[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.
O1, O2, O3 = the reflections of Oa, Ob, Oc in BC, CA, AB, resp.
Ma, Mb, Mc = the midpoints of NaO1, NbO2, NcO3, resp.
ABC, MaMbMc are circumcyclologic.
ie the circumcircles of AMbMc, BMcMa, CMaMb and ABC are concurrent
the circmcircles of MaBC, MbCA, McAB and MaMbMc are concurrent
Cyclologic centers?
[Peter Moses]:
Hi Antreas,
Also MaMbMc is perspective to:
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Excentral triangle at X(1)X(4127)∩X(9)X(19872)
= a*(9*a^3 + 9*a^2*b - 9*a*b^2 - 9*b^3 + 9*a^2*c - 13*a*b*c - 21*b^2*c - 9*a*c^2 - 21*b*c^2 - 9*c^3) : :
= lies on the Jerabek circumhyperbola of the excentral triangle and these lines: {1, 4127}, {9, 19872}, {40, 3627}, {191, 9780}, {2136, 4816}, {3219, 3337}, {3626, 5541}, {3646, 6763}, {3647, 13146}, {6326, 13624}
= excentral isogonal conjugate of X(31663)
= X(3634)-Ceva conjugate of X(1)
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Second circumperp triangle at X(1)X(4757)∩X(3)X(5260)
= a*(9*a^3 - 9*a*b^2 - a*b*c - 3*b^2*c - 9*a*c^2 - 3*b*c^2) : :
= lies on these lines {1, 4757}, {3, 5260}, {8, 19704}, {21, 19862}, {100, 3626}, {404, 19872}, {1001, 4189}, {1621, 19535}, {2975, 3621}, {3634, 13587}, {3736, 16477}, {4816, 5010}, {5251, 22266}, {5284, 7280}, {5550, 16370}, {6265, 13624}, {7173, 15680}, {19705, 19877}, {21161, 22936}, {22355, 27026}
= {X(7280),X(17574)}-harmonic conjugate of X(5284)
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Outer Garcia triangle at X(1)X(3968)∩X(8)X(381)
= a*(a^3 - 4*a^2*b - a*b^2 + 4*b^3 - 4*a^2*c + 23*a*b*c - 19*b^2*c - a*c^2 - 19*b*c^2 + 4*c^3) : :
= 2 X[8148] - 5 X[16615].
= lies on these lines: {1, 3968}, {8, 381}, {10, 13606}, {100, 13624}, {2550, 3621}, {3617, 3813}, {3626, 21630}, {4420, 10222}, {4853, 5303}, {5178, 12751}, {5223, 14923}, {10914, 11684}, {11024, 20050}
= reflection of X(13606) in X(10).
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Garcia reflection triangle at Q = X(1)X(748)∩X(2)X(3296)
= a*(a - b - c)*(a + 2*b + c)*(a + b + 2*c) : :
= X[1] - 3 X[5506],8 X[3634] - 3 X[5557],7 X[9780] - 3 X[9782].
= lies on Feuerbach circumhyperbola and these lines: {1, 748}, {2, 3296}, {4, 3617}, {7, 12}, {8, 3058}, {9, 3871}, {10, 79}, {21, 210}, {44, 2298}, {45, 941}, {80, 3626}, {84, 1796}, {100, 3065}, {104, 6986}, {191, 3956}, {256, 3214}, {314, 3701}, {354, 17546}, {404, 7284}, {484, 4540}, {517, 16615}, {518, 17536}, {519, 13606}, {943, 3935}, {958, 2320}, {960, 1320}, {1000, 3621}, {1171, 5297}, {1389, 11278}, {1392, 5330}, {2346, 15254}, {2478, 5686}, {2481, 6540}, {2891, 25479}, {3062, 7997}, {3219, 3579}, {3254, 6734}, {3475, 5550}, {3625, 5559}, {3634, 5557}, {3679, 5560}, {3680, 3877}, {3691, 4876}, {3740, 17531}, {3811, 16858}, {3869, 17098}, {3873, 17534}, {3885, 4900}, {3889, 7308}, {3983, 15481}, {4096, 27368}, {4430, 16842}, {4532, 16126}, {4537, 5425}, {4647, 4756}, {4661, 11108}, {4866, 5250}, {4943, 23836}, {5044, 15179}, {5251, 5424}, {5551, 5708}, {5665, 15556}, {5719, 17552}, {6595, 18259}, {6684, 13243}, {6763, 9342}, {7320, 20050}, {10266, 21031}, {11544, 17484}, {11604, 13272}, {11698, 16006}, {12019, 24298}, {17420, 23838}, {18483, 25006}, {21077, 31254}, {26792, 31419}, {27549, 30479}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {210, 5302, 4420}, {4420, 5302, 21}
= X(1268)-Ceva conjugate of X(1255)
= X(i)-cross conjugate of X(j) for these (i,j): {3737, 3699}, {4041, 644}, {4662, 8}
= X(i)-isoconjugate of X(j) for these (i,j): {6, 553}, {7, 2308}, {34, 3916}, {56, 1125}, {57, 1100}, {58, 3649}, {73, 31900}, {77, 2355}, {101, 30724}, {106, 5298}, {109, 4977}, {181, 30593}, {222, 1839}, {269, 3683}, {273, 23201}, {278, 22054}, {604, 4359}, {608, 4001}, {651, 4979}, {1014, 1962}, {1106, 3702}, {1213, 1412}, {1230, 16947}, {1269, 1397}, {1396, 3958}, {1400, 8025}, {1402, 16709}, {1407, 3686}, {1408, 4647}, {1411, 4973}, {1414, 4983}, {1415, 4978}, {1416, 4966}, {1417, 4975}, {1431, 4697}, {1434, 20970}, {1461, 4976}, {2163, 4870}, {2171, 30581}, {4565, 4988}, {4990, 6614}, {7341, 8013}
= cevapoint of X(i) and X(j) for these (i,j): {1, 3579}, {9, 210}
= crosspoint of X(1268) and X(4102)
= trilinear pole of line {650, 4501}
= barycentric product X(i) X(j) for these {i,j}: {1, 4102}, {8, 1255}, {9, 1268}, {21, 6539}, {55, 32018}, {210, 32014}, {312, 1126}, {318, 1796}, {643, 31010}, {644, 4608}, {650, 6540}, {1171, 3701}, {1320, 31011}, {2185, 6538}, {3596, 28615}, {3700, 4596}, {4041, 4632}, {4086, 4629}, {4391, 8701}
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Isogonal conjugate of Q: Q* = MIDPOINT OF X(3336) AND X(5563)
= a*(a + b - c)*(a - b + c)*(2*a + b + c) : :
= (r+3 R) X[1] - 3 r X[3], X[1] + 3 X[3336], X[1] - 3 X[5563], 2 X[1] - 3 X[20323], 3 X[404] - X[4420], 2 X[3336] + X[20323].
= lies on these lines: {1, 3}, {2, 5302}, {4, 17728}, {7, 5550}, {10, 5434}, {11, 1354}, {12, 3634}, {21, 3742}, {34, 17523}, {44, 583}, {45, 2285}, {58, 14158}, {60, 757}, {63, 25524}, {79, 3582}, {81, 4719}, {88, 961}, {89, 959}, {104, 7686}, {108, 1887}, {140, 13407}, {142, 24953}, {145, 9352}, {210, 474}, {226, 5433}, {229, 4228}, {244, 1104}, {329, 24954}, {355, 4317}, {388, 5435}, {404, 518}, {496, 1770}, {499, 17605}, {501, 1412}, {529, 24982}, {553, 1125}, {584, 604}, {595, 18360}, {603, 1418}, {614, 4252}, {631, 17718}, {749, 1469}, {758, 17614}, {896, 28352}, {908, 6691}, {910, 1475}, {936, 4005}, {946, 11246}, {950, 15326}, {956, 3698}, {958, 3306}, {960, 3218}, {976, 21342}, {978, 4641}, {993, 5439}, {997, 3962}, {1001, 4652}, {1042, 1450}, {1100, 17454}, {1106, 1427}, {1193, 1464}, {1203, 8614}, {1210, 7354}, {1357, 2842}, {1358, 10521}, {1386, 17092}, {1393, 1455}, {1398, 1452}, {1406, 16466}, {1434, 1447}, {1445, 5220}, {1458, 2594}, {1468, 3752}, {1478, 17606}, {1698, 11237}, {1708, 15650}, {1737, 18357}, {1768, 9856}, {1788, 3600}, {1836, 3086}, {1837, 4293}, {2096, 12679}, {2173, 2260}, {2306, 19373}, {2348, 4253}, {2975, 3812}, {3052, 28011}, {3058, 31730}, {3149, 12680}, {3296, 3524}, {3474, 12701}, {3475, 3523}, {3476, 3621}, {3485, 5265}, {3555, 3689}, {3556, 26866}, {3584, 5442}, {3585, 31776}, {3598, 24796}, {3616, 4640}, {3624, 4654}, {3625, 4315}, {3626, 10106}, {3669, 4782}, {3671, 4031}, {3674, 7181}, {3681, 17572}, {3720, 11553}, {3740, 17531}, {3753, 8666}, {3811, 16371}, {3848, 5047}, {3869, 23958}, {3873, 4188}, {3874, 5440}, {3925, 12436}, {3928, 8583}, {3983, 4413}, {4018, 30144}, {4295, 11376}, {4299, 5722}, {4301, 17613}, {4308, 20050}, {4311, 10950}, {4324, 31795}, {4325, 28160}, {4333, 9668}, {4340, 17723}, {4355, 5219}, {4383, 11512}, {4423, 31424}, {4650, 21214}, {4676, 26093}, {4857, 28146}, {4861, 10107}, {4999, 5249}, {5030, 16601}, {5044, 6763}, {5057, 20084}, {5083, 12432}, {5123, 20060}, {5218, 11037}, {5225, 10431}, {5229, 5704}, {5247, 16610}, {5267, 5427}, {5270, 9956}, {5290, 19872}, {5294, 25914}, {5393, 10910}, {5405, 10911}, {5432, 21620}, {5542, 15837}, {5586, 14150}, {5587, 9657}, {5714, 6832}, {5880, 10527}, {5905, 25681}, {6001, 26877}, {6284, 11019}, {6684, 15888}, {6692, 12527}, {6848, 12678}, {6904, 24477}, {6905, 12675}, {6906, 13374}, {6911, 14872}, {7173, 8226}, {7175, 16477}, {7198, 9436}, {7248, 27655}, {7989, 9656}, {8686, 8698}, {9579, 10896}, {9581, 12943}, {9655, 10826}, {9776, 30478}, {9965, 28647}, {10072, 12699}, {10074, 17636}, {10085, 19541}, {10090, 12738}, {10091, 11670}, {10308, 16141}, {10916, 11112}, {11038, 15717}, {11194, 19860}, {11544, 12047}, {12019, 20118}, {12688, 22753}, {12832, 18976}, {13369, 17637}, {13747, 21077}, {14027, 31524}, {15299, 31391}, {16736, 27660}, {17351, 25591}, {17736, 25066}, {17751, 24593}, {17863, 18661}, {18253, 24564}, {19861, 31165}, {20470, 22345}, {22344, 23383}, {22836, 24473}, {24628, 30038}, {31190, 31246}
= midpoint of X(3336) and X(5563)
= reflection of X(20323) in X(5563)
= X(2308)-cross conjugate of X(1100)
= X(i)-isoconjugate of X(j) for these (i,j): {6, 4102}, {8, 1126}, {9, 1255}, {41, 32018}, {55, 1268}, {60, 6538}, {281, 1796}, {284, 6539}, {312, 28615}, {522, 8701}, {663, 6540}, {1171, 2321}, {1334, 32014}, {2316, 31011}, {3700, 4629}, {3709, 4632}, {3939, 4608}, {4041, 4596}, {5546, 31010}, {5547, 31013}
= crosspoint of X(i) and X(j) for these (i,j): {1, 10308}, {57, 1014}
= crosssum of X(i) and X(j) for these (i,j): {1, 3579}, {9, 210}
= crossdifference of every pair of points on line {650, 4501}
= barycentric product X(i) X(j) for these {i,j}: {1, 553}, {7, 1100}, {12, 30581}, {34, 4001}, {56, 4359}, {57, 1125}, {65, 8025}, {77, 1839}, {81, 3649}, {85, 2308}, {88, 5298}, {89, 4870}, {100, 30724}, {109, 4978}, {269, 3686}, {273, 22054}, {278, 3916}, {279, 3683}, {331, 23201}, {348, 2355}, {552, 21816}, {604, 1269}, {651, 4977}, {664, 4979}, {934, 4976}, {1014, 1213}, {1214, 31900}, {1230, 1408}, {1400, 16709}, {1407, 3702}, {1412, 4647}, {1414, 4988}, {1432, 4697}, {1434, 1962}, {1461, 4985}, {1462, 4966}, {2006, 4973}, {2171, 30593}, {3669, 4427}, {4115, 7203}, {4565, 30591}, {4573, 4983}, {4617, 4990}, {4856, 19604}, {4970, 7153}, {4989, 21446}
= barycentric quotient X(i) / X(j) for these {i,j}: {1, 4102}, {7, 32018}, {56, 1255}, {57, 1268}, {65, 6539}, {553, 75}, {603, 1796}, {604, 1126}, {651, 6540}, {1014, 32014}, {1100, 8}, {1125, 312}, {1213, 3701}, {1269, 28659}, {1319, 31011}, {1397, 28615}, {1408, 1171}, {1414, 4632}, {1415, 8701}, {1839, 318}, {1962, 2321}, {2171, 6538}, {2308, 9}, {2355, 281}, {3649, 321}, {3669, 4608}, {3683, 346}, {3686, 341}, {3916, 345}, {3958, 3710}, {4001, 3718}, {4017, 31010}, {4359, 3596}, {4427, 646}, {4565, 4596}, {4647, 30713}, {4697, 17787}, {4870, 4671}, {4969, 4723}, {4970, 4110}, {4974, 3975}, {4976, 4397}, {4977, 4391}, {4979, 522}, {4983, 3700}, {4984, 4768}, {4988, 4086}, {4989, 30854}, {5298, 4358}, {8025, 314}, {16709, 28660}, {17454, 4420}, {20970, 210}, {21816, 6057}, {22054, 78}, {22080, 3694}, {23201, 219}, {30581, 261}, {30724, 693}, {31900, 31623}
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Best regards,
Peter Moses.
Let ABC be a triangle.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.
O1, O2, O3 = the reflections of Oa, Ob, Oc in BC, CA, AB, resp.
Ma, Mb, Mc = the midpoints of NaO1, NbO2, NcO3, resp.
ABC, MaMbMc are circumcyclologic.
ie the circumcircles of AMbMc, BMcMa, CMaMb and ABC are concurrent
the circmcircles of MaBC, MbCA, McAB and MaMbMc are concurrent
Cyclologic centers?
[Peter Moses]:
Hi Antreas,
Also MaMbMc is perspective to:
--------------------------------------------------------------------
Excentral triangle at X(1)X(4127)∩X(9)X(19872)
= a*(9*a^3 + 9*a^2*b - 9*a*b^2 - 9*b^3 + 9*a^2*c - 13*a*b*c - 21*b^2*c - 9*a*c^2 - 21*b*c^2 - 9*c^3) : :
= lies on the Jerabek circumhyperbola of the excentral triangle and these lines: {1, 4127}, {9, 19872}, {40, 3627}, {191, 9780}, {2136, 4816}, {3219, 3337}, {3626, 5541}, {3646, 6763}, {3647, 13146}, {6326, 13624}
= excentral isogonal conjugate of X(31663)
= X(3634)-Ceva conjugate of X(1)
--------------------------------------------------------------------
Second circumperp triangle at X(1)X(4757)∩X(3)X(5260)
= a*(9*a^3 - 9*a*b^2 - a*b*c - 3*b^2*c - 9*a*c^2 - 3*b*c^2) : :
= lies on these lines {1, 4757}, {3, 5260}, {8, 19704}, {21, 19862}, {100, 3626}, {404, 19872}, {1001, 4189}, {1621, 19535}, {2975, 3621}, {3634, 13587}, {3736, 16477}, {4816, 5010}, {5251, 22266}, {5284, 7280}, {5550, 16370}, {6265, 13624}, {7173, 15680}, {19705, 19877}, {21161, 22936}, {22355, 27026}
= {X(7280),X(17574)}-harmonic conjugate of X(5284)
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Outer Garcia triangle at X(1)X(3968)∩X(8)X(381)
= a*(a^3 - 4*a^2*b - a*b^2 + 4*b^3 - 4*a^2*c + 23*a*b*c - 19*b^2*c - a*c^2 - 19*b*c^2 + 4*c^3) : :
= 2 X[8148] - 5 X[16615].
= lies on these lines: {1, 3968}, {8, 381}, {10, 13606}, {100, 13624}, {2550, 3621}, {3617, 3813}, {3626, 21630}, {4420, 10222}, {4853, 5303}, {5178, 12751}, {5223, 14923}, {10914, 11684}, {11024, 20050}
= reflection of X(13606) in X(10).
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Garcia reflection triangle at Q = X(1)X(748)∩X(2)X(3296)
= a*(a - b - c)*(a + 2*b + c)*(a + b + 2*c) : :
= X[1] - 3 X[5506],8 X[3634] - 3 X[5557],7 X[9780] - 3 X[9782].
= lies on Feuerbach circumhyperbola and these lines: {1, 748}, {2, 3296}, {4, 3617}, {7, 12}, {8, 3058}, {9, 3871}, {10, 79}, {21, 210}, {44, 2298}, {45, 941}, {80, 3626}, {84, 1796}, {100, 3065}, {104, 6986}, {191, 3956}, {256, 3214}, {314, 3701}, {354, 17546}, {404, 7284}, {484, 4540}, {517, 16615}, {518, 17536}, {519, 13606}, {943, 3935}, {958, 2320}, {960, 1320}, {1000, 3621}, {1171, 5297}, {1389, 11278}, {1392, 5330}, {2346, 15254}, {2478, 5686}, {2481, 6540}, {2891, 25479}, {3062, 7997}, {3219, 3579}, {3254, 6734}, {3475, 5550}, {3625, 5559}, {3634, 5557}, {3679, 5560}, {3680, 3877}, {3691, 4876}, {3740, 17531}, {3811, 16858}, {3869, 17098}, {3873, 17534}, {3885, 4900}, {3889, 7308}, {3983, 15481}, {4096, 27368}, {4430, 16842}, {4532, 16126}, {4537, 5425}, {4647, 4756}, {4661, 11108}, {4866, 5250}, {4943, 23836}, {5044, 15179}, {5251, 5424}, {5551, 5708}, {5665, 15556}, {5719, 17552}, {6595, 18259}, {6684, 13243}, {6763, 9342}, {7320, 20050}, {10266, 21031}, {11544, 17484}, {11604, 13272}, {11698, 16006}, {12019, 24298}, {17420, 23838}, {18483, 25006}, {21077, 31254}, {26792, 31419}, {27549, 30479}
= isogonal conjugate of Q*
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {210, 5302, 4420}, {4420, 5302, 21}
= X(1268)-Ceva conjugate of X(1255)
= X(i)-cross conjugate of X(j) for these (i,j): {3737, 3699}, {4041, 644}, {4662, 8}
= X(i)-isoconjugate of X(j) for these (i,j): {6, 553}, {7, 2308}, {34, 3916}, {56, 1125}, {57, 1100}, {58, 3649}, {73, 31900}, {77, 2355}, {101, 30724}, {106, 5298}, {109, 4977}, {181, 30593}, {222, 1839}, {269, 3683}, {273, 23201}, {278, 22054}, {604, 4359}, {608, 4001}, {651, 4979}, {1014, 1962}, {1106, 3702}, {1213, 1412}, {1230, 16947}, {1269, 1397}, {1396, 3958}, {1400, 8025}, {1402, 16709}, {1407, 3686}, {1408, 4647}, {1411, 4973}, {1414, 4983}, {1415, 4978}, {1416, 4966}, {1417, 4975}, {1431, 4697}, {1434, 20970}, {1461, 4976}, {2163, 4870}, {2171, 30581}, {4565, 4988}, {4990, 6614}, {7341, 8013}
= cevapoint of X(i) and X(j) for these (i,j): {1, 3579}, {9, 210}
= crosspoint of X(1268) and X(4102)
= trilinear pole of line {650, 4501}
= barycentric product X(i) X(j) for these {i,j}: {1, 4102}, {8, 1255}, {9, 1268}, {21, 6539}, {55, 32018}, {210, 32014}, {312, 1126}, {318, 1796}, {643, 31010}, {644, 4608}, {650, 6540}, {1171, 3701}, {1320, 31011}, {2185, 6538}, {3596, 28615}, {3700, 4596}, {4041, 4632}, {4086, 4629}, {4391, 8701}
= barycentric quotient X(i) / X(j) for these {i,j}: {1, 553}, {8, 4359}, {9, 1125}, {21, 8025}, {33, 1839}, {37, 3649}, {41, 2308}, {44, 5298}, {45, 4870}, {55, 1100}, {60, 30581}, {78, 4001}, {200, 3686}, {210, 1213}, {212, 22054}, {219, 3916}, {220, 3683}, {312, 1269}, {333, 16709}, {346, 3702}, {513, 30724}, {522, 4978}, {607, 2355}, {644, 4427}, {650, 4977}, {663, 4979}, {1126, 57}, {1171, 1014}, {1172, 31900}, {1255, 7}, {1268, 85}, {1334, 1962}, {1796, 77}, {2185, 30593}, {2318, 3958}, {2321, 4647}, {2323, 4973}, {2325, 4975}, {2329, 4697}, {3158, 4856}, {3208, 4970}, {3239, 4985}, {3684, 4974}, {3686, 6533}, {3689, 4969}, {3693, 4966}, {3700, 30591}, {3701, 1230}, {3709, 4983}, {3900, 4976}, {4041, 4988}, {4069, 4115}, {4102, 75}, {4130, 4990}, {4420, 3578}, {4515, 4046}, {4526, 30592}, {4578, 30729}, {4596, 4573}, {4608, 24002}, {4629, 1414}, {4632, 4625}, {4873, 4717}, {4895, 4984}, {6538, 6358}, {6539, 1441}, {6540, 4554}, {7064, 21816}, {8701, 651}, {28615, 56}, {31010, 4077}, {32018, 6063}
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Isogonal conjugate of Q: Q* = MIDPOINT OF X(3336) AND X(5563)
= a*(a + b - c)*(a - b + c)*(2*a + b + c) : :
= (r+3 R) X[1] - 3 r X[3], X[1] + 3 X[3336], X[1] - 3 X[5563], 2 X[1] - 3 X[20323], 3 X[404] - X[4420], 2 X[3336] + X[20323].
= lies on these lines: {1, 3}, {2, 5302}, {4, 17728}, {7, 5550}, {10, 5434}, {11, 1354}, {12, 3634}, {21, 3742}, {34, 17523}, {44, 583}, {45, 2285}, {58, 14158}, {60, 757}, {63, 25524}, {79, 3582}, {81, 4719}, {88, 961}, {89, 959}, {104, 7686}, {108, 1887}, {140, 13407}, {142, 24953}, {145, 9352}, {210, 474}, {226, 5433}, {229, 4228}, {244, 1104}, {329, 24954}, {355, 4317}, {388, 5435}, {404, 518}, {496, 1770}, {499, 17605}, {501, 1412}, {529, 24982}, {553, 1125}, {584, 604}, {595, 18360}, {603, 1418}, {614, 4252}, {631, 17718}, {749, 1469}, {758, 17614}, {896, 28352}, {908, 6691}, {910, 1475}, {936, 4005}, {946, 11246}, {950, 15326}, {956, 3698}, {958, 3306}, {960, 3218}, {976, 21342}, {978, 4641}, {993, 5439}, {997, 3962}, {1001, 4652}, {1042, 1450}, {1100, 17454}, {1106, 1427}, {1193, 1464}, {1203, 8614}, {1210, 7354}, {1357, 2842}, {1358, 10521}, {1386, 17092}, {1393, 1455}, {1398, 1452}, {1406, 16466}, {1434, 1447}, {1445, 5220}, {1458, 2594}, {1468, 3752}, {1478, 17606}, {1698, 11237}, {1708, 15650}, {1737, 18357}, {1768, 9856}, {1788, 3600}, {1836, 3086}, {1837, 4293}, {2096, 12679}, {2173, 2260}, {2306, 19373}, {2348, 4253}, {2975, 3812}, {3052, 28011}, {3058, 31730}, {3149, 12680}, {3296, 3524}, {3474, 12701}, {3475, 3523}, {3476, 3621}, {3485, 5265}, {3555, 3689}, {3556, 26866}, {3584, 5442}, {3585, 31776}, {3598, 24796}, {3616, 4640}, {3624, 4654}, {3625, 4315}, {3626, 10106}, {3669, 4782}, {3671, 4031}, {3674, 7181}, {3681, 17572}, {3720, 11553}, {3740, 17531}, {3753, 8666}, {3811, 16371}, {3848, 5047}, {3869, 23958}, {3873, 4188}, {3874, 5440}, {3925, 12436}, {3928, 8583}, {3983, 4413}, {4018, 30144}, {4295, 11376}, {4299, 5722}, {4301, 17613}, {4308, 20050}, {4311, 10950}, {4324, 31795}, {4325, 28160}, {4333, 9668}, {4340, 17723}, {4355, 5219}, {4383, 11512}, {4423, 31424}, {4650, 21214}, {4676, 26093}, {4857, 28146}, {4861, 10107}, {4999, 5249}, {5030, 16601}, {5044, 6763}, {5057, 20084}, {5083, 12432}, {5123, 20060}, {5218, 11037}, {5225, 10431}, {5229, 5704}, {5247, 16610}, {5267, 5427}, {5270, 9956}, {5290, 19872}, {5294, 25914}, {5393, 10910}, {5405, 10911}, {5432, 21620}, {5542, 15837}, {5586, 14150}, {5587, 9657}, {5714, 6832}, {5880, 10527}, {5905, 25681}, {6001, 26877}, {6284, 11019}, {6684, 15888}, {6692, 12527}, {6848, 12678}, {6904, 24477}, {6905, 12675}, {6906, 13374}, {6911, 14872}, {7173, 8226}, {7175, 16477}, {7198, 9436}, {7248, 27655}, {7989, 9656}, {8686, 8698}, {9579, 10896}, {9581, 12943}, {9655, 10826}, {9776, 30478}, {9965, 28647}, {10072, 12699}, {10074, 17636}, {10085, 19541}, {10090, 12738}, {10091, 11670}, {10308, 16141}, {10916, 11112}, {11038, 15717}, {11194, 19860}, {11544, 12047}, {12019, 20118}, {12688, 22753}, {12832, 18976}, {13369, 17637}, {13747, 21077}, {14027, 31524}, {15299, 31391}, {16736, 27660}, {17351, 25591}, {17736, 25066}, {17751, 24593}, {17863, 18661}, {18253, 24564}, {19861, 31165}, {20470, 22345}, {22344, 23383}, {22836, 24473}, {24628, 30038}, {31190, 31246}
= isogonal conjugate of Q
= midpoint of X(3336) and X(5563)
= reflection of X(20323) in X(5563)
= X(i)-Ceva conjugate of X(j) for these (i,j): {553, 1100}, {1414, 3669}, {26700, 513}
= X(2308)-cross conjugate of X(1100)
= X(i)-isoconjugate of X(j) for these (i,j): {6, 4102}, {8, 1126}, {9, 1255}, {41, 32018}, {55, 1268}, {60, 6538}, {281, 1796}, {284, 6539}, {312, 28615}, {522, 8701}, {663, 6540}, {1171, 2321}, {1334, 32014}, {2316, 31011}, {3700, 4629}, {3709, 4632}, {3939, 4608}, {4041, 4596}, {5546, 31010}, {5547, 31013}
= crosspoint of X(i) and X(j) for these (i,j): {1, 10308}, {57, 1014}
= crosssum of X(i) and X(j) for these (i,j): {1, 3579}, {9, 210}
= crossdifference of every pair of points on line {650, 4501}
= barycentric product X(i) X(j) for these {i,j}: {1, 553}, {7, 1100}, {12, 30581}, {34, 4001}, {56, 4359}, {57, 1125}, {65, 8025}, {77, 1839}, {81, 3649}, {85, 2308}, {88, 5298}, {89, 4870}, {100, 30724}, {109, 4978}, {269, 3686}, {273, 22054}, {278, 3916}, {279, 3683}, {331, 23201}, {348, 2355}, {552, 21816}, {604, 1269}, {651, 4977}, {664, 4979}, {934, 4976}, {1014, 1213}, {1214, 31900}, {1230, 1408}, {1400, 16709}, {1407, 3702}, {1412, 4647}, {1414, 4988}, {1432, 4697}, {1434, 1962}, {1461, 4985}, {1462, 4966}, {2006, 4973}, {2171, 30593}, {3669, 4427}, {4115, 7203}, {4565, 30591}, {4573, 4983}, {4617, 4990}, {4856, 19604}, {4970, 7153}, {4989, 21446}
= barycentric quotient X(i) / X(j) for these {i,j}: {1, 4102}, {7, 32018}, {56, 1255}, {57, 1268}, {65, 6539}, {553, 75}, {603, 1796}, {604, 1126}, {651, 6540}, {1014, 32014}, {1100, 8}, {1125, 312}, {1213, 3701}, {1269, 28659}, {1319, 31011}, {1397, 28615}, {1408, 1171}, {1414, 4632}, {1415, 8701}, {1839, 318}, {1962, 2321}, {2171, 6538}, {2308, 9}, {2355, 281}, {3649, 321}, {3669, 4608}, {3683, 346}, {3686, 341}, {3916, 345}, {3958, 3710}, {4001, 3718}, {4017, 31010}, {4359, 3596}, {4427, 646}, {4565, 4596}, {4647, 30713}, {4697, 17787}, {4870, 4671}, {4969, 4723}, {4970, 4110}, {4974, 3975}, {4976, 4397}, {4977, 4391}, {4979, 522}, {4983, 3700}, {4984, 4768}, {4988, 4086}, {4989, 30854}, {5298, 4358}, {8025, 314}, {16709, 28660}, {17454, 4420}, {20970, 210}, {21816, 6057}, {22054, 78}, {22080, 3694}, {23201, 219}, {30581, 261}, {30724, 693}, {31900, 31623}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 36, 13624}, {1, 46, 12702}, {1, 57, 5221}, {1, 5221, 65}, {1, 5903, 11278}, {1, 11278, 5048}, {1, 12702, 3057}, {1, 13624, 2646}, {3, 3338, 354}, {3, 12704, 7957}, {7, 7288, 11375}, {35, 5045, 3748}, {36, 942, 2646}, {36, 3337, 942}, {40, 3304, 5919}, {46, 999, 3057}, {46, 3057, 5183}, {55, 3333, 17609}, {56, 57, 65}, {56, 65, 1319}, {56, 1388, 13462}, {56, 2099, 1420}, {56, 5221, 1}, {56, 11509, 1617}, {57, 1420, 3339}, {57, 3361, 56}, {63, 25524, 25917}, {65, 1319, 11011}, {79, 3582, 9955}, {388, 5435, 24914}, {553, 1125, 3649}, {553, 5298, 4870}, {942, 13624, 1}, {999, 12702, 1}, {1125, 3649, 4870}, {1125, 3916, 3683}, {1125, 4973, 3916}, {1418, 1471, 1456}, {1420, 3339, 2099}, {1434, 1447, 4059}, {1447, 4059, 24805}, {1788, 3600, 5252}, {2099, 3339, 65}, {2975, 27003, 3812}, {3218, 5253, 960}, {3333, 15803, 55}, {3340, 13462, 1388}, {3361, 15932, 13370}, {3474, 14986, 12701}, {3513, 3514, 3748}, {3555, 25440, 3689}, {3649, 5298, 1125}, {3746, 5131, 31663}, {3911, 4298, 12}, {4315, 4848, 10944}, {4860, 5204, 1}, {5045, 5122, 35}, {5049, 31663, 3746}, {5126, 31794, 1}, {5265, 21454, 3485}, {5903, 24928, 5048}, {7280, 18398, 24929}, {11278, 24928, 1}, {12009, 31794, 942}, {13388, 13389, 20182}, {15325, 24470, 12047}
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Best regards,
Peter Moses.
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