072}, {18726, 22097}, {19822, 19854}, {23681, 24554}, {24177, 24181}, {25074, 25096}
X(25081) lies on these lines: {1, 2287}, {2, 17861}, {9, 758}, {10, 37}, {19, 5248}, {21, 1781}, {45, 4016}, {55, 11221}, {71, 3754}, {142, 8680}, {200, 1962}, {281, 451}, {344, 18697}, {346, 4647}, {551, 1108}, {579, 5883}, {965, 22836}, {1125, 3002}, {1376, 10158}, {1743, 2650}, {1761, 3647}, {1766, 12567}, {1826, 3822}, {1901, 11263}, {1953, 3884}, {2171, 3294}, {2292, 3731}, {3161, 17164}, {3247, 3811}, {3678, 22021}, {3919, 21866}, {3949, 4015}, {4047, 4084}, {4068, 6600}, {4364, 16608}, {4698, 16578}, {4856, 4875}, {5251, 5279}, {5296, 18391}, {5333, 16585}, {5436, 18673}, {5750, 24036}, {6536, 21717}, {6708, 16579}, {8804, 12609}, {10180, 13405}, {11683, 16053}, {16601, 17355}, {17052, 25359}, {20228, 21332}, {21061, 21808}, {21811, 21921}, {25072, 25097}, {25085, 25086}
X(25081) = complement of X(18698)
X(25082) lies on these lines: {1, 644}, {2, 277}, {8, 1212}, {9, 21}, {37, 2275}, {39, 16614}, {45, 5782}, {100, 169}, {145, 3991}, {190, 18055}, {220, 4511}, {329, 14021}, {344, 348}, {346, 3702}, {391, 3694}, {672, 3868}, {728, 3872}, {964, 5283}, {993, 17744}, {1018, 14923}, {1334, 3061}, {1475, 3889}, {1759, 24047}, {2082, 3871}, {2141, 3681}, {2170, 3208}, {2329, 3897}, {2975, 17742}, {3219, 16367}, {3305, 15490}, {3501, 14439}, {3665, 16593}, {3705, 21795}, {3729, 24554}, {3730, 3869}, {3731, 7963}, {3870, 16572}, {3873, 3970}, {3912, 24635}, {3930, 21384}, {3976, 23649}, {4200, 5089}, {4251, 4587}, {4391, 6376}, {4513, 4861}, {4664, 25268}, {4666, 24771}, {4676, 5764}, {4687, 16705}, {4850, 16600}, {5030, 17736}, {5057, 17732}, {5179, 11681}, {5294, 17023}, {5333, 16831}, {5525, 8666}, {5526, 22836}, {5540, 8715}, {5552, 6554}, {5830, 17340}, {7191, 9605}, {7705, 21044}, {7741, 21090}, {8609, 11240}, {11109, 17916}, {11680, 21073}, {16583, 17756}, {16720, 17279}, {16728, 17169}, {17077, 20946}, {17092, 17234}, {17350, 17696}, {17754, 21808}, {20271, 20331}, {25001, 25252}
X(25083) lies on these lines: {2, 277}, {3, 63}, {8, 24635}, {9, 11343}, {32, 4641}, {37, 980}, {39, 712}, {57, 21477}, {69, 3694}, {75, 25252}, {77, 3692}, {81, 5266}, {99, 448}, {100, 7291}, {110, 2754}, {190, 1944}, {219, 326}, {222, 3719}, {241, 3693}, {304, 345}, {306, 3933}, {441, 525}, {517, 1959}, {518, 2223}, {536, 8609}, {910, 20602}, {942, 1009}, {988, 8583}, {1010, 16735}, {1026, 23102}, {1108, 3875}, {1155, 20715}, {1212, 4384}, {1229, 17077}, {1332, 6510}, {1418, 17298}, {1465, 17789}, {1742, 4073}, {2108, 18208}, {2325, 16578}, {2340, 4712}, {2795, 20544}, {3008, 24036}, {3218, 21495}, {3219, 21511}, {3263, 5089}, {3305, 21514}, {3306, 21526}, {3501, 7146}, {3509, 21989}, {3616, 6051}, {3663, 25078}, {3712, 8758}, {3739, 20236}, {3928, 16431}, {3929, 16436}, {3965, 4416}, {3990, 23124}, {3991, 17316}, {4001, 7767}, {4515, 17294}, {4574, 20744}, {4869, 17092}, {5256, 9605}, {5273, 24609}, {5294, 7819}, {5437, 21519}, {7308, 21496}, {17355, 25065}, {17754, 21985}, {20740, 20808}, {22134, 23125}, {22363, 23167}, {24203, 24559}, {24993, 25255}, {25067, 25101}, {25076, 25097}
X(25083) = isogonal conjugate of X(8751)
X(25083) = isotomic conjugate of polar conjugate of X(518)
X(25083) = complement of polar conjugate of X(15344)
X(25083) = complement of complement of X(25257)
X(25083) = X(19)-isoconjugate of X(105)
X(25084) lies on these lines: {1, 4524}, {2, 647}, {37, 17069}, {665, 3835}, {804, 25126}, {905, 3239}, {940, 9404}, {1021, 17022}, {2488, 24462}, {3709, 4369}, {4477, 5268}, {4789, 24948}, {4885, 6586}, {7658, 25076}, {8651, 24533}
X(25084) = complement of X(24622)
X(25085) lies on these lines: {2, 20890}, {37, 1738}, {16601, 25065}, {25068, 25093}, {25081, 25086}
X(25086) lies on these lines: {2, 277}, {9, 5439}, {37, 1698}, {169, 4423}, {451, 5089}, {517, 21921}, {551, 4875}, {910, 5259}, {1210, 5257}, {1212, 3624}, {1759, 15254}, {3208, 4002}, {3290, 16589}, {3294, 3812}, {3634, 3693}, {3691, 5045}, {3731, 24174}, {3739, 7264}, {3740, 3970}, {3742, 16552}, {3754, 4520}, {3826, 21073}, {3991, 9780}, {4009, 22011}, {4515, 19875}, {5044, 21808}, {5302, 17736}, {6051, 16583}, {6666, 17048}, {9331, 21896}, {16610, 25092}, {19878, 24036}, {25072, 25073}, {25081, 25085}
X(25087) lies on these lines: {1, 6}, {2, 23581}, {10, 17916}, {33, 169}, {40, 5089}, {201, 3730}, {225, 5179}, {255, 8558}, {1729, 1936}, {3220, 18596}, {6554, 7952}, {17860, 17911}, {25066, 25091}
X(25088) lies on these lines: {2, 23581}, {10, 37}, {33, 11221}, {201, 3294}, {220, 758}, {2294, 2324}, {2318, 3970}, {3191, 21808}, {16601, 25061}, {25068, 25096}
X(25089) lies on these lines: {2, 1930}, {37, 19862}, {1574, 4868}, {1575, 3743}, {1759, 17124}, {3290, 19878}, {3634, 16611}, {3678, 24512}, {3721, 3833}, {3727, 3918}, {3780, 3956}, {4015, 20963}, {5297, 5299}, {7832, 17263}, {10176, 17750}, {16589, 24036}, {16601, 25064}, {20331, 21816}, {25068, 25092}
X(25090) lies on these lines: {2, 24788}, {37, 442}, {38, 22126}, {1375, 3666}, {4245, 17451}, {4254, 18674}, {4267, 18669}, {16577, 16600}, {16601, 25061}, {25062, 25065}, {25081, 25085}
X(25091) lies on these lines: {1, 939}, {2, 37}, {3, 16389}, {9, 223}, {33, 1376}, {40, 13737}, {57, 2324}, {63, 220}, {198, 10319}, {201, 960}, {210, 8758}, {219, 1708}, {225, 1329}, {227, 2551}, {306, 3965}, {329, 1427}, {394, 2911}, {452, 15852}, {517, 1730}, {518, 2318}, {650, 24115}, {908, 6354}, {910, 3101}, {936, 17102}, {940, 3553}, {942, 3191}, {1211, 15526}, {1418, 9965}, {1465, 3452}, {1766, 11347}, {1818, 10391}, {1834, 24982}, {2003, 6510}, {2123, 7952}, {2287, 18603}, {2654, 5836}, {3190, 5728}, {3219, 6610}, {3753, 19257}, {4383, 8557}, {4875, 5278}, {5256, 17825}, {5294, 23292}, {5745, 16578}, {5784, 24430}, {6358, 6708}, {6666, 16579}, {15853, 24635}, {16601, 25061}, {16700, 24556}, {17044, 18652}, {17080, 18228}, {19542, 21062}, {20103, 24025}, {21871, 24310}, {25066, 25087}
X(25092) lies on these lines: {1, 672}, {2, 3760}, {6, 5248}, {9, 386}, {10, 2276}, {21, 5280}, {35, 5276}, {37, 39}, {38, 3970}, {42, 16552}, {44, 20970}, {145, 9331}, {169, 17594}, {171, 24047}, {172, 5267}, {192, 17030}, {218, 19765}, {519, 1107}, {551, 2275}, {609, 4189}, {941, 1743}, {978, 3731}, {1001, 9605}, {1015, 3636}, {1018, 10459}, {1193, 3294}, {1212, 3931}, {1573, 3626}, {1574, 3828}, {1575, 3634}, {1621, 5299}, {1655, 6381}, {1698, 17756}, {1739, 21921}, {1759, 4414}, {1962, 20456}, {2140, 3663}, {2277, 3986}, {2303, 4278}, {2345, 19858}, {2975, 16785}, {3008, 3666}, {3061, 15953}, {3244, 16975}, {3293, 3691}, {3635, 17448}, {3670, 21808}, {3693, 19868}, {3702, 4099}, {3707, 4277}, {3741, 21070}, {3743, 24036}, {3811, 16517}, {3815, 3825}, {3821, 22380}, {3822, 5254}, {3912, 16887}, {4261, 5257}, {4424, 17451}, {4745, 21868}, {5179, 5530}, {5286, 10198}, {5294, 17023}, {5305, 6690}, {6683, 20530}, {6685, 21838}, {7746, 20104}, {10448, 16788}, {13006, 13411}, {16610, 25086}, {16819, 17759}, {17717, 24045}, {17758, 24214}, {21101, 24068}, {21240, 25349}, {21820, 24051}, {21883, 24044}, {25068, 25089}
X(25092) = complement of X(20888)
X(25093) lies on these lines: {2, 21414}, {25068, 25085}, {25073, 25095}
X(25094) lies on these lines: {1, 89}, {2, 24209}, {37, 16586}, {226, 17080}, {3666, 8609}, {3977, 17760}
X(25095) lies on these lines: {1, 7259}, {2, 24791}, {9, 3708}, {37, 16613}, {523, 3039}, {4422, 16599}, {16598, 24036}, {25073, 25093}
X(25096) lies on these lines: {2, 21436}, {9, 7004}, {37, 3756}, {644, 3315}, {1212, 14936}, {1638, 16578}, {2284, 5083}, {3239, 24003}, {3310, 13006}, {4422, 23587}, {4850, 16600}, {5573, 24771}, {14740, 17435}, {15490, 23511}, {25062, 25066}, {25068, 25088}, {25070, 25075}, {25074, 25080}
X(25097) lies on these lines: {1, 4578}, {2, 20881}, {9, 3942}, {918, 3960}, {11716, 24820}, {17355, 25067}, {24003, 24025}, {25065, 25101}, {25072, 25081}, {25076, 25083}
X(25098) lies on these lines: {1, 1938}, {2, 21438}, {37, 4411}, {63, 22383}, {441, 525}, {650, 824}, {665, 3798}, {918, 6589}, {1946, 23093}, {3239, 24782}, {3310, 4468}, {3663, 23806}, {3931, 14077}, {4063, 8662}, {4083, 8640}, {4369, 21348}, {4458, 6129}, {6586, 17069}, {7658, 25076}, {9443, 17592}, {15280, 24210}, {15283, 24239}, {17921, 20906}
X(25098) = complement of X(21438)
X(25099) lies on these lines: {2, 37}, {9, 7146}, {44, 15988}, {65, 21371}, {241, 894}, {257, 17260}, {264, 14571}, {326, 5782}, {458, 1841}, {517, 19243}, {518, 25024}, {650, 24141}, {1001, 3057}, {1108, 3618}, {1155, 8424}, {1213, 25007}, {1284, 3812}, {1766, 11343}, {3589, 8609}, {3661, 3965}, {3694, 17286}, {3932, 24987}, {4026, 24982}, {4875, 17349}, {5750, 16578}, {16601, 17760}, {17355, 25065}, {25072, 25081}
X(25100) lies on these lines:
X(25101) lies on these lines: {1, 4899}, {2, 2415}, {7, 4480}, {9, 69}, {10, 3685}, {37, 3589}, {44, 3629}, {45, 3763}, {75, 2325}, {141, 16814}, {142, 190}, {144, 17298}, {165, 9801}, {192, 3008}, {193, 3973}, {238, 4078}, {239, 3950}, {319, 3707}, {345, 7308}, {346, 4384}, {391, 17294}, {519, 15485}, {524, 15492}, {527, 17234}, {536, 17337}, {597, 3723}, {894, 4473}, {966, 17286}, {1001, 3717}, {1026, 2293}, {1083, 2330}, {1125, 17368}, {1213, 17359}, {1265, 5436}, {1266, 17262}, {1654, 17268}, {1696, 21477}, {1743, 17316}, {2321, 5564}, {2345, 24603}, {2346, 4578}, {3241, 16487}, {3244, 17121}, {3247, 3618}, {3305, 3687}, {3664, 17244}, {3686, 17233}, {3710, 5047}, {3730, 21371}, {3739, 17340}, {3748, 4126}, {3757, 4082}, {3834, 17334}, {3883, 3932}, {3929, 18141}, {3943, 17348}, {3946, 4664}, {4021, 4704}, {4029, 4360}, {4098, 17319}, {4364, 17357}, {4370, 17245}, {4389, 17341}, {4395, 4718}, {4398, 17067}, {4419, 17282}, {4643, 17267}, {4657, 16675}, {4667, 17317}, {4681, 6687}, {4684, 5220}, {4687, 5750}, {4698, 17369}, {4700, 17377}, {4755, 17398}, {4851, 6144}, {4856, 17389}, {4862, 20073}, {4869, 6172}, {4966, 15481}, {4967, 17259}, {5224, 17342}, {5257, 17289}, {5296, 17308}, {5325, 14829}, {5745, 18743}, {5749, 16831}, {6390, 25066}, {6646, 17266}, {8236, 10005}, {8245, 9950}, {10030, 18140}, {15828, 17300}, {16601, 17760}, {16669, 17390}, {16676, 17321}, {17229, 17330}, {17230, 17331}, {17231, 17332}, {17232, 17333}, {17240, 17346}, {17241, 17347}, {17246, 17356}, {17248, 17358}, {17256, 17285}, {17257, 17284}, {17258, 17283}, {17265, 17276}, {17269, 17275}, {17312, 20072}, {17781, 18139}, {20236, 25001}, {20881, 20905}, {21283, 25006}, {21873, 22047}, {21879, 24090}, {22214, 24575}, {25065, 25097}, {25067, 25083}
X(25101) = complement of complement of X(25269)
Collineation mappings involving Gemini triangle 16: X(25102)-X(25146)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 16, as in centers X(25102)-X(25146). Then
m(X) = (b + c)(a b + a c - b c) + (a b c - a c^2 + b^2 c) y + (a b c - a b^2 + b c^2) z : :
(Clark Kimberling, October 13, 2018)
X(25102) lies on these lines: {1, 20530}, {2, 17448}, {8, 21264}, {10, 141}, {12, 20541}, {37, 2998}, {75, 21868}, {76, 536}, {192, 20943}, {519, 3934}, {535, 7830}, {668, 1107}, {712, 21067}, {742, 4095}, {1018, 4721}, {1084, 4755}, {1500, 4681}, {1575, 1909}, {2228, 7148}, {3208, 4713}, {3501, 17351}, {3661, 4417}, {3666, 18136}, {3734, 8715}, {3742, 20340}, {3752, 20917}, {3795, 9902}, {3912, 21025}, {4437, 21965}, {4726, 20888}, {7784, 11236}, {7815, 8666}, {9025, 24327}, {10449, 17372}, {10987, 16920}, {12513, 15271}, {17033, 21904}, {17050, 24770}, {17229, 21024}, {17290, 24418}, {17792, 25131}, {18194, 24679}, {20911, 21021}, {24215, 25350}, {25103, 25119}, {25105, 25110}, {25106, 25108}, {25111, 25123}, {25121, 25124}
X(25103) lies on these lines: {2, 24664}, {960, 21231}, {25102, 25119}, {25106, 25112}, {25107, 25139}, {25122, 25138}
X(25104) lies on these lines: {2, 24664}, {141, 1329}, {1714, 24174}, {1755, 18738}, {2083, 16560}, {14829, 20917}, {16599, 25071}, {25108, 25109}, {25120, 25122}
X(25105) lies on these lines: {2, 24664}, {25102, 25110}
X(25106) lies on these lines: {2, 3728}, {37, 6377}, {43, 20923}, {75, 16569}, {740, 3216}, {899, 20891}, {1215, 3739}, {1698, 4751}, {3752, 21080}, {16706, 17793}, {17337, 24327}, {17448, 24659}, {24655, 24735}, {25102, 25108}, {25103, 25112}, {25111, 25115}, {25121, 25125}, {25133, 25141}, {25140, 25146}
X(25107) lies on these lines: {1, 24745}, {2, 17448}, {8, 20530}, {9, 24343}, {10, 3934}, {141, 2885}, {194, 1575}, {350, 21868}, {536, 18135}, {668, 16604}, {1107, 7786}, {1329, 20541}, {1574, 6381}, {3214, 4852}, {3739, 5772}, {17792, 25120}, {18140, 20691}, {25103, 25139}, {25119, 25132}, {25131, 25137}, {25140, 25145}
X(25108) lies on these lines: {2, 3056}, {37, 4941}, {141, 3740}, {210, 17232}, {518, 17234}, {960, 3836}, {3742, 17245}, {3826, 5836}, {4085, 10179}, {9025, 17337}, {15587, 16593}, {25102, 25106}, {25104, 25109}, {25121, 25146}, {25125, 25140}
X(25109) lies on these lines: {2, 17448}, {10, 20530}, {536, 18140}, {984, 1698}, {1575, 1655}, {3760, 4726}, {3828, 3934}, {3831, 17239}, {4698, 20363}, {4852, 6048}, {5257, 25376}, {9780, 21264}, {25104, 25108}, {25121, 25122}, {25132, 25139}
X(25110) lies on these lines: {2, 25119}, {25102, 25105}
X(25111) lies on these lines: {2, 24525}, {141, 210}, {742, 18138}, {2176, 21590}, {3011, 4074}, {4437, 17056}, {25102, 25123}, {25106, 25115}, {25116, 25139}
X(25112) lies on these lines: {2, 24526}, {25103, 25106}, {25122, 25133}, {25138, 25141}
X(25113) lies on these lines: {2, 18194}, {10, 4684}, {726, 18143}, {3741, 17231}, {3840, 17241}, {3971, 18144}, {4871, 21238}, {17244, 21257}, {17245, 20340}, {20917, 21080}, {25102, 25106}, {25114, 25130}, {25124, 25144}
X(25114) lies on these lines: {2, 24527}, {17448, 25140}, {25103, 25106}, {25113, 25130}, {25133, 25138}
X(25115) lies on these lines: {2, 17448}, {42, 20530}, {210, 3739}, {536, 18152}, {1575, 3770}, {25106, 25111}, {25123, 25139}
X(25116) lies on these lines: {2, 17448}, {43, 4852}, {1215, 3739}, {25111, 25139}, {25119, 25129}
X(25117) lies on these lines: {2, 23633}, {21191, 25126}, {24742, 25279}, {25102, 25106}, {25125, 25146}
X(25118) lies on these lines: {2, 24671}, {4096, 18150}, {25102, 25106}
X(25119) lies on these lines: {2, 25110}, {25102, 25103}, {25106, 25111}, {25107, 25132}, {25116, 25129}
X(25120) lies on these lines: {2, 3728}, {10, 3688}, {75, 17793}, {141, 20339}, {740, 18147}, {992, 4039}, {1045, 18140}, {1213, 24327}, {1215, 17303}, {1575, 21080}, {1740, 6376}, {2234, 18133}, {2887, 23305}, {4967, 12263}, {17279, 24003}, {17334, 25382}, {17792, 25107}, {19856, 24325}, {24524, 24661}, {25104, 25122}
X(25121) lies on these lines: {2, 18194}, {10, 75}, {37, 23643}, {141, 20340}, {1084, 1213}, {3661, 21257}, {3741, 17239}, {3840, 17228}, {3844, 15985}, {4110, 24456}, {4871, 17231}, {4970, 21858}, {17792, 25107}, {18144, 24165}, {24575, 25280}, {25102, 25124}, {25106, 25125}, {25108, 25146}, {25109, 25122}
X(25122) lies on these lines: {2, 24527}, {10, 1015}, {141, 20338}, {714, 18148}, {1698, 18194}, {3831, 3844}, {10479, 24174}, {25103, 25138}, {25104, 25120}, {25109, 25121}, {25112, 25133}
X(25123) lies on these lines: {2, 3728}, {10, 12}, {43, 312}, {714, 3752}, {2092, 3985}, {2667, 18743}, {3936, 21688}, {4647, 6048}, {6685, 10180}, {17793, 19786}, {25102, 25111}, {25115, 25139}
X(25124) lies on these lines: {1, 75}, {2, 3728}, {10, 4111}, {37, 714}, {192, 1962}, {385, 3757}, {512, 21211}, {518, 15985}, {523, 24141}, {726, 3743}, {894, 3747}, {1221, 3510}, {1402, 4032}, {1486, 8424}, {2292, 24349}, {3720, 20891}, {3739, 3741}, {4022, 6682}, {4068, 4363}, {4698, 24003}, {4699, 10453}, {4772, 17163}, {7155, 20146}, {17322, 17793}, {17398, 24327}, {17792, 24656}, {23668, 23944}, {25102, 25121}, {25113, 25144}, {25129, 25130}
X(25125) lies on these lines: {2, 17448}, {5, 10}, {37, 4110}, {239, 20530}, {312, 21868}, {518, 20340}, {536, 3264}, {1575, 3975}, {3596, 21892}, {3662, 3739}, {3687, 21025}, {3752, 6376}, {3823, 20486}, {4394, 6002}, {16602, 20917}, {17490, 20943}, {23301, 25142}, {25106, 25121}, {25108, 25140}, {25117, 25146}, {25137, 25141}
X(25126) lies on these lines: {2, 669}, {523, 18154}, {650, 3837}, {804, 25084}, {4369, 21051}, {4507, 4992}, {14426, 23803}, {21128, 21349}, {21191, 25117}, {24674, 24749}
X(25126) = complement of X(25537)
X(25127) lies on these lines: {2, 24665}, {514, 6586}, {21191, 25129}
X(25128) lies on these lines: {2, 23655}, {10, 7234}, {38, 20909}, {514, 23768}, {521, 3716}, {522, 4087}, {650, 3907}, {663, 21300}, {788, 3741}, {982, 21197}, {2526, 24720}, {3221, 21191}, {4369, 8678}, {4874, 20316}, {5737, 23865}, {7662, 24718}, {21056, 23572}, {24533, 24755}
X(25129) lies on these lines: {2, 24525}, {75, 16969}, {698, 3290}, {742, 18157}, {3242, 15668}, {3739, 5836}, {21191, 25127}, {25102, 25106}, {25116, 25119}, {25124, 25130}
X(25130) lies on these lines: {1, 3696}, {2, 17448}, {37, 194}, {274, 536}, {330, 4687}, {730, 1125}, {1100, 16827}, {1107, 4698}, {2176, 4670}, {3230, 17175}, {3616, 21264}, {4364, 24215}, {4688, 17144}, {4739, 17143}, {4755, 5283}, {6706, 24331}, {7786, 16604}, {9534, 17372}, {10436, 16969}, {25113, 25114}, {25124, 25129}
X(25131) lies on these lines: {2, 24664}, {17792, 25102}, {25107, 25137}
X(25132) lies on these lines: {2, 24664}, {65, 21231}, {85, 8680}, {2294, 3212}, {16609, 17056}, {17090, 25255}, {17792, 24656}, {25102, 25111}, {25107, 25119}, {25109, 25139}
X(25133) lies on these lines: {2, 24664}, {1107, 21232}, {25106, 25141}, {25112, 25122}, {25114, 25138}
X(25134) lies on these lines: {2, 24670}, {9055, 18052}, {25102, 25111}
X(25135) lies on these lines: {2, 3056}, {5, 10}, {11, 24997}, {238, 1376}, {325, 24471}, {518, 3705}, {908, 20487}, {978, 1104}, {3035, 6679}, {3061, 20697}, {3123, 3666}, {3742, 3756}, {4640, 6210}, {16298, 17749}, {25102, 25103}
X(25136) lies on these lines: {2, 25307}, {758, 20255}, {960, 3739}, {24654, 25295}, {25102, 25111}, {25103, 25106}, {25124, 25129}
X(25137) lies on these lines: {2, 3056}, {10, 20256}, {120, 125}, {518, 18134}, {960, 2887}, {3925, 5836}, {25102, 25111}, {25107, 25131}, {25125, 25141}
X(25138) lies on these lines: {2, 24531}, {21232, 21254}, {25103, 25122}, {25112, 25141}, {25114, 25133}
X(25139) lies on these lines: {2, 25110}, {891, 4928}, {25103, 25107}, {25109, 25132}, {25111, 25116}, {25115, 25123}
X(25140) lies on these lines: {2, 18194}, {10, 141}, {37, 20532}, {726, 18144}, {1502, 6381}, {2228, 18040}, {3741, 17228}, {3840, 17231}, {3912, 21257}, {3971, 18133}, {4518, 18207}, {4871, 17241}, {6374, 6376}, {17229, 24688}, {17234, 20340}, {17308, 20139}, {17448, 25114}, {18143, 24165}, {25106, 25146}, {25107, 25145}, {25108, 25125}
X(25141) lies on these lines: {2, 24527}, {10, 141}, {6376, 6386}, {7148, 18040}, {20366, 24443}, {21024, 24688}, {25106, 25133}, {25112, 25138}, {25125, 25137}
X(25142) lies on these lines: {2, 20983}, {42, 23506}, {43, 8640}, {44, 513}, {693, 14404}, {756, 21350}, {3221, 24675}, {3808, 11068}, {3835, 4083}, {3837, 20316}, {4010, 4397}, {4383, 20473}, {4940, 9400}, {21191, 25117}, {23301, 25125}, {23655, 24532}, {24666, 24747}
X(25143) lies on these lines: {2, 23655}, {984, 21197}, {3835, 4507}, {4147, 4885}, {21191, 25117}
X(25143) = complement of X(24666)
X(25144) lies on these lines: {2, 3056}, {6, 16569}, {10, 141}, {37, 19584}, {210, 17238}, {354, 17232}, {511, 3634}, {613, 16862}, {978, 1386}, {1100, 17795}, {1428, 17531}, {1469, 9780}, {3589, 6686}, {3619, 3779}, {3705, 3742}, {3740, 5224}, {3799, 17324}, {3838, 20545}, {3841, 24206}, {3848, 17245}, {3880, 4085}, {3912, 4890}, {3925, 24997}, {4357, 7064}, {4429, 5836}, {4553, 17384}, {4699, 19586}, {5249, 20487}, {6210, 15254}, {8167, 10387}, {9342, 15988}, {15310, 24295}, {17235, 21865}, {17291, 20358}, {25106, 25121}, {25113, 25124}
X(25145) lies on these lines: {2, 24670}, {25102, 25106}, {25107, 25140}, {25109, 25121}
X(25146) lies on these lines: {2, 24671}, {537, 18150}, {3837, 14426}, {25106, 25140}, {25108, 25121}, {25113, 25124}, {25117, 25125}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.
X(25147) lies on these lines: {4,12026}, {5,128}, {381,9143}, {546,1141}, {930,3628}, {1656,6592}, {3090,13512}, {3327,10593}, {3459,14143}, {6343,8254}, {7159,10592}, {7741,14101}, {10126,20413}, {13406,15367}, {14652,18378}, {15307,22051}
X(25147) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,137,1263}, {5,1263,14072}, {1656,11671,6592}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.
X(25148) lies on the hyperbola {{A, B, C, X(4), X(5)}} and these lines: {5,930}, {137,252}, {1263,19552}, {10126,20413}, {15643,21394}, {20030,22335}
X(25148) = isogonal conjugate of X(6150)
X(25148) = reflection of X(i) in X(j) for these {i,j}: {930,21975}, {3459,137}
X(25148) = antigonal conjugate of X(3459)
X(25148) = antipode in hyperbola {{A, B, C, X(4), X(5)}} of X(3459)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.
X(25149) lies on these lines: {30,511}, {128,8562}, {137,8901}, {1141,10412}, {6132,14769}, {14225,15619}
X(25149) = isogonal conjugate of anticomplement of X(35591)
X(25149) = complementary conjugate of X(35591)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28460.
X(25150) lies on these lines: {1,3327}, {3,252}, {4,11671}, {5,128}, {12,14101}, {26,15959}, {30,511}, {54,14071}, {140,6592}, {156,6069}, {1157,19553}, {5576,14769}, {5889,13505}, {7387,15960}, {7512,14652}, {7568,13467}, {10205,14143}, {11412,13504}, {13160,15367}, {14118,14674}, {14865,14978}, {15425,15957}
X(25150) = isogonal conjugate of X(15907)
X(25150) = orthic isogonal conjugate of X(33333)
Related centers to Fermat-Dao equilateral triangles: X(25151)-X(25236)
This preamble and centers X(25151)-X(25236) were contributed by César Eliud Lozada, October 13, 2018.
For definitions and coordinates of all Fermat-Dao equilateral triangles, see Index of triangles referenced in ETC.
A complete list of orthologic and parallelogic triangles to Fermat-Dao triangles and centers can be downloaded from here.
The reciprocal orthologic center of these triangles is X(25152)
X(25151) lies on these lines: {17,14182}, {61,25180}, {396,23007}, {511,16529}, {512,5470}, {6787,22846}, {11542,22999}, {16267,25217}, {16808,23017}, {16960,25182}
X(25151) = reflection of X(25217) in X(16267)
X(25151) = {X(16960), X(25182)}-harmonic conjugate of X(25220)
The reciprocal orthologic center of these triangles is X(25151)
X(25152) lies on these lines: {13,11085}, {14,1916}, {531,25215}, {2782,16462}, {6033,11582}, {8015,25155}, {14181,16464}, {16639,25207}
The reciprocal orthologic center of these triangles is X(25151)
X(25153) lies on these lines: {6,23896}, {13,11085}, {39,395}, {523,11624}, {531,25221}, {2782,11626}, {2854,25232}, {11083,12205}, {11141,14181}, {16641,25207}
X(25153) = reflection of X(25231) in X(11626)
The reciprocal orthologic center of these triangles is X(5470)
X(25154) lies on these lines: {2,9736}, {3,5459}, {4,542}, {5,5463}, {13,15}, {20,20415}, {62,5469}, {115,10653}, {376,6771}, {381,530}, {395,23006}, {511,25223}, {512,25175}, {524,20428}, {531,6321}, {543,5613}, {549,5473}, {616,3545}, {618,5055}, {622,11178}, {2782,22694}, {3058,10062}, {3543,6770}, {3655,11705}, {3839,22796}, {5054,6669}, {5335,11179}, {5434,10078}, {5470,16631}, {5472,10654}, {5969,25195}, {6054,22797}, {6774,9166}, {6778,22515}, {8703,21156}, {8724,16627}, {9115,18581}, {9140,16770}, {9762,9771}, {9830,22575}, {10056,13076}, {10072,18974}, {12355,16629}, {16808,25156}, {16809,22998}
X(25154) = midpoint of X(i) and X(j) for these {i,j}: {381, 13103}, {3543, 6770}
X(25154) = reflection of X(i) in X(j) for these (i,j): (3, 5459), (376, 6771), (381, 5478), (549, 20252), (3655, 11705), (5463, 5), (5473, 549), (5617, 381), (6054, 22797), (25164, 9880)
X(25154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 20423, 25164), (13, 23005, 6772), (549, 20252, 22489), (5473, 22489, 549), (5478, 13103, 5617)
The reciprocal orthologic center of these triangles is X(25151)
X(25155) lies on these lines: {2,3107}, {6,23896}, {13,94}, {2088,6772}, {2782,5640}, {3129,14181}, {6582,11146}, {8015,25152}, {15018,22687}, {16637,25207}, {16770,23005}
X(25155) = reflection of X(25233) in X(5640)
The reciprocal orthologic center of these triangles is X(5470)
X(25156) lies on these lines: {3,13}, {14,530}, {16,5470}, {30,6778}, {99,21360}, {115,6779}, {148,532}, {511,25227}, {512,25177}, {524,22578}, {542,19107}, {543,22494}, {622,22850}, {2782,22696}, {5463,16967}, {5469,9115}, {5969,25203}, {6781,16962}, {8595,22490}, {9116,9886}, {10654,20088}, {12355,25166}, {16808,25154}, {19106,19924}
X(25156) = reflection of X(i) in X(j) for these (i,j): (14, 23005), (6779, 115), (25235, 14)
X(25156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 6779, 16963), (5469, 9115, 16961)
The reciprocal orthologic center of these triangles is X(6582)
X(25157) lies on these lines: {2,3106}, {13,76}, {14,24256}, {15,3734}, {16,183}, {17,6581}, {39,16644}, {61,7770}, {62,7751}, {141,23024}, {381,511}, {524,22701}, {538,3107}, {624,3314}, {633,16044}, {635,5025}, {2782,5464}, {3104,3934}, {5459,22688}, {5463,22712}, {5470,5969}, {5981,8289}, {6671,7835}, {6672,17004}, {6694,16895}, {6695,7856}, {7761,23005}, {11185,23004}, {11305,22846}, {11542,23000}, {16634,18906}, {16808,23018}, {16960,25199}, {22486,22493}, {22489,22691}
X(25157) = reflection of X(i) in X(j) for these (i,j): (3106, 2), (25167, 9466)
X(25157) = {X(76), X(11303)}-harmonic conjugate of X(25195)
The reciprocal orthologic center of these triangles is X(6298)
X(25158) lies on these lines: {2,22688}, {13,83}, {17,6296}, {61,25192}, {754,16267}, {5459,22510}, {11542,23001}, {11632,25168}, {16808,23019}, {16960,25200}
The reciprocal orthologic center of these triangles is X(13705)
X(25159) lies on these lines: {13,1327}, {17,13706}, {61,25193}, {530,22631}, {11542,23002}, {16808,23020}, {16960,25201}
The reciprocal orthologic center of these triangles is X(13825)
X(25160) lies on these lines: {13,1328}, {17,13826}, {61,25194}, {530,22602}, {11542,23003}, {16808,23021}, {16960,25202}
The reciprocal orthologic center of these triangles is X(25162)
X(25161) lies on these lines: {14,25173}, {18,14178}, {62,25175}, {395,23014}, {511,16530}, {512,5469}, {6787,22891}, {11543,23008}, {16268,25214}, {16809,23023}, {16961,25177}
X(25161) = reflection of X(25214) in X(16268)
X(25161) = {X(16961), X(25177)}-harmonic conjugate of X(25219)
The reciprocal orthologic center of these triangles is X(25161)
X(25162) lies on these lines: {14,11080}, {530,25218}, {2782,16461}, {6033,11581}, {8014,25165}, {14177,16463}, {16638,25208}
The reciprocal orthologic center of these triangles is X(25161)
X(25163) lies on these lines: {6,23895}, {14,11080}, {39,396}, {523,11626}, {530,25222}, {2782,11624}, {2854,25231}, {11088,12204}, {11142,14177}, {16640,25208}
X(25163) = reflection of X(25232) in X(11624)
The reciprocal orthologic center of these triangles is X(5469)
X(25164) lies on these lines: {2,9735}, {3,5460}, {4,542}, {5,5464}, {14,16}, {20,20416}, {61,5470}, {115,10654}, {376,6774}, {381,531}, {396,23013}, {511,25224}, {512,25180}, {524,20429}, {530,6321}, {543,5617}, {549,5474}, {617,3545}, {619,5055}, {621,11178}, {2782,22693}, {3058,10061}, {3543,6773}, {3655,11706}, {3839,22797}, {5054,6670}, {5334,11179}, {5434,10077}, {5469,16630}, {5471,10653}, {5969,25191}, {6054,22796}, {6771,9166}, {6777,22515}, {8703,21157}, {8724,16626}, {9117,18582}, {9140,16771}, {9760,9771}, {9830,22576}, {10056,13075}, {10072,18975}, {12355,16628}, {16808,22997}, {16809,25166}
X(25164) = midpoint of X(i) and X(j) for these {i,j}: {381, 13102}, {3543, 6773}
X(25164) = reflection of X(i) in X(j) for these (i,j): (3, 5460), (376, 6774), (381, 5479), (549, 20253), (3655, 11706), (5464, 5), (5474, 549), (5613, 381), (6054, 22796), (25154, 9880)
X(25164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 20423, 25154), (14, 23004, 6775), (549, 20253, 22490), (5474, 22490, 549), (5479, 13102, 5613)
The reciprocal orthologic center of these triangles is X(25161)
X(25165) lies on these lines: {2,3106}, {6,23895}, {14,94}, {530,25226}, {2088,6775}, {2782,5640}, {3130,14177}, {6295,11145}, {8014,25162}, {15018,22689}, {16636,25208}, {16771,23004}
X(25165) = reflection of X(25234) in X(5640)
The reciprocal orthologic center of these triangles is X(5469)
X(25166) lies on these lines: {3,14}, {13,531}, {15,5469}, {30,6777}, {99,21359}, {115,6780}, {148,533}, {511,25228}, {512,25182}, {524,22577}, {542,19106}, {543,22493}, {621,22894}, {2782,22695}, {5464,16966}, {5470,9117}, {5969,25199}, {6781,16963}, {8594,22489}, {9114,9885}, {10653,20088}, {12355,25156}, {16809,25164}, {19107,19924}
X(25166) = reflection of X(i) in X(j) for these (i,j): (13, 23004), (6780, 115), (25236, 13)
X(25166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 6780, 16962), (5470, 9117, 16960)
The reciprocal orthologic center of these triangles is X(6295)
X(25167) lies on these lines: {2,3107}, {13,24256}, {14,76}, {15,183}, {16,3734}, {18,6294}, {39,16645}, {61,7751}, {62,7770}, {141,23018}, {381,511}, {524,22702}, {538,3106}, {623,3314}, {634,16044}, {636,5025}, {2782,5463}, {3105,3934}, {5460,22690}, {5464,22712}, {5469,5969}, {5980,8289}, {6671,17004}, {6672,7835}, {6694,7856}, {6695,16895}, {7761,23004}, {11185,23005}, {11306,22891}, {11543,23009}, {16635,18906}, {16809,23024}, {16961,25203}, {22486,22494}, {22490,22692}
X(25167) = reflection of X(i) in X(j) for these (i,j): (3107, 2), (25157, 9466)
X(25167) = {X(76), X(11304)}-harmonic conjugate of X(25191)
The reciprocal orthologic center of these triangles is X(6299)
X(25168) lies on these lines: {2,22690}, {14,83}, {18,6297}, {62,25196}, {754,16268}, {5460,22511}, {11543,23010}, {11632,25158}, {16809,23025}, {16961,25204}
The reciprocal orthologic center of these triangles is X(13703)
X(25169) lies on these lines: {14,1327}, {18,13704}, {62,25197}, {531,22633}, {11543,23011}, {16809,23026}, {16961,25205}
The reciprocal orthologic center of these triangles is X(13823)
X(25170) lies on these lines: {14,1328}, {18,13824}, {62,25198}, {531,22604}, {11543,23012}, {16809,23027}, {16961,25206}
The reciprocal orthologic center of these triangles is X(25172)
X(25171) lies on these lines: {530,25230}, {8014,25181}, {11080,25179}, {14175,16463}, {16461,25218}, {16638,25209}
The reciprocal orthologic center of these triangles is X(25171)
X(25172) lies on these lines: {531,25229}, {8015,25176}, {11085,25174}, {14176,16464}, {16462,25215}, {16639,25210}
The reciprocal orthologic center of these triangles is X(25162)
X(25173) lies on these lines: {6,14186}, {14,25161}, {16,23014}, {115,512}, {381,23023}, {395,25219}, {511,9115}, {6785,22512}, {10653,25175}, {11486,23028}, {14178,16645}, {16268,23008}, {16961,25214}, {16963,25177}, {18581,25223}
X(25173) = midpoint of X(16) and X(23014)
X(25173) = reflection of X(i) in X(j) for these (i,j): (25178, 14113), (25219, 395)
X(25173) = {X(16961), X(25227)}-harmonic conjugate of X(25214)
The reciprocal orthologic center of these triangles is X(25171)
X(25174) lies on these lines: {6,691}, {512,11624}, {531,25231}, {11085,25172}, {11141,14176}, {11626,25221}, {16641,25210}
X(25174) = reflection of X(25221) in X(11626)
The reciprocal orthologic center of these triangles is X(25162)
X(25175) lies on these lines: {5,14178}, {13,23014}, {30,14186}, {62,25161}, {298,511}, {381,25223}, {395,23028}, {512,25154}, {5318,23023}, {6785,25180}, {10653,25173}, {11080,11624}, {16808,25177}, {16809,23008}, {18581,25219}
X(25175) = reflection of X(i) in X(j) for these (i,j): (14178, 5), (25223, 381)
The reciprocal orthologic center of these triangles is X(25171)
X(25176) lies on these lines: {6,691}, {511,8595}, {531,25233}, {3129,14176}, {5640,25225}, {8015,25172}, {10654,11002}, {14174,15018}, {16637,25210}
X(25176) = reflection of X(25225) in X(5640)
The reciprocal orthologic center of these triangles is X(25162)
X(25177) lies on these lines: {14,23008}, {511,25235}, {512,25156}, {14178,16967}, {14186,16241}, {16808,25175}, {16961,25161}, {16963,25173}, {16965,23023}
X(25177) = reflection of X(i) in X(j) for these (i,j): (14, 23014), (25227, 14)
X(25177) = {X(25161), X(25219)}-harmonic conjugate of X(16961)
The reciprocal orthologic center of these triangles is X(25152)
X(25178) lies on these lines: {6,14188}, {15,23007}, {115,512}, {381,23017}, {396,25220}, {511,9117}, {6785,22513}, {10654,25180}, {11081,16248}, {11485,23022}, {14182,16644}, {16267,22999}, {16960,25217}, {16962,25182}, {18582,25224}
X(25178) = midpoint of X(15) and X(23007)
X(25178) = reflection of X(i) in X(j) for these (i,j): (25173, 14113), (25220, 396)
X(25178) = {X(16960), X(25228)}-harmonic conjugate of X(25217)
The reciprocal orthologic center of these triangles is X(25172)
X(25179) lies on these lines: {6,691}, {512,11626}, {530,25232}, {11080,25171}, {11142,14175}, {11624,25222}, {16640,25209}
X(25179) = reflection of X(25222) in X(11624)
The reciprocal orthologic center of these triangles is X(25152)
X(25180) lies on these lines: {5,14182}, {14,23007}, {30,14188}, {61,25151}, {299,383}, {381,25224}, {396,23022}, {512,25164}, {5321,23017}, {6785,25175}, {10654,25178}, {11085,11626}, {16808,22999}, {16809,25182}, {18582,25220}
X(25180) = reflection of X(i) in X(j) for these (i,j): (14182, 5), (25224, 381)
The reciprocal orthologic center of these triangles is X(25172)
X(25181) lies on these lines: {6,691}, {511,8594}, {530,25234}, {3130,14175}, {5640,25226}, {8014,25171}, {10653,11002}, {14180,15018}, {16636,25209}
X(25181) = reflection of X(25226) in X(5640)
The reciprocal orthologic center of these triangles is X(25152)
X(25182) lies on these lines: {13,22999}, {511,25236}, {512,25166}, {14182,16966}, {14188,16242}, {16809,25180}, {16960,25151}, {16962,25178}, {16964,23017}
X(25182) = reflection of X(i) in X(j) for these (i,j): (13, 23007), (25228, 13)
X(25182) = {X(25151), X(25220)}-harmonic conjugate of X(16960)
The reciprocal orthologic center of these triangles is X(6582)
X(25183) lies on these lines: {13,76}, {39,22892}, {99,6295}, {115,141}, {183,6582}, {381,23018}, {395,18806}, {396,538}, {511,22796}, {616,6194}, {618,5976}, {3934,22847}, {6581,16644}, {6669,22691}, {10654,25191}, {13876,13877}, {13929,13930}, {16267,23000}, {16635,18906}, {16962,25199}
The reciprocal orthologic center of these triangles is X(6298)
X(25184) lies on these lines: {13,83}, {115,3589}, {381,23019}, {396,754}, {618,22691}, {3329,14904}, {6109,6669}, {6292,22892}, {6296,16644}, {6298,11174}, {6704,22847}, {10654,25192}, {13876,13878}, {13929,13931}, {16267,23001}, {16962,25200}
The reciprocal orthologic center of these triangles is X(13705)
X(25185) lies on these lines: {13,1327}, {115,13703}, {381,23020}, {530,10668}, {5459,13929}, {6302,13701}, {6306,22847}, {9112,22541}, {10654,25193}, {13706,16644}, {16267,23002}, {16962,25201}
The reciprocal orthologic center of these triangles is X(13825)
X(25186) lies on these lines: {13,1328}, {115,13823}, {381,23021}, {530,10672}, {5459,13850}, {6302,22847}, {6306,13821}, {9112,19101}, {10654,25194}, {13826,16644}, {16267,23003}, {16962,25202}
The reciprocal orthologic center of these triangles is X(6295)
X(25187) lies on these lines: {14,76}, {39,22848}, {99,6582}, {115,141}, {183,6295}, {381,23024}, {395,538}, {396,18806}, {511,22797}, {617,6194}, {619,5976}, {3934,22893}, {6294,16645}, {6670,22692}, {10653,25195}, {13875,13877}, {13928,13930}, {16268,23009}, {16634,18906}, {16963,25203}
The reciprocal orthologic center of these triangles is X(6299)
X(25188) lies on these lines: {14,83}, {115,3589}, {381,23025}, {395,754}, {619,22692}, {3329,14905}, {6108,6670}, {6292,22848}, {6297,16645}, {6299,11174}, {6704,22893}, {10653,25196}, {13875,13878}, {13928,13931}, {16268,23010}, {16963,25204}
The reciprocal orthologic center of these triangles is X(13703)
X(25189) lies on these lines: {14,1327}, {115,13703}, {381,23026}, {531,10667}, {5460,13928}, {6303,13701}, {6307,22893}, {9113,22541}, {10653,25197}, {13704,16645}, {16268,23011}, {16963,25205}
The reciprocal orthologic center of these triangles is X(13823)
X(25190) lies on these lines: {14,1328}, {115,13823}, {381,23027}, {531,10671}, {5460,13850}, {6303,22893}, {6307,13821}, {9113,19101}, {10653,25198}, {13824,16645}, {16268,23012}, {16963,25206}
The reciprocal orthologic center of these triangles is X(6582)
X(25191) lies on these lines: {5,6581}, {14,76}, {39,11305}, {61,7770}, {194,16634}, {381,538}, {533,7812}, {732,25196}, {2782,6294}, {3107,11303}, {5321,23018}, {5969,25164}, {9466,11297}, {9760,13085}, {10654,25183}, {16808,23000}, {16809,25199}
X(25191) = reflection of X(i) in X(j) for these (i,j): (6581, 5), (25195, 6248)
X(25191) = {X(76), X(11304)}-harmonic conjugate of X(25167)
The reciprocal orthologic center of these triangles is X(6298)
X(25192) lies on these lines: {5,6296}, {14,83}, {61,25158}, {316,25204}, {381,754}, {732,25195}, {5321,23019}, {6292,11306}, {7770,16964}, {10654,25184}, {16634,20088}, {16808,23001}, {16809,25200}
X(25192) = reflection of X(i) in X(j) for these (i,j): (6296, 5), (25196, 6249)
The reciprocal orthologic center of these triangles is X(13705)
X(25193) lies on these lines: {5,13706}, {14,1327}, {61,25159}, {381,13687}, {530,22634}, {5321,23020}, {10654,25185}, {16808,23002}, {16809,25201}
X(25193) = reflection of X(i) in X(j) for these (i,j): (13706, 5), (25197, 13687)
The reciprocal orthologic center of these triangles is X(13825)
X(25194) lies on these lines: {5,13826}, {14,1328}, {61,25160}, {381,13807}, {530,22605}, {5321,23021}, {10654,25186}, {16808,23003}, {16809,25202}
X(25194) = reflection of X(i) in X(j) for these (i,j): (13826, 5), (25198, 13807)
The reciprocal orthologic center of these triangles is X(6295)
X(25195) lies on these lines: {5,6294}, {13,76}, {39,11306}, {62,7770}, {194,16635}, {381,538}, {532,7812}, {732,25192}, {2782,6581}, {3106,11304}, {5318,23024}, {5969,25154}, {9466,11298}, {9762,13085}, {10653,25187}, {16808,25203}, {16809,23009}
X(25195) = reflection of X(i) in X(j) for these (i,j): (6294, 5), (25191, 6248)
X(25195) = {X(76), X(11303)}-harmonic conjugate of X(25157)
The reciprocal orthologic center of these triangles is X(6299)
X(25196) lies on these lines: {5,6297}, {13,83}, {62,25168}, {316,25200}, {381,754}, {732,25191}, {5318,23025}, {6292,11305}, {7770,16965}, {10653,25188}, {16635,20088}, {16808,25204}, {16809,23010}
X(25196) = reflection of X(i) in X(j) for these (i,j): (6297, 5), (25192, 6249)
The reciprocal orthologic center of these triangles is X(13703)
X(25197) lies on these lines: {5,13704}, {13,1327}, {62,25169}, {381,13687}, {531,22635}, {5318,23026}, {10653,25189}, {16808,25205}, {16809,23011}
X(25197) = reflection of X(i) in X(j) for these (i,j): (13704, 5), (25193, 13687)
The reciprocal orthologic center of these triangles is X(13823)
X(25198) lies on these lines: {5,13824}, {13,1328}, {62,25170}, {381,13807}, {531,22606}, {5318,23027}, {10653,25190}, {16808,25206}, {16809,23012}
X(25198) = reflection of X(i) in X(j) for these (i,j): (13824, 5), (25194, 13807)
The reciprocal orthologic center of these triangles is X(6582)
X(25199) lies on these lines: {13,538}, {18,76}, {621,22695}, {622,20081}, {732,23024}, {3818,13108}, {5969,25166}, {6581,16966}, {16809,25191}, {16960,25157}, {16962,25183}, {16964,23018}
The reciprocal orthologic center of these triangles is X(6298)
X(25200) lies on these lines: {13,754}, {18,83}, {316,25196}, {621,6777}, {624,2896}, {6296,16966}, {13111,19130}, {16809,25192}, {16960,25158}, {16962,25184}, {16964,23019}
The reciprocal orthologic center of these triangles is X(13705)
X(25201) lies on these lines: {13,23002}, {18,1327}, {530,22636}, {13706,16966}, {13713,25205}, {16809,25193}, {16960,25159}, {16962,25185}, {16964,23020}
The reciprocal orthologic center of these triangles is X(13825)
X(25202) lies on these lines: {13,23003}, {18,1328}, {530,22607}, {13826,16966}, {13836,25206}, {16809,25194}, {16960,25160}, {16962,25186}, {16964,23021}
The reciprocal orthologic center of these triangles is X(6295)
X(25203) lies on these lines: {14,538}, {17,76}, {621,20081}, {622,22696}, {732,23018}, {3818,13108}, {5969,25156}, {6294,16967}, {16808,25195}, {16961,25167}, {16963,25187}, {16965,23024}
The reciprocal orthologic center of these triangles is X(6299)
X(25204) lies on these lines: {14,754}, {17,83}, {316,25192}, {622,6778}, {623,2896}, {6297,16967}, {13111,19130}, {16808,25196}, {16961,25168}, {16963,25188}, {16965,23025}
The reciprocal orthologic center of these triangles is X(13703)
X(25205) lies on these lines: {14,23011}, {17,1327}, {531,22637}, {13704,16967}, {13713,25201}, {16808,25197}, {16961,25169}, {16963,25189}, {16965,23026}
The reciprocal orthologic center of these triangles is X(13823)
X(25206) lies on these lines: {14,23012}, {17,1328}, {531,22608}, {13824,16967}, {13836,25202}, {16808,25198}, {16961,25170}, {16963,25190}, {16965,23027}
The reciprocal orthologic center of these triangles is X(25151)
X(25207) lies on these lines: {13,99}, {530,16248}, {531,25212}, {2782,16260}, {14181,16643}, {16262,25208}, {16637,25155}, {16639,25152}, {16641,25153}
X(25207) = reflection of X(25216) in X(16260)
The reciprocal orthologic center of these triangles is X(25161)
X(25208) lies on these lines: {14,99}, {530,25211}, {531,16247}, {2782,16259}, {14177,16642}, {16262,25207}, {16636,25165}, {16638,25162}, {16640,25163}
X(25208) = reflection of X(25213) in X(16259)
The reciprocal orthologic center of these triangles is X(25210)
X(25209) lies on these lines: {530,25213}, {14175,16642}, {16259,25211}, {16636,25181}, {16638,25171}, {16640,25179}
X(25209) = reflection of X(25211) in X(16259)
The reciprocal orthologic center of these triangles is X(25209)
X(25210) lies on these lines: {531,25216}, {14176,16643}, {16260,25212}, {16637,25176}, {16639,25172}, {16641,25174}
X(25210) = reflection of X(25212) in X(16260)
The reciprocal parallelogic center of these triangles is X(25212)
X(25211) lies on these lines: {530,25208}, {14183,16642}, {16259,25209}, {16636,25226}, {16638,25218}, {16640,25222}
X(25211) = reflection of X(25209) in X(16259)
The reciprocal parallelogic center of these triangles is X(25211)
X(25212) lies on these lines: {531,25207}, {14184,16643}, {16260,25210}, {16637,25225}, {16639,25215}, {16641,25221}
X(25212) = reflection of X(25210) in X(16260)
The reciprocal parallelogic center of these triangles is X(25214)
X(25213) lies on these lines: {530,25209}, {2782,16259}, {14185,16642}, {16262,25216}, {16636,25234}, {16638,25230}, {16640,25232}
X(25213) = reflection of X(25208) in X(16259)
The reciprocal parallelogic center of these triangles is X(25213)
X(25214) lies on these lines: {14,25219}, {18,14186}, {62,25223}, {395,23008}, {511,5469}, {512,16530}, {11543,23014}, {16268,25161}, {16809,23028}, {16961,25173}
X(25214) = reflection of X(25161) in X(16268)
X(25214) = {X(16961), X(25227)}-harmonic conjugate of X(25173)
The reciprocal parallelogic center of these triangles is X(25211)
X(25215) lies on these lines: {531,25152}, {8015,25225}, {11085,25221}, {14184,16464}, {16462,25172}, {16639,25212}
The reciprocal parallelogic center of these triangles is X(25217)
X(25216) lies on these lines: {531,25210}, {2782,16260}, {14187,16643}, {16262,25213}, {16639,25229}, {16641,25231}
X(25216) = reflection of X(25207) in X(16260)
The reciprocal parallelogic center of these triangles is X(25216)
X(25217) lies on these lines: {13,25220}, {17,14188}, {61,25224}, {396,22999}, {511,5470}, {512,16529}, {11542,23007}, {16267,25151}, {16808,23022}, {16960,25178}
X(25217) = reflection of X(25151) in X(16267)
X(25217) = {X(16960), X(25228)}-harmonic conjugate of X(25178)
The reciprocal parallelogic center of these triangles is X(25212)
X(25218) lies on these lines: {530,25162}, {16461,25171}, {16638,25211}
The reciprocal parallelogic center of these triangles is X(25213)
X(25219) lies on these lines: {6,14178}, {14,25214}, {16,23008}, {115,511}, {381,23028}, {395,25173}, {512,9115}, {10653,25223}, {11486,23023}, {14186,16645}, {16268,23014}, {16961,25161}, {16963,25227}, {18581,25175}
X(25219) = midpoint of X(16) and X(23008)
X(25219) = reflection of X(25173) in X(395)
X(25219) = {X(16961), X(25177)}-harmonic conjugate of X(25161)
The reciprocal parallelogic center of these triangles is X(25216)
X(25220) lies on these lines: {6,14182}, {13,25217}, {15,22999}, {115,511}, {381,23022}, {396,25178}, {512,9117}, {10654,25224}, {11485,23017}, {14188,16644}, {16267,23007}, {16960,25151}, {16962,25228}, {18582,25180}
X(25220) = midpoint of X(15) and X(22999)
X(25220) = reflection of X(25178) in X(396)
X(25220) = {X(16960), X(25182)}-harmonic conjugate of X(25151)
The reciprocal parallelogic center of these triangles is X(25211)
X(25221) lies on these lines: {6,25225}, {531,25153}, {671,25222}, {11085,25215}, {11141,14184}, {11626,25174}, {16641,25212}
X(25221) = reflection of X(25174) in X(11626)
The reciprocal parallelogic center of these triangles is X(25212)
X(25222) lies on these lines: {6,25226}, {530,25163}, {671,25221}, {11080,25218}, {11142,14183}, {11624,25179}, {16640,25211}
X(25222) = reflection of X(25179) in X(11624)
The reciprocal parallelogic center of these triangles is X(25213)
X(25223) lies on these lines: {5,14186}, {13,23008}, {30,14178}, {62,25214}, {381,25175}, {395,23023}, {511,25154}, {512,5617}, {5318,23028}, {6787,11178}, {10653,25219}, {16808,25227}, {16809,23014}, {18581,25173}
X(25223) = reflection of X(i) in X(j) for these (i,j): (14186, 5), (25175, 381)
The reciprocal parallelogic center of these triangles is X(25216)
X(25224) lies on these lines: {5,14188}, {14,22999}, {30,14182}, {61,25217}, {381,25180}, {396,23017}, {511,25164}, {512,5613}, {5321,23022}, {6787,11178}, {10654,25220}, {16808,23007}, {16809,25228}, {18582,25178}
X(25224) = reflection of X(i) in X(j) for these (i,j): (14188, 5), (25180, 381)
The reciprocal parallelogic center of these triangles is X(25211)
X(25225) lies on these lines: {6,25221}, {23,14174}, {511,25154}, {621,11581}, {3129,14184}, {5640,25176}, {8015,25215}, {11317,25226}, {16637,25212}
X(25225) = reflection of X(25176) in X(5640)
The reciprocal parallelogic center of these triangles is X(25212)
X(25226) lies on these lines: {6,25222}, {23,14180}, {511,25164}, {530,25165}, {622,11582}, {3130,14183}, {5640,25181}, {8014,25218}, {11317,25225}, {16636,25211}
X(25226) = reflection of X(25181) in X(5640)
The reciprocal parallelogic center of these triangles is X(25213)
X(25227) lies on these lines: {14,23008}, {511,25156}, {512,25235}, {14178,16241}, {14186,16967}, {16808,25223}, {16961,25173}, {16963,25219}, {16965,23028}
X(25227) = reflection of X(i) in X(j) for these (i,j): (14, 23008), (25177, 14)
X(25227) = {X(25173), X(25214)}-harmonic conjugate of X(16961)
The reciprocal parallelogic center of these triangles is X(25216)
X(25228) lies on these lines: {13,22999}, {511,25166}, {512,25236}, {14182,16242}, {14188,16966}, {16809,25224}, {16960,25178}, {16962,25220}, {16964,23022}
X(25228) = reflection of X(i) in X(j) for these (i,j): (13, 22999), (25182, 13)
X(25228) = {X(25178), X(25217)}-harmonic conjugate of X(16960)
The reciprocal parallelogic center of these triangles is X(25217)
X(25229) lies on these lines: {531,25172}, {2782,16462}, {6321,11582}, {8015,25233}, {11085,25231}, {14187,16464}, {16639,25216}
The reciprocal parallelogic center of these triangles is X(25214)
X(25230) lies on these lines: {530,25171}, {2782,16461}, {6321,11581}, {8014,25234}, {11080,25232}, {14185,16463}, {16638,25213}
The reciprocal parallelogic center of these triangles is X(25217)
X(25231) lies on these lines: {6,25233}, {531,25174}, {2782,11626}, {2854,25163}, {11085,25229}, {11141,14187}, {16641,25216}
X(25231) = reflection of X(25153) in X(11626)
The reciprocal parallelogic center of these triangles is X(25214)
X(25232) lies on these lines: {6,25234}, {530,25179}, {2782,11624}, {2854,25153}, {11080,25230}, {11142,14185}, {16640,25213}
X(25232) = reflection of X(25163) in X(11624)
The reciprocal parallelogic center of these triangles is X(25234)
X(25233) lies on these lines: {6,25231}, {23,22687}, {531,25176}, {2782,5640}, {3129,14187}, {5617,11078}, {8015,25229}, {16637,25216}, {21466,22998}
X(25233) = reflection of X(25155) in X(5640)
The reciprocal parallelogic center of these triangles is X(25214)
X(25234) lies on these lines: {6,25232}, {23,22689}, {530,25181}, {2782,5640}, {3130,14185}, {5613,11092}, {8014,25230}, {16636,25213}, {21467,22997}
X(25234) = reflection of X(25165) in X(5640)
The reciprocal parallelogic center of these triangles is X(16529)
X(25235) lies on these lines: {13,5055}, {14,530}, {15,532}, {16,5613}, {17,5472}, {115,16530}, {187,16529}, {395,5470}, {511,25177}, {512,25227}, {524,6780}, {542,6779}, {618,21360}, {5463,5569}, {5464,8595}, {5617,16808}, {5965,6781}, {5980,7777}, {6108,23234}, {6770,10646}, {6777,13102}, {8860,9762}, {9114,10488}, {11489,22846}, {13103,16809}, {16965,23006}, {22496,22578}, {22507,23004}
X(25235) = reflection of X(i) in X(j) for these (i,j): (13, 9115), (14, 22998), (5464, 8595), (6778, 16), (19107, 6777), (22494, 5463), (22578, 22496), (23005, 6782), (25156, 14), (25236, 6781)
X(25235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 9115, 16963), (115, 16530, 16961), (6782, 23005, 14), (22998, 23005, 6782)
The reciprocal parallelogic center of these triangles is X(16530)
X(25236) lies on these lines: {13,531}, {14,5055}, {15,5617}, {16,533}, {18,5471}, {115,16529}, {187,16530}, {396,5469}, {511,25182}, {512,25228}, {524,6779}, {542,6780}, {619,21359}, {5463,8594}, {5464,5569}, {5613,16809}, {5965,6781}, {5981,7777}, {6109,23234}, {6773,10645}, {6778,13103}, {8860,9760}, {9116,10488}, {11488,22891}, {13102,16808}, {16964,23013}, {22495,22577}, {22509,23005}
X(25236) = reflection of X(i) in X(j) for these (i,j): (13, 22997), (14, 9117), (5463, 8594), (6777, 15), (19106, 6778), (22493, 5464), (22577, 22495), (23004, 6783), (25166, 13), (25235, 6781)
X(25236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 9117, 16962), (115, 16529, 16960), (6783, 23004, 13), (22997, 23004, 6783)
Collineation mappings involving Gemini triangle 17: X(25237)-X(25272)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 17, as in centers X(25237)-X(25272). Then
m(X) = a (b + c) (b + c - a) x + b (c - a)(c + a - b) y + c (b - a) (b + a - c) z : :
(Clark Kimberling, October 14, 2018)
X(25237) lies on these lines: {2, 277}, {144, 145}, {148, 1655}, {194, 16865}, {239, 3219}, {536, 4875}, {672, 20247}, {894, 11319}, {2269, 20248}, {2275, 24403}, {3208, 21272}, {3210, 24599}, {3729, 17164}, {3732, 3871}, {3970, 3995}, {4080, 17244}, {4513, 17262}, {4552, 8545}, {4936, 25268}, {7264, 24036}, {7754, 20045}, {9310, 17136}, {16920, 17350}, {17257, 17676}, {17261, 25241}, {17451, 20244}, {17760, 25253}, {20173, 24635}, {21029, 25353}, {25245, 25269}, {25246, 25254}
X(25237) = anticomplement of X(20880)
X(25238) lies on these lines: {2, 23581}, {192, 12649}, {239, 3219}, {17247, 25261}, {25241, 25247}, {25242, 25266}, {25253, 25265}
X(25239) lies on these lines: {2, 23581}, {192, 3100}, {3970, 3995}, {17489, 18359}, {25243, 25244}, {25252, 25253}
X(25240) lies on these lines: {2, 23581}, {4080, 17244}
X(25241) lies on these lines: {2, 17861}, {7, 192}, {1444, 19623}, {1764, 1999}, {2397, 17262}, {3210, 5744}, {3729, 25245}, {3995, 22019}, {4293, 20009}, {5088, 25264}, {11683, 21511}, {17033, 17489}, {17261, 25237}, {21023, 25348}, {25238, 25247}, {25246, 25249}, {25262, 25270}, {25268, 25269}
X(25242) lies on these lines: {1, 87}, {2, 277}, {8, 3177}, {9, 17691}, {20, 72}, {37, 4352}, {75, 1212}, {76, 6184}, {78, 10025}, {85, 3693}, {190, 220}, {218, 384}, {239, 16572}, {241, 312}, {279, 304}, {306, 18663}, {321, 24635}, {329, 6999}, {345, 948}, {480, 9305}, {517, 20535}, {536, 17158}, {664, 4513}, {728, 9312}, {2896, 4741}, {3008, 17490}, {3160, 4552}, {3210, 5222}, {3212, 3501}, {3474, 20715}, {3732, 5687}, {3880, 9311}, {3912, 10481}, {4201, 17257}, {4393, 7839}, {4416, 6743}, {4419, 7738}, {4515, 16284}, {4699, 20236}, {4740, 20881}, {5088, 17742}, {5308, 25263}, {5584, 11683}, {6542, 7960}, {6646, 7791}, {7229, 25255}, {7750, 17347}, {7787, 17745}, {7876, 17236}, {9607, 17246}, {17014, 17147}, {17090, 21232}, {17170, 20533}, {17181, 21073}, {17759, 21216}, {20065, 20072}, {21049, 25355}, {25238, 25266}, {25253, 25272}
X(25242) = anticomplement of X(3673)
X(25243) lies on these lines: {2, 37}, {8, 1736}, {9, 4552}, {145, 5728}, {594, 25000}, {883, 17350}, {1713, 19742}, {1766, 14953}, {2183, 21271}, {2267, 17221}, {2321, 25019}, {3161, 25252}, {3305, 18662}, {3695, 25017}, {3729, 25268}, {3731, 25255}, {3876, 20222}, {3969, 13567}, {11433, 20017}, {12848, 20110}, {16578, 17077}, {17220, 21801}, {17261, 25237}, {21273, 21371}, {24778, 25076}, {25239, 25244}, {25257, 25269}
X(25243) = anticomplement of X(20905)
X(25244) lies on these lines: {2, 277}, {37, 18600}, {144, 4190}, {192, 3622}, {894, 5764}, {2329, 17136}, {3061, 20244}, {3177, 3617}, {3219, 14953}, {3729, 19861}, {3970, 17169}, {3995, 16826}, {7229, 25252}, {16604, 24403}, {16919, 17350}, {17164, 17760}, {17355, 25260}, {17367, 17489}, {17496, 17741}, {17754, 20247}, {21029, 24318}, {25239, 25243}
X(25245) lies on these lines: {2, 37}, {190, 15988}, {740, 25024}, {894, 1442}, {3729, 25241}, {3731, 25251}, {3989, 24997}, {4451, 4651}, {11997, 20863}, {17116, 25250}, {17261, 25255}, {20881, 25065}, {25237, 25269}
X(25245) = anticomplement of X(26538)
X(25246) lies on these lines: {2, 20890}, {192, 3434}, {3995, 22015}, {25237, 25254}, {25241, 25249}
X(25246) = anticomplement of X(20890)
X(25247) lies on these lines: {2, 21414}, {192, 21285}, {25238, 25241}, {25253, 25262}, {25265, 25270}
X(25247) = anticomplement of X(21414)
X(25248) lies on these lines: {2, 986}, {192, 20247}, {384, 4427}, {846, 16931}, {1500, 17141}, {3730, 17489}, {3797, 17751}, {4414, 16822}, {4418, 16930}, {4642, 17755}, {7783, 17136}, {16549, 25263}, {16552, 17497}, {16705, 24254}, {16827, 17495}, {16910, 24248}, {17033, 17147}, {17261, 25261}, {21226, 21272}, {25238, 25241}, {25262, 25265}
X(25249) lies on these lines: {2, 277}, {37, 16708}, {192, 3873}, {194, 23407}, {335, 3995}, {1999, 18206}, {3177, 20012}, {17177, 22015}, {25241, 25246}, {25254, 25266}
X(25250) lies on these lines: {2, 24209}, {142, 192}, {239, 4552}, {4025, 25258}, {17116, 25245}, {17261, 25237}, {17339, 25252}, {25257, 25268}
X(25251) lies on these lines: {2, 20881}, {45, 2397}, {192, 4402}, {3731, 25245}, {17261, 25237}
X(25252) lies on these lines: {1, 87}, {2, 17861}, {3, 11683}, {9, 24268}, {75, 25083}, {92, 3998}, {144, 2801}, {190, 219}, {200, 21084}, {239, 1723}, {278, 345}, {306, 6360}, {312, 1214}, {322, 3694}, {326, 1944}, {329, 21078}, {346, 347}, {536, 1108}, {573, 22003}, {1278, 20881}, {1330, 18666}, {1943, 3719}, {1959, 10446}, {2256, 17262}, {2257, 3875}, {2414, 6601}, {2550, 21804}, {2893, 24316}, {3161, 25243}, {3177, 4416}, {3332, 24280}, {3668, 3912}, {4431, 4847}, {5279, 17134}, {7229, 25244}, {7560, 21376}, {17339, 25250}, {17740, 21442}, {20880, 24554}, {24179, 24559}, {25001, 25082}, {25239, 25253}
X(25252) = anticomplement of X(17861)
X(25253) lies on these lines: {1, 3159}, {2, 986}, {3, 4427}, {8, 80}, {65, 4358}, {72, 3702}, {100, 23844}, {145, 3994}, {190, 2975}, {304, 20347}, {312, 3869}, {313, 21271}, {321, 960}, {341, 14923}, {346, 21801}, {350, 20247}, {392, 4968}, {517, 3701}, {518, 22016}, {726, 1201}, {740, 25277}, {846, 16347}, {902, 8669}, {946, 3006}, {962, 10327}, {978, 17495}, {1001, 17142}, {1043, 7424}, {1125, 3977}, {1191, 3891}, {1193, 17147}, {1230, 22076}, {1265, 3434}, {1975, 17136}, {2415, 3445}, {2899, 5554}, {3057, 3967}, {3230, 22036}, {3263, 20244}, {3485, 17776}, {3555, 4742}, {3616, 17140}, {3622, 4704}, {3649, 18139}, {3681, 4673}, {3704, 5741}, {3729, 19861}, {3874, 4975}, {3876, 4651}, {3877, 4385}, {3884, 4692}, {3885, 4737}, {3886, 3984}, {3893, 4487}, {3915, 20045}, {3923, 11115}, {3924, 4011}, {3971, 10459}, {3976, 17154}, {3985, 17451}, {4009, 5836}, {4082, 4301}, {4115, 16552}, {4190, 24280}, {4387, 12635}, {4418, 19284}, {4485, 21272}, {4511, 7283}, {4552, 10571}, {4647, 10176}, {4723, 10914}, {5014, 12701}, {5016, 24703}, {5045, 17146}, {5057, 7270}, {5300, 12699}, {5687, 17780}, {6327, 11415}, {9534, 17163}, {11684, 14829}, {16466, 17150}, {16704, 17733}, {17155, 21214}, {17220, 20336}, {17760, 25237}, {18135, 24282}, {18137, 20718}, {20018, 25294}, {21674, 25385}, {25238, 25265}, {25239, 25252}, {25242, 25272}, {25247, 25262}
X(25253) = anticomplement of X(24443)
X(25254) lies on these lines: {2, 17861}, {63, 3187}, {145, 758}, {192, 3151}, {226, 3995}, {345, 20896}, {3743, 10587}, {3957, 6758}, {4137, 24248}, {5745, 17495}, {17164, 25257}, {25237, 25246}, {25249, 25266}
X(25255) lies on these lines: {2, 17861}, {7, 2294}, {8, 192}, {21, 11683}, {37, 1441}, {42, 21084}, {92, 18662}, {144, 758}, {284, 14543}, {306, 3950}, {322, 4664}, {346, 18697}, {523, 885}, {1781, 14953}, {1953, 24424}, {1959, 17183}, {1962, 10578}, {2650, 3177}, {3673, 24554}, {3729, 17164}, {3731, 25243}, {3958, 6172}, {4016, 4419}, {4032, 21808}, {4065, 6765}, {4294, 20061}, {4313, 18673}, {4431, 17163}, {4461, 4647}, {5271, 17147}, {7229, 25242}, {7672, 20718}, {7674, 24394}, {13576, 21804}, {16601, 25001}, {16609, 21811}, {16824, 25264}, {17090, 25132}, {17116, 25257}, {17139, 18714}, {17261, 25245}, {17776, 20896}, {21675, 25359}, {24987, 25260}, {24993, 25083}
X(25255) = anticomplement of X(18698)
X(25256) lies on these lines: {2, 277}, {144, 15680}, {192, 3623}, {3177, 3621}, {4393, 25264}, {17489, 25266}, {19582, 25272}
X(25257) lies on these lines: {2, 277}, {20, 912}, {192, 3945}, {194, 712}, {323, 401}, {1443, 4552}, {2795, 20556}, {3729, 25241}, {5440, 16380}, {6999, 17484}, {17116, 25255}, {17164, 25254}, {25243, 25269}, {25250, 25268}
X(25257) = anticomplement of anticomplement of X(25083)
X(25258) lies on these lines: {2, 647}, {37, 18155}, {192, 4467}, {321, 16751}, {693, 21225}, {804, 25299}, {905, 21438}, {1021, 1999}, {3210, 17069}, {3250, 20295}, {4024, 4560}, {4025, 25250}, {4524, 20012}, {4529, 24560}, {6332, 17496}, {6590, 11068}
X(25258) = anticomplement of X(24622)
X(25259) lies on these lines: {2, 2400}, {190, 1252}, {192, 23757}, {193, 9031}, {297, 525}, {513, 4122}, {514, 4024}, {522, 3935}, {523, 8663}, {649, 2786}, {650, 4467}, {652, 3219}, {661, 824}, {677, 1897}, {693, 918}, {768, 4079}, {900, 4380}, {1011, 23093}, {1491, 18004}, {1577, 23875}, {1639, 17069}, {2254, 4522}, {2340, 8714}, {2509, 16757}, {2785, 4474}, {3004, 4776}, {3151, 20298}, {3762, 23876}, {3776, 4728}, {3810, 23745}, {3835, 4120}, {3910, 4462}, {3952, 25266}, {4064, 20294}, {4130, 24562}, {4184, 22388}, {4453, 4885}, {4500, 4931}, {4656, 23801}, {4707, 4791}, {4762, 4820}, {4893, 21196}, {6332, 17496}, {6370, 23758}, {6586, 16751}, {6590, 7192}, {7253, 14954}, {17233, 21133}, {17280, 21202}, {17375, 23730}, {18155, 21611}
X(25259) = anticomplement of X(4025)
X(25259) = isotomic conjugate of trilinear pole of line X(3)X(142)
X(25259) = pole wrt polar circle of trilinear polar of X(26705) (line X(6)X(1836))
X(25259) = polar conjugate of X(26705)
X(25260) lies on these lines: {2, 20890}, {192, 2550}, {17261, 25237}, {17355, 25244}, {17489, 17752}, {24987, 25255}
X(25261) lies on these lines: {2, 277}, {9, 20247}, {37, 20448}, {192, 3617}, {1107, 24403}, {3177, 3622}, {3290, 18600}, {3488, 20071}, {3661, 3995}, {12649, 17257}, {14949, 17496}, {16815, 17495}, {17247, 25238}, {17261, 25248}, {17266, 25263}, {17494, 21201}, {20347, 21808}, {24987, 25255}
X(25262) lies on these lines: {2, 23581}, {25241, 25270}, {25247, 25253}, {25248, 25265}
X(25263) lies on these lines: {1, 4568}, {2, 1930}, {10, 17497}, {37, 16705}, {190, 17103}, {192, 3616}, {335, 17169}, {762, 8682}, {1655, 6625}, {3952, 17499}, {3995, 16826}, {4080, 17244}, {4552, 17084}, {5308, 25242}, {5692, 20109}, {9780, 21216}, {16549, 25248}, {17141, 24512}, {17147, 17397}, {17266, 25261}
X(25264) lies on these lines: {1, 87}, {2, 3760}, {10, 1655}, {35, 385}, {36, 7783}, {37, 274}, {39, 350}, {42, 17499}, {55, 7754}, {75, 5283}, {76, 2276}, {99, 172}, {144, 20018}, {148, 3585}, {190, 213}, {193, 4294}, {239, 3219}, {257, 4424}, {312, 980}, {315, 9598}, {321, 10471}, {335, 3970}, {346, 4352}, {384, 5280}, {386, 24514}, {519, 21226}, {536, 1107}, {538, 1500}, {609, 3552}, {668, 20691}, {672, 17034}, {1018, 17752}, {1045, 21080}, {1078, 4396}, {1266, 17050}, {1449, 20170}, {1479, 7774}, {1575, 18140}, {1914, 7760}, {1920, 21883}, {1930, 3797}, {1999, 18206}, {2176, 17262}, {2241, 7798}, {2242, 7781}, {2275, 7757}, {2397, 16820}, {2664, 3971}, {2996, 10590}, {3085, 6392}, {3159, 4568}, {3177, 12526}, {3210, 4384}, {3244, 9263}, {3294, 16827}, {3295, 22253}, {3583, 7785}, {3584, 19570}, {3644, 16975}, {3730, 17033}, {3761, 20081}, {3875, 21384}, {3891, 23407}, {3912, 24214}, {3920, 8267}, {3950, 24215}, {3972, 7296}, {3995, 16826}, {4037, 16720}, {4099, 7187}, {4253, 17027}, {4302, 20065}, {4324, 14712}, {4360, 20963}, {4366, 5299}, {4393, 25256}, {4441, 17030}, {4493, 14620}, {4552, 7176}, {4718, 17448}, {4857, 13571}, {5010, 7793}, {5088, 25241}, {5248, 16998}, {5259, 17000}, {5332, 7894}, {6284, 7762}, {6645, 16785}, {7031, 7766}, {7741, 7777}, {7759, 9664}, {9331, 20105}, {9534, 17257}, {9596, 11185}, {10754, 19369}, {16667, 20168}, {16815, 17495}, {16818, 17302}, {16824, 25255}, {16831, 24621}, {16832, 17490}, {16887, 21070}, {17756, 18135}, {21024, 25349}
X(25264) = anticomplement of X(20888)
X(25265) lies on these lines: {2, 24791}, {525, 644}, {6758, 24074}, {25238, 25253}, {25247, 25270}, {25248, 25262}, {25266, 25272}, {25267, 25268}
X(25266) lies on these lines: {2, 21436}, {643, 17498}, {644, 2414}, {1025, 4552}, {1331, 2398}, {3952, 25259}, {17489, 25256}, {25238, 25242}, {25249, 25254}, {25265, 25272}
X(25267) lies on these lines: {3882, 4552}, {25265, 25268}
X(25268) lies on these lines: {2, 20881}, {8, 2310}, {190, 644}, {192, 3161}, {522, 4069}, {645, 4560}, {646, 4526}, {3210, 8055}, {3729, 25243}, {4033, 4391}, {4587, 14543}, {4664, 25082}, {4750, 22003}, {4936, 25237}, {6557, 17490}, {8834, 17480}, {17183, 21809}, {17261, 25245}, {21272, 21362}, {25241, 25269}, {25250, 25257}, {25265, 25267}
X(25268) = anticomplement of isotomic conjugate of X(5382)
X(25269) lies on these lines: {2, 2415}, {6, 190}, {9, 1278}, {37, 24063}, {44, 3644}, {45, 4699}, {75, 16814}, {86, 16674}, {144, 6542}, {194, 3230}, {239, 3973}, {344, 4440}, {346, 3620}, {527, 17242}, {536, 15492}, {545, 17234}, {726, 15485}, {894, 3247}, {903, 17265}, {1266, 17338}, {1334, 20081}, {2321, 17333}, {2325, 3662}, {3159, 4257}, {3208, 21219}, {3619, 4419}, {3621, 5223}, {3630, 3943}, {3631, 4741}, {3685, 16496}, {3723, 4664}, {3758, 4681}, {3759, 4718}, {3950, 4480}, {3995, 14996}, {4000, 4473}, {4029, 17391}, {4363, 16677}, {4370, 17352}, {4384, 4821}, {4389, 17340}, {4398, 4422}, {4416, 20055}, {4431, 17331}, {4488, 17316}, {4659, 4772}, {4686, 17335}, {4715, 17386}, {4740, 17277}, {4764, 17348}, {4862, 17266}, {4873, 17287}, {4903, 17596}, {4908, 17240}, {11008, 17314}, {14997, 17147}, {16484, 24349}, {16486, 17480}, {16885, 17160}, {17229, 17329}, {17232, 17264}, {17235, 17342}, {17238, 17258}, {17246, 17354}, {17249, 17359}, {17254, 17286}, {17255, 17285}, {17268, 17274}, {17269, 17273}, {17300, 17487}, {17307, 24441}, {25237, 25245}, {25241, 25268}, {25243, 25257}
X(25269) = anticomplement of anticomplement of X(25101)
X(25270) lies on these lines: {2, 986}, {1655, 24282}, {3552, 4427}, {3797, 3869}, {3924, 6651}, {21219, 21272}, {21295, 25283}, {25237, 25245}, {25241, 25262}, {25247, 25265}
X(25271) lies on these lines: {2, 21438}, {145, 1938}, {192, 693}, {323, 401}, {650, 3210}, {824, 17147}, {3666, 21611}, {4025, 25250}, {4467, 21225}, {17132, 23733}
X(25271) = anticomplement of X(21438)
X(25272) lies on these lines: {2, 24795}, {37, 16811}, {145, 14759}, {190, 17136}, {1026, 3952}, {3732, 17780}, {4595, 21272}, {17164, 17760}, {19582, 25256}, {25242, 25253}, {25265, 25266}
Collineation mappings involving Gemini triangle 18: X(25273)-X(25313)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 18, as in centers X(25273)-X(25313). Then
m(X) = (b + c) (b c + c a + a b) x + (c - a)(b c + c a + a b) y + ( (b - a) (b c + c a + a b) z : :
(Clark Kimberling, October 14, 2018)
X(25273) lies on these lines: {2, 24527}, {7, 8}, {4598, 20996}, {25277, 25313}, {25283, 25309}, {25298, 25308}
X(25274) lies on these lines: {2, 24664}, {322, 3869}, {4485, 21272}, {4566, 7138}, {24524, 25290}, {25277, 25283}, {25278, 25310}, {25293, 25309}
X(25275) lies on these lines: {2, 24664}, {69, 313}, {279, 291}, {25279, 25280}, {25291, 25293}
X(25276) lies on these lines: {2, 24664}, {24524, 25281}
X(25277) lies on these lines: {2, 3728}, {8, 17153}, {43, 22024}, {75, 3681}, {192, 872}, {518, 20892}, {726, 3214}, {740, 25253}, {1278, 21805}, {1921, 20247}, {3264, 22277}, {3617, 4772}, {3739, 3999}, {3779, 20352}, {4361, 17142}, {4699, 17140}, {4704, 17592}, {4738, 5902}, {10009, 17141}, {17135, 20891}, {17147, 21080}, {17157, 17495}, {17349, 24351}, {17792, 25292}, {24524, 25279}, {25273, 25313}, {25274, 25283}, {25282, 25286}
X(25278) lies on these lines: {2, 17448}, {8, 76}, {69, 3264}, {75, 4678}, {144, 4110}, {145, 6376}, {304, 4723}, {350, 3621}, {391, 17786}, {519, 18135}, {646, 6172}, {1909, 3617}, {3241, 18140}, {3263, 10513}, {3266, 10327}, {3436, 20553}, {3625, 3760}, {3626, 3761}, {3632, 6381}, {3730, 23891}, {4033, 25333}, {4371, 18144}, {4668, 20888}, {4737, 20911}, {6555, 20935}, {6683, 16975}, {17756, 21226}, {17759, 20105}, {18133, 20047}, {20052, 20943}, {25274, 25310}, {25291, 25292}, {25293, 25312}
X(25279) lies on these lines: {2, 3056}, {8, 20892}, {9, 3888}, {69, 3263}, {75, 4553}, {100, 1253}, {210, 17343}, {513, 17336}, {518, 17375}, {674, 17234}, {1654, 3909}, {2550, 14923}, {3006, 7998}, {3271, 17338}, {3573, 24309}, {3662, 3688}, {3705, 3819}, {3729, 3799}, {3779, 3873}, {3781, 3869}, {3877, 4660}, {4110, 23354}, {4358, 21299}, {4517, 6646}, {6007, 17242}, {7064, 17333}, {9024, 17337}, {9025, 17349}, {17244, 21746}, {17278, 25048}, {17360, 22271}, {17364, 20683}, {17378, 22277}, {20923, 21278}, {22016, 24717}, {22294, 22301}, {24524, 25277}, {24742, 25117}, {25275, 25280}, {25290, 25297}, {25298, 25311}
X(25280) lies on these lines: {2, 17448}, {8, 350}, {10, 274}, {75, 3617}, {76, 3679}, {85, 24797}, {319, 17751}, {325, 21031}, {335, 21951}, {519, 18140}, {966, 17786}, {1078, 5258}, {1334, 4595}, {1575, 21226}, {1654, 17787}, {1655, 20691}, {2238, 17752}, {2329, 3570}, {3263, 20955}, {3264, 17271}, {3294, 23891}, {3596, 17270}, {3626, 6381}, {3661, 3975}, {3701, 17762}, {3760, 4668}, {3807, 17760}, {3934, 13466}, {4033, 17256}, {4104, 7018}, {4110, 17257}, {4441, 4678}, {4505, 24697}, {4669, 18145}, {4677, 18146}, {4690, 17790}, {4691, 20888}, {4723, 20911}, {5564, 18133}, {6626, 7257}, {9263, 16604}, {10987, 16914}, {17235, 24770}, {17759, 21868}, {18159, 20880}, {24575, 25121}, {25275, 25279}, {25292, 25293}
X(25281) lies on these lines: {2, 25110}, {24524, 25276}
X(25282) lies on these lines: {2, 24525}, {69, 4661}, {518, 18138}, {1909, 17165}, {3681, 16703}, {3952, 17149}, {24524, 25294}, {25277, 25286}, {25287, 25310}
X(25283) lies on these lines: {2, 24526}, {3681, 20955}, {21295, 25270}, {25273, 25309}, {25274, 25277}, {25293, 25313}
X(25284) lies on these lines: {2, 18194}, {8, 4772}, {10, 17178}, {319, 4651}, {894, 23354}, {4851, 17721}, {17135, 17373}, {17300, 20352}, {17792, 25295}, {24524, 25277}, {25285, 25303}
X(25285) lies on these lines: {2, 24527}, {24575, 25311}, {25274, 25277}, {25284, 25303}, {25309, 25313}
X(25286) lies on these lines: {2, 17448}, {42, 668}, {75, 4661}, {210, 18059}, {310, 4685}, {350, 20011}, {519, 18152}, {1909, 4651}, {1920, 21805}, {1965, 3935}, {1978, 4090}, {3240, 17149}, {6376, 17018}, {17752, 21753}, {20048, 20943}, {25277, 25282}, {25294, 25310}
X(25287) lies on these lines: {2, 17448}, {42, 6376}, {43, 668}, {75, 3681}, {76, 4685}, {85, 24800}, {200, 1965}, {350, 20012}, {561, 21805}, {899, 6384}, {1575, 21223}, {2238, 17786}, {3952, 8026}, {4033, 7109}, {4090, 6382}, {4110, 24514}, {17144, 18152}, {19998, 20943}, {25282, 25310}, {25290, 25302}
X(25288) lies on these lines: {2, 23633}, {20983, 25299}, {24524, 25277}
X(25289) lies on these lines: {2, 24671}, {4553, 17165}, {24524, 25277}
X(25290) lies on these lines: {2, 25110}, {8, 21416}, {3681, 16284}, {3930, 20248}, {24524, 25274}, {25277, 25282}, {25279, 25297}, {25287, 25302}
X(25291) lies on these lines: {2, 3728}, {8, 4710}, {69, 20351}, {346, 3949}, {1278, 17794}, {1654, 24351}, {3056, 25298}, {3681, 17787}, {6327, 11677}, {25275, 25293}, {25278, 25292}
X(25292) lies on these lines: {2, 18194}, {8, 726}, {9, 23354}, {10, 23579}, {69, 20352}, {319, 350}, {1654, 25054}, {3416, 15983}, {4651, 21223}, {8050, 17275}, {17792, 25277}, {20055, 21299}, {24524, 25295}, {25278, 25291}, {25280, 25293}
X(25293) lies on these lines: {2, 24527}, {8, 291}, {10, 20464}, {69, 20350}, {1837, 3056}, {4651, 24528}, {25274, 25309}, {25275, 25291}, {25278, 25312}, {25280, 25292}, {25283, 25313}
X(25294) lies on these lines: {2, 3728}, {8, 79}, {42, 3952}, {714, 3210}, {740, 20012}, {3293, 22024}, {20018, 25253}, {21020, 24349}, {24524, 25282}, {25286, 25310}
X(25295) lies on these lines: {2, 3728}, {37, 3121}, {75, 3873}, {145, 740}, {190, 4068}, {192, 714}, {354, 20892}, {518, 15983}, {1107, 23432}, {1215, 22167}, {1962, 4704}, {2234, 16710}, {3696, 3922}, {3747, 17350}, {3948, 4890}, {3995, 21080}, {4033, 22279}, {4360, 17142}, {4772, 21020}, {16738, 24437}, {17147, 17157}, {17379, 24351}, {17792, 25284}, {21100, 21803}, {24524, 25292}, {24654, 25136}, {25302, 25303}
X(25296) lies on these lines: {2, 17448}, {8, 3761}, {85, 4487}, {145, 668}, {274, 3617}, {350, 20014}, {1909, 4678}, {3621, 17144}, {3623, 6376}, {3760, 20053}, {4441, 20052}, {4935, 17158}, {6381, 20050}
X(25297) lies on these lines: {2, 17448}, {200, 668}, {350, 20015}, {519, 18153}, {3699, 20935}, {3870, 6376}, {4110, 10025}, {10327, 16284}, {25279, 25290}, {25308, 25310}
X(25298) lies on these lines: {2, 17448}, {4, 8}, {10, 20456}, {44, 4033}, {69, 20892}, {75, 4410}, {100, 20777}, {239, 668}, {312, 20055}, {313, 17362}, {319, 20891}, {350, 20016}, {518, 20352}, {519, 3948}, {524, 3264}, {752, 4783}, {1107, 20868}, {1269, 4399}, {1909, 4359}, {1931, 7257}, {3056, 25291}, {3210, 21219}, {3263, 7779}, {3596, 17363}, {3625, 4044}, {3686, 3963}, {3770, 5564}, {3975, 4358}, {4110, 17350}, {4380, 4462}, {4393, 6376}, {4852, 18133}, {4986, 20432}, {5904, 21435}, {16816, 20917}, {17148, 21857}, {17299, 22016}, {17348, 18040}, {17349, 17786}, {17372, 18137}, {17373, 20923}, {17792, 25277}, {25273, 25308}, {25279, 25311}
X(25299) lies on these lines: {2, 669}, {10, 18197}, {512, 24622}, {649, 17072}, {804, 25258}, {1491, 17494}, {2533, 7192}, {2978, 21302}, {8640, 21260}, {8646, 24601}, {20983, 25288}, {21191, 23655}
X(25299) = anticomplement of X(25537)
X(25300) lies on these lines: {2, 24665}, {330, 23572}, {514, 19565}, {693, 14404}, {3004, 24533}, {4083, 17217}, {21272, 25312}, {25302, 25305}
X(25301) lies on these lines: {2, 23655}, {8, 649}, {333, 23865}, {518, 20952}, {788, 17135}, {3221, 9493}, {3835, 10453}, {3907, 17494}, {4651, 7234}, {7192, 8678}, {20909, 24349}
X(25302) lies on these lines: {2, 24525}, {75, 14923}, {518, 3263}, {883, 1458}, {4576, 20045}, {24524, 25277}, {25287, 25290}, {25295, 25303}, {25300, 25305}
X(25303) lies on these lines: {1, 76}, {2, 17448}, {37, 21226}, {75, 145}, {83, 16784}, {85, 3476}, {86, 1222}, {99, 3746}, {183, 3304}, {192, 20105}, {257, 3726}, {274, 519}, {316, 5270}, {320, 17152}, {325, 15888}, {330, 2276}, {335, 3727}, {551, 18140}, {612, 11059}, {668, 1125}, {1015, 6683}, {1078, 5563}, {1107, 9263}, {1914, 6645}, {1975, 3303}, {2295, 10027}, {3230, 17499}, {3241, 17144}, {3244, 17143}, {3266, 3920}, {3552, 10987}, {3584, 7769}, {3616, 6376}, {3623, 4441}, {3635, 20888}, {3636, 6381}, {3723, 3770}, {3750, 8033}, {3780, 16827}, {3975, 16826}, {4317, 14907}, {4359, 20016}, {4389, 24418}, {4696, 20947}, {4968, 17762}, {5255, 17103}, {5434, 7750}, {5564, 16709}, {7187, 24326}, {7278, 20924}, {7321, 20244}, {7760, 16785}, {7763, 10056}, {7773, 11237}, {8025, 19811}, {12513, 16992}, {15589, 16284}, {16969, 24514}, {16971, 17034}, {17379, 17787}, {17752, 24512}, {25284, 25285}, {25295, 25302}
X(25304) lies on these lines: {2, 3056}, {6, 100}, {7, 3888}, {8, 511}, {55, 15988}, {69, 674}, {141, 11680}, {145, 1469}, {149, 12589}, {193, 3779}, {263, 7155}, {344, 4553}, {404, 613}, {517, 12530}, {518, 1278}, {611, 3871}, {766, 24282}, {1002, 20090}, {1332, 1486}, {1350, 2975}, {1351, 5687}, {1428, 4188}, {1621, 10387}, {1959, 4073}, {1992, 22277}, {2293, 22370}, {2475, 12588}, {2876, 12272}, {3060, 10327}, {3094, 23423}, {3161, 3799}, {3416, 5086}, {3688, 17257}, {3873, 24471}, {4000, 25048}, {4259, 20040}, {4488, 4499}, {4704, 19586}, {4855, 16475}, {5175, 10477}, {5369, 21216}, {5480, 11681}, {5552, 14853}, {7077, 9309}, {10446, 20556}, {10519, 10527}, {15310, 24280}, {17316, 21746}, {17757, 21850}, {17868, 24341}, {25278, 25291}
X(25305) lies on these lines: {350, 21272}, {517, 3262}, {17448, 25313}, {24524, 25274}, {25300, 25302}
X(25306) lies on these lines: {2, 3056}, {4, 8}, {31, 43}, {57, 3888}, {312, 21278}, {511, 3705}, {674, 4417}, {1193, 16478}, {1469, 3873}, {3006, 3060}, {3212, 20537}, {3741, 11680}, {3772, 25048}, {4553, 18743}, {6210, 11688}, {24524, 25274}
X(25307) lies on these lines: {2, 25136}, {75, 3869}, {758, 21281}, {4647, 24282}, {24524, 25282}, {25274, 25277}, {25295, 25302}
X(25308) lies on these lines: {2, 3056}, {63, 3888}, {100, 212}, {312, 4553}, {674, 18134}, {1026, 21361}, {2836, 2895}, {2979, 3006}, {3705, 3917}, {3779, 17778}, {3781, 4388}, {3792, 4865}, {3869, 6327}, {3873, 4259}, {4438, 7186}, {5794, 14923}, {24524, 25282}, {24789, 25048}, {25273, 25298}, {25297, 25310}
X(25309) lies on these lines: {2, 24531}, {6631, 23861}, {21272, 21295}, {25273, 25283}, {25274, 25293}, {25285, 25313}, {25310, 25312}
X(25310) lies on these lines: {2, 25110}, {668, 891}, {3676, 4551}, {25274, 25278}, {25282, 25287}, {25286, 25294}, {25297, 25308}, {25309, 25312}
X(25311) lies on these lines: {2, 18194}, {6, 20532}, {7, 8}, {100, 20794}, {932, 23857}, {1716, 10027}, {3240, 8050}, {4110, 24451}, {4851, 20530}, {5687, 22152}, {6542, 21299}, {7155, 9025}, {7779, 10327}, {10453, 17373}, {17299, 24717}, {17350, 23354}, {17375, 20352}, {21080, 21219}, {24575, 25285}, {25279, 25298}
X(25312) lies on these lines: {2, 24677}, {8, 17793}, {668, 3888}, {1018, 20979}, {3699, 23354}, {21272, 25300}, {25278, 25293}, {25309, 25310}
X(25313) lies on these lines: {2, 24664}, {193, 3212}, {15983, 20955}, {17448, 25305}, {21226, 21272}, {25273, 25277}, {25283, 25293}, {25285, 25309}
Lionheart-Moses images: X(25314)-X(25336)
Lionheart-Moses images are defined in the preamble just before X(25045).
X(25314) lies on these lines: {6, 17500}, {193, 1634}, {263, 1992}, {542, 3545}, {3618, 20021}
X(25314) = midpoint of X(25046) and X(25317)
X(25314) = reflection of X(i) in X(j) for these {i,j}: {25051, 25317}, {25317, 6}
X(25314) = {X(6),X(25046)}-harmonic conjugate of X(25051)
X(25315) lies on these lines: {6, 10330}, {69, 3124}, {1992, 2854}, {3618, 4576}, {5969, 13331}
X(25315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25047, 25052), (6, 25322, 25047), (6, 25334, 25325)
X(25316) lies on these lines: {6, 82}, {190, 20455}, {193, 4553}, {518, 17264}, {2876, 24482}
X(25316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25050, 25048), (6, 25323, 25050)
X(25317) lies on these lines: {2, 6784}, {6, 17500}, {69, 7668}, {193, 20021}, {542, 3839}, {1634, 3618}
X(25317) = midpoint of X(25051) and X(25314)
X(25317) = reflection of X(i) in X(j) for these {i,j}: {25046, 25314}, {25314, 6}
X(25317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25051, 25046), (6, 25324, 25051)
X(25318) lies on these lines: {6, 19585}, {69, 1084}, {193, 694}, {263, 1992}, {670, 3618}, {5032, 5969}
X(25318) = midpoint of X(25054) and X(25319)
X(25318) = reflection of X(i) in X(j) for these {i,j}: {25319, 6}, {25332, 25319}
X(25318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25054, 25332), (6, 25326, 25054)
X(25319) lies on these lines: {2, 6784}, {6, 19585}, {193, 670}, {694, 3618}, {5969, 13331}
X(25319) = midpoint of X(25318) and X(25332)
X(25319) = reflection of X(i) in X(j) for these {i,j}: {25054, 25318}, {25318, 6}
X(25319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25327, 25332), (6, 25332, 25054)
X(25320) lies on these lines: {2, 2854}, {4, 11579}, {6, 3448}, {67, 193}, {69, 125}, {110, 3618}, {146, 5480}, {182, 12383}, {265, 6776}, {399, 18583}, {542, 3545}, {597, 7605}, {1205, 11800}, {1351, 10264}, {1352, 9976}, {1992, 9140}, {2072, 3564}, {2892, 23300}, {2930, 3589}, {3091, 14982}, {3619, 5181}, {3620, 6698}, {3751, 13605}, {5476, 16261}, {5622, 17702}, {5663, 14853}, {5987, 16989}, {6391, 23296}, {6593, 14683}, {9970, 12317}, {10117, 15583}, {10519, 14984}, {10620, 21850}, {10733, 14927}, {10752, 16003}, {10754, 15357}, {11801, 18440}, {13198, 18935}
X(25320) = midpoint of X(i) and X(j) for these {i,j}: {6, 25330}, {3448, 25321}
X(25320) = reflection of X(i) in X(j) for these {i,j}: {3448, 25330}, {10519, 15061}, {11061, 25321}, {25321, 6}, {25330, 25328}
X(25320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3448, 11061), (6, 25328, 3448), (6, 25335, 25329), (110, 15118, 3618), (125, 895, 69), (1352, 20301, 15081), (5181, 15059, 3619), (5480, 16010, 146), (9976, 20301, 1352)
X(25321) lies on these lines: {2, 9769}, {6, 3448}, {20, 10752}, {67, 3618}, {69, 6593}, {110, 193}, {113, 5921}, {141, 16176}, {146, 6776}, {399, 1353}, {403, 3564}, {542, 3839}, {895, 14683}, {1351, 12383}, {1992, 2854}, {2914, 19139}, {2930, 3629}, {3580, 15471}, {3620, 5972}, {5181, 20080}, {5477, 15342}, {5596, 13248}, {5642, 11160}, {5663, 14912}, {6403, 11557}, {7731, 9967}, {9129, 10553}, {10272, 11898}, {10519, 15462}, {11179, 17853}, {11574, 13201}, {11597, 19154}, {11702, 19155}, {12007, 16010}, {12220, 13417}, {12270, 12294}, {14002, 16510}, {15081, 18583}, {20113, 23330}
X(25321) = midpoint of X(i) and X(j) for these {i,j}: {6, 25331}, {11061, 25320}
X(25321) = reflection of X(i) in X(j) for these {i,j}: {3448, 25320}, {10519, 15462}, {11061, 25331}, {25320, 6}, {25331, 25329}
X(25321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11061, 3448), (6, 25329, 11061), (6, 25336, 25328), (110, 5095, 193), (6776, 9970, 146)
X(25322) lies on the cubic K484 and these lines: {6, 10330}, {39, 597}, {141, 3124}, {524, 20977}, {1112, 1843}, {3589, 4576}, {8584, 20380}
X(25322) = midpoint of X(i) and X(j) for these {i,j}: {6, 25047}, {25052, 25334}
X(25322) = reflection of X(i) in X(j) for these {i,j}: {141, 3124}, {4576, 3589}, {25325, 6}
X(25322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25334, 25052), (25047, 25052, 25334), (25047, 25315, 6)
X(25322) = isotomic of the anticomplement X(7813)
X(25322) = X(i)-cross conjugate of X(j) for these (i,j): {7813, 2}, {21906, 523}
X(25322) = X(32)-isoconjugate of X(18075)
X(25322) = cevapoint of X(i) and X(j) for these (i,j): {524, 3589}, {690, 3124}, {826, 1648}
X(25322) = trilinear pole of line {1649, 3005}
X(25322) = barycentric quotient X(75)/X(18075)
X(25323) lies on these lines: {6, 82}, {10, 2836}, {518, 4439}, {524, 4553}, {4557, 22116}, {9055, 20455}
X(25323) = midpoint of X(6) and X(25050)
X(25323) = {X(25050),X(25316)}-harmonic conjugate of X(6)
X(25324) lies on these lines: {6, 17500}, {141, 7668}, {524, 20021}, {542, 1353}, {1634, 3589}
X(25324) = midpoint of X(6) and X(25051)
X(25324) = reflection of X(i) in X(j) for these {i,j}: {141, 7668}, {1634, 3589}
X(25324) = {X(25051),X(25317)}-harmonic conjugate of X(6)
X(25325) lies on these lines: {6, 10330}, {125, 126}, {524, 4576}, {597, 3124}, {1569, 5052}, {13196, 15826}
X(25325) = midpoint of X(6) and X(25052)
X(25325) = reflection of X(25322) in X(6)
X(25325) = {X(6),X(25334)}-harmonic conjugate of X(25315)
X(25326) lies on these lines: {6, 19585}, {141, 1084}, {524, 694}, {670, 3589}, {1569, 5052}
X(25326) = midpoint of X(6) and X(25054)
X(25326) = reflection of X(i) in X(j) for these {i,j}: {141, 1084}, {670, 3589}, {25327, 6}
X(25326) = {X(25054),X(25318)}-harmonic conjugate of X(6)
X(25327) lies on these lines: {6, 19585}, {39, 597}, {141, 7668}, {524, 670}, {694, 3589}, {1576, 19120}, {2086, 4576}
X(25327) = midpoint of X(i) and X(j) for these {i,j}: {6, 25332}
X(25327) = reflection of X(i) in X(j) for these {i,j}: {694, 3589}, {25326, 6}
X(25327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25319, 25332, 6)
X(25328) lies on these lines: {2, 2930}, {4, 16010}, {5, 542}, {6, 3448}, {10, 2836}, {20, 5621}, {52, 2781}, {67, 524}, {98, 24975}, {110, 3589}, {125, 126}, {140, 12584}, {195, 15141}, {265, 1177}, {399, 14561}, {427, 5095}, {511, 10264}, {518, 13605}, {631, 19468}, {1112, 3867}, {1205, 9019}, {1368, 22165}, {1853, 2892}, {1907, 13148}, {1992, 16176}, {2393, 8262}, {3564, 9976}, {3580, 8705}, {3618, 14683}, {3629, 23300}, {3818, 11801}, {5085, 12383}, {5480, 5663}, {5622, 6146}, {5648, 20582}, {5846, 13211}, {5969, 15357}, {5987, 7792}, {6034, 15342}, {6699, 21167}, {7984, 9053}, {8681, 19510}, {9927, 19127}, {10516, 15081}, {10627, 12359}, {12233, 15063}, +{12317, 14853}, {13353, 15462}, {14644, 14982}, {15069, 22533}, {15074, 15532}, {15116, 23332}, {16511, 21243}, {18583, 19140}
X(25328) = complement X[2930]
X(25328) = midpoint of X(i) and X(j) for these {i,j}: {4, 16010}, {6, 3448}, {67, 895}, {265, 11579}, {11061, 25335}, {25320, 25330}
X(25328) = reflection of X(i) in X(j) for these {i,j}: {5, 20301}, {110, 3589}, {141, 125}, {3818, 11801}, {5181, 6698}, {5648, 20582}, {6593, 15118}, {12584, 140}, {19140, 18583}, {23315, 23300}, {25329, 6}
X(25328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25330, 3448), (6, 25335, 11061), (6, 25336, 25321), (125, 5181, 6698), (895, 9140, 67), (3448, 11061, 25335), (3448, 25320, 6), (5181, 6698, 141), (6593, 15118, 597)
X(25328) = nine-point-circle-inverse of X(5461)
X(25328) = complement of X(2930)
X(25328) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 15899}, {75, 13140}, {13574, 10}, {22259, 37}
X(25328) = crosspoint of X(671) and X(18019)
X(25328) = crosssum of X(187) and X(18374)
X(25329) lies on these lines: {6, 3448}, {67, 3589}, {69, 16176}, {110, 524}, {125, 597}, {141, 5972}, {193, 2930}, {542, 1353}, {575, 10264}, {858, 19379}, {1112, 1843}, {1352, 10254}, {1503, 7728}, {1992, 14683}, {2781, 9967}, {3564, 19140}, {3630, 5181}, {5476, 11801}, {5642, 22165}, {5663, 8550}, {9019, 13417}, {9143, 15534}, {10620, 11179}, {11232, 22330}, {11477, 12383}, {11579, 12007}, {12902, 20423}, {13169, 20582}, {13413, 18583}, {14094, 16657}, {14912, 16010}
X(25329) = midpoint of X(i) and X(j) for these {i,j}: {6, 11061}, {69, 16176}, {193, 2930}, {3448, 25336}, {9143, 15534}, {11477, 12383}, {25321, 25331}
X(25329) = reflection of X(i) in X(j) for these {i,j}: {67, 3589}, {141, 6593}, {597, 15303}, {3629, 5095}, {3630, 5181}, {10264, 575}, {11579, 12007}, {13169, 20582}, {22165, 5642}, {25328, 6}
X(25329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25331, 11061), (6, 25335, 25320), (6, 25336, 3448), (3448, 11061, 25336), (11061, 25321, 6)
X(25330) lies on these lines: {6, 3448}, {67, 6391}, {125, 2930}, {265, 16010}, {399, 20301}, {542, 5050}, {599, 2854}, {895, 6144}, {1350, 10264}, {1853, 11216}, {2916, 12310}, {3242, 13605}, {3589, 14683}, {3867, 18947}, {5480, 12317}, {5621, 17702}, {11579, 14852}, {11646, 15538}, {12308, 19130}, {17847, 23300}
X(25330) = midpoint of X(i) and X(j) for these {i,j}: {3448, 25320}, {25331, 25335}
X(25330) = reflection of X(i) in X(j) for these {i,j}: {6, 25320}, {25320, 25328}, {25331, 6}, {25336, 25331}
X(25330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3448, 25335), (6, 25335, 25336), (125, 2930, 3763), (3448, 25328, 6)
X(25331) lies on these lines: {6, 3448}, {67, 6723}, {110, 16176}, {542, 5093}, {599, 5642}, {1539, 9970}, {2854, 3060}, {2930, 6144}, {3564, 11563}, {3629, 14683}, {3763, 6593}, {5095, 12167}, {9927, 14982}, {10516, 15046}, {12007, 12317}, {15069, 19140}
X(25331) = midpoint of X(i) and X(j) for these {i,j}: {11061, 25321}, {25330, 25336}
X(25331) = reflection of X(i) in X(j) for these {i,j}: {6, 25321}, {25321, 25329}, {25330, 6}, {25335, 25330}
X(25331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11061, 25336), (6, 25336, 25335), (11061, 25329, 6)
X(25332) lies on the cubics K355 and K738 and on these lines: {2, 694}, {6, 19585}, {69, 290}, {75, 3056}, {237, 19599}, {401, 12215}, {511, 3978}, {804, 25046}, {888, 25052}, {1084, 3618}, {1316, 19571}, {1369, 3448}, {1370, 2882}, {1469, 17149}, {3228, 11175}, {3506, 17941}, {5207, 14957}, {6393, 21531}, {16951, 20965}, {17135, 25049}, {20245, 20345}, {20351, 20352}
X(25332) = isotomic conjugate of isogonal conjugate of X(3511)
X(25332) = isotomic conjugate of anticomplement of X(39092)
X(25332) = anticomplement of X(694)
X(25333) lies on these lines: {76, 25048}, {320, 20345}, {668, 24482}, {1655, 4664}, {3448, 21287}, {3770, 25047}, {21278, 25049}
X(25333) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {82, 536}, {536, 21289}, {899, 2896}, {3230, 21217}, {6381, 1369}
X(25334) lies on these lines: {6, 10330}, {76, 338}, {2854, 6144}, {3763, 4576}, {8267, 15534}
X(25334) = reflection of X(i) in X(j) for these {i,j}: {6, 25047}, {25052, 25322}
X(25334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25047, 25052, 25322), (25052, 25322, 6), (25315, 25325, 6)
X(25335) lies on these lines: {3, 67}, {6, 3448}, {66, 17847}, {110, 3763}, {125, 19125}, {141, 14683}, {399, 10516}, {524, 5189}, {895, 8877}, {1181, 14789}, {1503, 12317}, {1853, 15141}, {2836, 5904}, {2854, 12220}, {2892, 8549}, {2916, 13171}, {3818, 12308}, {5085, 10264}, {5095, 11405}, {5987, 7778}, {6593, 9140}, {8262, 19596}, {9143, 21358}, {11562, 18440}, {11579, 12228}, {15462, 20379}
X(25335) = reflection of X(i) in X(j) for these {i,j}: {6, 3448}, {2930, 67}, {11061, 25328}, {12308, 3818}, {14683, 141}, {16176, 895}, {17847, 66}, {25331, 25330}, {25336, 6}
X(25335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3448, 25330), (6, 25336, 25331), (67, 2930, 599), (895, 16176, 15534), (3448, 11061, 25328), (11061, 18125, 5169), (11061, 25328, 6), (25320, 25329, 6), (25330, 25336, 6)
X(25336) lies on these lines: {6, 3448}, {67, 3763}, {110, 599}, {159, 2930}, {399, 15069}, {524, 14683}, {542, 1351}, {2781, 15102}, {2854, 6144}, {5663, 15074}, {5987, 9766}, {6593, 15059}, {8550, 12317}, {9143, 15533}, {9970, 10113}, {10516, 19140}
X(25336) = reflection of X(i) in X(j) for these {i,j}: {6, 11061}, {3448, 25329}, {6144, 16176}, {12317, 8550}, {15069, 399}, {15533, 9143}, {25330, 25331}, {25335, 6}
X(25336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11061, 25331), (6, 25335, 25330), (3448, 11061, 25329), (3448, 25329, 6), (25321, 25328, 6), (25331, 25335, 6)
As a point on the Euler line, X(25337) has Shinagawa coefficients (-5*E-12*F, 7*E+4*F)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28467.
X(25337) lies on these lines: {2, 3}, {523, 11619}, {2781, 10272}, {3564, 19127}, {3589, 13364}, {6689, 13598}, {8717, 23329}, {9019, 13451}, {9820, 10627}, {11591, 16252}, {13391, 23292}
X(25337) = midpoint of X(i) and X(j) for these {i,j}: {5, 22}, {6676, 16618}, {7502, 15760}
X(25337) = reflection of X(i) in X(j) for these (i,j): (140, 6676), (427, 3628), (18570, 3530)
X(25337) = complement of the complement of X(12083)
X(25337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10020, 140), (140, 10096, 6677), (550, 7542, 23336), (2937, 13160, 11819), (3530, 16238, 140), (3530, 18282, 16238), (3547, 10565, 18420), (6636, 7552, 2072), (6677, 13383, 10096), (7517, 7558, 5), (7542, 23336, 140), (10125, 16196, 140), (10565, 18420, 26), (13383, 16197, 140)
As a point on the Euler line, X(25338) has Shinagawa coefficients (-3*E-24*F, 17*E+8*F)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28467.
X(25338) lies on these lines: {2, 3}, {511, 10272}, {523, 6140}, {1154, 16534}, {1533, 12041}, {5305, 16308}, {5446, 15806}, {5462, 20193}, {7286, 15325}, {8157, 24305}, {8254, 10110}, {8705, 18583}, {11649, 13451}, {12002, 13446}, {14643, 15107}, {21660, 22051}
X(25338) = midpoint of X(i) and X(j) for these {i,j}: {5, 23}, {468, 16619}, {550, 18325}, {1533, 12041}, {2070, 11563}, {7575, 11799}
X(25338) = reflection of X(i) in X(j) for these (i,j): (3, 22249), (140, 468), (548, 18571), (858, 3628), (18323, 3861), (18572, 3850)
X(25338) = complement of complement of X(37924)
X(25338) = orthoptic-circle-of-Steiner inellipse-inverseof X(7533)
X(25338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 18282, 140), (23, 186, 2937), (140, 10096, 468), (186, 18325, 550), (403, 18572, 3850), (1656, 5899, 5189), (2070, 18378, 23), (7426, 11799, 7575), (7495, 7574, 15122), (7545, 7552, 5)
Barycentrics 20*S^4-(87*R^4+10*R^2*(SA-8*SW)-4*SA^2-12*SB*SC+20*SW^2)*S^2-(27*R^4-26*R^2*SW+8*SW^2)*SB*SC : :
X(25339) = X[137] + 3 X[547], 7 X[5] + X[1141], X[1263] + 7 X[3090], 5 X[1656] - X[6592], 15 X[1656] + X[11671], 3 X[6592] + X[11671], 3 X[1141] - 7 X[12026], 3 X[5] + X[12026], X[128] - 5 X[12812], 3 X[3628] - X[13372], 17 X[7486] - X[13512], 9 X[5055] - X[14072], X[930] - 9 X[15699], 11 X[5056] - 3 X[23237], X[140] + 3 X[23516], X[11671] - 9 X[25147], 5 X[1656] + 3 X[25147], X[6592] + 3 X[25147]
Barycentrics 2 a^16-15 a^14 b^2+47 a^12 b^4-87 a^10 b^6+115 a^8 b^8-117 a^6 b^10+85 a^4 b^12-37 a^2 b^14+7 b^16-15 a^14 c^2+62 a^12 b^2 c^2-91 a^10 b^4 c^2+38 a^8 b^6 c^2+81 a^6 b^8 c^2-176 a^4 b^10 c^2+145 a^2 b^12 c^2-44 b^14 c^2+47 a^12 c^4-91 a^10 b^2 c^4+48 a^8 b^4 c^4-27 a^6 b^6 c^4+112 a^4 b^8 c^4-213 a^2 b^10 c^4+124 b^12 c^4-87 a^10 c^6+38 a^8 b^2 c^6-27 a^6 b^4 c^6-42 a^4 b^6 c^6+105 a^2 b^8 c^6-212 b^10 c^6+115 a^8 c^8+81 a^6 b^2 c^8+112 a^4 b^4 c^8+105 a^2 b^6 c^8+250 b^8 c^8-117 a^6 c^10-176 a^4 b^2 c^10-213 a^2 b^4 c^10-212 b^6 c^10+85 a^4 c^12+145 a^2 b^2 c^12+124 b^4 c^12-37 a^2 c^14-44 b^2 c^14+7 c^16 : :
See Antreas Hatzipolakis, César Lozada, and Peter Moses, Hyacinthos 28465 and Hyacinthos 28470.
X(25339) lies on these lines: {5, 49}, {128, 12812}, {137, 547}, {140, 23516}, {930, 15699}, {1209, 20413}, {1263, 3090}, {1656, 6592}, {3628, 13372}, {5055, 14072}, {5056, 23237}, {7486, 13512}, {10096, 15366}, {14652, 21308}
X(25339) = {X(1656), X(25147)}-harmonic conjugate of X(6592)
Barycentrics (S^2+SB*SC)*(8*S^4+(R^2*(41*R^2+2*SA-26*SW)+4*SA^2+4*SB*SC)*S^2-90*R^8+6*(11*SA+28*SW)*R^6+(14*SA^2-84*SA*SW-121*SW^2)*R^4-2*(4*SA^2-17*SA*SW-19*SW^2)*SW*R^2-4*(SA+SW)*SW^3) : :
X(25340) = 3 X[547] - X[18807], X[25043] + 3 X[25147]
Barycentrics (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^22 b^2-8 a^20 b^4+31 a^18 b^6-80 a^16 b^8+154 a^14 b^10-224 a^12 b^12+238 a^10 b^14-176 a^8 b^16+85 a^6 b^18-24 a^4 b^20+3 a^2 b^22+a^22 c^2-10 a^20 b^2 c^2+34 a^18 b^4 c^2-51 a^16 b^6 c^2+4 a^14 b^8 c^2+156 a^12 b^10 c^2-390 a^10 b^12 c^2+526 a^8 b^14 c^2-429 a^6 b^16 c^2+206 a^4 b^18 c^2-52 a^2 b^20 c^2+5 b^22 c^2-8 a^20 c^4+34 a^18 b^2 c^4-36 a^16 b^4 c^4-35 a^14 b^6 c^4+96 a^12 b^8 c^4+24 a^10 b^10 c^4-370 a^8 b^12 c^4+665 a^6 b^14 c^4-576 a^4 b^16 c^4+248 a^2 b^18 c^4-42 b^20 c^4+31 a^18 c^6-51 a^16 b^2 c^6-35 a^14 b^4 c^6+106 a^12 b^6 c^6-61 a^10 b^8 c^6+93 a^8 b^10 c^6-406 a^6 b^12 c^6+727 a^4 b^14 c^6-565 a^2 b^16 c^6+161 b^18 c^6-80 a^16 c^8+4 a^14 b^2 c^8+96 a^12 b^4 c^8-61 a^10 b^6 c^8+16 a^8 b^8 c^8+85 a^6 b^10 c^8-420 a^4 b^12 c^8+682 a^2 b^14 c^8-376 b^16 c^8+154 a^14 c^10+156 a^12 b^2 c^10+24 a^10 b^4 c^10+93 a^8 b^6 c^10+85 a^6 b^8 c^10+174 a^4 b^10 c^10-316 a^2 b^12 c^10+602 b^14 c^10-224 a^12 c^12-390 a^10 b^2 c^12-370 a^8 b^4 c^12-406 a^6 b^6 c^12-420 a^4 b^8 c^12-316 a^2 b^10 c^12-700 b^12 c^12+238 a^10 c^14+526 a^8 b^2 c^14+665 a^6 b^4 c^14+727 a^4 b^6 c^14+682 a^2 b^8 c^14+602 b^10 c^14-176 a^8 c^16-429 a^6 b^2 c^16-576 a^4 b^4 c^16-565 a^2 b^6 c^16-376 b^8 c^16+85 a^6 c^18+206 a^4 b^2 c^18+248 a^2 b^4 c^18+161 b^6 c^18-24 a^4 c^20-52 a^2 b^2 c^20-42 b^4 c^20+3 a^2 c^22+5 b^2 c^22) : :
See Antreas Hatzipolakis, César Lozada, and Peter Moses, Hyacinthos 28465 and Hyacinthos 28470.
X(25340) lies on these lines: {137, 1209}, {547, 18807}, {3628, 25150}, {25043, 25147}
Collineation mappings involving Gemini triangle 19: X(25341)-X(25385)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 19, as in centers X(25341)-X(25385). Then
m(X) = (b + c) x + (a + b) y + (a + c) z : :
(Clark Kimberling, October 15, 2018)
X(25341) lies on these lines: {2, 1762}, {5, 24684}, {140, 8680}, {1656, 24682}, {3090, 24683}, {3634, 9028}, {4364, 25342}, {5432, 24424}, {7570, 24687}, {7571, 24686}, {24318, 24323}, {24319, 24324}
X(25341) = complement of X(24317)
X(25342) lies on these lines: {2, 7}, {11, 24685}, {514, 1125}, {544, 6702}, {3570, 3686}, {4364, 25341}, {4758, 25377}, {5845, 6667}, {6684, 24455}, {11814, 24358}, {15903, 21138}, {17023, 19955}, {24315, 25369}
X(25343) lies on these lines: {2, 1762}, {22, 24686}, {25, 24682}, {427, 8680}, {3923, 24337}, {4438, 24318}, {5089, 20305}, {6636, 24687}, {7494, 24683}, {7499, 24684}
X(25343) = complement of X(24321)
X(25344) lies on these lines: {2, 1762}, {23, 24687}, {25, 24686}, {858, 8680}, {1995, 24682}, {2254, 2786}, {7493, 24683}, {7495, 24684}
X(25344) = complement of X(24322)
X(25345) lies on these lines: {1, 4805}, {2, 4376}, {10, 712}, {31, 4799}, {37, 20541}, {171, 24699}, {213, 17211}, {315, 16974}, {333, 3772}, {524, 3791}, {536, 3703}, {626, 16600}, {742, 2887}, {1086, 24631}, {2486, 2886}, {3120, 24330}, {3329, 16706}, {3662, 24691}, {3721, 24995}, {3821, 25349}, {3925, 25384}, {3944, 4713}, {4045, 24036}, {4202, 16720}, {4357, 21264}, {4363, 17889}, {4429, 4518}, {4670, 5249}, {4797, 6679}, {4972, 24326}, {16886, 17489}, {17184, 24690}, {20255, 24211}, {24169, 24318}, {24325, 25373}, {25346, 25348}
X(25346) lies on these lines: {2, 4381}, {10, 696}, {32, 4837}, {626, 744}, {1078, 24707}, {3821, 24317}, {4364, 20531}, {4443, 24348}, {4836, 6680}, {5988, 24327}, {25345, 25348}
X(25346) = complement of X(4381)
X(25347) lies on these lines: {2, 3122}, {39, 24688}, {674, 1125}, {700, 3739}, {714, 3934}, {1281, 24340}, {3821, 24317}, {3842, 9055}, {4286, 24259}, {4735, 12263}, {6007, 6683}, {7786, 24696}, {24348, 25364}, {24463, 25382}
X(25347) = complement of X(24327)
X(25348) lies on these lines: {2, 24329}, {41, 24689}, {3821, 25369}, {3826, 25359}, {4364, 25379}, {5880, 6998}, {17047, 25065}, {20531, 24424}, {21023, 25241}, {25345, 25346}, {25349, 25360}
X(25348) = complement of X(24329)
X(25349) lies on these lines: {1, 24691}, {2, 45}, {10, 538}, {38, 9055}, {42, 524}, {43, 4643}, {86, 17735}, {141, 2276}, {291, 4026}, {350, 17246}, {527, 6685}, {536, 3741}, {594, 17759}, {672, 3589}, {742, 3666}, {846, 24358}, {982, 24357}, {1500, 16887}, {1575, 4357}, {1655, 21025}, {2227, 21035}, {2238, 17332}, {2292, 16720}, {2295, 16705}, {3663, 21264}, {3821, 25345}, {4022, 23462}, {4376, 4414}, {4395, 24592}, {4416, 21904}, {4425, 24318}, {4443, 24458}, {4640, 4670}, {4657, 17754}, {4665, 4884}, {4685, 4690}, {4868, 8682}, {4971, 17135}, {5283, 20255}, {5724, 25434}, {6682, 25368}, {10453, 17318}, {16589, 24170}, {17026, 17301}, {17027, 17395}, {17032, 17392}, {17045, 24512}, {17235, 20335}, {17334, 24514}, {17379, 21793}, {21024, 25264}, {21240, 25092}, {24169, 25357}, {24215, 24656}, {25348, 25360}, {25366, 25370}
X(25350) lies on these lines: {2, 45}, {39, 20255}, {43, 524}, {116, 4045}, {141, 1575}, {194, 21025}, {244, 24326}, {527, 6686}, {536, 3840}, {742, 3752}, {899, 24690}, {982, 9055}, {1155, 4797}, {1574, 16887}, {2140, 6683}, {2276, 17243}, {3097, 3932}, {3208, 24652}, {3589, 17754}, {3629, 21904}, {3663, 20530}, {3741, 4665}, {3795, 4966}, {3821, 25376}, {4395, 17026}, {4643, 16569}, {4670, 6685}, {4971, 10453}, {6682, 25384}, {7263, 21264}, {7786, 24190}, {7804, 17729}, {7808, 14377}, {16720, 24443}, {17063, 24357}, {17596, 24358}, {24169, 24318}, {24215, 25102}, {25353, 25357}
X(25351) lies on these lines: {2, 4432}, {10, 537}, {11, 25377}, {44, 24692}, {75, 4505}, {141, 4732}, {142, 4085}, {190, 1698}, {320, 4753}, {516, 19512}, {519, 3834}, {528, 1125}, {545, 3828}, {726, 3823}, {740, 1738}, {752, 3008}, {812, 3837}, {899, 4892}, {900, 6702}, {903, 17250}, {1266, 4439}, {1647, 24183}, {1739, 2228}, {2796, 3634}, {2802, 25031}, {3120, 24003}, {3579, 24827}, {3626, 9041}, {3679, 17227}, {3782, 4096}, {3821, 3826}, {3838, 6686}, {3840, 21949}, {3844, 9055}, {3919, 25391}, {3925, 6682}, {4429, 24325}, {4440, 9780}, {4454, 7613}, {4473, 19877}, {4645, 4974}, {4660, 17278}, {4672, 5880}, {4693, 17266}, {4709, 17231}, {4759, 6687}, {5587, 24813}, {5791, 24835}, {9708, 24826}, {9709, 24820}, {10175, 24828}, {13893, 24819}, {13947, 24818}, {16610, 21241}, {17495, 21026}, {17606, 24840}, {24188, 24222}, {24836, 24914}
X(25352) lies on these lines: {1, 2}, {45, 2796}, {3821, 3826}, {3822, 25353}, {3834, 4407}, {3836, 17237}, {3992, 20913}, {4078, 24357}, {4085, 4698}, {4125, 20888}, {4429, 25354}, {4439, 4688}, {4732, 17243}, {4753, 17392}, {5883, 20683}, {9330, 21093}, {16589, 23897}
X(25353) lies on these lines: {2, 7}, {10, 12933}, {37, 24241}, {55, 24694}, {984, 5988}, {993, 22780}, {1212, 17062}, {1621, 24712}, {1738, 3097}, {3816, 4364}, {3821, 25345}, {3822, 25352}, {3826, 25355}, {3842, 25360}, {3944, 4419}, {4363, 4438}, {4670, 6679}, {5283, 24211}, {5432, 24685}, {5845, 6690}, {6184, 20540}, {8680, 23305}, {16601, 17046}, {21029, 25237}, {24210, 24386}, {24330, 25385}, {25350, 25357}
X(25353) = complement of X(24333)
X(25354) lies on these lines: {1, 1654}, {2, 846}, {10, 37}, {58, 86}, {142, 3647}, {516, 6998}, {524, 551}, {975, 1716}, {984, 3122}, {1001, 4471}, {1010, 12579}, {1211, 10180}, {1698, 17280}, {1738, 3634}, {1962, 21085}, {2292, 22174}, {2486, 3826}, {3616, 20090}, {3624, 3662}, {3636, 4684}, {3685, 19856}, {3723, 17772}, {3741, 25058}, {3756, 6682}, {3775, 4708}, {3797, 24603}, {3828, 17264}, {3836, 4698}, {3844, 4755}, {3865, 4835}, {4260, 10176}, {4364, 24325}, {4429, 25352}, {4432, 25357}, {4649, 17256}, {4655, 15668}, {4672, 17398}, {4683, 5333}, {4703, 19701}, {4974, 17045}, {5248, 23868}, {5284, 8935}, {5550, 17236}, {7379, 8245}, {14007, 24851}, {16468, 17397}, {16822, 16904}, {16825, 17321}, {16830, 17766}, {17023, 20142}
X(25354) = complement of X(24342)
X(25355) lies on these lines: {2, 45}, {141, 1418}, {200, 524}, {527, 20103}, {536, 11019}, {1376, 5845}, {2886, 24318}, {3035, 24333}, {3693, 17243}, {3826, 25353}, {4390, 7181}, {4395, 24600}, {4643, 8580}, {4665, 4847}, {4670, 13405}, {5835, 16720}, {7263, 8568}, {10580, 17318}, {14548, 17390}, {21049, 25242}, {21258, 25066}
X(25355) = complement of X(24352)
X(25356) lies on these lines: {2, 24354}, {37, 21261}, {667, 24698}, {812, 25357}, {900, 25381}, {3768, 17248}, {4491, 24721}, {8632, 17322}, {17256, 23650}, {17392, 24673}, {21262, 21348}
X(25357) lies on these lines: {2, 4376}, {10, 9055}, {11, 25383}, {142, 3589}, {190, 17248}, {238, 24699}, {320, 20142}, {335, 5224}, {524, 4974}, {536, 1738}, {673, 4657}, {742, 3836}, {748, 4799}, {812, 25356}, {1086, 1213}, {1111, 25468}, {3120, 4465}, {3290, 20541}, {3662, 4643}, {3821, 3826}, {3925, 25368}, {4366, 17322}, {4422, 5257}, {4429, 24357}, {4432, 25354}, {4437, 17239}, {4472, 25373}, {4684, 4725}, {4698, 16593}, {4713, 17889}, {4748, 17236}, {4760, 24542}, {16720, 17674}, {24169, 25349}, {25350, 25353}, {25369, 25376}
X(25357) = complement of X(24358)
X(25358) lies on these lines: {2, 45}, {10, 4971}, {37, 24081}, {141, 16831}, {239, 1213}, {524, 1125}, {527, 19878}, {536, 3634}, {551, 4690}, {742, 4698}, {1698, 4665}, {3035, 24319}, {3589, 5257}, {3616, 17251}, {3624, 4643}, {3636, 4725}, {3723, 4478}, {3758, 17248}, {3842, 9055}, {3986, 17385}, {4357, 6707}, {4395, 24603}, {4432, 25354}, {4657, 16832}, {4659, 19872}, {4667, 19883}, {4670, 19862}, {4693, 19856}, {4715, 4758}, {4748, 5550}, {4760, 8040}, {5224, 17373}, {5845, 24317}, {6667, 25371}, {6690, 25369}, {9780, 17318}, {17119, 17321}, {17243, 17327}, {17245, 17326}, {17250, 17392}, {17330, 17397}, {17337, 17400}, {24685, 25359}
X(25359) lies on these lines: {2, 1762}, {405, 24682}, {442, 8680}, {497, 6536}, {524, 11281}, {527, 11263}, {960, 4708}, {1001, 4471}, {1213, 16609}, {2486, 2886}, {3120, 4419}, {3485, 4748}, {3826, 25348}, {4269, 4357}, {4363, 4438}, {4472, 24318}, {5224, 18714}, {5257, 18589}, {5327, 24697}, {5988, 24450}, {6857, 24683}, {7483, 24684}, {12635, 17251}, {17052, 25081}, {17322, 20769}, {21675, 25255}, {24685, 25358}
X(25360) lies on these lines: {2, 1762}, {1368, 8680}, {1995, 24686}, {3842, 25353}, {5020, 24682}, {6353, 24683}, {6676, 24684}, {13595, 24687}, {25348, 25349}
X(25361) lies on these lines: {2, 1762}, {27, 86}, {30, 551}, {37, 226}, {142, 6678}, {497, 3120}, {1001, 1626}, {2886, 20256}, {3151, 17134}, {3188, 17084}, {3782, 24424}, {4096, 21077}, {6358, 21091}, {7490, 24683}, {7522, 24682}, {7536, 24684}, {18673, 25015}
X(25361) = complement of X(1762)
X(25362) lies on these lines: {2, 1762}, {30, 24682}, {381, 8680}, {545, 3829}, {551, 9028}, {3524, 24683}, {3923, 24348}, {5054, 24684}, {11238, 24424}, {17250, 18735}, {17271, 18161}
X(25363) lies on these lines: {2, 1762}, {405, 8680}, {442, 24682}, {443, 24683}, {474, 24684}, {497, 4470}, {960, 4670}, {1001, 3923}, {2886, 4472}, {3980, 24335}, {4364, 24333}, {5750, 18589}
X(25364) lies on these lines: {2, 1762}, {37, 141}, {384, 24726}, {3821, 4443}, {5988, 24327}, {6656, 8680}, {7770, 24682}, {7807, 24684}, {14001, 24683}, {14829, 17305}, {16580, 17353}, {16887, 19623}, {17790, 24211}, {24295, 24340}, {24348, 25347}
X(25365) lies on these lines: {2, 1473}, {5, 142}, {12, 1406}, {19, 24701}, {226, 1368}, {427, 5249}, {442, 10436}, {511, 16608}, {518, 23305}, {1329, 3836}, {1503, 14529}, {1824, 18651}, {1892, 2002}, {1901, 21530}, {2886, 20256}, {3742, 17111}, {3821, 25371}, {3826, 4472}, {3838, 23304}, {3842, 25353}, {4187, 17282}, {5480, 21258}, {5750, 8728}, {5905, 21015}, {14019, 17353}, {15172, 18257}, {19544, 20266}, {20227, 23537}
X(25365) = complement of X(24320)
X(25366) lies on these lines: {2, 24323}, {545, 24319}, {3816, 4364}, {4472, 24318}, {4670, 6693}, {25349, 25370}
X(25367) lies on these lines: {2, 2161}, {545, 24318}, {812, 25356}, {3816, 4364}, {4419, 10584}, {4423, 24712}, {4472, 24319}, {4657, 24618}, {4670, 11230}, {8609, 21237}, {17793, 24348}, {24685, 25358}
X(25368) lies on these lines: {2, 37}, {38, 24330}, {141, 4071}, {354, 4670}, {517, 24455}, {712, 12263}, {742, 3741}, {756, 4465}, {982, 4363}, {984, 4713}, {1215, 9055}, {1281, 5263}, {2486, 2886}, {3920, 4396}, {3925, 25357}, {4357, 20541}, {4376, 24552}, {4388, 4643}, {4399, 4771}, {4659, 17591}, {6682, 25349}, {9507, 24325}, {10180, 17061}, {17318, 17592}, {17369, 24631}, {18610, 20834}, {24311, 24398}
X(25369) lies on these lines: {2, 24319}, {11, 24336}, {56, 24702}, {527, 10199}, {1001, 4471}, {1125, 13323}, {3816, 4364}, {3821, 25348}, {4419, 24318}, {4472, 6667}, {5433, 24700}, {5988, 24456}, {6690, 25358}, {17452, 24986}, {24315, 25342}, {25357, 25376}
X(25370) lies on these lines: {2, 24335}, {58, 24704}, {2486, 2886}, {3120, 4363}, {3821, 24317}, {4432, 25354}, {4466, 17290}, {4670, 11263}, {25349, 25366}
X(25371) lies on these lines: {2, 24336}, {11, 24319}, {21, 24700}, {37, 20258}, {55, 24334}, {65, 24705}, {86, 4032}, {142, 10445}, {527, 942}, {982, 4419}, {1001, 3923}, {2171, 17183}, {2269, 24993}, {2478, 24702}, {3663, 20227}, {3816, 4364}, {3821, 25365}, {4095, 17790}, {4472, 6690}, {4670, 8680}, {5436, 13735}, {6667, 25358}
X(25372) lies on these lines: {2, 24337}, {66, 24706}, {17453, 21407}, {24248, 24347}, {24295, 24317}
X(25373) lies on these lines: {2, 24339}, {120, 3842}, {3821, 24404}, {4472, 25357}, {24325, 25345}
X(25373) = complement of X(24339)
X(25374) lies on these lines: {2, 4381}, {83, 24707}, {744, 6292}, {1086, 1215}, {3821, 24327}, {4472, 25357}, {4836, 7889}, {4837, 7808}, {24348, 25347}
X(25374) = complement of X(24340)
X(25375) lies on these lines: {2, 2310}, {10, 12710}, {85, 24708}, {142, 9944}, {1001, 3923}, {2293, 25001}, {2486, 3816}, {4032, 20992}, {4422, 6690}, {6706, 15726}, {8299, 24336}, {17755, 24351}
X(25375) = complement of X(24341)
X(25376) lies on these lines: {2, 24343}, {141, 20363}, {984, 4026}, {3821, 25350}, {4110, 17248}, {4357, 17448}, {5257, 25109}, {25357, 25369}
X(25377) lies on these lines: {1, 2}, {11, 25351}, {88, 2796}, {142, 3835}, {214, 6715}, {244, 11814}, {537, 16594}, {764, 21204}, {867, 3825}, {946, 19515}, {3663, 24191}, {3756, 24003}, {3816, 24169}, {3821, 25378}, {3937, 12014}, {4011, 24395}, {4364, 24318}, {4454, 24398}, {4758, 25342}, {17063, 24399}, {17132, 24408}, {19945, 24456}
X(25378) lies on these lines: {2, 4432}, {11, 4364}, {89, 24710}, {244, 24456}, {1125, 10129}, {1647, 4389}, {3624, 16393}, {3821, 25377}, {6999, 9779}, {24408, 24441}
X(25379) lies on these lines: {2, 24346}, {10, 523}, {86, 17943}, {101, 24713}, {3753, 4026}, {3822, 3842}, {4364, 25348}, {4466, 4557}, {4552, 21045}, {4934, 21889}, {5848, 17044}, {16578, 21252}
X(25380) lies on these lines: {2, 2254}, {8, 14413}, {10, 3960}, {171, 22384}, {404, 8648}, {513, 6687}, {514, 2977}, {521, 24675}, {522, 676}, {523, 21212}, {650, 24720}, {659, 4763}, {693, 4913}, {812, 3837}, {891, 19947}, {900, 6667}, {905, 3907}, {1125, 3887}, {1491, 4369}, {1635, 4830}, {1638, 4458}, {1698, 3762}, {2814, 6684}, {2820, 6712}, {2832, 23814}, {2976, 6006}, {3035, 3042}, {3616, 4895}, {3622, 23057}, {3669, 4147}, {3756, 17071}, {3810, 14837}, {3842, 18004}, {4010, 4928}, {4025, 4522}, {4088, 4453}, {4778, 14425}, {4818, 6590}, {5711, 23141}, {6002, 21260}, {9780, 14430}, {17496, 21052}, {21105, 21952}, {21188, 23877}, {21204, 23770}
X(25381) lies on these lines: {1, 19949}, {2, 649}, {10, 514}, {513, 4708}, {659, 24721}, {661, 10196}, {665, 21261}, {693, 4359}, {812, 3837}, {900, 25356}, {1019, 17210}, {2254, 2786}, {3961, 23655}, {4010, 13246}, {4357, 21211}, {4728, 24183}, {5029, 21303}, {5224, 21143}, {6002, 15973}, {7658, 23801}, {14475, 24184}, {17181, 19974}, {17251, 23345}, {21192, 23804}
X(25382) lies on these lines: {2, 19945}, {10, 513}, {75, 24487}, {274, 9296}, {536, 551}, {545, 17793}, {668, 24722}, {730, 4506}, {894, 9421}, {1575, 4070}, {3923, 24346}, {4363, 24327}, {4370, 9283}, {4381, 24334}, {4443, 24451}, {16495, 24485}, {16726, 21100}, {17303, 24289}, {17334, 25120}, {17790, 24342}, {24318, 24348}, {24324, 24350}, {24463, 25347}
X(25383) lies on these lines: {2, 4760}, {11, 25357}, {524, 4892}, {536, 3006}, {625, 16611}, {662, 5333}, {742, 21241}, {812, 3837}, {896, 24724}, {2486, 2886}, {2642, 4728}, {2643, 4137}, {3772, 17778}, {3912, 20541}, {4465, 4708}, {4670, 24712}, {4725, 4938}, {4799, 24892}, {17237, 21264}, {17497, 21057}
X(25384) lies on these lines: {2, 37}, {10, 742}, {44, 14621}, {45, 17738}, {518, 4670}, {537, 10022}, {740, 4665}, {756, 24330}, {984, 4363}, {3416, 4690}, {3821, 3826}, {3823, 4708}, {3925, 25345}, {4396, 5297}, {4470, 24349}, {4472, 9055}, {4643, 4645}, {4981, 24690}, {5263, 24358}, {6682, 25350}, {8680, 23305}, {15569, 24331}, {17369, 17755}
X(25384) = complement of X(24357)
X(25385) lies on these lines: {2, 846}, {10, 908}, {11, 24325}, {75, 17717}, {141, 4892}, {142, 4871}, {226, 1469}, {518, 21242}, {726, 4054}, {740, 5718}, {940, 23812}, {964, 1125}, {984, 21093}, {1150, 17770}, {1215, 2886}, {2887, 3838}, {3589, 17070}, {3739, 5087}, {3782, 6682}, {3817, 12618}, {3826, 24003}, {3831, 12609}, {3840, 5249}, {3846, 17605}, {3914, 6685}, {4023, 4732}, {4090, 25006}, {4363, 24317}, {4413, 24693}, {4417, 21085}, {4647, 17748}, {4671, 6541}, {4689, 17764}, {4703, 5737}, {5121, 24199}, {5263, 17719}, {5741, 21020}, {7321, 18201}, {7483, 24850}, {12579, 16342}, {13740, 24161}, {19270, 24851}, {19945, 24456}, {21674, 25253}, {24165, 24239}, {24250, 24254}, {24330, 25353}
X(25385) = complement of X(4414)
Collineation mappings involving Gemini triangle 20: X(25386)-X(25400)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 20, as in centers X(25386)-X(25400). Then
m(X) = -2 (b + c) (2 a + 2 b - c)(2a - b + 2 c) x + b (a - 2 b - 2 c)(2 a + 2 b - c) y + c (a - 2 b - 2 c) (2 a + 2c - b) z : :
and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, October 15, 2018)
X(25386) lies on these lines: {1, 2}, {142, 25389}
X(25387) lies on this line: {2, 3}
X(25388) lies on this line: {2, 3}
X(25389) lies on these lines: {2, 3}, {142, 25386}
X(25390) lies on these lines: {1, 2}
X(25391) lies on these lines: {1, 2}, {3919, 25351}
X(25392) lies on these lines: {2, 37}, {4715, 20569}, {4777, 25393}
X(25393) lies on these lines: {2, 650}, {4411, 4791}, {4777, 25392}
X(25394) lies on this line: {2, 3}
X(25395) lies on this line: {2, 3}
X(25396) lies on this line: {2, 3}
X(25397) lies on this line: {2, 3}
X(25398) lies on these lines: {2, 45}
X(25399) lies on these lines: {2, 37}
X(25400) lies on these lines: {1, 2}
As a point on the Euler line, X(25401) has Shinagawa coefficients (3*E-136*F, -9*E+24*F)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28468.
X(25401) lies on this line: {2,3}
X(25401) = complement of the isogonal conjugate of X(21002)
X(25401) = midpoint of X(2) and X(5498)
X(25401) = reflection of X(2) in X(12043)
X(25401) = reflection of X(2) in the line X(523)X(12043)
As a point on the Euler line, X(25402) has Shinagawa coefficients (E-24*F, -3*E-56*F)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28468.
X(25402) lies on these lines: {2, 3}, {6153, 20584}, {13851, 15806}
X(25402) = midpoint of X(i) and X(j) for these {i,j}: {4, 5498}, {10125, 18567}, {18282, 18377} X(25402) = reflection of X(3) in X(12043) X(25402) = reflection of X(3) in the line X(523)X(12043) X(25402) = X(5498)-of-Euler triangle X(25402) = X(12043)-of-X3-ABC-reflections triangle X(25402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 3627, 14940), (5, 18403, 140), (5, 18567, 10125), (546, 1885, 3861), (1656, 18566, 15332), (3850, 3860, 13487), (3856, 12811, 23411), (12102, 22249, 6240)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28468.
X(25403) lies on this line: {2, 3}
X(25403) = midpoint of X(i) and X(j) for these {i,j}: {2, 5501}, {12100, 20030}
X(25403) = reflection of X(i) in X(j) for these (i,j): (2, 12056), (10109, 13469), (15334, 11540)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28468.
X(25404) lies on this line: {2, 3}
X(25404) = midpoint of X(i) and X(j) for these {i,j}: {4, 5501}, {546, 20030}
X(25404) = reflection of X(i) in X(j) for these (i,j): (3, 12056), (3530, 13469), (10289, 12811), (15327, 15957), (15335, 3856), (15336, 12108), (15957, 19940)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28477.
X(25405) lies on these lines: {1,3}, {4,6049}, {30,15368}, {214,3880}, {226,10283}, {355,6981}, {376,4345}, {495,551}, {496,5882}, {497,3655}, {515,1387}, {519,3035}, {912,19907}, {944,5274}, {1056,5080}, {1125,5123}, {1201,5399}, {1210,1483}, {1317,1737}, {1318,1870}, {1320,4881}, {1391,4511}, {1785,3319}, {2718,8699}, {3476,5886}, {3582,7972}, {3616,5176}, {3622,6919}, {3636,5087}, {3653,5218}, {3656,4293}, {3897,17624}, {3911,5844}, {3916,5330}, {4308,10595}, {4311,22791}, {4861,17614}, {5083,14988}, {5180,11037}, {5252,11230}, {5284,17615}, {5326,10039}, {5722,6969}, {5901,10106}, {6001,11715}, {8609,17455}, {9581,18526}, {9613,18493}, {9624,9654}, {9655,11522}, {9956,10944}, {10786,14986}, {11231,12647}, {11376,18480}, {11813,21620}, {12577,16137}, {13464,18990}, {20586,22935}
X(25405) = midpoint of X(i) and X/j) for these {i,j}: {1,1319}, {36,5048}, {1317,1737}, {1785,3319}, {12735,15325}
X(25405) = reflection of X(i) in X(j) for these {i,j}: {5122,5126}, {5123,1125}, {5126,1319}, {5570,5045}, {7743,1387}
X(25405) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,36,5048}, {1,56,10222}, {1,57,10247}, {1,1385,9957}, {1,1388,1385}, {1,1420,1482}, {1,5563,11011}, {1,13384,6767}, {1,13462,16200}, {1,20323,5045}, {1,21842,3057}, {36,9819,13528}, {1319,5048,36}, {1385,3660,5126}, {3057,21842,13624}, {5049,9957,12915}, {6767,10246,13384}, {7373,8171,999}
X(25406) lies on the cubic K1085 and these lines: {2,154}, {3,69}, {4,83}, {5,12017}, {6,20}, {22,11433}, {25,18928}, {30,5050}, {74,11061}, {140,18440}, {141,3523}, {146,6593}, {147,5026}, {159,17928}, {165,5847}, {184,7386}, {185,11574}, {193,1350}, {206,6816}, {376,511}, {382,18583}, {388,2330}, {497,1428}, {516,16475}, {518,5731}, {524,10304}, {542,3524}, {548,1353}, {549,11180}, {550,1351}, {575,3529}, {576,17538}, {597,3543}, {599,15692}, {611,4293}, {613,4294}, {631,1352}, {637,10784}, {638,10783}, {732,6194}, {895,16163}, {962,1386}, {1007,5207}, {1092,3538}, {1181,20806}, {1285,5039}, {1370,5012}, {1587,19145}, {1588,19146}, {1629,6819}, {1657,21850}, {1691,7735}, {1692,2549}, {1843,9729}, {1885,19118}, {1899,7494}, {2393,20791}, {2781,25321}, {2794,5182}, {2883,19132}, {3090,3818}, {3091,3589}, {3098,3528}, {3146,5480}, {3167,10691}, {3184,10762}, {3313,5889}, {3332,6996}, {3525,24206}, {3526,18358}, {3533,18553}, {3534,5093}, {3545,11645}, {3620,7891}, {3751,4297}, {3763,10303}, {3767,5033}, {3832,7923}, {4190,15988}, {4220,14555}, {4301,16491}, {4734,9778}, {5032,5102}, {5034,7737}, {5071,10168}, {5135,5800}, {5157,6815}, {5181,15051}, {5218,12588}, {5422,7500}, {5476,15682}, {5596,7503}, {5622,17702}, {5654,6643}, {5820,6966}, {5870,7389}, {5871,7388}, {5943,7714}, {5965,19708}, {5978,6773}, {5979,6770}, {5984,8290}, {5999,7736}, {6055,23053}, {6146,7400}, {6225,19149}, {6247,20079}, {6457,11513}, {6458,11514}, {6515,6636}, {6618,15466}, {6676,23291}, {6759,6804}, {6803,9833}, {6823,18945}, {6995,10601}, {7288,12589}, {7375,14242}, {7376,14227}, {7390,17277}, {7394,7605}, {7396,23292}, {7398,17825}, {7401,13336}, {7407,17381}, {7484,14826}, {7493,15080}, {7512,18916}, {7516,11487}, {7519,15018}, {7525,18951}, {7667,11402}, {7694,16041}, {7791,10131}, {8584,15697}, {8721,14001}, {9308,15258}, {9772,11177}, {9862,12177}, {9969,15043}, {9970,12244}, {10565,13567}, {10574,12220}, {10602,14914}, {10733,15118}, {10752,16111}, {10755,24466}, {10766,14689}, {10996,18935}, {11001,20423}, {11003,16063}, {11178,15702}, {11188,16836}, {11257,13354}, {11477,12007}, {11482,12103}, {11515,18930}, {11516,18929}, {11579,12383}, {11821,12164}, {12322,21737}, {12362,19125}, {13160,23300}, {13367,15812}, {13442,19766}, {14039,15428}, {14360,14688}, {14532,18907}, {14683,16010}, {14810,21735}, {14913,17704}, {15471,20725}, {15520,19924}, {15577,22467}, {15583,17845}, {15708,21358}, {15721,20582}, {15760,18918}, {18141,19649}, {20080,21734}, {24695,24728}
X(25406) = midpoint of X(i) and X(j) for these {i,j}: {376, 14912}, {3534, 5093}, {6776, 10519}
X(25406) = reflection of X(i) in X(j) for these {i,j}: {2, 5085}, {4, 14561}, {69, 10519}, {599, 21167}, {1992, 14912}, {10519, 3}, {14561, 182}, {14853, 5050}, {14912, 11179}, {21356, 3524}, {25320, 5622}
X(25406) = anticomplement X(10516)
X(25406) = barycentric product X(i)*X(j) for these {i,j}: {69, 5304}
X(25406) = barycentric quotient X(i)/X(j) for these {i,j}: {5304, 4}
X(25406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6776, 69), (4, 182, 3618), (182, 10984, 1176), (193, 3522, 1350), (376, 11179, 1992), (487, 488, 3933), (631, 1352, 3619), (1350, 8550, 193), (1352, 5092, 631), (1370, 5012, 11427), (1899, 22352, 7494), (2546, 2547, 12203), (3523, 5921, 141), (3618, 14927, 4), (6459, 6460, 7738), (10574, 12220, 19161), (11257, 13354, 18906), (15080, 18911, 7493)
X(25407) lies on the cubic K1085 and these lines: {2,24651}, {3,15460}, {20,2575}, {22,110}, {69,2574}, {511,1113}, {3917,13415}, {9019,15162}, {15461,20806}
X(25407) = reflection of X(24650) in X(13414)
X(25408) lies on the cubic K1085 and these lines: {2,24650}, {3,15461}, {20,2574}, {22,110}, {69,2575}, {511,1114}, {3917,13414}, {9019,15163}, {15460,20806}
X(25408) = reflection of X(24651) in X(13415)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28487.
X(25409) lies on these lines: {2,16939}, {8586,15533}
X(25409) = anticomplement of X(16939)
X(25409) = antigonal conjugate of X(34581)
X(25409) = antigonal image of isogonal conjugate of X(13492)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28487.
X(25410) lies on these lines: (none)
X(25410) = antigonal image of X(13493)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28487.
X(25411) lies on the cubics K220 and K950, and on these lines: {6,4845}, {9,1155}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28487.
X(25412) lies on this line: {3680,5048}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28489.
Let A'B'C' be the Fuhrmann triangle of the orthic triangle of the Fuhrmann triangle. A'B'C' is homothetic to the excentral triangle (and perspective to ABC) at X(1). X(25413) = X(20)-of-A'B'C'. (Randy Hutson, November 30, 2018)
X(25413) lies on these lines: {1,3}, {5,1537}, {8,6923}, {10,6980}, {119,8256}, {140,3877}, {145,6948}, {355,2800}, {381,12672}, {392,3526}, {404,10698}, {912,10914}, {946,6971}, {952,14923}, {962,6928}, {1071,18526}, {1145,10942}, {1483,3885}, {1656,3753}, {1837,10738}, {2771,5881}, {2802,5884}, {3617,6982}, {3656,10199}, {3679,5694}, {3754,5886}, {3843,9856}, {3868,5844}, {3869,5690}, {3919,13464}, {5330,6940}, {5450,12515}, {5554,6929}, {5603,6958}, {5657,6863}, {5790,5836}, {5840,10950}, {6001,18525}, {6261,18524}, {6797,9581}, {6850,12245}, {6868,20070}, {6882,22791}, {6961,10595}, {7489,19860}, {7741,12619}, {10052,12647}, {10107,18493}, {10404,13375}, {10525,10573}, {10826,17638}, {10948,12832}, {11362,21077}, {11373,12736}, {11376,12758}, {11729,13747}, {12747,17636}, {12773,17654}
X(25413) = reflection of X(i) in X(j) for these {i,j}: {1482,65}, {3869,5690}, {3885,1483}, {5697,1385}, {5887,5836}, {10247,10273}, {10284,5885}, {12645,10914}, {12747,17636}, {12773,17654}, {18526,1071}
X(25413) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {40,7982,5538}, {65,3057,5570}, {65,20323,5902}, {1482,5708,12001}, {1482,12702,10306}, {5836,5887,5790}, {5885,7957,11849}, {5885,10284,1}, {8148,12001,1482}, {11009,14800,1}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28489.
X(25414) lies on these lines: {1,3}, {5,12758}, {11,17665}, {12,1537}, {149,1837}, {355,17638}, {404,12740}, {1145,10958}, {1317,5884}, {1858,3880}, {2800,10944}, {2802,10950}, {3754,15558}, {3878,17757}, {3884,5432}, {3885,17637}, {5836,11680}, {6917,10043}, {8275,12409}, {10199,17622}, {10573,10947}, {10948,20118}
X(25414) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,5119,11849}, {1,5697,10284}, {55,5697,3057}, {65,3057,5048}, {999,5903,65}, {1837,14923,17636}, {5885,9957,1}, {5919,13751,1}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28489.
X(25415) lies on these lines: {1,3}, {8,6871}, {9,3899}, {10,6933}, {11,3656}, {33,1845}, {79,3633}, {80,1537}, {84,21398}, {90,1389}, {145,4295}, {200,4867}, {226,12647}, {392,8167}, {474,10107}, {497,11041}, {498,11362}, {499,4848}, {516,11526}, {519,1478}, {528,3243}, {674,16496}, {758,3872}, {944,1770}, {946,10573}, {952,1836}, {962,10572}, {968,17461}, {1000,3475}, {1022,4905}, {1056,11551}, {1317,11246}, {1320,3873}, {1387,17728}, {1479,4301}, {1572,5332}, {1698,5443}, {1709,2800}, {1723,1953}, {1728,15556}, {1737,5603}, {1756,7174}, {1788,10595}, {1837,22791}, {2161,16670}, {2364,16548}, {2802,3870}, {2807,15430}, {2886,3679}, {2900,3880}, {3085,4323}, {3086,5734}, {3158,5541}, {3241,4293}, {3244,4292}, {3306,3919}, {3419,3632}, {3474,7967}, {3485,8164}, {3555,10912}, {3583,5727}, {3585,5881}, {3624,5445}, {3626,3984}, {3635,4311}, {3654,5432}, {3655,15326}, {3753,5289}, {3754,19861}, {3811,14923}, {3877,5284}, {3878,19860}, {3898,4666}, {3901,6762}, {3922,16408}, {4018,12513}, {4084,22837}, {4299,5882}, {4305,20070}, {4333,18481}, {4338,7354}, {4677,11525}, {4845,10697}, {4853,5904}, {5252,5844}, {5258,12526}, {5274,18391}, {5326,15950}, {5441,8000}, {5554,21616}, {5559,5665}, {5561,13143}, {5587,18393}, {5690,11375}, {5691,7971}, {5692,9623}, {5730,5836}, {5790,17605}, {5837,19854}, {5842,10483}, {5854,12749}, {6264,11571}, {6834,15867}, {6950,14497}, {7160,15173}, {7162,17097}, {7672,12758}, {7680,7741}, {8168,10914}, {8227,18395}, {9814,15733}, {9897,10742}, {10390,13602}, {10525,18962}, {10950,12699}, {10954,15908}, {12650,15071}, {12943,18499}, {17606,18493}
X(25415) = reflection of X(i) in X(j) for these {i,j}: {1,2099}, {3632,3419}, {5119,1}, {7991,3428}, {10679,10222}, {12647,226}
X(25415) = X(378)-of-reflection-triangle-of-X(1)
X(25415) = (tangential-of-excentral)-to-ABC similarity image of X(12703)
X(25415) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,40,3612}, {1,65,3338}, {1,484,3576}, {1,2093,36}, {1,3336,1420}, {1,3339,5563}, {1,5010,13384}, {1,5903,46}, {1,7991,35}, {1,11010,3601}, {1,11280,7982}, {1,11531,5697}, {1,15803,21842}, {1,18421,5902}, {3,11011,1}, {8,12047,10827}, {36,2093,46}, {36,5903,2093}, {40,13384,5010}, {55,5173,10980}, {56,10222,1}, {57,16200,1}, {65,1482,1}, {65,5048,999}, {65,5570,5902}, {65,11011,18967}, {65,20323,5708}, {942,2098,1}, {942,11278,2098}, {946,10573,10826}, {999,1482,5048}, {999,5048,1}, {1319,10247,1}, {1482,12001,10222}, {3339,16189,1}, {3340,7962,11529}, {3340,7982,1}, {3474,7967,21578}, {3485,12245,10039}, {4848,13464,499}, {5010,13384,3612}, {5903,11009,1}, {5919,15934,1}, {7962,11529,1}, {7982,11529,7962}, {10306,22766,35}, {11224,18421,1}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28489.
X(25416) lies on these lines: {1,1145}, {3,13278}, {4,145}, {8,1387}, {11,519}, {65,1317}, {72,15558}, {80,3633}, {100,999}, {104,10306}, {119,10222}, {214,3635}, {392,14740}, {517,3937}, {518,12758}, {528,3243}, {1385,22082}, {1392,11681}, {1478,13271}, {1483,3885}, {2800,3555}, {2829,7982}, {3036,3632}, {3338,5541}, {3625,6702}, {3679,6667}, {3753,18240}, {3811,12740}, {3813,8068}, {3870,6265}, {3880,5570}, {3913,10090}, {4345,17556}, {4999,5559}, {5708,9945}, {6049,19537}, {7962,11113}, {7993,12127}, {9797,9803}, {10031,20095}, {10058,12513}, {10074,13205}, {10087,22560}, {10247,11729}, {10914,12736}, {10956,11011}, {11362,21154}, {12001,12331}, {12019,12531}, {12630,20119}, {12649,19914}, {12751,16200}, {21077,21630}
X(25416) = midpoint of X(i) and X(j) for these {i,j}: {80,3633}, {145,1320}, {3555,17652}, {7972,12653}, {12531,20050}, {12630,20119}
X(25416) = reflection of X(i) in X(j) for these {i,j}: {8,1387}, {72,15558}, {100,12735}, {119,10222}, {1145,1}, {1317,3244}, {1537,1482}, {3625,6702}, {3632,3036}, {10609,1317}, {10914,12736}, {12531,12019}, {12732,10609}, {13257,10698}, {13996,214}, {17757,5048}
X(25416) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {100,3241,12735}, {3632,16173,3036}
Centers associated with Gemini triangles 1-4: X(25417) - X(25431))
These centers were contributed by Randy Hutson, October 17, 2018. Gemini triangles are introduced in the preamble just before X(24537).
Let A1B1C1 be the Gemini triangle 1. Let LA be the line through A1 parallel to BC, and define LB and LC cyclically. Let A'1 = LB∩LC, and define B'1 and C'1 cyclically. A'1B'1C'1 is homothetic to ABC at X(25417).
Let A12B12C12 be the Gemini triangle 12. Let LA be the line through A12 parallel to BC, and define LB and LC cyclically. Let A'12 = LB∩LC, and define B'12 and C'12 cyclically. Triangle A'12B'12C'12 is homothetic to ABC at X(25417). (Randy Hutson, November 30, 2018)
X(25417) lies on these lines: {1, 2308}, {2, 319}, {6, 1255}, {8, 1224}, {28, 11396}, {57, 1442}, {81, 16884}, {88, 940}, {89, 3666}, {105, 8652}, {274, 4393}, {278, 6994}, {291, 3722}, {559, 2306}, {593, 2241}, {959, 1319}, {961, 2099}, {1002, 3748}, {1029, 1479}, {1219, 3623}, {1449,3305} et al
X(25417) = isogonal conjugate of X(16777)
Let A'1B'1C'1 be as at X(25417). Let A2B2C2 be the Gemini triangle 2. Let MA be the line through A2 parallel to BC, and define MB and MC cyclically. Let A'2 = MB∩MC, and define B'2 and C'2 cyclically. Triangles A'1B'1C'1 and A'2B'2C'2 are homothetic at X(25418).
X(25418) lies on these lines: {89, 3666}, {14996, 16816}
X(25419) lies on these lines: {551, 3158}, {3616, 4682}
X(25420) lies on these lines: {1, 966}, {37, 291}, {86, 7240}, {2663, 3986} et al
X(25420) = {X(86),X(25422)}-harmonic conjugate of X(25421)
X(25421) lies on these lines: {1, 193}, {2, 16571}, {37, 256}, {43, 5296}, {86, 7240}, {87, 3616}, {846, 1400}, {894, 4368}, {966, 4489}, {1045, 5257} et al
X(25421) = {X(86),X(25422)}-harmonic conjugate of X(25420)
X(25422) lies on these lines: {1, 4407}, {37, 3122}, {86, 7240}, {551, 3248}, {872, 3986}, {2667, 5257} et al
X(25422) = {X(25420),X(25421)}-harmonic conjugate of X(86)
X(25423) lies on these lines: {2, 669}, {30, 511} et al
X(25423) = isogonal conjugate of X(25424)
X(25423) = crossdifference of every pair of points on line X(6)X(3229)
X(25423) = infinite point of perspectrix of Gemini triangles 3 and 4
X(25424) lies on the circumcircle and these lines: {6, 699}, {98, 3098}, {111, 12149}, {574, 729}, {3222, 4576} et al
X(25424) = isogonal conjugate of X(25423)
X(25424) = trilinear pole of line X(6)X(3229)
X(25424) = Thomson-isogonal conjugate of X(32515)
X(25424) = Lucas-isogonal conjugate of X(32515)
X(25424) = Λ(PU(148))
X(25425) lies on these lines: {2280, 2344}, {3403, 4384}
X(25425) = isogonal conjugate of X(25429)
X(25425) = trilinear pole of line X(4806)X(25423)
X(25426) lies on these lines: {1, 1573}, {6, 2667}, {42, 292}, {56, 18755}, {58, 1914}, {86, 239}, {87, 1045}, {187, 2163}, {213, 1126}, {870, 4393}, {893, 25429}, {1107, 5331}, {2665, 25428} et al
X(25426) = isogonal conjugate of X(16826)
X(25427) lies on these lines: {1, 1573}, {2, 24578}, {9, 86}, {37, 1045}, {40, 6998}, {191, 3294}, {274, 3985}, {1050, 1107} et al
X(25428) lies on these lines: {2, 24578}, {2665, 25426}
X(25429) lies on these lines: {1, 1755}, {6, 3229}, {9, 86}, {58, 2176}, {81, 2311}, {171, 2196}, {893, 25426}, {942, 1107}, {3736, 3774} et al
X(25429) = isogonal conjugate of X(25425)
Let A3B3C3 be the Gemini triangle 3. Let LA be the line through A3 parallel to BC, and define LB and LC cyclically. Let A'3 = LB∩LC, and define B'3 and C'3 cyclically. A'3B'3C'3 is homothetic to ABC at X(25430).
X(25430) is the perspector of the ellipse defined at X(3646).
Let AABACA be the A-extouch triangle. Let AiBiCi be the intouch triangle. Let A' be the centroid of AABACAAi. Define B', C' cyclically. The lines AA', BB', CC' concur in X(25430).
X(25430) lies on these lines: {1, 210}, {2, 2321}, {9, 81}, {28, 33}, {37, 57}, {63, 89}, {88, 4606}, {105, 612}, {200, 15569}, {226, 279}, {274, 312}, {278, 1826}, {291, 3677}, {614, 1390}, {940, 3731}, {957, 7962}, {959, 3340}, {961, 1420}, {985, 1961}, {1002, 3720}, {1123, 5405}, {1219, 3616}, {1224, 3624}, {1255, 5256}, {1280, 4666}, {1336, 5393}, {1419, 1422}, {1449, 3305} et al
X(25430) = isogonal conjugate of X(1449)
X(25430) = isotomic conjugate of X(19804)
Let A'3B'3C'3 be as at X(25430). Let A4B4C4 be the Gemini triangle 4. Let MA be the line through A4 parallel to BC, and define MB and MC cyclically. Let A'4 = MB∩MC, and define B'4 and C'4 cyclically. Triangles A'3B'3C'3 and A'4B'4C'4 are homothetic at X(25431).
X(25431) lies on these lines: {1, 210}, {10, 1255}, {35, 199}, {37, 191}, {46, 16673}, {1698, 4007} et al
X(25432) lies on these lines: {30,511}, {81,21997}, {1211,17292}, {17799,33160}
X(25432) = isogonal conjugate of X(25433)
X(25433) lies on the circumcircle and the line {99,17280}
X(25433) = isogonal conjugate of X(25432)
X(25433) = intersection, other than A,B,C, of conic {{A, B, C, X(37), X(17280)}} and circumcircle
X(25434) lies on these lines: {30,511}, {141,24291}, {333,1931}, {4544,17351}, {5724,25349}, {7267,21044}, {16826,17056}, {17778,20349}
X(25434) = isogonal conjugate of X(25435)
X(25435) lies on the circumcircle and these lines: {99,6646}, {110,21008}
X(25435) = isogonal conjugate of X(25434)
X(25435) lies on the circumcircle and these lines: {99,6646}, {110,21008}
X(25435) = isogonal conjugate of X(25434)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28494.
X(25436) lies on these lines: {1, 10774}, {10, 900}, {11, 23869}, {12, 13604}, {80, 3120}, {214, 11814}, {355, 18342}, {546, 946}, {2802, 21093}, {5587, 18341}, {6788, 24188}
X(25436) = midpoint of X(i) and X(j) for these {i,j}: {80, 15343}, {355, 18342}
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28494.
X(25437) lies on these lines: {1, 18341}, {10, 140}, {496, 11798}, {900, 946}, {5533, 15529}, {5886, 18342}
X(25437) = midpoint of X(1) and X(18341)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28499.
X(25438) lies on these lines: {1,88}, {3,5854}, {5,13271}, {8,10058}, {11,5687}, {55,1145}, {78,12758}, {80,6735}, {104,519}, {119,381}, {145,10074}, {149,5154}, {200,18254}, {518,12515}, {952,3913}, {997,15558}, {1317,1470}, {1376,1387}, {1519,14217}, {1768,6765}, {2136,6264}, {2800,3811}, {2801,2950}, {2829,10306}, {2900,12691}, {3035,3295}, {3158,6326}, {3189,12247}, {3359,15528}, {3434,8068}, {3689,17638}, {3870,11570}, {3880,12737}, {3935,12532}, {4421,10269}, {5440,12740}, {5531,7995}, {5840,6256}, {5853,10265}, {6154,10956}, {6224,12648}, {6667,9709}, {8069,18802}, {10528,20095}, {10609,11509}, {10698,22836}, {10742,12607}, {10915,12751}, {10942,13272}, {11236,22799}, {11362,17009}, {12115,13199}, {12514,14740}, {12764,17757}
X(25438) = midpoint of X(i) and X(j) for these {i,j}: {3189,12247}
= reflection of X(i) in X(j) for these {i,j}: {100,8715}, {10698,22836}, {10742,12607}, {12751,10915}, {13271,5}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {100,1320,10090}, {100,3871,10087}, {100,13278,1}, {145,17100,10074}, {5440,17652,12740}, {10679,12331,119}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28499.
X(25439) lies on these lines: {1,88}, {3,3244}, {8,3746}, {10,1001}, {21,3632}, {35,145}, {36,3241}, {40,3243}, {41,1023}, {46,3881}, {55,519}, {56,3635}, {57,3892}, {78,3884}, {149,7951}, {165,4973}, {200,10176}, {392,3689}, {405,3626}, {411,11531}, {474,3636}, {484,3873}, {495,528}, {496,6667}, {497,3814}, {515,10679}, {535,4302}, {551,1376}, {595,2209}, {643,9275}, {758,3870}, {942,15570}, {943,12625}, {946,18491}, {950,10915}, {958,3625}, {997,3158}, {999,4421}, {1018,2280}, {1125,3303}, {1329,15172}, {1478,11239}, {1479,10528}, {1482,6796}, {1621,3679}, {1697,3811}, {1698,17546}, {2077,7967}, {2136,11525}, {2241,20691}, {2646,22837}, {2886,10197}, {2975,3633}, {3035,10199}, {3057,22836}, {3058,17757}, {3208,4251}, {3240,5315}, {3256,3476}, {3293,3915}, {3305,3956}, {3336,3889}, {3339,11526}, {3421,10385}, {3434,3822}, {3436,4309}, {3584,11680}, {3617,5259}, {3621,5258}, {3623,5563}, {3634,16856}, {3656,18524}, {3678,4917}, {3748,3753}, {3816,15170}, {3825,5552}, {3828,4423}, {3833,4666}, {3868,11010}, {3919,15934}, {3935,5692}, {3938,4424}, {3950,4254}, {3957,5902}, {4084,12702}, {4125,4387}, {4188,20057}, {4189,5288}, {4297,10306}, {4301,11500}, {4326,5223}, {4428,4669}, {4430,4880}, {4649,5145}, {4668,5260}, {4757,11520}, {4848,11510}, {4857,11681}, {5082,10198}, {5123,18527}, {5267,12513}, {5276,9331}, {5284,19875}, {5290,16133}, {5291,10987}, {5310,20020}, {5440,5919}, {5450,11849}, {5493,12631}, {5537,5731}, {5726,8543}, {5727,15863}, {5775,12630}, {5882,11248}, {5886,12331}, {6049,13370}, {6154,11112}, {6681,10072}, {6684,16202}, {6905,16200}, {7982,11491}, {7991,12511}, {8256,12433}, {8668,12437}, {9668,11236}, {9709,19862}, {10267,11362}, {10483,20066}, {10624,21077}, {10902,12245}, {10965,12053}, {11499,12000}, {12575,21616}, {12607,15171}, {12632,19843}, {12703,18446}, {15808,16408}, {16474,17126}, {16784,17756}, {18483,18518}
X(25439) = midpoint of X(i) and X(j) for these {i,j}: {1,3895}, {1478,20075}, {3870,5119}, {10087,13278}, {12703,18446}
X(25439) = reflection of X(i) in X(j) for these {i,j}: {993,55}, {3434,3822}
X(25439) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3871,8715}, {8,3746,5248}, {35,145,8666}, {1376,6767,551}, {1697,3811,3878}, {3295,3913,10}, {3303,5687,1125}, {3434,10056,3822}, {3871,13278,3895}, {3895,10087,8715}, {4189,20050,5288}, {4428,8168,9708}, {8168,9708,4669}, {11239,20075,1478}, {11499,12000,13464}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28499.
X(25440) lies on these lines: {1,88}, {2,35}, {3,10}, {4,2077}, {5,3035}, {6,9679}, {8,36}, {9,2173}, {11,13747}, {12,6174}, {20,8165}, {21,1698}, {24,1861}, {30,1329}, {31,3216}, {32,1575}, {39,4386}, {40,997}, {41,16549}, {43,58}, {46,78}, {55,474}, {56,519}, {57,3811}, {63,3678}, {65,5440}, {72,1155}, {75,1078}, {80,17100}, {81,5312}, {99,6376}, {101,3501}, {104,5881}, {140,2886}, {145,5563}, {165,411}, {171,386}, {182,17792}, {183,20888}, {187,1574}, {191,3876}, {198,17355}, {200,4973}, {210,3916}, {224,17700}, {226,7702}, {228,3980}, {238,17749}, {318,4242}, {333,4278}, {376,2551}, {377,498}, {405,3634}, {442,5432}, {443,5218}, {480,5850}, {484,3869}, {495,17563}, {496,528}, {497,10200}, {499,3434}, {516,3149}, {517,6924}, {535,3436}, {548,9711}, {549,4999}, {550,3820}, {551,3295}, {574,1107}, {595,978}, {601,5150}, {603,4551}, {631,2550}, {662,17104}, {851,3454}, {899,1724}, {908,1770}, {946,6911}, {950,11502}, {952,8256}, {956,3626}, {960,3579}, {962,5537}, {975,3743}, {976,3670}, {995,5255}, {999,3244}, {1001,16408}, {1004,1259}, {1010,4276}, {1012,19925}, {1018,9310}, {1030,17303}, {1145,10944}, {1151,1377}, {1152,1378}, {1158,5720}, {1193,5264}, {1203,17126}, {1210,8069}, {1319,10914}, {1385,5836}, {1387,13463}, {1402,17733}, {1403,8669}, {1444,17270}, {1454,15556}, {1465,4347}, {1466,4298}, {1468,3293}, {1470,10106}, {1478,4190}, {1490,10270}, {1532,11826}, {1621,3624}, {1697,3898}, {1699,6915}, {1706,3576}, {1739,3924}, {1740,4279}, {1754,3682}, {1768,12528}, {1771,22350}, {1816,2328}, {1837,2932}, {1975,6381}, {2049,19760}, {2057,20588}, {2078,5082}, {2178,2321}, {2209,18792}, {2222,2757}, {2223,16825}, {2241,16604}, {2242,20691}, {2276,5277}, {2475,7951}, {2478,4302}, {2646,3753}, {2801,17857}, {2975,3679}, {3008,21477}, {3057,17614}, {3061,5011}, {3085,6904}, {3086,17784}, {3158,3333}, {3218,4420}, {3256,3485}, {3303,3636}, {3304,3635}, {3336,3868}, {3337,3873}, {3338,3870}, {3339,12559}, {3359,6261}, {3361,6765}, {3416,5096}, {3419,5172}, {3430,20368}, {3452,6985}, {3523,15931}, {3530,9710}, {3555,3689}, {3560,10175}, {3583,4193}, {3585,11681}, {3612,3918}, {3616,3746}, {3617,5258}, {3625,12513}, {3661,19308}, {3681,5131}, {3684,4253}, {3724,4647}, {3741,4191}, {3749,11512}, {3752,5266}, {3771,16056}, {3788,20541}, {3813,15325}, {3816,15171}, {3817,6918}, {3826,6675}, {3828,16370}, {3831,13738}, {3836,16415}, {3840,16059}, {3846,19543}, {3877,11010}, {3884,5119}, {3911,10916}, {3912,11329}, {3923,19548}, {3925,7483}, {3927,4134}, {3930,17736}, {3938,3953}, {3940,4067}, {3941,4974}, {3951,3988}, {3984,4127}, {4015,4652}, {4023,15447}, {4084,12635}, {4090,20805}, {4187,6284}, {4189,5251}, {4203,16569}, {4251,17754}, {4255,5711}, {4257,5247}, {4293,7080}, {4301,10306}, {4304,8582}, {4306,9364}, {4308,13370}, {4311,6736}, {4314,9843}, {4324,11114}, {4384,21495}, {4423,16862}, {4428,19883}, {4511,5903}, {4557,22458}, {4640,5044}, {4660,19513}, {4669,11194}, {4689,6051}, {4745,19705}, {4847,7742}, {4861,4881}, {5019,21857}, {5030,21384}, {5047,9342}, {5080,10483}, {5086,18395}, {5087,22793}, {5123,18480}, {5124,17275}, {5126,11260}, {5205,7283}, {5260,17549}, {5280,17756}, {5281,17580}, {5284,17535}, {5289,12702}, {5293,17596}, {5322,10327}, {5493,6244}, {5542,6600}, {5554,14803}, {5584,12447}, {5587,6906}, {5657,6942}, {5691,6909}, {5694,10225}, {5705,6986}, {5777,18232}, {5818,6950}, {5842,6922}, {5880,11263}, {5882,10269}, {5886,11849}, {6224,14800}, {6256,6948}, {6358,22342}, {6516,9312}, {6644,9712}, {6667,10593}, {6679,16298}, {6685,11358}, {6690,8728}, {6702,10058}, {6713,10943}, {6835,12558}, {6845,18406}, {6876,7688}, {6910,19854}, {6912,7989}, {6914,9956}, {6946,7704}, {6955,10786}, {6959,10525}, {7295,17353}, {7330,15064}, {7354,17757}, {7411,8580}, {7580,12512}, {7786,20179}, {7815,21264}, {7824,17030}, {7987,9623}, {8053,16286}, {8167,16863}, {8668,21627}, {8671,17793}, {9367,16283}, {9458,13589}, {9655,11236}, {9701,22115}, {10039,14793}, {10165,10267}, {10479,16451}, {10527,14798}, {10609,10950}, {10679,13464}, {10948,15866}, {11113,15338}, {11249,11362}, {11285,20172}, {11347,20106}, {11349,17284}, {11373,13205}, {11375,14882}, {11507,12609}, {11599,13173}, {11680,17566}, {11813,12699}, {12178,21636}, {12332,21635}, {12436,13405}, {12607,18990}, {12629,13462}, {12672,13528}, {12688,17613}, {12751,18861}, {12953,17556}, {13204,13605}, {13607,16203}, {14377,20335}, {15326,21031}, {15587,18233}, {15625,16414}, {15888,17583}, {16062,19842}, {16342,16828}, {16347,19874}, {16611,16968}, {16819,17684}, {16858,19876}, {16865,19877}, {17279,20872}, {17281,19297}, {17299,21773}, {17308,21511}, {17536,19872}, {17750,18755}, {17886,20926}, {18492,21669}, {19270,19858}, {19278,19853}, {20833,20989}
X(25440) = midpoint of X(i) and X(j) for these {i,j}: {3,11499}, {46,78}, {56,5687}, {100,10090}, {3149,10310}, {3436,4299}, {4311,6736}
X(25440) = reflection of X(i) in X(j) for these {i,j}: {496,6691}, {1479,3825}, {10199,17564}, {12053,1125}, {12059,3678}, {21616,6700}
X(25440) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,100,8715}, {1,5541,3885}, {2,35,5248}, {2,1479,3825}, {3,10,993}, {3,355,5450}, {3,958,5267}, {3,1376,10}, {3,9709,958}, {3,11500,4297}, {3,18524,18481}, {8,36,8666}, {8,4188,36}, {10,5267,958}, {40,997,3878}, {40,5438,997}, {55,474,1125}, {57,3811,3874}, {65,5440,22836}, {100,404,1}, {100,5253,3871}, {165,411,12511}, {165,936,12514}, {187,1574,4426}, {377,498,3822}, {404,3871,5253}, {405,4413,3634}, {443,5218,10198}, {496,6691,10199}, {496,17564,6691}, {497,17567,10200}, {499,6921,6681}, {936,12514,10176}, {956,19537,5204}, {958,1376,9709}, {958,5267,993}, {958,9709,10}, {975,17594,3743}, {978,3550,595}, {999,3913,3244}, {1001,16408,19862}, {1004,1259,4292}, {1319,10914,22837}, {1470,11501,10106}, {1621,17531,3624}, {1698,5010,21}, {2975,13587,7280}, {3218,4420,5904}, {3338,3870,3881}, {3434,6921,499}, {3679,7280,2975}, {3868,9352,3336}, {3871,5253,1}, {4189,9780,5251}, {4190,5552,1478}, {4292,6745,21077}, {4413,5217,405}, {4421,16417,551}, {4423,16862,19878}, {4861,4881,21842}, {5119,19861,3884}, {5657,6942,11012}, {5687,16371,56}, {5880,11374,11263}, {6911,11248,946}, {6918,11496,3817}, {6940,11491,3576}, {10306,22753,4301}, {11681,17579,3585}, {12512,20103,12572}
Collineation mappings involving Gemini triangle 21: X(25441)-X(25473)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 21, as in centers X(25441)-X(25473). Then
m(X) = (a + b)(a + c)(a + b + c) x + b (a + b)(b + c) y + c (a + c)(b + c) z : :
A point X lies on the Euler line if and only m(X) lies on the Euler line. (Clark Kimberling, October 19, 2018)
X(25441) lies on these lines: {1, 2}, {5, 1746}, {21, 6693}, {58, 5051}, {79, 4697}, {81, 3454}, {191, 4425}, {267, 24169}, {442, 6703}, {474, 6739}, {986, 4736}, {1098, 1724}, {1100, 24935}, {1203, 3846}, {1329, 25680}, {1788, 24912}, {2363, 16062}, {2476, 24624}, {3336, 3821}, {3589, 4187}, {3615, 4193}, {3670, 19786}, {3812, 24881}, {3816, 25648}, {3833, 24904}, {3936, 4658}, {4220, 7683}, {4239, 5358}, {4257, 17676}, {4657, 25445}, {4894, 17716}, {5259, 6679}, {5437, 25443}, {6690, 24934}, {6691, 24925}, {6692, 24882}, {6856, 24937}, {7741, 25496}, {10974, 18417}, {16706, 24923}, {17194, 27531}, {17370, 24901}, {17384, 24890}, {18398, 26128}, {23903, 24275}, {24443, 24911}, {25444, 25448}, {25451, 25463}, {25461, 25462}
X(25442) lies on these lines: {2, 3}, {942, 24913}, {25445, 25452}
X(25443) lies on these lines: {2, 3}, {1210, 24884}, {1698, 25447}, {1714, 24174}, {2939, 25361}, {3911, 24880}, {5437, 25441}, {5438, 25645}, {5439, 16706}, {5440, 25650}, {9612, 25651}, {9895, 17923}, {19804, 25446}, {25452, 25468}, {25456, 25458}
X(25444) lies on these lines: {2, 3}, {12, 24880}, {495, 24883}, {496, 24936}, {2886, 25645}, {3017, 15888}, {3216, 24161}, {3925, 24931}, {5506, 24342}, {7951, 24902}, {17757, 25446}, {24390, 25650}, {25441, 25448}
X(25445) lies on these lines: {2, 6}, {58, 21245}, {1761, 25370}, {2308, 24941}, {3017, 4852}, {3739, 24880}, {4361, 24883}, {4395, 25459}, {4657, 25441}, {4670, 24937}, {4851, 25645}, {6693, 17052}, {17239, 24931}, {17296, 25669}, {17353, 25659}, {17356, 24884}, {17390, 25650}, {25442, 25452}, {25451, 25453}, {25457, 25460}, {25469, 25470}
X(25446) lies on these lines: {1, 2}, {5, 17277}, {58, 26051}, {75, 5791}, {320, 3824}, {333, 442}, {346, 24044}, {377, 24587}, {759, 4189}, {966, 3767}, {1043, 6675}, {1150, 4197}, {1247, 17596}, {1324, 16451}, {1654, 3454}, {1834, 11110}, {2476, 2651}, {3686, 24937}, {3841, 4645}, {4395, 25455}, {4658, 26109}, {5051, 5235}, {5737, 16062}, {6703, 14007}, {6856, 14555}, {7379, 7683}, {8258, 24342}, {8728, 14829}, {9711, 25448}, {16704, 26131}, {17606, 25677}, {17757, 25444}, {19804, 25443}, {24390, 25648}, {25456, 25457}, {25458, 25472}
X(25447) lies on these lines: {2, 7}, {10, 24435}, {21, 17052}, {37, 24890}, {141, 5135}, {474, 17327}, {631, 24931}, {662, 5224}, {857, 4877}, {1213, 1375}, {1698, 25443}, {1781, 25359}, {2345, 22003}, {2487, 14319}, {3946, 24883}, {4000, 24880}, {4643, 25649}, {4657, 25441}, {4859, 24902}, {5251, 20305}, {5439, 25498}, {5440, 17239}, {6675, 18635}, {6690, 25664}, {6852, 24220}, {16608, 24953}, {17237, 24935}, {17296, 25650}, {21239, 25648}, {25457, 25469}, {25460, 25470}
X(25448) lies on these lines: {2, 11}, {1329, 24880}, {3813, 25645}, {3847, 25648}, {4187, 24902}, {9508, 24920}, {9710, 24931}, {9711, 25446}, {12607, 24883}, {24390, 25669}, {25441, 25444}
X(25449) lies on these lines: {2, 3}
X(25450) lies on these lines: {2, 3}, {523, 25473}
X(25451) lies on these lines: {2, 31}, {3741, 24890}, {20083, 24914}, {25441, 25463}, {25445, 25453}
X(25452) lies on these lines: {2, 32}, {21240, 24888}, {25442, 25445}, {25443, 25468}, {25458, 25465}
X(25453) lies on these lines: {1, 2}, {6, 2887}, {9, 4425}, {19, 4212}, {31, 4450}, {44, 4703}, {46, 8258}, {55, 4085}, {57, 24169}, {63, 3821}, {81, 25957}, {171, 4429}, {197, 16059}, {321, 26061}, {345, 4970}, {518, 26128}, {726, 19785}, {894, 17889}, {940, 3836}, {982, 16706}, {984, 19786}, {1215, 3772}, {1386, 4865}, {1468, 4202}, {1738, 3980}, {1757, 27184}, {1836, 4672}, {2238, 2273}, {2293, 27521}, {2308, 6327}, {2886, 3589}, {3120, 26223}, {3416, 3791}, {3618, 21241}, {3666, 4438}, {3742, 17356}, {3751, 25527}, {3775, 4042}, {3823, 4682}, {3826, 6703}, {3846, 4383}, {3914, 3923}, {3944, 27064}, {3966, 4974}, {3993, 17776}, {4000, 24165}, {4011, 17353}, {4038, 17234}, {4071, 16972}, {4119, 16777}, {4385, 19803}, {4388, 16468}, {4641, 4655}, {4649, 18134}, {4670, 24694}, {4697, 5880}, {5014, 17469}, {5015, 16478}, {5247, 16062}, {6693, 25440}, {7262, 24723}, {7834, 20544}, {17064, 25385}, {17123, 17352}, {17124, 24988}, {17303, 27798}, {17380, 17600}, {18905, 23543}, {20064, 21747}, {20369, 21376}, {24248, 26065}, {24325, 24789}, {24597, 26034}, {25445, 25451}
X(25454) lies on these lines: {2, 44}, {141, 25669}, {4657, 25441}, {4851, 25650}, {17279, 24890}, {25460, 25469}
X(25455) lies on these lines: {2, 45}, {141, 19270}, {4357, 25649}, {4395, 25446}, {4657, 25441}, {4698, 24884}, {17235, 24937}, {17306, 25669}, {17313, 24936}, {17321, 24890}, {17382, 24880}
X(25456) lies on these lines: {2, 6}, {25443, 25458}, {25446, 25457}, {27473, 27474}
X(25457) lies on these lines: {2, 37}, {6, 1509}, {9, 25682}, {10, 24437}, {86, 4272}, {274, 1213}, {314, 6707}, {386, 20150}, {583, 17277}, {1125, 20174}, {1268, 3963}, {1333, 16917}, {1441, 24909}, {1654, 16709}, {1698, 25458}, {2092, 25508}, {2220, 17000}, {2663, 3216}, {5069, 27318}, {5224, 18827}, {16696, 26045}, {16819, 24897}, {25445, 25460}, {25446, 25456}, {25447, 25469}
X(25458) lies on these lines: {2, 39}, {349, 24915}, {1269, 24944}, {1698, 25457}, {3760, 25660}, {5437, 25467}, {10479, 24174}, {24530, 27020}, {25443, 25456}, {25446, 25472}, {25452, 25465}
X(25459) lies on these lines: {1, 2}, {4395, 25445}
X(25460) lies on these lines: {1, 2}, {17673, 24632}, {25445, 25457}, {25447, 25470}, {25454, 25469}
X(25461) lies on these lines: {2, 7}, {1014, 25004}, {2893, 25946}, {3672, 24195}, {4648, 17058}, {17169, 27524}, {25441, 25462}, {25446, 25456}
X(25462) lies on these lines: {2, 3}, {25441, 25461}
X(25463) lies on these lines: {2, 3}, {24530, 24880}, {24955, 26127}, {25441, 25451}
X(25464) lies on these lines: {2, 3}
X(25465) lies on these lines: {2, 3}, {25452, 25458}
Let DEF be the medial triangle. The circle that touches all three incircles of the triangles AEF, BFD, CDE and encompasses them touches in points that we denote by them in A', B', C', respectively. The lines DA', EB' , FC' concur in X(25466). See X(25466). (Angel Montesdeoca, November 29, 2019)
X(25466) lies on these lines: {1, 442}, {2, 12}, {3, 6690}, {4, 1001}, {5, 515}, {6, 26036}, {8, 3475}, {9, 5290}, {10, 141}, {11, 2476}, {21, 7354}, {30, 5248}, {34, 25985}, {35, 11112}, {36, 7483}, {37, 5254}, {38, 21674}, {40, 5735}, {55, 377}, {57, 26066}, {63, 10404}, {65, 5249}, {72, 13407}, {75, 3704}, {78, 17718}, {86, 21276}, {92, 25987}, {104, 6852}, {119, 1656}, {140, 26286}, {226, 960}, {236, 8088}, {278, 15832}, {341, 20917}, {354, 6734}, {355, 6881}, {392, 12047}, {404, 5432}, {405, 1478}, {427, 1891}, {429, 5307}, {443, 1376}, {452, 5229}, {474, 498}, {496, 551}, {497, 5177}, {517, 3824}, {519, 3841}, {527, 18249}, {528, 3295}, {632, 6681}, {758, 6147}, {908, 24564}, {940, 5230}, {944, 6829}, {946, 3838}, {956, 19854}, {971, 12617}, {976, 17724}, {986, 1086}, {993, 6675}, {997, 11374}, {999, 26363}, {1006, 11827}, {1046, 17365}, {1056, 12513}, {1058, 11235}, {1191, 26098}, {1193, 5718}, {1210, 3742}, {1319, 24541}, {1386, 5717}, {1388, 10957}, {1426, 5236}, {1437, 25526}, {1469, 26543}, {1479, 17532}, {1503, 14529}, {1532, 8227}, {1621, 2475}, {1698, 3338}, {1722, 17278}, {1737, 5439}, {1738, 4646}, {1788, 9776}, {1836, 5250}, {2260, 5742}, {2292, 3782}, {2478, 4423}, {2550, 2894}, {2829, 3560}, {3036, 5554}, {3058, 6175}, {3086, 6856}, {3090, 10785}, {3091, 7958}, {3146, 7965}, {3189, 10578}, {3303, 3434}, {3304, 10527}, {3306, 24914}, {3333, 5705}, {3339, 6173}, {3421, 19855}, {3428, 6889}, {3452, 3947}, {3485, 5289}, {3487, 12635}, {3576, 6831}, {3585, 5259}, {3614, 4193}, {3622, 11680}, {3624, 4187}, {3634, 3820}, {3636, 24387}, {3649, 3869}, {3660, 5123}, {3666, 23536}, {3671, 5837}, {3698, 6735}, {3703, 4968}, {3714, 3912}, {3720, 21935}, {3740, 21075}, {3744, 28027}, {3752, 5530}, {3753, 8256}, {3754, 5690}, {3757, 7270}, {3814, 5126}, {3815, 16604}, {3848, 9843}, {3868, 21677}, {3878, 11263}, {3884, 5499}, {3924, 5724}, {3927, 5852}, {3928, 4355}, {3931, 23537}, {3932, 4385}, {3957, 5178}, {4018, 11551}, {4026, 16062}, {4030, 5300}, {4189, 15326}, {4190, 5217}, {4202, 26115}, {4292, 4640}, {4293, 6857}, {4294, 4428}, {4297, 8727}, {4298, 5745}, {4299, 16370}, {4364, 8680}, {4413, 5552}, {4512, 9579}, {4654, 12526}, {4670, 9028}, {4860, 9780}, {4966, 10449}, {5044, 21077}, {5045, 10916}, {5046, 5284}, {5047, 5080}, {5084, 8167}, {5141, 7173}, {5142, 17917}, {5204, 6910}, {5218, 6904}, {5219, 8583}, {5227, 17303}, {5251, 5270}, {5252, 19860}, {5263, 26051}, {5271, 10371}, {5302, 12527}, {5326, 17566}, {5436, 5691}, {5603, 6937}, {5687, 10056}, {5719, 22836}, {5726, 20195}, {5731, 6828}, {5784, 8255}, {5786, 15668}, {5842, 6917}, {5886, 6842}, {5949, 23905}, {6067, 11038}, {6223, 16112}, {6253, 6839}, {6256, 6913}, {6656, 27255}, {6666, 18250}, {6684, 12436}, {6702, 11698}, {6824, 12114}, {6825, 22753}, {6826, 11500}, {6827, 10894}, {6832, 12115}, {6836, 8273}, {6841, 18481}, {6846, 12667}, {6850, 11496}, {6854, 10786}, {6861, 22758}, {6862, 10269}, {6871, 10896}, {6872, 12943}, {6877, 10805}, {6883, 10526}, {6897, 10310}, {6901, 11491}, {6911, 26487}, {6922, 10165}, {6933, 10958}, {6942, 21155}, {6947, 10599}, {6951, 11826}, {6982, 10893}, {6984, 12116}, {6989, 22667}, {7028, 8382}, {7080, 26040}, {7113, 17398}, {7181, 27187}, {7535, 9798}, {7741, 17530}, {8164, 17582}, {8261, 18389}, {9564, 10408}, {9581, 10582}, {9612, 24703}, {9654, 11108}, {9655, 16418}, {9712, 10831}, {9713, 10037}, {9840, 23383}, {9844, 10177}, {9940, 12616}, {9956, 10942}, {9957, 13463}, {10107, 11362}, {10179, 12053}, {10197, 25440}, {10246, 26470}, {10523, 15842}, {10593, 15808}, {10943, 15178}, {11037, 24477}, {11319, 24542}, {11375, 19861}, {11376, 15845}, {11684, 17483}, {12514, 17768}, {12572, 15254}, {13731, 20470}, {14011, 25507}, {15048, 25092}, {15267, 15282}, {15338, 17579}, {15674, 20067}, {15726, 21628}, {15733, 16201}, {16408, 26364}, {16458, 20266}, {16830, 19893}, {16844, 19754}, {16915, 26629}, {17044, 20531}, {17073, 21530}, {17140, 27690}, {17531, 27529}, {17609, 26015}, {17647, 24929}, {17670, 27020}, {17674, 26030}, {17683, 26007}, {17717, 21214}, {17721, 28011}, {17751, 18139}, {18231, 21454}, {18634, 19859}, {20271, 21965}, {20541, 24656}, {21625, 24386}, {22654, 25514}, {24537, 25968}, {24982, 26482}, {25441, 25444}
X(25467) lies on these lines: {2, 85}, {142, 25660}, {5437, 25458}, {25446, 25456}
X(25468) lies on these lines: {1, 24366}, {2, 99}, {10, 24185}, {120, 8728}, {141, 3679}, {274, 17673}, {377, 25497}, {754, 26079}, {1086, 21711}, {1111, 25357}, {3934, 17674}, {6739, 15668}, {7761, 17679}, {7813, 16711}, {7830, 17690}, {7834, 17683}, {14431, 24287}, {17389, 18139}, {21232, 25351}, {24364, 25650}, {24375, 25663}, {24378, 25665}, {24384, 25669}, {24988, 27076}, {25443, 25452}
X(25469) lies on these lines: {2, 45}, {75, 24922}, {99, 8287}, {620, 17058}, {662, 24636}, {2640, 24348}, {3662, 25662}, {4427, 24955}, {16726, 24962}, {17276, 25687}, {25445, 25470}, {25447, 25457}, {25454, 25460}
X(25470) lies on these lines: {2, 37}, {86, 16917}, {584, 662}, {982, 24520}, {3775, 9534}, {4272, 17778}, {4352, 26041}, {4360, 20255}, {4716, 10449}, {16705, 26045}, {16752, 26110}, {24174, 24372}, {24790, 25538}, {25445, 25469}, {25447, 25460}
X(25471) lies on these lines: {2, 31}, {75, 24880}, {740, 24899}, {3416, 25645}, {4657, 25441}, {24342, 24902}
X(25472) lies on these lines: {2, 668}, {10, 7170}, {115, 799}, {620, 7257}, {16613, 25530}, {20982, 27805}, {25446, 25458}
X(25473) lies on these lines: {2, 669}, {523, 25450}
Collineation mappings involving Gemini triangle 22: X(25474)-X(25484)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 22, as in centers X(25474)-X(25484). Then
m(X) = (a + b + c)(a + 2b + c)(a + b + 2 c) x - b (2 a + b + c)(a + b + 2 c) y - c (2 a + b + c)(a + 2 b + c) z : :
A point X lies on the Euler line if and only m(X) lies on the Euler line. (Clark Kimberling, October 19, 2018)
X(25474) lies on these lines: {1, 2}, {37, 24967}, {3770, 25480}, {3953, 17315}, {4851, 25478}
X(25475) lies on these lines: {2, 3}
X(25476) lies on these lines: {2, 3}
X(25477) lies on these lines: {2, 3}
X(25478) lies on these lines: {2, 6}, {3739, 24963}, {4851, 25474}, {17290, 24968}
X(25479) lies on these lines: {1, 2}, {45, 24967}, {1330, 25483}, {11544, 17297}
X(25480) lies on these lines: {2, 37}, {3770, 25474}
X(25481) lies on these lines: {2, 650}
X(25482) lies on these lines: {2, 3}
X(25483) lies on these lines: {2, 3}, {1330, 25479}
X(25484) lies on these lines: {2, 3}
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28502.
X(25485) lies on the circle (X(1), X(11715)), the circle (X(5), X(15863)), and on these lines: {1, 104}, {4, 7972}, {5, 15863}, {10, 11729}, {11, 11011}, {30, 11274}, {80, 5603}, {100, 7982}, {119, 519}, {145, 12751}, {149, 5734}, {153, 3241}, {214, 517}, {354, 17654}, {515, 1317}, {546, 946}, {551, 6713}, {962, 12119}, {999, 12332}, {1145, 6745}, {1320, 3577}, {1387, 9952}, {1389, 13143}, {1482, 2802}, {1484, 7680}, {2098, 12739}, {2099, 12736}, {2801, 10247}, {2829, 4342}, {3035, 11362}, {3036, 10175}, {3295, 22775}, {3632, 15017}, {3656, 10738}, {4301, 5840}, {4861, 20117}, {5330, 11014}, {5541, 11224}, {5563, 18861}, {5587, 12531}, {5660, 8166}, {5730, 14740}, {5851, 15570}, {5886, 6702}, {5901, 12619}, {6224, 14217}, {6264, 13227}, {6796, 10087}, {6797, 13374}, {7681, 11698}, {8196, 12461}, {8203, 12460}, {9809, 20057}, {9897, 11522}, {10031, 10724}, {10246, 12515}, {10283, 17051}, {10543, 13607}, {10595, 12247}, {11278, 22935}, {11366, 12463}, {11367, 12462}, {11496, 12773}, {11531, 15015}, {12653, 16189}, {12672, 17660}, {22799, 24042}
X(24585) = midpoint of X(i) and X(j) for these {i,j}: {1, 10698}, {4, 7972}, {100, 7982}, {104, 13253}, {145, 12751}, {962, 12119}, {1317, 1537}, {1320, 6326}, {1482, 6265}, {3244, 21635}, {6224, 14217}, {11278, 22935}, {12672, 17660}
X(24585) = reflection of X(i) in X(j) for these (i,j): (10, 11729), (11, 13464), (80, 16174), (214, 19907), (5882, 12735), (5884, 5083), (6797, 13374), (10265, 1387), (11362, 3035), (11715, 1), (12619, 5901), (15863, 5), (19914, 6702)
X(24585) = center of the circle {X(1), X(10698), X(10700)}
X(24585) = center of the circle {X(100), X(7982), X(8699)}
X(24585) = X(7972)-of-Euler-triangle
X(24585) = X(10698)-of-anti-Aquila-triangle
X(24585) = X(12295)-of-2nd circumperp-triangle
X(24585) = X(12302)-of-incircle-circles-triangle
X(24585) = X(12893)-of-Hutson intouch-triangle
X(24585) = X(12901)-of-intouch-triangle
X(24585) = X(15863)-of-Johnson-triangle
X(24585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 13253, 104), (80, 5603, 16174), (104, 10698, 13253), (2099, 12740, 12736), (5886, 19914, 6702), (6326, 16200, 1320), (10074, 12775, 5450), (10595, 12247, 16173)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28502.
X(25486) lies on these lines: {2, 5477}, {4, 543}, {99, 23334}, {114, 524}, {115, 8176}, {325, 2482}, {542, 9756}, {620, 8182}, {1007, 8593}, {1506, 7817}, {2794, 7618}, {3788, 5569}, {5286, 9166}, {5461, 7736}, {6033, 11165}, {6055, 8550}, {6248, 20112}, {7622, 9744}, {9742, 9877}, {9754, 23234}, {18800, 22110}
X(24586) = midpoint of X(i) and X(j) for these {i,j}: {99, 23334}, {6033, 11165}, {6054, 9770}
X(24586) = reflection of X(i) in X(j) for these (i,j): (115, 8176), (6055, 9771), (8182, 620)
X(24586) = X(9880)-of-Artzt-triangle
X(24586) = X(20112)-of-1st-anti-Brocard-triangle
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28502.
X(25487) lies on these lines: {3, 1986}, {26, 15035}, {30, 113}, {49, 17854}, {74, 18445}, {110, 12084}, {125, 12370}, {185, 10226}, {265, 18281}, {1147, 3357}, {2071, 3043}, {2777, 12038}, {3520, 7723}, {5972, 15761}, {6644, 15472}, {6699, 11430}, {7387, 15040}, {10111, 10264}, {10113, 10224}, {10721, 11449}, {12006, 22962}, {12121, 18569}, {12236, 13352}, {12295, 17701}, {12302, 15132}, {12358, 18570}, {12825, 22115}, {12893, 13346}, {13367, 16111}, {13371, 17702}, {15051, 18324}, {15061, 18912}, {15331, 16223}, {15647, 22802}, {16222, 22467}, {18390, 20304}
X(24587) = midpoint of X(i) and X(j) for these {i,j}: {110, 12084}, {1147, 13293}, {12121, 18569}, {12302, 15132}, {12893, 13346}
X(24587) = reflection of X(i) in X(j) for these (i,j): (125, 23336), (10113, 10224), (12041, 10226), (15761, 5972), (20773, 1511)
X(24587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15463, 14708), (1511, 1539, 20771)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28502.
X(25488) lies on these lines: {5, 524}, {67, 5133}, {140, 9019}, {373, 468}, {575, 23410}, {597, 1995}, {2854, 10272}, {3618, 8547}, {5486, 14561}, {5943, 15118}, {6593, 13366}, {7495, 9971}, {8262, 13857}, {8550, 12134}, {9976, 12007}, {10301, 19127}, {12039, 19130}
X(24588) = midpoint of X(12039) and X(19130)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28502.
X(25489) lies on these lines: {6, 12824}, {113, 524}, {576, 2854}, {1495, 6593}, {1511, 9019}, {2781, 4550}, {5505, 9970}, {10752, 11459}
X(25489) = reflection of X(125) in X(25488)
Collineation mappings involving Gemini triangle 23: X(25490)-X(25542)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 23, as in centers X(25490)-X(25542). Then
m(X) = b c (a + b + c) x + a c (a + c) y + a b ( a + b) z : :
Fixed points of m include X(i) for these i: 2, 36, 238, 667. A point X lies on the Euler line if and only m(X) lies on the Euler line. (Clark Kimberling, October 19, 2018)
X(25490) lies on these lines: {2, 3}, {47, 238}, {158, 6708}, {1125, 10571}, {1936, 26363}, {2122, 26126}, {2360, 13478}, {3085, 26095}, {3086, 16466}, {3916, 27339}, {10200, 25496}, {11374, 27287}, {17306, 19864}, {20256, 20805}, {25493, 25524}, {25504, 25506}
X(25491) lies on these lines: {2, 3}, {238, 2964}, {392, 19863}, {499, 16466}, {3624, 25493}
X(25492) lies on these lines: {1, 2}, {36, 17697}, {46, 27002}, {56, 13741}, {244, 25591}, {344, 7786}, {405, 25531}, {496, 4429}, {726, 27291}, {982, 25079}, {3338, 27064}, {3662, 21616}, {3670, 19582}, {3816, 16062}, {4193, 15666}, {4201, 26139}, {4423, 19270}, {4647, 24620}, {4903, 24068}, {5192, 5253}, {5204, 13735}, {5259, 19278}, {5263, 16408}, {5303, 11346}, {5433, 27339}, {6675, 26979}, {7288, 13742}, {8167, 11110}, {8227, 17282}, {10896, 17678}, {11680, 17674}, {13740, 25524}, {17531, 24552}, {17676, 26127}, {25504, 25505}, {25513, 25522}, {25535, 25539}, {26129, 27506}
X(25493) lies on these lines: {2, 11}, {5, 15622}, {3624, 25491}, {25490, 25524}
X(25494) lies on these lines: {2, 3}, {6, 5208}, {63, 238}, {69, 5324}, {105, 5273}, {394, 3794}, {958, 3757}, {1194, 22398}, {1259, 3705}, {1473, 3662}, {1724, 10461}, {1899, 26530}, {2053, 6384}, {2887, 7295}, {3220, 25527}, {3868, 7191}, {3917, 26657}, {4388, 7083}, {4423, 17322}, {4438, 26241}, {4512, 12652}, {5249, 25496}, {5279, 26242}, {5329, 6679}, {9306, 26625}, {11442, 26542}, {17123, 27680}, {24320, 27184}
X(25495) lies on these lines: {2, 3}, {105, 5211}, {238, 244}, {614, 1046}, {2650, 7191}, {3757, 5258}, {3837, 4057}, {4892, 24436}, {5324, 17778}, {5358, 25650}, {25496, 27186}
X(25496) lies on these lines: {1, 312}, {2, 31}, {6, 3741}, {10, 3966}, {11, 5150}, {36, 19260}, {37, 4011}, {38, 26223}, {42, 24552}, {43, 5263}, {55, 6685}, {56, 226}, {57, 4697}, {63, 4672}, {86, 87}, {142, 15497}, {192, 17600}, {321, 17017}, {333, 16468}, {499, 25519}, {516, 16435}, {603, 26126}, {614, 24325}, {726, 17599}, {740, 5256}, {846, 4676}, {870, 1920}, {894, 982}, {940, 3840}, {956, 20498}, {964, 1193}, {968, 4432}, {975, 25079}, {978, 1010}, {984, 27064}, {1001, 16058}, {1009, 21321}, {1150, 2308}, {1203, 10479}, {1386, 4362}, {1699, 6996}, {1724, 19863}, {1746, 10886}, {1836, 3821}, {1961, 18743}, {2267, 26105}, {2293, 27394}, {2886, 3589}, {3006, 26061}, {3305, 3842}, {3329, 3705}, {3434, 4085}, {3616, 17697}, {3618, 21242}, {3624, 4892}, {3666, 3923}, {3685, 17592}, {3703, 17726}, {3720, 19684}, {3742, 4670}, {3752, 3980}, {3771, 5718}, {3772, 25385}, {3775, 5739}, {3791, 11679}, {3816, 6703}, {3831, 5711}, {3915, 26115}, {3936, 24943}, {3944, 19786}, {3993, 4387}, {4038, 17379}, {4071, 17303}, {4104, 19868}, {4138, 19862}, {4357, 4703}, {4358, 5311}, {4363, 24165}, {4384, 27798}, {4413, 6686}, {4418, 4850}, {4425, 4657}, {4434, 5269}, {4438, 5294}, {4649, 10453}, {4650, 24627}, {4675, 23812}, {4680, 19867}, {4970, 5695}, {4974, 5271}, {5248, 16287}, {5249, 25494}, {5259, 16289}, {5268, 24003}, {5272, 10436}, {5550, 26132}, {5880, 24169}, {7081, 17716}, {7741, 25441}, {7808, 20544}, {8025, 9345}, {8053, 19341}, {8167, 15668}, {10200, 25490}, {10448, 11319}, {10601, 26013}, {16454, 27627}, {16469, 18229}, {16706, 17889}, {16777, 21101}, {17023, 24210}, {17184, 24725}, {17469, 26227}, {17598, 24349}, {17723, 24295}, {18067, 18140}, {18203, 20274}, {19269, 24789}, {19804, 24342}, {20083, 25639}, {20108, 25440}, {25495, 27186}, {25497, 25500}, {25502, 25531}
X(25497) lies on these lines: {2, 32}, {10, 4950}, {31, 21240}, {86, 5283}, {141, 1724}, {213, 24549}, {274, 17688}, {316, 16906}, {377, 25468}, {405, 3286}, {574, 27162}, {980, 16050}, {1759, 4797}, {1930, 16974}, {2241, 26965}, {2242, 27097}, {3589, 16783}, {3734, 11319}, {3924, 24254}, {3934, 5192}, {4056, 25345}, {4202, 7761}, {4376, 16600}, {4648, 13742}, {4657, 5259}, {4670, 16601}, {5264, 20255}, {5280, 17279}, {5291, 27248}, {5563, 24652}, {6679, 17046}, {6707, 13728}, {7747, 16910}, {7751, 27040}, {7768, 16991}, {7772, 27109}, {7781, 18600}, {7795, 17526}, {7798, 26770}, {7820, 16909}, {7826, 17007}, {7828, 17003}, {7832, 16905}, {7885, 16908}, {7931, 16907}, {13732, 24220}, {17245, 17540}, {19846, 20541}, {25496, 25500}, {25506, 25520}
X(25497) = complement of X(26085)
X(25498) lies on these lines: {1, 4445}, {2, 37}, {6, 4708}, {7, 4798}, {8, 4910}, {10, 4399}, {44, 17248}, {82, 10130}, {86, 17237}, {140, 12610}, {141, 1125}, {142, 3647}, {238, 3624}, {498, 20270}, {551, 17390}, {1001, 16852}, {1100, 4690}, {1213, 17023}, {1268, 17117}, {1418, 17095}, {1449, 17251}, {1654, 16666}, {1698, 4361}, {3244, 4478}, {3247, 17293}, {3589, 5257}, {3616, 4851}, {3634, 3773}, {3661, 3723}, {3663, 4472}, {3696, 19856}, {3707, 6329}, {3763, 16831}, {3986, 4422}, {4021, 4665}, {4205, 21245}, {4357, 4670}, {4364, 5750}, {4395, 6541}, {4423, 16352}, {4648, 5550}, {4715, 17253}, {4725, 16884}, {4859, 24784}, {4875, 27507}, {4967, 17395}, {5439, 25447}, {6651, 27191}, {6676, 18589}, {6706, 17073}, {8167, 25514}, {10436, 17235}, {11230, 24220}, {16604, 16696}, {16668, 17346}, {16669, 17256}, {16671, 17331}, {16672, 17286}, {16673, 17269}, {16723, 27195}, {16777, 17229}, {16814, 17368}, {16826, 17231}, {17210, 18166}, {17238, 17374}, {17250, 17344}, {17275, 26626}, {17296, 25055}, {17323, 25590}, {17366, 24603}, {19878, 24295}, {19883, 21255}, {24739, 27148}, {24774, 24780}, {24986, 26628}, {25107, 26115}, {25499, 25512}, {25508, 25534}, {26959, 27164}
X(25498) = complement of X(17303)
X(25499) lies on these lines: {1, 141}, {2, 39}, {37, 1930}, {120, 8728}, {213, 4357}, {334, 27255}, {350, 27274}, {405, 3286}, {626, 5051}, {940, 24632}, {964, 3734}, {1107, 16818}, {1125, 16850}, {1573, 26965}, {1574, 27026}, {1575, 25599}, {2176, 17325}, {2292, 24254}, {3294, 4364}, {3589, 16552}, {3739, 16828}, {4000, 19853}, {4045, 17672}, {4202, 16830}, {4648, 13725}, {4698, 16601}, {5277, 16060}, {5750, 24214}, {6682, 17048}, {7264, 25368}, {7761, 17676}, {7816, 11115}, {9840, 24220}, {16549, 25349}, {16706, 16819}, {16826, 18139}, {16827, 17326}, {16831, 25527}, {16887, 24512}, {16929, 16994}, {17023, 20963}, {17143, 17302}, {17144, 17399}, {17175, 17398}, {17289, 25264}, {17321, 27248}, {17322, 25538}, {17400, 18143}, {19863, 21264}, {19864, 20530}, {24652, 25055}, {25498, 25512}, {25520, 25530}, {26030, 27076}
X(25500) lies on these lines: {2, 41}, {3, 142}, {9, 17687}, {55, 17050}, {57, 16831}, {86, 27626}, {220, 5745}, {284, 25508}, {405, 20335}, {579, 20150}, {993, 17758}, {2271, 3008}, {3295, 20257}, {3601, 14942}, {3624, 25532}, {3664, 5021}, {3836, 7800}, {5249, 16367}, {5436, 13727}, {6626, 17227}, {6690, 21258}, {6700, 19319}, {6706, 24249}, {6996, 25525}, {10198, 17062}, {10436, 16061}, {10448, 26978}, {11110, 17306}, {13411, 19313}, {14828, 21384}, {16054, 25507}, {16060, 17282}, {16601, 24333}, {16917, 27147}, {17206, 17298}, {17278, 18755}, {17695, 26806}, {19320, 27385}, {19518, 21246}, {19860, 21232}, {22127, 25878}, {24331, 24929}, {25496, 25497}, {25501, 25514}
X(25501) lies on these lines: {1, 2}, {37, 24165}, {86, 17123}, {142, 4425}, {354, 3842}, {672, 3986}, {982, 4687}, {1001, 16059}, {1962, 24589}, {2887, 17245}, {3136, 3825}, {3742, 4698}, {3752, 10180}, {3971, 24325}, {3993, 4359}, {4011, 10436}, {4038, 17277}, {4191, 5248}, {4192, 10165}, {4675, 4703}, {4697, 15254}, {4713, 4798}, {4970, 19804}, {4981, 17450}, {5257, 24512}, {5278, 9345}, {6384, 18133}, {6707, 21264}, {6821, 26105}, {8167, 15668}, {12545, 13731}, {14969, 19750}, {16058, 25524}, {16604, 21838}, {16844, 19715}, {17125, 19684}, {17889, 27147}, {19341, 20470}, {23791, 27193}, {25500, 25514}, {25507, 25531}, {25526, 25542}
X(25502) lies on these lines: {1, 2}, {35, 16059}, {36, 16058}, {37, 3848}, {55, 9337}, {81, 17125}, {86, 18192}, {142, 3944}, {171, 4423}, {238, 8167}, {350, 25590}, {750, 5284}, {846, 3306}, {940, 9332}, {968, 1054}, {984, 3742}, {991, 10171}, {1001, 3550}, {1011, 7280}, {1376, 16484}, {1468, 17536}, {1478, 6822}, {1479, 6821}, {1575, 3247}, {1621, 17124}, {1724, 19715}, {1742, 3817}, {1743, 24512}, {1757, 7308}, {2177, 9342}, {2238, 16667}, {2276, 16673}, {3576, 19540}, {3583, 6817}, {3585, 6818}, {3681, 17450}, {3731, 17754}, {3746, 16421}, {3750, 4413}, {3816, 17245}, {3846, 17234}, {3925, 24217}, {3993, 17490}, {4000, 25420}, {4038, 4383}, {4191, 5010}, {4192, 7987}, {4334, 5226}, {4438, 17263}, {4650, 15254}, {4687, 6682}, {4970, 24620}, {5247, 16842}, {5275, 16779}, {5333, 18169}, {5437, 17596}, {6051, 24174}, {6384, 18140}, {7998, 20961}, {8731, 15803}, {9336, 22199}, {15569, 16602}, {15668, 20530}, {16345, 25542}, {16571, 21264}, {16610, 17592}, {17038, 21330}, {17064, 20195}, {18139, 25960}, {18743, 24325}, {18792, 25507}, {21001, 23660}, {24046, 27784}, {25496, 25531}
X(25503) lies on these lines: {2, 45}, {6, 17250}, {10, 3946}, {141, 3616}, {142, 19878}, {145, 4445}, {238, 3624}, {597, 4748}, {599, 17397}, {1125, 17313}, {1266, 17303}, {1698, 17382}, {3632, 17239}, {3636, 4851}, {3763, 17244}, {3943, 17293}, {4000, 19877}, {4029, 17269}, {4395, 9780}, {4478, 20053}, {4655, 19862}, {4668, 4852}, {5308, 20582}, {5750, 17255}, {7232, 17398}, {16672, 17292}, {16674, 17285}, {16675, 17371}, {16677, 17358}, {16777, 17230}, {16826, 21358}, {16884, 17238}, {17023, 17251}, {17118, 17324}, {17119, 17399}, {17253, 17381}, {17259, 17384}, {17262, 17385}, {17308, 17318}
X(25504) lies on these lines: {2, 6}, {1100, 27299}, {1444, 11342}, {1575, 27264}, {2345, 27097}, {5124, 17696}, {16685, 21281}, {17303, 27248}, {20255, 21769}, {24667, 26107}, {25490, 25506}, {25492, 25505}, {25518, 25530}, {26042, 27272}
X(25505) lies on these lines: {1, 21238}, {2, 37}, {6, 26959}, {82, 26232}, {583, 24514}, {594, 27091}, {966, 20561}, {1078, 5301}, {1125, 21257}, {1213, 16525}, {1269, 24598}, {1740, 24688}, {2178, 7770}, {2275, 3770}, {3624, 18792}, {3662, 26979}, {3726, 27285}, {3758, 26963}, {3759, 26772}, {3934, 17053}, {4272, 17027}, {4357, 24317}, {4384, 27111}, {4446, 12263}, {4675, 26149}, {15668, 25510}, {16515, 16819}, {16738, 17250}, {16777, 27020}, {17144, 21858}, {17178, 17361}, {17227, 27145}, {17228, 27095}, {17275, 26801}, {17299, 26752}, {17306, 25534}, {17360, 26756}, {17394, 27166}, {17397, 27042}, {18042, 26222}, {19864, 25506}, {24512, 27158}, {25492, 25504}, {25515, 25527}, {25523, 25536}
X(25506) lies on these lines: {2, 39}, {405, 25530}, {19864, 25505}, {25490, 25504}, {25497, 25520}
X(25507) lies on these lines: {1, 14007}, {2, 6}, {8, 17551}, {21, 4423}, {27, 19812}, {37, 16736}, {58, 17123}, {142, 17182}, {226, 1434}, {274, 312}, {314, 19804}, {354, 3786}, {978, 5331}, {1001, 13588}, {1010, 1125}, {1014, 5226}, {1043, 3616}, {1330, 17514}, {1699, 4229}, {1848, 14013}, {3286, 8167}, {3624, 4892}, {3634, 4658}, {3736, 26102}, {3742, 5208}, {3752, 25058}, {3816, 14009}, {3848, 18165}, {4021, 19820}, {4102, 17315}, {4184, 5284}, {4683, 8040}, {4698, 27064}, {4699, 20182}, {4751, 5256}, {4886, 24603}, {5249, 8822}, {5271, 17394}, {5437, 17185}, {7308, 18206}, {7415, 10165}, {8056, 10455}, {9534, 16456}, {9776, 17139}, {10180, 24342}, {14011, 25466}, {16054, 25500}, {16610, 25059}, {16709, 18743}, {17103, 26132}, {17169, 18228}, {17305, 27186}, {17397, 24789}, {18792, 25502}, {23812, 24697}, {25501, 25531}, {25515, 25521}, {25529, 25536}
X(25508) lies on these lines: {2, 6}, {9, 17175}, {21, 20470}, {37, 274}, {238, 25512}, {284, 25500}, {314, 3739}, {595, 16690}, {1001, 1010}, {1030, 16917}, {1100, 16819}, {1125, 3736}, {1434, 26125}, {2092, 25457}, {3286, 11110}, {3624, 18792}, {3770, 16589}, {3913, 14007}, {3948, 24944}, {3986, 17205}, {4446, 4476}, {4649, 16828}, {4687, 16709}, {4698, 16696}, {4699, 20170}, {5257, 16887}, {5263, 14005}, {5296, 17169}, {5949, 17669}, {7054, 27176}, {8053, 13588}, {10013, 10453}, {16503, 16818}, {16752, 17321}, {17202, 27147}, {17303, 27255}, {17551, 26115}, {19804, 25058}, {24450, 25124}, {24589, 25060}, {25498, 25534}
X(25509) lies on these lines: {1, 2}, {1699, 17282}, {3361, 13742}, {4011, 4862}, {4901, 4906}, {5290, 13741}, {5436, 25914}, {5573, 17279}, {8167, 17306}, {10980, 17353}, {16469, 18141}, {17123, 17272}, {25527, 25531}
X(25510) lies on these lines: {1, 2}, {56, 11353}, {86, 4503}, {1015, 3975}, {1019, 3835}, {1100, 27111}, {2325, 26076}, {3247, 26042}, {3760, 24621}, {3834, 25534}, {3986, 26082}, {6999, 26139}, {8610, 17790}, {10436, 26107}, {15668, 25505}, {17178, 17252}, {17260, 26963}, {17304, 26143}, {17306, 25535}, {17312, 27095}, {17319, 27102}, {17322, 26979}, {17324, 27017}, {17326, 27145}, {19224, 25542}, {24598, 25264}
X(25511) lies on these lines: {2, 650}, {37, 21438}, {86, 22383}, {513, 27293}, {647, 3261}, {652, 17215}, {905, 18155}, {1459, 24718}, {3239, 23785}, {3716, 23800}, {4106, 27345}, {4369, 14349}, {4449, 25637}, {4455, 23818}, {4657, 25603}, {4687, 21611}, {4776, 26983}, {6586, 25594}, {6589, 24622}, {7234, 21260}, {14077, 19853}, {19786, 24793}, {20952, 21348}
X(25512) lies on these lines: {1, 2}, {36, 11110}, {56, 16844}, {72, 13476}, {75, 27785}, {86, 1203}, {213, 17398}, {238, 25508}, {274, 1269}, {321, 27784}, {405, 20470}, {596, 3989}, {1001, 16458}, {1010, 5259}, {1213, 20963}, {1224, 17289}, {1486, 19286}, {2049, 4423}, {2260, 5257}, {3294, 5750}, {3583, 26051}, {3666, 6533}, {3671, 17077}, {3739, 4647}, {3743, 4359}, {3842, 4022}, {3881, 4981}, {4026, 17529}, {4299, 13736}, {4355, 26125}, {4357, 17175}, {4657, 24790}, {5248, 16454}, {5253, 17557}, {5267, 17588}, {5284, 14005}, {5303, 17553}, {5506, 27064}, {8193, 19313}, {11365, 16353}, {12410, 19319}, {13740, 25542}, {15668, 16466}, {16288, 16678}, {16357, 23383}, {19895, 19896}, {25498, 25499}
X(25512) = complement of X(19874)
X(25513) lies on these lines: {2, 3}, {238, 255}, {1125, 4306}, {2551, 26095}, {3562, 14986}, {3624, 25521}, {4357, 17181}, {10396, 27411}, {25492, 25522}
X(25514) lies on these lines: {2, 3}, {6, 18165}, {57, 238}, {81, 3167}, {142, 15497}, {197, 6690}, {226, 24320}, {261, 8033}, {284, 5275}, {612, 24929}, {614, 942}, {940, 2194}, {954, 20760}, {956, 3757}, {1001, 3185}, {1040, 1859}, {1062, 9895}, {1125, 21621}, {1376, 3823}, {1466, 5121}, {1473, 5249}, {1486, 2886}, {1875, 19372}, {2355, 24611}, {3220, 25525}, {3601, 5268}, {3705, 26241}, {3819, 25878}, {4260, 4383}, {4423, 15494}, {5324, 5712}, {5708, 7292}, {6703, 10192}, {7083, 26098}, {7191, 15934}, {8167, 25498}, {8193, 19854}, {9798, 10198}, {11365, 26363}, {12410, 19843}, {17123, 27626}, {22654, 25466}, {25500, 25501}
X(25515) lies on these lines: {2, 3}, {86, 222}, {219, 333}, {238, 17188}, {286, 1214}, {4357, 17182}, {8822, 27339}, {17185, 27164}, {23602, 27509}, {25505, 25527}, {25507, 25521}
X(25516) lies on these lines: {2, 3}, {9, 4269}, {58, 226}, {72, 333}, {81, 3157}, {86, 1437}, {238, 1780}, {272, 1175}, {283, 17167}, {284, 13411}, {386, 1751}, {946, 2328}, {950, 4653}, {975, 10393}, {1043, 3419}, {1125, 2360}, {1490, 17194}, {1778, 5746}, {1807, 24624}, {2194, 11375}, {2893, 25650}, {3485, 16466}, {3624, 25523}, {3916, 8822}, {4877, 8804}, {5208, 14054}, {5714, 16948}, {8227, 17188}, {8748, 17102}, {12514, 24310}, {17740, 19848}, {25525, 25526}
X(25517) lies on these lines: {2, 3}, {333, 7078}, {10461, 27411}
X(25518) lies on these lines: {2, 3}, {238, 10072}, {14986, 22117}, {25504, 25530}
X(25519) lies on these lines: {2, 3}, {238, 26363}, {333, 16471}, {499, 25496}, {4357, 21616}, {10527, 16466}, {11374, 27254}, {19864, 25527}, {25492, 25504}
X(25520) lies on these lines: {2, 3}, {238, 1923}, {385, 27262}, {17128, 27312}, {25497, 25506}, {25499, 25530}
X(25521) lies on these lines: {2, 7}, {37, 3673}, {71, 17753}, {75, 3694}, {150, 26063}, {198, 17682}, {239, 27267}, {281, 331}, {573, 2140}, {673, 4254}, {991, 1125}, {1001, 13727}, {1213, 21258}, {1441, 24554}, {1738, 3085}, {2092, 4000}, {2256, 24203}, {2335, 2481}, {3169, 20257}, {3247, 27253}, {3286, 11110}, {3487, 4260}, {3624, 25513}, {3686, 27304}, {3821, 10198}, {4026, 16062}, {4197, 23542}, {4270, 5222}, {4440, 25601}, {4659, 27544}, {5224, 16608}, {5756, 19858}, {6510, 17394}, {7380, 25365}, {10822, 19853}, {16578, 27268}, {16713, 21296}, {17049, 24752}, {17062, 27299}, {17073, 17322}, {17277, 23151}, {17278, 21892}, {17364, 27317}, {20358, 25613}, {20880, 27396}, {25492, 25504}, {25507, 25515}, {26106, 26959}
X(25522) lies on these lines: {1, 1329}, {2, 40}, {3, 1538}, {5, 8583}, {9, 499}, {10, 7962}, {11, 936}, {36, 405}, {57, 10200}, {140, 4512}, {165, 13747}, {200, 496}, {388, 1125}, {392, 1698}, {442, 7988}, {443, 3817}, {452, 10165}, {474, 1699}, {497, 6700}, {515, 6919}, {516, 17567}, {908, 3333}, {997, 3825}, {1058, 6745}, {1376, 9614}, {1479, 5438}, {1490, 14022}, {1512, 3090}, {1519, 6967}, {1697, 26364}, {1737, 15829}, {1842, 7498}, {2478, 3576}, {3086, 3452}, {3149, 25893}, {3485, 9843}, {3679, 10912}, {3753, 11522}, {3814, 9578}, {3820, 4853}, {3847, 5794}, {3944, 11512}, {4193, 5587}, {4295, 6692}, {4423, 16293}, {4679, 5433}, {5044, 5231}, {5082, 20103}, {5129, 5550}, {5187, 18492}, {5316, 19843}, {5328, 14986}, {5437, 12047}, {5554, 16200}, {5603, 8582}, {5691, 17556}, {5705, 25917}, {5745, 11023}, {5748, 21620}, {5880, 7483}, {5882, 24558}, {5886, 17527}, {6282, 7681}, {6667, 26066}, {6675, 16159}, {6691, 15803}, {6734, 10584}, {6762, 10072}, {6856, 10171}, {6857, 12436}, {6890, 11372}, {6891, 12705}, {6904, 18483}, {6926, 10860}, {6931, 24987}, {6933, 24564}, {7288, 12572}, {7308, 26363}, {7330, 26492}, {7743, 9709}, {7956, 12651}, {7982, 24982}, {7987, 11113}, {7989, 17533}, {8580, 24390}, {9580, 25440}, {9623, 11376}, {9624, 19860}, {9779, 17580}, {9955, 16408}, {10582, 11374}, {11012, 25875}, {11108, 11230}, {11236, 25055}, {12629, 21031}, {13411, 26105}, {16417, 22793}, {17306, 19864}, {18140, 20449}, {25492, 25513}
X(25523) lies on these lines: {1, 21231}, {2, 48}, {3, 142}, {19, 24684}, {27, 19812}, {86, 4269}, {140, 16608}, {141, 4999}, {379, 22054}, {405, 15669}, {614, 1964}, {1445, 27473}, {1471, 4648}, {2256, 3911}, {2260, 5736}, {3306, 7573}, {3624, 25516}, {3772, 5110}, {5433, 18635}, {5444, 24780}, {6707, 21239}, {17303, 21091}, {17306, 25532}, {18650, 24682}, {24179, 24424}, {25505, 25536}
X(25524) lies on these lines: {1, 474}, {2, 12}, {3, 142}, {4, 3816}, {5, 6256}, {6, 978}, {8, 3304}, {9, 3361}, {10, 999}, {11, 377}, {20, 26105}, {21, 4423}, {28, 17917}, {35, 4428}, {36, 405}, {37, 988}, {46, 392}, {55, 404}, {57, 960}, {63, 25917}, {65, 3306}, {72, 3338}, {78, 354}, {84, 16112}, {86, 4267}, {100, 3303}, {104, 3090}, {140, 10198}, {145, 8168}, {171, 1191}, {183, 20449}, {220, 17754}, {226, 25681}, {236, 7588}, {238, 4252}, {330, 16999}, {355, 16203}, {381, 3825}, {386, 24923}, {411, 8273}, {442, 499}, {443, 2886}, {468, 22479}, {475, 11399}, {480, 11038}, {495, 26364}, {496, 11235}, {497, 6904}, {498, 13747}, {515, 6918}, {518, 936}, {519, 7373}, {528, 1058}, {551, 3295}, {620, 22514}, {631, 3428}, {750, 1201}, {758, 5708}, {859, 25526}, {894, 25895}, {908, 10404}, {940, 1193}, {942, 997}, {944, 6946}, {956, 1698}, {964, 26094}, {965, 2260}, {976, 17597}, {993, 11108}, {995, 5711}, {1012, 3838}, {1038, 25947}, {1056, 12607}, {1078, 22520}, {1104, 5272}, {1106, 6180}, {1155, 5250}, {1210, 5794}, {1319, 19860}, {1385, 6911}, {1387, 13205}, {1388, 11501}, {1450, 25934}, {1455, 19372}, {1465, 15832}, {1466, 3485}, {1468, 4383}, {1470, 5249}, {1471, 25878}, {1478, 4187}, {1479, 11112}, {1482, 3754}, {1616, 5255}, {1621, 4188}, {1656, 3822}, {1696, 5749}, {1697, 10179}, {1722, 16602}, {1898, 17616}, {2077, 9624}, {2178, 17398}, {2275, 5275}, {2292, 17595}, {2334, 3240}, {2475, 10896}, {2476, 22760}, {2478, 7354}, {2550, 3813}, {2829, 6893}, {2932, 16173}, {3035, 3085}, {3068, 19013}, {3069, 19014}, {3091, 3847}, {3149, 3576}, {3189, 10580}, {3242, 3976}, {3286, 11110}, {3337, 5692}, {3339, 15829}, {3421, 9711}, {3434, 10586}, {3452, 4298}, {3475, 27383}, {3487, 25557}, {3523, 5584}, {3526, 6681}, {3556, 10192}, {3560, 3824}, {3585, 17556}, {3589, 22769}, {3601, 10582}, {3614, 6931}, {3617, 9342}, {3634, 8666}, {3636, 6767}, {3646, 15254}, {3655, 18518}, {3683, 4652}, {3698, 3872}, {3720, 19765}, {3744, 28011}, {3772, 24178}, {3811, 5045}, {3814, 9654}, {3826, 17582}, {3831, 5793}, {3833, 10246}, {3868, 4860}, {3869, 5221}, {3870, 17609}, {3874, 3940}, {3884, 12702}, {3889, 4420}, {3911, 26066}, {3918, 22837}, {3922, 5048}, {3925, 10527}, {3927, 10176}, {3934, 22779}, {4000, 11505}, {4004, 25415}, {4189, 5284}, {4190, 6284}, {4193, 10895}, {4225, 5333}, {4245, 15654}, {4257, 8692}, {4258, 16503}, {4281, 18166}, {4292, 24703}, {4293, 5084}, {4297, 19541}, {4299, 11113}, {4301, 6244}, {4309, 17583}, {4315, 5795}, {4317, 17575}, {4327, 25067}, {4361, 17733}, {4386, 16781}, {4422, 24826}, {4425, 20849}, {4640, 15803}, {4662, 6762}, {4666, 4855}, {4670, 24328}, {4697, 23085}, {5010, 19537}, {5020, 22654}, {5024, 25092}, {5044, 5220}, {5046, 12943}, {5055, 26321}, {5070, 20107}, {5080, 9657}, {5120, 5257}, {5123, 5193}, {5132, 20150}, {5177, 10589}, {5229, 6919}, {5251, 16842}, {5252, 24982}, {5258, 16864}, {5259, 7280}, {5264, 16483}, {5267, 16418}, {5277, 16502}, {5288, 19875}, {5290, 13370}, {5294, 25881}, {5302, 7308}, {5303, 16865}, {5316, 12527}, {5432, 6921}, {5436, 7580}, {5443, 19525}, {5449, 22659}, {5450, 6913}, {5461, 22565}, {5552, 15888}, {5554, 10944}, {5603, 6940}, {5720, 12675}, {5722, 17647}, {5730, 5902}, {5731, 6915}, {5744, 18253}, {5781, 22088}, {5842, 6885}, {5853, 21625}, {5901, 11248}, {5927, 10085}, {5972, 22586}, {6036, 22504}, {6118, 22624}, {6119, 22595}, {6174, 13279}, {6261, 9940}, {6669, 22773}, {6670, 22774}, {6673, 22772}, {6674, 22771}, {6675, 22667}, {6684, 22770}, {6685, 16409}, {6689, 22781}, {6696, 22778}, {6699, 22583}, {6700, 21620}, {6701, 13743}, {6702, 12773}, {6703, 15509}, {6704, 22780}, {6705, 18237}, {6706, 16852}, {6713, 6862}, {6720, 19162}, {6734, 17728}, {6744, 12437}, {6763, 15650}, {6837, 7958}, {6850, 7681}, {6854, 10785}, {6871, 7173}, {6872, 15326}, {6882, 10894}, {6883, 26286}, {6891, 7680}, {6897, 15908}, {6922, 26332}, {6923, 10893}, {6924, 10267}, {6944, 18242}, {6947, 11827}, {6955, 10531}, {6964, 12667}, {6967, 10532}, {6977, 21154}, {6983, 12115}, {6985, 13624}, {7028, 7587}, {7091, 18227}, {7223, 26563}, {7330, 15297}, {7741, 17532}, {7846, 22744}, {7982, 10107}, {8301, 17044}, {8582, 10106}, {8668, 11024}, {8728, 15325}, {9623, 11260}, {9710, 26040}, {9780, 17535}, {9841, 15726}, {9858, 15733}, {10072, 24390}, {10090, 11507}, {10178, 12565}, {10199, 17528}, {10306, 13464}, {10448, 16405}, {10459, 17124}, {10629, 13272}, {10680, 26446}, {10827, 17619}, {10855, 18251}, {10980, 11523}, {11023, 11281}, {11037, 25568}, {11114, 26127}, {11227, 12520}, {11246, 11415}, {11285, 27255}, {11321, 26959}, {11329, 17397}, {11358, 26102}, {11392, 26020}, {11509, 15950}, {11684, 23958}, {11723, 12327}, {11724, 12178}, {11725, 13173}, {11729, 12332}, {11735, 13204}, {12447, 24391}, {12577, 20103}, {12608, 12676}, {12705, 21164}, {12711, 17612}, {12953, 17579}, {13405, 16411}, {13411, 16410}, {13738, 19701}, {13740, 25492}, {13902, 19000}, {13906, 19024}, {13959, 18999}, {13964, 19023}, {14793, 16153}, {14803, 23708}, {15171, 17563}, {15183, 22755}, {15808, 17573}, {15934, 22836}, {16058, 25501}, {16209, 17613}, {16412, 17023}, {16429, 20268}, {16458, 19863}, {16823, 21010}, {16831, 21477}, {16844, 19762}, {16853, 19878}, {16917, 20172}, {17384, 19527}, {17527, 18990}, {17529, 19854}, {17697, 25531}, {17698, 27802}, {17718, 27385}, {17720, 23536}, {17721, 28018}, {17742, 25068}, {17798, 19314}, {19284, 24552}, {19478, 20304}, {19521, 20195}, {19684, 27639}, {19836, 20266}, {21049, 24247}, {21258, 24384}, {24365, 24432}, {24557, 27624}, {24596, 26964}, {24914, 24987}, {25490, 25493}, {25946, 26626}, {25962, 26481}
X(25525) lies on these lines: {1, 442}, {2, 7}, {3, 3824}, {4, 1125}, {5, 1490}, {8, 4035}, {10, 3487}, {11, 10382}, {21, 9579}, {36, 405}, {40, 6889}, {72, 1698}, {78, 4197}, {84, 6824}, {120, 5268}, {140, 5812}, {141, 18229}, {165, 5880}, {200, 3925}, {236, 8080}, {282, 13853}, {306, 4007}, {321, 4873}, {345, 4659}, {354, 5231}, {377, 3601}, {381, 13151}, {440, 17073}, {443, 5438}, {452, 5550}, {495, 9623}, {499, 10396}, {515, 6843}, {551, 3488}, {610, 6678}, {631, 12436}, {920, 12695}, {936, 8728}, {942, 5705}, {943, 25440}, {946, 6908}, {950, 3616}, {954, 1376}, {958, 5290}, {968, 3120}, {975, 24160}, {1001, 1699}, {1005, 5284}, {1210, 6856}, {1211, 16608}, {1260, 4413}, {1420, 24541}, {1449, 5712}, {1467, 15844}, {1621, 9580}, {1656, 5777}, {1706, 3085}, {1713, 19720}, {1750, 3816}, {1751, 2364}, {1763, 9816}, {1790, 5333}, {1836, 4512}, {2051, 2140}, {2184, 7110}, {2345, 20106}, {2476, 9581}, {2550, 3158}, {2551, 3947}, {2893, 26109}, {2999, 5718}, {3008, 26036}, {3011, 5269}, {3090, 9843}, {3220, 25514}, {3243, 3475}, {3333, 26363}, {3339, 26066}, {3340, 24987}, {3361, 4999}, {3434, 10389}, {3485, 5665}, {3586, 17532}, {3622, 5175}, {3646, 21616}, {3649, 12526}, {3651, 5248}, {3666, 23681}, {3679, 5425}, {3731, 4415}, {3742, 5728}, {3752, 4859}, {3811, 3841}, {3822, 5587}, {3825, 6990}, {3826, 8580}, {3848, 5927}, {3936, 4034}, {4054, 17776}, {4199, 16778}, {4208, 5703}, {4292, 6857}, {4312, 4640}, {4384, 4417}, {4423, 13615}, {4656, 16676}, {4657, 10888}, {4666, 11680}, {4853, 15888}, {4882, 9710}, {5176, 9578}, {5180, 5250}, {5261, 5795}, {5272, 17717}, {5542, 24477}, {5573, 24239}, {5658, 10171}, {5714, 12572}, {5720, 6881}, {5732, 8727}, {5737, 17272}, {5743, 16832}, {5758, 6684}, {5759, 10164}, {5784, 11018}, {5791, 6147}, {5853, 10578}, {5886, 6907}, {6245, 6855}, {6260, 6846}, {6667, 11219}, {6700, 17582}, {6706, 17284}, {6708, 23058}, {6734, 11518}, {6736, 11530}, {6745, 26040}, {6762, 19843}, {6828, 10884}, {6831, 8726}, {6832, 10200}, {6847, 9841}, {6861, 7330}, {6913, 10269}, {6933, 10395}, {6937, 9624}, {6987, 10165}, {6996, 25500}, {7028, 7593}, {7290, 26098}, {7483, 15803}, {7522, 19701}, {8582, 10588}, {8583, 11375}, {9119, 26958}, {9940, 12664}, {10319, 16548}, {10404, 24953}, {10580, 24386}, {10585, 24982}, {10980, 25557}, {11036, 24391}, {11263, 12514}, {11679, 17296}, {13407, 19854}, {14054, 18398}, {14829, 17298}, {15474, 25430}, {15650, 19872}, {16579, 17080}, {17022, 17720}, {17278, 23511}, {17528, 24929}, {17594, 17889}, {19786, 20268}, {19804, 20882}, {19855, 21075}, {20104, 26878}, {25516, 25526}
X(25526) lies on these lines: {1, 75}, {2, 58}, {3, 10478}, {6, 16458}, {8, 4658}, {10, 81}, {12, 1408}, {21, 36}, {28, 142}, {29, 17194}, {35, 13588}, {46, 17185}, {56, 19259}, {65, 18417}, {72, 4670}, {191, 4697}, {226, 5323}, {238, 25508}, {283, 10198}, {284, 443}, {333, 1698}, {334, 741}, {386, 16454}, {388, 1412}, {404, 4276}, {405, 3286}, {442, 6703}, {474, 4267}, {859, 25524}, {940, 2049}, {946, 4221}, {964, 10458}, {1001, 17524}, {1014, 4298}, {1089, 1961}, {1098, 6679}, {1126, 4651}, {1203, 27644}, {1326, 4201}, {1333, 13728}, {1434, 4355}, {1437, 25466}, {1468, 19858}, {1509, 19856}, {1746, 13323}, {1778, 5257}, {1817, 12436}, {2193, 18641}, {2287, 17745}, {2303, 5280}, {2363, 2887}, {2901, 17019}, {3110, 16377}, {3216, 4281}, {3244, 4720}, {3337, 6682}, {3338, 10461}, {3589, 17529}, {3616, 4653}, {3617, 26860}, {3621, 4803}, {3624, 4892}, {3634, 5235}, {3743, 4418}, {3753, 18178}, {3786, 5904}, {3812, 18180}, {3825, 14008}, {3828, 4921}, {3923, 27785}, {3936, 24931}, {3980, 25058}, {4184, 5248}, {4197, 9275}, {4234, 25055}, {4253, 19288}, {4255, 19290}, {4256, 19284}, {4257, 16342}, {4294, 17201}, {4295, 17183}, {4645, 19865}, {5009, 17381}, {5208, 18398}, {5247, 16828}, {5313, 5331}, {5324, 17581}, {5439, 18165}, {5550, 17588}, {5687, 18185}, {5886, 15952}, {6707, 17514}, {7283, 16826}, {7415, 7987}, {7741, 14009}, {7951, 14011}, {8583, 24556}, {9534, 17379}, {9780, 16704}, {10180, 24850}, {12609, 16049}, {14829, 19280}, {16050, 16831}, {16456, 19732}, {16457, 19749}, {16713, 19855}, {16738, 19853}, {16926, 17034}, {16948, 17557}, {17023, 26643}, {17056, 25645}, {17206, 17210}, {17521, 27186}, {17531, 20108}, {17553, 19883}, {19276, 19765}, {19283, 19762}, {19332, 19722}, {24046, 26627}, {24987, 26637}, {25501, 25542}, {25516, 25525}, {25650, 26109}, {26828, 27146}
X(25526) = complement of X(26064)
X(25527) lies on these lines: {1, 977}, {2, 7}, {10, 24159}, {75, 23681}, {78, 4202}, {86, 19812}, {141, 3772}, {200, 4429}, {278, 9312}, {306, 3875}, {312, 1930}, {314, 19803}, {321, 17286}, {333, 17189}, {344, 4656}, {345, 3663}, {612, 25957}, {614, 25760}, {936, 24781}, {940, 17298}, {1125, 4138}, {1211, 4384}, {1215, 1698}, {1220, 5290}, {1227, 27191}, {1330, 1453}, {1427, 25887}, {1449, 17778}, {1478, 19869}, {1699, 13727}, {1707, 4655}, {1999, 17296}, {2049, 3824}, {2999, 4417}, {3008, 14555}, {3011, 26034}, {3120, 24943}, {3220, 25494}, {3247, 4071}, {3416, 17061}, {3436, 25904}, {3586, 17677}, {3601, 4201}, {3624, 4892}, {3666, 17304}, {3677, 3705}, {3687, 4000}, {3729, 3782}, {3741, 17064}, {3749, 4660}, {3751, 25453}, {3752, 17290}, {3761, 18067}, {3771, 3821}, {3836, 5268}, {3846, 5272}, {3870, 4972}, {3886, 3914}, {3920, 25959}, {3936, 5256}, {3946, 4035}, {3954, 17308}, {4001, 24597}, {4195, 9579}, {4383, 23151}, {4388, 7290}, {4415, 17279}, {4512, 24723}, {4645, 5269}, {4657, 17056}, {4675, 6703}, {4708, 19744}, {4859, 19804}, {4886, 16833}, {5233, 23511}, {5271, 17270}, {5287, 18139}, {5307, 24291}, {5436, 26117}, {5712, 16780}, {5722, 16052}, {5737, 17237}, {5739, 26723}, {5743, 17278}, {6327, 26230}, {7191, 25958}, {7377, 10888}, {8557, 25364}, {9612, 13740}, {10444, 19542}, {10447, 19792}, {11359, 24929}, {12047, 19836}, {12572, 13742}, {13407, 19784}, {14829, 17227}, {16831, 25499}, {17022, 17234}, {17151, 19796}, {17160, 19830}, {17171, 17182}, {17244, 25430}, {17269, 22034}, {17283, 18743}, {17397, 26109}, {18140, 21590}, {18141, 21255}, {19793, 23537}, {19808, 25590}, {19861, 24984}, {19864, 25519}, {21384, 27319}, {25492, 25513}, {25505, 25515}, {25509, 25531}, {25681, 25914}
X(25528) lies on these lines: {1, 75}, {2, 87}, {6, 16569}, {8, 25573}, {10, 25311}, {36, 15485}, {43, 17379}, {238, 4252}, {1001, 8572}, {1125, 26143}, {1654, 25572}, {1698, 25121}, {2309, 26102}, {3248, 4751}, {3551, 24456}, {3616, 25570}, {3624, 25535}, {3739, 18194}, {3783, 3945}, {6685, 20146}, {7240, 17257}, {9780, 23579}, {15668, 20530}, {16468, 27623}, {16710, 17157}, {16815, 23524}, {16816, 23532}, {17303, 20532}, {18192, 27164}
X(25529) lies on these lines: {2, 45}, {10, 3699}, {11, 2476}, {86, 4604}, {145, 17724}, {312, 17785}, {664, 2006}, {3624, 13740}, {3996, 17064}, {11814, 19878}, {16816, 27739}, {17160, 27757}, {17244, 17720}, {17292, 27747}, {17379, 26137}, {25507, 25536}
X(25530) lies on these lines: {2, 99}, {405, 25506}, {741, 1125}, {799, 1015}, {5209, 20530}, {6626, 26959}, {7257, 27076}, {8849, 17557}, {16613, 25472}, {20982, 24505}, {25499, 25520}, {25504, 25518}, {25532, 25534}
X(25531) lies on these lines: {1, 1120}, {2, 11}, {8, 16486}, {86, 799}, {88, 4427}, {106, 1125}, {190, 244}, {238, 4871}, {312, 5272}, {333, 3840}, {341, 28011}, {405, 25492}, {614, 18743}, {748, 14829}, {940, 26103}, {958, 26093}, {964, 19749}, {1010, 19832}, {1043, 27627}, {1054, 4432}, {1086, 17777}, {1279, 5205}, {1421, 14594}, {3057, 25965}, {3120, 24709}, {3303, 26029}, {3304, 26111}, {3306, 4676}, {3315, 3952}, {3486, 25879}, {3616, 16594}, {3623, 4152}, {3624, 13740}, {3662, 4679}, {3685, 16610}, {3722, 9458}, {3742, 27064}, {3750, 6686}, {3756, 4422}, {3837, 4491}, {3873, 26688}, {3932, 5211}, {3937, 24482}, {3996, 16569}, {4003, 17261}, {4011, 17063}, {4202, 26127}, {4358, 7292}, {4387, 17490}, {4519, 17117}, {4640, 27002}, {4756, 17154}, {4860, 17350}, {4928, 25532}, {5047, 26094}, {5192, 5550}, {5259, 19847}, {5695, 24620}, {7988, 13727}, {8256, 25979}, {11110, 19864}, {11269, 17352}, {15254, 24627}, {16853, 19853}, {17054, 19582}, {17125, 17277}, {17244, 17723}, {17283, 25760}, {17546, 19874}, {17597, 27538}, {17697, 25524}, {17793, 26102}, {18228, 26112}, {23988, 24486}, {24358, 26273}, {24757, 26671}, {25496, 25502}, {25501, 25507}, {25509, 25527}, {25533, 25536}, {25914, 26117}
X(25531) = complement of X(26073)
X(25532) lies on these lines: {1, 17048}, {2, 101}, {36, 20335}, {86, 27195}, {88, 24185}, {100, 17761}, {105, 831}, {142, 10090}, {214, 9317}, {404, 2140}, {1018, 24203}, {3035, 4904}, {3624, 25500}, {4188, 14377}, {4928, 25531}, {5030, 20347}, {5253, 17758}, {5540, 24685}, {6706, 17614}, {8671, 24447}, {9318, 24036}, {10200, 26101}, {13587, 17729}, {13747, 21258}, {17189, 27633}, {17306, 25523}, {25530, 25534}
X(25532) = complement of X(26074)
X(25533) lies on these lines: {2, 98}, {11, 662}, {21, 10165}, {86, 9432}, {99, 17777}, {404, 3615}, {409, 759}, {423, 24559}, {425, 17923}, {501, 3825}, {643, 3035}, {1098, 5433}, {2185, 3816}, {2360, 14011}, {2607, 16598}, {2651, 3911}, {3120, 24617}, {4591, 19634}, {4881, 7424}, {5127, 6681}, {5886, 11116}, {9275, 10199}, {13588, 17188}, {17476, 24345}, {25531, 25536}
X(25533) = complement of X(26075)
X(25534) lies on these lines: {2, 45}, {86, 20332}, {334, 17322}, {1268, 26812}, {3763, 26107}, {3834, 25510}, {3837, 4491}, {4360, 27095}, {4395, 26048}, {4499, 24487}, {4657, 20333}, {4947, 25382}, {6386, 18140}, {15668, 25535}, {16706, 27111}, {16726, 27195}, {16777, 20532}, {17160, 27044}, {17237, 26959}, {17273, 26963}, {17297, 27106}, {17301, 27091}, {17306, 25505}, {17307, 26971}, {17323, 26042}, {17326, 27164}, {17395, 26752}, {17398, 26149}, {25498, 25508}, {25530, 25532}
X(25535) lies on these lines: {1, 25121}, {2, 37}, {330, 18133}, {1125, 14823}, {1269, 27318}, {3248, 26069}, {3616, 25311}, {3619, 26113}, {3624, 25528}, {5550, 26135}, {15668, 25534}, {17238, 27166}, {17306, 25510}, {25492, 25539}
X(25536) lies on these lines: {1, 18805}, {2, 662}, {21, 1633}, {37, 86}, {81, 1332}, {99, 1086}, {110, 24542}, {141, 261}, {287, 26543}, {320, 1931}, {333, 4690}, {593, 18139}, {620, 17058}, {645, 4422}, {671, 23905}, {757, 17300}, {1326, 3836}, {1509, 17392}, {1963, 17317}, {2640, 24714}, {3616, 16597}, {3834, 16702}, {3912, 19623}, {4675, 17103}, {5880, 11104}, {6626, 17237}, {14534, 17056}, {17279, 27958}, {17374, 17731}, {24617, 27191}, {25505, 25523}, {25507, 25529}, {25530, 25532}, {25531, 25533}
X(25536) = complement of X(26081)
X(25537) lies on these lines: {2, 669}, {512, 25084}, {523, 25258}, {659, 693}, {661, 4367}, {667, 3835}, {748, 15373}, {804, 24622}, {890, 27013}, {1491, 27674}, {2978, 27673}, {3287, 25854}, {3733, 4806}, {3837, 4057}, {4010, 8062}, {4369, 4455}, {5926, 19543}, {7234, 25666}, {8651, 24782}, {8654, 26249}, {16468, 23570}, {18197, 26102}, {20979, 24666}, {23655, 24532}, {25128, 25636}
X(25537) = complement of X(25299)
X(25537) = anticomplement of X(25126)
X(25538) lies on these lines: {2, 7}, {69, 17030}, {75, 3774}, {86, 23660}, {269, 26134}, {274, 27633}, {314, 21264}, {319, 16829}, {1125, 6196}, {1269, 25264}, {3624, 25528}, {3879, 26801}, {4967, 26752}, {5224, 16819}, {15668, 25505}, {16706, 27042}, {16738, 17288}, {16815, 26772}, {16826, 26971}, {16831, 26107}, {17023, 26110}, {17117, 26812}, {17237, 27164}, {17261, 26976}, {17266, 27261}, {17321, 27255}, {17322, 25499}, {24790, 25470}, {25498, 25508}, {25590, 26042}
X(25538) = complement of X(26082)
X(25539) lies on these lines: {1, 3763}, {2, 38}, {9, 3337}, {10, 17370}, {238, 3624}, {726, 17371}, {740, 17383}, {1125, 3883}, {1698, 4901}, {3220, 25542}, {3526, 6211}, {3775, 17367}, {3840, 19812}, {3923, 17305}, {3993, 17399}, {4357, 4896}, {4389, 24295}, {4423, 7295}, {4672, 17236}, {4974, 17238}, {4991, 17360}, {6210, 11230}, {8167, 24436}, {16468, 17237}, {16477, 17272}, {16825, 17307}, {17278, 19856}, {17290, 24342}, {17353, 19878}, {17592, 24943}, {19864, 25505}, {24248, 26104}, {25492, 25535}
X(25539) = complement of X(26083)
X(25540) lies on these lines: {2, 3}, {238, 920}
X(25640) = crosssum of circumcircle intercepts of line X(3)X(521)
X(25640) = orthopole of line X(3)X(521)
X(25541) lies on these lines: {2, 3}
X(25641) = crosssum of circumcircle intercepts of line X(3)X(526)
X(25641) = orthopole of line X(3)X(526)
X(25542) lies on these lines: {1, 210}, {2, 35}, {3, 7988}, {5, 15931}, {9, 18398}, {10, 5284}, {21, 19862}, {36, 405}, {55, 16853}, {56, 16857}, {72, 5506}, {79, 142}, {100, 17546}, {191, 5439}, {386, 17125}, {392, 11009}, {404, 19878}, {411, 10171}, {452, 10483}, {498, 17559}, {499, 16845}, {551, 5260}, {748, 1203}, {908, 1125}, {958, 17542}, {960, 5425}, {993, 5550}, {1001, 1698}, {1089, 16823}, {1376, 16854}, {1478, 5129}, {1621, 3634}, {1656, 10902}, {1724, 19714}, {2077, 3526}, {2886, 17590}, {3086, 17554}, {3220, 25539}, {3245, 5250}, {3293, 16484}, {3295, 19875}, {3305, 5904}, {3336, 3683}, {3576, 18761}, {3582, 24953}, {3583, 8728}, {3616, 5258}, {3647, 27003}, {3679, 19536}, {3742, 6763}, {3817, 6986}, {3828, 3871}, {3848, 3916}, {3874, 27065}, {3894, 15650}, {3925, 4857}, {3947, 7677}, {3957, 4015}, {4002, 5541}, {4187, 6668}, {4205, 19881}, {4302, 17582}, {4309, 26040}, {4413, 16855}, {4653, 27627}, {4668, 6767}, {4975, 16824}, {5010, 16408}, {5067, 6796}, {5084, 7951}, {5204, 16866}, {5217, 16863}, {5253, 16861}, {5262, 27784}, {5264, 15485}, {5267, 16858}, {5299, 16589}, {5303, 17543}, {5442, 6692}, {5692, 12559}, {5902, 12526}, {6533, 7283}, {6690, 17575}, {6691, 15670}, {6883, 8227}, {6913, 7987}, {6920, 10165}, {7280, 16418}, {7411, 12571}, {7688, 9955}, {8053, 16297}, {8715, 19877}, {9709, 19876}, {9779, 12511}, {10172, 11491}, {10826, 25893}, {11012, 11230}, {11110, 19864}, {13740, 25512}, {13745, 25914}, {16296, 23383}, {16345, 25502}, {16848, 17717}, {16912, 26959}, {17284, 21986}, {17528, 18514}, {17552, 19854}, {19224, 25510}, {19270, 19847}, {19861, 24926}, {21616, 26725}, {25501, 25526}
X(25542) = complement of X(26060)
X(25642) = crosssum of circumcircle intercepts of line X(3)X(674)
X(25642) = orthopole of line X(3)X(674)
Collineation mappings involving Gemini triangle 24: X(25543)-X(25554)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 24, as in centers X(25543)-X(25554). Then
m(X) = (a + b + c)(2a + b + c)(2a + b + 2c) x - (a + c)(2a + b + c) (a + 2b + 2c) y - (a + b)(2a + b + c)(a + 2b + 2c) : :
A point X lies on the Euler line if and only m(X) lies on the Euler line. (Clark Kimberling, October 19, 2018)
X(25543) lies on these lines: {1, 2}
X(25544) lies on these lines: {2, 3}
X(25545) lies on these lines: {2, 3}
X(25546) lies on these lines: {2, 3}
X(25547) lies on these lines: {2, 3}
X(25548) lies on these lines: {2, 3}
X(25549) lies on these lines: {2, 3}
X(25550) lies on these lines: {2, 3}
X(25551) lies on these lines: {2, 3}
X(25552) lies on these lines: {2, 3}
X(25553) lies on these lines: {2, 3}
X(25554) lies on these lines: {2, 3}
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28505.
X(25555) lies on these lines: {2, 576}, {3, 5476}, {4, 83}, {5, 542}, {6, 17}, {30, 20190}, {51, 7495}, {61, 9117}, {62, 9115}, {69, 15520}, {110, 7605}, {114, 3329}, {115, 5038}, {125, 15018}, {140, 143}, {141, 5097}, {262, 7875}, {373, 5972}, {382, 10541}, {384, 10992}, {395, 14136}, {396, 14137}, {468, 5943}, {524, 3628}, {546, 11645}, {550, 5092}, {599, 5070}, {631, 20423}, {1350, 15720}, {1352, 5056}, {1428, 5270}, {1503, 3850}, {1657, 5085}, {1992, 5067}, {2030, 7745}, {2330, 4857}, {2781, 12006}, {3090, 7856}, {3091, 11179}, {3095, 7889}, {3098, 3523}, {3398, 10991}, {3522, 17508}, {3526, 11477}, {3549, 11511}, {3564, 6329}, {3642, 22683}, {3643, 22685}, {3763, 5093}, {3815, 6721}, {3818, 3851}, {4045, 10796}, {4663, 11230}, {5012, 7533}, {5055, 15069}, {5068, 6776}, {5073, 12017}, {5094, 10601}, {5182, 16044}, {5355, 7697}, {5422, 21243}, {5486, 9813}, {5642, 16042}, {5892, 15122}, {6034, 7765}, {6036, 7792}, {6639, 8538}, {6688, 23292}, {6689, 15026}, {6759, 23327}, {7499, 21849}, {7505, 8541}, {7527, 10990}, {7570, 11225}, {7607, 7806}, {7749, 11261}, {7797, 12177}, {7804, 23698}, {7831, 22521}, {7907, 22486}, {8252, 9974}, {8253, 9975}, {8537, 14940}, {8584, 15699}, {8681, 9820}, {9019, 10095}, {9730, 20417}, {9976, 14643}, {10110, 16618}, {10112, 14788}, {10249, 22802}, {11180, 15022}, {11255, 16511}, {11303, 16002}, {11304, 16001}, {11451, 15073}, {12007, 18358}, {13353, 13419}, {13364, 25338}, {13434, 15462}, {14791, 19136}, {14810, 15712}, {14864, 23300}, {15000, 18114}, {15037, 19140}, {15074, 16776}, {15534, 15703}, {18381, 19153}
X(25555) = midpoint of X(i) and X(j) for these {i,j}: {5, 575}, {6, 24206}, {141, 5097}, {182, 19130}, {3589, 18583}, {4045, 10796}, {5092, 5480}, {5476, 10168}, {6593, 20301}, {8550, 18553}, {12007, 18358}, {14810, 21850}
X(25555) = reflection of X(15516) in X(6329)
X(25555) = complement of the complement of X(576)
X(25555) = complement of the complementary conjugate of X(15850)
X(25555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 597, 575), (5, 7817, 20398), (5, 8550, 18553), (17, 18, 1506), (182, 14561, 19130), (373, 14389, 5972), (575, 18553, 8550), (3526, 14848, 11477), (3618, 14561, 182), (5070, 11482, 599), (6694, 6695, 6680), (14632, 14633, 140)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28505.
X(25556) lies on the cubic K802 and these lines: {3, 19379}, {5, 25329}, {6, 13}, {74, 182}, {110, 576}, {125, 15018}, {146, 11179}, {323, 5642}, {511, 1511}, {524, 10272}, {575, 5663}, {597, 10264}, {729, 6236}, {895, 14491}, {1351, 12584}, {1352, 25321}, {1495, 11649}, {1539, 11645}, {1974, 3043}, {1992, 20125}, {2781, 5092}, {2854, 5097}, {2914, 5095}, {2930, 5093}, {3098, 10298}, {4663, 11699}, {5191, 18114}, {5505, 9813}, {5609, 22330}, {5972, 15066}, {6699, 10168}, {6723, 15106}, {6759, 13248}, {9969, 11702}, {10296, 13202}, {10541, 15041}, {11061, 14561}, {11470, 15463}, {11597, 18374}, {12041, 20190}, {12367, 18449}, {12383, 20423}, {12900, 24206}, {13289, 19153}, {14094, 22234}, {15032, 15063}, {15037, 16003}, {16163, 19924}, {18583, 20301}, {19150, 22336}
X(25556) = midpoint of X(i) and X(j) for these {i,j}: {5, 25329}, {6, 19140}, {110, 576}, {182, 9970}, {399, 9976}, {1351, 12584}, {3098, 10752}, {4663, 11699}, {6759, 13248}
X(25556) = reflection of X(i) in X(j) for these (i,j): (125, 25555), (12041, 20190), (20301, 18583),
X(25556) = X(575)-of-anti-orthocentroidal triangle
X(25556) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 399, 9976), (9976, 19140, 399), (10752, 15462, 3098)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28505.
Let A' be the nine-point center of the triangle IBC, where I = X(1), and define B' and C' cyclically. The triangle A'B'C' is homothetic to the Fuhrmann triangle at X(1), and X(25557) = X(6)-of-A'B'C'. See also X(5886) and X(5901). (Randy Hutson, July 11, 2019)
X(25557) lies on these lines: {1, 528}, {2, 3715}, {5, 2801}, {7, 21}, {8, 17297}, {9, 583}, {10, 141}, {11, 10129}, {12, 7705}, {37, 24231}, {48, 5829}, {57, 6690}, {75, 4966}, {145, 2550}, {226, 3660}, {238, 17365}, {244, 5718}, {320, 16823}, {329, 8167}, {335, 16593}, {354, 2886}, {392, 11551}, {442, 18398}, {496, 2486}, {499, 5729}, {516, 550}, {524, 16825}, {527, 1125}, {551, 5126}, {553, 4640}, {673, 20132}, {726, 17243}, {740, 7263}, {750, 17724}, {940, 17061}, {946, 15726}, {954, 8071}, {971, 9955}, {982, 17056}, {1329, 5439}, {1376, 3475}, {1386, 3664}, {1401, 18165}, {1445, 15296}, {1621, 11246}, {1757, 17337}, {1836, 4666}, {3008, 4663}, {3035, 3306}, {3058, 20292}, {3120, 17450}, {3243, 3632}, {3246, 4896}, {3296, 19843}, {3337, 7483}, {3416, 17298}, {3452, 3848}, {3474, 4428}, {3555, 9710}, {3576, 5735}, {3629, 4974}, {3634, 4407}, {3662, 4026}, {3663, 15569}, {3685, 7321}, {3696, 4684}, {3703, 17140}, {3720, 3782}, {3751, 17278}, {3756, 17717}, {3790, 17241}, {3813, 5045}, {3820, 3833}, {3824, 10916}, {3829, 3838}, {3871, 9782}, {3873, 3925}, {3914, 4883}, {3923, 7228}, {3932, 17234}, {3962, 24564}, {3976, 4443}, {4023, 24589}, {4133, 4726}, {4310, 4648}, {4355, 5436}, {4421, 10578}, {4423, 5905}, {4444, 4806}, {4644, 16020}, {4649, 17366}, {4654, 10582}, {4655, 7238}, {4699, 4733}, {4719, 24171}, {4743, 5853}, {4888, 7290}, {4995, 9352}, {5219, 6667}, {5223, 20195}, {5248, 24470}, {5284, 17483}, {5450, 5901}, {5550, 6172}, {5572, 16193}, {5696, 24390}, {5708, 10198}, {5794, 11518}, {5805, 18481}, {5847, 17376}, {5850, 6666}, {5851, 5886}, {5855, 11529}, {5904, 17529}, {6006, 23814}, {6361, 11495}, {6594, 17564}, {6691, 8257}, {6701, 20116}, {7277, 16468}, {7288, 12848}, {7354, 18450}, {7680, 10202}, {7958, 12528}, {7965, 11220}, {7988, 13257}, {8256, 15888}, {8301, 14828}, {8545, 11375}, {8581, 15844}, {10177, 12047}, {10580, 11235}, {10785, 16112}, {10861, 11025}, {10944, 14151}, {11036, 12635}, {11037, 12513}, {11260, 12577}, {11269, 17070}, {15298, 17437}, {17000, 20072}, {17307, 19877}
X(25557) = midpoint of X(i) and X(j) for these {i,j}: {1, 5880}, {7, 1001}, {142, 5542}
X(25557) = reflection of X(i) in X(j) for these (i,j): (3826, 142), (15254, 1125), (15481, 6666)
X(25557) = complement of X(5220)
X(25557) = X(5480)-of-Fuhrmann-triangle
X(25557) = X(5880)-of-anti-Aquila-triangle
X(25557) = X(8550)-of-Wasat-triangle
X(25557) = X(9970)-of-K798i-triangle
X(25557) = X(15069)-of-3rd-Euler-triangle
X(25557) = X(15581)-of-2nd Zaniah-triangle
X(25557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6173, 5880), (7, 3616, 5698), (226, 3742, 3816), (354, 5249, 2886), (3306, 17718, 3035), (3475, 9776, 1376), (3616, 5698, 1001), (3812, 21620, 12607), (3838, 11019, 3829), (4654, 10582, 24703), (4684, 24199, 3696), (5045, 12609, 3813), (5439, 13407, 1329), (17140, 18139, 3703), (17234, 24349, 3932)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28505.
X(25558) lies on these lines: {7, 528}, {11, 10129}, {80, 6173}, {104, 1001}, {116, 119}, {214, 527}, {518, 1145}, {908, 12831}, {952, 5880}, {971, 12611}, {1537, 15726}, {2550, 12531}, {3035, 5220}, {3218, 6174}, {4996, 5852}, {5542, 21630}, {5735, 12119}, {5770, 20400}, {5784, 17660}, {5850, 6594}, {17768, 18450}
X(25558) = midpoint of X(i) and X(j) for these {i,j}: {5735, 12119}, {5784, 17660}
X(25558) = reflection of X(i) in X(j) for these (i,j): (11, 25557), (5220, 3035)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28505.
X(25559) lies on these lines: {2, 98}, {3, 6778}, {5, 20378}, {13, 5611}, {61, 115}, {99, 628}, {303, 5983}, {511, 6783}, {575, 6114}, {619, 13349}, {1569, 3107}, {5464, 13188}, {5470, 13102}, {5474, 13103}, {5479, 20252}, {5965, 5982}, {6777, 20416}, {10723, 25154}, {14137, 15516}, {14185, 16770}, {16001, 22513}, {16530, 22848}, {22505, 22797}, {22686, 23024}
X(25559) = midpoint of X(i) and X(j) for these {i,j}: {3, 6778}, {98, 22509}, {5474, 13103}, {5613, 6770}
X(25559) = reflection of X(i) in X(j) for these (i,j): (115, 20415), (5479, 20252), (6774, 6771), (6777, 20416), (16002, 115), (25560, 6036)
X(25559) = complement of X(22507)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28505.
X(25560) lies on these lines: {2, 98}, {3, 6777}, {5, 20377}, {14, 5615}, {62, 115}, {99, 627}, {302, 5982}, {511, 6782}, {575, 6115}, {618, 13350}, {1569, 3106}, {5463, 13188}, {5469, 13103}, {5473, 13102}, {5478, 20253}, {5965, 5983}, {6778, 20415}, {10723, 25164}, {14136, 15516}, {14187, 16771}, {16002, 22512}, {16529, 22892}, {22505, 22796}, {22684, 23018}
X(25560) = midpoint of X(i) and X(j) for these {i,j}: {3, 6777}, {98, 22507}, {5473, 13102}, {5617, 6773}
X(25560) = reflection of X(i) in X(j) for these (i,j): (115, 20416), (5478, 20253), (6771, 6774), (6778, 20415), (16001, 115), (25559, 6036)
X(25560) = complement of X(22509)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28506.
X(25561) lies on these lines: {2, 1495}, {5, 542}, {6, 18362}, {30, 14810}, {141, 3845}, {182, 5055}, {373, 9140}, {381, 511}, {524, 5066}, {547, 1503}, {576, 3851}, {671, 3399}, {1350, 14269}, {1352, 1992}, {1656, 20190}, {1853, 10219}, {3091, 7946}, {3098, 3830}, {3363, 19662}, {3534, 3763}, {3564, 11737}, {3589, 10109}, {3619, 15682}, {3849, 9996}, {5031, 7880}, {5071, 11179}, {5072, 14848}, {5085, 15703}, {5169, 13857}, {5650, 10989}, {6287, 13335}, {7533, 15360}, {7603, 11646}, {7809, 14994}, {7841, 10356}, {9041, 18357}, {10706, 15030}, {11180, 14561}, {11182, 18309}, {11801, 20113}, {14892, 18583}, {14927, 15709}, {15060, 16776}, {15686, 21167}, {15694, 17508}, {16626, 22579}, {16627, 22580}, {21850, 22165}
X(25561) = midpoint of X(i) and X(j) for these {i,j}: {2, 3818}, {141, 3845}, {381, 11178}, {1352, 5476}, {3098, 3830}, {5066, 18358}, {11182, 18309}, {15060, 16776}, {21850, 22165}
X(25561) = reflection of X(i) in X(j) for these (i,j): (3589, 10109), (5092, 2), (5097, 5476), (10168, 547), (19130, 5066)
X(25561) = X(22566)-of-1st Brocard triangle
X(25561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 18553, 575), (381, 10516, 11178), (1352, 3545, 5476), (3830, 21358, 3098), (5459, 5460, 7817)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28506.
X(25562) lies on these lines: {2, 98}, {316, 19924}, {524, 22566}, {543, 3818}, {599, 6033}, {690, 18309}, {2482, 11645}, {3845, 5969}, {5024, 11646}, {5092, 9167}, {6034, 19709}, {9830, 12040}, {10516, 11632}, {12042, 20582}, {14830, 21358}, {14981, 18553}
X(25562) = midpoint of X(i) and X(j) for these {i,j}: {147, 19905}, {599, 6033}, {1352, 6054}, {11180, 12177}
X(25562) = reflection of X(i) in X(j) for these (i,j): (6055, 24206), (12042, 20582), (115, 25561)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28506.
X(25563) lies on these lines: {2, 3357}, {3, 161}, {4, 11204}, {5, 1539}, {20, 23325}, {30, 20191}, {49, 16003}, {54, 5900}, {64, 3526}, {66, 17508}, {74, 6143}, {125, 3520}, {140, 6000}, {154, 15720}, {185, 10294}, {186, 13419}, {381, 8567}, {468, 13474}, {542, 12038}, {547, 5893}, {549, 6247}, {550, 18383}, {578, 18916}, {631, 5651}, {632, 2883}, {1204, 18388}, {1216, 15122}, {1498, 5054}, {1503, 3530}, {1568, 11440}, {1594, 21663}, {1614, 13399}, {1620, 18494}, {1656, 10606}, {1657, 18376}, {1658, 6697}, {2781, 12006}, {3516, 18390}, {3521, 15041}, {3523, 11202}, {3524, 9833}, {3533, 6225}, {3541, 11438}, {3628, 15311}, {3851, 5925}, {4550, 18431}, {5055, 5895}, {5449, 11250}, {5498, 5663}, {5876, 14156}, {5890, 12242}, {5907, 10257}, {5972, 12162}, {7577, 11468}, {7687, 18560}, {7689, 18281}, {8254, 10628}, {8981, 13980}, {8991, 13966}, {9705, 12317}, {9919, 22462}, {10018, 11381}, {10116, 10264}, {10192, 14869}, {10226, 13561}, {10295, 11572}, {11231, 12262}, {11487, 22581}, {11793, 16196}, {12289, 23040}, {13093, 15694}, {13293, 14130}, {13382, 23292}, {13434, 15057}, {13754, 23336}, {14530, 15701}, {14810, 23300}, {14864, 15712}, {15055, 19506}, {15696, 18405}, {15704, 23324}, {21659, 23294}
X(25563) = midpoint of X(i) and X(j) for these {i,j}: {3, 20299}, {140, 6696}, {550, 18383}, {5449, 11250}, {6247, 10282}, {6697, 15578}, {10193, 23329}, {10226, 13561}, {14810, 23300}
X(25563) = complement of the complement of X(3357)
X(25563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 23329, 20299), (125, 3520, 13403), (549, 6247, 10282), (550, 23332, 18383), (631, 6759, 10182), (1656, 10606, 22802), (3523, 14216, 11202), (10193, 20299, 3)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28506.
X(25564) lies on these lines: {3, 113}, {20, 19506}, {30, 15090}, {74, 184}, {110, 3357}, {125, 3520}, {186, 13202}, {265, 23329}, {378, 7687}, {399, 10606}, {542, 12901}, {550, 23315}, {974, 11430}, {1177, 17508}, {1204, 15463}, {1498, 15040}, {1511, 6000}, {1539, 15646}, {1614, 17856}, {1986, 21663}, {2071, 12827}, {2778, 13624}, {2781, 5092}, {3528, 13203}, {5010, 19505}, {5663, 10226}, {6699, 10193}, {6723, 7526}, {6759, 15035}, {7280, 10118}, {7712, 9934}, {7722, 11468}, {8546, 15578}, {8567, 10620}, {9976, 10249}, {10114, 20417}, {10264, 23328}, {10272, 15311}, {10605, 12227}, {10610, 10628}, {10721, 21844}, {10733, 23325}, {11250, 17702}, {11410, 19457}, {11413, 22109}, {11438, 15472}, {11454, 12219}, {12084, 12893}, {12121, 18381}, {12244, 23040}, {12292, 17701}, {13367, 17854}, {14130, 23515}, {14915, 20773}, {15041, 17847}, {15042, 17812}, {16219, 17835}, {18281, 19479}
X(25564) = midpoint of X(i) and X(j) for these {i,j}: {3, 13293}, {20, 19506}, {110, 3357}, {550, 23315}, {1511, 11598}, {2935, 13289}, {12084, 12893}, {12121, 18381}
X(25564) = reflection of X(125) in X(25563)
X(25564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 2935, 13289), (9934, 15051, 11202), (13289, 13293, 2935)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28506.
X(25565) lies on these lines: {2, 3098}, {5, 542}, {182, 3545}, {373, 12824}, {381, 5085}, {511, 547}, {524, 10109}, {576, 5056}, {599, 1351}, {1503, 11737}, {1506, 6034}, {1992, 5071}, {3090, 20423}, {3543, 17508}, {3589, 5066}, {3818, 19709}, {3845, 5092}, {3850, 20190}, {5079, 14848}, {5480, 15699}, {7570, 15360}, {7605, 9140}, {7875, 10033}, {9830, 15092}, {11539, 14810}, {18583, 20583}
X(25565) = midpoint of X(i) and X(j) for these {i,j}: {2, 19130}, {381, 10168}, {3589, 5066}, {3845, 5092}, {5476, 24206}
X(25565) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5055, 5476, 24206), (5071, 14561, 11178)
See Antreas Hatzipolakis and César Lozada, Hyacinthos 28506.
X(25566) lies on these lines: {6, 13}, {23, 5642}, {182, 10706}, {541, 10168}, {5972, 21766}, {6593, 11645}, {9970, 11178}
X(25566) = midpoint of X(i) and X(j) for these {i,j}: {182, 10706}, {381, 19140}, {5476, 5655}, {9970, 11178}
X(25566) = reflection of X(125) in X(25565)
Centers associated with the Gemini triangles 3-6: X(25567)-X(25580)
These centers were contributed by Randy Hutson, October 22, 2018. The Gemini triangles are introduced in the preamble just before X(24537).
X(25567) lies on these lines: {2, 3158}, {5423, 9458}, {8056, 15590}
X(25568) lies on these lines: {1, 2551}, {2, 210}, {4, 2900}, {7, 480}, {8, 12}, {9, 13405}, {10, 3487}, {11, 5748}, {20, 12678}, {31, 23693}, {43, 4000}, {55, 329}, {57, 6745}, {63, 5218}, {65, 7080}, {69, 7081}, {72, 3085}, {78, 388}, {100, 3474}, {120, 3699}, {142, 8580}, {144, 4640}, {145, 1837}, {165, 527}, {171, 4644}, {193, 3769}, {200, 226}, {278, 3190}, {306, 3974}, {346, 3967}, {348, 4447}, {355, 5761}, {377, 4420}, {443, 13407}, {495, 3940}, {497, 908}, {498, 5904}, {516, 3158}, {519, 3545}, {528, 9812}, {529, 5731}, {612, 5712}, {664, 2898}, {758, 5657}, {912, 14647}, {936, 21620}, {938, 1329}, {944, 6903}, {946, 6765}, {948, 2340}, {958, 5703}, {962, 3913}, {966, 4104}, {976, 5716}, {997, 1056}, {1001, 10578}, {1058, 21616}, {1125, 6762}, {1155, 9965}, {1215, 2345}, {1482, 7956}, {1698, 24391}, {1699, 5853}, {1706, 3671}, {1750, 3174}, {1754, 3939}, {1788, 3868}, {1818, 7365}, {1826, 4028}, {1836, 3689}, {1857, 1897}, {2077, 2096} et al
X(25568) = reflection of X(24477) in X(2)
The side-triangle of the Gemini triangles 3 and 5 is degenerate, collinear on line X(513)X(663) (the trilinear polar of X(57)).
X(25569) lies on these lines: {1, 659}, {214, 3126}, {513, 663}, {667, 4879}, {1635, 3722} et al
X(25570) lies on these lines: {1, 4452}, {7, 1740}, {8, 7184}, {9, 25571}, {43, 7240}, {69, 2234}, {87, 5222}, {144, 2664}, {1045, 3945}, {4489, 25572}, {20050, 25573} et al
X(25571) lies on these lines: {1, 4402}, {2, 1740}, {7, 2664}, {9, 25570}, {43, 3945}, {193, 899} {344, 2234}, {346, 16571}, {390, 21214}, {391, 7184}, {978, 4307}, {1045, 5308} et al
X(25572) lies on these lines: {1, 6646}, {8, 7240}, {43, 193}, {69, 87}, {192, 25573}, {524, 1740}, {3783, 20080}, {4489, 25570} et al
X(25573) lies on these lines: {1, 69}, {87, 6542}, {145, 7184}, {192, 25572}, {519, 16571}, {5839, 16569}, {20050, 25570} et al
X(25574) is the infinite point of the line of the (degenerate) cross-triangle of the Gemini triangles 5 and 6.
X(25574) lies on these lines: {30, 511}, {1120, 8851} et al
X(25574) = isogonal conjugate of X(25575)
X(25575) lies on the circumcircle and these lines: {727, 1149}, {1463, 8686}, {7292, 9082}
X(25575) = isogonal conjugate of X(25574)
Line X(3756)X(25574) is the line of the (degenerate) cross-triangle of the Gemini triangles 5 and 6.
X(25576) lies on these lines: {659, 3667}, {812, 4462}
X(25576) = isogonal conjugate of X(25577)
X(25576) = trilinear pole of line X(3756)X(25574)
X(25577) lies on these lines: {100, 649}, {101, 1293}, {739, 8054}, {812, 4554}, {874, 4598}, {1908, 3736}, {1979, 6377} et al
X(25577) = isogonal conjugate of X(25576)
X(25577) = trilinear pole, wrt 1st circumperp triangle, of line X(4)X(9)
X(25577) = crossdifference of every pair of points on line X(3756)X(25574)
Let A5B5C5 be the Gemini triangle 5. Let LA be the line through A5 parallel to BC, and define LB, LC cyclically. Let A'5 = LB∩LC, and define B'5, C'5 cyclically. Let A'3B'3C'3 be as at X(25430). Triangles A'3B'3C'3 and A'5B'5C'5 are homothetic at X(25578).
X(25578) lies on these lines: {1255, 2136}, {3666, 8056}, {4686, 6557}, {25579, 25580} et al
Let A6B6C6 be the Gemini triangle 6. Let MA be the line through A6 parallel to BC, and define MB, MC cyclically. Let A'6 = MB∩MC, and define B'6, C'6 cyclically. Let A'3B'3C'3 be as at X(25430). Triangles A'3B'3C'3 and A'6B'6C'6 are homothetic at X(25579).
X(25579) lies on these lines: {37, 57}, {958, 3646}, {5903, 25431}, {25578, 25580} et al
Let A'5B'5C'5 be as at X(25578) and A'6B'6C'6 be as at X(25579). Triangles A'5B'5C'5 and A'6B'6C'6 are homothetic at X(25580).
X(25580) lies on these lines: {57, 1122}, {88, 908}, {905, 2516}, {1054, 5537}, {25578, 25579} et al
Collineation mappings involving Gemini triangle 25: X(25581)-X(25605)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 25, as in centers X(25581)-X(25605). Then
m(X) = a(a + b + c)(a - b - c) x - b (a + b)(a - b + c) y - c (a + c)(a + b - c) z : :
(Clark Kimberling, October 23, 2018)
X(25581) lies on these lines: {2, 277}, {7, 6910}, {21, 7179}, {41, 25353}, {140, 26229}, {498, 26563}, {1621, 17181}, {3616, 5543}, {3665, 6690}, {3877, 17084}, {3926, 4968}, {3933, 26227}, {4189, 4911}, {4357, 27385}, {4414, 4920}, {5267, 7272}, {7763, 26234}, {10448, 24211}, {11374, 20347}, {20267, 25092}, {21285, 24929}, {25582, 25584}, {25587, 25597}, {25598, 25599}
X(25582) lies on these lines: {2, 17861}, {35, 27127}, {2175, 25348}, {17023, 17073}, {17298, 27006}, {20269, 24199}, {25581, 25584}, {25590, 25593}, {25600, 25601}
X(25583) lies on these lines: {2, 277}, {7, 474}, {10, 348}, {75, 3086}, {220, 25355}, {329, 16412}, {443, 7179}, {936, 9436}, {956, 17081}, {997, 6604}, {1266, 3624}, {1375, 5273}, {1376, 17170}, {1447, 17567}, {1565, 9709}, {1997, 3760}, {2550, 17181}, {2551, 5088}, {3421, 7176}, {3663, 11512}, {3665, 4413}, {3811, 14548}, {4644, 5277}, {4911, 6904}, {5905, 25946}, {6349, 19822}, {7223, 21031}, {8580, 17093}, {10436, 13411}, {10481, 20103}, {14007, 16705}, {17095, 19843}, {18698, 19854}, {24318, 26036}, {25591, 25605}
X(25584) lies on these lines: {2, 37}, {7232, 25878}, {17287, 26540}, {17362, 25964}, {24773, 25072}, {25581, 25582}, {25590, 25600}, {25593, 25601}
X(25585) lies on these lines: {2, 277}, {10, 7278}, {75, 5550}, {85, 19877}, {86, 4420}, {210, 17169}, {274, 3701}, {321, 24051}, {1441, 17095}, {1447, 17535}, {1930, 24589}, {3616, 17158}, {3634, 26563}, {4002, 21272}, {4751, 20435}, {4872, 26060}, {7200, 25614}, {7264, 19878}, {10527, 20895}, {16862, 26229}, {24790, 25089}, {25590, 25591}
X(25586) lies on these lines: {2, 21436}, {16586, 17866}
X(25587) lies on these lines: {2, 20890}, {25581, 25597}
X(25588) lies on these lines: {2, 21414}
X(25589) lies on these lines: {2, 2415}, {2324, 16831}, {7283, 16828}, {10447, 17776}, {16601, 20236}, {24603, 27514}, {25081, 26665}, {25581, 25582}
X(25590) lies on these lines: {1, 75}, {2, 2415}, {6, 4688}, {7, 10}, {8, 3664}, {9, 3739}, {37, 4659}, {40, 10442}, {57, 18229}, {63, 1781}, {69, 3679}, {85, 7271}, {141, 6173}, {142, 2345}, {165, 3980}, {190, 4751}, {192, 16673}, {239, 4772}, {269, 996}, {306, 19825}, {307, 5705}, {310, 18078}, {319, 4668}, {320, 17270}, {321, 17022}, {350, 25502}, {519, 3945}, {524, 4034}, {527, 966}, {536, 3247}, {594, 4675}, {894, 1743}, {903, 17249}, {936, 4328}, {997, 7190}, {1014, 8666}, {1086, 17303}, {1100, 17119}, {1122, 3698}, {1125, 3672}, {1211, 4654}, {1213, 17276}, {1215, 7274}, {1266, 3624}, {1268, 17250}, {1278, 16826}, {1449, 4361}, {1698, 4357}, {1699, 7996}, {2321, 4648}, {2999, 4359}, {3008, 5749}, {3175, 25430}, {3187, 26860}, {3241, 4909}, {3263, 5268}, {3550, 16992}, {3589, 10022}, {3596, 3761}, {3616, 4021}, {3617, 4896}, {3629, 4795}, {3632, 3879}, {3634, 4346}, {3635, 4460}, {3661, 17298}, {3662, 17308}, {3668, 19843}, {3686, 4644}, {3741, 10980}, {3753, 24471}, {3834, 17293}, {3928, 5737}, {3929, 19732}, {3950, 4461}, {3973, 17277}, {4000, 4470}, {4007, 4665}, {4050, 24656}, {4310, 19868}, {4398, 17322}, {4418, 4512}, {4419, 5257}, {4431, 17316}, {4440, 17248}, {4441, 26102}, {4445, 17376}, {4454, 5296}, {4472, 4657}, {4667, 5839}, {4677, 5564}, {4686, 16777}, {4698, 16676}, {4708, 17255}, {4718, 16672}, {4726, 17318}, {4735, 7241}, {4740, 17319}, {4798, 17045}, {4853, 20895}, {4873, 17243}, {4887, 9780}, {4902, 5224}, {5228, 5783}, {5249, 19822}, {5271, 16704}, {5272, 26234}, {7201, 21078}, {7227, 17279}, {7232, 17239}, {7988, 25385}, {7991, 10446}, {8582, 24213}, {8583, 24179}, {9578, 10401}, {9746, 24728}, {10468, 18193}, {12526, 20245}, {15479, 21233}, {16394, 16485}, {16469, 16825}, {16503, 20181}, {16549, 27626}, {16670, 17348}, {16684, 16688}, {16738, 18186}, {16779, 20172}, {16815, 17350}, {16816, 17120}, {16834, 17117}, {17050, 20258}, {17065, 24463}, {17067, 26039}, {17183, 20244}, {17229, 17313}, {17234, 17286}, {17235, 17327}, {17245, 17281}, {17251, 17345}, {17257, 24603}, {17260, 25728}, {17265, 17359}, {17275, 17365}, {17278, 17369}, {17280, 27147}, {17282, 17289}, {17290, 17385}, {17294, 17300}, {17299, 17392}, {17301, 17398}, {17323, 25498}, {17371, 27191}, {17861, 20880}, {18065, 18143}, {18134, 19797}, {19804, 23511}, {19808, 25527}, {19836, 24162}, {19853, 24214}, {19860, 24209}, {20347, 26037}, {20907, 21173}, {22464, 26363}, {24202, 24540}, {24554, 25078}, {24774, 25971}, {25068, 25887}, {25538, 26042}, {25582, 25593}, {25584, 25600}, {25585, 25591}, {26149, 27091}
X(25590) = anticomplement of X(3986)
X(25591) lies on these lines: {1, 996}, {2, 986}, {10, 3877}, {21, 4011}, {43, 3702}, {75, 27627}, {244, 25492}, {312, 1193}, {321, 978}, {404, 3923}, {474, 4418}, {982, 26094}, {984, 27261}, {995, 1089}, {1125, 3971}, {1201, 4385}, {1211, 27555}, {1215, 3616}, {1468, 27064}, {2345, 21809}, {2901, 5313}, {3061, 27040}, {3142, 20545}, {3175, 4719}, {3214, 4673}, {3710, 24239}, {3741, 3876}, {3782, 25914}, {3831, 3869}, {3840, 3868}, {3915, 7081}, {3924, 13741}, {3944, 4202}, {3976, 17165}, {3980, 17531}, {4054, 24178}, {4188, 24850}, {4197, 25385}, {4201, 17777}, {4255, 4387}, {4383, 27368}, {4422, 24953}, {4647, 17749}, {4695, 26029}, {4713, 16720}, {4968, 21214}, {5233, 20653}, {5293, 24552}, {5550, 24325}, {6651, 17684}, {9458, 9709}, {9780, 24003}, {10176, 10479}, {11375, 17279}, {12047, 25957}, {12609, 25961}, {16466, 17763}, {16817, 17125}, {16822, 17541}, {16916, 27954}, {16969, 21021}, {17674, 17889}, {19847, 24046}, {19867, 25378}, {21616, 25760}, {22220, 24349}, {24958, 25688}, {25583, 25605}, {25585, 25590}, {26971, 27680}
X(25592) lies on these lines: {2, 277}, {348, 10528}, {4190, 7179}, {10529, 17095}, {17181, 20075}
X(25593) lies on these lines: {2, 277}, {474, 27186}, {1375, 3218}, {3661, 27006}, {4904, 26639}, {5249, 25946}, {17896, 25603}, {24199, 24780}, {25582, 25590}, {25584, 25601}
X(25594) lies on these lines: {2, 647}, {86, 9404}, {665, 27293}, {1021, 16831}, {3700, 4687}, {6586, 25511}, {8651, 25299}, {11176, 24533}, {25603, 25604}
X(25595) lies on these lines: {2, 24783}
X(25596) lies on these lines: {2, 20890}, {25581, 25582}
X(25597) lies on these lines: {2, 21404}, {75, 4854}, {25581, 25587}
X(25598) lies on these lines: {1, 24995}, {2, 1930}, {10, 3905}, {31, 17211}, {32, 25345}, {86, 17104}, {626, 16974}, {1125, 24241}, {3263, 19846}, {3616, 21285}, {3772, 20888}, {3933, 17061}, {4920, 6679}, {7786, 16706}, {7803, 24036}, {16887, 26128}, {17321, 26363}, {25581, 25599}
X(25599) lies on these lines: {1, 21281}, {2, 3760}, {10, 16705}, {35, 86}, {39, 16818}, {42, 16887}, {69, 5312}, {75, 19863}, {213, 25349}, {348, 3339}, {350, 19864}, {609, 22267}, {1125, 27162}, {1266, 3624}, {1575, 25499}, {1655, 27324}, {1909, 16712}, {1930, 3666}, {3216, 4357}, {3616, 20244}, {3761, 4352}, {3821, 17211}, {3926, 19784}, {3931, 14210}, {5280, 16060}, {6381, 26030}, {6685, 24214}, {7031, 17379}, {17023, 20367}, {17302, 26959}, {17759, 27274}, {18600, 26115}, {20970, 24690}, {25581, 25598}
X(25600) lies on these lines: {2, 20881}, {1111, 27543}, {3161, 24773}, {17132, 24781}, {17278, 21826}, {25582, 25601}, {25584, 25590}
X(25601) lies on these lines: {2, 2415}, {190, 27254}, {239, 26059}, {344, 27305}, {527, 27267}, {3661, 27544}, {3875, 27317}, {4440, 25521}, {5223, 10528}, {16826, 27334}, {17257, 27039}, {25082, 26538}, {25582, 25600}, {25584, 25593}
X(25602) lies on these lines: {2, 24792}, {3738, 4389}
X(25603) lies on these lines: {2, 21438}, {650, 16706}, {3662, 22383}, {4657, 25511}, {4885, 19786}, {17370, 21611}, {17383, 26114}, {17896, 25593}, {25594, 25604}
X(25604) lies on these lines: {2, 2400}, {142, 652}, {693, 14838}, {4468, 21188}, {6589, 17278}, {23806, 25924}, {24793, 25084}, {25594, 25603}
X(25605) lies on these lines: {2, 24795}, {1565, 9458}, {3699, 17213}, {25583, 25591}
See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28514.
X(25606) lies on these lines: {9, 119}, {11, 8257}, {57, 5856}, {100, 12848}, {527, 1155}, {528, 18391}, {1317, 15185}, {1376, 5851}
X(25606) = midpoint of X(100) and X(12848)
X(25606) = reflection of X(11) in X(8257)
See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28514.
X(25607) lies on these lines: {99, 1046}, {114, 124}, {115, 8258}, {758, 11711}, {896, 1281}, {4697, 5988}, {5429, 7983}, {10026, 17770}
X(25607) = midpoint of X(99) and X(1046)
X(25607) = reflection of X(115) in X(8258)
See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28514.
X(25608) lies on these lines: {114, 6109}, {115, 8259}
X(25608) = reflection of X(115 ) in X(8259)
See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28514.
X(25609) lies on these lines: {114, 6108}, {115, 8260}
X(25609) = reflection of X(115 ) in X(8260)
Collineation mappings involving Gemini triangle 26: X(25610)-X(25638)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 26, as in centers X(25610)-X(25638). Then
m(X) = (a + b + c)(a b + a c - b c) x + (b + c)(a b - a c + b c) y + (b + c)(a c - a b + b c) z : :
(Clark Kimberling, October 23, 2018)
X(25610) lies on these lines: {2, 17448}, {10, 2176}, {43, 21025}, {44, 966}, {992, 17275}, {1213, 9711}, {1575, 26029}, {1698, 21384}, {3214, 17299}, {6048, 21024}, {17278, 27019}, {17279, 26752}, {17281, 27040}, {17301, 18135}, {25611, 25613}, {25616, 25632}
X(25611) lies on these lines: {2, 3728}, {10, 2209}, {672, 17303}, {899, 17279}, {1698, 21371}, {4438, 27102}, {9780, 26685}, {17289, 26037}, {17358, 26038}, {25610, 25613}, {25616, 25620}, {25617, 25619}, {25634, 25635}, {26030, 27032}
X(25612) lies on these lines: {2, 17448}, {10, 3208}, {1575, 26046}, {2345, 16814}, {5123, 6554}, {9780, 20331}, {21951, 27538}, {25623, 25624}
X(25613) lies on these lines: {2, 3056}, {9, 20486}, {71, 1836}, {518, 27267}, {966, 2348}, {2228, 17278}, {2550, 5766}, {2886, 22370}, {3208, 21927}, {3779, 17718}, {3781, 11375}, {4645, 26066}, {4648, 17728}, {20358, 25521}, {24478, 24789}, {24736, 25113}, {25125, 25623}, {25610, 25611}, {25624, 25634}
X(25614) lies on these lines: {2, 17448}, {10, 37}, {39, 3828}, {76, 4688}, {210, 22293}, {536, 27269}, {756, 21951}, {1107, 1698}, {1212, 5123}, {1573, 3634}, {1575, 9780}, {2275, 19877}, {3501, 16814}, {3739, 6376}, {3753, 21879}, {3983, 20693}, {3987, 21816}, {4002, 21888}, {4699, 20943}, {4735, 21700}, {5283, 19875}, {7200, 25585}, {10987, 16859}, {16602, 24603}, {16669, 17750}, {17237, 20255}, {17384, 27299}, {25121, 25141}, {25384, 25994}, {25624, 25625}
X(25615) lies on these lines: {2, 25110}, {244, 21893}, {2170, 24003}, {3740, 22230}
X(25616) lies on these lines: {2, 24525}, {2225, 26061}, {25610, 25632}, {25611, 25620}
X(25617) lies on these lines: {2, 24526}, {25611, 25619}, {25625, 25630}
X(25618) lies on these lines: {2, 18194}, {10, 21803}, {42, 21238}, {1698, 17349}, {2238, 21026}, {4365, 21858}, {21330, 25102}, {25610, 25611}, {25619, 25629}
X(25619) lies on these lines: {2, 24527}, {1698, 16468}, {25611, 25617}, {25618, 25629}
X(25620) lies on these lines: {2, 17448}, {25611, 25616}
X(25621) lies on these lines: {2, 23633}, {44, 21026}, {20979, 23301}, {25610, 25611}
X(25622) lies on these lines: {2, 24671}, {25610, 25611}
X(25623) lies on these lines: {2, 3728}, {10, 22370}, {344, 899}, {2345, 26037}, {4488, 7229}, {17280, 26038}, {25125, 25613}, {25612, 25624}, {27687, 27688}
X(25624) lies on these lines: {2, 18194}, {10, 192}, {87, 1698}, {1575, 26037}, {6376, 17157}, {9780, 26069}, {17155, 18133}, {17238, 20340}, {17786, 22174}, {25612, 25623}, {25613, 25634}, {25614, 25625}
X(25625) lies on these lines: {2, 24527}, {10, 75}, {17459, 21868}, {25614, 25624}, {25617, 25630}
X(25626) lies on these lines: {2, 24665}, {20979, 25628}
X(25627) lies on these lines: {2, 23655}, {10, 3835}, {38, 21197}, {649, 26037}, {661, 17072}, {693, 4147}, {756, 20909}, {1491, 20316}, {3907, 27674}, {4449, 27193}, {9780, 27345}, {17259, 23865}, {20979, 25299}, {20983, 21191}, {23301, 25125}, {24533, 25636}
X(25628) lies on these lines: {2, 24525}, {2183, 17303}, {20979, 25626}, {25610, 25611}
X(25629) lies on these lines: {2, 17448}, {6, 1698}, {1100, 3831}, {1575, 26030}, {2329, 5123}, {6685, 21025}, {17303, 26029}, {17357, 27020}, {25618, 25619}
X(25630) lies on these lines: {2, 24664}, {25617, 25625}
X(25631) lies on these lines: {2, 3056}, {7, 20486}, {10, 1423}, {144, 5880}, {672, 966}, {1376, 26671}, {2228, 4000}, {3598, 5232}, {3620, 24477}, {3925, 5224}, {4413, 17277}, {4429, 25978}, {25612, 25623}, {26046, 26077}
X(25632) lies on these lines: {2, 24670}, {25610, 25616}
X(25633) lies on these lines: {2, 25110}, {25634, 25638}
X(25634) lies on these lines: {2, 24671}, {25611, 25635}, {25613, 25624}, {25633, 25638}
X(25635) lies on these lines: {2, 18194}, {8, 21238}, {44, 966}, {25611, 25634}, {26042, 26069}
X(25636) lies on these lines: {2, 20983}, {8, 23506}, {10, 8640}, {513, 26049}, {649, 21051}, {659, 20316}, {1980, 5278}, {2517, 23951}, {2978, 27045}, {4083, 27293}, {18197, 21260}, {19732, 21005}, {20979, 23301}, {24533, 25627}, {25128, 25537}
X(25637) lies on these lines: {2, 23655}, {650, 21052}, {4147, 26114}, {4449, 25511}, {17072, 27527}, {17418, 24718}, {19744, 23865}, {20979, 23301}, {24755, 25126}
X(25638) lies on these lines: {2, 23656}, {659, 21053}, {5750, 23569}, {20979, 23301}, {21834, 27293}, {25633, 25634}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28519.
X(25639) lies on these lines: {1,2476}, {2,35}, {4,993}, {5,10}, {8,4867}, {9,6990}, {11,214}, {12,519}, {21,3583}, {30,4999}, {36,2475}, {40,6830}, {56,17532}, {58,14009}, {72,17605}, {75,7752}, {79,3218}, {100,7504}, {113,124}, {115,1107}, {116,17050}, {119,15863}, {141,9047}, {149,3746}, {165,6943}, {191,5057}, {226,3874}, {238,24880}, {325,20888}, {355,6980}, {377,499}, {381,958}, {386,17717}, {404,6681}, {405,10896}, {427,1900}, {443,10200}, {495,3244}, {496,551}, {497,6856}, {498,3434}, {515,6842}, {516,6831}, {528,6668}, {535,2975}, {567,9701}, {626,21264}, {758,6734}, {908,3678}, {936,6991}, {942,3838}, {956,10895}, {997,6829}, {1001,9669}, {1086,24167}, {1089,3006}, {1100,5949}, {1203,24883}, {1210,5883}, {1376,1656}, {1478,6871}, {1484,15178}, {1506,1575}, {1532,19925}, {1574,7603}, {1594,1861}, {1621,4857}, {1698,4193}, {1699,5705}, {1706,6975}, {1724,24892}, {1737,3754}, {2077,6952}, {2140,17046}, {2478,19854}, {2550,3090}, {2551,3545}, {2599,6358}, {2802,8068}, {3035,3628}, {3086,5177}, {3091,19843}, {3120,3670}, {3136,3454}, {3295,10197}, {3336,20292}, {3419,11375}, {3576,6937}, {3582,5253}, {3584,3871}, {3614,3626}, {3624,4197}, {3625,10592}, {3634,3925}, {3635,15888}, {3647,5745}, {3671,15844}, {3679,11280}, {3698,17619}, {3703,4066}, {3704,4717}, {3706,21081}, {3742,3824}, {3743,24210}, {3753,17606}, {3811,5219}, {3816,8728}, {3826,3847}, {3828,17533}, {3831,21241}, {3833,12446}, {3851,9708}, {3868,10129}, {3869,18393}, {3872,10827}, {3881,13407}, {3884,24987}, {3892,21620}, {3898,12053}, {3918,6702}, {3934,20541}, {4015,25006}, {4119,21067}, {4202,19864}, {4292,4973}, {4297,6907}, {4302,6910}, {4316,5303}, {4324,17549}, {4386,7746}, {4426,5475}, {4511,5443}, {4640,22793}, {4691,21031}, {4847,21077}, {4868,5530}, {4880,14450}, {5025,17030}, {5046,5251}, {5051,19863}, {5055,9709}, {5080,5258}, {5082,10588}, {5154,9780}, {5192,19846}, {5225,6857}, {5231,9612}, {5249,6701}, {5252,22837}, {5255,17734}, {5280,17737}, {5288,20060}, {5289,18493}, {5432,20104}, {5433,11112}, {5450,6923}, {5499,13624}, {5587,6941}, {5603,6874}, {5691,6932}, {5697,17057}, {5714,24477}, {5726,12629}, {5777,21635}, {5791,24703}, {5794,5886}, {5832,18232}, {5850,6067}, {5882,10943}, {6224,24926}, {6256,6982}, {6284,7483}, {6684,6882}, {6690,15171}, {6700,10171}, {6796,6863}, {6832,10598}, {6836,12511}, {6853,10902}, {6862,10525}, {6903,7688}, {6913,10893}, {6919,19855}, {6922,10164}, {6945,7989}, {7280,17579}, {7514,9712}, {7828,20179}, {7887,20172}, {7958,12447}, {8087,11013}, {8226,12571}, {9397,15280}, {9654,12513}, {9655,11194}, {9678,23261}, {9702,18350}, {9713,13861}, {9957,21630}, {10056,10585}, {10106,10957}, {10172,23513}, {10199,17528}, {10572,24541}, {10826,19860}, {10894,22770}, {11019,16193}, {11113,24953}, {11114,18514}, {11281,12433}, {12953,16370}, {13747,20107}, {15310,24251}, {15792,19642}, {16601,21090}, {16819,17669}, {17062,17761}, {17173,20961}, {17443,21018}, {17529,19878}, {17674,19847}, {17792,24206}, {17866,21207}, {17889,24046}, {19861,23708}, {23537,24239}
X(25639) = midpoint of X(i) and X(j) for these {i,j}: {1,5086}, {8,11009}, {12,24390}, {2975,3585}, {5288,20060}, {6734,12047}
X(25639) = reflection of X(i) in X(j) for these {i,j}: {2646,1125}, {5267,4999}, {14526,6701}
complement of X(35)
X(25639) = complementary conjugate of X(3647)
X(25639) = isotomic conjugate of isogonal conjugate of X(20961)
X(25639) = polar conjugate of isogonal conjugate of X(22058)
X(25639) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,2476,3822}, {1,11680,24387}, {2,1479,5248}, {2,7741,3825}, {5,10,3814}, {5,2886,10}, {5,7681,3817}, {8,5141,7951}, {10,946,3878}, {10,3817,21616}, {10,11813,960}, {10,21616,10176}, {11,442,1125}, {226,10916,3874}, {443,10589,10200}, {495,3813,3244}, {497,6856,10198}, {498,3434,8715}, {942,3838,11263}, {960,9955,11813}, {1125,17647,214}, {1210,12609,5883}, {1478,10527,8666}, {1699,5705,12514}, {1699,6828,12558}, {2476,11680,1}, {2975,17577,3585}, {3419,11375,22836}, {3434,6933,498}, {3816,8728,19862}, {3820,9710,10}, {3822,24387,1}, {3825,3841,2}, {3826,3847,17527}, {3918,6702,24982}, {3925,4187,3634}, {3925,7173,4187}, {5836,9956,10}, {6871,10527,1478}, {8728,10593,3816}, {17530,24390,12}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28519.
X(25640) lies on the nine-point circle and these lines: {2,1295}, {3,6717}, {4,11}, {5,123}, {12,3318}, {113,2850}, {114,2798}, {115,1865}, {116,2823}, {117,2849}, {118,2812}, {119,2804}, {120,9521}, {121,9525}, {122,442}, {124,946}, {125,429}, {126,9531}, {133,2845}, {136,431}, {225,20620}, {235,5521}, {281,5514}, {381,10746}, {403,5520}, {406,11496}, {515,11719}, {1465,1532}, {1519,1875}, {1528,6001}, {1537,1845}, {1596,2834}, {1878,3259}, {1881,5190}, {2840,5510}, {2851,5512}, {3545,10715}, {4242,24466}, {5603,10702}, {5886,11733}, {6907,17073}, {7952,10271}, {10763,14853}, {15908,17555}
X(25640) = midpoint of X(4) and X(108)
X(25640) = reflection of X(i) in X(j) for these {i,j}: {3,6717}, {123,5}
X(25640) = isogonal conjugate of X(15405)
X(25640) = complement of X(1295)
X(25640) = complementary conjugate of X(6001)
X(25640) = perspector of circumconic centered at X(14571)
X(25640) = homothetic center of cyclic quadrilateral ABCX(108) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time
X(25640) = anticenter of cyclic quadrilateral ABCX(108)
X(25640) = Kirikami six circles image of X(108)
X(25640) = center of hyperbola {{A,B,C,X(4),X(108)}}
X(25640) = X(108)-of-Euler-triangle
X(25640) = X(2766)-of-(reflection of Euler triangle in Euler line)
X(25640) = antipode in nine-point circle of X(123)
X(25640) = X(2)-Ceva conjugate of X(14571)
X(25640) = inverse-in-polar-circle of X(104)
X(25640) = orthoptic-circle-of-Steiner-inellipse inverse of X(9107)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28519.
For another construction: See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29110.
X(25641) lies on the nine-point circle and these lines: {2,477}, {3,16177}, {4,476}, {5,3258}, {20,14989}, {30,125}, {113,523}, {115,3003}, {122,2072}, {127,10297}, {135,10151}, {136,403}, {137,11563}, {235,16178}, {381,2453}, {1522,1523}, {3091,14731}, {3154,23515}, {3448,5627}, {5446,16978}, {5520,6841}, {5972,14934}, {7471,15468}, {7728,14993}, {9179,23699}, {10024,12091}, {11251,18809}, {12079,16003}, {14611,16534}, {14644,17511}, {14915,16280}, {16188,18556}, {16340,20304}, {18402,23290}, {18403,20625}
X(25641) = midpoint of X(i) and X(j) for these {i,j}: {4,476}, {5,18319}, {20,14989}, {1553,6070}
X(25641) = reflection of X(i) in X(j) for these {i,j}: {3,22104}, {3258,5}, {10113,21316}, {14611,16534}, {14934,5972}, {16003,12079}, {16340,20304}, {16978,5446}
X(25641) = isogonal conjugate of X(15396)
X(25641) = complement of X(477)
X(25641) = complementary conjugate of X(5663)
X(25641) = polar-circle-inverse of X(32710)
X(25641) = reflection of X(32110) in the orthic axis
X(25641) = perspector of circumconic centered at X(3018)
X(25641) = anticenter of cyclic quadrilateral ABCX(476)
X(25641) = Kirikami six circles image of X(476)
X(25641) = center of hyperbola {{A,B,C,X(4),X(476)}}
X(25641) = X(476)-of-Euler-triangle
X(25641) = X(110)-of-(reflection of Euler triangle in Euler line)
X(25641) = reflection of X(113) in Euler line
X(25641) = antipode in nine-point circle of X(3258)
X(25641) = X(2)-Ceva conjugate of X(3018)
X(25641) = orthoptic-circle-of-Steiner-inellipse inverse of X(9060)
X(25641) = homothetic center of cyclic quadrilateral ABCX(476) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28519.
X(25642) lies on the nine-point circle and these lines: {4,675}, {5,5513}, {12,6025}, {117,5805}, {118,381}, {120,6881}, {132,15762}, {1596,20622}
X(25642) = midpoint of X(4) and X(675)
X(25642) = reflection of X(5513) in X(5)
X(25642) = homothetic center of cyclic quadrilateral ABCX(675) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time
X(25642) = anticenter of cyclic quadrilateral ABCX(675)
X(25642) = Kirikami six circles image of X(675)
X(25642) = center of hyperbola {{A,B,C,X(4),X(675)}}
X(25642) = X(675)-of-Euler-triangle
X(25642) = antipode in nine-point circle of X(5513)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28520.
X(25643) lies on this line: {1319,5901}
See Peter Moses, Hyacinthos 28525.
X(25644) lies on these lines: {3,2799}, {22,1637}, {32,2507}, {523,15646}, {804,12042}, {3268,15246}, {6636,9979}, {7485,14417}, {7630,8673}, {7663,9175}
Collineation mappings involving Gemini triangle 27: X(25645)-X(25689)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 27, as in centers X(25645)-X(25689). Then
m(X) = (a + b) (a + c)(a - b - c) x + b (b + c) (a + b) y + c (b + c) (a + c) z : :
The point m(X) is on the Euler line if and only if the point X is on the Euler line. (Clark Kimberling, October 24, 2018)
X(25645) lies on these lines: {1, 2}, {3, 23860}, {21, 3454}, {35, 2887}, {37, 24935}, {58, 3936}, {79, 4892}, {81, 6693}, {141, 5135}, {229, 404}, {321, 24160}, {540, 16948}, {1089, 17719}, {1203, 6679}, {1211, 6675}, {1213, 4273}, {1329, 25648}, {1724, 4417}, {1770, 4138}, {2345, 24937}, {2886, 25444}, {3035, 24925}, {3416, 25471}, {3430, 8229}, {3452, 24933}, {3647, 4683}, {3813, 25448}, {3846, 5259}, {3878, 16110}, {4193, 6740}, {4202, 4256}, {4262, 26085}, {4418, 11263}, {4438, 5904}, {4647, 24161}, {4653, 5051}, {4851, 25445}, {4999, 24934}, {5145, 27252}, {5248, 25760}, {5433, 24938}, {5438, 25443}, {5718, 17698}, {5956, 16610}, {7288, 24912}, {7945, 17232}, {9571, 27629}, {15674, 26064}, {16862, 17265}, {17056, 25526}, {17231, 24890}, {17279, 25649}, {17567, 24884}, {17737, 21070}, {17740, 24159}, {18417, 22076}, {20337, 24271}, {25079, 25655}, {25440, 25957}, {25646, 25680}, {25660, 25662}, {25670, 25679}
X(25645) = complement of X(24883)
X(25646) lies on these lines: {2, 3}, {17606, 24880}, {24931, 25917}, {25645, 25680}, {25649, 25656}
X(25647) lies on these lines: {2, 3}, {1940, 18588}, {2839, 11814}, {3061, 18669}, {4292, 24884}, {5316, 24931}, {18743, 25650}, {25656, 25683}
X(25648) lies on these lines: {2, 3}, {11, 24880}, {495, 24936}, {496, 24883}, {1329, 25645}, {2842, 11814}, {3816, 25441}, {3847, 25448}, {6284, 24885}, {7741, 24902}, {10176, 25079}, {10713, 21031}, {17757, 25650}, {21239, 25447}, {24390, 25446}, {25657, 25683}
X(25649) lies on these lines: {2, 6}, {42, 24941}, {284, 21245}, {3285, 21287}, {3739, 24937}, {3771, 25655}, {3834, 24884}, {4357, 25455}, {4643, 25447}, {8818, 24271}, {17243, 25650}, {17279, 25645}, {17348, 24880}, {17385, 24931}, {25646, 25656}, {25660, 25665}
X(25650) lies on these lines: {1, 2}, {3, 18134}, {21, 1330}, {35, 4645}, {58, 17778}, {69, 261}, {141, 19270}, {226, 7283}, {312, 11374}, {320, 3916}, {333, 6675}, {344, 24933}, {345, 3487}, {350, 17866}, {404, 18139}, {405, 4417}, {442, 1043}, {474, 17234}, {631, 18141}, {740, 24161}, {1010, 17056}, {1211, 11110}, {1434, 6390}, {2321, 24937}, {2891, 24953}, {2893, 25516}, {2895, 15674}, {2901, 24160}, {3210, 24159}, {3454, 4653}, {3523, 4869}, {3578, 15671}, {3648, 17491}, {3649, 3712}, {3685, 12047}, {3695, 5719}, {3704, 11281}, {3933, 14828}, {4188, 26141}, {4281, 27319}, {4385, 17718}, {4388, 5248}, {4427, 14450}, {4647, 26725}, {4658, 6693}, {4851, 25454}, {4892, 24851}, {4966, 4999}, {4975, 5443}, {5047, 5741}, {5208, 10974}, {5233, 11108}, {5358, 25495}, {5440, 25443}, {5718, 13740}, {6757, 14844}, {7270, 24929}, {7377, 19768}, {7483, 14829}, {7836, 17300}, {11115, 26131}, {12607, 21290}, {13571, 17379}, {14210, 17084}, {14555, 16845}, {15803, 17298}, {16062, 19765}, {16777, 24935}, {17243, 25649}, {17296, 25447}, {17311, 24890}, {17390, 25445}, {17588, 26064}, {17757, 25648}, {18743, 25647}, {24364, 25468}, {24390, 25444}, {24520, 27680}, {25079, 25688}, {25526, 26109}, {25660, 25677}, {25661, 25685}
X(25651) lies on these lines: {1, 25647}, {2, 7}, {6, 8287}, {44, 24890}, {284, 857}, {386, 6198}, {1375, 1901}, {2287, 17052}, {3019, 3090}, {3589, 4187}, {4000, 24224}, {4292, 24882}, {5091, 5954}, {7110, 16609}, {8680, 24780}, {9612, 25443}, {13411, 24933}, {17279, 25645}, {17357, 24935}, {24773, 27471}
X(25652) lies on these lines: {2, 11}, {1329, 25645}, {3813, 24880}, {4187, 25669}, {11814, 24959}, {12607, 21290}, {17056, 18169}, {24390, 24902}
X(25653) lies on these lines: {2, 3}
X(25654) lies on these lines: {2, 3}, {523, 25686}
X(25655) lies on these lines: {2, 31}, {213, 24942}, {3771, 25649}, {25079, 25645}
X(25656) lies on these lines: {2, 32}, {25646, 25649}, {25647, 25683}, {25661, 25678}
X(25657) lies on these lines: {2, 39}, {17279, 25669}, {25646, 25649}, {25648, 25683}
X(25658) lies on these lines: {2, 44}, {8062, 25666}, {17279, 25645}
X(25659) lies on these lines: {2, 45}, {3589, 13741}, {4193, 17259}, {6687, 24880}, {17279, 25645}, {17351, 24884}, {17353, 25445}, {24890, 26685}
X(25660) lies on these lines: {2, 37}, {6, 645}, {76, 15668}, {86, 3770}, {141, 18140}, {142, 25467}, {274, 6707}, {313, 16826}, {314, 1213}, {668, 17390}, {726, 24923}, {1100, 3975}, {1230, 5333}, {1655, 16696}, {2092, 27111}, {3264, 17319}, {3596, 16777}, {3760, 25458}, {3765, 17394}, {3912, 21076}, {4494, 16673}, {4648, 18135}, {4851, 6376}, {4916, 25278}, {8818, 15455}, {13476, 17794}, {16589, 27164}, {17034, 21788}, {17244, 18044}, {17300, 18133}, {17313, 18146}, {17372, 25280}, {17499, 18166}, {17669, 21245}, {20174, 24603}, {21858, 26048}, {24067, 24195}, {24897, 25264}, {24945, 26959}, {25645, 25662}, {25649, 25665}, {25650, 25677}, {26109, 27792}
X(25661) lies on these lines: {1, 25660}, {2, 39}, {75, 21816}, {313, 24944}, {1213, 10471}, {3061, 14210}, {3294, 17790}, {3770, 17175}, {3975, 20963}, {17762, 18137}, {24530, 25264}, {25650, 25685}, {25656, 25678}
X(25662) lies on these lines: {2, 6}, {3662, 25469}, {17227, 24922}, {17279, 25687}, {17280, 24957}, {25645, 25660}, {25668, 25669}, {25671, 25679}, {25675, 25682}
X(25663) lies on these lines: {1, 2}, {631, 17232}, {1654, 6675}, {3936, 16948}, {3945, 7796}, {7881, 17300}, {11374, 17280}, {16672, 24935}, {24375, 25468}
X(25664) lies on these lines: {1, 2}, {643, 1754}, {2328, 3936}, {6690, 25447}, {21075, 24933}
X(25665) lies on these lines: {1, 2}, {940, 7881}, {5337, 18134}, {24378, 25468}, {25649, 25660}
X(25666) lies on these lines: {2, 661}, {513, 6687}, {514, 4521}, {649, 4763}, {650, 812}, {663, 24749}, {693, 4893}, {788, 25142}, {824, 3239}, {918, 21212}, {1125, 4160}, {1491, 3716}, {1577, 25667}, {1635, 20295}, {1639, 3004}, {1698, 4761}, {2488, 10006}, {2516, 6008}, {2786, 14321}, {3005, 24003}, {3589, 9013}, {3700, 21196}, {3709, 24782}, {3776, 4468}, {3805, 6682}, {3907, 21051}, {4010, 4913}, {4025, 27484}, {4077, 5219}, {4120, 4467}, {4129, 6002}, {4367, 24747}, {4383, 7252}, {4394, 4785}, {4455, 24533}, {4728, 17494}, {4789, 4988}, {4806, 9508}, {4830, 24719}, {4833, 17259}, {4879, 4964}, {4931, 17161}, {4979, 27013}, {5745, 8774}, {6679, 9256}, {7180, 25084}, {7234, 25537}, {8062, 25658}, {8678, 24675}, {10278, 12072}, {16058, 23864}, {17072, 25143}, {20908, 21611}, {21104, 21204}, {21297, 26777}, {27014, 27293}
X(25667) lies on these lines: {2, 650}, {75, 21196}, {312, 4024}, {321, 17161}, {661, 18155}, {798, 4369}, {824, 21611}, {1577, 25666}, {3004, 20950}, {3798, 4406}, {3835, 20954}, {4374, 17069}, {4500, 18137}, {4940, 23794}, {4978, 21385}, {8062, 27400}, {10449, 14077}, {16751, 26983}
X(25668) lies on these lines: {2, 31}, {4974, 24899}, {17279, 25645}, {17371, 24931}, {25662, 25669}
X(25669) lies on these lines: {1, 2}, {9, 24935}, {140, 24938}, {141, 25454}, {3936, 6693}, {4187, 25652}, {17279, 25657}, {17296, 25445}, {17306, 25455}, {24384, 25468}, {24390, 25448}, {25662, 25668}
X(25670) lies on these lines: {2, 3}, {25645, 25679}
X(25671) lies on these lines: {2, 3}, {25079, 25645}, {25662, 25679}
X(25672) lies on these lines: {2, 3}
X(25673) lies on these lines: {2, 3}
X(25674) lies on these lines: {2, 3}
X(25675) lies on these lines: {2, 3}, {25662, 25682}
X(25676) lies on these lines: {2, 3}
X(25677) lies on these lines: {2, 3}, {17606, 25446}, {25079, 25120}, {25650, 25660}
X(25678) lies on these lines: {2, 3}, {25656, 25661}
X(25679) lies on these lines: {2, 7}, {2478, 27389}, {2893, 26019}, {4193, 27381}, {17183, 27524}, {25645, 25670}, {25650, 25660}, {25662, 25671}
X(25680) lies on these lines: {2, 7}, {1329, 25441}, {25645, 25646}
X(25681) lies on these lines: {1, 1329}, {2, 65}, {3, 12608}, {4, 5087}, {5, 997}, {8, 1392}, {9, 583}, {10, 1482}, {11, 78}, {12, 19861}, {21, 4679}, {40, 1537}, {46, 13747}, {55, 27385}, {56, 908}, {57, 6691}, {63, 5433}, {72, 499}, {100, 12701}, {123, 1062}, {140, 12514}, {200, 3813}, {210, 10527}, {214, 10742}, {226, 25524}, {329, 7288}, {355, 3814}, {377, 17605}, {381, 17647}, {388, 5748}, {392, 498}, {404, 1836}, {405, 22766}, {443, 3838}, {474, 5880}, {496, 3811}, {497, 26129}, {517, 6959}, {518, 3086}, {519, 11373}, {528, 9614}, {529, 1420}, {551, 5795}, {631, 4640}, {912, 26492}, {936, 2886}, {942, 10200}, {946, 1376}, {958, 999}, {976, 17721}, {978, 3772}, {988, 4415}, {1001, 13411}, {1086, 11512}, {1155, 6921}, {1193, 17720}, {1210, 12635}, {1319, 3436}, {1478, 17614}, {1479, 5440}, {1519, 10310}, {1698, 5443}, {1699, 5438}, {1706, 11522}, {1737, 5730}, {1770, 16371}, {1837, 4193}, {2098, 6735}, {2099, 24982}, {2264, 27395}, {2478, 2646}, {2551, 3616}, {2975, 27131}, {3036, 3632}, {3057, 5552}, {3185, 19513}, {3189, 5274}, {3305, 24953}, {3306, 3649}, {3419, 7741}, {3421, 11260}, {3475, 18227}, {3476, 24558}, {3486, 6919}, {3487, 3742}, {3523, 5698}, {3555, 10072}, {3556, 16434}, {3582, 5904}, {3612, 11113}, {3622, 8165}, {3634, 5837}, {3671, 6692}, {3683, 6910}, {3740, 19843}, {3820, 5901}, {3825, 5722}, {3847, 9581}, {3868, 17728}, {3872, 21031}, {3874, 10199}, {3877, 25414}, {3878, 25413}, {3880, 7080}, {3899, 5445}, {3913, 6745}, {3916, 10052}, {3940, 10916}, {4188, 5057}, {4255, 24210}, {4267, 17182}, {4295, 17567}, {4420, 4863}, {4421, 10624}, {4643, 24317}, {4657, 21246}, {4851, 21244}, {4855, 6284}, {5044, 11230}, {5083, 12059}, {5086, 5154}, {5121, 17054}, {5219, 8583}, {5250, 5432}, {5252, 11681}, {5253, 10404}, {5302, 5550}, {5554, 11011}, {5603, 5836}, {5703, 26105}, {5708, 5745}, {5741, 10371}, {5761, 13374}, {5784, 6837}, {5791, 10176}, {5887, 6958}, {5919, 10528}, {6001, 6891}, {6259, 21635}, {6261, 6922}, {6713, 24467}, {6736, 10912}, {6743, 24386}, {6831, 15842}, {6834, 14110}, {6857, 15254}, {6890, 12688}, {6909, 12679}, {6926, 9943}, {6944, 7686}, {6963, 21740}, {7171, 18243}, {7483, 17700}, {7515, 16596}, {7982, 8256}, {8168, 21627}, {8169, 13405}, {8580, 9710}, {9335, 26729}, {9564, 19858}, {9623, 9624}, {9708, 12001}, {9709, 18493}, {10085, 13257}, {10106, 11236}, {10165, 12572}, {10171, 12447}, {10393, 14022}, {10401, 24540}, {10572, 17556}, {10573, 17619}, {10584, 12649}, {10586, 17609}, {10785, 14872}, {10826, 17533}, {11194, 12527}, {11813, 12699}, {12513, 21075}, {12609, 16408}, {14986, 25568}, {15507, 19514}, {15803, 17768}, {15950, 19860}, {16579, 27784}, {17278, 24161}, {17279, 25079}, {17572, 20292}, {17597, 28018}, {17719, 21214}, {17724, 28011}, {17783, 28027}, {18249, 19878}, {18253, 26725}, {18967, 24541}, {23708, 24390}, {24789, 27627}, {25527, 25914}, {25645, 25646}, {25650, 25660}, {25912, 26132}
X(25681) = complement of X(1788)
X(25682) lies on these lines: {2, 85}, {9, 25457}, {1441, 24952}, {25650, 25660}, {25662, 25675}
X(25683) lies on these lines: {2, 99}, {5, 120}, {116, 4465}, {754, 26145}, {2789, 11814}, {5741, 13466}, {7861, 17674}, {24070, 24185}, {25647, 25656}, {25648, 25657}
X(25684) lies on these lines: {2, 650}, {1635, 24921}, {2485, 24782}, {4411, 24793}, {5271, 21721}, {8062, 25658}, {19792, 21438}
X(25685) lies on these lines: {2, 668}, {115, 7257}, {620, 799}, {4033, 24957}, {5209, 10026}, {17786, 25687}, {25650, 25661}
X(25686) lies on these lines: {2, 669}, {523, 25654}, {659, 27293}, {1491, 8062}
X(25687) lies on these lines: {2, 7}, {75, 24957}, {257, 7110}, {284, 17669}, {645, 27688}, {965, 7887}, {992, 7828}, {1901, 17694}, {17276, 25469}, {17279, 25662}, {17786, 25685}, {25649, 25660}
X(25688) lies on these lines: {1, 25660}, {2, 38}, {76, 2669}, {386, 3596}, {662, 24267}, {726, 24530}, {1125, 24944}, {1329, 10479}, {2607, 3923}, {3736, 3948}, {3770, 18792}, {17279, 25645}, {17289, 24931}, {17303, 21810}, {24161, 24746}, {24520, 26107}, {24958, 25591}, {25079, 25650}
X(25689) lies on these lines: {1, 2}, {24530, 24653}
Collineation mappings involving Gemini triangle 28: X(25690)-X(25710)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 28, as in centers X(25690)-X(25710). Then
m(X) = (a - b - c) (2 a + 2 b - c) (2 a - b + 2 c) x - (a + c)(a - 2 b - 2 c)(2 a + 2 b - c) y : (a + b)(a - 2 b - 2 c)(2 a + 2 c - b) z:
The point m(X) is on the Euler line if and only if the point X is on the Euler line. (Clark Kimberling, October 25, 2018)
X(25690) lies on these lines: {1, 2}, {9, 25693}, {1211, 25395}, {4363, 4867}, {5289, 25691}, {25699, 25702}
X(25691) lies on these lines: {2, 3}, {5289, 25690}
X(25692) lies on these lines: {2, 3}
X(25693) lies on these lines: {2, 3}, {9, 25690}
X(25694) lies on these lines: {2, 6}, {2163, 4715}, {4363, 4867}
X(25695) lies on these lines: {2, 3}
X(25696) lies on these lines: {2, 3}
X(25697) lies on these lines: {1, 2}, {80, 21251}, {993, 23170}
X(25698) lies on these lines: {1, 2}, {3899, 4432}, {3919, 24344}, {4363, 5425}
X(25699) lies on these lines: {2, 6}, {25690, 25702}
X(25700) lies on these lines: {2, 6}, {274, 25392}
X(25701) lies on these lines: {2, 3}
X(25702) lies on these lines: {2, 3}, {25690, 25699}
X(25703) lies on these lines: {2, 3}
X(25704) lies on these lines: {2, 3}
X(25705) lies on these lines: {2, 3}
X(25706) lies on these lines: {2, 3}
X(25707) lies on these lines: {2, 3}
X(25708) lies on these lines: {2, 3}
X(25709) lies on these lines: {2, 3}
X(25710) lies on these lines: {2, 7}, {5289, 25690}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28528.
X(25711) lies on these lines: {3,1177}, {4,12824}, {5,113}, {24,110}, {26,16165}, {30,16104}, {52,5095}, {74,7503}, {146,17854}, {186,15136}, {265,11746}, {389,542}, {399,7506}, {468,13148}, {568,2854}, {973,12236}, {1092,1511}, {1112,3575}, {1154,3292}, {1539,18377}, {1625,2493}, {1995,5890}, {2777,12605}, {2979,15020}, {3448,7544}, {3548,6293}, {5462,12099}, {5504,19504}, {5562,5642}, {5609,6102}, {5878,7728}, {5892,20397}, {5946,18474}, {5972,7542}, {6000,10297}, {6241,10706}, {6639,7723}, {6644,15132}, {7505,7722}, {7547,11439}, {7731,15035}, {9140,15043}, {9517,16230}, {9729,20417}, {10020,10272}, {10574,15054}, {10620,15151}, {11412,15034}, {11438,19140}, {11451,15025}, {11470,13352}, {11560,22970}, {11597,12606}, {12133,23047}, {12302,15472}, {12359,12827}, {12412,13198}, {13201,15051}, {13417,16163}, {14644,15102}, {14787,20126}, {14982,18917}, {15021,20791}, {15059,15100}, {16776,18440}, {21649,24981}
X(25711) = midpoint of X(i) and X(j) for these {i,j}: {110,1986}, {113,11562}, {185,15063}, {5562,14448}, {5609,6102}, {12270,12292}, {13417,16163}, {21649,24981}
X(25711) = reflection of X(i) in X(j) for these {i,j}: {125,9826}, {265,11746}, {974,14708}, {1112,11557}, {12358,5972}, {14708,11561}, {15738,5}, {16003,16270}, {20379,12006}, {20417,9729}
X(25711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125,16223,9826),(265,16222,11746),(5609,20772,10539),(5642,14448,5562),{9730,16003,16270),(9970,15462,15141)
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28528.
X(25712) lies on these lines: {3,3620}, {4,9820}, {110,3529}, {186,9937}, {376,1216}, {631,1209}, {3090,3431}, {3147,12383}, {3524,14516}, {3545,12278}, {6225,16163}, {6353,12118}, {7487,21850}, {8254,18420}, {8889,12038}, {8907,21844}, {9729,11179}, {10299,11442}, {11441,17538}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28528.
X(25713) lies on these lines: {1209,6905}, {1385,1798}, {6906,10575}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28528.
X(25714) lies on these lines: {3,2888}, {23,1154}, {24,9925}, {26,9143}, {52,110}, {54,575}, {186,539}, {195,9716}, {399,17714}, {511,12380}, {542,7512}, {576,7730}, {631,19468}, {973,6593}, {1092,12291}, {1147,12280}, {1173,11800}, {1209,7550}, {1493,11597}, {1511,10821}, {2917,2930}, {2931,6193}, {3519,7488}, {3628,11804}, {5642,12242}, {6152,14984}, {6288,7527}, {7464,18400}, {7545,20424}, {7691,10575}, {9545,13472}, {9920,12082}, {9934,10628}, {10272,18369}, {10594,12310}, {11803,13621}, {12226,15132}, {12273,15083}, {12899,18951}, {14865,17702}, {15133,23330}, {15246,20379}
X(25714) = midpoint of X(3519) and X(23236)
X(25714) = reflection of X(i) in X(j) for these {i,j}: {2914,110}, {15089,1511}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28528.
X(25715) lies on these lines: {3,18124}, {4,9932}, {22,7689}, {52,19141}, {578,9813}, {1209,7503}, {12293,23330}
Collineation mappings involving Gemini triangle 29: X(25716)-X(25726)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 29, as in centers X(25716)-X(25726). Then
m(X) = a(3 a^2 - b^2 - c^2 - 2 a b - 2 a c + 2 b c) x + (c - a) (a^2 - 3 b^2 + c^2 - 2 a b - 2 a c + 2 b c) y + (b - a) (a^2 - 3 c^2 + b^2 - 2 a c - 2 a b + 2 b c) z : :
(Clark Kimberling, October 25, 2018)
X(25716) lies on these lines: {1, 85}, {2, 25718}, {7, 3623}, {57, 4393}, {77, 3875}, {145, 3160}, {192, 1419}, {241, 16834}, {269, 4360}, {279, 3241}, {307, 20013}, {347, 3879}, {348, 519}, {934, 3871}, {1025, 3208}, {1323, 3244}, {1420, 3212}, {1434, 18421}, {1442, 10436}, {2099, 4955}, {3188, 11520}, {3243, 14189}, {3340, 7176}, {3476, 3674}, {3598, 6049}, {3635, 10481}, {3679, 17095}, {3889, 23839}, {3895, 7177}, {4328, 17393}, {4554, 24524}, {4848, 17081}, {4855, 21272}, {5088, 7982}, {5119, 7183}, {5881, 17181}, {5882, 17170}, {6063, 25303}, {6516, 8715}, {6610, 17318}, {7223, 11011}, {9578, 17084}, {16200, 17753}, {17086, 17296}
X(25717) lies on these lines: {25722, 25726}
X(25718) lies on these lines: {2, 25716}, {7, 145}, {8, 348}, {77, 4853}, {85, 3241}, {239, 5435}, {269, 11519}, {279, 519}, {306, 18624}, {347, 6737}, {651, 4513}, {934, 5687}, {1323, 3632}, {1419, 4461}, {1442, 19860}, {1447, 6049}, {1943, 5273}, {3177, 4552}, {3212, 4308}, {3339, 7268}, {3476, 7198}, {3555, 23839}, {3621, 9436}, {3633, 10481}, {4050, 6168}, {4350, 12629}, {4554, 25278}, {5226, 17316}, {5838, 9311}, {6604, 20050}
X(25719) lies on these lines: {2, 25716}, {7, 3621}, {8, 279}, {10, 664}, {75, 18811}, {77, 4967}, {85, 519}, {145, 5543}, {200, 1996}, {226, 6542}, {319, 3668}, {347, 17270}, {348, 3679}, {524, 14564}, {527, 20089}, {948, 17294}, {1323, 3626}, {1418, 4399}, {1441, 3879}, {3160, 3617}, {3212, 10106}, {3625, 10481}, {3632, 6604}, {3674, 5252}, {3911, 16816}, {4431, 6180}, {4464, 7190}, {4554, 25280}, {4665, 6610}, {4669, 17078}, {4677, 17079}, {4685, 7196}, {4816, 21314}, {4848, 7176}, {5088, 11362}, {5881, 17170}, {6063, 24524}, {6737, 16284}, {14189, 24393}, {17353, 26653}, {23058, 26658}
X(25720) lies on these lines: {2, 25716}, {42, 664}, {3160, 20012}, {4554, 25286}, {9312, 17018}, {9436, 20011}
X(25721) lies on these lines: {2, 25716}, {42, 9312}, {43, 664}, {348, 4685}, {1419, 17759}, {4554, 25287}, {9436, 20012}
X(25722) lies on these lines: {1, 9859}, {2, 14100}, {7, 3434}, {8, 971}, {9, 100}, {63, 2942}, {78, 11372}, {142, 7671}, {144, 3059}, {145, 8581}, {200, 3062}, {390, 3890}, {404, 15299}, {480, 16112}, {516, 3869}, {518, 1278}, {1445, 9352}, {1621, 4326}, {1742, 24635}, {2293, 24341}, {2310, 26669}, {2389, 23839}, {2550, 5086}, {2801, 12531}, {2975, 5732}, {3174, 8545}, {3419, 9803}, {3868, 4312}, {3870, 10865}, {3871, 15298}, {3877, 9963}, {4012, 5942}, {4208, 12710}, {4313, 18251}, {4499, 25731}, {4847, 11220}, {4907, 25930}, {5175, 5728}, {5178, 12669}, {5223, 11684}, {5250, 5785}, {5439, 15008}, {5552, 5817}, {5687, 5779}, {6173, 11025}, {10129, 21617}, {10384, 19861}, {10392, 24982}, {10527, 21151}, {12536, 12709}, {12688, 20007}, {14942, 26651}, {20533, 25279}, {21039, 24708}, {25717, 25726}
X(25723) lies on these lines: {1, 348}, {2, 25716}, {85, 551}, {150, 13607}, {519, 17095}, {664, 1125}, {1319, 3674}, {1323, 3636}, {1442, 4357}, {1565, 15178}, {3160, 3622}, {3340, 17081}, {3616, 9312}, {3668, 17394}, {3746, 6516}, {4554, 25303}, {5088, 13464}, {5882, 17181}, {7181, 11011}, {10106, 17084}
X(25724) lies on these lines: {664, 1897}
X(25725) lies on these lines: {100, 108}
X(25726) lies on these lines: {2, 25716}, {77, 17117}, {239, 241}, {347, 17363}, {948, 17389}, {1278, 1419}, {4393, 9312}, {4554, 25298}, {5723, 17266}, {6610, 17160}, {9436, 20016}, {17086, 17287}, {25717, 25722}
Collineation mappings involving Gemini triangle 30: X(25727)-X(25737)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 30, as in centers X(25727)-X(25737). Then
m(X) = a (a - b - c) (3 a - b - c) x + (c - a) (a - 3 b + c) (a - b + c) y + (b - a) (a - 3c + b) (a - c + b) z : :
(Clark Kimberling, October 25, 2018)
X(25727) lies on these lines: {63, 4980}, {25728, 25730}
X(25728) lies on these lines: {1, 4704}, {2, 4488}, {6, 4681}, {7, 4480}, {9, 75}, {10, 24280}, {44, 3875}, {45, 4698}, {63, 4358}, {69, 2325}, {86, 16676}, {144, 3161}, {165, 27538}, {192, 1743}, {193, 3950}, {239, 3973}, {312, 3929}, {319, 4873}, {344, 527}, {346, 4416}, {390, 4899}, {391, 4431}, {516, 27549}, {536, 16885}, {545, 17278}, {894, 3731}, {1026, 1742}, {1278, 16833}, {1449, 4664}, {1707, 3971}, {1766, 20602}, {3008, 15828}, {3177, 4936}, {3219, 4671}, {3247, 3758}, {3305, 24589}, {3662, 4473}, {3663, 20073}, {3679, 17331}, {3685, 5223}, {3717, 5698}, {3730, 6381}, {3886, 5220}, {3928, 18743}, {4007, 17346}, {4078, 24695}, {4360, 16670}, {4361, 15492}, {4363, 16814}, {4370, 17274}, {4419, 17304}, {4422, 17276}, {4432, 16496}, {4440, 4859}, {4454, 18230}, {4499, 25279}, {4518, 9746}, {4578, 7676}, {4643, 17286}, {4656, 26065}, {4666, 17146}, {4670, 16675}, {4676, 7174}, {4715, 17311}, {4738, 5119}, {4741, 17268}, {4888, 17244}, {4889, 15534}, {4908, 17309}, {5231, 17777}, {5695, 15481}, {6173, 17263}, {6646, 17284}, {7229, 24603}, {7321, 20195}, {14210, 17742}, {15485, 24821}, {15624, 23343}, {16667, 17319}, {16669, 17318}, {16673, 17379}, {16832, 17116}, {17151, 17349}, {17242, 20072}, {17253, 17359}, {17254, 17358}, {17255, 17357}, {17257, 17308}, {17258, 17306}, {17260, 25590}, {17264, 17296}, {17267, 17345}, {17269, 17344}, {17270, 17281}, {17272, 17280}, {17273, 17342}, {17285, 17329}, {17384, 24441}, {17776, 17781}, {19824, 26723}, {21061, 22016}, {21362, 22370}, {25727, 25730}, {25731, 25737}
X(25729) lies on these lines: {63, 5222}, {144, 3243}, {1959, 3970}, {3039, 24798}, {3890, 25731}, {5853, 25256}, {25732, 25735}
X(25730) lies on these lines: {2, 25731}, {144, 4699}, {894, 27823}, {25727, 25728}
X(25731) lies on these lines: {2, 25730}, {9, 19604}, {63, 3973}, {144, 145}, {3729, 9519}, {3890, 25729}, {4110, 4462}, {4488, 14923}, {4499, 25722}, {8051, 18228}, {25728, 25737}
X(25732) lies on these lines: {2, 25733}, {25729, 25735}
X(25733) lies on these lines: {2, 25732}
X(25734) lies on these lines: {1, 20068}, {2, 4488}, {9, 4359}, {40, 4723}, {63, 190}, {144, 306}, {165, 3952}, {200, 4427}, {321, 3929}, {329, 3977}, {345, 17781}, {346, 4001}, {527, 17776}, {545, 24789}, {1743, 17147}, {1897, 8765}, {1999, 25269}, {3161, 9965}, {3219, 3729}, {3305, 17336}, {3578, 4007}, {3912, 20078}, {3928, 4358}, {3951, 7283}, {4054, 5273}, {4113, 5220}, {4419, 5294}, {4450, 4901}, {4480, 5905}, {4499, 25308}, {4512, 17165}, {4641, 17262}, {4659, 5278}, {4686, 19723}, {4692, 12514}, {4899, 20075}, {5234, 17164}, {5256, 17350}, {5287, 17261}, {8025, 16673}, {8616, 24821}, {16570, 17763}, {17132, 19789}, {17151, 19742}, {17339, 26840}, {20073, 26065}, {20879, 20927}, {24280, 25006}
X(25735) lies on these lines: {4488, 14923}, {25729, 25732}
X(25736) lies on these lines: (none)
X(25737) lies on these lines: {144, 4473}, {190, 2415}, {3436, 6068}, {4767, 23831}, {25728, 25731}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28534.
X(25738) lies on these lines: {2,49}, {3,68}, {4,94}, {5,5422}, {6,5576}, {26,3580}, {30,11457}, {52,18381}, {54,23293}, {66,1351}, {69,3519}, {70,1993}, {125,1147}, {140,18911}, {155,2072}, {156,7505}, {182,1209}, {184,5449}, {185,9927}, {193,11255}, {195,15141}, {378,12370}, {381,11432}, {382,13093}, {389,18474}, {427,13292}, {539,1092}, {542,10539}, {569,21243}, {631,2888}, {858,16266}, {1181,10024}, {1204,17702}, {1352,1656}, {1498,11799}, {1503,7517}, {1568,15083}, {1594,12161}, {1614,10201}, {2070,9833}, {3090,3410}, {3357,16003}, {3521,15077}, {3542,10540}, {3548,6193}, {3549,6776}, {3564,11585}, {3567,11818}, {3767,22146}, {5094,15120}, {5446,11550}, {5654,10255}, {5889,18569}, {5946,7544}, {6143,9545}, {6243,6515}, {6288,9827}, {6643,23039}, {6644,14516}, {7383,13339}, {7391,10263}, {7394,10095}, {7401,18950}, {7506,12134}, {7526,12022}, {7528,11433}, {7529,18440}, {7530,16659}, {7689,21659}, {8541,12585}, {9140,18281}, {9544,14940}, {9707,10020}, {10055,18447}, {10071,18455}, {10112,13352}, {10264,11250}, {10298,12254}, {10605,12293}, {10620,20427}, {10627,16063}, {10984,18128}, {11264,13561}, {11411,18436}, {11412,14791}, {11456,15761}, {11572,14831}, {12121,18931}, {12162,18390}, {12163,18396}, {12421,23307}, {12902,18565}, {13579,14111}, {13598,14864}, {13754,18404}, {15027,15128}, {15089,16867}, {15749,17505}, {15760,18914}, {18394,18568}.
X(25738) = reflection of X(11441) in X(5)
X(25738) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,18951,568}, {5,18356,11442}, {68,1899,3}, {125,1147,6640}, {184,5449,6639}, {1181,14852,10024}, {3548,6193,22115}, {5449,10116,184}, {6102,6746,568}, {6146,12359,3}, {6193,23291,3548}, {6515,14790,6243}, {10112,20299,13352}, {11411,18531,18436}, {11442,18912,5}, {12134,13567,7506}, {12163,18396,18563}
X(25738) = X(11441)-of-Johnson-triangle
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28534.
X(25739) lies on these lines: {2,11464}, {3,12278}, {4,51}, {5,1614}, {20,9927}, {22,14852}, {30,74}, {49,10224}, {50,1157}, {54,1594}, {67,11564}, {68,70}, {110,2072}, {115,13509}, {125,186}, {140,6288}, {156,10255}, {184,7577}, {195,11264}, {232,15340}, {235,16659}, {323,539}, {378,1853}, {381,5422}, {403,1503}, {427,12022}, {542,1568}, {858,15133}, {924,5962}, {1141,3484}, {1154,7574}, {1181,7547}, {1199,3574}, {1216,2888}, {1352,12283}, {1478,19368}, {1479,11461}, {1495,15081}, {1531,12317}, {1596,16658}, {1650,6760}, {1656,9707}, {2071,17702}, {2777,13399}, {2914,10114}, {2979,14791}, {3153,3448}, {3410,5891}, {3520,20299}, {3541,15126}, {3545,3618}, {3564,11416}, {3843,12174}, {5159,15034}, {5449,7488}, {5576,13434}, {5640,11818}, {5663,18403}, {5889,18569}, {5899,13171}, {6030,25337}, {6143,12254}, {6240,15138}, {6247,18560}, {6640,11449}, {6643,7999}, {6759,16868}, {6776,7699}, {7401,11465}, {7505,9833}, {7507,7592}, {7533,14845}, {7544,15024}, {7575,15027}, {7576,13567}, {7579,18432}, {7687,10821}, {7722,19506}, {7741,9638}, {9705,9820}, {10113,17854}, {10149,12904}, {10257,15035}, {10282,14940}, {10296,16003}, {10297,14094}, {10304,18387}, {10605,18405}, {10610,11565}, {10721,15311}, {11413,12293}, {11438,18559}, {11440,18563}, {11442,11459}, {11466,18582}, {11467,18581}, {11563,11801}, {11585,14516}, {12006,22804}, {12111,18404}, {12225,12359}, {12241,15559}, {12902,18859}, {13403,14865}, {13491,18379}, {13619,21663}, {14483,15321}, {14561,19123}, {14917,18304}, {15032,18388}, {15036,16976}, {15043,18952}, {15045,18420}, {15054,18323}, {15061,15646}, {15072,18392}, {16072,18440}, {16661,17712}, {18356,18436}, {18533,23291}, {18914,23047}.
X(25739) = midpoint of X(i) and X(j) for these {i,j}: {3153,3448}, {10733,13445}
X(25739) = reflection of X(i) in X(j) for these {i,j}: {4,13851}, {110,2072}, {186,125}, {10540,5}, {11563,11801}, {13619,21663}, {14157,403}
X(25739) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,1899,5890}, {4,11457,6241}, {4,14216,12290}, {4,18912,3567}, {184,23325,7577}, {185,18383,4}, {389,11572,4}, {427,12022,15033}, {1594,6146,54}, {1853,18396,378}, {5449,11750,7488}, {6143,12254,13367}, {6241,18394,4}, {11442,18531,11459}, {11550,18390,4}, {12289,23294,3}, {14157,14644,403}, {18381,18390,11550}, {18420,18911,15045}, {20299,21659,3520}
See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28534.
X(25740) lies on these lines: {30,12219}, {2072,3564}, {3167,6640}, {9820,24572}, {12429,18404}
Collineation mappings involving Gemini triangle 31: X(25741)-X(25767)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 31, as in centers X(25741)-X(25767). Then
m(X) = b c (b ^2 - a c)(c^2 - a b) x + b^2 (a^2 - b c)(c^2 - a b) y + c^2 (a^2 - b c) (b^2 - a c) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 26, 2018)
X(25741) lies on these lines: {1, 2}, {11, 25754}, {75, 25758}, {172, 25810}, {244, 334}, {335, 20542}, {1107, 25818}, {2975, 25831}, {3756, 20532}, {3963, 19953}, {18787, 20554}, {19950, 20913}, {19951, 20917}, {25745, 25747}, {25751, 25763}
X(25742) lies on these lines: {2, 3}, {25745, 25752}
X(25743) lies on these lines: {2, 3}, {25747, 25748}
X(25744) lies on these lines: {2, 3}
X(25745) lies on these lines: {2, 6}, {3661, 25757}, {4357, 25817}, {25741, 25747}, {25742, 25752}, {25751, 25755}
X(25746) lies on these lines: {1, 2}, {1978, 21404}, {4124, 6376}, {25757, 25766}
X(25747) lies on these lines: {2, 7}, {141, 25756}, {334, 25757}, {3663, 25834}, {25741, 25745}, {25743, 25748}
X(25748) lies on these lines: {1, 2}, {38, 22116}, {244, 25811}, {1083, 5337}, {2112, 25820}, {3675, 3954}, {20356, 26582}, {25743, 25747}, {25757, 25758}
X(25749) lies on these lines: {2, 3}
X(25750) lies on these lines: {2, 3}
X(25751) lies on these lines: {2, 31}, {25741, 25763}, {25745, 25755}
X(25752) lies on these lines: {2, 32}, {17550, 25818}, {25742, 25745}, {25758, 25767}
X(25753) lies on these lines: {2, 37}, {141, 25755}, {3835, 25385}, {25741, 25745}, {25754, 25761}
X(25754) lies on these lines: {2, 39}, {11, 25741}, {25742, 25745}, {25753, 25761}
X(25755) lies on these lines: {1, 2}, {141, 25753}, {594, 25811}, {25745, 25751}
X(25756) lies on these lines: {1, 2}, {141, 25747}, {2321, 25834}
X(25757) lies on these lines: {2, 37}, {274, 19933}, {334, 25747}, {870, 5378}, {894, 25804}, {984, 4583}, {3661, 25745}, {17143, 19963}, {17793, 18895}, {18140, 19950}, {25746, 25766}, {25748, 25758}
X(25758) lies on these lines: {2, 39}, {75, 25741}, {1909, 25800}, {1978, 21404}, {4639, 24484}, {16915, 25810}, {25748, 25757}, {25752, 25767}
X(25759) lies on these lines: {2, 650}, {75, 4010}, {3766, 20913}, {3835, 20518}, {4369, 25836}
See the preamble just before X(25957).
X(25760) lies on these lines: {1,3454}, {2,31}, {8,21935}, {9,124}, {10,908}, {11,141}, {38,3705}, {42,4417}, {43,4972}, {63,1756}, {69,11269}, {75,3120}, {100,4660}, {125,17047}, {244,3662}, {305,21415}, {306,24210}, {312,15523}, {321,3944}, {333,24892}, {561,7018}, {614,25527}, {894,24725}, {899,4429}, {976,5015}, {978,4202}, {982,17184}, {984,3006}, {1193,16062}, {1203,20083}, {1211,2886}, {1330,1468}, {1621,3771}, {1647,17227}, {1836,4418}, {2651,5235}, {3011,3883}, {3218,4655}, {3219,4438}, {3240,4085}, {3259,24250}, {3416,17720}, {3550,4450}, {3687,3914}, {3703,4415}, {3720,18134}, {3735,4165}, {3741,11680}, {3772,3966}, {3773,4671}, {3775,21242}, {3782,17155}, {3790,3994}, {3821,4850}, {3826,5241}, {3831,4193}, {3844,5087}, {3912,25378}, {3920,4865}, {3923,5057}, {3925,5743}, {3938,4514}, {3961,5014}, {3980,20292}, {4003,17235}, {4026,5718}, {4104,25006}, {4138,5249}, {4359,17889}, {4414,24723}, {4519,17229}, {4643,9318}, {4657,17723}, {4679,17279}, {4860,7232}, {4892,24325}, {5231,17272}, {5248,25645}, {5271,17064}, {5293,5300}, {5301,24935}, {5360,20544}, {5793,10895}, {5846,17602}, {6210,8229}, {7897,17236}, {8286,17275}, {9284,18904}, {9345,17300}, {10479,25639}, {14434,21260}, {15485,24542}, {16703,17203}, {16739,16891}, {16827,16906}, {17017,19786}, {17280,17777}, {17283,25531}, {17284,24709}, {17725,20045}, {17740,24248}, {21616,25591}
X(25760) = complement X(17126)
X(25760) = isotomic conjugate of the isogonal conjugate of X(3764)
X(25760) = X(i)-complementary conjugate of X(j) for these (i,j): {4492, 10}, {8695, 4777}
X(25760) = barycentric product X(i)*X(j) for these {i,j}: {7, 4165}, {75, 3735}, {76, 3764}
X(25760) = barycentric quotient X(i)/X(j) for these {i,j}: {3735, 1}, {3764, 6}, {4165, 8}
X(25760) = trilinear product X(2)*X(3735)
X(25760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4388, 31), (2, 4645, 750), (2, 6327, 171), (2, 17127, 6679), (2887, 3846, 2), (3416, 17720, 17763), (4429, 5233, 899), (4438, 4703, 3219), (4972, 5741, 43)
X(25761) lies on these lines: {1, 2}, {244, 20333}, {25753, 25754}
X(25762) lies on these lines: {2, 3}
X(25763) lies on these lines: {2, 3}, {25741, 25751}
X(25764) lies on these lines: {2, 3}
X(25765) lies on these lines: {2, 3}
X(25766) lies on these lines: {2, 3}, {25746, 25757}
X(25767) lies on these lines: {2, 3}, {25752, 25758}
Collineation mappings involving Gemini triangle 32: X(25768)-X(25799)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 32, as in centers X(25768)-X(25799). Then
m(X) = b c (b ^2 + a c)(c^2 + a b) x - b^2 (a^2 + b c)(c^2 + a b) y - c^2 (a^2 + b c) (b^2 + a c) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 26, 2018)
X(25768) lies on these lines: {1, 2}, {100, 25872}, {1575, 25857}, {1914, 25849}, {6376, 25786}, {20255, 25781}, {21025, 25851}, {25769, 25957}, {25772, 25774}, {25778, 25792}, {25785, 25787}
X(25769) lies on these lines: {2, 3}, {25768, 25957}, {25772, 25779}
X(25770) lies on these lines: {2, 3}, {25774, 25775}, {25784, 25786}
X(25771) lies on these lines: {2, 3}
X(25772) lies on these lines: {2, 6}, {3662, 25785}, {3912, 25856}, {17296, 25860}, {20255, 25773}, {25768, 25774}, {25769, 25779}, {25778, 25782}, {25783, 25957}
X(25773) lies on these lines: {1, 2}, {6376, 25784}, {20255, 25772}, {20691, 25856}
X(25774) lies on these lines: {2, 7}, {25768, 25772}, {25770, 25775}
X(25775) lies on these lines: {1, 2}, {2275, 21725}, {2309, 23927}, {3959, 23929}, {18904, 21921}, {25770, 25774}, {25785, 25786}
X(25776) lies on these lines: {2, 3}
X(25777) lies on these lines: {2, 3}
X(26777) = anticomplement of X(26985)
X(25778) lies on these lines: {2, 31}, {25768, 25792}, {25772, 25782}
X(25779) lies on these lines: {2, 32}, {20541, 25838}, {25769, 25772}, {25786, 25799}
X(25780) lies on these lines: {2, 37}, {3662, 25787}, {4384, 25858}, {25768, 25772}, {25781, 25790}
X(25781) lies on these lines: {2, 39}, {20255, 25768}, {25769, 25772}, {25780, 25790}
X(25782) lies on these lines: {1, 2}, {6384, 21725}, {25772, 25778}
X(25783) lies on these lines: {1, 2}, {38, 25785}, {25772, 25957}
X(25784) lies on these lines: {2, 6}, {6376, 25773}, {25770, 25786}
X(25785) lies on these lines: {2, 37}, {38, 25783}, {239, 25842}, {3662, 25772}, {3875, 25860}, {6376, 25773}, {25768, 25787}, {25775, 25786}
X(25786) lies on these lines: {2, 39}, {350, 25838}, {6376, 25768}, {16916, 25849}, {25770, 25784}, {25775, 25785}, {25779, 25799}
X(25787) lies on these lines: {2, 6}, {3661, 25850}, {3662, 25780}, {25768, 25785}, {25789, 25790}
X(25788) lies on these lines: {1, 2}, {3662, 25772}, {17242, 25856}, {17363, 25842}
X(25789) lies on these lines: {2, 31}, {25768, 25772}, {25787, 25790}
X(25790) lies on these lines: {1, 2}, {25780, 25781}, {25787, 25789}
X(25791) lies on these lines: {2, 3}
X(25792) lies on these lines: {2, 3}, {25768, 25778}
X(25793) lies on these lines: {2, 3}
X(25794) lies on these lines: {2, 3}
X(25795) lies on these lines: {2, 3}
X(25796) lies on these lines: {2, 3}
X(25797) lies on these lines: {2, 3}
X(25798) lies on these lines: {2, 3}, {6376, 25773}
X(25799) lies on these lines: {2, 3}, {25779, 25786}
Collineation mappings involving Gemini triangle 33: X(25800)-X(25837)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 33, as in centers X(25800)-X(25837). Then
m(X) = a^2 (b^2 - a c) (c^2 - a b) x + a c (a^2 - b c) (c^2 - a b) y + a b (a^2 - b c) (b^2 - a c) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 27, 2018)
X(25800) lies on these lines: {1, 2}, {100, 3802}, {649, 4414}, {660, 984}, {673, 20356}, {750, 25804}, {870, 5378}, {1083, 3747}, {1909, 25758}, {1911, 17755}, {2239, 16514}, {2276, 25818}, {4386, 25810}, {4443, 24516}, {5091, 5277}, {6651, 18794}, {9310, 25801}, {17475, 25812}, {17738, 20681}, {25817, 25820}, {25834, 25835}
X(25801) lies on these lines: {2, 3}, {41, 25833}, {5276, 25835}, {9310, 25800}, {25804, 25810}
X(25802) lies on these lines: {2, 3}, {25816, 25818}
X(25803) lies on these lines: {2, 3}
X(25804) lies on these lines: {2, 6}, {100, 17475}, {750, 25800}, {799, 18268}, {894, 25757}, {1922, 17793}, {2235, 4396}, {2278, 25822}, {2280, 25814}, {4386, 25809}, {20603, 20704}, {25801, 25810}, {25831, 25835}
X(25805) lies on these lines: {1, 2}, {32, 5091}, {1429, 18266}, {18755, 25825}, {25816, 25817}, {25824, 25833}
X(25806) lies on these lines: {1, 2}, {20456, 20590}, {25817, 25818}, {25830, 25833}
X(25807) lies on these lines: {2, 3}
X(25808) lies on these lines: {2, 3}, {659, 5276}
X(25809) lies on these lines: {2, 31}, {4386, 25804}, {5091, 5277}, {14598, 20333}
X(25810) lies on these lines: {2, 32}, {172, 25741}, {4386, 25800}, {5276, 25818}, {16915, 25758}, {18897, 20542}, {25801, 25804}
X(25811) lies on these lines: {2, 37}, {244, 25748}, {594, 25755}, {750, 25800}, {1054, 24578}, {1491, 25837}, {5283, 19950}, {19962, 25092}, {24512, 25813}, {25812, 25823}
X(25812) lies on these lines: {2, 39}, {17475, 25800}, {25801, 25804}, {25811, 25823}
X(25813) lies on these lines: {1, 2}, {4386, 25804}, {24512, 25811}
X(25814) lies on these lines: {1, 2}, {2280, 25804}, {17754, 25834}, {21840, 25817}
X(25815) lies on these lines: {2, 44}, {659, 24512}, {750, 25800}
X(25816) lies on these lines: {2, 6}, {25802, 25818}, {25805, 25817}
X(25817) lies on these lines: {1, 25804}, {2, 37}, {244, 292}, {2177, 16526}, {3726, 24420}, {4357, 25745}, {21840, 25814}, {25800, 25820}, {25805, 25816}, {25806, 25818}, {25827, 25833}
X(25818) lies on these lines: {2, 39}, {1107, 25741}, {2276, 25800}, {5276, 25810}, {17550, 25752}, {25802, 25816}, {25806, 25817}
X(25819) lies on these lines: {2, 6}, {55, 669}, {171, 9421}, {190, 21783}, {645, 21755}, {5091, 5277}, {9509, 17475}, {17750, 25823}
X(25820) lies on these lines: {1, 9423}, {2, 6}, {37, 3571}, {38, 9506}, {45, 4784}, {100, 9509}, {750, 25823}, {1054, 24578}, {1083, 5277}, {2112, 25748}, {3110, 16589}, {4422, 21783}, {4589, 24505}, {16514, 24602}, {25800, 25817}
X(25821) lies on these lines: {1, 2}
X(25822) lies on these lines: {2, 661}, {39, 665}, {659, 24512}, {2278, 25804}
X(25823) lies on these lines: {1, 2}, {661, 876}, {750, 25820}, {756, 22116}, {2239, 16369}, {17750, 25819}, {24450, 24516}, {25811, 25812}
X(25824) lies on these lines: {2, 3}, {25805, 25833}
X(25825) lies on these lines: {2, 3}, {4413, 24533}, {5091, 5277}, {18755, 25805}
X(25826) lies on these lines: {2, 3}
X(25827) lies on these lines: {2, 3}, {25817, 25833}
X(25828) lies on these lines: {2, 3}
X(25829) lies on these lines: {2, 3}
X(25830) lies on these lines: {2, 3}, {25805, 25816}, {25806, 25833}
X(25831) lies on these lines: {2, 3}, {100, 3802}, {2975, 25741}, {5276, 25810}, {25804, 25835}
X(25832) lies on these lines: {2, 38}, {5091, 5277}
X(25833) lies on these lines: {2, 7}, {41, 25801}, {5091, 5277}, {25805, 25824}, {25806, 25830}, {25817, 25827}
X(25834) lies on these lines: {2, 37}, {100, 17475}, {244, 4876}, {982, 3252}, {2321, 25756}, {3663, 25747}, {17596, 18783}, {17754, 25814}, {25800, 25835}
X(25835) lies on these lines: {1, 25833}, {2, 39}, {5276, 25801}, {25800, 25834}, {25804, 25831}
X(25836) lies on these lines: {1, 5029}, {2, 649}, {659, 24512}, {661, 875}, {4369, 25759}
X(25837) lies on these lines: {2, 650}, {38, 661}, {659, 24512}, {1491, 25811}, {2276, 4010}, {3815, 3837}, {12782, 24290}, {14404, 19586}
Collineation mappings involving Gemini triangle 34: X(25838)-X(25874)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 34, as in centers X(25838)-X(25874). Then
m(X) = a^2 (b^2 + a c) (c^2 + a b) x - a c (a^2 + b c) (c^2 + a b) y - a b (a^2 + b c) (b^2 + a c) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 27, 2018)
X(25838) lies on these lines: {1, 2}, {350, 25786}, {748, 25839}, {2275, 25857}, {2975, 25872}, {4426, 25849}, {9310, 25842}, {20541, 25779}, {25848, 25866}, {25856, 25858}
X(25839) lies on these lines: {2, 3}, {748, 25838}, {25842, 25849}
X(25840) lies on these lines: {2, 3}, {25844, 25845}, {25855, 25857}
X(25841) lies on these lines: {2, 3}
X(25842) lies on these lines: {2, 6}, {9, 25856}, {239, 25785}, {748, 25853}, {9310, 25838}, {17363, 25788}, {25839, 25849}, {25848, 25852}
X(25843) lies on these lines: {1, 2}, {904, 17084}, {2275, 25855}
X(25844) lies on these lines: {2, 7}, {8245, 16366}, {9310, 25838}, {16968, 23493}, {25840, 25845}
X(25845) lies on these lines: {1, 2}, {17470, 22447}, {25840, 25844}, {25856, 25857}
X(25846) lies on these lines: {2, 3}
X(25847) lies on these lines: {2, 3}
X(25848) lies on these lines: {2, 31}, {25838, 25866}, {25842, 25852}
X(25849) lies on these lines: {2, 32}, {1914, 25768}, {4426, 25838}, {25839, 25842}, {25857, 25872}
X(25850) lies on these lines: {2, 37}, {9, 25858}, {3661, 25787}, {3662, 21823}, {4426, 25866}, {8424, 20998}, {9310, 25838}, {17276, 21887}, {18904, 21921}, {25851, 25864}
X(25851) lies on these lines: {2, 39}, {3865, 20859}, {17451, 18904}, {21025, 25768}, {25839, 25842}, {25850, 25864}
X(25852) lies on these lines: {1, 2}, {25842, 25848}
X(25853) lies on these lines: {1, 2}, {38, 25856}, {748, 25842}
X(25854) lies on these lines: {2, 44}, {3287, 25537}, {9310, 25838}
X(25855) lies on these lines: {2, 6}, {2275, 25843}, {25840, 25857}
X(25856) lies on these lines: {2, 37}, {9, 25842}, {38, 25853}, {2275, 25843}, {3912, 25772}, {17242, 25788}, {20691, 25773}, {25838, 25858}, {25845, 25857}
X(25857) lies on these lines: {2, 39}, {1575, 25768}, {2275, 25838}, {25840, 25855}, {25845, 25856}, {25849, 25872}
X(25858) lies on these lines: {2, 6}, {9, 25850}, {4384, 25780}, {25838, 25856}, {25863, 25864}
X(25859) lies on these lines: {1, 2}
X(25860) lies on these lines: {1, 2}, {9, 25842}, {1959, 18786}, {3875, 25785}, {17296, 25772}
X(25861) lies on these lines: {2, 661}, {647, 25862}, {3287, 25537}, {3805, 25873}
X(25862) lies on these lines: {2, 650}, {647, 25861}
X(25863) lies on these lines: {2, 31}, {9310, 25838}, {25858, 25864}
X(25864) lies on these lines: {1, 2}, {25850, 25851}, {25858, 25863}
X(25865) lies on these lines: {2, 3}
X(25866) lies on these lines: {2, 3}, {4426, 25850}, {25838, 25848}
X(25867) lies on these lines: {2, 3}
X(25868) lies on these lines: {2, 3}
X(25869) lies on these lines: {2, 3}
X(25870) lies on these lines: {2, 3}
X(25871) lies on these lines: {2, 3}, {2275, 25843}
X(25872) lies on these lines: {2, 3}, {100, 25768}, {2975, 25838}, {25849, 25857}
X(25873) lies on these lines: {2, 38}, {3805, 25861}, {25838, 25848}
X(25874) lies on these lines: {2, 38}, {9310, 25838}, {25845, 25856}
Collineation mappings involving Gemini triangle 35: X(25875)-X(25929)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 35, as in centers X(25875)-X(25929). Then
m(X) = (a - b - c)(a^2 - b^2 - c^2) x + 2 a c (a - b + c) y + 2 a b (a - c + b) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 28, 2018)
X(25875) lies on these lines: {1, 10601}, {2, 3}, {10, 11508}, {51, 19782}, {55, 24982}, {56, 908}, {224, 1864}, {394, 1724}, {956, 3876}, {958, 1319}, {1001, 3057}, {1125, 22767}, {1145, 3295}, {1210, 1259}, {1260, 12649}, {1376, 17606}, {1470, 6691}, {1617, 3436}, {1858, 15297}, {2975, 18228}, {3816, 26357}, {3984, 12513}, {4423, 10966}, {5119, 5259}, {5123, 11501}, {5226, 5253}, {5248, 8582}, {5251, 8583}, {5260, 18230}, {5687, 25005}, {5722, 11517}, {6284, 25973}, {8069, 26364}, {8071, 10200}, {8256, 26358}, {10396, 16465}, {10884, 18239}, {11012, 25522}, {11451, 19771}, {11496, 13528}, {11499, 17619}, {11510, 24987}, {17614, 22758}, {25878, 25886}, {25879, 25919}
X(25876) lies on these lines: {2, 3}, {8, 17102}, {58, 11427}, {73, 26871}, {216, 966}, {386, 11433}, {958, 25882}, {1038, 1457}, {1040, 19860}, {1809, 10527}, {2968, 3617}, {3086, 23518}, {4252, 23292}, {4255, 13567}, {4869, 18643}, {5217, 25968}, {6350, 24987}, {17073, 27006}, {22344, 26929}, {25880, 25881}, {25893, 25914}
X(25877) lies on these lines: {2, 3}, {343, 3216}, {19861, 25882}, {23541, 26094}
X(25878) lies on these lines: {1, 3059}, {2, 6}, {7, 220}, {9, 241}, {37, 25930}, {45, 26669}, {56, 27626}, {63, 1418}, {75, 4513}, {77, 1212}, {85, 27420}, {105, 10387}, {142, 219}, {218, 3664}, {319, 26531}, {386, 19521}, {405, 991}, {443, 3332}, {474, 13329}, {480, 9440}, {607, 25993}, {651, 18230}, {748, 20978}, {958, 1458}, {960, 2263}, {1001, 1818}, {1014, 5022}, {1038, 1456}, {1191, 4344}, {1253, 1376}, {1279, 19861}, {1350, 4223}, {1407, 5273}, {1419, 7308}, {1423, 1696}, {1471, 25524}, {1790, 16439}, {1936, 4413}, {2191, 17597}, {2256, 4000}, {2323, 20195}, {2911, 4675}, {3305, 6610}, {3713, 3912}, {3819, 25514}, {4319, 15587}, {4349, 16466}, {4350, 15853}, {4361, 20905}, {4363, 25001}, {4643, 25019}, {4657, 26006}, {4851, 25935}, {4888, 5526}, {5706, 8728}, {5733, 17529}, {5781, 18650}, {5782, 17279}, {5783, 17284}, {6603, 7190}, {6989, 17814}, {7074, 26040}, {7232, 25584}, {7290, 8583}, {7365, 18228}, {10446, 17682}, {16777, 24554}, {17262, 25243}, {17275, 26001}, {17298, 23151}, {17321, 26658}, {17595, 25939}, {19520, 19765}, {21239, 26063}, {22127, 25500}, {24179, 24774}, {24547, 26675}, {24993, 26653}, {25279, 26241}, {25875, 25886}, {25885, 25889}, {25895, 25899}, {25921, 25922}
X(25879) lies on these lines: {1, 2}, {56, 26685}, {1463, 17257}, {3486, 25531}, {3646, 13736}, {4422, 8572}, {5253, 26065}, {5423, 17480}, {5573, 11851}, {13742, 17614}, {20076, 26688}, {25875, 25919}, {25893, 25906}, {25894, 25905}, {25922, 25929}
X(25880) lies on these lines: {2, 7}, {269, 26690}, {614, 4073}, {728, 4452}, {1279, 19861}, {1723, 21255}, {2257, 4869}, {3242, 19860}, {3692, 4000}, {3731, 25097}, {3875, 26653}, {4202, 5250}, {16572, 21296}, {17270, 26526}, {25876, 25881}, {25895, 25921}, {25899, 25922}
X(25881) lies on these lines: {1, 2}, {946, 17674}, {3752, 4918}, {4054, 24178}, {5253, 17353}, {5294, 25524}, {12527, 26688}, {17164, 24175}, {17679, 18483}, {23536, 25079}, {24177, 25253}, {25876, 25880}, {25894, 25912}, {25895, 25896}, {25906, 25919}, {25914, 25917}
X(25882) lies on these lines: {2, 11}, {10, 2968}, {12, 24984}, {43, 13567}, {117, 122}, {171, 23292}, {197, 25947}, {475, 10310}, {899, 26005}, {958, 25876}, {960, 1361}, {1329, 1846}, {1738, 26011}, {1769, 14429}, {1861, 9371}, {3240, 26540}, {4551, 26932}, {5217, 24538}, {6253, 27378}, {6796, 7515}, {16586, 24987}, {17056, 25970}, {19861, 25877}, {25001, 26251}, {25017, 26030}
X(25883) lies on these lines: {2, 3}, {1001, 21333}, {7295, 24991}
X(25884) lies on these lines: {2, 3}, {25926, 25927}
X(25885) lies on these lines: {2, 31}, {212, 2886}, {255, 26363}, {394, 3720}, {405, 1064}, {442, 602}, {601, 7483}, {603, 4999}, {614, 11031}, {958, 1457}, {1001, 14547}, {1253, 3434}, {1468, 24541}, {1471, 5249}, {1496, 10527}, {1497, 10198}, {1699, 2328}, {2212, 25985}, {3072, 6856}, {3073, 6857}, {3074, 19843}, {3683, 25939}, {3915, 24987}, {4011, 26591}, {4224, 6210}, {4423, 17811}, {5278, 26013}, {5743, 25968}, {7299, 24953}, {8167, 25934}, {15254, 25091}, {16825, 17862}, {17337, 25973}, {19861, 25894}, {25878, 25889}, {25890, 25919}
X(25886) lies on these lines: {2, 32}, {2241, 26526}, {3934, 26654}, {7761, 26527}, {14827, 21285}, {25875, 25878}, {25896, 25913}
X(25887) lies on these lines: {2, 37}, {141, 1108}, {662, 24557}, {960, 27626}, {1001, 3601}, {1212, 4357}, {1279, 19861}, {1284, 25917}, {1418, 3662}, {1427, 25527}, {1766, 21526}, {2264, 25940}, {3061, 24471}, {3663, 25066}, {3694, 3946}, {3763, 8609}, {3875, 4515}, {3932, 8582}, {3965, 5222}, {3991, 4021}, {4875, 5232}, {8424, 15254}, {15492, 26699}, {15852, 16062}, {15988, 16666}, {17237, 26543}, {17304, 25083}, {17348, 26671}, {18698, 24774}, {24986, 26526}, {25068, 25590}, {25888, 25904}, {25893, 25907}, {25898, 25921}
X(25888) lies on these lines: {2, 39}, {1573, 26548}, {1574, 26676}, {25875, 25878}, {25887, 25904}
X(25889) lies on these lines: {1, 2}, {1458, 3305}, {1464, 25917}, {1818, 4423}, {2187, 16419}, {2269, 27639}, {2293, 5284}, {2308, 15066}, {2318, 3742}, {4210, 5888}, {4322, 5260}, {4878, 4883}, {8167, 14547}, {15254, 22053}, {25878, 25885}, {25897, 25919}
X(25890) lies on these lines: {1, 2}, {25885, 25919}, {25893, 25903}
X(25891) lies on these lines: {2, 44}, {9, 5575}, {513, 25900}, {518, 11518}, {1100, 26657}, {1279, 19861}, {3246, 8583}, {25899, 25921}
X(25892) lies on these lines: {2, 45}, {3, 2951}, {1279, 19861}, {3285, 24557}
X(25893) lies on these lines: {1, 17658}, {2, 11}, {9, 17625}, {31, 25934}, {56, 329}, {142, 15804}, {238, 17811}, {355, 11108}, {377, 7958}, {404, 9812}, {405, 1490}, {442, 10893}, {443, 11826}, {452, 8273}, {474, 2077}, {614, 25091}, {946, 16410}, {954, 10582}, {958, 1319}, {1260, 11019}, {1470, 5249}, {1617, 3452}, {1709, 10857}, {2078, 20196}, {3149, 25522}, {3286, 24556}, {3295, 8582}, {3304, 24558}, {3681, 12513}, {3753, 12703}, {3913, 24982}, {4187, 11500}, {4383, 25941}, {4387, 17862}, {4512, 17613}, {4966, 11433}, {5173, 8257}, {5259, 19520}, {5272, 25939}, {5919, 10912}, {6675, 26492}, {7191, 26669}, {7308, 18236}, {7580, 17618}, {7677, 18228}, {8726, 17649}, {8728, 10525}, {9954, 24928}, {10267, 17527}, {10310, 17567}, {10785, 16845}, {10826, 25542}, {10948, 19854}, {12915, 20588}, {16112, 17616}, {16842, 17619}, {16857, 18519}, {17265, 25970}, {20347, 24540}, {25876, 25914}, {25878, 25885}, {25879, 25906}, {25887, 25907}, {25890, 25903}
X(25894) lies on these lines: {2, 7}, {1711, 3771}, {1750, 11345}, {2257, 17778}, {3587, 17678}, {3719, 3772}, {3995, 26653}, {8557, 18134}, {13735, 18540}, {19861, 25885}, {25879, 25905}, {25881, 25912}, {25895, 25908}
X(25895) lies on these lines: {2, 37}, {894, 25524}, {1108, 26041}, {1463, 17257}, {1696, 27064}, {3718, 17053}, {3729, 11512}, {9311, 25978}, {17319, 25965}, {19861, 25898}, {25878, 25899}, {25880, 25921}, {25881, 25896}, {25894, 25908}, {26689, 27624}
X(25896) lies on these lines: {2, 39}, {25881, 25895}, {25886, 25913}
X(25897) lies on these lines: {2, 6}, {1172, 25986}, {5429, 8583}, {6349, 17257}, {17518, 22076}, {19861, 25885}, {25889, 25919}
X(25898) lies on these lines: {2, 6}, {65, 257}, {182, 19270}, {287, 1910}, {332, 2092}, {511, 1010}, {576, 19280}, {958, 1284}, {1038, 25910}, {1281, 4835}, {1351, 2049}, {1429, 4357}, {1503, 26117}, {3056, 5263}, {3564, 4205}, {3662, 10401}, {4416, 5247}, {5050, 19273}, {5085, 19278}, {5749, 5933}, {6776, 13725}, {9001, 25981}, {9002, 26652}, {9010, 24561}, {9013, 26545}, {10371, 17363}, {10381, 17499}, {10436, 16609}, {10441, 13740}, {10754, 26727}, {12017, 19279}, {19861, 25895}, {25887, 25921}, {25903, 25904}
X(25899) lies on these lines: {1, 2}, {25878, 25895}, {25880, 25922}, {25891, 25921}
X(25900) lies on these lines: {2, 661}, {142, 4077}, {513, 25891}, {514, 25009}, {649, 24562}, {650, 25955}, {657, 4131}, {693, 21127}, {1021, 14838}, {1635, 26641}, {2488, 3126}, {3835, 26695}, {4171, 4467}, {7252, 17811}, {8611, 17069}, {17420, 27674}, {17896, 21195}
X(25901) lies on these lines: {2, 667}, {405, 3309}, {513, 25903}, {663, 25924}, {958, 3669}, {1001, 4162}, {1946, 17072}, {4083, 19860}, {4367, 25926}, {4462, 5260}, {4705, 26641}, {4905, 5251}, {8641, 21302}, {8678, 24562}, {22091, 25380}
X(25902) lies on these lines: {2, 650}, {514, 25008}, {647, 4391}, {649, 24560}, {4017, 4468}, {4380, 26652}, {4776, 26596}, {8760, 13725}, {15411, 16754}
X(25903) lies on these lines: {2, 31}, {10, 25990}, {55, 25137}, {405, 15310}, {513, 25901}, {518, 11520}, {958, 1463}, {1279, 19861}, {5247, 17364}, {6210, 16048}, {17811, 24551}, {23693, 24982}, {25890, 25893}, {25898, 25904}
X(25904) lies on these lines: {1, 2}, {40, 17526}, {65, 5294}, {125, 4205}, {515, 4202}, {516, 11319}, {946, 5192}, {950, 4972}, {960, 25992}, {986, 3977}, {1220, 5249}, {1319, 25914}, {1842, 11109}, {3436, 25527}, {3671, 26223}, {3753, 17698}, {3869, 17353}, {4001, 5247}, {4357, 5260}, {5250, 13742}, {5484, 17291}, {5750, 17451}, {5793, 24789}, {12512, 17539}, {12527, 17184}, {17164, 17355}, {18250, 26580}, {25887, 25888}, {25898, 25903}
X(25905) lies on these lines: {2, 3}, {1458, 17257}, {18928, 19765}, {25879, 25894}
X(25906) lies on these lines: {1, 15988}, {2, 3}, {56, 26132}, {72, 26699}, {956, 11043}, {958, 1284}, {1001, 8240}, {1043, 26671}, {1468, 24551}, {1792, 1834}, {2292, 19860}, {2975, 27184}, {3303, 12642}, {5260, 11688}, {7283, 26665}, {16817, 26538}, {19861, 25885}, {25879, 25893}, {25881, 25919}, {27286, 27410}
X(25907) lies on these lines: {2, 3}, {577, 5275}, {614, 17102}, {956, 1060}, {958, 1038}, {1001, 1040}, {4265, 26958}, {5276, 15905}, {15812, 26543}, {18639, 26101}, {25887, 25893}
X(25908) lies on these lines: {2, 3}, {333, 18592}, {2322, 18667}, {6349, 17257}, {7011, 26125}, {17073, 27184}, {17778, 18643}, {18642, 25977}, {25894, 25895}
X(25909) lies on these lines: {2, 3}, {958, 6349}, {1038, 3305}, {1060, 3876}, {1214, 5260}, {2303, 3284}, {2975, 17073}, {19861, 25915}
X(25910) lies on these lines: {2, 3}, {1038, 25898}, {1220, 22341}
X(25911) lies on these lines: {2, 3}
X(25912) lies on these lines: {2, 3}, {1463, 17257}, {25681, 26132}, {25881, 25894}
X(25913) lies on these lines: {2, 3}, {25886, 25896}
X(25914) lies on these lines: {2, 12}, {10, 16602}, {11, 4202}, {141, 978}, {386, 4966}, {442, 19864}, {474, 8193}, {988, 17279}, {1125, 1279}, {1211, 27627}, {1213, 16604}, {1319, 25904}, {1698, 24222}, {1834, 3840}, {3624, 13728}, {3704, 3752}, {3710, 4003}, {3782, 25591}, {3813, 4429}, {3816, 16062}, {3825, 16052}, {3912, 4719}, {3925, 17674}, {4187, 19847}, {4205, 19862}, {4415, 25079}, {5192, 7354}, {5204, 17526}, {5436, 25509}, {5835, 24174}, {8167, 13725}, {8270, 8583}, {8362, 16593}, {11319, 15326}, {13745, 25542}, {15654, 19267}, {15888, 26030}, {17366, 17733}, {17529, 19863}, {17614, 19869}, {18253, 24627}, {19861, 25877}, {20541, 24668}, {25527, 25681}, {25531, 26117}, {25876, 25893}, {25881, 25917}
X(25915) lies on these lines: {2, 19}, {3, 2951}, {9, 1020}, {347, 7079}, {1038, 1456}, {1040, 4423}, {1214, 7308}, {1696, 7011}, {2257, 18643}, {3305, 6349}, {4466, 7289}, {5227, 26006}, {17022, 22119}, {18639, 25964}, {19861, 25909}, {25876, 25880}, {25887, 25893}, {25894, 25895}
X(25916) lies on these lines: {2, 38}, {993, 8583}, {19861, 25885}, {24443, 25982}
X(25917) lies on these lines: {1, 210}, {2, 65}, {3, 1709}, {4, 4679}, {5, 14110}, {8, 3740}, {9, 56}, {10, 11}, {12, 3452}, {21, 662}, {37, 992}, {40, 4413}, {44, 1468}, {46, 16408}, {55, 936}, {63, 25524}, {72, 354}, {73, 1212}, {75, 19582}, {78, 1001}, {140, 5887}, {142, 3649}, {145, 4662}, {165, 9856}, {188, 10506}, {200, 3303}, {329, 10404}, {375, 16980}, {377, 24703}, {386, 6051}, {388, 18228}, {404, 4640}, {405, 997}, {442, 17605}, {443, 1836}, {451, 1905}, {474, 1155}, {499, 5791}, {517, 1656}, {518, 3616}, {519, 3697}, {551, 3555}, {612, 1191}, {631, 6001}, {632, 14988}, {748, 1104}, {756, 1201}, {758, 5439}, {899, 4646}, {908, 24564}, {942, 3624}, {946, 3925}, {950, 12447}, {956, 20323}, {958, 1319}, {962, 26040}, {965, 2264}, {968, 4255}, {971, 7987}, {975, 3745}, {976, 1279}, {978, 3666}, {984, 21214}, {986, 16610}, {993, 17614}, {1006, 12664}, {1038, 1456}, {1058, 4863}, {1071, 10165}, {1122, 3665}, {1210, 21677}, {1213, 2262}, {1284, 25887}, {1329, 24987}, {1376, 5250}, {1385, 5251}, {1420, 5234}, {1463, 17257}, {1464, 25889}, {1469, 5296}, {1490, 8273}, {1697, 8580}, {1737, 17527}, {1837, 5084}, {1858, 6857}, {1859, 7498}, {2098, 9623}, {2099, 15829}, {2292, 3752}, {2476, 5087}, {2478, 5794}, {2550, 12701}, {2551, 5252}, {2594, 25941}, {2975, 5302}, {3035, 17638}, {3090, 7686}, {3216, 3931}, {3219, 5253}, {3242, 28011}, {3244, 4015}, {3294, 25066}, {3295, 3689}, {3304, 3715}, {3338, 3927}, {3361, 3929}, {3474, 17580}, {3476, 18227}, {3486, 5129}, {3501, 4520}, {3522, 15726}, {3523, 9943}, {3545, 16616}, {3556, 7484}, {3576, 5777}, {3601, 5696}, {3612, 16418}, {3617, 3880}, {3621, 4711}, {3622, 3681}, {3625, 3956}, {3626, 3898}, {3634, 3753}, {3636, 4533}, {3679, 3893}, {3696, 3702}, {3706, 9534}, {3711, 6765}, {3714, 4358}, {3727, 16605}, {3730, 25068}, {3742, 3868}, {3744, 5293}, {3748, 3811}, {3763, 3827}, {3782, 24178}, {3813, 25006}, {3816, 6734}, {3820, 10039}, {3828, 4002}, {3833, 4084}, {3838, 4197}, {3841, 11813}, {3877, 4731}, {3881, 4134}, {3892, 4539}, {3899, 19872}, {3911, 12709}, {3922, 5903}, {3923, 17635}, {3924, 17125}, {3967, 4968}, {3984, 4666}, {3997, 25089}, {4009, 4385}, {4018, 5883}, {4067, 19883}, {4101, 4966}, {4295, 17582}, {4297, 5927}, {4359, 25253}, {4415, 23536}, {4466, 17237}, {4511, 5047}, {4512, 5217}, {4518, 16823}, {4519, 5295}, {4540, 4669}, {4850, 27625}, {5045, 5904}, {5119, 9709}, {5221, 5437}, {5235, 18178}, {5248, 5440}, {5258, 24928}, {5259, 24929}, {5260, 12739}, {5268, 5710}, {5273, 7288}, {5289, 11011}, {5298, 5325}, {5313, 27785}, {5323, 24557}, {5328, 10588}, {5432, 6700}, {5433, 5745}, {5434, 12527}, {5450, 18239}, {5603, 19855}, {5651, 14529}, {5657, 6983}, {5691, 10157}, {5693, 9940}, {5694, 10202}, {5697, 19875}, {5698, 6904}, {5705, 25522}, {5730, 16842}, {5779, 10085}, {5780, 17857}, {5790, 18543}, {5795, 10944}, {5806, 7988}, {5811, 12678}, {5837, 8582}, {5880, 11415}, {5882, 18908}, {5886, 19854}, {5972, 10693}, {6666, 15950}, {6684, 12672}, {6690, 27385}, {6691, 18253}, {6735, 9711}, {6916, 12679}, {6959, 26446}, {6971, 9956}, {7082, 22768}, {7354, 12572}, {8167, 12635}, {8728, 12047}, {9961, 10178}, {10106, 18250}, {10371, 14555}, {10374, 18641}, {10391, 17558}, {10480, 18229}, {10582, 11523}, {10974, 17514}, {11113, 17647}, {11227, 15071}, {11230, 24474}, {11246, 12436}, {11376, 17642}, {11512, 17595}, {11684, 27003}, {12609, 17529}, {12721, 17355}, {13624, 18515}, {15569, 19767}, {15670, 17637}, {15888, 21075}, {16193, 18397}, {16471, 16843}, {16602, 24443}, {16604, 21879}, {17164, 24589}, {17303, 21871}, {17348, 27368}, {17441, 24943}, {17557, 18165}, {17559, 18391}, {17660, 18254}, {19732, 21334}, {19874, 22325}, {20227, 21810}, {21874, 24512}, {22065, 24511}, {24931, 25646}, {25881, 25914}
X(25918) lies on these lines: {2, 85}, {9, 1424}, {37, 24652}, {39, 304}, {56, 894}, {75, 2275}, {172, 3758}, {194, 312}, {322, 26042}, {341, 21226}, {982, 17760}, {1038, 25898}, {1463, 17257}, {1565, 8362}, {1655, 18743}, {2276, 18156}, {3662, 3665}, {3673, 26959}, {3752, 21216}, {3759, 4372}, {3934, 20925}, {3959, 25350}, {4673, 17759}, {4872, 7791}, {5088, 7770}, {5749, 17081}, {6656, 17181}, {7179, 26561}, {7181, 17368}, {7267, 7296}, {7270, 7774}, {7786, 20924}, {16043, 17170}, {16284, 26752}, {19804, 27318}, {20435, 27351}, {20927, 26979}, {25066, 27248}, {25074, 27264}, {25082, 27097}
X(25919) lies on these lines: {2, 11}, {2975, 8686}, {3544, 18524}, {3756, 4571}, {17460, 19860}, {25875, 25879}, {25881, 25906}, {25885, 25890}, {25889, 25897}, {25921, 25923}
X(25920) lies on these lines: {2, 101}, {106, 26698}, {1477, 2975}, {4587, 4904}
X(25921) lies on these lines: {2, 45}, {25878, 25922}, {25880, 25895}, {25887, 25898}, {25891, 25899}, {25919, 25923}
X(25922) lies on these lines: {2, 37}, {19861, 25928}, {25878, 25921}, {25879, 25929}, {25880, 25899}
X(25923) lies on these lines: {2, 900}, {392, 2815}, {525, 21189}, {1769, 14429}, {6129, 9031}, {25919, 25921}
X(25924) lies on these lines: {2, 649}, {9, 3676}, {513, 25891}, {652, 4885}, {657, 693}, {661, 24562}, {663, 25901}, {812, 25009}, {2520, 3126}, {3305, 4468}, {3669, 20318}, {4369, 26695}, {4382, 26546}, {4512, 15599}, {4521, 7308}, {4893, 26641}, {7216, 20296}, {17215, 27193}, {17418, 27674}, {20979, 25008}, {22108, 23813}, {23655, 25941}, {23806, 25604}
X(25925) lies on these lines: {2, 650}, {513, 25891}, {647, 7656}, {905, 3239}, {3126, 17115}, {3669, 4468}, {4025, 4130}, {4394, 25955}, {4423, 11934}, {4767, 26698}, {6129, 20315}, {6904, 8142}, {8760, 11108}, {17410, 20295}, {17811, 22383}, {25091, 25098}
X(25926) lies on these lines: {2, 659}, {513, 25891}, {1001, 4925}, {1491, 24562}, {1960, 8583}, {2826, 11108}, {4367, 25901}, {4782, 25955}, {4874, 26695}, {9511, 25380}, {19860, 21343}, {19861, 25569}, {21146, 26017}, {21173, 25084}, {25884, 25927}, {25919, 25921}
X(25927) lies on these lines: {2, 669}, {663, 25901}, {4455, 25008}, {25884, 25926}
X(25928) lies on these lines: {2, 7}, {15988, 20228}, {19861, 25922}, {25024, 25965}, {25878, 25895}, {25887, 25898}
X(25929) lies on these lines: {2, 38}, {1279, 19861}, {5204, 8544}, {8583, 15485}, {25879, 25922}, {25881, 25895}
Collineation mappings involving Gemini triangle 36: X(25930)-X(25956)
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 36, as in centers X(25930)-X(25956). Then
m(X) = (a + b - c) (a - b + c) (a^2 - b^2 - c^2) x + 2 a c (b + c - a) (a + b - c) y + 2 a b (b + c - a) (a - b + c) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line. (Clark Kimberling, October 28, 2018)
X(25930) lies on these lines: {1, 2}, {6, 25067}, {7, 2324}, {9, 77}, {34, 26003}, {37, 25878}, {40, 11349}, {57, 6602}, {63, 220}, {69, 25019}, {142, 7190}, {144, 269}, {218, 394}, {219, 1445}, {223, 3160}, {279, 329}, {326, 344}, {392, 21514}, {517, 24590}, {908, 948}, {1040, 25932}, {1212, 3305}, {1329, 26002}, {1376, 25931}, {1441, 27384}, {1442, 18230}, {1443, 6172}, {1696, 24471}, {1818, 7675}, {1993, 17745}, {2323, 8257}, {2991, 27819}, {3247, 24554}, {3306, 5228}, {3553, 4648}, {3729, 25243}, {3761, 26592}, {3875, 20905}, {4319, 20533}, {4766, 26607}, {4851, 25964}, {4907, 25722}, {5250, 11343}, {5526, 15066}, {5584, 11350}, {5750, 24553}, {5905, 10481}, {6282, 24604}, {6999, 27129}, {7271, 20059}, {7957, 16438}, {8831, 13389}, {8833, 13388}, {8917, 15856}, {9441, 11329}, {10436, 25001}, {10884, 14021}, {16593, 25933}, {17270, 25000}, {17296, 26540}, {17353, 26668}, {17484, 21314}, {17614, 21526}, {25938, 25944}, {27064, 27340}
X(25931) lies on these lines: {2, 3}, {56, 26001}, {1376, 25930}, {1604, 21279}, {1804, 20262}, {4254, 24553}, {5942, 7053}, {17316, 25954}, {22753, 24590}
X(25932) lies on these lines: {2, 3}, {216, 4648}, {1040, 25930}, {3306, 6350}, {4251, 11427}, {4253, 11433}, {4258, 23292}, {4855, 26006}, {5022, 13567}, {5308, 17102}, {23207, 26040}
X(25933) lies on these lines: {2, 3}, {343, 16552}, {16593, 25930}
X(25934) lies on these lines: {2, 6}, {31, 25893}, {57, 2324}, {154, 19649}, {219, 3911}, {220, 5744}, {222, 3452}, {329, 1407}, {474, 5706}, {613, 5121}, {651, 5328}, {908, 6180}, {990, 17612}, {1181, 6967}, {1329, 9370}, {1376, 25938}, {1450, 25524}, {1498, 6926}, {1616, 24558}, {2003, 20196}, {3218, 26669}, {3306, 5228}, {3713, 26011}, {3756, 12595}, {3819, 19544}, {4358, 26651}, {4363, 26591}, {4513, 17740}, {5707, 16408}, {5710, 19861}, {5711, 8583}, {5784, 9817}, {6700, 7078}, {6891, 17814}, {6983, 10982}, {8167, 25885}, {9306, 16434}, {16777, 26635}, {17021, 24554}, {17074, 18228}, {25001, 26627}, {25942, 25943}, {26612, 26659}
X(25934) = isotomic conjugate of polar conjugate of X(37391)
X(25935) lies on these lines: {1, 2}, {19, 18650}, {37, 25019}, {40, 14021}, {63, 6604}, {85, 92}, {86, 648}, {142, 1441}, {226, 17451}, {241, 21258}, {307, 16608}, {321, 25002}, {322, 17234}, {344, 3692}, {379, 515}, {728, 17776}, {857, 946}, {1108, 18635}, {1375, 1385}, {1449, 26668}, {1790, 14013}, {1953, 18589}, {2257, 5738}, {2293, 21931}, {2321, 25001}, {2991, 15988}, {3305, 18928}, {3306, 6350}, {3576, 24580}, {3663, 25255}, {3664, 26651}, {3950, 25243}, {4044, 26592}, {4297, 14953}, {4357, 24554}, {4851, 25878}, {5174, 26003}, {5257, 25000}, {5294, 7123}, {5731, 24604}, {6999, 27000}, {7190, 27509}, {8756, 24315}, {8804, 17220}, {9436, 24635}, {10165, 24581}, {10601, 22131}, {14361, 17911}, {16284, 18134}, {16580, 17443}, {17243, 25067}, {17858, 26165}, {17862, 20880}, {23536, 26550}, {24210, 26607}, {25946, 25954}, {27184, 27288}
X(25936) lies on these lines: {2, 3}
X(25937) lies on these lines: {2, 3}
X(25938) lies on these lines: {2, 31}, {212, 3035}, {255, 26364}, {329, 9364}, {394, 899}, {601, 4187}, {602, 13747}, {603, 1329}, {908, 9316}, {1106, 3436}, {1376, 25934}, {1395, 26020}, {1468, 24982}, {1496, 5552}, {1497, 10200}, {1771, 6700}, {1818, 11502}, {1961, 26635}, {3072, 17567}, {3831, 24537}, {3980, 26591}, {4413, 17811}, {9352, 26669}, {25930, 25944}
X(25939) lies on these lines: {2, 37}, {9, 1465}, {10, 2968}, {21, 15852}, {57, 219}, {63, 1427}, {227, 958}, {241, 5744}, {333, 18603}, {354, 25941}, {910, 24611}, {1040, 1376}, {1212, 5723}, {1214, 5745}, {1418, 3218}, {1818, 17603}, {2550, 9371}, {2999, 17825}, {3190, 11018}, {3306, 5228}, {3683, 25885}, {3687, 13567}, {3696, 26013}, {4646, 19765}, {5235, 16699}, {5272, 25893}, {5273, 17080}, {5554, 21896}, {5784, 7004}, {6610, 22129}, {10319, 15509}, {16696, 26638}, {16726, 26637}, {16736, 24556}, {17595, 25878}, {21892, 25023}, {25019, 26005}, {25062, 25066}
X(25940) lies on these lines: {2, 41}, {8, 1429}, {31, 5337}, {48, 141}, {57, 78}, {63, 21495}, {69, 604}, {75, 7225}, {101, 17284}, {142, 5736}, {218, 21526}, {284, 17306}, {379, 20335}, {572, 17272}, {584, 17384}, {599, 7113}, {662, 17227}, {672, 21477}, {748, 2210}, {750, 869}, {940, 1193}, {976, 3666}, {1376, 2340}, {1958, 3662}, {2174, 3763}, {2187, 7494}, {2208, 14826}, {2264, 25887}, {2267, 4643}, {2268, 4357}, {2278, 17237}, {2280, 17023}, {2287, 27626}, {2345, 18162}, {3061, 7291}, {3204, 17357}, {3217, 17353}, {3306, 25946}, {3419, 16377}, {3661, 4390}, {3684, 5222}, {3720, 16352}, {3912, 9310}, {4268, 17344}, {4503, 5114}, {4511, 7146}, {5782, 24328}, {7124, 17811}, {7175, 21296}, {16551, 25078}, {17228, 18042}, {20367, 25440}, {21244, 21270}, {24268, 26563}, {25941, 25947}, {26222, 27095}
X(25941) lies on these lines: {1, 2}, {3, 2187}, {31, 394}, {55, 1818}, {63, 1458}, {71, 16678}, {72, 1066}, {73, 960}, {81, 10460}, {103, 110}, {219, 604}, {323, 21747}, {350, 1226}, {354, 25939}, {392, 1064}, {517, 16056}, {518, 2318}, {579, 16878}, {674, 18613}, {740, 17862}, {851, 14110}, {902, 15066}, {991, 4512}, {1001, 14547}, {1042, 3869}, {1108, 2238}, {1215, 26591}, {1259, 1496}, {1376, 25934}, {1385, 8731}, {1457, 5289}, {1621, 2293}, {1993, 2308}, {2256, 2276}, {2594, 25917}, {2635, 24703}, {2654, 5794}, {2887, 23541}, {2975, 4322}, {3185, 22097}, {3191, 21620}, {3428, 4191}, {3452, 4551}, {3668, 20347}, {3681, 26669}, {3706, 26011}, {3781, 23853}, {3896, 20905}, {4300, 5250}, {4303, 12514}, {4306, 12526}, {4334, 9965}, {4383, 25893}, {4640, 22053}, {4878, 25067}, {5044, 5399}, {5741, 26010}, {6509, 23207}, {8012, 23988}, {10458, 26638}, {10476, 27621}, {10571, 15829}, {16059, 22770}, {17127, 20978}, {17165, 25243}, {17187, 26637}, {17592, 26635}, {18139, 25970}, {20470, 22276}, {20878, 22066}, {20999, 26885}, {23655, 25924}, {25940, 25947}
X(25942) lies on these lines: {2, 37}, {1088, 21218}, {7182, 16588}, {17316, 25952}, {25934, 25943}
X(25943) lies on these lines: {1, 2}, {1429, 23622}, {1959, 14189}, {25934, 25942}
X(25944) lies on these lines: {2, 38}, {25930, 25938}
X(25945) lies on these lines: {2, 3}
X(25946) lies on these lines: {2, 3}, {36, 24603}, {81, 5277}, {100, 16826}, {239, 5253}, {274, 26243}, {1014, 1654}, {1030, 6707}, {1213, 1444}, {1376, 4433}, {2303, 24530}, {2893, 25461}, {3306, 25940}, {4472, 19297}, {5249, 25593}, {5256, 11512}, {5276, 24598}, {5905, 25583}, {6645, 26048}, {16831, 25440}, {16997, 24621}, {17614, 26639}, {20269, 27186}, {25068, 27065}, {25524, 26626}, {25930, 25938}, {25935, 25954}
X(25947) lies on these lines: {2, 3}, {197, 25882}, {216, 5275}, {612, 17102}, {1038, 25524}, {1040, 1376}, {1062, 5687}, {2968, 10327}, {5096, 26958}, {25940, 25941}
X(25948) lies on these lines: {2, 3}
X(25949) lies on these lines: {2, 3}
X(25950) lies on these lines: {2, 3}, {86, 18591}, {1214, 16826}, {6350, 17316}, {6651, 27420}, {15831, 17350}
X(25951) lies on these lines: {2, 3}
X(25952) lies on these lines: {2, 3}, {17316, 25942}
X(25953) lies on these lines: {2, 3}, {17316, 25942}
X(25954) lies on these lines: {2, 11}, {63, 3119}, {404, 26531}, {650, 1814}, {1146, 6516}, {3580, 9090}, {3658, 24619}, {5687, 26658}, {6915, 27000}, {17316, 25931}, {21495, 26001}, {25935, 25946}
X(25955) lies on these lines: {2, 649}, {57, 4521}, {650, 25900}, {812, 26695}, {1635, 24562}, {3306, 4468}, {3676, 5437}, {4369, 25009}, {4379, 26546}, {4394, 25925}, {4782, 25926}, {17215, 27014}
X(25956) lies on these lines: {2, 3}
Points with 1st barcentric b3 + k a b c + c3: X(259576)-X(25961)
For each real number k, let P(k) = a (a^2 - b c) ((a^2 - b c) (a b^3 + a c^3 - 2 b^2 c^2) - b c (a c - b^2) (a b - c^2) k) : : . Then the point (A,B,C,X(2); A',B',C',X(2)) collineation image of P(k), where A'B'C' = Gemini triangle 31, is the point with barycentrics b^3 + k a b c + c^3 : : . Both P(k) and Q(k) are on the line X(2)X(31). (Peter Moses, October 28, 2018)
Q(-1) = X(25957)
Q(1) = X(25760)
Q(1/2) = X(25958)
Q(-1/2) = X(25959)
Q(2) = X(3846)
Q(-2) = X(3836)
Q(3) = X(25961)
Q(-3) = X(25962)
X(25957) lies on these lines: {1,4202}, {2,31}, {9,4683}, {10,3681}, {38,3662}, {42,4429}, {43,3936}, {58,19846}, {75,15523}, {76,21415}, {81,25453}, {100,3771}, {120,21015}, {141,3779}, {210,3823}, {244,3705}, {306,1738}, {312,3120}, {321,17889}, {334,561}, {354,3834}, {599,4042}, {612,25527}, {614,17282}, {894,16991}, {899,4417}, {908,4138}, {978,17674}, {982,3006}, {984,17184}, {1046,1698}, {1086,3703}, {1211,3826}, {1621,4660}, {1836,17279}, {2476,3831}, {3125,4165}, {3218,4438}, {3219,4655}, {3416,24789}, {3661,21020}, {3706,17231}, {3720,17234}, {3772,17763}, {3782,3932}, {3840,11680}, {3841,10479}, {3912,3914}, {3923,20292}, {3924,7270}, {3944,4358}, {3966,17278}, {3989,4389}, {4011,5057}, {4085,17018}, {4201,10448}, {4365,17233}, {4376,24699}, {4387,17267}, {4418,5880}, {4423,17265}, {4450,8616}, {4657,21249}, {4687,6536}, {4716,20017}, {4722,17364}, {4799,24358}, {4847,21255}, {4850,24169}, {4854,17243}, {4865,7191}, {5263,24943}, {5311,19786}, {5369,21240}, {5741,16569}, {8024,18067}, {8229,20368}, {10453,17232}, {11269,18141}, {12047,25591}, {14829,24892}, {16706,17017}, {16886,20271}, {16891,18057}, {17033,17673}, {17059,24392}, {17156,17296}, {17228,21027}, {17290,17599}, {17776,24248}, {17888,20237}, {19742,20290}, {20255,20486}, {25440,25645}
X(25957) = complement X(17127)
X(25957) = X(7241)-complementary conjugate of X(10)
X(25957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 2887, 25760), (2, 4388, 748), (2, 4645, 31), (2, 6327, 238), (2, 17126, 6679), (2, 25778, 25789), (1086, 3703, 17155), (2887, 3836, 2), (3840, 21241, 11680), (4429, 18134, 42), (4450, 24542, 8616), (4972, 18139, 1), (5741, 24988, 16569), (17231, 21949, 3706)
X(25958) lies on these lines: {2,31}, {8,3454}, {124,18228}, {141,11680}, {344,21249}, {3006,7226}, {3240,4417}, {3705,4392}, {3831,5154}, {3844,17605}, {3936,17018}, {3944,4671}, {4429,5741}, {4438,4683}, {5361,24892}, {7191,25527}, {9464,21415}, {17047,23293}, {20066,24946}
X(25958) = complement of X(30652)
X(25958) = {X(2),X(2887)}-harmonic conjugate of X(25959)
X(25959) lies on these lines: {2,31}, {8,24159}, {10,3901}, {124,5328}, {984,21026}, {3006,3662}, {3120,4671}, {3240,3936}, {3454,9780}, {3831,5141}, {3920,25527}, {4422,24710}, {4442,17233}, {4865,17024}, {4972,17018}, {5057,17279}, {5233,24988}, {5372,24892}, {7226,17184}, {7292,17282}, {15523,17889}, {16706,17025}, {17162,17373}, {17321,21249}, {17350,17491}, {18067,21415}
X(25959) = complement of X(30653)
X(25959) = {X(2),X(2887)}-harmonic conjugate of X(25958)
X(25960) lies on these lines: {1,5741}, {2,31}, {10,3877}, {11,5743}, {42,5233}, {57,4683}, {312,6535}, {756,3705}, {966,10589}, {978,5051}, {1211,3816}, {2886,5241}, {3120,19804}, {3218,4703}, {3454,3624}, {3662,22174}, {3720,4417}, {3739,17605}, {3771,5284}, {3825,10479}, {3944,4359}, {3966,17763}, {3980,5057}, {4415,17155}, {4418,24703}, {4425,4850}, {4643,17728}, {4865,5297}, {4972,16569}, {9345,17778}, {11269,14555}, {15523,18743}, {17063,17184}, {17135,24217}, {17277,24892}, {17889,24589}, {18139,25502}, {24210,25378}
X(25961 ) lies on these lines: {1,17674}, {2,31}, {10,3873}, {12,7248}, {42,17234}, {43,18139}, {55,17265}, {75,6535}, {210,3834}, {354,3823}, {612,17282}, {756,3662}, {899,18134}, {1699,24709}, {1962,17244}, {2308,17352}, {3006,17063}, {3120,18743}, {3158,17059}, {3266,18067}, {3305,4683}, {3550,24542}, {3705,21026}, {3715,7232}, {3720,4429}, {3745,17356}, {3831,4197}, {3932,17155}, {3936,16569}, {4011,20292}, {4358,17889}, {4418,17279}, {4422,11246}, {4450,15485}, {4660,5284}, {4865,7292}, {4871,11680}, {5311,16706}, {5332,17398}, {6057,7263}, {8013,17228}, {8027,21260}, {12609,25591}, {15523,19804}, {17283,24943}, {17763,24789}
Extending the preamble just before X(24537), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 37, as in centers X(25962)-X(25997). Then
m(X) = 2 b c (b + c - a) x + (a - b + c) (a^2 - b^2 + c^2) y + (a + b - c) (a^2 + b^2 - c^2) z : :
A point X lies on the Euler line if and only if m(X) also lies on the Euler line; examples: m(X(4)) = X(11109) and X(23) = X(11285). (Clark Kimberling, October 29, 2018)
X(25962) lies on these lines: {2, 3}, {7, 11681}, {10, 12736}, {12, 3812}, {63, 1329}, {119, 1071}, {224, 1837}, {394, 1714}, {495, 5554}, {1210, 16465}, {1259, 10523}, {1387, 24390}, {2886, 11376}, {3306, 15844}, {3814, 4292}, {3822, 8582}, {3825, 4304}, {3826, 5832}, {3841, 3884}, {3868, 17757}, {3925, 5836}, {5252, 19860}, {5784, 17606}, {8583, 23708}, {10391, 10958}, {10884, 18242}, {10957, 24541}, {11520, 12607}, {15908, 22835}, {17614, 26470}, {17616, 18238}, {25524, 26481}
X(25963) lies on these lines: {2, 3}, {343, 1724}, {24982, 25968}
X(25964) lies on these lines: {2, 6}, {7, 6554}, {9, 16608}, {10, 3059}, {37, 25019}, {75, 25002}, {120, 17792}, {142, 1439}, {198, 26130}, {226, 14524}, {307, 1212}, {440, 1730}, {572, 1375}, {594, 25001}, {910, 18650}, {1086, 20905}, {1100, 26006}, {1146, 1441}, {1442, 17044}, {1503, 4223}, {1746, 6678}, {1834, 25017}, {1876, 3812}, {2262, 18589}, {2310, 21931}, {3663, 4904}, {3739, 26001}, {3823, 24982}, {3912, 3965}, {3943, 25243}, {4364, 24554}, {4851, 25930}, {5053, 24884}, {5228, 27509}, {5305, 17189}, {5751, 8728}, {5803, 11108}, {5928, 9816}, {6247, 6846}, {6908, 15873}, {7745, 26678}, {10177, 24388}, {10361, 19372}, {10446, 17671}, {17220, 17747}, {17243, 26669}, {17362, 25584}, {17365, 26651}, {17810, 26052}, {17863, 21049}, {18639, 25915}, {24181, 24213}, {24547, 26526}, {24993, 26532}, {25970, 25972}, {25975, 25980}, {26538, 26570}
X(25964) = complement of X(37659)
X(25965) lies on these lines: {1, 2}, {244, 9369}, {341, 17054}, {894, 3812}, {986, 17261}, {1739, 7283}, {1878, 11109}, {2551, 3662}, {2975, 27002}, {3057, 25531}, {3339, 17350}, {3685, 24440}, {3704, 17268}, {3714, 17117}, {3753, 13741}, {5086, 24988}, {5260, 24627}, {5423, 11851}, {5573, 17480}, {6646, 18250}, {8165, 26132}, {12053, 26139}, {17319, 25895}, {17541, 27000}, {25024, 25928}, {25973, 25984}
X(25966) lies on these lines: {2, 7}, {3663, 25082}, {3668, 26690}, {3823, 24982}, {3879, 26653}, {4967, 26526}, {5782, 17278}, {5830, 17357}, {11109, 25967}, {20432, 20905}, {25007, 26530}, {25980, 25995}
X(25967) lies on these lines: {1, 2}, {515, 17674}, {3436, 17282}, {3812, 5294}, {3977, 24443}, {5750, 21921}, {11109, 25966}, {17184, 18250}, {25975, 25976}
X(25968) lies on these lines: {1, 2968}, {2, 11}, {12, 11109}, {31, 13567}, {40, 7515}, {42, 23292}, {65, 3040}, {109, 26932}, {238, 26005}, {388, 1359}, {406, 11496}, {441, 2223}, {475, 11500}, {860, 5842}, {939, 3085}, {946, 6711}, {958, 24538}, {1125, 24025}, {1411, 19860}, {1617, 20266}, {1897, 4081}, {2187, 10192}, {2342, 26956}, {3011, 26011}, {3052, 26958}, {4858, 15253}, {5125, 6253}, {5136, 7680}, {5217, 25876}, {5584, 27407}, {5743, 25885}, {6284, 17555}, {7536, 10434}, {7965, 14004}, {10902, 18641}, {13405, 24030}, {15252, 24026}, {16586, 24541}, {17061, 17862}, {17126, 26540}, {17724, 24353}, {17798, 21940}, {20905, 26230}, {20999, 26933}, {24537, 25466}, {24982, 25963}
X(25969) lies on these lines: {2, 3}
X(25970) lies on these lines: {2, 31}, {10, 343}, {1193, 24985}, {3720, 23541}, {3826, 13567}, {3925, 26013}, {17056, 25882}, {17265, 25893}, {18139, 25941}, {24982, 25977}, {25964, 25972}
X(25971) lies on these lines: {2, 37}, {44, 26671}, {1418, 27334}, {1874, 25992}, {3668, 8568}, {3723, 24559}, {3823, 24982}, {3826, 8582}, {4891, 24551}, {6706, 10436}, {17239, 25007}, {17861, 25066}, {18698, 25068}, {20258, 24471}, {24540, 26653}, {24774, 25590}, {24986, 26676}, {25023, 25357}, {25973, 25985}, {25978, 25997}
X(25972) lies on these lines: {1, 2}, {16409, 18518}, {25964, 25970}
X(25973) lies on these lines: {2, 11}, {5, 3359}, {10, 17658}, {12, 3812}, {142, 17625}, {165, 14022}, {226, 18236}, {329, 1329}, {355, 8728}, {405, 11826}, {442, 5927}, {443, 12114}, {1698, 10523}, {1738, 25091}, {2887, 26005}, {3305, 26066}, {3624, 10948}, {3681, 21031}, {3932, 17862}, {4187, 6684}, {4193, 9812}, {4847, 17626}, {5084, 10893}, {5554, 15888}, {6067, 25006}, {6284, 25875}, {6796, 13747}, {6991, 11024}, {8226, 17613}, {8583, 11373}, {9710, 10912}, {10525, 11108}, {10598, 17559}, {10785, 17582}, {10944, 19860}, {10949, 24541}, {11109, 25992}, {11928, 16853}, {13567, 25957}, {16408, 26492}, {17337, 25885}, {17528, 18516}, {17529, 17614}, {17757, 24473}, {20347, 24986}, {21955, 26669}, {25964, 25970}, {25965, 25984}, {25971, 25985}
X(25974) lies on these lines: {1, 25895}, {2, 6}, {894, 3812}, {10446, 11353}, {11109, 25976}
X(25975) lies on these lines: {2, 37}, {322, 26106}, {894, 3812}, {1766, 11353}, {3596, 20227}, {3729, 24174}, {3965, 26048}, {4719, 17319}, {24982, 25978}, {25964, 25980}, {25967, 25976}, {25986, 25993}
X(25976) lies on these lines: {2, 39}, {5294, 25994}, {11109, 25974}, {25967, 25975}
X(25977) lies on these lines: {2, 6}, {894, 25986}, {1899, 19310}, {13995, 25005}, {16352, 26869}, {16608, 27184}, {17167, 26019}, {17184, 26532}, {18642, 25908}, {24982, 25970}
X(25978) lies on these lines: {2, 6}, {12, 894}, {19, 297}, {192, 4918}, {257, 17248}, {313, 26665}, {1874, 25988}, {4357, 16609}, {4429, 25631}, {9311, 25895}, {11683, 17257}, {17353, 21244}, {22370, 26590}, {24982, 25975}, {25971, 25997}
X(25979) lies on these lines: {1, 2}, {65, 26791}, {1837, 26073}, {2551, 26840}, {2885, 3699}, {3340, 27130}, {5253, 8686}, {5795, 27002}, {8256, 25531}, {14923, 26139}
X(25980) lies on these lines: {1, 2}, {17862, 25021}, {25964, 25975}, {25966, 25995}
X(25981) lies on these lines: {2, 650}, {513, 26545}, {514, 24560}, {649, 25008}, {905, 24622}, {1010, 8760}, {2522, 4391}, {4106, 26596}, {5263, 11934}, {8678, 24561}, {9001, 25898}
X(25982) lies on these lines: {1, 2}, {12, 5294}, {1842, 17555}, {3947, 26223}, {3977, 13161}, {4202, 6684}, {4217, 18492}, {5123, 25992}, {5192, 10175}, {5587, 17526}, {11319, 19925}, {11681, 17353}, {18249, 26580}, {24443, 25916}
X(25983) lies on these lines: {2, 3}, {894, 24982}, {7270, 26011}
X(25984) lies on these lines: {2, 3}, {12, 894}, {3580, 25005}, {11681, 26065}, {15988, 26131}, {19861, 24161}, {24982, 25970}, {25965, 25973}
X(25985) lies on these lines: {2, 3}, {10, 1905}, {12, 1875}, {33, 2886}, {34, 25466}, {63, 1892}, {120, 20620}, {124, 226}, {208, 15844}, {317, 16992}, {342, 7179}, {343, 10477}, {607, 26036}, {958, 11392}, {1001, 11393}, {1452, 26066}, {1829, 24987}, {1843, 26543}, {1859, 1861}, {1876, 5249}, {2194, 23292}, {2212, 25885}, {3192, 5718}, {3195, 26098}, {3434, 7071}, {3757, 5081}, {5089, 8736}, {5090, 19860}, {5276, 16318}, {5283, 27376}, {6198, 24390}, {10198, 11398}, {11363, 24541}, {11399, 26363}, {16589, 27371}, {16998, 27377}, {20122, 26932}, {25971, 25973}
X(25986) lies on these lines: {2, 3}, {86, 18679}, {317, 19732}, {894, 25977}, {1172, 25897}, {2322, 17778}, {5278, 27377}, {17907, 19701}, {18643, 18667}, {25975, 25993}
X(25987) lies on these lines: {2, 3}, {10, 1844}, {12, 1940}, {92, 25466}, {240, 21674}, {323, 2906}, {495, 1148}, {1835, 5249}, {1848, 24564}, {3194, 26131}, {3580, 3868}, {3925, 5174}, {5081, 16817}, {5279, 7359}, {5523, 17916}, {24982, 25993}
X(25988) lies on these lines: {2, 3}, {1874, 25978}, {3194, 15988}
X(25989) lies on these lines: {2, 3}
X(25990) lies on these lines: {2, 3}, {10, 25903}, {894, 3812}, {1330, 26530}, {10449, 26657}, {26132, 27410}
X(25991) lies on these lines: {2, 3}
X(25992) lies on these lines: {2, 12}, {5, 19846}, {8, 17285}, {10, 3744}, {11, 13741}, {55, 13742}, {65, 17353}, {960, 25904}, {964, 3826}, {1125, 17724}, {1376, 17526}, {1698, 17698}, {1874, 25971}, {2292, 4422}, {2886, 5192}, {3035, 12746}, {3649, 27064}, {3701, 17061}, {3714, 26723}, {3812, 5294}, {3925, 13740}, {3932, 5262}, {4026, 5047}, {4429, 6284}, {5044, 19869}, {5123, 25982}, {5221, 26065}, {5750, 7359}, {6690, 26030}, {7819, 26007}, {9708, 19836}, {9710, 24552}, {10371, 17284}, {10404, 17282}, {10436, 24796}, {11108, 12410}, {11109, 25973}, {11115, 24988}, {13735, 15338}, {15325, 19847}, {17123, 19879}, {17356, 23536}, {18082, 19874}, {24982, 25963}
X(25993) lies on these lines: {2, 19}, {4, 2951}, {7, 7079}, {33, 26040}, {142, 281}, {273, 1855}, {451, 6684}, {607, 25878}, {653, 21617}, {1376, 7071}, {1783, 3664}, {1827, 1861}, {1839, 26003}, {1876, 3812}, {2331, 4648}, {3663, 17916}, {11024, 11471}, {11109, 25966}, {14571, 17245}, {17022, 17903}, {20883, 20905}, {24982, 25987}, {25971, 25973}, {25975, 25986}
X(25994) lies on these lines: {2, 85}, {9, 3503}, {37, 2998}, {65, 24514}, {75, 16605}, {76, 16583}, {169, 7770}, {183, 16968}, {192, 341}, {194, 3752}, {238, 17739}, {257, 960}, {312, 21216}, {349, 25021}, {384, 910}, {385, 1104}, {742, 21025}, {894, 3812}, {942, 17499}, {1108, 26107}, {1655, 3666}, {1874, 25978}, {1909, 3290}, {2551, 17257}, {2975, 26279}, {3125, 4721}, {3673, 27299}, {3693, 26752}, {3701, 17489}, {3702, 17497}, {3727, 4465}, {3729, 24440}, {3732, 13741}, {3959, 4713}, {4000, 6392}, {4357, 17062}, {4875, 26801}, {5016, 17007}, {5179, 6656}, {5294, 25976}, {6381, 16600}, {11324, 15487}, {16601, 27020}, {16602, 27318}, {16611, 20888}, {20347, 26562}, {20911, 27040}, {21044, 24995}, {21951, 24330}, {25066, 27091}, {25242, 26029}, {25384, 25614}
X(25995) lies on these lines: {2, 37}, {4518, 23677}, {24982, 25997}, {25966, 25980}
X(25996) lies on these lines: {2, 900}, {2815, 3753}, {6366, 23678}, {17724, 24353}
X(25997) lies on these lines: {2, 7}, {3914, 7155}, {20432, 26538}, {23675, 24349}, {24982, 25995}, {25964, 25975}, {25971, 25978}
X(25998) lies on these lines: {1, 17048}, {141, 2140}, {16609, 26243}, {24355, 24432}
X(25999) lies on these lines: {1, 474}, {63, 2640}, {846, 27798}, {1054, 14829}, {3666, 24372}, {5737, 17596}, {17490, 24377}, {19804, 24369}
X(26000) lies on these lines: {1, 2}, {31, 92}, {55, 7235}, {57, 24235}, {171, 1441}, {322, 3769}, {333, 11031}, {415, 774}, {846, 25255}, {983, 2997}, {1376, 24365}, {1400, 7009}, {1733, 4418}, {1754, 17861}, {1955, 2206}, {3772, 8286}, {4213, 21044}, {4220, 16609}, {5174, 21935}, {9317, 16056}
| 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) |
.png){kind=link}