leftri rightri

This is PART 19: Centers X(36001) - X(38000)

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

X(36001) = EULER LINE INTERCEPT OF X(100)X(477)

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

X(36001) lies on these lines: {2, 3}, {74, 517}, {98, 2691}, {100, 477}, {108, 2693}, {758, 33535}, {841, 9058}, {842, 1292}, {1294, 2766}, {1297, 10101}, {2077, 2687}, {2694, 30250}, {2697, 26706}, {2752, 30257}, {3336, 20129}, {5160, 5221}, {13397, 32710}

X(36002) = EULER LINE INTERCEPT OF X(100)X(516)

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

X(36002) lies on these lines: {2, 3}, {11, 7677}, {33, 17080}, {36, 28164}, {40, 3876}, {46, 9961}, {55, 5226}, {57, 8544}, {63, 1750}, {100, 516}, {104, 28160}, {165, 3305}, {191, 31871}, {200, 3869}, {243, 24032}, {497, 33925}, {515, 13279}, {517, 3935}, {644, 23691}, {651, 1936}, {661, 1021}, {750, 1742}, {896, 9355}, {899, 9441}, {946, 34486}, {962, 3871}, {970, 11381}, {971, 3218}, {990, 4850}, {1001, 9779}, {1155, 1156}, {1173, 34800}, {1376, 5698}, {1465, 3100}, {1490, 1998}, {1617, 5274}, {1621, 1699}, {1698, 12511}, {1745, 3562}, {1754, 32911}, {1758, 2310}, {1898, 7098}, {2077, 28150}, {2346, 17718}, {2801, 5536}, {2975, 5231}, {3000, 9364}, {3219, 5927}, {3303, 3485}, {3304, 3486}, {3306, 5732}, {3474, 11502}, {3621, 8158}, {3660, 18450}, {3740, 7964}, {3746, 10624}, {3817, 5284}, {3870, 3885}, {3957, 10222}, {4297, 5253}, {4311, 5563}, {4316, 10090}, {4413, 11495}, {4420, 7957}, {4847, 5086}, {5218, 7676}, {5229, 26357}, {5259, 12571}, {5260, 19925}, {5400, 13329}, {5435, 10430}, {5527, 35293}, {5550, 8273}, {5584, 9780}, {5657, 18491}, {5658, 5905}, {5687, 20070}, {5709, 12528}, {5731, 22753}, {5735, 31164}, {5752, 12111}, {5759, 31018}, {5762, 13257}, {5805, 31019}, {5818, 35239}, {6361, 11499}, {6690, 7965}, {7360, 30807}, {7688, 10175}, {7742, 10591}, {8580, 12446}, {8715, 9589}, {9342, 10164}, {9809, 17768}, {10157, 27065}, {10167, 27003}, {10382, 11020}, {10393, 11518}, {10582, 30389}, {10902, 18483}, {11012, 31673}, {11372, 35258}, {11491, 12699}, {11678, 20588}, {11684, 31803}, {12245, 18518}, {12331, 28212}, {12607, 34687}, {12618, 32779}, {14151, 18839}, {14459, 28870}, {14512, 25954}, {15178, 29817}, {16132, 31870}, {16870, 22464}, {17763, 28850}, {18524, 28174}, {18540, 21165}, {19862, 35202}, {22334, 34259}, {22765, 28186}, {28154, 34474}, {28178, 35000}, {28182, 33814}, {31658, 35595}

X(36002) = anticomplement of X(37374)
X(36002) = excentral-hexyl-ellipse-inverse of X(2)

X(36003) = EULER LINE INTERCEPT OF X(100)X(518)

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

X(36003) lies on these lines: {2, 3}, {36, 12750}, {46, 3870}, {55, 25557}, {65, 3957}, {100, 518}, {149, 7677}, {200, 6763}, {224, 11523}, {1260, 20078}, {1617, 20075}, {1621, 5880}, {1998, 15803}, {2078, 3254}, {2646, 29817}, {3219, 5784}, {3612, 4666}, {4015, 4652}, {5096, 15447}, {5126, 13279}, {5231, 7280}, {5258, 17647}, {10090, 17010}, {10427, 30295}, {10578, 11507}, {10580, 22766}, {11492, 26394}, {11493, 26418}, {12511, 19861}, {17603, 27003}, {24541, 35202}

X(36004) = EULER LINE INTERCEPT OF X(100)X(529)

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

X(36004) lies on these lines: {2, 3}, {36, 149}, {56, 20066}, {100, 529}, {145, 34607}, {214, 5180}, {484, 519}, {516, 4881}, {2099, 21454}, {2975, 34612}, {3241, 3881}, {3582, 17010}, {3655, 35004}, {3656, 26287}, {3679, 4652}, {4293, 11239}, {4299, 20060}, {4304, 27003}, {4316, 5080}, {4325, 34637}, {4421, 34605}, {4855, 28609}, {5010, 10197}, {5204, 11235}, {5253, 15338}, {5298, 10707}, {5303, 31157}, {5434, 14882}, {5440, 17484}, {5687, 34740}, {5841, 34474}, {8715, 34690}, {9782, 35016}, {10225, 12247}, {10385, 34471}, {10483, 27529}, {11248, 34617}, {11681, 34739}, {12248, 18524}, {13199, 22765}, {15933, 30274}, {17729, 26140}, {24929, 26842}, {25055, 27186}, {28146, 35271}, {28178, 34123}, {28190, 34122}, {30282, 31019}, {31145, 34610}, {32141, 34698}

X(36004) = anticomplement of X(37375)

X(36005) = EULER LINE INTERCEPT OF X(100)X(535)

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

X(36005) lies on these lines: {2, 3}, {8, 34620}, {36, 10707}, {100, 535}, {145, 34707}, {517, 10031}, {519, 3245}, {528, 15326}, {551, 20292}, {1320, 21578}, {3218, 9963}, {3241, 3474}, {3868, 34701}, {3871, 4299}, {3874, 3885}, {4311, 11009}, {4324, 5253}, {4511, 28534}, {4677, 6763}, {4881, 28146}, {5080, 6174}, {5204, 34706}, {5563, 34649}, {6781, 33854}, {9945, 17484}, {10624, 24926}, {25557, 30332}, {28154, 35271}, {28182, 34123}, {31145, 34740}

X(36006) = EULER LINE INTERCEPT OF X(100)X(551)

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

X(36006) lies on these lines: {2, 3}, {35, 19883}, {36, 3828}, {88, 30115}, {100, 551}, {519, 5253}, {993, 19876}, {999, 31145}, {1014, 17271}, {1376, 3241}, {2975, 19875}, {3584, 10090}, {3616, 4421}, {3634, 5303}, {3653, 11491}, {3833, 15015}, {3871, 25524}, {3876, 3928}, {4413, 11194}, {4669, 5563}, {4881, 31662}, {5008, 33854}, {5041, 5277}, {5096, 20582}, {5122, 27065}, {5204, 19877}, {5330, 11531}, {5362, 34755}, {5367, 34754}, {5433, 26060}, {6437, 31473}, {9843, 11015}, {10269, 34627}, {11230, 34474}, {17614, 33179}, {19723, 19769}, {19797, 19850}, {22753, 34632}, {24473, 27003}, {25055, 25440}, {30392, 35262}

X(36007) = EULER LINE INTERCEPT OF X(8)X(101)

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

X(36007) lies on these lines: {1, 26267}, {2, 3}, {8, 101}, {51, 19717}, {78, 26265}, {102, 9057}, {154, 5278}, {184, 19742}, {1503, 25000}, {2187, 4651}, {3220, 17077}, {5011, 16830}, {7191, 8555}, {9579, 30742}, {9777, 19743}, {17751, 26232}, {17810, 19684}, {19740, 34417}, {20245, 27401}

X(36008) = EULER LINE INTERCEPT OF X(11)X(101)

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

X(36008) lies on these lines: {2, 3}, {9, 24346}, {11, 101}, {1026, 3419}, {1083, 2886}, {2690, 5520}

X(36009) = EULER LINE INTERCEPT OF X(19)X(101)

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

X(36009) lies on these lines: {1, 1831}, {2, 3}, {19, 101}, {34, 3338}, {104, 26705}, {354, 1870}, {517, 2355}, {584, 1172}, {1068, 11399}, {1614, 2194}, {1827, 24929}, {1859, 6198}, {1871, 11363}, {1875, 32636}, {3193, 10539}, {5842, 20988}, {6197, 7957}, {7680, 20989}, {7713, 12704}, {8185, 26332}, {8192, 10597}, {9625, 18406}, {9798, 10532}, {11365, 12116}, {11496, 15494}

X(36010) = EULER LINE INTERCEPT OF X(33)X(101)

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

X(36010) lies on these lines: {2, 3}, {33, 101}, {169, 212}, {281, 12329}, {5179, 5285}

X(36110) = polar conjugate of isotomic conjugate of X(37136)

X(36011) = EULER LINE INTERCEPT OF X(37)X(101)

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

X(36011) lies on these lines: {1, 1762}, {2, 3}, {37, 101}, {58, 942}, {65, 1780}, {81, 15934}, {86, 1565}, {272, 17863}, {283, 18180}, {517, 2328}, {993, 8680}, {1043, 3695}, {1125, 25361}, {1305, 1441}, {1324, 6690}, {1385, 2360}, {1408, 34489}, {1790, 13151}, {1819, 33596}, {1859, 9895}, {1905, 2299}, {1935, 20122}, {2287, 3940}, {2690, 12030}, {3185, 5248}, {3418, 31019}, {4267, 5358}, {5251, 5285}, {5708, 16948}, {5886, 17188}, {9945, 31333}, {10198, 23843}, {17194, 18443}, {23850, 25466}

X(36012) = EULER LINE INTERCEPT OF X(40)X(101)

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

X(36012) lies on these lines: {2, 3}, {40, 101}, {198, 5759}, {1604, 35514}, {2550, 15817}, {4258, 5706}

X(36013) = EULER LINE INTERCEPT OF X(45)X(101)

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

X(36013) lies on these lines: {2, 3}, {45, 101}, {184, 10246}, {759, 5132}, {993, 32935}, {1001, 11734}, {2646, 24431}, {4653, 5135}, {5248, 23844}

X(36014) = EULER LINE INTERCEPT OF X(55)X(101)

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

X(36014) lies on these lines: {2, 3}, {55, 101}, {1486, 15817}, {2194, 4251}, {2329, 15621}, {5144, 16678}

X(36015) = EULER LINE INTERCEPT OF X(58)X(101)

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

X(36015) lies on these lines: {2, 3}, {58, 101}, {81, 20760}, {110, 29330}, {1423, 18792}, {2178, 3286}, {2277, 3736}, {3781, 4269}, {5327, 15507}, {23383, 23398}

X(36016) = EULER LINE INTERCEPT OF X(63)X(101)

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

X(36016) lies on these lines: {2, 3}, {7, 2178}, {36, 35290}, {41, 1708}, {63, 101}, {172, 241}, {1214, 1951}, {1305, 20624}, {2327, 16574}, {3002, 32911}, {10902, 25935}, {11012, 26006}

X(36017) = EULER LINE INTERCEPT OF X(72)X(101)

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

X(36017) lies on these lines: {1, 25090}, {2, 3}, {58, 241}, {72, 101}, {284, 5728}, {1014, 10004}, {1214, 2299}, {1437, 18206}, {1708, 2194}, {2328, 25091}, {8748, 14192}

X(36018) = EULER LINE INTERCEPT OF X(78)X(101)

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

X(36018) lies on these lines: {2, 3}, {37, 3100}, {78, 101}, {105, 1305}, {241, 1104}, {307, 3220}, {347, 7677}, {1001, 4329}, {1295, 9057}, {1621, 3101}, {4265, 18635}, {4298, 5322}, {4314, 5310}, {5703, 27802}, {16823, 17866}, {17000, 18666}

X(36019) = EULER LINE INTERCEPT OF X(101)X(226)

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

X(36019) lies on these lines: {2, 3}, {7, 3211}, {9, 8680}, {101, 226}, {169, 1708}, {218, 948}, {239, 14054}, {673, 5728}

X(36020) = EULER LINE INTERCEPT OF X(101)X(228)

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

X(36020) lies on these lines: {2, 3}, {41, 212}, {101, 228}, {241, 18165}, {1951, 2299}

X(36021) = EULER LINE INTERCEPT OF X(101)X(239)

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

X(36021) lies on these lines: {2, 3}, {41, 26267}, {101, 239}, {2112, 16609}, {5723, 17966}, {5826, 26626}, {9057, 12032}

X(36022) = EULER LINE INTERCEPT OF X(101)X(321)

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

X(36022) lies on these lines: {2, 3}, {48, 2997}, {101, 321}, {284, 17863}, {1214, 1305}, {1441, 1474}, {2172, 14213}, {2345, 30906}, {5016, 24632}, {18815, 34079}

X(36023) = EULER LINE INTERCEPT OF X(101)X(329)

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

X(36023) lies on these lines: {2, 3}, {6, 347}, {9, 17134}, {101, 329}, {226, 1055}, {239, 20222}, {1305, 1751}, {1708, 2082}, {1730, 3101}, {6360, 19742}, {15669, 22054}

X(36024) = EULER LINE INTERCEPT OF X(101)X(346)

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

X(36024) lies on these lines: {2, 3}, {101, 346}, {106, 347}, {551, 3007}, {3100, 30115}, {3164, 30933}, {4257, 5435}, {4296, 30117}, {22240, 30904}, {30737, 30893}

X(36025) = EULER LINE INTERCEPT OF X(101)X(386)

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

X(36025) lies on these lines: {2, 3}, {37, 5132}, {101, 386}, {579, 24320}, {5089, 9895}, {5248, 20875}, {5275, 19763}, {23383, 23851}

X(36026) = EULER LINE INTERCEPT OF X(101)X(477)

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

X(36026) lies on these lines: {2, 3}, {74, 516}, {101, 477}, {841, 9057}, {1305, 32710}, {1544, 10721}, {2693, 26705}, {15035, 18653}

X(36027) = EULER LINE INTERCEPT OF X(101)X(515)

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

X(36027) lies on these lines: {2, 3}, {101, 515}, {517, 1952}, {912, 10025}, {971, 1944}, {1737, 9441}, {4511, 30807}, {5762, 17950}

X(36028) = EULER LINE INTERCEPT OF X(101)X(516)

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

X(36028) lies on these lines: {2, 3}, {101, 516}, {103, 17729}, {118, 5134}, {1434, 14520}, {1530, 28146}, {1541, 28150}, {3509, 28850}

X(36029) = EULER LINE INTERCEPT OF X(101)X(610)

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

X(36029) lies on these lines: {2, 3}, {101, 610}, {104, 1305}, {321, 10538}, {347, 999}, {515, 5285}, {517, 3101}, {577, 5317}, {942, 4296}, {1610, 14110}, {2690, 2694}, {3100, 24929}, {3576, 30265}, {4294, 9911}, {4329, 5603}, {9537, 12702}

X(36030) = EULER LINE INTERCEPT OF X(101)X(649)

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

X(36030) lies on these lines: {2, 3}, {100, 21302}, {101, 649}, {1305, 1309}, {1633, 4057}, {2737, 9057}

X(36031) = EULER LINE INTERCEPT OF X(101)X(661)

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

X(36031) lies on these lines: {2, 3}, {100, 1577}, {101, 661}, {523, 4552}, {1305, 2766}, {2691, 9057}

X(36032) = EULER LINE INTERCEPT OF X(101)X(691)

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

X(36032) lies on these lines: {2, 3}, {99, 2690}, {101, 691}, {935, 1305}, {2696, 9057}

leftri

Centers associated with trilinear products of circumcircle-P-antipodes: X(36033)-X(36151)

rightri

Let P = p : q : r (trilinears). Then the locus of the trilinear product of circumcircle-P-antipodes is the circumconic with perspector the trilinear product X(6)*P = a p : b q : c r.

Let L be a line. The trilinear product of the (real or nonreal) circumcircle intercepts of L is the trilinear pole of the X(2)-isoconjugate of the isogonal conjugate of L (or equivalently, X(6)*L). These intercepts are also circumcircle-P-antipodes for all P on L.

Contributed by Randy Hutson, January 3, 2020.

X(36033) = CENTER OF LOCUS OF TRILINEAR PRODUCT OF CIRCUMCIRCLE ANTIPODES

Barycentrics    a^3 (a^2 - b^2 - c^2) (a^5 - a^3 (b^2 + c^2) + a^2 (b^3 + c^3) - b^5 + b^3 c^2 + b^2 c^3 - c^5) : :
Trilinears    sin 2A (a sin 2A - b sin 2B - c sin 2C) : :
Trilinears    sin 2A (a^2 cos A - b^2 cos B - c^2 cos C) : :

The locus of the trilinear product of circumcircle antipodes is a circumconic that is also the locus of the barycentric product of circumcircle-X(63)-antipodes, and the locus of trilinear poles of lines passing through X(48). The conic is the isogonal conjugate of line X(240)X(522), which is the Mimosa transform of the circumcircle, and passes through X(109), X(162), X(163), X(293), X(906), X(1331), X(1795), X(1822), X(1823), X(4575), X(4592), X(21180), X(35200), X(36034), X(36036), X(36037), X(36039)-X(36053), X(36055)-X(36062). The perspector of this conic is X(48).

X(36033) lies on these lines: {31, 65}, {32, 2253}, {47, 1724}, {48, 14585}, {72, 255}, {603, 1425}, {656, 25440}, {1399, 19366}, {3142, 5348}, {3781, 23116}

X(36033) = isogonal conjugate of polar conjugate of X(1726)
X(36033) = isotomic conjugate of polar conjugate of X(2908)
X(36033) = complement of isogonal conjugate of X(23843)
X(36033) = complement of isotomic conjugate of X(21270)
X(36033) = X(2)-Ceva conjugate of X(48)
X(36033) = perspector of circumconic centered at X(48)
X(36033) = X(i)-isoconjugate of X(j) for these {i,j}: {92, 7094}, {1969, 7139}
X(36033) = trilinear product X(i)*X(j) for these {i,j}: {3, 23843}, {6, 22130}, {48, 1726}, {63, 2908}, {184, 21270}, {577, 17902}, {9247, 20926}
X(36033) = trilinear quotient X(i)/X(j) for these (i,j): (48, 7094), (1726, 92), (2908, 19), (9247, 7139), (17902, 2052), (20926, 1969), (21270, 264), (22130, 2), (23843, 4)
X(36033) = barycentric product X(i)*X(j) for these {i,j}: {1, 22130}, {3, 1726}, {48, 21270}, {63, 23843}, {69, 2908}, {184, 20926}, {255, 17902}
X(36033) = barycentric quotient X(i)/X(j) for these (i,j): (184, 7094), (1726, 264), (2908, 4), (21270, 1969), (22130, 75), (23843, 92)

X(36034) = TRILINEAR PRODUCT X(74)*X(110)

Barycentrics    a^3/((2 a^4 - a^2 (b^2 + c^2) - (b^2 - c^2)^2) (b^2 - c^2)) : :
Trilinears    csc(B - C)/(cos A - 2 cos B cos C) : :

As the trilinear product of circumcircle antipodes, X(36034) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36034) lies on these lines: {1, 36062}, {74, 36069}, {109, 1304}, {110, 36064}, {162, 656}, {163, 822}, {293, 896}, {662, 36083}, {906, 32640}, {1101, 4575}, {1331, 4570}, {1725, 36053}, {1755, 2159}, {1795, 5127}, {4592, 24041}, {6149, 35200}, {36035, 36047}

X(36034) = isogonal conjugate of X(36035)
X(36034) = isotomic conjugate of polar conjugate of X(36131)
X(36034) = trilinear pole of line X(48)X(163)
X(36034) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 36035}, {2, 1637}, {4, 9033}, {30, 523}, {76, 14398}, {92, 2631}, {107, 1650}, {115, 2407}, {125, 4240}, {264, 9409}, {338, 2420}, {339, 23347}, {477, 3258}, {512, 3260}, {525, 1990}, {526, 14254}, {656, 1784}, {661, 14206}, {690, 9214}, {1577, 2173}, {2799, 35906}, {3284, 14618}, {16230, 35912}
X(36034) = trilinear product X(i)*X(j) for these {i,j}: {2, 32640}, {3, 1304}, {63, 36131}, {69, 32715}, {74, 110}, {112, 14919}, {162, 35200}, {163, 2349}, {184, 16077}, {249, 2433}, {250, 14380}, {476, 14385}, {526, 15395}, {648, 18877}, {662, 2159}, {691, 9717}, {1494, 1576}, {2394, 23357}, {2715, 35910}, {4558, 8749}, {4575, 36119}, {15066, 32681}, {16080, 32661}
X(36034) = trilinear quotient X(i)/X(j) for these (i,j): (1, 36035), (3, 9033), (6, 1637), (32, 14398), (48, 2631), (74, 523), (99, 3260), (110, 30), (112, 1990), (162, 1784), (163, 2173), (184, 9409), (249, 2407), (250, 4240), (476, 14254), (520, 1650), (526, 3258), (662, 14206), (691, 9214), (2715, 35906), (9717, 690), (14380, 125), (14385, 526), (14574, 9407), (14919, 525), (15395, 476), (16077, 264), (16080, 14618), (18877, 647), (23357, 2420), (32640, 6), (32661, 3284), (32715, 25), (34767, 339), (35200, 656), (35908, 16230), (35910, 2799), (36117, 36130), (36119, 24006), (36131, 19)
X(36034) = barycentric product X(i)*X(j) for these {i,j}: {48, 16077}, {63, 1304}, {69, 36131}, {74, 662}, {75, 32640}, {99, 2159}, {110, 2349} {162, 14919}, {163, 1494}, {304, 32715}, {648, 35200}, {811, 18877}, {1101, 2394}, {2433, 24041}, {4558, 36119}, {4575, 16080}, {4592, 8749}, {14385, 32680}, {15066, 36083}, {35910, 36084}
X(36034) = barycentric quotient X(i)/X(j) for these (i,j): (6, 36035), (31, 1637), (48, 9033), (74, 1577), (110, 14206), (112, 1784), (163, 30), (250, 24001), (662, 3260), (822, 1650), (1101, 2407), (1304, 92), (1494, 20948), (1576, 2173), (2159, 523), (2349, 850), (2394, 23994), (2433, 1109), (4575, 11064), (8749, 24006), (14380, 20902), (14385, 32679), (14919, 14208), (16077, 1969), (18877, 656), (32640, 1), (32712, 36130), (32715, 19), (35200, 525), (36119, 14618), (36131, 4), (36142, 9214)

X(36035) = ISOGONAL CONJUGATE OF X(36034)

Barycentrics    b c (b^2 - c^2) (2 a^4 - a^2 (b^2 + c^2) - (b^2 - c^2)^2) : :
Trilinears    sin(B - C) (cos A - 2 cos B cos C) : :

X(36035) lies on these lines: {79, 35053}, {162, 823}, {240, 522}, {442, 2804}, {897, 1821}, {1099, 6739}, {1109, 2632}, {2631, 14400}, {32679, 33593}, {32680, 36096}, {36034, 36047}

X(36035) = isogonal conjugate of X(36034)
X(36035) = polar conjugate of isogonal conjugate of X(2631)
X(36035) = crossdifference of every pair of points on line X(48)X(163)
X(36035) = circle-{{X(11),X(36),X(65)}}-inverse of X(656)
X(36035) = {X(2588),X(2599)}-harmonic conjugate of X(656)
X(36035) = intersection of tangents at X(1099) and X(1109) to the inellipse centered at X(10)
X(36035) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 36034}, {2, 32640}, {3, 1304}, {63, 36131}, {69, 32715}, {74, 110}, {112, 14919}, {162, 35200}, {163, 2349}, {184, 16077}, {249, 2433}, {250, 14380}, {476, 14385}, {526, 15395}, {648, 18877}, {662, 2159}, {691, 9717}, {1494, 1576}, {2394, 23357}, {4558, 8749}, {4575, 36119}, {16080, 32661}
X(36035) = trilinear product X(i)*X(j) for these {i,j}: {2, 1637}, {4, 9033}, {30, 523}, {76, 14398}, {92, 2631}, {107, 1650}, {115, 2407}, {125, 4240}, {264, 9409}, {338, 2420}, {339, 23347}, {476, 3258}, {512, 3260}, {525, 1990}, {526, 14254}, {656, 1784}, {661, 14206}, {690, 9214}, {850, 1495}, {1577, 2173}, {2394, 3163}, {2501, 11064}, {3284, 14618}, {3708, 24001}, {9406, 20948}
X(36035) = trilinear quotient X(i)/X(j) for these (i,j): (1, 36034), (4, 1304), (6, 32640), (19, 36131), (25, 32715), (30, 110), (115, 2433), (125, 14380), (264, 16077), (338, 2394), (339, 34767), (476, 15395), (523, 74), (525, 14919), (526, 14385), (647, 18877), (656, 35200), (661, 2159), (690, 9717), (850, 1494), (1495, 1576), (1577, 2349), (1637, 6), (1650, 520), (1784, 162), (1990, 112), (2173, 163), (2407, 249), (2420, 23357), (2501, 8749), (2631, 48), (3163, 2420), (3258, 526), (3260, 99), (3284, 32661), (4240, 250), (9033, 3), (9214, 691), (9407, 14574), (9409, 184), (11064, 4558), (14206, 662), (14254, 476), (14398, 32), (14618, 16080), (24006, 36119), (36130, 36117)
X(36035) = barycentric product X(i)*X(j) for these {i,j}: {30, 1577}, {75, 1637}, {92, 9033}, {125, 24001}, {264, 2631}, {523, 14206}, {525, 1784}, {561, 14398}, {661, 3260}, {823, 1650}, {850, 2173}, {1099, 2394}, {1109, 2407}, {1495, 20948}, {1969, 9409}, {1990, 14208}, {2420, 23994}, {3258, 32680}, {4240, 20902}, {11064, 24006}, {14254, 32679}
X(36035) = barycentric quotient X(i)/X(j) for these (i,j): (6, 36034), (25, 36131), (30, 662), (92, 16077), (512, 2159), (523, 2349), (647, 35200), (656, 14919), (661, 74), (1099, 2407), (1495, 163), (1577, 1494), (1637, 1), (1650, 24018), (1784, 648), (1990, 162), (2173, 110), (2420, 1101), (2501, 36119), (2631, 3), (3258, 32679), (3260, 799), (3284, 4575), (3708, 14380), (6739, 4585), (9033, 63), (9406, 1576), (9409, 48), (11064, 4592), (14206, 99), (14254, 32680), (14398, 31), (20902, 34767), (24001, 18020), (24006, 16080)

X(36036) = TRILINEAR PRODUCT X(98)*X(99)

Barycentrics    b c/((b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2)) : :
Barycentrics    csc A csc(B - C) sec(A + ω) : :
Barycentrics    (csc A)/(b^2 sin 2B - c^2 sin 2C) : :

As the trilinear product of circumcircle antipodes, X(36036) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48), and as the trilinear product of Steiner circumellipse antipodes, X(36036) also lies on conic {{A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75).

X(36036) lies on these lines: {98, 36066}, {99, 36065}, {109, 22456}, {162, 23999}, {163, 811}, {293, 1966}, {336, 14210}, {662, 36132}, {668, 906}, {789, 2715}, {799, 4575}, {1331, 1978}, {1733, 36051}, {1795, 5209}, {1821, 14206}, {1910, 36133}, {3401, 3404}, {4554, 17932}, {4592, 4602}, {4622, 20568}, {18031, 36057}

X(36036) = trilinear pole of line X(48)X(75)
X(36036) = trilinear product of Steiner circumellipse intercepts of line X(2)X(98)
X(36036) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 17994}, {6, 3569}, {25, 684}, {32, 2799}, {115, 14966}, {232, 647}, {237, 523}, {240, 810}, {297, 3049}, {511, 512}, {520, 34854}, {525, 2211}, {661, 1755}, {688, 20022}, {798, 1959}, {850, 9418}, {1084, 2396}, {1495, 32112}, {1577, 9417}, {1974, 6333}, {2084, 3405}, {2395, 11672}, {2421, 3124}, {2501, 3289}, {2643, 23997}, {4230, 20975}, {9409, 35908}, {14398, 35910}
X(36036) = trilinear product X(i)*X(j) for these {i,j}: {2, 2966}, {3, 22456}, {4, 17932}, {69, 685}, {75, 36084}, {76, 2715}, {98, 99}, {107, 6394}, {110, 290}, {162, 336}, {183, 6037}, {248, 6331}, {287, 648}, {293, 811}, {304, 36104}, {305, 32696}, {662, 1821}, {670, 1976}, {799, 1910}, {850, 868}, {879, 18020}, {1576, 18024}, {2395, 4590}, {2421, 34536}, {2422, 34537}, {3403, 36132}, {3404, 4593}, {4558, 16081}, {4563, 6531}, {4577, 20021}, {4592, 36120}, {16077, 35912}, {20023, 32716}
X(36036) = trilinear quotient X(i)/X(j) for these (i,j): (2, 3569), (4, 17994), (69, 684), (76, 2799), (98, 512), (99, 511), (107, 34854), (110, 237), (112, 2211), (163, 9417), (248, 3049), (249, 14966), (287, 647), (290, 523), (293, 810), (305, 6333), (336, 656), (648, 232), (662, 1755), (685, 25), (689, 20022), (799, 1959), (811, 240), (868, 1576), (879, 20975), (1494, 32112), (1576, 9418), (1821, 661), (1910, 798), (2395, 3124), (2421, 11672), (2422, 1084), (2715, 32), (2966, 6), (3404, 2084), (4558, 3289), (4590, 2421), (4593, 3405), (6037, 263), (6331, 297), (6394, 520), (6531, 2489), (16077, 35908), (16081, 2501), (17932, 3), (18020, 4230), (18024, 850), (20021, 3005), (22456, 4), (24041, 23997), (32696, 1974), (34536, 2395), (34537, 2396), (35906, 14398), (35912, 9409), (36084, 31), (36104, 1973), (36132, 3402)
X(36036) = barycentric product X(i)*X(j) for these {i,j}: {63, 22456}, {75, 2966}, {76, 36084}, {92, 17932}, {98, 799}, {99, 1821}, {163, 18024}, {287, 811}, {290, 662}, {293, 6331}, {304, 685}, {305, 36104}, {336, 648}, {561, 2715}, {670, 1910}, {689, 3404}, {823, 6394}, {1976, 4602}, {2395, 24037}, {3403, 6037}, {4563, 36120}, {4592, 16081}, {4593, 20021}, {20023, 36132}, {24041, 34536}
X(36036) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3569), (19, 17994), (31, 2491), (63, 684), (92, 16230), (98, 661), (99, 1959), (110, 1755), (162, 232), (163, 237), (248, 810), (249, 23997), (287, 656), (290, 1577), (293, 647), (304, 6333), (336, 525), (648, 240), (662, 511), (685, 19), (799, 325), (811, 297), (823, 6530), (879, 3708), (1101, 14966), (1577, 868), (1821, 523), (1910, 512), (1976, 798), (2349, 32112), (2395, 2643), (2421, 23996), (2715, 31), (2966, 1), (3404, 3005), (4575, 3289), (4593, 20022), (6037, 2186), (6331, 27818), (6394, 24018), (16081, 24006), (17932, 63), (18024, 20948), (20021, 8061), (22456, 92), (23997, 11672), (24019, 34854), (24037, 2396), (24041, 2421), (32696, 1973), (32716, 3402), (34536, 1109), (35912, 2631), (36084, 6), (36085, 5968), (36104, 25), (36120, 2501), (36132, 263)

X(36037) = TRILINEAR PRODUCT X(100)*X(104)

Barycentrics    a/((b - c) (a^2 (b + c) - 2 a b c - (b - c)^2 (b + c))) : :
Trilinears    1/(sin 2B cot(C/2) - sin 2C cot(B/2)) : :
Trilinears    directed distance from A to Sherman line : :

As the trilinear product of circumcircle antipodes, X(36037) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

Let A'B'C' be the circumcevian triangle of X(900). Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear on line X(519)X(4768). The lines AA", BB", CC" concur in X(36037).

X(36037) lies on these lines: {59, 100}, {101, 36137}, {104, 517}, {109, 522}, {163, 1021}, {190, 36090}, {293, 17763}, {519, 1795}, {643, 4570}, {644, 906}, {664, 4025}, {677, 3935}, {765, 1331}, {909, 34075}, {1737, 5081}, {1809, 4511}, {1936, 2342}, {2222, 3738}, {2250, 36060}, {2398, 2401}, {3075, 31680}, {4242, 36040}, {4585, 9268}, {4592, 4600}, {6740, 16704}, {8851, 34858}, {12649, 14266}, {24035, 36044}, {32669, 36147}, {33649, 34772}

X(36037) = isogonal conjugate of X(1769)
X(36037) = isotomic conjugate of X(36038)
X(36037) = cevapoint of X(i) and X(j) for these {i,j}: {31, 1635}, {522, 1737}, {656, 758}
X(36037) = trilinear pole of line X(9)X(48) (the Fermat axis of the excentral triangle and of the 2nd extouch triangle)
X(36037) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1769}, {2, 3310}, {4, 8677}, {6, 10015}, {11, 23981}, {31, 36038}, {56, 2804}, {264, 23220}, {513, 517}, {514, 2183}, {521, 1875}, {522, 1457}, {523, 859}, {649, 908}, {650, 1465}, {663, 22464}, {667, 3262}, {900, 14260}, {1015, 2397}, {1086, 2427}, {1145, 23345}, {1459, 1785}, {7649, 22350}
X(36037) = trilinear product X(i)*X(j) for these {i,j}: {2, 32641}, {3, 1309}, {6, 13136}, {8, 2720}, {69, 14776}, {78, 36110}, {100, 104}, {101, 34234}, {108, 1809}, {190, 909}, {312, 32669}, {662, 2250}, {664, 2342}, {668, 34858}, {692, 18816}, {906, 16082}, {997, 36090}, {1016, 2423}, {1252, 2401}, {1331, 36123}, {1795, 1897}, {2167, 35321}, {6099, 14266}, {6335, 14578}, {10428, 17780}, {17740, 32685}
X(36037) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1769), (2, 10015), (3, 8677), (6, 3310), (8, 2804), (59, 23981), (75, 36038), (100, 517), (101, 2183), (104, 513), (108, 1875), (109, 1457), (110, 859), (184, 23220), (190, 908), (651, 1465), (664, 22464), (668, 3262), (901, 14260), (909, 649), (1016, 2397), (1252, 2427), (1309, 4), (1331, 22350), (1795, 1459), (1809, 521), (1897, 1785), (2250, 661), (2342, 663), (2401, 1086), (2423, 1015), (2720, 56), (10428, 23345), (13136, 2), (14776, 25), (16082, 17924), (17780, 1145), (18816, 693), (32641, 6), (32669, 604), (34234, 514), (34858, 667), (35321, 1953), (36090, 998), (36110, 34)
X(36037) = barycentric product X(i)*X(j) for these {i,j}: {1, 13136}, {63, 1309}, {75, 32641}, {95, 35321}, {99, 2250}, {100, 34234}, {101, 18816}, {104, 190}, {304, 14776}, {312, 2720}, {345, 36110}, {653, 1809}, {668, 909}, {765, 2401}, {1331, 16082}, {1332, 36123}, {1795, 6335}, {1978, 34858}, {2342, 4554}, {2423, 7035}, {3596, 32669}, {10428, 24004}, {17740, 36090}
X(36037) = barycentric quotient X(i)/X(j) for these (i,j): (1, 10015), (2, 36038), (6, 1769), (9, 2804), (31, 3310), (44, 23757), (48, 8677), (59, 24029), (81, 23788), (100, 908), (101, 517), (104, 514), (109, 1465), (163, 859), (190, 3262), (644, 6735), (651, 22464), (662, 17139), (692, 2183), (765, 2397), (906, 22350), (909, 513), (1023, 1145), (1309, 92), (1635, 3259), (1795, 905), (1809, 6332), (2250, 523), (2342, 650), (2401, 1111), (2423, 244), (2720, 57), (10428, 1022), (13136, 75), (14578, 1459), (14776, 19), (18816, 3261), (32641, 1), (32665, 14260), (32669, 56), (32685, 998), (34234, 693), (34858, 649), (35321, 51), (36110, 278), (36123, 17924), (36137, 957)

X(36038) = TRILINEAR PRODUCT OF STEINER CIRCUMELLIPSE INTERCEPTS OF SHERMAN LINE

Barycentrics    b c (b - c) (a^2 (b + c) - 2 a b c - (b - c)^2 (b + c)) : :
Barycentrics    (csc A) (sin 2B cot(C/2) - sin 2C cot(B/2)) : :

X(36038) lies on these lines: {149, 150}, {522, 693}, {664, 1897}, {903, 18816}, {1111, 3120}, {1577, 2610}, {1769, 23788}, {2785, 3766}, {3762, 4080}, {4106, 15313}, {4978, 17496}, {5990, 5991}, {6332, 17924}, {14208, 20948}, {14304, 18815}, {17894, 35518}, {17898, 20294}, {23595, 24018}

X(36038) = isotomic conjugate of X(36037)
X(36038) = crossdifference of every pair of points on line X(41)X(9247)
X(36038) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 14776}, {6, 32641}, {9, 32669}, {31, 36037}, {32, 13136}, {55, 2720}, {100, 34858}, {101, 909}, {104, 692}, {109, 2342}, {184, 1309}, {212, 36110}, {1252, 2423}, {1783, 14578}, {32739, 34234}
X(36038) = trilinear product X(i)*X(j) for these {i,j}: {2, 10015}, {7, 2804}, {10, 23788}, {75, 1769}, {76, 3310}, {264, 8677}, {513, 3262}, {514, 908}, {517, 693}, {522, 22464}, {523, 17139}, {850, 859}, {903, 23757}, {1086, 2397}, {1457, 35519}, {1465, 4391}, {1875, 35518}, {2183, 3261}, {3259, 4555}, {3676, 6735}, {4858, 24029}, {18022, 23220}
X(36038) = trilinear quotient X(i)/X(j) for these (i,j): (2, 32641), (4, 14776), (7, 2720), (57, 32669), (75, 36037), (76, 13136), (264, 1309), (273, 36110), (514, 909), (517, 692), (522, 2342), (693, 104), (859, 951), (905, 14578), (908, 101), (1086, 2423), (1465, 1415), (1769, 31), (2183, 32739), (2397, 1252), (2804, 55), (3259, 1960), (3261, 34234), (3262, 100), (3310, 32), (6735, 3939), (8677, 184), (10015, 6), (17139, 110), (22350, 32656), (22464, 109), (23220, 14575), (23757, 902), (23788, 58), (24029, 2149), (35518, 1809)
X(36038) = barycentric product X(i)*X(j) for these {i,j}: {75, 10015}, {76, 1769}, {321, 23788}, {514, 3262}, {517, 3261}, {561, 3310}, {693, 908}, {859, 20948}, {1111, 2397}, {1465, 35519}, {1577, 17139}, {1969, 8677}, {4391, 22464}, {6735, 24002}, {20568, 23757}
X(36038) = barycentric quotient X(i)/X(j) for these (i,j): (1, 32641), (2, 36037), (19, 14776), (56, 32669), (57, 2720), (75, 13136), (92, 1309), (244, 2423), (278, 36110), (513, 909), (514, 104), (517, 101), (523, 2250), (650, 2342), (693, 34234), (859, 163), (905, 1795), (908, 100), (1111, 2401), (1457, 1415), (1459, 14578), (1465, 109), (1769, 6), (1875, 32674), (2183, 692), (2397, 765), (3259, 1635), (3261, 18816), (3262, 190), (3310, 31), (6332, 1809), (6735, 644), (8677, 48), (10015, 1), (17139, 662), (17924, 36123), (22350, 906), (22464, 651), (23220, 9247), (23757, 44), (23788, 81), (24029, 59)

X(36039) = TRILINEAR PRODUCT X(101)*X(103)

Barycentrics    a^3/((b - c) (2 a^3 - a^2 b - a^2 c - b^3 + b^2 c + b c^2 - c^3)) : :
Trilinears    a^2/((b - c) ((c - a) cot B - (a - b) cot C)) : :

As the trilinear product of circumcircle antipodes, X(36039) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36039) lies on these lines: {100, 32684}, {101, 1262}, {103, 672}, {109, 652}, {162, 1021}, {293, 1757}, {677, 1252}, {692, 32721}, {906, 1110}, {911, 32665}, {1734, 1783}, {1736, 8558}, {1795, 2338}, {2149, 36054}, {2424, 2427}, {4567, 4592}, {8693, 35184}, {32698, 36052}, {36087, 36101}

X(36039) = trilinear pole of line X(48)X(692)
X(36039) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 676}, {514, 516}, {649, 35517}, {693, 910}, {1086, 2398}, {1146, 23973}, {1886, 4025}, {2426, 23989}, {7649, 26006}
X(36039) = trilinear product X(i)*X(j) for these {i,j}: {2, 32642}, {6, 677}, {100, 911}, {101, 103}, {109, 2338}, {692, 36101}, {906, 36122}, {1252, 2424}, {1815, 8750}, {2400, 23990}, {3730, 35184}, {5223, 36136}, {18025, 32739}, {29616, 32721}
X(36039) = trilinear quotient X(i)/X(j) for these (i,j): (6, 676), (101, 516), (103, 514), (190, 35517), (677, 2), (692, 910), (911, 513), (1252, 2398), (1262, 23973), (1331, 26006), (1815, 4025), (2338, 522), (2400, 23989), (2424, 1086), (8750, 1886), (18025, 3261), (23990, 2426), (32642, 6), (35184, 14377), (36101, 693), (36122, 17924)
X(36039) = barycentric product X(i)*X(j) for these {i,j}: {1, 677}, {75, 32642}, {100, 103}, {101, 36101}, {190, 911}, {651, 2338}, {692, 18025}, {765, 2424}, {1110, 2400}, {1331, 36122}, {1783, 1815}, {3681, 35184}, {29616, 36136}
X(36039) = barycentric quotient X(i)/X(j) for these (i,j): (100, 35517), (103, 693), (677, 75), (692, 516), (906, 26006), (911, 514), (1110, 2398), (1262, 24015), (1815, 15413), (2338, 4391), (2424, 1111), (32642, 1), (32739, 910), (36101, 3261)

X(36040) = TRILINEAR PRODUCT X(102)*X(109)

Barycentrics    a^3/((b - c) (a - b - c) (2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2)) : :
Trilinears    a/((cos B - cos C) (sin B (sec A - sec B) + sin C (sec A - sec C))) : :

As the trilinear product of circumcircle antipodes, X(36040) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36040) lies on these lines: {36, 102}, {59, 1331}, {108, 21189}, {109, 36067}, {163, 32667}, {293, 1758}, {651, 36088}, {652, 32674}, {672, 15629}, {906, 2149}, {1415, 36135}, {1735, 15379}, {2361, 36055}, {3911, 15633}, {4242, 36037}, {6081, 8059}, {24027, 36059}, {32677, 36141}, {32735, 36057}, {36094, 36100}

X(36040) = isogonal conjugate of X(14304)
X(36040) = trilinear pole of line X(48)X(1415)
X(36040) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 14304}, {280, 6087}, {515, 522}, {663, 35516}, {2182, 4391}, {2425, 23978}, {3239, 34050}, {6332, 8755}
X(36040) = trilinear product X(i)*X(j) for these {i,j}: {2, 32643}, {3, 36067}, {63, 32667}, {102, 109}, {108, 36055}, {221, 6081}, {651, 32677}, {1262, 2432}, {1415, 36100}, {1461, 15629}, {2399, 23979}, {10571, 35183}
X(36040) = trilinear quotient X(i)/X(j) for these (i,j): (1, 14304), (102, 522), (109, 515), (221, 6087), (664, 35516), (1461, 34050), (2399, 23978), (6081, 280), (15629, 3239), (23979, 2425), (32643, 6), (32667, 19), (32674, 8755), (32677, 650), (34393, 35519), (35183, 10570), (36055, 521), (36067, 4), (36100, 4391)
X(36040) = barycentric product X(i)*X(j) for these {i,j}: {63, 36067}, {69, 32667}, {75, 32643}, {102, 651}, {109, 36100}, {223, 6081}, {653, 36055}, {664, 32677}, {934, 15629}, {1415, 34393}, {1813, 36121}, {2399, 24027}, {2432, 7045}, {17080, 35183}
X(36040) = barycentric quotient X(i)/X(j) for these (i,j): (6, 14304), (102, 4391), (651, 35516), (1415, 515), (2432, 24026), (15629, 4397), (24027, 2406), (32643, 1), (32667, 4), (32677, 522), (36055, 6332), (36067, 92), (36100, 35519)

X(36041) = TRILINEAR PRODUCT X(105)*X(1292)

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

As the trilinear product of circumcircle antipodes, X(36041) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36041) lies on these lines: {906, 919}, {1331, 36086}, {1738, 36124}, {21185, 36111}, {32735, 36059}

X(36041) = trilinear pole of line X(48)X(1438)
X(36041) = X(i)-isoconjugate of X(j) for these {i,j}: {218, 918}, {344, 665}, {518, 3309}, {672, 4468}, {2254, 3870}, {2402, 6184}
X(36041) = trilinear product X(i)*X(j) for these {i,j}: {2, 32644}, {105, 1292}, {277, 919}, {2191, 36086}, {2428, 6185}, {6601, 32735}
X(36041) = trilinear quotient X(i)/X(j) for these (i,j): (105, 3309), (277, 918), (666, 344), (673, 4468), (919, 218), (1292, 518), (2191, 2254), (2428, 6184), (6185, 2402), (32644, 6), (32735, 1617), (36086, 3870)
X(36041) = barycentric product X(i)*X(j) for these {i,j}: {75, 32644}, {277, 36086}, {666, 2191}, {673, 1292}, {6601, 36146}
X(36041) = barycentric quotient X(i)/X(j) for these (i,j): (105, 4468), (1292, 3912), (2191, 918), (32644, 1), (36086, 344), (36146, 6604)

X(36042) = TRILINEAR PRODUCT X(106)*X(1293)

Barycentrics    a^3/((b - c) (2 a - b - c) (3 a - b - c)) : :

As the trilinear product of circumcircle antipodes, X(36042) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36042) lies on these lines: {901, 1293}, {906, 32645}, {1739, 36125}, {1795, 10428}, {3445, 14260}, {4592, 4622}, {16944, 16945}, {27834, 36091}

X(36042) = trilinear pole of line X(48)X(9456)
X(36042) = X(i)-isoconjugate of X(j) for these {i,j}: {44, 4462}, {145, 900}, {513, 4487}, {519, 3667}, {1743, 3762}, {2403, 4370}, {3161, 30725}, {3756, 17780}, {3911, 4521}, {4358, 4394}, {6544, 31227}
X(36042) = trilinear product X(i)*X(j) for these {i,j}: {2, 32645}, {106, 1293}, {901, 3445}, {2226, 2429}, {4373, 32719}, {8056, 32665}, {9456, 27834}
X(36042) = trilinear quotient X(i)/X(j) for these (i,j): (88, 4462), (100, 4487), (106, 3667), (901, 145), (1293, 519), (2226, 2403), (2316, 4521), (2429, 4370), (3445, 900), (4638, 31227), (4674, 4404), (5382, 24004), (5548, 3161), (8056, 3762), (9456, 4394), (23345, 3756), (27834, 4358), (32645, 6), (32665, 1743), (32719, 3052)
X(36042) = barycentric product X(i)*X(j) for these {i,j}: {75, 32645}, {88, 1293}, {106, 27834}, {679, 2429}, {901, 8056}, {3257, 3445}, {4373, 32665}, {4582, 16945}, {5382, 23345}, {5548, 19604}
X(36042) = barycentric quotient X(i)/X(j) for these (i,j): (101, 4487), (106, 4462), (901, 18743), (1293, 4358), (2429, 4738), (3445, 3762), (16945, 30725), (27834, 3264), (32645, 1), (32665, 145), (32719, 1743)

X(36043) = TRILINEAR PRODUCT X(107)*X(1294)

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

As the trilinear product of circumcircle antipodes, X(36043) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36043) lies on these lines: {162, 24021}, {906, 32646}, {1294, 36068}, {1784, 35200}, {4575, 24000}, {4592, 23999}, {8720, 12271}, {17898, 36126}

X(36043) = trilinear pole of line X(48)X(24019)
X(36043) = X(520)-isoconjugate of X(6000)
X(36043) = trilinear product X(i)*X(j) for these {i,j}: {2, 32646}, {107, 1294}, {2416, 23590}
X(36043) = trilinear quotient X(i)/X(j) for these (i,j): (107, 6000), (1294, 520), (23590, 2442), (32646, 6)
X(36043) = barycentric product X(i)*X(j) for these {i,j}: {75, 32646}, {823, 1294}, {2416, 24021}
X(36043) = barycentric quotient X(i)/X(j) for these (i,j): (1294, 24018), (2416, 24020), (24019, 6000), (24021, 2404), (32646, 1)

X(36044) = TRILINEAR PRODUCT X(108)*X(1295)

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

As the trilinear product of circumcircle antipodes, X(36044) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36044) lies on these lines: {109, 24033}, {906, 7115}, {1295, 1465}, {1331, 7012}, {1785, 1795}, {21186, 36127}, {24035, 36037}

X(36044) = trilinear pole of line X(48)X(2331)
X(36044) = X(i)-isoconjugate of X(j) for these {i,j}: {521, 6001}, {2405, 35072}, {2443, 23983}
X(36044) = trilinear product X(i)*X(j) for these {i,j}: {2, 32647}, {108, 1295}, {2417, 23985}, {2431, 23984}
X(36044) = trilinear quotient X(i)/X(j) for these (i,j): (108, 6001), (1295, 521), (2417, 23983), (2431, 35072), (23984, 2405), (23985, 2443), (32647, 6)
X(36044) = barycentric product X(i)*X(j) for these {i,j}: {75, 32647}, {653, 1295}, {2417, 24033}, {2431, 24032}
X(36044) = barycentric quotient X(i)/X(j) for these (i,j): (1295, 6332), (2431, 24031), (24033, 2405), (32647, 1)

X(36045) = TRILINEAR PRODUCT X(111)*X(1296)

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

As the trilinear product of circumcircle antipodes, X(36045) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36045) lies on these lines: {293, 17955}, {906, 32648}, {1296, 36070}, {4575, 36142}, {4592, 36085}

X(36045) = trilinear pole of line X(48)X(923)
X(36045) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 9125}, {524, 1499}, {690, 1992}, {896, 14207}, {1384, 35522}, {2408, 2482}, {3266, 8644}
X(36045) = trilinear product X(i)*X(j) for these {i,j}: {2, 32648}, {111, 1296}, {691, 21448}, {2434, 10630), {5485, 32729}, {32740, 35179}
X(36045) = trilinear quotient X(i)/X(j) for these (i,j): (6, 9125), (111, 1499), (691, 1992), (897, 14207), (1296, 524), (2434, 2482), (5485, 35522), (10630, 2408), (21448, 690), (32648, 6), (32729, 1384), (32740, 8644), (35179, 3266)
X(36045) = barycentric product X(i)*X(j) for these {i,j}: {75, 32648}, {897, 1296}, {923, 35179}, {5485, 36142}, {21448, 36085}
X(36045) = barycentric quotient X(i)/X(j) for these (i,j): (1296, 14210), (32648, 1), (36085, 11059), (36142, 1992)

X(36046) = TRILINEAR PRODUCT X(112)*X(1297)

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

As the trilinear product of circumcircle antipodes, X(36046) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36046) lies on these lines: {109, 32687}, {162, 36092}, {240, 293}, {906, 32649}, {1297, 36071}, {35200, 36131}

X(36046) = isogonal conjugate of polar conjugate of X(36092)
X(36046) = trilinear pole of line X(48)X(32676)
X(36046) = X(i)-isoconjugate of X(j) for these {i,j}: {441, 523}, {525, 1503}, {850, 8779}, {1577, 8766}, {2409, 15526}, {2419, 23976}, {3265, 16318}
X(36046) = trilinear product X(i)*X(j) for these {i,j}: {2, 32649}, {3, 32687}, {48, 36092}, {112, 1297}, {163, 8767}, {1576, 6330}, {2435, 23964}
X(36046) = trilinear quotient X(i)/X(j) for these (i,j): (110, 441), (112, 1503), (163, 8766), (1297, 525), (1576, 8779), (2435, 15526), (2445, 23976), (6330, 850), (8767, 1577), (23964, 2409), (32649, 6), (32687, 4), (32713, 16318), (35140, 3267), (36092, 92)
X(36046) = barycentric product X(i)*X(j) for these {i,j}: {3, 36092}, {63, 32687}, {75, 32649}, {110, 8767}, {162, 1297}, {163, 6330}, {2435, 24000}, {32676, 35140}
X(36046) = barycentric quotient X(i)/X(j) for these (i,j): (162, 30737), (163, 441), (1297, 14208), (6330, 20948), (8767, 850), (32649, 1), (32676, 1503), (32687, 92), (36092, 264)

X(36047) = TRILINEAR PRODUCT X(476)*X(477)

Barycentrics    b c/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2) (a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4))) : :
Trilinears    1/((1 + 2 cos 2A) sin(B - C) (4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C))) : :

As the trilinear product of circumcircle antipodes, X(36047) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36047) lies on these lines: {906, 32650}, {2166, 35200}, {32680, 36097}, {36034, 36035}, {36096, 36102}, {36116, 36130}, {36143, 36151}

X(36047) = trilinear pole of line X(48)X(32678)
X(36047) = X(i)-isoconjugate of X(j) for these {i,j}: {526, 5663}, {14270, 35520}
X(36047) = trilinear product X(i)*X(j) for these {i,j}: {2, 32650}, {476, 477}, {1989, 30528}, {32678, 36102}, {32680, 36151}, {36061, 36130}, {36062, 36129}
X(36047) = trilinear quotient X(i)/X(j) for these (i,j): (476, 5663), (477, 526), (30528, 323), (32650, 6), (35139, 35520), (36102, 32679), (36129, 36063), (36151, 2624)
X(36047) = barycentric product X(i)*X(j) for these {i,j}: {75, 32650}, {476, 36102}, {477, 32680}, {2166, 30528}, {35139, 36151}
X(36047) = barycentric quotient X(i)/X(j) for these (i,j): (32650, 1), (32680, 35520), (36102, 3268), (36151, 526)

X(36048) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(7)

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

As the trilinear product of circumcircle antipodes, X(36048) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48), and as the trilinear product of circumcircle-X(7)-antipodes, X(36048) also lies on conic {{A,B,C,X(651),X(664),X(934)}} with center X(223) and perspector X(57).

X(36048) lies on these lines: {109, 36118}, {163, 1020}, {651, 906}, {664, 1331}, {934, 15439}, {943, 36056}, {1414, 4575}, {1795, 3664}, {1847, 3215}, {2982, 34056}, {4592, 4625}

X(36048) = trilinear pole of line X(48)X(57)
X(36048) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 33525}, {521, 1859}, {522, 14547}, {523, 8021}, {657, 5249}, {942, 3900}
X(36048) = trilinear product X(i)*X(j) for these {i,j}: {2, 32651}, {7, 15439}, {651, 2982}, {658, 2259}, {934, 943}, {1794, 36118}
X(36048) = trilinear quotient X(i)/X(j) for these (i,j): (6, 33525), (108, 1859), (109, 14547), (110, 8021), (658, 5249), (934, 942), (943, 3900), (2259, 657), (2982, 650), (15439, 55), (32651, 6), (36059, 23207), (36118, 1838)
X(36048) = barycentric product X(i)*X(j) for these {i,j}: {85, 15439}, {658, 943}, {664, 2982}, {1794, 13149}, {2259, 4569}
X(36048) = barycentric quotient X(i)/X(j) for these (i,j): (163, 8021), (651, 6734), (934, 5249), (943, 3239), (1020, 442), (2259, 3900), (2982, 522), (15439, 9)

X(36049) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(9)

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

As the trilinear product of circumcircle antipodes, X(36049) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36049) lies on these lines: {6, 268}, {31, 24010}, {84, 294}, {101, 2425}, {108, 8064}, {109, 1783}, {218, 1433}, {255, 3341}, {282, 1743}, {293, 5247}, {579, 1436}, {644, 1331}, {645, 4592}, {652, 32674}, {905, 6614}, {906, 3939}, {1422, 1708}, {1723, 7129}, {1903, 2341}, {2192, 4845}, {2357, 5547}, {4575, 5546}, {6081, 26715}, {7078, 8886}, {14331, 36044}, {14837, 36118}, {15291, 15627}

X(36049) = isogonal conjugate of X(14837)
X(36049) = cevapoint of X(i) and X(j) for these {i,j}: {6, 652}, {31, 657}, {650, 1108}
X(36049) = crosssum of X(i) and X(j) for these {i,j}: {656, 6587}, {6129, 14298}
X(36049) = trilinear pole of line X(48)X(55)
X(36049) = crossdifference of every pair of points on line X(3318)X(6087)
X(36049) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 14837}, {2, 6129}, {6, 17896}, {40, 514}, {57, 8058}, {196, 521}, {198, 693}, {208, 6332}, {221, 4391}, {223, 522}, {273, 10397}, {322, 649}, {329, 513}, {342, 652}, {347, 650}, {523, 1817}, {661, 8822}, {905, 7952}, {1577, 2360}, {2187, 3261}, {2199, 35519}, {2324, 3676}, {3345, 8063}, {3669, 7080}, {3676, 4876}, {3900, 14256}, {6087, 36100}, {7074, 24002}, {7078, 17924}
X(36049) = trilinear product X(i)*X(j) for these {i,j}: {2, 32652}, {6, 13138}, {9, 8059}, {84, 101}, {100, 1436}, {108, 268}, {109, 282}, {110, 1903}, {189, 692}, {190, 2208}, {271, 32674}, {280, 1415}, {309, 32739}, {644, 1413}, {651, 2192}, {653, 2188}, {662, 2357}, {664, 7118}, {934, 7367}, {1331, 7129}, {1332, 7151}, {1422, 3939}, {1433, 1783}, {1490, 8064}, {2182, 6081}, {7003, 36059}, {7020, 32660}
X(36049) = trilinear quotient X(i)/X(j) for these (i,j): (1, 14837), (2, 17896), (6, 6129), (9, 8058), (84, 514), (100, 329), (101, 40), (108, 196), (109, 223), (110, 1817), (163, 2360), (189, 693), (190, 322), (212, 10397), (268, 521), (271, 6332), (280, 4391), (282, 522), (309, 3261), (644, 7080), (651, 347), (653, 342), (662, 8822), (692, 198), (906, 7078), (934, 14256), (1413, 3669), (1415, 221), (1429, 3676), (1433, 905), (1436, 513), (1440, 24002), (1490, 8063), (1783, 7952), (1903, 523), (2182, 6087), (2188, 652), (2192, 650), (2208, 649), (2357, 661), (3939, 2324), (6081, 36100), (7118, 663), (7129, 7649), (7151, 6591), (7367, 3900), (8059, 57), (8064, 3345), (8808, 4077), (13138, 2), (32652, 6), (32660, 7114), (32674, 208), (32739, 2187), (34404, 35519), (36059, 7011)
X(36049) = barycentric product X(i)*X(j) for these {i,j}: {1, 13138}, {8, 8059}, {75, 32652}, {84, 100}, {99, 662}, {101, 189}, {108, 271}, {109, 280}, {190, 1436}, {268, 653}, {282, 651}, {309, 692}, {644, 1422}, {658, 7367}, {662, 1903}, {664, 2192}, {668, 2208}, {1332, 7129}, {1413, 3699}, {1415, 34404}, {1433, 1897}, {1440, 3939}, {2188, 18026}, {4554, 7118}, {4561, 7151}, {5546, 8808}, {7020, 36059}
X(36049) = barycentric quotient X(i)/X(j) for these (i,j): (1, 17896), (6, 14837), (84, 693), (99, 1577), (100, 322), (101, 329), (108, 342), (109, 347), (110, 8822), (163, 1817), (189, 3261), (268, 6332), (271, 35518), (280, 35519), (282, 4391), (662, 523), (692, 40), (1413, 3676), (1415, 223), (1422, 24002), (1433, 4025), (1436, 514), (1903, 1577), (2188, 521), (2192, 522), (2208, 513), (3939, 7080), (7118, 650), (7129, 17924), (7151, 7649), (7367, 3239), (8059, 7), (13138, 75), (32652, 1), (32660, 7011), (32674, 196), (36059, 7013)

X(36050) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(10)

Barycentrics    a/(b^2 (a - b) (a + b - c) - c^2 (a - c) (a - b + c)) : :
Barycentrics    a/((b - c) (b^3 + c^3 - a^2 b - a^2 c + a b c)) : :
Trilinears    1/(b^2 (cos A - cos B) - c^2 (cos A - cos C)) : :

As the trilinear product of circumcircle antipodes, X(36050) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36050) lies on these lines: {1, 34588}, {31, 24026}, {46, 2217}, {80, 5247}, {100, 4575}, {108, 21189}, {109, 23987}, {163, 1783}, {255, 24034}, {668, 4592}, {906, 1018}, {1331, 3952}, {1724, 36052}, {1754, 13478}, {1771, 1795}, {3751, 36056}, {4551, 36059}

X(36050) = isogonal conjugate of X(21189)
X(36050) = trilinear pole of line X(37)X(48)
X(36050) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 21189}, {2, 6589}, {513, 3869}, {514, 573}, {522, 10571}, {523, 4225}, {693, 3185}
X(36050) = trilinear product X(i)*X(j) for these {i,j}: {2, 32653}, {3, 26704}, {100, 2217}, {101, 13478}, {109, 10570}, {110, 15232}, {515, 35183}, {692, 2995}, {4559, 19607}, {8687, 19608}
X(36050) = trilinear quotient X(i)/X(j) for these (i,j): (1, 21189), (6, 6589), (100, 3869), (101, 573), (109, 10571), (110, 4225), (692, 3185), (2217, 513), (2995, 693), (10570, 522), (13478, 514), (15232, 523), (19607, 4560), (19608, 3910), (26704, 4), (32653, 6), (35183, 102), (36108, 36121)
X(36050) = barycentric product X(i)*X(j) for these {i,j}: {63, 26704}, {75, 32653}, {100, 13478}, {101, 2995}, {190, 2217}, {651, 10570}, {662, 15232}, {4551, 19607}
X(36050) = barycentric quotient X(i)/X(j) for these (i,j): (6, 21189), (31, 6589), (100, 4417), (101, 3869), (692, 573), (2217, 514), (2995, 3261), (10570, 4391), (13478, 693), (15232, 1577), (19607, 18155), (26704, 92), (32653, 1), (32700, 36121)

X(36051) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(512)

Barycentrics    a^3/(2 a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2) : :
Trilinears    1/(cos A cos(A + ω) - cos B cos(B + ω) - cos C cos(C + ω)) : :

As the trilinear product of circumcircle antipodes, X(36051) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48), and as the trilinear product of circumcircle-X(512)-antipodes, X(36051) also lies on conic {{A,B,C,X(1),X(31)}} with perspector X(798).

Let A'B'C' be the circumcevian triangle of X(511). Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear with the trilinear product X(3)*X(511) and the trilinear product X(511)*X(512). The lines AA", BB", CC" concur in X(36051).

X(36051) lies on these lines: {1, 4592}, {31, 4575}, {42, 1331}, {47, 163}, {109, 3563}, {162, 1096}, {213, 906}, {741, 10425}, {896, 36061}, {923, 6149}, {1402, 36059}, {1733, 36036}, {2624, 36060}

X(36051) = isogonal conjugate of X(1733)
X(36051) = cevapoint of X(31) and X(1755)
X(36051) = trilinear pole of line X(48)X(798)
X(36051) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1733}, {2, 230}, {4, 3564}, {69, 460}, {75, 8772}, {523, 4226}, {671, 5477}
X(36051) = trilinear product X(i)*X(j) for these {i,j}: {2, 32654}, {3, 3563}, {6, 2987}, {31, 8773}, {32, 8781}, {110, 35364}, {512, 10425}, {810, 36105}
X(36051) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1733), (3, 3564), (6, 230), (25, 460), (31, 8772), (110, 4226), (187, 5477), (2987, 2), (3563, 4), (8773, 75), (10425, 99), (32654, 6), (35364, 523), (36105, 811)
X(36051) = barycentric product X(i)*X(j) for these {i,j}: {1, 2987}, {6, 8773}, {31, 8781}, {48, 35142}, {63, 3563}, {75, 32654}, {647, 36105}, {661, 10425}, {662, 35364}
X(36051) = barycentric quotient X(i)/X(j) for these (i,j): (6, 1733), (31, 230), (48, 3564), (163, 4226), (1755, 114), (1973, 460), (2987, 75), (3563, 92), (8773, 76), (8781, 561), (10425, 799), (32654, 1), (35142, 1969), (35364, 1577), (36105, 6331)

X(36052) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(513)

Barycentrics    a^2/(a^3 (b + c) - a^2 (b^2 + c^2) - a (b - c)^2 (b + c) + (b^2 - c^2)^2) : :
Trilinears    (sin A)/(sin B (cos A + cos B - 1) + sin C (cos A + cos C - 1)) : :

As the trilinear product of circumcircle antipodes, X(36052) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

Let A'B'C' be the circumcevian triangle of X(517). Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear on line X(1769)X(22350). The lines AA", BB", CC" concur in X(36052).

X(36052) lies on these lines: {1, 1331}, {6, 906}, {31, 998}, {34, 46}, {36, 10692}, {56, 215}, {58, 1800}, {80, 2342}, {86, 4592}, {106, 6099}, {162, 1780}, {163, 913}, {244, 255}, {283, 21616}, {517, 1411}, {996, 12647}, {1027, 36057}, {1220, 10039}, {1421, 12704}, {1723, 7129}, {1724, 36050}, {1737, 5081}, {1795, 3738}, {2424, 36056}, {2774, 10091}, {3074, 17719}, {5127, 36061}, {7078, 34430}, {21180, 36053}, {23345, 36058}, {32698, 36039}

X(36052) = isogonal conjugate of X(1737)
X(36052) = cevapoint of X(i) and X(j) for these {i,j}: {6, 2316}, {31, 2183}
X(36052) = crosssum of X(46) and X(1718)
X(36052) = trilinear pole of line X(48)X(649)
X(36052) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1737}, {4, 912}, {2, 8609}, {80, 11570}, {92, 2252}, {19, 914}, {523, 3658}, {104, 119}
X(36052) = trilinear product X(i)*X(j) for these {i,j}: {2, 32655}, {3, 915}, {6, 2990}, {63, 913}, {110, 3657}, {513, 6099}, {517, 15381}, {905, 32698}, {1459, 36106}
X(36052) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1737), (3, 912), (6, 8609), (36, 11570), (48, 2252), (63, 914), (110, 3658), (517, 119), (913, 19), (915, 4), (2990, 2), (3657, 523), (6099, 100), (15381, 104), (32655, 6), (32698, 1783), (36106, 1897)
X(36052) = barycentric product X(i)*X(j) for these {i,j}: {1, 2990}, {63, 915}, {69, 913}, {75, 32655}, {514, 6099}, {662, 3657}, {905, 36106}, {908, 15381}, {4025, 32698}
X(36052) = barycentric quotient X(i)/X(j) for these (i,j): (3, 914), (6, 1737), (48, 912), (913, 4), (915, 92), (2183, 119), (2990, 75), (3657, 1577), (6099, 190), (15381, 34234), (32655, 1), (32698, 1897), (36106, 6335)

X(36053) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(523)

Barycentrics    a/(a^4 (b^2 + c^2) - 2 a^2 (b^4 - b^2 c^2 + c^4) + (b^2 - c^2)^2 (b^2 + c^2)) : :
Trilinears    1/(1 + cos 2B + cos 2C) : :

As the trilinear product of circumcircle antipodes, X(36053) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48), and as the trilinear product of circumcircle-X(523)-antipodes, X(36053) also lies on conic {{A,B,C,X(1),X(10)}} with center X(244) and perspector X(661).

Let A'B'C' be the circumcevian triangle of X(30). Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear with X(36035). The lines AA", BB", CC" concur in X(36053).

X(36053) lies on these lines: {1, 4575}, {10, 1331}, {19, 163}, {31, 1099}, {37, 906}, {47, 158}, {65, 5504}, {75, 4592}, {91, 255}, {109, 225}, {759, 10420}, {920, 36063}, {1725, 36034}, {1822, 2589}, {1823, 2588}, {2166, 6149}, {2190, 36134}, {4354, 10058}, {10090, 36055}, {10419, 18593}, {17898, 36062}, {18827, 18878}, {21180, 36052}, {23894, 36060}, {32679, 35200}

X(36053) = isogonal conjugate of X(1725)
X(36053) = cevapoint of X(31) and X(2173)
X(36053) = crosspoint of X(i) and X(j) for these {i,j}: {1, 6149}, {31, 2173}
X(36053) = crosssum of X(i) and X(j) for these {i,j}: {1, 2166}, {75, 2349}
X(36053) = trilinear pole of line X(48)X(661)
X(36053) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1725}, {2, 3003}, {3, 403}, {4, 13754}, {6, 3580}, {30, 14264}, {265, 1986}, {648, 686}
X(36053) = trilinear product X(i)*X(j) for these {i,j}: {2, 14910}, {3, 1300}, {4, 5504}, {6, 2986}, {74, 15454}, {112, 15421}, {186, 12028}, {647, 687}, {656, 36114}
X(36053) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1725), (2, 3580), (3, 13754), (4, 403), (6, 3003), (74, 14264), (186, 1986), (647, 686), (687, 648), (1300, 4), (2986, 2), (5504, 3), (12028, 265), (14910, 6), (15421, 525), (15454, 30), (36114, 162)
X(36053) = barycentric product X(i)*X(j) for these {i,j}: {1, 2986}, {63, 1300}, {75, 14910}, {92, 5504}, {162, 15421}, {525, 36114}, {656, 687}, {661, 18878}, {2349, 15454}
X(36053) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3580), (6, 1725), (19, 403), (31, 3003), (48, 13754), (162, 16237), (656, 6334), (687, 811), (1300, 92), (2173, 113), (2986, 75), (5504, 63), (14910, 1), (15421, 14208), (15454, 14206), (18878, 799), (36114, 648)

X(36054) = CROSSDIFFERENCE OF X(4) AND X(65)

Barycentrics    a^3 (a^2 - b^2 - c^2)^2 (a - b - c) (b - c) : :
Trilinears    (cos^2 A) (cos B - cos C) : :
Trilinears    (cos A) (sec B - sec C) : :

The trilinear polar of X(36054) passes through X(2638).

X(36054) lies on these lines: {6, 2431}, {112, 2761}, {212, 23614}, {394, 4131}, {649, 8677}, {520, 647}, {521, 650}, {680, 822}, {1491, 9253}, {1812, 35518}, {2149, 36039}, {3287, 23874}, {4378, 9391}, {6787, 15276}, {9337, 24279}

X(36054) = isogonal conjugate of polar conjugate of X(521)
X(36054) = isotomic conjugate of polar conjugate of X(1946)
X(36054) = perspector of hyperbola {{A,B,C,X(3),X(21)}}
X(36054) = intersection of trilinear polars of X(3) and X(21)
X(36054) = crossdifference of every pair of points on line X(4)X(65)
X(36054) = crosssum of Feuerbach hyperbola intercepts of orthic axis
X(36054) = X(i)-isoconjugate of X(j) for these {i,j}: {4, 653}, {19, 18026}, {33, 13149}, {34, 6335}, {65, 823}, {92, 108}, {107, 226}, {109, 2052}, {158, 651}, {190, 1118}, {225, 648}, {264, 32674}, {273, 1783}, {278, 1897}, {281, 36118}, {318, 32714}, {331, 8750}, {349, 32713}, {393, 664}, {650, 24032}, {658, 1857}, {811, 1880}, {1096, 4554}, {1214, 36126}, {1400, 6528}, {1441, 24019}, {1978, 7337}, {2207, 4572}, {2212, 4569}
X(36054) = trilinear product X(i)*X(j) for these {i,j}: {3, 652}, {9, 23224}, {21, 822}, {41, 4131}, {48, 521}, {55, 4091}, {63, 1946}, {78, 22383}, {184, 6332}, {212, 905}, {219, 1459}, {255, 650}, {283, 647}, {284, 520}, {326, 3063}, {332, 3049}, {394, 663}, {513, 2289}, {514, 6056}, {522, 577}, {649, 1259}, {651, 2638}, {656, 2193}, {657, 1804}, {667, 3719}, {810, 1812}, {1264, 1919}, {2175, 30805}, {2194, 24018}, {2968, 32660}, {7128, 23614}, {7182, 8641}, {9247, 35518}
X(36054) = trilinear quotient X(i)/X(j) for these (i,j): (3, 653), (21, 823), (48, 108), (63, 18026), (77, 13149), (78, 6335), (184, 32674), (212, 1783), (219, 1897), (222, 36118), (255, 651), (283, 648), (284, 107), (326, 4554), (332, 6331), (333, 6528), (394, 664), (520, 226), (521, 92), (522, 2052), (577, 109), (603, 32714), (647, 225), (649, 1118), (650, 158), (657, 1857), (651, 24032), (652, 4), (663, 393), (810, 1880), (822, 65), (905, 273), (1172, 36126), (1259, 190), (1264, 1978), (1459, 278), (1804, 658), (1812, 811), (1919, 7337), (1946, 19), (2193, 162), (2194, 24019), (2289, 100), (2638, 650), (3063, 1096), (3265, 349), (3719, 668), (3926, 4572), (4025, 331), (4091, 7), (4131, 85), (6056, 101), (6332, 264), (7182, 4569), (8641, 2212), (22383, 34), (23224, 57), (24018, 1441), (30805, 6063), (35518, 1969), (36059, 7128)
X(36054) = barycentric product X(i)*X(j) for these {i,j}: {3, 521}, {8, 23224}, {9, 4091}, {21, 520}, {41, 30805}, {48, 6332}, {55, 4131}, {63, 652}, {69, 1946}, {78, 1459}, {184, 35518}, {212, 4025}, {219, 905}, {255, 522}, {283, 656}, {284, 24018}, {326, 663}, {332, 810}, {333, 822}, {345, 22383}, {394, 650}, {513, 1259}, {525, 2193}, {577, 4391}, {647, 1812}, {649, 3719}, {657, 7182}, {664, 2638}, {667, 1264}, {693, 6056}, {1809, 8677}, {2194, 3265}, {2289, 514}, {2968, 36059}, {3063, 3926}
X(36054) = barycentric quotient X(i)/X(j) for these (i,j): (3, 18026), (21, 6528), (48, 653), (184, 108), (212, 1897), (219, 6335), (222, 32714), (255, 664), (283, 811), (284, 823), (326, 4572), (394, 4554), (520, 1441), (521, 264), (577, 651), (603, 36118), (650, 2052), (652, 92), (657, 33), (663, 158), (667, 1118), (810, 225), (822, 226), (905, 331), (1259, 668), (1264, 6386), (1459, 273), (1812, 6331), (1946, 4), (2193, 648), (2194, 107), (2289, 190), (2299, 36126), (2638, 522), (3049, 1880), (3063, 393), (3719, 1978), (4091, 85), (4131, 6063), (4391, 18027), (6056, 100), (6332, 1969), (8641, 1857), (9247, 32674), (14585, 1415), (22383, 278), (23224, 7), (23614, 2968), (24018, 349), (30805, 20567), (32660, 7128), (35518, 18022)

X(36055) = TRILINEAR PRODUCT X(3)*X(102)

Barycentrics    a^3 (a^2 - b^2 - c^2)/(2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2) : :
Trilinears    (cos A)/((b + c) sec A - b sec B - c sec C) : :

X(36055) is the trilinear product of the circumcircle intercepts of the line through X(3) and the trilinear product X(2)*X(36054). As the trilinear product of circumcircle antipodes, X(36055) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36055) lies on these lines: {1, 34588}, {3, 102}, {6, 268}, {21, 162}, {163, 2193}, {255, 7114}, {906, 2289}, {1259, 1331}, {1295, 1465}, {1411, 35014}, {1809, 4511}, {2361, 36040}, {10017, 10746}, {10090, 36053}

X(36055) = isogonal conjugate of polar conjugate of X(36100)
X(36055) = isotomic conjugate of polar conjugate of X(32677)
X(36055) = trilinear pole of line X(48)X(36054)
X(36055) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 8755}, {4, 515}, {25, 35516}, {92, 2182}, {108, 14304}, {281, 34050}, {522, 23987}, {650, 24035}
X(36055) = trilinear product X(i)*X(j) for these {i,j}: {3, 102}, {48, 36100}, {63, 32677}, {222, 15629}, {255, 36121}, {521, 36040}
X(36055) = trilinear quotient X(i)/X(j) for these (i,j): (3, 515), (6, 8755), (69, 35516), (102, 4), (109, 23987), (222, 34050), (521, 14304), (651, 24035), (15629, 281), (32677, 19), (36040, 108), (36067, 36127), (36100, 92), (36121, 158)
X(36055) = barycentric product X(i)*X(j) for these {i,j}: {3, 36100}, {63, 102}, {69, 32677}, {77, 15629}, {394, 36121}, {2399, 36059}, {6332, 36040}
X(36055) = barycentric quotient X(i)/X(j) for these (i,j): (63, 35516), (102, 92), (109, 24035), (603, 34050), (15629, 318), (32667, 36127), (32677, 4), (36040, 653), (36059, 2406), (36100, 264), (36121, 2052)

X(36056) = TRILINEAR PRODUCT X(3)*X(103)

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

X(36056) is the trilinear product of the circumcircle intercepts of line X(3)X(4091). As the trilinear product of circumcircle antipodes, X(36056) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36056) lies on these lines: {3, 7215}, {55, 103}, {81, 162}, {163, 911}, {212, 7125}, {218, 1433}, {255, 906}, {394, 1260}, {677, 3935}, {943, 36048}, {972, 24016}, {1792, 4592}, {2424, 36052}, {3751, 36050}

X(36056) = isogonal conjugate of polar conjugate of X(36101)
X(36056) = isotomic conjugate of polar conjugate of X(911)
X(36056) = trilinear pole of line X(48)X(23224)
X(36056) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 1886}, {4, 516}, {25, 35517}, {92, 910}, {118, 917}, {393, 26006}, {676, 1897}, {2398, 7649}
X(36056) = trilinear product X(i)*X(j) for these {i,j}: {2, 32657}, {3, 103}, {6, 1815}, {48, 36101}, {63, 911}, {184, 18025}, {222, 2338}, {255, 36122}, {677, 1459}, {905, 36039}, {916, 15380}, {1331, 2424}, {2400, 32656}, {4025, 32642}
X(36056) = trilinear quotient X(i)/X(j) for these (i,j): (3, 516), (6, 1886), (48, 910), (69, 35517), (103, 4), (394, 26006), (677, 1897), (911, 19), (916, 118), (1331, 2398), (1459, 676), (1815, 2), (2338, 281), (2424, 7649), (15380, 917), (18025, 264), (24016, 36118), (32642, 8750), (32656, 2426), (32657, 6), (32668, 32714), (36039, 1783), (36101, 92), (36122, 158)
X(36056) = barycentric product X(i)*X(j) for these {i,j}: {1, 1815}, {3, 36101}, {48, 18025}, {63, 103}, {69, 911}, {75, 32657}, {77, 2338}, {394, 36122}, {677, 905}, {906, 2400}, {1332, 2424}, {4025, 36039}, {15413, 32642}
X(36056) = barycentric quotient X(i)/X(j) for these (i,j): (48, 516), (63, 35517), (103, 92), (163, 4241), (184, 910), (255, 26006), (677, 6335), (906, 2398), (911, 4), (1815, 75), (2253, 118), (2338, 318), (2424, 17924), (18025, 1969), (24016, 13149), (32642, 1783), (32657, 1), (32668, 36118), (36039, 1897), (36101, 264), (36122, 2052)

X(36057) = TRILINEAR PRODUCT X(3)*X(105)

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

X(36057) is the trilinear product of the circumcircle intercepts of line X(3)X(905). As the trilinear product of circumcircle antipodes, X(36057) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36057) lies on these lines: {3, 906}, {6, 3423}, {27, 162}, {31, 57}, {36, 32666}, {58, 163}, {63, 212}, {73, 1803}, {84, 294}, {103, 672}, {184, 222}, {283, 4592}, {295, 7193}, {603, 7177}, {654, 35365}, {1027, 36052}, {1041, 1445}, {1707, 1768}, {1754, 13478}, {1777, 14377}, {1790, 4575}, {1795, 23696}, {1796, 5314}, {1810, 1818}, {1861, 36111}, {1936, 2342}, {3939, 4712}, {7070, 28071}, {8750, 16560}, {18031, 36036}, {20793, 22148}, {32735, 36040}, {34078, 36146}

X(36057) = isogonal conjugate of X(1861)
X(36057) = isotomic conjugate of polar conjugate of X(1438)
X(36057) = trilinear pole of line X(48)X(1459)
X(36057) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1861}, {2, 5089}, {4, 518}, {19, 3912}, {25, 3263}, {33, 9436}, {34, 3717}, {75, 2356}, {92, 672}, {158, 1818}, {241, 281}, {264, 2223}, {278, 3693}, {318, 1458}, {393, 25083}, {523, 4238}, {607, 27818}, {665, 6335}, {883, 18344}, {918, 1783}, {926, 18026}, {1025, 3064}, {1026, 7649}, {1824, 30941}, {1826, 18206}, {1897, 2254}, {1969, 9454}, {2052, 20752}, {4712, 36124}, {7046, 34855}
X(36057) = trilinear product X(i)*X(j) for these {i,j}: {2, 32658}, {3, 105}, {6, 1814}, {31, 31637}, {48, 673}, {63, 1438}, {77, 2195}, {78, 1416}, {109, 23696}, {110, 10099}, {184, 2481}, {219, 1462}, {222, 294}, {255, 36124}, {394, 8751}, {521, 32735}, {603, 14942}, {652, 36146}, {666, 22383}, {884, 6516}, {885, 36059}, {905, 919}, {927, 1946}, {1024, 1813}, {1027, 1331}, {1437, 13576}, {1459, 36086}, {1790, 18785}, {2196, 6654}, {4025, 32666}, {6185, 20752}, {6559, 7099}, {7053, 28071}, {9247, 18031}
X(36057) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1861), (3, 518), (6, 5089), (31, 2356), (48, 672), (63, 3912), (69, 3263), (77, 9436), (78, 3717), (105, 4), (110, 4238), (184, 2223), (219, 3693), (222, 241), (255, 1818), (294, 281), (348, 27818), (394, 25083), (577, 20752), (603, 1458), (666, 6335), (673, 92), (884, 18344), (905, 918), (919, 1783), (927, 18026), (1024, 3064), (1027, 7649), (1331, 1026), (1416, 34), (1437, 3286), (1438, 19), (1444, 30941), (1459, 2254), (1462, 278), (1790, 18206), (1813, 1025), (1814, 2), (1818, 4712), (1946, 926), (2195, 33), (2196, 3252), (2481, 264), (6516, 883), (6559, 7101), (7053, 34855), (8751, 393), (9247, 9454), (10099, 523), (14942, 318), (18031, 1969), (18785, 1826), (20752, 6184), (22383, 665), (23696, 522), (28071, 7046), (31637, 75), (32658, 6), (32666, 8750), (32735, 108), (34018, 331), (36059, 2283), (36086, 1897), (36124, 158), (36146, 653)
X(36057) = barycentric product X(i)*X(j) for these {i,j}: {1, 1814}, {3, 673}, {6, 31637}, {48, 2481}, {63, 105}, {69, 1438}, {75, 32658}, {77, 294}, {78, 1462}, {184, 18031}, {212, 34018}, {222, 14942}, {295, 6654}, {326, 8751}, {345, 1416}, {348, 2195}, {394, 36124}, {521, 36146}, {651, 23696}, {652, 927}, {662, 10099}, {666, 1459}, {885, 1813}, {905, 36086}, {919, 4025}, {1024, 6516}, {1027, 1332}, {1444, 18785}, {1790, 13576}, {1818, 6185}, {1946, 34085}, {6332, 32735}, {6559, 7053}, {7177, 28071}, {15413, 32666}
X(36057) = barycentric quotient X(i)/X(j) for these (i,j): (3, 3912), (6, 1861), (31, 5089), (48, 518), (63, 3263), (77, 27818), (105, 92), (163, 4238), (184, 672), (212, 3693), (222, 9436), (255, 25083), (294, 318), (577, 1818), (603, 241), (673, 264), (884, 3064), (919, 1897), (1027, 17924), (1416, 278), (1438, 4), (1444, 18157), (1459, 918), (1462, 273), (1790, 30941), (1813, 883), (1814, 75), (1818, 4437), (2195, 281), (2481, 1969), (7099, 34855), (7193, 17755), (8751, 158), (9247, 2223), (10099, 1577), (14942, 7017), (18031, 18022), (22383, 2254), (23696, 4391), (28071, 7101), (31637, 76), (32658, 1), (32666, 1783), (32735, 653), (36059, 1025), (36086, 6335), (36124, 2052), (36146, 18026)

X(36058) = TRILINEAR PRODUCT X(3)*X(106)

Barycentrics    a^3 (a^2 - b^2 - c^2)/(2 a - b - c) : :

X(36058) is the trilinear product of the circumcircle intercepts of line X(3)X(1459). As the trilinear product of circumcircle antipodes, X(36058) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36058) lies on these lines: {3, 1331}, {28, 88}, {36, 2390}, {46, 2217}, {48, 906}, {56, 106}, {57, 15906}, {104, 517}, {163, 1333}, {513, 10090}, {579, 1436}, {603, 36059}, {911, 32665}, {963, 10310}, {1437, 4575}, {1444, 4592}, {1791, 3916}, {1795, 8677}, {1811, 5440}, {2810, 2932}, {3417, 32612}, {3433, 8069}, {4622, 20568}, {6075, 10738}, {8679, 33844}, {23345, 36052}

X(36058) = isogonal conjugate of X(38462)
X(36058) = isogonal conjugate of polar conjugate of X(88)
X(36058) = isotomic conjugate of polar conjugate of X(9456)
X(36058) = trilinear pole of line X(48)X(22383)
X(36058) = crossdifference of every pair of points on the line through X(1639) and the polar conjugates of PU(50)
X(36058) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 8756}, {4, 519}, {8, 1877}, {19, 4358}, {25, 3264}, {34, 4723}, {44, 92}, {158, 5440}, {264, 902}, {281, 3911}, {318, 1319}, {393, 3977}, {648, 4120}, {811, 4730}, {900, 1897}, {1023, 17924}, {1309, 23757}, {1404, 7017}, {1635, 6335}, {1783, 3762}, {1824, 30939}, {1969, 2251}, {2052, 22356}, {4370, 6336}, {4738, 36125}, {6331, 14407}, {6591, 24004}, {7649, 17780}, {9459, 18022}
X(36058) = trilinear product X(i)*X(j) for these {i,j}: {2, 32659}, {3, 106}, {6, 1797}, {48, 88}, {63, 9456}, {78, 1417}, {184, 903}, {222, 2316}, {255, 36125}, {394, 8752}, {577, 6336}, {603, 1320}, {647, 4591}, {679, 23202}, {810, 4622}, {901, 1459}, {905, 32665}, {906, 1022}, {1331, 23345}, {1437, 4674}, {1795, 14260}, {1807, 16944}, {1811, 17109}, {2226, 22356}, {3049, 4615}, {3257, 22383}, {4025, 32719}, {4049, 32661}, {6548, 32656}, {9247, 20568}, {10428, 22350}, {23838, 36059}
X(36058) = trilinear quotient X(i)/X(j) for these (i,j): (3, 519), (6, 8756), (56, 1877), (69, 3264), (48, 44), (63, 4358), (78, 4723), (88, 92), (106, 4), (184, 902), (222, 3911), (255, 5440), (394, 3977), (577, 22356), (603, 1319), (647, 4120), (810, 4730), (901, 1897), (903, 264), (905, 3762), (906, 1023), (1022, 17924), (1320, 318), (1331, 17780), (1332, 24004), (1417, 34), (1444, 30939), (1459, 900), (1797, 2), (2226, 6336), (2316, 281), (3049, 14407), (3257, 6335), (4049, 14618), (4591, 648), (4615, 6331), (4622, 811), (4997, 7017), (5440, 4738), (6336, 2052), (8677, 23757), (8752, 393), (9247, 2251), (9456, 19), (10428, 36123), (14260, 1785), (14575, 9459), (16944, 1870), (17109, 1878), (20568, 1969), (22350, 1145), (22356, 4370), (22383, 1635), (23202, 678), (23345, 7649), (32656, 23344), (32659, 6), (32665, 1783), (32719, 8750), (34230, 1861), (36059, 23703), (36125, 158)
X(36058) = barycentric product X(i)*X(j) for these {i,j}: {1, 1797}, {3, 88}, {48, 903}, {63, 106}, {69, 9456}, {75, 32659}, {77, 2316}, {184, 20568}, {222, 1320}, {255, 6336}, {326, 8752}, {345, 1417}, {394, 36125}, {603, 4997}, {647, 4622}, {679, 22356}, {810, 4615}, {901, 905}, {906, 6548}, {1022, 1331}, {1332, 23345}, {1437, 4080}, {1459, 3257}, {1790, 4674}, {1813, 23838}, {1814, 34230}, {2226, 5440}, {3049, 4634}, {4025, 32665}, {4049, 4575}, {4555, 22383}, {15413, 32719}
X(36058) = barycentric quotient X(i)/X(j) for these (i,j): (3, 4358), (31, 8756), (48, 519), (63, 3264), (88, 264), (106, 92), (184, 44), (255, 3977), (577, 5440), (603, 3911), (810, 4120), (901, 6335), (903, 1969), (906, 17780), (1320, 7017), (1331, 24004), (1417, 278), (1437, 16704), (1459, 3762), (1790, 30939), (1797, 75), (2316, 318), (3049, 4730), (4622, 6331), (8752, 158), (9247, 902), (9456, 4), (10428, 16082), (16944, 17923), (20568, 18022), (22356, 4738), (22383, 900), (23202, 4370), (23345, 17924), (32659, 1), (32665, 1897), (32719, 1783), (36125, 2052)

X(36059) = TRILINEAR PRODUCT X(3)*X(109)

Barycentrics    a^3 (a^2 - b^2 - c^2)/((b - c) (a - b - c)) : :
Trilinears    (sin 2A)/(cos B - cos C) : :

X(36059) is the trilinear product of the circumcircle intercepts of line X(3)X(73). As the trilinear product of circumcircle antipodes, X(36059) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36059) lies on these lines: {3, 1364}, {34, 15906}, {49, 23070}, {56, 215}, {59, 100}, {63, 34588}, {65, 5504}, {73, 1437}, {101, 2425}, {108, 110}, {109, 692}, {163, 1415}, {184, 222}, {212, 7125}, {221, 2841}, {255, 7114}, {293, 1214}, {603, 36058}, {906, 32660}, {912, 1319}, {916, 2078}, {934, 15439}, {942, 19365}, {971, 10535}, {1069, 11510}, {1092, 7078}, {1331, 1813}, {1362, 1397}, {1402, 36051}, {1409, 36060}, {1428, 3660}, {1459, 35350}, {1465, 26884}, {1949, 2193}, {2003, 2194}, {2149, 36039}, {2342, 15626}, {2406, 14544}, {2477, 8614}, {3562, 34148}, {3564, 5061}, {4551, 36050}, {4554, 17932}, {4579, 14594}, {4592, 6516}, {5012, 17074}, {5091, 24465}, {5172, 13754}, {6056, 7011}, {8757, 10539}, {9306, 34048}, {11214, 26888}, {12118, 18961}, {13273, 17702}, {14529, 21147}, {15958, 36134}, {18026, 18831}, {20986, 22130}, {20999, 34858}, {22115, 23071}, {22341, 35200}, {23353, 36127}, {24027, 36040}, {32735, 36041}

X(36059) = isogonal conjugate of polar conjugate of X(651)
X(36059) = isotomic conjugate of polar conjugate of X(1415)
X(36059) = trilinear pole of line X(48)X(577)
X(36059) = crossdifference of every pair of points on line X(1146)X(8735)
X(36059) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 3064}, {4, 522}, {8, 7649}, {9, 17924}, {11, 1897}, {19, 4391}, {21, 24006}, {27, 3700}, {28, 4086}, {29, 523}, {33, 693}, {34, 4397}, {75, 18344}, {92, 650}, {108, 24026}, {158, 521}, {190, 8735}, {225, 7253}, {226, 17926}, {264, 663}, {273, 3900}, {278, 3239}, {281, 514}, {284, 14618}, {286, 4041}, {312, 6591}, {318, 513}, {331, 657}, {333, 2501}, {393, 6332}, {525, 8748}, {607, 3261}, {649, 7017}, {652, 2052}, {653, 1146}, {656, 1896}, {661, 31623}, {850, 2299}, {885, 1861}, {1172, 1577}, {1783, 4858}, {1824, 18155}, {1826, 4560}, {1857, 4025}, {1969, 3063}, {2170, 6335}, {2204, 20948}, {2310, 18026}, {2399, 8755}, {2489, 28660}, {6129, 7020}, {7003, 14837}, {7008, 17896}, {8750, 34387}, {23978, 32674}
X(36059) = trilinear product X(i)*X(j) for these {i,j}: {2, 32660}, {3, 109}, {6, 1813}, {7, 32656}, {31, 6516}, {48, 651}, {56, 1331}, {57, 906}, {58, 23067}, {59, 1459}, {63, 1415}, {65, 4575}, {71, 4565}, {73, 110}, {77, 692}, {100, 603}, {101, 222}, {108, 255}, {162, 22341}, {163, 1214}, {184, 664}, {212, 934}, {219, 1461}, {226, 32661}, {228, 1414}, {307, 1576}, {348, 32739}, {394, 32674}, {521, 24027}, {577, 653}, {604, 1332}, {652, 1262}, {662, 1409}, {905, 2149}, {1020, 2193}, {1025, 32658}, {1092, 36127}, {1106, 4571}, {1397, 4561}, {1400, 4558}, {1402, 4592}, {1407, 4587}, {1437, 4551}, {1783, 7125}, {1790, 4559}, {1795, 23981}, {1804, 8750}, {1818, 32735), (1897, 7335}, {1946, 7045}, {2200, 4573}, {2283, 36057}, {2720, 22350}, {3157, 36082}, {4303, 15439}, {4554, 9247}, {4564, 22383}, {4572, 14575}, {6056, 36118}, {6332, 23979}, {7011, 36049}, {7013, 32652}, {7078, 8059}, {7114, 13138}, {7128, 36054}, {14578, 24029}, {23207, 36048}, {23703, 36058}
X(36059) = trilinear quotient X(i)/X(j) for these (i,j): (3, 522), (6, 3064), (31, 18344), (48, 650), (56, 7649), (57, 17924), (59, 1897), (63, 4391), (65, 24006), (69, 35519), (71, 3700), (72, 4086), (73, 523), (77, 693), (78, 4397), (100, 318), (101, 281), (108, 158), (109, 4), (110, 29), (112, 8748), (162, 1896), (163, 1172), (184, 663), (190, 7017), (212, 3900), (219, 3239), (222, 514), (226, 14618), (228, 4041), (255, 521), (283, 7253), (284, 17926), (307, 850), (326, 35518), (348, 3261), (394, 6332), (521, 24026), (577, 652), (603, 513), (604, 6591), (649, 8735), (651, 92), (652, 1146), (653, 2052), (658, 331), (662, 31623), (664, 264), (692, 33), (905, 4858), (906, 9), (934, 273), (1214, 1577), (1231, 20948), (1262, 653), (1331, 8), (1332, 312), (1400, 2501), (1409, 661), (1414, 286), (1415, 19), (1437, 3737), (1444, 18155), (1459, 11), (1461, 278), (1576, 2299), (1790, 4560), (1804, 4025), (1813, 2), (1946, 2310), (2149, 1783), (2193, 1021), (2200, 3709), (2283, 1861), (2425, 8755), (4025, 34387), (4554, 1969), (4558, 333), (4559, 1826), (4561, 3596), (4563, 28660), (4564, 6335), (4565, 27), (4571, 341), (4572, 18022), (4575, 21), (4587, 346), (4592, 314), (6332, 23978), (6516, 75), (7011, 14837), (7013, 17896), (7114, 6129), (7045, 18026), (7078, 8058), (7125, 905), (7183, 15413), (7335, 1459), (8750, 1857), (9247, 3063), (13138, 7020), (22341, 656), (22350, 2804), (22383, 2170), (23067, 10), (23979, 32674), (23981, 1785), (24027, 108), (32652, 7008), (32656, 55), (32658, 1024), (32661, 284), (32674, 393), (32735, 36124), (32739, 607), (36049, 7003), (36057, 885), (36058, 23838), (36082, 7040), (36127, 1093)
X(36059) = barycentric product X(i)*X(j) for these {i,j}: {1, 1813}, {3, 651}, {6, 6516}, {7, 906}, {48, 664}, {56, 1332}, {57, 1331}, {59, 905}, {63, 109}, {65, 4558}, {69, 1415}, {71, 1414}, {72, 4565}, {73, 662}, {75, 32660}, {77, 101}, {78, 1461}, {81, 23067}, {85, 32656}, {99, 1409}, {100, 222}, {108, 394}, {110, 1214}, {163, 307}, {184, 4554}, {190, 603}, {212, 658}, {219, 934}, {226, 4575}, {228, 4573}, {255, 653}, {269, 4587}, {283, 1020}, {326, 32674}, {348, 692}, {521, 1262}, {577, 18026}, {604, 4561}, {648, 22341}, {652, 7045}, {883, 32658}, {1025, 36057}, {1231, 1576}, {1275, 1946}, {1400, 4592}, {1402, 4563}, {1407, 4571}, {1437, 4552}, {1441, 32661}, {1444, 4559}, {1459, 4564}, {1783, 1804}, {1790, 4551}, {1795, 24029}, {1797, 23703}, {1897, 7125}, {2149, 4025}, {2193, 4566}, {2200, 4625}, {2289, 36118}, {2406, 36055}, {4565, 26884}, {4572, 9247}, {4998, 22383}, {6332, 24027}, {6335, 7335}, {6505, 36082}, {6507, 36127}, {7011, 13138}, {7013, 36049}, {7182, 32739}, {7183, 8750}, {11214, 26888}, {15439, 18607}, {25083, 32735}
X(36059) = barycentric quotient X(i)/X(j) for these (i,j): (3, 4391), (31, 3064), (32, 18344), (48, 522), (56, 17924), (59, 6335), (63, 35519), (65, 14618), (71, 4086), (73, 1577), (77, 3261), (100, 7017), (101, 318), (108, 2052), (109, 92), (110, 31623), (112, 1896), (163, 29), (184, 650), (212, 3239), (219, 4397), (222, 693), (228, 3700), (255, 6332), (307, 20948), (394, 35518), (521, 23978), (577, 521), (603, 514), (604, 7649), (651, 264), (652, 24026), (664, 1969), (667, 8735), (692, 281), (905, 34387), (906, 8), (934, 331), (1214, 850), (1262, 18026), (1331, 312), (1332, 3596), (1397, 6591), (1400, 24006), (1402, 2501), (1409, 523), (1415, 4), (1437, 4560), (1459, 4858), (1461, 273), (1576, 1172), (1790, 18155), (1804, 15413), (1813, 75), (1946, 1146), (2149, 1897), (2193, 7253), (2194, 17926), (2200, 4041), (2720, 16082), (4554, 18022), (4558, 314), (4561, 28659), (4565, 286), (4575, 333), (4587, 341), (4592, 28660), (6516, 76), (7011, 17896), (7114, 14837), (7125, 4025), (7335, 905), (9247, 663), (14575, 3063), (18026, 18027), (22341, 525), (22383, 11), (23067, 321), (23979, 108), (24027, 653), (32652, 7003), (32656, 9), (32658, 885), (32659, 23838), (32660, 1), (32661, 21), (32674, 158), (32676, 8748), (32739, 33), (36049, 7020), (36054, 2968), (36055, 2399), (36127, 6521)

X(36060) = TRILINEAR PRODUCT X(3)*X(111)

Barycentrics    a^3 (a^2 - b^2 - c^2)/(2 a^2 - b^2 - c^2) : :

X(36060) is the trilinear product of the circumcircle intercepts of line X(3)X(647). As the trilinear product of circumcircle antipodes, X(36060) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36060) lies on these lines: {19, 162}, {31, 163}, {48, 4575}, {63, 3708}, {71, 895}, {109, 111}, {228, 906}, {293, 2631}, {691, 2249}, {896, 2157}, {1409, 36059}, {1707, 2156}, {1755, 2159}, {1821, 14206}, {2148, 36134}, {2250, 36037}, {2281, 32740}, {2357, 5547}, {2624, 36051}, {7902, 18268}, {23894, 36053}

X(36060) = isogonal conjugate of polar conjugate of X(897)
X(36060) = isotomic conjugate of polar conjugate of X(923)
X(36060) = trilinear pole of line X(48)X(810)
X(36060) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 468}, {4, 524}, {19, 14210}, {25, 3266}, {27, 4062}, {92, 896}, {187, 264}, {281, 7181}, {297, 5967}, {351, 6331}, {393, 6390}, {523, 4235}, {648, 690}, {811, 2642}, {897, 671}, {922, 1969}, {2052, 3292}, {2501, 5468}, {5467, 14618}, {14567, 18022}, {23889, 24006}, {24006, 23889}
X(36060) = trilinear product X(i)*X(j) for these {i,j}: {2, 14908}, {3, 111}, {6, 895}, {32, 30786}, {48, 897}, {63, 923}, {69, 32740}, {110, 10097}, {184, 671}, {187, 896}, {222, 5547}, {248, 5968}, {255, 36128}, {394, 8753}, {525, 32729}, {577, 17983}, {647, 691}, {656, 36142}, {810, 36085}, {892, 3049}, {1576, 14977}, {4558, 9178}, {4575, 23894}, {5466, 32661}, {14575, 18023}
X(36060) = trilinear quotient X(i)/X(j) for these (i,j): (3, 524), (6, 468), (48, 896), (63, 14210), (69, 3266), (71, 4062), (110, 4235), (111, 4), (184, 187), (187, 897), (222, 7181), (248, 5967), (394, 6390), (525, 35522), (577, 3292), (647, 690), (671, 264), (691, 648), (810, 2642), (892, 6331), (895, 2), (896, 671), (897, 92), (923, 19), (3049, 351), (4558, 5468), (4575, 23889), (4592, 24039), (5466, 14618), (5547, 281), (5968, 297), (8753, 393), (9178, 2501), (9247, 922), (10097, 523), (14575, 14567), (14908, 6), (14977, 850), (17983, 2052), (18023, 18022), (23894, 24006), (30786, 76), (32661, 5467), (32729, 112), (32740, 25), (36085, 811), (36128, 158), (36142, 162)
X(36060) = barycentric product X(i)*X(j) for these {i,j}: {1, 895}, {3, 897}, {31, 30786}, {48, 671}, {63, 111}, {69, 923}, {75, 14908}, {77, 5547}, {163, 14977}, {255, 17983}, {293, 5968}, {304, 32740}, {326, 8753}, {394, 36128}, {525, 36142}, {647, 36085}, {656, 691}, {662, 10097}, {810, 892}, {4558, 23894}, {4575, 5466}, {4592, 9178}, {9247, 18023}, {14208, 32729}
X(36060) = barycentric quotient X(i)/X(j) for these (i,j): (3, 14210), (31, 468), (48, 524), (63, 3266), (111, 92), (163, 4235), (184, 896), (228, 4062), (255, 6390), (603, 7181), (656, 35522), (671, 1969), (691, 811), (810, 690), (895, 75), (897, 264), (923, 4), (3049, 2642), (4558, 24039), (4575, 5468), (5547, 318), (8753, 158), (9178, 24006), (9247, 187), (10097, 1577), (14575, 922), (14908, 1), (14977, 20948), (23894, 14618), (30786, 561), (32661, 23889), (32729, 162), (32740, 19), (36085, 6331), (36128, 2052), (36142, 648)

X(36061) = TRILINEAR PRODUCT X(3)*X(476)

Barycentrics    a (a^2 - b^2 - c^2)/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2)) : :
Trilinears    (cos A)/((1 + 2 cos 2A) sin(B - C)) : :

X(36061) is the trilinear product of the circumcircle intercepts of line X(3)X(125). As the trilinear product of circumcircle antipodes, X(36061) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36061) lies on these lines: {109, 476}, {162, 24006}, {163, 661}, {255, 36062}, {656, 4575}, {896, 36051}, {906, 32662}, {1101, 2616}, {1331, 4064}, {1793, 35200}, {2166, 6149}, {4592, 14208}, {5127, 36052}, {36085, 36096}, {36114, 36116}

X(36061) = isogonal conjugate of polar conjugate of X(32680)
X(36061) = isotomic conjugate of polar conjugate of X(32678)
X(36061) = X(92)-isoconjugate of X(2624)
X(36061) = trilinear pole of line X(48)X(3708)
X(36061) = X(i)-isoconjugate of X(j) for these {i,j}: {4, 526}, {19, 32679}, {25, 3268}, {50, 14618}, {92, 2624}, {115, 14590}, {186, 523}, {264, 14270}, {323, 2501}, {338, 14591}, {340, 512}, {393, 8552}, {647, 14165}, {850, 34397}, {924, 5962}, {1835, 35057}, {2489, 7799}, {6149, 24006}
X(36061) = trilinear product X(i)*X(j) for these {i,j}: {2, 32662}, {3, 476}, {48, 32680}, {63, 32678}, {69, 14560}, {94, 32661}, {110, 265}, {184, 35139}, {249, 14582}, {255, 36129}, {328, 1576}, {925, 5961}, {1141, 23181}, {1793, 26700}, {1989, 4558}, {2166, 4575}, {2410, 32663}, {2166, 4575}, {4563, 11060}, {8552, 23588}, {14592, 23357}
X(36061) = trilinear quotient X(i)/X(j) for these (i,j): (3, 526), (48, 2624), (63, 32679), (69, 3268), (94, 14618), (99, 340), (110, 186), (184, 14270), (249, 14590), (265, 523), (328, 850), (394, 8552), (476, 4), (648, 14165), (925, 5962), (1576, 34397), (1793, 35057), (1989, 2501), (2166, 24006), (4558, 323), (4563, 7799), (4575, 6149), (5961, 924), (11060, 2489), (14560, 25), (14582, 115), (14592, 338), (23181, 1154), (23357, 14591), (26700, 1835), (32661, 50), (32662, 6), (32663, 2436), (32678, 19), (32680, 92), (35139, 264), (36047, 36130), (36129, 158)
X(36061) = barycentric product X(i)*X(j) for these {i,j}: {3, 32680}, {48, 35139}, {63, 476}, {69, 32678}, {75, 32662}, {94, 4575}, {163, 328}, {265, 662}, {304, 14560}, {394, 36129}, {1101, 14592}, {1989, 4592}, {2166, 4558}, {2410, 36062}, {14582, 24041}
X(36061) = barycentric quotient X(i)/X(j) for these (i,j): (3, 32679), (48, 526), (63, 3268), (162, 14165), (163, 186), (184, 2624), (255, 8552), (265, 1577), (328, 20948), (476, 92), (662, 340), (1101, 14590), (1989, 24006), (2166, 14618), (4575, 323), (4592, 7799), (9247, 14270), (14560, 19), (14582, 1109), (14592, 23994), (23588, 36129), (23995, 14591), (32650, 36130), (32661, 6149), (32662, 1), (32678, 4), (32680, 264), (35139, 1969), (36062, 2411), (36129, 2052), (36145, 5962)

X(36062) = TRILINEAR PRODUCT X(3)*X(477)

Barycentrics    a (a^2 - b^2 - c^2)/(a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (b^2 + c^2) (3 b^4 - 5 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4)) : :
Trilinears    (cos A)/(4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)) : :

X(36062) is the trilinear product of the circumcircle intercepts of the line through X(3) and the trilinear product X(2)*X(2631). As the trilinear product of circumcircle antipodes, X(36062) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36062) lies on these lines: {1, 36034}, {109, 477}, {162, 1784}, {163, 2173}, {255, 36061}, {656, 35200}, {906, 32663}, {17898, 36053}, {24000, 35201}, {36063, 36117}

X(36062) = isogonal conjugate of X(36063)
X(36062) = isotomic conjugate of polar conjugate of X(36151)
X(36062) = trilinear pole of line X(48)X(2631)
X(36062) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 36063}, {4, 5663}, {74, 11251}, {523, 7480}
X(36062) = trilinear product X(i)*X(j) for these {i,j}: {2, 32663}, {3, 477}, {48, 36102}, {63, 36151}, {110, 14220}, {255, 36130}, {647, 30528}, {2411, 32662}
X(36062) = trilinear quotient X(i)/X(j) for these (i,j): (1, 36063), (3, 5663), (30, 11251), (69, 35520), (110, 7480), (477, 4), (14220, 523), (30528, 648), (32662, 2437), (32663, 6), (36047, 36129), (36102, 92), (36130, 158), (36151, 19)
X(36062) = barycentric product X(i)*X(j) for these {i,j}: {3, 36102}, {63, 477}, {69, 36151}, {75, 32663}, {394, 36130}, {656, 30528}, {2411, 36061}
X(36062) = barycentric quotient X(i)/X(j) for these (i,j): (6, 36063), (48, 5663), (63, 35520), (163, 7480), (477, 92), (30528, 811), (32650, 36129), (32663, 1), (36061, 2410), (36102, 264), (36117, 15459), (36130, 2052), (36151, 4)

X(36063) = ISOGONAL CONJUGATE OF X(36062)

Barycentrics    a (a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (b^2 + c^2) (3 b^4 - 5 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4))/(a^2 - b^2 - c^2) : :
Trilinears    (sec A) (4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)) : :

X(36063) lies on these lines: {1, 162}, {158, 2166}, {240, 522}, {774, 1109}, {920, 36053}, {1830, 1844}, {16562, 24019}, {36062, 36117}

X(36063) = isogonal conjugate of X(36062)
X(36063) = polar conjugate of X(36102)
X(36063) = pole wrt polar circle of trilinear polar of X(36102) (line X(1)X(36035))
X(36063) = crossdifference of every pair of points on line X(48)X(2631)
X(36063) = circle-{{X(11),X(36),X(65)}}-inverse of X(1784)
X(36063) = {X(1),X(2629)}-harmonic conjugate of X(35200)
X(36063) = {X(162),X(36119)}-harmonic conjugate of X(1)
X(36063) = {X(2588),X(2599)}-harmonic conjugate of X(1784)
X(36063) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 36062}, {2, 32663}, {3, 477}, {48, 36102}, {63, 36151}, {110, 14220}, {255, 36130}, {647, 30528}, {2411, 32662}
X(36063) = trilinear product X(i)*X(j) for these {i,j}: {4, 5663}, {25, 35520}, {74, 11251}, {523, 7480}
X(36063) = trilinear quotient X(i)/X(j) for these (i,j): (1, 36062), (4, 477), (6, 32663), (19, 36151), (92, 36102), (158, 36130), (523, 14220), (648, 30528), (2437, 32662), (5663, 3), (7480, 110), (11251, 30), (35520, 69), (36129, 36047)
X(36063) = barycentric product X(i)*X(j) for these {i,j}: {19, 35520}, {1577, 7480}, {2349, 11251}
X(36063) = barycentric quotient X(i)/X(j) for these (i,j): (4, 36102), (6, 36062), (19, 477), (25, 36151), (162, 30528), (393, 36130), (7480, 662), (11251, 14206), (35520, 304)

X(36064) = TRILINEAR PRODUCT X(74)*X(26700)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36064) lies on the circumcircle.

X(36064) lies on the circumcircle and these lines: {74, 1464}, {79, 2695}, {110, 36034}, {112, 36131}, {759, 14158}, {1302, 36083}, {1406, 14264}, {2694, 7100}, {2738, 8606}, {32640, 36075}

X(36064) = trilinear pole of line X(6)X(2159)
X(36064) = Ψ(X(i), X(j)) for these (i,j): (1, 74), (6, 2159), (30, 1), (14206, 2)
X(36064) = X(i)-isoconjugate of X(j) for these {i,j}: {9033, 11107}, {9404, 14206}, {35193, 36035}
X(36064) = trilinear product X(74)*X(26700)
X(36064) = trilinear quotient X(i)/X(j) for these (i,j): (74, 35057), (1304, 11107), (2159, 9404), (32640, 35192), (36034, 35193)
X(36064) = barycentric product X(2349)*X(26700)
X(36064) = barycentric quotient X(i)/X(j) for these (i,j): (2159, 35057), (26700, 14206), (32640, 35193), (36131, 11107)

X(36065) = TRILINEAR PRODUCT X(98)*X(29055)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36065) lies on the circumcircle.

X(36065) lies on the circumcircle and these lines: {98, 1284}, {99, 36036}, {110, 36084}, {112, 36104}, {256, 2708}, {1431, 2699}, {1432, 2700}, {2707, 7015}, {26714, 36132}

X(36065) = trilinear pole of line X(6)X(1910)
X(36065) = Ψ(X(i), X(j)) for these (i,j): (1, 98), (6, 1910), (511, 1), (1959, 2)
X(36065) = X(i)-isoconjugate of X(j) for these {i,j}: {511, 3907}, {684, 14006}, {1959, 3287}
X(36065) = trilinear product X(98)*X(29055)
X(36065) = trilinear quotient X(i)/X(j) for these (i,j): (98, 3907), (685, 14006), (1910, 3287), (29055, 511)
X(36065) = barycentric product X(1821)*X(29055)
X(36065) = barycentric quotient X(i)/X(j) for these (i,j): (1910, 3907), (1976, 3287), (29055, 1959), (36104, 14006)

X(36066) = TRILINEAR PRODUCT X(99)*X(741)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36066) lies on the circumcircle.

X(36066) lies on the circumcircle and these lines: {1, 12031}, {98, 36036}, {99, 4367}, {100, 4589}, {101, 4567}, {106, 4622}, {107, 23999}, {110, 24041}, {111, 36085}, {261, 2726}, {291, 28482}, {292, 2375}, {334, 2372}, {593, 9111}, {660, 8701}, {691, 876}, {729, 18268}, {739, 30576}, {757, 2382}, {759, 18827}, {789, 4623}, {805, 875}, {825, 4556}, {835, 4639}, {873, 9073}, {1284, 35108}, {1414, 29055}, {1509, 14665}, {2311, 35106}, {3563, 36105}, {3733, 4590}, {4583, 8707}, {4592, 6010}, {4612, 28847}, {8708, 34067}

X(36066) = isogonal conjugate of X(4155)
X(36066) = trilinear pole of line X(6)X(662)
X(36066) = Λ(PU(79))
X(36066) = Λ(X(i), X(j)) for these {i,j}: {351, 1635}, {2642, 2643}
X(36066) = Ψ(X(i), X(j)) for these (i,j): (1, 99), (6, 662), (512, 1), (661, 2), (740, 1)
X(36066) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(78)}}
X(36066) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 4155}, {10, 4455}, {37, 21832}, {42, 4010}, {238, 4705}, {239, 4079}, {512, 740}, {523, 3747}, {594, 8632}, {656, 862}, {659, 756}, {661, 2238}, {669, 35544}, {798, 3948}, {812, 1500}, {872, 3766}, {1084, 27853}, {1914, 4024}, {2086, 27805}, {2210, 4036}, {2643, 3573}, {3124, 3570}, {3709, 16609}
X(36066) = trilinear product X(i)*X(j) for these {i,j}: {58, 4589}, {81, 4584}, {99, 741}, {110, 18827}, {249, 4444}, {292, 4610}, {335, 4556}, {593, 4562}, {660, 757}, {799, 18268}, {813, 1509}, {849, 4583}, {873, 34067}, {875, 24037}, {876, 24041}, {1333, 4639}, {1911, 4623}, {2311, 4573}, {3572, 4590}, {4636, 7233}
X(36066) = trilinear quotient X(i)/X(j) for these (i,j): (1, 4155), (58, 4455), (81, 21832), (86, 4010), (99, 740), (110, 3747), (162, 862), (291, 4705), (292, 4079), (334, 4036), (335, 4024), (593, 8632), (660, 756), (662, 2238), (670, 35544), (741, 512), (757, 659), (799, 3948), (813, 1500), (873, 3766), (876, 2643), (1509, 812), (2311, 3709), (3572, 3124), (4444, 115), (4556, 1914), (4562, 594), (4573, 16609), (4583, 1089), (4584, 37), (4589, 10), (4590, 3570), (4610, 239), (4612, 3694), (4623, 350), (4639, 321), (18268, 798), (18827, 523), (20981, 2086), (24037, 874), (24041, 3573), (34067, 872), (34537, 27853)
X(36066) = barycentric product X(i)*X(j) for these {i,j}: {58, 4639}, {81, 4589}, {86, 4584}, {291, 4610}, {292, 4623}, {334, 4556}, {335, 4036}, {593, 4583}, {660, 1509}, {662, 18827}, {670, 18268}, {741, 799}, {757, 4562}, {813, 873}, {875, 34537}, {876, 4590}, {2311, 4625}, {3572, 24037}, {4444, 24041}, {4612, 7233}
X(36066) = barycentric quotient X(i)/X(j) for these (i,j): (6, 4155), (58, 21832), (81, 4010), (99, 3948), (110, 2238), (112, 862), (163, 3747), (249, 3573), (291, 4024), (292, 4705), (335, 4036), (593, 659), (660, 594), (662, 740), (741, 661), (757, 812), (799, 35544), (813, 756), (849, 8632), (875, 3124), (876, 115), (1333, 4455), (1414, 16609), (1509, 3766), (1911, 4079), (2311, 4041), (3572, 2643), (4036, 238), (4444, 1109), (4562, 1089), (4583, 28654), (4584, 10), (4589, 321), (4590, 874), (4610, 350), (4612, 3685), (4623, 1921), (4639, 313), (18268, 512), (18827, 1577), (24037, 27853), (24041, 3570), (34067, 1500)

X(36067) = TRILINEAR PRODUCT X(102)*X(108)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 - b^2 - c^2) (2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2)) : :
Trilinears    a/((sec B - sec C) ((b + c) sec A - b sec B - c sec C)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36067) lies on the circumcircle.

Let A', B', C' be the intersections of line X(1)X(4) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(36067).

X(36067) lies on the circumcircle and these lines: {1, 2733}, {4, 2734}, {24, 953}, {25, 35012}, {34, 2716}, {73, 2732}, {100, 7012}, {101, 7115}, {102, 1457}, {104, 1455}, {108, 6129}, {109, 36040}, {112, 32643}, {225, 2695}, {278, 2723}, {653, 9056, 36088}, {1295, 1465}, {1309, 2405}, {1459, 8059}, {2222, 23706}, {2726, 6353}, {2745, 34040}, {6589, 23985}, {26703, 36093}, {26704, 36108}, {26715, 32674}, {32677, 32726}

X(36067) = polar conjugate of isogonal conjugate of X(32643)
X(36067) = trilinear pole of line X(6)X(3209)
X(36067) = polar-circle-inverse of X(10017)
X(36067) = Ψ(X(i), X(j)) for these (i,j): (1, 102), (6, 3209), (63, 651), (515, 1), (521, 1), (650, 19)
X(36067) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 14304}, {515, 521}, {1946, 35516}, {2182, 6332}, {2406, 34591}, {23987, 24031}, {24035, 35072}
X(36067) = trilinear product X(i)*X(j) for these {i,j}: {2, 32667}, {4, 36040}, {92, 32643}, {102, 108}, {109, 36121}, {653, 32677}, {2432, 7128}, {10571, 36108}, {17080, 32700}, {32674, 36100}, {36055, 36127}
X(36067) = trilinear quotient X(i)/X(j) for these (i,j): (4, 14304), (102, 521), (108, 515), (2432, 34591), (7128, 2406), (18026, 35516), (23984, 24035), (24033, 23987), (32643, 48), (32667, 6), (32674, 2182), (32677, 652), (34393, 35518), (36040, 3), (36100, 6332), (36108, 10570), (36121, 522)
X(36067) = barycentric product X(i)*X(j) for these {i,j}: {75, 32667}, {92, 36040}, {102, 653}, {108, 36100}, {264, 32643}, {651, 36121}, {17080, 36108}, {18026, 32677}, {32674, 34393}
X(36067) = barycentric quotient X(i)/X(j) for these (i,j): (19, 14304), (102, 6332), (653, 35516), (2432, 2968), (23985, 23987), (24033, 24035), (32643, 3), (32667, 1), (32674, 515), (32677, 521), (32700, 10570), (36040, 63), (36100, 35518), (36121, 4391)

X(36068) = TRILINEAR PRODUCT X(107)*X(26701)

Barycentrics    a/((b + c) (b^2 - c^2) (a^2 - b^2 - c^2)^2 (a^6 - a^4 (2 b^2 - 3 b c + 2 c^2) + a^2 (b - c)^2 (b^2 + c^2) - b c (b^2 - c^2)^2)) : :
Barycentrics    (tan A)/((tan B - tan C) ((b + c) tan A - b tan B - c tan C)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36068) lies on the circumcircle.

X(36068) lies on the circumcircle and these lines: {99, 23999}, {101, 32670}, {107, 24021}, {110, 24000}, {1294, 36043}, {1297, 36092}, {26717, 36139}

X(36068) = trilinear pole of line X(6)X(24019)
X(36068) = Ψ(X(i), X(j)) for these (i,j): (1, 107), (6, 24019), (520, 1), (24018, 2)
X(36068) = X(i)-isoconjugate of X(j) for these {i,j}: {656, 856}, {24018, 3330}
X(36068) = trilinear product X(i)*X(j) for these {i,j}: {2, 32670}, {107, 26701}
X(36068) = trilinear quotient X(i)/X(j) for these (i,j): (162, 856), (24019, 3330), (26701, 520), (32670, 6)
X(36068) = barycentric product X(i)*X(j) for these {i,j}: {75, 32670}, {823, 26701}
X(36068) = barycentric quotient X(i)/X(j) for these (i,j): (112, 856), (26701, 24018), (32670, 1), (32713, 3330)

X(36069) = TRILINEAR PRODUCT X(110)*X(759)

Barycentrics    a^2/((b + c) (b^2 - c^2) (a^2 - b^2 - c^2 + b c)) : :
Barycentrics    a csc(B - C)/((b + c) cot A - b cot B - c cot C) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36069) lies on the circumcircle.

X(36069) lies on the circumcircle and these lines: {1, 12030}, {60, 953}, {74, 36034}, {80, 2372}, {98, 8229}, {99, 4467}, {100, 4570}, {101, 32671}, {104, 30576}, {107, 24000}, {110, 1101}, {111, 34079}, {593, 840}, {662, 9070}, {691, 9273}, {849, 2718}, {1300, 36114}, {2750, 6061}, {4017, 26700}, {4556, 4588}, {4575, 6011}, {6187, 28482}, {7252, 23357}, {8687, 32675}

X(36069) = isogonal conjugate of X(6370)
X(36069) = trilinear pole of line X(6)X(163)
X(36069) = Λ(X(i), X(j)) for these {i,j}: {351, 4809}, {643, 4427}, {1109, 2632}, {1637, 1639}, {1769, 2292}, {3268, 4453}, {3569, 4016}, {4036, 4064}, {4647, 4768}, {4707, 4736}
X(36069) = Ψ(X(i), X(j)) for these (i,j): (1, 60), (6, 163), (12, 1), (523, 1), (758, 1), (1577, 2)
X(36069) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 6370}, {12, 3738}, {36, 4036}, {115, 4585}, {320, 4705}, {338, 1983}, {512, 35550}, {523, 758}, {594, 3960}, {654, 6358}, {656, 860}, {661, 3936}, {756, 4453}, {850, 3724}, {1577, 2245}, {2171, 3904}, {3218, 4024}, {8648, 34388}
X(36069) = trilinear product X(i)*X(j) for these {i,j}: {2, 32671}, {60, 2222}, {110, 759}, {163, 24624}, {523, 9274}, {655, 2150}, {661, 9273}, {662, 34079}, {1576, 14616}, {2161, 4556}, {2185, 32675}
X(36069) = trilinear quotient X(i)/X(j) for these (i,j): (1, 6370), (60, 3738), (80, 4036), (99, 35550), (110, 758), (162, 860), (163, 2245), (249, 4585), (593, 3960), (655, 6358), (662, 3936), (757, 4453), (759, 523), (1576, 3724), (2150, 654), (2161, 4024), (2185, 3904), (2222, 12), (4556, 3218), (6187, 4705), (9273, 662), (9274, 110), (14616, 850), (23357, 1983), (24624, 1577), (32671, 6), (32675, 2171), (34079, 661), (35174, 34388)
X(36069) = barycentric product X(i)*X(j) for these {i,j}: {60, 655}, {75, 32671}, {80, 4556}, {99, 34079}, {110, 24624}, {163, 14616}, {261, 32675}, {523, 9273}, {662, 759}, {1577, 9274}, {2150, 35174}, {2185, 2222}, {4610, 6187}
X(36069) = barycentric quotient X(i)/X(j) for these (i,j): (6, 6370), (60, 3904), (110, 3936), (112, 860), (163, 758), (593, 4453), (655, 34388), (662, 35550), (759, 1577), (849, 3960), (1101, 4585), (1576, 2245), (2150, 3738), (2161, 4036), (2222, 6358), (4556, 320), (6187, 4024), (9273, 99), (9274, 662), (14616, 20948), (23995, 1983), (24624, 850), (32671, 1), (32675, 12), (34079, 523)

X(36070) = TRILINEAR PRODUCT X(111)*X(8691)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36070) lies on the circumcircle.

X(36070) lies on the circumcircle and these lines: {99, 36085}, {101, 32672}, {110, 36142}, {1296, 36045}, {2721, 34916}, {2729, 34914}, {30247, 36115}

X(36070) = trilinear pole of line X(6)X(923)
X(36070) = Ψ(X(i), X(j)) for these (i,j): (1, 111), (6, 923), (524, 1), (14210, 2)
X(36070) = X(i)-isoconjugate of X(j) for these {i,j}: {524, 4160}, {896, 4789}, {4750, 5297}
X(36070) = trilinear product X(i)*X(j) for these {i,j}: {2, 32672}, {111, 8691}, {32740, 35181}
X(36070) = trilinear quotient X(i)/X(j) for these (i,j): (111, 4160), (897, 4789), (8691, 524), (32672, 6), (34916, 4750), (35181, 3266)
X(36070) = barycentric product X(i)*X(j) for these {i,j}: {75, 32672}, {897, 8691}, {923, 35181}, {5380, 34916}
X(36070) = barycentric quotient X(i)/X(j) for these (i,j): (111, 4789), (923, 4160), (8691, 14210), (32672, 1)

X(36071) = TRILINEAR PRODUCT X(112)*X(26702)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36071) lies on the circumcircle.

X(36071) lies on the circumcircle and these lines: {74, 36131}, {98, 36104}, {101, 32673}, {1297, 36046}, {2373, 36095}, {4556, 6183}

X(36071) = trilinear pole of line X(6)X(32676)
X(36071) = Ψ(X(i), X(j)) for these (i,j): (1, 112), (6, 32676), (525, 1), (14208, 2)
X(36071) = X(656)-isoconjugate of X(857)
X(36071) = trilinear product X(i)*X(j) for these {i,j}: {2, 32673}, {112, 26702}
X(36071) = trilinear quotient X(i)/X(j) for these (i,j): (162, 857), (26702, 525), (32673, 6)
X(36071) = barycentric product X(i)*X(j) for these {i,j}: {75, 32673}, {162, 26702}
X(36071) = barycentric quotient X(i)/X(j) for these (i,j): (112, 857), (26702, 14208), (32673, 1)

X(36072) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(15)

Barycentrics    a^2/((b - c) (Sqrt[3] (a - b - c) (a + b + c) + 2 S)) : :
Barycentrics    a^2 (cos A/2) csc(A/2 - π/3)/(cos B - cos C) : :
Barycentrics    a^3 sec(A/2) csc(A/2 - π/3)/(b - c) : :
Barycentrics    a^2/(sin(B + π/3) - sin(C + π/3)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36072) lies on the circumcircle.

X(36072) lies on the circumcircle and these lines: {74, 10638}, {104, 1251}, {105, 2306}, {163, 5995}, {675, 1081}, {692, 36073}, {759, 7052}, {2153, 2170}

X(36072) = trilinear pole of line X(6)X(2151)
X(36072) = Ψ(X(i), X(j)) for these (i,j): (1, 15), (6, 2151), (13, 1)
X(36072) = {X(692),X(36074)}-harmonic conjugate of X(36073)
X(36072) = X(36075)-cross conjugate of X(36073)
X(36072) = X(i)-isoconjugate of X(j) for these {i,j}: {522, 1082}, {693, 1250}
X(36072) = trilinear product X(i)*X(j) for these {i,j}: {101, 2306}, {109, 1251}, {692, 1081}, {10638, 26700}
X(36072) = trilinear quotient X(i)/X(j) for these (i,j): (109, 1082), (692, 1250), (1081, 693), (1251, 522), (2306, 514), (10638, 35057)
X(36072) = barycentric product X(i)*X(j) for these {i,j}: {100, 2306}, {101, 1081}, {651, 1251}
X(36072) = barycentric quotient X(i)/X(j) for these (i,j): (1081, 3261), (1251, 4391), (1415, 1082), (2306, 693), (32739, 1250)

X(36073) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(16)

Barycentrics    a^2/((b - c) (Sqrt[3] (a - b - c) (a + b + c) - 2 S)) : :
Barycentrics    a^2 cos(A/2) csc(A/2 + π/3)/(cos B - cos C) : :
Barycentrics    a^3 sec(A/2) csc(A/2 + π/3)/(b - c) : :
Barycentrics    a^2/(sin(B - π/3) - sin(C - π/3)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36073) lies on the circumcircle.

X(36073) lies on the circumcircle and these lines: {74, 1250}, {104, 18469}, {105, 33654}, {163, 5994}, {554, 675}, {692, 36072}, {759, 33655}, {2154, 2170}, {2307, 28471}

X(36073) = trilinear pole of line X(6)X(2152)
X(36073) = Ψ(X(i), X(j)) for these (i,j): (1, 16), (6, 2152), (14, 1)
X(36073) = {X(692),X(36074)}-harmonic conjugate of X(36072)
X(36073) = X(36075)-cross conjugate of X(36072)
X(36073) = X(i)-isoconjugate of X(j) for these {i,j}: {522, 559}, {693, 10638}, {1251, 4467}
X(36073) = trilinear product X(i)*X(j) for these {i,j}: {101, 33654}, {109, 33653}, {554, 692}, {1250, 26700}
X(36073) = trilinear quotient X(i)/X(j) for these (i,j): (109, 559), (554, 693), (692, 10638), (1082, 4467), (1250, 35057), (26700, 1081), (33653, 522), (33654, 514)
X(36073) = barycentric product X(i)*X(j) for these {i,j}: {100, 33654}, {101, 554}, {651, 33653}, {2307, 6742}
X(36073) = barycentric quotient X(i)/X(j) for these (i,j): (554, 3261), (1415, 559), (2307, 4467), (32739, 10638), (33653, 4391), (33654, 693)

X(36074) = {X(36072),X(36073)}-HARMONIC CONJUGATE OF X(692)

Barycentrics    a^2 (3 a^4 + 6 a^3 (b + c) + 12 a^2 b c - 6 a (b^2 - c^2) (b - c) - 3 (b^2 - c^2)^2 + 4 S^2)/(b - c) : :
Barycentrics    a^2 (3 cos^2 A + sin^2 A + 3 cos B cos C + sin B sin C - 3 cos C cos A - sin C sin A - 3 cos A cos B - sin A sin B) : :

X(36074) lies on these lines: {100, 26733}, {101, 109}, {213, 18360}, {692, 36072}, {901, 32693}, {1406, 14974}, {1464, 17735}, {1500, 8614}, {2099, 9346}, {4588, 8687}

X(36074) = {X(101),X(109)}-harmonic conjugate of X(36075)
X(36074) = {X(36072),X(36073)}-harmonic conjugate of X(692)

X(36075) = CROSSPOINT OF X(36072) AND X(36073)

Barycentrics    a^2 (3 a^4 + 6 a^3 (b + c) + 12 a^2 b c - 6 a (b^2 - c^2) (b - c) - 3 (b^2 - c^2)^2 - 4 S^2)/(b - c) : :
Barycentrics    a^2 (3 cos^2 A - sin^2 A + 3 cos B cos C - sin B sin C - 3 cos C cos A + sin C sin A - 3 cos A cos B + sin A sin B) : :

X(36075) lies on these lines: {6, 19302}, {56, 17962}, {101, 109}, {110, 26733}, {163, 2420}, {172, 18360}, {187, 1464}, {603, 2280}, {901, 8687}, {1406, 3053}, {2199, 2267}, {4556, 4565}, {4588, 32693}, {6076, 13562}, {8614, 18755}, {32640, 36064}

X(36075) = crosspoint of X(36072) and X(36073)
X(36075) = intersection of tangents to circumcircle at X(36072) and X(36073)
X(36075) = pole wrt circumcircle of line X(692)X(36072)
X(36075) = trilinear pole of line X(2308)X(23201)
X(36075) = {X(101),X(109)}-harmonic conjugate of X(36074)

X(36076) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(24)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 - b^2 - c^2) (a^4 + 2 a^2 b c - (b^2 - c^2)^2)) : :
Barycentrics    a^2/((cos B - cos C) (2 + sec B sec C)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36076) lies on the circumcircle.

X(36076) lies on the circumcircle and these lines: {4, 38964}, {102, 3422}, {104, 1061}, {162, 925}, {759, 2299}, {2222, 8750}

X(36076) = Ψ(X(i), X(j)) for these (i,j): (1, 24), (3, 47), (68, 1), (91, 4)
X(36076) = X(i)-isoconjugate of X(j) for these {i,j}: {521, 1478}, {522, 1060}
X(36076) = polar-circle-inverse of X(38964)
X(36076) = trilinear product X(i)*X(j) for these {i,j}: {108, 3422}, {109, 1061}
X(36076) = trilinear quotient X(i)/X(j) for these (i,j): (108, 1478), (109, 1060), (1061, 522), (3422, 521)
X(36076) = barycentric product X(i)*X(j) for these {i,j}: {651, 1061}, {653, 3422}
X(36076) = barycentric quotient X(i)/X(j) for these (i,j): (1061, 4391), (1415, 1060), (3422, 6332), (32674, 1478)

X(36077) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(27)

Barycentrics  &nbsnbsp; a/((b^2 - c^2) (a^2 - b^2 - c^2) (a^3 - a (b + c)^2 - 2 b c (b + c))) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36077) lies on the circumcircle.

X(36077) lies on the circumcircle and these lines: {4, 38967}, {100, 648}, {101, 162}, {2215, 2249}, {26703, 32958}

X(36077) = trilinear pole of line X(6)X(28)
X(36077) = Ψ(X(i), X(j)) for these (i,j): (1, 27), (6, 28), (71, 1), (72, 2)
X(36077) = polar-circle-inverse of X(38967)
X(36077) = X(i)-isoconjugate of X(j) for these {i,j}: {405, 656}, {647, 5271}, (2335, 8611), {5320, 14208}
X(36077) = trilinear product X(i)*X(j) for these {i,j}: {27, 36080}, {648, 2215}
X(36077) = trilinear quotient X(i)/X(j) for these (i,j): (162, 405), (648, 5271), (2215, 647), (2335, 8611), (32676, 5320), (36080, 71)
X(36077) = barycentric product X(i)*X(j) for these {i,j}: {286, 36080}, {811, 2215}
X(36077) = barycentric quotient X(i)/X(j) for these (i,j): (112, 405), (162, 5271), (2215, 656), (36080, 72)

X(36078) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(54)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 - b^2 - c^2 + b c) (a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :
Barycentrics    a^2 sec(B - C)/(cos(C - A) - cos(A - B)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36078) lies on the circumcircle.

X(36078) lies on the circumcircle and these lines: {54, 953}, {110, 36134}, {112, 32675}, {655, 925}, {1411, 26707}, {2006, 26708}, {2169, 2716}, {2594, 14979}, {4559, 14586}, {8685, 8744}, {24027, 26700}

X(36078) = isogonal conjugate of X(6369)
X(36078) = trilinear pole of line X(6)X(2148)
X(36078) = Ψ(X(i), X(j)) for these (i,j): (1, 54), (5, 1), (6, 2148), (14213, 2)
X(36078) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 6369}, {5, 3738}, {311, 8648}, {654, 14213}, {1953, 3904}, {4282, 18314}
X(36078) = trilinear product X(i)*X(j) for these {i,j}: {54, 2222}, {655, 2148}, {2167, 32675}
X(36078) = trilinear quotient X(i)/X(j) for these (i,j): (1, 6369), (54, 3738), (655, 14213), (2148, 654), (2167, 3904), (2222, 5), (14586, 4282), (21741, 2081), (32675, 1953), (35174, 311)
X(36078) = barycentric product X(i)*X(j) for these {i,j}: {54, 655}, {95, 32675}, {2148, 35174}, {2167, 2222}
X(36078) = barycentric quotient X(i)/X(j) for these (i,j): (6, 6369), (54, 3904), (655, 311), (2148, 3738), (2222, 14213), (32675, 5)

X(36079) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(64)

Barycentrics    a^2/((b - c) (a - b - c)^2 (3 a^4 - 2 a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :
Barycentrics    a^2/((1 + cos A) (cos B - cos C) (cos A - cos B cos C))

As the trilinear product of circumcircle-X(1)-antipodes, X(36079) lies on the circumcircle.

X(36079) lies on the circumcircle and these lines: {64, 103}, {104, 4341}, {107, 36118}, {112, 1461}, {253, 2370}, {269, 1295}, {972, 19614}, {1042, 1297}, {1073, 1407}, {1294, 3668}, {1305, 23973}, {2371, 30457}, {2738, 11589}, {4350, 26703}, {6614, 8059}

X(36079) = trilinear pole of line X(6)X(2155)
X(36079) = Ψ(X(i), X(j)) for these (i,j): (1, 64), (6, 2155), (20, 1), (18750, 2)
X(36079) = X(i)-isoconjugate of X(j) for these {i,j}: {20, 3900}, {154, 4397}, {522, 7070}, {610, 3239}, {657, 18750}, {1394, 4163}, {2287, 6587}, {2328, 17898}, {3172, 15416}, {4105, 33673}, {4130, 18623}, {8641, 14615}
X(36079) = trilinear product X(i)*X(j) for these {i,j}: {64, 934}, {109, 8809}, {658, 2155}, {1073, 32714}, {1461, 2184}, {4569, 33581}, {4617, 30457}, {13149, 14642}, {19614, 36118}
X(36079) = trilinear quotient X(i)/X(j) for these (i,j): (64, 3900), (109, 7070), (253, 4397), (658, 18750), (934, 20), (1427, 6587), (1461, 610), (2155, 657), (2184, 3239), (3668, 17898), (4569, 14615), (4617, 18623), (4626, 33673), (6614, 1394), (8809, 522), (13149, 15466), (30457, 4130), (32714, 1249), (33581, 8641), (34403, 15416), (36118, 1895)
X(36079) = barycentric product X(i)*X(j) for these {i,j}: {64, 658}, {253, 1461}, {651, 8809}, {934, 2184}, {1073, 36118}, {2155, 4569}, {4626, 30457}, {13149, 19614}, {19611, 32714}
X(36079) = barycentric quotient X(i)/X(j) for these (i,j): (64, 3239), (658, 14615), (934, 18750), (1042, 6587), (1415, 7070), (1427, 17898), (1461, 20), (2155, 3900), (2184, 4397), (4617, 33673), (6614, 18623), (8809, 4391), (19611, 15416), (30457, 4163), (32714, 1895), (33581, 657), (36118, 15466)

X(36080) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(71)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36080) lies on the circumcircle.

X(36080) lies on the circumcircle and these lines: {99, 1332}, {100, 4574}, {104, 2256}, {106, 2215}, {107, 1783}, {108, 4559}, {110, 906}, {112, 692}, {644, 835}, {651, 1305}, {741, 2196}, {1415, 15439}, {6013, 35338}

X(36080) = trilinear pole of line X(6)X(228)
X(36080) = Ψ(X(i), X(j)) for these (i,j): (1, 71), (2, 72), (6, 228), (27, 1), (28, 6), (286, 2)
X(36080) = X(i)-isoconjugate of X(j) for these {i,j}: {513, 5271}, {514, 405}
X(36080) = trilinear product X(i)*X(j) for these {i,j}: {71, 36077}, {100, 2215}
X(36080) = trilinear quotient X(i)/X(j) for these (i,j): (100, 5271), (101, 405), (2215, 513), (36077, 27)
X(36080) = barycentric product X(i)*X(j) for these {i,j}: {72, 36077}, {190, 2215}, {651, 2335}
X(36080) = barycentric quotient X(i)/X(j) for these (i,j): (101, 5271), (692, 405), (2215, 514), (2335, 4391), (36077, 286)

X(36081) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(83)

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

As the trilinear product of circumcircle-X(1)-antipodes, X(36081) lies on the circumcircle.

X(36081) lies on the circumcircle and these lines: {82, 2382}, {83, 14665}, {99, 4583}, {110, 660}, {251, 9111}, {291, 28485}, {741, 30663}, {825, 4628}, {831, 4562}, {1799, 2862}, {3112, 9073}, {4557, 4577}, {4564, 29055}, {12032, 34055}

X(36081) = trilinear pole of line X(6)X(82)
X(36081) = Ψ(X(i), X(j)) for these (i,j): (1, 83), (6, 82), (38, 2), (39, 1)
X(36081) = X(i)-isoconjugate of X(j) for these {i,j}: {38, 659}, {39, 812}, {141, 8632}, {826, 5009}, {1914, 16892}, {1964, 3766}, {2084, 30940}, {3005, 33295}
X(36081) = trilinear product X(i)*X(j) for these {i,j}: {82, 660}, {83, 813}, {251, 4562}, {335, 4628}, {733, 18047}, {3112, 34067}
X(36081) = trilinear quotient X(i)/X(j) for these (i,j): (82, 659), (83, 812), (251, 8632), (335, 16892), (660, 38), (813, 39), (827, 5009), (3112, 3766), (4562, 141), (4577, 33295), (4579, 2236), (4583, 1930), (4593, 30940), (4628, 1914), (18047, 732), (34067, 1964)
X(36081) = barycentric product X(i)*X(j) for these {i,j}: {82, 4562}, {83, 660}, {251, 4583}, {308, 34067}, {334, 4628}, {813, 3112}, {4579, 14970}
X(36081) = barycentric quotient X(i)/X(j) for these (i,j): (82, 812), (83, 3766), (251, 659), (291, 16892), (660, 141), (813, 38), (4562, 1930), (4577, 30940), (4579, 732), (4583, 8024), (4599, 33295), (4628, 238), (34067, 39), (34072, 5009)

X(36082) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(1)X(90)

Barycentrics    a^2/((b - c) (a - b - c) (a^3 + a^2 (b + c) - a (b^2 + c^2) - (b - c)^2 (b + c))) : :
Barycentrics    a^2/((cos B - cos C) (cos A - cos B - cos C)) : :

As the trilinear product of circumcircle-X(1)-antipodes, X(36082) lies on the circumcircle.

X(36082) lies on the circumcircle and these lines: {1, 915}, {48, 20624}, {90, 104}, {100, 13256}, {102, 1069}, {103, 7072}, {106, 10571}, {497, 917}, {675, 7318}, {759, 2360}, {944, 7040}, {1300, 10572}, {1311, 2994}, {1331, 6099}, {1461, 26700}, {1630, 2164}, {1813, 13397}, {2365, 6512}, {2376, 34036}, {2717, 18839}, {4551, 9058}, {4575, 13398}, {6513, 26703}

X(36082) = trilinear pole of line X(6)X(1195)
X(36082) = Ψ(X(i), X(j)) for these (i,j): (1, 90), (4, 46), (6, 1195), (46, 1), (90, 3)
X(36082) = Λ(X(1), X(7649))
X(36082) = X(i)-isoconjugate of X(j) for these {i,j}: {46, 522}, {521, 1068}, {650, 5905}, {663, 20930}, {2178, 4391}, {3064, 6505}
X(36082) = trilinear product X(i)*X(j) for these {i,j}: {90, 109}, {108, 1069}, {651, 2164}, {1415, 2994}, {2349, 36149}, {6513, 32674}, {7040, 36059}
X(36082) = trilinear quotient X(i)/X(j) for these (i,j): (90, 522), (108, 1068), (109, 46), (651, 5905), (664, 20930), (1069, 521), (1415, 2178), (1813, 6505), (2164, 650), (2994, 4391), (6513, 6332), (20570, 35519), (36059, 3157), (36149, 2173)
X(36082) = barycentric product X(i)*X(j) for these {i,j}: {90, 651}, {108, 6513}, {109, 2994}, {653, 1069}, {664, 2164}, {1415, 20570}, {1494, 36149}, {1813, 7040}, {6512, 36127}
X(36082) = barycentric quotient X(i)/X(j) for these (i,j): (90, 4391), (109, 5905), (651, 20930), (1069, 6332), (1415, 46), (2164, 522), (2994, 35519), (6513, 35518), (32660, 3157), (32674, 1068), (36059, 6505), (36149, 30)

X(36083) = TRILINEAR PRODUCT X(74)*X(1302)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36083) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36083) lies on these lines: {100, 32681}, {162, 36131}, {662, 36034}, {1302, 36064}, {1725, 2349}

X(36083) = trilinear pole of line X(1)X(2159)
X(36083) = X(i)-isoconjugate of X(j) for these {i,j}: {30, 8675}, {378, 9033}, {1495, 30474}
X(36083) = trilinear product X(i)*X(j) for these {i,j}: {2, 32681}, {74, 1302}, {1304, 4846}, {1494, 32738}, {2349, 36149}
X(36083) = trilinear quotient X(i)/X(j) for these (i,j): (74, 8675), (1302, 30), (1304, 378), (1494, 30474), (32640, 5063), (32681, 6), (32738, 1495), (36149, 2173)
X(36083) = barycentric product X(i)*X(j) for these {i,j}: {75, 32681}, {1302, 2349}, {1494, 36149}, {32738, 33805}
X(36083) = barycentric quotient X(i)/X(j) for these (i,j): (1302, 14206), (2159, 8675), (2349, 30474), (32681, 1), (32738, 2173), (36034, 15066), (36131, 378), (36149, 30)

X(36084) = TRILINEAR PRODUCT X(98)*X(110)

Barycentrics    a/((b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2)) : :
Trilinears    csc(B - C) sec(A + ω) : :

As the trilinear product of circumcircle-X(2)-antipodes, X(36084) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36084) lies on these lines: {98, 8229}, {100, 2715}, {110, 36065}, {162, 661}, {163, 36132}, {190, 2966}, {240, 8772}, {293, 896}, {653, 685}, {656, 662}, {673, 23692}, {799, 4575}, {823, 24000}, {897, 1910}, {1580, 1733}, {1931, 36101}, {2617, 4599}, {2651, 36100}, {6037, 8685}, {15440, 22456}, {32678, 36096}, {32696, 36099}, {36114, 36120}

X(36084) = trilinear pole of line X(1)X(163)
X(36084) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 3569}, {3, 16230}, {4, 684}, {6, 2799}, {25, 6333}, {69, 17994}, {76, 2491}, {110, 868}, {115, 2421}, {125, 4230}, {232, 525}, {237, 850}, {240, 656}, {297, 647}, {325, 512}, {338, 14966}, {511, 523}, {520, 6530}, {526, 14356}, {661, 1959}, {690, 5968}, {877, 20975}, {1109, 23997}, {1577, 1755}, {1637, 35910}, {2396, 3124}, {2489, 6393}, {3005, 20022}, {3289, 14618}, {3405, 8061}, {6368, 19189}, {9033, 35908}
X(36084) = trilinear product X(i)*X(j) for these {i,j}: {2, 2715}, {3, 685}, {6, 2966}, {25, 17932}, {31, 36036}, {63, 36104}, {69, 32696}, {98, 110}, {99, 1976}, {107, 17974}, {112, 287}, {162, 293}, {163, 1821}, {182, 6037}, {183, 32716}, {184, 22456}, {248, 648}, {249, 2395}, {250, 879}, {290, 1576}, {336, 32676}, {476, 14355}, {662, 1910}, {691, 5967}, {827, 20021}, {878, 18020}, {1304, 35912}, {2422, 4590}, {3404, 4599}, {4558, 6531}, {4575, 36120}, {16081, 32661}
X(36084) = trilinear quotient X(i)/X(j) for these (i,j): (2, 2799), (3, 684), (4, 16230), (6, 3569), (25, 17994), (32, 2491), (69, 6333), (98, 523), (99, 325), (107, 6530), (110, 511), (112, 232), (162, 240), (163, 1755), (248, 647), (249, 2421), (250, 4230), (287, 525), (290, 850), (293, 656), (336, 14208), (476, 14356), (523, 868), (648, 297), (662, 1959), (685, 4), (691, 5968), (878, 20975), (879, 125), (933, 19189), (1101, 23997), (1304, 35908), (1576, 237), (1821, 1577), (1910, 661), (1976, 512), (2395, 115), (2422, 3124), (2715, 6), (2966, 2), (3404, 8061), (4563, 6393), (4577, 20022), (4590, 2396), (4599, 3405), (5967, 690), (6037, 262), (6531, 2501), (14355, 526), (16081, 14618), (17932, 69), (17974, 520), (18020, 877), (20021, 826), (22456, 264), (23357, 14966), (32661, 3289), (32696, 25), (32716, 263), (35906, 1637), (35912, 9033), (36036, 75), (36104, 19), (36120, 24006), (36132, 2186)
X(36084) = barycentric product X(i)*X(j) for these {i,j}: {1, 2966}, {6, 36036}, {19, 17932}, {48, 22456}, {63, 685}, {69, 36104}, {75, 2715}, {98, 662}, {99, 1910}, {110, 1821}, {112, 336}, {162, 287}, {163, 290}, {183, 36132}, {248, 811}, {293, 648}, {304, 32696}, {799, 1976}, {823, 17974}, {2395, 24041}, {2422, 24037}, {3403, 32716}, {3404, 4577}, {4558, 36120}, {4575, 16081}, {4592, 6531}, {4599, 20021}, {14355, 32680}, {27958, 36065}
X(36084) = barycentric quotient X(i)/X(j) for these (i,j): (1, 2799), (19, 16230), (31, 3569), (48, 684), (63, 6333), (98, 1577), (110, 1959), (112, 240), (162, 297), (163, 511), (248, 656), (287, 14208), (290, 20948), (293, 525), (336, 3267), (560, 2491), (661, 868), (662, 325), (685, 92), (827, 3405), (878, 3708), (879, 20902), (1101, 2421), (1576, 1755), (1821, 850), (1910, 523), (1973, 17994), (1976, 661), (2395, 1109), (2422, 2643), (2715, 1), (2966, 75), (3404, 826), (4592, 6393), (4599, 20022), (6531, 24006), (14355, 32679), (17932, 304), (17974, 24018), (22456, 1969), (23357, 23997), (23995, 14966), (24019, 6530), (24041, 2396), (32676, 232), (32678, 14356), (32696, 19), (32716, 2186), (36034, 35910), (36036, 76), (36104, 4), (36120, 14618), (36131, 35908), (36132, 262), (36142, 5968)

X(36085) = TRILINEAR PRODUCT X(99)*X(111)

Barycentrics    a/((b^2 - c^2) (2 a^2 - b^2 - c^2)) : :
Trilinears    1/((cot B - cot C) (2 cot A - cot B - cot C)) : :

As the trilinear product of circumcircle-X(2)-antipodes, X(36085) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36085) lies on these lines: {88, 1931}, {99, 36070}, {100, 691}, {111, 36066}, {190, 892}, {661, 662}, {671, 24624}, {799, 1577}, {811, 36115}, {823, 23999}, {896, 897}, {923, 1580}, {1156, 2651}, {1492, 32729}, {1821, 14206}, {1959, 2349}, {4592, 36045}, {23695, 23707}, {23889, 23894}, {36061, 36096}, {36105, 36128}

X(36085) = isogonal conjugate of X(2642)
X(36085) = isotomic conjugate of isogonal conjugate of X(36142)
X(36085) = trilinear pole of line X(1)X(662)
X(36085) = barycentric square root of X(34539)
X(36085) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 2642}, {2, 351}, {6, 690}, {25, 14417}, {32, 35522}, {110, 1648}, {111, 1649}, {115, 5467}, {187, 523}, {468, 647}, {512, 524}, {649, 4062}, {661, 896}, {669, 3266}, {694, 11183}, {798, 14210}, {850, 14567}, {888, 14608}, {922, 1577}, {2482, 9178}, {2489, 6390}, {2501, 3292}, {2643, 23889}, {3124, 5468}, {3569, 5967}, {4235, 20975}, {9125, 21448}
X(36085) = trilinear product X(i)*X(j) for these {i,j}: {2, 691}, {6, 892}, {75, 36142}, {76, 32729}, {99, 111}, {110, 671}, {112, 30786}, {249, 5466}, {250, 14977}, {524, 34574}, {648, 895}, {662, 897}, {670, 32740}, {799, 923}, {811, 36060}, {1576, 18023}, {2966, 5968}, {4558, 17983}, {4563, 8753}, {4590, 9178}, {4592, 36128}, {5468, 10630}, {6331, 14908}, {9150, 14609}, {10097, 18020}, {11059, 32648}, {24041, 23894}
X(36085) = trilinear quotient X(i)/X(j) for these (i,j): (1, 2642), (2, 690), (6, 351), (69, 14417), (76, 35522), (99, 524), (110, 187), (111, 512), (163, 922), (190, 4062), (249, 5467), (385, 11183), (523, 1648), (524, 1649), (648, 468), (662, 896), (670, 3266), (671, 523), (691, 6), (799, 14210), (892, 2), (895, 647), (897, 661), (923, 798), (1576, 14567), (1992, 9125), (2966, 5967), (4558, 3292), (4563, 6390), (4590, 5468), (5466, 115), (5468, 2482), (5968, 3569), (8753, 2489), (9150, 14608), (9178, 3124), (10097, 20975), (10630, 9178), (14908, 3049), (14977, 125), (17983, 2501), (18020, 4235), (18023, 850), (23894, 2643), (24037, 24039), (24039, 24038), (24041, 23889), (30786, 525), (32729, 32), (32740, 669), (34574, 111), (36060, 810), (36142, 31)
X(36085) = barycentric product X(i)*X(j) for these {i,j}: {1, 892}, {75, 691}, {76, 36142}, {99, 897}, {111, 799}, {162, 30786}, {163, 18023}, {561, 32729}, {670, 923}, {662, 671}, {811, 895}, {4563, 36128}, {4590, 23894}, {4592, 17983}, {4602, 32740}, {5466, 24041}, {5968, 36036}, {6331, 36060}, {9178, 24037}, {10630, 24039}, {11059, 36045}, {14210, 34574}
X(36085) = barycentric quotient X(i)/X(j) for these (i,j): (1, 690), (6, 2642), (31, 351), (63, 14417), (75, 35522), (99, 14210), (100, 4062), (110, 896), (111, 661), (162, 468), (163, 187), (249, 23889), (661, 1648), (662, 524), (671, 1577), (691, 1), (799, 3266), (892, 75), (895, 656), (896, 1649), (897, 523), (923, 512), (1101, 5467), (1576, 922), (1580, 11183), (4575, 3292), (4590, 24039), (4592, 6390), (5466, 1109), (5468, 24038), (9178, 2643), (10097, 3708), (10630, 23894), (14908, 810), (14977, 20902), (17983, 24006), (18023, 20948), (23889, 2482), (23894, 115), (24041, 5468), (30786, 14208), (32729, 31), (32740, 798), (34574, 897), (36045, 21448), (36060, 647), (36084, 5967), (36128, 2501), (36133, 14608), (36142, 6)

X(36086) = TRILINEAR PRODUCT X(100)*X(105)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36086) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36086) lies on these lines: {31, 6654}, {44, 294}, {59, 513}, {88, 105}, {100, 650}, {101, 4794}, {109, 658}, {190, 522}, {238, 516}, {241, 1279}, {320, 31637}, {518, 677}, {643, 799}, {653, 7012}, {655, 885}, {660, 2284}, {662, 3737}, {663, 4564}, {884, 23832}, {897, 16548}, {1026, 23704}, {1027, 3257}, {1110, 4040}, {1308, 3960}, {1331, 36041}, {1416, 9364}, {1438, 5053}, {1897, 36111}, {1936, 2342}, {2398, 2402}, {3286, 17798}, {6163, 27834}, {9371, 36100}, {13397, 35185}, {13576, 14956}, {13589, 36087}, {23703, 35340}, {36106, 36124}

X(36086) = isogonal conjugate of X(2254)
X(36086) = cevapoint of X(i) and X(j) for these {i,j}: {1, 2254}, {105, 1027}, {513, 1279}, {663, 672}, {1024, 2195}
X(36086) = crosspoint of X(666) and X(927)
X(36086) = crosssum of X(665) and X(926)
X(36086) = trilinear pole of line X(1)X(41)
X(36086) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 2254}, {2, 665}, {6, 918}, {7, 926}, {11, 2283}, {100, 3675}, {105, 3126}, {241, 650}, {244, 1026}, {512, 30941}, {513, 518}, {514, 672}, {522, 1458}, {523, 3286}, {649, 3912}, {651, 17435}, {652, 5236}, {661, 18206}, {663, 9436}, {667, 3263}, {693, 2223}, {798, 18157}, {812, 3252}, {883, 3271}, {905, 5089}, {1025, 2170}, {1027, 4712}, {1086, 2284}, {1459, 1861}, {1818, 7649}, {2340, 3676}, {2356, 4025}, {3063, 27818}, {3261, 9454}, {3669, 3693}, {3900, 34855}, {6591, 25083}, {17924, 20752}
X(36086) = trilinear product X(i)*X(j) for these {i,j}: {2, 919}, {6, 666}, {8, 32735}, {9, 36146}, {41, 34085}, {55, 927}, {59, 885}, {75, 32666}, {82, 35333}, {100, 105}, {101, 673}, {109, 14942}, {110, 13576}, {190, 1438}, {294, 651}, {344, 32644}, {513, 5377}, {644, 1462}, {662, 18785}, {664, 2195}, {692, 2481}, {765, 1027}, {813, 6654}, {884, 4998}, {934, 28071}, {1024, 4564}, {1331, 36124}, {1332, 8751}, {1416, 3699}, {1461, 6559}, {1783, 1814}, {1897, 36057}, {2284, 6185}, {3434, 35185}, {3870, 36041}, {4384, 36138}, {4441, 32724}, {6335, 32658}, {7012, 23696}, {8750, 31637}, {18031, 32739}, {28420, 32703}
X(36086) = trilinear quotient X(i)/X(j) for these (i,j): (1, 2254), (2, 918), (6, 665), (55, 926), (59, 2283), (99, 30941), (100, 518), (101, 672), (105, 513), (109, 1458), (110, 3286), (190, 3912), (294, 650), (513, 3675), (518, 3126), (644, 3693), (650, 17435), (651, 241), (653, 5236), (662, 18206), (664, 9436), (666, 2), (668, 3263), (673, 514), (692, 2223), (765, 1026), (799, 18157), (813, 3252), (884, 3271), (885, 11), (906, 20752), (919, 6), (927, 7), (934, 34855), (1024, 2170), (1026, 4712), (1027, 244), (1252, 2284), (1331, 1818), (1332, 25083), (1438, 649), (1462, 3669), (1783, 5089), (1814, 905), (1897, 1861), (2195, 663), (2284, 6184), (2481, 693), (3699, 3717), (3939, 2340), (4554, 27818), (4564, 1025), (4998, 883), (5377, 100), (6559, 3239), (6654, 812), (8750, 2356), (8751, 6591), (13576, 523), (14942, 522), (18031, 3261), (18785, 661), (23696, 7004), (28071, 3900), (31637, 4025), (32658, 22383), (32666, 31), (32735, 56), (32739, 9454), (34018, 24002), (34085, 85), (35185, 3433), (35333, 38), (36041, 2191), (36057, 1459), (36124, 7649), (36138, 2279), (36146, 57)
X(36086) = barycentric product X(i)*X(j) for these {i,j}: {1, 666}, {8, 36146}, {9, 927}, {55, 34085}, {75, 919}, {76, 32666}, {83, 35333}, {99, 18785}, {100, 673}, {101, 2481}, {105, 190}, {294, 664}, {312, 32735}, {344, 36041}, {646, 1416}, {651, 14942}, {658, 28071}, {660, 6654}, {662, 13576}, {668, 1438}, {692, 18031}, {885, 4564}, {934, 6559}, {1016, 1027}, {1024, 4998}, {1026, 6185}, {1332, 36124}, {1462, 3699}, {1783, 31637}, {1814, 1897}, {2195, 4554}, {2398, 9503}, {3939, 34018}, {4441, 36138}, {4561, 8751}, {6335, 36057}, {20927, 35185}, {21615, 32724}, {28420, 36111}
X(36086) = barycentric quotient X(i)/X(j) for these (i,j): (1, 918), (6, 2254), (31, 665), (41, 926), (59, 1025), (99, 18157), (100, 3912), (101, 518), (105, 514), (108, 5236), (109, 241), (110, 18206), (163, 3286), (190, 3263), (294, 522), (644, 3717), (649, 3675), (651, 9436), (662, 30941), (663, 17435), (664, 27818), (666, 75), (672, 3126), (673, 693), (692, 672), (884, 2170), (885, 4858), (906, 1818), (919, 1), (927, 85), (1024, 11), (1026, 4437), (1027, 1086), (1110, 2284), (1331, 25083), (1252, 1026), (1415, 1458), (1416, 3669), (1438, 513), (1461, 34855), (1462, 3676), (1783, 1861), (1814, 4025), (2149, 2283), (2195, 650), (2284, 4712), (2481, 3261), (3939, 3693), (4564, 883), (6559, 4397), (6654, 3766), (8750, 5089), (8751, 7649), (9503, 2400), (13576, 1577), (14942, 4391), (18785, 523), (21615, 4441), (23696, 26932), (28071, 3239), (31637, 15413), (32644, 2191), (32656, 20752), (32658, 1459), (32666, 6), (32724, 2279), (32735, 57), (32739, 2223), (34067, 3252), (34085, 6063), (35333, 141), (36041, 277), (36057, 905), (36124, 17924), (36138, 1002), (36146, 7)

X(36087) = TRILINEAR PRODUCT X(101)*X(675)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36087) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36087) lies on these lines: {88, 2224}, {100, 1110}, {190, 1252}, {651, 2149}, {653, 7115}, {658, 1262}, {673, 675}, {799, 4567}, {13589, 36086}, {32641, 34234}, {36039, 36101}

X(36087) = trilinear pole of line X(1)X(692)
X(36087) = X(i)-isoconjugate of X(j) for these {i,j}: {514, 674}, {523, 14964}, {649, 3006}, {693, 2225}, {3261, 8618}
X(36087) = trilinear product X(i)*X(j) for these {i,j}: {2, 32682}, {100, 2224}, {101, 675}
X(36087) = trilinear quotient X(i)/X(j) for these (i,j): (101, 674), (110, 14964), (190, 3006), (675, 514), (692, 2225), (2224, 513), (32682, 6), (32739, 8618)
X(36087) = barycentric product X(i)*X(j) for these {i,j}: {75, 32682}, {100, 675}, {190, 2224}
X(36087) = barycentric quotient X(i)/X(j) for these (i,j): (100, 3006), (163, 14964), (675, 693), (692, 674), (2224, 514), (32682, 1), (32739, 2225)

X(36088) = TRILINEAR PRODUCT X(102)*X(9056)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36088) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36088) lies on these lines: {100, 32683}, {651, 36040}, {653, 9056}, {1735, 36100}

X(36088) = trilinear pole of line X(1)X(32677)
X(36088) = X(515)-isoconjugate of X(8999)
X(36088) = trilinear product X(i)*X(j) for these {i,j}: {2, 32683}, {102, 9056}
X(36088) = trilinear quotient X(i)/X(j) for these (i,j): (102, 8999), (9056, 515), (32683, 6)
X(36088) = barycentric product X(i)*X(j) for these {i,j}: {75, 32683}, {9056, 36100}
X(36088) = barycentric quotient X(i)/X(j) for these (i,j): (32677, 8999), (32683, 1)

X(36089) = TRILINEAR PRODUCT X(103)*X(9057)

Barycentrics    a/((b - c) (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) (a^3 (b + c) - a^2 (b^2 + b c + c^2) - a (b^3 + b^2 c + b c^2 + c^3) + b^4 + b^3 c + 4 b^2 c^2 + b c^3 + c^4)) : :
Barycentrics    a^2/(((c - a) cot B - (a - b) cot C) ((b^3 - c^3) cos A + a^2 (b cos B - c cos C) + b c (b - c))) : :

As the trilinear product of circumcircle-X(2)-antipodes, X(36089) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36089) lies on these lines: {100, 32684}, {190, 677}, {658, 9057}, {1736, 36101}

X(36089) = trilinear pole of line X(1)X(911)
X(36089) = X(516)-isoconjugate of X(9000)
X(36089) = trilinear product X(i)*X(j) for these {i,j}: {2, 32684}, {103, 9057}
X(36089) = trilinear quotient X(i)/X(j) for these (i,j): (103, 9000), (9057, 516), (32684, 6)
X(36089) = barycentric product X(i)*X(j) for these {i,j}: {75, 32684}, {9057, 36101}
X(36089) = barycentric quotient X(i)/X(j) for these (i,j): (911, 9000), (32684, 1)

X(36090) = TRILINEAR PRODUCT X(104)*X(9058)

Barycentrics    a/((b - c) (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c) (a^3- a^2 (b + c) - a (b - c)^2 + (b + c) (b^2 + c^2))) : :
Barycentrics    a/((cos B + cos C - 1) (2 (sin B - sin C) (cos A - 1) + sin 2B + sin 2C)) : :

As the trilinear product of circumcircle-X(2)-antipodes, X(36090) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36090) lies on these lines: {88, 10428}, {100, 32641}, {190, 36037}, {651, 2720}, {653, 36110}, {1737, 34234}

X(36090) = trilinear pole of line X(1)X(909)
X(36090) = X(i)-isoconjugate of X(j) for these {i,j}: {517, 9001}, {650, 2804}, {997, 1769}, {1795, 2252}, {3310, 17740}
X(36090) = trilinear product X(i)*X(j) for these {i,j}: {2, 32685}, {104, 9058}, {998, 36037}, {2720, 30513}
X(36090) = trilinear quotient X(i)/X(j) for these (i,j): (651, 2804), (998, 1769), (1737, 2183), (1785, 2252), (2720, 650), (9058, 517), (13136, 17740), (32685, 6), (36037, 997)
X(36090) = barycentric product X(i)*X(j) for these {i,j}: {75, 32685}, {998, 13136}, {9058, 34234}
X(36090) = barycentric quotient X(i)/X(j) for these (i,j): (998, 10015), (9058, 908), (32641, 997), (32685, 1)

X(36091) = TRILINEAR PRODUCT X(106)*X(9059)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36091) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36091) lies on these lines: {88, 1739}, {100, 32665}, {190, 901}, {799, 4622}, {5176, 10428}, {27834, 36042}

X(36091) = trilinear pole of line X(1)X(9456)
X(36091) = X(i)-isoconjugate of X(j) for these {i,j}: {519, 9002}, {900, 995}, {1960, 4389}
X(36091) = trilinear product X(i)*X(j) for these {i,j}: {2, 32686}, {106, 9059}, {901, 996}
X(36091) = trilinear quotient X(i)/X(j) for these (i,j): (901, 995), (996, 900), (4555, 4389), (9059, 519), (32686, 6)
X(36091) = barycentric product X(i)*X(j) for these {i,j}: {75, 32686}, {88, 9059}, {996, 3257}
X(36091) = barycentric quotient X(i)/X(j) for these (i,j): (996, 3762), (3257, 4389), (9059, 4358), (32665, 995), (32686, 1)

X(36092) = TRILINEAR PRODUCT X(107)*X(1297)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36092) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36092) lies on these lines: {100, 32687}, {162, 36046}, {662, 24000}, {799, 23999}, {823, 17898}, {1297, 36068}, {2349, 8767}

X(36092) = polar conjugate of isogonal conjugate of X(36046)
X(36092) = trilinear pole of line X(1)X(8767)
X(36092) = X(i)-isoconjugate of X(j) for these {i,j}: {441, 647}, {520, 1503}, {525, 8779}, {656, 8766}, {2312, 24018}, {3269, 34211}
X(36092) = trilinear product X(i)*X(j) for these {i,j}: {2, 32687}, {92, 36046}, {107, 1297}, {112, 6330}, {162, 8767}, {264, 32649}, {23964, 34212}, {32713, 35140}
X(36092) = trilinear quotient X(i)/X(j) for these (i,j): (107, 1503), (112, 8779), (162, 8766), (648, 441), (1297, 520), (6330, 525), (6528, 30737), (8767, 656), (23582, 34211), (24019, 2312), (32649, 184), (32687, 6), (34212, 15526), (35140, 3265), (36046, 48)
X(36092) = barycentric product X(i)*X(j) for these {i,j}: {75, 32687}, {162, 6330}, {264, 36046}, {648, 8767}, {823, 1297}, {1969, 32649}, {23999, 34212}, {24019, 35140}
X(36092) = barycentric quotient X(i)/X(j) for these (i,j): (162, 441), (823, 30737), (1297, 24018), (6330, 14208), (8767, 525), (24000, 34211), (24019, 1503), (32649, 48), (32676, 8779), (32687, 1), (32713, 2312), (34212, 2632), (36046, 3)

X(36093) = TRILINEAR PRODUCT X(108)*X(26703)

Barycentrics    a/((b - c) (a - b - c) (a^2 - b^2 - c^2) (a^4 (b + c) - 2 a^3 b c - (b - c)^2 (b + c) (b^2 + c^2))) : :
Barycentrics    a/((sec B - sec C) (b^2 sec B + c^2 sec C - (b^2 + c^2) sec A)) : :

As the trilinear product of circumcircle-X(2)-antipodes, X(36093) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36093) lies on these lines: {100, 7115}, {190, 7012}, {653, 21186}, {23707, 36140}, {26703, 36067}, {34234, 36110}

X(36093) = trilinear pole of line X(1)X(20613)
X(36093) = X(521)-isoconjugate of X(3827)
X(36093) = trilinear product X(i)*X(j) for these {i,j}: {2, 32688}, {108, 26703}
X(36093) = trilinear quotient X(i)/X(j) for these (i,j): (108, 3827), (26703, 521), (32688, 6)
X(36093) = barycentric product X(i)*X(j) for these {i,j}: {75, 32688}, {653, 26703}
X(36093) = barycentric quotient X(i)/X(j) for these (i,j): (26703, 6332), (32674, 3827), (32688, 1)

X(36094) = TRILINEAR PRODUCT X(109)*X(1311)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36094) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36094) lies on these lines: {59, 190}, {100, 2149}, {651, 21189}, {658, 7339}, {673, 32735}, {1156, 36141}, {1311, 2720}, {7677, 36101}, {36040, 36100}

X(36094) = trilinear pole of line X(1)X(1415)
X(36094) = X(i)-isoconjugate of X(j) for these {i,j}: {522, 8679}, {663, 33864}
X(36094) = trilinear product X(i)*X(j) for these {i,j}: {2, 32689}, {109, 1311}
X(36094) = trilinear quotient X(i)/X(j) for these (i,j): (109, 8679), (664, 33864), (1311, 522), (32689, 6)
X(36094) = barycentric product X(i)*X(j) for these {i,j}: {75, 32689}, {651, 1311}
X(36094) = barycentric quotient X(i)/X(j) for these (i,j): (651, 33864), (1311, 650), (1415, 8679), (32689, 1)

X(36095) = TRILINEAR PRODUCT X(112)*X(2373)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36095) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36095) lies on these lines: {100, 10423}, {1821, 36104}, {2349, 36131}, {2373, 36071}

X(36095) = trilinear pole of line X(1)X(32676)
X(36095) = X(i)-isoconjugate of X(j) for these {i,j}: {265, 14208}, {520, 5523}, {525, 2393}, {647, 858}
X(36095) = trilinear product X(i)*X(j) for these {i,j}: {2, 10423}, {107, 18876}, {112, 2373}, {186, 32676}, {648, 1177}
X(36095) = trilinear quotient X(i)/X(j) for these (i,j): (107, 5523), (112, 2393), (186, 14208), (648, 858), (1177, 647), (2373, 525), (10423, 6), (18876, 520), (32676, 265)
X(36095) = barycentric product X(i)*X(j) for these {i,j}: {75, 10423}, {162, 2373}, {823, 18876}
X(36095) = barycentric quotient X(i)/X(j) for these (i,j): (162, 858), (2373, 14208), (10423, 1), (18876, 24018), (24019, 5523), (32676, 2393)

X(36096) = TRILINEAR PRODUCT X(476)*X(842)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36096) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36096) lies on these lines: {100, 23969}, {1821, 2166}, {32678, 36084}, {32680, 36035}, {36047, 36102}, {36061, 36085}

X(36096) = trilinear pole of line X(1)X(32678)
X(36096) = X(i)-isoconjugate of X(j) for these {i,j}: {50, 18312}, {323, 1640}, {526, 542}, {2247, 32679}, {3268, 5191}
X(36096) = trilinear product X(i)*X(j) for these {i,j}: {2, 23969}, {476, 842}, {1989, 5649}, {5641, 14560}
X(36096) = trilinear quotient X(i)/X(j) for these (i,j): (94, 18312), (476, 542), (842, 526), (1989, 1640), (5641, 3268), (5649, 323), (23969, 6), (32678, 2247)
X(36096) = barycentric product X(i)*X(j) for these {i,j}: {75, 23969}, {842, 32680}, {2166, 5649}, {5641, 32678}
X(36096) = barycentric quotient X(i)/X(j) for these (i,j): (842, 32679), (2166, 18312), (14560, 2247), (23969, 1), (32678, 542)

X(36097) = TRILINEAR PRODUCT X(477)*X(9060)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36097) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36097) lies on these lines: {100, 32690}, {32680, 36047}

X(36097) = trilinear pole of line X(1)X(36151)
X(36097) = X(5663)-isoconjugate of X(9003)
X(36097) = trilinear product X(i)*X(j) for these {i,j}: {2, 32690}, {477, 9060}
X(36097) = trilinear quotient X(i)/X(j) for these (i,j): (477, 9003), (9060, 5663), (32690, 6)
X(36097) = barycentric product X(i)*X(j) for these {i,j}: {75, 32690}, {9060, 36102}
X(36097) = barycentric quotient X(32690)/X(1)

X(36098) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(2)X(12)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36098) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36098) lies on these lines: {88, 961}, {100, 1415}, {108, 36099}, {109, 190}, {478, 26264}, {651, 32736}, {655, 4581}, {658, 6614}, {662, 4551}, {673, 1416}, {799, 1414}, {1156, 2298}, {1220, 34234}, {1395, 3769}, {1791, 36100}, {2359, 23707}, {2363, 24624}, {14544, 15420}

X(36098) = isogonal conjugate of X(17420)
X(36098) = trilinear pole of line X(1)X(572)
X(36098) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 17420}, {6, 3910}, {8, 6371}, {41, 4509}, {55, 3004}, {513, 960}, {514, 2269}, {521, 1829}, {522, 1193}, {649, 3687}, {650, 3666}, {652, 1848}, {663, 4357}, {693, 20967}, {1211, 7252}, {2092, 4560}, {2170, 3882}, {2292, 3737}, {2300, 4391}, {2354, 6332}, {3063, 20911}, {3709, 16705}
X(36098) = trilinear product X(i)*X(j) for these {i,j}: {2, 8687}, {6, 6648}, {7, 32736}, {56, 8707}, {57, 36147}, {59, 4581}, {100, 961}, {108, 1791}, {109, 1220}, {651, 2298}, {653, 2359}, {692, 31643}, {1169, 4552}, {1415, 30710}, {2363, 4551}, {4559, 14534}, {4565, 14624}, {7115, 15420}
X(36098) = trilinear quotient X(i)/X(j) for these (i,j): (1, 17420), (2, 3910), (7, 3004), (56, 6371), (85, 4509), (100, 960), (101, 2269), (108, 1829), (109, 1193), (190, 3687), (651, 3666), (653, 1848), (664, 4357), (692, 20967), (961, 513), (1169, 7252), (1220, 522), (1240, 35519), (1415, 2300), (1791, 521), (2298, 650), (2359, 652), (2363, 3737), (4551, 2292), (4552, 1211), (4554, 20911), (4559, 2092), (4564, 3882), (4573, 16705), (4581, 11), (6648, 2), (8687, 6), (8707, 8), (14534, 4560), (14624, 3700), (15420, 26932), (30710, 4391), (31643, 693), (32674, 2354), (32736, 55), (36147, 9)
X(36098) = barycentric product X(i)*X(j) for these {i,j}: {1, 6648}, {7, 36147}, {57, 8707}, {75, 8687}, {85, 32736}, {101, 31643}, {109, 30710}, {190, 961}, {653, 1791}, {664, 2298}, {651, 1220}, {1240, 1415}, {1414, 14624}, {2359, 18026}, {2363, 4552}, {4551, 14534}, {4564, 4581}, {7012, 15420}
X(36098) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3910), (6, 17420), (7, 4509), (57, 3004), (59, 3882), (100, 3687), (101, 960), (108, 1848), (109, 3666), (604, 6371), (651, 4357), (664, 20911), (692, 2269), (961, 514), (1169, 3737), (1414, 16705), (1415, 1193), (1220, 4391), (1791, 6332), (2298, 522), (2359, 521), (2363, 4560), (4551, 1211), (4552, 18697), (4559, 2292), (4581, 4858), (6648, 75), (8687, 1), (8707, 312), (14534, 18155), (14624, 4086), (15420, 17880), (30710, 35519), (31643, 3261), (32674, 1829), (32736, 9), (32739, 20967), (36147, 8)

X(36099) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(2)X(19)

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

As the trilinear product of circumcircle-X(2)-antipodes, X(36099) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36099) lies on these lines: {69, 21148}, {88, 8752}, {100, 8750}, {108, 36098}, {112, 662}, {162, 1633}, {190, 1783}, {648, 799}, {651, 32674}, {658, 32714}, {673, 8751}, {823, 6529}, {897, 8753}, {1036, 23707}, {1039, 1156}, {1821, 2281}, {2221, 34234}, {2339, 36100}, {2349, 8749}, {8743, 14258}, {17602, 17726}, {32696, 36084}

X(36099) = isogonal conjugate of X(2522)
X(36099) = trilinear pole of line X(1)X(25)
X(36099) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 2522}, {3, 6590}, {6, 23874}, {48, 2517}, {63, 8678}, {69, 2484}, {304, 8646}, {388, 652}, {513, 5227}, {521, 2285}, {612, 905}, {647, 1010}, {650, 1038}, {1459, 2345}, {1460, 6332}, {2300, 6332}, {4385, 22383}
X(36099) = trilinear product X(i)*X(j) for these {i,j}: {2, 32691}, {19, 1310}, {108, 2339}, {648, 1245}, {651, 1039}, {653, 1036}, {811, 2281}, {1472, 6335}, {1897, 2221}, {30479, 32674}
X(36099) = trilinear quotient X(i)/X(j) for these (i,j): (1, 2522), (2, 23874), (4, 6590), (19, 8678), (25, 2484), (92, 2517), (100, 5227), (108, 2285), (648, 1010), (651, 1038), (653, 388), (1036, 652), (1039, 650), (1245, 647), (1310, 63), (1472, 22383), (1783, 612), (1897, 2345), (1973, 8646), (2221, 1459), (2281, 810), (2339, 521), (3732, 7386), (6335, 4385), (30479, 6332), (32674, 1460), (32691, 6), (32714, 4320)
X(36099) = barycentric product X(i)*X(j) for these {i,j}: {4, 1310}, {75, 32691}, {108, 30479}, {653, 2339}, {664, 1039}, {811, 1245}, {1036, 18026}, {2221, 6335}, {2281, 6331}
X(36099) = barycentric quotient X(i)/X(j) for these (i,j): (1, 23874), (4, 2517), (6, 2522), (19, 6590), (25, 8678), (101, 5227), (108, 388), (109, 1038), (112, 2303), (162, 1010), (190, 19799), (1036, 521), (1039, 522), (1245, 656), (1310, 69), (1472, 1459), (1633, 7386), (1783, 2345), (1897, 4385), (1973, 2484), (1974, 8646), (2221, 905), (2281, 647), (2339, 6332), (8750, 612), (30479, 35518), (32674, 2285), (32691, 1)

X(36100) = TRILINEAR PRODUCT X(2)*X(102)

Barycentrics    a/(2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2) : :
Barycentrics    1/((b + c) sec A - b sec B - c sec C) : :

X(36100) is the trilinear product of the circumcircle intercepts of line X(2)X(2399). As the trilinear product of circumcircle-X(2)-antipodes, X(36100) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36100) lies on these lines: {2, 196}, {4, 280}, {21, 162}, {40, 78}, {63, 223}, {108, 34588}, {144, 30680}, {190, 329}, {348, 658}, {655, 908}, {660, 23691}, {662, 1812}, {673, 2432}, {823, 31623}, {972, 6081}, {1735, 36088}, {1791, 36098}, {2339, 36099}, {2399, 3904}, {2651, 36084}, {5057, 15633}, {9371, 36086}, {12514, 4025}, {26703, 36067}, {36040, 36094}

X(36100) = isogonal conjugate of X(2182)
X(36100) = isotomic conjugate of isogonal conjugate of X(32677)
X(36100) = polar conjugate of isogonal conjugate of X(36055)
X(36100) = cevapoint of X(i) and X(j) for these {i,j}: {1, 2182}, {3, 2323}, {9, 517}
X(36100) = trilinear pole of line X(1)X(521)
X(36100) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 2182}, {3, 8755}, {6, 515}, {32, 35516}, {522, 2425}, {652, 23987}, {663, 2406}, {1415, 14304}, {6087, 36049}
X(36100) = trilinear product X(i)*X(j) for these {i,j}: {2, 102}, {6, 34393}, {63, 36121}, {75, 32677}, {92, 36055}, {109, 2399}, {664, 2432}, {1262, 15633}, {4391, 36040}, {6081, 14837}, {6332, 36067}, {32643, 35519}, {32667, 35518}
X(36100) = trilinear quotient X(i)/X(j) for these (i,j): (1, 2182), (2, 515), (4, 8755), (76, 35516), (102, 6), (109, 2425), (653, 23987), (664, 2406), (2399, 522), (2432, 663), (4391, 14304), (6081, 36049), (14837, 6087), (15633, 1146), (32677, 31), (34393, 2), (36040, 1415), (36055, 48), (36067, 32674), (36121, 19)
X(36100) = barycentric product X(i)*X(j) for these {i,j}: {1, 34393}, {69, 36121}, {75, 102}, {76, 32677}, {264, 36055}, {651, 2399}, {6081, 17896}, {7045, 15633}, {35518, 36067}, {35519, 36040}
X(36100) = barycentric quotient X(i)/X(j) for these (i,j): (1, 515), (6, 2182), (19, 8755), (75, 35516), (102, 1), (108, 23987), (651, 2406), (1735, 117), (2399, 4391), (6081, 13138), (15633, 24026), (32643, 1415), (32667, 32674), (32677, 6), (34393, 75), (36040, 109), (36055, 3), (36067, 108), (36088, 9056), (36108, 26704), (36121, 4)

X(36101) = TRILINEAR PRODUCT X(2)*X(103)

Barycentrics    a/(2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) : :
Barycentrics    1/(a^2 - b^2 cos C - c^2 cos B) : :
Trilinears    1/((a - b) cot C + (a - c) cot B) : :

X(36101) is the trilinear product of the circumcircle intercepts of line X(2)X(2400). As the trilinear product of circumcircle-X(2)-antipodes, X(36101) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36101) lies on these lines: {2, 658}, {7, 281}, {9, 77}, {57, 3119}, {63, 100}, {69, 144}, {81, 162}, {142, 7110}, {282, 1445}, {286, 823}, {329, 30622}, {518, 677}, {527, 655}, {662, 911}, {673, 918}, {908, 15634}, {934, 34591}, {971, 7291}, {1156, 3738}, {1462, 17435}, {1492, 2975}, {1736, 36089}, {1931, 36084}, {3219, 6605}, {5819, 5942}, {7112, 27818}, {7677, 36094}, {13577, 26871}, {30565, 34234}, {36039, 36087}

X(36101) = isogonal conjugate of X(910)
X(36101) = isotomic conjugate of X(30807)
X(36101) = anticomplement of X(39063)
X(36101) = isotomic conjugate of isogonal conjugate of X(911)
X(36101) = polar conjugate of isogonal conjugate of X(36056)
X(36101) = cevapoint of X(i) and X(j) for these {i,j}: {1, 910}, {9, 518}
X(36101) = trilinear pole of line X(1)X(905)
X(36101) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 910}, {3, 1886}, {6, 516}, {9, 1456}, {25, 26006}, {32, 35517}, {101, 676}, {105, 9502}, {514, 2426}, {647, 4241}, {649, 2398}, {657, 23973}
X(36101) = trilinear product X(i)*X(j) for these {i,j}: {2, 103}, {4, 1815}, {6, 18025}, {7, 2338}, {63, 36122}, {75, 911}, {92, 36056}, {101, 2400}, {190, 2424}, {264, 32657}, {514, 677}, {518, 9503}, {693, 36039}, {1252, 15634}, {3239, 24016}, {3261, 32642}, {4397, 32668}
X(36101) = trilinear quotient X(i)/X(j) for these (i,j): (1, 910), (2, 516), (4, 1886), (57, 1456), (69, 26006), (76, 35517), (101, 2426), (103, 6), (190, 2398), (514, 676), (518, 9502), (648, 4241), (658, 23973), (677, 101), (911, 31), (1815, 3), (2338, 55), (2400, 514), (2424, 649), (9503, 105), (15634, 1086), (18025, 2), (24016, 1461), (32642, 32739), (32657, 184), (36039, 692), (36056, 48), (36122, 19)
X(36101) = barycentric product X(i)*X(j) for these {i,j}: {1, 18025}, {69, 36122}, {75, 103}, {76, 911}, {85, 2338}, {92, 1815}, {100, 2400}, {264, 36056}, {668, 2424}, {677, 693}, {765, 15634}, {1969, 32657}, {3261, 36039}, {3912, 9503}, {4397, 24016}
X(36101) = barycentric quotient X(i)/X(j) for these (i,j): (1, 516), (6, 910), (19, 1886), (56, 1456), (63, 26006), (75, 35517), (100, 2398), (103, 1), (162, 4241), (658, 24015), (677, 100), (911, 6), (934, 23973), (1736, 118), (1815, 63), (2338, 9), (2400, 693), (2424, 513), (9503, 673), (15634, 1111), (18025, 75), (24016, 934), (32642, 692), (32657, 48), (32668, 1461), (36039, 101), (36056, 3), (36089, 9057), (36109, 26705), (36122, 4), (36136, 26716)

X(36102) = TRILINEAR PRODUCT X(2)*X(477)

Barycentrics    b c/(a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4)) : :
Barycentrics    1/(4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)) : :

X(36102) is the trilinear product of the circumcircle intercepts of line X(2)X(2411). As the trilinear product of circumcircle-X(2)-antipodes, X(36102) lies on ellipse {{A,B,C,X(88),X(100)}} with center X(9) and perspector X(1).

X(36102) lies on these lines: {63, 32680}, {100, 477}, {162, 1784}, {662, 14206}, {1577, 2349}, {36047, 36096}

X(36102) = polar conjugate of X(36063)
X(36102) = trilinear pole of line X(1)X(36035) (the polar of X(36063) wrt polar circle)
X(36102) = X(i)-isoconjugate of X(j) for these {i,j}: {6, 5663}, {32, 35520}, {48, 36063}, {526, 2437}, {647, 7480}, {524, 2437}, {2410, 14270}
X(36102) = trilinear product X(i)*X(j) for these {i,j}: {2, 477}, {63, 36130}, {75, 36151}, {92, 36062}, {264, 32663}, {476, 2411}, {648, 14220}, {2436, 35139}, {3268, 32650}, {32679, 36047}
X(36102) = trilinear quotient X(i)/X(j) for these (i,j): (2, 5663), (76, 35520), (92, 36063), (476, 2437), (477, 6), (648, 7480), (2411, 524), (2436, 14270), (14220, 647), (32650, 14560), (32663, 184), (32712, 32715), (35139, 2410), (36047, 32678), (36062, 48), (36117, 36131) (36130, 19), (36151, 31)
X(36102) = barycentric product X(i)*X(j) for these {i,j}: {69, 36130}, {75, 477}, {76, 36151}, {264, 36062}, {811, 14220}, {1577, 30528}, {1969, 32663}, {2411, 32680}, {3268, 36047}
X(36102) = barycentric quotient X(i)/X(j) for these (i,j): (1, 5663), (4, 36063), (75, 35520), (162, 7480), (477, 1), (1784, 11251), (2411, 32679), (2436, 2624), (14220, 656), (30528, 662), (32650, 32678), (32663, 48), (32678, 2437), (32680, 2410), (32712, 36131), (36047, 476), (36062, 3), (36097, 9060), (36117, 1304), (36130, 4), (36144, 32732), (36151, 6)

X(36103) = CENTER OF LOCUS OF TRILINEAR PRODUCT OF CIRCUMCIRCLE-X(4)-ANTIPODES

Barycentrics    a (a^5 + (a^4 - 2 a^2 b c) (b + c) - a (b^2 - c^2)^2 - (b^4 - c^4) (b - c))/(a^2 - b^2 - c^2) : :
Trilinears    tan A (a tan A - b tan B - c tan C) : :

The locus of the trilinear product of circumcircle-X(4)-antipodes is a circumconic that is also the locus of the barycentric product of circumcircle-X(92)-antipodes, and the locus of trilinear poles of lines passing through X(19). The conic is the isogonal conjugate of line X(521)X(656), and passes through X(108), X(162), X(811), X(1783), X(1897), X(8767), X(24019), X(36104)-X(36130). The perspector of this conic is X(19).

X(36103) lies on these lines: {1, 406}, {2, 7219}, {4, 990}, {10, 23050}, {19, 2207}, {25, 34}, {31, 1452}, {33, 429}, {37, 2331}, {40, 8750}, {65, 3195}, {108, 21147}, {169, 8743}, {205, 32674}, {225, 1096}, {232, 16968}, {235, 3772}, {341, 1897}, {405, 1712}, {451, 975}, {910, 3172}, {1039, 1848}, {1191, 1829}, {1249, 6554}, {1593, 3752}, {1722, 1861}, {1783, 17742}, {1876, 17054}, {2551, 7952}, {2883, 23982}, {3162, 15487}, {3556, 8900}, {4194, 5262}, {4205, 18643}, {4646, 7071}, {5336, 7106}, {7290, 7713}, {17602, 17726}

X(36103) = polar conjugate of isotomic conjugate of X(1763)
X(36103) = complement of X(7219)
X(36103) = X(2)-Ceva conjugate of X(19)
X(36103) = perspector of circumconic centered at X(19)
X(36103) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 7219}, {63, 7097}, {69, 7169}
X(36103) = trilinear product X(i)*X(j) for these {i,j}: {2, 21148}, {4, 3556}, {6, 17903}, {19, 1763}, {25, 4329}, {346, 405}, {608, 27540}, {1039, 8900}, {1973, 20914}, {8750, 21174}
X(36103) = trilinear quotient X(i)/X(j) for these (i,j): (4, 7219), (19, 7097), (25, 7169), (405, 1407), (1763, 63), (3556, 3), (4329, 69), (8900, 1038), (17903, 2), (20914, 304), (21148, 6), (21174, 4025), (27540, 345)
X(36103) = barycentric product X(i)*X(j) for these {i,j}: {1, 17903}, {4, 1763}, {19, 4329}, {25, 20914}, {34, 27540}, {75, 21148}, {92, 3556}, {341, 405}, {1783, 21174}
X(36103) = barycentric quotient X(i)/X(j) for these (i,j): (19, 7219), (25, 7097), (405, 269), (1763, 69), (1973, 7169), (3556, 63), (4329, 304), (17903, 75), (20914, 305), (21148, 1), (21174, 15413), (27540, 3718)

X(36104) = TRILINEAR PRODUCT X(98)*X(112)

Barycentrics    a/((b^2 - c^2) (a^2 - b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2)) : :
Trilinears    tan A sec(A + ω) csc(B - C) : :

As the trilinear product of circumcircle-X(4)-antipodes, X(36104) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36104) lies on these lines: {98, 36071}, {108, 2715}, {112, 36065}, {162, 661}, {163, 811}, {240, 293}, {685, 1897}, {798, 24000}, {825, 22456}, {1783, 4705}, {1821, 36095}, {1910, 36119}, {20031, 36127}, {32676, 36132}

X(36104) = polar conjugate of X(36084)
X(36104) = pole wrt polar circle of trilinear polar of X(36084) (line X(1)X(163))
X(36104) = trilinear pole of line X(19)X(560) (the polar of X(36084) wrt polar circle)
X(36104) = barycentric product of circumcircle intercepts of line X(31)X(92)
X(36104) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 684}, {3, 2799}, {69, 3569}, {125, 2421}, {232, 3265}, {240, 24018}, {297, 520}, {305, 2491}, {325, 647}, {511, 525}, {512, 6393}, {656, 1959}, {850, 3289}, {877, 3269}, {1755, 14208}, {2396, 20975}, {2419, 9475}, {3926, 17994}, {4230, 15526}, {8673, 34138}, {9033, 35910}
X(36104) = trilinear product X(i)*X(j) for these {i,j}: {3, 20031}, {4, 2715}, {6, 685}, {19, 36084}, {25, 2966}, {32, 22456}, {98, 112}, {107, 248}, {110, 6531}, {162, 1910}, {163, 36120}, {250, 2395}, {287, 32713}, {293, 24019}, {458, 32716}, {648, 1976}, {878, 23582}, {879, 23964}, {1289, 11610}, {1304, 35906}, {1576, 16081}, {1821, 32676}, {1973, 36036}, {2207, 17932}, {2422, 18020}, {2445, 9476}, {6037, 10311}, {6331, 14601}, {6528, 14600}, {6529, 17974}, {32695, 35912}
X(36104) = trilinear quotient X(i)/X(j) for these (i,j): (4, 2799), (6, 684), (25, 3569), (98, 525), (99, 6393), (107, 297), (112, 511), (162, 1959), (248, 520), (250, 2421), (293, 24018), (648, 325), (685, 2), (878, 3269), (879, 15526), (1289, 34138), (1304, 35910), (1576, 3289), (1821, 14208), (1910, 656), (1974, 2491), (1976, 647), (2207, 17994), (2395, 125), (2422, 20975), (2445, 9475), (2715, 3), (2966, 69), (6529, 6530), (6531, 523), (9476, 2419), (11610, 8673), (14601, 3049), (16081, 850), (17932, 3926), (18020, 2396), (20031, 4), (22456, 76), (23582, 877), (23964, 4230), (24019, 240), (32676, 1755), (32695, 35908), (32713, 232), (35906, 9033), (36036, 304), (36084, 63), (36120, 1577)
X(36104) = barycentric product X(i)*X(j) for these {i,j}: {1, 685}, {4, 36084}, {19, 2966}, {25, 36036}, {31, 22456}, {63, 20031}, {92, 2715}, {98, 162}, {107, 293}, {110, 36120}, {112, 1821}, {163, 16081}, {248, 823}, {287, 24019}, {290, 32676}, {336, 32713}, {458, 36132}, {648, 1910}, {662, 6531}, {811, 1976}, {878, 23999}, {879, 24000}, {1096, 17932}, {14006, 36065}
X(36104) = barycentric quotient X(i)/X(j) for these (i,j): (1, 6333), (19, 2799), (31, 684), (98, 14208), (112, 1959), (162, 325), (248, 24018), (293, 3265), (662, 6393), (685, 75), (878, 2632), (879, 17879), (1096, 16230), (1821, 3267), (1910, 525), (1973, 3569), (1976, 656), (2395, 20902), (2422, 3708), (2715, 63), (2966, 304), (6531, 1577), (14600, 822), (14601, 810), (16081, 20948), (20031, 92), (22456, 561), (24000, 877), (24019, 297), (32676, 511), (32713, 240), (36036, 305), (36084, 69), (36120, 850), (36131, 35910)

X(36105) = TRILINEAR PRODUCT X(99)*X(3563)

Barycentrics    a/((b^2 - c^2) (a^2 - b^2 - c^2) (2 a^4 - a^2 (b^2 + c^2) + (b^2 - c^2)^2)) : :
Barycentrics    (csc 2A) csc(B - C)/(b^2 cos^2 C + c^2 cos^2 B - b c cos A) : :

As the trilinear product of circumcircle-X(4)-antipodes, X(36105) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36105) lies on these lines: {108, 10425}, {162, 24041}, {811, 24037}, {1733, 36036}, {1783, 4567}, {1897, 4600}, {3563, 36066}, {4622, 36125}, {8773, 36119}, {23999, 36126}, {36085, 36128}

X(36105) = trilinear pole of line X(19)X(662)
X(36105) = X(i)-isoconjugate of X(j) for these {i,j}: {230, 647}, {512, 3564}, {525, 1692}, {656, 8772}, {810, 1733}
X(36105) = trilinear product X(i)*X(j) for these {i,j}: {2, 32697}, {4, 10425}, {99, 3563}, {112, 8781}, {162, 8773}, {648, 2987}, {811, 36051}, {6331, 32654}
X(36105) = trilinear quotient X(i)/X(j) for these (i,j): (99, 3564), (112, 1692), (162, 8772), (648, 230), (811, 1733), (2987, 647), (3563, 512), (8773, 656), (8781, 525), (10425, 3), (32654, 3049), (32697, 6), (36051, 810)
X(36105) = barycentric product X(i)*X(j) for these {i,j}: {75, 32697}, {92, 10425}, {162, 8781}, {648, 8773}, {662, 35142}, {799, 3563}, {811, 2987}, {6331, 36051}
X(36105) = barycentric quotient X(i)/X(j) for these (i,j): (162, 230), (648, 1733), (662, 3564), (2987, 656), (3563, 661), (8773, 525), (8781, 14208), (10425, 48), (24019, 460), (32654, 810), (32697, 1), (35142, 1577), (36051, 647)

X(36106) = TRILINEAR PRODUCT X(100)*X(915)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36106) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36106) lies on these lines: {59, 108}, {162, 4570}, {677, 2990}, {765, 1897}, {811, 4600}, {901, 915}, {913, 34075}, {1252, 1783}, {1331, 7649}, {1737, 5081}, {2319, 16671}, {4242, 36110}, {7012, 36127}, {7045, 36118}, {36086, 36124}

X(36106) = trilinear pole of line X(19)X(101)
X(36106) = X(i)-isoconjugate of X(j) for these {i,j}: {513, 912}, {514, 2252}, {649, 914}, {905, 8609}, {1459, 1737}
X(36106) = trilinear product X(i)*X(j) for these {i,j}: {2, 32698}, {4, 6099}, {100, 915}, {190, 913}, {1783, 2990}, {1897, 36052}, {6335, 32655}
X(36106) = trilinear quotient X(i)/X(j) for these (i,j): (100, 912), (101, 2252), (190, 914), (913, 649), (915, 513), (1783, 8609), (1897, 1737), (2990, 905), (4242, 11570), (6099, 3), (32655, 22383), (32698, 6), (36052, 1459)
X(36106) = barycentric product X(i)*X(j) for these {i,j}: {75, 32698}, {92, 6099}, {190, 915}, {668, 913}, {1897, 2990}, {6335, 36052}
X(36106) = barycentric quotient X(i)/X(j) for these (i,j): (100, 914), (101, 912), (692, 2252), (913, 513), (915, 514), (1783, 1737), (2990, 4025), (6099, 63), (8750, 8609), (32655, 1459), (32698, 1), (36052, 905)

X(36107) = TRILINEAR PRODUCT X(101)*X(917)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36107) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36107) lies on these lines: {108, 2149}, {811, 4567}, {906, 17924}, {917, 919}, {1110, 1783}, {1252, 1897}, {1262, 36118}, {1736, 8558}, {7115, 36127}, {32641, 36123}, {32665, 36125}

X(36107) = trilinear pole of line X(19)X(692)
X(36107) = X(i)-isoconjugate of X(j) for these {i,j}: {514, 916}, {905, 1736}, {4025, 8608}
X(36107) = trilinear product X(i)*X(j) for these {i,j}: {2, 32699}, {4, 35182}, {101, 917}, {2989, 8750}
X(36107) = trilinear quotient X(i)/X(j) for these (i,j): (101, 916), (917, 514), (1783, 1736), (2989, 4025), (8750, 8608), (32699, 6), (35182, 3)
X(36107) = barycentric product X(i)*X(j) for these {i,j}: {75, 32699}, {92, 35182}, {100, 917}, {1783, 2989}
X(36107) = barycentric quotient X(i)/X(j) for these (i,j): (692, 916), (917, 693), (2989, 15413), (8750, 1736), (32699, 1), (35182, 63)

X(36108) = TRILINEAR PRODUCT X(102)*X(26704)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36108) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36108) lies on these lines: {108, 21189}, {1783, 32700}, {26704, 36067}

X(36108) = trilinear pole of line X(19)X(32677)
X(36108) = trilinear product X(i)*X(j) for these {i,j}: {2, 32700}, {4, 35183}, {102, 26704}, {10570, 36067}, {36050, 36121}
X(36108) = trilinear quotient X(i)/X(j) for these (i,j): (26704, 515), (32700, 6), (35183, 3), (36067, 10571), (36121, 21189)
X(36108) = barycentric product X(i)*X(j) for these {i,j}: {75, 32700}, {92, 35183}, {26704, 36100}
X(36108) = barycentric quotient X(i)/X(j) for these (i,j): (32667, 10571), (32700, 1), (35183, 63)

X(36109) = TRILINEAR PRODUCT X(103)*X(26705)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36109) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36109) lies on these lines: {108, 35184}, {677, 1897}, {1734, 1783}, {24016, 26705}

X(36109) = trilinear pole of line X(19)X(911)
X(36109) = trilinear product X(i)*X(j) for these {i,j}: {2, 32701}, {4, 35184}, {103, 26705}
X(36109) = trilinear quotient X(i)/X(j) for these (i,j): (26705, 516), (32701, 6), (35184, 3)
X(36109) = barycentric product X(i)*X(j) for these {i,j}: {75, 32701}, {92, 35184}, {26705, 36101}
X(36109) = barycentric quotient X(i)/X(j) for these (i,j): (32701, 1), (35184, 63)

X(36110) = TRILINEAR PRODUCT X(104)*X(108)

Barycentrics    a/((b - c) (a - b - c) (a^2 - b^2 - c^2) (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)) : :
Trilinears    1/((1 - cos B - cos C) (sec B - sec C)) : :

As the trilinear product of circumcircle-X(4)-antipodes, X(36110) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36110) lies on these lines: {104, 1455}, {108, 513}, {109, 522}, {162, 3737}, {650, 1415}, {653, 36090}, {811, 1414}, {909, 36140}, {1416, 1430}, {1417, 1875}, {1785, 1795}, {1876, 15635}, {1877, 36123}, {2342, 23710}, {3676, 6614}, {4242, 36106}, {6001, 15500}, {7649, 24033}, {23706, 23838}, {24019, 32669}, {32674, 36137}, {34051, 36122}, {34234, 36093}

X(36110) = polar conjugate of isogonal conjugate of X(32669)
X(36110) = trilinear pole of line X(19)X(604)
X(36110) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 2804}, {8, 8677}, {78, 1769}, {212, 36038}, {219, 10015}, {345, 3310}, {517, 521}, {522, 22350}, {652, 908}, {1459, 6735}, {1946, 3262}, {2183, 6332}, {3596, 23220}, {23706, 24031}
X(36110) = trilinear product X(i)*X(j) for these {i,j}: {2, 32702}, {4, 2720}, {7, 14776}, {34, 36037}, {56, 1309}, {92, 32669}, {104, 108}, {109, 36123}, {278, 32641}, {608, 13136}, {653, 909}, {1415, 16082}, {1783, 34051}, {1795, 36127}, {2342, 36118}, {34234, 32674}
X(36110) = trilinear quotient X(i)/X(j) for these (i,j): (4, 2804), (34, 1769), (56, 8677), (104, 521), (108, 517), (109, 22350), (273, 36038), (278, 10015), (608, 3310), (653, 908), (909, 652), (1309, 8), (1397, 23220), (1877, 23757), (1897, 6735), (2720, 3), (13136, 345), (14776, 55), (16082, 4391), (18026, 3262), (18816, 35518), (24033, 23706), (32641, 219), (32669, 48), (32674, 2183), (32702, 6), (32714, 1465), (34051, 905), (34234, 6332), (36037, 78), (36118, 22464), (36123, 522), (36127, 1785)
X(36110) = barycentric product X(i)*X(j) for these {i,j}: {34, 13136}, {57, 1309}, {75, 32702}, {85, 14776}, {92, 2720}, {104, 653}, {108, 34234}, {109, 16082}, {264, 32669}, {273, 32641}, {278, 36037}, {651, 36123}, {909, 18026}, {1897, 34051}, {2342, 13149}, {18816, 32674}
X(36110) = barycentric quotient X(i)/X(j) for these (i,j): (19, 2804), (34, 10015), (104, 6332), (108, 908), (278, 36038), (604, 8677), (608, 1769), (653, 3262), (909, 521), (1309, 312), (1395, 3310), (1415, 22350), (1783, 6735), (2720, 63), 13136, 3718), (14776, 9), (16082, 35519), (23985, 23706), (32641, 78), (32669, 3), (32674, 517), (32702, 1), (32714, 22464), (34051, 4025), (34234, 35518), (36037, 345), (36123, 4391)

X(36111) = TRILINEAR PRODUCT X(105)*X(26706)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36111) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36111) lies on these lines: {108, 32735}, {919, 1783}, {1861, 36057}, {1897, 36086}, {21185, 36041}

X(36111) = trilinear pole of line X(19)X(1438)
X(36111) = trilinear product X(i)*X(j) for these {i,j}: {2, 32703}, {4, 35185}, {105, 26706}
X(36111) = trilinear quotient X(i)/X(j) for these (i,j): (26706, 518), (32703, 6), (35185, 3)
X(36111) = barycentric product X(i)*X(j) for these {i,j}: {75, 32703}, {92, 35185}, {673, 26706}
X(36111) = barycentric quotient X(i)/X(j) for these (i,j): (26706, 3912), (32703, 1), (35185, 63)

X(36112) = TRILINEAR PRODUCT X(106)*X(32704)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36112) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36112) lies on these lines: {108, 35186}, {811, 4622}, {901, 1897}, {1783, 32665}, {10428, 36123}

X(36112) = trilinear pole of line X(19)X(9456)
X(36112) = X(519)-isoconjugate of X(32475)
X(36112) = trilinear product X(i)*X(j) for these {i,j}: {2, 32705}, {4, 35186}, {106, 32704}
X(36112) = trilinear quotient X(i)/X(j) for these (i,j): (106, 32475), (32704, 519), (32705, 6), (35186, 3)
X(36112) = barycentric product X(i)*X(j) for these {i,j}: {75, 32705}, {88, 32704}, {92, 35186}
X(36112) = barycentric quotient X(i)/X(j) for these (i,j): (9456, 32475), (32704, 4358), (32705, 1), (35186, 63)

X(36113) = TRILINEAR PRODUCT X(109)*X(32706)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36113) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36113) lies on these lines: {59, 1897}, {108, 24027}, {1735, 15379}, {1783, 2149}, {2720, 32706}, {7339, 36118}, {32735, 36124}

X(36113) = trilinear pole of line X(19)X(1415)
X(36113) = X(i)-isoconjugate of X(j) for these {i,j}: {521, 1735}, {6332, 8607}
X(36113) = trilinear product X(i)*X(j) for these {i,j}: {2, 32707}, {4, 35187}, {109, 32706}, {2988, 32674}, {15379, 23987}
X(36113) = trilinear quotient X(i)/X(j) for these (i,j): (108, 1735), (2988, 6332), (23987, 117), (32674, 8607), (32706, 522), (32707, 6), (35187, 3)
X(36113) = barycentric product X(i)*X(j) for these {i,j}: {75, 32707}, {92, 35187}, {108, 2988}, {651, 32706}, {24027, 24035}
X(36113) = barycentric quotient X(i)/X(j) for these (i,j): (2988, 35518), (32674, 1735), (32706, 4391), (32707, 1), (35187, 63)

X(36114) = TRILINEAR PRODUCT X(110)*X(1300)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36114) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36114) lies on these lines: {108, 7477}, {162, 1101}, {687, 1897}, {811, 24041}, {1300, 36069}, {1725, 36034}, {1783, 32708}, {1895, 36130}, {4575, 24006}, {24000, 36126}, {24001, 36129}, {36061, 36116}, {36084, 36120}, {36128, 36142}

X(36114) = trilinear pole of line X(19)X(163)
X(36114) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 686}, {6, 6334}, {69, 21731}, {113, 14380}, {125, 15329}, {403, 520}, {523, 13754}, {525, 3003}, {647, 3580}, {656, 1725}, {1577, 2315}, {3269, 16237}
X(36114) = trilinear product X(i)*X(j) for these {i,j}: {2, 32708}, {4, 10420}, {6, 687}, {25, 18878}, {107, 5504}, {110, 1300}, {112, 2986}, {162, 36053}, {250, 15328}, {648, 14910}, {2501, 18879}, {4240, 10419}, {15421, 23964}
X(36114) = trilinear quotient X(i)/X(j) for these (i,j): (2, 6334), (6, 686), (25, 21731), (107, 403), (110, 13754), (112, 3003), (162, 1725), (163, 2315), (250, 15329), (648, 3580), (687, 2), (1300, 523), (2986, 525), (4240, 113), (5504, 520), (10419, 14380), (10420, 3), (14910, 647), (15328, 125), (15421, 15526), (18878, 69), (18879, 4558), (23582, 16237), (32708, 6), (36053, 656)
X(36114) = barycentric product X(i)*X(j) for these {i,j}: {1, 687}, {19, 18878}, {75, 32708}, {92, 10420}, {162, 2986}, {648, 36053}, {662, 1300}, {811, 14910}, {823, 5504}, {10419, 24001}, {15421, 24000}, {18879, 24006}
X(36114) = barycentric quotient X(i)/X(j) for these (i,j): (1, 6334), (112, 1725), (162, 3580), (163, 13754), (687, 75), (1300, 1577), (1576, 2315), (1973, 21731), (2986, 14208), (5504, 24018), (10420, 63), (14910, 656), (15328, 20902), (15421, 17879), (18878, 304), (18879, 4592), (24000, 16237), (24019, 403), (32676, 3003), (32708, 1), (36053, 525)

X(36115) = TRILINEAR PRODUCT X(111)*X(30247)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36115) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36115) lies on these lines: {108, 35188}, {162, 36142}, {811, 36085}, {1783, 32709}, {30247, 36070}

X(36115) = trilinear pole of line X(19)X(923)
X(36115) = X(i)-isoconjugate of X(j) for these {i,j}: {524, 30209}, {1995, 14417}
X(36115) = trilinear product X(i)*X(j) for these {i,j}: {2, 32709}, {4, 35188}, {111, 30247}
X(36115) = trilinear quotient X(i)/X(j) for these (i,j): (111, 30209), (5486, 14417), (30247, 524), (32709, 2), (35188, 3)
X(36115) = barycentric product X(i)*X(j) for these {i,j}: {75, 32709}, {92, 35188}, 897, 30247}
X(36115) = barycentric quotient X(i)/X(j) for these (i,j): (923, 30209), (30247, 14210), (32709, 1), (35188, 63)

X(36116) = TRILINEAR PRODUCT X(476)*X(32710)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36116) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36116) lies on these lines: {108, 35189}, {1783, 32711}, {36047, 36130}, {36061, 36114}

X(36116) = trilinear pole of line X(19)X(32678)
X(36116) = X(526)-isoconjugate of X(17702)
X(36116) = trilinear product X(i)*X(j) for these {i,j}: {2, 32711}, {4, 35189}, {476, 32710}
X(36116) = trilinear quotient X(i)/X(j) for these (i,j): (476, 17702), (32710, 526), (32711, 6), (35189, 3)
X(36116) = barycentric product X(i)*X(j) for these {i,j}: {75, 32711}, {92, 35189}, {32680, 32710}
X(36116) = barycentric quotient X(i)/X(j) for these (i,j): (32678, 17702), (32710, 32679), (32711, 1), (35189, 63)

X(36117) = TRILINEAR PRODUCT X(477)*X(1304)

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

As the trilinear product of circumcircle-X(4)-antipodes, X(36117) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36117) lies on these lines: {1783, 32712}, {36034, 36035}, {36062, 36063}, {36131, 36144}

X(36117) = X(i)-isoconjugate of X(j) for these {i,j}: {5663, 9033}, {9409, 35520}
X(36117) = trilinear product X(i)*X(j) for these {i,j}: {2, 32712}, {477, 1304}, {15459, 32663}, {36034, 36130}, {36102, 36131}
X(36117) = trilinear quotient X(i)/X(j) for these (i,j): (477, 9033), (1304, 5663), (16077, 35520), (32663, 1636), (32712, 6), (36130, 36035), (36151, 2631)
X(36117) = barycentric product X(i)*X(j) for these {i,j}: {75, 32712}, {1304, 36102}, {15459, 36062}, {16077, 36151}
X(36117) = barycentric quotient X(i)/X(j) for these (i,j): (32712, 1), (36151, 9033)

X(36118) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(4)X(7)

Barycentrics    1/((a - b - c)^2 (b - c) (a^2 - b^2 - c^2)) : :
Barycentrics    (1 - sec A)/(cos B - cos C) : :
Barycentrics    a (tan A/2)/(sec B - sec C) : :
Barycentrics    a (tan A)/((1 + cos A) (cos B - cos C)) : :
Barycentrics    a tan A sec^2(A/2)/(cos B - cos C) : :

As the trilinear product of circumcircle-X(4)-antipodes, X(36118) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19), and as the trilinear product of circumcircle-X(7)-antipodes, X(36118) also lies on conic {{A,B,C,X(651),X(664),X(934)}} with center X(223) and perspector X(57).

X(36118) lies on these lines: {4, 279}, {7, 34231}, {27, 1427}, {29, 1446}, {34, 1847}, {77, 342}, {85, 11109}, {107, 36079}, {108, 934}, {109, 36048}, {162, 658}, {208, 7177}, {269, 273}, {278, 8735}, {318, 9312}, {347, 6925}, {348, 17555}, {412, 3188}, {469, 7365}, {514, 23984}, {651, 653}, {664, 1897}, {811, 4625}, {1042, 36120}, {1088, 14004}, {1119, 36125}, {1262, 36107}, {1323, 1785}, {1448, 7513}, {1461, 24019}, {1875, 34855}, {1895, 34059}, {3160, 7952}, {3668, 7282}, {3676, 6614}, {4242, 6516}, {5081, 9436}, {7045, 36106}, {7128, 32674}, {7339, 36113}, {8767, 14944}, {14837, 36049}, {24016, 26705}

X(36118) = polar conjugate of X(3239)
X(36118) = pole wrt polar circle of trilinear polar of X(3239) (line X(1146)X(2310))
X(36118) = trilinear pole of line X(19)X(57) (the polar of X(3239) wrt polar circle)
X(36118) = X(7649)-cross conjugate of X(278)
X(36118) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 3900}, {8, 1946}, {9, 652}, {32, 15416}, {41, 6332}, {48, 3239}, {55, 521}, {63, 657}, {69, 8641}, {71, 1021}, {72, 21789}, {77, 4105}, {78, 663}, {101, 34591}, {184, 4397}, {200, 1459}, {212, 522}, {219, 650}, {220, 905}, {222, 4130}, {228, 7253}, {268, 14298}, {281, 36054}, {283, 4041}, {284, 8611}, {345, 3063}, {346, 22383}, {512, 1792}, {513, 1260}, {514, 1802}, {520, 4183}, {603, 4163}, {644, 7117}, {647, 2287}, {649, 3692}, {656, 2328}, {661, 2327}, {667, 1265}, {692, 2968}, {810, 1043}, {822, 2322}, {906, 1146}, {1253, 4025}, {1259, 18344}, {1331, 2310}, {1332, 14936}, {1783, 35072}, {1812, 3709}, {1897, 2638}, {2170, 4587}, {2175, 35518}, {2188, 8058}, {2193, 3700}, {2289, 3064}, {2332, 24018}, {3271, 4571}, {3939, 7004}, {4091, 7079}, {4131, 7071}, {4845, 14414}, {7046, 23224}, {8606, 35057}, {8750, 24031}, {24026, 32656}
X(36118) = trilinear product X(i)*X(j) for these {i,j}: {2, 32714}, {4, 934}, {6, 13149}, {7, 108}, {19, 658}, {25, 4569}, {27, 1020}, {28, 4566}, {33, 4626}, {34, 664}, {56, 18026}, {57, 653}, {77, 36127}, {85, 32674}, {92, 1461}, {99, 1426}, {100, 1119}, {101, 1847}, {107, 1439}, {109, 273}, {112, 1446}, {162, 3668}, {190, 1435}, {225, 1414}, {269, 1897}, {278, 651}, {279, 1783}, {281, 4617}, {318, 6614}, {331, 1415}, {342, 8059}, {514, 7128}, {608, 4554}, {648, 1427}, {668, 1398}, {811, 1042}, {905, 23984}, {927, 1876}, {1088, 8750}, {1118, 6516}, {1262, 17924}, {1275, 6591}, {1395, 4572}, {1407, 6335}, {1459, 24032}, {1838, 36048}, {1880, 4573}, {1895, 36079}, {3676, 7012}, {4025, 24033}, {5236, 36146}, {7045, 7649}, {7115, 24002}, {7282, 26700}, {15413, 23985}, {22464, 36110}, {23973, 36122}
X(36118) = trilinear quotient X(i)/X(j) for these (i,j): (4, 3900), (7, 521), (19, 657), (25, 8641), (27, 1021), (28, 21789), (33, 4105), (34, 663), (56, 1946), (57, 652), (76, 15416), (85, 6332), (92, 3239), (99, 1792), (100, 1260), (101, 1802), (107, 4183), (108, 55), (109, 212), (162, 2328), (190, 3692), (196, 14298), (222, 36054), (225, 4041), (226, 8611), (264, 4397), (269, 1459), (273, 522), (278, 650), (279, 905), (281, 4130), (286, 7253), (318, 4163), (331, 4391), (342, 8058), (608, 3063), (648, 2287), (651, 219), (653, 9), (658, 63), (662, 2327), (664, 78), (668, 1265), (693, 2968), (811, 1043), (823, 2322), (905, 35072), (934, 3), (1020, 71), (1042, 810), (1088, 4025), (1118, 18344), (1119, 513), (1262, 906), (1275, 1332), (1323, 14414), (1398, 667), (1407, 22383), (1414, 283), (1426, 512), (1427, 647), (1435, 649), (1439, 520), (1446, 525), (1459, 2638), (1461, 48), (1783, 220), (1813, 2289), (1847, 514), (1876, 926), (1880, 3709), (1897, 200), (3668, 656), (3669, 7117), (3676, 7004), (4025, 24031), (4242, 856), (4554, 345), (4564, 4587), (4565, 2193), (4566, 72), (4569, 69), (4572, 3718), (4573, 1812), (4617, 222), (4625, 332), (4626, 77), (4998, 4571), (6063, 35518), (6335, 346), (6516, 1259), (6591, 14936), (6614, 603), (7012, 3939), (7045, 1331), (7053, 23224), (7056, 4131), (7128, 101), (7177, 4091), (7282, 35057), (7365, 2522), (7649, 2310), (8059, 2188), (8750, 1253), (13149, 2), (15413, 23983), (17924, 1146), (18026, 8), (23984, 1783), (24002, 26932), (24015, 26006), (24016, 36056), (24019, 2332), (24027, 32656), (24032, 1897), (24033, 8750), (26700, 8606), (32674, 41), (32714, 6), (36048, 1794), (36059, 6056), (36079, 19614), (36110, 2342), (36127, 33)
X(36118) = barycentric product X(i)*X(j) for these {i,j}: {1, 13149}, {4, 658}, {7, 653}, {19, 4569}, {27, 4566}, {34, 4554}, {57, 18026}, {75, 32714}, {85, 108}, {92, 934}, {100, 1847}, {109, 331}, {162, 1446}, {190, 1119}, {225, 4573}, {264, 1461}, {269, 6335}, {273, 651}, {278, 664}, {279, 1897}, {281, 4626}, {286, 1020}, {318, 4617}, {348, 36127}, {608, 4572}, {648, 3668}, {668, 1435}, {693, 7128}, {799, 1426}, {811, 1427}, {823, 1439}, {905, 24032}, {927, 5236}, {1042, 6331}, {1088, 1783}, {1275, 7649}, {1398, 1978}, {1876, 34085}, {1880, 4625}, {4025, 23984}, {6063, 32674}, {6614, 7017}, {7012, 24002}, {7045, 17924}, {15413, 24033}, {15466, 36079}, {24015, 36122}
X(36118) = barycentric quotient X(i)/X(j) for these (i,j): (4, 3239), (7, 6332), (19, 3900), (25, 657), (27, 7253), (28, 1021), (33, 4130), (34, 650), (56, 652), (57, 521), (59, 4587), (65, 8611), (75, 15416), (85, 35518), (92, 4397), (100, 3692), (101, 1260), (107, 2322), (108, 9), (109, 219), (110, 2327), (112, 2328), (162, 2287), (190, 1265), (196, 8058), (208, 14298), (225, 3700), (269, 905), (273, 4391), (278, 522), (279, 4025), (281, 4163), (331, 35519), (514, 2968), (603, 36054), (604, 1946), (607, 4105), (608, 663), (648, 1043), (651, 78), (653, 8), (658, 69), (662, 1792), (664, 345), (692, 1802), (693, 33), (905, 24031), (934, 63), (1020, 72), (1042, 647), (1088, 15413), (1106, 22383), (1118, 3064), (1119, 514), (1262, 1331), (1275, 4561), (1395, 3063), (1398, 649), (1407, 1459), (1414, 1812), (1415, 212), (1426, 661), (1427, 656), (1435, 513), (1439, 24018), (1446, 14208), (1459, 35072), (1461, 3), (1474, 21789), (1783, 200), (1813, 1259), (1847, 693), (1880, 4041), (1897, 346), (1973, 8641), (3668, 525), (3669, 7004), (3676, 26932), (4025, 23983), (4320, 2522), (4554, 3718), (4564, 4571), (4565, 283), (4566, 306), (4569, 304), (4573, 332), (4617, 77), (4626, 348), (6335, 341), (6516, 3719), (6591, 2310), (6610, 14414), (6614, 222), (7012, 644), (7045, 1332), (7053, 4091), (7056, 30805), (7099, 23224), (7115, 3939), (7128, 100), (7177, 4131), (7339, 1813), (7365, 23874), (7649, 1146), (8059, 268), (8735, 23615), (8750, 220), (13149, 75), (17924, 24026), (18026, 312), (22383, 2638), (23973, 26006), (23979, 32656), (23984, 1897), (23985, 8750), (24002, 17880), (24016, 1815), (24019, 4183), (24027, 906), (24032, 6335), (24033, 1783), (32651, 1794), (32660, 6056), (32668, 36056), (32674, 55), (32702, 2342), (32713, 2332), (32714, 1), (36059, 2289), (36079, 1073), (36124, 28132), (36127, 281)

X(36119) = TRILINEAR PRODUCT X(4)*X(74)

Barycentrics    a/((a^2 - b^2 - c^2) (a^2 (2 a^2 - b^2 - c^2) - (b^2 - c^2)^2)) : :
Trilinears    1/(tan B tan C - 3) : :
Trilinears    (sec A)/(cos A - 2 cos B cos C) : :
Trilinears    (sec A)/(3 cos A - 2 sin B sin C) : :

X(36119) is the trilinear product of the (real or nonreal) circumcircle intercepts of line X(4)X(523). As the trilinear product of circumcircle-X(4)-antipodes, X(36119) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19), and as the trilinear product of circumcircle-X(523)-antipodes, X(36119) also lies on conic {{A,B,C,X(1),X(10)}} with center X(244) and perspector X(661).

Let A'B'C' and A"B"C" be the orthocentroidal and anti-orthocentroidal triangles, resp. Let A* be the trilinear product A'*A", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(36119).

X(36119) lies on these lines: {1, 162}, {10, 1897}, {19, 2159}, {37, 1783}, {65, 74}, {75, 811}, {91, 1895}, {158, 1109}, {225, 36127}, {240, 897}, {759, 1304}, {774, 2190}, {1725, 36034}, {1784, 2166}, {1785, 5620}, {1910, 36104}, {2586, 2589}, {2587, 2588}, {3668, 7282}, {4246, 7984}, {8773, 36105}, {16077, 18827}, {23894, 36128}, {24006, 36130}

X(36119) = polar conjugate of X(14206)
X(36119) = pole wrt polar circle of trilinear polar of X(14206) (line X(1099)X(6739))
X(36119) = trilinear pole of line X(19)X(661) (the polar of X(14206) wrt polar circle)
X(36119) = crossdifference of every pair of points on line X(2631)X(14395)
X(36119) = {X(1),X(36063)}-harmonic conjugate of X(162)
X(36119) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 3284}, {3, 30}, {6, 11064}, {48, 14206}, {63, 2173}, {69, 1495}, {110, 9033}, {184, 3260}, {255, 1784}, {265, 1511}, {304, 9406}, {305, 9407}, {394, 1990}, {511, 35912}, {520, 4240}, {525, 2420}, {647, 2407}, {651, 14395}, {662, 2631}, {822, 24001}, {1099, 35200}, {1637, 4558}, {3163, 14919}, {3292, 9214}, {4575, 36035}
X(36119) = trilinear product X(i)*X(j) for these {i,j}: {2, 8749}, {4, 74}, {6, 16080}, {19, 2349}, {25, 1494}, {92, 2159}, {98, 35908}, {107, 14380}, {110, 18808}, {112, 2394}, {158, 35200}, {186, 5627}, {393, 14919}, {523, 1304}, {648, 2433}, {685, 32112}, {850, 32715}, {1577, 36131}, {1973, 33805}, {2052, 18877}, {6531, 35910}, {9717, 17983}, {14618, 32640}, {24006, 36034}
X(36119) = trilinear quotient X(i)/X(j) for these (i,j): (2, 11064), (4, 30), (6, 3284), (19, 2173), (25, 1495), (74, 3), (92, 14206), (98, 35912), (107, 4240), (112, 2420), (158, 1784), (186, 1511), (264, 3260), (393, 1990), (523, 9033), (648, 2407), (650, 14395), (661, 2631), (823, 24001), (860, 6739), (1304, 110), (1494, 69), (1784, 1099), (1973, 9406), (1974, 9407), (1990, 3163), (2159, 48), (2349, 63), (2394, 525), (2433, 647), (2501, 1637), (5627, 265), (6531, 35906), (8749, 6), (9717, 3292), (14380, 520), (14919, 394), (16080, 2), (17983, 9214), (18808, 523), (18877, 577), (24006, 36035), (32112, 684), (32640, 32661), (32715, 1576), (33805, 304), (35200, 255), (35908, 511), (36034, 4575), (36131, 163)
X(36119) = barycentric product X(i)*X(j) for these {i,j}: {1, 16080}, {4, 2349}, {19, 1494}, {25, 33805}, {74, 92}, {75, 8749}, {158, 14919}, {162, 2394}, {264, 2159}, {661, 16077}, {662, 18808}, {811, 2433}, {823, 14380}, {850, 36131}, {1304, 1577}, {1821, 35908}, {2052, 35200}, {14618, 36034}, {20948, 32715}, {35910, 36120}
X(36119) = barycentric quotient X(i)/X(j) for these (i,j): (1, 11064), (4, 14206), (19, 30), (25, 2173), (74, 63), (92, 3260), (107, 24001), (162, 2407), (393, 1784), (661, 9033), (663, 14395), (1096, 1990), (1304, 662), (1494, 304), (1910, 35912), (1973, 1495), (1974, 9406), (1990, 1099), (2159, 3), (2349, 69), (2394, 14208), (2433, 656), (2501, 36035), (8749, 1), (14380, 24018), (14919, 326), (16077, 799), (16080, 75), (18808, 1577), (18877, 255), (24019, 4240), (32640, 4575), (32676, 2420), (32715, 163), (33805, 305), (35200, 394), (36034, 4558), (36128, 9214), (36131, 110)

X(36120) = TRILINEAR PRODUCT X(4)*X(98)

Barycentrics    1/(a (a^2 - b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2)) : :
Trilinears    sec A sec(A + ω) : :

X(36120) is the trilinear product of the (real or nonreal) circumcircle intercepts of line X(4)X(512). As the trilinear product of circumcircle-X(4)-antipodes, X(36120) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19), and as the trilinear product of circumcircle-X(512)-antipodes, X(36120) also lies on conic {{A,B,C,X(1),X(31)}} with perspector X(798).

X(36120) lies on these lines: {1, 336}, {19, 3402}, {31, 92}, {42, 1897}, {98, 108}, {158, 1910}, {213, 1783}, {240, 1967}, {290, 1245}, {741, 22456}, {823, 2643}, {897, 24001}, {923, 1784}, {1042, 36118}, {1096, 6521}, {1733, 36036}, {9252, 17872}, {36084, 36114}

X(36120) = polar conjugate of X(1959)
X(36120) = trilinear pole of line X(19)X(798) (the polar of X(1959) wrt polar circle)
X(36120) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 3289}, {3, 511}, {32, 6393}, {48, 1959}, {63, 1755}, {69, 237}, {110, 684}, {184, 325}, {232, 394}, {240, 255}, {287, 11672}, {293, 23996}, {297, 577}, {304, 9417}, {305, 9418}, {520, 4230}, {525, 14966}, {656, 23997}, {647, 2421}, {1092, 6530}, {1576, 6333}, {2211, 3926}, {2396, 3049}, {2799, 32661}, {3284, 35910}, {3569, 4558}, {3964, 34854}
X(36120) = trilinear product X(i)*X(j) for these {i,j}: {2, 6531}, {4, 98}, {6, 16081}, {19, 1821}, {25, 290}, {92, 1910}, {107, 879}, {158, 293}, {232, 34536}, {248, 2052}, {264, 1976}, {287, 393}, {336, 1096}, {512, 22456}, {523, 685}, {648, 2395}, {850, 32696}, {1093, 17974}, {1577, 36104}, {1974, 18024}, {2422, 6331}, {2501, 2966}, {2715, 14618}, {6394, 6524}, {6528, 878}, {16080, 35906}, {24006, 36084}
X(36120) = trilinear quotient X(i)/X(j) for these (i,j): (4, 511), (6, 3289), (19, 1755), (25, 237), (76, 6393), (92, 1959), (98, 3), (107, 4230), (112, 14966), (158, 240), (162, 23997), (232, 11672), (240, 23996), (248, 577), (264, 325), (287, 394), (290, 69), (293, 255), (336, 326), (393, 232), (523, 684), (648, 2421), (685, 110), (850, 6333), (879, 520), (1093, 6530), (1821, 63), (1910, 48), (1973, 9417), (1974, 9418), (1976, 184), (2052, 297), (2207, 2211), (2395, 647), (2422, 3049), (2501, 3569), (2715, 32661), (2966, 4558), (6331, 2396), (6394, 3964), (6524, 34854), (6528, 877), (6531, 6), (14618, 2799), (16080, 35910), (16081, 2), (17974, 1092), (18024, 305), (22456, 99), (32696, 1576), (34536, 287), (35906, 3284), (36036, 4592), (36084, 4575), (36104, 163)
X(36120) = barycentric product X(i)*X(j) for these {i,j}: {1, 16081}, {4, 1821}, {19, 290}, {75, 6531}, {92, 98}, {158, 287}, {240, 34536}, {264, 1910}, {293, 2052}, {336, 393}, {685, 1577}, {811, 2395}, {823, 879}, {850, 36104}, {1969, 1976}, {1973, 18024}, {2501, 36036}, {2966, 24006}, {6394, 6520}, {6521, 17974}, {14618, 36084}, {20948, 32696}
X(36120) = barycentric quotient X(i)/X(j) for these (i,j): (4, 1959), (19, 511), (25, 1755), (31, 3289), (75, 6393), (92, 325), (98, 63), (112, 23997), (158, 297), (162, 2421), (232, 23996), (248, 255), (287, 326), (290, 304), (293, 394), (336, 3926), (393, 240), (685, 662), (811, 2396), (823, 877), (879, 24018), (1096, 232), (1577, 6333), (1821, 69), (1910, 3), (1973, 237), (1974, 9417), (1976, 48), (2395, 656), (2422, 810), (2715, 4575), (2966, 4592), (6394, 1102), (6520, 6530), (6531, 1), (16081, 75), (17974, 6507), (24006, 2799), (24019, 4230), (32676, 14966), (32696, 163), (34536, 336), (36036, 4563), (36084, 4558), (36104, 110), (36119, 35910)

X(36121) = TRILINEAR PRODUCT X(4)*X(102)

Barycentrics    a/((a^2 - b^2 - c^2) (2 a^5 + a^4 (b + c) - 2 a^3 (b^2 + c^2) - (b + c) (b^2 - c^2)^2)) : :
Trilinears    (sec A)/((b + c) sec A - b sec B - c sec C) : :

X(36121) is the trilinear product of the (real or nonreal) circumcircle intercepts of line X(4)X(522). As the trilinear product of circumcircle-X(4)-antipodes, X(36121) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36121) lies on the Feuerbach hyperbola and these lines: {1, 102}, {4, 1854}, {7, 34231}, {8, 1897}, {9, 1783}, {21, 162}, {33, 3577}, {34, 84}, {80, 1785}, {90, 1718}, {104, 1455}, {240, 2648}, {278, 3427}, {314, 811}, {393, 1146}, {885, 36124}, {1172, 1905}, {1320, 15500}, {1389, 6198}, {1735, 15379}, {1876, 9372}, {1896, 36126}, {10703, 23706}, {12016, 32714}, {23838, 36125}, {30479, 34393}

X(36121) = isogonal conjugate of complement of X(5081)
X(36121) = trilinear pole of line X(19)X(650)
X(36121) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 515}, {63, 2182}, {78, 1455}, {184, 35516}, {219, 34050}, {394, 8755}, {652, 2406}, {2425, 6332}
X(36121) = trilinear product X(i)*X(j) for these {i,j}: {4, 102}, {19, 36100}, {25, 34393}, {92, 32677}, {158, 36055}, {278, 15629}, {522, 36067}, {653, 2432}, {2399, 32674}, {4391, 32667}, {21189, 36108}
X(36121) = trilinear quotient X(i)/X(j) for these (i,j): (4, 515), (19, 2182), (34, 1455), (102, 3), (264, 35516), (278, 34050), (393, 8755), (653, 2406), (2399, 6332), (2432, 652), (15629, 219), (32643, 32660), (32667, 1415), (32674, 2425), (32677, 48), (34393, 69), (36055, 255), (36067, 109), (36100, 63), (36108, 36050), (36127, 23987)
X(36121) = barycentric product X(i)*X(j) for these {i,j}: {4, 36100}, {19, 34393}, {92, 102}, {108, 2399}, {264, 32677}, {273, 15629}, {2052, 36055}, {4391, 36067}, {32667, 35519}
X(36121) = barycentric quotient X(i)/X(j) for these (i,j): (19, 515), (25, 2182), (34, 34050), (92, 35516), (102, 63), (108, 2406), (608, 1455), (1096, 8755), (2399, 35518), (15629, 78), (32667, 109), (32677, 3), (32700, 36050), (34393, 304), (36040, 1813), (36055, 394), (36067, 651), (36100, 69), (36127, 24035)

X(36122) = TRILINEAR PRODUCT X(4)*X(103)

Barycentrics    a/((a^2 - b^2 - c^2) (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c))) : :
Trilinears    (tan A)/((a - b) cot C + (a - c) cot B) : :
Trilinears    (sec A)/(a^2 - b^2 cos C - c^2 cos B) : :

X(36122) is the trilinear product of the circumcircle intercepts of line X(4)X(514). As the trilinear product of circumcircle-X(4)-antipodes, X(36122) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36122) lies on these lines: {1, 1783}, {2, 1897}, {4, 279}, {11, 278}, {28, 911}, {33, 57}, {81, 162}, {89, 1013}, {105, 5089}, {240, 1929}, {274, 811}, {277, 7952}, {677, 2990}, {1022, 36125}, {1170, 6198}, {1422, 7008}, {1736, 8558}, {1785, 34578}, {1870, 34056}, {2006, 23710}, {2310, 32714}, {2401, 2424}, {8056, 23052}, {34051, 36110}

X(36122) = polar conjugate of isogonal conjugate of X(911)
X(36122) = polar conjugate of anticomplement of X(241)
X(36122) = trilinear pole of line X(19)X(513)
X(36122) = X(i)-isoconjugate of X(j) for these {i,j}: {6, 26006}, {3, 516}, {63, 910}, {78, 1456}, {184, 35517}, {394, 1886}, {676, 1331}, {1815, 23972}, {1459, 2398}, {2426, 4025}, {1814, 9502}
X(36122) = trilinear product X(i)*X(j) for these {i,j}: {4, 103}, {19, 36101}, {25, 18025}, {92, 911}, {158, 36056}, {278, 2338}, {393, 1815}, {677, 7649}, {1897, 2424}, {2052, 32657}, {2400, 8750}, {5089, 9503}, {17924, 36039}
X(36122) = trilinear quotient X(i)/X(j) for these (i,j): (2, 26006), (4, 516), (19, 910), (34, 1456), (103, 3), (264, 35517), (393, 1886), (677, 1331), (911, 48), (1815, 394), (1886, 23972), (1897, 2398), (2338, 219), (2400, 4025), (2424, 1459), (5089, 9502), (7649, 676), (8750, 2426), (9503, 1814), (13149, 24015), (18025, 69), (32642, 32656), (32657, 577), (36039, 906), (36056, 255), (36101, 63), (36118, 23973)
X(36122) = barycentric product X(i)*X(j) for these {i,j}: {4, 36101}, {19, 18025}, {92, 103}, {158, 1815}, {264, 911}, {273, 2338}, {677, 17924}, {1783, 2400}, {1861, 9503}, {2052, 36056}, {2424, 6335}
X(36122) = barycentric quotient X(i)/X(j) for these (i,j): (1, 26006), (19, 516), (25, 910), (92, 35517), (103, 63), (608, 1456), (677, 1332), (911, 3), (1096, 1886), (1783, 2398), (1815, 326), (1886, 24014), (2338, 78), (2356, 9502), (2400, 15413), (2424, 905), (6591, 676), (9503, 31637), (18025, 304), (32642, 906), (32657, 255), (32714, 23973), (36039, 1331), (36056, 394), (36101, 69), (36118, 24015)

X(36123) = TRILINEAR PRODUCT X(4)*X(104)

Barycentrics    1/((a^2 - b^2 - c^2) ((a - b + c) (a + b - c) (b + c) - 2 a b c)) : :
Trilinears    (sec A)/(cos B + cos C - 1) : :

X(36123) is the trilinear product of the circumcircle intercepts of line X(4)X(513). As the trilinear product of circumcircle-X(4)-antipodes, X(36123) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36123) lies on these lines: {1, 318}, {2, 1809}, {4, 11}, {6, 281}, {10, 1167}, {29, 58}, {34, 158}, {86, 811}, {92, 998}, {106, 1309}, {240, 17954}, {242, 9432}, {269, 273}, {475, 34430}, {499, 17555}, {653, 1845}, {673, 1981}, {759, 7452}, {909, 1474}, {939, 3085}, {1027, 36124}, {1387, 21664}, {1411, 1870}, {1413, 34051}, {1431, 1905}, {1438, 2202}, {1737, 5081}, {1877, 36110}, {1878, 15635}, {2163, 14004}, {2215, 2250}, {2342, 3072}, {2401, 2424}, {2720, 32706}, {3445, 7952}, {4242, 10090}, {7040, 14266}, {8747, 36126}, {10428, 36112}, {11398, 16066}, {23345, 36125}, {23706, 32486}, {32641, 36107}

X(36123) = isogonal conjugate of X(22350)
X(36123) = polar conjugate of X(908)
X(36123) = pole wrt polar circle of trilinear polar of X(908) (line X(1145)X(1769))
X(36123) = trilinear pole of line X(19)X(649) (the polar of X(908) wrt polar circle)
X(36123) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 22350}, {3, 517}, {48, 908}, {63, 2183}, {72, 859}, {78, 1457}, {100, 8677}, {184, 3262}, {212, 22464}, {219, 1465}, {228, 17139}, {255, 1785}, {394, 14571}, {520, 4246}, {521, 23981}, {603, 6735}, {652, 24029}, {905, 2427}, {906, 10015}, {1145, 36058}, {1259, 1875}, {1331, 1769}, {1332, 3310}, {1807, 34586}, {2397, 22383}, {5440, 14260}
X(36123) = trilinear product X(i)*X(j) for these {i,j}: {4, 104}, {6, 16082}, {19, 34234}, {25, 18816}, {27, 2250}, {92, 909}, {158, 1795}, {264, 34858}, {273, 2342}, {281, 34051}, {513, 1309}, {522, 36110}, {693, 14776}, {915, 14266}, {1118, 1809}, {1783, 2401}, {2052, 14578}, {2423, 6335}, {4391, 32702}, {6591, 13136}, {7649, 36037}, {15635, 15742}, {17924, 32641}
X(36123) = trilinear quotient X(i)/X(j) for these (i,j): (1, 22350), (4, 517), (19, 2183), (28, 859), (34, 1457), (92, 908), (104, 3), (107, 4246), (108, 23981), (158, 1785), (264, 3262), (273, 22464), (278, 1465), (286, 17139), (318, 6735), (393, 14571), (513, 8677), (653, 24029), (909, 48), (1118, 1875), (1309, 100), (1783, 2427), (1795, 255), (1809, 1259), (1870, 34586), (2250, 71), (2342, 212), (2401, 905), (2423, 22383), (6335, 2397), (6591, 3310), (7649, 1769), (10428, 36058), (13136, 1332), (14266, 912), (14578, 577), (14776, 692), (15635, 3937), (16082, 2), (17923, 16586), (17924, 10015), (18816, 69), (32641, 906), (32669, 32660), (32702, 1415), (34051, 222), (34234, 63), (34858, 184), (36037, 1331), (36110, 109), (36125, 14260), (36127, 23706)
X(36123) = barycentric product X(i)*X(j) for these {i,j}: {1, 16082}, {4, 34234}, {19, 18816}, {92, 104}, {264, 909}, {286, 2250}, {318, 34051}, {331, 2342}, {514, 1309}, {1795, 2052}, {1897, 2401}, {1969, 34858}, {3261, 14776}, {4391, 36110}, {7649, 13136}, {17924, 36037}, {32702, 35519}
X(36123) = barycentric quotient X(i)/X(j) for these (i,j): (4, 908), (6, 22350), (19, 517), (25, 2183), (27, 17139), (34, 1465), (92, 3262), (104, 63), (108, 24029), (278, 22464), (281, 6735), (393, 1785), (608, 1457), (649, 8677), (909, 3), (1096, 14571), (1309, 190), (1474, 859), (1795, 394), (1809, 3719), (1870, 16586), (1897, 2397), (2250, 72), (2324, 1459), (2342, 219), (2401, 4025), (2720, 1813), (3064, 2804), (6591, 1769), (7649, 10015), (8750, 2427), (8756, 1145), (10428, 1797), (13136, 4561), (14266, 914), (14578, 255), (14776, 101), (15635, 3942), (16082, 75), (17924, 36038), (18816, 304), (24019, 4246), (32641, 1331), (32674, 23981), (32702, 109), (34051, 77), (34234, 69), (34858, 48), (36037, 1332), (36110, 651)

X(36124) = TRILINEAR PRODUCT X(4)*X(105)

Barycentrics    1/((a^2 - b^2 - c^2) (b^2 + c^2 - a b - a c)) : :

X(36124) is the trilinear product of the circumcircle intercepts of line X(4)X(885). As the trilinear product of circumcircle-X(4)-antipodes, X(36124) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36124) lies on hyperbola {{A,B,C,X(4),X(27)}} and these lines: {4, 218}, {25, 105}, {27, 162}, {29, 811}, {33, 92}, {34, 1847}, {242, 5089}, {273, 34018}, {281, 34337}, {666, 5081}, {885, 36121}, {917, 919}, {1027, 36123}, {1039, 2481}, {1096, 36127}, {1174, 3755}, {1416, 1430}, {1438, 8747}, {1462, 7151}, {1738, 36041}, {1860, 2195}, {1862, 7102}, {1886, 2201}, {2550, 7123}, {3423, 4000}, {5125, 31638}, {6336, 23710}, {7713, 18785}, {32735, 36113}, {36086, 36106}

X(36124) = isogonal conjugate of X(1818)
X(36124) = polar conjugate of X(3912)
X(36124) = pole wrt polar circle of trilinear polar of X(3912) (line X(918)X(2254))
X(36124) = cevapoint of X(i) and X(j) for these {i,j}: {1, 1738}, {4, 242}
X(36124) = trilinear pole of line X(19)X(1024) (the polar of X(3912) wrt polar circle)
X(36124) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1818}, {3, 518}, {6, 25083}, {63, 672}, {48, 3912}, {69, 2223}, {71, 18206}, {72, 3286}, {77, 2340}, {78, 1458}, {184, 3263}, {212, 9436}, {219, 241}, {222, 3693}, {228, 30941}, {255, 1861}, {304, 9454}, {305, 9455}, {326, 2356}, {394, 5089}, {520, 4238}, {521, 2283}, {603, 3717}, {652, 1025}, {665, 1332}, {883, 1946}, {905, 2284}, {906, 918}, {926, 6516}, {1026, 1459}, {1259, 1876}, {1260, 34855}, {1331, 2254}, {1814, 6184}, {1815, 9502}, {2991, 20728}
X(36124) = trilinear product X(i)*X(j) for these {i,j}: {2, 8751}, {4, 105}, {19, 673}, {25, 2481}, {27, 18785}, {28, 13576}, {34, 14942}, {92, 1438}, {107, 10099}, {108, 885}, {158, 36057}, {273, 2195}, {278, 294}, {281, 1462}, {318, 1416}, {393, 1814}, {607, 34018}, {653, 1024}, {666, 6591}, {884, 18026}, {919, 17924}, {927, 18344}, {1027, 1897}, {1096, 31637}, {1119, 28071}, {1435, 6559}, {1886, 9503}, {1973, 18031}, {2052, 32658}, {3064, 36146}, {5089, 6185}, {7649, 36086}, {23696, 36127}, {28132, 32714}
X(36124) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1818), (2, 25083), (4, 518), (19, 672), (25, 2223), (27, 18206), (28, 3286), (33, 2340), (34, 1458), (92, 3912), (105, 3), (107, 4238), (108, 2283), (158, 1861), (264, 3263), (273, 9436), (278, 241), (281, 3693), (286, 30941), (294, 219), (318, 3717), (331, 27818), (393, 5089), (653, 1025), (666, 1332), (673, 63), (884, 1946), (885, 521), (919, 906), (927, 6516), (1024, 652), (1027, 1459), (1096, 2356), (1118, 1876), (1119, 34855), (1416, 603), (1438, 48), (1462, 222), (1783, 2284), (1814, 394), (1861, 4712), (1886, 9502), (1897, 1026), (1973, 9454), (1974, 9455), (2195, 212), (2481, 69), (3290, 20728), (5089, 6184), (6185, 1814), (6559, 3692), (6591, 665), (7649, 2254), (8735, 17435), (8751, 6), (9503, 1815), (10099, 520), (13576, 72), (14942, 78), (17924, 918), (18026, 883), (18031, 304), (18344, 926), (18785, 71), (28071, 1260), (31637, 326), (32658, 577), (32666, 32656), (34018, 348), (36057, 255), (36086, 1331), (36146, 1813)
X(36124) = barycentric product X(i)*X(j) for these {i,j}: {4, 673}, {19, 2481}, {25, 18031}, {27, 13576}, {33, 34018}, {75, 8751}, {92, 105}, {158, 1814}, {264, 1438}, {273, 294}, {278, 14942}, {286, 18785}, {318, 1462}, {331, 2195}, {393, 31637}, {653, 885}, {666, 7649}, {823, 10099}, {927, 3064}, {1024, 18026}, {1027, 6335}, {1119, 6559}, {1416, 7017}, {1847, 28071}, {1861, 6185}, {2052, 36057}, {17924, 36086}, {18344, 34085}, {28132, 36118}
X(36124) = barycentric quotient X(i)/X(j) for these (i,j): (1, 25083), (4, 3912), (6, 1818), (19, 518), (25, 672), (27, 30941), (28, 18206), (33, 3693), (34, 241), (92, 3263), (105, 63), (108, 1025), (242, 17755), (273, 27818), (278, 9436), (281, 3717), (286, 18157), (294, 78), (393, 1861), (607, 2340), (608, 1458), (653, 883), (666, 4561), (673, 69), (884, 652), (885, 6332), (919, 1331), (1024, 521), (1027, 905), (1096, 5089), (1118, 5236), (1416, 222), (1435, 34855), (1438, 3), (1462, 77), (1474, 3286), (1783, 1026), (1814, 326), (1861, 4437), (1973, 2223), (1974, 9454), (2195, 219), (2207, 2356), (2356, 6184), (2481, 304), (5089, 4712), (6185, 31637), (6559, 1265), (6591, 2254), (7649, 918), (8750, 2284), (8751, 1), (10099, 24018), (13576, 306), (14942, 345), (18031, 305), (18785, 72), (24019, 4238), (28071, 3692), (31637, 3926), (32658, 255), (32666, 906), (32674, 2283), (32735, 1813), (34018, 7182), (36057, 394), (36086, 1332), (36146, 6516)

X(36125) = TRILINEAR PRODUCT X(4)*X(106)

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

X(36125) is the trilinear product of the circumcircle intercepts of line X(4)X(2457). As the trilinear product of circumcircle-X(4)-antipodes, X(36125) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36125) lies on these lines: {4, 145}, {8, 12876}, {19, 1743}, {28, 88}, {34, 106}, {278, 1846}, {286, 811}, {651, 15906}, {901, 915, 36106}, {1022, 36122}, {1118, 36127}, {1119, 36118}, {1168, 1877}, {1318, 1870}, {1417, 1875}, {1633, 1718}, {1739, 36042}, {1830, 4792}, {3753, 23617}, {4622, 36105}, {4714, 10912}, {4945, 5155}, {4997, 5142}, {5317, 9456}, {10702, 35015}, {11400, 35502}, {23345, 36123}, {23838, 36121}, {32665, 36107}, {34230, 34231}

X(36125) = isogonal conjugate of X(5440)
X(36125) = polar conjugate of X(4358)
X(36125) = pole wrt polar circle of trilinear polar of X(4358) (line X(1145)X(3762))
X(36125) = trilinear pole of line X(19)X(4394) (the polar of X(4358) wrt polar circle)
X(36125) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 5440}, {2, 22356}, {3, 519}, {6, 3977}, {44, 63}, {48, 4358}, {69, 902}, {71, 16704}, {78, 1319}, {184, 3264}, {214, 1807}, {219, 3911}, {228, 30939}, {304, 2251}, {305, 9459}, {306, 3285}, {345, 1404}, {394, 8756}, {521, 23703}, {603, 4723}, {900, 1331}, {905, 1023}, {906, 3762}, {1145, 1795}, {1259, 1877}, {1332, 1635}, {1459, 17780}, {1960, 4561}, {4025, 23344}, {4120, 4558}, {4563, 14407}, {4592, 4730}, {22383, 24004}
X(36125) = trilinear product X(i)*X(j) for these {i,j}: {2, 8752}, {4, 106}, {6, 6336}, {19, 88}, {25, 903}, {28, 4674}, {34, 1320}, {92, 9456}, {108, 23838}, {112, 4049}, {158, 36058}, {278, 2316}, {318, 1417}, {393, 1797}, {608, 4997}, {901, 7649}, {1022, 1783}, {1168, 1870}, {1318, 1877}, {1474, 4080}, {1785, 10428}, {1897, 23345}, {1973, 20568}, {2052, 32659}, {2489, 4615}, {2501, 4591}, {3257, 6591}, {6548, 8750}, {14260, 36123}, {17924, 32665}
X(36125) = trilinear quotient X(i)/X(j) for these (i,j): (1, 5440), (2, 3977), (4, 519), (6, 22356), (19, 44), (25, 902), (27, 16704), (34, 1319), (88, 63), (92, 4358), (106, 3), (108, 23703), (264, 3264), (278, 3911), (286, 30939), (318, 4723), (393, 8756), (608, 1404), (901, 1331), (903, 69), (1022, 905), (1118, 1877), (1168, 1807), (1320, 78), (1417, 603), (1474, 3285), (1783, 1023), (1785, 1145), (1797, 394), (1870, 214), (1877, 1317), (1897, 17780), (1973, 2251), (1974, 9459), (2316, 219), (2489, 14407), (2501, 4120), (3257, 1332), (4049, 525), (4080, 306), (4555, 4561), (4591, 4558), (4615, 4563), (4622, 4592), (4674, 72), (4792, 3940), (4997, 345), (6335, 24004), (6336, 2), (6548, 4025), (6591, 1635), (7649, 900), (8610, 22428), (8750, 23344), (8752, 6), (9456, 48), (10428, 1795), (14260, 22350), (17924, 3762), (20568, 304), (23345, 1459), (23838, 521), (32659, 577), (32665, 906), (32719, 32656), (34230, 1818), (36058, 255)
X(36125) = barycentric product X(i)*X(j) for these {i,j}: {1, 6336}, {4, 88}, {19, 903}, {25, 20568}, {27, 4674}, {28, 4080}, {34, 4997}, {75, 8752}, {92, 106}, {158, 1797}, {162, 4049}, {264, 9456}, {273, 2316}, {278, 1320}, {653, 23838}, {901, 17924}, {1022, 1897}, {1168, 17923}, {1417, 7017}, {1783, 6548}, {2052, 36058}, {2489, 4634}, {2501, 4622}, {3257, 7649}, {4555, 6591}, {4591, 24006}, {6335, 23345}, {14260, 16082}
X(36125) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3977), (4, 4358), (6, 5440), (19, 519), (25, 44), (27, 30939), (28, 16704), (31, 22356), (34, 3911), (88, 69), (92, 3264), (106, 63), (281, 4723), (608, 1319), (901, 1332), (903, 304), (1022, 4025), (1096, 8756), (1320, 345), (1395, 1404), (1417, 222), (1783, 17780), (1797, 326), (1897, 24004), (1973, 902), (1974, 2251), (2203, 3285), (2316, 78), (2489, 4730), (3257, 4561), (4049, 14208), (4080, 20336), (4591, 4592), (4622, 4563), (4674, 306), (4997, 3718), (6336, 75), (6548, 15413), (6591, 900), (7649, 3762), (8750, 1023), (8752, 1), (9456, 3), (14571, 1145), (17923, 1227), (20568, 305), (23345, 905), (23838, 6332), (32659, 255), (32665, 1331), (32674, 23703), (32719, 906), (34230, 25083), (36058, 394)

X(36126) = TRILINEAR PRODUCT X(4)*X(107)

Barycentrics    b c/((b^2 - c^2) (a^2 - b^2 - c^2)^3) : :
Barycentrics    (sec A) (tan A)/(tan B - tan C) : :
Trilinears    (sec^2 A)/(tan B - tan C) : :
Trilinears    (sec^2 A)/(sin 2B - sin 2C) : :

X(36126) is the trilinear product of the circumcircle intercepts of line X(4)X(51). As the trilinear product of circumcircle-X(4)-antipodes, X(36126) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36126) lies on these lines: {92, 8767}, {107, 108}, {112, 681}, {158, 1109}, {162, 823}, {811, 2617}, {1096, 6521}, {1783, 6529}, {1896, 36121}, {1897, 15352}, {6520, 36128}, {8747, 36123}, {17898, 36043}, {23999, 36105}, {24000, 36114}, {24006, 24021}, {24019, 24024}

X(36126) = isogonal conjugate of isotomic conjugate of polar conjugate of X(822)
X(36126) = polar conjugate of X(24018)
X(36126) = trilinear pole of line X(19)X(158) (the polar of X(24018) wrt polar circle)
X(36126) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 32320}, {3, 520}, {32, 4143}, {48, 24018}, {63, 822}, {71, 4091}, {72, 23224}, {184, 3265}, {228, 4131}, {255, 656}, {326, 810}, {394, 647}, {512, 3964}, {521, 22341}, {523, 1092}, {525, 577}, {648, 35071}, {661, 6507}, {669, 4176}, {798, 1102}, {850, 23606}, {905, 3990}, {924, 16391}, {1214, 36054}, {1459, 3682}, {1577, 4100}, {2632, 4575}, {3049, 3926}, {3267, 14585}, {3269, 4558}, {3998, 22383}, {4025, 4055}, {15526, 32661}, {23357, 23616}, {24020, 32676}
X(36126) = trilinear product X(i)*X(j) for these {i,j}: {2, 6529}, {4, 107}, {6, 15352}, {19, 823}, {25, 6528}, {29, 36127}, {92, 24019}, {99, 6524}, {108, 1896}, {110, 1093}, {112, 2052}, {158, 162}, {163, 6521}, {264, 32713}, {393, 648}, {525, 23590}, {647, 34538}, {653, 8748}, {656, 24021}, {662, 6520}, {811, 1096}, {1625, 8794}, {1897, 8747}, {2501, 23582}, {2207, 6331}, {3267, 23975}, {5317, 6335}, {8767, 24024}, {14208, 24022}, {14618, 23964}, {24000, 24006}
X(36126) = trilinear quotient X(i)/X(j) for these (i,j): (4, 520), (6, 32320), (19, 822), (27, 4091), (28, 23224), (76, 4143), (92, 24018), (99, 3964), (107, 3), (108, 22341), (110, 1092), (112, 577), (158, 656), (162, 255), (163, 4100), (264, 3265), (286, 4131), (338, 23616), (393, 647), (647, 35071), (648, 394), (662, 6507), (670, 4176), (799, 1102), (811, 326), (823, 63), (925, 16391), (1093, 523), (1096, 810), (1172, 36054), (1576, 23606), (1783, 3990), (1896, 521), (1897, 3682), (2052, 525), (2207, 3049), (2501, 3269), (3267, 23974), (5317, 22383), (6331, 3926), (6335, 3998), (6520, 661), (6521, 1577), (6524, 512), (6528, 69), (6529, 6), (8747, 1459), (8748, 652), (8750, 4055), (8794, 15412), (14208, 24020), (14618, 15526), (15352, 2), (14165, 8552), (18027, 3267), (23582, 4558), (23590, 112), (23964, 32661), (23999, 4592), (24000, 4575), (24006, 2632), (24019, 48), (24021, 162), (24022, 32676), (24024, 8766), (32713, 184), (34538, 648), (36127, 73)
X(36126) = barycentric product X(i)*X(j) for these {i,j}: {1, 15352}, {4, 823}, {19, 6528}, {75, 6529}, {92, 107}, {99, 6520}, {110, 6521}, {158, 648}, {162, 2052}, {264, 24019}, {393, 811}, {525, 24021}, {653, 1896}, {656, 34538}, {662, 1093}, {799, 6524}, {1096, 6331}, {1969, 32713}, {2501, 23999}, {2617, 8794}, {3267, 24022}, {6335, 8747}, {8748, 18026}, {14165, 36129}, {14208, 23590}, {14618, 24000}, {18027, 32676}, {23582, 24006}, {31623, 36127}
X(36126) = barycentric quotient X(i)/X(j) for these (i,j): (4, 24018), (19, 520), (25, 822), (27, 4131), (28, 4091), (75, 4143), (92, 3265), (99, 1102), (107, 63), (110, 6507), (112, 255), (158, 525), (162, 394), (163, 1092), (286, 30805), (393, 656), (525, 24020), (648, 326), (662, 3964), (799, 4176), (810, 35071), (811, 3926), (823, 69), (1093, 1577), (1096, 647), (1109, 23616), (1474, 23224), (1576, 4100), (1783, 3682), (1896, 6332), (1897, 3998), (2052, 14208), (2207, 810), (2299, 36054), (2501, 2632), (5317, 1459), (6520, 523), (6521, 850), (6524, 661), (6528, 304), (6529, 1), (8747, 905), (8748, 521), (8750, 3990), (14208, 23974), (14618, 17879), (15352, 75), (23582, 4592), (23590, 162), (23964, 4575), (23975, 32676), (23999, 4563), (24000, 4558), (24006, 15526), (24019, 3), (24021, 648), (24022, 112), (32676, 577), (32713, 48), (34538, 811), (32674, 22341), (36127, 1214), (36145, 16391)

X(36127) = TRILINEAR PRODUCT X(4)*X(108)

Barycentrics    1/((a^2 - b^2 - c^2)^2 (b - c) (a - b - c)) : :
Trilinears    (tan A)/(sec B - sec C) : :

X(36127) is the trilinear product of the circumcircle intercepts of line X(4)X(65). As the trilinear product of circumcircle-X(4)-antipodes, X(36127) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36127) lies on these lines: {4, 1854}, {11, 278}, {34, 158}, {92, 34036}, {107, 109}, {108, 676}, {196, 6525}, {221, 1148}, {223, 24030}, {225, 36119}, {226, 8767}, {243, 1465}, {664, 811}, {1096, 36124}, {1118, 36125}, {1411, 8747}, {1783, 4559}, {1880, 1945}, {1895, 21147}, {1897, 4551}, {3176, 6523}, {6335, 14594}, {6529, 24019}, {7012, 36106}, {7115, 36107}, {7649, 24033}, {7952, 10271}, {10571, 14249}, {20031, 36104}, {21186, 36044}, {23353, 36059}, {26704, 36067}

X(36127) = isogonal conjugate of isotomic conjugate of polar conjugate of X(652)
X(36127) = polar conjugate of X(6332)
X(36127) = pole wrt polar circle of trilinear polar of X(6332) (line X(2968)X(4082))
X(36127) = trilinear pole of line X(19)X(208) (the polar of X(6332) wrt polar circle)
X(36127) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 521}, {21, 520), {31, 520}, {48, 6332}, {63, 652}, {69, 1946}, {78, 1459}, {109, 24031}, {184, 35518}, {212, 4025}, {219, 905}, {255, 522}, {283, 656}, {284, 24018}, {326, 663}, {332, 810}, {333, 822}, {345, 22383}, {394, 650}, {513, 1259}, {514, 2289}, {525, 2193}, {577, 4391}, {647, 1812}, {649, 3719}, {651, 35072}, {657, 7183}, {661, 6514}, {664, 2638}, {667, 1264}, {693, 6056}, {906, 26932}, {1331, 7004}, {1332, 7117}, {1415, 23983}, {1804, 3900}, {1813, 34591}, {1818, 23696}, {2194, 3265}, {2204, 4143}, {3063, 3926}, {3064, 6507}, {3239, 7125}, {3682, 3737}, {3964, 18344}, {3990, 4560}, {3998, 7252}, {7055, 8641}, {31623, 32320}
X(36127) = trilinear product X(i)*X(j) for these {i,j}: {4, 108}, {19, 653}, {25, 18026}, {33, 36118}, {34, 1897}, {65, 107}, {73, 36126}, {92, 32674}, {100, 1118}, {109, 158}, {162, 225}, {226, 24019}, {273, 8750}, {278, 1783}, {281, 32714}, {393, 651}, {522, 24033}, {607, 13149}, {608, 6335}, {648, 1880}, {650, 23984}, {663, 24032}, {664, 1096}, {668, 7337}, {823, 1400}, {934, 1857}, {1093, 36059}, {1214, 6529}, {1402, 6528}, {1409, 15352}, {1415, 2052}, {1441, 32713}, {1785, 36110}, {1813, 6520}, {2207, 4554}, {3064, 7128}, {4391, 23985}, {4551, 8747}, {4552, 5317}, {4569, 6059}, {6516, 6524}, {6521, 32660}, {7012, 7649}, {7115, 17924}, {23706, 36123}, {23987, 36121}
X(36127) = trilinear quotient X(i)/X(j) for these (i,j): (4, 521), (19, 652), (25, 1946), (34, 1459), (65, 520), (92, 6332), (100, 1259), (101, 2289), (107, 21), (108, 3), (109, 255), (112, 2193), (158, 522), (162, 283), (190, 3719), (225, 656), (226, 24018), (264, 35518), (273, 4025), (278, 905), (331, 15413), (393, 650), (522, 24031), (608, 22383), (648, 1812), (650, 35072), (651, 394), (653, 63), (658, 7183), (662, 6514), (663, 2638), (664, 326), (668, 1264), (692, 6056), (811, 332), (823, 333), (934, 1804), (1096, 663), (1118, 513), (1231, 4143), (1400, 822), (1409, 32320), (1415, 577), (1441, 3265), (1461, 7125), (1783, 219), (1813, 6507), (1857, 3900), (1880, 647), (1897, 78), (1981, 6518), (2052, 4391), (2207, 3063), (3064, 34591), (4391, 23983), (4551, 3682), (4552, 3998), (4554, 3926), (4559, 3990), (4569, 7055), (5317, 7252), (6059, 8641), (6335, 345), (6516, 3964), (6520, 3064), (6524, 18344), (6528, 314), (6529, 1172), (6591, 7117), (7012, 1331), (7017, 15416), (7115, 906), (7128, 1813), (7337, 667), (7649, 7004), (8747, 3737), (8750, 212), (13149, 348), (15352, 31623), (17924, 26932), (18026, 69), (23706, 22350), (23984, 651), (23985, 1415), (24019, 284), (24032, 664), (24033, 109), (32660, 4100), (32674, 48), (32713, 2194), (32714, 222), (36059, 1092), (36067, 36055), (36110, 1795), (36118, 77), (36124, 23696), (36126, 29)
X(36127) = barycentric product X(i)*X(j) for these {i,j}: {4, 653}, {19, 18026}, {33, 13149}, {34, 6335}, {65, 823}, {73, 15352}, {92, 108}, {107, 226}, {109, 2052}, {158, 651}, {190, 1118}, {225, 648}, {264, 32674}, {273, 1783}, {278, 1897}, {281, 36118}, {307, 6529}, {318, 32714}, {331, 8750}, {349, 32713}, {393, 664}, {522, 23984}, {650, 24032}, {658, 1857}, {811, 1880}, {1093, 1813}, {1096, 4554}, {1214, 36126}, {1400, 6528}, {1441, 24019}, {1978, 7337}, {2207, 4572}, {4391, 24033}, {4552, 8747}, {6516, 6520}, {6521, 36059}, {7012, 17924}, {16082, 23706}, {23985, 35519}, {24035, 36121}
X(36127) = barycentric quotient X(i)/X(j) for these (i,j): (4, 6332), (19, 521), (25, 652), (34, 905), (65, 24018), (92, 35518), (100, 3719), (101, 1259), (107, 333), (108, 63), (109, 394), (110, 6514), (112, 283), (158, 4391), (162, 1812), (190, 1264), (225, 525), (226, 3265), (273, 15413), (278, 4025), (307, 4143), (318, 15416), (393, 522), (522, 23983), (608, 1459), (648, 332), (650, 24031), (651, 326), (653, 69), (658, 7055), (663, 35072), (664, 3926), (692, 2289), (823, 314), (934, 7183), (1096, 650), (1118, 514), (1395, 22383), (1400, 520), (1402, 822), (1415, 255), (1461, 1804), (1783, 78), (1813, 3964), (1857, 3239), (1880, 656), (1897, 345), (1973, 1946), (2052, 35519), (2207, 663), (3063, 2638), (3064, 2968), (4551, 3998), (4559, 3682), (5317, 3737), (6335, 3718), (6516, 1102), (6524, 3064), (6525, 14331), (6528, 28660), (6529, 29), (6591, 7004), (7012, 1332), (7115, 1331), (7128, 6516), (7337, 649), (7649, 26932), (8747, 4560), (8750, 219), (8751, 23696), (13149, 7182), (17924, 17880), (18026, 304), (18344, 34591), (23353, 6518), (23984, 664), (23985, 109), (24019, 21), (24032, 4554), (24033, 651), (32660, 1092), (32667, 36055), (32674, 3), (32676, 2193), (32702, 1795), (32713, 284), (32714, 77), (32739, 6056), (36059, 6507), (36082, 6512), (36118, 348), (36126, 31623)

X(36128) = TRILINEAR PRODUCT X(4)*X(111)

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

X(36128) is the trilinear product of the circumcircle intercepts of line X(4)X(1499). As the trilinear product of circumcircle-X(4)-antipodes, X(36128) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36128) lies on these lines: {19, 162}, {92, 811}, {108, 111}, {240, 17955}, {923, 1096}, {1783, 1824}, {1826, 1897}, {2358, 7316}, {6520, 36126}, {8749, 9214}, {23894, 36119}, {36085, 36105}, {36114, 36142}

X(36128) = isogonal conjugate of isotomic conjugate of polar conjugate of X(896)
X(36128) = polar conjugate of X(14210)
X(36128) = pole wrt polar circle of trilinear polar of X(14210) (line X(2642)X(4750))
X(36128) = trilinear pole of line X(19)X(23894) (the polar of X(14210) wrt polar circle)
X(36128) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 3292}, {3, 524}, {6, 6390}, {48, 14210}, {63, 896}, {69, 187}, {76, 23200}, {184, 3266}, {222, 3712}, {287, 9155}, {304, 922} {305, 14567}, {351, 4563}, {394, 468}, {520, 4235}, {525, 5467}, {647, 5468}, {656, 23889}, {690, 4558}, {810, 24039}, {895, 2482}, {1331, 4750}, {1332, 14419}, {1444, 21839}, {1790, 4062}, {2642, 4592}, {8552, 14559}, {9717, 11064}, {32661, 35522}
X(36128) = trilinear product X(i)*X(j) for these {i,j}: {2, 8753}, {4, 111}, {6, 17983}, {19, 897}, {25, 671}, {92, 923}, {107, 10097}, {112, 5466}, {158, 36060}, {162, 23894}, {232, 9154}, {264, 32740}, {281, 7316}, {393, 895}, {468, 10630}, {648, 9178}, {691, 2501}, {892, 2489}, {1974, 18023}, {2052, 14908}, {2207, 30786}, {5380, 6591}, {8749, 9214}, {14618, 32729}, {14977, 32713}, {24006, 36142}
X(36128) = trilinear quotient X(i)/X(j) for these (i,j): (2, 6390), (4, 524), (6, 3292), (19, 896), (25, 187), (32, 23200), (92, 14210), (107, 4235), (111, 3), (112, 5467), (162, 23889), (232, 9155), (264, 3266), (281, 3712), (393, 468), (468, 2482), (648, 5468), (671, 69), (691, 4558), (811, 24039), (892, 4563), (895, 394), (897, 63), (923, 48), (1824, 21839), (1826, 4062), (1973, 922), (1974, 14567), (2489, 351), (2501, 690), (5380, 1332), (5466, 525), (6591, 14419), (7316, 222), (7649, 4750), (8749, 9717), (8753, 6), (9154, 287), (9178, 647), (9213, 8552), (9214, 11064), (10097, 520), (10630, 895), (14618, 35522), (14908, 577), (14977, 3265), (17983, 2), (18023, 305), (23894, 656), (30786, 3926), (32729, 32661), (32740, 184), (36060, 255), (36085, 4592), (36142, 4575)
X(36128) = barycentric product X(i)*X(j) for these {i,j}: {1, 17983}, {4, 897}, {19, 671}, {75, 8753}, {92, 111}, {158, 895}, {162, 5466}, {240, 9154}, {264, 923}, {318, 7316}, {648, 23894}, {691, 24006}, {811, 9178}, {823, 10097}, {1096, 30786}, {1969, 32740}, {1973, 18023}, {2052, 36060}, {2501, 36085}, {5380, 7649}, {9213, 36129}, {9214, 36119}, {14618, 36142}, {14977, 24019}
X(36128) = barycentric quotient X(i)/X(j) for these (i,j): (1, 6390), (4, 14210), (19, 524), (25, 896), (31, 3292), (33, 3712), (92, 3266), (111, 63), (112, 23889), (162, 5468), (468, 24038), (560, 23200), (648, 24039), (671, 304), (691, 4592), (895, 326), (897, 69), (923, 3), (1096, 468), (1824, 4062), (1973, 187), (1974, 922), (2333, 21839), (2489, 2642), (5380, 4561), (5466, 14208), (6591, 4750), (7316, 77), (8753, 1), (9154, 336), (9178, 656), (10097, 24018), (14908, 255), (17983, 75), (23894, 525), (24006, 35522), (24019, 4235), (32676, 5467), (32729, 4575), (32740, 48), (36060, 394), (36085, 4563), (36142, 4558)

X(36129) = TRILINEAR PRODUCT X(4)*X(476)

Barycentrics    b c/((a^2 - b^2 - c^2) (b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2)) : :
Trilinears    (sec A csc(B - C))/(1 + 2 cos 2A) : :

X(36129) is the trilinear product of the circumcircle intercepts of line X(4)X(94). As the trilinear product of circumcircle-X(4)-antipodes, X(36129) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36129) lies on these lines: {1, 36130}, {108, 476}, {162, 24006}, {1784, 2166}, {24001, 36114}, {24019, 32678}, {36034, 36035}

X(36129) = isogonal conjugate of isotomic conjugate of polar conjugate of X(2624)
X(36129) = polar conjugate of X(32679)
X(36129) = trilinear pole of line X(19)X(2166) (the polar of X(32679) wrt polar circle)
X(36129) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 526}, {6, 8552}, {48, 32679}, {50, 525}, {63, 2624}, {69, 14270}, {184, 3268}, {186, 520}, {323, 647}, {523, 22115}, {656, 6149}, {1511, 14380}, {2088, 4558}, {3049, 7799}, {3265, 34397}, {3267, 19627}, {3269, 14590}, {3292, 9213}, {14165, 32320}, {14591, 15526}
X(36129) = trilinear product X(i)*X(j) for these {i,j}: {4, 476}, {19, 32680}, {25, 35139}, {92, 32678}, {94, 112}, {99, 18384}, {107, 265}, {110, 6344}, {158, 36061}, {162, 2166}, {264, 14560}, {328, 32713}, {648, 1989}, {1576, 18817}, {2052, 32662}, {4240, 5627}, {6331, 11060}, {14559, 17983}, {14582, 23582}, {14592, 23964}, {36047, 36063}
X(36129) = trilinear quotient X(i)/X(j) for these (i,j): (2, 8552), (4, 526), (19, 2624), (25, 14270), (92, 32679), (94, 525), (107, 186), (110, 22115), (112, 50), (162, 6149), (264, 3268), (265, 520), (328, 3265), (476, 3), (648, 323), (1989, 647), (2166, 656), (2501, 2088), (4240, 1511), (5627, 14380), (6331, 7799), (6344, 523), (6528, 340), (11060, 3049), (14559, 3292), (14560, 184), (14582, 3269), (14592, 15526), (15352, 14165), (17983, 9213), (18384, 512), (18817, 850), (20573, 3267), (23582, 14590), (23588, 32662), (23964, 14591), (32650, 32663), (32662, 577), (32678, 48), (32680, 63), (32713, 34397), (35139, 69), (36047, 36062), (36061, 255)
X(36129) = barycentric product X(i)*X(j) for these {i,j}: {4, 32680}, {19, 35139}, {92, 476}, {94, 162}, {163, 18817}, {264, 32678}, {265, 823}, {328, 24019}, {648, 2166}, {662, 6344}, {799, 18384}, {811, 1989}, {1969, 14560}, {2052, 36061}, {2410, 36130}, {5627, 24001}, {14582, 23999}, {14592, 24000}, {20573, 32676}
X(36129) = barycentric quotient X(i)/X(j) for these (i,j): (1, 8552), (4, 32679), (19, 526), (25, 2624), (92, 3268), (94, 14208), (112, 6149), (162, 323), (163, 22115), (265, 24018), (476, 63), (811, 7799), (823, 340), (1784, 5664), (1973, 14270), (1989, 656), (2166, 525), (6344, 1577), (11060, 810), (14560, 48), (14582, 2632), (14592, 17879), (18384, 661), (18817, 20948), (23588, 36061), (24000, 14590), (24001, 6148), (24019, 186), (32650, 36062), (32662, 255), (32676, 50), (32678, 3), (32680, 69), (35139, 304), (36061, 394), (36126, 14165), (36128, 9213), (36130, 2411)

X(36130) = TRILINEAR PRODUCT X(4)*X(477)

Barycentrics    b c/((a^2 - b^2 - c^2) (a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4))) : :
Trilinears    (sec A)/(4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)) : :

X(36130) is the trilinear product of the circumcircle intercepts of line X(4)X(526). As the trilinear product of circumcircle-X(4)-antipodes, X(36130) lies on the conic {{A,B,C,X(108),X(162)}} with center X(36103) and perspector X(19).

X(36130) lies on these lines: {1, 36129}, {108, 477}, {162, 1784}, {1099, 1101}, {1895, 36114}, {24006, 36119}, {24019, 36151}, {36047, 36116}

X(36130) = trilinear pole of line X(19)X(2624)
X(36130) = X(i)-isoconjugate of X(j) for these {i,j}: {3, 5663}, {184, 35520}, {255, 36063}, {520, 7480}, {2437, 8552}
X(36130) = trilinear product X(i)*X(j) for these {i,j}: {4, 477}, {19, 36102}, {92, 36151}, {107, 14220}, {158, 36062}, {2052, 32663}, {2501, 30528}, {36035, 36117}
X(36130) = trilinear quotient X(i)/X(j) for these (i,j): (4, 5663), (107, 7480), (158, 36063), (264, 35520), (477, 3), (2411, 8552), (14220, 520), (30528, 4558), (32650, 32662), (32663, 577), (32712, 32640), (36047, 36061), (36062, 255), (36102, 63), (36117, 36034), (36151, 48)
X(36130) = barycentric product X(i)*X(j) for these {i,j}: {4, 36102}, {92, 477}, {264, 36151}, {823, 14220}, {2052, 36062}, {2411, 36129}, {24006, 30528}
X(36130) = barycentric quotient X(i)/X(j) for these (i,j): (19, 5663), (92, 35520), (393, 36063), (477, 63), (14220, 24018), (24019, 7480), (30528, 4592), (32650, 36061), (32663, 255), (32712, 36034), (36062, 394), (36102, 69), (36129, 2410), (36151, 3)

X(36131) = TRILINEAR PRODUCT X(74)*X(112)

Barycentrics    a^3/((b^2 - c^2) (a^2 - b^2 - c^2) (2 a^4 - a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :
Barycentrics    a^3/(SA (SB - SC) (SA SB + SA SC - 2 SB SC)) : :
Barycentrics    a^3/((tan B - tan C) (2 tan A - tan B - tan C)) : :
Barycentrics    a^3/((sin 2B - sin 2C) (2 sin 2A - sin 2B - sin 2C)) : :
Barycentrics    a^2/((sin 2B - sin 2C) (cos A - 2 cos B cos C)) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36131) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36131) lies on these lines: {19, 36151}, {74, 36071}, {101, 1304}, {112, 36064}, {162, 36083}, {163, 822}, {661, 24019}, {662, 24018}, {692, 32715}, {810, 32676}, {1415, 32640}, {1910, 36104}, {2159, 9406}, {2349, 36095}, {4586, 16077}, {8749, 34079}, {16080, 33665}, {32674, 32695}, {35200, 36046}, {36117, 36144}

X(36131) = polar conjugate of isotomic conjugate of X(36034)
X(36131) = trilinear pole of line X(31)X(2159)
X(36131) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 9033}, {30, 525}, {63, 36035}, {69, 1637}, {75, 2631}, {76, 9409}, {125, 2407}, {264, 1636}, {339, 2420}, {523, 11064}, {647, 3260}, {648, 1650}, {656, 14206}, {850, 3284}, {1494, 14401}, {1495, 3267}, {1784, 24018}, {1990, 3265}, {2173, 14208}, {2799, 35912}, {6333, 35906}
X(36131) = trilinear product X(i)*X(j) for these {i,j}: {2, 32715}, {3, 32695}, {4, 32640}, {6, 1304}, {19, 36034}, {32, 16077}, {74, 112}, {107, 18877}, {110, 8749}, {162, 2159}, {163, 36119}, {184, 15459}, {250, 2433}, {378, 32681}, {1301, 15291}, {1495, 34568}, {1576, 16080}, {2349, 32676}, {2715, 35908}, {14919, 32713}, {24019, 35200}, {32696, 35910}
X(36131) = trilinear quotient X(i)/X(j) for these (i,j): (6, 9033), (19, 36035), (25, 1637), (31, 2631), (32, 9409), (74, 525), (110, 11064), (112, 30), (162, 14206), (184, 1636), (250, 2407), (647, 1650), (648, 3260), (1304, 2), (1494, 3267), (1495, 14401), (1576, 3284), (2159, 656), (2349, 14208), (2394, 339), (2433, 125), (2715, 35912), (8749, 523), (14919, 3265), (15291, 8057), (15459, 264), (16077, 76), (16080, 850), (18877, 520), (24019, 1784), (32676, 2173), (32681, 4846), (32695, 4), (32696, 35906), (32713, 1990), (32715, 6), (34568, 1494), (35200, 24018), (35908, 2799), (35910, 6333), (36034, 63), (36117, 36102), (36119, 1577)
X(36131) = barycentric product X(i)*X(j) for these {i,j}: {1, 1304}, {4, 36034}, {31, 16077}, {48, 15459}, {63, 32695}, {74, 162}, {75, 32715}, {92, 32640}, {107, 35200}, {110, 36119}, {112, 2349}, {163, 16080}, {378, 36083}, {648, 2159}, {662, 8749}, {823, 18877}, {1494, 32676}, {2173, 34568}, {11107, 36064}, {14919, 24019}, {35908, 36084}, {35910, 36104}
X(36131) = barycentric quotient X(i)/X(j) for these (i,j): (25, 36035), (31, 9033), (32, 2631), (74, 14208), (112, 14206), (162, 3260), (163, 11064), (560, 9409), (810, 1650), (1304, 75), (1973, 1637), (2159, 525), (2349, 3267), (2433, 20902), (8749, 1577), (9247, 1636), (9406, 14401), (15459, 1969), (16077, 561), (16080, 20948), (18877, 24018), (32640, 63), (32676, 30), (32695, 92), (32712, 36102), (32713, 1784), (32715, 1), (34568, 33805), (35200, 3265), (36034, 69), (36119, 850)

X(36132) = TRILINEAR PRODUCT X(98)*X(26714)

Barycentrics    a/((b^2 - c^2) (b^4 + c^4 - a^2 b^2 - a^2 c^2) (a^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2)) : :
Barycentrics    a^2 sec(A + ω)/((b^2 - c^2) sin 2A + b^2 sin 2C - c^2 sin 2B) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36132) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36132) lies on these lines: {101, 6037}, {163, 36084}, {662, 36036}, {692, 32716}, {1910, 9417}, {26714, 36065}, {32676, 36104}

X(36132) = trilinear pole of line X(31)X(1910)
X(36132) = X(i)-isoconjugate of X(j) for these {i,j}: {182, 2799}, {183, 3569}, {325, 3288}, {458, 684}, {511, 23878}, {6333, 10311}
X(36132) = trilinear product X(i)*X(j) for these {i,j}: {2, 32716}, {6, 6037}, {98, 26714}, {262, 2715}, {263, 2966}, {2186, 36084}, {3402, 36036}
X(36132) = trilinear quotient X(i)/X(j) for these (i,j): (98, 23878), (262, 2799), (263, 3569), (685, 458), (1976, 3288), (2715, 182), (2966, 183), (6037, 2), (26714, 511), (32696, 10311), (32716, 6), (36036, 3403)
X(36132) = barycentric product X(i)*X(j) for these {i,j}: {1, 6037}, {75, 32716}, {262, 36084}, {263, 36036}, {1821, 26714}, {2186, 2966}
X(36132) = barycentric quotient X(i)/X(j) for these (i,j): (1910, 1976), (2186, 2799), (2966, 3403), (3402, 3569), (6037, 75), (26714, 1959), (32716, 1), (36036, 20023), (36084, 183), (36104, 458)

X(36133) = TRILINEAR PRODUCT X(99)*X(729)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36133) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36133) lies on these lines: {101, 4600}, {163, 24041}, {662, 1924, 24037}, {692, 4567}, {729, 18268}, {886, 4586}, {923, 1580}, {1910, 36036}, {3228, 34079}, {4622, 9456}, {23999, 24019}, {33665, 34087}

X(36133) = trilinear pole of line X(31)X(662)
X(36133) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 888}, {6, 9148}, {76, 887}, {512, 538}, {523, 3231}, {661, 2234}, {690, 14609}, {850, 33875}
X(36133) = trilinear product X(i)*X(j) for these {i,j}: {2, 32717}, {6, 9150}, {32, 886}, {99, 729}, {110, 3228}, {691, 14608}, {1576, 34087}
X(36133) = trilinear quotient X(i)/X(j) for these (i,j): (2, 9148), (6, 888), (32, 887), (99, 538), (110, 3231), (662, 2234), (691, 14609), (729, 512), (886, 76), (1576, 33875), (3228, 523), (9150, 2), (14608, 690), (32717, 6), (34087, 850)
X(36133) = barycentric product X(i)*X(j) for these {i,j}: {1, 9150}, {31, 886}, {75, 32717}, {163, 34087}, {662, 3228}, {729, 799}, {14608, 36085}
X(36133) = barycentric quotient X(i)/X(j) for these (i,j): (1, 9148), (31, 888), (110, 2234), (163, 3231), (560, 887), (662, 538), (729, 661), (799, 30736), (886, 561), (3228, 1577), (9150, 75), (32717, 1), (34087, 20948), (36142, 14609)

X(36134) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(3)X(54)

Barycentrics    a^3/((b^2 - c^2) (a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :
Trilinears    csc(2B - 2C) : :

As the trilinear product of circumcircle antipodes, X(36134) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with center X(36033) and perspector X(48).

X(36134) lies on these lines: {11, 2602}, {12, 2601}, {38, 293}, {54, 60}, {109, 933}, {110, 36078}, {820, 2169}, {906, 14586}, {1101, 2616}, {1109, 2619}, {1331, 18315}, {2148, 36060}, {2190, 36053}, {15958, 36059}

X(36134) = isogonal conjugate of X(2618)
X(36134) = cevapoint of X(1) and X(2616)
X(36134) = trilinear pole of line X(47)X(48)
X(36134) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 2618}, {2, 12077}, {3, 23290}, {4, 6368}, {5, 523}, {6, 18314}, {32, 15415}, {51, 850}, {53, 525}, {115, 14570}, {216, 14618}, {264, 15451}, {311, 512}, {324, 647}, {338, 1625}, {343, 2501}, {520, 13450}, {661, 14213}, {1109, 2617}, {1577, 1953}, {2181, 14208}, {2489, 28706}, {3199, 3267}
X(36134) = trilinear product X(i)*X(j) for these {i,j}: {2, 14586}, {3, 933}, {4, 15958}, {6, 18315}, {54, 110}, {95, 1576}, {97, 112}, {107, 19210}, {109, 35196}, {162, 2169}, {163, 2167}, {184, 18831}, {249, 2623}, {275, 32661}, {648, 14533}, {662, 2148}, {1101, 2616}, {2190, 4575}, {4558, 8882}, {15412, 23357}
X(36134) = trilinear quotient X(i)/X(j) for these (i,j): (1, 2618), (2, 18314), (3, 6368), (4, 23290), (6, 12077), (54, 523), (76, 15415), (95, 850), (97, 525), (99, 311), (107, 13450), (110, 5), (112, 53), (163, 1953), (184, 15451), (249, 14570), (275, 14618), (648, 324), (662, 14213), (933, 4), (1101, 2617), (1576, 51), (2148, 661), (2167, 1577), (2169, 656), (2190, 24006), (2616, 1109), (2623, 115), (4282, 2600), (4558, 343), (4563, 28706), (4592, 18695), (8882, 2501), (14533, 647), (14586, 6), (15412, 338), (15958, 3), (18315, 2), (18831, 264), (19210, 520), (23357, 1625), (32661, 216), (32676, 2181), (34386, 3267), (35196, 522)
X(36134) = barycentric product X(i)*X(j) for these {i,j}: {1, 18315}, {48, 18831}, {54, 662}, {63, 933}, {75, 14586}, {92, 15958}, {95, 163}, {97, 162}, {99, 2148}, {110, 2167}, {249, 2616}, {275, 4575}, {648, 2169}, {651, 35196}, {811, 14533}, {823, 19210}, {1101, 15412}, {2190, 4558}, {2623, 24041}, {4592, 8882}, {32676, 34386}
X(36134) = barycentric quotient X(i)/X(j) for these (i,j): (1, 18314), (6, 2618), (19, 23290), (31, 12077), (48, 6368), (54, 1577), (75, 15415), (95, 20948), (97, 14208), (110, 14213), (162, 324), (163, 5), (249, 343), (662, 311), (933, 92), (1101, 14570), (1576, 1953), (2148, 523), (2167, 850), (2169, 525), (2190, 14618), (2616, 338), (2623, 1109), (4558, 18695), (4592, 28706), (8882, 24006), (9247, 15451), (14533, 656), (14586, 1), (15412, 23994), (15958, 63), (18315, 75), (18831, 1969), (19210, 24018), (23357, 2617), (23995, 1625), (24019, 13450), (32676, 53), (35196, 4391)

X(36135) = TRILINEAR PRODUCT X(102)*X(26715)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36135) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36135) lies on these lines: {692, 32720}, {1415, 36040}, {26715, 32674}

X(36135) = trilinear pole of line X(31)X(32677)
X(36135) = trilinear product X(i)*X(j) for these {i,j}: {2, 32720}, {102, 26715}
X(36135) = trilinear quotient X(i)/X(j) for these (i,j): (26715, 515), (32720, 6)
X(36135) = barycentric product X(i)*X(j) for these {i,j}: {75, 32720}, {26715, 36100}
X(36135) = barycentric quotient X(32720)/X(1)

X(36136) = TRILINEAR PRODUCT X(103)*X(26716)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36136) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36136) lies on these lines: {101, 677}, {692, 32721}, {1461, 24016}

X(36136) = trilinear pole of line X(31)X(911)
X(36136) = X(676)-isoconjugate of X(29616)
X(36136) = trilinear product X(i)*X(j) for these {i,j}: {2, 32721}, {103, 26716}
X(36136) = trilinear quotient X(i)/X(j) for these (i,j): (677, 29616), (24016, 10004), (26716, 516), (32721, 6), (36039, 5223)
X(36136) = barycentric product X(i)*X(j) for these {i,j}: {75, 32721}, {911, 32040}, {26716, 36101}
X(36136) = barycentric quotient X(i)/X(j) for these (i,j): (32642, 5223), (32668, 10004), (32721, 1), (36039, 29616)

X(36137) = TRILINEAR PRODUCT X(104)*X(32722)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36137) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36137) lies on these lines: {101, 36037}, {692, 32641}, {1415, 2720}, {9456, 10428}, {32674, 36110}

X(36137) = trilinear pole of line X(31)X(909)
X(36137) = X(956)-isoconjugate of X(10015)
X(36137) = trilinear product X(i)*X(j) for these {i,j}: {2, 32723}, {104, 32722}, {957, 32641}
X(36137) = trilinear quotient X(i)/X(j) for these (i,j): (957, 10015), (32641, 956), (32722, 517), (32723, 6)
X(36137) = barycentric product X(i)*X(j) for these {i,j}: {75, 32723}, {957, 36037}, {32722, 34234}
X(36137) = barycentric quotient X(i)/X(j) for these (i,j): (957, 36038), (32722, 908), (32723, 1)

X(36138) = TRILINEAR PRODUCT X(105)*X(8693)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36138) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36138) lies on these lines: {101, 4794}, {692, 919}, {1415, 32735}, {1438, 9454}

X(36138) = trilinear pole of line X(31)X(1438)
X(36138) = X(i)-isoconjugate of X(j) for these {i,j}: {518, 4762}, {665, 4441}, {918, 1001}, {2254, 4384}, {2481, 33570}, {3912, 4724}
X(36138) = trilinear product X(i)*X(j) for these {i,j}: {2, 32724}, {105, 8693}, {919, 1002}, {2279, 36086}, {27475, 32666}
X(36138) = trilinear quotient X(i)/X(j) for these (i,j): (105, 4762), (666, 4441), (919, 1001), (1002, 918), (1438, 4724), (2223, 33570), (2279, 2254), (8693, 518), (32041, 3263), (32666, 2280), (32724, 6), (32735, 5228), (36086, 4384)
X(36138) = barycentric product X(i)*X(j) for these {i,j}: {75, 32724}, {666, 2279}, {673, 8693}, {919, 27475}, {1002, 36086}, {1438, 32041}
X(36138) = barycentric quotient X(i)/X(j) for these (i,j): (666, 21615), (919, 4384), (1438, 4762), (2279, 918), (8693, 3912), (9454, 33570), (32666, 1001), (32724, 1), (36086, 4441)

X(36139) = TRILINEAR PRODUCT X(107)*X(26717)

Barycentrics    a/((b^2 - c^2) (a^2 - b^2 - c^2)^2 (a^6 (b^2 + c^2) - 2 a^4 (b^4 - b^2 c^2 + c^4) + a^2 (b^2 - c^2)^2 (b^2 + c^2) - 2 b^2 c^2 (b^2 - c^2)^2)) : :
Barycentrics    a^3/((csc 2B - csc 2C) (2 csc 2A - csc 2B - csc 2C)) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36139) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36139) lies on these lines: {163, 24000}, {662, 23999}, {692, 32725}, {23348, 34030}, {24019, 24021}, {26717, 36068}

X(36139) = trilinear pole of line X(31)X(24019)
X(36139) = X(i)-isoconjugate of X(j) for these {i,j}: {3265, 3331}, {24018, 32713}
X(36139) = trilinear product X(i)*X(j) for these {i,j}: {2, 32725}, {107, 26717}
X(36139) = trilinear quotient X(i)/X(j) for these (i,j): (24019, 32713), (26717, 520), (32713, 3331), (32725, 6)
X(36139) = barycentric product X(i)*X(j) for these {i,j}: {75, 32725}, {823, 26717}
X(36139) = barycentric quotient X(i)/X(j) for these (i,j): (26717, 24018), (32725, 1)

X(36140) = TRILINEAR PRODUCT X(108)*X(32726)

Barycentrics    a^2/((b - c) (a - b - c) (a^2 - b^2 - c^2) (a^4 (b + c) - a^3 (b^2 + c^2) - a^2 (b - c)^2 (b + c) + a (b^2 - c^2)^2 - 2 b c (b - c)^2 (b + c))) : :
Barycentrics    a^3/((sec B - sec C) (2 sec A - sec B - sec C)) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36140) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36140) lies on these lines: {101, 7012}, {692, 7115}, {909, 36110}, {8122, 18340}, {23707, 36093}, {24033, 32674}, {32677, 32726}

X(36140) = isogonal conjugate of isotomic conjugate of isogonal conjugate of X(2637)
X(36140) = trilinear pole of line X(31)X(32674)
X(36140) = X(i)-isoconjugate of X(j) for these {i,j}: {75, 2637}, {2635, 6332}
X(36140) = trilinear product X(i)*X(j) for these {i,j}: {2, 32727}, {108, 32726}, {653, 34078}, {23707, 32674}
X(36140) = trilinear quotient X(i)/X(j) for these (i,j): (31, 2637), (23707, 6332), (32674, 2635), (32726, 521), (32727, 6), (34078, 652)
X(36140) = barycentric product X(i)*X(j) for these {i,j}: {75, 32727}, {108, 23707}, {653, 32726}, {18026, 34078}
X(36140) = barycentric quotient X(i)/X(j) for these (i,j): (32, 2637), (23707, 35518), (32726, 6332), (32727, 1), (34078, 521)

X(36141) = TRILINEAR PRODUCT X(109)*X(2291)

Barycentrics    a^3/((b - c) (a - b - c) (a (2 a - b - c) - (b - c)^2)) : :
Barycentrics    a^3/((cos B - cos C) (2 cos A - cos B - cos C)) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36141) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36141) lies on these lines: {59, 101}, {649, 1461}, {662, 1021}, {692, 2149}, {909, 2272}, {911, 7113}, {1024, 36146}, {1156, 36094}, {1404, 1438}, {1415, 3063}, {2224, 34056}, {4586, 35157}, {32677, 36040}

X(36141) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(1)X(651)
X(36141) = trilinear pole of line X(31)X(1415)
X(36141) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 6366}, {8, 1638}, {76, 6139}, {312, 14413}, {514, 6745}, {522, 527}, {650, 30806}, {693, 6603}, {1155, 4391}), {1323, 3239}, {6332, 23710}, {23346, 23978}
X(36141) = trilinear product X(i)*X(j) for these {i,j}: {2, 32728}, {6, 14733}, {32, 35157}, {109, 2291}, {651, 34068}, {692, 34056}, {934, 18889}, {1156, 1415}, {1262, 23351}, {1461, 4845}, {23893, 24027}, {32735, 36146}
X(36141) = trilinear quotient X(i)/X(j) for these (i,j): (6, 6366), (32, 6139), (56, 1638), (101, 6745), (109, 527), (604, 14413), (651, 30806), (692, 6603), (1156, 4391), (1415, 1155), (1461, 1323), (2291, 522), (4845, 3239), (14733, 2), (18889, 3900), (23351, 1146), (23893, 24026), (23979, 23346), (32674, 23710), (32728, 6), (34056, 693), (34068, 650), (35157, 76)
X(36141) = barycentric product X(i)*X(j) for these {i,j}: {1, 14733}, {31, 35157}, {75, 32728}, {101, 34056}, {109, 1156}, {651, 2291}, {658, 18889}, {664, 34068}, {934, 4845}, {1121, 1415}, {1262, 23893}, {7045, 23351}
X(36141) = barycentric quotient X(i)/X(j) for these (i,j): (31, 6366), (109, 30806), (560, 6139), (604, 1638), (692, 6745), (1156, 35519), (1397, 14413), (1415, 527), (2291, 4391), (4845, 4397), (14733, 75), (18889, 3239), (23351, 24026), (23893, 23978), (32728, 1), (32739, 6603), (34056, 3261), (34068, 522), (35157, 561)

X(36141) = isogonal conjugate of isotomic conjugate of X(37139)

X(36142) = TRILINEAR PRODUCT X(110)*X(111)

Barycentrics a^3/((b^2 - c^2) (2 a^2 - b^2 - c^2)) : :
Barycentrics a^3/((cot B - cot C) (2 cot A - cot B - cot C)) : :
Barycentrics b c (SB + SC)^2/((SB - SC) (2 SA - SB - SC)) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36142) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

Let A1B1C1 and A2B2C2 be the 1st and 2nd Parry triangles. Let A' be the trilinear product A1*A2, and define B', C' cyclically. The lines AA', BB', CC' concur in X(36142).

X(36142) lies on these lines: {1, 36150}, {101, 691}, {110, 36070}, {111, 34079}, {162, 36115}, {163, 798}, {661, 662}, {671, 33665}, {692, 32729}, {892, 4586}, {897, 1910}, {922, 923}, {1755, 2159}, {4575, 36045}, {9274, 32671}, {18268, 32740}, {23894, 32678}, {24000, 24019}, {36114, 36128}

X(36142) = isogonal conjugate of isotomic conjugate of X(36085)
X(36142) = trilinear pole of line X(31)X(163)
X(36142) = trilinear product of PU(62)
X(36142) = trilinear product of circumcircle intercepts of Parry circle
X(36142) = barycentric product of circumcircle intercepts of line X(1)X(662)
X(36142) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 690}, {4, 14417}, {6, 35522}, {10, 4750}, {13, 9204}, {14, 9205}, {75, 2642}, {76, 351}, {115, 5468}, {125, 4235}, {187, 850}, {321, 14419}, {338, 5467}, {468, 525}, {512, 3266}, {514, 4062}, {523, 524}, {661, 14210}, {671, 1649}, {693, 21839}, {896, 1577}, {1109, 23889}, {2394, 5642}, {2501, 6390}, {2643, 24039}, {2799, 5967}, {3292, 14618}
X(36142) = trilinear product X(i)*X(j) for these {i,j}: {2, 32729}, {6, 691}, {15, 9206}, {16, 9207}, {31, 36085}, {32, 892}, {99, 32740}, {110, 111}, {112, 895}, {162, 36060}, {163, 897}, {187, 34574}, {249, 9178}, {250, 10097}, {648, 14908}, {662, 923}, {670, 19626}, {671, 1576}, {1101, 23894}, {1333, 5380}, {1992, 32648}, {2420, 9139}, {2715, 5968}, {4558, 8753}, {4575, 36128}, {5466, 23357}, {9214, 32640}, {11634, 15387}, {14574, 18023}, {17983, 32661}
X(36142) = trilinear quotient X(i)/X(j) for these (i,j): (2, 35522), (3, 14417), (6, 690), (15, 9204), (16, 9205), (31, 2642), (32, 351), (58, 4750), (99, 3266), (101, 4062), (110, 524), (111, 523), (112, 468), (163, 896), (187, 1649), (249, 5468), (250, 4235), (662, 14210), (671, 850), (691, 2), (692, 21839), (892, 76), (895, 525), (897, 1577), (923, 661), (1101, 23889), (1333, 14419), (1576, 187), (2420, 5642), (2715, 5967), (4558, 6390), (5380, 321), (5466, 338), (5968, 2799), (8753, 2501), (9139, 2394), (9178, 115), (9206, 13), (9207, 14), (10097, 125), (11634, 126), (14574, 14567), (14908, 647), (14977, 339), (17983, 14618), (19626, 669), (23357, 5467), (23894, 1109), (24041, 24039), (30786, 3267), (32640, 9717), (32648, 21448), (32661, 3292), (32729, 6), (32740, 512), (34574, 671), (36060, 656), (36085, 75), (36128, 24006)
X(36142) = barycentric product X(i)*X(j) for these {i,j}: {1, 691}, {6, 36085}, {31, 892}, {58, 5380}, {75, 32729}, {99, 923}, {110, 897}, {111, 662}, {162, 895}, {163, 671}, {249, 23894}, {648, 36060}, {799, 32740}, {811, 14908}, {1101, 5466}, {1992, 36045}, {4558, 36128}, {4575, 17983}, {4592, 8753}, {4602, 19626}, {5968, 36084}, {9178, 24041}, {9214, 36034}, {14609, 36133}, {30786, 32676}
X(36142) = barycentric quotient X(i)/X(j) for these (i,j): (1, 35522), (31, 690), (32, 2642), (48, 14417), (110, 14210), (111, 1577), (163, 524), (249, 24039), (560, 351), (662, 3266), (671, 20948), (691, 75), (692, 4062), (798, 1648), (892, 561), (895, 14208), (897, 850), (922, 1649), (923, 523), (1101, 5468), (1333, 4750), (1576, 896), (2151, 9204), (2152, 9205), (2206, 14419), (4575, 6390), (5380, 313), (5466, 23994), (8753, 24006), (9178, 1109), (10097, 20902), (14574, 922), (14908, 656), (19626, 798), (23357, 23889), (23894, 338), (23995, 5467), (32676, 468), (32729, 1), (32739, 21839), (32740, 661), (36045, 5485), (36060, 525), (36085, 76), (36128, 14618)

X(36143) = TRILINEAR PRODUCT X(476)*X(32730)

Barycentrics    a/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2) (a^6 (b^2 + c^2) - 2 a^4 (b^4 + c^4) + a^2 (b^6 + c^6) - 2 b^2 c^2 (b^2 - c^2)^2)) : :
Barycentrics    a^2/(sin(B - C) (1 + 2 cos 2A) (b sin(C - A) (1 + 2 cos 2B) - c sin(A - B) (1 + 2 cos 2C))) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36143) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36143) lies on these lines: {692, 32731}, {36047, 36151}

X(36143) = trilinear pole of line X(31)X(32678)
X(36143) = X(3016)-isoconjugate of X(3268)
X(36143) = trilinear product X(i)*X(j) for these {i,j}: {2, 32731}, {476, 32730}
X(36143) = trilinear quotient X(i)/X(j) for these (i,j): (14560, 3016), (32730, 526), (32731, 6)
X(36143) = barycentric product X(i)*X(j) for these {i,j}: {75, 32731}, {32680, 32730}
X(36143) = barycentric quotient X(i)/X(j) for these (i,j): (32730, 32679), (32731, 1)

X(36144) = TRILINEAR PRODUCT X(477)*X(32732)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36144) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36144) lies on these lines: {692, 32733}, {32678, 36047}, {36117, 36131}

X(36144) = trilinear pole of line X(31)X(36151)
X(36144) = trilinear product X(i)*X(j) for these {i,j}: {2, 32733}, {477, 32732}
X(36144) = trilinear quotient X(i)/X(j) for these (i,j): (32732, 5663), (32733, 6)
X(36144) = barycentric product X(i)*X(j) for these {i,j}: {75, 32733}, {32732, 36102}
X(36144) = barycentric quotient X(32733)/X(1)

X(36145) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(5)X(6)

Barycentrics    a/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)) : :
Barycentrics    a^3/(sec 2B - sec 2C) : :
Trilinears    a/(b cos(A - B) - c cos(A - C)) : :

As the trilinear product of circumcircle-X(5)-antipodes, X(36145) lies on conic {{A,B,C,X(162),X(1956),X(2166),X(2222),X(2617)}} with perspector X(1953), and as the trilinear product of circumcircle-X(6)-antipodes, X(36145) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36145) lies on these lines: {91, 1910}, {101, 925}, {163, 2617}, {284, 913}, {610, 1820}, {692, 32734}, {2165, 34079}, {2166, 2168}, {2222, 32692}, {5392, 33665}

X(36145) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(1)X(91)
X(36145) = isogonal conjugate of polar conjugate of trilinear pole of line X(19)X(91)
X(36145) = trilinear pole of line X(31)X(1820)
X(36145) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 924}, {6, 6563}, {24, 525}, {47, 1577}, {52, 15412}, {69, 6753}, {76, 34952}, {264, 30451}, {317, 647}, {512, 7763}, {520, 11547}, {523, 1993}, {571, 850}, {656, 1748}, {1147, 14618}, {2501, 9723}, {3265, 8745}
X(36145) = trilinear product X(i)*X(j) for these {i,j}: {2, 32734}, {5, 32692}, {6, 925}, {68, 112}, {91, 163}, {96, 1625}, {110, 2165}, {162, 1820}, {184, 30450}, {648, 2351}, {847, 32661}, {1576, 5392}, {2168, 2617}, {4558, 14593}, {5962, 32662}, {6529, 16391}
X(36145) = trilinear quotient X(i)/X(j) for these (i,j): (2, 6563), (6, 924), (25, 6753), (32, 34952), (68, 525), (91, 1577), (96, 15412), (99, 7763), (107, 11547), (110, 1993), (112, 24), (162, 1748), (163, 47), (184, 30451), (648, 317), (847, 14618), (925, 2), (1576, 571), (1625, 52), (1820, 656), (2165, 523), (2168, 2616), (2351, 647), (4558, 9723), (5392, 850), (14593, 2501), (20563, 3267), (20571, 20948), (30450, 264), (32661, 1147), (32662, 5961), (32692, 54), (32713, 8745), (32734, 6)
X(36145) = barycentric product X(i)*X(j) for these {i,j}: {1, 925}, {48, 30450}, {68, 162}, {75, 32734}, {91, 110}, {96, 2617}, {163, 5392}, {648, 1820}, {662, 2165}, {811, 2351}, {847, 4575}, {1576, 20571}, {2168, 14570}, {4592, 14593}, {5962, 36061}, {14213, 32692}, {16391, 36126}, {20563, 32676}
X(36145) = barycentric quotient X(i)/X(j) for these (i,j): (1, 6563), (31, 924), (68, 14208), (91, 850), (112, 1748), (162, 317), (163, 1993), (560, 34952), (662, 7763), (925, 75), (1576, 47), (1820, 525), (1973, 6753), (2165, 1577), (2168, 15412), (2351, 656), (4575, 9723), (5392, 20948), (9247, 30451), (14593, 24006), (24019, 11547), (30450, 1969), (32676, 24), (32692, 2167), (32734, 1)

X(36146) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(7)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36146) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31), and as the trilinear product of circumcircle-X(7)-antipodes, X(36146) also lies on conic {{A,B,C,X(651),X(664),X(934)}} with center X(223) and perspector X(57).

X(36146) lies on these lines: {59, 513}, {101, 514}, {105, 1319}, {163, 1019}, {241, 294}, {515, 14942}, {649, 7045}, {662, 4620}, {673, 909}, {876, 2283}, {919, 934}, {1022, 32665}, {1024, 36141}, {1025, 5377}, {1027, 32666}, {1429, 1438}, {1461, 4626}, {1462, 6610}, {1814, 32677}, {2201, 5236}, {2224, 34018}, {2402, 2406}, {4394, 9358}, {4551, 35333}, {4872, 6996}, {7128, 32674}, {15726, 28071}, {34078, 36057}

X(36146) = isogonal conjugate of isotomic conjugate of X(34085)
X(36146) = trilinear pole of line X(31)X(57)
X(36146) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 926}, {8, 665}, {9, 2254}, {11, 2284}, {55, 918}, {76, 927}, {100, 17435}, {241, 3900}, {294, 3126}, {513, 3693}, {514, 2340}, {518, 650}, {521, 5089}, {522, 672}, {644, 3675}, {649, 3717}, {652, 1861}, {657, 9436}, {663, 3912}, {883 14936}, {884, 4437}, {885, 6184}, {1024, 4712}, {1025, 2310}, {1026, 2170}, {1146, 2283}, {1458, 3239}, {1818, 3064}, {2223, 4391}, {2356, 6332}, {3063, 3263}, {4130, 34855}, {8641, 27818}, {18344, 25083}
X(36146) = trilinear product X(i)*X(j) for these {i,j}: {2, 32735}, {6, 927}, {7, 919}, {31, 34085}, {32, 926}, {56, 666}, {57, 36086}, {85, 32666}, {100, 1462}, {105, 651}, {108, 1814}, {109, 673}, {190, 1416}, {294, 934}, {653, 36057}, {658, 2195}, {664, 1438}, {692, 34018}, {884, 1275}, {885, 1262}, {1024, 7045}, {1027, 4564}, {1415, 2481}, {1461, 14942}, {1813, 36124}, {2283, 6185}, {3669, 5377}, {4617, 28071}, {6516, 8751}, {6559, 6614}, {7128, 23696}, {7339, 28132}, {18026, 32658}, {31637, 32674}
X(36146) = trilinear quotient X(i)/X(j) for these (i,j): (6, 926), (7, 918), (32, 927), (56, 665), (57, 2254), (59, 2284), (100, 3693), (101, 2340), (105, 650), (108, 5089), (109, 672), (190, 3717), (241, 3126), (294, 3900), (513, 17435), (651, 518), (653, 1861), (658, 9436), (664, 3912), (666, 8), (673, 522), (883, 4437), (884, 14936), (885, 1146), (919, 55), (926, 76), (927, 2), (934, 241), (1024, 2310), (1025, 4712), (1027, 2170), (1262, 2283), (1275, 883), (1415, 2223), (1416, 649), (1438, 663), (1461, 1458), (1462, 513), (1813, 1818), (1814, 521), (2195, 657), (2283, 6184), (2481, 4391), (3669, 3675), (4554, 3263), (4564, 1026), (4569, 27818), (4617, 34855), (5377, 644), (6185, 885), (6516, 25083), (6559, 4163), (7045, 1025), (8751, 18344), (14942, 3239), (23696, 34591), (28071, 4130), (28132, 4081), (31637, 6332), (32658, 1946), (32666, 41), (32674, 2356), (32735, 6), (34018, 693), (34085, 75), (36057, 652), (36086, 9), (36118, 5236), (36124, 3064), (36141, 32735)
X(36146) = barycentric product X(i)*X(j) for these {i,j}: {1, 927}, {6, 34085}, {7, 36086}, {57, 666}, {75, 32735}, {85, 919}, {101, 34018}, {105, 664}, {108, 31637}, {109, 2481}, {190, 1462}, {294, 658}, {651, 673}, {653, 1814}, {668, 1416}, {885, 7045}, {934, 14942}, {1024, 1275}, {1025, 6185}, {1027, 4998}, {1415, 18031}, {1438, 4554}, {2195, 4569}, {3676, 5377}, {4617, 6559}, {4626, 28071}, {6063, 32666}, {6516, 36124}, {6604, 36041}, {18026, 36057}
X(36146) = barycentric quotient X(i)/X(j) for these (i,j): (31, 926), (56, 2254), (57, 918), (59, 1026), (100, 3717), (101, 3693), (105, 522), (108, 1861), (109, 518), (294, 3239), (604, 665), (649, 17435), (651, 3912), (658, 27818), (664, 3263), (666, 312), (673, 4391), (692, 2340), (884, 2310), (885, 24026), (919, 9), (927, 75), (934, 9436), (1024, 1146), (1025, 4437), (1027, 11), (1262, 1025), (1415, 672), (1416, 513), (1438, 650), (1458, 3126), (1461, 241), (1462, 514), (1813, 25083), (1814, 6332), (2149, 2284), (2195, 3900), (2283, 4712), (2481, 35519), (5377, 3699), (6614, 34855), (7045, 883), (8751, 3064), (14942, 4397), (23696, 2968), (24027, 2283), (28071, 4163), (31637, 35518), (32658, 652), (32666, 55), (32674, 5089), (32714, 5236), (32735, 1), (34018, 3261), (34085, 76), (36041, 6601), (36057, 521), (36086, 8)

X(36147) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(8)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36147) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31), and as the trilinear product of circumcircle-X(8)-antipodes, X(36147) lies on conic {{A,B,C,X(100),X(664)}} with center X(1) and perspector X(9).

X(36147) lies on these lines: {100, 1415}, {101, 3699}, {163, 643}, {205, 341}, {644, 692}, {662, 4033}, {664, 1461}, {830, 831}, {909, 2359}, {911, 1791}, {950, 1220}, {961, 1280}, {1100, 1320}, {1897, 32674}, {2224, 30710}, {2363, 18268}, {3903, 4559}, {6740, 14624}, {8851, 34077}, {15420, 17136}, {32665, 35342}, {32669, 36037}

X(36147) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(1)X(312)
X(36147) = isotomic conjugate of X(4509)
X(36147) = trilinear pole of line X(9)X(31)
X(36147) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 6371}, {6, 3004}, {31, 4509}, 56, 3910}, {57, 17420}, {58, 21124}, {244, 3882}, {512, 16705}, {513, 3666}, {514, 1193}, {649, 4357}, {667, 20911}, {693, 2300}, {798, 16739}, {905, 1829}, {960, 3669}, {1019, 2292}, {1211, 3733}, {1459, 1848}, {2092, 7192}, {2269, 3676}, {2354, 4025}, {4017, 17185}, {4267, 7178}
X(36147) = trilinear product X(i)*X(j) for these {i,j}: {2, 32736}, {6, 8707}, {8, 8687}, {9, 36098}, {55, 6648}, {82, 35334}, {100, 2298}, {101, 1220}, {110, 14624}, {644, 961}, {692, 30710}, {1018, 2363}, {1169, 3952}, {1240, 32739}, {1252, 4581}, {1783, 1791}, {1897, 2359}, {4557, 14534}
X(36147) = trilinear quotient X(i)/X(j) for these (i,j): (2, 3004), (6, 6371), (8, 3910), (9, 17420), (10, 21124), (75, 4509), (99, 16705), (100, 3666), (101, 1193), (190, 4357), (643, 17185), (644, 960), (668, 20911), (692, 2300), (765, 3882), (799, 16739), (961, 3669), (1018, 2292), (1169, 3733), (1220, 514), (1240, 3261), (1783, 1829), (1791, 905), (1897, 1848), (2298, 513), (2359, 1459), (2363, 1019), (3939, 2269), (3699, 3687), (3952, 1211), (4033, 18697), (4557, 2092), (4581, 1086), (5546, 4267), (6648, 7), (8687, 56), (8707, 2), (8750, 2354), (14534, 7192), (14624, 523), (15420, 1565), (27808, 1228), (30710, 693), (32736, 6), (35334, 38), (36098, 57)
X(36147) = barycentric product X(i)*X(j) for these {i,j}: {1, 8707}, {8, 36098}, {9, 6648}, {75, 32736}, {83, 35334}, {100, 1220}, {101, 30710}, {190, 2298}, {312, 8687}, {662, 14624}, {692, 1240}, {765, 4581}, {961, 3699}, {1018, 14534}, {1169, 4033}, {1791, 1897}, {2359, 6335}, {2363, 3952}, {3939, 31643}
X(36147) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3004), (2, 4509), (9, 3910), (31, 6371), (37, 21124), (55, 17420), (99, 16739), (100, 4357), (101, 3666), (190, 20911), (644, 3687), (662, 16705), (692, 1193), (961, 3676), (1018, 1211), (1169, 1019), (1220, 693), (1252, 3882), (1783, 1848), (1791, 4025), (2298, 514), (2359, 905), (2363, 7192), (3939, 960), (3952, 18697), (4033, 1228), (4557, 2292), (4581, 1111), (5546, 17185), (6648, 85), (8687, 57), (8707, 75), (8750, 1829), (14534, 7199), (14624, 1577), (30710, 3261), (32736, 1), (32739, 2300), (35334, 141), (36098, 7)

X(36148) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF NAPOLEON AXIS

Barycentrics    a/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2)) : :
Barycentrics    a^2/((3 sin^2 A - cos^2 A)(cos B sin C - sin B cos C)) : :
Barycentrics    a/(sin^2 A sin(2B - 2C) - sin^2 B sin(2C - 2A) - sin^2 C sin(2A - 2B)) : :
Barycentrics    csc(B - C)/(3 - cot^2 A) : :
Trilinears    1/[directed distance from A to Napoleon axis] : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36148) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36148) lies on these lines: {101, 930}, {692, 32737}, {1910, 2962}, {2963, 34079}, {11140, 33665}

X(36148) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 1510}, {54, 20577}, {143, 15412}, {512, 7769}, {523, 1994}, {525, 3518}, {647, 32002}, {850, 2965}, {1577, 2964}, {18314, 25044}
X(36148) = trilinear product X(i)*X(j) for these {i,j}: {2, 32737}, {6, 930}, {17, 16807}, {18, 16806}, {110, 2963}, {112, 3519}, {163, 2962}, {252, 1625}, {1576, 11140}, {14586, 25043}
X(36148) = trilinear quotient X(i)/X(j) for these (i,j): (5, 20577), (6, 1510), (99, 7769), (110, 1994), (112, 3518), (163, 2964), (252, 15412), (648, 32002), (930, 2), (1576, 2965), (1625, 143), (2962, 1577), (2963, 523), (3519, 525), (11140, 850), (14586, 25044), (16806, 62), (16807, 61), (25043, 18314), (32737, 6)
X(36148) = barycentric product X(i)*X(j) for these {i,j}: {1, 930}, {75, 32737}, {110, 2962}, {162, 3519}, {163, 11140}, {252, 2617}, {662, 2963}
X(36148) = barycentric quotient X(i)/X(j) for these (i,j): (31, 1510), (162, 32002), (163, 1994), (662, 7769), (930, 75), (1576, 2964), (1953, 20577), (2962, 850), (2963, 1577), (3519, 14208), (11140, 20948), (32676, 3518), (32737, 1)

X(36149) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(30)

Barycentrics    a/((b^2 - c^2) (a^4 - 2 a^2 (b^2 + c^2) + b^4 + 4 b^2 c^2 + c^4)) : :
Trilinears    a/((cos A) (sin 2B - sin 2C) + cos B sin C - sin B cos C) : :

As the trilinear product of circumcircle-X(6)-antipodes, X(36149) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31), and as the trilinear product of circumcircle-X(30)-antipodes, X(36149) lies on conic {{A,B,C,X(1),X(162),X(26700)}} with perspector X(2173).

X(36149) lies on these lines: {1, 2159}, {101, 1302}, {162, 36083}, {692, 32738}, {26700, 32681}, {33665, 34289}, {34079, 34288}

X(36149) = trilinear pole of line X(31)X(2173)
X(36149) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 8675}, {6, 30474}, {378, 525}, {512, 32833}, {523, 15066}, {850, 5063}
X(36149) = trilinear product X(i)*X(j) for these {i,j}: {2, 32738}, {6, 1302}, {30, 32681}, {110, 34288}, {112, 4846}, {1576, 34289}, {2173, 36083}
X(36149) = trilinear quotient X(i)/X(j) for these (i,j): (2, 30474), (6, 8675), (99, 32833), (110, 15066), (112, 378), (1302, 2), (1576, 5063), (4846, 525), (32681, 74), (32738, 6), (34288, 523), (34289, 850), (36083, 2349)
X(36149) = barycentric product X(i)*X(j) for these {i,j}: {1, 1302}, {30, 36083}, {75, 32738}, {162, 4846}, {163, 34289}, {662, 34288}, {14206, 32681}
X(36149) = barycentric quotient X(i)/X(j) for these (i,j): (1, 30474), (31, 8675), (163, 15066), (662, 32833), (1302, 75), (4846, 14208), (32676, 378), (32681, 2349), (32738, 1), (34288, 1577), (34289, 20948), (36083, 1494)

X(36150) = TRILINEAR PRODUCT OF CIRCUMCIRCLE INTERCEPTS OF LINE X(6)X(690)

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

As the trilinear product of circumcircle-X(6)-antipodes, X(36150) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36150) lies on these lines: {1, 36142}, {101, 2770}, {163, 896}, {661, 923}, {662, 14210}, {692, 21839}

X(36150) = trilinear pole of line X(31)X(2642)
X(36150) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 2854}, {671, 9177}
X(36150) = trilinear product X(i)*X(j) for these {i,j}: {2, 32741}, {6, 2770}
X(36150) = trilinear quotient X(i)/X(j) for these (i,j): (6, 2854), (187, 9177), (2770, 2), (32741, 6)
X(36150) = barycentric product X(i)*X(j) for these {i,j}: {1, 2770}, {75, 32741}
X(36150) = barycentric quotient X(i)/X(j) for these (i,j): (31, 2854), (922, 9177), (2770, 75), (32741, 1)

X(36151) = TRILINEAR PRODUCT X(6)*X(477)

Barycentrics    a/(a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4)) : :
Trilinears    a/(4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)) : :

X(36151) is the trilinear product of the circumcircle intercepts of line X(6)X(1637). As the trilinear product of circumcircle-X(6)-antipodes, X(36151) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31).

X(36151) lies on these lines: {19, 36131}, {48, 32678}, {101, 477}, {163, 2173}, {661, 2159}, {662, 14206}, {1415, 32663}, {12211, 20977}, {24019, 36130}, {36047, 36143}

X(36151) = polar conjugate of isotomic conjugate of X(36062)
X(36151) = X(i)-isoconjugate of X(j) for these {i,j}: {2, 5663}, {6, 35520}, {63, 36063}, {526, 2410}, {2437, 3268}
X(36151) = trilinear product X(i)*X(j) for these {i,j}: {4, 32663}, {6, 477}, {19, 36062}, {31, 36102}, {48, 36130}, {476, 2436}, {526, 32650}, {2411, 14560}, {2624, 36047}, {2631, 36117}
X(36151) = trilinear quotient X(i)/X(j) for these (i,j): (2, 35520), (6, 5663), (19, 36063), (476, 2410), (477, 2), (2411, 3268), (2436, 526), (14560, 2437), (32650, 476), (32663, 3), (36047, 32680), (36062, 63), (36102, 75), (36130, 92)
X(36151) = barycentric product X(i)*X(j) for these {i,j}: {1, 477}, {3, 36130}, {4, 36062}, {6, 36102}, {92, 32663}, {526, 36047}, {661, 30528}, {2411, 32678}, {2436, 32680}, {32650, 32679}
X(36151) = barycentric quotient X(i)/X(j) for these (i,j): (1, 35520), (25, 36063), (31, 5663), (477, 75), (2436, 32679), (30528, 799), (32650, 32680), (32663, 63), (32678, 2410), (36047, 35139), (36062, 69), (36102, 76), (36130, 264)

X(36152) = X(1)X(3)∩X(47)X(73)

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

See Kadir Altintas and César Lozada, Euclid 469 .

X(36152) lies on the conics {{A, B, C, X(40), X(20419)}}, {{A, B, C, X(65), X(7163)}} and these lines: {1, 3}, {8, 27086}, {20, 10058}, {21, 1478}, {22, 3011}, {24, 225}, {47, 73}, {80, 11500}, {90, 1490}, {140, 26481}, {186, 1068}, {199, 29658}, {279, 32624}, {283, 4278}, {388, 6875}, {404, 26363}, {405, 7951}, {411, 1479}, {495, 5428}, {497, 6876}, {498, 1006}, {499, 6905}, {579, 2302}, {601, 4337}, {859, 8185}, {920, 18446}, {993, 24987}, {1001, 5443}, {1011, 29640}, {1012, 10483}, {1066, 22361}, {1072, 10323}, {1259, 5904}, {1324, 13738}, {1376, 5445}, {1433, 20419}, {1609, 8557}, {1626, 7428}, {1727, 15071}, {1737, 6796}, {1756, 7295}, {1781, 2178}, {1838, 14017}, {2006, 10260}, {2594, 5398}, {2932, 12750}, {2933, 16453}, {2939, 21381}, {3145, 30362}, {3149, 7741}, {3157, 6149}, {3560, 3585}, {3583, 6985}, {3584, 28466}, {3651, 4302}, {4188, 10527}, {4189, 4293}, {4191, 33140}, {4210, 11269}, {4225, 5230}, {4297, 17010}, {4299, 6906}, {4311, 5267}, {4996, 10074}, {5248, 12047}, {5251, 10827}, {5259, 11344}, {5292, 16451}, {5298, 10959}, {5396, 16472}, {5427, 10950}, {5433, 6924}, {5450, 21578}, {5541, 8668}, {6097, 13408}, {6636, 26228}, {6734, 25440}, {6827, 10320}, {6863, 8070}, {6914, 7354}, {6928, 8068}, {6942, 7288}, {6950, 10532}, {6987, 10321}, {7485, 29639}, {7489, 10895}, {7508, 18990}, {7972, 12513}, {8553, 8609}, {8618, 23852}, {10056, 21161}, {10087, 12245}, {10523, 31789}, {10529, 17100}, {10573, 11491}, {10587, 17548}, {10590, 16865}, {11237, 28443}, {11334, 23383}, {11499, 18395}, {12114, 15446}, {12943, 13743}, {15325, 26475}, {15646, 16272}, {15654, 20999}, {16064, 29675}, {16293, 25542}, {16370, 34620}, {19524, 25524}, {20470, 20842}, {23710, 32534}, {23850, 28348}

X(36152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7280, 11012), (1, 10268, 5119), (1, 15932, 5902), (3, 1617, 8071), (3, 6585, 11012), (3, 7742, 36), (3, 8069, 35), (35, 36, 46), (36, 14795, 5903), (36, 14798, 1), (36, 21842, 56), (55, 35239, 11010), (56, 10267, 1), (56, 11507, 5902), (65, 24299, 1), (1617, 8071, 5563), (2078, 11012, 1), (3428, 11508, 5697), (5903, 14795, 55), (11249, 11510, 1), (16202, 26437, 1)

X(36153) = ISOGONAL CONJUGATE OF X(34110)

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

See Kadir Altintas and César Lozada, Euclid 470 .

X(36153) lies on the conics {{A, B, C, X(6), X(11817)}}, {{A, B, C, X(54), X(20188)}} and these lines: {2, 32136}, {3, 34567}, {4, 11565}, {5, 542}, {6, 9683}, {24, 5944}, {49, 13363}, {54, 1511}, {110, 15047}, {125, 8254}, {140, 1493}, {143, 2937}, {156, 5422}, {182, 6101}, {184, 15026}, {195, 32142}, {389, 10610}, {399, 11017}, {567, 1986}, {569, 6102}, {632, 34986}, {1154, 1199}, {1173, 5899}, {1209, 32165}, {1593, 13491}, {1598, 19118}, {1614, 13364}, {1656, 11423}, {1994, 10627}, {3292, 16239}, {3523, 13472}, {3526, 11422}, {5050, 7393}, {5446, 15516}, {5622, 19362}, {5663, 13434}, {5876, 7592}, {6146, 22804}, {6636, 13421}, {9703, 15028}, {9704, 15024}, {10095, 11817}, {10540, 18874}, {11245, 34826}, {11591, 15087}, {12007, 32358}, {13368, 19468}, {13391, 14627}, {13561, 14389}, {13564, 16982}, {14449, 22352}, {15004, 17714}, {15012, 15646}, {15018, 18350}, {15019, 18378}, {15043, 32171}, {18128, 33332}

X(36153) = midpoint of X(1199) and X(13353)
X(36153) = reflection of X(1511) in X(27866)
X(36153) = isogonal conjugate of X(34110)
X(36153) = crosspoint of X(250) and X(35311)
X(36153) = X(249)-Ceva conjugate of-X(35324)
X(36153) = X(6)-reciprocal conjugate of-X(34110)
X(36153) = barycentric product X(i)*X(j) for these {i, j}: {140, 34545}, {249, 11792}
X(36153) = barycentric quotient X(6)/X(34110)
X(36153) = trilinear product X(1101)*X(11792)
X(36153) = antipode of X(1511) in conic described at X(27866)
X(36153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 12006, 1511), (54, 15037, 12006), (110, 15047, 32205), (140, 13366, 1493), (5946, 32046, 5944)

X(36154) = ORTHOGONAL PROJECTION OF X(8) ON THE EULER LINE

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

X(36154) lies on the Fuhrmann circle and these lines: {2, 3}, {8, 523}, {110, 6739}, {125, 6740}, {145, 13869}, {664, 21294}, {759, 24916}, {952, 3448}, {1054, 10774}, {1290, 2975}, {1329, 5520}, {1330, 6790}, {1793, 6011}, {3258, 35193}, {5433, 31524}, {5690, 14731}, {6788, 24443}, {10778, 13605}

X(36154) = Euler line intercept, other than X(4), of Fuhrmann circle

X(36155) = ORTHOGONAL PROJECTION OF X(10) ON THE EULER LINE

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

X(36155) lies on these lines: {2, 3}, {8, 13869}, {10, 523}, {125, 952}, {1290, 5260}, {1387, 8286}, {1565, 23674}, {1834, 6788}, {3035, 31845}, {3258, 31841}, {3454, 6789}, {6740, 15059}, {7294, 31524}, {8287, 12019}, {17044, 21253}

X(36156) = ORTHOGONAL PROJECTION OF X(32) ON THE EULER LINE

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

X(36156) lies on these lines: {2, 3}, {32, 523}, {842, 12110}, {935, 10312}, {1632, 14908}, {2452, 30435}, {2453, 3053}, {3111, 16776}, {3455, 7668}, {5099, 7747}, {5969, 6593}, {7745, 16320}, {16316, 18907}

X(36156) = {X(3),X(1316)}-harmonic conjugate of X(36157)

X(36157) = ORTHOGONAL PROJECTION OF X(39) ON THE EULER LINE

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

X(36157) lies on these lines: {2, 3}, {39, 523}, {83, 691}, {141, 5118}, {682, 34978}, {1499, 10568}, {1506, 5099}, {2452, 9605}, {2453, 5013}, {2882, 15118}, {3111, 3589}, {3933, 23342}, {5254, 14609}, {5305, 14700}, {5652, 10097}, {6390, 30736}, {7827, 16092}, {16316, 34235}

X(36157) = {X(3),X(1316)}-harmonic conjugate of X(36156)

X(36158) = ORTHOGONAL PROJECTION OF X(40) ON THE EULER LINE

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

X(36158) lies on these lines: {2, 3}, {40, 523}, {74, 952}, {477, 901}, {517, 13868}, {842, 2737}, {1309, 2693}, {2716, 30264}, {2777, 6739}, {6011, 24466}, {6740, 15055}

X(36159) = ORTHOGONAL PROJECTION OF X(49) ON THE EULER LINE

Barycentrics    2*a^16 - 8*a^14*b^2 + 11*a^12*b^4 - 5*a^10*b^6 - 2*a^6*b^10 + 3*a^4*b^12 - a^2*b^14 - 8*a^14*c^2 + 22*a^12*b^2*c^2 - 21*a^10*b^4*c^2 + 7*a^8*b^6*c^2 + 4*a^6*b^8*c^2 - 6*a^4*b^10*c^2 + a^2*b^12*c^2 + b^14*c^2 + 11*a^12*c^4 - 21*a^10*b^2*c^4 + 12*a^8*b^4*c^4 - 4*a^6*b^6*c^4 + 5*a^4*b^8*c^4 + 3*a^2*b^10*c^4 - 6*b^12*c^4 - 5*a^10*c^6 + 7*a^8*b^2*c^6 - 4*a^6*b^4*c^6 - 4*a^4*b^6*c^6 - 3*a^2*b^8*c^6 + 15*b^10*c^6 + 4*a^6*b^2*c^8 + 5*a^4*b^4*c^8 - 3*a^2*b^6*c^8 - 20*b^8*c^8 - 2*a^6*c^10 - 6*a^4*b^2*c^10 + 3*a^2*b^4*c^10 + 15*b^6*c^10 + 3*a^4*c^12 + a^2*b^2*c^12 - 6*b^4*c^12 - a^2*c^14 + b^2*c^14 : :

X(36159) lies on these lines: {2, 3}, {49, 523}, {54, 476}, {3233, 18350}, {6070, 10116}, {9705, 14480}, {11449, 15111}, {11464, 15112}, {12038, 14934}, {13403, 25641}

X(36160) = ORTHOGONAL PROJECTION OF X(52) ON THE EULER LINE

Barycentrics    2*a^14*b^2 - 9*a^12*b^4 + 15*a^10*b^6 - 10*a^8*b^8 + 3*a^4*b^12 - a^2*b^14 + 2*a^14*c^2 - 6*a^12*b^2*c^2 + 7*a^10*b^4*c^2 - 7*a^8*b^6*c^2 + 10*a^6*b^8*c^2 - 10*a^4*b^10*c^2 + 5*a^2*b^12*c^2 - b^14*c^2 - 9*a^12*c^4 + 7*a^10*b^2*c^4 + 2*a^8*b^4*c^4 - 6*a^6*b^6*c^4 + 9*a^4*b^8*c^4 - 9*a^2*b^10*c^4 + 6*b^12*c^4 + 15*a^10*c^6 - 7*a^8*b^2*c^6 - 6*a^6*b^4*c^6 - 4*a^4*b^6*c^6 + 5*a^2*b^8*c^6 - 15*b^10*c^6 - 10*a^8*c^8 + 10*a^6*b^2*c^8 + 9*a^4*b^4*c^8 + 5*a^2*b^6*c^8 + 20*b^8*c^8 - 10*a^4*b^2*c^10 - 9*a^2*b^4*c^10 - 15*b^6*c^10 + 3*a^4*c^12 + 5*a^2*b^2*c^12 + 6*b^4*c^12 - a^2*c^14 - b^2*c^14 : :

X(36160) lies on these lines: {2, 3}, {52, 523}, {250, 8884}, {2453, 17834}, {3060, 15112}, {9159, 15028}, {9820, 16319}, {14480, 15801}, {15800, 20957}

X(36161) = ORTHOGONAL PROJECTION OF X(54) ON THE EULER LINE

Barycentrics    2*a^16 - 8*a^14*b^2 + 11*a^12*b^4 - 5*a^10*b^6 - 2*a^6*b^10 + 3*a^4*b^12 - a^2*b^14 - 8*a^14*c^2 + 22*a^12*b^2*c^2 - 19*a^10*b^4*c^2 + 3*a^8*b^6*c^2 + 4*a^6*b^8*c^2 - 2*a^4*b^10*c^2 - a^2*b^12*c^2 + b^14*c^2 + 11*a^12*c^4 - 19*a^10*b^2*c^4 + 12*a^8*b^4*c^4 - 2*a^6*b^6*c^4 - 5*a^4*b^8*c^4 + 9*a^2*b^10*c^4 - 6*b^12*c^4 - 5*a^10*c^6 + 3*a^8*b^2*c^6 - 2*a^6*b^4*c^6 + 8*a^4*b^6*c^6 - 7*a^2*b^8*c^6 + 15*b^10*c^6 + 4*a^6*b^2*c^8 - 5*a^4*b^4*c^8 - 7*a^2*b^6*c^8 - 20*b^8*c^8 - 2*a^6*c^10 - 2*a^4*b^2*c^10 + 9*a^2*b^4*c^10 + 15*b^6*c^10 + 3*a^4*c^12 - a^2*b^2*c^12 - 6*b^4*c^12 - a^2*c^14 + b^2*c^14 : :

X(36161) lies on these lines: {2, 3}, {49, 14611}, {54, 523}, {476, 13434}, {1141, 8902}, {1511, 25150}, {2453, 19357}, {5892, 15537}, {9706, 14480}, {13403, 34150}, {30504, 32744}

X(36162) = ORTHOGONAL PROJECTION OF X(64) ON THE EULER LINE

Barycentrics    a^16 - a^14*b^2 - 8*a^12*b^4 + 20*a^10*b^6 - 15*a^8*b^8 - a^6*b^10 + 6*a^4*b^12 - 2*a^2*b^14 - a^14*c^2 + 17*a^12*b^2*c^2 - 20*a^10*b^4*c^2 - 27*a^8*b^6*c^2 + 47*a^6*b^8*c^2 - 13*a^4*b^10*c^2 - 2*a^2*b^12*c^2 - b^14*c^2 - 8*a^12*c^4 - 20*a^10*b^2*c^4 + 84*a^8*b^4*c^4 - 46*a^6*b^6*c^4 - 34*a^4*b^8*c^4 + 18*a^2*b^10*c^4 + 6*b^12*c^4 + 20*a^10*c^6 - 27*a^8*b^2*c^6 - 46*a^6*b^4*c^6 + 82*a^4*b^6*c^6 - 14*a^2*b^8*c^6 - 15*b^10*c^6 - 15*a^8*c^8 + 47*a^6*b^2*c^8 - 34*a^4*b^4*c^8 - 14*a^2*b^6*c^8 + 20*b^8*c^8 - a^6*c^10 - 13*a^4*b^2*c^10 + 18*a^2*b^4*c^10 - 15*b^6*c^10 + 6*a^4*c^12 - 2*a^2*b^2*c^12 + 6*b^4*c^12 - 2*a^2*c^14 - b^2*c^14 : :

X(36162) lies on these lines: {2, 3}, {64, 523}, {185, 2452}, {477, 6080}, {1093, 2693}, {1294, 2972}, {5889, 14508}, {6662, 33541}, {9530, 10990}, {13997, 18381}, {14989, 18394}, {17703, 34178}

X(36163) = ORTHOGONAL PROJECTION OF X(69) ON THE EULER LINE

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

X(36163) lies on these lines: {2, 3}, {69, 523}, {110, 2794}, {125, 23698}, {141, 2453}, {193, 2452}, {246, 2782}, {247, 33511}, {250, 17907}, {315, 2396}, {476, 2710}, {543, 9140}, {691, 2857}, {754, 23061}, {1648, 15538}, {1899, 18347}, {2088, 2549}, {2395, 35902}, {3014, 9145}, {3233, 35260}, {3917, 31848}, {6033, 9155}, {6776, 6795}, {6787, 7761}, {7737, 32761}, {7778, 16320}, {7802, 17941}, {7842, 11052}, {11057, 22254}, {11442, 18337}, {13172, 31127}, {14731, 33884}, {14916, 34312}, {16303, 32220}, {18343, 33102}, {24270, 32815}

X(36163) = isotomic conjugate of anticomplement of X(39078)
X(36163) = complement of X(36181)
X(36163) = anticomplement of X(1316)

X(36164) = ORTHOGONAL PROJECTION OF X(74) ON THE EULER LINE

Barycentrics    2*a^16 - 4*a^14*b^2 - 7*a^12*b^4 + 25*a^10*b^6 - 20*a^8*b^8 - 2*a^6*b^10 + 9*a^4*b^12 - 3*a^2*b^14 - 4*a^14*c^2 + 26*a^12*b^2*c^2 - 29*a^10*b^4*c^2 - 27*a^8*b^6*c^2 + 56*a^6*b^8*c^2 - 22*a^4*b^10*c^2 + a^2*b^12*c^2 - b^14*c^2 - 7*a^12*c^4 - 29*a^10*b^2*c^4 + 96*a^8*b^4*c^4 - 54*a^6*b^6*c^4 - 27*a^4*b^8*c^4 + 15*a^2*b^10*c^4 + 6*b^12*c^4 + 25*a^10*c^6 - 27*a^8*b^2*c^6 - 54*a^6*b^4*c^6 + 80*a^4*b^6*c^6 - 13*a^2*b^8*c^6 - 15*b^10*c^6 - 20*a^8*c^8 + 56*a^6*b^2*c^8 - 27*a^4*b^4*c^8 - 13*a^2*b^6*c^8 + 20*b^8*c^8 - 2*a^6*c^10 - 22*a^4*b^2*c^10 + 15*a^2*b^4*c^10 - 15*b^6*c^10 + 9*a^4*c^12 + a^2*b^2*c^12 + 6*b^4*c^12 - 3*a^2*c^14 - b^2*c^14 : :

X(36164) lies on these lines: {2, 3}, {74, 477}, {98, 841}, {110, 14508}, {113, 31379}, {125, 34150}, {476, 15055}, {1294, 32710}, {1300, 2693}, {1553, 5972}, {2777, 3258}, {3233, 15035}, {5663, 14611}, {6070, 20417}, {6699, 25641}, {10990, 32417}, {11801, 21269}, {12041, 16168}, {14480, 15054}, {14644, 14989}, {14851, 20127}, {16319, 32111}, {16534, 31378}

X(36165) = ORTHOGONAL PROJECTION OF X(76) ON THE EULER LINE

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

X(36165) lies on these lines: {2, 3}, {76, 523}, {115, 9828}, {141, 6787}, {315, 23342}, {626, 5099}, {671, 7668}, {691, 1078}, {2452, 7754}, {2882, 5181}, {3767, 14898}, {5254, 14700}, {7748, 14609}, {7789, 16320}, {26869, 32463}

X(36166) = ORTHOGONAL PROJECTION OF X(98) ON THE EULER LINE

Barycentrics    2*a^14 - 4*a^12*b^2 + 3*a^10*b^4 - 2*a^8*b^6 - 2*a^6*b^8 + 6*a^4*b^10 - 3*a^2*b^12 - 4*a^12*c^2 + 6*a^10*b^2*c^2 - 2*a^8*b^4*c^2 + 3*a^6*b^6*c^2 - 9*a^4*b^8*c^2 + 7*a^2*b^10*c^2 - b^12*c^2 + 3*a^10*c^4 - 2*a^8*b^2*c^4 + 3*a^4*b^6*c^4 - 11*a^2*b^8*c^4 + 3*b^10*c^4 - 2*a^8*c^6 + 3*a^6*b^2*c^6 + 3*a^4*b^4*c^6 + 14*a^2*b^6*c^6 - 2*b^8*c^6 - 2*a^6*c^8 - 9*a^4*b^2*c^8 - 11*a^2*b^4*c^8 - 2*b^6*c^8 + 6*a^4*c^10 + 7*a^2*b^2*c^10 + 3*b^4*c^10 - 3*a^2*c^12 - b^2*c^12 : :

X(36166) lies on these lines: {2, 3}, {74, 1499}, {98, 523}, {111, 477}, {114, 16760}, {115, 34366}, {125, 1550}, {232, 35907}, {511, 14999}, {525, 22265}, {691, 34473}, {841, 9084}, {1503, 11005}, {2373, 32710}, {2374, 2693}, {2452, 9755}, {2453, 9756}, {2682, 2777}, {2697, 3563}, {2794, 5099}, {5913, 8429}, {6036, 16188}, {6055, 16092}, {9076, 14979}, {19165, 30715}

X(36166) = midpoint of X(98) and X(842)
X(36166) = complement of X(36173)
X(36166) = anticomplement of X(36170)
X(36166) = Thomson-isogonal conjugate of X(5653)
X(36166) = radical trace of circumcircle and circle O' as described at X(6039)

X(36167) = ORTHOGONAL PROJECTION OF X(100) ON THE EULER LINE

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

X(36167) lies on these lines: {2, 3}, {100, 523}, {110, 6003}, {476, 6011}, {517, 31525}, {691, 9070}, {2687, 34474}, {2689, 33637}, {2691, 9058}, {2766, 13397}, {3035, 5520}, {3871, 13869}, {9060, 30257}, {10420, 30250}

X(36168) = ORTHOGONAL PROJECTION OF X(111) ON THE EULER LINE

Barycentrics    2*a^12 - 4*a^10*b^2 - 5*a^8*b^4 + 7*a^6*b^6 + 3*a^4*b^8 - 3*a^2*b^10 - 4*a^10*c^2 + 22*a^8*b^2*c^2 - 11*a^6*b^4*c^2 - 23*a^4*b^6*c^2 + 13*a^2*b^8*c^2 - b^10*c^2 - 5*a^8*c^4 - 11*a^6*b^2*c^4 + 42*a^4*b^4*c^4 - 10*a^2*b^6*c^4 + 7*a^6*c^6 - 23*a^4*b^2*c^6 - 10*a^2*b^4*c^6 + 2*b^6*c^6 + 3*a^4*c^8 + 13*a^2*b^2*c^8 - 3*a^2*c^10 - b^2*c^10 : :

X(36168) lies on these lines: {2, 3}, {98, 10102}, {111, 523}, {524, 9129}, {542, 32222}, {842, 9084}, {2686, 2777}, {5099, 10418}, {6719, 31655}, {7664, 14588}, {9182, 26276}

X(36169) = ORTHOGONAL PROJECTION OF X(113) ON THE EULER LINE

Barycentrics    4*a^14*b^2 - 15*a^12*b^4 + 17*a^10*b^6 - 10*a^6*b^10 + a^4*b^12 + 5*a^2*b^14 - 2*b^16 + 4*a^14*c^2 - 2*a^12*b^2*c^2 + 7*a^10*b^4*c^2 - 43*a^8*b^6*c^2 + 40*a^6*b^8*c^2 + 14*a^4*b^10*c^2 - 27*a^2*b^12*c^2 + 7*b^14*c^2 - 15*a^12*c^4 + 7*a^10*b^2*c^4 + 54*a^8*b^4*c^4 - 26*a^6*b^6*c^4 - 69*a^4*b^8*c^4 + 51*a^2*b^10*c^4 - 2*b^12*c^4 + 17*a^10*c^6 - 43*a^8*b^2*c^6 - 26*a^6*b^4*c^6 + 108*a^4*b^6*c^6 - 29*a^2*b^8*c^6 - 23*b^10*c^6 + 40*a^6*b^2*c^8 - 69*a^4*b^4*c^8 - 29*a^2*b^6*c^8 + 40*b^8*c^8 - 10*a^6*c^10 + 14*a^4*b^2*c^10 + 51*a^2*b^4*c^10 - 23*b^6*c^10 + a^4*c^12 - 27*a^2*b^2*c^12 - 2*b^4*c^12 + 5*a^2*c^14 + 7*b^2*c^14 - 2*c^16 : :

X(36169) lies on these lines: {2, 3}, {110, 34150}, {113, 523}, {125, 1553}, {131, 18809}, {250, 16934}, {2777, 22104}, {3233, 17702}, {5627, 14094}, {5663, 12079}, {6070, 15063}, {10264, 21315}, {12900, 31379}, {14508, 15059}, {14643, 14934}, {14989, 15035}, {15044, 31876}, {21269, 34153}, {21316, 32423}

X(36169) = Euler line intercept of minor axis of hyperbola {{A,B,C,X(4),X(476)}}

X(36170) = ORTHOGONAL PROJECTION OF X(114) ON THE EULER LINE

Barycentrics    4*a^12*b^2 - 9*a^10*b^4 + 4*a^8*b^6 + 4*a^6*b^8 - 6*a^4*b^10 + 5*a^2*b^12 - 2*b^14 + 4*a^12*c^2 - 14*a^10*b^2*c^2 + 20*a^8*b^4*c^2 - 17*a^6*b^6*c^2 + 13*a^4*b^8*c^2 - 13*a^2*b^10*c^2 + 7*b^12*c^2 - 9*a^10*c^4 + 20*a^8*b^2*c^4 - 6*a^6*b^4*c^4 - 3*a^4*b^6*c^4 + 11*a^2*b^8*c^4 - 9*b^10*c^4 + 4*a^8*c^6 - 17*a^6*b^2*c^6 - 3*a^4*b^4*c^6 - 6*a^2*b^6*c^6 + 4*b^8*c^6 + 4*a^6*c^8 + 13*a^4*b^2*c^8 + 11*a^2*b^4*c^8 + 4*b^6*c^8 - 6*a^4*c^10 - 13*a^2*b^2*c^10 - 9*b^4*c^10 + 5*a^2*c^12 + 7*b^2*c^12 - 2*c^14 : :

X(36170) lies on these lines: {2, 3}, {110, 1550}, {113, 1499}, {114, 523}, {126, 25641}, {325, 14221}, {3564, 11005}, {6054, 16092}, {6721, 16760}, {8791, 16934}, {11184, 16279}, {15535, 34953}

X(36170) = complement of X(36166)

X(36171) = ORTHOGONAL PROJECTION OF X(145) ON THE EULER LINE

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

X(36171) lies on these lines: {2, 3}, {56, 1290}, {145, 523}, {952, 14683}, {2687, 11248}, {3336, 6788}, {3448, 6740}, {3623, 13869}, {5520, 11681}, {6790, 25253}, {20066, 23844}

X(36172) = ORTHOGONAL PROJECTION OF X(146) ON THE EULER LINE

Barycentrics    a^16 - 11*a^12*b^4 + 21*a^10*b^6 - 10*a^8*b^8 - 6*a^6*b^10 + 5*a^4*b^12 + a^2*b^14 - b^16 + 12*a^12*b^2*c^2 - 11*a^10*b^4*c^2 - 35*a^8*b^6*c^2 + 48*a^6*b^8*c^2 - 4*a^4*b^10*c^2 - 13*a^2*b^12*c^2 + 3*b^14*c^2 - 11*a^12*c^4 - 11*a^10*b^2*c^4 + 75*a^8*b^4*c^4 - 40*a^6*b^6*c^4 - 48*a^4*b^8*c^4 + 33*a^2*b^10*c^4 + 2*b^12*c^4 + 21*a^10*c^6 - 35*a^8*b^2*c^6 - 40*a^6*b^4*c^6 + 94*a^4*b^6*c^6 - 21*a^2*b^8*c^6 - 19*b^10*c^6 - 10*a^8*c^8 + 48*a^6*b^2*c^8 - 48*a^4*b^4*c^8 - 21*a^2*b^6*c^8 + 30*b^8*c^8 - 6*a^6*c^10 - 4*a^4*b^2*c^10 + 33*a^2*b^4*c^10 - 19*b^6*c^10 + 5*a^4*c^12 - 13*a^2*b^2*c^12 + 2*b^4*c^12 + a^2*c^14 + 3*b^2*c^14 - c^16 : :

X(36172) lies on these lines: {2, 3}, {74, 25641}, {110, 1553}, {113, 477}, {125, 14508}, {146, 523}, {476, 2777}, {1539, 20957}, {3448, 34150}, {5627, 16003}, {6070, 15054}, {7728, 16168}, {10152, 30716}, {10620, 34209}, {12373, 33965}, {12374, 33964}, {12902, 21269}, {14480, 15063}, {14989, 17702}, {15055, 22104}

X(36173) = ORTHOGONAL PROJECTION OF X(147) ON THE EULER LINE

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

X(36173) lies on these lines: {2, 3}, {98, 16188}, {114, 842}, {132, 250}, {146, 1499}, {147, 523}, {511, 11005}, {691, 2794}, {1503, 14999}, {1550, 3448}, {6023, 12185}, {6027, 12184}, {9749, 11629}, {9750, 11630}, {11177, 16092}, {14360, 30474}

X(36173) = anticomplement of X(36166)

X(36174) = ORTHOGONAL PROJECTION OF X(148) ON THE EULER LINE

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

X(36174) lies on these lines: {2, 3}, {99, 5099}, {110, 2682}, {115, 691}, {148, 523}, {842, 23698}, {1499, 3448}, {6023, 13182}, {6027, 13183}, {14639, 16188}, {16760, 21166}

X(36175) = ORTHOGONAL PROJECTION OF X(149) ON THE EULER LINE

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

X(36175) lies on these lines: {2, 3}, {11, 1290}, {100, 5520}, {149, 523}, {513, 10778}, {517, 10767}, {2687, 5840}, {3448, 6003}, {3583, 13604}, {5080, 25436}, {13273, 31524}, {13274, 31522}

X(36176) = ORTHOGONAL PROJECTION OF X(157) ON THE EULER LINE

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

X(36176) lies on these lines: {2, 3}, {6, 250}, {107, 34473}, {112, 8429}, {157, 523}, {264, 2453}, {648, 7669}, {935, 2857}, {1304, 2710}, {2967, 5191}, {5152, 6331}, {14060, 32713}, {14687, 35908}, {16328, 32217}, {19165, 35278}, {34217, 35282}

X(36177) = ORTHOGONAL PROJECTION OF X(182) ON THE EULER LINE

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

X(36177) lies on these lines: {2, 3}, {182, 523}, {543, 33509}, {1511, 2782}, {1561, 16111}, {2452, 5050}, {2453, 5085}, {2794, 6699}, {3111, 15536}, {3233, 5651}, {4045, 31379}, {5012, 14480}, {5946, 31850}, {10264, 11005}, {11003, 14611}, {12079, 18911}, {13394, 16319}, {14805, 14934}, {14999, 32515}

X(36177) = midpoint of X(3) and X(1316)
X(36177) = Brocard-circle-inverse of X(8723)

X(36178) = ORTHOGONAL PROJECTION OF X(184) ON THE EULER LINE

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

X(36178) lies on these lines: {2, 3}, {154, 2453}, {184, 523}, {476, 5012}, {1899, 12079}, {2452, 11402}, {3233, 9306}, {3796, 6795}, {8901, 13558}, {9544, 14611}, {10192, 16319}, {11464, 15111}, {11657, 13567}

X(36179) = ORTHOGONAL PROJECTION OF X(185) ON THE EULER LINE

Barycentrics    2*a^14*b^2 - 9*a^12*b^4 + 15*a^10*b^6 - 10*a^8*b^8 + 3*a^4*b^12 - a^2*b^14 + 2*a^14*c^2 + 2*a^12*b^2*c^2 - 5*a^10*b^4*c^2 - 15*a^8*b^6*c^2 + 26*a^6*b^8*c^2 - 10*a^4*b^10*c^2 + a^2*b^12*c^2 - b^14*c^2 - 9*a^12*c^4 - 5*a^10*b^2*c^4 + 42*a^8*b^4*c^4 - 26*a^6*b^6*c^4 - 11*a^4*b^8*c^4 + 3*a^2*b^10*c^4 + 6*b^12*c^4 + 15*a^10*c^6 - 15*a^8*b^2*c^6 - 26*a^6*b^4*c^6 + 36*a^4*b^6*c^6 - 3*a^2*b^8*c^6 - 15*b^10*c^6 - 10*a^8*c^8 + 26*a^6*b^2*c^8 - 11*a^4*b^4*c^8 - 3*a^2*b^6*c^8 + 20*b^8*c^8 - 10*a^4*b^2*c^10 + 3*a^2*b^4*c^10 - 15*b^6*c^10 + 3*a^4*c^12 + a^2*b^2*c^12 + 6*b^4*c^12 - a^2*c^14 - b^2*c^14 : :

X(36179) lies on these lines: {2, 3}, {49, 14934}, {54, 477}, {64, 2453}, {185, 523}, {250, 1105}, {3521, 20957}, {6241, 15111}, {13630, 16168}, {14508, 15062}, {15072, 15112}, {16252, 16319}

X(36180) = ORTHOGONAL PROJECTION OF X(187) ON THE EULER LINE

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

X(36180) lies on these lines: {2, 3}, {187, 523}, {1384, 2452}, {1495, 1499}, {2453, 5210}, {3111, 8705}, {5099, 6781}, {5118, 32217}, {5191, 9123}, {6390, 14588}, {9177, 35345}, {16092, 26613}

X(36181) = ORTHOGONAL PROJECTION OF X(193) ON THE EULER LINE

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

X(36181) lies on these lines: {2, 3}, {51, 11554}, {69, 2453}, {110, 23698}, {193, 523}, {250, 393}, {317, 30716}, {476, 23700}, {543, 9143}, {1007, 16320}, {2549, 11003}, {2782, 14683}, {2794, 3448}, {5099, 32827}, {5191, 6321}, {5967, 31670}, {7605, 7804}, {7737, 11002}, {9752, 16188}, {10723, 35278}, {16978, 16981}

X(36181) = anticomplement of X(36163)

X(36182) = ORTHOGONAL PROJECTION OF X(194) ON THE EULER LINE

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

X(36182) lies on the circle {{X(4),X(194),X(3557),X(3558)}} and these lines: {2, 3}, {32, 691}, {148, 9149}, {194, 523}, {250, 1968}, {511, 10568}, {842, 9737}, {895, 2882}, {1634, 8591}, {1975, 2453}, {2452, 7839}, {3053, 14898}, {3972, 32531}, {5099, 7752}, {5118, 6787}, {5201, 14712}, {9716, 31962}

X(36182) = 2nd-Brocard-circle-inverse of X(2)

X(36183) = ORTHOGONAL PROJECTION OF X(262) ON THE EULER LINE

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

X(36183) lies on these lines: {2, 3}, {98, 1576}, {262, 523}, {525, 18304}, {842, 3613}, {1352, 2421}, {1503, 15920}, {2679, 16188}, {3815, 34235}, {5476, 16092}, {5480, 6785}, {6795, 11174}, {7699, 32120}, {9755, 34978}

X(36184) = ORTHOGONAL PROJECTION OF X(265) ON THE EULER LINE

Barycentrics    2*a^16 - 6*a^14*b^2 + 5*a^12*b^4 - 3*a^10*b^6 + 10*a^8*b^8 - 12*a^6*b^10 + a^4*b^12 + 5*a^2*b^14 - 2*b^16 - 6*a^14*c^2 + 18*a^12*b^2*c^2 - 13*a^10*b^4*c^2 - 13*a^8*b^6*c^2 + 18*a^6*b^8*c^2 + 10*a^4*b^10*c^2 - 23*a^2*b^12*c^2 + 9*b^14*c^2 + 5*a^12*c^4 - 13*a^10*b^2*c^4 + 24*a^8*b^4*c^4 - 8*a^6*b^6*c^4 - 33*a^4*b^8*c^4 + 39*a^2*b^10*c^4 - 14*b^12*c^4 - 3*a^10*c^6 - 13*a^8*b^2*c^6 - 8*a^6*b^4*c^6 + 44*a^4*b^6*c^6 - 21*a^2*b^8*c^6 + 7*b^10*c^6 + 10*a^8*c^8 + 18*a^6*b^2*c^8 - 33*a^4*b^4*c^8 - 21*a^2*b^6*c^8 - 12*a^6*c^10 + 10*a^4*b^2*c^10 + 39*a^2*b^4*c^10 + 7*b^6*c^10 + a^4*c^12 - 23*a^2*b^2*c^12 - 14*b^4*c^12 + 5*a^2*c^14 + 9*b^2*c^14 - 2*c^16 : :

X(36184) lies on these lines: {2, 3}, {265, 523}, {476, 14644}, {477, 10733}, {3233, 14643}, {3258, 14934}, {5663, 10689}, {7687, 25641}, {10113, 16168}, {10721, 14508}, {11801, 34209}, {12052, 16222}, {14611, 32423}, {15111, 18392}, {15112, 18394}, {16003, 32417}, {16163, 31379}, {18319, 21316}, {22104, 23515}, {31945, 32609}

X(36185) = ORTHOGONAL PROJECTION OF X(298) ON THE EULER LINE

Barycentrics    a^14 - 2*a^12*b^2 - 3*a^10*b^4 + 10*a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10 + 7*a^2*b^12 - 2*b^14 - 2*a^12*c^2 + 3*a^10*b^2*c^2 - 3*a^8*b^4*c^2 + a^6*b^6*c^2 + 15*a^4*b^8*c^2 - 24*a^2*b^10*c^2 + 10*b^12*c^2 - 3*a^10*c^4 - 3*a^8*b^2*c^4 - 9*a^4*b^6*c^4 + 33*a^2*b^8*c^4 - 18*b^10*c^4 + 10*a^8*c^6 + a^6*b^2*c^6 - 9*a^4*b^4*c^6 - 32*a^2*b^6*c^6 + 10*b^8*c^6 - 5*a^6*c^8 + 15*a^4*b^2*c^8 + 33*a^2*b^4*c^8 + 10*b^6*c^8 - 6*a^4*c^10 - 24*a^2*b^2*c^10 - 18*b^4*c^10 + 7*a^2*c^12 + 10*b^2*c^12 - 2*c^14 - 2*Sqrt[3]*(a^12 - 5*a^10*b^2 + 8*a^8*b^4 - 4*a^6*b^6 - a^4*b^8 + a^2*b^10 - 5*a^10*c^2 + 11*a^8*b^2*c^2 - 10*a^6*b^4*c^2 - a^4*b^6*c^2 + 3*a^2*b^8*c^2 + 2*b^10*c^2 + 8*a^8*c^4 - 10*a^6*b^2*c^4 + 14*a^4*b^4*c^4 - 4*a^2*b^6*c^4 - 8*b^8*c^4 - 4*a^6*c^6 - a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 12*b^6*c^6 - a^4*c^8 + 3*a^2*b^2*c^8 - 8*b^4*c^8 + a^2*c^10 + 2*b^2*c^10)*S : :

X(36185) lies on these lines: {2, 3}, {14, 9159}, {15, 30468}, {125, 11092}, {298, 523}, {511, 11078}, {531, 9140}, {532, 23061}, {2452, 3181}, {3258, 11131}, {3643, 7998}, {5463, 34312}, {5473, 14187}, {5613, 11130}, {5978, 11629}, {6151, 14137}, {6670, 15289}, {6773, 6795}, {6774, 8836}, {8015, 16964}, {10654, 18911}, {11549, 16645}, {13102, 16771}, {15743, 16242}, {16770, 20425}, {23004, 30465}, {30460, 34220}

X(36186) = ORTHOGONAL PROJECTION OF X(299) ON THE EULER LINE

Barycentrics    a^14 - 2*a^12*b^2 - 3*a^10*b^4 + 10*a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10 + 7*a^2*b^12 - 2*b^14 - 2*a^12*c^2 + 3*a^10*b^2*c^2 - 3*a^8*b^4*c^2 + a^6*b^6*c^2 + 15*a^4*b^8*c^2 - 24*a^2*b^10*c^2 + 10*b^12*c^2 - 3*a^10*c^4 - 3*a^8*b^2*c^4 - 9*a^4*b^6*c^4 + 33*a^2*b^8*c^4 - 18*b^10*c^4 + 10*a^8*c^6 + a^6*b^2*c^6 - 9*a^4*b^4*c^6 - 32*a^2*b^6*c^6 + 10*b^8*c^6 - 5*a^6*c^8 + 15*a^4*b^2*c^8 + 33*a^2*b^4*c^8 + 10*b^6*c^8 - 6*a^4*c^10 - 24*a^2*b^2*c^10 - 18*b^4*c^10 + 7*a^2*c^12 + 10*b^2*c^12 - 2*c^14 + 2*Sqrt[3]*(a^12 - 5*a^10*b^2 + 8*a^8*b^4 - 4*a^6*b^6 - a^4*b^8 + a^2*b^10 - 5*a^10*c^2 + 11*a^8*b^2*c^2 - 10*a^6*b^4*c^2 - a^4*b^6*c^2 + 3*a^2*b^8*c^2 + 2*b^10*c^2 + 8*a^8*c^4 - 10*a^6*b^2*c^4 + 14*a^4*b^4*c^4 - 4*a^2*b^6*c^4 - 8*b^8*c^4 - 4*a^6*c^6 - a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 12*b^6*c^6 - a^4*c^8 + 3*a^2*b^2*c^8 - 8*b^4*c^8 + a^2*c^10 + 2*b^2*c^10)*S : :

X(36186) lies on these lines: {2, 3}, {13, 9159}, {16, 30465}, {125, 11078}, {299, 523}, {511, 11092}, {530, 9140}, {533, 23061}, {2452, 3180}, {2981, 14136}, {3258, 11130}, {3642, 7998}, {5464, 34312}, {5474, 14185}, {5617, 11131}, {5979, 11630}, {6669, 15290}, {6770, 6795}, {6771, 8838}, {8014, 16965}, {10653, 18911}, {11537, 16644}, {11586, 16241}, {13103, 16770}, {16771, 20426}, {23005, 30468}, {30463, 34219}

X(36187) = ORTHOGONAL PROJECTION OF X(315) ON THE EULER LINE

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

X(36187) lies on these lines: {2, 3}, {67, 5969}, {315, 523}, {316, 3001}, {691, 7802}, {2452, 7762}, {2453, 7784}, {5099, 7825}, {5171, 16188}

X(36188) = ORTHOGONAL PROJECTION OF X(323) ON THE EULER LINE

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

X(36188) lies on these lines: {2, 3}, {182, 9159}, {323, 523}, {476, 511}, {477, 10564}, {1092, 15112}, {2452, 11004}, {2453, 15066}, {3233, 35265}, {3292, 14480}, {5642, 9158}, {6795, 11003}, {13857, 34312}

X(36189) = ORTHOGONAL PROJECTION OF X(338) ON THE EULER LINE

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

X(36189) lies on these lines: {2, 3}, {115, 647}, {125, 512}, {127, 16221}, {246, 3566}, {338, 523}, {339, 6563}, {842, 34175}, {5139, 16177}, {8901, 9213}, {12188, 14611}

X(36190) = ORTHOGONAL PROJECTION OF X(343) ON THE EULER LINE

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

X(36190) lies on these lines: {2, 3}, {343, 523}, {1899, 6795}, {2452, 6515}, {2790, 12827}, {3233, 10192}, {3258, 3917}, {9159, 26913}, {9306, 16319}, {10278, 10412}

X(36191) = ORTHOGONAL PROJECTION OF X(393) ON THE EULER LINE

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

X(36191) lies on these lines: {2, 3}, {53, 2453}, {250, 317}, {393, 523}, {1249, 2452}, {2710, 22239}, {2857, 10423}, {5191, 13200}, {13567, 18338}, {32649, 35088}

X(36192) = ORTHOGONAL PROJECTION OF X(394) ON THE EULER LINE

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

X(36192) lies on these lines: {2, 3}, {110, 2790}, {154, 3233}, {184, 6795}, {250, 15466}, {394, 523}, {476, 2706}, {925, 2972}, {1993, 2452}, {2453, 17811}, {2986, 20975}, {5012, 9159}, {8029, 15328}, {10420, 23606}

X(36193) = ORTHOGONAL PROJECTION OF X(399) ON THE EULER LINE

Barycentrics    a^16 - 5*a^14*b^2 + 10*a^12*b^4 - 10*a^10*b^6 + 5*a^8*b^8 - a^6*b^10 - 5*a^14*c^2 + 10*a^12*b^2*c^2 - 10*a^10*b^4*c^2 + 15*a^8*b^6*c^2 - 11*a^6*b^8*c^2 - 2*a^4*b^10*c^2 + 2*a^2*b^12*c^2 + b^14*c^2 + 10*a^12*c^4 - 10*a^10*b^2*c^4 - 15*a^8*b^4*c^4 + 9*a^6*b^6*c^4 + 18*a^4*b^8*c^4 - 6*a^2*b^10*c^4 - 6*b^12*c^4 - 10*a^10*c^6 + 15*a^8*b^2*c^6 + 9*a^6*b^4*c^6 - 32*a^4*b^6*c^6 + 4*a^2*b^8*c^6 + 15*b^10*c^6 + 5*a^8*c^8 - 11*a^6*b^2*c^8 + 18*a^4*b^4*c^8 + 4*a^2*b^6*c^8 - 20*b^8*c^8 - a^6*c^10 - 2*a^4*b^2*c^10 - 6*a^2*b^4*c^10 + 15*b^6*c^10 + 2*a^2*b^2*c^12 - 6*b^4*c^12 + b^2*c^14 : :

X(36193) lies on these lines: {2, 3}, {110, 16168}, {113, 20957}, {250, 34334}, {265, 25641}, {399, 523}, {476, 5663}, {477, 1511}, {1553, 7728}, {2970, 10688}, {3233, 14934}, {3258, 14643}, {3448, 34209}, {5609, 14480}, {6070, 14993}, {10088, 33965}, {10091, 33964}, {12041, 14508}, {12383, 34193}, {12902, 34150}, {14851, 31379}, {14989, 15468}, {15061, 22104}, {15081, 21315}, {18319, 32423}, {20125, 33505}

X(36194) = ORTHOGONAL PROJECTION OF X(599) ON THE EULER LINE

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

X(36194) lies on these lines: {2, 3}, {125, 543}, {183, 16092}, {511, 16279}, {523, 599}, {524, 2452}, {542, 6795}, {804, 5653}, {1648, 2549}, {2396, 7788}, {2453, 21358}, {2782, 9140}, {2794, 5642}, {3258, 5108}, {3569, 34359}, {3734, 34512}, {3849, 9181}, {6054, 9155}, {6772, 30468}, {6775, 30465}, {6792, 15048}, {7811, 22254}, {7998, 34312}, {9158, 9996}, {11057, 17941}, {11594, 13377}, {13188, 31127}, {30789, 33813}

X(36194) = circumcircle-inverse of X(37916)
X(36194) = Artzt-to-McCay similarity image of X(110)

X(36195) = ORTHOGONAL PROJECTION OF X(656) ON THE EULER LINE

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

X(36195) lies on these lines: {2, 3}, {12, 22342}, {117, 3258}, {125, 515}, {517, 6739}, {523, 656}, {1319, 8286}, {1425, 15556}, {1558, 2777}, {1735, 22094}, {1834, 24443}, {3585, 14873}, {3833, 24169}, {5088, 23674}, {5529, 6127}, {10149, 16332}, {17647, 34829}, {19925, 30436}

X(36195) = polar-circle-inverse of X(29)

X(36196) = ORTHOGONAL PROJECTION OF X(671) ON THE EULER LINE

Barycentrics    2*a^10 - 4*a^8*b^2 - 7*a^6*b^4 + 8*a^4*b^6 + 5*a^2*b^8 - 4*b^10 - 4*a^8*c^2 + 26*a^6*b^2*c^2 - 12*a^4*b^4*c^2 - 25*a^2*b^6*c^2 + 11*b^8*c^2 - 7*a^6*c^4 - 12*a^4*b^2*c^4 + 42*a^2*b^4*c^4 - 7*b^6*c^4 + 8*a^4*c^6 - 25*a^2*b^2*c^6 - 7*b^4*c^6 + 5*a^2*c^8 + 11*b^2*c^8 - 4*c^10 : :

X(36196) lies on these lines: {2, 3}, {115, 5912}, {316, 9182}, {523, 671}, {524, 9144}, {542, 2682}, {543, 5099}, {691, 9166}, {1499, 9140}, {5465, 14999}, {14832, 23992}, {23004, 34316}, {23005, 34315}

X(36197) =  X(2)X(24274)∩X(115)X(125)

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

X(36197) lies on these lines: {2, 24274}, {9, 2648}, {11, 17435}, {37, 4551}, {115, 125}, {661, 18210}, {756, 21795}, {762, 1334}, {1146, 7358}, {1864, 20229}, {2310, 3119}, {2801, 25069}, {3954, 22032}, {4466, 10933}, {7069, 16588}, {20230, 20311}, {20684, 20689}

X(36197) = X(i)-Ceva conjugate of X(j) for these (i,j): {37, 4041}, {210, 4524}, {1446, 523}, {1826, 4705}, {1903, 512}, {2250, 4730}, {2287, 3900}, {4183, 8641}, {4515, 4171}, {21044, 4516}
X(36197) = X(i)-isoconjugate of X(j) for these (i,j): {56, 4620}, {58, 1275}, {59, 1434}, {81, 7045}, {86, 1262}, {99, 1461}, {100, 4637}, {101, 4616}, {109, 4573}, {110, 658}, {163, 4569}, {249, 3668}, {269, 4567}, {274, 24027}, {279, 4570}, {310, 23979}, {333, 7339}, {643, 4617}, {645, 6614}, {651, 1414}, {662, 934}, {664, 4565}, {692, 4635}, {1014, 4564}, {1042, 4590}, {1043, 23971}, {1101, 1446}, {1106, 4601}, {1400, 7340}, {1407, 4600}, {1412, 4998}, {1415, 4625}, {1427, 24041}, {1442, 35049}, {1444, 7128}, {2287, 24013}, {2328, 23586}, {4556, 4566}, {4558, 36118}, {4575, 13149}, {4592, 32714}, {4619, 7192}, {4626, 5546}, {5379, 7177}, {7203, 31615}, {18604, 24032}
X(36197) = crosspoint of X(i) and X(j) for these (i,j): {37, 4041}, {210, 3700}, {523, 1446}, {650, 7073}, {657, 1334}, {661, 1824}, {1146, 2310}, {2287, 3900}, {3239, 4082}, {4171, 4515}, {4391, 4451}
X(36197) = crosssum of X(i) and X(j) for these (i,j): {81, 1414}, {651, 1442}, {658, 1434}, {662, 1444}, {934, 1427}, {1014, 4565}, {1262, 7045}
X(36197) = crossdifference of every pair of points on line {110, 934}
X(36197) = barycentric product X(i)*X(j) for these {i,j}: {8, 4516}, {9, 21044}, {10, 2310}, {11, 210}, {21, 4092}, {37, 1146}, {42, 24026}, {65, 4081}, {115, 2287}, {125, 4183}, {200, 3120}, {213, 23978}, {220, 16732}, {226, 3119}, {228, 21666}, {244, 4082}, {321, 14936}, {341, 3122}, {346, 3125}, {512, 4397}, {514, 4171}, {522, 4041}, {523, 3900}, {650, 3700}, {657, 1577}, {661, 3239}, {663, 4086}, {693, 4524}, {762, 26856}, {850, 8641}, {1021, 4024}, {1043, 2643}, {1086, 4515}, {1098, 21043}, {1109, 2328}, {1253, 21207}, {1334, 4858}, {1427, 23970}, {1441, 3022}, {1446, 35508}, {1792, 8754}, {1824, 2968}, {1826, 34591}, {1903, 5514}, {2170, 2321}, {2322, 3708}, {2326, 21046}, {2332, 20902}, {2489, 15416}, {3064, 8611}, {3271, 3701}, {3668, 24010}, {3694, 8735}, {3709, 4391}, {4017, 4163}, {4036, 21789}, {4049, 14427}, {4069, 21132}, {4077, 4105}, {4130, 7178}, {4466, 7079}, {4551, 23615}, {4705, 7253}, {6057, 18191}, {6741, 7073}, {7046, 18210}, {7058, 21833}, {7259, 21131}, {21889, 34896}, {24290, 28132}
X(36197) = barycentric quotient X (i)/X(j) for these {i,j}: {9, 4620}, {21, 7340}, {37, 1275}, {42, 7045}, {115, 1446}, {200, 4600}, {210, 4998}, {213, 1262}, {220, 4567}, {346, 4601}, {512, 934}, {513, 4616}, {514, 4635}, {522, 4625}, {523, 4569}, {649, 4637}, {650, 4573}, {657, 662}, {661, 658}, {663, 1414}, {798, 1461}, {1021, 4610}, {1042, 24013}, {1043, 24037}, {1146, 274}, {1253, 4570}, {1334, 4564}, {1402, 7339}, {1427, 23586}, {1918, 24027}, {2170, 1434}, {2205, 23979}, {2287, 4590}, {2310, 86}, {2328, 24041}, {2333, 7128}, {2489, 32714}, {2501, 13149}, {2643, 3668}, {2971, 1426}, {3022, 21}, {3063, 4565}, {3119, 333}, {3120, 1088}, {3121, 1407}, {3122, 269}, {3124, 1427}, {3125, 279}, {3239, 799}, {3270, 1444}, {3271, 1014}, {3668, 24011}, {3700, 4554}, {3709, 651}, {3900, 99}, {4017, 4626}, {4041, 664}, {4079, 1020}, {4081, 314}, {4082, 7035}, {4086, 4572}, {4092, 1441}, {4105, 643}, {4130, 645}, {4163, 7257}, {4171, 190}, {4183, 18020}, {4397, 670}, {4515, 1016}, {4516, 7}, {4524, 100}, {4705, 4566}, {7063, 1402}, {7071, 5379}, {7180, 4617}, {7253, 4623}, {8641, 110}, {14936, 81}, {18191, 552}, {18210, 7056}, {20975, 1439}, {21044, 85}, {21833, 6354}, {23615, 18155}, {23978, 6385}, {24010, 1043}, {24012, 2328}, {24026, 310}, {34591, 17206}, {35508, 2287}
X(36197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1864, 20310, 20229}, {2310, 3119, 14936}

X(36198) =  X(338)-CROSS CONJUGATE OF X(6)

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

X(36198) lies on the cubic K1144 and these lines: {3049, 7668}, {3050, 14575}, {9418, 21646}

X(36198) = isogonal conjugate of the anticomplement of X(23962)
X(36198) = X(338)-cross conjugate of X(6)
X(36198) = X(662)-isoconjugate of X(34845)
X(36198) = barycentric quotient X(512)/X(34845)

X(36199) =  X(6)-CEVA CONJUGATE OF X(338)

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

X(36199) lies on the cubic K1144 and these lines: {523, 3613}, {924, 21646}, {2485, 18314}, {22456, 23963}

X(36199) = X(6)-Ceva conjugate of X (338)
X(36199) = barycentric product X(850)*X (34845)
X(36199) = barycentric quotient X (34845)/X(110)

X(36200) =  X(6)-CEVA CONJUGATE OF X(3613)

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

X(36200) lies on the cubic K1144 and these lines: {338, 3613}, {7755, 8265}

X(36200) = X(6)-Ceva conjugate of X(3613)

X(36201) =  X(30)X(511)∩X(64)X(67)

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

X(36201) lies on these lines: {2, 15113}, {3, 15116}, {4, 1177}, {6, 1562}, {20, 1632}, {22, 12827}, {25, 125}, {30, 511}, {64, 67}, {66, 74}, {110, 1370}, {112, 35902}, {113, 206}, {141, 11598}, {146, 5596}, {154, 5642}, {159, 2935}, {182, 19506}, {247, 1976}, {265, 9919}, {381, 10249}, {382, 8549}, {428, 12099}, {468, 15126}, {576, 12897}, {895, 3146}, {974, 9969}, {1112, 11245}, {1205, 11381}, {1352, 16111}, {1368, 5972}, {1495, 32125}, {1498, 15063}, {1560, 35901}, {1596, 7687}, {1657, 34787}, {1899, 12828}, {2883, 6593}, {2930, 17845}, {3313, 12825}, {3357, 34118}, {3424, 9769}, {3448, 7500}, {3575, 31978}, {3818, 6644}, {3830, 23049}, {5085, 16072}, {5092, 12900}, {5095, 5895}, {5480, 10169}, {5505, 35512}, {5655, 32063}, {5656, 10706}, {5878, 8538}, {5893, 32300}, {5894, 8263}, {5925, 15069}, {6145, 34437}, {6146, 16105}, {6225, 11061}, {6241, 32317}, {6247, 13419}, {6266, 32281}, {6267, 32280}, {6285, 32243}, {6293, 14448}, {6677, 6723}, {6696, 6698}, {6756, 16270}, {6759, 14791}, {6776, 10721}, {7355, 32297}, {7530, 18381}, {7728, 19149}, {7729, 9971}, {7973, 32298}, {8550, 13403}, {8991, 32303}, {9140, 32064}, {9833, 30714}, {9899, 32261}, {9914, 16010}, {9924, 32114}, {10060, 32307}, {10076, 32308}, {10193, 15578}, {10282, 17712}, {10297, 15125}, {10991, 14908}, {11178, 11204}, {11579, 12295}, {11748, 32191}, {12106, 15579}, {12173, 32251}, {12202, 32242}, {12250, 32247}, {12262, 32238}, {12278, 32244}, {12289, 32234}, {12324, 15054}, {12335, 32256}, {12367, 12379}, {12468, 32265}, {12469, 32266}, {12502, 32268}, {12584, 15581}, {12779, 32278}, {12791, 32279}, {12920, 32287}, {12930, 32288}, {12940, 32289}, {12950, 32290}, {12986, 32295}, {12987, 32296}, {13093, 32306}, {13094, 32309}, {13095, 32310}, {13148, 32392}, {13293, 15577}, {13383, 15114}, {13568, 22967}, {13980, 32304}, {14094, 34781}, {14216, 16003}, {14643, 23041}, {15115, 23335}, {15738, 16655}, {17847, 24981}, {17854, 19161}, {18383, 20301}, {18440, 20127}, {19087, 32252}, {19088, 32253}, {19140, 34776}, {19467, 32245}, {20427, 32275}, {22778, 32270}, {22802, 32271}, {25564, 35228}, {30443, 32260}, {31383, 34944}, {32273, 34786}, {35864, 35876}, {35865, 35877}

X(36201) = Thomson-isogonal conjugate of X(10423)
X(36201) = crossdifference of every pair of points on line {6, 14396}
X(36201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {154, 15131, 5642}, {1368, 20772, 5972}, {10990, 32250, 67}, {15647, 23315, 5972}, {34135, 34136, 125}

X(36202) =  X(6)X(64)∩X(691)X(11413)

Barycentrics    a^2*(3*a^18 - 8*a^16*b^2 + a^14*b^4 + 13*a^12*b^6 - 9*a^10*b^8 - 3*a^8*b^10 + 3*a^6*b^12 - a^4*b^14 + 2*a^2*b^16 - b^18 - 8*a^16*c^2 + 41*a^14*b^2*c^2 - 46*a^12*b^4*c^2 - 7*a^10*b^6*c^2 + 34*a^8*b^8*c^2 - 37*a^6*b^10*c^2 + 30*a^4*b^12*c^2 + 3*a^2*b^14*c^2 - 10*b^16*c^2 + a^14*c^4 - 46*a^12*b^2*c^4 + 84*a^10*b^4*c^4 - 39*a^8*b^6*c^4 + 65*a^6*b^8*c^4 - 36*a^4*b^10*c^4 - 54*a^2*b^12*c^4 + 25*b^14*c^4 + 13*a^12*c^6 - 7*a^10*b^2*c^6 - 39*a^8*b^4*c^6 - 62*a^6*b^6*c^6 + 7*a^4*b^8*c^6 + 93*a^2*b^10*c^6 - 5*b^12*c^6 - 9*a^10*c^8 + 34*a^8*b^2*c^8 + 65*a^6*b^4*c^8 + 7*a^4*b^6*c^8 - 88*a^2*b^8*c^8 - 9*b^10*c^8 - 3*a^8*c^10 - 37*a^6*b^2*c^10 - 36*a^4*b^4*c^10 + 93*a^2*b^6*c^10 - 9*b^8*c^10 + 3*a^6*c^12 + 30*a^4*b^2*c^12 - 54*a^2*b^4*c^12 - 5*b^6*c^12 - a^4*c^14 + 3*a^2*b^2*c^14 + 25*b^4*c^14 + 2*a^2*c^16 - 10*b^2*c^16 - c^18) : :

X(36202) lies on the cubic K1142 and these lines: {6, 64}, {691, 11413}, {8673, 21733}, {9175, 35905}, {11479, 11637}

X(36202) = reflection of X(35905) in X(9175)
X(36202) = psi-transform of X(35904)

X(36203) =  X(2)X(6)∩X(843)X(22239)

Barycentrics    a^12 + 4*a^10*b^2 + a^8*b^4 - 14*a^6*b^6 + a^4*b^8 + 10*a^2*b^10 - 3*b^12 + 4*a^10*c^2 - 29*a^8*b^2*c^2 + 27*a^6*b^4*c^2 + 55*a^4*b^6*c^2 - 47*a^2*b^8*c^2 + 6*b^10*c^2 + a^8*c^4 + 27*a^6*b^2*c^4 - 120*a^4*b^4*c^4 + 37*a^2*b^6*c^4 + 3*b^8*c^4 - 14*a^6*c^6 + 55*a^4*b^2*c^6 + 37*a^2*b^4*c^6 - 12*b^6*c^6 + a^4*c^8 - 47*a^2*b^2*c^8 + 3*b^4*c^8 + 10*a^2*c^10 + 6*b^2*c^10 - 3*c^12 : :

X(36203) lies on the cubic K1142 and these lines: {2, 6}, {843, 22239}

X(36203) = psi-transform of X (35903)

X(36204) =  X(4)X(32)∩X(125)X(10418)

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

X(36204) lies on the cubic K1142 and these lines: {4, 32}, {125, 10418}, {804, 1637}, {3163, 11177}, {6034, 6793}

X(36204) = {X(98),X(115)}-harmonic conjugate of X(6103)

X(36205) =  X(1)X(4124)∩X(3)X(514)

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

X(36205) lies on these lines: {1,4124), (3,514), (10,19884), (239,379), (355,381), (386,24281), (3008,31184), (3912,30808), (6542,31014), (6547,24159), (34362,35085}

X(36205) = E(X(3),X(514)-antipode of X(3)

X(36206) =  X(63)X(514)∩X(726)X(1478)

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

X(36206) lies on these lines: {63,514), (726,1478}

X(36206) = E(X(3),X(513)-antipode of X(3)

X(36207) =  X(2)X(2452)∩X(3)X(523)

Barycentrics    a^8 - 2*a^6*b^2 + a^2*b^6 - 2*a^6*c^2 + 5*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + a^2*c^6 - 2*b^2*c^6 : :
X(36207) = 5 X[1656] - 4 X[18122]

X(36207) lies on these lines: {2,2452), (3,523), (67,3014), (69,868), (76,31998), (183,892), (325,5094), (338,22143), (381,524), (385,1995), (511,15928), (520,31848), (599,14995), (924,18321), (940,24345), (1316,2407), (1656,18122), (1975,4590), (2453,5467), (2854,12188), (3018,5181), (3163,5972), (4230,9308), (5077,17948), (5108,8371), (5169,7779), (5737,24348), (5912,21448), (6090,14999), (7493,16316), (7697,8542), (8860,18823), (9003,18332), (9035,31953), (9145,13188), (9214,36194), (13881,23991), (14356,34507), (16092,32216), (30882,35148), (31861,32515}

X(36207) = reflection of X(13188) in X(9145)
X(36207) = E(X(3),X(523)-antipode of X(3)

X(36208) =  X(6)X(13)∩X(15)X(74)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - 4*a^2*c^6 + 2*b^2*c^6 + c^8 - 2*Sqrt[3]*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)*S) : :
Barycentrics    Cos[A + Pi/6]^2*Sec[A - Pi/6]*Sin[A] : :

X(36208) lies on the cubics K261a and K390, and on these lines: {6, 13}, {15, 74}, {16, 1511}, {17, 125}, {18, 14643}, {61, 5663}, {62, 110}, {146, 10654}, {202, 10091}, {203, 3028}, {323, 532}, {395, 10272}, {396, 10264}, {397, 32423}, {619, 14972}, {895, 16461}, {1525, 12112}, {1986, 8740}, {2307, 19470}, {2914, 6116}, {3024, 7005}, {3043, 3201}, {3047, 3205}, {3107, 13858}, {3412, 16003}, {5237, 15035}, {5238, 12041}, {5340, 12902}, {5352, 15055}, {5353, 6126}, {5357, 7343}, {5612, 19295}, {5642, 16963}, {6107, 6113}, {6699, 16241}, {7006, 10088}, {7728, 16964}, {8838, 15018}, {9140, 16267}, {10620, 22236}, {10646, 15051}, {10653, 12383}, {10677, 11139}, {11004, 16770}, {11134, 11597}, {11142, 11486}, {11243, 13289}, {12900, 16967}, {13202, 19107}, {15081, 18582}, {16962, 20126}, {16965, 17702}, {22238, 32609}

X(36208) = isogonal conjugate of X(36210)
X(36208) = X(13)-Ceva conjugate of X(6104)
X(36208) = X(i)-isoconjugate of X(j) for these (i,j): {2154, 11092}, {3376, 11600}, {3384, 11582}, {5619, 32679}
X(36208) = crosssum of X(23284) and X(30465)
X(36208) = crossdifference of every pair of points on line {526, 14447}
X(36208) = barycentric product X(i)*X(j) for these {i,j}: {13, 11130}, {16, 11078}, {249, 30460}, {299, 11081}, {3457, 11128}, {6104, 19779}, {11145, 11601}, {17403, 23283}
X(36208) = barycentric quotient X(i)/X(j) for these {i,j}: {16, 11092}, {2088, 30463}, {3201, 11146}, {3457, 11085}, {6104, 16771}, {6138, 23284}, {11078, 301}, {11081, 14}, {11130, 298}, {11134, 6105}, {11142, 11582}, {14560, 5619}, {30460, 338}, {34395, 11086}
X(36208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 10657, 10658}, {16, 11081, 6104}, {12375, 12376, 10657}

X(36209) =  X(6)X(13)∩X(16)X(74)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - 4*a^2*c^6 + 2*b^2*c^6 + c^8 + 2*Sqrt[3]*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)*S) : :
Barycentrics    Cos[A - Pi/6]^2*Sec[A + Pi/6]*Sin[A] : :

X(36209) lies on the cubics K261b and K390 and on these lines: {6, 13}, {15, 1511}, {16, 74}, {17, 14643}, {18, 125}, {61, 110}, {62, 5663}, {146, 10653}, {202, 3028}, {203, 10091}, {323, 533}, {395, 10264}, {396, 10272}, {398, 32423}, {618, 14972}, {895, 16462}, {1524, 12112}, {1986, 8739}, {2914, 6117}, {3024, 7006}, {3043, 3200}, {3047, 3206}, {3106, 13859}, {3411, 16003}, {5237, 12041}, {5238, 15035}, {5339, 12902}, {5351, 15055}, {5353, 7343}, {5357, 6126}, {5616, 19294}, {5642, 16962}, {6106, 6112}, {6699, 16242}, {7005, 10088}, {7127, 7727}, {7728, 16965}, {8836, 15018}, {9140, 16268}, {10620, 22238}, {10645, 15051}, {10654, 12383}, {10678, 11138}, {11004, 16771}, {11137, 11597}, {11141, 11485}, {11244, 13289}, {12900, 16966}, {13202, 19106}, {15081, 18581}, {16963, 20126}, {16964, 17702}, {22236, 32609}

X(36209) = isogonal conjugate of X(36211)
X(36209) = X(14)-Ceva conjugate of X(6105)
X(36209) = X(i)-isoconjugate of X(j) for these (i,j): {2153, 11078}, {3375, 11581}, {3383, 11601}, {5618, 32679}
X(36209) = crosssum of X(23283) and X(30468)
X(36209) = crossdifference of every pair of points on line {526, 14446}
X(36209) = barycentric product X(i)*X(j) for these {i,j}: {14, 11131}, {15, 11092}, {249, 30463}, {298, 11086}, {3458, 11129}, {6105, 19778}, {11146, 11600}, {17402, 23284}
X(36209) = barycentric quotient X(i)/X(j) for these {i,j}: {15, 11078}, {2088, 30460}, {3200, 11145}, {3458, 11080}, {6105, 16770}, {6137, 23283}, {11086, 13}, {11092, 300}, {11131, 299}, {11137, 6104}, {11141, 11581}, {14560, 5618}, {30463, 338}, {34394, 11081}
X(36209) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 10658, 10657}, {15, 11086, 6105}, {12375, 12376, 10658}

X(36210) =  X(13)X(5627)∩X(14)X(16)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - 4*a^2*c^6 + 2*b^2*c^6 + c^8 + 2*Sqrt[3]*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)*S) : :
Barycentrics    4 - 3*Sec[A + Pi/6]^2 : :
X(36210) = X[14] + 2 X[11549], 2 X[14] + X[15743], 4 X[11549] - X[15743]

X(36210) is the perspector of ABC and the reflection of the anticevian triangle of X(14) in the trilinear polar of X(14). (Randy Hutson, January 17, 2020)

X(36210) lies on the cubics K261b, K278, K438, and these lines: {13, 5627}, {14, 16}, {15, 31378}, {18, 36185}, {23, 16464}, {61, 8918}, {186, 6105}, {249, 531}, {323, 533}, {396, 18285}, {398, 8930}, {403, 8738}, {476, 2380}, {523, 22510}, {532, 23896}, {842, 5619}, {1989, 3003}, {2070, 11141}, {3431, 10654}, {5616, 11600}, {5899, 21311}, {6034, 25152}, {6670, 11120}, {8015, 34394}, {14568, 21468}, {16267, 18776}

X(36210) = isogonal conjugate of X(36208)
X(36210) = X(i)-cross conjugate of X(j) for these (i,j): {15, 11600}, {30465, 23284}, {35443, 23896}
X(36210) = X(i)-isoconjugate of X(j) for these (i,j): {13, 1095}, {1101, 30460}, {2152, 11078}, {2153, 11130}, {3375, 6104}
X(36210) = cevapoint of X(23284) and X(30465)
X(36210) = trilinear pole of line {526, 14447}
X(36210) = Kosnita(X(14),X(15)) point
X(36210) = barycentric product X(i)*X(j) for these {i,j}: {14, 11092}, {298, 11085}, {301, 11086}, {470, 10218}, {3268, 5619}, {11582, 19778}, {11600, 16771}, {23284, 23896}
X(36210) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 11078}, {15, 11130}, {115, 30460}, {298, 11128}, {2151, 1095}, {3458, 11081}, {5619, 476}, {6105, 11145}, {11085, 13}, {11086, 16}, {11092, 299}, {11137, 3201}, {11138, 11601}, {11141, 6104}, {11582, 16770}, {11600, 19779}, {16464, 11142}, {20579, 23283}, {23284, 23871}
X(36210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 11085, 11582}, {14, 11549, 15743}, {10218, 11085, 14}, {11543, 34326, 14}

X(36211) =  X(13)X(15)∩X(14)X(5627)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - 4*a^2*c^6 + 2*b^2*c^6 + c^8 - 2*Sqrt[3]*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)*S) : :
Barycentrics    4 - 3*Sec[A - Pi/6]^2 : :
X(36211) = X[13] + 2 X[11537], 2 X[13] + X[11586], 4 X[11537] - X[11586]

X(36211) is the perspector of ABC and the reflection of the anticevian triangle of X(13) in the trilinear polar of X(13). (Randy Hutson, January 17, 2020)

X(36211) lies on the cubics K261a, K278, K438a, and these lines: {13, 15}, {14, 5627}, {16, 31378}, {17, 36186}, {23, 16463}, {62, 8919}, {186, 6104}, {249, 530}, {323, 532}, {395, 18285}, {397, 8929}, {403, 8737}, {476, 2381}, {523, 22511}, {533, 23895}, {842, 5618}, {1989, 3003}, {2070, 11142}, {3431, 10653}, {5612, 11601}, {5899, 21310}, {6034, 25162}, {6669, 11119}, {8014, 34395}, {14568, 21469}, {16268, 18777}

X(36211) = isogonal conjugate of X(36209)
X(36211) = X(i)-cross conjugate of X(j) for these (i,j): {16, 11601}, {30468, 23283}, {35444, 23895}
X(36211) = X(i)-isoconjugate of X(j) for these (i,j): {14, 1094}, {1101, 30463}, {2151, 11092}, {2154, 11131}, {3384, 6105}
X(36211) = cevapoint of X(23283) and X(30468)
X(36211) = trilinear pole of line {526, 14446}
X(36211) = Kosnita(X(13),X(16)) point
X(36211) = barycentric product X(i)*X(j) for these {i,j}: {13, 11078}, {299, 11080}, {300, 11081}, {471, 10217}, {3268, 5618}, {11581, 19779}, {11601, 16770}, {23283, 23895}
X(36211) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 11092}, {16, 11131}, {115, 30463}, {299, 11129}, {2152, 1094}, {3457, 11086}, {5618, 476}, {6104, 11146}, {11078, 298}, {11080, 14}, {11081, 15}, {11134, 3200}, {11139, 11600}, {11142, 6105}, {11581, 16771}, {11601, 19778}, {16463, 11141}, {20578, 23284}, {23283, 23870}
X(36211) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 11080, 11581}, {13, 11537, 11586}, {10217, 11080, 13}, {11542, 34325, 13}

X(36212) =  ISOGONAL CONJUGATE OF X(6531)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
Barycentrics    cos A cos(A + ω) : :
X(36212) = X[237] - 3 X[9155], X[20975] - 3 X[22087]

X(36212) lies on these lines: {2, 39}, {3, 49}, {6, 3964}, {22, 30270}, {32, 1993}, {51, 3095}, {63, 22070}, {69, 216}, {97, 28724}, {99, 401}, {110, 2710}, {114, 2450}, {141, 570}, {147, 8841}, {160, 3313}, {187, 249}, {193, 800}, {232, 297}, {237, 511}, {263, 35439}, {287, 12215}, {311, 14767}, {343, 3933}, {373, 32447}, {441, 525}, {458, 1975}, {491, 8963}, {524, 3003}, {566, 599}, {574, 15066}, {577, 9723}, {801, 9290}, {858, 14981}, {906, 20808}, {1015, 26639}, {1370, 8721}, {1444, 18591}, {1495, 6660}, {1570, 2987}, {1575, 25007}, {1634, 2393}, {1994, 5007}, {2021, 8623}, {2092, 15988}, {2421, 35910}, {2482, 18334}, {2493, 22110}, {2782, 21531}, {2967, 15143}, {2979, 5188}, {3002, 18206}, {3051, 13357}, {3053, 35302}, {3148, 9306}, {3164, 14615}, {3199, 32816}, {3260, 14570}, {3284, 4558}, {3398, 13366}, {3580, 7813}, {3589, 5421}, {3763, 13351}, {3819, 13334}, {3912, 13006}, {4074, 7789}, {4159, 7816}, {5008, 11004}, {5013, 9225}, {5020, 10983}, {5041, 34545}, {5065, 26206}, {5422, 7772}, {5650, 11171}, {5891, 35934}, {6337, 6509}, {6389, 28419}, {6461, 6617}, {6467, 20794}, {6503, 23115}, {6515, 7758}, {6803, 13599}, {7117, 20769}, {7399, 27354}, {7750, 35937}, {7820, 14389}, {7998, 21163}, {8115, 15167}, {8116, 15166}, {8681, 20975}, {8724, 13857}, {9177, 33927}, {9605, 10601}, {10607, 15905}, {11331, 32821}, {11381, 31952}, {11427, 14001}, {11574, 20775}, {12110, 35919}, {13335, 34396}, {14913, 23635}, {15030, 32444}, {15533, 18573}, {15595, 34138}, {16696, 26543}, {18592, 28754}, {18604, 23130}, {20732, 20755}, {20777, 20821}, {21444, 35060}, {21639, 22143}, {21796, 26699}, {22071, 22370}, {22085, 23200}, {22097, 22447}, {22424, 23210}, {23061, 35298}, {24530, 26671}, {32815, 33843}, {32827, 33842}

X(36212) = midpoint of X(i) and X(j) for these {i,j}: {1634, 3001}, {3260, 14570}
X(36212) = reflection of X(3003) in X(34990)
X(36212) = isogonal conjugate of X(6531)
X(36212) = isotomic conjugate of X(16081)
X(36212) = complement of the isogonal conjugate of X(32654)
X(36212) = complement of the isotomic conjugate of X(2987)
X(36212) = isotomic conjugate of the isogonal conjugate of X(3289)
X(36212) = isogonal conjugate of the isotomic conjugate of X(6393)
X(36212) = isotomic conjugate of the polar conjugate of X(511)
X(36212) = isogonal conjugate of the polar conjugate of X(325)
X(36212) = X(i)-complementary conjugate of X(j) for these (i,j): {48, 31842}, {2987, 2887}, {3563, 20305}, {8773, 626}, {8781, 21235}, {9247, 35067}, {32654, 10}, {32697, 21259}, {35364, 21253}, {36051, 141}
X(36212) = X(i)-Ceva conjugate of X(j) for these (i,j): {325, 511}, {2396, 6333}, {4590, 15631}, {17932, 520}
X(36212) = X(3289)-cross conjugate of X(511)
X(36212) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6531}, {4, 1910}, {6, 36120}, {19, 98}, {25, 1821}, {31, 16081}, {34, 15628}, {92, 1976}, {158, 248}, {162, 2395}, {287, 1096}, {290, 1973}, {293, 393}, {336, 2207}, {523, 36104}, {656, 20031}, {661, 685}, {798, 22456}, {811, 2422}, {823, 878}, {879, 24019}, {1577, 32696}, {1969, 14601}, {2489, 36036}, {2501, 36084}, {2715, 24006}, {3404, 32085}, {5967, 36128}, {6520, 17974}, {35906, 36119}
X(36212) = crosspoint of X(i) and X(j) for these (i,j): {2, 2987}, {325, 6393}, {4590, 17932}
X(36212) = crosssum of X(i) and X(j) for these (i,j): {4, 419}, {6, 230}, {25, 2211}, {607, 862}, {3124, 17994}
X(36212) = crossdifference of every pair of points on line {25, 669}
X(36212) = barycentric product X(i)*X(j) for these {i,j}: {3, 325}, {6, 6393}, {63, 1959}, {69, 511}, {76, 3289}, {99, 684}, {110, 6333}, {232, 3926}, {237, 305}, {240, 326}, {248, 32458}, {297, 394}, {304, 1755}, {306, 17209}, {336, 23996}, {520, 877}, {525, 2421}, {647, 2396}, {879, 15631}, {2799, 4558}, {2967, 6394}, {3265, 4230}, {3267, 14966}, {3569, 4563}, {3917, 20022}, {3964, 6530}, {4176, 34854}, {5968, 6390}, {9155, 30786}, {11064, 35910}, {14208, 23997}, {20806, 34138}
X(36212) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36120}, {2, 16081}, {3, 98}, {6, 6531}, {48, 1910}, {63, 1821}, {69, 290}, {99, 22456}, {110, 685}, {112, 20031}, {163, 36104}, {184, 1976}, {219, 15628}, {232, 393}, {237, 25}, {240, 158}, {255, 293}, {287, 34536}, {297, 2052}, {305, 18024}, {325, 264}, {326, 336}, {394, 287}, {446, 12131}, {511, 4}, {520, 879}, {577, 248}, {647, 2395}, {684, 523}, {868, 2970}, {877, 6528}, {895, 9154}, {1092, 17974}, {1576, 32696}, {1755, 19}, {1959, 92}, {2211, 2207}, {2396, 6331}, {2421, 648}, {2491, 2489}, {2799, 14618}, {2967, 6530}, {3049, 2422}, {3284, 35906}, {3289, 6}, {3292, 5967}, {3564, 14265}, {3569, 2501}, {3917, 20021}, {3964, 6394}, {4020, 3404}, {4230, 107}, {4558, 2966}, {4575, 36084}, {4592, 36036}, {5360, 1824}, {5968, 17983}, {5976, 17984}, {6333, 850}, {6393, 76}, {6530, 1093}, {9155, 468}, {9417, 1973}, {9418, 1974}, {9419, 2211}, {9475, 16318}, {9723, 31635}, {10316, 11610}, {11672, 232}, {12215, 14382}, {14251, 17980}, {14356, 6344}, {14575, 14601}, {14585, 14600}, {14966, 112}, {14984, 34175}, {15631, 877}, {17209, 27}, {17970, 34238}, {19189, 8884}, {20806, 31636}, {22115, 14355}, {23098, 2967}, {23996, 240}, {23997, 162}, {32112, 18808}, {32661, 2715}, {34157, 3563}, {34854, 6524}, {35910, 16080}
X(36212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 3117, 1194}, {2967, 15143, 34854}, {3095, 11328, 51}, {3819, 13334, 14096}, {3926, 28710, 39}, {4558, 22151, 3284}, {5408, 5409, 184}, {9306, 9737, 3148}, {9723, 20806, 577}, {13335, 34986, 34396}, {20775, 20819, 11574}, {28417, 28706, 3934}, {28441, 28728, 76}

X(36213) =  X(2)X(98)∩X(3)X(8925)

Barycentrics    a^2*(a^2 - b*c)*(a^2 + b*c)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)
X(36213) = 3 X[2] + X[25046], X[69] + 3 X[25314], 3 X[597] - X[25324], 5 X[3618] - X[25051]

X(36213) lies on the cubic K252 and these lines: {2, 98}, {3, 8925}, {6, 694}, {23, 33873}, {69, 25314}, {99, 25332}, {111, 11175}, {217, 3491}, {237, 511}, {238, 1284}, {263, 576}, {317, 1974}, {323, 7711}, {325, 8840}, {394, 20885}, {419, 3978}, {420, 19128}, {526, 6593}, {575, 34236}, {597, 25324}, {804, 4107}, {1193, 20663}, {1194, 3124}, {1503, 21531}, {1613, 3167}, {1691, 8623}, {1692, 3229}, {1993, 35431}, {1994, 35426}, {2030, 3231}, {2211, 15143}, {2308, 8054}, {2421, 6786}, {2482, 5118}, {2502, 17413}, {2871, 34990}, {3051, 20976}, {3202, 3788}, {3203, 6680}, {3589, 7668}, {3618, 25051}, {4048, 4159}, {5020, 20998}, {5092, 5191}, {5202, 21352}, {5989, 8842}, {6656, 14133}, {6784, 9149}, {7664, 32223}, {8290, 9469}, {11286, 35399}, {14913, 15450}, {14957, 29012}, {16069, 17941}, {18374, 35088}, {20854, 35458}, {35296, 35375}

X(36213) = isogonal conjugate of X(36897)
X(36213) = complement of X(20021)
X(36213) = midpoint of X(i) and X(j) for these {i,j}: {6, 1634}, {20021, 25046}
X(36213) = reflection of X(7668) in X(3589)
X(36213) = complement of the isotomic conjugate of X(20022)
X(36213) = isogonal conjugate of the isotomic conjugate of X(5976)
X(36213) = psi-transform of X(15915)
X(36213) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 8623}, {82, 511}, {237, 16587}, {251, 16609}, {511, 21249}, {1755, 6292}, {1959, 21248}, {3112, 21531}, {3405, 141}, {4599, 24284}, {20022, 2887}, {23997, 3005}, {34072, 2799}
X(36213) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8623}, {6, 511}, {110, 5027}, {4577, 2799}, {4590, 14966}, {14382, 385}, {18020, 17941}
X(36213) = X(i)-isoconjugate of X(j) for these (i,j): {75, 34238}, {92, 15391}, {98, 1581}, {290, 1967}, {336, 17980}, {694, 1821}, {882, 36036}, {1910, 1916}, {1927, 18024}, {1934, 1976}, {3404, 14970}
X(36213) = crosspoint of X(i) and X(j) for these (i,j): {2, 20022}, {6, 1691}, {385, 14382}, {880, 4590}, {4230, 18020}
X(36213) = crosssum of X(i) and X(j) for these (i,j): {2, 1916}, {694, 14251}, {879, 20975}, {881, 3124}, {15391, 34238}
X(36213) = crossdifference of every pair of points on line {694, 804}
X(36213) = X(237)-of-1st-Brocard-triangle
X(36213) = 1st-Brocard-isogonal conjugate of X(34359)
X(36213) = barycentric product X(i)*X(j) for these {i,j}: {6, 5976}, {232, 12215}, {237, 3978}, {325, 1691}, {385, 511}, {804, 2421}, {880, 2491}, {1580, 1959}, {1755, 1966}, {1926, 9417}, {2236, 3405}, {2396, 5027}, {2679, 4590}, {3289, 17984}, {3569, 17941}, {4039, 17209}, {4230, 24284}, {5026, 5968}, {8623, 20022}, {9418, 14603}, {11672, 14382}, {14295, 14966}
X(36213) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34238}, {184, 15391}, {237, 694}, {325, 18896}, {385, 290}, {419, 16081}, {511, 1916}, {1580, 1821}, {1691, 98}, {1755, 1581}, {1933, 1910}, {1959, 1934}, {2211, 17980}, {2421, 18829}, {2491, 882}, {2679, 115}, {2715, 18858}, {3978, 18024}, {4027, 14382}, {5027, 2395}, {5976, 76}, {8623, 20021}, {9417, 1967}, {9418, 9468}, {9419, 14251}, {12829, 14265}, {14602, 1976}, {14966, 805}, {18902, 14601}
X(36213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 25046, 20021}, {110, 1976, 3506}, {182, 3506, 1976}

X(36214) =  ISOGONAL CONJUGATE OF X(419)

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

X(36214) lies on the Jerabek circumhyperbola, the cubics K354, K484, K779, and these lines: {2, 19222}, {3, 1808}, {4, 147}, {6, 694}, {39, 695}, {54, 3398}, {64, 31952}, {65, 291}, {67, 3001}, {69, 20819}, {71, 7015}, {73, 295}, {74, 805}, {98, 23098}, {110, 19576}, {248, 3289}, {290, 325}, {337, 7019}, {511, 8841}, {684, 879}, {733, 907}, {882, 35364}, {1176, 4558}, {1177, 17938}, {1245, 1967}, {2196, 3955}, {2456, 34238}, {3431, 26316}, {6391, 22152}, {8569, 32748}, {10342, 10349}, {14060, 22062}

X(36214) = reflection of X(3511) in X(11672)
X(36214) = isogonal conjugate of X(419)
X(36214) = isotomic conjugate of X(17984)
X(36214) = antitomic image of X(3504)
X(36214) = isotomic conjugate of the isogonal conjugate of X(17970)
X(36214) = isotomic conjugate of the polar conjugate of X(694)
X(36214) = isogonal conjugate of the polar conjugate of X(1916)
X(36214) = X(i)-Ceva conjugate of X(j) for these (i,j): {1916, 694}, {15391, 3}
X(36214) = X(i)-cross conjugate of X(j) for these (i,j): {287, 14941}, {17970, 694}
X(36214) = cevapoint of X(i) and X(j) for these (i,j): {684, 20975}, {3289, 20775}
X(36214) = crosspoint of X(287) and X(8858)
X(36214) = trilinear pole of line {647, 3917}
X(36214) = crossdifference of every pair of points on line {804, 12829}
X(36214) = X(i)-isoconjugate of X(j) for these (i,j): {1, 419}, {4, 1580}, {19, 385}, {25, 1966}, {28, 4039}, {31, 17984}, {92, 1691}, {162, 804}, {171, 242}, {238, 7009}, {239, 7119}, {264, 1933}, {811, 5027}, {862, 17103}, {894, 2201}, {1096, 12215}, {1284, 14006}, {1783, 4107}, {1897, 4164}, {1926, 1974}, {1969, 14602}, {1973, 3978}, {2236, 32085}, {2295, 31905}, {5026, 36128}, {8750, 14296}, {14295, 32676}, {24019, 24284}
X(36214) = barycentric product X(i)*X(j) for these {i,j}: {3, 1916}, {48, 1934}, {63, 1581}, {69, 694}, {76, 17970}, {184, 18896}, {257, 295}, {292, 7019}, {304, 1967}, {305, 9468}, {325, 15391}, {334, 7116}, {335, 7015}, {337, 893}, {525, 805}, {647, 18829}, {733, 3933}, {882, 4563}, {2196, 7018}, {3267, 17938}, {3917, 14970}, {3926, 17980}, {6393, 34238}, {18872, 30786}
X(36214) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17984}, {3, 385}, {6, 419}, {48, 1580}, {63, 1966}, {69, 3978}, {71, 4039}, {184, 1691}, {287, 14382}, {292, 7009}, {295, 894}, {304, 1926}, {305, 14603}, {337, 1920}, {394, 12215}, {520, 24284}, {525, 14295}, {647, 804}, {694, 4}, {733, 32085}, {805, 648}, {881, 2489}, {882, 2501}, {893, 242}, {904, 2201}, {905, 14296}, {1178, 31905}, {1459, 4107}, {1581, 92}, {1808, 27958}, {1911, 7119}, {1916, 264}, {1927, 1973}, {1934, 1969}, {1967, 19}, {2196, 171}, {2311, 14006}, {3049, 5027}, {3292, 5026}, {3917, 732}, {3933, 35540}, {3955, 27982}, {4020, 2236}, {4558, 17941}, {4563, 880}, {7015, 239}, {7019, 1921}, {7116, 238}, {8789, 1974}, {9247, 1933}, {9468, 25}, {14251, 232}, {14575, 14602}, {15391, 98}, {17938, 112}, {17970, 6}, {17980, 393}, {18829, 6331}, {18872, 468}, {18896, 18022}, {20775, 8623}, {22383, 4164}, {34238, 6531}
X(36214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1916, 14251, 6234}, {3493, 6234, 17980}

leftri

Points on permutation ellipses: X(36215)-X(36240)

rightri

Contributed by Clark Kimberling and Peter Moses, January 7, 2020.

Suppose that P = p : q : r (barycentrics) is a point other than X(2) = 1 : 1 : 1 in the plane of a triangle ABC. Let T denote the triangle with vertices

p : q : r
q : r : p
r : p : q

Let T' denote the obverse of T, defined in the preamble just before X(24307) by vertices

p : r : q
q : p : r
r : q : p

The six points, corresponding to the permutations pqr, qrp, rpq, prq, qpr, rqp, lie on the permutation ellipse of P, as defined in the preamble just before X(34341), given by

(q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

If P' lies on E(P), then E(P') = E(P). Moreover, if U = u : v : w is a point, other than P, then the point, other than P', in which the line UP' meets E(P), is the E(P,U)-antipode of P', as defined and formulated in the preamble just before X(35025).

X(36215) =  E(X(69),X(75))-ANTIPODE OF X(69)

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

X(36215) lies on these lines: {2, 35963}, {7, 8}, {57, 27919}, {664, 9263}, {1992, 4762}, {3618, 5701}, {18906, 24280}, {24247, 24282}, {24248, 30228}

X(36216) =  E(X(69),X(513))-ANTIPODE OF X(69)

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

X(36216) lies on these lines: {2, 24289}, {7, 350}, {69, 513}, {75, 4124}, {337, 2310}, {346, 4562}, {536, 1992}, {668, 29349}, {883, 25718}, {1575, 26685}, {3253, 9295}, {3596, 9296}, {3735, 34344}, {17321, 24338}, {17350, 17759}, {18906, 24280}, {24282, 34342}

X(36217) =  E(X(10),X(75))-ANTIPODE OF X(10)

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

X(36217) lies on these lines: {2, 35032}, {10, 75}, {551, 4785}, {1125, 24502}, {3123, 20366}, {4364, 25382}, {9791, 30649}, {24325, 25376}, {24348, 25370}

X(36218) =  E(X(10),X(37))-ANTIPODE OF X(10)

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

X(36218) lies on these lines: {1, 24505}, {2, 35025}, {10, 37}, {86, 9505}, {190, 291}, {551, 28840}, {1086, 11599}, {1125, 23822}, {2054, 4366}, {2486, 20531}, {3923, 24923}, {4364, 24348}, {4472, 25382}, {30571, 35166}

X(36219) =  E(X(10),X(141))-ANTIPODE OF X(10)

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

X(36219) lies on these lines: {2, 35026}, {10, 141}, {86, 33674}, {551, 4762}, {1125, 5701}, {4310, 24418}, {4363, 24327}, {4364, 25375}, {4422, 17793}, {5263, 10030}, {16706, 33676}, {24315, 24346}, {24348, 25359}, {25303, 33677}

X(36220) =  E(X(10),X(4363))-ANTIPODE OF X(10)

Barycentrics    4*a^5 - 3*a^4*b - 7*a^3*b^2 + 8*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c + 14*a^3*b*c - 7*a^2*b^2*c - 10*a*b^3*c + 6*b^4*c - 7*a^3*c^2 - 7*a^2*b*c^2 + 26*a*b^2*c^2 - 7*b^3*c^2 + 8*a^2*c^3 - 10*a*b*c^3 - 7*b^2*c^3 - 3*a*c^4 + 6*b*c^4 + c^5 : :

X(36220) lies on these lines: {2, 24411}, {10, 527}, {522, 551}, {18821, 35154}, {24461, 30331}

X(36221) =  E(X(8),X(75))-ANTIPODE OF X(8)

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

X(36221) lies on these lines: {2, 35026}, {6, 33674}, {7, 8}, {190, 8299}, {1362, 4569}, {3241, 4762}, {3616, 5701}, {4000, 33676}, {4454, 24351}

X(36222) =  E(X(8),X(513))-ANTIPODE OF X(8)

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

X(36222) lies on these lines: {2, 19945}, {8, 513}, {75, 4124}, {76, 9296}, {350, 30947}, {536, 3241}, {545, 17794}, {646, 4014}, {889, 4441}, {995, 3923}, {1026, 3729}, {2230, 3240}, {2345, 24289}, {3123, 26076}, {3685, 24409}, {3952, 17487}, {4419, 24451}, {4454, 24351}, {4947, 26142}, {5695, 34230}, {7283, 24395}, {9263, 24722}, {16495, 24507}, {24485, 27846}, {34363, 35119}

X(36223) =  E(X(8),X(523))-ANTIPODE OF X(8)

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

X(36223) lies on these lines: {2, 24345}, {8, 523}, {45, 10026}, {69, 35960}, {148, 24711}, {325, 30741}, {385, 26228}, {524, 3241}, {599, 35080}, {645, 4934}, {1150, 36207}, {3017, 3821}, {4389, 17731}, {6646, 24338}, {7779, 29832}, {17346, 35148}, {19945, 26840}, {24316, 24351}, {31998, 34016}, {35147, 35152}, {35150, 35154}

X(36224) =  E(X(8),X(4363))-ANTIPODE OF X(8)

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

X(36224) lies on these lines: {2, 24345}, {6, 35085}, {8, 524}, {409, 3304}, {523, 3241}, {17378, 35153}

X(36225) =  E(X(37),X(1))-ANTIPODE OF X(37)

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

X(36225) lies on these lines: {2, 35956}, {10, 37}, {75, 24505}, {335, 668}, {4688, 28840}, {27483, 35173}

X(36226) =  E(X(37),X(514))-ANTIPODE OF X(37)

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

X(36226) lies on these lines: {1, 16377}, {2, 2087}, {10, 19895}, {37, 514}, {86, 6631}, {142, 6547}, {187, 6647}, {239, 16971}, {335, 35103}, {519, 3696}, {551, 35119}, {894, 1016}, {975, 36205}, {1015, 21232}, {1018, 7200}, {1086, 2802}, {2295, 7278}, {3008, 31197}, {3125, 21272}, {3230, 30806}, {3570, 16820}, {3758, 25036}, {3912, 5718}, {4366, 24261}, {4482, 24358}, {4555, 16826}, {4670, 6633}, {4675, 24864}, {6542, 31025}, {7208, 20331}, {8649, 24685}, {9460, 31332}, {10027, 20924}, {13466, 24003}, {14475, 30573}, {16720, 29699}, {17205, 21888}, {17213, 21013}, {17237, 25031}, {17261, 32028}, {17316, 30225}, {17351, 32094}, {17755, 33908}, {18061, 27295}, {24254, 31317}, {29571, 35092}, {29573, 34362}, {31397, 35094}

X(36227) =  E(X(37),X(523))-ANTIPODE OF X(37)

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

X(36227) lies on these lines: {2, 35960}, {6, 24345}, {37, 523}, {86, 35148}, {230, 1108}, {325, 30748}, {385, 26234}, {524, 4688}, {594, 1215}, {1086, 9278}, {1109, 21341}, {1213, 24348}, {2481, 16732}, {3121, 26278}, {5949, 23991}, {7779, 31077}, {17245, 35080}, {21254, 35068}, {35079, 35085}, {35083, 35086}, {35146, 35147}

X(36228) =  E(X(37),X(2))-ANTIPODE OF X(36226)

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

X(36228) lies on these lines: {2, 35960}, {37, 524}, {523, 4688}, {594, 24318}, {18829, 35155}, {24348, 35080}, {31144, 35153}

X(36229) =  E(X(141),X(76))-ANTIPODE OF X(141)

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

X(36229) lies on these lines: {76, 141}, {597, 25423}

X(36230) =  E(X(141),X(514))-ANTIPODE OF X(141)

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

X(37230) lies on these lines: {1, 7829}, {2, 24262}, {10, 19942}, {75, 6547}, {115, 18061}, {141, 514}, {239, 32779}, {257, 6292}, {519, 597}, {543, 17738}, {626, 3061}, {952, 4422}, {1016, 17280}, {1146, 27076}, {3496, 7830}, {3661, 35092}, {3734, 24247}, {3735, 4045}, {3912, 5718}, {4437, 33908}, {4555, 29587}, {6631, 17285}, {6646, 32028}, {6683, 21965}, {17269, 24864}, {17334, 32106}, {17340, 32094}, {29577, 35962}

X(36231) =  E(X(141),X(10))-ANTIPODE OF X(141)

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

X(36231) lies on these lines: {2, 35963}, {10, 141}, {182, 24279}, {597, 4762}, {894, 10030}, {1015, 17044}, {3023, 9317}, {3589, 5701}, {3923, 24256}, {35120, 35961}

X(36232) =  E(X(141),X(513))-ANTIPODE OF X(141)

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

X(36232) lies on these lines: {2, 24289}, {75, 35119}, {76, 24502}, {141, 513}, {142, 20530}, {350, 894}, {536, 597}, {1575, 17353}, {3734, 24279}, {3739, 25382}, {3923, 24256}, {4048, 24265}, {4074, 32930}, {4422, 6184}, {4562, 17280}, {4657, 24338}, {5026, 5150}, {17342, 35123}, {27076, 29349}

X(36233) =  E(X(141),X(37))-ANTIPODE OF X(141)

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

X(36233 lies on these lines: {2, 35964}, {37, 141}

X(36234) =  E(X(141),X(2))-ANTIPODE OF X(36230)

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

X(36234) lies on these lines: {2, 24262}, {141, 519}, {514, 597}, {2796, 24261}, {6633, 17395}, {12035, 27076}, {17367, 35092}, {35121, 35962}

X(36235) =  E(X(141),X(2))-ANTIPODE OF X(36232)

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

X(36235) lies on these lines: {2, 24289}, {141, 536}, {513, 597}, {4664, 35123}

X(36236) =  E(X(100),X(513))-ANTIPODE OF X(100)

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

X(36236) lies on these lines: {8, 36205}, {10, 19893}, {75, 16504}, {80, 519}, {100, 514}, {190, 4777}, {239, 335}, {522, 6163}, {523, 765}, {594, 35085}, {666, 885}, {813, 21832}, {900, 3257}, {901, 6550}, {1016, 3952}, {1897, 18344}, {3008, 31226}, {3240, 24281}, {3699, 6631}, {3799, 14077}, {3888, 9001}, {3912, 30857}, {4360, 18822}, {4427, 32028}, {4555, 4618}, {4562, 35148}, {4756, 32094}, {4767, 6633}, {6542, 31058}, {6547, 33148}, {10695, 30993}, {25725, 27834}, {26227, 35957}

X(36237) =  E(X(100),X(190))-ANTIPODE OF X(100)

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

X(36237) lies on these lines: {11, 4440}, {80, 2796}, {100, 190}, {104, 932}, {144, 528}, {522, 6163}, {537, 1320}, {545, 10707}, {651, 30572}, {765, 4926}, {952, 24844}, {1086, 31272}, {2802, 24821}, {2827, 3888}, {3035, 4473}, {3257, 4777}, {3315, 24416}, {3738, 4499}, {5840, 24817}, {5851, 20533}, {9055, 10755}, {9458, 14193}, {10724, 29243}, {27074, 28743}

X(36238) =  E(X(100),X(513))-ANTIPODE OF X(100)

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

X(36238) lies on these lines: {100, 513}, {190, 1491}, {350, 30993}, {536, 4956}, {660, 2254}, {764, 4555}, {876, 4562}, {889, 35147}, {898, 14419}, {905, 9266}, {1016, 2530}, {1916, 5992}, {3777, 6631}, {3935, 9025}, {4499, 4724}, {4705, 32028}, {4850, 24338}, {17756, 24289}, {25382, 32779}

X(36239) =  E(X(100),X(523))-ANTIPODE OF X(100)

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

X(36239) lies on these lines: {100, 523}, {105, 385}, {325, 30787}, {524, 10707}, {799, 31998}, {891, 2703}, {892, 35147}, {1150, 36207}, {3570, 18014}, {5235, 24348}, {5380, 14431}, {7779, 31126}, {17731, 20347}, {18013, 35154}

X(36240) =  E(X(100),X(2))-ANTIPODE OF X(36236)

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

X(36240) lies on these lines: {36, 100}, {514, 10707}, {518, 3799}, {903, 4777}, {1121, 35167}, {3912, 4767}, {16504, 17342}, {18822, 35153}, {30580, 31992}

leftri

Suren-Moses equilateral-triangle circumcevian-inversion points: X(36241)-X(36244)

rightri

Contributed by Peter Moses, January 9, 2020.

Suren asked Peter Moses for the locus of a point P such that the circumcevian-inversion triangle of P is equilateral. Moses found that the locus consists of four points, all on the Brocard axis, X(3)X(6).

X(36241) =  1st SUREN-MOSES EQUILATERAL-TRIANGLE CIRCUMCEVIAN-INVERSION POINT

Barycentrics    a^2*(4*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 5*b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 8*a^2*c^4 - b^2*c^4 - 5*c^6 - (a^2*b^2 - b^4 + a^2*c^2 - c^4)*(2*Sqrt[3]*S + Sqrt[2*(3*(5*a^4 - 4*a^2*b^2 + 5*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 5*c^4) - 2*Sqrt[3]*(a^2 + b^2 + c^2)*S)])) : :
Barycentrics    Sin[A]^2*(2*Cot[A]*(-4 + Csc[ω^2) + Cos[A + w]*Csc[A]*Csc[ω*(-Sqrt[3] + Cot[ω - Csc[ω*Sqrt[-6 + 15*Cos[2*w] - Sqrt[3]*Sin[2*w]])) : :

The circumcevian-inversion perspector of X(36241) is X(16)

If you have GeoGebra, you can view X(36241).

X(36241) lies on this line: {3,6}

X(36241) = {X(16),X(61)}-harmonic conjugate of X(36242)

X(36242) =  2nd SUREN-MOSES EQUILATERAL-TRIANGLE CIRCUMCEVIAN-INVERSION POINT

Barycentrics    a^2*(4*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 5*b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 8*a^2*c^4 - b^2*c^4 - 5*c^6 - (a^2*b^2 - b^4 + a^2*c^2 - c^4)*(2*Sqrt[3]*S - Sqrt[2*(3*(5*a^4 - 4*a^2*b^2 + 5*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 5*c^4) - 2*Sqrt[3]*(a^2 + b^2 + c^2)*S)])) : :
Barycentrics    Sin[A]^2*(2*Cot[A]*(-4 + Csc[ω^2) + Cos[A + w]*Csc[A]*Csc[ω*(-Sqrt[3] + Cot[ω + Csc[ω*Sqrt[-6 + 15*Cos[2*w] - Sqrt[3]*Sin[2*w]])) : :

The circumcevian-inversion perspector of X(36242) is X(16)

X(36242) lies on this line: {3,6}

X(36242) = {X(16),X(61)}-harmonic conjugate of X(36241)

X(36243) =  3rd SUREN-MOSES EQUILATERAL-TRIANGLE CIRCUMCEVIAN-INVERSION POINT

Barycentrics    a^2*(4*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 5*b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 8*a^2*c^4 - b^2*c^4 - 5*c^6 - (a^2*b^2 - b^4 + a^2*c^2 - c^4)*(-2*Sqrt[3]*S + Sqrt[2*(3*(5*a^4 - 4*a^2*b^2 + 5*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 5*c^4) + 2*Sqrt[3]*(a^2 + b^2 + c^2)*S)])) : :
Barycentrics    Sin[A]^2*(2*Cot[A]*(-4 + Csc[ω]^2) + Cos[A + ω]*Csc[A]*Csc[ω]*(Sqrt[3] + Cot[ω] - Csc[ω]*Sqrt[-6 + 15*Cos[2*ω] + Sqrt[3]*Sin[2*ω]])) : :

The circumcevian-inversion perspector of X(36243) is X(15)

X(36243) lies on this line: {3,6}

X(36243) = {X(15),X(62)}-harmonic conjugate of X(36244)

X(36244) =  4th SUREN-MOSES EQUILATERAL-TRIANGLE CIRCUMCEVIAN-INVERSION POINT

Barycentrics    a^2*(4*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 5*b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 8*a^2*c^4 - b^2*c^4 - 5*c^6 - (a^2*b^2 - b^4 + a^2*c^2 - c^4)*(-2*Sqrt[3]*S - Sqrt[2*(3*(5*a^4 - 4*a^2*b^2 + 5*b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 5*c^4) + 2*Sqrt[3]*(a^2 + b^2 + c^2)*S)])) : :
Barycentrics    Sin[A]^2*(2*Cot[A]*(-4 + Csc[ω]^2) + Cos[A + ω]*Csc[A]*Csc[ω]*(Sqrt[3] + Cot[ω] + Csc[ω]*Sqrt[-6 + 15*Cos[2*ω] + Sqrt[3]*Sin[2*ω]])) : :

The circumcevian-inversion perspector of X(36244) is X(15)

X(36244) lies on this line: {3,6}

X(36244) = {X(15),X(62)}-harmonic conjugate of X(36243)

X(36245) =  X(3)X(161)∩X(4)X(3164)

Barycentrics    SA*((24*R^2-7*SA-SW)*S^2+(SB+SC)*(4*(16*R^2-4*SA-11*SW)*R^2+6*SA^2-6*SB*SC+7*SW^2)) : :
X(36245) = 3*X(381)-2*X(6750)

See Kadir Altintas and César Lozada, Euclid 496 .

X(36245) lies on these lines: {3, 161}, {4, 3164}, {5, 8884}, {381, 6750}, {2888, 31388}, {6638, 15653}, {7691, 35442}, {10539, 18464}, {10745, 31656}, {18403, 24573}, {19206, 32438}, {19210, 32423}, {23606, 34799}

X(36245) = reflection of X(i) in X(j) for these (i,j): (3, 10600), (8884, 5)
X(36245) = anticomplement of X(37846)
X(36245) = X(8884)-of-Johnson-triangle

X(36246) =  X(3)X(618)∩X(13)X(9159)

Barycentrics    S^4+3*(3*(SA+SW)*R^2-SA^2-SB*SC)*S^2-3*SB*SC*SW^2-sqrt(3)*S*((SA-15*R^2)*S^2+3*(3*R^2+SW)*SB*SC) : :

See Kadir Altintas and César Lozada, Euclid 496 .

X(36246) lies on these lines: {3, 618}, {13, 9159}, {616, 3448}

X(36246) = outer-Napoleon-isogonal conjugate of X(62)

X(36247) =  X(3)X(619)∩X(14)X(9159)

Barycentrics    S^4+3*(3*(SA+SW)*R^2-SA^2-SB*SC)*S^2-3*SB*SC*SW^2+sqrt(3)*S*((SA-15*R^2)*S^2+3*(3*R^2+SW)*SB*SC) : :

See Kadir Altintas and César Lozada, Euclid 496 .

X(36247) lies on these lines: {3, 619}, {14, 9159}, {617, 3448}

X(36247) = inner-Napoleon-isogonal conjugate of X(61)

X(36248) =  X(3)X(623)∩X(15)X(1337)

Barycentrics    (SB+SC)*(3*(2*R^2+2*SA-SW)*S^2-3*(SA-2*SW)*SA*SW+S*sqrt(3)*(SA*(6*R^2-3*SA+8*SW)-3*S^2)) : :

See Kadir Altintas and César Lozada, Euclid 496 .

X(36248) lies on these lines: {3, 623}, {15, 1337}, {531, 3439}, {532, 2925}, {3130, 6671}, {5978, 6636}

X(36249) =  X(3)X(624)∩X(16)X(1338)

Barycentrics    (SB+SC)*(3*(2*R^2+2*SA-SW)*S^2-3*(SA-2*SW)*SA*SW-S*sqrt(3)*(SA*(6*R^2-3*SA+8*SW)-3*S^2)) : :

See Kadir Altintas and César Lozada, Euclid 496 .

X(36249) lies on these lines: {3, 624}, {16, 1338}, {530, 3438}, {533, 2926}, {3129, 6672}, {5979, 6636}

X(36250) =  MIDPOINT OF X(58) AND X(24851)

Barycentrics    (b + c)*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c - a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :
X(36250) = X[58] - 3 X[33135], X[1046] - 3 X[3017], X[3704] - 3 X[16052], X[24851] + 3 X[33135]

See Kadir Altintas and Peter Moses, Euclid 517 .

X(36250) lies on these lines: {1, 149}, {5, 2486}, {10, 321}, {12, 4868}, {35, 33133}, {37, 3841}, {58, 24851}, {65, 17705}, {79, 81}, {83, 11599}, {191, 24883}, {192, 30172}, {225, 3671}, {226, 2594}, {386, 3944}, {442, 3743}, {497, 30148}, {516, 3072}, {519, 5015}, {522, 21203}, {536, 25370}, {551, 23536}, {595, 33095}, {596, 29655}, {740, 3454}, {758, 1834}, {846, 24880}, {1010, 1125}, {1046, 3017}, {1070, 5542}, {1072, 4301}, {1193, 11813}, {1203, 5057}, {1210, 1725}, {1386, 22793}, {1479, 1717}, {1698, 6536}, {1724, 33128}, {1738, 3634}, {1785, 6738}, {2392, 18178}, {2796, 8258}, {2887, 2901}, {3122, 24046}, {3125, 23903}, {3178, 4065}, {3336, 33102}, {3434, 30145}, {3583, 5262}, {3585, 17016}, {3647, 35466}, {3663, 10916}, {3666, 25639}, {3670, 33145}, {3672, 31418}, {3673, 21207}, {3678, 4415}, {3704, 16052}, {3752, 3825}, {3755, 21077}, {3772, 5248}, {3782, 3874}, {3822, 3931}, {3824, 15569}, {3891, 4894}, {3923, 20083}, {3946, 18483}, {4197, 27785}, {4276, 30362}, {4309, 26228}, {4418, 25441}, {4658, 33097}, {4719, 9955}, {4850, 7741}, {4857, 5189}, {5259, 33129}, {5264, 33094}, {5270, 17015}, {5292, 16566}, {5312, 31053}, {5721, 31803}, {5904, 33151}, {6675, 17070}, {6693, 24850}, {6701, 17056}, {6757, 16732}, {6763, 33142}, {7683, 29057}, {8728, 27784}, {9664, 16974}, {9791, 25446}, {10479, 32776}, {13408, 16125}, {14815, 21963}, {15171, 17061}, {16600, 21090}, {17147, 30171}, {17719, 33771}, {17720, 25440}, {17733, 31964}, {18398, 33146}, {21956, 28594}, {23542, 24026}, {24068, 29673}, {25270, 30165}, {25542, 26724}

X(36250) = midpoint of X(58) and X(24851)
X(36250) = reflection of X(i) in X(j) for these {i,j}: {21081, 3454}, {24850, 6693}
X(36250) = crosspoint of X(75) and X(1029)
X(36250) = crosssum of X(31) and X(1030)
X(36250) = barycentric product X(10)*X(33150)
X(36250) = barycentric quotient X(33150)/X(86)
X(36250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3120, 11263}, {442, 4854, 3743}, {1089, 4972, 10}, {4424, 21935, 10}, {4442, 5051, 4647}, {4647, 5051, 10}, {4653, 24161, 1125}, {23537, 24210, 1125}, {24851, 33135, 58}, {24883, 33100, 191}

X(36251) =  X(4)X(13)∩X(5)X(39)

Barycentrics    (a^2 + b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) + 2*Sqrt[3]*(a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(36251) = 3 X[13] + X[16964]

See Kadir Altintas and Peter Moses, Euclid 517 .

X(36251) lies on these lines: {2, 33410}, {3, 6108}, {4, 13}, {5, 39}, {6, 20429}, {17, 671}, {62, 530}, {76, 635}, {83, 11603}, {99, 11308}, {382, 22513}, {396, 20415}, {397, 575}, {398, 542}, {597, 31695}, {618, 11290}, {624, 3107}, {627, 14904}, {630, 6669}, {636, 6656}, {3106, 7685}, {5025, 25195}, {5286, 6782}, {5318, 29012}, {5340, 25154}, {5469, 12243}, {5472, 7745}, {5523, 6117}, {5980, 7797}, {6109, 11623}, {6302, 7388}, {6306, 7389}, {6771, 16772}, {7803, 22687}, {7828, 11307}, {7841, 34509}, {11298, 35697}, {11302, 35696}, {11305, 34505}, {12203, 16965}, {16630, 25191}, {16631, 22688}, {22574, 34508}, {33229, 33465}

X(36251) = {X(5),X(5254)}-harmonic conjugate of X(36252)

X(36252) =  X(4)X(14)∩X(5)X(39)

Barycentrics    (a^2 + b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) - 2*Sqrt[3]*(a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*S : :
X(36252) = 3 X[14] + X[16965]

See Kadir Altintas and Peter Moses, Euclid 517 .

X(36252) lies on these lines: {2, 33411}, {3, 6109}, {4, 14}, {5, 39}, {6, 20428}, {18, 671}, {61, 531}, {76, 636}, {83, 11602}, {99, 11307}, {382, 22512}, {395, 20416}, {397, 542}, {398, 575}, {597, 31696}, {619, 11289}, {623, 3106}, {628, 14905}, {629, 6670}, {635, 6656}, {3107, 7684}, {5025, 25191}, {5286, 6783}, {5321, 29012}, {5339, 25164}, {5470, 12243}, {5471, 7745}, {5523, 6116}, {5981, 7797}, {6108, 11623}, {6303, 7388}, {6307, 7389}, {6774, 16773}, {7803, 22689}, {7828, 11308}, {7841, 34508}, {11297, 35693}, {11301, 35692}, {11306, 34505}, {12203, 16964}, {16630, 22690}, {16631, 25195}, {22573, 34509}, {33229, 33464}

X(36252) = {X(5),X(5254)}-harmonic conjugate of X(36251)

X(36253) =  COMPLEMENT OF X(30714)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^8 - 2*a^6*b^2 + a^4*b^4 - 4*a^2*b^6 + 3*b^8 - 2*a^6*c^2 + 4*a^2*b^4*c^2 - 12*b^6*c^2 + a^4*c^4 + 4*a^2*b^2*c^4 + 18*b^4*c^4 - 4*a^2*c^6 - 12*b^2*c^6 + 3*c^8) : :
X(36253) = 5 X[2] - 3 X[11693], 9 X[2] - 5 X[15034], X[3] - 3 X[125], X[3] + 3 X[265], 2 X[3] - 3 X[6699], 7 X[3] - 3 X[12121], 5 X[3] + 3 X[12902], X[3] - 5 X[15027], 5 X[3] - 9 X[15061], 5 X[3] - 3 X[16163], X[4] + 3 X[9140], 3 X[4] - 7 X[15044], 3 X[4] + X[15054], 3 X[5] - X[5609], X[23] + 3 X[25739], 3 X[67] + X[11477], 3 X[74] + X[3146], 3 X[110] - 7 X[3090], 3 X[110] - 11 X[15025], X[110] - 5 X[15081], X[110] - 3 X[23515], 3 X[113] - 5 X[3091], 3 X[113] - X[14094], X[113] - 3 X[14644], 3 X[115] - X[31854], 7 X[125] - X[12121], 5 X[125] + X[12902], 3 X[125] - 5 X[15027], 5 X[125] - 3 X[15061], 5 X[125] - X[16163], 3 X[125] - 2 X[20397], 2 X[265] + X[6699], 7 X[265] + X[12121], 5 X[265] - X[12902], 3 X[265] + 5 X[15027], 5 X[265] + 3 X[15061], 5 X[265] + X[16163], 3 X[265] + 2 X[20397], 3 X[376] - 7 X[15057], 3 X[381] - X[15063], X[382] + 3 X[20126], 3 X[399] - 11 X[5072], 2 X[546] - 3 X[7687], X[546] - 3 X[11801], 3 X[568] - X[14448], 2 X[575] - 3 X[15118], X[575] - 3 X[20301], 5 X[632] - 3 X[1511], 5 X[632] - 6 X[6723], 3 X[1209] - X[25714], X[1493] - 3 X[11804], 5 X[1656] - 3 X[5642], 15 X[1656] - 7 X[15039], 5 X[1656] - X[23236], 3 X[2072] - X[3292], 7 X[3090] - 6 X[12900], 7 X[3090] - 11 X[15025], 7 X[3090] - 15 X[15081], 7 X[3090] - 9 X[23515], 5 X[3091] + 3 X[3448], 5 X[3091] - X[14094], 5 X[3091] - 9 X[14644], X[3146] - 3 X[12295], 3 X[3448] + X[14094], X[3448] + 3 X[14644], 11 X[3525] - 3 X[12383], 11 X[3525] - 7 X[15020], 11 X[3525] - 15 X[15059], X[3529] + 3 X[10733], X[3529] - 5 X[15021], X[3529] - 3 X[16111], X[3627] - 3 X[10113], X[3627] + 3 X[10264], 4 X[3628] - 3 X[5972], 5 X[3628] - 3 X[13392], 2 X[3628] - 3 X[20304], 7 X[3832] - 3 X[10706], 7 X[3851] - 3 X[5655], 11 X[5056] - 3 X[9143], 5 X[5076] + 3 X[10620], 5 X[5076] - 3 X[13202], 13 X[5079] - 9 X[14643], 13 X[5079] - 3 X[24981], 3 X[5095] - 5 X[11482], 4 X[5159] - 3 X[14156], 2 X[5462] - 3 X[12099], 2 X[5609] - 3 X[16534], 3 X[5627] + X[17511], 9 X[5642] - 7 X[15039], 3 X[5642] - X[23236], 5 X[5972] - 4 X[13392], 3 X[6053] - 8 X[12811], 7 X[6699] - 2 X[12121], 5 X[6699] + 2 X[12902], 3 X[6699] - 10 X[15027], 5 X[6699] - 6 X[15061], 5 X[6699] - 2 X[16163], 3 X[6699] - 4 X[20397], X[7982] + 3 X[13211], 9 X[9140] + 7 X[15044], 9 X[9140] - X[15054], 3 X[9140] - X[16003], X[9716] + 3 X[11564], X[10222] - 3 X[12261], 3 X[10272] - 5 X[12812], 13 X[10303] - 9 X[15035], 7 X[10541] - 3 X[32233], 3 X[10733] + 5 X[15021], X[10990] - 3 X[20126], 5 X[11482] + 3 X[32306], 27 X[11693] - 25 X[15034], 9 X[11693] - 5 X[30714], 3 X[11735] - 2 X[15178], 3 X[12041] - X[15704], 3 X[12099] - X[25711], 2 X[12105] - 3 X[32223], 5 X[12121] + 7 X[12902], 3 X[12121] - 35 X[15027], 5 X[12121] - 21 X[15061], 5 X[12121] - 7 X[16163], 3 X[12121] - 14 X[20397], 3 X[12236] - 2 X[16625], 3 X[12383] - 7 X[15020], X[12383] - 5 X[15059], 5 X[12812] - 6 X[15088], 6 X[12900] - 11 X[15025], 2 X[12900] - 5 X[15081], 2 X[12900] - 3 X[23515], 3 X[12902] + 25 X[15027], X[12902] + 3 X[15061], 3 X[12902] + 10 X[20397], 2 X[13392] - 5 X[20304], 3 X[13851] - X[18323], X[14094] - 9 X[14644], 3 X[14643] - X[24981], 3 X[14683] - 19 X[15022], 3 X[14708] - 4 X[15012], 3 X[14852] + X[15133], 7 X[14869] - 9 X[34128], 7 X[14869] - 3 X[34153], X[14982] + 3 X[25330], 7 X[15020] - 15 X[15059], 5 X[15021] - 3 X[16111], 11 X[15025] - 15 X[15081], 11 X[15025] - 9 X[23515], 25 X[15027] - 9 X[15061], 25 X[15027] - 3 X[16163], 5 X[15027] - 2 X[20397], 5 X[15034] - 3 X[30714], 7 X[15039] - 3 X[23236], 7 X[15044] + X[15054], 7 X[15044] + 3 X[16003], X[15054] - 3 X[16003], 9 X[15055] - 5 X[17538], 3 X[15061] - X[16163], 9 X[15061] - 10 X[20397], 5 X[15081] - 3 X[23515], 3 X[15113] - 4 X[32767], 3 X[15359] - 2 X[20398], X[15801] + 3 X[33565], 3 X[16163] - 10 X[20397], 4 X[20398] - 3 X[33511], 4 X[20399] - 3 X[33512], 3 X[34128] - X[34153]

See Kadir Altintas and Peter Moses, Euclid 517 .

X(36253) lies on these lines: {2, 11693}, {3, 125}, {4, 541}, {5, 542}, {23, 25739}, {30, 15153}, {67, 11477}, {68, 895}, {74, 3146}, {110, 569}, {113, 3091}, {115, 31854}, {140, 20396}, {155, 32272}, {376, 15057}, {381, 15063}, {382, 10990}, {389, 546}, {399, 5072}, {539, 2072}, {567, 32235}, {568, 14448}, {576, 12585}, {578, 15132}, {632, 1511}, {690, 24978}, {1209, 7550}, {1216, 14984}, {1493, 11804}, {1656, 5642}, {1995, 18474}, {2777, 3627}, {2781, 5446}, {2854, 10170}, {3303, 12904}, {3304, 12903}, {3518, 12140}, {3525, 12383}, {3529, 10733}, {3628, 5972}, {3832, 10706}, {3851, 5655}, {3861, 13393}, {5056, 9143}, {5076, 10620}, {5079, 14643}, {5095, 11482}, {5159, 14156}, {5448, 25738}, {5462, 12099}, {5504, 15077}, {5621, 7387}, {5627, 17511}, {5654, 25320}, {6053, 12811}, {6070, 36184}, {6425, 8994}, {6426, 13969}, {6427, 19051}, {6428, 19052}, {6453, 35835}, {6454, 35834}, {7530, 18381}, {7556, 11750}, {7575, 18400}, {7577, 11422}, {7706, 26869}, {7723, 21649}, {7982, 13211}, {9716, 11564}, {9976, 15068}, {10024, 18128}, {10112, 10224}, {10114, 32136}, {10222, 12261}, {10272, 12812}, {10297, 13754}, {10303, 15035}, {10541, 32233}, {10628, 12236}, {11178, 16511}, {11559, 17505}, {11565, 34577}, {11645, 16619}, {11704, 34799}, {11735, 15178}, {12041, 15704}, {12105, 32223}, {12241, 15114}, {12319, 32263}, {12370, 15113}, {12897, 20299}, {13148, 23047}, {13399, 31726}, {13403, 13561}, {13851, 18323}, {14683, 15022}, {14708, 15012}, {14791, 32273}, {14869, 34128}, {14982, 25330}, {15055, 17538}, {15545, 16278}, {15801, 33565}, {18390, 31861}, {18571, 30522}, {20303, 33547}, {20399, 33512}, {23306, 23307}

X(36253) = complement of X(30714)
X(36253) = midpoint of X(i) and X(j) for these {i,j}: {4, 16003}, {74, 12295}, {113, 3448}, {125, 265}, {382, 10990}, {895, 32275}, {3861, 13393}, {5095, 32306}, {6070, 36184}, {7723, 21649}, {10113, 10264}, {10620, 13202}, {10733, 16111}, {12319, 32263}, {12902, 16163}, {13399, 31726}, {15545, 16278}, {25328, 32274}
X(36253) = reflection of X(i) in X(j) for these {i,j}: {3, 20397}, {110, 12900}, {140, 20396}, {1511, 6723}, {5972, 20304}, {6699, 125}, {7687, 11801}, {10272, 15088}, {11557, 11746}, {15118, 20301}, {16534, 5}, {20417, 20379}, {25711, 5462}, {33511, 15359}
X(36253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 125, 20397}, {3, 15027, 125}, {3, 20397, 6699}, {4, 9140, 16003}, {110, 15025, 3090}, {110, 15081, 23515}, {110, 23515, 12900}, {125, 16163, 15061}, {265, 15027, 3}, {265, 15061, 12902}, {382, 20126, 10990}, {1656, 23236, 5642}, {3090, 15025, 23515}, {3090, 15081, 15025}, {3091, 3448, 14094}, {3091, 14094, 113}, {3448, 14644, 113}, {3525, 12383, 15020}, {3529, 15021, 16111}, {9140, 15044, 15054}, {10733, 15021, 3529}, {12099, 25711, 5462}, {12902, 15061, 16163}, {14094, 14644, 3091}, {15020, 15059, 3525}, {15044, 15054, 4}

X(36254) =  X(110)X(15766)∩X(399)X(14354)

Barycentrics    a^2*(5*a^26 - 47*a^24*b^2 + 186*a^22*b^4 - 374*a^20*b^6 + 275*a^18*b^8 + 495*a^16*b^10 - 1716*a^14*b^12 + 2508*a^12*b^14 - 2277*a^10*b^16 + 1375*a^8*b^18 - 550*a^6*b^20 + 138*a^4*b^22 - 19*a^2*b^24 + b^26 - 47*a^24*c^2 + 360*a^22*b^2*c^2 - 1164*a^20*b^4*c^2 + 2030*a^18*b^6*c^2 - 2025*a^16*b^8*c^2 + 1224*a^14*b^10*c^2 - 840*a^12*b^12*c^2 + 1188*a^10*b^14*c^2 - 1305*a^8*b^16*c^2 + 800*a^6*b^18*c^2 - 252*a^4*b^20*c^2 + 30*a^2*b^22*c^2 + b^24*c^2 + 186*a^22*c^4 - 1164*a^20*b^2*c^4 + 3108*a^18*b^4*c^4 - 4530*a^16*b^6*c^4 + 3834*a^14*b^8*c^4 - 1860*a^12*b^10*c^4 + 438*a^10*b^12*c^4 + 72*a^8*b^14*c^4 - 60*a^6*b^16*c^4 - 144*a^4*b^18*c^4 + 174*a^2*b^20*c^4 - 54*b^22*c^4 - 374*a^20*c^6 + 2030*a^18*b^2*c^6 - 4530*a^16*b^4*c^6 + 5360*a^14*b^6*c^6 - 3494*a^12*b^8*c^6 + 1026*a^10*b^10*c^6 + 26*a^8*b^12*c^6 - 206*a^6*b^14*c^6 + 642*a^4*b^16*c^6 - 746*a^2*b^18*c^6 + 266*b^20*c^6 + 275*a^18*c^8 - 2025*a^16*b^2*c^8 + 3834*a^14*b^4*c^8 - 3494*a^12*b^6*c^8 + 1791*a^10*b^8*c^8 - 357*a^8*b^10*c^8 - 278*a^6*b^12*c^8 - 612*a^4*b^14*c^8 + 1491*a^2*b^16*c^8 - 625*b^18*c^8 + 495*a^16*c^10 + 1224*a^14*b^2*c^10 - 1860*a^12*b^4*c^10 + 1026*a^10*b^6*c^10 - 357*a^8*b^8*c^10 + 588*a^6*b^10*c^10 + 228*a^4*b^12*c^10 - 2100*a^2*b^14*c^10 + 783*b^16*c^10 - 1716*a^14*c^12 - 840*a^12*b^2*c^12 + 438*a^10*b^4*c^12 + 26*a^8*b^6*c^12 - 278*a^6*b^8*c^12 + 228*a^4*b^10*c^12 + 2340*a^2*b^12*c^12 - 372*b^14*c^12 + 2508*a^12*c^14 + 1188*a^10*b^2*c^14 + 72*a^8*b^4*c^14 - 206*a^6*b^6*c^14 - 612*a^4*b^8*c^14 - 2100*a^2*b^10*c^14 - 372*b^12*c^14 - 2277*a^10*c^16 - 1305*a^8*b^2*c^16 - 60*a^6*b^4*c^16 + 642*a^4*b^6*c^16 + 1491*a^2*b^8*c^16 + 783*b^10*c^16 + 1375*a^8*c^18 + 800*a^6*b^2*c^18 - 144*a^4*b^4*c^18 - 746*a^2*b^6*c^18 - 625*b^8*c^18 - 550*a^6*c^20 - 252*a^4*b^2*c^20 + 174*a^2*b^4*c^20 + 266*b^6*c^20 + 138*a^4*c^22 + 30*a^2*b^2*c^22 - 54*b^4*c^22 - 19*a^2*c^24 + b^2*c^24 + c^26) : :

See Kadir Altintas and Peter Moses, Euclid 518 .

X(36254) lies on these lines: {110, 15766}, {399, 14354}

X(36255) =  X(2)X(526)∩X(115)X(125)

Barycentrics    (b^2 - c^2)*(a^8*b^2 - 4*a^6*b^4 + 5*a^4*b^6 - 2*a^2*b^8 + a^8*c^2 + 2*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - a^2*b^6*c^2 + b^8*c^2 - 4*a^6*c^4 - 2*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - b^6*c^4 + 5*a^4*c^6 - a^2*b^2*c^6 - b^4*c^6 - 2*a^2*c^8 + b^2*c^8) : :
X(36255) = 3 X[15061] - 2 X[16235]

See Minh Trịnh Xuân and Peter Moses, Euclid 519 .

X(36255) lies on these lines: {2, 526}, {30, 19902}, {94, 5466}, {110, 11176}, {114, 9189}, {115, 125}, {351, 542}, {512, 32225}, {523, 3580}, {684, 1649}, {804, 9138}, {1499, 11799}, {2492, 6792}, {2780, 20126}, {3049, 3231}, {3448, 9147}, {5652, 9517}, {5653, 9169}, {5663, 19912}, {9033, 11123}, {9188, 34319}, {12828, 17994}, {15061, 16235}

X(36255) = midpoint of X(i) and X(j) for these {i,j}: {3448, 9147}, {9138, 9140}
X(36255) = reflection of X(i) in X(j) for these {i,j}: {110, 11176}, {9148, 125}, {13291, 1637}, {34319, 9188}
X(36255) = crossdifference of every pair of points on line {110, 3016}.

leftri

TC(X(i),X(j)-antipodes: X(36256)-X(36295)

rightri

This preamble and centers X(36256)-X(36295) were contributed by Clark Kimberling and Peter Moses, January 13, 2020.

In this paragraph, all coordinates are trilinears. Suppose that P = p : q : r. The trilinear permutation conic denoted by TC(P), is the conic that passes through the six points

p : q : r,    q : r : p,    r : p : q,    p : r : q,    q : p : r,   r : q : p.

An equation for TC(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

Thus, TC(P) is analogous, and symbolically identical to, the permutation ellipse E(P), defined in the preamble just before X(34341).

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Suppose that P = p : q : r (trilinears). For the rest of this paragraph, p:q:r are trilinears, but all other coordinates, and the equation for TC(P), are in barycentric coordinates. The conic TC(P) passes through the six points

ap : bq : cr,    aq : br : cp,    ar : bp : cq,    ap : br : cq,    aq : bp : cr,   ar : bq : cp.

(Note that these six points are not six permutations of ap: bq : cr.)

An equation for TC(P) is (q r + r p + p q)(b^2 c^2 x^2 + c^2 a^2 y^2 + a^2 b^2 z^2) - abc(p^2 + q^2 + r^2)(ayz + bzx + cxy) = 0.

**************************

The TC(P,U)-antipode of P is the point, other than P, in which the line PU meets TC(P), where

P = p : q : r (trilinears) = ap : bq : cr (barycentrics)
U = u : v : w (trilinears) = au : bv : cw (barycentrics)

Barycentrics for TC(P,U)-antipode of P are f(a,b,c,p,q,r,u,v,w) : f(b,c,a,q,r,p,v,w,u) : f(c,a,b,r,p,q,w,u,v), where

f(a,b,c,p,q,r,u,v,w) = a (b^2 c^2 q^3 u^2 - b^2 c^2 p q r u^2 + b^2 c^2 q^2 r u^2 + b^2 c^2 q r^2 u^2 + b^2 c^2 r^3 u^2 - 2 a b c^2 p q^2 u v + a b c^2 p^2 r u v - 2 a b c^2 p q r u v - a b c^2 q^2 r u v + a b c^2 r^3 u v + a^2 c^2 p^2 q v^2 + a^2 c^2 p^2 r v^2 + a^2 c^2 p q r v^2 + a b^2 c p^2 q u w + a b^2 c q^3 u w - 2 a b^2 c p q r u w - 2 a b^2 c p r^2 u w - a b^2 c q r^2 u w - a^2 b c p^3 v w - a^2 b c p q^2 v w - a^2 b c p r^2 v w + a^2 b^2 p^2 q w^2 + a^2 b^2 p^2 r w^2 + a^2 b^2 p q r w^2)

X(36256) =  TC(X(1),X(75))-ANTIPODE OF X(1)

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

X(36256) lies on these lines: {6, 75}, {660, 29936}

X(36257) =  TC(X(1),X(76))-ANTIPODE OF X(1)

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

X(36257) lies on this line: {6, 76}

X(36258) =  TC(X(1),X(31))-ANTIPODE OF X(1)

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

X(36258) lies on these lines: {1, 9321}, {6, 31}, {37, 9318}, {45, 4724}, {100, 294}, {244, 292}, {813, 24484}, {1447, 26242}, {10025, 17261}, {20672, 20999}

X(36259) =  TC(X(1),X(514))-ANTIPODE OF X(1)

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

X(36259) lies on this line: {6, 514}

X(36260) =  TC(X(1),X(10))-ANTIPODE OF X(1)

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

X(36260) lies on this line: {6, 10}

X(36261) =  TC(X(1),X(3))-ANTIPODE OF X(1)

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

X(36261) lies on these lines: {1, 2609}, {3, 6}, {37, 2607}, {45, 3709}, {662, 34990}, {692, 7669}, {13006, 21004}

X(36262) =  TC(X(6),X(2))-ANTIPODE OF X(6)

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

X(36262) lies one this line: {2, 31}

X(36263) =  TC(X(6),X(1))-ANTIPODE OF X(6)

Barycentrics    a*(a^2 - 2*b^2 - 2*c^2) : :
Trilinears    4 SA + SB + SC : :
Trilinears    4 cot A + cot B + cot C : :

X(36263) lies on these lines: {1, 21}, {2, 18201}, {8, 32845}, {9, 244}, {10, 17679}, {11, 17334}, {37, 9345}, {44, 4003}, {45, 672}, {48, 2157}, {57, 756}, {69, 32848}, {75, 799}, {100, 28563}, {141, 33161}, {171, 7226}, {190, 30942}, {192, 32919}, {201, 1106}, {210, 9350}, {238, 4392}, {320, 29643}, {333, 17155}, {345, 33081}, {518, 2177}, {527, 24725}, {537, 26227}, {599, 4141}, {612, 3928}, {614, 3929}, {678, 35445}, {726, 1150}, {748, 982}, {750, 984}, {752, 29832}, {899, 5220}, {902, 3242}, {940, 3989}, {960, 32577}, {976, 3916}, {988, 3951}, {1001, 17449}, {1193, 3927}, {1253, 7004}, {1357, 5650}, {1401, 3690}, {1473, 5217}, {1647, 4679}, {1742, 13243}, {1757, 4850}, {1836, 29690}, {1958, 16556}, {2173, 16567}, {2234, 3116}, {2239, 3240}, {2308, 17599}, {2886, 33098}, {2895, 32855}, {3006, 4655}, {3120, 17276}, {3210, 32864}, {3220, 7302}, {3305, 18193}, {3315, 15485}, {3617, 26034}, {3626, 33074}, {3634, 26061}, {3662, 33115}, {3663, 33128}, {3666, 4663}, {3681, 5524}, {3683, 21342}, {3688, 3937}, {3703, 33080}, {3705, 4683}, {3722, 16496}, {3741, 32933}, {3750, 4430}, {3782, 17070}, {3821, 33114}, {3842, 26627}, {3870, 17782}, {3920, 4650}, {3935, 17601}, {3938, 4640}, {3977, 33156}, {3980, 4981}, {3999, 15254}, {4001, 32852}, {4363, 30970}, {4389, 29631}, {4390, 4475}, {4415, 29662}, {4419, 11269}, {4427, 32941}, {4438, 17184}, {4641, 17017}, {4722, 5256}, {4847, 33094}, {4880, 30116}, {4884, 32854}, {5057, 29676}, {5204, 7085}, {5223, 21805}, {5269, 9340}, {5278, 24165}, {5285, 5370}, {5294, 19862}, {5550, 26065}, {5695, 31136}, {5708, 28274}, {5718, 5852}, {5745, 33127}, {5905, 33105}, {6646, 25760}, {6682, 26223}, {7191, 7262}, {7225, 7237}, {9780, 32781}, {10404, 21674}, {10453, 32936}, {11680, 33099}, {14829, 32925}, {15481, 16610}, {15650, 27627}, {16477, 17025}, {16704, 32921}, {17063, 27065}, {17122, 23958}, {17127, 17598}, {17135, 32934}, {17147, 17162}, {17149, 18075}, {17165, 32916}, {17274, 29857}, {17347, 32843}, {17350, 32944}, {17435, 32578}, {17483, 33111}, {17484, 17717}, {17591, 32911}, {17764, 21283}, {17767, 21242}, {17768, 33104}, {17770, 33070}, {17771, 31034}, {17772, 31303}, {17781, 24239}, {17897, 20879}, {18249, 23675}, {20068, 32920}, {20078, 26098}, {21320, 30944}, {21582, 23665}, {21808, 31442}, {24248, 33136}, {24349, 32917}, {24627, 32931}, {24723, 33120}, {25957, 26840}, {26102, 33761}, {26279, 30800}, {27184, 33119}, {28082, 31445}, {29641, 33067}, {29664, 33097}, {29671, 32859}, {29673, 32950}, {29680, 33096}, {29828, 31161}, {29849, 33066}, {31302, 32927}, {31330, 32939}, {32776, 33121}, {32782, 33167}, {32784, 33170}, {32849, 33087}, {32851, 33065}, {32857, 33108}, {32862, 33085}, {32863, 33092}, {32865, 33102}, {32918, 32937}, {33064, 33113}, {33068, 33117}, {33069, 33116}, {33082, 33089}, {33083, 33169}, {33084, 33168}, {33086, 33165}, {33100, 33141}, {33118, 33125}, {33137, 33145}, {33138, 33146}, {33139, 33149}, {33140, 33151}, {33142, 33154}, {33143, 35466}, {33164, 33172}, {33166, 33174}

X(36263) = {X(1),X(63)}-harmonic conjugate of X(896)

X(36264) =  TC(X(6),X(75))-ANTIPODE OF X(6)

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

X(36264) lies on this line: {31, 75}

X(36265) =  TC(X(6),X(6))-ANTIPODE OF X(6)

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

X(36265) lies on these lines: {6, 31}, {38, 9318}, {190, 2310}, {982, 1447}

X(36266) =  TC(X(6),X(561))-ANTIPODE OF X(6)

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

X(36266) lies on these lines: {31, 561}

X(36267) =  TC(X(6),X(513))-ANTIPODE OF X(6)

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

X(36267) lies on these lines: {1, 4585}, {6, 24405}, {31, 513}, {43, 765}, {44, 4003}, {190, 24294}, {238, 993}, {244, 2265}, {320, 29658}, {560, 1423}, {651, 3248}, {692, 3123}, {748, 16482}, {1964, 9414}, {2161, 4475}, {3257, 16468}, {3573, 24338}, {5091, 19945}, {6163, 32911}, {15988, 17445}

X(36268) =  TC(X(75),X(75))-ANTIPODE OF X(75)

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

X(36268) lies on these lines: {2, 37}, {190, 20671}, {291, 3248}, {3097, 3764}, {3226, 20467}, {3240, 4782}

X(36269) =  TC(X(75),X(6))-ANTIPODE OF X(75)

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

X(36269) lies on these lines: {2, 6}, {42, 3571}, {190, 25054}, {2108, 3882}, {2276, 24504}, {3240, 4784}, {13576, 25051}

X(36270) =  TC(X(75),X(76))-ANTIPODE OF X(75)

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

X(36270) lies on these lines: {2, 39}, {24482, 24513}

X(36271) =  TC(X(75),X(31))-ANTIPODE OF X(75)

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

X(36271) lies on this line: {2, 31}

X(36272) =  TC(X(75),X(561))-ANTIPODE OF X(75)

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

X(36272) lies on this line: {2, 561}

X(36273) =  TC(X(75),X(514))-ANTIPODE OF X(75)

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

X(36273) lies on these lines: {2, 514}, {3097, 3240}

X(36274) =  TC(X(75),X(9))-ANTIPODE OF X(75)

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

X(36274) lies on these lines: {1, 3}, {31, 2111}, {100, 23622}, {1026, 3501}, {3573, 9310}

X(36275) =  TC(X(75),X(3758))-ANTIPODE OF X(75)

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

X(36275) lies on these lines: {1, 24482}, {2, 44}, {6, 3257}, {190, 33908}, {513, 3240}, {903, 20972}, {1017, 4604}, {2087, 24874}, {16505, 24405}, {24004, 24524}

X(36276) =  TC(X(69),X(2))-ANTIPODE OF X(69)

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

X(36276) lies on these lines: {2, 7}, {651, 9359}

X(36277) =  TC(X(69),X(1))-ANTIPODE OF X(69)

Barycentrics    a*(5*a^2 - b^2 - c^2) : :
Trilinears    cot A' : cot B' : cot C', where A'B'C' is the circumsymmedial triangle

X(36277) lies on these lines: {1, 21}, {2, 15601}, {6, 4689}, {9, 5297}, {19, 162}, {44, 4386}, {55, 4663}, {57, 7292}, {69, 35263}, {92, 8765}, {100, 1743}, {109, 1445}, {110, 7341}, {165, 32911}, {171, 3305}, {193, 35261}, {204, 1748}, {238, 3306}, {516, 24597}, {527, 26228}, {560, 4575}, {612, 7262}, {614, 4650}, {672, 3240}, {748, 9340}, {752, 29857}, {902, 3751}, {1331, 21059}, {1740, 2617}, {1836, 17070}, {2234, 19591}, {2308, 17594}, {3011, 24695}, {3052, 3870}, {3218, 7290}, {3219, 5269}, {3246, 4860}, {3315, 16487}, {3474, 26723}, {3550, 5524}, {3617, 26065}, {3626, 33163}, {3634, 26034}, {3683, 5287}, {3731, 9347}, {3749, 32912}, {3875, 4427}, {3886, 16704}, {3920, 3929}, {3928, 7191}, {3951, 5266}, {4008, 14206}, {4252, 19861}, {4257, 35262}, {4312, 33129}, {4414, 16475}, {4640, 5256}, {4652, 16466}, {4654, 29681}, {4655, 29855}, {4672, 29828}, {4850, 16469}, {5204, 7293}, {5217, 5314}, {5282, 16676}, {5294, 9780}, {5329, 5370}, {5363, 24436}, {5573, 23958}, {7295, 7302}, {9352, 23511}, {9580, 33142}, {14212, 17871}, {16467, 23832}, {16468, 17779}, {16477, 17601}, {17162, 32929}, {17274, 26230}, {17298, 24542}, {17884, 18750}, {19872, 32781}, {25734, 32926}, {28570, 30811}, {28609, 29665}

X(36277) = trilinear product X(2)*X(1384)

X(36278) =  TC(X(69),X(513))-ANTIPODE OF X(69)

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

X(36278) lies on these lines: {1, 4585}, {8, 24410}, {9, 1026}, {44, 4386}, {63, 513}, {75, 24411}, {78, 7299}, {100, 2265}, {190, 9355}, {200, 765}, {238, 997}, {320, 5231}, {518, 2099}, {644, 1156}, {752, 1757}, {1052, 16569}, {1332, 2310}, {1743, 33760}, {1776, 23691}, {2161, 9024}, {2170, 10755}, {3257, 5223}, {3305, 16482}, {3681, 6163}, {3799, 16561}, {3888, 16560}, {8245, 18042}, {9282, 30721}

X(36279) =  TC(X(63),X(1))-ANTIPODE OF X(63)

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

X(36279) lies on these lines: {1, 3}, {4, 653}, {5, 1788}, {6, 5011}, {7, 495}, {8, 2094}, {10, 527}, {29, 8762}, {30, 3474}, {44, 169}, {45, 2245}, {47, 18360}, {63, 3753}, {72, 9709}, {75, 5774}, {78, 4018}, {79, 10895}, {80, 12943}, {88, 957}, {109, 5398}, {140, 3485}, {145, 10031}, {208, 1872}, {218, 2246}, {226, 26446}, {244, 16483}, {329, 442}, {355, 4292}, {377, 3421}, {381, 1737}, {382, 1770}, {388, 5690}, {392, 3306}, {404, 5730}, {412, 1148}, {474, 3869}, {496, 962}, {497, 28174}, {498, 3649}, {499, 18493}, {516, 5722}, {548, 4305}, {550, 3486}, {553, 3654}, {595, 17054}, {758, 1376}, {851, 3240}, {938, 6361}, {952, 4293}, {956, 3218}, {958, 3754}, {959, 19513}, {960, 16408}, {971, 30353}, {993, 3919}, {997, 16417}, {1001, 5883}, {1004, 3868}, {1046, 2640}, {1056, 21454}, {1058, 20070}, {1071, 8544}, {1158, 7686}, {1191, 24046}, {1210, 9669}, {1254, 3157}, {1330, 5827}, {1387, 6966}, {1393, 34040}, {1406, 23070}, {1448, 23072}, {1452, 1598}, {1478, 5790}, {1571, 31461}, {1597, 1905}, {1656, 12047}, {1657, 10572}, {1698, 31142}, {1706, 5784}, {1708, 6913}, {1721, 21848}, {1739, 4383}, {1768, 6797}, {1854, 3357}, {1940, 7524}, {2096, 9799}, {2097, 2810}, {2160, 2911}, {2173, 19350}, {2178, 21863}, {2182, 16670}, {2362, 3311}, {2651, 11116}, {2771, 18397}, {2800, 22753}, {3052, 30117}, {3058, 18530}, {3085, 6147}, {3086, 22791}, {3214, 7352}, {3242, 33844}, {3244, 34639}, {3297, 35610}, {3298, 35611}, {3312, 16232}, {3452, 3634}, {3476, 5844}, {3488, 9778}, {3526, 11375}, {3560, 7098}, {3586, 28146}, {3600, 6955}, {3621, 4190}, {3625, 17647}, {3626, 5794}, {3671, 6684}, {3679, 4880}, {3683, 16857}, {3697, 3951}, {3715, 19875}, {3812, 8257}, {3833, 8167}, {3843, 4338}, {3851, 17606}, {3870, 24473}, {3871, 36003}, {3874, 3913}, {3877, 27003}, {3878, 25524}, {3911, 5886}, {3916, 4004}, {3928, 9623}, {3959, 5021}, {4042, 4714}, {4084, 12635}, {4125, 4942}, {4187, 11415}, {4294, 12433}, {4298, 11362}, {4299, 10950}, {4301, 11373}, {4306, 5399}, {4312, 5587}, {4314, 17706}, {4315, 28234}, {4317, 10944}, {4333, 17800}, {4413, 5692}, {4511, 4930}, {4513, 17736}, {4640, 16418}, {4654, 31434}, {4663, 34371}, {4695, 32912}, {4757, 22836}, {4784, 29126}, {4792, 16944}, {4870, 15694}, {4887, 10521}, {4973, 11194}, {5030, 34522}, {5044, 12526}, {5055, 17605}, {5057, 17556}, {5218, 5719}, {5219, 11231}, {5229, 6917}, {5248, 33815}, {5250, 5439}, {5265, 10595}, {5289, 35272}, {5434, 12647}, {5435, 5603}, {5530, 9566}, {5550, 7483}, {5691, 12684}, {5694, 5780}, {5703, 16137}, {5704, 6831}, {5721, 5753}, {5727, 28160}, {5731, 11041}, {5806, 12705}, {5837, 12436}, {5884, 11500}, {5887, 6918}, {5901, 7288}, {5905, 17757}, {6001, 19541}, {6675, 28629}, {6692, 19862}, {6738, 31730}, {6875, 17097}, {6911, 14988}, {7319, 10308}, {7354, 10573}, {7672, 18450}, {7682, 10893}, {7702, 11929}, {7743, 31162}, {7951, 11552}, {8147, 15852}, {8614, 16473}, {8727, 14647}, {8732, 20330}, {9579, 18480}, {9580, 18527}, {9581, 22793}, {9612, 9956}, {9613, 31776}, {9948, 31673}, {10039, 10404}, {10044, 34502}, {10427, 34619}, {10580, 15170}, {10738, 12832}, {11019, 28194}, {11230, 31231}, {11359, 33068}, {11495, 30329}, {11496, 31870}, {11499, 12738}, {11551, 17718}, {11570, 12331}, {11670, 12308}, {11682, 17614}, {12515, 12736}, {12560, 31658}, {12664, 15239}, {12709, 31837}, {13996, 34749}, {14974, 20271}, {16466, 24443}, {16863, 25917}, {17532, 20292}, {17634, 31937}, {17728, 30384}, {17732, 21049}, {18467, 34474}, {19872, 20196}, {20214, 32635}, {28212, 30305}, {28349, 28370}, {33298, 33865}, {34637, 34717}

X(36279) = {X(1),X(3)}-harmonic conjugate of X(37606)

X(36280) =  TC(X(63),X(513))-ANTIPODE OF X(63)

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

X(36280) lies on these lines; {3, 513}, {8, 6163}, {11, 22148}, {44, 169}, {56, 87}, {320, 17181}, {405, 24482}, {518, 1351}, {522, 36205}, {651, 15507}, {752, 11236}, {764, 1083}, {765, 5687}, {956, 3257}, {1052, 24174}, {1757, 5903}, {3939, 29349}, {4124, 24395}, {4357, 19927}, {4645, 11681}, {11108, 16482}, {11248, 15310}, {17770, 24220}, {22161, 24703}, {23772, 24846}, {23981, 34048}

X(36281) =  TC(X(10),X(2))-ANTIPODE OF X(10)

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

X(36281) lies on these lines: {2, 37}, {2109, 29821}, {4782, 16666}, {6377, 20467}

X(36282) =  TC(X(31),X(2))-ANTIPODE OF X(31)

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

X(36282) lies on this line: {2, 32}

X(36283) =  TC(X(31),X(1))-ANTIPODE OF X(31)

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

X(36283) lies on these lines: {1, 32}, {39, 17595}, {44, 16583}, {45, 2245}, {712, 24586}, {993, 21331}, {1015, 22448}, {1500, 22426}, {2240, 3240}, {3125, 5282}, {3617, 26085}, {3670, 7772}, {3782, 5309}, {3878, 9351}, {4346, 5286}, {4797, 30105}, {4799, 7818}, {5289, 8649}, {7815, 18055}, {7867, 17211}, {9650, 21965}, {17451, 31456}, {17601, 31451}, {26770, 30579}, {30945, 33952}

X(36284) =  TC(X(31),X(75))-ANTIPODE OF X(31)

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

X(36284) lies on this line: {32, 75}

X(36285) =  TC(X(31),X(6))-ANTIPODE OF X(31)

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

X(36285) lies on these lines: {3, 6}, {2607, 3721}

X(36286) =  TC(X(31),X(76))-ANTIPODE OF X(31)

Barycentrics    a*(-(a^12*b^9) + a^15*b^3*c^3 - a^12*b^6*c^3 - a^9*b^9*c^3 - a^3*b^15*c^3 - a^12*b^3*c^6 + 4*a^6*b^9*c^6 - b^15*c^6 - a^12*c^9 - a^9*b^3*c^9 + 4*a^6*b^6*c^9 + 3*a^3*b^9*c^9 - b^12*c^9 - b^9*c^12 - a^3*b^3*c^15 - b^6*c^15) : :

X(36286) lies on this line: {32, 76}

X(36287) =  TC(X(31),X(513))-ANTIPODE OF X(31)

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

X(36287) lies on these lines: {32, 513}, {44, 16583}, {238, 2275}, {518, 5028}

X(36288) =  TC(X(76),X(2))-ANTIPODE OF X(76)

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

X(36288) lies on these lines: {2, 37}, {7032, 34252}

X(36289) =  TC(X(76),X(1))-ANTIPODE OF X(76)

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

X(36289) lies on these lines: {1, 75}, {31, 662}, {43, 3248}, {45, 2235}, {48, 33760}, {82, 1958}, {87, 872}, {674, 24598}, {749, 22277}, {869, 3758}, {896, 3116}, {897, 2186}, {1575, 3240}, {2279, 16670}, {2309, 4687}, {2664, 17335}, {3009, 4664}, {3056, 24530}, {3264, 7976}, {3617, 26042}, {3759, 7032}, {3783, 17360}, {3809, 17369}, {3873, 16726}, {3941, 27644}, {7184, 17361}, {7189, 17366}, {7321, 25570}, {17872, 18041}, {28358, 28370}

X(36290) =  TC(X(76),X(6))-ANTIPODE OF X(76)

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

X(36290) lies on this line: {6, 75}

X(36291) =  TC(X(76),X(76))-ANTIPODE OF X(76)

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

X(36291) lies on this line: {10, 75}

X(36292) =  TC(X(76),X(31))-ANTIPODE OF X(76)

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

X(36292) lies on this line: {31, 75}

X(36293) =  TC(X(76),X(561))-ANTIPODE OF X(76)

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

X(36293) lies on these lines: {38, 75}, {1581, 4117}

X(36294) =  TC(X(76),X(513))-ANTIPODE OF X(76)

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

X(36294) lies on these lines: {9, 660}, {37, 24482}, {75, 513}, {87, 30663}, {144, 145}, {190, 9016}, {238, 7032}, {335, 3271}, {512, 35957}, {765, 15624}, {1654, 2113}, {2975, 9414}, {3248, 20332}, {3797, 9025}, {3873, 24403}, {3888, 17755}, {4687, 16482}, {9470, 17000}

X(36295) =  TC(X(76),X(1577))-ANTIPODE OF X(76)

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

X(36295) lies on this line: {75, 1577}

X(36296) =  X(4)X(13)∩X(15)X(74)

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

X(36296) lies on the Jerabek circumhyperbola, the cubic K1145a, and these lines: {2, 2993}, {4, 13}, {6, 3130}, {15, 74}, {16, 3431}, {54, 62}, {64, 22236}, {65, 2153}, {67, 30454}, {184, 5158}, {216, 21647}, {265, 10217}, {290, 300}, {577, 32585}, {1989, 8015}, {2992, 3180}, {3003, 34394}, {3129, 11243}, {3426, 11485}, {3527, 21310}, {5238, 11270}, {5612, 33565}, {6138, 15453}, {8603, 11080}, {10645, 20421}, {11063, 11136}, {11738, 34754}, {14528, 22238}, {15328, 20578}, {15851, 19364}

X(36296) = isogonal conjugate of X(470)
X(36296) = isogonal conjugate of the complement of X(19772)
X(36296) = isotomic conjugate of the polar conjugate of X(3457)
X(36296) = isogonal conjugate of the polar conjugate of X(13)
X(36296) = X(13)-Ceva conjugate of X(3457)
X(36296) = X(i)-isoconjugate of X(j) for these (i,j): {1, 470}, {15, 92}, {19, 298}, {75, 8739}, {162, 23870}, {264, 2151}, {340, 2154}, {472, 3384}, {811, 6137}, {1969, 34394}, {2167, 6117}, {2190, 33529}, {2349, 6110}, {17402, 24006}
X(36296) = cevapoint of X(6) and X(11243)
X(36296) = crosssum of X(i) and X(j) for these (i,j): {15, 8739}, {6111, 23714}, {6137, 30465}
X(36296) = crossdifference of every pair of points on line {6110, 6782}
X(36296) = X(11079)-Ceva conjugate of X(36297)
X(36296) = X(3284)-cross conjugate of X(36297)
X(36296) = homothetic center of X(14)- and X(15)-Ehrmann triangles
X(36296) = barycentric product X(i)*X(j) for these {i,j}: {3, 13}, {15, 10217}, {16, 265}, {63, 2153}, {69, 3457}, {184, 300}, {328, 34395}, {394, 8737}, {525, 5995}, {647, 23895}, {4558, 20578}, {5612, 15392}, {8838, 32585}, {9206, 14417}, {10218, 36208}, {11077, 33530}, {14582, 17403}, {15398, 30454}, {16770, 32586}, {23871, 32662}
X(36296) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 298}, {6, 470}, {13, 264}, {16, 340}, {32, 8739}, {51, 6117}, {184, 15}, {216, 33529}, {265, 301}, {300, 18022}, {647, 23870}, {1495, 6110}, {2153, 92}, {3049, 6137}, {3130, 11094}, {3457, 4}, {5995, 648}, {8737, 2052}, {8740, 14165}, {9247, 2151}, {10217, 300}, {11060, 8738}, {11081, 471}, {11083, 473}, {11142, 472}, {14575, 34394}, {20578, 14618}, {20975, 30465}, {23895, 6331}, {30452, 2970}, {30454, 34336}, {32586, 19778}, {32661, 17402}, {32662, 23896}, {34395, 186}
X(36296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 11081, 3457}, {6, 11142, 11083}, {15, 5668, 35469}, {184, 5158, 36297}, {11081, 11083, 11142}, {11083, 11142, 3457}

X(36297) =  X(4)X(14)∩X(16)X(74)

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

X(36297) lies on the Jerabek circumhyperbola, the cubic K1145b, and these lines: {2, 2992}, {4, 14}, {6, 3129}, {15, 3431}, {16, 74}, {54, 61}, {64, 22238}, {65, 2154}, {67, 30455}, {184, 5158}, {216, 21648}, {265, 10218}, {290, 301}, {577, 32586}, {1989, 8014}, {2993, 3181}, {3003, 34395}, {3130, 11244}, {3426, 11486}, {3527, 21311}, {5237, 11270}, {5616, 33565}, {6137, 15453}, {8604, 11085}, {10646, 20421}, {11063, 11135}, {11738, 34755}, {14528, 22236}, {15328, 20579}, {15851, 19363}

X(36297) = isogonal conjugate of X(471)
X(36297) = isogonal conjugate of the complement of X(19773)
X(36297) = isotomic conjugate of the polar conjugate of X(3458)
X(36297) = isogonal conjugate of the polar conjugate of X(14)
X(36297) = X(14)-Ceva conjugate of X(3458)
X(36297) = X(11079)-Ceva conjugate of X(36296)
X(36297) = X(3284)-cross conjugate of X(36296)
X(36297) = homothetic center of X(13)- and X(16)-Ehrmann triangles
X(36297) = X(i)-isoconjugate of X(j) for these (i,j): {1, 471}, {16, 92}, {19, 299}, {75, 8740}, {162, 23871}, {264, 2152}, {340, 2153}, {473, 3375}, {811, 6138}, {1969, 34395}, {2167, 6116}, {2190, 33530}, {2349, 6111}, {17403, 24006}
X(36297) = cevapoint of X(6) and X(11244)
X(36297) = crosssum of X(i) and X(j) for these (i,j): {16, 8740}, {6110, 23715}, {6138, 30468}
X(36297) = crossdifference of every pair of points on line {6111, 6783}
X(36297) = barycentric product X(i)*X(j) for these {i,j}: {3, 14}, {15, 265}, {16, 10218}, {63, 2154}, {69, 3458}, {184, 301}, {328, 34394}, {394, 8738}, {525, 5994}, {647, 23896}, {4558, 20579}, {5616, 15392}, {8836, 32586}, {9207, 14417}, {10217, 36209}, {11077, 33529}, {14582, 17402}, {15398, 30455}, {16771, 32585}, {23870, 32662}
X(36297) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 299}, {6, 471}, {14, 264}, {15, 340}, {32, 8740}, {51, 6116}, {184, 16}, {216, 33530}, {265, 300}, {301, 18022}, {647, 23871}, {1495, 6111}, {2154, 92}, {3049, 6138}, {3129, 11093}, {3458, 4}, {5994, 648}, {8738, 2052}, {8739, 14165}, {9247, 2152}, {10218, 301}, {11060, 8737}, {11086, 470}, {11088, 472}, {11141, 473}, {14575, 34395}, {20579, 14618}, {20975, 30468}, {23896, 6331}, {30453, 2970}, {30455, 34336}, {32585, 19779}, {32661, 17403}, {32662, 23895}, {34394, 186}
X(36297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 11086, 3458}, {6, 11141, 11088}, {16, 5669, 35470}, {184, 5158, 36296}, {11086, 11088, 11141}, {11088, 11141, 3458}

X(36298) =  X(2)X(19777)∩X(4)X(14)

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

X(36298) lies on the cubic K1145a and these lines: {2, 19777}, {4, 14}, {6, 8015}, {13, 5627}, {15, 1138}, {298, 5641}, {301, 14387}, {381, 10218}, {396, 523}, {477, 5994}, {1495, 3081}, {1989, 3457}, {2154, 2306}, {3180, 11092}, {3458, 34288}, {9154, 9207}, {10217, 14993}, {10654, 15442}

X(36298) = X(13)-Ceva conjugate of X(8015)
X(36298) = X(1989)-Ceva conjugate of X(36299)
X(36298) = X(3163)-cross conjugate of X(36299)
X(36298) = {X(1495),X(18487)}-harmonic conjugate of X(36299)
X(36298) = X(i)-isoconjugate of X(j) for these (i,j): {16, 2349}, {299, 2159}, {471, 35200}, {1494, 2152}, {23871, 36034}, {33805, 34395}
X(36298) = crosspoint of X(1989) and X(11085)
X(36298) = crosssum of X(323) and X(11130)
X(36298) = barycentric product X(i)*X(j) for these {i,j}: {14, 30}, {15, 14254}, {265, 6110}, {298, 14583}, {301, 1495}, {1637, 23896}, {2154, 14206}, {2407, 20579}, {3260, 3458}, {6111, 10218}, {8738, 11064}
X(36298) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 1494}, {30, 299}, {1495, 16}, {1637, 23871}, {1990, 471}, {2154, 2349}, {2420, 17403}, {3458, 74}, {6110, 340}, {8738, 16080}, {9406, 2152}, {9407, 34395}, {14254, 300}, {14398, 6138}, {14581, 8740}, {14583, 13}, {16240, 6111}, {20579, 2394}, {30453, 12079}, {34394, 14385}
X(36298) = {X(6),X(11085)}-harmonic conjugate of X(8015)

X(36299) =  X(2)X(19776)∩X(4)X(13)

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

X(36299) lies on the cubic K1145b and these lines: {2, 19776}, {4, 13}, {6, 8014}, {14, 5627}, {16, 1138}, {299, 5641}, {300, 14387}, {381, 10217}, {395, 523}, {477, 5995}, {1495, 3081}, {1989, 3458}, {2153, 33654}, {3181, 11078}, {3457, 34288}, {9154, 9206}, {10218, 14993}, {10653, 15441}

X(36299) = X(14)-Ceva conjugate of X(8014)
X(36299) = X(1989)-Ceva conjugate of X(36298)
X(36299) = X(3163)-cross conjugate of X(36298)
X(36299) = {X(1495),X(18487)}-harmonic conjugate of X(36298)
X(36299) = X(i)-isoconjugate of X(j) for these (i,j): {15, 2349}, {298, 2159}, {470, 35200}, {1494, 2151}, {23870, 36034}, {33805, 34394}
X(36299) = crosspoint of X(1989) and X(11080)
X(36299) = crosssum of X(323) and X(11131)
X(36299) = barycentric product X(i)*X(j) for these {i,j}: {13, 30}, {16, 14254}, {265, 6111}, {299, 14583}, {300, 1495}, {1637, 23895}, {2153, 14206}, {2407, 20578}, {3260, 3457}, {6110, 10217}, {8737, 11064}
X(36299) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 1494}, {30, 298}, {1495, 15}, {1637, 23870}, {1990, 470}, {2153, 2349}, {2420, 17402}, {3457, 74}, {6111, 340}, {8737, 16080}, {9406, 2151}, {9407, 34394}, {14254, 301}, {14398, 6137}, {14581, 8739}, {14583, 14}, {16240, 6110}, {20578, 2394}, {30452, 12079}, {34395, 14385}
X(36299) = {X(6),X(11080)}-harmonic conjugate of X(8014)

X(36300) =  X(2)X(19712)∩X(4)X(15)

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

X(36300) lies on the conic {{A,B,C,X(4),X(5)}}, the cubic K1145a, and these lines: {2, 19712}, {4, 15}, {6, 11087}, {13, 1141}, {18, 1487}, {51, 3078}, {62, 3459}, {252, 8175}, {303, 32036}, {327, 34389}, {2165, 21461}, {2963, 3457}, {3180, 11144}, {6117, 13450}, {8018, 8603}, {8172, 10646}, {11600, 16809}

X(36300) = X(233)-cross conjugate of X(36301)
X(36300) = X(i)-isoconjugate of X(j) for these (i,j): {61, 2167}, {302, 2148}, {473, 2169}, {23872, 36134}
X(36300) = barycentric product X(i)*X(j) for these {i,j}: {5, 17}, {51, 34389}, {62, 25043}, {311, 21461}, {324, 32585}, {343, 8741}, {11087, 33530}, {11139, 33529}, {12077, 32036}, {16806, 18314}
X(36300) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 302}, {17, 95}, {51, 61}, {53, 473}, {3199, 10642}, {8741, 275}, {12077, 23872}, {16806, 18315}, {21461, 54}, {25043, 34390}, {32585, 97}, {33530, 11132}, {34389, 34384}
X(36300) = {X(17),X(8741)}-harmonic conjugate of X(32585)

X(36301) =  X(2)X(19713)∩X(4)X(16)

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

X(36301) lies on the conic {{A,B,C,X(4),X(5)}}, the cubic K1145b, and these lines: {2, 19713}, {4, 16}, {6, 11082}, {14, 1141}, {17, 1487}, {51, 3078}, {61, 3459}, {252, 8174}, {302, 32037}, {327, 34390}, {2165, 21462}, {2963, 3458}, {3181, 11143}, {6116, 13450}, {8019, 8604}, {8173, 10645}, {11601, 16808}

X(36301) = X(233)-cross conjugate of X(36300)
X(36301) = X(i)-isoconjugate of X(j) for these (i,j): {62, 2167}, {303, 2148}, {472, 2169}, {23873, 36134}
X(36301) = barycentric product X(i)*X(j) for these {i,j}: {5, 18}, {51, 34390}, {61, 25043}, {311, 21462}, {324, 32586}, {343, 8742}, {11082, 33529}, {11138, 33530}, {12077, 32037}, {16807, 18314}
X(36301) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 303}, {18, 95}, {51, 62}, {53, 472}, {3199, 10641}, {8742, 275}, {12077, 23873}, {16807, 18315}, {21462, 54}, {25043, 34389}, {32586, 97}, {33529, 11133}, {34390, 34384}
X(36301) = {X(18),X(8742)}-harmonic conjugate of X(32586)

X(36302) =  X(2)X(19775)∩X(4)X(6)

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

X)36302) lies on the cubic K1145a and these lines: {2, 19775}, {4, 6}, {13, 6117}, {15, 5667}, {62, 3462}, {470, 11542}, {471, 11486}, {621, 648}, {622, 17907}, {1080, 16318}, {3180, 11093}, {3457, 10633}, {6110, 19106}, {9308, 11303}

X(36302) = polar conjugate of X(19774)
X(36302) = polar conjugate of the isotomic conjugate of X(19772)
X(36302) = polar conjugate of the isogonal conjugate of X(11243)
X(36302) = X(13)-Ceva conjugate of X(4)
X(36302) = X(11243)-cross conjugate of X(19772)
X(36302) = X(48)-isoconjugate of X(19774)
X(36302) = crosspoint of X(13) and X(8919)
X(36302) = barycentric product X(i)*X(j) for these {i,j}: {4, 19772}, {264, 11243}, {470, 8919}, {472, 8175}
X(36302) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 19774}, {10641, 8479}, {11243, 3}, {19772, 69}
X(36302) = {X(393),X(5335)}-harmonic conjugate of X(4)
X(36302) = {X(4),X(1249)}-harmonic conjugate of X(36303)

X(36303) =  X(2)X(19774)∩X(4)X(6)

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

X(36303) lies on the cubic K1145b and these lines: {2, 19774}, {4, 6}, {14, 6116}, {16, 5667}, {61, 3462}, {383, 16318}, {470, 11485}, {471, 11543}, {621, 17907}, {622, 648}, {3181, 11094}, {3458, 10632}, {6111, 19107}, {9308, 11304}

X(36303) = polar conjugate of X(19775)
X(36303) = polar conjugate of the isotomic conjugate of X(19773)
X(36303) = polar conjugate of the isogonal conjugate of X(11244)
X(36303) = X(14)-Ceva conjugate of X(4)
X(36303) = X(11244)-cross conjugate of X(19773)
X(36303) = X(48)-isoconjugate of X(19775)
X(36303) = crosspoint of X(14) and X(8918)
X(36303) = barycentric product X(i)*X(j) for these {i,j}: {4, 19773}, {264, 11244}, {471, 8918}, {473, 8174}
X(36303) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 19775}, {10642, 8471}, {11244, 3}, {19773, 69}
X(36303) = {X(393),X(5334)}-harmonic conjugate of X(4)
X(36303) = {X(4),X(1249)}-harmonic conjugate of X(36302)

X(36304) =  X(2)X(17)∩X(6)X(11087)

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

X(36304) lies on the cubic k1145a and these lines: {2, 17}, {6, 11087}, {13, 11600}, {15, 8172}, {396, 15802}, {2981, 22900}, {3087, 8741}, {5335, 8174}, {5472, 12077}, {8603, 11080}, {9112, 16806}, {32627, 34008}, {34565, 36305}

X(36304) = X(13)-Ceva conjugate of X(36300)
X(36304) = crosspoint of X(17) and X(11139)
X(36304) = crosssum of X(i) and X(j) for these (i,j): {61, 11146}, {11126, 33526}
X(36304) = barycentric product X(i)*X(j) for these {i,j}: {17, 396}, {532, 11087}, {618, 11139}
X(36304) = barycentric quotient X(i)/X(j) for these {i,j}: {396, 302}, {463, 473}, {532, 11132}, {8014, 8838}, {11087, 11117}, {11139, 11119}, {16806, 10409}, {21461, 2981}

X(36305) =  X(2)X(18)∩X(6)X(11082)

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

X(36305) lies on the cubic K1145b and these lines: {2, 18}, {6, 11082}, {14, 11601}, {16, 8173}, {395, 15778}, {3087, 8742}, {5334, 8175}, {5471, 12077}, {6151, 22856}, {8604, 11085}, {9113, 16807}, {32628, 34009}, {34565, 36304}

X(36305) = X(14)-Ceva conjugate of X(36301)
X(36305) = crosspoint of X(18) and X(11138)
X(36305) = crosssum of X(i) and X(j) for these (i,j): {62, 11145}, {11127, 33527}
X(36305) = barycentric product X(i)*X(j) for these {i,j}: {18, 395}, {533, 11082}, {619, 11138}
X(36305) = barycentric quotient X(i)/X(j) for these {i,j}: {395, 303}, {462, 472}, {533, 11133}, {8015, 8836}, {11082, 11118}, {11138, 11120}, {16807, 10410}, {21462, 6151}

X(36306) =  TRILINEAR POLE OF LINE X(4)X(13)

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

X(36306) lies on the Simmons circumconic (perspector X(13)) and these lines: {4, 21466}, {13, 470}, {107, 5995}, {112, 476}, {297, 11078}, {463, 18384}, {473, 8014}, {648, 23895}, {685, 20578}, {1304, 5618}, {1990, 11537}, {3457, 16081}, {6110, 11586}, {6117, 11581}, {8737, 17983}, {10217, 36302}

X(36306) = polar conjugate of X(23870)
X(36306) = polar conjugate of the isotomic conjugate of X(23895)
X(36306) = polar conjugate of the isogonal conjugate of X(5995)
X(36306) = X(i)-cross conjugate of X(j) for these (i,j): {4240, 36309}, {5995, 23895}, {10633, 250}, {36302, 32230}
X(36306) = X(i)-isoconjugate of X(j) for these (i,j): {15, 656}, {48, 23870}, {63, 6137}, {298, 810}, {470, 822}, {525, 2151}, {2154, 8552}, {3708, 17402}, {4575, 30465}, {8739, 24018}, {9204, 36060}, {14208, 34394}, {32679, 36297}
X(36306) = cevapoint of X(i) and X(j) for these (i,j): {463, 2501}, {523, 5318}, {3457, 20578}
X(36306) = trilinear pole of line {4, 13}
X(36306) = barycentric product X(i)*X(j) for these {i,j}: {4, 23895}, {13, 648}, {99, 8737}, {112, 300}, {264, 5995}, {471, 476}, {811, 2153}, {3457, 6331}, {6344, 17403}, {6528, 36296}, {8740, 35139}, {16077, 36299}, {18020, 20578}
X(36306) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 23870}, {13, 525}, {16, 8552}, {25, 6137}, {107, 470}, {112, 15}, {250, 17402}, {300, 3267}, {462, 14447}, {463, 35443}, {468, 9204}, {471, 3268}, {648, 298}, {2153, 656}, {2501, 30465}, {3457, 647}, {5618, 10217}, {5995, 3}, {6111, 5664}, {6138, 16186}, {8737, 523}, {8738, 23284}, {8740, 526}, {9206, 895}, {14560, 36297}, {18384, 20579}, {20578, 125}, {23895, 69}, {32676, 2151}, {32713, 8739}, {35360, 33529}, {36296, 520}, {36299, 9033}

X(36307) =  TRILINEAR POLE OF LINE X(13)X(5466)

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

X(36307) lies on the Simmons circumconic (perspector X(13)) and these lines: {13, 531}, {111, 230}, {300, 18023}, {691, 11586}, {892, 11118}, {895, 11139}, {3457, 18818}, {5032, 22826}, {5466, 23283}, {8737, 17983}, {9154, 9206}, {9214, 11080}, {11078, 17948}

X(36307) = X(9214)-cross conjugate of X(36310)
X(36307) = X(i)-isoconjugate of X(j) for these (i,j): {15, 896}, {163, 9204}, {298, 922}, {524, 2151}, {2642, 17402}, {6137, 23889}, {14210, 34394}
X(36307) = trilinear pole of line {13, 5466}
X(36307) = barycentric product X(i)*X(j) for these {i,j}: {13, 671}, {111, 300}, {850, 9206}, {892, 20578}, {3457, 18023}, {5466, 23895}, {8737, 30786}
X(36307) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 524}, {111, 15}, {300, 3266}, {523, 9204}, {671, 298}, {691, 17402}, {923, 2151}, {2153, 896}, {3457, 187}, {5466, 23870}, {5995, 5467}, {8014, 9115}, {8737, 468}, {8753, 8739}, {9178, 6137}, {9206, 110}, {17983, 470}, {20578, 690}, {23283, 9205}, {23895, 5468}, {30452, 1648}, {30454, 8030}, {32740, 34394}, {36296, 3292}, {36299, 5642}

X(36308) =  TRILINEAR POLE OF LINE X(13)X(2394)

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

X(36307) lies on the Simmons circumconic (perspector X(13)), the cubic K419a, and these lines: {2, 10217}, {13, 470}, {30, 74}, {298, 1494}, {395, 11079}, {471, 36299}, {11080, 19776}, {19772, 19778}

X(36308) = X(2)-cross conjugate of X(36311)
X(36308) = polar conjugate of X(6110)
X(36308) = antitomic image of X(19776)
X(36308) = isotomic conjugate of the complement of X(11078)
X(36308) = X(i)-cross conjugate of X(j) for these (i,j): {14, 11118}, {23871, 23895}, {36211, 11119}, {36299, 13}
X(36308) = X(i)-isoconjugate of X(j) for these (i,j): {15, 2173}, {30, 2151}, {48, 6110}, {298, 9406}, {1094, 36299}, {1511, 2154}, {6149, 36298}, {14206, 34394}, {35201, 36297}
X(36308) = cevapoint of X(i) and X(j) for these (i,j): {2, 11078}, {13, 36299}, {16, 36296}, {20578, 30468}
X(36308) = trilinear pole of line {13, 2394}
X(36308) = barycentric product X(i)*X(j) for these {i,j}: {13, 1494}, {74, 300}, {299, 5627}, {2153, 33805}, {2394, 23895}, {31621, 36299}
X(36308) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6110}, {13, 30}, {16, 1511}, {74, 15}, {299, 6148}, {300, 3260}, {471, 14920}, {1494, 298}, {1989, 36298}, {2153, 2173}, {2159, 2151}, {2394, 23870}, {2433, 6137}, {3457, 1495}, {3470, 5616}, {5627, 14}, {5995, 2420}, {8737, 1990}, {8749, 8739}, {11079, 36297}, {11080, 36299}, {12079, 30465}, {16080, 470}, {20578, 1637}, {23871, 5664}, {23895, 2407}, {30468, 3258}, {36296, 3284}, {36299, 3163}

X(36309) =  TRILINEAR POLE OF LINE X(4)X(14)

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

X(36309) lies on the Simmons circumconic (perspector X(14)) and these lines: {4, 21467}, {14, 471}, {107, 5994}, {112, 476}, {297, 11092}, {462, 18384}, {472, 8015}, {648, 23896}, {685, 20579}, {1304, 5619}, {1990, 11549}, {3458, 16081}, {6111, 15743}, {6116, 11582}, {8738, 17983}, {10218, 36303}

X(36309) = polar conjugate of X(23871)
X(36309) = polar conjugate of the isotomic conjugate of X(23896)
X(36309) = polar conjugate of the isogonal conjugate of X(5994)
X(36309) = X(i)-cross conjugate of X(j) for these (i,j): {4240, 36306}, {5994, 23896}, {10632, 250}, {36303, 32230}
X(36309) = X(i)-isoconjugate of X(j) for these (i,j): {16, 656}, {48, 23871}, {63, 6138}, {299, 810}, {471, 822}, {525, 2152}, {2153, 8552}, {3708, 17403}, {4575, 30468}, {8740, 24018}, {9205, 36060}, {14208, 34395}, {32679, 36296}
X(36309) = cevapoint of X(i) and X(j) for these (i,j): {462, 2501}, {523, 5321}, {3458, 20579}
X(36309) = trilinear pole of line {4, 14}
X(36309) = barycentric product X(i)*X(j) for these {i,j}: {4, 23896}, {14, 648}, {99, 8738}, {112, 301}, {264, 5994}, {470, 476}, {811, 2154}, {3458, 6331}, {6344, 17402}, {6528, 36297}, {8739, 35139}, {16077, 36298}, {18020, 20579}
X(36309) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 23871}, {14, 525}, {15, 8552}, {25, 6138}, {107, 471}, {112, 16}, {250, 17403}, {301, 3267}, {462, 35444}, {463, 14446}, {468, 9205}, {470, 3268}, {648, 299}, {2154, 656}, {2501, 30468}, {3458, 647}, {5619, 10218}, {5994, 3}, {6110, 5664}, {6137, 16186}, {8737, 23283}, {8738, 523}, {8739, 526}, {9207, 895}, {14560, 36296}, {18384, 20578}, {20579, 125}, {23896, 69}, {32676, 2152}, {32713, 8740}, {35360, 33530}, {36297, 520}, {36298, 9033}

X(36310) =  TRILINEAR POLE OF LINE X(14)X(5466)

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

X(36310) lies on the Simmons circumconic (perspector X(14)) and these lines: {14, 530}, {111, 230}, {301, 18023}, {691, 15743}, {892, 11117}, {895, 11138}, {3458, 18818}, {5032, 22827}, {5466, 23284}, {8738, 17983}, {9154, 9207}, {9214, 11085}, {11092, 17948}

X(36310) = X(9214)-cross conjugate of X(36307)
X(36310) = X(i)-isoconjugate of X(j) for these (i,j): {16, 896}, {163, 9205}, {299, 922}, {524, 2152}, {2642, 17403}, {6138, 23889}, {14210, 34395}
X(36310) = trilinear pole of line {14, 5466}
X(36310) = barycentric product X(i)*X(j) for these {i,j}: {14, 671}, {111, 301}, {850, 9207}, {892, 20579}, {3458, 18023}, {5466, 23896}, {8738, 30786}
X(36310) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 524}, {111, 16}, {301, 3266}, {523, 9205}, {671, 299}, {691, 17403}, {923, 2152}, {2154, 896}, {3458, 187}, {5466, 23871}, {5994, 5467}, {8015, 9117}, {8738, 468}, {8753, 8740}, {9178, 6138}, {9207, 110}, {17983, 471}, {20579, 690}, {23284, 9204}, {23896, 5468}, {30453, 1648}, {30455, 8030}, {32740, 34395}, {36297, 3292}, {36298, 5642}

X(36311) =  TRILINEAR POLE OF LINE X(14)X(2394)

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

X(36311) lies on the Simmons circumconic (perspetor X(14)), the cubic K419b, and these lines: {2, 10218}, {14, 471}, {30, 74}, {299, 1494}, {396, 11079}, {470, 36298}, {11085, 19777}, {19773, 19779}

X(36311) = X(2)-cross conjugate of X(36308)
X(36311) = polar conjugate of X(6111)
X(36311) = antitomic image of X(19777)
X(36311) = isotomic conjugate of the complement of X(11092)
X(36311) = X(i)-cross conjugate of X(j) for these (i,j): {13, 11117}, {23870, 23896}, {36210, 11120}, {36298, 14}
X(36311) = X(i)-isoconjugate of X(j) for these (i,j): {16, 2173}, {30, 2152}, {48, 6111}, {299, 9406}, {1095, 36298}, {1511, 2153}, {6149, 36299}, {14206, 34395}, {35201, 36296}
X(36311) = cevapoint of X(i) and X(j) for these (i,j): {2, 11092}, {14, 36298}, {15, 36297}, {20579, 30465}
X(36311) = trilinear pole of line {14, 2394}
X(36311) = barycentric product X(i)*X(j) for these {i,j}: {14, 1494}, {74, 301}, {298, 5627}, {2154, 33805}, {2394, 23896}, {31621, 36298}
X(36311) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6111}, {14, 30}, {15, 1511}, {74, 16}, {298, 6148}, {301, 3260}, {470, 14920}, {1494, 299}, {1989, 36299}, {2154, 2173}, {2159, 2152}, {2394, 23871}, {2433, 6138}, {3458, 1495}, {3470, 5612}, {5627, 13}, {5994, 2420}, {8738, 1990}, {8749, 8740}, {11079, 36296}, {11085, 36298}, {12079, 30468}, {16080, 471}, {20579, 1637}, {23870, 5664}, {23896, 2407}, {30465, 3258}, {36297, 3284}, {36298, 3163}

X(36312) =  MIDPOINT OF X(11146) AND X(11581)

Barycentrics    3*a^8*b^2 - 8*a^6*b^4 + 6*a^4*b^6 - b^10 + 3*a^8*c^2 + 4*a^6*b^2*c^2 - 21*a^4*b^4*c^2 + 11*a^2*b^6*c^2 + 3*b^8*c^2 - 8*a^6*c^4 - 21*a^4*b^2*c^4 - 22*a^2*b^4*c^4 - 2*b^6*c^4 + 6*a^4*c^6 + 11*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 + 2*Sqrt[3]*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 6*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 - 6*b^4*c^4 + 3*a^2*c^6 + 4*b^2*c^6 - c^8)*S : :
X(36312) = 2 X[6671] + X[11581]

See Kadir Altintas and Peter Moses, Euclid 528 .

X(36312) lies on these lines: {30, 5459}, {623, 3580}, {6104, 6671}

X(36312) = midpoint of X(11146) and X(11581)
X(36312) = reflection of X(11146) in X(6671)

X(36313) =  MIDPOINT OF X(11145) AND X(11582)

Barycentrics    3*a^8*b^2 - 8*a^6*b^4 + 6*a^4*b^6 - b^10 + 3*a^8*c^2 + 4*a^6*b^2*c^2 - 21*a^4*b^4*c^2 + 11*a^2*b^6*c^2 + 3*b^8*c^2 - 8*a^6*c^4 - 21*a^4*b^2*c^4 - 22*a^2*b^4*c^4 - 2*b^6*c^4 + 6*a^4*c^6 + 11*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 - 2*Sqrt[3]*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 6*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 - 6*b^4*c^4 + 3*a^2*c^6 + 4*b^2*c^6 - c^8)*S : :
X(36313) = 2 X[6672] + X[11582]

See Kadir Altintas and Peter Moses, Euclid 528 .

X(36313) lies on these lines: {30, 5460}, {624, 3580}, {6105, 6672}

X(36313) = midpoint of X(11145) and X(11582)
X(36313) = reflection of X(11145) in X(6672)

X(36314) =  MIDPOINT OF X(14451) AND X(15766)

Barycentrics    (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(3*a^14*b^2 - 17*a^12*b^4 + 39*a^10*b^6 - 45*a^8*b^8 + 25*a^6*b^10 - 3*a^4*b^12 - 3*a^2*b^14 + b^16 + 3*a^14*c^2 + 10*a^12*b^2*c^2 - 21*a^10*b^4*c^2 - 12*a^8*b^6*c^2 + 25*a^6*b^8*c^2 - 6*a^4*b^10*c^2 + 9*a^2*b^12*c^2 - 8*b^14*c^2 - 17*a^12*c^4 - 21*a^10*b^2*c^4 + 90*a^8*b^4*c^4 - 47*a^6*b^6*c^4 - 24*a^4*b^8*c^4 - 9*a^2*b^10*c^4 + 28*b^12*c^4 + 39*a^10*c^6 - 12*a^8*b^2*c^6 - 47*a^6*b^4*c^6 + 66*a^4*b^6*c^6 + 3*a^2*b^8*c^6 - 56*b^10*c^6 - 45*a^8*c^8 + 25*a^6*b^2*c^8 - 24*a^4*b^4*c^8 + 3*a^2*b^6*c^8 + 70*b^8*c^8 + 25*a^6*c^10 - 6*a^4*b^2*c^10 - 9*a^2*b^4*c^10 - 56*b^6*c^10 - 3*a^4*c^12 + 9*a^2*b^2*c^12 + 28*b^4*c^12 - 3*a^2*c^14 - 8*b^2*c^14 + c^16) : :
X(36314) = 2 X[10272] + X[14451]

See Kadir Altintas and Peter Moses, Euclid 528 .

X(36314) lies on these lines: {30, 110}, {10272, 14354}

X(36314) = midpoint of X(14451) and X(15766)
X(36314) = reflection of X(15766) in X(10272)
X(36314) = barycentric quotient X(11749)/X(13582)

X(36315) =  MIDPOINT OF X(14452) AND X(15767)

Barycentrics    (a^3 + a^2*b - a*b^2 - b^3 + a^2*c + a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)*(a^9*b - a^8*b^2 - 4*a^7*b^3 + 4*a^6*b^4 + 6*a^5*b^5 - 6*a^4*b^6 - 4*a^3*b^7 + 4*a^2*b^8 + a*b^9 - b^10 + a^9*c - 6*a^8*b*c + 5*a^7*b^2*c + 6*a^6*b^3*c - 11*a^5*b^4*c + 4*a^4*b^5*c + 3*a^3*b^6*c - 2*a^2*b^7*c + 2*a*b^8*c - 2*b^9*c - a^8*c^2 + 5*a^7*b*c^2 - 4*a^6*b^2*c^2 + 5*a^4*b^4*c^2 - 3*a^2*b^6*c^2 - 5*a*b^7*c^2 + 3*b^8*c^2 - 4*a^7*c^3 + 6*a^6*b*c^3 - 14*a^4*b^3*c^3 + 4*a^3*b^4*c^3 + 2*a^2*b^5*c^3 - 7*a*b^6*c^3 + 8*b^7*c^3 + 4*a^6*c^4 - 11*a^5*b*c^4 + 5*a^4*b^2*c^4 + 4*a^3*b^3*c^4 - 2*a^2*b^4*c^4 + 9*a*b^5*c^4 - 2*b^6*c^4 + 6*a^5*c^5 + 4*a^4*b*c^5 + 2*a^2*b^3*c^5 + 9*a*b^4*c^5 - 12*b^5*c^5 - 6*a^4*c^6 + 3*a^3*b*c^6 - 3*a^2*b^2*c^6 - 7*a*b^3*c^6 - 2*b^4*c^6 - 4*a^3*c^7 - 2*a^2*b*c^7 - 5*a*b^2*c^7 + 8*b^3*c^7 + 4*a^2*c^8 + 2*a*b*c^8 + 3*b^2*c^8 + a*c^9 - 2*b*c^9 - c^10) : :

See Kadir Altintas and Peter Moses, Euclid 528 .

X(36315) lies on these lines: {30, 6246}, {14452, 15767}

X(36315) = midpoint of X(14452) and X(15767)

X(36316) =  TRILINEAR POLE OF X(13)X(523)

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

X(36316) lies on the Kiepert circumhyperbola, the Simmons circumconic (perspector X(13)), and these lines: {2, 18777}, {4, 21466}, {13, 5916}, {14, 476}, {671, 11078}, {1648, 1989}, {5466, 23283}, {8014, 12816}, {11658, 36186}

X(36316) = X(11537)-cross conjugate of X(13)
X(36316) = X(i)-isoconjugate of X(j) for these (i,j): {530, 2151}, {1094, 11537}, {6149, 18776}
X(36316) = cevapoint of X(i) and X(j) for these (i,j): {13, 11537}, {9200, 30465}
X(36316) = trilinear pole of line {13, 523}
X(36316) = barycentric product X(i)*X(j) for these {i,j}: {300, 2378}, {11119, 16256}
X(36316) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 530}, {1989, 18776}, {2378, 15}, {8737, 23712}, {11080, 11537}, {16256, 618}, {20578, 9200}

X(36317) =  TRILINEAR POLE OF X(14)X(523)

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

X(36317) lies on the Kiepert circumhyperbola, the Simmons circumconic (perspector X(14)), and these lines: {2, 18776}, {4, 21467}, {13, 476}, {14, 5917}, {671, 11092}, {1648, 1989}, {5466, 23284}, {8015, 12817}, {11659, 36185}

X(36317) = X(11549)-cross conjugate of X(14)
X(36317) = X(i)-isoconjugate of X(j) for these (i,j): {531, 2152}, {1095, 11549}, {6149, 18777}
X(36317) = cevapoint of X(i) and X(j) for these (i,j): {14, 11549}, {9201, 30468}
X(36317) = trilinear pole of line {14, 523}
X(36317) = barycentric product X(i)*X(j) for these {i,j}: {301, 2379}, {11120, 16255}
X(36317) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 531}, {1989, 18777}, {2379, 16}, {8738, 23713}, {11085, 11549}, {16255, 619}, {20579, 9201}

leftri

Orthologic centers related to Fermat-Dao-Nhi triangles: X(36318)-X(36402)

rightri

This preamble and centers X(36318)-X(36402) were contributed by César Eliud Lozada, January 15, 2020.

Fermat-Dao-Nhi equilateral triangles were introduced in the preamble just before X(33602).

These triangles have these properties:

A complete list of orthologic and parallelogic centers related to these triangles can be seen here.

X(36318) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    -sqrt(3)*S^3+(9*SA-7*SW)*S^2+9*SW*SB*SC : :
X(36318) = 7*X(2)-6*X(5617) = 2*X(2)-3*X(6770) = 11*X(2)-12*X(6771) = 4*X(5617)-7*X(6770) = 11*X(5617)-14*X(6771) = 12*X(5617)-7*X(36344) = 9*X(5617)-7*X(36363) = 3*X(5617)-7*X(36383) = 11*X(6770)-8*X(6771) = 3*X(6770)-X(36344) = 9*X(6770)-4*X(36363) = 3*X(6770)-4*X(36383) = 24*X(6771)-11*X(36344) = 18*X(6771)-11*X(36363) = 6*X(6771)-11*X(36383) = 3*X(11177)-2*X(36382) = 3*X(36344)-4*X(36363) = X(36344)-4*X(36383)

The reciprocal orthologic center of these triangles is X(5978)

X(36318) lies on these lines: {2,98}, {13,33603}, {30,35749}, {376,35751}, {530,5863}, {531,35690}, {616,8703}, {618,33615}, {2782,36331}, {3180,11645}, {3534,33624}, {3830,33625}, {3839,32907}, {5463,15698}, {12243,36330}, {13103,33699}, {15682,35752}

X(36318) = reflection of X(i) in X(j) for these (i,j): (2, 36383), (15682, 35752), (35750, 3534), (36344, 2)
X(36318) = anticomplement of X(36363)
X(36318) = {X(6770), X(36344)}-harmonic conjugate of X(2)

X(36319) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 1st BROCARD

Barycentrics    -sqrt(3)*S^3+9*SB*SC*SW+(9*SA-5*SW)*S^2 : :
X(36319) = 5*X(2)-6*X(5613) = 4*X(2)-3*X(6773) = 13*X(2)-12*X(6774) = 3*X(147)-2*X(36363) = 8*X(5613)-5*X(6773) = 13*X(5613)-10*X(6774) = 12*X(5613)-5*X(36320) = 3*X(5613)-5*X(36362) = 9*X(5613)-5*X(36382) = 13*X(6773)-16*X(6774) = 3*X(6773)-2*X(36320) = 3*X(6773)-8*X(36362) = 9*X(6773)-8*X(36382) = 24*X(6774)-13*X(36320) = 6*X(6774)-13*X(36362) = 18*X(6774)-13*X(36382)

X(36319) lies on these lines: {2,98}, {4,22495}, {14,33604}, {30,36331}, {530,35695}, {531,5863}, {617,3534}, {619,15719}, {2782,35749}, {3091,32909}, {3830,33626}, {5464,19708}, {11001,36329}, {12101,13102}, {13858,35473}, {15640,36346}, {22532,35931}, {35737,35742}

X(36319) = reflection of X(i) in X(j) for these (i,j): (2, 36362), (11001, 36329), (11177, 22509), (36320, 2), (36327, 3830)
X(36319) = anticomplement of X(36382)
X(36319) = {X(2), X(36320)}-harmonic conjugate of X(6773)

X(36320) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    sqrt(3)*S^3+(9*SA-7*SW)*S^2+9*SW*SB*SC : :
X(36320) = 7*X(2)-6*X(5613) = 2*X(2)-3*X(6773) = 11*X(2)-12*X(6774) = 4*X(5613)-7*X(6773) = 11*X(5613)-14*X(6774) = 12*X(5613)-7*X(36319) = 9*X(5613)-7*X(36362) = 3*X(5613)-7*X(36382) = 11*X(6773)-8*X(6774) = 3*X(6773)-X(36319) = 9*X(6773)-4*X(36362) = 3*X(6773)-4*X(36382) = 24*X(6774)-11*X(36319) = 18*X(6774)-11*X(36362) = 6*X(6774)-11*X(36382) = 3*X(11177)-2*X(36383) = 3*X(36319)-4*X(36362) = X(36319)-4*X(36382)

The reciprocal orthologic center of these triangles is X(5979)

X(36320) lies on these lines: {2,98}, {14,33602}, {30,36327}, {376,36329}, {530,35694}, {531,5862}, {617,8703}, {619,33614}, {2782,35750}, {3181,11645}, {3534,33622}, {3830,33623}, {3839,32909}, {5464,15698}, {12243,35752}, {13102,33699}, {15682,36330}, {35736,35742}

X(36320) = reflection of X(i) in X(j) for these (i,j): (2, 36382), (15682, 36330), (36319, 2), (36331, 3534)
X(36320) = anticomplement of X(36362)
X(36320) = {X(6773), X(36319)}-harmonic conjugate of X(2)

X(36321) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 2nd FERMAT-DAO

Barycentrics    a^2*(-sqrt(3)*(a^6+17*(b^2+c^2)*a^4-(19*b^4+6*b^2*c^2+19*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b^2*c^2+2*((12*b^4+b^2*c^2+12*c^4)*a^4-2*(b^2+c^2)*(6*b^4-7*b^2*c^2+6*c^4)*a^2+(b^4-16*b^2*c^2+c^4)*b^2*c^2)*S) : :
X(36321) = 5*X(2)-6*X(14182) = 7*X(2)-6*X(14188) = 13*X(2)-12*X(33481) = 11*X(2)-12*X(33491) = 7*X(14182)-5*X(14188) = 13*X(14182)-10*X(33481) = 11*X(14182)-10*X(33491) = 12*X(14182)-5*X(36325) = 3*X(14182)-5*X(36367) = 9*X(14182)-5*X(36387) = 13*X(14188)-14*X(33481) = 11*X(14188)-14*X(33491) = 12*X(14188)-7*X(36325) = 3*X(14188)-7*X(36367) = 9*X(14188)-7*X(36387) = 11*X(33481)-13*X(33491) = 24*X(33481)-13*X(36325) = 6*X(33481)-13*X(36367) = 18*X(33481)-13*X(36387)

The reciprocal orthologic center of these triangles is X(25207)

X(36321) lies on these lines: {2,14182}, {511,36327}, {512,36331}, {35736,35760}, {35737,35761}

X(36321) = reflection of X(i) in X(j) for these (i,j): (2, 36367), (36325, 2)
X(36321) = anticomplement of X(36387)

X(36322) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    6*sqrt(3)*((b^2+c^2)*a^2-b^4-c^4)*a^2*S+2*(b^2+c^2)*a^6+(2*b^4+9*b^2*c^2+2*c^4)*a^4-4*(b^6+c^6)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(36322) = 7*X(2)-6*X(22714) = 11*X(2)-12*X(33479) = 11*X(22714)-14*X(33479) = 12*X(22714)-7*X(36347) = 9*X(22714)-7*X(36365) = 3*X(22714)-7*X(36385) = 24*X(33479)-11*X(36347) = 18*X(33479)-11*X(36365) = 6*X(33479)-11*X(36385) = 3*X(36347)-4*X(36365) = X(36347)-4*X(36385) = X(36365)-3*X(36385)

The reciprocal orthologic center of these triangles is X(22689)

X(36322) lies on these lines: {2,51}, {376,3105}, {532,34623}, {2782,35749}, {5865,35918}

X(36322) = reflection of X(i) in X(j) for these (i,j): (2, 36385), (376, 3105), (36347, 2)
X(36322) = anticomplement of X(36365)

X(36323) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    -6*sqrt(3)*((b^2+c^2)*a^2-b^4-c^4)*a^2*S+2*(b^2+c^2)*a^6+(2*b^4+9*b^2*c^2+2*c^4)*a^4-4*(b^6+c^6)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(36323) = 7*X(2)-6*X(22715) = 11*X(2)-12*X(33478) = 11*X(22715)-14*X(33478) = 12*X(22715)-7*X(36345) = 9*X(22715)-7*X(36364) = 3*X(22715)-7*X(36384) = 24*X(33478)-11*X(36345) = 18*X(33478)-11*X(36364) = 6*X(33478)-11*X(36384) = 3*X(36345)-4*X(36364) = X(36345)-4*X(36384) = X(36364)-3*X(36384)

The reciprocal orthologic center of these triangles is X(22687)

X(36323) lies on these lines: {2,51}, {376,3104}, {533,34623}, {2782,36327}, {5864,35917}, {35736,35745}

X(36323) = reflection of X(i) in X(j) for these (i,j): (2, 36384), (376, 3104), (36345, 2)
X(36323) = anticomplement of X(36364)
X(36323) = lies on the circumconic with center X(1364))

X(36324) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO INNER-FERMAT

Barycentrics    -2*sqrt(3)*(25*a^2-11*c^2-11*b^2)*S+19*a^4-2*(b^2+c^2)*a^2-17*(b^2-c^2)^2 : :
X(36324) = 11*X(2)-12*X(18) = 7*X(2)-6*X(628) = 2*X(2)-3*X(22114) = 17*X(2)-12*X(22845) = 3*X(2)-4*X(36368) = 5*X(2)-4*X(36388) = 14*X(18)-11*X(628) = 8*X(18)-11*X(22114) = 17*X(18)-11*X(22845) = 18*X(18)-11*X(33624) = 6*X(18)-11*X(33627) = 24*X(18)-11*X(36346) = 9*X(18)-11*X(36368) = 15*X(18)-11*X(36388) = 4*X(628)-7*X(22114) = 17*X(628)-14*X(22845) = 9*X(628)-7*X(33624) = 3*X(628)-7*X(33627) = 12*X(628)-7*X(36346) = 9*X(628)-14*X(36368) = 15*X(628)-14*X(36388)

The reciprocal orthologic center of these triangles is X(616)

X(36324) lies on these lines: {2,18}, {5488,22494}, {5965,36326}, {11121,33602}, {15697,22531}, {30471,33614}, {35736,35746}

X(36324) = reflection of X(i) in X(j) for these (i,j): (2, 33627), (33624, 36368), (36346, 2)
X(36324) = anticomplement of X(33624)
X(36324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 33627, 22114), (22114, 36346, 2), (33624, 33627, 36368), (33624, 36368, 2)

X(36325) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 2nd FERMAT-DAO

Barycentrics    a^2*(sqrt(3)*(a^6-19*(b^2+c^2)*a^4+(17*b^4-6*b^2*c^2+17*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b^2*c^2+2*((12*b^4-b^2*c^2+12*c^4)*a^4-2*(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*b^2*c^2)*S) : :
X(36325) = 7*X(2)-6*X(14182) = 5*X(2)-6*X(14188) = 11*X(2)-12*X(33481) = 13*X(2)-12*X(33491) = 5*X(14182)-7*X(14188) = 11*X(14182)-14*X(33481) = 13*X(14182)-14*X(33491) = 12*X(14182)-7*X(36321) = 9*X(14182)-7*X(36367) = 3*X(14182)-7*X(36387) = 11*X(14188)-10*X(33481) = 13*X(14188)-10*X(33491) = 12*X(14188)-5*X(36321) = 9*X(14188)-5*X(36367) = 3*X(14188)-5*X(36387) = 13*X(33481)-11*X(33491) = 24*X(33481)-11*X(36321) = 18*X(33481)-11*X(36367) = 6*X(33481)-11*X(36387)

The reciprocal orthologic center of these triangles is X(25207)

X(36325) lies on these lines: {2,14182}, {511,36331}, {512,36327}, {35736,35761}, {35737,35760}

X(36325) = reflection of X(i) in X(j) for these (i,j): (2, 36387), (36321, 2)
X(36325) = anticomplement of X(36367)

X(36326) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO OUTER-FERMAT

Barycentrics    2*sqrt(3)*(25*a^2-11*c^2-11*b^2)*S+19*a^4-2*(b^2+c^2)*a^2-17*(b^2-c^2)^2 : :
X(36326) = 11*X(2)-12*X(17) = 7*X(2)-6*X(627) = 2*X(2)-3*X(22113) = 17*X(2)-12*X(22844) = 3*X(2)-4*X(36366) = 5*X(2)-4*X(36386) = 14*X(17)-11*X(627) = 8*X(17)-11*X(22113) = 17*X(17)-11*X(22844) = 18*X(17)-11*X(33622) = 6*X(17)-11*X(33626) = 24*X(17)-11*X(36352) = 9*X(17)-11*X(36366) = 15*X(17)-11*X(36386) = 4*X(627)-7*X(22113) = 17*X(627)-14*X(22844) = 9*X(627)-7*X(33622) = 3*X(627)-7*X(33626) = 12*X(627)-7*X(36352) = 9*X(627)-14*X(36366) = 15*X(627)-14*X(36386)

The reciprocal orthologic center of these triangles is X(617)

X(36326) lies on these lines: {2,17}, {5487,22493}, {5965,36324}, {11122,33603}, {15697,22532}, {30472,33615}

X(36326) = reflection of X(i) in X(j) for these (i,j): (2, 33626), (33622, 36366), (36352, 2)
X(36326) = anticomplement of X(33622)
X(36326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 33626, 22113), (22113, 36352, 2), (33622, 33626, 36366), (33622, 36366, 2)

X(36327) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 4th FERMAT-DAO

Barycentrics    6*S^2-sqrt(3)*(9*SA-4*SW)*S-27*SB*SC : :
X(36327) = 5*X(2)-6*X(14) = 4*X(2)-3*X(617) = 13*X(2)-12*X(619) = 11*X(2)-12*X(5460) = 7*X(2)-6*X(5464) = 17*X(2)-18*X(22490) = 8*X(14)-5*X(617) = 13*X(14)-10*X(619) = 11*X(14)-10*X(5460) = 7*X(14)-5*X(5464) = 23*X(14)-20*X(6670) = 17*X(14)-15*X(22490) = 9*X(14)-5*X(36329) = 3*X(14)-5*X(36330) = 12*X(14)-5*X(36331) = 13*X(617)-16*X(619) = 11*X(617)-16*X(5460) = 7*X(617)-8*X(5464) = 9*X(617)-8*X(36329) = 3*X(617)-8*X(36330) = 3*X(617)-2*X(36331)

The reciprocal orthologic center of these triangles is X(5469)

X(36327) lies on these lines: {2,14}, {4,36362}, {30,36320}, {148,33623}, {511,36321}, {512,36325}, {524,35694}, {542,10721}, {543,5862}, {627,35931}, {633,11295}, {671,33602}, {2482,33614}, {2782,36323}, {3524,32909}, {3534,6773}, {3545,16002}, {3830,33626}, {3845,13102}, {4669,9900}, {5858,33610}, {5863,33625}, {5969,36338}, {6774,15719}, {6777,8591}, {9830,35690}, {11001,33627}, {12243,36383}, {15702,20416}, {33603,33609}, {33622,35751}, {35736,35748}, {35737,35759}

X(36327) = reflection of X(i) in X(j) for these (i,j): (2, 36330), (5863, 35693), (11001, 36382), (35695, 5858), (36319, 3830), (36331, 2)
X(36327) = anticomplement of X(36329)
X(36327) = {X(2), X(36331)}-harmonic conjugate of X(617)

X(36328) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 1st FERMAT-DAO

Barycentrics    a^2*(-sqrt(3)*(a^6-19*(b^2+c^2)*a^4+(17*b^4-6*b^2*c^2+17*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b^2*c^2+2*((12*b^4-b^2*c^2+12*c^4)*a^4-2*(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*b^2*c^2)*S) : :
X(36328) = 7*X(2)-6*X(14178) = 5*X(2)-6*X(14186) = 11*X(2)-12*X(33480) = 13*X(2)-12*X(33490) = 5*X(14178)-7*X(14186) = 11*X(14178)-14*X(33480) = 13*X(14178)-14*X(33490) = 12*X(14178)-7*X(36354) = 9*X(14178)-7*X(36369) = 3*X(14178)-7*X(36389) = 11*X(14186)-10*X(33480) = 13*X(14186)-10*X(33490) = 12*X(14186)-5*X(36354) = 9*X(14186)-5*X(36369) = 3*X(14186)-5*X(36389) = 13*X(33480)-11*X(33490) = 24*X(33480)-11*X(36354) = 18*X(33480)-11*X(36369) = 6*X(33480)-11*X(36389)

The reciprocal orthologic center of these triangles is X(25208)

X(36328) lies on these lines: {2,14178}, {511,35750}, {512,35749}

X(36328) = reflection of X(i) in X(j) for these (i,j): (2, 36389), (36354, 2)
X(36328) = anticomplement of X(36369)

X(36329) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 1st INNER-FERMAT-DAO-NHI

Barycentrics    -2*(5*a^2-4*b^2-4*c^2)*S+(7*a^4-5*(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3) : :
X(36329) = 4*X(2)-3*X(14) = X(2)-3*X(617) = 5*X(2)-6*X(619) = 7*X(2)-6*X(5460) = 2*X(2)-3*X(5464) = 13*X(2)-12*X(6670) = 10*X(2)-9*X(22490) = X(14)-4*X(617) = 5*X(14)-8*X(619) = 7*X(14)-8*X(5460) = 13*X(14)-16*X(6670) = 5*X(14)-6*X(22490) = 9*X(14)-4*X(36327) = 3*X(14)-2*X(36330) = 3*X(14)+4*X(36331) = 5*X(617)-2*X(619) = 7*X(617)-2*X(5460) = 13*X(617)-4*X(6670) = 10*X(617)-3*X(22490) = 9*X(617)-X(36327) = 6*X(617)-X(36330) = 3*X(617)+X(36331)

The reciprocal orthologic center of these triangles is X(35749)

X(36329) lies on these lines: {2,14}, {3,22493}, {13,22578}, {17,31693}, {22,13858}, {30,22494}, {376,36320}, {394,10658}, {511,36387}, {512,36367}, {524,35692}, {533,35932}, {542,1350}, {543,5859}, {671,32552}, {1384,9113}, {2482,6777}, {2782,36364}, {3412,11303}, {3830,36388}, {3845,5613}, {4669,12780}, {5054,32909}, {5055,16002}, {5066,25164}, {5863,35695}, {5969,36373}, {6773,19708}, {6774,15701}, {8591,33611}, {8703,36382}, {9763,16808}, {9830,35696}, {9886,33459}, {11001,36319}, {12792,34582}, {12816,33458}, {13102,19709}, {15534,25236}, {15693,21157}, {15694,20416}, {16963,22496}, {16965,22495}, {22489,31696}, {33606,33618}, {33609,33625}, {33624,35749}, {35734,35748}, {35735,35759}

X(36329) = midpoint of X(i) and X(j) for these {i,j}: {2, 36331}, {5863, 35695}, {11001, 36319}
X(36329) = reflection of X(i) in X(j) for these (i,j): (14, 5464), (671, 32552), (5464, 617), (9116, 9114), (22496, 35303), (22578, 13), (35693, 33458), (35751, 15300), (36330, 2), (36382, 8703)
X(36329) = complement of X(36327)
X(36329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36330, 14), (617, 36331, 2), (3534, 15533, 35751), (5464, 22490, 619), (5464, 36330, 2), (9114, 35751, 15300), (15300, 35751, 9116), (22496, 35303, 16963), (33442, 33443, 6670)

X(36330) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 1st INNER-FERMAT-DAO-NHI

Barycentrics    -2*(7*a^2-2*b^2-2*c^2)*S+(5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2)*sqrt(3) : :
X(36330) = 2*X(2)-3*X(14) = 5*X(2)-3*X(617) = 7*X(2)-6*X(619) = 5*X(2)-6*X(5460) = 4*X(2)-3*X(5464) = 11*X(2)-12*X(6670) = 8*X(2)-9*X(22490) = 5*X(14)-2*X(617) = 7*X(14)-4*X(619) = 5*X(14)-4*X(5460) = 11*X(14)-8*X(6670) = 4*X(14)-3*X(22490) = 19*X(14)-16*X(35020) = 3*X(14)+2*X(36327) = 3*X(14)-X(36329) = 9*X(14)-2*X(36331)

The reciprocal orthologic center of these triangles is X(35749)

X(36330) lies on these lines: {2,14}, {3,32909}, {4,22495}, {13,31696}, {25,13858}, {30,22496}, {61,31693}, {148,33625}, {381,16002}, {511,36367}, {512,36387}, {524,35693}, {530,19107}, {533,22494}, {542,1351}, {543,5858}, {671,6778}, {1993,10658}, {2482,33621}, {2782,36363}, {3534,36368}, {3845,25164}, {4677,9900}, {4745,12780}, {5054,20416}, {5066,5613}, {5463,15300}, {5470,14136}, {5474,8703}, {5859,12817}, {5862,35694}, {5969,36393}, {6108,22572}, {6773,11001}, {6774,15693}, {6775,9113}, {8584,22579}, {8591,32553}, {9116,36386}, {9166,32552}, {9763,16809}, {9830,35697}, {10645,33474}, {11295,16964}, {11485,22489}, {11603,17503}, {12100,21157}, {12243,36318}, {15682,36320}, {16268,35303}, {16963,35230}, {21360,31694}, {22236,33414}, {31684,33603}, {33459,35692}, {33627,35750}, {34508,35931}, {35734,35759}, {35735,35748}

X(36330) = midpoint of X(i) and X(j) for these {i,j}: {2, 36327}, {5862, 35694}, {15682, 36320}
X(36330) = reflection of X(i) in X(j) for these (i,j): (3, 32909), (13, 31696), (381, 16002), (617, 5460), (5464, 14), (8591, 32553), (35692, 33459), (36329, 2), (36362, 3845)
X(36330) = complement of X(36331)
X(36330) = outer-Napoleon circle-inverse of-X(22490)
X(36330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36329, 5464), (14, 5464, 22490), (14, 36329, 2), (3830, 15534, 35752), (6780, 9760, 5464), (10654, 33518, 14), (16964, 22493, 11295), (25164, 36362, 3845)

X(36331) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 1st INNER-FERMAT-DAO-NHI

Barycentrics    -2*(11*a^2-7*b^2-7*c^2)*S+(13*a^4-8*(b^2+c^2)*a^2-5*(b^2-c^2)^2)*sqrt(3) : :
X(36331) = 7*X(2)-6*X(14) = 2*X(2)-3*X(617) = 11*X(2)-12*X(619) = 13*X(2)-12*X(5460) = 5*X(2)-6*X(5464) = 19*X(2)-18*X(22490) = 4*X(14)-7*X(617) = 11*X(14)-14*X(619) = 13*X(14)-14*X(5460) = 5*X(14)-7*X(5464) = 12*X(14)-7*X(36327) = 3*X(14)-7*X(36329) = 9*X(14)-7*X(36330) = 11*X(617)-8*X(619) = 13*X(617)-8*X(5460) = 5*X(617)-4*X(5464) = 25*X(617)-16*X(6670) = 19*X(617)-12*X(22490) = 3*X(617)-X(36327) = 3*X(617)-4*X(36329) = 9*X(617)-4*X(36330)

The reciprocal orthologic center of these triangles is X(35749)

X(36331) lies on these lines: {2,14}, {30,36319}, {376,36382}, {511,36325}, {512,36321}, {524,35695}, {542,11001}, {543,5863}, {616,15300}, {627,22493}, {631,32909}, {633,35931}, {671,33604}, {2482,33616}, {2782,36318}, {3534,33622}, {4745,9900}, {5066,13102}, {5071,16002}, {5474,15697}, {5859,33623}, {5862,33611}, {5969,36350}, {6773,8703}, {6778,8596}, {8591,33610}, {9830,35691}, {12817,33612}, {15682,33624}, {15709,20416}, {31693,33413}, {33626,35752}, {35736,35759}, {35737,35748}

X(36331) = reflection of X(i) in X(j) for these (i,j): (2, 36329), (5862, 35692), (15682, 36362), (35694, 5859), (36320, 3534), (36327, 2)
X(36331) = anticomplement of X(36330)
X(36331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36329, 617), (617, 36327, 2)

X(36332) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    (5-2*sqrt(3))*S^2+(9*SA-(2+sqrt(3))*SW)*S-9*SB*SC : :
X(36332) = 7*X(2)-6*X(6305) = 11*X(2)-12*X(33446) = 11*X(6305)-14*X(33446) = 12*X(6305)-7*X(36356) = 9*X(6305)-7*X(36372) = 3*X(6305)-7*X(36392) = 24*X(33446)-11*X(36356) = 18*X(33446)-11*X(36372) = 6*X(33446)-11*X(36392) = 3*X(36356)-4*X(36372) = X(36356)-4*X(36392) = X(36372)-3*X(36392)

The reciprocal orthologic center of these triangles is X(33442)

X(36332) lies on these lines: {2,372}, {530,36360}, {531,36341}, {3564,36335}

X(36332) = reflection of X(i) in X(j) for these (i,j): (2, 36392), (36334, 22485), (36356, 2)
X(36332) = anticomplement of X(36372)
X(36332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22485, 36370, 36348), (36370, 36392, 22485)

X(36333) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    (5+2*sqrt(3))*S^2-(9*SA-(2-sqrt(3))*SW)*S-9*SB*SC : :
X(36333) = 7*X(2)-6*X(6301) = 11*X(2)-12*X(33447) = 11*X(6301)-14*X(33447) = 12*X(6301)-7*X(36357) = 9*X(6301)-7*X(36374) = 3*X(6301)-7*X(36394) = 24*X(33447)-11*X(36357) = 18*X(33447)-11*X(36374) = 6*X(33447)-11*X(36394) = 3*X(36357)-4*X(36374) = X(36357)-4*X(36394) = X(36374)-3*X(36394)

The reciprocal orthologic center of these triangles is X(33443)

X(36333) lies on these lines: {2,371}, {530,36361}, {531,36343}, {3564,36334}

X(36333) = reflection of X(i) in X(j) for these (i,j): (2, 36394), (36335, 22484), (36357, 2)
X(36333) = anticomplement of X(36374)
X(36333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22484, 36371, 36349), (36371, 36394, 22484)

X(36334) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    (5+2*sqrt(3))*S^2+(9*SA-(2-sqrt(3))*SW)*S-9*SB*SC : :
X(36334) = 7*X(2)-6*X(6304) = 11*X(2)-12*X(33444) = 11*X(6304)-14*X(33444) = 12*X(6304)-7*X(36348) = 9*X(6304)-7*X(36370) = 3*X(6304)-7*X(36390) = 24*X(33444)-11*X(36348) = 18*X(33444)-11*X(36370) = 6*X(33444)-11*X(36390) = 3*X(36348)-4*X(36370) = X(36348)-4*X(36390) = X(36370)-3*X(36390)

The reciprocal orthologic center of these triangles is X(33440)

X(36334) lies on these lines: {2,372}, {530,36340}, {531,36353}, {3564,36333}, {35736,35744}

X(36334) = reflection of X(i) in X(j) for these (i,j): (2, 36390), (36332, 22485), (36348, 2)
X(36334) = anticomplement of X(36370)
X(36334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22485, 36372, 36356), (36372, 36390, 22485)

X(36335) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    (5-2*sqrt(3))*S^2-(9*SA-(2+sqrt(3))*SW)*S-9*SB*SC : :
X(36335) = 7*X(2)-6*X(6300) = 11*X(2)-12*X(33445) = 11*X(6300)-14*X(33445) = 12*X(6300)-7*X(36349) = 9*X(6300)-7*X(36371) = 3*X(6300)-7*X(36391) = 24*X(33445)-11*X(36349) = 18*X(33445)-11*X(36371) = 6*X(33445)-11*X(36391) = 3*X(36349)-4*X(36371) = X(36349)-4*X(36391) = X(36371)-3*X(36391)

The reciprocal orthologic center of these triangles is X(33441)

X(36335) lies on these lines: {2,371}, {530,36342}, {531,36355}, {3564,36332}, {35736,35743}

X(36335) = reflection of X(i) in X(j) for these (i,j): (2, 36391), (36333, 22484), (36349, 2)
X(36335) = anticomplement of X(36371)
X(36335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22484, 36374, 36357), (36374, 36391, 22484)

X(36336) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    -2*(2*(b^2+c^2)*a^2-7*b^2*c^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*((b^2+c^2)*a^2-2*b^2*c^2) : :
X(36336) = 7*X(2)-6*X(6294) = 11*X(2)-12*X(33483) = 11*X(6294)-14*X(33483) = 12*X(6294)-7*X(36358) = 9*X(6294)-7*X(36378) = 3*X(6294)-7*X(36398) = 24*X(33483)-11*X(36358) = 18*X(33483)-11*X(36378) = 6*X(33483)-11*X(36398) = 3*X(36358)-4*X(36378) = X(36358)-4*X(36398) = X(36378)-3*X(36398)

The reciprocal orthologic center of these triangles is X(6295)

X(36336) lies on these lines: {2,39}, {732,36339}, {5969,35749}, {33625,35694}

X(36336) = reflection of X(i) in X(j) for these (i,j): (2, 36398), (36358, 2)
X(36336) = anticomplement of X(36378)
X(36336) = {X(7795), X(32879)}-harmonic conjugate of X(30599)

X(36337) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    2*(7*a^4+3*(b^2+c^2)*a^2-2*b^4+3*b^2*c^2-2*c^4)*S+3*sqrt(3)*(2*a^4-b^4-c^4)*(a^2+b^2+c^2) : :
X(36337) = 7*X(2)-6*X(6297) = 11*X(2)-12*X(33485) = 11*X(6297)-14*X(33485) = 12*X(6297)-7*X(36359) = 9*X(6297)-7*X(36379) = 3*X(6297)-7*X(36399) = 24*X(33485)-11*X(36359) = 18*X(33485)-11*X(36379) = 6*X(33485)-11*X(36399) = 3*X(36359)-4*X(36379) = X(36359)-4*X(36399) = X(36379)-3*X(36399)

The reciprocal orthologic center of these triangles is X(6299)

X(36337) lies on these lines: {2,32}, {732,36338}

X(36337) = reflection of X(i) in X(j) for these (i,j): (2, 36399), (36339, 12156), (36359, 2)
X(36337) = anticomplement of X(36379)

X(36338) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    2*(2*(b^2+c^2)*a^2-7*b^2*c^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*((b^2+c^2)*a^2-2*b^2*c^2) : :
X(36338) = 7*X(2)-6*X(6581) = 11*X(2)-12*X(33482) = 11*X(6581)-14*X(33482) = 12*X(6581)-7*X(36350) = 9*X(6581)-7*X(36373) = 3*X(6581)-7*X(36393) = 24*X(33482)-11*X(36350) = 18*X(33482)-11*X(36373) = 6*X(33482)-11*X(36393) = 3*X(36350)-4*X(36373) = X(36350)-4*X(36393) = X(36373)-3*X(36393)

The reciprocal orthologic center of these triangles is X(6582)

X(36338) lies on these lines: {2,39}, {732,36337}, {5969,36327}, {33623,35690}, {35736,35755}

X(36338) = reflection of X(i) in X(j) for these (i,j): (2, 36393), (36350, 2)
X(36338) = anticomplement of X(36373)
X(36338) = {X(7803), X(32869)}-harmonic conjugate of X(34284)

X(36339) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    -2*(7*a^4+3*(b^2+c^2)*a^2-2*b^4+3*b^2*c^2-2*c^4)*S+3*sqrt(3)*(2*a^4-b^4-c^4)*(a^2+b^2+c^2) : :
X(36339) = 7*X(2)-6*X(6296) = 11*X(2)-12*X(33484) = 11*X(6296)-14*X(33484) = 12*X(6296)-7*X(36351) = 9*X(6296)-7*X(36375) = 3*X(6296)-7*X(36395) = 24*X(33484)-11*X(36351) = 18*X(33484)-11*X(36375) = 6*X(33484)-11*X(36395) = 3*X(36351)-4*X(36375) = X(36351)-4*X(36395) = X(36375)-3*X(36395)

The reciprocal orthologic center of these triangles is X(6298)

X(36339) lies on these lines: {2,32}, {732,36336}, {35736,35756}

X(36339) = reflection of X(i) in X(j) for these (i,j): (2, 36395), (36337, 12156), (36351, 2)
X(36339) = anticomplement of X(36375)

X(36340) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    S*((a^2+b^2+c^2+4*S)*sqrt(3)+15*a^2-3*b^2-3*c^2)+15*a^4-3*(b^2+c^2)*a^2-12*(b^2-c^2)^2 : :
X(36340) = 7*X(2)-6*X(13706) = 11*X(2)-12*X(33486) = 11*X(13706)-14*X(33486) = 12*X(13706)-7*X(36353) = 9*X(13706)-7*X(36376) = 3*X(13706)-7*X(36396) = 3*X(33456)-2*X(36400) = 24*X(33486)-11*X(36353) = 18*X(33486)-11*X(36376) = 6*X(33486)-11*X(36396) = 3*X(36353)-4*X(36376) = X(36353)-4*X(36396) = X(36376)-3*X(36396)

The reciprocal orthologic center of these triangles is X(13705)

X(36340) lies on these lines: {2,1327}, {530,36334}, {531,36348}, {35736,35757}

X(36340) = reflection of X(i) in X(j) for these (i,j): (2, 36396), (36353, 2)
X(36340) = anticomplement of X(36376)
X(36340) = {X(2), X(33456)}-harmonic conjugate of X(36341)

X(36341) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    S*(-(a^2+b^2+c^2+4*S)*sqrt(3)+15*a^2-3*b^2-3*c^2)+15*a^4-3*(b^2+c^2)*a^2-12*(b^2-c^2)^2 : :
X(36341) = 7*X(2)-6*X(13704) = 11*X(2)-12*X(33487) = 11*X(13704)-14*X(33487) = 12*X(13704)-7*X(36360) = 9*X(13704)-7*X(36380) = 3*X(13704)-7*X(36400) = 3*X(33456)-2*X(36396) = 24*X(33487)-11*X(36360) = 18*X(33487)-11*X(36380) = 6*X(33487)-11*X(36400) = 3*X(36360)-4*X(36380) = X(36360)-4*X(36400) = X(36380)-3*X(36400)

The reciprocal orthologic center of these triangles is X(13703)

X(36341) lies on these lines: {2,1327}, {530,36356}, {531,36332}

X(36341) = reflection of X(i) in X(j) for these (i,j): (2, 36400), (36360, 2)
X(36341) = anticomplement of X(36380)
X(36341) = {X(2), X(33456)}-harmonic conjugate of X(36340)

X(36342) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -S*(-(a^2+b^2+c^2-4*S)*sqrt(3)+15*a^2-3*b^2-3*c^2)+15*a^4-3*(b^2+c^2)*a^2-12*(b^2-c^2)^2 : :
X(36342) = 11*X(2)-12*X(33488) = 3*X(33457)-2*X(36401) = X(36377)-3*X(36397)

The reciprocal orthologic center of these triangles is X(13825)

X(36342) lies on these lines: {2,1328}, {530,36335}, {531,36349}, {35736,35758}

X(36342) = reflection of X(i) in X(j) for these (i,j): (2, 36397), (36355, 2)
X(36342) = anticomplement of X(36377)
X(36342) = {X(2), X(33457)}-harmonic conjugate of X(36343)

X(36343) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -S*((a^2+b^2+c^2-4*S)*sqrt(3)+15*a^2-3*b^2-3*c^2)+15*a^4-3*(b^2+c^2)*a^2-12*(b^2-c^2)^2 : :
X(36343) = 7*X(2)-6*X(13824) = 11*X(2)-12*X(33489) = 11*X(13824)-14*X(33489) = 12*X(13824)-7*X(36361) = 9*X(13824)-7*X(36381) = 3*X(13824)-7*X(36401) = 3*X(33457)-2*X(36397) = 24*X(33489)-11*X(36361) = 6*X(33489)-11*X(36401) = 3*X(36361)-4*X(36381) = X(36361)-4*X(36401) = X(36381)-3*X(36401)

The reciprocal orthologic center of these triangles is X(13823)

X(36343) lies on these lines: {2,1328}, {530,36357}, {531,36333}

X(36343) = reflection of X(i) in X(j) for these (i,j): (2, 36401), (36361, 2)
X(36343) = anticomplement of X(36381)
X(36343) = {X(2), X(33457)}-harmonic conjugate of X(36342)

X(36344) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    sqrt(3)*S^3+9*SB*SC*SW+(9*SA-5*SW)*S^2 : :
X(36344) = 5*X(2)-6*X(5617) = 4*X(2)-3*X(6770) = 13*X(2)-12*X(6771) = 3*X(147)-2*X(36362) = 8*X(5617)-5*X(6770) = 13*X(5617)-10*X(6771) = 12*X(5617)-5*X(36318) = 3*X(5617)-5*X(36363) = 9*X(5617)-5*X(36383) = 13*X(6770)-16*X(6771) = 3*X(6770)-2*X(36318) = 3*X(6770)-8*X(36363) = 9*X(6770)-8*X(36383) = 24*X(6771)-13*X(36318) = 6*X(6771)-13*X(36363) = 18*X(6771)-13*X(36383)

The reciprocal orthologic center of these triangles is X(5978)

X(36344) lies on these lines: {2,98}, {4,22496}, {13,33605}, {30,35750}, {530,5862}, {531,35691}, {616,3534}, {618,15719}, {2782,36323}, {3091,32907}, {3830,33627}, {5463,19708}, {11001,35751}, {12101,13103}, {13859,35473}, {15640,36352}, {22531,35932}

X(36344) = reflection of X(i) in X(j) for these (i,j): (2, 36363), (11001, 35751), (11177, 22507), (35749, 3830), (36318, 2)
X(36344) = anticomplement of X(36383)
X(36344) = {X(2), X(36318)}-harmonic conjugate of X(6770)

X(36345) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    4*S^4+(4*(3*SA-2*SW))*SW*S^2+(12*(SB+SC))*SA*SW^2+12*S*sqrt(3)*(S^2-SA*SW)*(SB+SC) : :
X(36345) = 5*X(2)-6*X(22715) = 13*X(2)-12*X(33478) = 2*X(3104)-3*X(3524) = 3*X(6194)-2*X(36365) = 13*X(22715)-10*X(33478) = 12*X(22715)-5*X(36323) = 3*X(22715)-5*X(36364) = 9*X(22715)-5*X(36384) = 24*X(33478)-13*X(36323) = 6*X(33478)-13*X(36364) = 18*X(33478)-13*X(36384) = X(36323)-4*X(36364) = 3*X(36323)-4*X(36384) = 3*X(36364)-X(36384)

The reciprocal orthologic center of these triangles is X(22687)

X(36345) lies on these lines: {2,51}, {2782,36318}, {3104,3524}, {5865,35917}, {5969,6770}, {12251,25195}, {35737,35745}

X(36345) = reflection of X(i) in X(j) for these (i,j): (2, 36364), (36323, 2), (36347, 33706)
X(36345) = anticomplement of X(36384)

X(36346) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO INNER-FERMAT

Barycentrics    -2*sqrt(3)*(23*a^2-13*c^2-13*b^2)*S+29*a^4-22*(b^2+c^2)*a^2-7*(b^2-c^2)^2 : :
X(36346) = 13*X(2)-12*X(18) = 5*X(2)-6*X(628) = 4*X(2)-3*X(22114) = 7*X(2)-12*X(22845) = 5*X(2)-4*X(36368) = 3*X(2)-4*X(36388) = 10*X(18)-13*X(628) = 16*X(18)-13*X(22114) = 7*X(18)-13*X(22845) = 6*X(18)-13*X(33624) = 18*X(18)-13*X(33627) = 24*X(18)-13*X(36324) = 15*X(18)-13*X(36368) = 9*X(18)-13*X(36388) = 23*X(628)-20*X(630) = 8*X(628)-5*X(22114) = 7*X(628)-10*X(22845) = 29*X(628)-20*X(33464) = 3*X(628)-5*X(33624) = 9*X(628)-5*X(33627) = 12*X(628)-5*X(36324) = 3*X(628)-2*X(36368) = 9*X(628)-10*X(36388)

The reciprocal orthologic center of these triangles is X(616)

X(36346) lies on these lines: {2,18}, {5965,15697}, {11121,33604}, {15640,36319}, {30471,33616}, {35737,35746}

X(36346) = reflection of X(i) in X(j) for these (i,j): (2, 33624), (33627, 36388), (36324, 2)
X(36346) = anticomplement of X(33627)
X(36346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36324, 22114), (628, 36368, 2), (33624, 33627, 36388), (33627, 36388, 2)

X(36347) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    4*S^4+4*(3*SA-2*SW)*SW*S^2+12*(SB+SC)*SA*SW^2-12*S*sqrt(3)*(S^2-SA*SW)*(SB+SC) : :
X(36347) = 5*X(2)-6*X(22714) = 13*X(2)-12*X(33479) = 2*X(3105)-3*X(3524) = 3*X(6194)-2*X(36364) = 13*X(22714)-10*X(33479) = 12*X(22714)-5*X(36322) = 3*X(22714)-5*X(36365) = 9*X(22714)-5*X(36385) = 24*X(33479)-13*X(36322) = 6*X(33479)-13*X(36365) = 18*X(33479)-13*X(36385) = X(36322)-4*X(36365) = 3*X(36322)-4*X(36385) = 3*X(36365)-X(36385)

The reciprocal orthologic center of these triangles is X(22689)

X(36347) lies on these lines: {2,51}, {2782,35750}, {3105,3524}, {5864,35918}, {5969,6773}, {12251,25191}

X(36347) = reflection of X(i) in X(j) for these (i,j): (2, 36365), (36322, 2), (36345, 33706)
X(36347) = anticomplement of X(36385)

X(36348) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    S*((a^2+b^2+c^2+4*S)*sqrt(3)+13*a^2-5*b^2-5*c^2)+5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2 : :
X(36348) = 5*X(2)-6*X(6304) = 13*X(2)-12*X(33444) = 3*X(488)-2*X(36372) = 13*X(6304)-10*X(33444) = 12*X(6304)-5*X(36334) = 3*X(6304)-5*X(36370) = 9*X(6304)-5*X(36390) = 24*X(33444)-13*X(36334) = 6*X(33444)-13*X(36370) = 18*X(33444)-13*X(36390) = X(36334)-4*X(36370) = 3*X(36334)-4*X(36390) = 3*X(36370)-X(36390)

The reciprocal orthologic center of these triangles is X(33440)

X(36348) lies on these lines: {2,372}, {530,36353}, {531,36340}, {3564,36357}, {35737,35744}

X(36348) = reflection of X(i) in X(j) for these (i,j): (2, 36370), (36334, 2)
X(36348) = anticomplement of X(36390)
X(36348) = {X(22485), X(36370)}-harmonic conjugate of X(36332)

X(36349) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    -S*(-(a^2+b^2+c^2-4*S)*sqrt(3)+13*a^2-5*b^2-5*c^2)+5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2 : :
X(36349) = 5*X(2)-6*X(6300) = 13*X(2)-12*X(33445) = 3*X(487)-2*X(36374) = 13*X(6300)-10*X(33445) = 12*X(6300)-5*X(36335) = 3*X(6300)-5*X(36371) = 9*X(6300)-5*X(36391) = 24*X(33445)-13*X(36335) = 6*X(33445)-13*X(36371) = 18*X(33445)-13*X(36391) = X(36335)-4*X(36371) = 3*X(36335)-4*X(36391) = 3*X(36371)-X(36391)

The reciprocal orthologic center of these triangles is X(33441)

X(36349) lies on these lines: {2,371}, {530,36355}, {531,36342}, {3564,36356}, {35737,35743}

X(36349) = reflection of X(i) in X(j) for these (i,j): (2, 36371), (36335, 2)
X(36349) = anticomplement of X(36391)
X(36349) = {X(22484), X(36371)}-harmonic conjugate of X(36333)

X(36350) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    2*(4*(b^2+c^2)*a^2-5*b^2*c^2)*S+3*sqrt(3)*(a^2+c^2+b^2)*((b^2+c^2)*a^2-2*b^2*c^2) : :
X(36350) = 5*X(2)-6*X(6581) = 13*X(2)-12*X(33482) = 3*X(194)-2*X(36378) = 13*X(6581)-10*X(33482) = 12*X(6581)-5*X(36338) = 3*X(6581)-5*X(36373) = 9*X(6581)-5*X(36393) = 24*X(33482)-13*X(36338) = 6*X(33482)-13*X(36373) = 18*X(33482)-13*X(36393) = X(36338)-4*X(36373) = 3*X(36338)-4*X(36393) = 3*X(36373)-X(36393)

The reciprocal orthologic center of these triangles is X(6582)

X(36350) lies on these lines: {2,39}, {732,36359}, {5969,36331}, {33610,35691}, {35737,35755}

X(36350) = reflection of X(i) in X(j) for these (i,j): (2, 36373), (36338, 2), (36358, 11055)
X(36350) = anticomplement of X(36393)

X(36351) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    -2*(5*a^4-3*(b^2+c^2)*a^2-4*b^4-3*b^2*c^2-4*c^4)*S+3*sqrt(3)*(a^2+b^2+c^2)*(2*a^4-b^4-c^4) : :
X(36351) = 5*X(2)-6*X(6296) = 13*X(2)-12*X(33484) = 3*X(2896)-2*X(36379) = 13*X(6296)-10*X(33484) = 12*X(6296)-5*X(36339) = 3*X(6296)-5*X(36375) = 9*X(6296)-5*X(36395) = 24*X(33484)-13*X(36339) = 6*X(33484)-13*X(36375) = 18*X(33484)-13*X(36395) = X(36339)-4*X(36375) = 3*X(36339)-4*X(36395) = 3*X(36375)-X(36395)

The reciprocal orthologic center of these triangles is X(6298)

X(36351) lies on these lines: {2,32}, {732,36358}, {35737,35756}

X(36351) = reflection of X(i) in X(j) for these (i,j): (2, 36375), (36339, 2)
X(36351) = anticomplement of X(36395)

X(36352) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO OUTER-FERMAT

Barycentrics    2*sqrt(3)*(-13*c^2-13*b^2+23*a^2)*S+29*a^4-22*(b^2+c^2)*a^2-7*(b^2-c^2)^2 : :
X(36352) = 13*X(2)-12*X(17) = 5*X(2)-6*X(627) = 4*X(2)-3*X(22113) = 7*X(2)-12*X(22844) = 5*X(2)-4*X(36366) = 3*X(2)-4*X(36386) = 10*X(17)-13*X(627) = 16*X(17)-13*X(22113) = 7*X(17)-13*X(22844) = 6*X(17)-13*X(33622) = 18*X(17)-13*X(33626) = 24*X(17)-13*X(36326) = 15*X(17)-13*X(36366) = 9*X(17)-13*X(36386) = 23*X(627)-20*X(629) = 8*X(627)-5*X(22113) = 7*X(627)-10*X(22844) = 29*X(627)-20*X(33465) = 3*X(627)-5*X(33622) = 9*X(627)-5*X(33626) = 12*X(627)-5*X(36326) = 3*X(627)-2*X(36366) = 9*X(627)-10*X(36386)

The reciprocal orthologic center of these triangles is X(617)

X(36352) lies on these lines: {2,17}, {5965,15697}, {11122,33605}, {15640,36344}, {30472,33617}

X(36352) = reflection of X(i) in X(j) for these (i,j): (2, 33622), (33626, 36386), (36326, 2)
X(36352) = anticomplement of X(33626)
X(36352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36326, 22113), (627, 36366, 2), (33622, 33626, 36386), (33626, 36386, 2)

X(36353) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (-(a^2+b^2+c^2+4*S)*sqrt(3)+9*a^2-9*b^2-9*c^2)*S+21*a^4-15*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : :
X(36353) = 5*X(2)-6*X(13706) = 13*X(2)-12*X(33486) = 3*X(13678)-2*X(36380) = 13*X(13706)-10*X(33486) = 12*X(13706)-5*X(36340) = 3*X(13706)-5*X(36376) = 9*X(13706)-5*X(36396) = 24*X(33486)-13*X(36340) = 6*X(33486)-13*X(36376) = 18*X(33486)-13*X(36396) = X(36340)-4*X(36376) = 3*X(36340)-4*X(36396) = 3*X(36376)-X(36396)

The reciprocal orthologic center of these triangles is X(13705)

X(36353) lies on these lines: {2,1327}, {530,36348}, {531,36334}, {35737,35757}

X(36353) = reflection of X(i) in X(j) for these (i,j): (2, 36376), (33456, 22917), (36340, 2)
X(36353) = anticomplement of X(36396)
X(36353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13678, 36360), (1327, 13712, 13704), (13701, 36400, 2)

X(36354) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 1st FERMAT-DAO

Barycentrics    a^2*(sqrt(3)*(a^6+17*(b^2+c^2)*a^4-(19*b^4+6*b^2*c^2+19*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b^2*c^2+2*((12*b^4+b^2*c^2+12*c^4)*a^4-2*(b^2+c^2)*(6*b^4-7*b^2*c^2+6*c^4)*a^2+(b^4-16*b^2*c^2+c^4)*b^2*c^2)*S) : :
X(36354) = 5*X(2)-6*X(14178) = 7*X(2)-6*X(14186) = 13*X(2)-12*X(33480) = 11*X(2)-12*X(33490) = 7*X(14178)-5*X(14186) = 13*X(14178)-10*X(33480) = 11*X(14178)-10*X(33490) = 12*X(14178)-5*X(36328) = 3*X(14178)-5*X(36369) = 9*X(14178)-5*X(36389) = 13*X(14186)-14*X(33480) = 11*X(14186)-14*X(33490) = 12*X(14186)-7*X(36328) = 3*X(14186)-7*X(36369) = 9*X(14186)-7*X(36389) = 11*X(33480)-13*X(33490) = 24*X(33480)-13*X(36328) = 6*X(33480)-13*X(36369) = 18*X(33480)-13*X(36389)

The reciprocal orthologic center of these triangles is X(25208)

X(36354) lies on these lines: {2,14178}, {511,35749}, {512,35750}

X(36354) = reflection of X(i) in X(j) for these (i,j): (2, 36369), (36328, 2)
X(36354) = anticomplement of X(36389)

X(36355) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -((a^2+b^2+c^2-4*S)*sqrt(3)+9*a^2-9*b^2-9*c^2)*S+21*a^4-15*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(13825)

X(36355) lies on these lines: {2,1328}, {530,36349}, {531,36335}, {35737,35758}

X(36355) = reflection of X(i) in X(j) for these (i,j): (2, 36377), (33457, 22919), (36342, 2)
X(36355) = anticomplement of X(36397)
X(36355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13798, 36361), (1328, 13835, 13824), (13821, 36401, 2)

X(36356) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    S*(-(a^2+b^2+c^2+4*S)*sqrt(3)+13*a^2-5*b^2-5*c^2)+5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2 : :
X(36356) = 5*X(2)-6*X(6305) = 13*X(2)-12*X(33446) = 3*X(488)-2*X(36370) = 13*X(6305)-10*X(33446) = 12*X(6305)-5*X(36332) = 3*X(6305)-5*X(36372) = 9*X(6305)-5*X(36392) = 24*X(33446)-13*X(36332) = 6*X(33446)-13*X(36372) = 18*X(33446)-13*X(36392) = X(36332)-4*X(36372) = 3*X(36332)-4*X(36392) = 3*X(36372)-X(36392)

The reciprocal orthologic center of these triangles is X(33442)

X(36356) lies on these lines: {2,372}, {530,36341}, {531,36360}, {3564,36349}

X(36356) = reflection of X(i) in X(j) for these (i,j): (2, 36372), (36332, 2)
X(36356) = anticomplement of X(36392)
X(36356) = {X(22485), X(36372)}-harmonic conjugate of X(36334)

X(36357) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    -S*((a^2+b^2+c^2-4*S)*sqrt(3)+13*a^2-5*b^2-5*c^2)+5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2 : :
X(36357) = 5*X(2)-6*X(6301) = 13*X(2)-12*X(33447) = 3*X(487)-2*X(36371) = 13*X(6301)-10*X(33447) = 12*X(6301)-5*X(36333) = 3*X(6301)-5*X(36374) = 9*X(6301)-5*X(36394) = 24*X(33447)-13*X(36333) = 6*X(33447)-13*X(36374) = 18*X(33447)-13*X(36394) = X(36333)-4*X(36374) = 3*X(36333)-4*X(36394) = 3*X(36374)-X(36394)

The reciprocal orthologic center of these triangles is X(33443)

X(36357) lies on these lines: {2,371}, {530,36343}, {531,36361}, {3564,36348}

X(36357) = reflection of X(i) in X(j) for these (i,j): (2, 36374), (36333, 2)
X(36357) = anticomplement of X(36394)
X(36357) = {X(22484), X(36374)}-harmonic conjugate of X(36335)

X(36358) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    -2*(4*(b^2+c^2)*a^2-5*b^2*c^2)*S+3*sqrt(3)*((b^2+c^2)*a^2-2*b^2*c^2)*(a^2+c^2+b^2) : :
X(36358) = 5*X(2)-6*X(6294) = 13*X(2)-12*X(33483) = 3*X(194)-2*X(36373) = 13*X(6294)-10*X(33483) = 12*X(6294)-5*X(36336) = 3*X(6294)-5*X(36378) = 9*X(6294)-5*X(36398) = 24*X(33483)-13*X(36336) = 6*X(33483)-13*X(36378) = 18*X(33483)-13*X(36398) = X(36336)-4*X(36378) = 3*X(36336)-4*X(36398) = 3*X(36378)-X(36398)

The reciprocal orthologic center of these triangles is X(6295)

X(36358) lies on these lines: {2,39}, {732,36351}, {5969,35750}, {33611,35695}

X(36358) = reflection of X(i) in X(j) for these (i,j): (2, 36378), (36336, 2), (36350, 11055)
X(36358) = anticomplement of X(36398)

X(36359) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    2*(5*a^4-3*(b^2+c^2)*a^2-4*b^4-3*b^2*c^2-4*c^4)*S+3*sqrt(3)*(a^2+c^2+b^2)*(2*a^4-b^4-c^4) : :
X(36359) = 5*X(2)-6*X(6297) = 13*X(2)-12*X(33485) = 3*X(2896)-2*X(36375) = 13*X(6297)-10*X(33485) = 12*X(6297)-5*X(36337) = 3*X(6297)-5*X(36379) = 9*X(6297)-5*X(36399) = 24*X(33485)-13*X(36337) = 6*X(33485)-13*X(36379) = 18*X(33485)-13*X(36399) = X(36337)-4*X(36379) = 3*X(36337)-4*X(36399) = 3*X(36379)-X(36399)

The reciprocal orthologic center of these triangles is X(6299)

X(36359) lies on these lines: {2,32}, {732,36350}

X(36359) = reflection of X(i) in X(j) for these (i,j): (2, 36379), (36337, 2)
X(36359) = anticomplement of X(36399)

X(36360) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    ((a^2+b^2+c^2+4*S)*sqrt(3)+9*a^2-9*b^2-9*c^2)*S+21*a^4-15*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : :
X(36360) = 5*X(2)-6*X(13704) = 13*X(2)-12*X(33487) = 3*X(13678)-2*X(36376) = 13*X(13704)-10*X(33487) = 12*X(13704)-5*X(36341) = 3*X(13704)-5*X(36380) = 9*X(13704)-5*X(36400) = 24*X(33487)-13*X(36341) = 6*X(33487)-13*X(36380) = 18*X(33487)-13*X(36400) = X(36341)-4*X(36380) = 3*X(36341)-4*X(36400) = 3*X(36380)-X(36400)

The reciprocal orthologic center of these triangles is X(13703)

X(36360) lies on these lines: {2,1327}, {530,36332}, {531,36356}

X(36360) = reflection of X(i) in X(j) for these (i,j): (2, 36380), (33456, 22872), (36341, 2)
X(36360) = anticomplement of X(36400)
X(36360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13678, 36353), (1327, 13712, 13706), (13701, 36396, 2)

X(36361) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -(-(a^2+b^2+c^2-4*S)*sqrt(3)+9*a^2-9*b^2-9*c^2)*S+21*a^4-15*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : :
X(36361) = 5*X(2)-6*X(13824) = 13*X(2)-12*X(33489) = 3*X(13798)-2*X(36377) = 13*X(13824)-10*X(33489) = 12*X(13824)-5*X(36343) = 3*X(13824)-5*X(36381) = 9*X(13824)-5*X(36401) = 18*X(33489)-13*X(36401) = X(36343)-4*X(36381) = 3*X(36343)-4*X(36401) = 3*X(36381)-X(36401)

The reciprocal orthologic center of these triangles is X(13823)

X(36361) lies on these lines: {2,1328}, {530,36333}, {531,36357}

X(36361) = reflection of X(i) in X(j) for these (i,j): (2, 36381), (33457, 22874), (36343, 2)
X(36361) = anticomplement of X(36401)
X(36361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13798, 36355), (1328, 13835, 13826), (13821, 36397, 2)

X(36362) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    -8*sqrt(3)*S^3+11*a^6-11*(b^2+c^2)*a^4+(7*b^4+6*b^2*c^2+7*c^4)*a^2-7*(b^4-c^4)*(b^2-c^2) : :
X(36362) = 2*X(2)-3*X(5613) = 5*X(2)-3*X(6773) = 7*X(2)-6*X(6774) = 3*X(147)-X(36344) = 5*X(5613)-2*X(6773) = 7*X(5613)-4*X(6774) = 3*X(5613)+2*X(36319) = 9*X(5613)-2*X(36320) = 3*X(5613)-X(36382) = 7*X(6773)-10*X(6774) = 3*X(6773)+5*X(36319) = 9*X(6773)-5*X(36320) = 6*X(6773)-5*X(36382) = 6*X(6774)+7*X(36319) = 18*X(6774)-7*X(36320) = 12*X(6774)-7*X(36382)

The reciprocal orthologic center of these triangles is X(5979)

X(36362) lies on these lines: {2,98}, {4,36327}, {14,5066}, {30,22494}, {299,11645}, {530,35692}, {531,3830}, {617,11001}, {619,15693}, {2782,35752}, {3545,32909}, {3839,16002}, {3845,25164}, {5071,20416}, {5464,8703}, {5471,18362}, {5474,19710}, {5873,22495}, {6775,6778}, {6777,22566}, {9760,33459}, {9763,18440}, {12816,23004}, {13858,18570}, {14830,32552}, {15682,33624}, {15713,21157}, {22512,22997}, {35734,35742}

X(36362) = midpoint of X(i) and X(j) for these {i,j}: {2, 36319}, {15682, 36331}
X(36362) = reflection of X(i) in X(j) for these (i,j): (11177, 25559), (14830, 32552), (22507, 6054), (36330, 3845), (36382, 2)
X(36362) = complement of X(36320)
X(36362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3845, 36330, 25164), (5613, 36382, 2)

X(36363) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    8*sqrt(3)*S^3+11*a^6-11*(b^2+c^2)*a^4+(7*b^4+6*b^2*c^2+7*c^4)*a^2-7*(b^4-c^4)*(b^2-c^2) : :
X(36363) = 2*X(2)-3*X(5617) = 5*X(2)-3*X(6770) = 7*X(2)-6*X(6771) = 3*X(147)-X(36319) = 5*X(5617)-2*X(6770) = 7*X(5617)-4*X(6771) = 9*X(5617)-2*X(36318) = 3*X(5617)+2*X(36344) = 3*X(5617)-X(36383) = 7*X(6770)-10*X(6771) = 9*X(6770)-5*X(36318) = 3*X(6770)+5*X(36344) = 6*X(6770)-5*X(36383) = 18*X(6771)-7*X(36318) = 6*X(6771)+7*X(36344) = 12*X(6771)-7*X(36383)

The reciprocal orthologic center of these triangles is X(5978)

X(36363) lies on these lines: {2,98}, {4,35749}, {13,5066}, {30,22493}, {298,11645}, {530,3830}, {531,35696}, {616,11001}, {618,15693}, {2782,36330}, {3545,32907}, {3839,16001}, {3845,25154}, {5071,20415}, {5463,8703}, {5472,18362}, {5473,19710}, {5872,22496}, {6772,6777}, {6778,22566}, {9761,18440}, {9762,33458}, {12817,23005}, {13859,18570}, {14830,32553}, {15682,33622}, {15713,21156}, {22513,22998}

X(36363) = midpoint of X(i) and X(j) for these {i,j}: {2, 36344}, {15682, 35750}
X(36363) = reflection of X(i) in X(j) for these (i,j): (11177, 25560), (14830, 32553), (22509, 6054), (35752, 3845), (36383, 2)
X(36363) = complement of X(36318)
X(36363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3845, 35752, 25154), (5617, 36383, 2)

X(36364) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    -6*sqrt(3)*S*((b^2+c^2)*a^2-b^4-c^4)*a^2+5*(b^2+c^2)*a^6-4*(b^4+c^4)*a^4-(b^2+c^2)*(b^4+8*b^2*c^2+c^4)*a^2+2*(b^2-c^2)^2*b^2*c^2 : :
X(36364) = 2*X(2)-3*X(22715) = 7*X(2)-6*X(33478) = 4*X(5066)-3*X(22693) = 3*X(6194)-X(36347) = 3*X(22686)-X(36385) = 7*X(22715)-4*X(33478) = 9*X(22715)-2*X(36323) = 3*X(22715)+2*X(36345) = 3*X(22715)-X(36384) = 18*X(33478)-7*X(36323) = 6*X(33478)+7*X(36345) = 12*X(33478)-7*X(36384) = X(36323)+3*X(36345) = 2*X(36323)-3*X(36384) = 2*X(36345)+X(36384)

The reciprocal orthologic center of these triangles is X(22687)

X(36364) lies on these lines: {2,51}, {299,23018}, {549,3104}, {2782,36329}, {5066,22693}, {35734,35745}

X(36364) = midpoint of X(2) and X(36345)
X(36364) = reflection of X(i) in X(j) for these (i,j): (3104, 549), (22684, 22712), (36384, 2)
X(36364) = complement of X(36323)
X(36364) = {X(22715), X(36384)}-harmonic conjugate of X(2)

X(36365) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    6*sqrt(3)*S*((b^2+c^2)*a^2-b^4-c^4)*a^2+5*(b^2+c^2)*a^6-4*(b^4+c^4)*a^4-(b^2+c^2)*(b^4+8*b^2*c^2+c^4)*a^2+2*(b^2-c^2)^2*b^2*c^2 : :
X(36365) = 2*X(2)-3*X(22714) = 7*X(2)-6*X(33479) = 4*X(5066)-3*X(22694) = 3*X(6194)-X(36345) = 3*X(22684)-X(36384) = 7*X(22714)-4*X(33479) = 9*X(22714)-2*X(36322) = 3*X(22714)+2*X(36347) = 3*X(22714)-X(36385) = 18*X(33479)-7*X(36322) = 6*X(33479)+7*X(36347) = 12*X(33479)-7*X(36385) = X(36322)+3*X(36347) = 2*X(36322)-3*X(36385) = 2*X(36347)+X(36385)

The reciprocal orthologic center of these triangles is X(22689)

X(36365) lies on these lines: {2,51}, {298,23024}, {549,3105}, {2782,35751}, {5066,22694}

X(36365) = midpoint of X(2) and X(36347)
X(36365) = reflection of X(i) in X(j) for these (i,j): (3105, 549), (22686, 22712), (36385, 2)
X(36365) = complement of X(36322)
X(36365) = {X(22714), X(36385)}-harmonic conjugate of X(2)

X(36366) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO OUTER-FERMAT

Barycentrics    2*sqrt(3)*(7*a^2-2*c^2-2*b^2)*S+a^4+7*(b^2+c^2)*a^2-8*(b^2-c^2)^2 : :
X(36366) = 2*X(2)-3*X(17) = 5*X(2)-3*X(627) = 7*X(2)-6*X(629) = 11*X(2)-12*X(6673) = X(2)+3*X(22113) = 8*X(2)-3*X(22844) = X(2)-6*X(33465) = 3*X(2)+X(36326) = 5*X(2)-X(36352) = 5*X(17)-2*X(627) = 7*X(17)-4*X(629) = 11*X(17)-8*X(6673) = X(17)+2*X(22113) = 4*X(17)-X(22844) = X(17)-4*X(33465) = 9*X(17)-2*X(33622) = 3*X(17)+2*X(33626) = 9*X(17)+2*X(36326) = 15*X(17)-2*X(36352) = 3*X(17)-X(36386)

The reciprocal orthologic center of these triangles is X(13)

X(36366) lies on these lines: {2,17}, {13,5859}, {530,22895}, {531,11122}, {3180,16808}, {3412,11299}, {3534,35752}, {3830,16629}, {3845,25164}, {4677,22652}, {4745,22896}, {5066,16626}, {5093,5476}, {5459,33607}, {5463,22892}, {5464,22900}, {5858,22894}, {5862,18582}, {5863,31705}, {8584,22891}, {8703,22890}, {10611,15533}, {11001,22532}, {11296,36388}, {11300,35689}, {12100,21159}, {22490,22893}, {22492,31704}, {22997,35693}, {35734,35747}

X(36366) = midpoint of X(i) and X(j) for these {i,j}: {2, 33626}, {33622, 36326}
X(36366) = reflection of X(i) in X(j) for these (i,j): (5463, 22892), (36386, 2)
X(36366) = complement of X(33622)
X(36366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22113, 33626), (2, 36326, 33622), (2, 36352, 627), (17, 36386, 2), (15534, 19709, 36368), (22113, 33465, 17), (22488, 22666, 36386), (33622, 33626, 36326)

X(36367) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 2nd FERMAT-DAO

Barycentrics    a^2*(6*b^2*c^2*(a^6+8*(b^2+c^2)*a^4-2*(5*b^4+3*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2))-4*S*sqrt(3)*((6*b^4+b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(6*b^4-5*b^2*c^2+6*c^4)*a^2+(b^4-10*b^2*c^2+c^4)*b^2*c^2)) : :
X(36367) = 2*X(2)-3*X(14182) = 4*X(2)-3*X(14188) = 7*X(2)-6*X(33481) = 5*X(2)-6*X(33491) = 7*X(14182)-4*X(33481) = 5*X(14182)-4*X(33491) = 3*X(14182)+2*X(36321) = 9*X(14182)-2*X(36325) = 3*X(14182)-X(36387) = 7*X(14188)-8*X(33481) = 5*X(14188)-8*X(33491) = 3*X(14188)+4*X(36321) = 9*X(14188)-4*X(36325) = 3*X(14188)-2*X(36387) = 5*X(33481)-7*X(33491) = 6*X(33481)+7*X(36321) = 18*X(33481)-7*X(36325) = 12*X(33481)-7*X(36387)

The reciprocal orthologic center of these triangles is X(25207)

X(36367) lies on these lines: {2,14182}, {511,36330}, {512,36329}, {3845,25224}, {5066,25180}, {35734,35761}, {35735,35760}

X(36367) = midpoint of X(2) and X(36321)
X(36367) = reflection of X(i) in X(j) for these (i,j): (14188, 14182), (36387, 2)
X(36367) = complement of X(36325)
X(36367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36387, 14188), (14182, 36387, 2)

X(36368) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO INNER-FERMAT

Barycentrics    -2*sqrt(3)*(7*a^2-2*c^2-2*b^2)*S+a^4+7*(b^2+c^2)*a^2-8*(b^2-c^2)^2 : :
X(36368) = 2*X(2)-3*X(18) = 5*X(2)-3*X(628) = 7*X(2)-6*X(630) = 11*X(2)-12*X(6674) = X(2)+3*X(22114) = 8*X(2)-3*X(22845) = X(2)-6*X(33464) = 3*X(2)+X(36324) = 5*X(2)-X(36346) = 5*X(18)-2*X(628) = 7*X(18)-4*X(630) = 11*X(18)-8*X(6674) = X(18)+2*X(22114) = 4*X(18)-X(22845) = X(18)-4*X(33464) = 9*X(18)-2*X(33624) = 3*X(18)+2*X(33627) = 9*X(18)+2*X(36324) = 15*X(18)-2*X(36346) = 3*X(18)-X(36388)

The reciprocal orthologic center of these triangles is X(14)

X(36368) lies on these lines: {2,18}, {14,5858}, {530,11121}, {531,22849}, {3181,16809}, {3411,11300}, {3534,36330}, {3830,16628}, {3845,25154}, {4677,22651}, {4745,22851}, {5066,16627}, {5093,5476}, {5460,33606}, {5463,22856}, {5464,22848}, {5859,22850}, {5862,31706}, {5863,18581}, {8584,22846}, {8703,22843}, {10612,15533}, {11001,22531}, {11295,36386}, {11299,35688}, {12100,21158}, {22489,22847}, {22491,31703}, {22998,35697}

X(36368) = midpoint of X(i) and X(j) for these {i,j}: {2, 33627}, {33624, 36324}
X(36368) = reflection of X(i) in X(j) for these (i,j): (5464, 22848), (36388, 2)
X(36368) = complement of X(33624)
X(36368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22114, 33627), (2, 36324, 33624), (2, 36346, 628), (18, 36388, 2), (15534, 19709, 36366), (22114, 33464, 18), (22487, 22665, 36388), (33624, 33627, 36324)

X(36369) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st FERMAT-DAO

Barycentrics    a^2*(6*b^2*c^2*(a^6+8*(b^2+c^2)*a^4-2*(5*b^4+3*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2))+4*S*sqrt(3)*((6*b^4+b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(6*b^4-5*b^2*c^2+6*c^4)*a^2+(b^4-10*b^2*c^2+c^4)*b^2*c^2)) : :
X(36369) = 2*X(2)-3*X(14178) = 4*X(2)-3*X(14186) = 7*X(2)-6*X(33480) = 5*X(2)-6*X(33490) = 7*X(14178)-4*X(33480) = 5*X(14178)-4*X(33490) = 9*X(14178)-2*X(36328) = 3*X(14178)+2*X(36354) = 3*X(14178)-X(36389) = 7*X(14186)-8*X(33480) = 5*X(14186)-8*X(33490) = 9*X(14186)-4*X(36328) = 3*X(14186)+4*X(36354) = 3*X(14186)-2*X(36389) = 5*X(33480)-7*X(33490) = 18*X(33480)-7*X(36328) = 6*X(33480)+7*X(36354) = 12*X(33480)-7*X(36389)

The reciprocal orthologic center of these triangles is X(25208)

X(36369) lies on these lines: {2,14178}, {511,35752}, {512,35751}, {3845,25223}, {5066,25175}

X(36369) = midpoint of X(2) and X(36354)
X(36369) = reflection of X(i) in X(j) for these (i,j): (14186, 14178), (36389, 2)
X(36369) = complement of X(36328)
X(36369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36389, 14186), (14178, 36389, 2)

X(36370) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    (2*(a^2+b^2+c^2+4*S)*sqrt(3)+14*a^2-4*c^2-4*b^2)*S+4*a^4+(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(36370) = 2*X(2)-3*X(6304) = 7*X(2)-6*X(33444) = 3*X(488)-X(36356) = 4*X(5066)-3*X(22634) = 7*X(6304)-4*X(33444) = 9*X(6304)-2*X(36334) = 3*X(6304)+2*X(36348) = 3*X(6304)-X(36390) = 3*X(22629)-X(36392) = 18*X(33444)-7*X(36334) = 6*X(33444)+7*X(36348) = 12*X(33444)-7*X(36390) = X(36334)+3*X(36348) = 2*X(36334)-3*X(36390) = 2*X(36348)+X(36390)

The reciprocal orthologic center of these triangles is X(33440)

X(36370) lies on these lines: {2,372}, {524,22917}, {530,36376}, {531,36396}, {3564,36374}, {5066,22634}, {31699,36397}, {35734,35744}

X(36370) = midpoint of X(2) and X(36348)
X(36370) = reflection of X(36390) in X(2)
X(36370) = complement of X(36334)
X(36370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22485, 36372), (488, 33446, 6304), (6304, 36390, 2), (22485, 36332, 36392), (36332, 36348, 22485)

X(36371) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    -(-2*(a^2+b^2+c^2-4*S)*sqrt(3)+14*a^2-4*c^2-4*b^2)*S+4*a^4+(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(36371) = 2*X(2)-3*X(6300) = 7*X(2)-6*X(33445) = 3*X(487)-X(36357) = 4*X(5066)-3*X(22605) = 7*X(6300)-4*X(33445) = 9*X(6300)-2*X(36335) = 3*X(6300)+2*X(36349) = 3*X(6300)-X(36391) = 3*X(22600)-X(36394) = 18*X(33445)-7*X(36335) = 6*X(33445)+7*X(36349) = 12*X(33445)-7*X(36391) = X(36335)+3*X(36349) = 2*X(36335)-3*X(36391) = 2*X(36349)+X(36391)

The reciprocal orthologic center of these triangles is X(33441)

X(36371) lies on these lines: {2,371}, {524,22919}, {530,36377}, {531,36397}, {3564,36372}, {5066,22605}, {31697,36396}, {35734,35743}

X(36371) = midpoint of X(2) and X(36349)
X(36371) = reflection of X(36391) in X(2)
X(36371) = complement of X(36335)
X(36371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22484, 36374), (487, 33447, 6300), (6300, 36391, 2), (22484, 36333, 36394), (36333, 36349, 22484)

X(36372) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    (-2*(a^2+b^2+c^2+4*S)*sqrt(3)+14*a^2-4*c^2-4*b^2)*S+4*a^4+(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(36372) = 2*X(2)-3*X(6305) = 7*X(2)-6*X(33446) = 3*X(488)-X(36348) = 4*X(5066)-3*X(22635) = 7*X(6305)-4*X(33446) = 9*X(6305)-2*X(36332) = 3*X(6305)+2*X(36356) = 3*X(6305)-X(36392) = 3*X(22627)-X(36390) = 18*X(33446)-7*X(36332) = 6*X(33446)+7*X(36356) = 12*X(33446)-7*X(36392) = X(36332)+3*X(36356) = 2*X(36332)-3*X(36392) = 2*X(36356)+X(36392)

The reciprocal orthologic center of these triangles is X(33442)

X(36372) lies on these lines: {2,372}, {524,22872}, {530,36400}, {531,36380}, {3564,36371}, {5066,22635}, {31700,36401}

X(36372) = midpoint of X(2) and X(36356)
X(36372) = reflection of X(36392) in X(2)
X(36372) = complement of X(36332)
X(36372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22485, 36370), (488, 33444, 6305), (6305, 36392, 2), (22485, 36334, 36390), (36334, 36356, 22485)

X(36373) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    2*(5*(b^2+c^2)*a^2-4*b^2*c^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*((b^2+c^2)*a^2-2*b^2*c^2) : :
X(36373) = 2*X(2)-3*X(6581) = 7*X(2)-6*X(33482) = 3*X(194)-X(36358) = 4*X(5066)-3*X(25191) = 7*X(6581)-4*X(33482) = 9*X(6581)-2*X(36338) = 3*X(6581)+2*X(36350) = 3*X(6581)-X(36393) = 3*X(22913)-X(36398) = 18*X(33482)-7*X(36338) = 6*X(33482)+7*X(36350) = 12*X(33482)-7*X(36393) = X(36338)+3*X(36350) = 2*X(36338)-3*X(36393) = 2*X(36350)+X(36393)

The reciprocal orthologic center of these triangles is X(6582)

X(36373) lies on these lines: {2,39}, {732,36379}, {5066,25191}, {5969,36329}, {35734,35755}

X(36373) = midpoint of X(2) and X(36350)
X(36373) = reflection of X(i) in X(j) for these (i,j): (22868, 7757), (36393, 2)
X(36373) = complement of X(36338)
X(36373) = {X(6581), X(36393)}-harmonic conjugate of X(2)

X(36374) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    (-2*(a^2+b^2+c^2-4*S)*sqrt(3)-14*a^2+4*c^2+4*b^2)*S+4*a^4+(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(36374) = 2*X(2)-3*X(6301) = 7*X(2)-6*X(33447) = 3*X(487)-X(36349) = 4*X(5066)-3*X(22606) = 7*X(6301)-4*X(33447) = 9*X(6301)-2*X(36333) = 3*X(6301)+2*X(36357) = 3*X(6301)-X(36394) = 3*X(22598)-X(36391) = 18*X(33447)-7*X(36333) = 6*X(33447)+7*X(36357) = 12*X(33447)-7*X(36394) = X(36333)+3*X(36357) = 2*X(36333)-3*X(36394) = 2*X(36357)+X(36394)

The reciprocal orthologic center of these triangles is X(33443)

X(36374) lies on these lines: {2,371}, {524,22874}, {530,36401}, {531,36381}, {3564,36370}, {5066,22606}, {31698,36400}

X(36374) = midpoint of X(2) and X(36357)
X(36374) = reflection of X(36394) in X(2)
X(36374) = complement of X(36333)
X(36374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22484, 36371), (487, 33445, 6301), (6301, 36394, 2), (22484, 36335, 36391), (36335, 36357, 22484)

X(36375) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    -2*(4*a^4-6*(b^2+c^2)*a^2-5*b^4-6*b^2*c^2-5*c^4)*S+3*sqrt(3)*(2*a^4-b^4-c^4)*(a^2+b^2+c^2) : :
X(36375) = 2*X(2)-3*X(6296) = 7*X(2)-6*X(33484) = 3*X(2896)-X(36359) = 4*X(5066)-3*X(25192) = 7*X(6296)-4*X(33484) = 9*X(6296)-2*X(36339) = 3*X(6296)+2*X(36351) = 3*X(6296)-X(36395) = 3*X(22915)-X(36399) = 18*X(33484)-7*X(36339) = 6*X(33484)+7*X(36351) = 12*X(33484)-7*X(36395) = X(36339)+3*X(36351) = 2*X(36339)-3*X(36395) = 2*X(36351)+X(36395)

The reciprocal orthologic center of these triangles is X(6298)

X(36375) lies on these lines: {2,32}, {732,36378}, {5066,25192}, {33611,35696}, {35734,35756}

X(36375) = midpoint of X(2) and X(36351)
X(36375) = reflection of X(i) in X(j) for these (i,j): (22870, 31168), (36395, 2)
X(36375) = complement of X(36339)
X(36375) = {X(6296), X(36395)}-harmonic conjugate of X(2)

X(36376) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    2*(-(a^2+b^2+c^2+4*S)*sqrt(3)+3*a^2-6*b^2-6*c^2)*S+24*a^4-21*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(36376) = 2*X(2)-3*X(13706) = 7*X(2)-6*X(33486) = 4*X(5066)-3*X(25193) = 3*X(13678)-X(36360) = 7*X(13706)-4*X(33486) = 9*X(13706)-2*X(36340) = 3*X(13706)+2*X(36353) = 3*X(13706)-X(36396) = 3*X(22917)-X(36400) = 18*X(33486)-7*X(36340) = 6*X(33486)+7*X(36353) = 12*X(33486)-7*X(36396) = X(36340)+3*X(36353) = 2*X(36340)-3*X(36396) = 2*X(36353)+X(36396)

The reciprocal orthologic center of these triangles is X(13705)

X(36376) lies on these lines: {2,1327}, {530,36370}, {531,36390}, {5066,25193}, {35734,35757}

X(36376) = midpoint of X(2) and X(36353)
X(36376) = reflection of X(i) in X(j) for these (i,j): (22872, 13712), (33456, 33470), (36396, 2)
X(36376) = complement of X(36340)
X(36376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13712, 36380), (13701, 13712, 13706), (13704, 22917, 13712), (13706, 36396, 2)

X(36377) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -2*((a^2+b^2+c^2-4*S)*sqrt(3)+3*a^2-6*b^2-6*c^2)*S+24*a^4-21*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(36377) = 2*X(2)-3*X(13826) = 7*X(2)-6*X(33488) = 4*X(5066)-3*X(25194) = 3*X(13798)-X(36361) = 3*X(13826)-X(36397) = 3*X(22919)-X(36401) = 2*X(36355)+X(36397)

The reciprocal orthologic center of these triangles is X(13825)

X(36377) lies on these lines: {2,1328}, {530,36371}, {531,36391}, {5066,25194}, {35734,35758}

X(36377) = midpoint of X(2) and X(36355)
X(36377) = reflection of X(i) in X(j) for these (i,j): (22874, 13835), (33457, 33472), (36397, 2)
X(36377) = complement of X(36342)
X(36377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13835, 36381), (13821, 13835, 13826), (13824, 22919, 13835), (13826, 36397, 2)

X(36378) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    -2*(5*(b^2+c^2)*a^2-4*b^2*c^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*((b^2+c^2)*a^2-2*b^2*c^2) : :
X(36378) = 2*X(2)-3*X(6294) = 7*X(2)-6*X(33483) = 3*X(194)-X(36350) = 4*X(5066)-3*X(25195) = 7*X(6294)-4*X(33483) = 9*X(6294)-2*X(36336) = 3*X(6294)+2*X(36358) = 3*X(6294)-X(36398) = 3*X(22868)-X(36393) = 18*X(33483)-7*X(36336) = 6*X(33483)+7*X(36358) = 12*X(33483)-7*X(36398) = X(36336)+3*X(36358) = 2*X(36336)-3*X(36398) = 2*X(36358)+X(36398)

The reciprocal orthologic center of these triangles is X(6295)

X(36378) lies on these lines: {2,39}, {732,36375}, {5066,25195}, {5969,35751}

X(36378) = midpoint of X(2) and X(36358)
X(36378) = reflection of X(i) in X(j) for these (i,j): (22913, 7757), (36398, 2)
X(36378) = complement of X(36336)
X(36378) = {X(6294), X(36398)}-harmonic conjugate of X(2)

X(36379) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    2*(4*a^4-6*(b^2+c^2)*a^2-5*b^4-6*b^2*c^2-5*c^4)*S+3*sqrt(3)*(2*a^4-b^4-c^4)*(a^2+b^2+c^2) : :
X(36379) = 2*X(2)-3*X(6297) = 7*X(2)-6*X(33485) = 3*X(2896)-X(36351) = 4*X(5066)-3*X(25196) = 7*X(6297)-4*X(33485) = 9*X(6297)-2*X(36337) = 3*X(6297)+2*X(36359) = 3*X(6297)-X(36399) = 3*X(22870)-X(36395) = 18*X(33485)-7*X(36337) = 6*X(33485)+7*X(36359) = 12*X(33485)-7*X(36399) = X(36337)+3*X(36359) = 2*X(36337)-3*X(36399) = 2*X(36359)+X(36399)

The reciprocal orthologic center of these triangles is X(6299)

X(36379) lies on these lines: {2,32}, {732,36373}, {5066,25196}, {33610,35692}

X(36379) = midpoint of X(2) and X(36359)
X(36379) = reflection of X(i) in X(j) for these (i,j): (22915, 31168), (36399, 2)
X(36379) = complement of X(36337)
X(36379) = {X(6297), X(36399)}-harmonic conjugate of X(2)

X(36380) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    2*((a^2+b^2+c^2+4*S)*sqrt(3)+3*a^2-6*b^2-6*c^2)*S+24*a^4-21*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(36380) = 2*X(2)-3*X(13704) = 7*X(2)-6*X(33487) = 4*X(5066)-3*X(25197) = 3*X(13678)-X(36353) = 7*X(13704)-4*X(33487) = 9*X(13704)-2*X(36341) = 3*X(13704)+2*X(36360) = 3*X(13704)-X(36400) = 3*X(22872)-X(36396) = 18*X(33487)-7*X(36341) = 6*X(33487)+7*X(36360) = 12*X(33487)-7*X(36400) = X(36341)+3*X(36360) = 2*X(36341)-3*X(36400) = 2*X(36360)+X(36400)

The reciprocal orthologic center of these triangles is X(13703)

X(36380) lies on these lines: {2,1327}, {530,36392}, {531,36372}, {5066,25197}

X(36380) = midpoint of X(2) and X(36360)
X(36380) = reflection of X(i) in X(j) for these (i,j): (22917, 13712), (33456, 33471), (36400, 2)
X(36380) = complement of X(36341)
X(36380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13712, 36376), (13701, 13712, 13704), (13704, 36400, 2), (13706, 22872, 13712)

X(36381) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -2*(-(a^2+b^2+c^2-4*S)*sqrt(3)+3*a^2-6*b^2-6*c^2)*S+24*a^4-21*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(36381) = 2*X(2)-3*X(13824) = 7*X(2)-6*X(33489) = 4*X(5066)-3*X(25198) = 3*X(13798)-X(36355) = 7*X(13824)-4*X(33489) = 9*X(13824)-2*X(36343) = 3*X(13824)+2*X(36361) = 3*X(13824)-X(36401) = 3*X(22874)-X(36397) = 18*X(33489)-7*X(36343) = 6*X(33489)+7*X(36361) = 12*X(33489)-7*X(36401) = X(36343)+3*X(36361) = 2*X(36343)-3*X(36401) = 2*X(36361)+X(36401)

The reciprocal orthologic center of these triangles is X(13823)

X(36381) lies on these lines: {2,1328}, {530,36394}, {531,36374}, {5066,25198}

X(36381) = midpoint of X(2) and X(36361)
X(36381) = reflection of X(i) in X(j) for these (i,j): (22919, 13835), (33457, 33473), (36401, 2)
X(36381) = complement of X(36343)
X(36381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13835, 36377), (13821, 13835, 13824), (13824, 36401, 2), (13826, 22874, 13835)

X(36382) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    8*sqrt(3)*S^3+13*a^6-13*(b^2+c^2)*a^4+(5*b^4-6*b^2*c^2+5*c^4)*a^2-5*(b^4-c^4)*(b^2-c^2) : :
X(36382) = 4*X(2)-3*X(5613) = X(2)-3*X(6773) = 5*X(2)-6*X(6774) = X(5613)-4*X(6773) = 5*X(5613)-8*X(6774) = 9*X(5613)-4*X(36319) = 3*X(5613)+4*X(36320) = 3*X(5613)-2*X(36362) = 5*X(6773)-2*X(6774) = 9*X(6773)-X(36319) = 3*X(6773)+X(36320) = 6*X(6773)-X(36362) = 18*X(6774)-5*X(36319) = 6*X(6774)+5*X(36320) = 12*X(6774)-5*X(36362)

The reciprocal orthologic center of these triangles is X(5979)

X(36382) lies on these lines: {2,98}, {4,32909}, {14,3845}, {30,22496}, {376,36331}, {530,35693}, {531,3534}, {617,19708}, {619,15701}, {2782,35751}, {3181,19924}, {3543,16002}, {3545,20416}, {3830,25164}, {5460,19709}, {5464,12100}, {5471,11648}, {5474,15690}, {5872,22493}, {6775,6777}, {8703,36329}, {9886,14830}, {10109,22490}, {11001,33627}, {11812,21157}, {12243,35749}, {15682,33625}, {16626,31693}, {35735,35742}

X(36382) = midpoint of X(i) and X(j) for these {i,j}: {2, 36320}, {11001, 36327}
X(36382) = reflection of X(i) in X(j) for these (i,j): (4, 32909), (3543, 16002), (6054, 25560), (22509, 6055), (36329, 8703), (36362, 2)
X(36382) = complement of X(36319)
X(36382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36362, 5613), (6773, 36320, 2)

X(36383) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st ANTI-BROCARD

Barycentrics    -8*sqrt(3)*S^3+13*a^6-13*(b^2+c^2)*a^4+(5*b^4-6*b^2*c^2+5*c^4)*a^2-5*(b^4-c^4)*(b^2-c^2) : :
X(36383) = 4*X(2)-3*X(5617) = X(2)-3*X(6770) = 5*X(2)-6*X(6771) = X(5617)-4*X(6770) = 5*X(5617)-8*X(6771) = 3*X(5617)+4*X(36318) = 9*X(5617)-4*X(36344) = 3*X(5617)-2*X(36363) = 5*X(6770)-2*X(6771) = 3*X(6770)+X(36318) = 9*X(6770)-X(36344) = 6*X(6770)-X(36363) = 6*X(6771)+5*X(36318) = 18*X(6771)-5*X(36344) = 12*X(6771)-5*X(36363)

The reciprocal orthologic center of these triangles is X(5978)

X(36383) lies on these lines: {2,98}, {4,32907}, {13,3845}, {30,22495}, {376,35750}, {530,3534}, {531,35697}, {616,19708}, {618,15701}, {2782,36329}, {3180,19924}, {3543,16001}, {3545,20415}, {3830,25154}, {5459,19709}, {5463,12100}, {5472,11648}, {5473,15690}, {5873,22494}, {6772,6778}, {8703,35751}, {9885,14830}, {10109,22489}, {11001,33626}, {11812,21156}, {12243,36327}, {15682,33623}, {16627,31694}

X(36383) = midpoint of X(i) and X(j) for these {i,j}: {2, 36318}, {11001, 35749}
X(36383) = reflection of X(i) in X(j) for these (i,j): (4, 32907), (3543, 16001), (6054, 25559), (22507, 6055), (35751, 8703), (36363, 2)
X(36383) = complement of X(36344)
X(36383) = {X(2), X(36363)}-harmonic conjugate of X(5617)

X(36384) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    -6*sqrt(3)*S*((b^2+c^2)*a^2-b^4-c^4)*a^2+(b^2+c^2)*a^6+4*(b^4+3*b^2*c^2+c^4)*a^4-(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :
X(36384) = 4*X(2)-3*X(22715) = 5*X(2)-6*X(33478) = 2*X(3845)-3*X(22693) = 3*X(22684)-2*X(36365) = 5*X(22715)-8*X(33478) = 3*X(22715)+4*X(36323) = 9*X(22715)-4*X(36345) = 3*X(22715)-2*X(36364) = 6*X(33478)+5*X(36323) = 18*X(33478)-5*X(36345) = 12*X(33478)-5*X(36364) = 3*X(36323)+X(36345) = 2*X(36323)+X(36364) = 2*X(36345)-3*X(36364)

The reciprocal orthologic center of these triangles is X(22687)

X(36384) lies on these lines: {2,51}, {30,3104}, {298,23018}, {533,6581}, {2782,36330}, {3105,16963}, {3845,22693}, {5617,5969}, {11055,33627}, {12816,25154}, {25164,35693}, {35735,35745}

X(36384) = midpoint of X(2) and X(36323)
X(36384) = reflection of X(i) in X(j) for these (i,j): (22712, 33463), (36364, 2)
X(36384) = complement of X(36345)
X(36384) = {X(2), X(36364)}-harmonic conjugate of X(22715)

X(36385) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st BROCARD-REFLECTED

Barycentrics    6*sqrt(3)*S*((b^2+c^2)*a^2-b^4-c^4)*a^2+(b^2+c^2)*a^6+4*(b^4+3*b^2*c^2+c^4)*a^4-(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :
X(36385) = 4*X(2)-3*X(22714) = 5*X(2)-6*X(33479) = 2*X(3845)-3*X(22694) = 3*X(22686)-2*X(36364) = 5*X(22714)-8*X(33479) = 3*X(22714)+4*X(36322) = 9*X(22714)-4*X(36347) = 3*X(22714)-2*X(36365) = 6*X(33479)+5*X(36322) = 18*X(33479)-5*X(36347) = 12*X(33479)-5*X(36365) = 3*X(36322)+X(36347) = 2*X(36322)+X(36365) = 2*X(36347)-3*X(36365)

The reciprocal orthologic center of these triangles is X(22689)

X(36385) lies on these lines: {2,51}, {30,3105}, {299,23024}, {532,6294}, {2782,35752}, {3104,16962}, {3845,22694}, {5613,5969}, {11055,33626}, {12817,25164}, {25154,35697}

X(36385) = midpoint of X(2) and X(36322)
X(36385) = reflection of X(i) in X(j) for these (i,j): (22712, 33462), (36365, 2)
X(36385) = complement of X(36347)
X(36385) = {X(2), X(36365)}-harmonic conjugate of X(22714)

X(36386) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO OUTER-FERMAT

Barycentrics    2*sqrt(3)*(5*a^2-4*c^2-4*b^2)*S+11*a^4-13*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
X(36386) = 4*X(2)-3*X(17) = X(2)-3*X(627) = 5*X(2)-6*X(629) = 13*X(2)-12*X(6673) = 7*X(2)-3*X(22113) = 2*X(2)+3*X(22844) = 11*X(2)-6*X(33465) = 5*X(2)-X(36326) = 3*X(2)+X(36352) = X(17)-4*X(627) = 5*X(17)-8*X(629) = 13*X(17)-16*X(6673) = 7*X(17)-4*X(22113) = X(17)+2*X(22844) = 11*X(17)-8*X(33465) = 3*X(17)+4*X(33622) = 9*X(17)-4*X(33626) = 15*X(17)-4*X(36326) = 9*X(17)+4*X(36352) = 3*X(17)-2*X(36366)

The reciprocal orthologic center of these triangles is X(13)

X(36386) lies on these lines: {2,17}, {14,33459}, {15,5862}, {298,11057}, {530,12816}, {616,19107}, {3534,22493}, {3643,16961}, {3830,35751}, {3845,16626}, {4669,22896}, {5085,5965}, {5460,11122}, {5463,5858}, {5464,30472}, {5859,16241}, {8703,36329}, {9116,36330}, {11132,13084}, {11295,36368}, {16629,19709}, {16965,21359}, {19708,22532}, {22892,33621}, {22897,34582}, {35735,35747}

X(36386) = midpoint of X(i) and X(j) for these {i,j}: {2, 33622}, {33626, 36352}
X(36386) = reflection of X(i) in X(j) for these (i,j): (5464, 30472), (11122, 5460), (36366, 2)
X(36386) = complement of X(33626)
X(36386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36352, 33626), (2, 36366, 17), (627, 22844, 17), (627, 33622, 2), (15533, 15693, 36388), (22488, 22666, 36366), (33622, 33626, 36352)

X(36387) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 2nd FERMAT-DAO

Barycentrics    a^2*(2*sqrt(3)*b^2*c^2*(a^6-10*(b^2+c^2)*a^4+2*(4*b^4-3*b^2*c^2+4*c^4)*a^2+(b^4-c^4)*(b^2-c^2))+4*S*((6*b^4-b^2*c^2+6*c^4)*a^4-(3*b^2-2*c^2)*(2*b^2-3*c^2)*(b^2+c^2)*a^2-(b^2+c^2)^2*b^2*c^2)) : :
X(36387) = 4*X(2)-3*X(14182) = 2*X(2)-3*X(14188) = 5*X(2)-6*X(33481) = 7*X(2)-6*X(33491) = 5*X(14182)-8*X(33481) = 7*X(14182)-8*X(33491) = 9*X(14182)-4*X(36321) = 3*X(14182)+4*X(36325) = 3*X(14182)-2*X(36367) = 5*X(14188)-4*X(33481) = 7*X(14188)-4*X(33491) = 9*X(14188)-2*X(36321) = 3*X(14188)+2*X(36325) = 3*X(14188)-X(36367) = 7*X(33481)-5*X(33491) = 18*X(33481)-5*X(36321) = 6*X(33481)+5*X(36325) = 12*X(33481)-5*X(36367)

The reciprocal orthologic center of these triangles is X(25207)

X(36387) lies on these lines: {2,14182}, {511,36329}, {512,36330}, {3845,25180}, {5066,25224}, {35734,35760}, {35735,35761}

X(36387) = midpoint of X(2) and X(36325)
X(36387) = reflection of X(i) in X(j) for these (i,j): (14182, 14188), (36367, 2)
X(36387) = complement of X(36321)
X(36387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36367, 14182), (14188, 36367, 2)

X(36388) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO INNER-FERMAT

Barycentrics    -2*sqrt(3)*(5*a^2-4*c^2-4*b^2)*S+11*a^4-13*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
X(36388) = 4*X(2)-3*X(18) = X(2)-3*X(628) = 5*X(2)-6*X(630) = 13*X(2)-12*X(6674) = 7*X(2)-3*X(22114) = 2*X(2)+3*X(22845) = 11*X(2)-6*X(33464) = 5*X(2)-X(36324) = 3*X(2)+X(36346) = X(18)-4*X(628) = 5*X(18)-8*X(630) = 13*X(18)-16*X(6674) = 7*X(18)-4*X(22114) = X(18)+2*X(22845) = 11*X(18)-8*X(33464) = 3*X(18)+4*X(33624) = 9*X(18)-4*X(33627) = 15*X(18)-4*X(36324) = 9*X(18)+4*X(36346) = 3*X(18)-2*X(36368)

The reciprocal orthologic center of these triangles is X(14)

X(36388) lies on these lines: {2,18}, {13,33458}, {16,5863}, {299,11057}, {531,12817}, {617,19106}, {3534,22494}, {3642,16960}, {3830,36329}, {3845,16627}, {4669,22851}, {5085,5965}, {5459,11121}, {5463,30471}, {5464,5859}, {5858,16242}, {8703,35751}, {9114,35752}, {11133,13083}, {11296,36366}, {16628,19709}, {16964,21360}, {19708,22531}, {22848,33620}, {22852,34582}

X(36388) = midpoint of X(i) and X(j) for these {i,j}: {2, 33624}, {33627, 36346}
X(36388) = reflection of X(i) in X(j) for these (i,j): (5463, 30471), (11121, 5459), (36368, 2)
X(36388) = complement of X(33627)
X(36388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36346, 33627), (2, 36368, 18), (628, 22845, 18), (628, 33624, 2), (15533, 15693, 36386), (22487, 22665, 36368), (33624, 33627, 36346)

X(36389) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st FERMAT-DAO

Barycentrics    a^2*(2*sqrt(3)*b^2*c^2*(a^6-10*(b^2+c^2)*a^4+2*(4*b^4-3*b^2*c^2+4*c^4)*a^2+(b^4-c^4)*(b^2-c^2))-4*S*((6*b^4-b^2*c^2+6*c^4)*a^4-(3*b^2-2*c^2)*(2*b^2-3*c^2)*(b^2+c^2)*a^2-(b^2+c^2)^2*b^2*c^2)) : :
X(36389) = 4*X(2)-3*X(14178) = 2*X(2)-3*X(14186) = 5*X(2)-6*X(33480) = 7*X(2)-6*X(33490) = 5*X(14178)-8*X(33480) = 7*X(14178)-8*X(33490) = 3*X(14178)+4*X(36328) = 9*X(14178)-4*X(36354) = 3*X(14178)-2*X(36369) = 5*X(14186)-4*X(33480) = 7*X(14186)-4*X(33490) = 3*X(14186)+2*X(36328) = 9*X(14186)-2*X(36354) = 3*X(14186)-X(36369) = 7*X(33480)-5*X(33490) = 6*X(33480)+5*X(36328) = 18*X(33480)-5*X(36354) = 12*X(33480)-5*X(36369)

The reciprocal orthologic center of these triangles is X(25208)

X(36389) lies on these lines: {2,14178}, {511,35751}, {512,35752}, {3845,25175}, {5066,25223}

X(36389) = midpoint of X(2) and X(36328)
X(36389) = reflection of X(i) in X(j) for these (i,j): (14178, 14186), (36369, 2)
X(36389) = complement of X(36354)
X(36389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36369, 14178), (14186, 36369, 2)

X(36390) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    2*(-(a^2+b^2+c^2+4*S)*sqrt(3)-4*b^2-4*c^2+5*a^2)*S+8*a^4-7*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(36390) = 4*X(2)-3*X(6304) = 5*X(2)-6*X(33444) = 2*X(3845)-3*X(22634) = 5*X(6304)-8*X(33444) = 3*X(6304)+4*X(36334) = 9*X(6304)-4*X(36348) = 3*X(6304)-2*X(36370) = 3*X(22627)-2*X(36372) = 6*X(33444)+5*X(36334) = 18*X(33444)-5*X(36348) = 12*X(33444)-5*X(36370) = 3*X(36334)+X(36348) = 2*X(36334)+X(36370) = 2*X(36348)-3*X(36370)

The reciprocal orthologic center of these triangles is X(33440)

X(36390) lies on these lines: {2,372}, {530,36396}, {531,36376}, {3564,36394}, {3845,22634}, {13706,13846}, {13824,22645}, {35735,35744}

X(36390) = midpoint of X(2) and X(36334)
X(36390) = reflection of X(36370) in X(2)
X(36390) = complement of X(36348)
X(36390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36370, 6304), (22485, 36334, 36372)

X(36391) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    -2*((a^2+b^2+c^2-4*S)*sqrt(3)+5*a^2-4*c^2-4*b^2)*S+8*a^4-7*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(36391) = 4*X(2)-3*X(6300) = 5*X(2)-6*X(33445) = 2*X(3845)-3*X(22605) = 5*X(6300)-8*X(33445) = 3*X(6300)+4*X(36335) = 9*X(6300)-4*X(36349) = 3*X(6300)-2*X(36371) = 3*X(22598)-2*X(36374) = 6*X(33445)+5*X(36335) = 18*X(33445)-5*X(36349) = 12*X(33445)-5*X(36371) = 3*X(36335)+X(36349) = 2*X(36335)+X(36371) = 2*X(36349)-3*X(36371)

The reciprocal orthologic center of these triangles is X(33441)

X(36391) lies on these lines: {2,371}, {530,36397}, {531,36377}, {3564,36392}, {3845,22605}, {13704,22616}, {13826,13847}, {35735,35743}

X(36391) = midpoint of X(2) and X(36335)
X(36391) = reflection of X(36371) in X(2)
X(36391) = complement of X(36349)
X(36391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36371, 6300), (22484, 36335, 36374)

X(36392) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st HALF-SQUARES

Barycentrics    2*((a^2+b^2+c^2+4*S)*sqrt(3)-4*b^2-4*c^2+5*a^2)*S+8*a^4-7*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(36392) = 4*X(2)-3*X(6305) = 5*X(2)-6*X(33446) = 2*X(3845)-3*X(22635) = 5*X(6305)-8*X(33446) = 3*X(6305)+4*X(36332) = 9*X(6305)-4*X(36356) = 3*X(6305)-2*X(36372) = 3*X(22629)-2*X(36370) = 6*X(33446)+5*X(36332) = 18*X(33446)-5*X(36356) = 12*X(33446)-5*X(36372) = 3*X(36332)+X(36356) = 2*X(36332)+X(36372) = 2*X(36356)-3*X(36372)

The reciprocal orthologic center of these triangles is X(33442)

X(36392) lies on these lines: {2,372}, {530,36380}, {531,36400}, {3564,36391}, {3845,22635}, {13704,13846}, {13826,22645}

X(36392) = midpoint of X(2) and X(36332)
X(36392) = reflection of X(36372) in X(2)
X(36392) = complement of X(36356)
X(36392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36372, 6305), (22485, 36332, 36370)

X(36393) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    2*((b^2+c^2)*a^2-8*b^2*c^2)*S+3*sqrt(3)*((b^2+c^2)*a^2-2*b^2*c^2)*(a^2+b^2+c^2) : :
X(36393) = 4*X(2)-3*X(6581) = 5*X(2)-6*X(33482) = 2*X(3845)-3*X(25191) = 5*X(6581)-8*X(33482) = 3*X(6581)+4*X(36338) = 9*X(6581)-4*X(36350) = 3*X(6581)-2*X(36373) = 3*X(22868)-2*X(36378) = 6*X(33482)+5*X(36338) = 18*X(33482)-5*X(36350) = 12*X(33482)-5*X(36373) = 3*X(36338)+X(36350) = 2*X(36338)+X(36373) = 2*X(36350)-3*X(36373)

The reciprocal orthologic center of these triangles is X(6582)

X(36393) lies on these lines: {2,39}, {732,36399}, {3845,25191}, {5969,36330}, {12816,35697}, {35735,35755}

X(36393) = midpoint of X(2) and X(36338)
X(36393) = reflection of X(i) in X(j) for these (i,j): (7757, 33467), (22913, 9466), (36373, 2), (36398, 14711)
X(36393) = complement of X(36350)
X(36393) = {X(2), X(36373)}-harmonic conjugate of X(6581)

X(36394) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 2nd HALF-SQUARES

Barycentrics    -2*(-(a^2+b^2+c^2-4*S)*sqrt(3)+5*a^2-4*c^2-4*b^2)*S+8*a^4-7*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(36394) = 4*X(2)-3*X(6301) = 5*X(2)-6*X(33447) = 2*X(3845)-3*X(22606) = 5*X(6301)-8*X(33447) = 3*X(6301)+4*X(36333) = 9*X(6301)-4*X(36357) = 3*X(6301)-2*X(36374) = 3*X(22600)-2*X(36371) = 6*X(33447)+5*X(36333) = 18*X(33447)-5*X(36357) = 12*X(33447)-5*X(36374) = 3*X(36333)+X(36357) = 2*X(36333)+X(36374) = 2*X(36357)-3*X(36374)

The reciprocal orthologic center of these triangles is X(33443)

X(36394) lies on these lines: {2,371}, {530,36381}, {531,36401}, {3564,36390}, {3845,22606}, {13706,22616}, {13824,13847}

X(36394) = midpoint of X(2) and X(36333)
X(36394) = reflection of X(36374) in X(2)
X(36394) = complement of X(36357)
X(36394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 36374, 6301), (22484, 36333, 36371)

X(36395) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    -2*(8*a^4+6*(b^2+c^2)*a^2+4*b^2*c^2-(b^2-c^2)^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*(2*a^4-b^4-c^4) : :
X(36395) = 4*X(2)-3*X(6296) = 5*X(2)-6*X(33484) = 2*X(3845)-3*X(25192) = 5*X(6296)-8*X(33484) = 3*X(6296)+4*X(36339) = 9*X(6296)-4*X(36351) = 3*X(6296)-2*X(36375) = 3*X(22870)-2*X(36379) = 6*X(33484)+5*X(36339) = 18*X(33484)-5*X(36351) = 12*X(33484)-5*X(36375) = 3*X(36339)+X(36351) = 2*X(36339)+X(36375) = 2*X(36351)-3*X(36375)

The reciprocal orthologic center of these triangles is X(6298)

X(36395) lies on these lines: {2,32}, {732,36398}, {3845,25192}, {33625,35697}, {35735,35756}

X(36395) = midpoint of X(2) and X(36339)
X(36395) = reflection of X(i) in X(j) for these (i,j): (31168, 33469), (36375, 2)
X(36395) = complement of X(36351)
X(36395) = {X(2), X(36375)}-harmonic conjugate of X(6296)

X(36396) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    2*((a^2+b^2+c^2+4*S)*sqrt(3)+9*a^2)*S+12*a^4+3*(b^2+c^2)*a^2-15*(b^2-c^2)^2 : :
X(36396) = 4*X(2)-3*X(13706) = 5*X(2)-6*X(33486) = 2*X(3845)-3*X(25193) = 5*X(13706)-8*X(33486) = 3*X(13706)+4*X(36340) = 9*X(13706)-4*X(36353) = 3*X(13706)-2*X(36376) = 3*X(22872)-2*X(36380) = 3*X(33456)-X(36341) = 6*X(33486)+5*X(36340) = 18*X(33486)-5*X(36353) = 12*X(33486)-5*X(36376) = 3*X(36340)+X(36353) = 2*X(36340)+X(36376) = 2*X(36353)-3*X(36376)

The reciprocal orthologic center of these triangles is X(13705)

X(36396) lies on these lines: {2,1327}, {530,36390}, {531,36370}, {3845,25193}, {6304,23251}, {25185,25186}, {31697,36371}, {35735,35757}

X(36396) = midpoint of X(2) and X(36340)
X(36396) = reflection of X(i) in X(j) for these (i,j): (13712, 33471), (36376, 2)
X(36396) = complement of X(36353)
X(36396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1327, 36400), (2, 36360, 13701), (2, 36376, 13706), (2, 36400, 22872), (1327, 22917, 22872), (22917, 36400, 2)

X(36397) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -2*(-(a^2+b^2+c^2-4*S)*sqrt(3)+9*a^2)*S+12*a^4+3*(b^2+c^2)*a^2-15*(b^2-c^2)^2 : :
X(36397) = 4*X(2)-3*X(13826) = 5*X(2)-6*X(33488) = 2*X(3845)-3*X(25194) = 9*X(13826)-4*X(36355) = 3*X(13826)-2*X(36377) = 3*X(22874)-2*X(36381) = 3*X(33457)-X(36343) = 12*X(33488)-5*X(36377) = 2*X(36355)-3*X(36377)

The reciprocal orthologic center of these triangles is X(13825)

X(36397) lies on these lines: {2,1328}, {530,36391}, {531,36371}, {3845,25194}, {6300,23261}, {25185,25186}, {31699,36370}, {35735,35758}

X(36397) = midpoint of X(2) and X(36342)
X(36397) = reflection of X(i) in X(j) for these (i,j): (13835, 33473), (36377, 2)
X(36397) = complement of X(36355)
X(36397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1328, 36401), (2, 36361, 13821), (2, 36377, 13826), (2, 36401, 22874), (1328, 22919, 22874), (22919, 36401, 2)

X(36398) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st NEUBERG

Barycentrics    -2*((b^2+c^2)*a^2-8*b^2*c^2)*S+3*sqrt(3)*((b^2+c^2)*a^2-2*b^2*c^2)*(a^2+b^2+c^2) : :
X(36398) = 4*X(2)-3*X(6294) = 5*X(2)-6*X(33483) = 2*X(3845)-3*X(25195) = 5*X(6294)-8*X(33483) = 3*X(6294)+4*X(36336) = 9*X(6294)-4*X(36358) = 3*X(6294)-2*X(36378) = 3*X(22913)-2*X(36373) = 6*X(33483)+5*X(36336) = 18*X(33483)-5*X(36358) = 12*X(33483)-5*X(36378) = 3*X(36336)+X(36358) = 2*X(36336)+X(36378) = 2*X(36358)-3*X(36378)

The reciprocal orthologic center of these triangles is X(6295)

X(36398) lies on these lines: {2,39}, {732,36395}, {3845,25195}, {5969,35752}, {12817,35693}

X(36398) = midpoint of X(2) and X(36336)
X(36398) = reflection of X(i) in X(j) for these (i,j): (7757, 33466), (22868, 9466), (36378, 2), (36393, 14711)
X(36398) = complement of X(36358)
X(36398) = {X(2), X(36378)}-harmonic conjugate of X(6294)

X(36399) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 2nd NEUBERG

Barycentrics    2*(8*a^4+6*(b^2+c^2)*a^2+4*b^2*c^2-(b^2-c^2)^2)*S+3*sqrt(3)*(a^2+b^2+c^2)*(2*a^4-b^4-c^4) : :
X(36399) = 4*X(2)-3*X(6297) = 5*X(2)-6*X(33485) = 2*X(3845)-3*X(25196) = 5*X(6297)-8*X(33485) = 3*X(6297)+4*X(36337) = 9*X(6297)-4*X(36359) = 3*X(6297)-2*X(36379) = 3*X(22915)-2*X(36375) = 6*X(33485)+5*X(36337) = 18*X(33485)-5*X(36359) = 12*X(33485)-5*X(36379) = 3*X(36337)+X(36359) = 2*X(36337)+X(36379) = 2*X(36359)-3*X(36379)

The reciprocal orthologic center of these triangles is X(6299)

X(36399) lies on these lines: {2,32}, {732,36393}, {3845,25196}, {33623,35693}

X(36399) = midpoint of X(2) and X(36337)
X(36399) = reflection of X(i) in X(j) for these (i,j): (31168, 33468), (36379, 2)
X(36399) = complement of X(36359)
X(36399) = {X(2), X(36379)}-harmonic conjugate of X(6297)

X(36400) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st TRI-SQUARES-CENTRAL

Barycentrics    2*(-(a^2+b^2+c^2+4*S)*sqrt(3)+9*a^2)*S+12*a^4+3*(b^2+c^2)*a^2-15*(b^2-c^2)^2 : :
X(36400) = 4*X(2)-3*X(13704) = 5*X(2)-6*X(33487) = 2*X(3845)-3*X(25197) = 5*X(13704)-8*X(33487) = 3*X(13704)+4*X(36341) = 9*X(13704)-4*X(36360) = 3*X(13704)-2*X(36380) = 3*X(22917)-2*X(36376) = 3*X(33456)-X(36340) = 6*X(33487)+5*X(36341) = 18*X(33487)-5*X(36360) = 12*X(33487)-5*X(36380) = 3*X(36341)+X(36360) = 2*X(36341)+X(36380) = 2*X(36360)-3*X(36380)

The reciprocal orthologic center of these triangles is X(13703)

X(36400) lies on these lines: {2,1327}, {530,36372}, {531,36392}, {3845,25197}, {6305,23251}, {25189,25190}, {31698,36374}

X(36400) = midpoint of X(2) and X(36341)
X(36400) = reflection of X(i) in X(j) for these (i,j): (13712, 33470), (36380, 2)
X(36400) = complement of X(36360)
X(36400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1327, 36396), (2, 36353, 13701), (2, 36380, 13704), (2, 36396, 22917), (1327, 22872, 22917), (22872, 36396, 2)

X(36401) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -2*((a^2+b^2+c^2-4*S)*sqrt(3)+9*a^2)*S+12*a^4+3*(b^2+c^2)*a^2-15*(b^2-c^2)^2 : :
X(36401) = 4*X(2)-3*X(13824) = 5*X(2)-6*X(33489) = 2*X(3845)-3*X(25198) = 5*X(13824)-8*X(33489) = 3*X(13824)+4*X(36343) = 9*X(13824)-4*X(36361) = 3*X(13824)-2*X(36381) = 3*X(22919)-2*X(36377) = 3*X(33457)-X(36342) = 6*X(33489)+5*X(36343) = 18*X(33489)-5*X(36361) = 12*X(33489)-5*X(36381) = 3*X(36343)+X(36361) = 2*X(36343)+X(36381) = 2*X(36361)-3*X(36381)

The reciprocal orthologic center of these triangles is X(13823)

X(36401) lies on these lines: {2,1328}, {530,36374}, {531,36394}, {3845,25198}, {6301,23261}, {25189,25190}, {31700,36372}

X(36401) = midpoint of X(2) and X(36343)
X(36401) = reflection of X(i) in X(j) for these (i,j): (13835, 33472), (36381, 2)
X(36401) = complement of X(36361)
X(36401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1328, 36397), (2, 36355, 13821), (2, 36381, 13824), (2, 36397, 22919), (1328, 22874, 22919), (22874, 36397, 2)

X(36402) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO 1st MORLEY

Barycentrics    56*a*sin(2*A/3+B/3)*sin((B-C)/3)^2*sin(C/3+2*A/3)-16*b*cos(A/3-B/3+Pi/6)*sin(A/3+2*B/3)*sin(-C/3+A/3)^2-16*c*cos(A/3-C/3+Pi/6)*sin(A/3-B/3)^2*sin(2*C/3+A/3)+(-7*a*sin(A)+2*b*sin(B)+2*c*sin(C))*sqrt(3) : :
X(36402) = 2*X(2)-3*X(8010), 4*X(2)-3*X(8011), 5*X(2)-6*X(33492), 7*X(2)-6*X(33493), 5*X(8010)-4*X(33492), 7*X(8010)-4*X(33493), 5*X(8011)-8*X(33492), 7*X(8011)-8*X(33493), 7*X(33492)-5*X(33493)

X(36402) lies on the line {2,8010}

X(36402) = reflection of X(8011) in X(8010)

X(36403) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO 1st MORLEY

Barycentrics    8*(4*Cos[A/3 - B/3 + Pi/6]*Sin[A/3 + (2*B)/3]*Sin[B]*Sin[A/3 - C/3]^2 - 5*Sin[A]*Sin[(2*A)/3 + B/3]*Sin[(B - C)/3]^2*Sin[(2*A)/3 + C/3] + 4*Cos[A/3 - C/3 + Pi/6]*Sin[A/3 - B/3]^2*Sin[A/3 + (2*C)/3]*Sin[C]) - Sqrt[3]*(-5*Sin[A]^2 + 4*Sin[B]^2 + 4*Sin[C]^2) : :
X(36403) = 4 X[2] - 3 X[8010], 2 X[2] - 3 X[8011], 7 X[2] - 6 X[33492], 5 X[2] - 6 X[33493], 7 X[8010] - 8 X[33492], 5 X[8010] - 8 X[33493], 3 X[8010] - 2 X[36402], 7 X[8011] - 4 X[33492], 5 X[8011] - 4 X[33493], 3 X[8011] - X[36402], 5 X[33492] - 7 X[33493], 12 X[33492] - 7 X[36402], 12 X[33493] - 5 X[36402]

X(36403) lies on the line {2,8010}

X(36403) = reflection of X(i) in X(j) for these {i,j}: {8010, 8011}, {36402, 2}
X(36403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36402, 8010}, {8011, 36402, 2}

leftri

Centers of TC conics: X(36404)-X(36411)

rightri

This preamble and centers X(36404)-X(36411) were contributed by Clark Kimberling and Peter Moses, January 16, 2020.

Trilinear permutation conics TC(P) are defined in the preamble just before X(36256). Briefly, if P = p : q : r (trilinears), then TC(P), is the conic that passes through the six points

p : q : r,    q : r : p,    r : p : q,    p : r : q,    q : p : r,   r : q : p.

An equation for TC(P) is (q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

For the equivalent formulation using barycentrics, see the aforementioned preamble. If P = p : q : r (barycentrics), then the center of TC(P) is given by

a(2a(q r + r p + p q) + (-a + b + c)(p^2 + q^2 + r^2)) : :

X(36404) = CENTER OF THE CONIC TC(X(1))

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

X(36404) lies on these lines: {1, 6}, {69, 17244}, {141, 31285}, {169, 17750}, {182, 990}, {193, 29569}, {545, 597}, {599, 16590}, {672, 4414}, {742, 4384}, {750, 2246}, {894, 3618}, {1054, 17754}, {1428, 4327}, {1572, 3997}, {1766, 29309}, {2082, 2295}, {2177, 14439}, {2271, 25066}, {2276, 23988}, {2280, 3722}, {2330, 4319}, {2348, 5275}, {3315, 26242}, {3416, 17330}, {3589, 10436}, {3672, 17350}, {3707, 5847}, {3755, 3923}, {3886, 17281}, {4664, 32029}, {4675, 5845}, {5165, 16574}, {5276, 9347}, {8545, 34253}, {11529, 21331}, {15988, 26669}

X(36405) = CENTER OF THE CONIC TC(X(6))

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

X(36405) lies on these lines: {1, 6}, {63, 9284}, {325, 4643}, {1708, 18905}, {2260, 25845}, {7778, 17237}

X(36406) = CENTER OF THE CONIC TC(X(75))

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

X(36406) lies on these lines: {1, 6}, {43, 1908}, {3208, 4676}, {3226, 4664}, {3501, 4672}, {3758, 17754}, {5749, 26752}, {5750, 27091}, {7075, 27064}, {9025, 19584}, {10436, 20148}

X(36407) = CENTER OF THE CONIC TC(X(69))

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

X(36406) lies on these lines: {1, 6}, {1707, 9257}

X(36408) = CENTER OF THE CONIC TC(X(63))

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

X(36408) lies on these lines: {1, 6}, {5, 20623}, {46, 1939}, {912, 5452}, {9367, 17437}

X(36409) = CENTER OF THE CONIC TC(X(10))

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

X(36409) lies on these lines; {1, 6}, {239, 25384}, {244, 21840}, {536, 31317}, {597, 17755}, {740, 17369}, {742, 17023}, {872, 4285}, {1654, 4687}, {2276, 3121}, {2278, 19554}, {3739, 17367}, {3842, 17330}, {4473, 31308}, {4664, 33888}, {4698, 5224}, {4755, 16590}, {6155, 16549}, {14439, 21806}, {17027, 25368}, {24357, 26626}, {27487, 29630}

X(36410) = CENTER OF THE CONIC TC(X(31))

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

X(36410) lies on this line: {1, 6}

X(36411) = CENTER OF THE CONIC TC(X(76))

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

X(36411) lies on these lines: {1, 6}, {1740, 9287}

leftri

Points on the barycentric square of the Euler line: X(36412)-X(36432)

rightri

This preamble and centers X(36412)-X(36432) were contributed by Clark Kimberling and Peter Moses, January 16, 2020.

Let L denote the Euler line and L^2 the set of barycentric squares of points on L, as in the preamble just before X(23582). The set L^2 is here named the barycentric Euler inellipse. It has perspector X(23582) and center X(23583), and it passes through X(i) for these 26 indices i: 2,393,577,3163,7054, and 36412, 36413, 36414, ..., 36432.

X(36412) = BARYCENTRIC SQUARE OF X(5)

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

X(36412) lies on these lines: {2, 10979}, {3, 14938}, {4, 577}, {5, 53}, {6, 13}, {30, 22052}, {32, 2165}, {39, 7403}, {137, 35319}, {231, 2965}, {232, 5133}, {264, 1972}, {297, 14767}, {324, 34836}, {393, 3091}, {546, 3284}, {570, 1506}, {571, 7747}, {648, 17035}, {800, 9722}, {1249, 3855}, {1532, 1865}, {1595, 22401}, {1609, 7529}, {1953, 35307}, {1968, 7544}, {1990, 3850}, {2963, 11063}, {3078, 23607}, {3087, 3832}, {3129, 8742}, {3130, 8741}, {3148, 35067}, {3574, 31353}, {3613, 11672}, {3843, 15905}, {3857, 15860}, {3858, 6749}, {5046, 7054}, {5066, 18487}, {5169, 15355}, {5421, 7765}, {5596, 7694}, {6103, 7533}, {6842, 18591}, {6997, 10314}, {7394, 10311}, {7506, 7749}, {7755, 13345}, {8573, 13881}, {8754, 23635}, {8882, 9380}, {8963, 15233}, {9224, 35133}, {9698, 13351}, {11574, 15980}, {14130, 15109}, {15760, 33842}, {17849, 18381}, {18531, 26899}, {21354, 33664}, {23261, 26868}, {35322, 36300}, {35323, 36301}

X(36412) = complement of isotomic conjugate of X(6662)

X(36413) = BARYCENTRIC SQUARE OF X(20)

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

X(36413) lies on these lines: {2, 6}, {20, 1249}, {30, 33630}, {53, 17578}, {115, 34570}, {216, 15717}, {253, 441}, {393, 3146}, {577, 3522}, {610, 18623}, {631, 15851}, {648, 6527}, {1033, 11413}, {1990, 5059}, {2060, 3344}, {2322, 24565}, {2331, 4296}, {3079, 23608}, {3087, 3832}, {3088, 22120}, {3100, 7129}, {3163, 15683}, {3424, 15583}, {3523, 5702}, {3562, 22124}, {3854, 6749}, {6616, 14365}, {7054, 17576}, {7396, 16318}, {8573, 22467}, {8744, 34621}, {8969, 19039}, {10979, 15705}, {11348, 32000}, {13341, 26216}, {15526, 35510}

X(36414) = BARYCENTRIC SQUARE OF X(22)

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

X(36414) lies on these lines: {2, 32}, {22, 8743}, {25, 10317}, {112, 1370}, {393, 7500}, {577, 6636}, {1968, 7391}, {2172, 21749}, {3162, 26283}, {4611, 34254}, {6997, 10312}, {7394, 10311}, {7750, 26159}, {13575, 28696}, {15013, 18018}

X(36415) = BARYCENTRIC SQUARE OF X(23)

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

X(36415) lies on these lines: {2, 112}, {23, 8744}, {32, 111}, {115, 251}, {393, 7519}, {577, 7492}, {1627, 10418}, {1637, 13195}, {1968, 31857}, {2493, 2965}, {3163, 10313}, {5169, 8791}, {6103, 7533}, {8428, 8743}, {12824, 28343}

X(36416) = BARYCENTRIC SQUARE OF X(24)

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

X(36416) lies on these lines: {6, 18532}, {24, 571}, {32, 393}, {577, 7488}, {1968, 7544}, {2207, 2965}, {14517, 35603}

X(36417) = BARYCENTRIC SQUARE OF X(25)

Barycentrics    a^4*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2 : :
Trilinears    sin^3 A tan^2 A : :

The trilinear polar of X(36417) passes through X(9426) and the polar conjugate of X(4609).

X(36417) lies on these lines: {2, 1968}, {6, 1619}, {22, 232}, {25, 32}, {107, 699}, {112, 2374}, {115, 13854}, {184, 2211}, {251, 393}, {264, 16950}, {305, 15014}, {385, 21447}, {427, 5475}, {428, 5309}, {1180, 33871}, {1194, 8743}, {1501, 1974}, {1627, 4232}, {1973, 21750}, {2052, 3407}, {3115, 18027}, {3767, 8879}, {6997, 10314}, {7714, 10312}, {7745, 15809}, {9909, 10316}, {10317, 20850}, {13575, 15526}, {15369, 19118}, {19124, 20965}, {21775, 32691}

X(36417) = isogonal conjugate of isotomic conjugate of X(2207)
X(36417) = X(63)-isoconjugate of X(305)
X(36417) = polar conjugate of isotomic conjugate of X(1974)

X(36418) = BARYCENTRIC SQUARE OF X(26)

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

X(36418) lies on these lines: {26, 8746}, {32, 2165}, {571, 9699}, {577, 7512}, {2965, 7506}

X(36419) = BARYCENTRIC SQUARE OF X(27)

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

X(36419) lies on these lines: {27, 58}, {29, 5717}, {81, 286}, {107, 741}, {278, 2189}, {306, 447}, {393, 1171}, {577, 7560}, {648, 3187}, {2052, 14534}, {2352, 36077}, {15376, 30117}

X(36420) = BARYCENTRIC SQUARE OF X(28)

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

X(36420) lies on these lines: {28, 1104}, {112, 5301}, {393, 1169}, {577, 7520}, {1474, 2206}, {2303, 2326}, {3269, 34440}, {7054, 17521}

X(36421) = BARYCENTRIC SQUARE OF X(29)

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

X(36421) lies on these lines: {19, 107}, {29, 284}, {81, 286}, {393, 7518}, {577, 7538}, {1172, 1896}, {2322, 2328}, {8558, 15393}, {15946, 34170}, {26165, 31623}

X(36422) = BARYCENTRIC SQUARE OF X(140)

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

X(36422) lies on these lines: {3, 14938}, {6, 15720}, {53, 15712}, {95, 15526}, {115, 2963}, {125, 34520}, {140, 233}, {216, 549}, {393, 3523}, {401, 6709}, {577, 631}, {2165, 15515}, {2965, 9698}, {3284, 12108}, {14096, 35067}, {15701, 15905}

X(36423) = BARYCENTRIC SQUARE OF X(186)

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

X(36423) lies on these lines: {32, 8749}, {50, 186}, {112, 393}, {115, 8882}, {577, 10298}, {1627, 6103}, {1989, 18559}, {3163, 9380}, {6128, 10312}

X(36424) = BARYCENTRIC SQUARE OF X(235)

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

X(36424) lies on these lines: {235, 800}, {1609, 1624}, {3163, 8745}

X(36425) = BARYCENTRIC SQUARE OF X(237)

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

X(36425) lies on these lines: {32, 263}, {237, 2211}, {577, 19121}, {1501, 8023}, {1613, 1624}, {8623, 35282}, {9419, 23611}, {10684, 18024}, {14966, 36213}

X(36426) = BARYCENTRIC SQUARE OF X(297)

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

X(36426) lies on these lines: {2, 107}, {4, 287}, {115, 6528}, {193, 317}, {264, 1972}, {297, 511}, {324, 23962}, {439, 34286}, {458, 19130}, {542, 33971}, {577, 17907}, {1916, 2052}, {5025, 14249}, {6523, 32972}, {6526, 32980}, {14041, 34170}, {18027, 27371}

X(36426) = reflection of X(577) in X(23583)
X(36426) = antipode of X(577) in barycentric Euler inellipse

X(36427) = BARYCENTRIC SQUARE OF X(376)

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

X(36427) lies on these lines: {2, 340}, {6, 3524}, {20, 3163}, {30, 393}, {216, 15705}, {577, 10304}, {1627, 33871}, {1990, 11001}, {3087, 3545}, {5055, 33636}, {5071, 6749}, {5158, 15692}, {5702, 19708}, {7735, 32216}, {15683, 18487}, {15706, 15851}, {15717, 15860}

X(36428) = BARYCENTRIC SQUARE OF X(377)

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

X(36428) lies on these lines: {2, 286}, {20, 7054}, {69, 26605}, {346, 2064}, {393, 2475}, {394, 1901}, {577, 4190}, {2345, 21582}

X(36429) = BARYCENTRIC SQUARE OF X(378)

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

X(36429) lies on these lines: {6, 18532}, {32, 8749}, {378, 5063}, {393, 2549}, {577, 2071}, {1180, 33871}, {1968, 3163}

X(36430) = BARYCENTRIC SQUARE OF X(381)

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

X(36430) lies on these lines: {4, 3163}, {6, 14269}, {30, 53}, {32, 1989}, {115, 34288}, {216, 5055}, {381, 5158}, {393, 3839}, {1990, 3845}, {3003, 18362}, {3284, 3830}, {3843, 15860}, {5054, 10979}, {5475, 14836}, {6749, 14893}, {11648, 33871}, {13342, 18367}, {15689, 22052}, {16303, 18424}, {18479, 34417}, {23607, 26880}

X(36431) = BARYCENTRIC SQUARE OF X(382)

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

X(36431) lies on these lines: {6, 14269}, {550, 577}, {1249, 3855}, {3163, 33630}, {5079, 5158}, {10979, 15700}

X(36432) = BARYCENTRIC SQUARE OF X(384)

Barycentrics    (a^4 + b^2*c^2)^2 : :

X(36432) lies on these lines: {2, 1974}, {6, 1916}, {32, 2998}, {264, 33336}, {384, 11380}, {393, 14035}, {577, 3552}, {3163, 19686}, {6660, 9229}, {7054, 17692}, {9230, 16985}, {10997, 11574}

X(36433) = BARYCENTRIC 4TH POWER OF X(3)

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

X(36433) lies on these lines: {115, 22261}, {216, 32046}, {577, 1147}, {1092, 35071}, {1970, 14152}, {1971, 2055}, {3284, 12106}, {10316, 11672}, {14585, 23606}, {15075, 15454}

X(36433) = isogonal conjugate of the polar conjugate of X(23606)
X(36433) = X(i)-isoconjugate of X(j) for these (i,j): {158, 18027}, {264, 6521}, {1093, 1969}, {6520, 18022}, {23962, 24021}, {23994, 34538}
X(36433) = barycentric product X(i)*X(j) for these {i,j}: {3, 23606}, {48, 4100}, {184, 1092}, {394, 14585}, {418, 19210}, {577, 577}, {3964, 14575}, {6056, 7335}, {6507, 9247}, {23357, 35071}, {32320, 32661}
X(36433) = barycentric quotient X(i)/X(j) for these {i,j}: {577, 18027}, {1092, 18022}, {4100, 1969}, {9247, 6521}, {14575, 1093}, {14585, 2052}, {23606, 264}, {23963, 34538}, {35071, 23962}

X(36434) = BARYCENTRIC 4TH POWER OF X(4)

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

X(36434) lies on these lines: {32, 6525}, {115, 6526}, {393, 800}, {2207, 6524}, {2548, 10002}, {3346, 35071}, {5286, 14249}, {6392, 6528}

X(36434) = polar conjugate of X(4176)
X(36434) = polar conjugate of the isotomic conjugate of X(6524)
X(36434) = perspector of ABC and orthoanticevian triangle of X(6524)
X(36434) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1102}, {48, 4176}, {63, 3964}, {69, 6507}, {249, 24020}, {255, 3926}, {304, 1092}, {305, 4100}, {326, 394}, {1101, 23974}, {1259, 7183}, {1264, 7125}, {1804, 3719}, {2289, 7055}, {4143, 4575}, {4600, 16730}, {24037, 35071}
X(36434) = barycentric product X(i)*X(j) for these {i,j}: {4, 6524}, {19, 6520}, {25, 1093}, {115, 23590}, {158, 1096}, {338, 23975}, {393, 393}, {1109, 24022}, {1118, 1857}, {1973, 6521}, {2052, 2207}, {2489, 15352}, {2501, 6529}, {2643, 24021}, {3124, 34538}, {3199, 8794}, {6525, 6526}, {8754, 32230}, {8884, 14569}
X(36434) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 4176}, {19, 1102}, {25, 3964}, {115, 23974}, {393, 3926}, {1084, 35071}, {1093, 305}, {1096, 326}, {1118, 7055}, {1356, 1363}, {1857, 1264}, {1973, 6507}, {1974, 1092}, {2207, 394}, {2501, 4143}, {2643, 24020}, {2971, 2972}, {3121, 16730}, {6059, 1259}, {6520, 304}, {6524, 69}, {6529, 4563}, {7063, 7065}, {7337, 1804}, {15422, 15414}, {23590, 4590}, {23975, 249}, {24021, 24037}, {24022, 24041}, {34538, 34537}
X(36434) = {X(393),X(6523)}-harmonic conjugate of X(3767)

X(36435) = BARYCENTRIC 4TH POWER OF X(30)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^4 : :
X(36435) = 3 X[648] + X[9410]

X(36435) lies on these lines: {30, 1990}, {648, 9410}, {14993, 23967}

X(36435) = X(3163)-Ceva conjugate of X(3081)
X(36435) = crosspoint of X(3081) and X(3163)
X(36435) = barycentric product X(i)*X(j) for these {i,j}: {30, 3081}, {1354, 6062}, {1495, 23097}, {3163, 3163}, {16163, 16240}
X(36435) = barycentric quotient X(i)/X(j) for these {i,j}: {3081, 1494}, {3163, 31621}

leftri

Homothetors involving triangles T(k): X(36436)-X(36472)

rightri

This preamble and centers X(36436)-X(36472) were contributed by Clark Kimberling and Peter Moses, January 17, 2020, and Randy Hutson, January 29, 2020.

Suppose that ABC is a triangle. The trisectors of segment BC are 0:1:2 and 0:2:1; these are two of the points on the permutation ellipse E(0:1:2), here named the trisection ellipse, given by the equation

5(x^2 + y^2 + z^2) - 2(y z + z x + x y) = 0.

For every real number k, let T(k) denote the central triangle with A-vertex 1 : k : k. The line AG, where G = 1:1:1 = X(2) meets the trisection ellipse in two points, 1 : k : k, where k = sqrt(27) - 5 and k = - sqrt(27) - 5. For these two values of k, the triangle T(k) is homothetic to many triangles, of which 17 for each k give homothetors (centers of homothety) included in this section:

Euler; (a^2+b^2-c^2) (a^2-b^2+c^2)-2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) k : :
reflection of ABC in X(3): 2 a^2 (a^2-b^2-c^2)-(a^2+b^2-c^2) (a^2-b^2+c^2) k : :
reflection of X(3) in ABC; a^2 (a^2-b^2-c^2)+(a^4-3 a^2 b^2+2 b^4-3 a^2 c^2-4 b^2 c^2+2 c^4) k : :
reflection of ABC in X(5) (aka Carnot, Johnson); a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4-a^2 (a^2-b^2-c^2) k : :
outer Garcia; b+c+a k : :
Mandart-incircle triangle; (a-b-c) (a^2-(b-c)^2 k) : :
inner Yff; 2 a^2 b c+(a^4-2 a^2 b^2+b^4-2 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) k : :
outer Yff; 2 a^2 b c-(a^4-2 a^2 b^2+b^4-2 a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4) k : :
anti-Aquila; a-(2 a+b+c) k : :
infinite altitude; (a^2+b^2-c^2) (a^2-b^2+c^2)-2 a^2 (a^2-b^2-c^2) k : :
3rd tri-squares central; a^2+S-k (a^2+2 S) : :
4th tri-squares central; a^2-S-k (a^2+2 S) : :
Ehrmann mid-triangle; a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4-(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) k : :
anti-inner-Grebe; a^2-k (a^2-S) : :
anti-outer-Grebe; a^2-k (a^2+S) : :
1st Kenmotu free-vertices triangle; a^2 (a^2-b^2-c^2+2 S)+(a^4-3 a^2 b^2+2 b^4-3 a^2 c^2-4 b^2 c^2+2 c^4-2 a^2 S) k : :
2nd Kenmotu free-vertices triangle; a^2 (a^2-b^2-c^2-2 S)+(a^4-3 a^2 b^2+2 b^4-3 a^2 c^2-4 b^2 c^2+2 c^4+2 a^2 S) k : :

For barycentrics and references for the various triangles, see Index of Triangles Referenced in ETC, by .

For every k, the homothetor of T(k) with each of the following triangles lies on the Euler line: Euler, reflection of X(3) in ABC, reflection of ABC in X(5), infinite altitude.
For every k, the homothetor of T(k) with each of the following triangles lies on the line X(2)X(6): 3rd tri-squares central, 4th tri-squares central, anti-inner Grebe, anti-outer Grebe.
For every k, the homothetor of T(k) with each of the following triangles lies on the line X(1)X(2): outer Garcia, inner Yff, outer Yff, anti-Aquila.

For k a nonconstant function symmetric in a,b,c, see the preamble just before X(36473).

The trisection ellipse is also the conic Cpar(X(2)); see the preamble before X(10001). (Randy Hutson, March 29, 2020)

X(36436) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND EULER

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

X(36436) lies on these lines: {2, 3}, {13, 3068}, {14, 3069}, {6278, 36372}, {6281, 36371}, {6564, 11488}, {6565, 11489}, {6770, 13674}, {6773, 13794}, {12256, 22605}, {12257, 22635}, {13666, 13704}, {13786, 13826}, {16808, 32785}, {16809, 32786}

X(36436) = {X(2),X(381)}-harmonic conjugate of X(36454)

X(36437) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND REFLECTION-OF-ABC IN X(3)

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

X(36437) lies on these lines: {2, 3}, {13, 6560}, {14, 6561}, {485, 35731}, {491, 616}, {492, 617}, {542, 33440}, {1327, 3366}, {1328, 3392}, {9541, 11489}, {10653, 35822}, {10654, 35823}, {12123, 22605}, {12124, 22635}, {34560, 34632}

X(36437) = {X(2),X(376)}-harmonic conjugate of X(36455)

X(36438) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND REFLECTION-OF-X(3) IN ABC

Barycentrics    11*a^4-25*(b^2+c^2)*a^2+(6*(b^2+c^2)*a^2-6*b^4+12*b^2*c^2-6*c^4)*sqrt(3)+14*(b^2-c^2)^2 : :

X(36438) lies on these lines: {2, 3}, {6,36452}, {13, 8253}, {14, 8252}, {11488, 18512}, {11489, 18510}, {13665, 23302}, {13785, 23303}, {16644, 35822}, {16645, 35823}, {31162, 34560}

X(36438) = {X(2),X(381)}-harmonic conjugate of X(36456)
X(36438) = {X(36452),X(36453)}-harmonic conjugate of X(6)

X(36439) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND REFLECTION-OF-ABC IN X(5)

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

X(36439) lies on these lines: {2, 3}, {13, 615}, {14, 590}, {395, 35822}, {396, 35823}, {542, 6303}, {3071, 35731}, {3364, 32787}, {3367, 35733}, {3390, 32788}, {6301, 22917}, {6304, 22874}, {6564, 23303}, {6565, 23302}, {11488, 13785}, {11489, 13665}, {13821, 35758}, {16808, 32790}, {16809, 32789}, {32419, 35741}, {32909, 35759}

X(36439) = {X(2),X(381)}-harmonic conjugate of X(36457)

X(36440) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND OUTER GARCIA

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

X(36440) lies on these lines: {1, 2}, {2042, 5882}, {2044, 28194}, {2046, 11362}, {3656, 18586}, {4301, 35732}, {10222, 14813}, {15765, 28204}

X(36440) = {X(2),X(3679)}-harmonic conjugate of X(36458)

X(36441) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND MANDART-INCIRCLE TRIANGLE

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

X(36441) lies on these lines: lies on these lines: {2, 11}, {2043, 11237}, {2045, 3303}, {2046, 9670}, {3584, 18587}, {9671, 35732}, {10072, 15765}, {14814, 31452}

X(36441) = {X(2),X(3058)}-harmonic conjugate of X(36459)

X(36442) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND INNER YFF

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

X(36442) lies on these lines: {1, 2}, {2041, 5563}, {2045, 3746}, {4995, 15765}, {5434, 18585}, {11238, 18587}

X(36442) = {X(1),X(2)}-harmonic conjugate of X(36443)
X(36442) = {X(2),X(10056)}-harmonic conjugate of X(36460)

X(36443) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND OUTER YFF

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

X(36443) lies on these lines: {1, 2}, {2041, 3746}, {2045, 5563}, {3058, 18585}, {5298, 15765}, {11237, 18587}, {14814, 15888}

X(36443) = {X(1),X(2)}-harmonic conjugate of X(36442)
X(36443) = {X(2),X(10072)}-harmonic conjugate of X(36461)

X(36444) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND ANTI-AQUILA

Barycentrics    (-3*b-3*c)*sqrt(3)+13*a+b+c : :

X(36444) lies on these lines: {1, 2}, {2041, 7982}, {2046, 9624}, {3653, 15765}, {3656, 18585}, {13688, 13704}, {13808, 13826}, {18587, 28204}

X(36444) = {X(2),X(551)}-harmonic conjugate of X(36462)

X(36445) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND INFINITE ALTITUDE

Barycentrics    13*a^4-2*(b^2+c^2)*a^2+(6*(b^2+c^2)*a^2-6*b^4+12*b^2*c^2-6*c^4)*sqrt(3)-11*(b^2-c^2)^2 : :

X(36445) lies on these lines: {2, 3}, {532, 5860}, {533, 5861}, {3068, 10654}, {3069, 10653}, {3364, 19054}, {3390, 19053}, {3642, 5591}, {3643, 5590}, {6278, 36392}, {6281, 36391}, {6459, 16962}, {6460, 16963}, {6560, 11489}, {6561, 11488}

X(36445) = reflection of X(36463) in X(2)
X(36445) = X(4)-of-triangle-T(sqrt(27)-5)
X(36445) = {X(3),X(10304)}-harmonic conjugate of X(36463)
X(36445) = {X(4),X(3545)}-harmonic conjugate of X(36463)
X(36445) = {X(20),X(5054)}-harmonic conjugate of X(36463)
X(36445) = {X(376),X(3524)}-harmonic conjugate of X(36463)
X(36445) = {X(381),X(3839)}-harmonic conjugate of X(36463)

X(36446) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND 3RD TRI-SQUARES CENTRAL

Barycentrics    3*(4+sqrt(3))*a^2+13*S : :

X(36446) lies on these lines: {2, 6}, {3524, 35739}, {13650, 36371}, {13651, 36392}, {13704, 22541}, {13711, 36391}, {13826, 19100}, {13833, 22919}, {14814, 31487}, {25189, 33442}

X(36446) = {X(2),X(6)}-harmonic conjugate of X(36447)
X(36446) = {X(2),X(13846)}-harmonic conjugate of X(36464)

X(36447) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND 4TH TRI-SQUARES CENTRAL

Barycentrics    3*(4+sqrt(3))*a^2-13*S : :

X(36447) lies on these lines: {2, 6}, {376, 35739}, {13704, 19099}, {13769, 22872}, {13770, 36391}, {13771, 36372}, {13826, 19101}, {13834, 36392}, {25186, 33441}

X(36447) = {X(2),X(6)}-harmonic conjugate of X(36446)
X(36447) = {X(2),X(13847)}-harmonic conjugate of X(36465)

X(36448) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND EHRMANN MID-TRIANGLE

Barycentrics    13*a^4+10*(b^2+c^2)*a^2+(9*(b^2+c^2)*a^2-9*b^4+18*b^2*c^2-9*c^4)*sqrt(3)-23*(b^2-c^2)^2 : :

X(36448) lies on these lines: {2, 3}, {395, 1327}, {396, 1328}, {6289, 36392}, {6290, 36391}, {12601, 36349}, {12602, 36356}, {36362, 36370}, {36363, 36374}, {36382, 36396}, {36383, 36401}

X(36448) = {X(2),X(3845)}-harmonic conjugate of X(36466)

X(36449) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND ANTI-INNER GREBE

Barycentrics    3*a^2+3*sqrt(3)*a^2+2*S : :

X(36449) lies on these lines: {2, 6}, {13, 1328}, {61, 2043}, {486, 16267}, {2041, 6419}, {2042, 3412}, {2044, 35823}, {2045, 6420}, {3390, 16962}, {5418, 16963}, {13929, 19073}

X(36449) = {X(2),X(6)}-harmonic conjugate of X(36450)
X(36449) = {X(2),X(19053)}-harmonic conjugate of X(36467)
X(36449) = {X(6),X(32787)}-harmonic conjugate of X(36467)

X(36450) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND ANTI-OUTER GREBE

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

X(36450) lies on these lines: {2, 6}, {14, 1327}, {62, 2043}, {485, 16268}, {2041, 6420}, {2042, 3411}, {2044, 35822}, {2045, 6419}, {3364, 16963}, {5420, 16962}, {13875, 19076}

X(36450) = {X(2),X(6)}-harmonic conjugate of X(36449)
X(36450) = {X(2),X(19054)}-harmonic conjugate of X(36468)
X(36450) = {X(6),X(32788)}-harmonic conjugate of X(36468)

X(36451) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND 2ND ANTI-CIRCUMPERP-TANGENTIAL

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

X(36451) lies on these lines: {2, 12}, {6,36438}, {2043, 11238}, {2045, 3304}, {2046, 9657}, {3582, 18587}, {9656, 35732}, {10056, 15765}

X(36452) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND 1ST KENMOTU FREE-VERTICES TRIANGLE

Barycentrics    (1+2*sqrt(3))*a^4+6*S*a^2-(-1+4*sqrt(3))*(b^2+c^2)*a^2+(b^2-c^2)^2*(-2+2*sqrt(3)) : :

X(36452) lies on these lines: {2, 372}, {14, 6396}, {16, 381}, {18, 18587}, {62, 13846}, {371, 16963}, {395, 35823}, {2045, 6419}, {3365, 13847}, {3390, 5054}, {3412, 6420}, {6564, 23303}, {6565, 11489}, {8976, 16267}, {11304, 22872}, {14814, 35813}, {15765, 16773}

X(36452) = {X(2),X(35822)}-harmonic conjugate of X(36469)
X(36452) = {X(6),X(36438)}-harmonic conjugate of X(36453)
X(36452) = {X(381),X(16645)}-harmonic conjugate of X(36470)

X(36453) = HOMOTHETOR OF THESE TRIANGLES: T(SQRT(27)-5) AND 2ND KENMOTU FREE-VERTICES TRIANGLE

Barycentrics    (1+2*sqrt(3))*a^4-6*S*a^2-(-1+4*sqrt(3))*(b^2+c^2)*a^2+(b^2-c^2)^2*(-2+2*sqrt(3)) : :

X(36453) lies on these lines: {2, 371}, {6, 36438}, {13, 6200}, {15, 381}, {17, 18587}, {61, 13847}, {372, 16962}, {396, 35822}, {2045, 6420}, {3364, 5054}, {3367, 35731}, {3389, 13846}, {3411, 6419}, {6564, 11488}, {6565, 23302}, {11303, 22919}, {13951, 16268}, {14814, 35812}, {15765, 16772}

X(36453) = {X(2),X(35823)}-harmonic conjugate of X(36470)
X(36453) = {X(6),X(36438)}-harmonic conjugate of X(36452)
X(36453) = {X(381),X(16644)}-harmonic conjugate of X(36469)

X(36454) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND EULER

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

X(36454) lies on these lines: {2, 3}, {13, 3069}, {14, 3068}, {6278, 36370}, {6281, 36374}, {6459, 35731}, {6564, 11489}, {6565, 11488}, {6770, 13794}, {6773, 13674}, {12256, 22606}, {12257, 22634}, {13666, 13706}, {13786, 13824}, {16808, 32786}, {16809, 32785}, {32787, 35740}

X(36454) = {X(2),X(381)}-harmonic conjugate of X(36436)

X(36455) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND REFLECTION OF ABC In X(3)

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

X(36455) lies on these lines: {2, 3}, {13, 6561}, {14, 6560}, {491, 617}, {492, 616}, {542, 33441}, {1327, 3391}, {1328, 3367}, {9541, 11488}, {10653, 35823}, {10654, 35822}, {12123, 22606}, {12124, 22634}

X(36455) = {X(2),X(376)}-harmonic conjugate of X(36437)

X(36456) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND REFLECTION OF X(3) IN ABC

Barycentrics    11*a^4-25*(b^2+c^2)*a^2+(-6*(b^2+c^2)*a^2+6*b^4-12*b^2*c^2+6*c^4)*sqrt(3)+14*(b^2-c^2)^2 : :

X(36456) lies on these lines: {2, 3}, {6, 36469}, {13, 8252}, {14, 8253}, {11488, 18510}, {11489, 18512}, {13665, 23303}, {13785, 23302}, {16644, 35823}, {16645, 35822}

X(36456) = {X(2),X(381)}-harmonic conjugate of X(36438)
X(36456) = {X(36469),X(36470)}-harmonic conjugate of X(6)

X(36457) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND REFLECTION OF ABC IN X(5)

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

X(36457) lies on these lines: {2, 3}, {13, 590}, {14, 615}, {395, 35823}, {396, 35822}, {542, 6302}, {3365, 32788}, {3389, 32787}, {6300, 22872}, {6305, 22919}, {6564, 23302}, {6565, 23303}, {11488, 13665}, {11489, 13785}, {16808, 32789}, {16809, 32790}

X(36457) = {X(2),X(381)}-harmonic conjugate of X(36439)

X(36458) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND OUTER GARCIA

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

X(36458) lies on these lines: {1, 2}, {2041, 5882}, {2043, 28194}, {2045, 11362}, {3656, 18587}, {10222, 14814}, {18585, 28204}

X(36458) = {X(2),X(3679)}-harmonic conjugate of X(36440)

X(36459) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND MANDART-INCIRCLE

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

X(36459) lies on these lines: {2, 11}, {2044, 11237}, {2045, 9670}, {2046, 3303}, {3584, 18586}, {10072, 18585}, {14813, 31452}

X(36459) = {X(2),X(3058)}-harmonic conjugate of X(36441)

X(36460) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND INNER YFF

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

X(36460) lies on these lines: {1, 2}, {2042, 5563}, {2046, 3746}, {4857, 35732}, {4995, 18585}, {5434, 15765}, {11238, 18586}

X(36460) = {X(1),X(2)}-harmonic conjugate of X(36461)
X(36460) = {X(2),X(10056)}-harmonic conjugate of X(36442)

X(36461) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND OUTER YFF

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

X(36461) lies on these lines: {1, 2}, {2042, 3746}, {2046, 5563}, {3058, 15765}, {5270, 35732}, {5298, 18585}, {11237, 18586}, {14813, 15888}

X(36461) = {X(1),X(2)}-harmonic conjugate of X(36460)
X(36461) = {X(2),X(10072)}-harmonic conjugate of X(36443)

X(36462) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND ANTI-AQUILA

Barycentrics    13*a+b+c+3*(b+c)*sqrt(3) : :

X(36462) lies on these lines: {1, 2}, {2042, 7982}, {2045, 9624}, {3653, 18585}, {3656, 15765}, {13688, 13706}, {13808, 13824}, {18586, 28204}

X(36462) = {X(2),X(551)}-harmonic conjugate of X(36444)

X(36463) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND INFINITE ALTITUDE

Barycentrics    13*a^4-2*(b^2+c^2)*a^2+(-6*(b^2+c^2)*a^2+6*b^4-12*b^2*c^2+6*c^4)*sqrt(3)-11*(b^2-c^2)^2 : :

X(36463) lies on these lines: {2, 3}, {532, 5861}, {533, 5860}, {3068, 10653}, {3069, 10654}, {3365, 19053}, {3389, 19054}, {3642, 5590}, {3643, 5591}, {6278, 36390}, {6281, 36394}, {6459, 16963}, {6460, 16962}, {6560, 11488}, {6561, 11489}

X(36463) = reflection of X(36445) in X(2)
X(36463) = X(4)-of-triangle-T(-sqrt(27)-5)
X(36463) = {X(3),X(10304)}-harmonic conjugate of X(36445)
X(36463) = {X(4),X(3545)}-harmonic conjugate of X(36445)
X(36463) = {X(20),X(5054)}-harmonic conjugate of X(36445)
X(36463) = {X(376),X(3524)}-harmonic conjugate of X(36445)
X(36463) = {X(381),X(3839)}-harmonic conjugate of X(36445)

X(36464) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND 3RD TRI-SQUARES CENTRAL

Barycentrics    3*(sqrt(3)-4)*a^2-13*S : :

X(36464) lies on these lines: {2, 6}, {4, 35730}, {13650, 36374}, {13651, 36390}, {13706, 22541}, {13711, 36394}, {13824, 19100}, {13833, 22874}, {14813, 31487}, {25185, 33440}

X(36464) = {X(2),X(6)}-harmonic conjugate of X(36465)
X(36464) = {X(2),X(13846)}-harmonic conjugate of X(36446)

X(36465) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND 4TH TRI-SQUARES CENTRAL

Barycentrics    3*(sqrt(3)-4)*a^2+13*S : :

X(36465) lies on these lines: {2, 6}, {13706, 19099}, {13769, 22917}, {13770, 36394}, {13771, 36370}, {13824, 19101}, {13834, 36390}, {25190, 33443}

X(36465) = {X(2),X(6)}-harmonic conjugate of X(36464)
X(36465) = {X(2),X(13847)}-harmonic conjugate of X(36447)

X(36466) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND EHRMANN MID-TRIANGLE

Barycentrics    13*a^4+10*(b^2+c^2)*a^2+(-9*(b^2+c^2)*a^2+9*b^4-18*b^2*c^2+9*c^4)*sqrt(3)-23*(b^2-c^2)^2 : :

X(36466) lies on these lines: {2, 3}, {395, 1328}, {396, 1327}, {6289, 36390}, {6290, 36394}, {12601, 36357}, {12602, 36348}, {36362, 36371}, {36363, 36372}, {36382, 36397}, {36383, 36400}

X(36466) = {X(2),X(3845)}-harmonic conjugate of X(36448)

X(36467) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND ANTI-INNER GREBE

Barycentrics    3*(sqrt(3)-1)*a^2-2*S : :

X(36467) lies on these lines: {2, 6}, {14, 1328}, {62, 2044}, {486, 16268}, {2041, 3411}, {2042, 6419}, {2043, 35823}, {2046, 6420}, {3365, 16963}, {5418, 16962}, {13928, 19075}

X(36467) = {X(2),X(6)}-harmonic conjugate of X(36468)
X(36467) = {X(2),X(19053)}-harmonic conjugate of X(36449)
X(36467) = {X(6),X(32787)}-harmonic conjugate of X(36449)

X(36468) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND ANTI-OUTER GREBE

Barycentrics    3*(sqrt(3)-1)*a^2+2*S : :

X(36468) lies on these lines: {2, 6}, {13, 1327}, {61, 2044}, {485, 16267}, {2041, 3412}, {2042, 6420}, {2043, 35822}, {2046, 6419}, {3389, 16962}, {5420, 16963}, {13876, 19074}

X(36468) = {X(2),X(6)}-harmonic conjugate of X(36467)
X(36468) = {X(2),X(19054)}-harmonic conjugate of X(36450)
X(36468) = {X(6),X(32788)}-harmonic conjugate of X(36450)

X(36469) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND 1ST KENMOTU FREE-VERTICES TRIANGLE

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

X(36469) lies on these lines: {2, 372}, {6, 36456}, {13, 6396}, {15, 381}, {17, 18586}, {61, 13846}, {371, 16962}, {396, 35823}, {2046, 6419}, {3365, 5054}, {3390, 13847}, {3411, 6420}, {6564, 23302}, {6565, 11488}, {8976, 16268}, {11303, 22917}, {14813, 35813}, {15764, 35740}, {16772, 18585}, {35734, 35739}

X(36469) = {X(2),X(35822)}-harmonic conjugate of X(36452)
X(36469) = {X(6),X(36456)}-harmonic conjugate of X(36470)
X(36469) = {X(381),X(16644)}-harmonic conjugate of X(36453)

X(36470) = HOMOTHETOR OF THESE TRIANGLES: T(-SQRT(27)-5) AND 2ND KENMOTU FREE-VERTICES TRIANGLE

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

X(36470) lies on these lines: {2, 371}, {6, 36456}, {14, 6200}, {16, 381}, {18, 18586}, {62, 13847}, {372, 16963}, {395, 35822}, {2046, 6420}, {3364, 13846}, {3389, 5054}, {3412, 6419}, {6564, 11489}, {6565, 23303}, {11304, 22874}, {13951, 16267}, {14813, 35812}, {16773, 18585}

X(36470) = {X(2),X(35823)}-harmonic conjugate of X(36453)
X(36470) = {X(6),X(36456)}-harmonic conjugate of X(36469)
X(36470) = {X(381),X(16645)}-harmonic conjugate of X(36452)

X(36471) =  MIDPOINT OF X(4) AND X(2710)

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

See Vu Thanh Tung and César Lozada, Euclid 537 .

X(36471) lies on the nine-point circle and these lines: {2, 2715}, {4, 2710}, {113, 625}, {114, 1503}, {115, 525}, {118, 20546}, {125, 15630}, {127, 512}, {132, 511}, {138, 14918}, {141, 16188}, {626, 31848}, {1560, 11064}, {3734, 25641}, {3818, 16760}, {5108, 31655}

X(36471) = midpoint of X(4) and X(2710)
X(36471) = complement of X(2715)
X(36471) = complementary conjugate of X(2799)
X(36471) = X(4)-Ceva conjugate of-X(2799)
X(36471) = X(i)-complementary conjugate of-X(j) for these (i,j): (1, 2799), (75, 24284), (82, 14316)
X(36471) = center of the circumconic {{A, B, C, X(4), X(2065), X(2710), X(15388)}}
X(36471) = orthoptic-circle-of-Steiner-inellipse-inverse of X(2857)

X(36472) =  MIDPOINT OF X(4) AND X(23700)

Barycentrics    (b^2-c^2)^2*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2)*(a^6-3*(b^2+c^2)*a^4+(3*b^4+b^2*c^2+3*c^4)*a^2-b^6-c^6) : :
X(36472) = X(187)+2*X(9721)

See Vu Thanh Tung and César Lozada, Euclid 537 .

X(36472) lies on the nine-point circle and these lines: {2, 10425}, {4, 14384}, {114, 230}, {115, 3566}, {126, 3580}, {127, 14113}, {131, 187}, {136, 2501}, {137, 3124}, {511, 31842}, {512, 5139}, {1648, 3258}, {3767, 18347}, {5099, 10413}, {6792, 31655}, {13881, 18348}, {15538, 25641}, {16188, 21850}, {31850, 33330}

X(36472) = midpoint of X(4) and X(23700)
X(36472) = complement of X(10425)
X(36472) = complementary conjugate of the isogonal conjugate of X(10425)
X(36472) = crosspoint of X(230) and X(2501)
X(36472) = X(2)-Ceva conjugate of-X(6132)
X(36472) = X(i)-complementary conjugate of-X(j) for these (i,j): (31, 6132), (230, 4369), (798, 36212)
X(36472) = center of the circumconic {{A, B, C, X(4), X(249), X(14253), X(23700), X(35296)}}
X(36472) = Dou-circles-radical-circle-inverse of X(136)

leftri

Homothetors involving triangles T(k): X(36473)-X(36513)

rightri

This preamble and centers X(36473)-X(36513) were contributed by Clark Kimberling and Peter Moses, January 18, 2020.

In the preamble just before X(36436), the triangle T(k), for k any real number, is defined as the central triangle with A-vertex 1 : k : k. This definition includes triangles for which k is a function symmetric in a,b,c and homogeneous of degree 0, such as (a^2+b^2+c^2)/(bc+ca+ab) and (a^3+b^3+c^3)/(abc).

For barycentrics and references for the various triangles mentioned in this section, see Index of Triangles Referenced in ETC, by .

X(36473) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND EULER

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

X(36473) lies on these lines: {2, 3}, {515, 29660}, {946, 29659}, {1072, 29676}, {4389, 24828}, {8227, 24331}, {10446, 19130}, {17230, 29331}, {17290, 24813}, {17354, 29243}

X(36474) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND REFLECTION OF ABC IN X(3)

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

(X(36474) lies on these lines: {2, 3}, {8, 29331}, {40, 29659}, {45, 24828}, {497, 20256}, {515, 25353}, {517, 24326}, {519, 7758}, {540, 14023}, {573, 31670}, {944, 29081}, {946, 24331}, {952, 11200}, {991, 1352}, {1001, 25365}, {1060, 2356}, {1478, 2223}, {1790, 31383}, {3419, 25083}, {3576, 29660}, {3938, 9933}, {3961, 5534}, {4363, 29243}, {4389, 24813}, {4459, 24248}, {4660, 20258}, {5731, 29373}, {13329, 14561}, {18506, 29820}, {19925, 25352}, {20073, 24844}, {25066, 26036}

X(36475) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND TANGENTIAL OF 1ST CIRCUMPERP

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

X(36475) lies on these lines: {2, 11}, {3, 29659}, {474, 24331}, {1403, 29670}, {3295, 29660}, {3666, 3689}, {3744, 17122}, {3911, 29655}, {3938, 17599}, {4030, 14829}, {4389, 24820}, {5737, 33117}, {7322, 17594}, {9318, 24326}, {16675, 31477}, {21010, 21976}, {24357, 34247}, {29673, 32916}

X(36476) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND TANGENTIAL OF 2ND CIRCUMPERP

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

X(36476) lies on these lines: {2, 12}, {3, 29659}, {172, 3691}, {190, 1001}, {405, 24331}, {950, 29655}, {956, 21010}, {993, 17798}, {999, 29660}, {1319, 2329}, {1959, 2099}, {4389, 24826}, {4423, 33144}, {16064, 29685}

X(36477) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND REFLECTION OF ABC IN X(5)

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

X(36477) lies on these lines: {1, 29331}, {2, 3}, {10, 24264}, {355, 29081}, {517, 24455}, {519, 7751}, {540, 7759}, {912, 24333}, {993, 20544}, {1385, 24331}, {1478, 17798}, {2271, 5305}, {2548, 33863}, {3098, 24220}, {3419, 20769}, {3767, 18755}, {4364, 29243}, {4389, 24833}, {5587, 29373}, {5886, 29660}, {6684, 25352}, {7776, 17206}, {9441, 26446}, {17325, 24827}, {17369, 24828}, {24357, 29010}, {29652, 35631}

X(36478) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND AQUILLA

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

X(36478) lies on these lines: {1, 2}, {484, 3496}, {537, 17305}, {846, 26061}, {894, 24692}, {984, 17325}, {1051, 32852}, {1757, 4643}, {2329, 5123}, {2640, 4429}, {3242, 25539}, {3589, 33076}, {3760, 33941}, {3821, 4440}, {3844, 4649}, {3923, 26083}, {3992, 18140}, {4026, 4422}, {4085, 17289}, {4389, 24821}, {4439, 17320}, {4657, 33165}, {4660, 17368}, {4670, 31151}, {4693, 17359}, {4753, 17271}, {5587, 29373}, {5902, 20715}, {16484, 17357}, {16788, 17057}, {17290, 31178}, {17369, 24715}, {17371, 32941}, {17596, 32780}, {18788, 26446}, {24325, 27191}, {32781, 32913}

X(36479) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND 5TH MIXTILINEAR

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

X(36479) lies on these lines: {1, 2}, {7, 4660}, {11, 30824}, {30, 4754}, {69, 33076}, {344, 16484}, {345, 3750}, {390, 3923}, {392, 4517}, {497, 1215}, {515, 24333}, {517, 24326}, {518, 4643}, {528, 4363}, {537, 4419}, {752, 4644}, {940, 4030}, {996, 1438}, {999, 4447}, {1000, 4876}, {1001, 4422}, {1429, 3476}, {1478, 20556}, {1621, 33163}, {1909, 3673}, {2177, 17740}, {2295, 16502}, {2329, 6554}, {2345, 16503}, {2550, 24325}, {2796, 4454}, {2887, 3475}, {3208, 31393}, {3242, 4026}, {3416, 17374}, {3434, 32771}, {3674, 18421}, {3691, 17742}, {3711, 5241}, {3748, 32777}, {3751, 3883}, {3821, 4310}, {3844, 15570}, {3846, 25568}, {3873, 26034}, {3975, 4737}, {3980, 17784}, {4000, 4085}, {4078, 4901}, {4090, 18228}, {4307, 4747}, {4344, 33682}, {4357, 16496}, {4364, 9041}, {4389, 24841}, {4407, 4748}, {4418, 20075}, {4429, 27191}, {4430, 33083}, {4440, 24248}, {4514, 26098}, {4659, 28580}, {4702, 17281}, {4709, 32087}, {4780, 17151}, {4796, 28566}, {4863, 31993}, {4865, 5712}, {5698, 32935}, {5749, 16779}, {5772, 8236}, {5905, 32947}, {6706, 24656}, {8193, 16064}, {8616, 26065}, {9052, 35628}, {9791, 31302}, {10944, 16377}, {12410, 20834}, {15485, 26685}, {16783, 26035}, {17318, 28503}, {17355, 30331}, {17484, 24710}, {17715, 32780}, {17718, 30823}, {17776, 33162}, {18141, 33079}, {19785, 32923}, {19822, 32945}, {20073, 24821}, {20894, 34284}, {21283, 31025}, {24217, 28808}, {24318, 28234}, {24477, 32916}, {24709, 32931}, {24715, 31178}, {25384, 28581}, {28562, 35578}, {32773, 33144}

X(36480) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND OUTER GARCIA

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

X(36480) lies on these lines: {1, 2}, {6, 4753}, {9, 4759}, {31, 4981}, {37, 4702}, {38, 3980}, {45, 4432}, {75, 4495}, {101, 2344}, {190, 984}, {210, 25496}, {238, 17335}, {244, 24594}, {274, 33937}, {292, 996}, {333, 17716}, {515, 25353}, {517, 24455}, {518, 4670}, {528, 4364}, {537, 4363}, {668, 870}, {726, 4659}, {740, 17318}, {750, 24593}, {752, 4407}, {756, 4011}, {758, 24333}, {903, 24452}, {956, 21010}, {993, 2223}, {1001, 3842}, {1086, 24693}, {1107, 25066}, {1211, 4865}, {1376, 6682}, {2099, 16609}, {2550, 3821}, {2796, 4419}, {3242, 24325}, {3416, 3775}, {3434, 4425}, {3488, 26036}, {3681, 32772}, {3686, 16972}, {3696, 32921}, {3745, 32853}, {3747, 27917}, {3751, 33682}, {3753, 20358}, {3791, 4042}, {3875, 4709}, {3886, 3993}, {3891, 21020}, {3925, 26128}, {3986, 30331}, {3989, 32929}, {3996, 17592}, {4021, 4780}, {4023, 17726}, {4085, 4657}, {4160, 4444}, {4294, 12579}, {4307, 17770}, {4349, 34379}, {4357, 4660}, {4361, 4732}, {4375, 29350}, {4386, 8624}, {4389, 24715}, {4414, 4781}, {4418, 7226}, {4439, 16521}, {4448, 4775}, {4472, 9041}, {4664, 4693}, {4665, 28503}, {4672, 5220}, {4687, 16484}, {4690, 28538}, {4748, 17766}, {4767, 32931}, {4792, 27922}, {5224, 33076}, {5233, 17722}, {5251, 23407}, {5252, 16603}, {5278, 17469}, {5283, 28594}, {5750, 16973}, {9347, 32919}, {9798, 12567}, {9997, 24254}, {10436, 16496}, {10707, 25378}, {14621, 27495}, {15485, 17260}, {16517, 17355}, {17126, 24616}, {17143, 33945}, {17227, 31151}, {17274, 24692}, {17289, 33165}, {17290, 25351}, {17449, 26627}, {17461, 24428}, {17598, 19804}, {17720, 21242}, {19786, 32865}, {19808, 33169}, {24342, 24349}, {24841, 31178}, {25342, 28234}, {26580, 33104}, {27184, 33109}, {28606, 32945}, {31993, 32920}, {32775, 33108}, {32776, 33110}, {32782, 33072}, {32784, 32850}, {33065, 33112}, {33073, 33084}, {33111, 33126}

X(36481) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND MANDART-INCIRCLE TRIANGLE

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

X(36481) lies on these lines: {2, 11}, {2646, 29660}, {3057, 29659}, {3666, 24217}, {3744, 17717}, {4030, 5233}, {4363, 24837}, {4389, 24840}, {10832, 20834}, {11376, 24331}, {21242, 32777}, {22706, 24210}

X(36482) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND ORTHIC OF INTOUCH

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

X(36482) lies on these lines: {2, 7}, {11, 982}, {56, 29660}, {65, 29659}, {75, 16888}, {85, 16603}, {150, 24268}, {171, 17718}, {241, 24798}, {291, 17889}, {984, 25365}, {1429, 30617}, {3434, 9451}, {3485, 24331}, {3661, 7185}, {3665, 7146}, {3673, 26012}, {3677, 24210}, {3711, 33079}, {3947, 25352}, {3961, 5018}, {4077, 4444}, {4389, 25371}, {4419, 4466}, {5988, 33144}, {6063, 6358}, {6354, 7204}, {7182, 30545}, {7201, 24357}, {7988, 18193}, {16609, 33298}, {17090, 29593}, {18343, 28125}, {24586, 33066}, {25257, 31033}

X(36483) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND TANGENTIAL OF EXCENTRAL

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

X(36483) lies on these lines: {2, 7}, {40, 29659}, {81, 3930}, {171, 210}, {190, 25371}, {291, 846}, {497, 3923}, {982, 4423}, {1621, 9451}, {1699, 21375}, {1762, 3925}, {2003, 20741}, {2999, 36403}, {3097, 17594}, {3333, 29660}, {3475, 29656}, {3677, 29820}, {3751, 3961}, {4384, 17789}, {4418, 13576}, {4863, 33169}, {5285, 20834}, {16560, 17369}, {17738, 24631}, {20601, 25365}, {21984, 25083}, {22116, 24479}, {24331, 31435}

X(36484) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND ANTI-1ST-EULER

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

X(36484) lies on these lines: {2, 3}, {944, 29659}, {4389, 24817}, {4911, 5122}, {5603, 18788}, {17305, 29243}, {17369, 24813}, {21165, 25353}, {25352, 31423}

X(36485) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND INNER JOHNSON

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

X(36485) lies on these lines: {2, 11}, {183, 4030}, {355, 29081}, {2201, 5101}, {2980, 21011}, {3938, 27918}, {4389, 24834}, {5014, 26250}, {5137, 33120}, {10944, 16377}, {11373, 29660}, {17122, 17721}, {17614, 24331}, {29652, 35626}

X(36486) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND OUTER JOHNSON

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

X(36486) lies on these lines: {2, 12}, {65, 24333}, {355, 29081}, {2329, 5252}, {4389, 24835}, {9708, 20486}, {11374, 29660}, {12527, 25353}

X(36487) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND 1ST JOHNSON-YFF

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

X(36487) lies on these lines: {1, 29331}, {2, 12}, {956, 20486}, {1319, 24331}, {1429, 5252}, {2099, 3212}, {3911, 25352}, {4363, 24816}, {4389, 24836}, {10944, 16377}, {11375, 29660}, {18954, 20834}

X(36488) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND 2ND JOHNSON-YFF

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

X(36488) lies on these lines: {1, 29331}, {2, 11}, {171, 17721}, {1837, 29659}, {1936, 29676}, {2646, 24331}, {3550, 31231}, {3684, 4863}, {3750, 17720}, {4124, 28125}, {4363, 24840}, {4389, 24837}, {9599, 17735}, {10833, 20834}, {11376, 29660}, {11997, 25384}, {20359, 29668}, {21334, 29652}, {24431, 36265}

X(36489) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND INFINITE ALTITUDE TRIANGLE

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

X(36489) lies on these lines: {1, 20731}, {2, 3}, {515, 29659}, {946, 29660}, {1064, 32462}, {1072, 29675}, {3098, 10446}, {3332, 10519}, {3576, 24331}, {3673, 24929}, {4293, 17798}, {4363, 24813}, {4389, 29243}, {4393, 29331}, {4419, 24817}, {5286, 18755}, {5603, 28885}, {5657, 9441}, {10164, 25352}, {10476, 29652}, {17354, 24828}, {18446, 24333}, {19557, 24247}, {24357, 30273}, {25384, 30271}

X(36489) = X(4)-of-triangle-T((a^2+b^2+c^2)/(bc+ca+ab))

X(36490) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND EHRMANN MID-TRIANGLE

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

X(36490) lies on these lines: {2, 3}, {115, 4262}, {3654, 28854}, {4251, 5309}, {4253, 7753}, {4389, 24827}, {5030, 5475}, {9955, 29660}, {17389, 29331}, {18480, 29659}, {18481, 24331}, {25352, 31730}

X(36491) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND ANTI-INNER-GREBE

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

X(36491) lies on these lines: {2, 6}, {45, 24843}, {4258, 32494}, {4389, 24818}, {5405, 36403}, {13971, 24331}, {17354, 24819}, {18991, 29660}, {18992, 29659}

X(36492) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND ANTI-OUTER-GREBE

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

X(36492) lies on these lines: {2, 6}, {45, 24842}, {4258, 32497}, {4389, 24819}, {5393, 36403}, {8983, 24331}, {17354, 24818}, {18991, 29659}, {18992, 29660}

X(36493) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND 2ND ANTI-CIRCUMPERP-TANGENTIAL

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

X(36493) lies on these lines: {2, 12}, {65, 29659}, {495, 21010}, {1319, 29660}, {1463, 32784}, {1478, 17798}, {3212, 3665}, {4363, 24836}, {4389, 24816}, {5252, 16603}, {7248, 32781}, {10831, 20834}, {11375, 24331}

X(36494) =  HOMOTHETOR OF THESE TRIANGLES: T((a^2+b^2+c^2)/(bc+ca+ab)) AND 2ND GEMINI 19

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

lies on these lines: {2, 3807}, {6, 31314}, {37, 17339}, {45, 33888}, {75, 142}, {86, 192}, {190, 1001}, {335, 4389}, {518, 17346}, {551, 726}, {594, 31329}, {742, 17378}, {984, 3122}, {1278, 4648}, {2276, 31348}, {2345, 31347}, {3739, 29613}, {4033, 10009}, {4360, 20159}, {4417, 27491}, {4686, 29623}, {4699, 17283}, {4740, 31139}, {5224, 27495}, {17277, 27484}, {17303, 31335}, {17318, 20131}, {18134, 27476}, {21101, 30963}, {24325, 29660}, {24403, 31063}, {24656, 33890}, {25361, 27479}, {27268, 31333}, {27958, 34053}

X(36495) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND EULER

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

X(36495) lies on these lines: {2, 3}, {115, 22407}, {976, 5587}, {5293, 7989}, {8227, 28082}, {17605, 28109}

X(36496) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND REFLECTION OF ABC IN X(3)

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

X(36496) lies on these lines: {2, 3}, {355, 4723}, {388, 17724}, {497, 1854}, {515, 976}, {946, 28082}, {952, 20035}, {1473, 23542}, {1479, 3670}, {1535, 12672}, {1754, 7683}, {1836, 28109}, {1842, 34822}, {5225, 17595}, {5293, 5691}, {5906, 26892}, {26333, 28074}

X(36497) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND TANGENTIAL OF 1ST CIRCUMPERP

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

X(36497) lies on these lines: {2, 11}, {3, 33119}, {10, 3145}, {474, 28082}, {976, 4642}, {4424, 5293}, {4438, 16064}, {4812, 26263}, {11358, 19729}, {11499, 19548}, {13589, 33166}, {20834, 33115}, {20999, 29673}, {24820, 33153}

X(36498) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND TANGENTIAL OF 2ND CIRCUMPERP

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

X(36498) lies on these lines: {2, 12}, {3, 33119}, {10, 20999}, {38, 405}, {244, 19529}, {956, 976}, {993, 3145}, {3953, 5251}, {5258, 5293}, {9708, 16422}, {11108, 32775}, {13732, 33163}, {19548, 22758}

X(36499) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND AQUILLA

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

X(36499) lies on these lines: {1, 2}, {5, 6211}, {335, 16908}, {442, 32780}, {1046, 2887}, {1247, 19808}, {1268, 35550}, {1757, 3454}, {2476, 26061}, {3145, 5251}, {3695, 33135}, {3841, 24342}, {3868, 31237}, {4202, 33119}, {4413, 16422}, {4438, 16062}, {4645, 8258}, {5015, 6679}, {5051, 33115}, {5219, 28109}, {5429, 7270}, {5587, 19548}, {5791, 32784}, {7683, 18788}, {23537, 33167}

X(36500) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND 5TH MIXTILINEAR

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

X(36500) lies on these lines: {1, 2}, {4, 17165}, {38, 17676}, {65, 5014}, {69, 20247}, {75, 5178}, {315, 17141}, {335, 16910}, {377, 17140}, {497, 25253}, {518, 4812}, {758, 4894}, {942, 5300}, {952, 19548}, {956, 3145}, {1043, 33089}, {1046, 20064}, {1104, 33114}, {1215, 28086}, {1626, 2975}, {1834, 3891}, {1837, 4696}, {2280, 4136}, {2475, 24349}, {2478, 3952}, {2650, 4865}, {3189, 17740}, {3419, 4968}, {3434, 17164}, {3701, 5722}, {3868, 5015}, {3869, 4514}, {3873, 7270}, {3874, 4680}, {3977, 4314}, {4168, 9310}, {4195, 33170}, {4201, 4392}, {4294, 4427}, {4295, 21282}, {5046, 32937}, {5252, 28109}, {5596, 16799}, {6284, 32933}, {7226, 26117}, {11319, 33163}, {16924, 31052}, {17697, 33166}, {21935, 32920}, {33824, 33888}

X(36501) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND MANDART-INCIRCLE TRIANGLE

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

X(36501) lies on these lines: {2, 11}, {946, 28109}, {976, 1837}, {1479, 19548}, {5293, 9581}, {8727, 28108}, {11376, 28082}

X(36502) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND 5TH BROCARD

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

X(36502) lies on these lines: {2, 32}, {976, 9857}, {3314, 22380}, {7761, 19674}, {7796, 22385}, {7849, 22425}, {7876, 22398}, {7906, 22393}, {9996, 19548}

X(36503) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND ORTHIC OF INTOUCH

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

X(36503) lies on these lines: {2, 7}, {388, 976}, {982, 8229}, {1401, 33069}, {3485, 28082}, {3649, 28109}, {3665, 17056}, {3772, 30617}, {4417, 33930}, {4812, 7201}, {4952, 5252}, {5261, 6555}, {5290, 5293}, {7225, 19786}, {16888, 17080}, {26118, 33144}

X(36504) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND TANGENTIAL OF EXCENTRAL

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

X(36504) lies on these lines: {2, 7}, {3145, 31424}, {5044, 16422}, {6211, 26118}, {7330, 19548}, {8229, 33119}, {16560, 32777}, {21367, 32779}, {28082, 31435}

X(36505) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND ANTI-AQUILLA

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

X(36505) lies on these lines: {1, 2}, {3, 33123}, {21, 26128}, {35, 33125}, {58, 33069}, {335, 16905}, {405, 32775}, {964, 33130}, {1001, 3145}, {1468, 33124}, {1724, 33065}, {1962, 19805}, {3701, 17725}, {3868, 6679}, {3936, 16478}, {4418, 24159}, {5015, 31237}, {5192, 17719}, {5217, 17290}, {5247, 33122}, {5248, 32776}, {5266, 25957}, {5886, 19548}, {6693, 18398}, {7283, 33143}, {11374, 32944}, {13740, 33127}, {15950, 28109}, {17526, 33144}, {17698, 32771}, {19278, 26150}, {24161, 24552}, {24850, 33146}, {25598, 33953}

X(36506) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND INNER JOHNSON

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

X(36506) lies on these lines: {2, 11}, {10, 28077}, {355, 19548}, {474, 28074}, {976, 10914}, {17614, 28082}

X(36507) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND OUTER JOHNSON

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

X(36507) lies on these lines: {2, 12}, {3, 33163}, {21, 32937}, {31, 72}, {355, 19548}, {1215, 13733}, {3145, 8193}, {5220, 16948}, {5791, 32918}, {11374, 32944}, {13732, 32931}

X(36508) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND 1ST JOHNSON-YFF

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

X(36508) lies on these lines: {1, 19548}, {2, 12}, {3, 28108}, {57, 5293}, {65, 976}, {404, 18048}, {603, 1463}, {1086, 2933}, {1259, 21320}, {1284, 1486}, {1319, 28077}, {1324, 24159}, {1466, 28079}, {1470, 28107}, {3556, 28353}, {4306, 5061}, {4346, 5217}, {13732, 17719}, {13733, 33127}, {28080, 28083}

X(36509) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND 2ND JOHNSON-YFF

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

X(36509) lies on these lines: {1, 19548}, {2, 11}, {33, 21333}, {976, 3057}, {1283, 29676}, {1697, 5293}, {1936, 3056}, {2330, 16793}, {2646, 28082}, {3145, 8240}, {3601, 11512}, {3915, 28275}, {4392, 13589}, {7071, 28106}, {33849, 34247}

X(36510) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND INFINITE ALTITUDE TRIANGLE

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

X(36510) lies on these lines: {2, 3}, {35, 24248}, {40, 976}, {41, 17756}, {165, 5293}, {1261, 5687}, {1626, 30478}, {1754, 3430}, {2550, 23843}, {2646, 28109}, {3072, 30269}, {3576, 28082}, {4812, 30273}, {9538, 20254}, {12245, 20035}, {19843, 23850}

X(36510) = X(4)-of-triangle-T((a^3+b^3+c^3)/(abc))

X(36511) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND ANTI-5TH-BROCARD

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

X(36511) lies on these lines: {2, 32}, {99, 22407}, {384, 22398}, {976, 10791}, {3329, 22380}, {7783, 22408}, {7804, 22442}, {10796, 19548}

X(36512) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND EHRMANN MID-TRIANGLE

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

X(36512) lies on these lines: {2, 3}, {976, 12699}, {5434, 17597}, {8148, 20035}, {10572, 28109}, {14537, 22442}, {18481, 28082}

X(36513) =  HOMOTHETOR OF THESE TRIANGLES: T((a^3+b^3+c^3)/(abc)) AND 2ND ANTI-CIRCUMPERP-TANGENTIAL

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

X(36513) lies on these lines: {2, 12}, {5, 33144}, {226, 28109}, {976, 5252}, {1478, 19548}, {2476, 24349}, {3953, 7951}, {5293, 9578}, {11375, 28082}, {17111, 26481}, {24995, 30617}

leftri

V transforms on the circumcircle: X(36514)-X(36517)

rightri

This preamble was contributed by Vu Thanh Tung, and centers X(36436)-X(36472) by Peter Moses, January 20, 2020.

Let X = x : y : z be a point in the plane of a triangle ABC, let A'B'C'= circumcevian triangle of X, and let OA = circumcenter of triangle XBC; define OB and OC cyclically.

The triangles OAOBOC and A'B'C' are perspective, and their perspector, on the circumcircle, is given by

V(X) = a^2 / (a^4 y (y - z) z + (b^2 - c^2) x^2 (c^2 y + b^2 z) + a^2 (-b^2 z (x^2 + 2 x z + y (y + z)) + c^2 y (x^2 + 2 x y + z (y + z)))) : :

Let X* denote the isogonal conjugate of X; then V(X*) = V(X), as in the following examples:

V(X(2)) = V(X(6)) = X(1296)
V(X(3)) = V(X(4)) = X(110)
V(X(5)) = V(X(54)) = X(1291)
V(X(7)) = V(X(55)) = X(20219)
V(X(9)) = V(X(57)) = X(28291)
V(X(17)) = V(X(61)) = X(36514)
V(X(18)) = V(X(62)) = X(36515)
V(X(19)) = V(X(63)) = X(36516)
V(X(39)) = V(X(83)) = X(36517)

V(X) is the perspector of ABC and the triangle formed by reflecting line XX' in the sides of ABC, where X' denotes the isogonal conjugate of X. (Randy Hutson, March 29, 2020)

X(36514) =  V(X(17))

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 16*b^2*c^2 - 5*c^4 + 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

X(36514) lies on the circumcircle and these lines: {16, 1337}, {98, 11122}, {512, 10409}, {622, 33500}, {2378, 33957}, {2379, 10645}, {2381, 6104}, {4558, 36515}, {5966, 13349}, {5994, 14183}, {5995, 9218}, {14658, 19780}, {30215, 32036}, {30559, 32627}

X(36514) = reflection of X(i) in X(j) for these {i,j}: {622, 33500}, {1337, 16}
X(36514) = reflection of X(10409) in the Brocard axis
X(36514) = Collings transform of X(i) for these i: {16, 33500}
X(36514) = X(17403)-cross conjugate of X(110)
X(36514) = X(i)-isoconjugate of X(j) for these (i,j): {661, 3181}, {1577, 19781}
X(36514) = cevapoint of X(16) and X(512)
X(36514) = trilinear pole of line X(6)X(3171)
X(36514) = perspector of ABC and the triangle formed by reflecting line X(5)X(14) in the sides of ABC
X(36514) = barycentric product X(110)*X(11122)
X(36514) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 3181}, {1576, 19781}, {11122, 850}, {17403, 30472}

X(36515) =  V(X(18))

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^4 - 2*a^2*b^2 - 5*b^4 - 2*a^2*c^2 + 16*b^2*c^2 - 5*c^4 - 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :

X(36515) lies on the circumcircle and these lines: {15, 1338}, {98, 11121}, {512, 10410}, {621, 33498}, {2378, 10646}, {2379, 33958}, {2380, 6105}, {4558, 36514}, {5966, 13350}, {5994, 9218}, {5995, 14184}, {14658, 19781}, {30216, 32037}, {30560, 32628}

X(36515) = reflection of X(i) in X(j) for these {i,j}: {621, 33498}, {1338, 15}
X(36515) = reflection of X(10410) in the Brocard axis
X(36515) = Collings transform of X(i) for these i: {15, 33498}
X(36515) = X(17402)-cross conjugate of X(110)
X(36515) = X(i)-isoconjugate of X(j) for these (i,j): {661, 3180}, {1577, 19780}
X(36515) = cevapoint of X(15) and X(512)
X(36515) = trilinear pole of line X(6)X(3170)
X(36515) = perspector of ABC and the triangle formed by reflecting line X(5)X(13) in the sides of ABC
X(36515) = barycentric product X(110)*X(11121)
X(36515) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 3180}, {1576, 19780}, {11121, 850}, {17402, 30471}

X(36516) =  V(X(19))

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

X(36516) lies on the circumcircle and these lines: {3, 2249}, {56, 35504}, {98, 18446}, {105, 1064}, {107, 1981}, {675, 18444}, {759, 991}, {929, 35338}, {1294, 30265}, {1297, 30269}

X(36516) = reflection of X(2249) in X(3)
X(36516) = Thomson-isogonal conjugate of X(8680)
X(36516) = perspector of ABC and the triangle formed by reflecting line X(19)X(63) in the sides of ABC

X(36517) =  V(X(39))

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

X(36517) lies on the circumcircle and these lines: {98, 732}, {511, 733}, {729, 9301}, {755, 35002}, {2698, 5188}, {5092, 5970}

X(36517) = trilinear pole of line X(6)X(8570)
X(36517) = perspector of ABC and the triangle formed by reflecting line X(39)X(83) in the sides of ABC

X(36518) =  COMPLEMENT OF X(15055)

Barycentrics    4*(b^2+c^2)*a^8-(9*b^4-2*b^2*c^2+9*c^4)*a^6+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^4+(5*b^4-6*b^2*c^2+5*c^4)*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (24*R^2+SA-6*SW)*S^2+(36*R^2-7*SW)*SB*SC : :
X(36518) = X(3)-4*X(12900), 2*X(3)+X(13202), X(4)+2*X(5972), 2*X(4)+X(16163), 2*X(5)+X(113), 4*X(5)-X(125), 7*X(5)-X(10264), 8*X(5)+X(15063), 7*X(5)-4*X(15088), 10*X(5)-X(16003), 5*X(5)-2*X(20304), 11*X(5)-2*X(20379), 13*X(5)-4*X(20396), 2*X(113)+X(125), 7*X(113)+2*X(10264), 4*X(113)-X(15063), 7*X(113)+8*X(15088), 5*X(113)+X(16003), 5*X(113)+4*X(20304), 11*X(113)+4*X(20379), 7*X(125)-4*X(10264), 2*X(125)+X(15063), 7*X(125)-16*X(15088), 5*X(125)-2*X(16003), 5*X(125)-8*X(20304), 11*X(125)-8*X(20379), 13*X(125)-16*X(20396), 4*X(5972)-X(16163), 8*X(12900)+X(13202)

See Kadir Altintas and César Lozada, Euclid 549 .

Let OA be the circle centered at A and tangent to the Euler line. Define OB and OC cyclically. Let LA be the polar of X(4) wrt OA, and define LB and LC cyclically. (Note: X(4) is the perspector of every circle centered at a vertex of ABC.) Let A' = LB∩LC, and define B' and C' cyclically. A'B'C' is the reflection of ABC in X(5972) (the radical center of OA, OB, OC), and is homothetic to the Euler triangle at X(36518). (Randy Hutson, March 29, 2020)

X(36518) lies on these lines: {2, 2777}, {3, 12900}, {4, 5972}, {5, 113}, {25, 22109}, {74, 3090}, {110, 578}, {114, 16278}, {140, 1539}, {146, 5056}, {247, 9155}, {265, 3851}, {355, 11723}, {378, 18418}, {381, 5642}, {389, 12825}, {399, 5072}, {403, 511}, {468, 1531}, {541, 5055}, {542, 3545}, {546, 1511}, {547, 34128}, {549, 34584}, {568, 5448}, {631, 10721}, {858, 1533}, {1092, 15472}, {1112, 5562}, {1312, 14499}, {1313, 14500}, {1352, 5095}, {1495, 10297}, {1514, 5159}, {1553, 3154}, {1561, 11007}, {1656, 6699}, {1986, 5907}, {2072, 14915}, {2682, 36170}, {2771, 23513}, {2931, 7529}, {3024, 3614}, {3028, 7173}, {3047, 13434}, {3070, 13990}, {3071, 8998}, {3146, 15051}, {3258, 36169}, {3448, 5068}, {3526, 20127}, {3529, 15036}, {3542, 15473}, {3544, 14094}, {3574, 6153}, {3628, 12041}, {3818, 32250}, {3832, 10733}, {3843, 12121}, {3850, 10113}, {3855, 12383}, {3856, 13392}, {3858, 34153}, {3860, 11694}, {5066, 23516}, {5067, 12244}, {5071, 10706}, {5079, 10620}, {5085, 16072}, {5181, 5480}, {5504, 11424}, {5576, 33547}, {5609, 11801}, {5640, 12827}, {5644, 5655}, {5650, 15760}, {5818, 7978}, {5892, 17853}, {5893, 11598}, {6033, 33511}, {6288, 14049}, {6321, 33512}, {6776, 32300}, {6804, 13203}, {7395, 10117}, {7403, 23306}, {7503, 13289}, {7506, 12893}, {7547, 12140}, {7577, 16261}, {7699, 11188}, {7722, 15058}, {7723, 11557}, {7989, 13211}, {8227, 11735}, {8994, 10576}, {9033, 11897}, {9306, 15463}, {9729, 17854}, {9818, 32607}, {9934, 10984}, {10020, 35240}, {10024, 10170}, {10539, 12228}, {10577, 13969}, {10628, 12824}, {10752, 32257}, {11441, 12227}, {11459, 16868}, {11479, 19457}, {11656, 22566}, {11693, 23046}, {11695, 17855}, {11720, 19925}, {11746, 21649}, {11793, 11807}, {11805, 13565}, {12219, 15056}, {12308, 15027}, {12358, 13417}, {13293, 17928}, {13358, 18874}, {13367, 20771}, {13406, 15067}, {13416, 16105}, {13754, 16222}, {14156, 31726}, {14982, 15118}, {15022, 15054}, {15072, 18504}, {15092, 15535}, {15125, 32125}, {15462, 19124}, {16836, 32743}, {17814, 19504}, {17835, 33537}, {18358, 32275}, {18376, 35264}, {18400, 35265}, {18531, 35268}, {19110, 31412}, {20773, 32340}, {22467, 25564}, {22804, 25402}, {24206, 32271}

X(36518) = midpoint of X(i) and X(j) for these {i,j}: {4, 15035}, {113, 23515}, {381, 14643}, {7728, 15041}, {15030, 16223}
X(36518) = reflection of X(i) in X(j) for these (i,j): (125, 23515), (5642, 14643), (10990, 15041), (15035, 5972), (15041, 6699), (16163, 15035), (23515, 5), (34128, 547)
X(36518) = complement of X(15055)
X(36518) = nine-point circle-inverse of-X(15063)
X(36518) = X(23515)-of-Johnson-triangle
X(36518) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5972, 16163), (5, 113, 125), (5, 10264, 15088), (74, 3090, 6723), (110, 3091, 7687), (113, 125, 15063), (140, 1539, 16111), (146, 5056, 15059), (146, 15059, 20417), (265, 16534, 24981), (381, 15046, 14643), (546, 1511, 12295), (1656, 7728, 6699), (3850, 10272, 10113), (6699, 7728, 10990), (7723, 11557, 14448), (8227, 12368, 11735), (10113, 10272, 30714), (16003, 20304, 125)

X(36519) =  COMPLEMENT OF X(34473)

Barycentrics    4*(b^2+c^2)*a^6-(7*b^4+2*b^2*c^2+7*c^4)*a^4+2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2-(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2 : :
Barycentrics    4*S^4-(SA^2-4*SB*SC+SW^2)*S^2-2*SB*SC*SW^2 : :
X(36519) = X(3)-4*X(6721), 2*X(3)-5*X(31274), X(4)+2*X(620), 2*X(5)+X(114), 4*X(5)-X(115), 8*X(5)+X(14981), 7*X(5)-4*X(15092), 2*X(114)+X(115), 4*X(114)-X(14981), 7*X(114)+8*X(15092), 2*X(115)+X(14981), 7*X(115)-16*X(15092), X(1569)+2*X(6248), 8*X(6721)-5*X(31274), X(14981)+4*X(23514), 8*X(15092)-7*X(23514)

See Kadir Altintas and César Lozada, Euclid 549 .

Let A'B'C' be the mid-triangle of the antipedal triangles of X(13) and X(14). A'B'C' is homothetic to the Euler triangle at X(36519). (Randy Hutson, March 29, 2020)

X(36519) lies on these lines: {2, 2794}, {3, 6721}, {4, 620}, {5, 39}, {30, 9167}, {98, 3090}, {99, 3091}, {113, 15357}, {140, 22505}, {147, 5056}, {148, 5068}, {187, 10011}, {355, 11724}, {376, 22247}, {381, 2482}, {542, 5050}, {543, 3545}, {546, 33813}, {547, 6055}, {625, 1513}, {626, 22712}, {631, 10722}, {1352, 5477}, {1656, 6033}, {2039, 14501}, {2040, 14502}, {2783, 23513}, {2784, 10171}, {3023, 3614}, {3027, 7173}, {3044, 13434}, {3070, 13989}, {3071, 8997}, {3544, 23235}, {3628, 12042}, {3832, 10723}, {3850, 10992}, {3851, 6321}, {3855, 13172}, {5066, 9880}, {5067, 9862}, {5071, 5461}, {5072, 13188}, {5079, 12188}, {5085, 33240}, {5099, 36170}, {5149, 13860}, {5818, 7970}, {6230, 10515}, {6231, 10514}, {6781, 13449}, {7617, 25486}, {7752, 32458}, {7764, 18768}, {7775, 9753}, {7844, 9744}, {7989, 13178}, {8227, 9864}, {8721, 32972}, {8724, 19709}, {8980, 10576}, {9749, 11306}, {9750, 11305}, {9881, 30308}, {10352, 32961}, {10577, 13967}, {11711, 19925}, {14645, 14853}, {14830, 15703}, {15088, 15535}, {19108, 31412}, {21163, 33184}

X(36519) = midpoint of X(i) and X(j) for these {i,j}: {4, 21166}, {114, 23514}, {381, 15561}, {3545, 23234}, {6054, 14651}, {22566, 34127}
X(36519) = reflection of X(i) in X(j) for these (i,j): (115, 23514), (2482, 15561), (6055, 34127), (14651, 5461), (14971, 5055), (21166, 620), (23514, 5), (34127, 547)
X(36519) = complement of X(34473)
X(36519) = nine-point circle-inverse of-X(14981)
X(36519) = X(23514)-of-Johnson-triangle
X(36519) = orthoptic circle of Steiner inellipse-inverse of-X(9157)
X(36519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6721, 31274), (5, 114, 115), (98, 3090, 6722), (114, 115, 14981), (147, 5056, 14061), (147, 14061, 11623), (547, 22566, 6055), (1656, 6033, 6036), (5071, 6054, 5461), (6033, 6036, 10991), (8227, 9864, 11725)

X(36520) =  COMPLEMENT OF X(23239)

Barycentrics    2*S^4+(4*R^2-SW)*(60*R^2+SA-8*SW)*S^2+(4*R^2-SW)*(36*R^2-7*SW)*SB*SC : :
X(36520) = X(4)+2*X(34842), 2*X(5)+X(122), 4*X(5)-X(133), X(107)-7*X(3090), 2*X(122)+X(133), 4*X(140)-X(3184), X(355)+2*X(11732), 5*X(631)+X(10152), X(1294)+5*X(3091), 5*X(1656)-2*X(6716), 5*X(1656)+X(10745), 7*X(3526)-X(23240), 7*X(3851)-X(22337), 11*X(5056)+X(34186), 13*X(5067)-X(5667), 13*X(5068)-X(34549), 5*X(5071)+X(10714), 5*X(5818)+X(10701), 2*X(6716)+X(10745)

See Kadir Altintas and César Lozada, Euclid 549 .

X(36520) lies on these lines: {2, 2777}, {3, 33892}, {4, 34842}, {5, 122}, {107, 3090}, {140, 3184}, {355, 11732}, {631, 10152}, {1294, 3091}, {1656, 6716}, {2797, 23514}, {2803, 23513}, {2816, 10172}, {3324, 3614}, {3526, 23240}, {3851, 22337}, {5055, 9530}, {5056, 34186}, {5067, 5667}, {5068, 34549}, {5071, 10714}, {5818, 10701}, {7158, 7173}, {7395, 14703}, {9033, 23515}

X(36520) = complement of X(23239)
X(36520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 122, 133), (1656, 10745, 6716)

leftri

Points on the Steiner Midellipse: X(36521)-X(36525)

rightri

This preamble and centers X(36521)-X(36525) were contributed by Clark Kimberling and Peter Moses, January 21, 2020.

The ellipse midway between the Steiner inellipse (SIE) and the Steiner circumellipse (SCE) is here named the Steiner Midellipse (SME). Specifically, for any X on SCE, let U = segment GX ^ CIE
Let M = midpoint of XU
Then SME is the locus of M as X goes around SCE MSE is, like SIE and SCE, a permutation ellipse; ie., if P = pqr = p : q : r is on SME, then all six permutations, pqr, qrp, rpq, prq, qpr, rqp are on SME.

Let G = X(2) = centroid of ABC. If P is on SCE then the point given by the combo G + 3 P is on SME; likewise, if P is on SIE, then G - 3 P is on SIE. An equation for SME follows:

7 (x^2 + y^2 + z^2) - 34 (y z + z x + x y) = 0.

X(36521) =  MIDPOINT OF X(99) AND X(2482)

Barycentrics    10*a^4 - 10*a^2*b^2 + b^4 - 10*a^2*c^2 + 8*b^2*c^2 + c^4 : :
X(36521) = X[2] + 3 X[99], 5 X[2] - 3 X[115], 11 X[2] - 3 X[148], 2 X[2] - 3 X[620], 7 X[2] - 3 X[671], X[2] - 3 X[2482], 3 X[98] - 7 X[15698], 5 X[99] + X[115], 11 X[99] + X[148], 2 X[99] + X[620], 7 X[99] + X[671], 4 X[99] + X[5461], 7 X[99] + 2 X[6722], 5 X[99] - X[8591], 19 X[99] + X[8596], 13 X[99] + 3 X[9166], 7 X[99] + 3 X[9167], 3 X[114] - X[3830], 11 X[115] - 5 X[148], 2 X[115] - 5 X[620], 7 X[115] - 5 X[671], X[115] - 5 X[2482], 4 X[115] - 5 X[5461], 7 X[115] - 10 X[6722]

X(36521) lies on the Steiner midellipse and these lines: {2, 99}, {22, 34013}, {39, 35954}, {98, 15698}, {114, 3830}, {147, 15697}, {376, 14981}, {381, 10992}, {524, 14148}, {538, 27088}, {542, 8703}, {549, 11623}, {599, 14928}, {626, 5077}, {690, 10190}, {754, 8598}, {1569, 11055}, {1975, 34506}, {2782, 12100}, {2794, 3534}, {2795, 15673}, {2796, 11725}, {3524, 23235}, {3845, 14160}, {3849, 6390}, {5017, 14645}, {5026, 8584}, {5463, 36329}, {5464, 35751}, {5475, 11164}, {5976, 14711}, {6033, 15685}, {6036, 11812}, {6054, 11001}, {6055, 13188}, {6337, 7775}, {6781, 7840}, {7484, 13233}, {7751, 35287}, {7756, 7870}, {7764, 33007}, {7781, 32985}, {7782, 7810}, {7798, 9741}, {7799, 9855}, {7801, 7830}, {7816, 31406}, {7829, 8369}, {7833, 7863}, {7838, 34511}, {7888, 33192}, {7902, 33197}, {8182, 32817}, {8358, 19662}, {8592, 11057}, {9114, 36330}, {9116, 35752}, {9880, 15561}, {10304, 10991}, {11147, 21843}, {11149, 17004}, {11539, 20398}, {11632, 15701}, {11693, 31854}, {12042, 15711}, {12117, 15682}, {12188, 15716}, {12243, 15719}, {12355, 23514}, {13172, 23234}, {15301, 32459}, {16508, 33894}, {19708, 21166}, {22110, 32479}, {22566, 33699}

X(36521) = midpoint of X(i) and X(j) for these {i,j}: {2, 15300}, {99, 2482}, {115, 8591}, {376, 14981}, {381, 10992}, {599, 14928}, {6055, 13188}, {6781, 7840}
X(36521) = reflection of X(i) in X(j) for these {i,j}: {115, 22247}, {381, 20399}, {620, 2482}, {671, 6722}, {2482, 35022}, {5461, 620}, {11623, 549}
X(36521) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 15300}, {99, 35022, 620}, {671, 9167, 6722}, {2482, 8591, 22247}, {2482, 15300, 2}

X(36522) =  MIDPOINT OF X(190) AND X(4370)

Barycentrics    10*a^2 - 10*a*b + b^2 - 10*a*c + 8*b*c + c^2 : :
X(36522) = X[2] + 3 X[190], 7 X[2] - 3 X[903], 5 X[2] - 3 X[1086], X[2] - 3 X[4370], 17 X[2] - 3 X[4409], 2 X[2] - 3 X[4422], 11 X[2] - 3 X[4440], 7 X[2] - 15 X[4473], 7 X[190] + X[903], 5 X[190] + X[1086], 17 X[190] + X[4409], 2 X[190] + X[4422], 11 X[190] + X[4440]]

X(36522) lies on the Steiner midellipse and these lines: {2, 45}, {44, 28309}, {524, 4908}, {528, 4669}, {537, 15569}, {900, 10196}, {2325, 4715}, {3161, 17313}, {3751, 8584}, {3830, 24828}, {3845, 29243}, {3929, 16561}, {4395, 28301}, {4437, 15533}, {4480, 31138}, {5845, 22165}, {15693, 24844}, {15698, 24813}, {17228, 17333}, {17264, 28333}, {17270, 17281}, {17330, 17336}, {17334, 17342}

X(36522) = midpoint of X(i) and X(j) for these {i,j}: {190, 4370}, {1086, 17487}, {4480, 31138}
X(36522) = reflection of X(4422) in X (4370)

X(36523) =  MIDPOINT OF X(115) AND X(671)

Barycentrics    2*a^4 - 2*a^2*b^2 - 7*b^4 - 2*a^2*c^2 + 16*b^2*c^2 - 7*c^4 : :
X(36523) = 7 X[2] - 3 X[99], X[2] - 3 X[115], 5 X[2] + 3 X[148], 4 X[2] - 3 X[620], X[2] + 3 X[671], 5 X[2] - 3 X[2482], 2 X[2] - 3 X[5461], 5 X[2] - 6 X[6722], 11 X[2] - 3 X[8591], 13 X[2] + 3 X[8596], 3 X[13] + X[36330], 3 X[14] + X[35752], 3 X[98] + X[15682], X[99] - 7 X[115], 5 X[99] + 7 X[148], 4 X[99] - 7 X[620], X[99] + 7 X[671], 5 X[99] - 7 X[2482], 2 X[99] - 7 X[5461], 3 X[114] - 5 X[19709], 5 X[115] + X[148], 4 X[115] - X[620], 5 X[115] - X[2482], 5 X[115] - 2 X[6722], 11 X[115] - X[8591], 13 X[115] + X[8596], 5 X[115] - 3 X[9166], 11 X[115] - 3 X[9167], 11 X[115] - 5 X[14061], 7 X[115] - 3 X[14971], 9 X[115] - X[15300], 4 X[148] + 5 X[620], X[148] - 5 X[671], 2 X[148] + 5 X[5461], X[148] + 2 X[6722], 11 X[148] + 5 X[8591], 13 X[148] - 5 X[8596], X[620] + 4 X[671], 5 X[620] - 4 X[2482], 5 X[620] - 8 X[6722], 11 X[620] - 4 X[8591], 13 X[620] + 4 X[8596], 5 X[671] + X[2482], 2 X[671] + X[5461], 5 X[671] + 2 X[6722], 11 X[671] + X[8591], 13 X[671] - X[8596]

X(36523) lies on the Steiner midellipse and these lines: {2 99}, {13, 31696}, {14, 31695}, {25, 13233}, {30, 11623}, {98, 15682}, {114, 19709}, {230, 32479}, {524, 32457}, {530, 11543}, {531, 11542}, {542, 1353}, {549, 20398}, {598, 5355}, {626, 34505}, {690, 10278}, {754, 8352}, {2782, 5066}, {2794, 3830}, {3363, 5254}, {3534, 6055}, {3543, 10991}, {3545, 14981}, {3793, 3849}, {4669, 11599}, {5054, 10992}, {5055, 20399}, {5071, 23235}, {5077, 7830}, {5309, 11317}, {5469, 35751}, {5470, 36329}, {5969, 19662}, {6034, 18800}, {6036, 12100}, {6680, 35954}, {6781, 8859}, {7748, 34506}, {7764, 33006}, {7765, 33013}, {7781, 32984}, {7810, 7910}, {7829, 8370}, {7841, 7854}, {8029, 18007}, {8597, 14568}, {8703, 23698}, {8724, 23514}, {9183, 10190}, {10150, 15301}, {11001, 14651}, {11054, 14041}, {11163, 18424}, {11602, 36366}, {11603, 36368}, {11606, 12156}, {11646, 15534}, {12042, 19710}, {12117, 15698}, {12243, 14639}, {12355, 15693}, {13172, 15719}, {13881, 34504}, {14148, 22110}, {14645, 15533}, {15048, 20112}, {15697, 34473}, {15713, 33813}, {16001, 32909}, {16002, 32907}, {19711, 26614}, {22489, 22578}, {22490, 22577}, {22515, 33699}, {25154, 36382}, {25164, 36383}, {31693, 36251}, {31694, 36252}

X(36523) = complement of X(15300)
X(36523) = midpoint of X(i) and X(j) for these {i,j}: {13, 31696}, {14, 31695}, {115, 671}, {148, 2482}, {3543, 10991}, {6055, 6321}, {9880, 11632}, {16001, 32909}, {16002, 32907}
X(36523) = reflection of X(i) in X(j) for these {i,j}: {99, 22247}, {549, 20398}, {620, 5461}, {2482, 6722}, {5461, 115}, {8591, 35022}, {14148, 22110}
X(36523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 14971, 22247}, {115, 148, 6722}, {115, 2482, 9166}, {148, 9166, 2482}, {671, 9166, 148}, {2482, 9166, 6722}, {6722, 9166, 5461}, {8591, 9167, 35022}, {8591, 14061, 9167}

X(36524) =  MIDPOINT OF X(668) AND X(13466)

Barycentrics    a^2*b^2 + 8*a^2*b*c - 10*a*b^2*c + a^2*c^2 - 10*a*b*c^2 + 10*b^2*c^2 : :
X(36524) = X[2] + 3 X[668], 5 X[2] - 3 X[1015], 7 X[2] - 3 X[3227], 11 X[2] - 3 X[9263], X[2] - 3 X[13466], 2 X[2] - 3 X[27076], 19 X[2] - 15 X[27195], 13 X[2] + 3 X[31298], 5 X[668] + X[1015], 7 X[668] + X[3227], 11 X[668] + X[9263], 2 X[668] + X[27076], 19 X[668] + 5 X[27195], 13 X[668] - X[31298], 7 X[1015] - 5 X[3227], 11 X[1015] - 5 X[9263], X[1015] - 5 X[13466], 2 X[1015] - 5 X[27076], 19 X[1015] - 25 X[27195]

X(36524) lies on the Steiner midellipse and these lines: {2, 668}, {537, 4745}, {2810, 22165}, {4482, 21781}, {4669, 14839}, {11055, 20671}

X(36524) = midpoint of X(668) and X(13466)
X(36524) = reflection of X(27076) in X(13466)

X(36525) =  MIDPOINT OF X(903) AND X(1086)

Barycentrics    2*a^2 - 2*a*b - 7*b^2 - 2*a*c + 16*b*c - 7*c^2 : :
X(36525) = 7 X[2] - 3 X[190], X[2] + 3 X[903], X[2] - 3 X[1086], 5 X[2] - 3 X[4370], 11 X[2] + 3 X[4409], 4 X[2] - 3 X[4422], 5 X[2] + 3 X[4440], 23 X[2] - 15 X[4473], X[190] + 7 X[903], X[190] - 7 X[1086], 5 X[190] - 7 X[4370], 11 X[190] + 7 X[4409], 4 X[190] - 7 X[4422], 5 X[190] + 7 X[4440], 23 X[190] - 35 X[4473], 5 X[903] + X[4370], 11 X[903] - X[4409], 4 X[903] + X[4422], 5 X[903] - X[4440], 5 X[1086] - X[4370], 11 X[1086] + X[4409], 4 X[1086] - X[4422], 5 X[1086] + X[4440], 23 X[1086] - 5 X[4473], 5 X[1266] + X[4727], X[3534] + 3 X[24833], 11 X[4370] + 5 X[4409], 4 X[4370] - 5 X[4422]

X(36525) lies on the Steiner midellipse and these lines: {2, 45}, {519, 7238}, {528, 5542}, {537, 4745}, {900, 21204}, {1266, 4727}, {2796, 11725}, {3416, 4677}, {3534, 24833}, {3629, 4902}, {3834, 28301}, {4395, 4700}, {4862, 17332}, {4908, 28297}, {5845, 8584}, {7263, 17274}, {8028, 24131}, {8703, 29243}, {15682, 24813}, {15719, 24817}, {17329, 17330}, {19709, 24828}, {24827, 33699}

X(36525) = midpoint of X(i) and X(j) for these {i,j}: {903, 1086}, {1266, 31138}, {4370, 4440}, {4409, 17487}
X(36525) = {X(1086),X(4409)}-harmonic conjugate of X(27191)

leftri

Homothetors involving triangles T(k): X(36526)-X(36587)

rightri

This preamble and centers X(36526)-X(36587) were contributed by Clark Kimberling and Peter Moses, January 22, 2020.

In the preamble just before X(36436), the triangle T(k), for k any real number, is defined as the central triangle with A-vertex 1 : k : k. This definition includes triangles for which k is a function symmetric in a,b,c and homogeneous of degree 0, such as (bc+ca+ab)/(a^2+b^2+c^2) and abc/(a^3+b^3+c^3).

See also the preamble just before X(36473).

For barycentrics and references for the various triangles mentioned in this section, see Index of Triangles Referenced in ETC, by .

X(36526) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND EULER

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

X(36526) lies on these lines: {2, 3}, {11, 36487}, {12, 36488}, {45, 29243}, {355, 36479}, {485, 36492}, {486, 36491}, {515, 24331}, {516, 25352}, {517, 4517}, {576, 5733}, {946, 36480}, {2548, 4253}, {2550, 15507}, {3017, 5319}, {3767, 4251}, {3818, 5816}, {4258, 13881}, {4363, 24828}, {4419, 24833}, {4454, 24844}, {5030, 31415}, {5587, 29659}, {7989, 36478}, {8227, 29660}, {9612, 36482}, {10246, 15251}, {10895, 36493}, {10896, 36481}, {17316, 29331}, {24220, 31670}

X(36527) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND REFLECTION OF ABC IN X(3)EULER

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

X(36527) lies on these lines: {2, 3}, {999, 36487}, {1482, 36480}, {1506, 33863}, {3017, 7755}, {3295, 36488}, {4363, 24844}, {4364, 24833}, {4472, 24828}, {5790, 29659}, {7746, 18755}, {7951, 17798}, {9441, 11231}, {9654, 36493}, {9669, 36481}, {9956, 36478}, {10246, 24331}, {12645, 36479}, {16826, 29331}, {20430, 25384}, {25352, 26446}

X(36528) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND TANGENTIAL TRIANGLE of 1ST CIRCUMPERP

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

X(36528) lies on these lines: {2, 11}, {3, 36480}, {8, 36476}, {9, 3550}, {142, 29656}, {171, 518}, {197, 20834}, {200, 36483}, {474, 29660}, {896, 5220}, {940, 3938}, {1259, 36486}, {1575, 16503}, {3295, 24331}, {3750, 3752}, {3913, 36479}, {4038, 15570}, {4068, 23944}, {4363, 24820}, {5248, 25352}, {5687, 29659}, {5853, 29655}, {6600, 29670}, {8424, 34247}, {10310, 36489}, {11248, 36477}, {11500, 36474}, {11501, 36493}, {11509, 36487}, {15624, 25384}, {23853, 29652}

X(36529) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND TANGENTIAL TRIANGLE of 2ND CIRCUMPERP

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

X(36529) lies on these lines: {2, 12}, {3, 36480}, {8, 36475}, {405, 29660}, {956, 29659}, {993, 22780}, {999, 24331}, {1001, 4364}, {1104, 29820}, {1107, 21008}, {1959, 5289}, {3428, 36489}, {3938, 19765}, {4363, 24826}, {5258, 36478}, {10966, 36488}, {11249, 36477}, {12114, 36474}, {12513, 36479}, {20834, 22654}, {22760, 36481}

X(36530) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND REFLECTION OF ABC IN X(5)

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

X(36530) lies on these lines: {1, 36481}, {2, 3}, {40, 36478}, {355, 36480}, {517, 29659}, {572, 3818}, {573, 19130}, {942, 36482}, {991, 24206}, {1385, 29660}, {1478, 36487}, {1479, 36488}, {1482, 36479}, {2223, 7951}, {3017, 7772}, {3311, 36492}, {3312, 36491}, {3661, 29331}, {4363, 24833}, {4364, 24828}, {4419, 24844}, {5709, 36483}, {5722, 26012}, {5886, 24331}, {10175, 25352}, {10525, 36485}, {10526, 36486}, {11248, 36475}, {11249, 36476}, {12618, 20430}, {15310, 32784}, {17305, 24813}, {17369, 29243}, {26446, 28885}

X(36531) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND AQUILLA

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

X(36531) lies on these lines: {1, 2}, {37, 4693}, {40, 36477}, {57, 36487}, {165, 36489}, {274, 4692}, {292, 1573}, {320, 4407}, {752, 17256}, {984, 4363}, {1107, 25068}, {1213, 33076}, {1478, 25353}, {1697, 36488}, {1757, 3758}, {2223, 5251}, {2550, 24319}, {3294, 3496}, {3842, 4432}, {3996, 10180}, {4026, 25358}, {4085, 17322}, {4360, 4732}, {4364, 24715}, {4389, 24693}, {4414, 24344}, {4424, 25368}, {4660, 17248}, {4674, 27922}, {4687, 32941}, {4698, 16484}, {4702, 4755}, {4709, 17319}, {4714, 17143}, {4737, 31997}, {4761, 27929}, {4981, 32913}, {5241, 17722}, {5258, 36476}, {5290, 36482}, {5541, 25427}, {5691, 36474}, {5692, 20715}, {6536, 33110}, {6684, 36484}, {7989, 36473}, {8185, 20834}, {9578, 36493}, {9581, 36481}, {9708, 21010}, {15485, 20179}, {17237, 31151}, {17254, 24692}, {17303, 33165}, {17305, 25351}, {17716, 19732}, {21027, 33155}, {24325, 24841}, {24366, 25499}, {24441, 24452}, {27798, 32926}, {27949, 31323}, {32092, 33941}, {36483, 36486}

X(36532) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 1ST AURIGA

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

X(36532) lies on these lines: {2, 5597}, {55, 36480}, {4363, 24823}, {8190, 20834}, {8197, 29659}, {9834, 36474}, {11252, 36477}, {11366, 24331}, {11822, 36489}, {11869, 36493}, {11871, 36481}, {11873, 36488}, {12454, 36479}, {18955, 36487}

X(36533) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 2ND AURIGA

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

X(36533) lies on these lines: {2, 5598}, {55, 36480}, {4363, 24824}, {8191, 20834}, {8204, 29659}, {9835, 36474}, {11253, 36477}, {11367, 24331}, {11823, 36489}, {11870, 36493}, {11872, 36481}, {11874, 36488}, {12455, 36479}, {18956, 36487}

X(36534) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 5TH MIXTILINEAR

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

X(36534) lies on these lines: {1, 2}, {192, 4693}, {194, 28598}, {348, 3476}, {355, 36473}, {390, 9791}, {517, 36489}, {518, 3758}, {528, 4389}, {752, 4741}, {894, 16496}, {944, 29081}, {984, 4432}, {1107, 25082}, {1319, 31225}, {1320, 27922}, {1482, 36477}, {1621, 20760}, {2098, 36488}, {2099, 3212}, {2320, 5773}, {3161, 16517}, {3210, 32945}, {3242, 4363}, {3246, 17335}, {3685, 7174}, {3902, 17144}, {3913, 36475}, {3923, 24821}, {3996, 4734}, {4085, 17383}, {4392, 24344}, {4429, 26150}, {4660, 17236}, {4664, 4702}, {4720, 33296}, {4863, 19786}, {5749, 16973}, {6762, 36483}, {7962, 24460}, {8192, 20834}, {9041, 17369}, {9997, 24282}, {10106, 36482}, {10912, 36485}, {10944, 36493}, {10950, 36481}, {12245, 36484}, {12513, 36476}, {12635, 36486}, {14996, 17145}, {16484, 27268}, {16491, 17121}, {17119, 32922}, {17238, 33076}, {17358, 33165}, {17360, 28538}, {17490, 17598}, {17725, 21242}, {17765, 32784}, {19065, 36491}, {19066, 36492}, {19822, 30614}, {21216, 32095}, {21283, 33155}, {22791, 36490}, {24552, 32937}, {27538, 32942}

X(36535) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND INNER GREBE

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

X(36535) lies on these lines: {2, 6}, {1161, 36477}, {3641, 36480}, {4363, 24831}, {5595, 20834}, {5689, 29659}, {5871, 36474}, {10514, 36473}, {10517, 36484}, {10923, 36493}, {10925, 36481}, {10927, 36488}, {11370, 24331}, {11824, 36489}, {12627, 36479}, {18959, 36487}

X(36536) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND OUTER GREBE

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

X(36536) lies on these lines: {2, 6}, {1160, 36477}, {3640, 36480}, {4363, 24832}, {5594, 20834}, {5688, 29659}, {5870, 36474}, {10515, 36473}, {10518, 36484}, {10924, 36493}, {10926, 36481}, {10928, 36488}, {11371, 24331}, {11825, 36489}, {12628, 36479}, {18960, 36487}

X(36537) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 5TH BROCARD

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

X(36537) lies on these lines: {1, 20924}, {2, 32}, {58, 7768}, {76, 33954}, {3098, 10446}, {3961, 33941}, {4251, 7859}, {4363, 24825}, {5180, 33867}, {9821, 36477}, {9857, 29659}, {9873, 36474}, {9941, 36480}, {9997, 24282}, {10356, 36473}, {10357, 36484}, {10828, 20834}, {10873, 36493}, {10874, 36481}, {10877, 36488}, {11368, 24331}, {12495, 36479}, {18957, 36487}

X(36538) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND ORTHIC OF INTOUCH

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

X(36538) lies on these lines: {1, 20731}, {2, 7}, {55, 24283}, {56, 24331}, {65, 36480}, {85, 1429}, {241, 24455}, {354, 36488}, {388, 29659}, {942, 36477}, {1086, 24315}, {1111, 24268}, {1215, 4413}, {1402, 29651}, {1403, 29670}, {1441, 7225}, {1758, 29675}, {1836, 36481}, {3485, 29660}, {3487, 36484}, {3676, 4375}, {3689, 32920}, {3923, 30982}, {3980, 8850}, {4032, 24357}, {4292, 36474}, {4393, 17090}, {4657, 16888}, {5018, 29820}, {5218, 17596}, {5228, 16609}, {5290, 36478}, {5722, 36490}, {6654, 34018}, {7182, 27916}, {7185, 17397}, {9612, 36473}, {10106, 36479}, {10404, 36493}, {10473, 29652}, {12588, 29673}, {14439, 32933}, {16603, 30617}, {17095, 24803}, {17290, 24323}, {17325, 25363}, {17625, 36485}, {20880, 25940}

X(36539) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND INTOUCH OF ORTHIC

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

X(36539) lies on these lines: {2, 3}, {33, 36488}, {34, 36487}, {1829, 36480}, {1892, 36482}, {2201, 5101}, {4363, 24814}, {5090, 29659}, {5130, 36486}, {11363, 24331}, {11392, 36493}, {11393, 36481}, {12135, 36479}

X(36540) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND TANGENTIAL OF EXCENTRAL

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

X(36540) lies on these lines: {2, 7}, {38, 9451}, {40, 36480}, {84, 36474}, {200, 36475}, {238, 3742}, {261, 18164}, {846, 8299}, {984, 1376}, {1764, 25368}, {2114, 17074}, {2550, 24283}, {3158, 3961}, {3220, 20834}, {3333, 24331}, {3666, 3684}, {4364, 16560}, {4384, 33944}, {4603, 6654}, {5709, 36477}, {6211, 10164}, {6626, 18206}, {6762, 36479}, {7290, 29820}, {21976, 25083}, {24477, 29655}, {26934, 29656}, {29660, 31435}, {30223, 36481}

X(36541) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND INNER YFF

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

X(36541) lies on these lines: {1, 2}, {3, 36487}, {5, 36481}, {35, 36489}, {45, 24846}, {55, 36477}, {484, 17753}, {495, 21010}, {1478, 2223}, {3295, 36488}, {4363, 24845}, {5218, 36484}, {6284, 36490}, {7951, 36473}, {8616, 27287}, {10037, 20834}, {13407, 36482}, {13905, 36492}, {13963, 36491}, {20358, 26446}, {24464, 33144}

X(36542) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND OUTER YFF

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

X(36542) lies on these lines: {1, 2}, {3, 36488}, {5, 36493}, {36, 36489}, {45, 24845}, {56, 36477}, {496, 36481}, {999, 36487}, {1479, 36474}, {4363, 24846}, {4660, 27305}, {5886, 20358}, {7288, 36484}, {7354, 36490}, {7741, 36473}, {8624, 9599}, {10046, 20834}, {12047, 36482}, {13904, 36492}, {13962, 36491}, {15325, 21010}, {17753, 18393}, {27339, 33106}

X(36543) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND ANTI-1ST EULER

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

X(36543) lies on these lines: {2, 3}, {944, 36480}, {1056, 21010}, {3085, 36493}, {3086, 36481}, {3487, 36482}, {4293, 36487}, {4294, 36488}, {4363, 24817}, {4364, 24813}, {5587, 25352}, {5603, 24331}, {5657, 18788}, {6684, 36478}, {12245, 36479}, {18446, 25353}, {25384, 30273}

X(36544) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND INNER JOHNSON

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

X(36544) lies on these lines: {2, 11}, {355, 36480}, {1575, 33141}, {3752, 24217}, {4363, 24834}, {4386, 33106}, {10522, 36486}, {10525, 36477}, {10829, 20834}, {10912, 36479}, {10914, 29659}, {10944, 36493}, {11373, 24331}, {11826, 36489}, {12114, 36474}, {17073, 29668}, {17614, 29660}, {17625, 36482}, {18961, 36487}, {24388, 29655}, {24837, 26659}

X(36545) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND OUTER JOHNSON

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

X(36545) lies on these lines: {2, 12}, {10, 12933}, {72, 29659}, {355, 36480}, {1104, 29675}, {1259, 36475}, {4363, 24835}, {10522, 36485}, {10526, 36477}, {10830, 20834}, {10950, 36481}, {10953, 36488}, {11374, 24331}, {11500, 36474}, {11827, 36489}, {12635, 36479}

X(36546) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND INNER YFF TANGENTS

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

X(36546) lies on these lines: {1, 2}, {119, 36473}, {3749, 27286}, {3913, 36485}, {4363, 24847}, {10679, 36477}, {10834, 20834}, {10956, 36493}, {10958, 36481}, {10965, 36488}, {11248, 36489}, {11509, 36487}, {12115, 36474}

X(36547) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND OUTER YFF TANGENTS

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

X(36547) lies on these lines: {1, 2}, {4363, 24848}, {10680, 36477}, {10835, 20834}, {10957, 36493}, {10959, 36481}, {10966, 36488}, {11249, 36489}, {12053, 25353}, {12116, 36474}, {12513, 36486}, {16496, 27254}, {18967, 36487}, {26470, 36473}

X(36548) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND ANTI-5TH BROCARD

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

X(36548) lies on these lines: {2, 32}, {58, 7878}, {76, 4251}, {182, 36489}, {3398, 36477}, {4363, 24815}, {10358, 36473}, {10359, 36484}, {10790, 20834}, {10791, 29659}, {10797, 36493}, {10798, 36481}, {10799, 36488}, {11364, 24331}, {12110, 36474}, {12194, 36480}, {12195, 36479}, {12835, 36487}, {14880, 36490}, {24333, 33940}

X(36549) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 3RD TRI-SQUARES CENTRAL

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

X(36549) lies on these lines: {2, 6}, {485, 36474}, {4363, 24842}, {6221, 36490}, {8981, 36477}, {8983, 36480}, {9540, 36489}, {13883, 24331}, {13889, 20834}, {13893, 29659}, {13897, 36493}, {13898, 36481}, {13901, 36488}, {13911, 36479}, {18965, 36487}

X(36550) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 4TH TRI-SQUARES CENTRAL

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

X(36550) lies on these lines: {2, 6}, {486, 36474}, {4363, 24843}, {6398, 36490}, {13935, 36489}, {13936, 24331}, {13943, 20834}, {13947, 29659}, {13954, 36493}, {13955, 36481}, {13958, 36488}, {13966, 36477}, {13971, 36480}, {13973, 36479}, {18966, 36487}

X(36551) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND EHRMANN MID-TRIANGLE

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

X(36551) lies on these lines: {2, 3}, {1478, 36481}, {1479, 36493}, {3583, 36488}, {3585, 36487}, {4363, 24827}, {5587, 29365}, {5722, 36482}, {9955, 24331}, {12699, 29659}, {17294, 29331}, {18480, 36480}, {18481, 29660}, {22791, 36479}

X(36552) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND ANTI-INNER GREBE

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

X(36552) lies on these lines: {2, 6}, {45, 24819}, {372, 36489}, {486, 36473}, {1588, 36474}, {3312, 36477}, {4363, 24818}, {6351, 16503}, {13936, 29659}, {13971, 29660}, {17354, 24843}, {18991, 24331}, {18992, 36480}, {18995, 36487}, {19005, 20834}, {19027, 36493}, {19029, 36481}, {19037, 36488}, {19065, 36479}

X(36553) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND ANTI-OUTER GREBE

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

X(36553) lies on these lines: {2, 6}, {45, 24818}, {371, 36489}, {485, 36473}, {1587, 36474}, {3311, 36477}, {4363, 24819}, {6352, 16503}, {8983, 29660}, {13883, 29659}, {17354, 24842}, {18991, 36480}, {18992, 24331}, {18996, 36487}, {19006, 20834}, {19028, 36493}, {19030, 36481}, {19038, 36488}, {19066, 36479}

X(36554) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND GEMINI 4

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

X(36554) lies on these lines: {1, 742}, {2, 31}, {6, 29659}, {86, 24331}, {190, 984}, {192, 4693}, {1001, 4265}, {1423, 36487}, {2345, 33165}, {3883, 33682}, {4649, 36479}, {4660, 20179}, {6210, 36477}, {15485, 17306}, {16468, 36478}, {16484, 17321}, {17353, 25352}, {20172, 24715}, {29652, 35623}

X(36555) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 1st KENMOTU-FREE-VERTICES TRIANGLE

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

X(36555) lies on these lines: {2, 372}, {6, 36556}, {371, 36477}, {5418, 36484}, {6200, 36489}, {20834, 35776}, {24331, 35762}, {29659, 35788}, {35641, 36480}, {35768, 36487}, {35800, 36493}, {35802, 36481}, {35808, 36488}, {35820, 36474}, {35821, 36490}, {35842, 36479}

X(36556) =  HOMOTHETOR OF THESE TRIANGLES: T((bc+ca+ab)/(a^2+b^2+c^2)) AND 2ND KENMOTU-FREE-VERTICES TRIANGLE

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

X(36556) lies on these lines: {2, 371}, {6, 36555}, {372, 36477}, {5420, 36484}, {6396, 36489}, {8960, 36492}, {20834, 35777}, {24331, 35763}, {29659, 35789}, {35642, 36480}, {35769, 36487}, {35801, 36493}, {35803, 36481}, {35809, 36488}, {35820, 36490}, {35821, 36474}, {35843, 36479}

X(36557) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND EULER

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

X(36557) lies on these lines: {2, 3}, {11, 36508}, {12, 36509}, {355, 36500}, {388, 17597}, {515, 23675}, {946, 976}, {1478, 3953}, {1482, 20035}, {1699, 5293}, {1837, 28109}, {5101, 23661}, {5475, 22442}, {6256, 28074}, {7989, 36499}, {8227, 36505}, {9612, 36503}, {10356, 36502}, {10358, 36511}, {10895, 36513}, {10896, 36501}, {12667, 28080}

X(36558) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND REFLECTION OF ABC IN X(3)

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

X(36558) lies on these lines: {2, 3}, {517, 5293}, {976, 1482}, {999, 36508}, {1324, 25639}, {1376, 23844}, {1385, 28083}, {3295, 36509}, {3814, 34868}, {5255, 28389}, {5706, 9567}, {6796, 31394}, {9654, 36513}, {9669, 36501}, {9956, 36499}, {10246, 28082}, {12645, 36500}, {16203, 28074}, {24295, 24309}

X(36559) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND TANGENTIAL TRIANGLE OF 1ST CIRCUMPERP

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

X(36559) lies on these lines: {2, 11}, {3, 38}, {8, 36498}, {22, 34247}, {35, 228}, {200, 36504}, {474, 36505}, {756, 20834}, {984, 16064}, {1030, 21817}, {1259, 36507}, {1739, 3746}, {2205, 17735}, {2223, 5314}, {3295, 4642}, {3913, 36500}, {3961, 20999}, {5096, 16687}, {5347, 20990}, {10310, 36510}, {11248, 19548}, {11500, 36496}, {11501, 36513}, {11509, 36508}, {12513, 20035}, {17165, 24820}, {18755, 21814}, {20068, 24826}, {24169, 25440}

X(36560) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND TANGENTIAL TRIANGLE OF 2ND CIRCUMPERP

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

X(36560) lies on these lines: {1, 1283}, {2, 12}, {3, 38}, {8, 36497}, {36, 5293}, {405, 32775}, {474, 33119}, {999, 28082}, {1324, 3670}, {2933, 17595}, {3428, 36510}, {3913, 20035}, {5258, 36499}, {10966, 36509}, {11101, 33148}, {11249, 19548}, {11399, 28076}, {12114, 36496}, {12513, 36500}, {13733, 33144}, {14455, 24159}, {19529, 33123}, {22760, 36501}, {22767, 28077}, {26437, 28109}

X(36561) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND REFLECTION OF ABC ABOUT X(5)

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

X(36561) lies on these lines: {1, 36501}, {2, 3}, {12, 17783}, {40, 36499}, {355, 976}, {942, 36503}, {1385, 36505}, {1478, 36508}, {1479, 36509}, {1482, 36500}, {3398, 36511}, {3670, 3944}, {3847, 17290}, {3923, 25639}, {4812, 20430}, {5293, 5587}, {5709, 36504}, {5886, 28082}, {7173, 17595}, {7683, 10441}, {9821, 36502}, {10525, 36506}, {10526, 36507}, {11248, 36497}, {11249, 36498}, {12047, 28109}, {12645, 20035}, {20805, 23542}

X(36562) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND ECENTRAL OF TANGENTIAL

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

X(36562) lies on these lines: {2, 3}, {976, 9798}, {3011, 23850}, {5285, 36504}, {5293, 8185}, {10831, 36513}, {10832, 36501}, {10833, 36509}, {11365, 28082}, {12410, 36500}, {18954, 36508}, {20999, 26228}

X(36563) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 1ST AURIGA

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

X(36563) lies on these lines: {2, 5597}, {55, 976}, {5293, 8186}, {9834, 36496}, {11252, 19548}, {11366, 28082}, {11822, 36510}, {11869, 36513}, {11871, 36501}, {11873, 36509}, {12454, 36500}, {12455, 20035}, {18955, 36508}

X(36564) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 2ND AURIGA

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

X(36564) lies on these lines: {2, 5598}, {55, 976}, {5293, 8187}, {9835, 36496}, {11253, 19548}, {11367, 28082}, {11823, 36510}, {11870, 36513}, {11872, 36501}, {11874, 36509}, {12454, 20035}, {12455, 36500}, {18956, 36508}

X(36565) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 5TH MIXTILINEAR

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

X(36565) lies on these lines: {1, 2}, {3, 4392}, {4, 33153}, {21, 7226}, {38, 4189}, {72, 17127}, {244, 17572}, {335, 16919}, {355, 36495}, {377, 33148}, {474, 9335}, {517, 36510}, {756, 16859}, {944, 36496}, {982, 4188}, {984, 16865}, {1043, 3891}, {1046, 30652}, {1058, 1807}, {1104, 3681}, {1468, 4430}, {1482, 19548}, {1731, 3247}, {2098, 36509}, {2099, 36508}, {2475, 33144}, {2650, 17716}, {2975, 3242}, {3052, 11684}, {3145, 3295}, {3189, 19785}, {3315, 25524}, {3487, 33112}, {3677, 4855}, {3744, 3869}, {3772, 5178}, {3868, 5266}, {3896, 19805}, {3913, 36497}, {3952, 17697}, {3984, 7290}, {4190, 4310}, {4195, 17165}, {4294, 33100}, {4339, 5905}, {4661, 5247}, {5015, 25958}, {5016, 33126}, {5047, 9330}, {5141, 17719}, {5253, 17597}, {5269, 11520}, {5300, 25959}, {5710, 34195}, {5880, 26729}, {6198, 28076}, {6284, 33151}, {6762, 36504}, {7270, 33122}, {7373, 16422}, {10106, 36503}, {10912, 36506}, {10944, 36513}, {10950, 36501}, {11011, 28109}, {11115, 24349}, {11319, 32937}, {12195, 36511}, {12495, 36502}, {12513, 36498}, {12635, 36507}, {14997, 16498}, {16787, 33299}, {17526, 33166}, {17539, 20068}, {17725, 21935}, {20066, 24248}, {22791, 36512}, {24549, 31130}, {33134, 34937}

X(36566) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND INNER GREBE

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

X(36566) lies on these lines: {2, 6}, {976, 3641}, {1161, 19548}, {5293, 5589}, {5871, 36496}, {10514, 36495}, {10923, 36513}, {10925, 36501}, {10927, 36509}, {11370, 28082}, {11824, 36510}, {12627, 36500}, {18959, 36508}

X(36567) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND OUTER GREBE

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

X(36567) lies on these lines: {2, 6}, {976, 3640}, {1160, 19548}, {5293, 5588}, {5870, 36496}, {10515, 36495}, {10924, 36513}, {10926, 36501}, {10928, 36509}, {11371, 28082}, {11825, 36510}, {12628, 36500}, {18960, 36508}

X(36568) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND OUTER GARCIA

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

X(36568) lies on these lines: {1, 2}, {3, 33119}, {4, 33163}, {5, 32931}, {6, 16886}, {11, 25591}, {12, 28109}, {21, 4438}, {31, 5015}, {38, 16062}, {40, 36496}, {46, 32948}, {58, 4680}, {65, 36503}, {72, 25760}, {75, 24995}, {171, 5300}, {244, 33833}, {281, 28102}, {335, 16906}, {355, 19548}, {405, 33115}, {442, 32771}, {515, 36510}, {595, 4894}, {942, 25957}, {946, 36495}, {958, 3145}, {964, 32780}, {982, 4202}, {983, 5192}, {984, 4812}, {986, 4972}, {1046, 6327}, {1215, 2476}, {1330, 32912}, {1468, 7270}, {1479, 32930}, {1710, 4418}, {1834, 3703}, {1837, 36509}, {2049, 19729}, {2292, 32773}, {2475, 33170}, {2887, 3868}, {3057, 36501}, {3454, 5904}, {3509, 26085}, {3670, 33125}, {3695, 32915}, {3701, 33165}, {3702, 33141}, {3710, 24210}, {3846, 3876}, {3874, 33069}, {3915, 4514}, {3927, 4683}, {4197, 24325}, {4385, 21935}, {4429, 4446}, {4968, 33169}, {5014, 5255}, {5016, 5247}, {5044, 25960}, {5046, 33166}, {5178, 32779}, {5252, 36508}, {5439, 25961}, {5711, 33072}, {5737, 16356}, {5791, 32917}, {5814, 32864}, {7235, 35552}, {7283, 33161}, {8258, 17126}, {9709, 16422}, {9941, 36502}, {12194, 36511}, {12514, 32947}, {12588, 16799}, {13740, 26061}, {16466, 32844}, {17063, 17674}, {17155, 23537}, {17550, 17755}, {17670, 24629}, {18480, 36512}, {20963, 34542}, {24211, 31130}, {24631, 33840}, {24851, 32933}, {26064, 34997}, {31317, 33841}

X(36569) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 5TH BROCARD

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

X(36569) lies on these lines: {2, 32}, {76, 19674}, {141, 33762}, {976, 9941}, {3098, 36510}, {3099, 5293}, {9821, 19548}, {9873, 36496}, {10356, 36495}, {10873, 36513}, {10874, 36501}, {10877, 36509}, {11368, 28082}, {12495, 36500}, {18957, 36508}

X(36570) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND ORTHIC OF INTOUCH

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

X(36570) lies on these lines: {1, 36510}, {2, 7}, {25, 244}, {41, 3752}, {56, 1626}, {65, 976}, {208, 1877}, {223, 1404}, {354, 36509}, {604, 1427}, {942, 19548}, {982, 4220}, {1106, 1426}, {1111, 13478}, {1396, 16947}, {1407, 7147}, {1425, 17114}, {1429, 17080}, {1836, 36501}, {3339, 5293}, {3485, 36505}, {3665, 6703}, {3666, 7225}, {3772, 26934}, {4032, 4812}, {4223, 17063}, {4292, 36496}, {5137, 20277}, {5290, 36499}, {5722, 36512}, {9612, 36495}, {10106, 36500}, {10404, 36513}, {14829, 33930}, {17625, 36506}

X(36571) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND INTOUCH OF ORTHIC

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

X(36571) lies on these lines: {2, 3}, {11, 23847}, {33, 21333}, {34, 36508}, {242, 7081}, {976, 1829}, {1892, 36503}, {2201, 2276}, {5101, 36506}, {5130, 36507}, {5293, 7713}, {11363, 28082}, {11392, 36513}, {11393, 36501}, {12135, 36500}

X(36572) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND TANGENTIAL OF EXCENTRAL

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

X(36572) lies on these lines: {1, 1283}, {2, 7}, {25, 982}, {38, 4220}, {40, 976}, {41, 4850}, {46, 5293}, {84, 36496}, {198, 17595}, {200, 36497}, {228, 17596}, {244, 4223}, {1762, 3772}, {1782, 34937}, {1936, 24476}, {2136, 20035}, {3333, 16485}, {3666, 18162}, {4386, 24310}, {4392, 35988}, {5320, 29821}, {5709, 19548}, {6762, 36500}, {7225, 26635}, {14455, 17889}, {16560, 17720}, {16888, 17923}, {21367, 33133}, {21368, 33151}, {30223, 36501}, {31435, 36505}

X(36573) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND INNER YFF

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

X(36573) lies on these lines: {1, 2}, {3, 28108}, {4, 17719}, {5, 36501}, {12, 17783}, {35, 24248}, {36, 28107}, {38, 6910}, {55, 19548}, {56, 17724}, {171, 3487}, {244, 6921}, {335, 16925}, {377, 33127}, {443, 33130}, {495, 36513}, {517, 28109}, {631, 982}, {902, 11415}, {946, 3749}, {984, 6857}, {986, 5218}, {1058, 17715}, {1279, 25681}, {1478, 36496}, {1785, 28076}, {2550, 24161}, {3035, 17054}, {3072, 5761}, {3145, 8069}, {3242, 4999}, {3295, 36509}, {3485, 5255}, {3523, 4310}, {3550, 4295}, {3601, 13161}, {3744, 11375}, {3782, 5217}, {3944, 4294}, {3976, 7288}, {4188, 33148}, {4189, 33153}, {4255, 17061}, {4339, 5226}, {4424, 31452}, {4855, 23536}, {4862, 16192}, {5247, 25568}, {5266, 11374}, {5433, 17597}, {5438, 24178}, {5711, 5719}, {6284, 36512}, {7736, 16787}, {7951, 36495}, {9352, 26729}, {11508, 28077}, {13407, 36503}, {15803, 24231}, {16045, 30869}, {16909, 31052}, {17063, 17567}, {17526, 32931}, {17594, 34937}, {17602, 19765}, {18048, 19270}, {20805, 21320}, {23675, 35262}, {24159, 25440}, {28258, 34247}

X(36574) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND OUTER YFF

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

X(36574) lies on these lines: {1, 2}, {3, 36509}, {4, 982}, {5, 33144}, {7, 24172}, {12, 17597}, {36, 36510}, {38, 2478}, {56, 19548}, {65, 17721}, {244, 377}, {335, 16924}, {388, 3976}, {443, 17063}, {496, 36501}, {497, 986}, {942, 26098}, {950, 988}, {984, 5084}, {999, 36508}, {1265, 25079}, {1279, 26066}, {1329, 3242}, {1478, 3953}, {1479, 3670}, {1788, 5255}, {1844, 28101}, {1854, 15845}, {1905, 28099}, {2275, 24247}, {2550, 24174}, {2886, 17054}, {3073, 5770}, {3090, 17719}, {3091, 4310}, {3145, 8071}, {3290, 26036}, {3304, 5724}, {3333, 24216}, {3434, 24443}, {3487, 17717}, {3672, 24211}, {3677, 9581}, {3721, 9599}, {3726, 9596}, {3744, 24914}, {3749, 6684}, {3756, 25524}, {3782, 10896}, {3944, 10591}, {3945, 24240}, {3999, 10404}, {4000, 17046}, {4292, 18193}, {4294, 17596}, {4295, 33106}, {4339, 5435}, {4392, 5046}, {4438, 13742}, {4648, 17048}, {5045, 5725}, {5082, 24440}, {5141, 33148}, {5154, 33153}, {5192, 33163}, {5225, 24851}, {5247, 24477}, {5439, 14523}, {5573, 24178}, {5714, 33103}, {5728, 28078}, {5902, 28107}, {6284, 17595}, {6682, 13725}, {6856, 33130}, {6933, 33127}, {7354, 36512}, {7735, 16787}, {7741, 36495}, {9612, 24231}, {10129, 26729}, {10473, 12109}, {11010, 24223}, {12047, 36503}, {16496, 21075}, {16781, 21965}, {17526, 33119}, {17889, 31418}, {18389, 28086}, {18398, 28081}, {22767, 28077}, {24159, 25639}, {30869, 32957}

X(36575) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND ANTI-1ST EULER

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

X(36575) lies on these lines: {2, 3}, {497, 986}, {515, 5293}, {944, 976}, {1076, 7009}, {1479, 17596}, {3085, 36513}, {3086, 36501}, {3487, 36503}, {4293, 36508}, {4294, 36509}, {4295, 28109}, {5603, 28082}, {6684, 36499}, {7683, 13329}, {7736, 22380}, {10357, 36502}, {10359, 36511}, {10531, 28074}, {12245, 36500}

X(36576) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND INNER JOHNSON

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

X(36576) lies on these lines: {2, 11}, {355, 976}, {496, 28074}, {867, 33144}, {1324, 1479}, {5293, 10826}, {10522, 36507}, {10525, 19548}, {10912, 36500}, {10944, 36513}, {11373, 28082}, {11826, 36510}, {12114, 36496}, {17614, 36505}, {17625, 36503}, {18961, 36508}

X(36577) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND OUTER JOHNSON

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

X(36577) lies on these lines: {2, 12}, {72, 25760}, {355, 976}, {442, 33163}, {498, 23850}, {1259, 36497}, {1478, 34868}, {2476, 32937}, {5293, 10827}, {5791, 32781}, {10522, 36506}, {10526, 19548}, {10950, 36501}, {10953, 36509}, {11374, 28082}, {11500, 36496}, {11827, 36510}, {12635, 36500}

X(36578) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND INNER YFF

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

X(36578) lies on these lines: {1, 2}, {20, 33153}, {119, 36495}, {3145, 13097}, {3189, 33133}, {3295, 28077}, {3523, 4392}, {3913, 36506}, {4188, 4310}, {4190, 33144}, {4339, 31053}, {6871, 17719}, {6904, 33148}, {10585, 17783}, {10679, 19548}, {10956, 36513}, {10958, 36501}, {10965, 36509}, {11248, 36510}, {11509, 36508}, {12115, 36496}, {24929, 28104}, {27655, 34247}

X(36579) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND OUTER YFF TANGENTS

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

X(36579) lies on these lines: {1, 2}, {20, 4392}, {38, 6872}, {452, 7226}, {982, 4190}, {986, 20075}, {2475, 4310}, {3056, 24476}, {3091, 33153}, {3189, 4850}, {3218, 4339}, {3242, 3436}, {3340, 28107}, {3672, 21285}, {3873, 5716}, {3945, 20247}, {4000, 5178}, {5177, 33148}, {5225, 33151}, {6871, 33144}, {9335, 17580}, {10585, 17724}, {10680, 19548}, {10957, 36513}, {10959, 36501}, {10966, 36509}, {11036, 33112}, {11249, 36510}, {11396, 28099}, {11851, 17164}, {12116, 36496}, {12513, 36507}, {18967, 36508}, {26470, 36495}

X(36580) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND ANTI-5TH BROCARD

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

X(36580) lies on these lines: {2, 32}, {6, 33762}, {182, 36510}, {976, 12194}, {1780, 4279}, {1914, 18055}, {3398, 19548}, {5293, 10789}, {10358, 36495}, {10797, 36513}, {10798, 36501}, {10799, 36509}, {11364, 28082}, {12110, 36496}, {12195, 36500}, {12835, 36508}, {14880, 36512}

X(36581) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 3RD TRI-SQUARES CENTRAL

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

X(36581) lies on these lines: {2, 6}, {485, 36496}, {976, 8983}, {5293, 13888}, {6221, 36512}, {8981, 19548}, {9540, 36510}, {13883, 28082}, {13897, 36513}, {13898, 36501}, {13901, 36509}, {13911, 36500}, {18965, 36508}

X(36582) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 4TH TRI-SQUARES CENTRAL

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

X(36582) lies on these lines: {2, 6}, {486, 36496}, {976, 13971}, {5293, 13942}, {6398, 36512}, {13935, 36510}, {13936, 28082}, {13954, 36513}, {13955, 36501}, {13958, 36509}, {13966, 19548}, {13973, 36500}, {18966, 36508}

X(36583) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND EHRMANN MID-TRIANGLE

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

X(36583) lies on these lines: {2, 3}, {976, 18480}, {1478, 36501}, {1479, 36513}, {3583, 36509}, {3585, 36508}, {3782, 10896}, {5293, 18492}, {5722, 36503}, {9955, 28082}, {11237, 17724}, {11648, 22407}, {14880, 36511}, {18481, 36505}, {22791, 36500}

X(36584) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND ANTI-INNER GREBE

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

X(36584) lies on these lines: {2, 6}, {372, 36510}, {486, 36495}, {976, 18992}, {1588, 36496}, {3312, 19548}, {5293, 19003}, {13971, 36505}, {18991, 28082}, {18995, 36508}, {19027, 36513}, {19029, 36501}, {19037, 36509}, {19065, 36500}, {26465, 28074}

X(36584) = {X(2),X(6)}-harmonic conjugate of X(36585)

X(36585) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND ANTI-OUTER GREBE

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

X(36585) lies on these lines: {2, 6}, {371, 36510}, {485, 36495}, {976, 18991}, {1587, 36496}, {3311, 19548}, {5293, 19004}, {8983, 36505}, {18992, 28082}, {18996, 36508}, {19028, 36513}, {19030, 36501}, {19038, 36509}, {19066, 36500}, {26459, 28074}

X(36585) = {X(2),X(6)}-harmonic conjugate of X(36584)

X(36586) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 1ST KENMOTU FREE-VERTICES TRIANGLE

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

X(36586) lies on these lines: {2, 372}, {6, 36587}, {371, 19548}, {976, 35641}, {5293, 35774}, {6200, 36510}, {28082, 35762}, {35768, 36508}, {35800, 36513}, {35802, 36501}, {35808, 36509}, {35820, 36496}, {35821, 36512}, {35842, 36500}

X(36587) =  HOMOTHETOR OF THESE TRIANGLES: T(abc/(a^3+b^3+c^3)) AND 2ND KENMOTU FREE-VERTICES TRIANGLE

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

X(36587) lies on these lines: {2, 371}, {6, 36586}, {372, 19548}, {976, 35642}, {5293, 35775}, {6396, 36510}, {28082, 35763}, {35769, 36508}, {35801, 36513}, {35803, 36501}, {35809, 36509}, {35820, 36512}, {35821, 36496}, {35843, 36500}

X(36588) =  X(2)X(1266)∩X(7)X(519)

Barycentrics    (a + b - 5*c)*(a - 5*b + c) : :
X(36588) = 5 X[7] + 4 X[17151]

X(36588) lies on the conic {{A,B,C,X(2),X(7)}} and these lines: {2, 1266}, {7, 519}, {8, 903}, {27, 4921}, {75, 4723}, {86, 16711}, {335, 4740}, {522, 6548}, {536, 27475}, {545, 673}, {675, 6014}, {1268, 4398}, {3663, 5936}, {3672, 25055}, {3679, 4346}, {4373, 17274}, {4419, 16590}, {4440, 17488}, {4441, 31002}, {4452, 30712}, {4460, 17378}, {4677, 4887}, {4896, 34747}, {4945, 5328}, {5308, 31139}, {7229, 17382}, {8236, 28580}, {14621, 35578}, {17320, 30598}, {19883, 25590}

X(36588) = isogonal conjugate of polar conjugate of isotomic conjugate of X(23073)
X(36588) = isotomic conjugate of X(3241)
X(36588) = isotomic conjugate of the anticomplement of X(3679)
X(36588) = isotomic conjugate of the complement of X(31145)
X(36588) = anticomplement of X(36911)
X(36588) = X(i)-cross conjugate of X(j) for these (i,j): {3679, 2}, {4346, 7}, {5316, 85}
X(36588) = X(i)-isoconjugate of X(j) for these (i,j): {6, 16670}, {19, 23073}, {31, 3241}, {32, 30829}, {55, 13462}, {58, 21870}, {100, 8656}, {692, 6006}, {1333, 4029}, {4982, 28615}
X(36588) = cevapoint of X(i) and X(j) for these (i,j): {2, 31145}, {1086, 4777}
X(36588) = trilinear pole of line {514, 1639}
X(36588) = barycentric product X(i)*X(j) for these {i,j}: {85, 4900}, {3261, 6014}
X(36588) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16670}, {2, 3241}, {3, 23073}, {10, 4029}, {37, 21870}, {57, 13462}, {75, 30829}, {514, 6006}, {649, 8656}, {1125, 4982}, {4900, 9}, {5219, 16236}, {6014, 101}

X(36589) =  X(2)X(7)∩X(241)X(31138)

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

X(36589) lies on these lines: {2, 7}, {241, 31138}, {320, 1443}, {519, 22464}, {651, 4715}, {752, 4318}, {903, 18815}, {1441, 17271}, {1442, 17378}, {1737, 4887}, {4001, 18625}, {4346, 18391}, {4552, 17310}, {4707, 30181}, {4862, 24208}, {4870, 17196}, {7232, 17092}, {7269, 17320}, {17075, 17364}

X(36589) = X(4867)-cross conjugate of X(27757)
X(36589) = X(i)-isoconjugate of X(j) for these (i,j): {2161, 2364}, {2320, 6187}, {2341, 28658}
X(36589) = barycentric product X(i)*X(j) for these {i,j}: {7, 27757}, {85, 4867}, {320, 5219}, {664, 23884}, {1443, 4671}, {2099, 20924}, {3679, 17078}
X(36589) = barycentric quotient X(i)/X(j) for these {i,j}: {36, 2364}, {320, 30608}, {1405, 6187}, {1443, 89}, {1464, 28658}, {2099, 2161}, {3218, 2320}, {4653, 2341}, {4867, 9}, {5219, 80}, {5235, 6740}, {23884, 522}, {27757, 8}

X(36590) =  X(11)X(1318)∩X(30)X(901)

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

X(36590) lies on these lines: {11, 1318}, {30, 901}, {80, 519}, {88, 1737}, {106, 3582}, {381, 14260}, {522, 14629}, {903, 18815}, {2166, 4674}, {2222, 13587}, {2325, 6735}, {3241, 14584}, {3257, 5080}, {4511, 4997}, {4555, 7809}, {5627, 18357}, {6336, 7541}, {6740, 28828}, {11237, 34230}, {24624, 36091}

X(36590) = X(i)-cross conjugate of X(j) for these (i,j): {8, 6740}, {9, 4997}, {3036, 8}, {3700, 4582}, {21031, 15065}
X(36590) = X(i)-isoconjugate of X(j) for these (i,j): {36, 1319}, {56, 214}, {57, 17455}, {902, 1443}, {1227, 1397}, {1317, 16944}, {1400, 17191}, {1404, 3218}, {1983, 30725}, {2251, 17078}, {3285, 18593}, {3911, 7113}
X(36590) = cevapoint of X(i) and X(j) for these (i,j): {1, 12515}, {11, 23838}, {7026, 7043}
X(36590) = trilinear pole of line {1639, 2804}
X(36590) = barycentric product X(i)*X(j) for these {i,j}: {80, 4997}, {312, 1168}, {1320, 18359}, {2316, 20566}, {4080, 6740}
X(36590) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 214}, {21, 17191}, {55, 17455}, {80, 3911}, {88, 1443}, {312, 1227}, {903, 17078}, {1168, 57}, {1320, 3218}, {2161, 1319}, {2316, 36}, {4674, 18593}, {4997, 320}, {6187, 1404}, {6740, 16704}, {23838, 3960}

X(36591) =  X(2)X(1266)∩X(514)X(4120)

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

X(36591) lies one these lines: {2, 1266}, {514, 4120}, {519, 30578}, {536, 31171}, {903, 4358}, {1644, 24428}, {3218, 16561}, {3679, 4125}, {4370, 35466}, {4908, 4945}, {17294, 31172}, {27752, 27754}, {31018, 31145}

X(36591) = X(i)-Ceva conjugate of X(j) for these (i,j): {903, 3679}, {4358, 27757}
X(36591) = X(24858)-isoconjugate of X(28607)
X(36591) = crossdifference of every pair of points on line {8656, 21747}
X(36591) = barycentric quotient X(i)/X(j) for these {i,j}: {3679, 24858}, {16489, 2163}
X(36591) = {X(4908),X(4945)}-harmonic conjugate of X(27757)

X(36592) =  X(2)X(679)∩X(88)X(519)

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

X(36592) lies on these lines: {2, 679}, {88, 519}, {903, 4358}, {2226, 4370}

X(36592) = X(i)-cross conjugate of X(j) for these (i,j): {3679, 903}, {28209, 4555}
X(36592) = X(44)-isoconjugate of X(16489)
X(36592) = cevapoint of X(1086) and X(23598)
X(36592) = trilinear pole of line {900, 1022}
X(36592) = barycentric product X(903)*X(24858)
X(36592) = barycentric quotient X(i)/X(j) for these {i,j}: {106, 16489}, {24858, 519}

X(36593) =  X(2)X(1000)∩X(8)X(903)

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

X(36593) liesw on these lines: {2, 1000}, {8, 903}, {88, 519}, {3679, 4767}, {4669, 31143}, {4674, 4677}, {10031, 14193}

X(36593) = reflection of X(4767) in X(3679)
X(36593) = X(903)-Ceva conjugate of X(4945)
X(36593) = barycentric product X(i)*X(j) for these {i,j}: {3241, 4945}, {4792, 30829}, {4997, 16236}
X(36593) = barycentric quotient X(16236)/X(3911)
X(36593) = {X(3679),X(4792)}-harmonic conjugate of X(4945)

X(36594) =  X(320)X(519)∩X(536)X(30575)

Barycentrics    (a - 2*b - 2*c)*(a + b - 2*c)^2*(a - 2*b + c)^2 : :
X(36594) = 2 X[4908] - 3 X[27757]

X((36594) lies on these lines: on lines {320, 519}, {536, 30575}, {4908, 4945}

X(36594) = X(3679)-cross conjugate of X(4945)
X(36594) = X(i)-isoconjugate of X(j) for these (i,j): {89, 1017}, {678, 2163}, {3251, 4588}, {4370, 28607}, {6544, 34073}
X(36594) = cevapoint of X(3679) and X(4945)
X(36594) = trilinear pole of line {4945, 23598}
X(36594) = barycentric product X(i)*X(j) for these {i,j}: {679, 4671}, {903, 4945}, {4555, 23598}, {4618, 4791}, {4792, 20568}
X(36594) = barycentric quotient X(i)/X(j) for these {i,j}: {45, 678}, {679, 89}, {1318, 2364}, {2177, 1017}, {2226, 2163}, {3679, 4370}, {4510, 29908}, {4618, 4604}, {4638, 4588}, {4671, 4738}, {4777, 6544}, {4792, 44}, {4873, 4152}, {4893, 3251}, {4908, 8028}, {4944, 4543}, {4945, 519}, {5219, 1317}, {23352, 1635}, {23598, 900}
X(36594) = {X(903),X(9460)}-harmonic conjugate of X(320)

X(36595) =  X(2)X(20223)∩X(7)X(519)

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

X(36595) lies on these lines: {2, 20223}, {7, 519}, {85, 903}, {226, 28301}, {241, 31139}, {545, 8545}, {553, 19819}, {1266, 3870}, {1441, 17274}, {3663, 10056}, {3872, 17079}, {4346, 31397}, {4887, 12647}, {4945, 5219}

X(36595) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 5219}, {664, 21183}
X(36595) = X(2320)-isoconjugate of X(34446)
X(36595) = barycentric product X(i)*X(j) for these {i,j}: {2099, 20925}, {3679, 17079}
X(36595) = barycentric quotient X(i)/X(j) for these {i,j}: {999, 2364}, {1405, 34446}, {3306, 2320}, {5219, 1000}, {35281, 5549}

X(36596) =  X(2)X(1000)∩X(29)X(12640)

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

X(36596) lies on thesse lines: {2, 1000}, {29, 12640}, {85, 903}, {519, 34234}, {3872, 30608}, {3895, 16561}, {4792, 31397}, {4997, 6735}, {30680, 31145}

X(36596) = X(i)-isoconjugate of X(j) for these (i,j): {999, 1319}, {1404, 3306}, {2251, 17079}
X(36596) = cevapoint of X(4530) and X(4814)
X(36596) = barycentric product X(1000)*X(4997)
X(36596) = barycentric quotient X(i)/X(j) for these {i,j}: {903, 17079}, {1000, 3911}, {1320, 3306}, {2316, 999}, {5548, 35281}, {34446, 1404}

X(36597) =  X(2)X(249)∩X(6036)X(20304)

Barycentrics    2*a^16 - 6*a^14*b^2 + 8*a^12*b^4 - 11*a^10*b^6 + 15*a^8*b^8 - 12*a^6*b^10 + 6*a^4*b^12 - 3*a^2*b^14 + b^16 - 6*a^14*c^2 + 14*a^12*b^2*c^2 - 9*a^10*b^4*c^2 - 2*a^8*b^6*c^2 + 2*a^6*b^8*c^2 + a^4*b^10*c^2 + 3*a^2*b^12*c^2 - 3*b^14*c^2 + 8*a^12*c^4 - 9*a^10*b^2*c^4 + 4*a^8*b^4*c^4 + 4*a^6*b^6*c^4 - 16*a^4*b^8*c^4 + 4*a^2*b^10*c^4 + 6*b^12*c^4 - 11*a^10*c^6 - 2*a^8*b^2*c^6 + 4*a^6*b^4*c^6 + 20*a^4*b^6*c^6 - 4*a^2*b^8*c^6 - 13*b^10*c^6 + 15*a^8*c^8 + 2*a^6*b^2*c^8 - 16*a^4*b^4*c^8 - 4*a^2*b^6*c^8 + 18*b^8*c^8 - 12*a^6*c^10 + a^4*b^2*c^10 + 4*a^2*b^4*c^10 - 13*b^6*c^10 + 6*a^4*c^12 + 3*a^2*b^2*c^12 + 6*b^4*c^12 - 3*a^2*c^14 - 3*b^2*c^14 + c^16 : :

X(36597) lies on these lines: {2, 249}, {6036, 20304}, {14566, 24975}, {16188, 34365}

leftri

Cevian-circumconic triangles: X(36598)-X(36650)

rightri

This preamble and centers X(36598)-X(36650) were contributed by César Eliud Lozada, January 23, 2020.

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and K a conic through A', B', C'. If A", B", C" are the points, others than A', B', C', at which K cuts BC, CA, AB, respectively, then AA", BB", CC" are concurrent.

If Pk is the perspector of K with respect to ABC, the triangle A"B"C" is named here the (P, Pk)-cevian-circumconic triangle.

If P = x : y : z and Pk = xk : yk : zk (barycentrics) then:

A" = 0 : 1/(z*(xk*y*z+x*y*zk-3*x*yk*z)) : 1/(y*(x*yk*z+xk*y*z-3*x*y*zk))

The perspector Q(P, Pk) of ABC and A"B"C" is:

Q(P, Pk) = x*(xk*y*z+x*y*zk-3*x*yk*z)*(x*yk*z+xk*y*z-3*x*y*zk) : :

If Pk = P then Q(P, Pk) = P = Pk.

If P = X(2) then Q(P, Pk) is the isotomic conjugate-of-the anticomplement-of-the anticomplement-of-Pk.

As a cevian triangle with respect to ABC, A"B"C" is perspective to these named anticevian triangles: anticomplementary, Bevan antipodal, excentral, Pelletier, Schroeter, Soddy, tangential, X-parabola-tangential.

X(36598) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(2))-CEVIAN-CIRCUMCONIC

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

X(36598) lies on these lines: {1,4704}, {6,3550}, {56,16468}, {86,18192}, {87,16569}, {106,29227}, {238,3445}, {292,1743}, {870,25590}, {1126,25439}, {1222,32941}, {3216,36602}, {3226,3875}, {4253,20667}, {16667,25426}, {17259,25528}, {17351,18194}, {23572,23892}

X(36598) = isogonal conjugate of X(16569)
X(36598) = isotomic conjugate of X(20943)
X(36598) = barycentric product X(i)*X(j) for these {i, j}: {7, 36630}, {75, 36614}, {514, 29227}
X(36598) = barycentric quotient X(i)/X(j) for these (i, j): (1, 1278), (6, 16569), (9, 4903), (31, 16969), (37, 4135), (42, 21868)
X(36598) = trilinear product X(i)*X(j) for these {i, j}: {2, 36614}, {57, 36630}, {513, 29227}
X(36598) = trilinear quotient X(i)/X(j) for these (i, j): (2, 1278), (3, 22149), (6, 16969), (7, 17090), (8, 4903), (9, 4050)
X(36598) = trilinear pole of the line {649, 4879}
X(36598) = lies on the circumconic with center X(8054))
X(36598) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(6)}} and {{A, B, C, X(2), X(4704)}}
X(36598) = cevapoint of X(2) and X(32005)
X(36598) = X(43)-cross conjugate of-X(1)
X(36598) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 16969}, {4, 22149}, {6, 1278}
X(36598) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 1278), (2, 20943), (6, 16569)

X(36599) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(3))-CEVIAN-CIRCUMCONIC

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

X(36599) lies on the Feuerbach hyperbola and these lines: {1,1898}, {4,4338}, {7,10591}, {79,9581}, {90,6985}, {165,3467}, {920,1156}, {1000,10572}, {1079,2310}, {1858,17098}, {2093,5560}, {3062,3336}, {3065,15803}, {3296,12047}, {3339,5561}, {3485,18490}, {3680,5904}, {4654,5557}, {4866,11010}, {5553,16127}, {5559,5881}, {5665,10399}, {5720,15175}, {6264,24302}, {6856,34919}, {6871,10940}, {7082,16117}, {8759,9355}, {9897,12641}, {11531,13143}, {12514,32635}

X(36599) = barycentric product X(63)*X(36610)
X(36599) = barycentric quotient X(1)/X(20078)
X(36599) = trilinear product X(3)*X(36610)
X(36599) = trilinear quotient X(2)/X(20078)
X(36599) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(35), X(2364)}}
X(36599) = X(46)-cross conjugate of-X(1)
X(36599) = X(6)-isoconjugate-of-X(20078)
X(36599) = X(1)-reciprocal conjugate of-X(20078)

X(36600) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(4))-CEVIAN-CIRCUMCONIC

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

X(36600) lies on these lines: {1,8762}, {3,36607}, {78,25734}, {2636,8764}

X(36600) = barycentric product X(92)*X(36607)
X(36600) = trilinear product X(4)*X(36607)
X(36600) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3)}} and {{A, B, C, X(46), X(2648)}}
X(36600) = X(1745)-cross conjugate of-X(1)

X(36601) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(7))-CEVIAN-CIRCUMCONIC

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

X(36601) lies on these lines: {200,25728}, {220,4421}

X(36601) = barycentric product X(7)*X(36628)
X(36601) = barycentric quotient X(1)/X(20089)
X(36601) = trilinear product X(57)*X(36628)
X(36601) = trilinear quotient X(2)/X(20089)
X(36601) = lies on the circumconic with center X(14714))
X(36601) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(33)}} and {{A, B, C, X(80), X(34409)}}
X(36601) = X(1742)-cross conjugate of-X(1)
X(36601) = X(6)-isoconjugate-of-X(20089)
X(36601) = X(1)-reciprocal conjugate of-X(20089)

X(36602) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(8))-CEVIAN-CIRCUMCONIC

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

X(36602) lies on these lines: {6,36619}, {979,6048}, {1126,8715}, {2334,5255}, {3216,36598}, {3445,11194}, {9432,15803}

X(36602) = isogonal conjugate of X(6048)
X(36602) = barycentric product X(75)*X(36619)
X(36602) = trilinear product X(2)*X(36619)
X(36602) = lies on the circumconic with center X(8054))
X(36602) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(6)}} and {{A, B, C, X(79), X(34399)}}
X(36602) = X(978)-cross conjugate of-X(1)

X(36603) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(9))-CEVIAN-CIRCUMCONIC

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

X(36603) lies on these lines: {1,4004}, {2,4488}, {88,23511}, {89,2999}, {105,8699}, {279,36621}, {330,16833}, {516,26718}, {519,6553}, {545,28655}, {1054,1280}, {1219,3679}, {1224,19876}, {1255,3306}, {3241,35577}, {3928,3973}, {14997,26745}, {25417,27003}, {30198,35348}

X(36603) = isogonal conjugate of X(3973)
X(36603) = isotomic conjugate of X(20942)
X(36603) = barycentric product X(i)*X(j) for these {i, j}: {1, 36606}, {9, 36621}, {693, 8699}
X(36603) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3621), (6, 3973), (31, 21000), (37, 4072), (48, 22147), (513, 4962)
X(36603) = trilinear product X(i)*X(j) for these {i, j}: {6, 36606}, {55, 36621}, {514, 8699}
X(36603) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3621), (3, 22147), (6, 21000), (10, 4072), (75, 20942), (513, 2516)
X(36603) = lies on the circumconic with center X(1015))
X(36603) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(4), X(8051)}}
X(36603) = cevapoint of X(244) and X(4394)
X(36603) = X(1743)-cross conjugate of-X(1)
X(36603) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 21000}, {4, 22147}, {6, 3621}
X(36603) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 3621), (2, 20942), (6, 3973)
X(36603) = X(513)-Zayin conjugate of-X(2516)
X(36603) = {X(8056), X(33795)}-harmonic conjugate of X(3973)

X(36604) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(1), X(10))-CEVIAN-CIRCUMCONIC

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

X(36604) lies on these lines: {1,20068}, {979,31855}, {1126,3871}

X(36604) = lies on the circumconic with center X(8054))
X(36604) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(6)}} and {{A, B, C, X(9), X(3871)}}

X(36605) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(2), X(7))-CEVIAN-CIRCUMCONIC

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

X(36605) lies on these lines: {2,25716}, {8,25728}, {29,20008}, {312,25278}, {1121,30695}, {3621,14942}, {6557,29616}, {10405,20059}

X(36605) = isotomic conjugate of X(20059)
X(36605) = barycentric product X(i)*X(j) for these {i, j}: {7, 36625}, {85, 36627}
X(36605) = barycentric quotient X(57)/X(33633)
X(36605) = trilinear product X(i)*X(j) for these {i, j}: {7, 36627}, {57, 36625}
X(36605) = trilinear quotient X(i)/X(j) for these (i, j): (7, 33633), (75, 20059)
X(36605) = trilinear pole of the line {522, 31287}
X(36605) = lies on the circumconic with center X(1146))
X(36605) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(8)}} and {{A, B, C, X(80), X(279)}}
X(36605) = X(144)-cross conjugate of-X(2)
X(36605) = X(i)-isoconjugate-of-X(j) for these {i,j}: {31, 20059}, {55, 33633}
X(36605) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (2, 20059), (57, 33633)

X(36606) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(2), X(8))-CEVIAN-CIRCUMCONIC

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

X(36606) lies on the circumhyperbola dual of Yff parabola and these lines: {2,4488}, {7,3623}, {27,19824}, {75,4678}, {86,4346}, {145,4902}, {335,4788}, {673,20059}, {675,8699}, {903,4452}, {3621,4373}, {3663,30712}, {4704,27475}, {7321,30598}

X(36606) = isogonal conjugate of X(21000)
X(36606) = isotomic conjugate of X(3621)
X(36606) = barycentric product X(i)*X(j) for these {i, j}: {8, 36621}, {75, 36603}
X(36606) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3973), (3, 22147), (6, 21000), (10, 4072), (75, 20942), (513, 2516)
X(36606) = trilinear product X(i)*X(j) for these {i, j}: {2, 36603}, {9, 36621}, {693, 8699}
X(36606) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3973), (63, 22147), (75, 3621), (76, 20942), (321, 4072), (514, 2516)
X(36606) = trilinear pole of the line {514, 2490}
X(36606) = intersection, other than A,B,C, of circumhyperbola dual of Yff parabola and conic {{A, B, C, X(4), X(6553)}}
X(36606) = cevapoint of X(1086) and X(3667)
X(36606) = X(145)-cross conjugate of-X(2)
X(36606) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 3973}, {19, 22147}, {31, 3621}
X(36606) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 3973), (2, 3621), (3, 22147)
X(36606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4373, 4887, 33800), (4373, 33800, 3621)

X(36607) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(3), X(1))-CEVIAN-CIRCUMCONIC

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

X(36607) lies on the line {3,36600}

X(36607) = barycentric product X(63)*X(36600)
X(36607) = trilinear product X(3)*X(36600)
X(36607) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(21)}} and {{A, B, C, X(296), X(36599)}}

X(36608) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(3), X(2))-CEVIAN-CIRCUMCONIC

Barycentrics    (S^2-SB*SC)*(3*S^2-4*R^2*(4*SB-SW)+4*SB^2-SW^2)*(3*S^2-4*R^2*(4*SC-SW)+4*SC^2-SW^2) : :

X(36608) lies on the line {577,36617}

X(36608) = isogonal conjugate of polar conjugate of X(38256)
X(36608) = isotomic conjugate of the polar conjugate of X(36617)
X(36608) = X(92)-isoconjugate of X(38297)
X(36608) = barycentric product X(69)*X(36617)
X(36608) = trilinear product X(63)*X(36617)
X(36608) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(95)}} and {{A, B, C, X(253), X(14941)}}

X(36609) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(3), X(6))-CEVIAN-CIRCUMCONIC

Barycentrics    (S^2-SB*SC)*(S^2-4*SA*SC)*(S^2-4*SA*SB) : :

X(36609) lies on these lines: {2,15851}, {3,3532}, {30,3346}, {381,1217}, {1073,15400}, {1214,1419}, {1297,9909}, {3682,22117}, {6617,14919}, {14938,15703}, {15066,31626}, {15685,18317}, {15694,22270}

X(36609) = isogonal conjugate of X(33630)
X(36609) = isotomic conjugate of the polar conjugate of X(3532)
X(36609) = barycentric product X(i)*X(j) for these {i, j}: {3, 35510}, {20, 15400}, {69, 3532}
X(36609) = barycentric quotient X(i)/X(j) for these (i, j): (3, 3146), (6, 33630), (48, 18594), (222, 18624), (1073, 14572)
X(36609) = trilinear product X(i)*X(j) for these {i, j}: {48, 35510}, {63, 3532}, {610, 15400}
X(36609) = trilinear quotient X(i)/X(j) for these (i, j): (3, 18594), (63, 3146), (77, 18624)
X(36609) = lies on the circumconic with center X(35071))
X(36609) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3)}} and {{A, B, C, X(30), X(6617)}}
X(36609) = pole of the trilinear polar of X(15400) with respect to MacBeath circumconic
X(36609) = cevapoint of X(3) and X(33636)
X(36609) = X(i)-isoconjugate-of-X(j) for these {i,j}: {4, 18594}, {19, 3146}, {33, 18624}, {204, 14572}
X(36609) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (3, 3146), (6, 33630), (48, 18594)

X(36610) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(4), X(1))-CEVIAN-CIRCUMCONIC

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

X(36610) lies on the line {4,4338}

X(36610) = polar conjugate of X(20078)
X(36610) = barycentric product X(92)*X(36599)
X(36610) = barycentric quotient X(4)/X(20078)
X(36610) = trilinear product X(4)*X(36599)
X(36610) = trilinear quotient X(92)/X(20078)
X(36610) = lies on the circumconic with center X(20620))
X(36610) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(29)}} and {{A, B, C, X(7), X(4338)}}
X(36610) = X(1068)-cross conjugate of-X(4)
X(36610) = X(48)-isoconjugate-of-X(20078)
X(36610) = X(4)-reciprocal conjugate of-X(20078)

X(36611) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(4), X(2))-CEVIAN-CIRCUMCONIC

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

X(36611) lies on these lines: {4,1353}, {93,5067}, {264,34803}, {393,36616}, {6531,33630}, {32001,35142}

X(36611) = polar conjugate of X(20080)
X(36611) = barycentric product X(264)*X(36616)
X(36611) = barycentric quotient X(i)/X(j) for these (i, j): (4, 20080), (19, 16570), (25, 5023)
X(36611) = trilinear product X(92)*X(36616)
X(36611) = trilinear quotient X(i)/X(j) for these (i, j): (4, 16570), (19, 5023), (92, 20080)
X(36611) = lies on the circumconic with center X(136))
X(36611) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(93)}} and {{A, B, C, X(6), X(5093)}}
X(36611) = X(i)-isoconjugate-of-X(j) for these {i,j}: {3, 16570}, {48, 20080}, {63, 5023}
X(36611) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 20080), (19, 16570), (25, 5023)

X(36612) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(4), X(3))-CEVIAN-CIRCUMCONIC

Barycentrics    SB*SC*(2*SB+2*R^2-SW)*(2*SC+2*R^2-SW) : :

X(36612) lies on these lines: {4,13292}, {93,3090}, {254,3147}, {3520,18852}, {6344,6622}, {7505,34208}, {7577,18854}, {18851,35481}

X(36612) = polar conjugate of the anticomplement of X(6515)
X(36612) = barycentric quotient X(393)/X(3147)
X(36612) = trilinear quotient X(158)/X(3147)
X(36612) = lies on the circumconic with center X(136))
X(36612) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(93)}} and {{A, B, C, X(5), X(34288)}}
X(36612) = X(255)-isoconjugate-of-X(3147)
X(36612) = X(393)-reciprocal conjugate of-X(3147)

X(36613) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(4), X(10))-CEVIAN-CIRCUMCONIC

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

X(36613) lies on these lines: {4,29219}, {917,29217}

X(36613) = polar conjugate of X(20017)
X(36613) = barycentric quotient X(4)/X(20017)
X(36613) = trilinear quotient X(92)/X(20017)
X(36613) = lies on the circumconic with center X(5190))
X(36613) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 20017}, {906, 29216}
X(36613) = X(4)-reciprocal conjugate of-X(20017)

X(36614) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(6), X(1))-CEVIAN-CIRCUMCONIC

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

X(36614) lies on these lines: {6,3550}, {604,21793}, {739,1613}, {1911,3052}, {1979,4383}, {2162,8616}, {16685,36619}

X(36614) = isogonal conjugate of X(1278)
X(36614) = isotomic conjugate of complement of X(36645)
X(36614) = anticomplement of the complementary conjugate of X(192)
X(36614) = complement of the anticomplementary conjugate of X(4788)
X(36614) = barycentric product X(i)*X(j) for these {i, j}: {1, 36598}, {57, 36630}, {513, 29227}
X(36614) = barycentric quotient X(i)/X(j) for these (i, j): (1, 20943), (6, 1278), (31, 16569), (32, 16969), (41, 4050), (42, 4135)
X(36614) = trilinear product X(i)*X(j) for these {i, j}: {6, 36598}, {56, 36630}, {649, 29227}
X(36614) = trilinear quotient X(i)/X(j) for these (i, j): (2, 20943), (6, 16569), (9, 4903), (31, 16969), (37, 4135), (42, 21868)
X(36614) = trilinear pole of the line {667, 23472}
X(36614) = lies on the circumconic with center X(23571))
X(36614) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(28523)}} and {{A, B, C, X(6), X(31)}}
X(36614) = cevapoint of X(i) and X(j) for these (i,j): (512, 23571), (649, 23470), (667, 23560)
X(36614) = X(2176)-cross conjugate of-X(6)
X(36614) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 16569}, {6, 20943}, {7, 4050}, {92, 22152}
X(36614) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 20943), (6, 1278), (31, 16569)
X(36614) = X(2162)-vertex conjugate of-X(2162)

X(36615) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(6), X(2))-CEVIAN-CIRCUMCONIC

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

X(36615) lies on these lines: {6,3552}, {32,2056}, {83,11333}, {213,3550}, {729,33786}, {3053,9468}, {3224,7793}, {3225,3360}, {17105,21759}

X(36615) = isogonal conjugate of X(20081)
X(36615) = isotomic conjugate of complement of X(36648)
X(36615) = anticomplement of the complementary conjugate of X(194)
X(36615) = complement of the anticomplementary conjugate of X(20105)
X(36615) = barycentric quotient X(i)/X(j) for these (i, j): (1, 20945), (6, 20081), (31, 16571), (32, 21001), (42, 21095), (56, 17091)
X(36615) = trilinear quotient X(i)/X(j) for these (i, j): (2, 20945), (6, 16571), (31, 21001), (37, 21095), (48, 22152), (57, 17091)
X(36615) = trilinear pole of the line {669, 23472}
X(36615) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(32)}} and {{A, B, C, X(25), X(699)}}
X(36615) = X(1613)-cross conjugate of-X(6)
X(36615) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 16571}, {6, 20945}, {9, 17091}, {92, 22152}
X(36615) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 20945), (6, 20081), (31, 16571)

X(36616) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(6), X(3))-CEVIAN-CIRCUMCONIC

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

X(36616) lies on the circumconic with center X(1084) and these lines: {2,15815}, {6,8780}, {25,22331}, {37,4421}, {111,1611}, {115,21974}, {308,8556}, {393,36611}, {524,6339}, {1184,1383}, {1995,3108}, {2165,10154}, {2493,34570}, {2987,20998}, {2998,8667}, {5020,22332}, {5023,8770}, {5585,21448}, {9465,34572}

X(36616) = isogonal conjugate of X(20080)
X(36616) = anticomplement of the complementary conjugate of X(193)
X(36616) = barycentric product X(3)*X(36611)
X(36616) = barycentric quotient X(i)/X(j) for these (i, j): (6, 20080), (31, 16570), (32, 5023)
X(36616) = trilinear product X(48)*X(36611)
X(36616) = trilinear quotient X(i)/X(j) for these (i, j): (6, 16570), (31, 5023)
X(36616) = polar conjugate of isotomic conjugate of X(38263)
X(36616) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(6)}} and {{A, B, C, X(3), X(22331)}}
X(36616) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 16570}, {63, 38282}, {75, 5023}
X(36616) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (6, 20080), (31, 16570), (32, 5023)

X(36617) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(6), X(4))-CEVIAN-CIRCUMCONIC

Barycentrics    (SB+SC)*(3*S^2-4*R^2*(4*SB-SW)+4*SB^2-SW^2)*(3*S^2-4*R^2*(4*SC-SW)+4*SC^2-SW^2) : :

X(36617) lies on these lines: {577,36608}, {1971,14642}

X(36617) = isogonal conjugate of isotomic conjugate of X(38256)
X(36617) = isogonal conjugate of polar conjugate of X(38264)
X(36617) = isogonal conjugate of the anticomplement of X(3164)
X(36617) = polar conjugate of the isotomic conjugate of X(36608)
X(36617) = anticomplement of the complementary conjugate of X(3164)
X(36617) = barycentric product X(4)*X(36608)
X(36617) = trilinear product X(19)*X(36608)
X(36617) = X(92)-isoconjugate of X(38283)
X(36617) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(184)}} and {{A, B, C, X(64), X(1298)}}
X(36617) = X(1988)-vertex conjugate of-X(1988)

X(36618) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(6), X(8))-CEVIAN-CIRCUMCONIC

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

X(36618) lies on the line {2300,3550}

X(36618) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(1397)}} and {{A, B, C, X(56), X(727)}}

X(36619) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(6), X(9))-CEVIAN-CIRCUMCONIC

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

X(36619) lies on these lines: {6,36602}, {3204,28615}, {16685,36614}, {20332,27623}

X(36619) = isogonal conjugate of the anticomplement of X(3210)
X(36619) = anticomplement of the complementary conjugate of X(3210)
X(36619) = barycentric product X(1)*X(36602)
X(36619) = trilinear product X(6)*X(36602)
X(36619) = trilinear pole of the line {667, 23570}
X(36619) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(31)}} and {{A, B, C, X(87), X(28523)}}

X(36620) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(7), X(1))-CEVIAN-CIRCUMCONIC

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

X(36620) lies on the circumhyperbola dual of Yff parabola and these lines: {2,3160}, {7,1699}, {11,479}, {75,31627}, {77,28626}, {347,5936}, {497,3599}, {658,9812}, {673,2898}, {934,19541}, {1223,18230}, {1659,16662}, {1996,5543}, {2400,3676}, {3817,10004}, {4373,9436}, {5226,27475}, {5748,35312}, {7056,9779}, {8727,14256}, {10431,34402}, {10578,31721}, {13390,16663}

X(36620) = barycentric product X(i)*X(j) for these {i, j}: {7, 10405}, {85, 3062}, {1088, 19605}
X(36620) = barycentric quotient X(i)/X(j) for these (i, j): (7, 144), (11, 13609), (56, 3207), (57, 165), (65, 21872), (85, 16284)
X(36620) = trilinear product X(i)*X(j) for these {i, j}: {7, 3062}, {57, 10405}, {85, 11051}, {279, 19605}
X(36620) = trilinear quotient X(i)/X(j) for these (i, j): (7, 165), (77, 22117), (85, 144), (226, 21872), (279, 1419), (479, 17106)
X(36620) = intersection, other than A,B,C, of conic {{A, B, C, X(1), X(10980)}} and circumhyperbola dual of Yff parabola
X(36620) = cevapoint of X(11) and X(3676)
X(36620) = X(i)-cross conjugate of-X(j) for these (i,j): (279, 7), (1146, 24002)
X(36620) = X(i)-isoconjugate-of-X(j) for these {i,j}: {33, 22117}, {41, 144}, {55, 165}
X(36620) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (7, 144), (11, 13609), (56, 3207)
X(36620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10405, 19605), (2, 31527, 3160), (7, 32079, 9533), (1699, 9533, 7), (9533, 15511, 1699), (31994, 34060, 3160)

X(36621) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(7), X(2))-CEVIAN-CIRCUMCONIC

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

X(36621) lies on these lines: {7,3623}, {279,36603}, {1358,6049}, {2369,8699}, {17089,32003}

X(36621) = barycentric product X(i)*X(j) for these {i, j}: {7, 36606}, {85, 36603}
X(36621) = barycentric quotient X(i)/X(j) for these (i, j): (7, 3621), (56, 21000), (57, 3973), (85, 20942), (222, 22147), (226, 4072)
X(36621) = trilinear product X(i)*X(j) for these {i, j}: {7, 36603}, {57, 36606}
X(36621) = trilinear quotient X(i)/X(j) for these (i, j): (7, 3973), (57, 21000), (77, 22147), (85, 3621), (1441, 4072)
X(36621) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3623)}} and {{A, B, C, X(7), X(85)}}
X(36621) = cevapoint of X(1358) and X(30719)
X(36621) = X(i)-isoconjugate-of-X(j) for these {i,j}: {9, 21000}, {33, 22147}, {41, 3621}
X(36621) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (7, 3621), (56, 21000), (57, 3973)

X(36622) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(7), X(4))-CEVIAN-CIRCUMCONIC

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

X(36622) lies on these lines: {63,3160}, {69,31627}, {77,9533}, {347,30304}, {18623,36101}

X(36622) = barycentric quotient X(57)/X(1750)
X(36622) = trilinear quotient X(7)/X(1750)
X(36622) = lies on the circumconic with center X(26932))
X(36622) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(10857)}} and {{A, B, C, X(7), X(63)}}
X(36622) = X(55)-isoconjugate-of-X(1750)
X(36622) = X(57)-reciprocal conjugate of-X(1750)

X(36623) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(7), X(10))-CEVIAN-CIRCUMCONIC

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

X(36623) lies on these lines: {4393,7176}, {7196,30963}

X(36623) = trilinear pole of the line {4785, 30723}
X(36623) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3226)}} and {{A, B, C, X(7), X(21454)}}

X(36624) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(8), X(1))-CEVIAN-CIRCUMCONIC

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

X(36624) lies on these lines: {8,3586}, {280,27383}, {346,36629}, {2322,3161}, {3710,6556}

X(36624) = barycentric product X(75)*X(36629)
X(36624) = barycentric quotient X(i)/X(j) for these (i, j): (8, 9965), (9, 15803), (210, 21866), (219, 23072), (346, 27383)
X(36624) = trilinear product X(2)*X(36629)
X(36624) = trilinear quotient X(i)/X(j) for these (i, j): (8, 15803), (78, 23072), (312, 9965), (341, 27383), (2321, 21866)
X(36624) = lies on the circumconic with center X(2968))
X(36624) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(34863)}} and {{A, B, C, X(4), X(3586)}}
X(36624) = X(i)-isoconjugate-of-X(j) for these {i,j}: {34, 23072}, {56, 15803}, {604, 9965}
X(36624) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (8, 9965), (9, 15803), (210, 21866)

X(36625) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(8), X(2))-CEVIAN-CIRCUMCONIC

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

X(36625) lies on these lines: {8,25728}, {346,36627}, {18025,21296}

X(36625) = barycentric product X(i)*X(j) for these {i, j}: {8, 36605}, {75, 36627}
X(36625) = barycentric quotient X(1)/X(33633)
X(36625) = trilinear product X(i)*X(j) for these {i, j}: {2, 36627}, {9, 36605}
X(36625) = trilinear quotient X(i)/X(j) for these (i, j): (2, 33633), (312, 20059)
X(36625) = lies on the circumconic with center X(2968))
X(36625) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(75)}} and {{A, B, C, X(281), X(28626)}}
X(36625) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 33633}, {604, 20059}
X(36625) = X(1)-reciprocal conjugate of-X(33633)

X(36626) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(8), X(10))-CEVIAN-CIRCUMCONIC

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

X(36626) lies on these lines: {8,90}, {75,7318}, {78,4354}, {280,4511}, {318,406}, {346,4420}, {1043,7072}, {1219,4861}, {1259,4081}, {1748,18533}, {2322,11107}, {2370,36082}, {7219,10538}

X(36626) = isogonal conjugate of X(1406)
X(36626) = isotomic conjugate of the isogonal conjugate of X(7072)
X(36626) = barycentric product X(i)*X(j) for these {i, j}: {8, 2994}, {9, 20570}, {76, 7072}, {90, 312}, {318, 6513}, {345, 7040}
X(36626) = barycentric quotient X(i)/X(j) for these (i, j): (6, 1406), (8, 5905), (9, 46), (55, 2178), (78, 6505), (90, 57)
X(36626) = trilinear product X(i)*X(j) for these {i, j}: {8, 90}, {9, 2994}, {55, 20570}, {75, 7072}, {78, 7040}, {200, 7318}
X(36626) = trilinear quotient X(i)/X(j) for these (i, j): (8, 46), (9, 2178), (78, 3157), (90, 56), (312, 5905), (318, 1068)
X(36626) = trilinear pole of the line {3239, 35057}
X(36626) = lies on the circumconic with center X(2968))
X(36626) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(5119)}} and {{A, B, C, X(2), X(2349)}}
X(36626) = X(78)-cross conjugate of-X(8)
X(36626) = X(i)-isoconjugate-of-X(j) for these {i,j}: {34, 3157}, {46, 56}, {57, 2178}
X(36626) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (6, 1406), (8, 5905), (9, 46)

X(36627) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(9), X(1))-CEVIAN-CIRCUMCONIC

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

X(36627) lies on these lines: {2,25716}, {9,4421}, {346,36625}, {3928,36101}

X(36627) = isogonal conjugate of X(33633)
X(36627) = barycentric product X(i)*X(j) for these {i, j}: {1, 36625}, {9, 36605}
X(36627) = barycentric quotient X(i)/X(j) for these (i, j): (6, 33633), (9, 20059)
X(36627) = trilinear product X(i)*X(j) for these {i, j}: {6, 36625}, {55, 36605}
X(36627) = lies on the circumconic with center X(35508))
X(36627) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(31508)}} and {{A, B, C, X(2), X(9)}}
X(36627) = X(56)-isoconjugate-of-X(20059)
X(36627) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (6, 33633), (9, 20059)

X(36628) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(9), X(2))-CEVIAN-CIRCUMCONIC

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

X(36628) lies on these lines: {9,25716}, {220,4421}

X(36628) = barycentric product X(8)*X(36601)
X(36628) = barycentric quotient X(9)/X(20089)
X(36628) = trilinear product X(9)*X(36601)
X(36628) = trilinear quotient X(8)/X(20089)
X(36628) = X(56)-isoconjugate-of-X(20089)
X(36628) = X(9)-reciprocal conjugate of-X(20089)

X(36629) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(9), X(6))-CEVIAN-CIRCUMCONIC

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

X(36629) lies on these lines: {9,3697}, {281,3950}, {282,6765}, {346,36624}, {3158,4183}, {3247,7110}

X(36629) = barycentric product X(1)*X(36624)
X(36629) = barycentric quotient X(i)/X(j) for these (i, j): (9, 9965), (55, 15803), (200, 27383), (212, 23072), (1334, 21866)
X(36629) = trilinear product X(6)*X(36624)
X(36629) = trilinear quotient X(i)/X(j) for these (i, j): (8, 9965), (9, 15803), (210, 21866), (219, 23072), (346, 27383)
X(36629) = lies on the circumconic with center X(35508))
X(36629) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(9)}} and {{A, B, C, X(10), X(3697)}}
X(36629) = X(2324)-cross conjugate of-X(9)
X(36629) = X(i)-isoconjugate-of-X(j) for these {i,j}: {56, 9965}, {57, 15803}, {278, 23072}
X(36629) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (9, 9965), (55, 15803), (200, 27383)

X(36630) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(9), X(8))-CEVIAN-CIRCUMCONIC

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

X(36630) lies on these lines: {6,3550}, {43,23470}, {57,4393}, {893,3749}, {2291,29227}, {2319,4050}, {3158,7077}

X(36630) = barycentric product X(i)*X(j) for these {i, j}: {8, 36598}, {312, 36614}, {522, 29227}
X(36630) = barycentric quotient X(i)/X(j) for these (i, j): (1, 17090), (8, 20943), (9, 1278), (41, 16969), (55, 16569), (200, 4903)
X(36630) = trilinear product X(i)*X(j) for these {i, j}: {8, 36614}, {9, 36598}, {650, 29227}
X(36630) = trilinear quotient X(i)/X(j) for these (i, j): (2, 17090), (8, 1278), (9, 16569), (55, 16969), (200, 4050), (210, 21868)
X(36630) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3550)}} and {{A, B, C, X(6), X(9)}}
X(36630) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 17090}, {7, 16969}, {56, 1278}
X(36630) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 17090), (8, 20943), (9, 1278)

X(36631) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(9), X(10))-CEVIAN-CIRCUMCONIC

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

X(36631) lies on these lines: {63,25716}, {219,4421}

X(36632) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(10), X(2))-CEVIAN-CIRCUMCONIC

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

X(36632) lies on these lines: {10,9791}, {12,6541}, {319,21089}, {594,4096}, {2321,6543}, {21081,36633}

X(36632) = barycentric quotient X(i)/X(j) for these (i, j): (10, 20090), (1089, 27705)
X(36632) = trilinear quotient X(321)/X(20090)
X(36632) = lies on the circumconic with center X(23943))
X(36632) = intersection, other than A,B,C, of conics {{A, B, C, X(7), X(9791)}} and {{A, B, C, X(10), X(12)}}
X(36632) = cevapoint of X(i) and X(j) for these (i,j): (523, 23943), (661, 23953)
X(36632) = X(1333)-isoconjugate-of-X(20090)
X(36632) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (10, 20090), (1089, 27705)

X(36633) = PERSPECTOR OF THESE TRIANGLES: ABC AND (X(10), X(8))-CEVIAN-CIRCUMCONIC

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

X(36633) lies on these lines: {10,33099}, {21081,36632}

X(36634) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND (X(1), X(2))-CEVIAN-CIRCUMCONIC

Barycentrics    a*(3*(b+c)*a-5*b*c) : :
X(36634) = X(1)-4*X(978) = 2*X(10)+X(20036)

X(36634) lies on these lines: {1,2}, {165,15310}, {210,17591}, {238,4421}, {291,36603}, {740,20942}, {1376,16468}, {1468,36006}, {1757,3928}, {3550,4383}, {3711,17598}, {3715,17593}, {3929,17596}, {4023,33174}, {4090,17490}, {4413,9332}, {4428,15485}, {4479,17151}, {4849,17063}, {4857,6822}, {4903,28522}, {4921,18192}, {4980,32931}, {5247,13566}, {5270,6821}, {5563,16409}, {7991,19540}, {9350,32911}, {16667,21904}, {20669,21780}, {21760,36650}, {24174,24473}, {26073,32946}, {36598,36646}

X(36634) = barycentric product X(i)*X(j) for these {i, j}: {1, 4788}, {75, 36647}
X(36634) = trilinear product X(i)*X(j) for these {i, j}: {2, 36647}, {6, 4788}
X(36634) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(36647)}} and {{A, B, C, X(2), X(4788)}}
X(36634) = X(43)-Zayin conjugate of-X(1)
X(36634) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (42, 25502, 1), (43, 899, 16569), (43, 16569, 1), (43, 26102, 3240), (3216, 6048, 1), (6685, 26038, 1698), (25889, 29986, 26237)

X(36635) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND (X(1), X(2))-CEVIAN-CIRCUMCONIC

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

X(36635) lies on these lines: {3,16468}, {6,31}, {9,21010}, {11,30971}, {45,16679}, {56,87}, {86,4423}, {144,1001}, {193,8299}, {329,24669}, {405,33682}, {579,3271}, {999,15485}, {1376,17349}, {1403,1707}, {1740,4383}, {1743,2223}, {2175,5053}, {2176,7032}, {2886,27317}, {3248,21769}, {3286,4225}, {3303,4649}, {3747,21785}, {3915,23579}, {3941,34247}, {4253,21746}, {4413,17277}, {4447,26685}, {4471,5124}, {4497,16686}, {4749,5069}, {5021,23660}, {5022,20459}, {5120,7083}, {5145,16466}, {5217,16477}, {14974,21760}, {15624,16669}, {15668,16355}, {16059,36646}, {16670,16688}, {16885,20990}, {17120,23407}, {17123,25528}, {17259,32918}, {20332,20676}, {23404,34445}, {23863,33863}, {27623,27636}

X(36635) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(7153)}} and {{A, B, C, X(42), X(9309)}}
X(36635) = pole of the trilinear polar of X(36598) with respect to circumcircle
X(36635) = crossdifference of every pair of points on line {X(514), X(23744)}
X(36635) = crosspoint of X(1252) and X(29227)
X(36635) = crosssum of X(1086) and X(29226)
X(36635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3052, 2209), (6, 20992, 55), (44, 3941, 34247), (87, 238, 28365), (672, 20978, 3056), (3747, 23524, 21785), (5120, 7083, 17798)

X(36636) = PERSPECTOR OF THESE TRIANGLES: BEVAN ANTIPODAL AND (X(1), X(6))-CEVIAN-CIRCUMCONIC

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

X(36636) lies on these lines: {1,5806}, {2,3160}, {6,57}, {9,17080}, {34,3601}, {42,3340}, {73,11518}, {77,5437}, {165,1456}, {221,5128}, {226,3672}, {227,1697}, {241,2124}, {278,1826}, {281,20201}, {347,3452}, {614,1420}, {651,3928}, {664,30567}, {1193,7273}, {1214,7308}, {1457,7962}, {1458,5573}, {2114,8830}, {3158,4318}, {3247,5226}, {3445,34039}, {3666,34991}, {3911,18623}, {3929,34048}, {4296,5438}, {5396,11529}, {5665,19767}, {5709,23071}, {5930,9581}, {6546,30719}, {8583,15832}, {10389,34036}, {10860,24025}, {15851,18594}, {15881,17102}, {16610,34488}, {22464,28609}, {31231,34050}

X(36636) = barycentric product X(i)*X(j) for these {i, j}: {1, 36640}, {7, 7991}
X(36636) = trilinear product X(i)*X(j) for these {i, j}: {6, 36640}, {57, 7991}
X(36636) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(1419)}} and {{A, B, C, X(6), X(19605)}}
X(36636) = X(662)-Beth conjugate of-X(3928)
X(36636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57, 223, 1419), (57, 33633, 1407), (223, 1465, 57), (269, 3752, 57), (1427, 2999, 57)

X(36637) = PERSPECTOR OF THESE TRIANGLES: SCHROETER AND (X(1), X(6))-CEVIAN-CIRCUMCONIC

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

X(36637) lies on these lines: {11,115}, {12,1018}, {1358,1577}, {4129,4904}, {4370,5949}, {16593,26794}

X(36638) = PERSPECTOR OF THESE TRIANGLES: SODDY AND (X(1), X(6))-CEVIAN-CIRCUMCONIC

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

X(36638) lies on these lines: {7,3057}, {9,3177}, {241,2275}, {347,9368}, {664,12513}, {2898,30854}, {9436,12640}, {27818,36640}

X(36638) = {X(3160), X(3212)}-harmonic conjugate of X(31526)

X(36639) = PERSPECTOR OF THESE TRIANGLES: PELLETIER AND (X(2), X(1))-CEVIAN-CIRCUMCONIC

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

X(36639) lies on these lines: {11,116}, {55,644}, {918,24840}, {1015,4162}, {1280,2098}, {1358,3309}, {3021,14839}, {3056,9041}, {3900,4534}

X(36639) = barycentric product X(55)*X(26572)
X(36639) = trilinear product X(41)*X(26572)
X(36639) = pole of the trilinear polar of X(4373) with respect to Feuerbach hyperbola

X(36640) = PERSPECTOR OF THESE TRIANGLES: SODDY AND (X(2), X(1))-CEVIAN-CIRCUMCONIC

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

X(36640) lies on these lines: {1,7}, {37,5226}, {63,18624}, {69,25718}, {75,31627}, {273,5936}, {278,5273}, {307,32087}, {348,30543}, {664,21296}, {1214,24554}, {1266,34060}, {1419,20059}, {1465,5328}, {3875,20008}, {4000,5435}, {4357,31994}, {4452,9436}, {5749,17086}, {17278,31188}, {18623,28610}, {20080,25726}, {27818,36638}

X(36640) = barycentric product X(i)*X(j) for these {i, j}: {75, 36636}, {85, 7991}
X(36640) = trilinear product X(i)*X(j) for these {i, j}: {2, 36636}, {7, 7991}
X(36640) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(7991)}} and {{A, B, C, X(8), X(4297)}}
X(36640) = X(99)-Beth conjugate of-X(21296)
X(36640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 347, 3160), (175, 176, 4297), (269, 4346, 7), (279, 3663, 7), (347, 22464, 7), (3668, 3672, 7)

X(36641) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND (X(2), X(1))-CEVIAN-CIRCUMCONIC

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

X(36641) lies on these lines: {3,142}, {22,347}, {25,281}, {36,15287}, {55,3247}, {71,35273}, {198,6600}, {610,15733}, {674,20818}, {1260,15494}, {1375,11677}, {1622,11414}, {2293,35267}, {3052,17053}, {4254,23868}, {5120,7083}, {5204,16688}, {5285,37519}, {7742,9591}, {8193,13737}, {9909,20875}, {10934,20833}, {12410,20836}, {13615,20988}, {18621,22770}, {22147,35327}, {23305,31184}

X(36641) = Stammler circle-inverse of-X(18327)
X(36641) = pole of the trilinear polar of X(4373) with respect to circumcircle
X(36641) = crosspoint of X(1293) and X(15378)
X(36641) = crosssum of X(116) and X(3667)
X(36641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1486, 1631, 3), (1631, 23854, 1486), (7083, 17798, 5120)

X(36642) = PERSPECTOR OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL AND (X(2), X(1))-CEVIAN-CIRCUMCONIC

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

X(36642) lies on these lines: {523,2487}, {3700,8029}, {4843,12069}

X(36643) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND (X(2), X(6))-CEVIAN-CIRCUMCONIC

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

X(36643) lies on these lines: {1,3053}, {9,46}, {40,3208}, {57,348}, {63,3691}, {484,17742}, {1055,11682}, {1449,3670}, {1707,16583}, {1711,21366}, {2082,3218}, {2083,12717}, {2093,2329}, {3061,15803}, {3247,5264}, {3501,5128}, {3928,5792}, {4262,12559}, {4513,5183}, {4652,17451}, {5119,17736}, {18786,36649}, {21808,35258}

X(36643) = X(i)-Zayin conjugate of-X(j) for these (i,j): (69, 9), (1368, 1726)
X(36643) = {X(46), X(1759)}-harmonic conjugate of X(9)

X(36644) = PERSPECTOR OF THESE TRIANGLES: SODDY AND (X(2), X(6))-CEVIAN-CIRCUMCONIC

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

X(36644) lies on these lines: {7,24210}, {75,31627}, {857,948}, {3772,17081}, {31038,31527}

X(36644) = barycentric product X(85)*X(7996)
X(36644) = trilinear product X(7)*X(7996)

X(36645) = PERSPECTOR OF THESE TRIANGLES: ANTICOMPLEMENTARY AND (X(6), X(1))-CEVIAN-CIRCUMCONIC

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

X(36645) lies on these lines: {194,712}, {8264,32033}, {17026,17490}, {21327,31276}

X(36645) = anticomplement of the isotomic conjugate of X(36614)
X(36645) = pole of the trilinear polar of X(36614) with respect to Steiner circumellipse

X(36646) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND (X(6), X(1))-CEVIAN-CIRCUMCONIC

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

X(36646) lies on these lines: {1,1278}, {6,43}, {21,3551}, {86,4479}, {238,5204}, {404,16468}, {536,24766}, {978,4257}, {2234,18194}, {2309,25528}, {2664,3973}, {3009,25269}, {3620,7184}, {3723,24661}, {3783,20080}, {7032,16571}, {8616,28365}, {16059,36635}, {24669,33147}, {36598,36634}

X(36646) = barycentric product X(1)*X(32005)
X(36646) = trilinear product X(6)*X(32005)
X(36646) = X(2176)-Zayin conjugate of-X(43)
X(36646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (87, 1740, 43), (2309, 25528, 26102)

X(36647) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND (X(6), X(1))-CEVIAN-CIRCUMCONIC

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

X(36647) lies on these lines: {1,6}, {190,32005}, {239,20942}, {595,9351}, {2238,3621}, {2295,5550}, {3009,3052}, {3747,21000}, {3943,20036}, {4346,28350}, {5023,8624}, {5204,17735}, {5217,21008}, {9259,14974}, {9310,21793}, {15808,17750}, {16827,17119}, {17160,27623}, {17262,32107}, {17365,24654}, {21358,27248}

X(36647) = isogonal conjugate of the isotomic conjugate of X(4788)
X(36647) = barycentric product X(i)*X(j) for these {i, j}: {1, 36634}, {6, 4788}
X(36647) = trilinear product X(i)*X(j) for these {i, j}: {6, 36634}, {31, 4788}
X(36647) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(36634)}} and {{A, B, C, X(37), X(4788)}}
X(36647) = pole of the trilinear polar of X(36614) with respect to circumcircle
X(36647) = crosspoint of X(1016) and X(29227)
X(36647) = crosssum of X(1015) and X(29226)
X(36647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2176, 3230, 16969), (2176, 16969, 6)

X(36648) = PERSPECTOR OF THESE TRIANGLES: ANTICOMPLEMENTARY AND (X(6), X(2))-CEVIAN-CIRCUMCONIC

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

X(36648) lies on these lines: {6,194}, {385,9909}, {3229,32746}, {7766,16276}, {11008,25054}

X(36648) = anticomplement of the isotomic conjugate of X(36615)
X(36648) = pole of the trilinear polar of X(36615) with respect to Steiner circumellipse
X(36648) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2998, 8264, 194), (3229, 32747, 32746)

X(36649) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND (X(6), X(2))-CEVIAN-CIRCUMCONIC

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

X(36649) lies on these lines: {1,20081}, {43,16549}, {87,6196}, {978,4257}, {3208,3510}, {16571,23652}, {18786,36643}, {21214,28397}, {26102,31000}

X(36649) = X(1613)-Zayin conjugate of-X(43)

X(36650) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND (X(6), X(2))-CEVIAN-CIRCUMCONIC

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

X(36650) lies on these lines: {2,6}, {694,36616}, {732,35294}, {2076,9909}, {3009,3052}, {3053,3229}, {3117,15815}, {3167,35006}, {3291,21969}, {3360,13586}, {3787,21849}, {3928,16514}, {4421,21780}, {4428,16969}, {5023,8623}, {5104,34481}, {8622,21000}, {11328,22331}, {21760,36634}

X(36650) = isogonal conjugate of the isotomic conjugate of X(20105)
X(36650) = barycentric product X(6)*X(20105)
X(36650) = pole of the trilinear polar of X(36615) with respect to circumcircle
X(36650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1613, 3231, 21001), (1613, 21001, 6)

leftri

Homothetors involving the Euler triangle and triangles T(k): X(36651)-X(36667)

rightri

This preamble and centers X(36651)-X(36666) were contributed by Clark Kimberling and Peter Moses, January 24, 2020.

In this section, k is a quotient of symmetric functions of homogeneity degree 2. The Euler triangle is homothetic to each triangle T(1 : k : k), with homothetor on the Euler line. See the preambles just before X(36436) and X(36473).

X(36651) = HOMOTHETOR OF THESE TRIANGLES: T(-(a^2+b^2+c^2)/(bc + ca + ab)) AND EULER

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

X(36651) lies on these lines: {2, 3}, {515, 29646}, {946, 29674}, {1072, 29657}, {5587, 16825}, {10446, 24206}, {10531, 20539}, {17236, 29369}, {17380, 29235}

X(36652) = HOMOTHETOR OF THESE TRIANGLES: T(-(a^2+b^2+c^2)/(bc + ca + ab)) AND EULER

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

X(36652) lies on these lines: {2, 3}, {10, 30854}, {75, 12618}, {76, 18738}, {200, 5015}, {516, 4429}, {894, 5805}, {971, 3662}, {990, 16706}, {991, 17234}, {1072, 26015}, {1210, 3673}, {1479, 14942}, {1699, 32773}, {1709, 33068}, {1738, 21629}, {1742, 3836}, {1750, 25527}, {2481, 33298}, {3332, 3618}, {3976, 11019}, {4385, 4847}, {4655, 9355}, {4911, 9612}, {4972, 9812}, {5233, 5400}, {5480, 10446}, {5658, 26132}, {5729, 17950}, {5732, 17282}, {5759, 26685}, {5762, 17350}, {5779, 6646}, {5817, 17257}, {5927, 27184}, {9581, 26012}, {11681, 20556}, {13329, 17352}, {17233, 29016}, {17291, 31672}, {17338, 31658}, {17368, 18482}, {19868, 19925}, {24283, 24851}, {28850, 29674}

X(36653) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a^2 + b^2 + c^2))/(a*b + a*c + b*c); ) AND EULER

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

X(36653) lies on these lines: {2, 3}, {5603, 32847}

X(36654) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + b^2 + c^2)/(4*(a*b + a*c + b*c)) AND EULER

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

X(36654) lies on these lines: {2, 3}, {516, 4759}, {517, 20683}, {3332, 5050}, {3932, 29073}, {3943, 29343}, {4251, 7745}, {4253, 5254}, {5045, 13161}, {10446, 21850}, {17757, 20556}, {24828, 29069}

X(36655) = HOMOTHETOR OF THESE TRIANGLES: T(S/(a^2 + b^2 + c^2)) AND EULER

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

X(36655) lies on these lines: {2, 3}, {69, 6215}, {141, 10514}, {193, 5875}, {372, 13749}, {485, 5480}, {486, 1503}, {491, 1161}, {492, 18509}, {511, 6290}, {524, 6281}, {542, 13927}, {626, 11825}, {638, 1160}, {639, 10516}, {640, 1350}, {642, 12306}, {1152, 13934}, {1588, 5305}, {2460, 35821}, {2548, 3070}, {2794, 13926}, {3068, 6202}, {3069, 5871}, {3071, 3767}, {3102, 6564}, {3629, 6279}, {3818, 6289}, {5874, 5921}, {6201, 31412}, {6222, 6251}, {6398, 8982}, {6405, 12948}, {6560, 14230}, {6565, 13748}, {6776, 7584}, {7353, 12958}, {7583, 14853}, {7586, 10783}, {9757, 32497}, {10515, 23311}, {10534, 13960}, {10846, 13758}, {12256, 13966}, {12322, 32828}, {12323, 32816}, {13931, 29012}, {13939, 14242}, {14235, 22644}, {14912, 19116}, {22725, 35830}, {23259, 26331}, {23312, 29181}, {32421, 35684}

X(36656) = HOMOTHETOR OF THESE TRIANGLES: T(-(S/(a^2 + b^2 + c^2))) AND EULER

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

X(36656) lies on these lines: {2, 3}, {69, 6214}, {141, 10515}, {154, 8968}, {193, 5874}, {371, 13748}, {485, 1503}, {486, 5480}, {491, 18511}, {492, 1160}, {511, 6289}, {524, 6278}, {542, 13874}, {626, 11824}, {637, 1161}, {639, 1350}, {640, 10516}, {641, 12305}, {1151, 13882}, {1587, 5305}, {2459, 35820}, {2548, 3071}, {2794, 13873}, {3068, 5870}, {3069, 6201}, {3070, 3767}, {3103, 6565}, {3629, 6280}, {3818, 6290}, {5871, 31412}, {5875, 5921}, {6221, 26441}, {6250, 6399}, {6283, 12949}, {6561, 14233}, {6564, 13749}, {6776, 7583}, {7362, 12959}, {7584, 14853}, {7585, 10784}, {8966, 10533}, {8981, 12257}, {9600, 31415}, {9758, 32494}, {10514, 23312}, {10845, 13638}, {12322, 32816}, {12323, 32828}, {13878, 29012}, {13886, 14227}, {14239, 22615}, {14912, 19117}, {22724, 35831}, {23249, 26330}, {23311, 29181}, {32419, 35685}

X(36657) = HOMOTHETOR OF THESE TRIANGLES: T(S/(2*(a^2 + b^2 + c^2))) AND EULER

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

X(36657) lies on these lines: {2, 3}, {511, 6215}, {640, 29181}, {1132, 14243}, {1160, 18509}, {1350, 10514}, {1351, 5875}, {1503, 7584}, {2548, 23251}, {3071, 5305}, {3311, 6202}, {3312, 5871}, {3767, 23261}, {3818, 6214}, {5102, 6279}, {5480, 7583}, {5870, 13785}, {5874, 18440}, {6201, 13665}, {6251, 14233}, {6281, 11477}, {6290, 31670}, {6418, 10783}, {6776, 19116}, {7586, 14242}, {7776, 12323}, {10784, 18510}, {11917, 26336}, {14853, 19117}, {18511, 26468}

X(36658) = HOMOTHETOR OF THESE TRIANGLES: T(-S/(2*(a^2 + b^2 + c^2))) AND EULER

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

X(36658) lies on these lines: {2, 3}, {511, 6214}, {639, 29181}, {1131, 14228}, {1161, 18511}, {1350, 10515}, {1351, 5874}, {1503, 7583}, {2548, 23261}, {3070, 5305}, {3311, 5870}, {3312, 6201}, {3767, 23251}, {3818, 6215}, {5102, 6280}, {5480, 7584}, {5871, 13665}, {5875, 18440}, {6202, 13785}, {6250, 14230}, {6278, 11477}, {6289, 31670}, {6417, 10784}, {6776, 19117}, {7585, 14227}, {7776, 12322}, {10783, 18512}, {11916, 26346}, {14853, 19116}, {18509, 26469}

X(36659) = HOMOTHETOR OF THESE TRIANGLES: T(-(a*b + a*c + b*c)/(a^2 + b^2 + c^2)) AND EULER

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

X(36659) lies on these lines: {2, 3}, {58, 3767}, {386, 2548}, {511, 5816}, {542, 5733}, {946, 16825}, {985, 3073}, {991, 32431}, {1352, 24220}, {1899, 17167}, {1961, 18528}, {3509, 7330}, {3781, 26063}, {4252, 13881}, {4256, 31415}, {5044, 26036}, {5138, 5747}, {5587, 29674}, {5707, 9958}, {5886, 28901}, {8227, 29646}, {8301, 11496}, {15251, 18493}, {16777, 29235}, {17257, 29369}

X(36660) = HOMOTHETOR OF THESE TRIANGLES: T(2*(a*b + a*c + b*c)/(a^2 + b^2 + c^2)) AND EULER

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

X(36660) lies on these lines: {2, 3}, {10, 28827}, {894, 5817}, {1992, 5733}, {2356, 19372}, {2551, 20544}, {2886, 27539}, {3085, 14942}, {3332, 17277}, {3622, 15251}, {3817, 19868}, {5759, 17260}, {5805, 17257}, {9436, 9612}, {21151, 27147}

X(36661) = HOMOTHETOR OF THESE TRIANGLES: T((a*b + a*c + b*c)/(2*(a^2 + b^2 + c^2))) AND EULER

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

X(36661) lies on these lines: {2, 3}, {9, 29085}, {3826, 29291}, {5733, 20423}, {5805, 29369}, {15251, 34773}

X(36662) = HOMOTHETOR OF THESE TRIANGLES: T(-2*(a*b + a*c + b*c)/(a^2 + b^2 + c^2)) AND EULER

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

X(36662) lies on these lines: {2, 3}, {40, 24603}, {69, 5816}, {169, 28827}, {239, 5603}, {345, 21073}, {355, 17316}, {515, 16831}, {944, 16826}, {946, 4384}, {948, 17181}, {952, 29585}, {966, 10446}, {1699, 16832}, {2051, 32022}, {2356, 9817}, {2548, 20970}, {2550, 20544}, {3008, 3817}, {3436, 28797}, {3452, 26036}, {3661, 5818}, {3912, 5587}, {4393, 10595}, {4417, 32816}, {5257, 10444}, {5657, 29576}, {5794, 30812}, {5881, 29574}, {5882, 29597}, {5886, 26626}, {7967, 29570}, {7988, 29598}, {7989, 17284}, {8227, 17023}, {9779, 28913}, {10175, 17308}, {10478, 14555}, {10527, 24612}, {11415, 24633}, {11433, 17167}, {11522, 16833}, {11679, 21075}, {11681, 28795}, {12571, 31211}, {13464, 16834}, {14829, 32828}, {15251, 17014}, {18141, 29456}, {18357, 29583}, {19925, 29571}, {24316, 27471}, {24817, 27949}

X(36663) = HOMOTHETOR OF THESE TRIANGLES: T((-a*b - a*c - b*c)/(2*(a^2 + b^2 + c^2))) AND EULER

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

X(36663) lies on these lines: {1, 29081}, {2, 3}, {355, 29331}, {519, 7759}, {540, 7751}, {1478, 21010}, {3818, 24220}, {5587, 18788}, {5816, 31670}, {5886, 29373}, {11550, 17167}

X(36664) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + b^2 + c^2)/S) AND EULER

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

X(36664) lies on these lines: {2, 3}, {141, 6202}, {182, 14242}, {639, 18840}, {1352, 7582}, {1587, 13972}, {1588, 10516}, {3068, 10514}, {3589, 5871}, {3618, 10783}, {3818, 14227}, {6215, 7585}, {6281, 19054}, {6290, 13886}, {7581, 14561}, {14237, 18841}

X(36665) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/S) AND EULER

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

X(36665) lies on these lines: {2, 3}, {141, 6201}, {182, 14227}, {640, 18840}, {1352, 7581}, {1587, 10516}, {1588, 13910}, {3069, 10515}, {3589, 5870}, {3618, 10784}, {3818, 14242}, {6214, 7586}, {6278, 19053}, {6289, 13939}, {7582, 14561}, {10514, 31412}, {11917, 32814}, {14232, 18841}

X(36666) = HOMOTHETOR OF THESE TRIANGLES: T(2*(a^2 + b^2 + c^2))/S) AND EULER

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

X(36666) lies on these lines: {2, 3}, {3589, 14242}, {7582, 10516}, {10514, 13886}, {14243, 18841}, {18840, 23311}

X(36667) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a^2 + b^2 + c^2))/S) AND EULER

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

X(36667) lies on these lines: {2, 3}, {3589, 14227}, {7581, 10516}, {10515, 13939}, {14228, 18841}, {18840, 23312}

X(36668) = X(2)X(1082)∩X(214)X(519)

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

X(36668) lies on the cubic K1148 and these lines: {2, 1082}, {214, 519}, {299, 320}, {619, 3666}, {3639, 27751}, {7026, 34234}

X(36668) = reflection of X(36669) in X(3911)
X(36668) = X(i)-complementary conjugate of X(j) for these (i,j): {11073, 624}, {14358, 21237}
X(36668) = X(i)-isoconjugate of X(j) for these (i,j): {106, 7126}, {2316, 7052}, {7043, 9456}
X(36668) = barycentric product X(i)*X(j) for these {i,j}: {1227, 33655}, {3264, 7051}
X(36668) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 7126}, {214, 5240}, {519, 7043}, {1319, 7052}, {5239, 1320}, {7026, 36590}, {7051, 106}, {7127, 2316}, {33655, 1168}

X(36669) = X(2)X(559)∩X(214)X(519)

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

X(36669) lies on the cubic K1148 and these lines: {2, 559}, {214, 519}, {298, 320}, {618, 3666}, {3638, 27751}, {7043, 34234}

X(36669) = reflection of X(36668) in X(3911)
X(36669) = X(i)-complementary conjugate of X(j) for these (i,j): {11072, 623}, {14359, 21237}
X(36669) = X(i)-isoconjugate of X(j) for these (i,j): {106, 19551}, {1168, 7127}, {2316, 33655}, {7026, 9456}
X(36669) = barycentric product X(i)*X(j) for these {i,j}: {1227, 7052}, {3264, 19373}
X(36669) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 19551}, {214, 5239}, {519, 7026}, {1319, 33655}, {5240, 1320}, {7043, 36590}, {7052, 1168}, {17455, 7127}, {19373, 106}

leftri

Homothetors involving the Euler triangle and triangles T(k): X(36670)-X(36695)

rightri

This preamble and centers X(36670)-X(36695) were contributed by Clark Kimberling and Peter Moses, January 25, 2020.

In this section, k is a quotient of symmetric functions of homogeneity degree 2. The Euler triangle is homothetic to each triangle T(1 : k : k), with homothetor on the Euler line. See the preambles just before X(36436), X(36473), and X(36651).

X(36670) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + b^2 + c^2)/(a + b + c)^2) AND EULER

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

X(36670) lies on these lines: {2, 3}, {43, 6769}, {182, 3332}, {344, 30273}, {346, 29010}, {387, 15488}, {672, 5709}, {1072, 11269}, {1699, 29633}, {1738, 12717}, {3333, 13161}, {3672, 20430}, {4253, 5286}, {4258, 7745}, {4911, 5714}, {5022, 5254}, {5552, 20556}, {5691, 29637}, {5811, 30946}, {6260, 20335}, {7989, 19856}, {8299, 11500}, {10446, 14853}, {17278, 30271}

X(36671) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/(a + b + c)^2) AND EULER

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

X(36671) lies on these lines: {2, 3}, {1699, 29674}, {3332, 3818}, {3673, 5714}, {5282, 5709}, {5691, 29646}, {16825, 19925}, {17017, 18528}, {18529, 29821}, {19768, 30828}

X(36672) = HOMOTHETOR OF THESE TRIANGLES: T((a + b + c)^2/(a^2 + b^2 + c^2)) AND EULER

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

X(36672) lies on these lines: {2, 3}, {387, 3767}, {1330, 32816}, {1699, 19856}, {1834, 13881}, {2238, 5706}, {5816, 6776}, {5818, 17737}, {7988, 29637}, {7989, 29633}, {10449, 32828}, {10519, 24220}, {19843, 20544}, {24248, 27691}

X(36673) = HOMOTHETOR OF THESE TRIANGLES: T(-((a + b + c)^2/(a^2 + b^2 + c^2))) AND EULER

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

X(36673) lies on these lines: {2, 3}, {387, 2548}, {1330, 32828}, {3817, 16825}, {5733, 11180}, {5816, 14853}, {7988, 29646}, {7989, 29674}, {10449, 32816}, {20544, 31418}

X(36674) = HOMOTHETOR OF THESE TRIANGLES: T((a*b + a*c + b*c)/(a + b + c)^2) AND EULER

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

X(36674) lies on these lines: {2, 3}, {7, 29369}, {32, 5292}, {40, 29674}, {241, 24701}, {515, 16825}, {572, 14561}, {573, 1352}, {985, 3072}, {991, 31670}, {1062, 2356}, {1479, 2223}, {3454, 7795}, {3509, 5709}, {3576, 29646}, {4271, 5820}, {4361, 29235}, {5282, 26921}, {5718, 19758}, {5759, 29085}, {5791, 26036}, {6184, 17732}, {8301, 11500}, {9548, 32778}, {11495, 29291}, {14826, 22139}

X(36675) = HOMOTHETOR OF THESE TRIANGLES: T((-(a*b) - a*c - b*c)/(a + b + c)^2) AND EULER

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

X(36675) lies on these lines: {2, 3}, {115, 5292}, {1352, 32431}, {5475, 20970}, {7694, 12571}, {18483, 28881}

X(36676) = HOMOTHETOR OF THESE TRIANGLES: T((a + b + c)^2/(a*b + a*c + b*c)) AND EULER

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

X(36676) lies on these lines: {2, 3}, {387, 31404}, {1330, 32832}, {3487, 26012}, {7752, 10449}, {8227, 16825}, {10175, 29674}

X(36677) = HOMOTHETOR OF THESE TRIANGLES: T(-((a + b + c)^2/(a*b + a*c + b*c))) AND EULER

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

X(36677) lies on these lines: {2, 3}, {5714, 26012}

X(36678) = HOMOTHETOR OF THESE TRIANGLES: T(S/(a + b + c)^2) AND EULER

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

X(36678) lies on these lines: {2, 3}, {1685, 6564}, {3071, 5292}, {5791, 31562}, {6565, 13333}, {7596, 18483}

X(36679) = HOMOTHETOR OF THESE TRIANGLES: T(-(S/(a + b + c)^2)) AND EULER

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

X(36679) lies on these lines: {2, 3}, {1686, 6565}, {3070, 5292}, {5791, 31561}, {6564, 13332}

X(36680) = HOMOTHETOR OF THESE TRIANGLES: T((a + b + c)^2/S) AND EULER

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

X(36680) lies on these lines: {2, 3}, {5816, 7582}

X(36681) = HOMOTHETOR OF THESE TRIANGLES: T(-((a + b + c)^2/S)) AND EULER

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

X(36681) lies on these lines: {2, 3}, {5816, 7581}

X(36682) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a^2 + b^2 + c^2))/(a + b + c)^2) AND EULER

Barycentrics    a^6 + 2*a^5*b - 3*a^4*b^2 - a^2*b^4 - 2*a*b^5 + 3*b^6 + 2*a^5*c + 2*a^4*b*c - 2*a*b^4*c - 2*b^5*c - 3*a^4*c^2 - 14*a^2*b^2*c^2 + 4*a*b^3*c^2 - 3*b^4*c^2 + 4*a*b^2*c^3 + 4*b^3*c^3 - a^2*c^4 - 2*a*b*c^4 - 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5 + 3*c^6 : :

X(36682) lies on these lines: {2, 3}, {1072, 5231}, {3332, 3589}, {3673, 5704}, {4000, 12618}, {4357, 5817}, {4429, 35514}, {5587, 19868}, {5658, 25527}, {5749, 5805}, {5759, 17353}, {14646, 33068}, {17282, 21151}, {21168, 26685}

X(36683) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a^2 + b^2 + c^2))/(a + b + c)^2) AND EULER

Barycentrics    -a^6 - 2*a^5*b - 5*a^4*b^2 + a^2*b^4 + 2*a*b^5 + 5*b^6 - 2*a^5*c - 2*a^4*b*c + 2*a*b^4*c + 2*b^5*c - 5*a^4*c^2 - 18*a^2*b^2*c^2 - 4*a*b^3*c^2 - 5*b^4*c^2 - 4*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + 2*a*b*c^4 - 5*b^2*c^4 + 2*a*c^5 + 2*b*c^5 + 5*c^6 : :

X(36683) lies on these lines: {2, 3}, {3454, 18840}, {4358, 9779}, {4911, 5704}, {18841, 20083}

X(36684) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a + b + c)^2)/(a^2 + b^2 + c^2)) AND EULER

Barycentrics    a^6 - 3*a^4*b^2 - 8*a^3*b^3 - a^2*b^4 + 8*a*b^5 + 3*b^6 - 8*a^3*b^2*c - 8*a^2*b^3*c + 8*a*b^4*c + 8*b^5*c - 3*a^4*c^2 - 8*a^3*b*c^2 - 14*a^2*b^2*c^2 - 16*a*b^3*c^2 - 3*b^4*c^2 - 8*a^3*c^3 - 8*a^2*b*c^3 - 16*a*b^2*c^3 - 16*b^3*c^3 - a^2*c^4 + 8*a*b*c^4 - 3*b^2*c^4 + 8*a*c^5 + 8*b*c^5 + 3*c^6 : :

X(36684) lies on these lines: {2, 3}, {387, 13881}, {5816, 14912}

X(36685) = HOMOTHETOR OF THESE TRIANGLES: T((a*b + a*c + b*c)/(2*(a + b + c)^2)) AND EULER

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

X(36685) lies on these lines: {2, 3}, {516, 3818}, {990, 29077}, {1479, 21010}, {2356, 8144}, {3454, 3734}, {3579, 28897}, {5292, 7737}, {7804, 20083}, {12699, 29365}

X(36686) = HOMOTHETOR OF THESE TRIANGLES: T( (-(a*b) - a*c - b*c)/(2*(a + b + c)^2)) AND EULER

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

X(36686) lies on these lines: {2, 3}, {3454, 7825}, {7861, 20083}, {18483, 28849}

X(36687) = HOMOTHETOR OF THESE TRIANGLES: T(-(a + b + c)^2/(2*(a*b + a*c + b*c)) AND EULER

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

X(36687) lies on these lines: {2, 3}, {10, 24045}, {1330, 7773}, {5587, 28870}, {9612, 26012}

X(36688) = HOMOTHETOR OF THESE TRIANGLES: T((2*S)/(a + b + c)^2) AND EULER

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

X(36688) lies on these lines: {2, 3}, {1588, 5292}, {5705, 31562}, {5715, 30324}, {6202, 7683}, {6245, 30276}, {6260, 30381}, {8233, 8957}

X(36689) = HOMOTHETOR OF THESE TRIANGLES: T( (-2*S)/(a + b + c)^2) AND EULER

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

X(36689) lies on these lines: {2, 3}, {1587, 5292}, {5705, 31561}, {5715, 30325}, {6201, 7683}, {6245, 30277}, {6260, 30380}

X(36690) = HOMOTHETOR OF THESE TRIANGLES: T((a + b + c)^2/(2*S)) AND EULER

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

X(36690) lies on these lines: {2, 3}, {387, 486}, {1588, 5816}

X(36691) = HOMOTHETOR OF THESE TRIANGLES: T(-(a + b + c)^2/(2*S)) AND EULER

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

X(36691) lies on these lines: {2, 3}, {387, 485}, {1587, 5816}

X(36692) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 - a*b + b^2 - a*c - b*c + c^2)/(a^2 + a*b + b^2 + a*c + b*c + c^2)) AND EULER

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

X(36692) lies on these lines: {2, 3}, {3817, 29633}, {12651, 16569}, {19925, 29637}

X(36693) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + a*b + b^2 + a*c + b*c + c^2)/(a^2 - a*b + b^2 - a*c - b*c + c^2)) AND EULER

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

X(36693) lies on these lines: {2, 3}, {516, 19856}, {3817, 29637}, {7785, 20077}, {12680, 28600}, {19925, 29633}, {28653, 30271}

X(36694) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 - 2*a*b + b^2 - 2*a*c - 2*b*c + c^2)/(a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2)) AND EULER

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

X(36694) lies on these lines: {2, 3}, {1002, 13374}, {4452, 20430}, {5811, 20347}, {6260, 30949}, {9842, 20335}

X(36695) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2)/(a^2 - 2*a*b + b^2 - 2*a*c - 2*b*c + c^2)) AND EULER

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

X(36695) lies on these lines: {2, 3}, {347, 1893}, {946, 30961}, {1002, 14872}, {3332, 5816}, {4461, 20430}, {5534, 17018}, {10449, 32834}

X(36696) =  X(2)X(6096)∩X(6)X(110)

Barycentrics    a^8-2 a^6 (b^2+c^2)-a^2 b^2 c^2 (b^2+c^2)+a^4 (-3 b^4+11 b^2 c^2-3 c^4) : :
Barycentrics    (18 R^2 SB+18 R^2 SC+6 R^2 SW-2 SB SW-2 SC SW-SW^2)S^2+SB SC SW^2-SB SW^3-SC SW^3 : :
X(36696) = 2*X(6)+X(111),4*X(6)-X(10765),X(6)+2*X(28662),X(69)-4*X(6719),2*X(111)+X(10765),X(111)-4*X(28662),2*X(126)-5*X(3618),4*X(182)-X(1296),4*X(597)-X(10717),X(895)+2*X(9129),X(1351)+2*X(14650),4*X(1386)-X(10704),X(1992)+2*X(9172),X(3751)+2*X(11721),4*X(5480)-X(10734),2*X(5512)+X(6776),X(9156)-4*X(9188),X(10748)-4*X(18583),X(10765)+8*X(28662)

See Tran Quang Hung and Ercole Suppa, Euclid 560 .

X(36696) lies on these lines: {2,6096}, {6,110}, {69,6387}, {115,25320}, {126,3618}, {182,729}, {352,34015}, {511,5166}, {512,15387}, {543,5034}, {597,10717}, {1084,4558}, {1351,14650}, {1386,10704}, {1570,15560}, {1691,9218}, {1976,14948}, {1992,9172}, {2780,5622}, {2847,35906}, {3751,11721}, {5050,33962}, {5052,34010}, {5480,10734}, {5512,6776}, {6094,30535}, {6794,16278}, {9156,9188}, {10748,18583}, {10754,25315}, {11579,14700}, {14848,32424}, {14853,23699}, {15566,21906}, {30435,34106}

X(36696) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {6,111,10765), (6,3124,895), (6,28662,111), (6,32740,110), (3124,9129,111}

leftri

Homothetors involving the infinite altitude triangle and triangles T(k): X(36697)-X(36716)

rightri

This preamble and centers X(36697)-X(36716) were contributed by Clark Kimberling and Peter Moses, January 27, 2020.

In this section, k is a quotient of symmetric functions of homogeneity degree 2. The infinite altitude triangle is homothetic to each triangle T(1 : k : k), with homothetor on the Euler line. See the preambles just before X(36436), X(36473), and X(36651).

X(36697) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/(a*b + a*c + b*c)) AND INFINITE ALTITUDE TRIANGLE

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

X(36697) lies on these lines: {2, 3}, {40, 16825}, {182, 10446}, {515, 29674}, {601, 985}, {602, 6361}, {946, 29646}, {1072, 29658}, {1745, 20731}, {3086, 17798}, {4362, 10476}, {4911, 11374}, {5286, 33863}, {8301, 10310}, {12116, 20539}, {17233, 29235}, {17350, 29369}

X(36698) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a*b + a*c + b*c))/(a^2 + b^2 + c^2)) AND INFINITE ALTITUDE TRIANGLE

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

X(36698) lies on these lines: {2, 3}, {8, 24633}, {40, 3912}, {69, 573}, {100, 28795}, {142, 10444}, {165, 17284}, {198, 27509}, {239, 944}, {241, 17170}, {329, 25083}, {344, 1766}, {345, 17742}, {497, 2223}, {515, 4384}, {516, 29571}, {517, 17316}, {572, 3618}, {946, 16831}, {948, 5088}, {962, 5308}, {980, 5712}, {988, 4298}, {1040, 2356}, {1376, 30847}, {1385, 26626}, {1445, 18650}, {1482, 29585}, {1764, 18141}, {1790, 11427}, {3008, 4297}, {3035, 30826}, {3434, 28797}, {3576, 17023}, {3579, 29579}, {3661, 5657}, {3664, 10443}, {3687, 9548}, {3785, 14829}, {3926, 4417}, {4000, 5336}, {4393, 7967}, {4648, 10446}, {4872, 31225}, {5132, 5800}, {5179, 28827}, {5222, 5731}, {5493, 29600}, {5587, 24603}, {5603, 16826}, {5691, 16832}, {5745, 26036}, {5813, 24635}, {5818, 29576}, {5834, 34522}, {5882, 16834}, {6350, 24611}, {6361, 17244}, {6542, 12245}, {6604, 20367}, {6684, 17308}, {7982, 29574}, {7987, 29598}, {7991, 29573}, {8804, 27384}, {8965, 31552}, {9778, 29627}, {10164, 29604}, {10165, 29603}, {10436, 10445}, {10595, 29570}, {11362, 17294}, {11495, 16593}, {11531, 29602}, {12610, 17321}, {12651, 17022}, {12702, 29583}, {13329, 25406}, {13464, 29597}, {13478, 32022}, {14555, 16552}, {16560, 24683}, {17077, 21279}, {18228, 25066}, {18655, 21617}, {20070, 29621}, {20533, 35514}, {24590, 25935}, {24682, 27473}, {24703, 30812}, {28164, 31211}, {28228, 29606}, {28234, 29605}, {29596, 35242}

X(36698) = {X(3),X(4)}-harmonic conjugate of X(36706)

X(36699) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a^2 + b^2 + c^2))/(a*b + a*c + b*c)) AND INFINITE ALTITUDE TRIANGLE

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

X(36699) lies on these lines: {2, 3}, {944, 32847}, {3332, 33750}, {3673, 5122}, {5092, 10446}, {17230, 29081}

X(36700) = HOMOTHETOR OF THESE TRIANGLES: T((-a*b - a*c - b*c)/S) AND INFINITE ALTITUDE TRIANGLE

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

X(36700) lies on these lines: {2, 3}, {26036, 32555}

X(36701) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/S) AND INFINITE ALTITUDE TRIANGLE

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

X(36701) lies on these lines: {2, 3}, {69, 10784}, {141, 5870}, {182, 7581}, {487, 10517}, {492, 10518}, {511, 7582}, {638, 10783}, {1152, 5286}, {1160, 7586}, {1249, 11514}, {1350, 1588}, {1352, 14227}, {1579, 8743}, {1587, 5085}, {3069, 11825}, {3071, 31884}, {3098, 23273}, {3589, 6201}, {5092, 23267}, {5254, 6410}, {5874, 32814}, {6202, 29181}, {6396, 26294}, {6409, 7745}, {6425, 31465}, {6459, 11824}, {7585, 26348}, {9541, 12306}, {12251, 13766}, {12305, 13935}, {13941, 35247}, {14232, 18840}, {14810, 23275}, {17508, 23269}

X(36702) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a^2 + b^2 + c^2))/S) AND INFINITE ALTITUDE TRIANGLE

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

X(36702) lies on these lines: {2, 3}, {141, 14242}, {1350, 7581}, {1587, 31884}, {5085, 7582}, {5286, 6409}, {10518, 12256}, {10519, 10783}, {14243, 18840}

X(36703) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + b^2 + c^2)/S) AND INFINITE ALTITUDE TRIANGLE

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

X(36703) lies on these lines: {2, 3}, {69, 10783}, {141, 5871}, {182, 7582}, {488, 10518}, {491, 10517}, {511, 7581}, {637, 10784}, {1151, 5286}, {1161, 7585}, {1249, 11513}, {1350, 1587}, {1352, 14242}, {1578, 8743}, {1588, 5085}, {3068, 11824}, {3070, 31884}, {3098, 23267}, {3589, 6202}, {5092, 23273}, {5254, 6409}, {6200, 26295}, {6201, 29181}, {6410, 7745}, {6460, 11825}, {7586, 26341}, {8972, 35246}, {9540, 12306}, {12251, 13647}, {14237, 18840}, {14810, 23269}, {17508, 23275}

X(36704) = HOMOTHETOR OF THESE TRIANGLES: T((a*b + a*c + b*c)/S) AND INFINITE ALTITUDE TRIANGLE

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

X(36704) lies on these lines: {2, 3}, {26036, 32556}

X(36705) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a^2 + b^2 + c^2))/(a*b + a*c + b*c)) AND INFINITE ALTITUDE TRIANGLE

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

X(36705) lies on these lines: {2, 3}, {10446, 14810}, {17236, 29085}

X(36706) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a*b + a*c + b*c))/(a^2 + b^2 + c^2)) AND INFINITE ALTITUDE TRIANGLE

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

X(36706) lies on these lines: {1, 348}, {2, 3}, {8, 24635}, {69, 991}, {86, 3332}, {307, 7675}, {344, 12618}, {388, 2223}, {497, 27339}, {516, 10436}, {572, 25406}, {894, 5759}, {971, 17257}, {988, 11019}, {990, 17321}, {1038, 2356}, {1043, 3926}, {1448, 17093}, {1790, 11206}, {1944, 5698}, {3286, 5800}, {3618, 13329}, {3662, 21151}, {4026, 11495}, {4294, 14942}, {4297, 19868}, {4340, 14828}, {4357, 5732}, {5250, 6225}, {5266, 10578}, {5731, 28901}, {5817, 17260}, {16020, 24781}, {17350, 21168}, {17353, 21153}, {19836, 35202}, {26685, 31658}, {27334, 35514}

X(36706) = {X(3),X(4)}-harmonic conjugate of X(36698)

X(36707) = HOMOTHETOR OF THESE TRIANGLES: T((a*b + a*c + b*c)/(2*(a^2 + b^2 + c^2))) AND INFINITE ALTITUDE TRIANGLE

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

X(36707) lies on these lines: {2, 3}, {355, 28850}, {572, 29012}, {894, 29085}, {991, 3818}, {2223, 3585}, {2356, 18447}, {4026, 29291}, {13329, 19130}

X(36708) = HOMOTHETOR OF THESE TRIANGLES: T(S/(a*b + a*c + b*c)) AND INFINITE ALTITUDE TRIANGLE

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

X(36708) lies on these lines: {2, 3}, {638, 17206}, {1587, 5021}, {1588, 2271}, {2200, 32590}, {3070, 33863}, {3071, 18755}, {3332, 12256}, {9733, 10446}

X(36709) = HOMOTHETOR OF THESE TRIANGLES: T(S/(a^2 + b^2 + c^2)) AND INFINITE ALTITUDE TRIANGLE

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

X(36709) lies on these lines: {2, 3}, {32, 3070}, {39, 3071}, {69, 1161}, {141, 11824}, {147, 8304}, {187, 6250}, {193, 11916}, {371, 1503}, {372, 5480}, {485, 6222}, {488, 1160}, {491, 6215}, {637, 3933}, {638, 7767}, {639, 30270}, {1352, 9732}, {1384, 23249}, {1587, 30435}, {1588, 9605}, {1991, 6281}, {3053, 23251}, {3068, 5871}, {3069, 6202}, {3092, 23115}, {3095, 13766}, {3311, 6776}, {3312, 12256}, {3618, 26341}, {3619, 35246}, {3785, 12323}, {3818, 9738}, {3867, 11514}, {3926, 12322}, {5013, 23261}, {5024, 23259}, {5870, 6459}, {6201, 6460}, {6221, 12257}, {6251, 32494}, {6278, 7758}, {6337, 12296}, {6417, 14912}, {6419, 8550}, {6561, 8721}, {6564, 14230}, {7585, 10783}, {7710, 10839}, {7735, 10846}, {7795, 10515}, {9733, 31670}, {10132, 31383}, {10516, 12306}, {10983, 12601}, {10991, 13908}, {11090, 33586}, {11825, 29181}, {12007, 35771}, {12313, 18440}, {14233, 35821}, {14235, 35786}, {18509, 32806}, {18860, 32435}, {21309, 23267}, {23253, 26330}, {25066, 31562}, {25406, 26348}

X(36710) = HOMOTHETOR OF THESE TRIANGLES: T(S/(2*a*b + 2*a*c + 2*b*c)) AND INFINITE ALTITUDE TRIANGLE

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

X(36710) lies on these lines: {2, 3}, {1160, 10446}, {2271, 3071}, {3070, 5021}, {12323, 17206}, {18755, 23261}, {23251, 33863}

X(36711) = HOMOTHETOR OF THESE TRIANGLES: T(S/(2*(a^2 + b^2 + c^2))) AND INFINITE ALTITUDE TRIANGLE

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

X(36711) lies on these lines: {2, 3}, {32, 23251}, {39, 23261}, {1131, 14243}, {1160, 31670}, {1161, 1352}, {1384, 13711}, {1503, 3311}, {3070, 30435}, {3071, 9605}, {3092, 22120}, {3095, 12601}, {3312, 5480}, {3564, 11916}, {3818, 9732}, {3933, 12322}, {5871, 7583}, {6202, 7584}, {6290, 18509}, {6395, 12256}, {6417, 6776}, {6418, 14853}, {6427, 8550}, {6500, 14912}, {6561, 22537}, {7585, 14242}, {7767, 12323}, {8721, 14233}, {9753, 10846}, {10516, 11824}, {10783, 19117}, {11917, 21850}, {13665, 13749}, {14561, 26341}, {21309, 23249}, {22246, 23273}, {22682, 22725}

X(36712) = HOMOTHETOR OF THESE TRIANGLES: T(-S/(2*(a^2 + b^2 + c^2))) AND INFINITE ALTITUDE TRIANGLE

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

X(36712) lies on these lines: {2, 3}, {32, 23261}, {39, 23251}, {1132, 14228}, {1160, 1352}, {1161, 31670}, {1384, 13834}, {1503, 3312}, {3070, 9605}, {3071, 30435}, {3093, 22120}, {3095, 12602}, {3311, 5480}, {3564, 11917}, {3818, 9733}, {3933, 12323}, {5870, 7584}, {6199, 12257}, {6201, 7583}, {6289, 18511}, {6417, 14853}, {6418, 6776}, {6428, 8550}, {6501, 14912}, {6560, 22536}, {7586, 14227}, {7767, 12322}, {8721, 14230}, {9753, 10845}, {10516, 11825}, {10784, 19116}, {11916, 21850}, {13748, 13785}, {14561, 26348}, {21309, 23259}, {22246, 23267}, {22682, 22724}

X(36713) = HOMOTHETOR OF THESE TRIANGLES: T(-(S/(2*a*b + 2*a*c + 2*b*c))) AND INFINITE ALTITUDE TRIANGLE

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

X(36713) lies on these lines: {2, 3}, {1161, 10446}, {2271, 3070}, {3071, 5021}, {12322, 17206}, {18755, 23251}, {23261, 33863}

X(36714) = HOMOTHETOR OF THESE TRIANGLES: T(-(S/(a^2 + b^2 + c^2))) AND INFINITE ALTITUDE TRIANGLE

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

X(36714) lies on these lines: {2, 3}, {32, 3071}, {39, 3070}, {69, 1160}, {141, 11825}, {147, 8305}, {187, 6251}, {193, 11917}, {371, 5480}, {372, 1503}, {486, 6399}, {487, 1161}, {492, 6214}, {591, 6278}, {637, 7767}, {638, 3933}, {640, 30270}, {1352, 9733}, {1384, 23259}, {1587, 9605}, {1588, 30435}, {3053, 23261}, {3068, 6201}, {3069, 5870}, {3093, 23115}, {3095, 13647}, {3311, 12257}, {3312, 6776}, {3618, 26348}, {3619, 35247}, {3785, 12322}, {3818, 9739}, {3867, 11513}, {3926, 12323}, {5013, 23251}, {5024, 23249}, {5871, 6460}, {6202, 6459}, {6250, 32497}, {6281, 7758}, {6337, 12297}, {6398, 12256}, {6418, 14912}, {6420, 8550}, {6560, 8721}, {6565, 14233}, {7586, 10784}, {7710, 10840}, {7735, 10845}, {7795, 10514}, {9732, 31670}, {10133, 31383}, {10516, 12305}, {10983, 12602}, {10991, 13968}, {11091, 33586}, {11824, 29181}, {12007, 35770}, {12314, 18440}, {14230, 35820}, {14239, 35787}, {18511, 32805}, {18860, 32432}, {21309, 23273}, {23263, 26331}, {25066, 31561}, {25406, 26341}

X(36715) = HOMOTHETOR OF THESE TRIANGLES: T(-(S/(a*b + a*c + b*c))) AND INFINITE ALTITUDE TRIANGLE

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

X(36715) lies on these lines: {2, 3}, {637, 17206}, {1587, 2271}, {1588, 5021}, {2200, 32592}, {3070, 18755}, {3071, 33863}, {3332, 12257}, {9732, 10446}

X(36716) = HOMOTHETOR OF THESE TRIANGLES: T((-(a*b) - a*c - b*c)/(2*(a^2 + b^2 + c^2))) AND INFINITE ALTITUDE TRIANGLE

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

X(36716) lies on these lines: {2, 3}, {238, 29020}, {239, 29081}, {517, 20715}, {572, 19130}, {573, 3818}, {2223, 3583}, {2356, 18455}, {3017, 5007}, {5134, 6184}, {5266, 31795}, {13329, 29012}, {24833, 29069}, {26446, 28866}

X(36717) = HOMOTHETOR OF THESE TRIANGLES: T(-S/(2*(a^2 + b^2 + c^2)); ) AND REFLECTION OF ABC IN X(3)

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

X(36717) lies on these lines: {2, 3}, {141, 14227}, {1350, 7582}, {1588, 31884}, {5085, 7581}, {5286, 6410}, {10517, 12257}, {10519, 10784}, {14228, 18840}

leftri

Homothetors involving the Ehrmann mid-triangle and triangles T(k): X(36718)-X(367)

rightri

This preamble and centers X(36718)-X(367XX) were contributed by Clark Kimberling and Peter Moses, January 28, 2020.

In this section, k is a quotient of symmetric functions of homogeneity degree 2. The Ehrmann mid-triangle is homothetic to each triangle T(1 : k : k), with homothetor on the Euler line. See the preambles just before X(36436), X(36473), and X(36651).

X(36718) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a^2 + b^2 + c^2))/S) AND EHRMANN MID-TRIANGLE

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

X(38718) lies on these lines: {2, 3}, {3098, 18509}, {18512, 31670}, {26336, 33878}

X(36719) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + b^2 + c^2)/S) AND EHRMANN MID-TRIANGLE

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

X(38719) lies on these lines: {2, 3}, {69, 26336}, {141, 18509}, {1160, 13749}, {1161, 1991}, {5309, 6424}, {5860, 5871}, {5874, 14242}, {6421, 7753}, {6560, 15484}, {7818, 11825}, {8396, 32787}, {13665, 31670}, {13785, 13972}, {18512, 21850}, {19146, 35823}

X(36720) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a^2 + b^2 + c^2))/(a*b + a*c + b*c)) AND EHRMANN MID-TRIANGLE

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

X(38720) lies on these lines: {2, 3}, {12702, 28854}, {17333, 29085}

X(36721) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a*b + a*c + b*c))/(a^2 + b^2 + c^2)) AND EHRMANN MID-TRIANGLE

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

X(38721) lies on these lines: {2, 3}, {894, 31671}, {971, 17274}, {990, 17382}, {991, 17313}, {1742, 31151}, {1834, 7739}, {4357, 31672}, {5657, 28915}, {5722, 9436}, {5779, 17333}, {5790, 29365}, {5807, 15956}, {9668, 14942}, {10394, 36589}, {10436, 18482}, {12618, 17281}, {19868, 31673}, {21629, 28580}

X(36722) = HOMOTHETOR OF THESE TRIANGLES: T((a^2 + b^2 + c^2)/(2*(a*b + a*c + b*c))) AND EHRMANN MID-TRIANGLE

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

X(38722) lies on these lines: {2, 3}, {10, 28854}, {495, 14942}, {1834, 5309}, {3019, 3629}, {3753, 32062}, {4357, 18482}, {5762, 17333}, {5805, 17274}, {10436, 31672}, {17257, 31671}, {18483, 19868}, {19870, 34618}

X(36723) = HOMOTHETOR OF THESE TRIANGLES: T(S/(a^2 + b^2 + c^2)) AND EHRMANN MID-TRIANGLE

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

X(38723) lies on these lines: {2, 3}, {1991, 6215}, {3068, 18509}, {5480, 32421}, {5860, 6202}, {6564, 18907}, {6565, 15048}, {7585, 26336}, {13763, 23249}, {13972, 19130}, {18511, 23259}, {23273, 26346}

X(36724) = HOMOTHETOR OF THESE TRIANGLES: T(S/(2*(a^2 + b^2 + c^2))) AND EHRMANN MID-TRIANGLE

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

X(38724) lies on these lines: {2, 3}, {13711, 18907}

X(36725) = HOMOTHETOR OF THESE TRIANGLES: T(-S/(2*(a^2 + b^2 + c^2))) AND EHRMANN MID-TRIANGLE

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

X(38725) lies on these lines: {2, 3}, {13834, 18907}

X(36726) = HOMOTHETOR OF THESE TRIANGLES: T(-(S/(a^2 + b^2 + c^2))) AND EHRMANN MID-TRIANGLE

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

X(38726) lies on these lines: {2, 3}, {591, 6214}, {3069, 18511}, {5480, 32419}, {5861, 6201}, {6564, 15048}, {6565, 18907}, {7586, 26346}, {13644, 23259}, {13910, 19130}, {18509, 23249}, {23267, 26336}

X(36727) = HOMOTHETOR OF THESE TRIANGLES: T((-(a*b) - a*c - b*c)/(2*(a^2 + b^2 + c^2))) AND EHRMANN MID-TRIANGLE

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

X(38727) lies on these lines: {2, 3}, {22791, 32847}

X(36728) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/(2*(a*b + a*c + b*c))) AND EHRMANN MID-TRIANGLE

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

X(38728) lies on these lines: {2, 3}, {141, 32431}, {239, 22791}, {355, 17294}, {952, 17389}, {1699, 33132}, {3008, 18483}, {3579, 24603}, {3655, 29597}, {3656, 16834}, {3661, 18357}, {3817, 28845}, {3912, 18480}, {4384, 12699}, {4654, 5928}, {5511, 7965}, {5816, 17251}, {7753, 20970}, {9955, 17023}, {10446, 17346}, {10888, 31142}, {16826, 34773}, {16831, 18481}, {16833, 31162}, {17284, 18492}, {17316, 18525}, {17392, 24220}, {17647, 30812}, {18493, 26626}, {18526, 29585}, {24390, 24612}, {28204, 29574}, {29571, 31673}, {29600, 34648}

X(36729) = HOMOTHETOR OF THESE TRIANGLES: T((-(a*b) - a*c - b*c)/(a^2 + b^2 + c^2)) AND EHRMANN MID-TRIANGLE

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

X(38729) lies on these lines: {2, 3}, {1699, 29365}, {8301, 18491}, {11231, 28897}, {12699, 29674}, {16825, 18480}, {17274, 29369}, {18481, 29646}

X(36730) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/(a*b + a*c + b*c)) AND EHRMANN MID-TRIANGLE

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

X(38730) lies on these lines: {2, 3}, {58, 5309}, {115, 4257}, {386, 7753}, {3098, 32431}, {4256, 5475}, {5886, 28845}, {9955, 29646}, {12699, 16825}, {17333, 29369}, {18480, 29674}

X(36731) = HOMOTHETOR OF THESE TRIANGLES: T((2*(a^2 + b^2 + c^2))/S; ) AND EHRMANN MID-TRIANGLE

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

X(38731) lies on these lines: {2, 3}, {165, 28897}, {239, 18525}, {516, 10516}, {517, 17294}, {527, 10445}, {573, 17251}, {1482, 17389}, {1766, 17359}, {3008, 31673}, {3579, 17308}, {3656, 29574}, {3661, 12702}, {3912, 12699}, {4384, 18480}, {4393, 18526}, {6361, 29611}, {6542, 8148}, {9812, 28915}, {9955, 16831}, {10446, 17297}, {10609, 28922}, {11238, 21010}, {11278, 29605}, {12610, 17301}, {13624, 29603}, {16826, 18493}, {16832, 18492}, {16834, 28204}, {17023, 18481}, {17259, 32431}, {17316, 22791}, {18483, 29571}, {25440, 30826}, {26626, 34773}, {28194, 29594}, {29365, 29575}, {29573, 31162}, {29604, 31730}, {29615, 34718}

X(36732) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a^2 + b^2 + c^2))/(a*b + a*c + b*c)) AND EHRMANN MID-TRIANGLE

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

X(38732) lies on these lines: {2, 3}, {10246, 28845}, {17389, 29081}, {18525, 32847}

X(36733) = HOMOTHETOR OF THESE TRIANGLES: T((-a^2 - b^2 - c^2)/S) AND EHRMANN MID-TRIANGLE

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

X(38733) lies on these lines: {2, 3}, {69, 26346}, {141, 18511}, {591, 1160}, {1161, 13748}, {5309, 6423}, {5475, 9600}, {5861, 5870}, {5875, 14227}, {6422, 7753}, {6561, 15484}, {7818, 11824}, {8416, 32788}, {13665, 13910}, {13785, 31670}, {18510, 21850}, {19145, 35822}

X(36734) = HOMOTHETOR OF THESE TRIANGLES: T((-2*(a^2 + b^2 + c^2))/S) AND EHRMANN MID-TRIANGLE

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

X(38734) lies on these lines: {2, 3}, {3098, 18511}, {18510, 31670}, {26346, 33878}

X(36735) = X(100)X(3413)∩X(104)X(3414)

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

X(36735) lies on the circumcircle and these lines: {3, 36736}, {100, 3413}, {104, 3414}, {513, 1379}, {517, 1380}, {1341, 5091}

X(36735) = reflection of X(36736) in X(3)
X(36735) = reflection of X(1379) in the line X(1)X(3)

X(36736) = X(100)X(3414)∩X(104)X(3413)

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

X(36736) lies on the circumcircle and these lines: {3, 36735}, {100, 3414}, {104, 3413}, {513, 1380}, {517, 1379}, {1340, 5091}

X(36736) = reflection of X(36735) in X(3)
X(36736) = reflection of X(1380) in the line X(1)X(3)

X(36737) =  ISOGONAL CONJUGATE OF X(3638)

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

See Kadir Altintas and César Lozada, Euclid 583 .

X(36737) lies on the conics {{A, B, C, X(1), X(33)}}, {{A, B, C, X(9), X(7059)}}, the cubics K206, K523 and these lines: {16, 55}, {101, 10638}, {203, 3022}, {2293, 2772}, {3639, 21453}

X(36737) = isogonal conjugate of X(3638)
X(36737) = X(6)-reciprocal conjugate of-X(3638)
X(36737) = lies on the circumconic with center X(14714))
X(36737) = trilinear pole of the line {657, 7127}
X(36737) = barycentric quotient X(6)/X(3638)
X(36737) = X(2925)-of-intouch triangle

X(36738) =  ISOGONAL CONJUGATE OF X(3639)

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

See Kadir Altintas and César Lozada, Euclid 583 .

X(36738) lies on the conics {{A, B, C, X(1), X(33)}}, {{A, B, C, X(9), X(7060)}}, the cubics K206, K523 and these lines: {15, 55}, {101, 1250}, {202, 3022}, {2293, 2772}, {3638, 21453}

X(36738) = isogonal conjugate of X(3639)
X(36738) = X(6)-reciprocal conjugate of-X(3639)
X(36738) = lies on the circumconic with center X(14714))
X(36738) = barycentric quotient X(6)/X(3639)
X(36738) = X(2926)-of-intouch triangle

X(36739) =  MIDPOINT OF X(15357) AND X(19598)

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

See Angel Montesdeoca, Euclid 589 and HG300120 .

X(36739) lies on the these lines: {2, 12064}, {110, 11123}, {125, 523}, {1511, 32204}, {5663, 8151}, {5972, 10190}, {6723, 10278}, {8029, 15059}, {9168, 13291}, {10279, 34128}, {15357, 19598}, {16220, 38728}

X(36739) = midpoint of X(15357) and X(19598)
X(36739) = reflection of X(1511) in X(32204)

X(36740) =  X(1)X(159)∩X(3)X(6)

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

X(36740) lies on these lines: {1, 159}, {2, 5324}, {3, 6}, {20, 5800}, {21, 69}, {22, 81}, {23, 14996}, {25, 940}, {28, 4340}, {31, 22097}, {35, 3751}, {36, 16475}, {37, 24320}, {41, 1818}, {55, 63}, {56, 77}, {60, 20806}, {86, 19310}, {141, 405}, {171, 197}, {193, 4189}, {206, 1437}, {380, 5732}, {394, 2194}, {404, 3618}, {474, 3589}, {524, 16370}, {542, 28444}, {597, 16371}, {599, 16418}, {604, 22390}, {611, 8069}, {613, 8071}, {954, 5845}, {956, 5846}, {958, 3416}, {993, 5847}, {999, 2097}, {1001, 4357}, {1006, 10519}, {1012, 1503}, {1213, 16849}, {1352, 3560}, {1428, 1470}, {1469, 19133}, {1473, 3666}, {1754, 18163}, {1992, 17549}, {2178, 16972}, {2264, 5784}, {2810, 12594}, {2911, 3781}, {3056, 26357}, {3149, 5480}, {3216, 31521}, {3242, 3295}, {3564, 6914}, {3619, 5047}, {3620, 16865}, {3629, 19535}, {3631, 19526}, {3746, 16496}, {3755, 24309}, {3763, 11108}, {3917, 5320}, {4223, 4648}, {4224, 5712}, {4383, 7484}, {4471, 20470}, {4641, 7085}, {4663, 5217}, {4996, 10755}, {5172, 9037}, {5227, 31424}, {5256, 7293}, {5323, 7520}, {5327, 10446}, {5358, 7535}, {5563, 16491}, {5706, 11414}, {5707, 7387}, {5710, 8192}, {5711, 9798}, {5738, 36018}, {5848, 10058}, {6329, 19537}, {6391, 34435}, {6518, 9025}, {6776, 6906}, {6905, 14853}, {6909, 25406}, {6911, 14561}, {6913, 10516}, {6924, 18583}, {6950, 14912}, {6985, 31670}, {7301, 16484}, {7465, 24597}, {7485, 32911}, {7496, 14997}, {7508, 34380}, {7580, 29181}, {8584, 19704}, {10829, 20986}, {11008, 17574}, {11031, 26934}, {11180, 28461}, {13204, 32278}, {13211, 32256}, {13567, 25907}, {13743, 18440}, {15668, 19309}, {15985, 19533}, {16048, 17234}, {16067, 28793}, {16352, 19701}, {16353, 19732}, {16580, 24701}, {16696, 19758}, {16842, 34573}, {16852, 17398}, {16857, 21358}, {16858, 21356}, {17056, 25514}, {17259, 19313}, {17277, 19314}, {17300, 17522}, {17542, 20582}, {17595, 26866}, {18134, 25494}, {18144, 19768}, {19285, 25526}, {19311, 27164}, {19459, 19765}, {20139, 33047}, {20589, 20678}, {20831, 20987}, {20834, 35623}, {23292, 25947}, {24264, 35104}, {24929, 34381}, {28348, 28369}, {34183, 34230}

X(36740) = reflection of X(6) in X(5138)
X(36740) = crossdifference of every pair of points on line {523, 2509}
X(36740) = Brocard-circle-inverse of X(36741)
X(36740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7289, 24476}, {1, 7295, 1486}, {3, 6, 36741}, {3, 4254, 5132}, {6, 1350, 4259}, {6, 4265, 3}, {35, 3751, 12329}

X(36741) =  X(1)X(12329)∩X(3)X(6)

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

X(36741) lies on these lines: {1, 12329}, {2, 5800}, {3, 6}, {21, 3618}, {22, 32911}, {23, 14997}, {25, 4383}, {35, 16475}, {36, 3751}, {41, 22390}, {43, 197}, {44, 24320}, {46, 3827}, {55, 1386}, {56, 78}, {57, 24476}, {69, 404}, {81, 7485}, {86, 19314}, {141, 474}, {159, 3216}, {193, 4188}, {206, 16471}, {212, 28274}, {218, 2172}, {238, 1486}, {405, 3589}, {411, 25406}, {524, 16371}, {597, 16370}, {599, 16417}, {604, 1818}, {611, 8071}, {613, 674}, {936, 5227}, {940, 7484}, {999, 3242}, {1001, 17023}, {1012, 5480}, {1155, 24611}, {1191, 12410}, {1213, 16852}, {1352, 6911}, {1376, 1460}, {1423, 23693}, {1428, 3779}, {1466, 7013}, {1469, 1470}, {1473, 4641}, {1503, 3149}, {1617, 3190}, {1724, 13730}, {1743, 3220}, {1754, 33811}, {1992, 13587}, {2194, 3796}, {2330, 26357}, {2339, 4640}, {2911, 7193}, {2932, 9024}, {2999, 5285}, {3560, 14561}, {3564, 6924}, {3619, 17531}, {3620, 17572}, {3629, 19537}, {3666, 7085}, {3746, 16491}, {3763, 16408}, {3844, 4413}, {4497, 20470}, {4663, 5204}, {5044, 27802}, {5247, 22654}, {5320, 22352}, {5323, 6904}, {5364, 20778}, {5476, 28444}, {5563, 16496}, {5687, 5846}, {5695, 24269}, {5706, 7395}, {5707, 7393}, {5847, 25440}, {5848, 10090}, {6007, 24265}, {6329, 19535}, {6776, 6905}, {6906, 14853}, {6914, 18583}, {6918, 10516}, {6940, 10519}, {6942, 14912}, {7074, 16541}, {7083, 20872}, {7289, 15803}, {7295, 16468}, {7496, 14996}, {7742, 22277}, {8193, 16466}, {8584, 19705}, {9052, 12595}, {10755, 17100}, {10759, 18861}, {12589, 27657}, {13211, 32270}, {13411, 25523}, {13567, 25947}, {14927, 36002}, {15668, 19313}, {16048, 17352}, {16352, 19732}, {16353, 19701}, {16849, 17398}, {16862, 34573}, {16917, 20139}, {17259, 19309}, {17277, 19310}, {19286, 25526}, {21356, 36006}, {22586, 32278}, {23292, 25907}

X(36741) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36740}, {3, 5120, 3286}, {6, 5085, 5135}, {6, 5096, 3}, {36, 3751, 22769}, {43, 5329, 197}, {182, 4260, 6}, {4383, 5347, 25}, {5256, 5314, 55}

X(36741) = Brocard-circle-inverse of X(36740)

X(36742) =  X(1)X(90)∩X(3)X(6)

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c + 2*a^3*b*c - 2*a^2*b^2*c - 2*a*b^3*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5) : :
Barycentrics    a^2*(s*S / R + SA) : :
Trilinears    cos A + sin A (sin A + sin B + sin C) : :
Trilinears    R cos A + s sin A : :

X(36742) lies on these lines: {1, 90}, {3, 6}, {4, 81}, {5, 940}, {21, 1993}, {24, 60}, {25, 1437}, {30, 5706}, {31, 10267}, {34, 222}, {35, 16473}, {36, 16472}, {42, 601}, {47, 55}, {56, 20122}, {84, 1449}, {140, 4383}, {154, 20831}, {171, 11499}, {184, 13730}, {219, 31445}, {226, 8757}, {255, 14547}, {285, 461}, {323, 16865}, {355, 5711}, {387, 6850}, {394, 405}, {404, 5422}, {474, 10601}, {495, 9370}, {595, 16202}, {602, 2308}, {608, 1871}, {611, 5266}, {614, 13373}, {631, 32911}, {651, 3487}, {946, 4667}, {952, 5710}, {995, 16203}, {999, 10571}, {1012, 1181}, {1062, 10391}, {1064, 1468}, {1092, 5320}, {1126, 35448}, {1147, 2194}, {1191, 10246}, {1193, 10269}, {1199, 6950}, {1203, 3576}, {1335, 7133}, {1385, 16466}, {1386, 12675}, {1399, 11507}, {1406, 5902}, {1407, 5708}, {1408, 5446}, {1419, 3333}, {1433, 7008}, {1451, 4303}, {1453, 18443}, {1480, 7982}, {1724, 6883}, {1834, 6923}, {1838, 7534}, {1994, 4189}, {2077, 5312}, {2303, 5778}, {2323, 31424}, {2594, 8069}, {2915, 33586}, {3060, 11337}, {3091, 14996}, {3149, 10982}, {3193, 6872}, {3216, 15805}, {3295, 22117}, {3488, 3562}, {3666, 24467}, {3745, 14872}, {3796, 20833}, {3945, 6846}, {4185, 18180}, {4188, 34545}, {4300, 35239}, {4340, 6826}, {4641, 26921}, {4648, 6887}, {4658, 18451}, {4850, 26877}, {5047, 15066}, {5228, 24470}, {5269, 5534}, {5292, 6842}, {5453, 16266}, {5482, 16434}, {5687, 17977}, {5712, 6824}, {5713, 6841}, {5718, 6862}, {5721, 6917}, {5722, 7524}, {5767, 15971}, {5800, 34938}, {5803, 15763}, {6147, 6180}, {6198, 10394}, {6829, 26131}, {6861, 17056}, {6875, 16948}, {6889, 24597}, {6906, 7592}, {6912, 11441}, {6913, 17814}, {6914, 12161}, {6937, 24883}, {7078, 24929}, {7171, 16667}, {7986, 15071}, {8760, 22383}, {9798, 20986}, {10303, 14997}, {11108, 17811}, {11374, 34048}, {11456, 21669}, {11491, 17126}, {11529, 34043}, {13567, 34120}, {13743, 18445}, {15018, 17572}, {15178, 16483}, {15317, 34435}, {15934, 23070}, {16408, 17825}, {16418, 22136}, {17527, 25934}, {22479, 26892}, {26098, 26470}

X(36742) = Brocard-circle-inverse of X(36754)
X(36742) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36744)
X(36742) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2003, 3157}, {3, 6, 36754}, {3, 1351, 5752}, {4, 81, 5707}, {42, 601, 11248}, {58, 581, 3}, {371, 372, 36744}, {500, 5398, 3}, {580, 991, 3}, {1064, 1468, 11249}

X(36743) =  X(1)X(21853)∩X(3)X(6)

Barycentrics    a^2*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :
Barycentrics    a^2*(SA - R*S / s) : :
Trilinears    cos A (sin A + sin B + sin C) - sin A : :
Trilinears    s cos A - R sin A : :
Trilinears    a - 2 s cos A : :

X(36743) lies on these lines: {1, 21853}, {2, 1444}, {3, 6}, {9, 36}, {19, 22479}, {22, 33854}, {35, 1449}, {37, 56}, {40, 3554}, {41, 22054}, {44, 198}, {45, 21773}, {48, 672}, {55, 1100}, {69, 21495}, {71, 604}, {86, 16367}, {100, 5839}, {141, 21477}, {183, 3770}, {193, 21537}, {197, 1575}, {218, 2174}, {219, 7113}, {220, 37519}, {226, 25523}, {230, 16434}, {241, 1804}, {378, 1172}, {380, 7688}, {391, 4188}, {395, 21476}, {396, 21475}, {404, 966}, {405, 1901}, {474, 1213}, {524, 16431}, {590, 16432}, {594, 956}, {597, 16436}, {599, 21539}, {615, 16433}, {836, 7114}, {910, 1436}, {940, 16696}, {958, 17303}, {992, 3330}, {993, 5750}, {999, 16777}, {1004, 15447}, {1006, 5746}, {1011, 24512}, {1014, 4648}, {1078, 34283}, {1108, 3428}, {1155, 2262}, {1319, 21871}, {1376, 17275}, {1388, 21864}, {1400, 1470}, {1486, 17798}, {1583, 31473}, {1617, 2256}, {1631, 7083}, {1696, 16814}, {1743, 7280}, {1761, 3061}, {1766, 8609}, {1778, 4225}, {1865, 4185}, {2099, 21863}, {2171, 26437}, {2223, 12329}, {2238, 4191}, {2260, 2268}, {2321, 8666}, {2323, 36152}, {2345, 2975}, {2352, 7085}, {2509, 23224}, {3068, 16440}, {3069, 16441}, {3087, 7412}, {3204, 3207}, {3218, 28936}, {3247, 5563}, {3295, 16884}, {3304, 3723}, {3435, 28266}, {3553, 3576}, {3580, 21478}, {3589, 11343}, {3618, 21511}, {3619, 21540}, {3629, 21524}, {3630, 21538}, {3631, 21532}, {3651, 5802}, {3686, 25440}, {3724, 3958}, {3763, 21526}, {3815, 19544}, {3911, 24005}, {3936, 21488}, {3964, 16728}, {4007, 5288}, {4220, 7736}, {4383, 11350}, {4426, 22654}, {4497, 8053}, {5010, 16667}, {5217, 16666}, {5275, 7484}, {5276, 7485}, {5301, 8193}, {5306, 21487}, {5329, 17754}, {5347, 20835}, {5364, 22099}, {5450, 10445}, {5687, 17362}, {5747, 6883}, {5816, 6911}, {6329, 21518}, {6882, 9722}, {7585, 21567}, {7586, 21566}, {7735, 19649}, {7792, 21485}, {8252, 21547}, {8253, 21548}, {8557, 11012}, {8584, 21497}, {8818, 11108}, {8972, 21568}, {11064, 21494}, {11194, 17281}, {11320, 26963}, {11329, 17277}, {11340, 32911}, {12410, 16781}, {12513, 17299}, {13006, 22132}, {13846, 21561}, {13847, 21558}, {13941, 21565}, {14974, 16685}, {15803, 32561}, {16371, 17330}, {16412, 17259}, {16686, 36641}, {16726, 28014}, {16885, 19297}, {17056, 21483}, {17349, 19308}, {17443, 34522}, {17444, 22770}, {17684, 26110}, {17696, 26106}, {17735, 21769}, {17796, 20818}, {19547, 31401}, {20146, 33063}, {20331, 20999}, {20582, 21533}, {20775, 33718}, {21480, 23302}, {21481, 23303}, {21482, 23292}, {21492, 32785}, {21507, 32455}, {21519, 34573}, {21546, 32790}, {21549, 32789}, {21553, 32786}, {21559, 32788}, {21560, 32787}, {23868, 36635}, {25504, 33821}, {25508, 33036}, {31449, 34261}

X(36743) = isogonal conjugate of the polar conjugate of X(475)
X(36743) = X(10623)-Ceva conjugate of X(55)
X(36743) = crosspoint of X(249) and X(8690)
X(36743) = crosssum of X(115) and X(4139)
X(36743) = crossdifference of every pair of points on line {523, 21185}
X(36743) = barycentric product X(3)*X(475)
X(36743) = barycentric quotient X(475)/X(264)
X(36743) = Brocard-circle-inverse of X(36744)
X(36743) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36754)
X(36743) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36744}, {3, 4254, 1030}, {3, 5120, 6}, {6, 1030, 4254}, {6, 3053, 2220}, {6, 4252, 4275}, {6, 4255, 4272}, {6, 5013, 4261}, {6, 5022, 583}, {6, 5124, 3}, {9, 36, 2178}, {39, 5019, 6}, {48, 672, 2911}, {58, 5105, 6}, {284, 4253, 6}, {371, 372, 36754}, {572, 579, 6}, {572, 5030, 579}, {573, 5053, 6}, {574, 5042, 2092}, {583, 2278, 6}, {1333, 5069, 6}, {2092, 5042, 6}, {2245, 4268, 6}, {4261, 5035, 6}, {4275, 5109, 6}, {4287, 5043, 6}, {5115, 5153, 6}, {17798, 20992, 1486}

X(36744) =  X(1)(X(2178)∩X(3)X(6)

Barycentrics    a^2*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :
Barycentrics    a^2*(R*S / s + SA) : :
Trilinears    cos A (sin A + sin B + sin C) + sin A : :
Trilinears    s cos A + R sin A : :
Trilinears    a + 2 s cos A : :

X(36744) lies on these lines: {1, 2178}, {3, 6}, {9, 35}, {19, 25}, {21, 966}, {22, 5276}, {24, 1172}, {36, 1449}, {40, 3553}, {41, 71}, {44, 5217}, {48, 836}, {56, 1100}, {69, 21511}, {73, 2199}, {81, 11340}, {86, 11329}, {99, 34283}, {100, 2345}, {141, 11343}, {193, 1444}, {219, 2174}, {220, 2301}, {230, 19544}, {325, 21485}, {380, 8557}, {390, 36007}, {391, 4189}, {393, 7412}, {395, 21475}, {396, 21476}, {405, 1213}, {474, 17398}, {478, 1950}, {524, 16436}, {590, 16433}, {594, 5687}, {597, 16431}, {599, 21509}, {604, 1470}, {615, 16432}, {759, 15322}, {940, 11350}, {941, 2303}, {950, 24005}, {956, 17362}, {958, 17275}, {965, 11344}, {993, 3686}, {999, 16884}, {1001, 19309}, {1006, 5802}, {1011, 2238}, {1036, 2281}, {1107, 22654}, {1185, 20848}, {1211, 16368}, {1259, 3965}, {1376, 17303}, {1415, 2286}, {1460, 2214}, {1584, 31473}, {1604, 1630}, {1613, 35216}, {1743, 5010}, {1759, 22021}, {1766, 11248}, {1778, 4184}, {1817, 5712}, {1841, 11398}, {1901, 7580}, {1914, 2277}, {1975, 3770}, {1992, 35276}, {2161, 2337}, {2183, 2268}, {2197, 10831}, {2223, 16972}, {2241, 17053}, {2251, 2273}, {2257, 15931}, {2260, 2280}, {2262, 2646}, {2267, 2347}, {2270, 3601}, {2285, 11509}, {2287, 20846}, {2288, 22074}, {2291, 8694}, {2321, 8715}, {2975, 5839}, {3068, 16441}, {3069, 16440}, {3220, 16517}, {3247, 3746}, {3295, 5011}, {3303, 3723}, {3332, 36012}, {3554, 3576}, {3560, 5816}, {3589, 21477}, {3618, 21495}, {3619, 21516}, {3629, 21518}, {3630, 21517}, {3631, 21510}, {3651, 5746}, {3666, 24611}, {3763, 21514}, {3815, 16434}, {3871, 17314}, {3913, 17299}, {3949, 5282}, {4034, 5258}, {4191, 24512}, {4220, 7735}, {4304, 20262}, {4366, 26107}, {4421, 17281}, {4426, 21857}, {4471, 7083}, {4557, 20678}, {4648, 11349}, {5046, 27524}, {5204, 16666}, {5248, 5257}, {5283, 13730}, {5320, 22080}, {5540, 26744}, {5584, 21866}, {5739, 27174}, {5747, 6985}, {5750, 25440}, {5949, 17532}, {6329, 21524}, {6767, 20997}, {6796, 10445}, {6842, 9722}, {6872, 27522}, {7031, 16470}, {7113, 8071}, {7280, 16667}, {7485, 33854}, {7585, 21566}, {7586, 21567}, {7736, 19649}, {8252, 21548}, {8253, 21547}, {8584, 21498}, {8609, 10267}, {8972, 21565}, {9300, 21487}, {11320, 26772}, {11347, 17056}, {11353, 27111}, {13567, 21482}, {13846, 21558}, {13847, 21561}, {13941, 21568}, {14389, 21478}, {15668, 16412}, {16367, 17277}, {16370, 17330}, {16519, 21771}, {16915, 26110}, {16973, 22769}, {17276, 24328}, {17379, 19308}, {17452, 26358}, {19281, 27042}, {20146, 33062}, {20582, 21515}, {21008, 21769}, {21480, 23303}, {21481, 23302}, {21492, 32786}, {21496, 34573}, {21523, 32455}, {21546, 32789}, {21549, 32790}, {21553, 32785}, {21559, 32787}, {21560, 32788}, {21997, 27252}, {22369, 33718}, {24682, 27691}, {25504, 33828}, {25508, 33035}, {27785, 27787}, {28476, 28847}

X(36744) = isogonal conjugate of the isotomic conjugate of X(5739)
X(36744) = isogonal conjugate of the polar conjugate of X(406)
X(36744) = X(i)-Ceva conjugate of X(j) for these (i,j): {941, 6}, {27174, 12514}
X(36744) = crosspoint of X(i) and X(j) for these (i,j): {249, 931}, {406, 5739}, {7115, 32693}
X(36744) = crosssum of X(i) and X(j) for these (i,j): {6, 13730}, {11, 13401}, {115, 8672}, {23880, 26932}
X(36744) = crossdifference of every pair of points on line {523, 905}
X(36744) = Brocard-circle-inverse of X(36743)
X(36744) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36742)
X(36744) = barycentric product X(i)*X(j) for these {i,j}: {1, 12514}, {3, 406}, {6, 5739}, {37, 27174}, {78, 1452}
X(36744) = barycentric quotient X(i)/X(j) for these {i,j}: {406, 264}, {1452, 273}, {5739, 76}, {12514, 75}, {27174, 274}
X(36744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36743}, {3, 4254, 6}, {3, 5120, 5124}, {6, 1030, 3}, {6, 3053, 1333}, {6, 4252, 5115}, {6, 4255, 5153}, {6, 4258, 584}, {6, 5013, 5069}, {6, 5124, 5120}, {32, 2092, 6}, {39, 16946, 6}, {40, 3553, 21853}, {41, 71, 2911}, {55, 198, 37}, {55, 15494, 968}, {55, 23868, 1486}, {58, 4270, 6}, {187, 4263, 5019}, {193, 21508, 1444}, {198, 11434, 19}, {284, 573, 6}, {284, 4288, 1333}, {371, 372, 36742}, {386, 4264, 6}, {572, 4266, 6}, {573, 1182, 2245}, {573, 4262, 284}, {579, 4251, 6}, {584, 2245, 6}, {1333, 4277, 6}, {2220, 4261, 6}, {2278, 4271, 6}, {4254, 8573, 584}, {4263, 5019, 6}, {4271, 17454, 2278}, {4272, 4275, 6}, {4285, 5115, 6}, {4289, 5036, 6}, {4290, 5153, 6}, {4471, 8053, 7083}, {5069, 33882, 6}, {16884, 21773, 999}, {34121, 34125, 3185}

X(36745) =  X(1)X(5920)∩X(3)X(6)

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 4*a^3*b*c - 2*a^2*b^2*c + 4*a*b^3*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 + 4*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5) : :
Barycentrics    a^2*(2*SA - s*S / R) : :

X(36745) lies on these lines: {1, 5920}, {2, 5706}, {3, 6}, {4, 4383}, {20, 32911}, {21, 10601}, {24, 5347}, {31, 10310}, {36, 34046}, {40, 2999}, {43, 11500}, {44, 7330}, {46, 221}, {55, 602}, {56, 1066}, {57, 7078}, {81, 3523}, {84, 1743}, {140, 5707}, {155, 6924}, {165, 1203}, {218, 1490}, {219, 936}, {220, 5044}, {222, 15803}, {238, 11496}, {255, 1466}, {387, 6865}, {394, 404}, {405, 17825}, {406, 26005}, {474, 17811}, {517, 1191}, {595, 10306}, {611, 988}, {631, 940}, {946, 3008}, {975, 2256}, {978, 22753}, {990, 5777}, {995, 22770}, {1006, 19765}, {1012, 1724}, {1064, 5584}, {1181, 6905}, {1193, 3428}, {1376, 3072}, {1407, 3157}, {1437, 17809}, {1451, 22072}, {1453, 6282}, {1482, 1616}, {1498, 1754}, {1708, 17102}, {1714, 6831}, {1722, 7686}, {1753, 3195}, {1834, 6827}, {1993, 4188}, {2093, 34040}, {2095, 24046}, {2187, 28270}, {2194, 10984}, {2323, 5438}, {2328, 16293}, {2361, 11509}, {2814, 23141}, {2911, 5720}, {3052, 11248}, {3146, 14997}, {3193, 6921}, {3332, 6864}, {3487, 5228}, {3562, 5435}, {3587, 15852}, {3751, 12675}, {3752, 5709}, {3796, 11337}, {4000, 5758}, {4189, 5422}, {4292, 34048}, {4293, 9370}, {4849, 5534}, {5010, 16472}, {5247, 12114}, {5272, 13374}, {5292, 6922}, {5312, 15931}, {5315, 7991}, {5452, 35599}, {5526, 8951}, {5657, 5710}, {5694, 7986}, {5711, 6684}, {5713, 8728}, {5718, 6889}, {5721, 6836}, {5800, 6803}, {5812, 23537}, {5956, 6911}, {6769, 7290}, {6833, 35466}, {6862, 31187}, {6887, 17337}, {6890, 24597}, {6906, 10982}, {6918, 17749}, {6942, 7592}, {6943, 24883}, {6986, 19767}, {6989, 17056}, {7171, 16669}, {7280, 16473}, {7484, 22076}, {7982, 16483}, {8572, 22765}, {9841, 16670}, {10222, 16486}, {10441, 16434}, {11849, 21000}, {12705, 14550}, {13730, 17810}, {15066, 17572}, {15811, 19541}, {16189, 16489}, {16417, 22136}, {16474, 30389}, {17054, 24474}, {17548, 34545}, {17567, 25934}, {17582, 25878}, {19349, 34042}, {23154, 26866}

X(36745) = crossdifference of every pair of points on line {523, 14300}
X(36745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5050, 13323}, {3, 5398, 4252}, {3, 5752, 1350}, {182, 15489, 3}, {371, 372, 5120}, {386, 13329, 3}, {1151, 1152, 5124}, {1754, 3216, 3149}, {1754, 16471, 1498}

X(36746) =  X(1)X(84)∩X(3)X(6)

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c + 4*a^3*b*c - 2*a^2*b^2*c - 4*a*b^3*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 4*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5) : :
Barycentrics    a^2*(2*SA + s*S / R) : :

X(36746) lies on these lines: {1, 84}, {3, 6}, {4, 940}, {20, 81}, {21, 394}, {30, 5707}, {34, 34042}, {35, 7074}, {37, 7330}, {42, 10310}, {55, 255}, {56, 1064}, {57, 9122}, {60, 35602}, {154, 1437}, {155, 5453}, {171, 11500}, {219, 31424}, {220, 31445}, {269, 3333}, {283, 20835}, {377, 5721}, {387, 6916}, {404, 10601}, {405, 17194}, {474, 17825}, {478, 12664}, {515, 5711}, {602, 8273}, {603, 14547}, {608, 12671}, {612, 14872}, {613, 988}, {631, 4383}, {651, 5703}, {938, 17074}, {942, 1407}, {944, 5710}, {946, 3664}, {975, 5777}, {999, 4306}, {1001, 3073}, {1072, 10404}, {1092, 2194}, {1100, 7171}, {1104, 18443}, {1158, 3931}, {1181, 6906}, {1191, 1385}, {1203, 7987}, {1408, 11414}, {1449, 9841}, {1451, 22053}, {1453, 8726}, {1468, 3428}, {1480, 10222}, {1496, 2293}, {1616, 10246}, {1715, 18163}, {1834, 6850}, {1993, 4189}, {1994, 17548}, {2003, 3601}, {2303, 5776}, {2801, 30142}, {3052, 10267}, {3085, 9370}, {3146, 14996}, {3157, 24929}, {3295, 23072}, {3359, 4646}, {3487, 6180}, {3523, 32911}, {3560, 17814}, {3562, 4313}, {3576, 16466}, {3745, 12680}, {3868, 22129}, {4188, 5422}, {4644, 5758}, {4648, 6846}, {5010, 16473}, {5084, 25934}, {5292, 6907}, {5315, 30389}, {5323, 36029}, {5347, 10323}, {5709, 15852}, {5712, 6847}, {5713, 8727}, {5716, 5768}, {5717, 6245}, {5718, 6833}, {5725, 12616}, {5788, 15973}, {5820, 14216}, {6769, 35658}, {6824, 17056}, {6828, 26131}, {6887, 17245}, {6889, 35466}, {6905, 10982}, {6909, 19767}, {6950, 7592}, {7280, 16472}, {7497, 18165}, {7508, 16266}, {7991, 16474}, {8757, 11374}, {9440, 12260}, {10202, 17054}, {11269, 15908}, {11337, 33586}, {13411, 34048}, {13743, 18451}, {15066, 16865}, {15068, 31649}, {15178, 16486}, {15316, 34435}, {15592, 22276}, {16189, 16490}, {16193, 34036}, {16845, 25878}, {17571, 22136}, {17595, 26877}, {19727, 25526}, {20986, 22654}, {26117, 26625}, {26958, 34120}

X(36746) = crossdifference of every pair of points on line {523, 14298}
X(36746) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(4254)
X(36746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1351, 970}, {3, 5396, 4255}, {20, 81, 5706}, {58, 991, 3}, {371, 372, 4254}, {1151, 1152, 1030}, {1413, 34046, 222}, {1437, 13730, 154}, {1468, 4300, 3428}, {2003, 3601, 7078}

X(36747) =  X(3)X(6)∩X(4)X(155)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 8*a^4*b^2*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :
Barycentrics    a^2*(2*SA - S^2/R^2) : :
X(36747) = 3 X[3796] - 4 X[32046], 3 X[11402] - X[11414]

X(36747) lies on these lines: {3, 6}, {4, 155}, {5, 394}, {20, 1994}, {22, 54}, {23, 9545}, {24, 3060}, {25, 1147}, {26, 5944}, {30, 1181}, {33, 1069}, {34, 3157}, {40, 16473}, {47, 11248}, {49, 154}, {51, 1092}, {64, 15317}, {68, 427}, {69, 7404}, {81, 6825}, {110, 10594}, {140, 10601}, {141, 14786}, {143, 6644}, {156, 7530}, {184, 7387}, {185, 12085}, {193, 3088}, {195, 382}, {215, 9658}, {235, 5654}, {265, 17847}, {323, 3091}, {376, 1199}, {378, 5889}, {381, 17814}, {399, 5076}, {539, 5064}, {546, 15068}, {599, 14787}, {631, 5422}, {858, 18912}, {940, 6863}, {1112, 5504}, {1154, 7526}, {1173, 15024}, {1204, 14831}, {1216, 7395}, {1217, 3087}, {1352, 7403}, {1353, 18914}, {1482, 23071}, {1593, 12160}, {1594, 14852}, {1595, 3564}, {1596, 34966}, {1597, 12162}, {1598, 3167}, {1656, 17811}, {1657, 15087}, {1658, 14449}, {1829, 9928}, {1838, 3173}, {1853, 25738}, {1899, 13292}, {1907, 9936}, {1986, 12302}, {1992, 18909}, {1995, 9781}, {2003, 5709}, {2070, 17821}, {2095, 23070}, {2323, 7330}, {2477, 9673}, {2777, 19456}, {2888, 5169}, {2904, 6240}, {2931, 15463}, {2979, 7509}, {3066, 10095}, {3090, 15066}, {3092, 10666}, {3093, 10665}, {3146, 11004}, {3515, 12038}, {3516, 7689}, {3523, 34545}, {3526, 15038}, {3527, 5020}, {3529, 15032}, {3532, 15002}, {3541, 6515}, {3542, 9820}, {3546, 11433}, {3547, 11427}, {3548, 13567}, {3549, 23292}, {3567, 17928}, {3575, 12118}, {3576, 16472}, {3627, 32139}, {3629, 6247}, {3796, 13391}, {3830, 15811}, {3851, 18555}, {3917, 7393}, {4383, 6958}, {5012, 10323}, {5094, 5449}, {5101, 12422}, {5130, 12423}, {5412, 8909}, {5447, 7484}, {5462, 9777}, {5480, 7528}, {5562, 9818}, {5706, 6923}, {5707, 6842}, {5876, 31861}, {5890, 11413}, {5891, 11479}, {5899, 9704}, {6101, 7514}, {6102, 10605}, {6146, 14790}, {6640, 26958}, {6756, 19139}, {6759, 13598}, {6776, 34938}, {6800, 12088}, {6803, 26206}, {6891, 32911}, {6913, 22136}, {7074, 11849}, {7391, 34224}, {7401, 14853}, {7405, 14561}, {7503, 11412}, {7506, 17810}, {7507, 9927}, {7516, 10627}, {7529, 9306}, {7553, 9833}, {7558, 14389}, {7728, 17838}, {8541, 21651}, {8549, 14216}, {9703, 18378}, {9706, 26881}, {9714, 10282}, {9715, 18475}, {9933, 12135}, {10112, 18381}, {10303, 15018}, {10571, 10680}, {10602, 32284}, {10984, 13366}, {11392, 18970}, {11393, 12428}, {11402, 11414}, {11403, 15083}, {11422, 12082}, {11423, 33524}, {11442, 15559}, {11444, 23061}, {11472, 12111}, {11484, 14845}, {12083, 17809}, {12166, 12167}, {12168, 12175}, {12173, 17702}, {12174, 14915}, {12241, 18531}, {12308, 22334}, {12370, 18396}, {12412, 13417}, {12429, 18474}, {12828, 15115}, {13367, 14070}, {13861, 35259}, {15019, 15028}, {15047, 15720}, {15106, 36253}, {16003, 17822}, {16982, 32171}, {18405, 31724}, {18533, 35603}, {18925, 31305}, {19360, 26937}, {22765, 34046}, {22800, 22971}, {22972, 22979}, {23236, 32271}, {26917, 30744}, {32140, 32358}, {32620, 35500}, {34117, 34782}, {34484, 35264}

X(36747) = midpoint of X(1593) and X(12160)
X(36747) = reflection of X(i) in X(j) for these {i,j}: {3, 578}, {1181, 12161}
X(36747) = isogonal conjugate of polar conjugate of X(37192)
X(36747) = Brocard-circle-inverse of X(36752)
X(36747) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(1609)
X(36747) = X(12705)-of-orthic-triangle if ABC is acute
X(36747) = X(i)-Ceva conjugate of X(j) for these (i,j): {1217, 3}, {3087, 5020}
X(36747) = cevapoint of X(155) and X(15805)
X(36747) = crosssum of X(3) and X(19458)
X(36747) = crossdifference of every pair of points on line {523, 14346}
X(36747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36752}, {3, 568, 9786}, {3, 1351, 52}, {3, 5050, 13336}, {3, 5093, 11432}, {3, 6243, 17834}, {3, 11426, 569}, {3, 11432, 9730}, {3, 13353, 5085}, {4, 155, 18451}, {4, 1993, 155}, {4, 6193, 12134}, {5, 16266, 394}, {20, 1994, 7592}, {23, 9545, 9707}, {26, 10263, 33586}, {49, 7517, 154}, {51, 1092, 6642}, {52, 13352, 3}, {182, 15644, 3}, {193, 3088, 11411}, {195, 382, 18445}, {195, 15800, 17824}, {371, 372, 1609}, {378, 5889, 12163}, {382, 18445, 1498}, {389, 13346, 3}, {394, 10982, 5}, {569, 10625, 3}, {576, 13346, 389}, {631, 5422, 15805}, {1147, 5446, 25}, {1597, 12164, 12162}, {1598, 3167, 10539}, {2979, 13434, 7509}, {3060, 34148, 24}, {3541, 6515, 12359}, {5562, 11424, 9818}, {6102, 12084, 10605}, {6759, 13598, 18534}, {9306, 10110, 7529}, {9833, 31670, 7553}, {11412, 15033, 7503}, {11425, 11477, 17834}, {11425, 17834, 3}, {11477, 17834, 6243}, {12111, 35502, 11472}, {12370, 18569, 18396}, {13292, 23335, 1899}, {13340, 13353, 3}, {13598, 34986, 6759}, {19357, 33586, 26}, {22236, 22238, 11063}

X(36748) =  X(2)X(6748)∩X(3)X(6)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    a^2*((r + 2*R)^2 - s^2 + 2*SA) : :

X(36748) lies on these lines: {2, 6748}, {3, 6}, {20, 53}, {22, 14577}, {64, 17849}, {69, 10607}, {95, 458}, {97, 394}, {115, 18536}, {154, 160}, {157, 9924}, {184, 26865}, {230, 7386}, {233, 3526}, {248, 22085}, {264, 35941}, {376, 393}, {382, 36412}, {441, 3763}, {465, 16644}, {466, 16645}, {590, 1589}, {599, 6389}, {615, 1590}, {631, 3087}, {1172, 6950}, {1249, 3528}, {1368, 9722}, {1576, 19132}, {1600, 26912}, {1809, 3713}, {1865, 6934}, {1971, 17811}, {1990, 3522}, {2165, 12362}, {2207, 10323}, {2548, 16197}, {3052, 23207}, {3054, 16051}, {3093, 26916}, {3148, 7716}, {3156, 26953}, {3163, 14093}, {3289, 3796}, {3523, 6749}, {3815, 7494}, {5054, 36422}, {6636, 8746}, {6641, 17810}, {6643, 13881}, {6676, 31489}, {7400, 7745}, {7484, 10311}, {7485, 10313}, {7509, 8882}, {7512, 8745}, {7999, 33629}, {8550, 26870}, {8911, 10133}, {9715, 14576}, {10132, 26920}, {10314, 16419}, {10608, 15073}, {10985, 11284}, {11402, 26907}, {11414, 34818}, {14578, 22055}, {14910, 34866}, {15695, 18487}, {15710, 36427}, {15846, 17819}, {15847, 17820}, {16884, 17102}, {17259, 21940}, {17337, 25932}, {17398, 25876}, {17809, 23606}, {17907, 35937}, {19355, 19446}, {19356, 19447}, {19357, 26876}, {37519, 22341}, {21734, 36413}, {26206, 35296}

X(36748) = isogonal conjugate of X(8796)
X(36748) = isotomic conjugate of the polar conjugate of X(11402)
X(36748) = isogonal conjugate of the polar conjugate of X(631)
X(36748) = X(i)-Ceva conjugate of X(j) for these (i,j): {631, 11402}, {5395, 3167}, {6570, 32320}
X(36748) = X(26907)-cross conjugate of X(631)
X(36748) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8796}, {19, 8797}, {75, 34818}, {92, 3527}
X(36748) = crosssum of X(i) and X(j) for these (i,j): {6, 3517}, {53, 8887}, {3527, 34818}
X(36748) = Brocard-circle-inverse of X(36751)
X(36748) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(11432)
X(36748) = barycentric product X(i)*X(j) for these {i,j}: {3, 631}, {69, 11402}, {95, 26907}, {394, 3087}
X(36748) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 8797}, {6, 8796}, {32, 34818}, {184, 3527}, {631, 264}, {3087, 2052}, {6755, 13450}, {11402, 4}, {17809, 11282}, {26907, 5}, {32078, 31505}
X(36748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36751}, {3, 577, 6}, {3, 15905, 216}, {3, 22401, 15815}, {6, 5023, 1609}, {6, 15815, 570}, {6, 22331, 13345}, {184, 26865, 26909}, {187, 5065, 8573}, {216, 577, 15905}, {216, 14961, 13351}, {216, 15905, 6}, {371, 372, 11432}, {372, 26868, 6}, {577, 10979, 3284}, {577, 22052, 3}, {577, 26899, 5063}, {1151, 1152, 9786}, {1609, 5063, 6}, {5065, 8573, 6}, {6409, 6410, 1620}, {19408, 19409, 394}, {23606, 26898, 17809}

X(36749) =  X(2)X(16266)∩X(3)X(6)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 6*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 2*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :
Barycentrics    a^2*(SA - S^2 / R^2) : :
X(36749) = 3 X[5050] - 2 X[5157]

X(36749) lies on these lines: {2, 16266}, {3, 6}, {4, 1994}, {5, 1993}, {20, 1199}, {22, 10263}, {24, 143}, {25, 49}, {26, 54}, {30, 7592}, {51, 1147}, {64, 15002}, {68, 5576}, {69, 14786}, {81, 6863}, {110, 9781}, {140, 5422}, {154, 9704}, {155, 195}, {156, 1493}, {184, 5446}, {193, 7404}, {265, 7507}, {323, 3090}, {378, 6102}, {382, 1181}, {394, 1656}, {427, 13292}, {517, 16473}, {524, 14787}, {546, 11441}, {631, 34545}, {858, 18952}, {1092, 5462}, {1154, 7503}, {1173, 5640}, {1353, 1595}, {1385, 16472}, {1498, 3830}, {1593, 34783}, {1597, 18439}, {1598, 10540}, {1614, 7530}, {1992, 8548}, {1995, 10095}, {2070, 19357}, {2937, 33586}, {2979, 7516}, {3088, 18917}, {3091, 11004}, {3146, 15032}, {3167, 3527}, {3193, 6929}, {3518, 9545}, {3525, 15018}, {3526, 10601}, {3541, 18951}, {3548, 11433}, {3549, 11427}, {3564, 7403}, {3567, 6644}, {3575, 35603}, {3627, 11456}, {3628, 15066}, {3796, 13564}, {3843, 18451}, {3851, 17814}, {5054, 15047}, {5070, 17811}, {5480, 12134}, {5504, 16222}, {5663, 35502}, {5707, 6980}, {5889, 7526}, {5890, 12084}, {5899, 17809}, {5946, 17928}, {6101, 7509}, {6146, 31723}, {6193, 7528}, {6247, 8584}, {6639, 23292}, {6640, 13567}, {6641, 19210}, {6642, 9777}, {6800, 17714}, {6958, 32911}, {7387, 11402}, {7395, 23039}, {7405, 18583}, {7485, 10627}, {7502, 14449}, {7514, 11412}, {7540, 9833}, {7553, 21850}, {7689, 14831}, {7728, 19456}, {7998, 13154}, {7999, 23061}, {8541, 32284}, {8549, 34780}, {9308, 14978}, {9544, 34484}, {9703, 13621}, {9706, 26882}, {9818, 12160}, {9925, 11188}, {10110, 10539}, {10112, 18474}, {10116, 11550}, {10282, 21849}, {10323, 13391}, {10620, 14448}, {11003, 12088}, {11225, 20299}, {11245, 23335}, {11264, 34514}, {11413, 13630}, {11424, 13754}, {11442, 32358}, {11459, 15801}, {11465, 12834}, {11597, 12310}, {11818, 14516}, {12022, 18569}, {12083, 13366}, {12111, 31861}, {12163, 14130}, {12164, 18435}, {12225, 31815}, {12227, 12295}, {12236, 15463}, {12241, 18404}, {12412, 15089}, {12605, 31802}, {13142, 15760}, {13371, 18912}, {13406, 22051}, {13451, 35264}, {14912, 34938}, {15019, 15024}, {15027, 15106}, {15030, 15083}, {15559, 32140}, {15800, 32341}, {16657, 22660}, {17824, 18405}, {17847, 32743}, {18281, 26879}, {18356, 33332}, {18369, 35259}, {18377, 20424}, {18396, 31724}, {18534, 19347}, {19360, 19361}, {22146, 35716}, {26917, 31283}, {30714, 34155}, {31236, 34826}

X(36749) = reflection of X(3) in X(569)
X(36749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36753}, {3, 1351, 6243}, {3, 11426, 567}, {3, 14627, 6}, {4, 1994, 12161}, {4, 12161, 18445}, {51, 1147, 7506}, {52, 578, 3}, {54, 3060, 26}, {110, 9781, 13861}, {155, 10982, 381}, {182, 10625, 3}, {184, 5446, 7517}, {195, 381, 155}, {371, 372, 8553}, {381, 12429, 6288}, {382, 15087, 1181}, {389, 13352, 3}, {427, 13292, 25738}, {567, 6243, 3}, {575, 15644, 13336}, {576, 578, 52}, {1092, 15004, 5462}, {1351, 11426, 3}, {2055, 30258, 3}, {2904, 12370, 18445}, {3167, 3527, 7529}, {3167, 7529, 18350}, {3311, 3312, 8573}, {3567, 34148, 6644}, {3574, 9927, 381}, {5054, 15047, 15805}, {5889, 15033, 7526}, {6193, 14853, 7528}, {9545, 11002, 3518}, {9704, 18378, 154}, {9730, 13346, 3}, {9818, 12160, 18436}, {10110, 34986, 10539}, {10263, 32046, 22}, {11412, 13434, 7514}, {11536, 21659, 15087}, {13336, 15644, 3}, {21850, 31804, 7553}

X(36749) = Brocard-circle-inverse of X(36753)
X(36749) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(8553)

X(36750) =  X(1)X(195)∩X(3)X(6)

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c + a^3*b*c - 2*a^2*b^2*c - a*b^3*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5) : :
Barycentrics    a^2*(2*s*S / R + SA) : :

X(36750) lies on these lines: {1, 195}, {3, 6}, {4, 1029}, {5, 81}, {21, 1994}, {42, 11849}, {49, 2194}, {51, 1437}, {55, 2964}, {56, 16472}, {60, 143}, {140, 32911}, {155, 6913}, {184, 20831}, {221, 1159}, {222, 1393}, {323, 5047}, {381, 5707}, {382, 5706}, {387, 6923}, {394, 11108}, {404, 34545}, {405, 1993}, {474, 5422}, {601, 35000}, {651, 6147}, {940, 1656}, {942, 2003}, {1006, 5453}, {1012, 7592}, {1147, 5320}, {1172, 7546}, {1199, 6906}, {1203, 1385}, {1449, 7330}, {1468, 22765}, {1754, 16117}, {2323, 31445}, {2915, 3060}, {3073, 4649}, {3090, 14996}, {3157, 15934}, {3193, 11113}, {3194, 7510}, {3216, 15047}, {3525, 14997}, {3526, 4383}, {3560, 12161}, {3562, 12433}, {3945, 6887}, {5012, 20833}, {5256, 24467}, {5262, 24475}, {5264, 12331}, {5292, 6980}, {5312, 26285}, {5313, 32612}, {5315, 15178}, {5347, 13564}, {5439, 22128}, {5710, 12645}, {5711, 5790}, {5712, 6861}, {5902, 8614}, {6883, 16266}, {6914, 19767}, {7508, 16948}, {8144, 10394}, {9653, 14667}, {10222, 16474}, {10246, 16466}, {10601, 16408}, {10982, 19541}, {11004, 16865}, {11402, 13730}, {11433, 34120}, {13621, 17104}, {13743, 15087}, {14988, 17016}, {15002, 34435}, {15018, 17531}, {15019, 16427}, {15032, 21669}, {15066, 16842}, {15988, 17698}, {16853, 17811}, {16863, 17825}, {17012, 26877}, {17074, 34753}, {17126, 32141}, {17379, 20746}, {19365, 20122}, {31794, 34043}

X(36750) = Brocard-circle-inverse of X(37509)
X(36750) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(1030)
X(36750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 37509}, {58, 5396, 3}, {371, 372, 1030}, {405, 1993, 22136}, {500, 580, 3}, {576, 13323, 5752}, {581, 5398, 3}, {582, 991, 3}, {942, 2003, 23070}, {3311, 3312, 4254}, {5752, 13323, 3}

X(36751) =  X(2)X(53)∩X(3)X(6)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 6*b^2*c^2 + 3*c^4) : :
Barycentrics    a^2*(2*SA - (r + 2*R)^2 + s^2) : :

X(36751) lies on these lines: {2, 53}, {3, 6}, {20, 6748}, {25, 26907}, {51, 26865}, {64, 26897}, {95, 9308}, {154, 157}, {160, 9924}, {230, 7494}, {232, 7484}, {233, 381}, {237, 7716}, {317, 35937}, {376, 3087}, {393, 631}, {394, 31626}, {418, 15649}, {465, 16645}, {466, 16644}, {590, 1590}, {615, 1589}, {1040, 31477}, {1172, 6942}, {1213, 25876}, {1249, 3524}, {1368, 15880}, {1503, 26870}, {1583, 8963}, {1656, 36412}, {1853, 26905}, {1865, 6833}, {1990, 3523}, {1995, 26895}, {2071, 16328}, {2165, 6676}, {2207, 7509}, {3055, 16051}, {3163, 15700}, {3522, 6749}, {3538, 31400}, {3547, 13881}, {3553, 31448}, {3763, 6389}, {3767, 16197}, {3815, 7386}, {5254, 7400}, {5406, 19408}, {5407, 19409}, {5475, 18536}, {5650, 33924}, {6413, 10133}, {6414, 10132}, {6638, 35222}, {6823, 9722}, {7383, 27376}, {7395, 14576}, {7485, 22240}, {7503, 11062}, {8745, 35921}, {9909, 10314}, {10594, 26896}, {12114, 21854}, {13006, 15817}, {15668, 21940}, {15701, 18487}, {16303, 16976}, {16777, 17102}, {17245, 25932}, {17818, 31364}, {20208, 21358}, {22062, 34817}, {26874, 33586}, {26906, 26958}, {31490, 34823}

X(36751) = isotomic conjugate of the polar conjugate of X(9777)
X(36751) = isogonal conjugate of the polar conjugate of X(3090)
X(36751) = X(3090)-Ceva conjugate of X(9777)
X(36751) = crosssum of X(6) and X(1598)
X(36751) = barycentric product X(i)*X(j) for these {i,j}: {3, 3090}, {69, 9777}
X(36751) = barycentric quotient X(i)/X(j) for these {i,j}: {3090, 264}, {9777, 4}
X(36751) = Brocard-circle-inverse of X(36748)
X(36751) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(11426)
X(36751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36748}, {3, 216, 6}, {3, 8961, 1579}, {3, 15905, 22052}, {6, 5023, 571}, {6, 8553, 3053}, {6, 22332, 5421}, {25, 26907, 26909}, {39, 8573, 6}, {216, 10979, 3}, {216, 22052, 5158}, {371, 372, 11426}, {566, 8553, 6}, {570, 1609, 6}, {1151, 1152, 11425}, {3284, 15851, 6}, {5158, 15905, 6}, {5158, 22052, 15905}, {6641, 26898, 154}, {6641, 32078, 26898}, {15109, 18573, 6}, {24245, 24246, 6676}

X(36752) =  X(2)X(155)∩X(3)X(6)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 8*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 8*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :
Barycentrics    a^2*(S^2 / (2*R^2) + SA) : :

X(36752) lies on these lines: {2, 155}, {3, 6}, {4, 5422}, {5, 1181}, {20, 34545}, {22, 3567}, {24, 5012}, {25, 5462}, {26, 3796}, {30, 10982}, {40, 16472}, {49, 17809}, {51, 7387}, {54, 9932}, {68, 7399}, {81, 6891}, {110, 15028}, {140, 394}, {143, 33586}, {154, 7506}, {156, 13363}, {184, 6642}, {185, 9818}, {195, 5054}, {323, 10303}, {343, 18951}, {378, 10574}, {381, 1498}, {399, 5079}, {549, 16266}, {597, 6247}, {631, 1199}, {940, 6958}, {1092, 13366}, {1147, 5892}, {1154, 7516}, {1173, 33524}, {1216, 7484}, {1352, 7405}, {1503, 7528}, {1594, 18911}, {1595, 18583}, {1597, 10575}, {1614, 1995}, {1656, 17814}, {1657, 15038}, {1714, 6842}, {1853, 5576}, {1885, 4846}, {1994, 3523}, {3060, 10323}, {3066, 13861}, {3090, 11441}, {3091, 11456}, {3146, 35237}, {3157, 10202}, {3167, 17836}, {3193, 6947}, {3515, 18475}, {3518, 6800}, {3525, 15066}, {3526, 15087}, {3541, 35603}, {3546, 11427}, {3547, 11433}, {3548, 23292}, {3549, 13567}, {3576, 16473}, {3580, 7558}, {3589, 14786}, {3618, 7404}, {3628, 15068}, {3843, 15811}, {4383, 6863}, {5020, 10539}, {5133, 11457}, {5446, 9777}, {5448, 16072}, {5449, 26869}, {5562, 7393}, {5622, 25711}, {5640, 10594}, {5644, 14845}, {5706, 6928}, {5707, 6882}, {5889, 7509}, {5890, 7503}, {5891, 12164}, {5943, 6759}, {5972, 19456}, {6102, 7514}, {6146, 18420}, {6193, 6803}, {6241, 11472}, {6461, 32177}, {6515, 7383}, {6639, 26958}, {6644, 12006}, {6689, 32341}, {6699, 19504}, {6723, 12227}, {6746, 21213}, {6776, 7401}, {6795, 36160}, {6816, 22660}, {6825, 32911}, {7394, 16659}, {7395, 13754}, {7403, 14216}, {7485, 11412}, {7517, 17810}, {7525, 16881}, {7526, 10605}, {7530, 10095}, {7544, 34224}, {7569, 23293}, {7706, 12173}, {7998, 15801}, {8547, 12061}, {8549, 9815}, {9306, 11695}, {9707, 11003}, {9723, 18939}, {9825, 31804}, {9826, 13198}, {10110, 18534}, {10117, 16222}, {10170, 15083}, {10606, 14130}, {10610, 18324}, {11403, 14915}, {11413, 15033}, {11424, 12085}, {11442, 14788}, {11479, 12162}, {11557, 13171}, {11750, 18494}, {11802, 32333}, {11806, 12168}, {12022, 12293}, {12082, 15019}, {12233, 18531}, {12315, 16194}, {12412, 16223}, {13154, 15067}, {13160, 14852}, {13321, 13564}, {13491, 31861}, {14070, 16226}, {14528, 15317}, {14643, 17838}, {14708, 19457}, {14853, 34938}, {15004, 35243}, {15022, 15052}, {15053, 32534}, {15061, 17847}, {15072, 35502}, {15106, 20397}, {15135, 20191}, {15581, 16776}, {15681, 16936}, {16003, 32300}, {17822, 18488}, {18435, 33537}, {19156, 20993}, {19467, 31833}, {20417, 34155}, {23294, 31236}, {25406, 31305}, {32322, 34114}, {35602, 36153}

X(36752) = Brocard-circle-inverse of X(36747)
X(36752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7592, 155}, {2, 18916, 12359}, {3, 6, 36747}, {3, 567, 11425}, {3, 568, 17834}, {3, 1351, 10625}, {3, 5050, 569}, {3, 6243, 1350}, {3, 11426, 13352}, {3, 11432, 52}, {5, 1181, 18451}, {51, 10984, 7387}, {52, 13336, 3}, {54, 15045, 17928}, {140, 12161, 394}, {155, 15805, 2}, {182, 389, 3}, {569, 9730, 3}, {575, 9729, 578}, {576, 13347, 15644}, {578, 9729, 3}, {631, 1199, 1993}, {1151, 1152, 15109}, {1181, 10601, 5}, {1181, 19360, 1899}, {1614, 15024, 1995}, {1656, 18445, 17814}, {3090, 15032, 11441}, {3618, 18909, 7404}, {5012, 15043, 24}, {5020, 19347, 10539}, {5085, 17834, 3}, {5644, 32063, 14845}, {5890, 7503, 12163}, {5943, 6759, 7529}, {6243, 13339, 3}, {6644, 32046, 19357}, {6776, 7401, 12134}, {6803, 14912, 6193}, {7399, 11245, 68}, {7484, 12160, 1216}, {7526, 13630, 10605}, {9777, 11414, 5446}, {9815, 11179, 9833}, {10574, 13434, 378}, {12006, 32046, 6644}, {13160, 18912, 14852}, {13346, 16836, 3}, {13347, 15644, 3}, {13861, 15026, 3066}, {14216, 14561, 7403}, {17814, 17825, 1656}

X(36753) =  X(2)X(1199)∩X(3)X(6)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 2*a^4*b^2*c^2 + 6*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 6*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :
Barycentrics    a^2*(S^2 / R^2 + SA) : :

X(36753) lies on these lines: {2, 1199}, {3, 6}, {4, 34545}, {5, 5422}, {22, 143}, {24, 5944}, {26, 3567}, {49, 6642}, {51, 7517}, {54, 6644}, {81, 6958}, {110, 11423}, {140, 1993}, {154, 13621}, {155, 1656}, {156, 1995}, {184, 5462}, {195, 394}, {323, 3525}, {378, 13630}, {381, 1181}, {382, 10982}, {399, 5072}, {427, 35603}, {458, 14978}, {517, 16472}, {546, 11456}, {597, 14787}, {631, 1994}, {632, 15066}, {1092, 5892}, {1147, 13366}, {1154, 7509}, {1385, 16473}, {1498, 3843}, {1594, 18952}, {1614, 5640}, {1899, 5576}, {2888, 14789}, {2904, 34115}, {2937, 3796}, {3066, 18369}, {3090, 15018}, {3091, 15032}, {3167, 19458}, {3518, 11003}, {3527, 18534}, {3548, 11427}, {3549, 11433}, {3564, 7405}, {3618, 11411}, {3851, 18451}, {5020, 18350}, {5055, 17814}, {5070, 17825}, {5133, 32140}, {5446, 10984}, {5644, 11484}, {5707, 6971}, {5710, 19914}, {5889, 7514}, {5890, 7526}, {5943, 10539}, {6101, 7485}, {6102, 7503}, {6193, 8548}, {6241, 31861}, {6639, 13567}, {6640, 23292}, {6776, 7528}, {6863, 32911}, {7387, 9777}, {7393, 12160}, {7395, 18436}, {7399, 13292}, {7401, 14912}, {7403, 18583}, {7404, 18917}, {7487, 33748}, {7502, 16881}, {7516, 11412}, {7529, 10540}, {7530, 9781}, {7540, 11179}, {7564, 25739}, {7566, 34514}, {7569, 34826}, {7579, 17824}, {7706, 21659}, {7999, 13154}, {8546, 12061}, {8547, 11663}, {8550, 12134}, {9704, 17809}, {9707, 12106}, {9818, 34783}, {10095, 10594}, {10263, 10323}, {10303, 11004}, {10574, 12084}, {10605, 14130}, {11002, 12088}, {11422, 15028}, {11479, 18435}, {11550, 18128}, {11695, 34986}, {11818, 34224}, {11898, 12585}, {12006, 17928}, {12227, 23515}, {12233, 18404}, {13198, 16222}, {13363, 32136}, {13367, 16226}, {13371, 18911}, {13491, 35502}, {13561, 31236}, {13564, 33586}, {14216, 23327}, {14269, 15811}, {14389, 26879}, {14528, 15002}, {14643, 19456}, {14763, 17822}, {15045, 34148}, {15061, 19504}, {16003, 34155}, {16657, 31725}, {17810, 18378}, {19149, 34780}, {19360, 19362}, {20126, 34470}, {26869, 32341}, {26913, 31283}

X(36753) = X(22270)-Ceva conjugate of X(3)
X(36753) = Brocard-circle-inverse of X(36749)
X(36753) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1199, 12161}, {3, 6, 36749}, {3, 5050, 13353}, {3, 11432, 568}, {5, 7592, 18445}, {5, 11245, 25738}, {52, 182, 3}, {54, 15043, 6644}, {155, 10601, 1656}, {156, 15026, 1995}, {184, 5462, 7506}, {195, 3526, 394}, {382, 15038, 10982}, {389, 569, 3}, {389, 575, 569}, {394, 15805, 3526}, {568, 13353, 3}, {578, 9730, 3}, {631, 1994, 16266}, {1614, 5640, 13861}, {1656, 15047, 10601}, {1656, 15087, 155}, {3091, 15032, 32139}, {3567, 5012, 26}, {3618, 11411, 14786}, {5050, 11432, 3}, {5422, 7592, 5}, {5446, 10984, 12083}, {5890, 13434, 7526}, {5946, 32046, 24}, {5946, 36153, 32046}, {6642, 11402, 49}, {7393, 12160, 23039}, {7529, 19347, 10540}, {9729, 13352, 3}, {10574, 15033, 12084}, {10984, 15004, 5446}, {11423, 15024, 110}, {15047, 15087, 1656}, {18583, 18914, 7403}

X(36754) =  X(1)X(6883)∩X(3)X(6)

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 2*a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 + 2*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5) : :
Barycentrics    a^2*(SA - s*S / R) : :
Trilinears    cos A - sin A (sin A + sin B + sin C) : :
Trilinears    R cos A - s sin A : :

X(36754) lies on these lines: {1, 6883}, {2, 3193}, {3, 6}, {4, 32911}, {5, 1714}, {21, 5422}, {26, 5347}, {31, 11248}, {35, 16472}, {36, 16473}, {40, 1203}, {42, 602}, {43, 3072}, {47, 11509}, {51, 13730}, {56, 7130}, {57, 3157}, {81, 631}, {84, 16670}, {140, 940}, {155, 3216}, {218, 5777}, {219, 5044}, {221, 36279}, {323, 17572}, {387, 6827}, {394, 474}, {404, 1993}, {405, 10601}, {517, 16466}, {595, 10679}, {601, 2308}, {607, 1871}, {613, 5266}, {692, 11365}, {936, 2323}, {942, 7078}, {995, 10680}, {1006, 19767}, {1012, 10982}, {1064, 35239}, {1066, 1471}, {1181, 3149}, {1191, 1482}, {1193, 11249}, {1199, 6942}, {1406, 3336}, {1407, 23070}, {1437, 11402}, {1451, 22350}, {1465, 19349}, {1468, 10269}, {1480, 7991}, {1498, 19541}, {1616, 10247}, {1617, 5399}, {1656, 24880}, {1724, 3560}, {1743, 7330}, {1754, 6985}, {1834, 6928}, {1872, 3195}, {1994, 4188}, {2003, 15803}, {2361, 11507}, {2594, 7742}, {2915, 3796}, {2999, 5709}, {3052, 11849}, {3073, 16468}, {3091, 14997}, {3240, 11491}, {3295, 7074}, {3332, 6849}, {3666, 26921}, {4189, 34545}, {4292, 8757}, {4641, 24467}, {4663, 12675}, {5012, 11337}, {5091, 31847}, {5222, 5758}, {5228, 6147}, {5247, 22758}, {5292, 6882}, {5312, 10902}, {5313, 11012}, {5315, 7982}, {5320, 10984}, {5530, 5711}, {5690, 5710}, {5693, 7986}, {5708, 23071}, {5712, 6989}, {5713, 6881}, {5800, 7401}, {6180, 24470}, {6769, 16469}, {6826, 13408}, {6830, 24883}, {6833, 24597}, {6862, 35466}, {6905, 7592}, {6915, 11441}, {6918, 17814}, {6924, 12161}, {6950, 16948}, {9370, 18990}, {10222, 16483}, {10303, 14996}, {11108, 17825}, {15018, 16865}, {15066, 17531}, {16408, 17811}, {17810, 20831}, {18397, 33178}, {19544, 34466}, {20833, 33586}, {23292, 34120}, {26470, 33137}, {26878, 28606}, {34043, 36636}

X(36754) = crossdifference of every pair of points on line {523, 13401}
X(36754) = Brocard-circle-inverse of X(36742)
X(36754) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36743)
X(36754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36742}, {42, 602, 10267}, {43, 3072, 11499}, {182, 970, 3}, {371, 372, 36743}, {386, 580, 3}, {575, 15489, 13323}, {581, 13329, 3}, {582, 5396, 3}, {4383, 5706, 5}, {13323, 15489, 3}, {16408, 22136, 17811}

X(36755) =  MIDPOINT OF X(36241) AND X(36242)

Barycentrics    a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 2*b^2*c^2 - 2*c^4 + 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :
Barycentrics    Sin[A]^2*(1 - Cot[A]*(2*Sqrt[3] + 3*Cot[w])) : :
X(36755) = 3 X[3] - X[15], 5 X[3] - X[5611], 5 X[15] - 3 X[5611], 3 X[376] + X[621], 3 X[549] - 2 X[6671], 3 X[5473] + X[19106], 3 X[6771] - 2 X[11542], 5 X[16960] - 3 X[20425], 5 X[16960] - 9 X[21156], X[20425] - 3 X[21156], 3 X[22843] + X[22849]

X(36755) lies on these lines: {3, 6}, {20, 20428}, {30, 618}, {74, 10409}, {140, 7684}, {376, 621}, {531, 8703}, {538, 6582}, {549, 6671}, {550, 35725}, {616, 7799}, {842, 36514}, {1495, 11131}, {3132, 3819}, {3292, 14170}, {3643, 7880}, {3917, 11130}, {5318, 6115}, {5463, 11645}, {5473, 19106}, {5978, 30472}, {5980, 31711}, {6000, 24303}, {6109, 10617}, {6771, 11542}, {11146, 15107}, {11707, 13624}, {14880, 33467}, {16960, 20425}, {19924, 35304}, {22843, 22849}

X(36755) = midpoint of X(i) and X(j) for these {i,j}: {3, 14538}, {16, 35002}, {20,20428}, {36241,36242}
X(36755) = reflection of X(i) in X(j) for these {i,j}: {13350, 3}, {7684, 140), (11707, 13624}, {5611, 21401}
X(36755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16, 5092}, {3, 1350, 9735}, {3, 3098, 36756}, {3, 5611, 21158}, {3, 9736, 13349}, {3, 33878, 11480}, {5611, 21158, 21401}, {11131, 34008, 1495}, {21158, 21401, 13350}

X(36756) =  MIDPOINT OF X(36243) AND X(36244)

Barycentrics    a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 - 2*b^2*c^2 - 2*c^4 - 2*Sqrt[3]*(a^2 - b^2 - c^2)*S) : :
Barycentrics    Sin[A]^2*(1 + Cot[A]*(2*Sqrt[3] - 3*Cot[w])) : :
X(36756) = 3 X[3] - X[16], 5 X[3] - X[5615], 5 X[16] - 3 X[5615], 3 X[376] + X[622], 3 X[549] - 2 X[6672], 3 X[5474] + X[19107], 3 X[6774] - 2 X[11543], 5 X[16961] - 3 X[20426], 5 X[16961] - 9 X[21157], X[20426] - 3 X[21157], 3 X[22890] + X[22895]

X(36756) lies on these lines: {3, 6}, {20, 20429}, {30, 619}, {74, 10410}, {140, 7685}, {376, 622}, {530, 8703}, {538, 6295}, {549, 6672}, {550, 35726}, {617, 7799}, {842, 36515}, {1495, 11130}, {3131, 3819}, {3292, 14169}, {3642, 7880}, {3917, 11131}, {5321, 6114}, {5464, 11645}, {5474, 19107}, {5979, 30471}, {5981, 31712}, {6108, 10616}, {6774, 11543}, {11145, 15107}, {11708, 13624}, {14880, 33466}, {16961, 20426}, {19924, 35303}, {22890, 22895}

X(36756) = midpoint of X(i) and X(j) for these {i,j}: {3, 14539}, {15, 35002}, {20, 20429}, {36243, 36244}
X(36756) = reflection of X(i) in X(j) for these {i,j}: {13349, 3}, {7685, 140}, {11708, 13624}, {5615,21402}
X(36756) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 15, 5092}, {3, 1350, 9736}, {3, 3098, 36755}, {3, 5615, 21159}, {3, 9735, 13350}, {3, 33878, 11481}, {5615, 21159, 21402}, {11130, 34009, 1495}, {21159, 21402, 13349}

X(36757) =  X(3)X(6)∩X(13)X(1503)

Barycentrics    a^2*(-2*sqrt(3)*(a^2+b^2+c^2)*S+3*a^4-4*(b^2+c^2)*a^2+b^4-6*b^2*c^2+c^4) : :
Barycentrics    Sin[A]^2*(2 + (Sqrt[3] + Cot[A])*Cot[w]) : :
X(36757) = 2*X(6)+X(15), X(16)-4*X(2030), X(69)-4*X(6671), 4*X(182)-X(14538), 2*X(623)-5*X(3618), X(1351)+2*X(13350), X(3751)+2*X(11707), X(6776)+2*X(7684), X(9162)-4*X(9188), 5*X(11482)+4*X(21401), 5*X(12017)-2*X(36755), 4*X(18583)-X(20428)

See Tran Quang Hung and César Lozada, Euclid 607 .

X(36757) lies on the conics {{A, B, C, X(13), X(14538)}}, {{A, B, C, X(16), X(2065)}} and these lines: {3, 6}, {13, 1503}, {14, 14561}, {17, 1352}, {18, 3589}, {69, 6671}, {147, 6783}, {193, 627}, {202, 1428}, {203, 611}, {396, 3564}, {398, 18583}, {524, 16530}, {542, 16267}, {613, 7005}, {623, 3618}, {698, 32465}, {1080, 14136}, {2211, 23023}, {2330, 7006}, {3091, 31705}, {3751, 11707}, {5477, 25560}, {5480, 16964}, {5965, 22892}, {5978, 14137}, {6109, 6773}, {6593, 36209}, {6776, 7684}, {8739, 19128}, {9162, 9188}, {10617, 16772}, {10653, 25406}, {10654, 14853}, {11244, 23042}, {11579, 36208}, {19130, 22795}, {21462, 30535}

X(36757) = reflection of X(36758) in X(1692)
X(36757) = 1st-Lemoine-circle-inverse of X(36760)
X(36757) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10613, 15), (6, 182, 62), (6, 5050, 36758), (6, 19145, 3389), (6, 19146, 3390), (6, 22236, 1351), (15, 62, 14538), (187, 11485, 15), (1662, 1663, 36760), (13350, 22236, 15), (19780, 22236, 13350)

X(36758) =  X(3)X(6)∩X(14)X(1503)

Barycentrics    a^2*( 2*sqrt(3)*(a^2+b^2+c^2)*S+3*a^4-4*(b^2+c^2)*a^2+b^4-6*b^2*c^2+c^4) : :
Barycentrics    Sin[A]^2*(2 - (Sqrt[3] - Cot[A])*Cot[w]) : :
X(36758) = 2*X(6)+X(16), X(15)-4*X(2030), X(69)-4*X(6672), 4*X(182)-X(14539), 2*X(624)-5*X(3618), X(1351)+2*X(13349), X(3751)+2*X(11708), X(6776)+2*X(7685), X(9163)-4*X(9188), 5*X(11482)+4*X(21402), 5*X(12017)-2*X(36756), 4*X(18583)-X(20429)

See Tran Quang Hung and César Lozada, Euclid 607 .

X(36758) lies on the conics {{A, B, C, X(14), X(14539)}} and {{A, B, C, X(15), X(2065)}} and these lines: {3, 6}, {13, 14561}, {14, 1503}, {17, 3589}, {18, 1352}, {69, 6672}, {147, 6782}, {193, 628}, {202, 611}, {203, 1428}, {383, 14137}, {395, 3564}, {397, 18583}, {524, 16529}, {542, 16268}, {613, 7006}, {624, 3618}, {698, 32466}, {2211, 23017}, {2330, 7005}, {3091, 31706}, {3751, 11708}, {5477, 25559}, {5480, 16965}, {5965, 22848}, {5979, 14136}, {6108, 6770}, {6593, 36208}, {6776, 7685}, {8740, 19128}, {9163, 9188}, {10616, 16773}, {10653, 14853}, {10654, 25406}, {11243, 23042}, {11579, 36209}, {19130, 22794}, {21461, 30535}

X(36758) = reflection of X(36757) in X(1692)
X(36758) = 1st-Lemoine-circle-inverse of X(36759)
X(36758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10614, 16), (6, 182, 61), (6, 5050, 36757), (6, 19145, 3364), (6, 19146, 3365), (6, 22238, 1351), (16, 61, 14539), (187, 11486, 16), (1662, 1663, 36759), (13349, 22238, 16), (19781, 22238, 13349)

X(36759) = MIDPOINT OF X(15) AND X(62)

Barycentrics    a^2 ((a^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2) - 2 Sqrt[3] S a^2) : :
Trilinears    sin(A - ω + π/6) : :
Trilinears    cos(A - ω - π/3) : :

X(36759) lies on these lines: {3, 6}, {13, 98}, {14, 10796}, {17, 10104}, {18, 83}, {23, 3457}, {25, 2004}, {30, 12205}, {99, 32465}, {110, 34394}, {140, 10617}, {202, 12835}, {203, 10802}, {298, 16530}, {303, 636}, {385, 22687}, {396, 11136}, {398, 32134}, {531, 11300}, {532, 5463}, {621, 7787}, {628, 33225}, {634, 7793}, {635, 10333}, {729, 9202}, {1576, 14186}, {2005, 5943}, {2378, 11636}, {2379, 32694}, {2698, 5994}, {3060, 21461}, {3129, 14704}, {3170, 3292}, {3180, 12214}, {3203, 3206}, {3407, 5981}, {3458, 5640}, {3972, 22689}, {5339, 18501}, {5970, 9203}, {6105, 16257}, {6774, 7753}, {7005, 10801}, {7006, 10799}, {7808, 11311}, {8260, 22522}, {9117, 32135}, {9763, 11301}, {10654, 10788}, {11003, 34395}, {11298, 16268}, {11364, 11707}, {12110, 16964}, {12177, 22997}, {13193, 36209}, {14880, 16965}, {21462, 34545}

X(36759) = midpoint of X(15) and X(62)
X(36759) = center of circle {{X(15),X(62),PU(1)}}
X(36759) = circumcircle-inverse of X(36760)
X(36759) = Brocard-circle-inverse of X(3107)
X(36759) = 1st-Lemoine-circle-inverse of X(36758)
X(36759) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(3104)
X(36759) = circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse of X(15)
X(36759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 3107), (6, 11842, 36760), (32, 182, 36760), (371, 372, 3104), (1379, 1380, 36760), (1662, 1663, 36758), (1687, 1688, 15)

X(36760) = MIDPOINT OF X(16) AND X(61)

Barycentrics    a^2 ((a^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2) + 2 Sqrt[3] S a^2) : :
Trilinears    sin(A - ω - π/6) : :
Trilinears    cos(A - ω + π/3) : :

X(36760) lies on these lines: {3, 6}, {13, 10796}, {14, 98}, {17, 83}, {18, 10104}, {23, 3458}, {25, 2005}, {30, 12204}, {99, 32466}, {110, 34395}, {140, 10616}, {202, 10802}, {203, 12835}, {299, 16529}, {302, 635}, {385, 22689}, {395, 11135}, {397, 32134}, {530, 11299}, {533, 5464}, {622, 7787}, {627, 33225}, {633, 7793}, {636, 10333}, {729, 9203}, {1576, 14188}, {2004, 5943}, {2378, 32694}, {2379, 11636}, {2698, 5995}, {3060, 21462}, {3130, 14705}, {3171, 3292}, {3181, 12213}, {3203, 3205}, {3407, 5980}, {3457, 5640}, {3972, 22687}, {5340, 18501}, {5970, 9202}, {6104, 16258}, {6771, 7753}, {7005, 10799}, {7006, 10801}, {7808, 11312}, {8259, 22523}, {9115, 32135}, {9761, 11302}, {10653, 10788}, {11003, 34394}, {11297, 16267}, {11364, 11708}, {12110, 16965}, {12177, 22998}, {13193, 36208}, {14880, 16964}, {21461, 34545}

X(36760) = midpoint of X(16) and X(61)
X(36760) = center of circle {{X(16),X(61),PU(1)}}
X(36760) = circumcircle-inverse of X(36759)
X(36760) = Brocard-circle-inverse of X(3106)
X(36760) = 1st-Lemoine-circle-inverse of X(36757)
X(36760) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(3105)
X(36760) = circle-{{X(1687),X(1688),PU(1),PU(2)}}-inverse of X(16)
X(36760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 3106), (6, 11842, 36759), (32, 182, 36759), (371, 372, 3105), (1379, 1380, 36759), (1662, 1663, 36757), (1687, 1688, 16)

leftri

Largest-circumscribed-equilateral triangle: X(36761)-X(36788)

rightri

This preamble and centers X(36761)-X(36788) were contributed by César Eliud Lozada, February 6, 2020.

Considere all equilateral triangles AeBeCe circumscribing ABC and such that A lies between Be and Ce (1), B lies between Ce and Ae (2) and C lies between Ae and Be (3) (see note at the end of this preamble). The A-vertex of the largest AeBeCe is the antipode of X(13) in the circle {{X(13), B, C}}, and the other two vertices are found cyclically. If this triangle is denoted as A'B'C' then A' has barycentric coordinates:

A' = -6*sqrt(3)*S*a^2 - (3*(a^2 + b^2 + c^2))*a^2 + 2*(b^2 - c^2)^2 :
  (7*b^2 + 2*c^2)*a^2 - 3*b^4 + 5*b^2*c^2 - 2*c^4 + 2*sqrt(3)*S*(2*a^2 + b^2) :
  (7*c^2 + 2*b^2)*a^2 - 3*c^4 + 5*c^2*b^2 - 2*b^4 + 2*sqrt(3)*S*(2*a^2 + c^2)

The center of A'B'C' is X(5463) and its squared-sidelength is 4*S*(cot(ω)+sqrt(3))/3, where S and ω are double-area and Brocard angle of ABC, respectively.

A'B'C' is perspective to the ABC-X(3)-reflections-triangle and it is also homothetic to the other triangles in the following list, where the given number n means that the respective homothetor is X(n):

(ABC-X3 reflections, 36761), (Bankoff, 36762), (3rd Fermat-Dao, 36763), (7th Fermat-Dao, 36764), (11th Fermat-Dao, 36765), (15th Fermat-Dao, 36766), (3rd inner-Fermat-Dao-Nhi, 35751), (4th inner-Fermat-Dao-Nhi, 36767), (1st outer-Fermat-Dao-Nhi, 36768), (2nd outer-Fermat-Dao-Nhi, 36769), (1st half-diamonds-central, 36770), (1st isodynamic-Dao, 23006), (3rd isodynamic-Dao, 36771), (1st Lemoine-Dao, 36772), (inner-Napoleon, 5463), (outer-Napoleon, 13)

Orthologic triangles to A'B'C' and orthologic centers:

(ABC, 5473, 13), (ABC-X3 reflections, 5473, 5473), (anti-Aquila, 5473, 11705), (anti-Ara, 5473, 12142), (anti-Artzt, 36775, 12155), (1st anti-Brocard, 36776, 5979), (5th anti-Brocard, 5473, 12205), (6th anti-Brocard, 36776, 12214), (2nd anti-circumperp-tangential, 5473, 18974), (anti-Euler, 5473, 6770), (anti-inner-Grebe, 5473, 19073), (anti-outer-Grebe, 5473, 19074), (anti-Mandart-incircle, 5473, 12337), (anti-McCay, 36777, 8595), (3rd anti-tri-squares, 36778, 22601), (4th anti-tri-squares, 36779, 22630), (anticomplementary, 5473, 616), (Aquila, 5473, 9901), (Ara, 5473, 9916), (Artzt, 36775, 9762), (1st Auriga, 5473, 12472), (2nd Auriga, 5473, 12473), (Bankoff, 5463, 34551), (1st Brocard-reflected, 36780, 22687), (1st Brocard, 36776, 3643), (5th Brocard, 5473, 9982), (6th Brocard, 36776, 9989), (2nd circumperp tangential, 5473, 22773), (Ehrmann-mid, 5473, 22796), (Euler, 5473, 5478), (inner-Fermat, 36781, 616), (outer-Fermat, 36782, 13), (2nd Fermat-Dao, 36783, 25207), (3rd Fermat-Dao, 5463, 16267), (4th Fermat-Dao, 9114, 5469), (6th Fermat-Dao, 36783, 25152), (7th Fermat-Dao, 5463, 396), (8th Fermat-Dao, 9114, 115), (10th Fermat-Dao, 36783, 25153), (11th Fermat-Dao, 5463, 381), (12th Fermat-Dao, 9114, 25154), (14th Fermat-Dao, 36783, 25155), (15th Fermat-Dao, 5463, 13), (16th Fermat-Dao, 9114, 25156), (1st inner-Fermat-Dao-Nhi, 9114, 35749), (2nd inner-Fermat-Dao-Nhi, 9114, 35750), (3rd inner-Fermat-Dao-Nhi, 5463, 2), (4th inner-Fermat-Dao-Nhi, 5463, 2), (1st outer-Fermat-Dao-Nhi, 5463, 2), (2nd outer-Fermat-Dao-Nhi, 5463, 2), (3rd outer-Fermat-Dao-Nhi, 9114, 35751), (4th outer-Fermat-Dao-Nhi, 9114, 35752), (outer-Garcia, 5473, 12781), (Gossard, 5473, 12793), (inner-Grebe, 5473, 6270), (outer-Grebe, 5473, 6268), (1st half-diamonds-central, 5463, 2), (2nd half-diamonds-central, 9114, 5459), (1st half-diamonds, 36781, 13), (2nd half-diamonds, 36782, 618), (1st half-squares, 36779, 33440), (2nd half-squares, 36778, 33441), (1st isodynamic-Dao, 5463, 13), (2nd isodynamic-Dao, 9114, 22998), (3rd isodynamic-Dao, 5463, 13), (4th isodynamic-Dao, 9114, 31710), (Johnson, 5473, 5617), (inner-Johnson, 5473, 12922), (outer-Johnson, 5473, 12932), (1st Johnson-Yff, 5473, 12942), (2nd Johnson-Yff, 5473, 12952), (1st Kenmotu-free-vertices, 5473, 35753), (2nd Kenmotu-free-vertices, 5473, 35754), (1st Lemoine-Dao, 5463, 10654), (2nd Lemoine-Dao, 9114, 13), (inner-Le Viet An, 36783, 14181), (Lucas homothetic, 5473, 12990), (Lucas(-1) homothetic, 5473, 12991), (Mandart-incircle, 5473, 13076), (McCay, 36777, 13084), (medial, 5473, 618), (5th mixtilinear, 5473, 7975), (Moses-Steiner osculatory, 36777, 34509), (inner-Napoleon, 9114, 5463), (outer-Napoleon, 5463, 2), (1st Neuberg, 36784, 6582), (2nd Neuberg, 36785, 6298), (1st tri-squares-central, 36786, 13705), (2nd tri-squares-central, 36787, 13825), (3rd tri-squares-central, 5473, 13917), (4th tri-squares-central, 5473, 13982), (1st tri-squares, 36775, 13646), (2nd tri-squares, 36775, 13765), (3rd tri-squares, 36779, 13876), (4th tri-squares, 36778, 13929), (inner-Vecten, 36778, 6302), (outer-Vecten, 36779, 6306), (Vu-Dao-X(16)-isodynamic, 36788, 13), (X3-ABC reflections, 5473, 13103), (inner-Yff, 5473, 10062), (outer-Yff, 5473, 10078), (inner-Yff tangents, 5473, 13105), (outer-Yff tangents, 5473, 13107)

Parallelogic triangles to A'B'C' and parallelogic centers:

(2nd Fermat-Dao, 36773, 25216), (4th Fermat-Dao, 6777, 16530), (6th Fermat-Dao, 36773, 25229), (8th Fermat-Dao, 6777, 9115), (10th Fermat-Dao, 36773, 25231), (12th Fermat-Dao, 6777, 5617), (14th Fermat-Dao, 36773, 25233), (16th Fermat-Dao, 6777, 25235), (1st inner-Fermat-Dao-Nhi, 6777, 35750), (2nd inner-Fermat-Dao-Nhi, 6777, 35749), (3rd outer-Fermat-Dao-Nhi, 6777, 35752), (4th outer-Fermat-Dao-Nhi, 6777, 35751), (2nd half-diamonds-central, 6777, 618), (2nd isodynamic-Dao, 6777, 23005), (4th isodynamic-Dao, 6777, 6782), (2nd Lemoine-Dao, 6777, 23006), (inner-Le Viet An, 36773, 14187), (inner-Napoleon, 6777, 13), (1st Parry, 5473, 13305), (2nd Parry, 5473, 9200), (Vu-Dao-X(16)-isodynamic, 36774, 4)

Definitions of all mentioned triangles can be seen here.

Note: The centers of circles {{X(13), B, C}}, {{X(13), C, A}} and {{X(13), A, B}} are the vertices of outer-Napoleon triangle, i.e., A'B'C' is the reflection triangle of X(13) in the vertices of the outer-Napoleon triangle. A similar construction can be made using X(14) and the inner-Napoleon triangle, but in this case, conditions (1), (2), (3) are not all satisfied at the same time.

X(36761) = PERSPECTOR OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND ABC-X3 REFLECTIONS

Barycentrics    3*S^4-(2*SA^2+7*SB*SC-2*SW^2)*S^2-4*SB*SC*SW^2-S*sqrt(3)*((SA-3*SW)*S^2+6*SW*SB*SC) : :

X(36761) lies on these lines: {376,5463}, {1080,9749}, {1503,5473}, {2794,5474}, {3105,11257}, {9114,12117}

X(36762) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND BANKOFF

Barycentrics    ((SB+SC)*(9*S^2+2*SA^2-2*SB*SC)+8*S*(S^2+SB*SC))*sqrt(3)+4*(SB+SC)*(4*S^2+SA^2-SB*SC)+S*(13*S^2+15*SB*SC) : :

X(36762) lies on these lines: {3,13}, {3390,9112}, {5463,34551}, {6777,35759}, {9114,35748}, {23006,35739}, {35734,35751}

X(36763) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 3rd FERMAT-DAO

Barycentrics    ((6*SW-SA)*S^2+(SA^2-SB*SC-SW^2)*SA)*sqrt(3)+S*(8*S^2-3*SA*SW+3*SW^2) : :

X(36763) lies on these lines: {2,14136}, {3,13}, {14,36519}, {396,3564}, {5463,16267}, {5470,9114}, {5478,18582}, {6115,6770}, {6302,22631}, {6306,22602}, {6772,20252}, {6777,16529}, {9112,16960}, {10613,16962}, {11542,23006}, {22236,22796}, {22489,22846}

X(36763) = {X(10611), X(13103)}-harmonic conjugate of X(13)

X(36764) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 7th FERMAT-DAO

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

X(36764) lies on these lines: {2,22574}, {3,13}, {115,9114}, {395,31406}, {396,5463}, {530,11488}, {5472,35751}, {6772,22489}, {6777,9117}, {6779,16960}, {9885,32459}, {16267,23006}

X(36764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 16241, 21156), (396, 5463, 9112)

X(36765) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 11th FERMAT-DAO

Barycentrics    (a^6+(b^4+6*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2))*sqrt(3)-2*S*(a^4-5*(b^2+c^2)*a^2+4*(b^2-c^2)^2) : :
X(36765) = X(3)+2*X(22796) = X(4)+2*X(618) = 2*X(4)+X(5473) = 4*X(5)-X(13) = 2*X(5)+X(5617) = 5*X(5)-2*X(20252) = X(13)+2*X(5617) = 5*X(13)-8*X(20252) = X(14)+2*X(114) = X(98)-4*X(6670) = X(99)+2*X(5479) = X(298)+2*X(7684) = 2*X(355)+X(7975) = 2*X(381)+X(5463) = X(616)+5*X(3091) = X(616)+2*X(5478) = 4*X(618)-X(5473) = 5*X(5617)+4*X(20252)

X(36765) lies on these lines: {2,9749}, {3,22796}, {4,618}, {5,13}, {11,12942}, {12,12952}, {14,114}, {98,6670}, {99,5479}, {119,13105}, {262,9762}, {298,7684}, {355,7975}, {381,5463}, {485,19074}, {486,19073}, {511,21359}, {530,3545}, {531,23234}, {542,5050}, {616,3091}, {620,5474}, {623,1080}, {946,12781}, {1656,6771}, {2782,5469}, {2794,11298}, {3090,6669}, {3564,16267}, {3851,13103}, {5066,25154}, {5071,5459}, {5072,16001}, {5079,20415}, {5460,6054}, {5470,23514}, {5613,6777}, {5979,7685}, {6033,6774}, {6115,7736}, {6268,10515}, {6270,10514}, {6779,20429}, {6782,9112}, {7395,9916}, {7507,12142}, {7741,10078}, {7814,11129}, {7951,10062}, {7974,11724}, {7989,9901}, {8227,11705}, {9114,25164}, {9116,9880}, {9864,11706}, {9982,10356}, {10109,36363}, {10358,12205}, {10895,18974}, {10896,13076}, {13107,26470}, {13916,19056}, {13981,19055}, {14145,22831}, {14561,16268}, {16626,16964}, {16808,23006}, {19709,35752}, {22236,22892}, {22513,23303}, {33412,33422}

X(36765) = reflection of X(i) in X(j) for these (i,j): (5470, 23514), (21156, 2), (22489, 5055)
X(36765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 618, 5473), (5, 5617, 13), (616, 3091, 5478), (3090, 6770, 6669)

X(36766) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 15th FERMAT-DAO

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

X(36766) lies on these lines: {2,13}, {5,23005}, {6,16530}, {14,8724}, {15,5617}, {17,5472}, {18,39}, {61,6782}, {99,623}, {115,16967}, {396,22998}, {542,16241}, {635,7836}, {3106,33391}, {3131,8174}, {5092,21156}, {5116,11646}, {5469,6772}, {5473,19106}, {5873,10104}, {6671,6783}, {6771,6778}, {7799,21359}, {8290,8291}, {9112,16960}, {9114,9885}, {10646,22513}, {11602,13188}, {14145,31703}, {19107,22796}, {22511,23303}

X(36766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 5463, 6779), (16, 6115, 13), (17, 25235, 5472), (618, 5979, 5463), (618, 6115, 16), (5463, 22489, 12155), (6778, 33417, 6771)

X(36767) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 4th INNER-FERMAT-DAO-NHI

Barycentrics    2*(a^2-8*b^2-8*c^2)*S+(11*a^4-13*(b^2+c^2)*a^2+2*(b^2-c^2)^2)*sqrt(3) : :
X(36767) = 8*X(2)-3*X(13) = 7*X(2)+3*X(616) = X(2)-6*X(618) = 11*X(2)-6*X(5459) = 2*X(2)+3*X(5463) = 17*X(2)-12*X(6669) = 14*X(2)-9*X(22489) = 11*X(2)-X(35749) = 9*X(2)+X(35750) = 4*X(2)+X(35751) = 6*X(2)-X(35752) = 7*X(13)+8*X(616) = X(13)-16*X(618) = 11*X(13)-16*X(5459) = X(13)+4*X(5463) = 7*X(13)-12*X(22489) = 3*X(13)+2*X(35751) = 9*X(13)-4*X(35752) = X(616)+14*X(618) = 11*X(616)+14*X(5459) = 2*X(616)-7*X(5463) = 2*X(616)+3*X(22489) = 7*X(616)+8*X(35019) = 12*X(616)-7*X(35751) = 18*X(616)+7*X(35752)

X(36767) lies on these lines: {2,13}, {14,15300}, {542,15040}, {549,36383}, {2482,6777}, {3524,36344}, {3526,32907}, {3830,5473}, {4677,7975}, {5469,9116}, {5617,8703}, {5863,33616}, {6778,9167}, {7485,13859}, {9114,36330}, {9885,33459}, {10109,25154}, {10657,17811}, {12100,36363}, {12816,33621}, {15534,36386}, {15701,21156}, {15703,16001}, {15719,36318}, {21359,35931}, {22165,36388}, {22490,22577}, {22493,35304}, {33474,35697}, {33606,35696}

X(36767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5463, 35751), (2, 35749, 5459), (2, 35751, 13), (5463, 22489, 616), (36330, 36521, 9114)

X(36768) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 1st OUTER-FERMAT-DAO-NHI

Barycentrics    2*(2*a^2-7*b^2-7*c^2)*S+(10*a^4-11*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3) : :
X(36768) = 7*X(2)-3*X(13) = 5*X(2)+3*X(616) = X(2)-3*X(618) = 5*X(2)-3*X(5459) = X(2)+3*X(5463) = 4*X(2)-3*X(6669) = 13*X(2)-9*X(22489) = 11*X(2)-6*X(35019) = 9*X(2)-X(35749) = 7*X(2)+X(35750) = 3*X(2)+X(35751) = 5*X(2)-X(35752) = 5*X(13)+7*X(616) = X(13)-7*X(618) = 5*X(13)-7*X(5459) = X(13)+7*X(5463) = 4*X(13)-7*X(6669) = 11*X(13)-14*X(35019) = 3*X(13)+X(35750) = 9*X(13)+7*X(35751) = 15*X(13)-7*X(35752)

X(36768) lies on these lines: {2,13}, {3,36363}, {99,36330}, {531,36521}, {533,35304}, {542,12100}, {627,22496}, {635,35932}, {2482,32553}, {3524,36318}, {3534,5617}, {5473,15682}, {5478,19709}, {5858,9885}, {5859,33619}, {6770,15719}, {6771,15713}, {6777,36331}, {8724,36382}, {9114,36327}, {9761,35696}, {10124,20415}, {10190,27551}, {11539,32907}, {13859,15246}, {14145,36368}, {15693,36383}, {15698,36344}, {15699,16001}, {22796,33699}, {33474,33561}, {33602,33614}, {33605,35691}, {36376,36377}, {36390,36391}

X(36768) = midpoint of X(i) and X(j) for these {i,j}: {616, 5459}, {618, 5463}, {2482, 32553}
X(36768) = reflection of X(20415) in X(10124)
X(36768) = complement of the complement of X(35751)
X(36768) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 616, 35752), (2, 35750, 13), (2, 35752, 5459)

X(36769) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 2nd OUTER-FERMAT-DAO-NHI

Barycentrics    2*(4*a^2-5*b^2-5*c^2)*S+(8*a^4-7*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3) : :
X(36769) = 5*X(2)-3*X(13) = X(2)+3*X(616) = 2*X(2)-3*X(618) = 4*X(2)-3*X(5459) = X(2)-3*X(5463) = 7*X(2)-6*X(6669) = 11*X(2)-9*X(22489) = 17*X(2)-12*X(35019) = 5*X(2)-X(35749) = 3*X(2)+X(35750) = X(13)+5*X(616) = 2*X(13)-5*X(618) = 4*X(13)-5*X(5459) = X(13)-5*X(5463) = 7*X(13)-10*X(6669) = 11*X(13)-15*X(22489) = 17*X(13)-20*X(35019) = 3*X(13)-X(35749) = 9*X(13)+5*X(35750) = 3*X(13)+5*X(35751) = 9*X(13)-5*X(35752)

X(36769) lies on these lines: {2,13}, {3,36383}, {99,36329}, {140,32907}, {376,36344}, {531,15300}, {532,35304}, {533,35931}, {542,8703}, {543,32553}, {547,16001}, {635,11296}, {2482,32552}, {3081,12793}, {3534,36363}, {3830,5617}, {4677,12781}, {5066,5478}, {5460,36523}, {5473,11001}, {5858,35696}, {5859,9885}, {6636,13859}, {6670,31695}, {6770,15698}, {6771,11812}, {6777,8591}, {8724,36362}, {9114,36331}, {9116,36330}, {9761,35697}, {11123,27551}, {11539,20415}, {12101,22796}, {14145,36388}, {15719,21156}, {16963,36251}, {19708,36318}, {19709,25154}, {33603,35690}, {33604,33616}, {33613,33625}, {36334,36335}, {36353,36355}

X(36769) = midpoint of X(i) and X(j) for these {i,j}: {2, 35751}, {616, 5463}, {3534, 36363}, {5858, 35696}, {35750, 35752}
X(36769) = reflection of X(i) in X(j) for these (i,j): (618, 5463), (5459, 618), (16001, 547), (31695, 6670), (32552, 2482), (32907, 140)
X(36769) = complement of X(35752)
X(36769) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 616, 35751), (2, 35749, 13), (2, 35750, 35752), (5463, 35751, 2), (35751, 35752, 35750)

X(36770) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 1st HALF-DIAMONDS-CENTRAL

Barycentrics    6*(a^2+2*b^2+2*c^2)*S-(7*a^4-11*(b^2+c^2)*a^2+4*(b^2-c^2)^2)*sqrt(3) : :
X(36770) = 6*X(2)-X(13) = 9*X(2)+X(616) = 3*X(2)+2*X(618) = 7*X(2)-2*X(5459) = 4*X(2)+X(5463) = 9*X(2)-4*X(6669) = 8*X(2)-3*X(22489) = 14*X(2)+X(35751) = 16*X(2)-X(35752) = 3*X(13)+2*X(616) = X(13)+4*X(618) = 7*X(13)-12*X(5459) = 2*X(13)+3*X(5463) = 3*X(13)-8*X(6669) = 4*X(13)-9*X(22489) = 11*X(13)-16*X(35019) = 7*X(13)+3*X(35751) = 8*X(13)-3*X(35752) = X(616)-6*X(618) = 7*X(616)+18*X(5459) = 4*X(616)-9*X(5463) = X(616)+4*X(6669) = 14*X(616)-9*X(35751) = 16*X(616)+9*X(35752)

X(36770) lies on these lines: {2,13}, {3,22796}, {5,5473}, {6,22892}, {10,7975}, {14,620}, {15,21359}, {17,11309}, {18,629}, {99,5469}, {114,21157}, {140,5617}, {298,6671}, {299,16530}, {302,11129}, {395,31406}, {532,16960}, {542,3763}, {590,19074}, {615,19073}, {619,6777}, {635,33367}, {1125,12781}, {2482,22490}, {3090,5478}, {3106,25183}, {3411,6694}, {3525,6770}, {3526,6771}, {3624,11705}, {3788,16241}, {4413,12337}, {5070,13103}, {5094,12142}, {5432,12952}, {5433,12942}, {5460,9114}, {5461,9116}, {5464,9167}, {5470,6722}, {5479,21166}, {6034,22848}, {6674,14145}, {6772,22847}, {6774,15561}, {7484,9916}, {7749,16644}, {7808,12205}, {7914,9982}, {8997,19075}, {9112,23302}, {9115,21360}, {9166,22577}, {9749,9751}, {11297,16967}, {11299,16809}, {11540,36383}, {12793,15184}, {12932,24953}, {13105,26364}, {13107,26363}, {13916,19108}, {13917,32785}, {13981,19109}, {13982,32786}, {13989,19076}, {15699,25154}, {16772,22845}, {22578,36521}

X(36770) = intersection, other than A,B,C, of conics {{A, B, C, X(18), X(8014)}} and {{A, B, C, X(298), X(6669)}}
X(36770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 616, 6669), (2, 618, 13), (2, 3643, 16966), (2, 5463, 22489), (13, 618, 5463), (99, 6670, 5469), (140, 5617, 21156), (298, 6671, 16962), (616, 6669, 13), (618, 6669, 616), (629, 11307, 18), (5463, 22489, 35752)

X(36771) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 3rd ISODYNAMIC-DAO

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

X(36771) lies on these lines: {2,13}, {17,5868}, {5318,5473}, {5472,7746}, {5617,9112}, {6777,6783}, {7684,9749}, {9114,31709}, {11480,22892}, {11485,22796}, {14136,18581}, {19106,31705}, {21156,22513}, {22688,32465}

X(36771) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5617, 11542, 9112), (6115, 18582, 13), (22513, 23302, 21156)

X(36772) = HOMOTHETIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL AND 1st LEMOINE-DAO

Barycentrics    4*S^3-(SB+SC)*(9*SA+2*SW)*S+sqrt(3)*((SA-3*SW)*S^2+2*SW*SB*SC) : :

X(36772) lies on these lines: {6,5473}, {13,15}, {14,2482}, {61,23006}, {115,5474}, {230,21158}, {618,5334}, {3104,32465}, {3389,12123}, {3390,12124}, {5463,6782}, {5472,22236}, {5478,11488}, {6777,23013}, {9112,11485}, {15534,25235}, {16809,16942}, {21359,33518}, {22489,31710}

X(36772) = reflection of X(9112) in X(11485)
X(36772) = {X(115), X(11480)}-harmonic conjugate of X(21156)

X(36773) = PARALLELOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO LE VIET AN-INNER

Barycentrics    a^2*((2*(b^2+2*c^2)*(2*b^2+c^2)*a^6-18*(b^2+c^2)*b^2*c^2*a^4-2*(2*b^8+2*c^8-(8*b^4-3*b^2*c^2+8*c^4)*b^2*c^2)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2)*S-((2*b^4-b^2*c^2+2*c^4)*a^8-6*(b^4-c^4)*(b^2-c^2)*a^6+3*(b^4+b^2*c^2+c^4)*(b^2-2*c^2)*(2*b^2-c^2)*a^4-(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4)*b^2*c^2)*sqrt(3)) : :
X(36773) = X(13)-2*X(14188) = 3*X(22489)-4*X(33481) = X(35751)+2*X(36387)

The reciprocal parallelogic center of these triangles is X(14187)

X(36773) lies on these lines: {13,14188}, {511,9114}, {512,6777}, {6779,25182}, {9112,25178}, {22489,33481}, {23006,23007}, {35751,36387}

X(36773) = reflection of X(13) in X(14188)

X(36774) = PARALLELOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO VU-DAO-X(16)-ISODYNAMIC

Barycentrics    15*sqrt(3)*S^4+3*(48*R^2+SA-9*SW)*S^3-sqrt(3)*(36*R^2*(2*SA+SW)-20*SA^2+11*SB*SC-7*SW^2)*S^2+3*(4*R^2*(18*SA^2-27*SA*SW+SW^2)-SW*(12*SA^2-21*SA*SW+SW^2))*S+sqrt(3)*(108*R^2-23*SW)*SB*SC*SW : :

The reciprocal parallelogic center of these triangles is X(4)

X(36774) lies on these lines: {6777,23871}, {9114,12117}

X(36775) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO ANTI-ARTZT

Barycentrics    -6*sqrt(3)*S*a^2+(a^2-2*b^2-2*c^2)*(5*a^2-b^2-c^2) : :
X(36775) = 3*X(5469)-4*X(33476) = 2*X(5859)+X(35751) = 4*X(13083)-3*X(21156) = 3*X(22489)-4*X(33475) = 2*X(33458)+X(35696) = 4*X(33458)-X(35752) = 2*X(35696)+X(35752)

The reciprocal orthologic center of these triangles is X(12155)

X(36775) lies on these lines: {2,22574}, {6,2482}, {13,543}, {14,11184}, {15,524}, {16,7618}, {17,34505}, {61,34511}, {99,9112}, {303,671}, {376,530}, {395,12040}, {531,1080}, {538,16962}, {1992,11153}, {3107,5969}, {3180,8595}, {5469,33476}, {5472,15300}, {5485,11488}, {5611,8724}, {5859,35751}, {6093,9203}, {7610,16241}, {7617,16966}, {7619,33416}, {7620,18582}, {7622,16242}, {7775,16964}, {8176,16809}, {8182,10645}, {8591,33376}, {8592,8594}, {9115,15534}, {9168,23870}, {9761,11301}, {9770,10654}, {9830,22568}, {11163,12154}, {13083,21156}, {16509,23302}, {19106,32479}, {22489,22846}, {22495,23006}, {22580,35303}, {33458,35696}

X(36775) = reflection of X(i) in X(j) for these (i,j): (13, 9763), (5463, 9885)
X(36775) = X(524)-Hirst inverse of-X(5463)
X(36775) = {X(33458), X(35696)}-harmonic conjugate of X(35752)

X(36776) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 1st ANTI-BROCARD

Barycentrics    2*S*(2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))+(4*(b^2+c^2)*a^6-(5*b^4+6*b^2*c^2+5*c^4)*a^4+2*(b^2+c^2)*(b^4+c^4)*a^2-(b^4-c^4)^2)*sqrt(3) : :
X(36776) = 4*X(5)-3*X(5469) = 2*X(98)-3*X(21156) = 4*X(619)-3*X(21156) = 4*X(620)-3*X(21157) = 2*X(5460)-3*X(23234) = 4*X(6669)-3*X(14651) = 2*X(6774)-3*X(15561) = 2*X(11632)-3*X(22489) = X(14692)+2*X(25559) = X(35751)+2*X(36362)

The reciprocal orthologic center of these triangles is X(5979)

X(36776) lies on these lines: {3,67}, {4,35689}, {5,5469}, {13,2782}, {14,114}, {15,5617}, {30,9114}, {98,619}, {99,5473}, {147,617}, {148,5478}, {383,6299}, {531,1080}, {616,14144}, {618,6773}, {620,21157}, {2794,5474}, {3023,12941}, {3027,12951}, {3564,22998}, {5459,12243}, {5460,23234}, {5978,9749}, {6321,22797}, {6669,14651}, {6770,22687}, {6771,12188}, {6774,15561}, {6778,23006}, {6779,22509}, {6783,9112}, {9880,22578}, {11300,12155}, {11311,11623}, {11632,22489}, {12177,22997}, {12184,18975}, {12185,13075}, {13102,22796}, {14692,25559}, {20428,25166}, {22566,25164}, {22577,25154}, {35751,36362}

X(36776) = midpoint of X(147) and X(617)
X(36776) = reflection of X(i) in X(j) for these (i,j): (13, 5613), (14, 114), (98, 619), (148, 5478), (5463, 8724), (5473, 99), (6321, 22797), (6770, 32552), (6773, 618), (6777, 5617), (12188, 6771), (12243, 5459), (13102, 22796), (22577, 25154), (22578, 9880), (25164, 22566), (25166, 20428)
X(36776) = {X(98), X(619)}-harmonic conjugate of X(21156)

X(36777) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO ANTI-MCCAY

Barycentrics    -2*sqrt(3)*(8*a^4-8*(b^2+c^2)*a^2-b^4+10*b^2*c^2-c^4)*S+2*a^6-6*(b^2+c^2)*a^4+9*(b^4+c^4)*a^2-(b^2+c^2)^3 : :
X(36777) = 3*X(5469)-4*X(33474) = 2*X(35692)+X(35751)

The reciprocal orthologic center of these triangles is X(8595)

X(36777) lies on these lines: {13,9886}, {14,543}, {99,9112}, {524,6779}, {531,5473}, {2482,16644}, {3181,8591}, {5464,23006}, {5469,33474}, {5477,11173}, {5969,32465}, {8724,13103}, {9762,23234}, {9830,35692}

X(36777) = reflection of X(13) in X(9886)

X(36778) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 3rd ANTI-TRI-SQUARES

Barycentrics    (4*((b^2+c^2)*a^2+10*b^2*c^2)*sqrt(3)+66*a^4-90*(b^2+c^2)*a^2+24*b^2*c^2)*S-18*(b^2+c^2)*a^4+6*(5*b^4+8*b^2*c^2+5*c^4)*a^2-12*(b^4-c^4)*(b^2-c^2)+sqrt(3)*(11*a^6-8*(b^2+c^2)*a^4-5*(b^4+6*b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)) : :
X(36778) = X(35751)+2*X(36371)

The reciprocal orthologic center of these triangles is X(22601)

X(36778) lies on these lines: {13,6300}, {487,33441}, {618,6337}, {642,3366}, {2043,5473}, {5463,32419}, {6306,22602}, {13926,35878}, {22609,23006}, {35751,36371}

X(36778) = reflection of X(13) in X(6300)

X(36779) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 4th ANTI-TRI-SQUARES

Barycentrics    -(-4*((b^2+c^2)*a^2+10*b^2*c^2)*sqrt(3)+66*a^4-90*(b^2+c^2)*a^2+24*b^2*c^2)*S-18*(b^2+c^2)*a^4+6*(5*b^4+8*b^2*c^2+5*c^4)*a^2-12*(b^4-c^4)*(b^2-c^2)-sqrt(3)*(11*a^6-8*(b^2+c^2)*a^4-5*(b^4+6*b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)) : :
X(36779) = X(35751)+2*X(36370)

The reciprocal orthologic center of these triangles is X(22630)

X(36779) lies on these lines: {13,6304}, {488,33440}, {618,6337}, {641,3367}, {2044,5473}, {5463,32421}, {6302,22631}, {13873,35879}, {22638,23006}, {35751,36370}

X(36779) = reflection of X(13) in X(6304)

X(36780) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 1st BROCARD-REFLECTED

Barycentrics    6*sqrt(3)*a^2*(a^2-b^2-c^2)*((b^2+c^2)*a^2+b^2*c^2)*S+3*(b^2+c^2)*a^8-(4*b^4+b^2*c^2+4*c^4)*a^6+3*(b^4-c^4)*(b^2-c^2)*a^4-(2*b^8+2*c^8+(b^4+18*b^2*c^2+c^4)*b^2*c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(36780) = X(35751)+2*X(36364)

The reciprocal orthologic center of these triangles is X(22687)

X(36780) lies on these lines: {13,22715}, {15,32465}, {16,31958}, {511,5463}, {698,21158}, {2782,9114}, {3105,22677}, {3107,16242}, {5473,22676}, {5978,9749}, {5980,6194}, {13083,21156}, {22701,23006}, {35751,36364}

X(36780) = reflection of X(13) in X(22715)

X(36781) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO INNER-FERMAT

Barycentrics    2*sqrt(3)*(7*a^4-11*(b^2+c^2)*a^2+b^4+10*b^2*c^2+c^4)*S+9*a^6-18*(b^2+c^2)*a^4+6*(2*b^4-b^2*c^2+2*c^4)*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(36781) = 3*X(5463)-4*X(14145) = 9*X(22489)-8*X(22846) = 2*X(33624)+X(35751) = X(35752)-4*X(36388)

The reciprocal orthologic center of these triangles is X(616)

X(36781) lies on these lines: {13,628}, {18,629}, {533,5463}, {550,5473}, {618,22114}, {630,10188}, {5979,33960}, {5983,22665}, {9114,35752}, {16530,35688}, {16964,22871}, {22489,22846}, {22511,33386}, {22855,23006}, {33624,35751}

X(36781) = reflection of X(i) in X(j) for these (i,j): (13, 628), (18, 30471), (11121, 630), (22114, 618)

X(36782) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO OUTER-FERMAT

Barycentrics    2*sqrt(3)*(4*a^4-8*(b^2+c^2)*a^2+(b^2-c^2)^2)*S+6*a^6-6*(b^2+c^2)*a^4-3*(b^4+8*b^2*c^2+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2) : :
X(36782) = 3*X(13)-4*X(10611) = X(13)-4*X(22892) = 2*X(15)+X(22894) = 3*X(17)-2*X(10611) = 3*X(5469)-2*X(11602) = 3*X(5469)-4*X(22891) = X(10611)-3*X(22892) = 4*X(14138)-3*X(16962) = 5*X(16960)-2*X(22900) = X(19106)-4*X(31705) = X(35751)+2*X(36366)

The reciprocal orthologic center of these triangles is X(13)

X(36782) lies on these lines: {2,5469}, {3,13}, {14,15561}, {15,5617}, {16,14136}, {18,629}, {61,618}, {62,8259}, {99,6671}, {115,33417}, {140,22511}, {299,22737}, {396,6779}, {530,30559}, {532,5463}, {616,22113}, {630,33021}, {632,22847}, {1080,9749}, {3106,30472}, {3107,16242}, {3166,6770}, {5238,22532}, {5472,22895}, {6115,10645}, {6673,11289}, {6694,33225}, {6772,22846}, {6782,34754}, {9112,19780}, {9761,11301}, {11300,22489}, {11485,22901}, {14145,22687}, {15534,36386}, {16626,16964}, {16809,16942}, {16960,22900}, {16966,31704}, {19106,31705}, {20252,23005}, {25560,25608}, {35751,36366}

X(36782) = midpoint of X(616) and X(22113)
X(36782) = reflection of X(i) in X(j) for these (i,j): (13, 17), (17, 22892), (627, 618), (11602, 22891)
X(36782) = intersection, other than A,B,C, of conics {{A, B, C, X(13), X(34219)}} and {{A, B, C, X(15), X(32627)}}
X(36782) = circumcircle-inverse of-X(31939)
X(36782) = inner-Napoleon circle-inverse of-X(22739)
X(36782) = outer-Napoleon-to-inner-Napoleon similarity image of X(17)
X(36782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (61, 618, 16530), (11602, 22891, 5469)

X(36783) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 2nd FERMAT-DAO

Barycentrics    ((2*(2*b^2-c^2)*(b^2-2*c^2)*a^6-6*(b^2+c^2)*b^2*c^2*a^4-2*(2*b^8+2*c^8-(8*b^4+3*b^2*c^2+8*c^4)*b^2*c^2)*a^2+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*b^2*c^2)*S-((2*b^4+b^2*c^2+2*c^4)*a^8-6*(b^2+c^2)*(b^4+c^4)*a^6+(b^4+b^2*c^2+c^4)*(6*b^4-b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(2*b^8+2*c^8-(b^4-5*b^2*c^2+c^4)*b^2*c^2)*a^2+(b^2-c^2)^2*(b^4+c^4)*b^2*c^2)*sqrt(3))*a^2 : :
X(36783) = 3*X(22489)-4*X(33491) = X(35751)+2*X(36367)

The reciprocal orthologic center of these triangles is X(25207)

X(36783) lies on these lines: {13,14182}, {16,9998}, {511,6777}, {512,9114}, {6779,25228}, {9112,25220}, {9203,32730}, {22489,33491}, {22999,23006}, {35751,36367}

X(36783) = reflection of X(13) in X(14182)

X(36784) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 1st NEUBERG

Barycentrics    -2*sqrt(3)*(2*(4*b^4+5*b^2*c^2+4*c^4)*a^4+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(3*b^2+c^2)*(b^2+3*c^2)*b^2*c^2)*S+6*(b^2+c^2)*a^8-6*(b^2+c^2)^2*a^6-3*(b^2+c^2)*(5*b^4+8*b^2*c^2+5*c^4)*a^4+3*(b^8+c^8+(b^4+10*b^2*c^2+c^4)*b^2*c^2)*a^2-3*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*b^2*c^2 : :
X(36784) = X(35751)+2*X(36373)

The reciprocal orthologic center of these triangles is X(6582)

X(36784) lies on these lines: {13,6581}, {538,5463}, {698,32465}, {5473,11257}, {5969,9114}, {23000,23006}, {35751,36373}

X(36784) = reflection of X(13) in X(6581)

X(36785) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 2nd NEUBERG

Barycentrics    2*sqrt(3)*(a^6+(b^2+c^2)*a^4+(b^4+3*b^2*c^2+c^4)*a^2+2*(b^2+c^2)*b^2*c^2)*S+a^8-6*(b^2+c^2)*a^6+12*(b^2+c^2)*b^2*c^2*a^2+(8*b^4+9*b^2*c^2+8*c^4)*a^4+3*(b^2+c^2)*(b^4-b^2*c^2+c^4)*a^2-2*(b^4+c^4)^2 : :
X(36785) = X(35751)+2*X(36375)

The reciprocal orthologic center of these triangles is X(6298)

X(36785) lies on these lines: {13,6296}, {732,32465}, {754,5463}, {2896,14904}, {5473,12122}, {9114,35696}, {23001,23006}, {35751,36375}

X(36785) = reflection of X(13) in X(6296)

X(36786) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (195*a^6+240*(b^2+c^2)*a^4-3*(151*b^4+154*b^2*c^2+151*c^4)*a^2+18*(b^4-c^4)*(b^2-c^2)+12*S*(3*(b^2+c^2)*a^2-4*b^4-58*b^2*c^2-4*c^4))*sqrt(3)-48*a^6+462*(b^2+c^2)*a^4-6*(103*b^4+120*b^2*c^2+103*c^4)*a^2+204*(b^4-c^4)*(b^2-c^2)+2*S*(1129*a^4-925*(b^2+c^2)*a^2-272*b^4+572*b^2*c^2-272*c^4) : :
X(36786) = X(35751)+2*X(36376)

The reciprocal orthologic center of these triangles is X(13705)

X(36786) lies on these lines: {13,13706}, {5473,13666}, {23002,23006}, {35751,36376}, {36353,36355}

X(36786) = reflection of X(13) in X(13706)

X(36787) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -(195*a^6+240*(b^2+c^2)*a^4-3*(151*b^4+154*b^2*c^2+151*c^4)*a^2+18*(b^4-c^4)*(b^2-c^2)-12*S*(3*(b^2+c^2)*a^2-4*b^4-58*b^2*c^2-4*c^4))*sqrt(3)-48*a^6+462*(b^2+c^2)*a^4-6*(103*b^4+120*b^2*c^2+103*c^4)*a^2+204*(b^4-c^4)*(b^2-c^2)-2*S*(1129*a^4-925*(b^2+c^2)*a^2-272*b^4+572*b^2*c^2-272*c^4) : :
X(36787) = X(35751)+2*X(36377)

The reciprocal orthologic center of these triangles is X(13825)

X(36787) lies on these lines: {13,13826}, {5473,13786}, {23003,23006}, {35751,36377}, {36353,36355}

X(36787) = reflection of X(13) in X(13826)

X(36788) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LARGEST-CIRCUMSCRIBED-EQUILATERAL TO VU-DAO-X(16)-ISODYNAMIC

Barycentrics    2*S*(6*a^18-39*(b^2+c^2)*a^16+5*(13*b^4+16*b^2*c^2+13*c^4)*a^14+6*(b^2+c^2)*(4*b^4-23*b^2*c^2+4*c^4)*a^12-6*(28*b^8+28*c^8-(5*b^2-7*b*c+5*c^2)*(5*b^2+7*b*c+5*c^2)*b^2*c^2)*a^10+(b^2+c^2)*(156*b^8+156*c^8-(203*b^4-151*b^2*c^2+203*c^4)*b^2*c^2)*a^8-(24*b^12+24*c^12+(97*b^8+97*c^8-5*(23*b^4-6*b^2*c^2+23*c^4)*b^2*c^2)*b^2*c^2)*a^6-3*(b^4-c^4)*(b^2-c^2)*(11*b^8+11*c^8-2*(8*b^4-b^2*c^2+8*c^4)*b^2*c^2)*a^4+(13*b^12+13*c^12+(b^2-c^2)^2*(13*b^4+36*b^2*c^2+13*c^4)*b^2*c^2)*(b^2-c^2)^2*a^2-(b^6+c^6)*b^2*c^2*(b^2-c^2)^4)*sqrt(3)+14*a^20-31*(b^2+c^2)*a^18-6*(14*b^4-5*b^2*c^2+14*c^4)*a^16+(b^2+c^2)*(361*b^4-292*b^2*c^2+361*c^4)*a^14-2*(208*b^8+208*c^8+(106*b^4+27*b^2*c^2+106*c^4)*b^2*c^2)*a^12+3*(b^2+c^2)*(8*b^8+8*c^8+5*(29*b^4-35*b^2*c^2+29*c^4)*b^2*c^2)*a^10+2*(170*b^12+170*c^12-3*(108*b^8+108*c^8-(59*b^4-35*b^2*c^2+59*c^4)*b^2*c^2)*b^2*c^2)*a^8-(b^2+c^2)*(287*b^12+287*c^12-3*(247*b^8+247*c^8-4*(82*b^4-87*b^2*c^2+82*c^4)*b^2*c^2)*b^2*c^2)*a^6+3*(b^2-c^2)^2*(26*b^12+26*c^12+(21*b^8+21*c^8-(23*b^4-74*b^2*c^2+23*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)^3*(5*b^8+5*c^8-4*(9*b^4-26*b^2*c^2+9*c^4)*b^2*c^2)*a^2-(b^2-c^2)^6*(4*b^4+5*b^2*c^2+4*c^4)*(b^4-b^2*c^2+c^4) : :

The reciprocal orthologic center of these triangles is X(13).

X(36788) lies on these lines: {13,125}, {5668,6779}, {6111,16256}, {6777,10722}, {9114,23871}

leftri

Points on the dual of the circumcircle: X(36789)-X(36793)

rightri
Contributed by Clark Kimberling and Peter Moses, January 9, 2020.

Suppose that P = p : q : r (barycentrics) is a point on the line at infinity, and U = u : v : w is a point. Then the point D(P,U) = p2u : q2v : r2w is on the inconic with perspector U. In particular, if U = X(76), then D(P,U) lies on the inellipse having perspector X(76) and center X(141). This inellipse is the dual of the circumcircle.

Also, D(P,X(76)) is the barycentric quotient P*/P, where P*, the isogonal conjugate of P, lies on the circumcircle.

The appearance of (i,j) in the following list means that D(X(i),X(76)) = X(j): (30,36789), (511,36790), (512, 3124), (513,1086), (514,23989), (517,26611), (518,4437), (519,36791), (521,23983), (522,23978), (523,338), (524,36792), (525,36793), (3900,23970)

The dual of the circumcircle is the barycentric square of line X(514)X(661) (the trilinear polar of X(75)). (Randy Hutson, March 29, 2020)

X(36789) = BARYCENTRIC QUOTIENT X(30)/X(74)

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

X(36789) lies on the dual of the circumcircle (an inellipse) and these lines: {2, 94}, {6, 2986}, {76, 6331}, {394, 648}, {1511, 14254}, {1576, 36192}, {1637, 5664}, {1990, 3260}, {2781, 25045}, {2970, 5972}, {3124, 5254}, {4240, 16165}, {4359, 23978}, {6148, 11070}, {7998, 15363}, {15066, 19221}, {16163, 16240}, {18314, 18557}

X(36789) = isotomic conjugate of the isogonal conjugate of X(3163)
X(36789) = isotomic conjugate of the polar conjugate of X(34334)
X(36789) = polar conjugate of the isogonal conjugate of X(16163)
X(36789) = X(i)-Ceva conjugate of X(j) for these (i,j): {76, 3260}, {3260, 23097}
X(36789) = X(i)-cross conjugate of X(j) for these (i,j): {3163, 34334}, {14401, 3233}, {23097, 3260}
X(36789) = X(i)-isoconjugate of X(j) for these (i,j): {74, 2159}, {560, 31621}, {810, 34568}, {2433, 36034}, {8749, 35200}, {14380, 36131}, {18877, 36119}
X(36789) = cevapoint of X(3163) and X(16163)
X(36789) = crosspoint of X(76) and X(3260)
X(36789) = trilinear pole of line {1553, 23097}
X(36789) = barycentric square of X(14206)
X(36789) = barycentric product X(i)*X(j) for these {i,j}: {30, 3260}, {69, 34334}, {75, 1099}, {76, 3163}, {264, 16163}, {305, 16240}, {850, 3233}, {1354, 3596}, {1494, 23097}, {1502, 9408}, {6062, 6063}, {6148, 14254}, {6331, 14401}, {14206, 14206}
X(36789) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 74}, {76, 31621}, {113, 14264}, {648, 34568}, {1099, 1}, {1354, 56}, {1511, 14385}, {1553, 5663}, {1637, 2433}, {1784, 36119}, {1990, 8749}, {2173, 2159}, {2420, 32640}, {3081, 1495}, {3163, 6}, {3233, 110}, {3260, 1494}, {3284, 18877}, {4240, 1304}, {5642, 9717}, {6062, 55}, {7359, 15627}, {9033, 14380}, {9214, 9139}, {9408, 32}, {10272, 3470}, {11064, 14919}, {14206, 2349}, {14254, 5627}, {14401, 647}, {15454, 10419}, {16163, 3}, {16240, 25}, {23097, 30}, {23347, 32715}, {34334, 4}, {36435, 9408}

X(36790) = BARYCENTRIC QUOTIENT X(511)/X(98)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)^2 : :
Barycentrics    cos^2(A + ω) : :
X(36790) = 3 X[9155] - 2 X[36213]

X(36790) lies on the dual of the circumcircle (an inellipse), the cubic K783, and these lines: {2, 694}, {3, 1976}, {6, 2987}, {22, 110}, {69, 1972}, {76, 18024}, {99, 287}, {141, 311}, {184, 35387}, {237, 511}, {246, 542}, {263, 3095}, {297, 6393}, {323, 5104}, {343, 8024}, {401, 19571}, {524, 6148}, {1086, 18179}, {1501, 1993}, {1583, 7598}, {1584, 7599}, {1634, 2871}, {1691, 35296}, {1959, 16591}, {1975, 22416}, {1994, 12212}, {2088, 2482}, {2502, 15066}, {2782, 20021}, {2799, 3569}, {2967, 23611}, {3098, 3506}, {3218, 34253}, {3917, 7467}, {5147, 25941}, {5967, 33813}, {6660, 35456}, {7664, 11064}, {11672, 16725}, {14602, 35374}, {17184, 23989}, {17811, 20998}, {20891, 23978}, {20975, 34383}, {34396, 35424}

X(36790) = midpoint of X(69) and X(14570)
X(36790) = reflection of X(i) in X(j) for these {i,j}: {6, 34990}, {338, 141}
X(36790) = isotomic conjugate of X(34536)
X(36790) = isotomic conjugate of the isogonal conjugate of X(11672)
X(36790) = isogonal conjugate of the isotomic conjugate of X(32458)
X(36790) = isotomic conjugate of the polar conjugate of X(2967)
X(36790) = X(i)-Ceva conjugate of X(j) for these (i,j): {76, 325}, {99, 684}, {249, 2421}, {325, 23098}
X(36790) = X(i)-cross conjugate of X(j) for these (i,j): {11672, 2967}, {23098, 325}
X(36790) = X(i)-isoconjugate of X(j) for these (i,j): {31, 34536}, {98, 1910}, {248, 36120}, {293, 6531}, {879, 36104}, {1821, 1976}, {2395, 36084}, {2422, 36036}
X(36790) = crosspoint of X(i) and X(j) for these (i,j): {76, 325}, {249, 2421}
X(36790) = crosssum of X(i) and X(j) for these (i,j): {32, 1976}, {115, 2395}
X(36790) = trilinear pole of line {6072, 23098}
X(36790) = crossdifference of every pair of points on line {1976, 2395}
X(36790) = barycentric square of X(1959)
X(36790) = barycentric product X(i)*X(j) for these {i,j}: {6, 32458}, {69, 2967}, {75, 23996}, {76, 11672}, {232, 6393}, {249, 35088}, {290, 23098}, {297, 36212}, {321, 16725}, {325, 511}, {394, 36426}, {523, 15631}, {684, 877}, {1355, 3596}, {1502, 9419}, {1959, 1959}, {2396, 3569}, {2421, 2799}, {4230, 6333}, {6063, 7062}, {18024, 23611}
X(36790) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34536}, {114, 14265}, {232, 6531}, {237, 1976}, {240, 36120}, {297, 16081}, {325, 290}, {511, 98}, {684, 879}, {805, 18858}, {877, 22456}, {1355, 56}, {1755, 1910}, {1959, 1821}, {2421, 2966}, {2491, 2422}, {2967, 4}, {3289, 248}, {3569, 2395}, {4230, 685}, {5968, 9154}, {5976, 14382}, {6072, 2782}, {7062, 55}, {9155, 5967}, {9418, 14601}, {9419, 32}, {11672, 6}, {14251, 34238}, {14966, 2715}, {15631, 99}, {16725, 81}, {23098, 511}, {23611, 237}, {23996, 1}, {23997, 36084}, {32458, 76}, {33569, 3288}, {34157, 2065}, {35088, 338}, {36212, 287}, {36425, 1501}, {36426, 2052}

X(36791) = BARYCENTRIC QUOTIENT X(519)/X(106)

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

X(36791) lies on the dual of the circumcircle (an inellipse) and these lines: {2, 646}, {75, 24183}, {76, 1978}, {312, 3969}, {321, 1086}, {338, 1230}, {346, 30680}, {519, 23644}, {668, 30578}, {1015, 27070}, {1500, 31035}, {2321, 14554}, {3124, 21024}, {3264, 3943}, {3266, 18035}, {3762, 4120}, {3948, 13466}, {4370, 16729}, {9059, 23858}, {21070, 22032}, {25278, 31018}, {25280, 27776}, {26526, 26591}

X(36791) = isotomic conjugate of X(2226)
X(36791) = isotomic conjugate of the isogonal conjugate of X(4370)
X(36791) = X(76)-Ceva conjugate of X(3264)
X(36791) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2226}, {32, 679}, {106, 9456}, {604, 1318}, {667, 4638}, {1022, 32719}, {1417, 2316}, {1919, 4618}, {2206, 30575}, {2441, 36042}, {8752, 36058}, {23345, 32665}, {32659, 36125}
X(36791) = crosspoint of X(76) and X(3264)
X(36791) = barycentric square of X(4358)
X(36791) = barycentric product X(i)*X(j) for these {i,j}: {75, 4738}, {76, 4370}, {321, 16729}, {519, 3264}, {561, 678}, {1017, 1502}, {1317, 3596}, {1978, 6544}, {3251, 6386}, {3762, 24004}, {3992, 30939}, {4152, 6063}, {4358, 4358}, {4543, 4572}, {6385, 21821}, {18022, 22371}, {31625, 35092}
X(36791) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2226}, {8, 1318}, {44, 9456}, {75, 679}, {190, 4638}, {214, 16944}, {321, 30575}, {519, 106}, {668, 4618}, {678, 31}, {900, 23345}, {1017, 32}, {1023, 32665}, {1145, 14260}, {1317, 56}, {1319, 1417}, {2325, 2316}, {2429, 32645}, {3251, 667}, {3264, 903}, {3762, 1022}, {3977, 1797}, {3992, 4674}, {4152, 55}, {4358, 88}, {4370, 6}, {4542, 3271}, {4543, 663}, {4723, 1320}, {4738, 1}, {4768, 23838}, {5440, 36058}, {6544, 649}, {8028, 902}, {8756, 8752}, {14027, 1357}, {14425, 2441}, {14442, 21143}, {16729, 81}, {17460, 17109}, {17780, 901}, {21821, 213}, {22356, 32659}, {22371, 184}, {23344, 32719}, {24004, 3257}, {30731, 5548}, {33922, 1960}, {35092, 1015}
X(36791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {312, 4033, 30566}, {312, 21600, 18359}

X(36792) = BARYCENTRIC QUOTIENT X(524)/X(111)

Barycentrics    b^2*c^2*(-2*a^2 + b^2 + c^2)^2 : :
X(36792) = 3 X[69] + X[25052], 3 X[141] - X[25322], 9 X[599] - X[25334], 3 X[3124] - 2 X[25322], 7 X[3619] - 3 X[25315], 5 X[3620] - X[25047], 3 X[4576] - X[25052]

X(36792) lies on the dual of the circumcircle (an inellipse) and these lines: {2, 34898}, {6, 4563}, {67, 69}, {76, 338}, {99, 2930}, {141, 3124}, {305, 15533}, {339, 32257}, {524, 3266}, {690, 5181}, {895, 9146}, {1086, 20911}, {1269, 23989}, {2482, 16733}, {3619, 25315}, {3620, 25047}, {3630, 25325}, {5095, 34336}, {5108, 32740}, {5468, 6593}, {5976, 23342}, {8024, 22165}, {8030, 20380}, {14210, 16597}, {15993, 30736}

X(36792) = midpoint of X(i) and X(j) for these {i,j}: {69, 4576}, {3630, 25325}
X(36792) = reflection of X(3124) in X(141)
X(36792) = isotomic conjugate of X(10630)
X(36792) = isotomic conjugate of the isogonal conjugate of X(2482)
X(36792) = isotomic conjugate of the polar conjugate of X(34336)
X(36792) = X(i)-Ceva conjugate of X(j) for these (i,j): {76, 3266}, {670, 35522}, {3266, 23106}
X(36792) = X(i)-cross conjugate of X(j) for these (i,j): {2482, 34336}, {23106, 3266}
X(36792) = X(i)-isoconjugate of X(j) for these (i,j): {31, 10630}, {111, 923}, {798, 34574}, {897, 32740}, {1973, 15398}, {2444, 36045}, {8753, 36060}, {9178, 36142}, {14908, 36128}, {23894, 32729}
X(36792) = crosspoint of X(76) and X(3266)
X(36792) = crosssum of X(32) and X(32740)
X(36792) = trilinear pole of line {1649, 6077}
X(36792) = barycentric square of X(14210)
X(36792) = barycentric product X(i)*X(j) for these {i,j}: {69, 34336}, {75, 24038}, {76, 2482}, {305, 5095}, {321, 16733}, {524, 3266}, {670, 1649}, {671, 23106}, {1366, 3596}, {5468, 35522}, {6063, 7067}, {8030, 18023}, {9464, 20380}, {14210, 14210}, {23992, 34537}
X(36792) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 10630}, {69, 15398}, {99, 34574}, {126, 14263}, {187, 32740}, {468, 8753}, {524, 111}, {690, 9178}, {896, 923}, {1366, 56}, {1641, 17964}, {1649, 512}, {2434, 32648}, {2482, 6}, {3266, 671}, {3292, 14908}, {3712, 5547}, {4590, 34539}, {5095, 25}, {5467, 32729}, {5468, 691}, {6077, 33962}, {6390, 895}, {7067, 55}, {7181, 7316}, {7664, 14246}, {8030, 187}, {9125, 2444}, {14210, 897}, {14417, 10097}, {14443, 22260}, {14444, 21906}, {14567, 19626}, {16733, 81}, {18311, 10561}, {20380, 1383}, {23106, 524}, {23889, 36142}, {23992, 3124}, {24038, 1}, {24039, 36085}, {30454, 3457}, {30455, 3458}, {33915, 351}, {33921, 17993}, {34161, 15387}, {34336, 4}, {35522, 5466}

X(36793) = BARYCENTRIC QUOTIENT X(525)/X(112)

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

X(36793) lies on the dual of the circumcircle (an inellipse) and these lines: {67, 69}, {76, 6331}, {125, 339}, {287, 305}, {338, 23962}, {343, 8024}, {1228, 26611}, {1853, 18018}, {2373, 10117}, {2781, 25053}, {2972, 3265}, {3266, 11064}, {5972, 34336}, {5986, 5989}, {10330, 16165}, {13203, 13219}, {13575, 34944}, {13854, 34129}, {14208, 34846}, {15526, 23974}

X(36793) = isotomic conjugate of X(23964)
X(36793) = isotomic conjugate of the isogonal conjugate of X(15526)
X(36793) = isotomic conjugate of the polar conjugate of X(339)
X(36793) = X(i)-Ceva conjugate of X(j) for these (i,j): {76, 3267}, {305, 3265}, {3267, 23107}
X(36793) = X(i)-cross conjugate of X(j) for these (i,j): {15526, 339}, {23107, 3267}, {23616, 3265}
X(36793) = X(i)-isoconjugate of X(j) for these (i,j): {31, 23964}, {32, 24000}, {112, 32676}, {158, 23963}, {163, 32713}, {250, 1973}, {255, 23975}, {393, 23995}, {560, 23582}, {577, 24022}, {823, 14574}, {1096, 23357}, {1101, 2207}, {1110, 36420}, {1501, 23999}, {1576, 24019}, {2445, 36046}, {9247, 32230}, {14585, 24021}, {23347, 36131}, {24041, 36417}, {34859, 36084}
X(36793) = crosspoint of X(i) and X(j) for these (i,j): {76, 3267}, {850, 18018}, {3926, 15414}
X(36793) = crosssum of X(206) and X(1576)
X(36793) = trilinear pole of line {5489, 23107}
X(36793) = crossdifference of every pair of points on line {14574, 34859}
X(36793) = barycentric square of X(14208)
X(36793) = barycentric product X(i)*X(j) for these {i,j}: {69, 339}, {75, 17879}, {76, 15526}, {125, 305}, {304, 20902}, {313, 17216}, {326, 23994}, {338, 3926}, {394, 23962}, {525, 3267}, {561, 2632}, {648, 23107}, {670, 5489}, {850, 3265}, {1367, 3596}, {1502, 3269}, {2052, 23974}, {2970, 4176}, {2972, 18022}, {4143, 14618}, {6063, 7068}, {6331, 23616}, {14208, 14208}, {15414, 18314}, {20948, 24018}, {34384, 35442}
X(36793) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 23964}, {69, 250}, {75, 24000}, {76, 23582}, {115, 2207}, {122, 154}, {125, 25}, {127, 8743}, {158, 24022}, {253, 15384}, {255, 23995}, {264, 32230}, {305, 18020}, {326, 1101}, {338, 393}, {339, 4}, {393, 23975}, {394, 23357}, {520, 1576}, {523, 32713}, {525, 112}, {561, 23999}, {577, 23963}, {656, 32676}, {850, 107}, {868, 34854}, {879, 32696}, {1086, 36420}, {1109, 1096}, {1365, 7337}, {1367, 56}, {1562, 3172}, {1577, 24019}, {1650, 1495}, {2052, 23590}, {2394, 32695}, {2435, 32649}, {2525, 35325}, {2632, 31}, {2970, 6524}, {2972, 184}, {3124, 36417}, {3265, 110}, {3267, 648}, {3269, 32}, {3569, 34859}, {3708, 1973}, {3926, 249}, {4064, 8750}, {4092, 6059}, {4143, 4558}, {4466, 1474}, {5489, 512}, {6333, 4230}, {6354, 23985}, {7068, 55}, {8552, 14591}, {9033, 23347}, {14208, 162}, {14376, 15388}, {14380, 32715}, {14618, 6529}, {15414, 18315}, {15421, 32708}, {15526, 6}, {16186, 34397}, {16732, 5317}, {17216, 58}, {17879, 1}, {17880, 270}, {18027, 34538}, {18210, 2203}, {18312, 35907}, {20336, 5379}, {20902, 19}, {20948, 823}, {20975, 1974}, {21046, 2333}, {21207, 8747}, {23107, 525}, {23616, 647}, {23962, 2052}, {23974, 394}, {23978, 36421}, {23983, 7054}, {23989, 36419}, {23994, 158}, {24018, 163}, {24020, 255}, {26932, 2189}, {26942, 7115}, {30805, 4556}, {34767, 1304}, {34980, 14575}, {35071, 14585}, {35442, 51}

X(36794) =  X(2)X(95)∩X(4)X(83)

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

See Francisco Javier García Capitán, Euclid 624 .

X(36794) lies on these lines: {2, 95}, {3, 10003}, {4, 83}, {5, 14152}, {6, 264}, {24, 7786}, {25, 11174}, {53, 597}, {76, 20806}, {86, 26003}, {98, 14575}, {107, 5640}, {112, 12150}, {141, 340}, {157, 35278}, {216, 401}, {232, 3329}, {250, 3613}, {273, 3758}, {297, 3589}, {311, 22151}, {318, 3759}, {324, 34545}, {344, 34231}, {373, 450}, {378, 3972}, {393, 30535}, {419, 1843}, {427, 7792}, {436, 5943}, {569, 8884}, {1078, 10312}, {1105, 11424}, {1235, 7760}, {1316, 23635}, {1576, 34845}, {1594, 7828}, {1629, 5012}, {1861, 20179}, {1968, 7787}, {1990, 6329}, {1992, 32000}, {2052, 5422}, {3164, 5158}, {3186, 8541}, {3284, 14767}, {3619, 32001}, {4230, 35222}, {4240, 10545}, {5050, 33971}, {5081, 17289}, {5092, 35474}, {5523, 7827}, {6240, 7847}, {6530, 18583}, {6819, 11427}, {7282, 16706}, {7507, 7851}, {7577, 14061}, {7578, 16080}, {7804, 15014}, {7829, 27371}, {7878, 8743}, {8739, 16250}, {9307, 13479}, {10601, 15466}, {11109, 17277}, {14165, 14389}, {14957, 19121}, {15019, 35360}, {15258, 33748}, {17381, 17555}, {17983, 21460}, {23583, 36412}, {26212, 32971}, {28704, 32828}, {31623, 32911}, {35941, 36751}

X(36794) = polar conjugate of X(3613)
X(36794) = isotomic conjugate of X(36952)
X(36794) = isotomic conjugate of the isogonal conjugate of X(10312)
X(36794) = isotomic conjugate of the polar conjugate of X(1629)
X(36794) = polar conjugate of the isotomic conjugate of X(1078)
X(36794) = polar conjugate of the isogonal conjugate of X(5012)
X(36794) = X(250)-Ceva conjugate of X(648)
X(36794) = X(i)-cross conjugate of X(j) for these (i,j): {5012, 1078}, {10312, 1629}, {11450, 99}
X(36794) = X(i)-isoconjugate of X(j) for these (i,j): {48, 3613}, {63, 27375}, {810, 11794}, {3708, 27867}, {4020, 30505}
X(36794) = cevapoint of X(i) and X(j) for these (i,j): {6, 34845}, {3050, 7668}, {5012, 10312}
X(36794) = barycentric product X(i)*X(j) for these {i,j}: {4, 1078}, {19, 33764}, {25, 33769}, {69, 1629}, {76, 10312}, {92, 18042}, {95, 30506}, {264, 5012}, {648, 31296}, {1973, 33778}, {3050, 6331}, {7668, 18020}
X(36794) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 3613}, {25, 27375}, {250, 27867}, {648, 11794}, {1078, 69}, {1629, 4}, {3050, 647}, {3203, 20775}, {5012, 3}, {7668, 125}, {10312, 6}, {16245, 5403}, {18042, 63}, {27010, 26932}, {30506, 5}, {31296, 525}, {32085, 30505}, {33764, 304}, {33769, 305}
X(36794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 577, 95}, {2, 3087, 317}, {4, 1974, 32085}, {4, 3618, 17907}, {6, 264, 648}, {6, 458, 264}, {141, 6749, 27377}, {141, 27377, 340}, {297, 6748, 32002}, {3589, 6748, 297}, {5012, 30506, 1629}, {7804, 33843, 15014}

leftri

Points on the dual of the incircle: X(36795)-X(36807)

rightri

Contributed by Clark Kimberling and Peter Moses, February 15, 2020.

Suppose that P = p : q : r (barycentrics) is a point on the line at infinity, and U = u : v : w is a point. Then the point D(P,U) = q r u : r p v : p q w lies on the circumconic with perspector U. In particular, if U = X(8), then D(P,U) lies on the circumconic having perspector X(8) and center X(3161). This circumconic is the dual of the incircle.

Also, D(P,X(8)) is the barycentric quotient X(8)/P.

The appearance of (i,j) in the following list means that D(X(i),X(8)) = X(j): (pending)

X(36795) =  BARYCENTRIC QUOTIENT X(8)/X(517)

Barycentrics    b*c*(-a + b + c)*(a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(-a^3 + a*b^2 + a^2*c - 2*a*b*c + b^2*c + a*c^2 - c^3) : :
Barycentrics    1/(2 a sec A + (b - c - a) sec B + (c - a - b) sec C) :

X(36795) lies on these lines: {2, 6335}, {21, 1809}, {63, 190}, {76, 348}, {78, 341}, {104, 1791}, {280, 27506}, {345, 646}, {346, 30680}, {645, 1812}, {1309, 26703}, {3264, 32851}, {4358, 13136}, {5205, 14198}, {6952, 7141}, {10449, 34259}, {13740, 36123}, {18743, 34404}, {32017, 34051}

X(36795) = isotomic conjugate of X(1465)
X(36795) = polar conjugate of X(1875)
X(36795) = polar conjugate of the isogonal conjugate of X(1809)
X(36795) = X(i)-cross conjugate of X(j) for these (i,j): {4358, 312}, {4511, 314}
X(36795) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1457}, {31, 1465}, {32, 22464}, {48, 1875}, {56, 2183}, {109, 3310}, {517, 604}, {603, 14571}, {608, 22350}, {649, 23981}, {653, 23220}, {667, 24029}, {859, 1400}, {908, 1397}, {909, 1361}, {1404, 14260}, {1408, 21801}, {1415, 1769}, {1846, 32659}, {8677, 32674}, {16947, 17757}, {22383, 23706}, {23979, 35015}
X(36795) = cevapoint of X(i) and X(j) for these (i,j): {312, 32851}, {346, 4723}
X(36795) = trilinear pole of line {8, 521}
X(36795) = barycentric product X(i)*X(j) for these {i,j}: {8, 18816}, {104, 3596}, {264, 1809}, {312, 34234}, {345, 16082}, {561, 2342}, {646, 2401}, {909, 28659}, {1309, 35518}, {2250, 28660}, {3718, 36123}, {4391, 13136}, {35519, 36037}
X(36795) = +barycentric quotient X(i)/X(j) for these {i,j}: {1, 1457}, {2, 1465}, {4, 1875}, {8, 517}, {9, 2183}, {21, 859}, {75, 22464}, {78, 22350}, {100, 23981}, {104, 56}, {190, 24029}, {281, 14571}, {312, 908}, {314, 17139}, {318, 1785}, {341, 6735}, {517, 1361}, {521, 8677}, {522, 1769}, {644, 2427}, {646, 2397}, {650, 3310}, {909, 604}, {1309, 108}, {1320, 14260}, {1795, 603}, {1809, 3}, {1897, 23706}, {1946, 23220}, {2250, 1400}, {2321, 21801}, {2342, 31}, {2401, 3669}, {2968, 35014}, {3596, 3262}, {3685, 15507}, {3701, 17757}, {4391, 10015}, {4397, 2804}, {4511, 34586}, {4723, 1145}, {4768, 23757}, {5081, 1845}, {6735, 24028}, {10428, 1417}, {13136, 651}, {14266, 18838}, {15501, 221}, {15635, 1357}, {16082, 278}, {17100, 34346}, {18155, 23788}, {18816, 7}, {24026, 35015}, {32641, 1415}, {32851, 16586}, {34051, 1407}, {34234, 57}, {34858, 1397}, {35014, 35012}, {35519, 36038}, {36037, 109}, {36123, 34}

X(36796) =  BARYCENTRIC QUOTIENT X(8)/X(518)

Barycentrics    b*c*(-a + b + c)*(a^2 + b^2 - a*c - b*c)*(-a^2 + a*b + b*c - c^2) : :
Barycentrics    1/((b^2 + c^2) (1 - cos A) - a^2 (cos B + cos C)) : :

X(36796) lies on the cubic K996 and these lines: {2, 4554}, {9, 75}, {76, 6554}, {85, 21446}, {105, 3757}, {200, 312}, {220, 17143}, {264, 281}, {274, 1212}, {282, 309}, {294, 314}, {321, 6605}, {335, 17435}, {346, 646}, {650, 28798}, {666, 1814}, {668, 1146}, {693, 30857}, {894, 1462}, {948, 30705}, {1921, 34852}, {2184, 20921}, {2297, 10436}, {3030, 3038}, {3041, 17794}, {3452, 7018}, {3685, 28058}, {3717, 3975}, {3912, 14943}, {3948, 18036}, {4124, 7077}, {4183, 31623}, {4621, 6654}, {4998, 25954}, {5199, 6381}, {5205, 14197}, {5452, 28934}, {6376, 23058}, {6996, 20605}, {7110, 15455}, {7112, 30807}, {8012, 24592}, {9367, 26959}, {15288, 16992}, {17260, 25001}, {17264, 20566}, {17350, 18811}, {18061, 34591}, {18135, 27541}, {18743, 19605}, {20942, 36627}, {26541, 26793}

X(36796) = isotomic conjugate of X(241)
X(36796) = polar conjugate of X(1876)
X(36796) = isotomic conjugate of the anticomplement of X(34852)
X(36796) = isotomic conjugate of the complement of X(30807)
X(36796) = isotomic conjugate of the isogonal conjugate of X(294)
X(36796) = X(2111)-complementary conjugate of X(2886)
X(36796) = X(18031)-Ceva conjugate of X(2481)
X(36796) = X(i)-cross conjugate of X(j) for these (i,j): {3685, 314}, {3693, 8}, {14942, 2481}, {34852, 2}
X(36796) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1458}, {7, 9454}, {31, 241}, {32, 9436}, {34, 20752}, {41, 34855}, {48, 1876}, {56, 672}, {57, 2223}, {85, 9455}, {109, 665}, {184, 5236}, {222, 2356}, {518, 604}, {603, 5089}, {608, 1818}, {649, 2283}, {653, 23225}, {658, 8638}, {667, 1025}, {883, 1919}, {926, 1461}, {1106, 3693}, {1362, 1438}, {1395, 25083}, {1397, 3912}, {1400, 3286}, {1402, 18206}, {1404, 34230}, {1407, 2340}, {1408, 3930}, {1412, 20683}, {1415, 2254}, {1416, 6184}, {1417, 14439}, {1428, 3252}, {1477, 20662}, {1911, 34253}, {2149, 3675}, {3932, 16947}, {17435, 24027}
X(36796) = cevapoint of X(i) and X(j) for these (i,j): {2, 30807}, {8, 3693}, {75, 33677}, {312, 3975}, {650, 4124}, {6559, 14942}
X(36796) = trilinear pole of line {8, 885}
X(36796) = crossdifference of every pair of points on line {8638, 23225}
X(36796) = barycentric product X(i)*X(j) for these {i,j}: {8, 2481}, {9, 18031}, {75, 14942}, {76, 294}, {85, 6559}, {105, 3596}, {312, 673}, {314, 13576}, {318, 31637}, {346, 34018}, {350, 33676}, {561, 2195}, {666, 4391}, {668, 885}, {884, 6386}, {927, 4397}, {1024, 1978}, {1438, 28659}, {1814, 7017}, {3239, 34085}, {3718, 36124}, {4554, 28132}, {5377, 34387}, {6063, 28071}, {18785, 28660}, {35519, 36086}
X(36796) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1458}, {2, 241}, {4, 1876}, {7, 34855}, {8, 518}, {9, 672}, {11, 3675}, {21, 3286}, {33, 2356}, {41, 9454}, {55, 2223}, {75, 9436}, {78, 1818}, {92, 5236}, {100, 2283}, {105, 56}, {190, 1025}, {200, 2340}, {210, 20683}, {219, 20752}, {239, 34253}, {281, 5089}, {294, 6}, {312, 3912}, {314, 30941}, {318, 1861}, {333, 18206}, {341, 3717}, {345, 25083}, {346, 3693}, {518, 1362}, {522, 2254}, {644, 2284}, {650, 665}, {666, 651}, {668, 883}, {673, 57}, {884, 667}, {885, 513}, {919, 1415}, {927, 934}, {1024, 649}, {1146, 17435}, {1320, 34230}, {1416, 1106}, {1438, 604}, {1462, 1407}, {1814, 222}, {1946, 23225}, {2175, 9455}, {2195, 31}, {2321, 3930}, {2325, 14439}, {2348, 20662}, {2481, 7}, {3596, 3263}, {3685, 8299}, {3693, 6184}, {3699, 1026}, {3700, 24290}, {3701, 3932}, {3702, 4966}, {3717, 4712}, {3729, 6168}, {3900, 926}, {3975, 17755}, {4086, 4088}, {4391, 918}, {4518, 22116}, {4673, 4684}, {4866, 14626}, {4876, 3252}, {5377, 59}, {6169, 9315}, {6185, 1462}, {6559, 9}, {6654, 1429}, {6745, 35293}, {7081, 4447}, {8641, 8638}, {8751, 608}, {13576, 65}, {14942, 1}, {17435, 35505}, {18031, 85}, {18155, 23829}, {18785, 1400}, {23601, 7124}, {23696, 1459}, {28071, 55}, {28132, 650}, {28660, 18157}, {31623, 15149}, {31637, 77}, {31638, 1445}, {33676, 291}, {34018, 279}, {34085, 658}, {36057, 603}, {36086, 109}, {36124, 34}, {36146, 1461}

X(36797) =  BARYCENTRIC QUOTIENT X(8)/X(525)

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

X(36797) lies on these lines: {4, 25650}, {8, 6062}, {21, 1809}, {27, 1810}, {28, 1811}, {29, 4997}, {33, 1808}, {55, 7017}, {99, 108}, {100, 107}, {110, 1309}, {112, 8707}, {162, 190}, {243, 3685}, {264, 1013}, {318, 1793}, {415, 1861}, {447, 1785}, {646, 4571}, {662, 7452}, {833, 1289}, {835, 36077}, {931, 26704}, {2659, 23693}, {3699, 4587}, {4183, 31623}, {4242, 15455}, {4736, 36063}, {5205, 14192}, {8690, 32704}, {10538, 15776}, {27396, 36421}, {34360, 35075}

X(36797) = isotomic conjugate of X(17094)
X(36797) = pole wrt polar circle of trilinear polar of X(7178) (line X(1365)X(2611))
X(36797) = polar conjugate of X(7178)
X(36797) = polar conjugate of the isotomic conjugate of X(645)
X(36797) = polar conjugate of the isogonal conjugate of X(5546)
X(36797) = X(811)-Ceva conjugate of X(648)
X(36797) = X(i)-cross conjugate of X(j) for these (i,j): {21, 5379}, {100, 643}, {318, 15742}, {2804, 6740}, {3700, 7017}, {5546, 645}, {17926, 31623}, {27396, 1016}
X(36797) = cevapoint of X(i) and X(j) for these (i,j): {21, 7253}, {55, 3700}, {100, 1897}, {522, 950}, {4183, 17926}, {4391, 26165}
X(36797) = trilinear pole of line {8, 29}
X(36797) = X(i)-isoconjugate of X(j) for these (i,j): {3, 4017}, {7, 810}, {31, 17094}, {34, 520}, {48, 7178}, {56, 656}, {57, 647}, {63, 7180}, {65, 1459}, {71, 3669}, {73, 513}, {77, 512}, {78, 7250}, {85, 3049}, {109, 18210}, {184, 4077}, {201, 3733}, {219, 7216}, {222, 661}, {225, 23224}, {226, 22383}, {228, 3676}, {244, 23067}, {278, 822}, {307, 667}, {348, 798}, {514, 1409}, {521, 1042}, {522, 1410}, {523, 603}, {525, 604}, {608, 24018}, {649, 1214}, {652, 1427}, {663, 1439}, {669, 7182}, {905, 1400}, {1019, 2197}, {1020, 7117}, {1231, 1919}, {1254, 23189}, {1363, 36126}, {1365, 4575}, {1367, 32676}, {1393, 23286}, {1395, 3265}, {1397, 14208}, {1402, 4025}, {1407, 8611}, {1408, 4064}, {1414, 20975}, {1415, 4466}, {1417, 14429}, {1425, 3737}, {1458, 10099}, {1813, 3125}, {1880, 4091}, {1946, 3668}, {2171, 7254}, {2196, 7212}, {2200, 24002}, {2489, 7183}, {2501, 7125}, {2616, 30493}, {3120, 36059}, {3122, 6516}, {3690, 7203}, {3700, 7099}, {3708, 4565}, {3709, 7177}, {3937, 4551}, {3942, 4559}, {4041, 7053}, {7147, 23090}, {7335, 24006}, {7649, 22341}, {16732, 32660}, {22094, 26700}, {30572, 36058}
X(36797) = barycentric product X(i)*X(j) for these {i,j}: {4, 645}, {8, 648}, {9, 811}, {19, 7257}, {21, 6335}, {27, 3699}, {28, 646}, {29, 190}, {33, 799}, {34, 7258}, {55, 6331}, {78, 823}, {92, 643}, {99, 281}, {100, 31623}, {107, 345}, {110, 7017}, {112, 3596}, {162, 312}, {219, 6528}, {264, 5546}, {270, 4033}, {273, 7259}, {278, 7256}, {286, 644}, {314, 1783}, {318, 662}, {333, 1897}, {461, 4633}, {607, 670}, {653, 1043}, {664, 2322}, {668, 1172}, {877, 15628}, {1259, 15352}, {1264, 6529}, {1332, 1896}, {1414, 7101}, {1824, 4631}, {1857, 4563}, {1978, 2299}, {2189, 27808}, {2204, 6386}, {2212, 4602}, {2287, 18026}, {2332, 4572}, {2501, 6064}, {3064, 4600}, {3700, 18020}, {3718, 24019}, {3719, 36126}, {4076, 17925}, {4183, 4554}, {4391, 5379}, {4560, 15742}, {4561, 8748}, {4562, 14024}, {4573, 7046}, {4601, 18344}, {4625, 7079}, {4998, 17926}, {7359, 16077}, {8611, 23999}, {8750, 28660}, {11107, 15455}, {14006, 27805}, {28659, 32676}
X(36797) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17094}, {4, 7178}, {8, 525}, {9, 656}, {19, 4017}, {21, 905}, {25, 7180}, {27, 3676}, {28, 3669}, {29, 514}, {33, 661}, {34, 7216}, {41, 810}, {55, 647}, {60, 7254}, {78, 24018}, {92, 4077}, {99, 348}, {100, 1214}, {101, 73}, {107, 278}, {108, 1427}, {110, 222}, {112, 56}, {162, 57}, {163, 603}, {190, 307}, {200, 8611}, {212, 822}, {219, 520}, {242, 7212}, {250, 4565}, {261, 15419}, {270, 1019}, {281, 523}, {283, 4091}, {284, 1459}, {286, 24002}, {294, 10099}, {312, 14208}, {314, 15413}, {318, 1577}, {332, 30805}, {333, 4025}, {345, 3265}, {461, 4841}, {497, 21107}, {522, 4466}, {525, 1367}, {607, 512}, {608, 7250}, {643, 63}, {644, 72}, {645, 69}, {646, 20336}, {648, 7}, {650, 18210}, {651, 1439}, {653, 3668}, {662, 77}, {668, 1231}, {692, 1409}, {799, 7182}, {811, 85}, {823, 273}, {906, 22341}, {1018, 201}, {1021, 7004}, {1043, 6332}, {1172, 513}, {1252, 23067}, {1264, 4143}, {1414, 7177}, {1415, 1410}, {1625, 30493}, {1783, 65}, {1812, 4131}, {1857, 2501}, {1896, 17924}, {1897, 226}, {2175, 3049}, {2189, 3733}, {2193, 23224}, {2194, 22383}, {2204, 667}, {2212, 798}, {2287, 521}, {2299, 649}, {2321, 4064}, {2322, 522}, {2325, 14429}, {2326, 3737}, {2328, 652}, {2332, 663}, {2501, 1365}, {3064, 3120}, {3559, 21188}, {3596, 3267}, {3685, 24459}, {3699, 306}, {3700, 125}, {3703, 2525}, {3709, 20975}, {3712, 14417}, {3737, 3942}, {3939, 71}, {3952, 26942}, {4041, 3708}, {4069, 3949}, {4086, 20902}, {4183, 650}, {4235, 7181}, {4238, 241}, {4240, 6357}, {4242, 18593}, {4246, 1465}, {4248, 30719}, {4282, 22379}, {4552, 6356}, {4557, 2197}, {4558, 1804}, {4559, 1425}, {4560, 1565}, {4563, 7055}, {4565, 7053}, {4566, 20618}, {4567, 6516}, {4570, 1813}, {4571, 3998}, {4573, 7056}, {4574, 7066}, {4575, 7125}, {4578, 3694}, {4587, 3682}, {4592, 7183}, {4612, 1444}, {4616, 30682}, {4636, 1790}, {5081, 4707}, {5379, 651}, {5546, 3}, {5547, 10097}, {6056, 32320}, {6059, 2489}, {6061, 23090}, {6062, 14401}, {6064, 4563}, {6065, 4574}, {6331, 6063}, {6332, 17216}, {6335, 1441}, {6528, 331}, {6529, 1118}, {6558, 3710}, {7012, 1020}, {7017, 850}, {7046, 3700}, {7054, 23189}, {7068, 23616}, {7071, 3709}, {7079, 4041}, {7101, 4086}, {7252, 3937}, {7253, 26932}, {7256, 345}, {7257, 304}, {7258, 3718}, {7259, 78}, {7359, 9033}, {7452, 34050}, {8611, 2632}, {8748, 7649}, {8750, 1400}, {8756, 30572}, {9404, 22094}, {11107, 14838}, {14006, 4369}, {14024, 812}, {14308, 1562}, {15627, 14380}, {15628, 879}, {15742, 4552}, {17188, 23727}, {17515, 3960}, {17925, 1358}, {17926, 11}, {18020, 4573}, {18026, 1446}, {18344, 3125}, {21044, 21134}, {21789, 7117}, {23090, 1364}, {24019, 34}, {27382, 8057}, {30728, 4101}, {30730, 3695}, {31623, 693}, {31900, 30724}, {31903, 30723}, {32320, 1363}, {32661, 7335}, {32674, 1042}, {32676, 604}, {32713, 608}, {35192, 23226}, {35325, 1401}
X(36797) = {X(162),X(1897)}-harmonic conjugate of X(648)

X(36798) =  BARYCENTRIC QUOTIENT X(8)/X(536)

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

X(36798) lis on the Feuerbach circumhyperbola and these lines: {1, 190}, {4, 6335}, {7, 1357}, {8, 646}, {9, 3699}, {21, 645}, {79, 15455}, {80, 32850}, {104, 898}, {256, 17353}, {314, 4519}, {513, 30866}, {668, 24482}, {739, 2298}, {889, 2481}, {894, 3315}, {941, 23988}, {983, 4621}, {1015, 24485}, {1016, 10755}, {1086, 34363}, {1156, 4607}, {1320, 3685}, {2325, 4876}, {3551, 17282}, {3712, 11609}, {3716, 4997}, {3886, 4900}, {4871, 30997}, {5377, 5381}, {6745, 9365}, {9263, 24507}, {24487, 27195}

X(36798) = X(31002)-Ceva conjugate of X(3227)
X(36798) = X(i)-cross conjugate of X(j) for these (i,j): {4009, 8}, {4526, 646}
X(36798) = X(i)-isoconjugate of X(j) for these (i,j): {56, 899}, {57, 3230}, {59, 19945}, {109, 891}, {536, 604}, {651, 3768}, {664, 890}, {1106, 4009}, {1397, 6381}, {1408, 3994}, {1414, 14404}, {1415, 4728}, {1461, 4526}, {1646, 4564}
X(36798) = cevapoint of X(i) and X(j) for these (i,j): {8, 4009}, {3271, 4526}
X(36798) = crosspoint of X(889) and X(5381)
X(36798) = crosssum of X(890) and X(1646)
X(36798) = trilinear pole of line {8, 650}
X(36798) = barycentric product X(i)*X(j) for these {i,j}: {8, 3227}, {9, 31002}, {11, 5381}, {522, 4607}, {645, 35353}, {650, 889}, {739, 3596}, {898, 4391}, {34075, 35519}
X(36798) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 536}, {9, 899}, {55, 3230}, {312, 6381}, {346, 4009}, {391, 4706}, {522, 4728}, {644, 23343}, {650, 891}, {663, 3768}, {739, 56}, {889, 4554}, {898, 651}, {1639, 30583}, {2170, 19945}, {2321, 3994}, {3063, 890}, {3227, 7}, {3239, 14430}, {3271, 1646}, {3596, 35543}, {3685, 4465}, {3699, 23891}, {3700, 14431}, {3709, 14404}, {3716, 14433}, {3900, 4526}, {4009, 13466}, {4526, 14434}, {4607, 664}, {4873, 4937}, {4895, 14437}, {4944, 28603}, {4976, 30592}, {5381, 4998}, {31002, 85}, {32718, 1415}, {34075, 109}, {35353, 7178}

X(36799) =  BARYCENTRIC QUOTIENT X(8)/X(726)

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

X(36799) lies on hyperbola {{A,B,C,X(6),X(9)}} and these lines: {2, 24343}, {6, 190}, {9, 646}, {19, 6335}, {55, 3699}, {57, 4554}, {284, 645}, {312, 2319}, {335, 20363}, {727, 8707}, {893, 6651}, {900, 23355}, {909, 13136}, {2160, 15455}, {2258, 18793}, {2291, 8709}, {2316, 4582}, {3685, 7077}, {4997, 28798}, {7155, 24840}, {24358, 28358}, {30568, 36630}

X(36799) = X(32020)-Ceva conjugate of X(3226)
X(36799) = X(i)-cross conjugate of X(j) for these (i,j): {3975, 333}, {4435, 3699}, {4876, 14942}, {8851, 3226}
X(36799) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1463}, {7, 21760}, {34, 20785}, {56, 1575}, {57, 3009}, {108, 22092}, {278, 20777}, {292, 8850}, {604, 726}, {651, 6373}, {1014, 21830}, {1400, 18792}, {1415, 3837}, {2149, 21140}, {9456, 24816}
X(36799) = cevapoint of X(9) and X(3685)
X(36799) = trilinear pole of line {8, 663}
X(36799) = barycentric product X(i)*X(j) for these {i,j}: {8, 3226}, {9, 32020}, {75, 8851}, {312, 20332}, {314, 18793}, {333, 27809}, {522, 8709}, {727, 3596}, {3253, 4518}, {28659, 34077}
X(36799) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1463}, {8, 726}, {9, 1575}, {11, 21140}, {21, 18792}, {41, 21760}, {55, 3009}, {212, 20777}, {219, 20785}, {238, 8850}, {519, 24816}, {522, 3837}, {652, 22092}, {663, 6373}, {727, 56}, {1334, 21830}, {3226, 7}, {3253, 1447}, {3596, 35538}, {3684, 17475}, {3685, 17793}, {3699, 23354}, {3700, 21053}, {4391, 20908}, {4433, 20681}, {8709, 664}, {8851, 1}, {18793, 65}, {20332, 57}, {27809, 226}, {32020, 85}, {34077, 604}
X(36799) = {X(20332),X(27809)}-harmonic conjugate of X(3226)

X(36800) =  BARYCENTRIC QUOTIENT X(8)/X(740)

Barycentrics    (a + b)*(a - b - c)*(a + c)*(-b^2 + a*c)*(a*b - c^2) : :
Barycentrics    1/(cos(B + ω) + cos(C + ω)) : :

X(36800) lies on these lines: {1, 4154}, {2, 694}, {9, 261}, {33, 1808}, {37, 86}, {81, 4621}, {99, 17738}, {210, 333}, {226, 4554}, {239, 24383}, {274, 17760}, {286, 334}, {291, 35623}, {312, 18021}, {314, 646}, {741, 1961}, {1821, 1934}, {2064, 18895}, {2170, 7257}, {2250, 4584}, {2303, 18268}, {3508, 18206}, {3862, 27164}, {3912, 24479}, {4444, 35354}, {4562, 14616}, {4633, 25430}, {4639, 17789}, {5205, 14196}, {8818, 15455}, {14195, 20593}, {17103, 18055}, {17777, 19635}, {20982, 25685}

X(36800) = isotomic conjugate of X(16609)
X(36800) = polar conjugate of X(1874)
X(36800) = isotomic conjugate of the complement of X(1959)
X(36800) = isotomic conjugate of the isogonal conjugate of X(2311)
X(36800) = polar conjugate of the isogonal conjugate of X(1808)
X(36800) = X(i)-cross conjugate of X(j) for these (i,j): {9, 33676}, {3716, 7257}, {3985, 8}
X(36800) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1284}, {31, 16609}, {37, 1428}, {42, 1429}, {48, 1874}, {56, 2238}, {57, 3747}, {65, 1914}, {73, 2201}, {109, 21832}, {213, 1447}, {222, 862}, {226, 2210}, {238, 1400}, {239, 1402}, {242, 1409}, {349, 18892}, {604, 740}, {651, 4455}, {659, 4559}, {692, 7212}, {1042, 3684}, {1106, 3985}, {1333, 7235}, {1397, 3948}, {1407, 4433}, {1408, 4037}, {1415, 4010}, {1441, 14599}, {1880, 7193}, {1918, 10030}, {1976, 16591}, {2171, 5009}, {2205, 18033}, {3027, 18268}, {3573, 7180}, {4155, 4565}, {4551, 8632}
X(36800) = cevapoint of X(i) and X(j) for these (i,j): {2, 1959}, {8, 3985}, {1808, 2311}, {2170, 3716}, {4518, 4876}
X(36800) = trilinear pole of line {8, 3907}
X(36800) = crossdifference of every pair of points on line {4455, 5027}
X(36800) = barycentric product X(i)*X(j) for these {i,j}: {8, 18827}, {21, 334}, {29, 337}, {76, 2311}, {86, 4518}, {264, 1808}, {274, 4876}, {284, 18895}, {291, 314}, {292, 28660}, {310, 7077}, {333, 335}, {522, 4589}, {645, 4444}, {650, 4639}, {660, 18155}, {741, 3596}, {876, 7257}, {1043, 7233}, {1916, 27958}, {3737, 4583}, {3907, 18829}, {4086, 36066}, {4391, 4584}, {4560, 4562}, {18268, 28659}, {30941, 33676}
X(36800) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1284}, {2, 16609}, {4, 1874}, {8, 740}, {9, 2238}, {10, 7235}, {21, 238}, {29, 242}, {33, 862}, {55, 3747}, {58, 1428}, {60, 5009}, {81, 1429}, {86, 1447}, {200, 4433}, {261, 33295}, {274, 10030}, {283, 7193}, {284, 1914}, {291, 65}, {292, 1400}, {295, 73}, {310, 18033}, {312, 3948}, {314, 350}, {333, 239}, {334, 1441}, {335, 226}, {337, 307}, {346, 3985}, {391, 4771}, {514, 7212}, {522, 4010}, {643, 3573}, {645, 3570}, {650, 21832}, {660, 4551}, {663, 4455}, {740, 3027}, {741, 56}, {805, 29055}, {813, 4559}, {876, 4017}, {1021, 4435}, {1043, 3685}, {1172, 2201}, {1808, 3}, {1812, 20769}, {1911, 1402}, {1959, 16591}, {2194, 2210}, {2196, 1409}, {2287, 3684}, {2311, 6}, {2321, 4037}, {3061, 18904}, {3572, 7180}, {3596, 35544}, {3685, 4368}, {3688, 4093}, {3737, 659}, {3786, 3783}, {3907, 804}, {3985, 35068}, {4041, 4155}, {4061, 4829}, {4433, 4094}, {4444, 7178}, {4518, 10}, {4560, 812}, {4562, 4552}, {4584, 651}, {4589, 664}, {4639, 4554}, {4720, 4693}, {4723, 4783}, {4765, 4839}, {4876, 37}, {5009, 12835}, {6332, 24459}, {7077, 42}, {7081, 4039}, {7233, 3668}, {7252, 8632}, {7253, 3716}, {7257, 874}, {14006, 419}, {17197, 27918}, {18155, 3766}, {18191, 27846}, {18206, 34253}, {18265, 1918}, {18268, 604}, {18792, 8850}, {18827, 7}, {18895, 349}, {23189, 22384}, {27958, 385}, {28660, 1921}, {30669, 4032}, {33676, 13576}, {36066, 1414}

X(36801) =  BARYCENTRIC QUOTIENT X(8)/X(812)

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

X(36801) lies on these lines: {100, 4621}, {190, 513}, {291, 4871}, {335, 17266}, {522, 646}, {645, 3737}, {650, 3699}, {813, 8707}, {2325, 4876}, {3676, 4554}, {3685, 7077}, {3717, 3975}, {3952, 27805}, {4009, 4518}, {4582, 23838}, {4589, 4633}, {5205, 14200}, {5378, 13136}, {6335, 7649}

X(36801) = X(4583)-Ceva conjugate of X(4562)
X(36801) = X(i)-cross conjugate of X(j) for these (i,j): {522, 33676}, {3716, 8}
X(36801) = X(i)-isoconjugate of X(j) for these (i,j): {34, 22384}, {56, 659}, {57, 8632}, {109, 27846}, {513, 1428}, {604, 812}, {649, 1429}, {667, 1447}, {876, 12835}, {1014, 4455}, {1106, 3716}, {1284, 3733}, {1333, 7212}, {1357, 3573}, {1397, 3766}, {1407, 4435}, {1408, 4010}, {1412, 21832}, {1415, 27918}, {1417, 4448}, {1431, 4164}, {1914, 3669}, {1919, 10030}, {1980, 18033}, {2210, 3676}, {3747, 7203}, {4017, 5009}, {4148, 7366}, {4155, 7341}, {8850, 23355}, {14599, 24002}
X(36801) = cevapoint of X(i) and X(j) for these (i,j): {8, 3716}, {522, 3717}, {3985, 4041}
X(36801) = trilinear pole of line {8, 2170}
X(36801) = barycentric product X(i)*X(j) for these {i,j}: {8, 4562}, {9, 4583}, {190, 4518}, {210, 4639}, {291, 646}, {312, 660}, {334, 644}, {335, 3699}, {668, 4876}, {813, 3596}, {1978, 7077}, {2311, 27808}, {2321, 4589}, {3701, 4584}, {3939, 18895}, {4076, 4444}, {4095, 18829}, {4391, 5378}, {6558, 7233}, {18827, 30730}, {28659, 34067}
X(36801) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 812}, {9, 659}, {10, 7212}, {55, 8632}, {100, 1429}, {101, 1428}, {190, 1447}, {200, 4435}, {210, 21832}, {219, 22384}, {291, 3669}, {312, 3766}, {334, 24002}, {335, 3676}, {346, 3716}, {391, 4830}, {522, 27918}, {644, 238}, {645, 33295}, {646, 350}, {650, 27846}, {660, 57}, {668, 10030}, {813, 56}, {1018, 1284}, {1026, 34253}, {1334, 4455}, {1808, 7254}, {1978, 18033}, {2311, 3733}, {2321, 4010}, {2325, 4448}, {2329, 4164}, {3061, 3808}, {3239, 4124}, {3572, 1357}, {3685, 4375}, {3699, 239}, {3710, 24459}, {3716, 35119}, {3790, 4486}, {3939, 1914}, {3952, 16609}, {3975, 27855}, {4007, 4810}, {4009, 14433}, {4061, 4839}, {4069, 2238}, {4076, 3570}, {4095, 804}, {4103, 7235}, {4444, 1358}, {4518, 514}, {4562, 7}, {4571, 20769}, {4578, 3684}, {4582, 27922}, {4583, 85}, {4584, 1014}, {4587, 7193}, {4589, 1434}, {4873, 4800}, {4876, 513}, {5378, 651}, {5423, 4148}, {5546, 5009}, {6558, 3685}, {7077, 649}, {7081, 4107}, {7257, 30940}, {8684, 7132}, {17787, 14296}, {18265, 1919}, {18827, 17096}, {30729, 4974}, {30730, 740}, {30731, 4432}, {34067, 604}

X(36802) =  BARYCENTRIC QUOTIENT X(8)/X(918)

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

X(36802) lies on these lines: on lines {100, 693}, {105, 5205}, {190, 522}, {645, 7253}, {646, 4076}, {660, 926}, {673, 3912}, {677, 883}, {885, 4582}, {919, 8707}, {1016, 3900}, {1738, 24781}, {2325, 6559}, {2402, 2414}, {2481, 3262}, {3158, 6654}, {3239, 3699}, {3685, 28058}, {4555, 6366}, {4569, 8269}, {4579, 32735}, {4583, 35574}, {4997, 6745}, {5091, 36221}, {5377, 13136}, {6335, 15742}, {6362, 32028}, {15633, 20130}

X(36802) = X(i)-cross conjugate of X(j) for these (i,j): {884, 294}, {885, 14942}, {3685, 4076}, {3693, 1016}, {28058, 4998}
X(36802) = X(i)-isoconjugate of X(j) for these (i,j): {56, 2254}, {57, 665}, {109, 3675}, {241, 649}, {244, 2283}, {269, 926}, {273, 23225}, {513, 1458}, {604, 918}, {663, 34855}, {667, 9436}, {672, 3669}, {883, 3248}, {1015, 1025}, {1026, 1357}, {1027, 1362}, {1088, 8638}, {1402, 23829}, {1408, 4088}, {1412, 24290}, {1416, 3126}, {1459, 1876}, {1461, 17435}, {1566, 32668}, {2223, 3676}, {3286, 4017}, {3323, 32666}, {3572, 34253}, {4925, 16945}, {5236, 22383}, {7180, 18206}, {7203, 20683}, {9454, 24002}, {35505, 36146}
X(36802) = cevapoint of X(i) and X(j) for these (i,j): {55, 4435}, {100, 2398}, {294, 884}, {522, 5853}, {673, 2402}, {885, 14942}, {3693, 3900}, {3700, 4433}, {4000, 6084}, {28071, 28132}
X(36802) = trilinear pole of line {8, 220}
X(36802) = barycentric product X(i)*X(j) for these {i,j}: {8, 666}, {105, 646}, {190, 14942}, {200, 34085}, {294, 668}, {312, 36086}, {341, 36146}, {346, 927}, {644, 2481}, {645, 13576}, {664, 6559}, {673, 3699}, {884, 31625}, {885, 1016}, {919, 3596}, {1024, 7035}, {1978, 2195}, {3570, 33676}, {3939, 18031}, {4391, 5377}, {4513, 14727}, {4554, 28071}, {4578, 34018}, {4998, 28132}, {7257, 18785}, {28659, 32666}
X(36802) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 918}, {9, 2254}, {55, 665}, {100, 241}, {101, 1458}, {105, 3669}, {190, 9436}, {210, 24290}, {220, 926}, {294, 513}, {333, 23829}, {643, 18206}, {644, 518}, {645, 30941}, {646, 3263}, {650, 3675}, {651, 34855}, {666, 7}, {673, 3676}, {765, 1025}, {884, 1015}, {885, 1086}, {918, 3323}, {919, 56}, {926, 35505}, {927, 279}, {1016, 883}, {1024, 244}, {1252, 2283}, {1783, 1876}, {1897, 5236}, {2195, 649}, {2284, 1362}, {2321, 4088}, {2481, 24002}, {3161, 4925}, {3573, 34253}, {3693, 3126}, {3699, 3912}, {3900, 17435}, {3939, 672}, {4069, 3930}, {4147, 23773}, {4571, 25083}, {4578, 3693}, {4587, 1818}, {5377, 651}, {5546, 3286}, {5548, 34230}, {6065, 2284}, {6558, 3717}, {6559, 522}, {7257, 18157}, {13576, 7178}, {14827, 8638}, {14942, 514}, {18785, 4017}, {23696, 3942}, {28071, 650}, {28132, 11}, {30720, 4899}, {30728, 4684}, {30729, 4966}, {30730, 3932}, {31638, 31605}, {32666, 604}, {32735, 1407}, {33676, 4444}, {34085, 1088}, {36041, 17107}, {36086, 57}, {36146, 269}

X(36803) =  BARYCENTRIC QUOTIENT X(8)/X(926)

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

X(36803) lies on these lines: {190, 3261}, {645, 666}, {646, 6386}, {660, 3766}, {689, 919}, {889, 2481}, {927, 8707}, {1921, 34852}, {1978, 3699}, {4554, 4885}, {4639, 17789}, {4997, 18031}, {6063, 30825}, {13136, 15418}, {13576, 34087}, {14727, 18830}, {21580, 27805}

X(36803) = isotomic conjugate of X(665)
X(36803) = isotomic conjugate of the complement of X(3766)
X(36803) = isotomic conjugate of the isogonal conjugate of X(666)
X(36803) = X(i)-cross conjugate of X(j) for these (i,j): {874, 670}, {1921, 31625}
X(36803) = X(i)-isoconjugate of X(j) for these (i,j): {19, 23225}, {31, 665}, {32, 2254}, {57, 8638}, {513, 9454}, {514, 9455}, {518, 1919}, {560, 918}, {604, 926}, {649, 2223}, {667, 672}, {669, 18206}, {798, 3286}, {1026, 1977}, {1458, 3063}, {1924, 30941}, {1980, 3912}, {2205, 23829}, {2206, 24290}, {2284, 3248}, {2356, 22383}, {3675, 32739}, {9426, 18157}, {15615, 36146}, {32666, 35505}
X(36803) = cevapoint of X(i) and X(j) for these (i,j): {2, 3766}, {850, 3948}, {1978, 27853}, {3263, 4391}
X(36803) = trilinear pole of line {8, 76}
X(36803) = barycentric product X(i)*X(j) for these {i,j}: {76, 666}, {105, 6386}, {190, 18031}, {312, 34085}, {561, 36086}, {646, 34018}, {668, 2481}, {670, 13576}, {673, 1978}, {919, 1502}, {927, 3596}, {1928, 32666}, {4572, 14942}, {4602, 18785}, {18833, 35333}, {28659, 36146}
X(36803) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 665}, {3, 23225}, {8, 926}, {55, 8638}, {75, 2254}, {76, 918}, {99, 3286}, {100, 2223}, {101, 9454}, {105, 667}, {190, 672}, {294, 3063}, {310, 23829}, {313, 4088}, {321, 24290}, {646, 3693}, {664, 1458}, {666, 6}, {668, 518}, {670, 30941}, {673, 649}, {692, 9455}, {693, 3675}, {799, 18206}, {874, 8299}, {883, 1362}, {885, 3271}, {918, 35505}, {919, 32}, {926, 15615}, {927, 56}, {1016, 2284}, {1027, 3248}, {1332, 20752}, {1438, 1919}, {1814, 22383}, {1897, 2356}, {1978, 3912}, {2481, 513}, {3263, 3126}, {3699, 2340}, {3952, 20683}, {4033, 3930}, {4391, 17435}, {4554, 241}, {4555, 34230}, {4561, 1818}, {4562, 3252}, {4569, 34855}, {4572, 9436}, {4583, 22116}, {4602, 18157}, {4998, 2283}, {5377, 692}, {6331, 15149}, {6335, 5089}, {6386, 3263}, {6559, 657}, {6654, 8632}, {7035, 1026}, {13576, 512}, {14625, 4832}, {14727, 9309}, {14942, 663}, {18026, 1876}, {18031, 514}, {18785, 798}, {24004, 14439}, {27808, 3932}, {27853, 17755}, {28071, 8641}, {28132, 14936}, {31637, 1459}, {32666, 560}, {32735, 1397}, {34018, 3669}, {34085, 57}, {35313, 20958}, {35333, 1964}, {35574, 34159}, {36086, 31}, {36146, 604}

X(36804) =  BARYCENTRIC QUOTIENT X(8)/X(3738)

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

X(36804) lies on these lines: {80, 32850}, {190, 655}, {645, 1016}, {765, 4086}, {2222, 8707}, {3257, 3762}, {3261, 4554}, {3262, 4358}, {3264, 32851}, {3699, 4397}, {4552, 15455}, {4562, 14616}, {4582, 24004}, {4585, 13136}, {5205, 14204}, {14628, 18743}, {17264, 20566}

X(36804) = isogonal conjugate of X(21758)
X(36804) = isotomic conjugate of X(3960)
X(36804) = isotomic conjugate of the complement of X(3762)
X(36804) = X(i)-cross conjugate of X(j) for these (i,j): {2397, 190}, {3992, 7035}, {4768, 75}, {17780, 668}, {26144, 86}, {32849, 1016}
X(36804) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21758}, {19, 22379}, {31, 3960}, {32, 4453}, {36, 649}, {56, 654}, {57, 8648}, {58, 21828}, {244, 1983}, {320, 1919}, {513, 7113}, {604, 3738}, {667, 3218}, {849, 2610}, {875, 27950}, {1019, 3724}, {1397, 3904}, {1443, 3063}, {1464, 7252}, {1635, 16944}, {1870, 22383}, {1980, 20924}, {2206, 4707}, {2245, 3733}, {2361, 3669}, {2423, 34586}, {3025, 32675}, {3248, 4585}, {4017, 4282}, {4089, 32739}, {17455, 23345}
X(36804) = cevapoint of X(i) and X(j) for these (i,j): {2, 3762}, {3239, 6735}, {4033, 24004}, {4358, 4391}
X(36804) = trilinear pole of line {8, 80}
X(36804) = barycentric product X(i)*X(j) for these {i,j}: {8, 35174}, {80, 668}, {99, 15065}, {100, 20566}, {190, 18359}, {312, 655}, {646, 2006}, {670, 34857}, {759, 27808}, {1978, 2161}, {2222, 3596}, {3678, 35139}, {3699, 18815}, {3952, 14616}, {3969, 32680}, {4033, 24624}, {4582, 14628}, {6187, 6386}, {28659, 32675}
X(36804) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3960}, {3, 22379}, {6, 21758}, {8, 3738}, {9, 654}, {37, 21828}, {55, 8648}, {75, 4453}, {80, 513}, {100, 36}, {101, 7113}, {190, 3218}, {312, 3904}, {321, 4707}, {594, 2610}, {644, 2323}, {646, 32851}, {655, 57}, {664, 1443}, {668, 320}, {693, 4089}, {759, 3733}, {901, 16944}, {1016, 4585}, {1018, 2245}, {1023, 17455}, {1089, 6370}, {1168, 23345}, {1252, 1983}, {1332, 22128}, {1793, 23189}, {1807, 1459}, {1897, 1870}, {1978, 20924}, {2006, 3669}, {2161, 649}, {2222, 56}, {2341, 7252}, {2397, 16586}, {3570, 27950}, {3678, 526}, {3699, 4511}, {3738, 3025}, {3799, 3792}, {3939, 2361}, {3952, 758}, {3969, 32679}, {4033, 3936}, {4103, 4053}, {4427, 4973}, {4551, 1464}, {4552, 18593}, {4554, 17078}, {4557, 3724}, {4671, 23884}, {4756, 4880}, {4767, 4867}, {5546, 4282}, {6187, 667}, {6335, 17923}, {6740, 3737}, {14147, 19302}, {14616, 7192}, {14628, 30725}, {15065, 523}, {15742, 4242}, {17780, 214}, {18359, 514}, {18815, 3676}, {20566, 693}, {24624, 1019}, {27808, 35550}, {32675, 604}, {34857, 512}, {35174, 7}, {36069, 849}, {36590, 23838}

X(36805) =  BARYCENTRIC QUOTIENT X(8)/X(3880)

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

X(36805) lies on the conic {{A,B,C,X(1),X(2) and these lines: {1, 1120}, {2, 646}, {28, 1811}, {57, 190}, {81, 645}, {88, 4358}, {89, 17350}, {105, 5205}, {274, 30818}, {277, 28808}, {278, 1997}, {279, 4554}, {291, 4871}, {312, 8056}, {330, 30861}, {668, 16594}, {961, 8686}, {1002, 30947}, {1022, 3762}, {1219, 26093}, {1224, 19847}, {1432, 27805}, {4621, 7132}, {5316, 25280}, {13136, 34051}, {24004, 31227}

X(36805) = isotomic conjugate of X(16610)
X(36805) = polar conjugate of X(1878)
X(36805) = isotomic conjugate of the complement of X(4358)
X(36805) = polar conjugate of the isogonal conjugate of X(1811)
X(36805) = X(i)-cross conjugate of X(j) for these (i,j): {900, 668}, {1320, 18816}, {4723, 75}, {21222, 190}, {21343, 18830}, {24841, 2481}
X(36805) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1149}, {19, 23205}, {31, 16610}, {32, 1266}, {44, 17109}, {48, 1878}, {101, 6085}, {106, 20972}, {190, 8660}, {604, 3880}, {649, 23832}, {1333, 4695}, {1918, 16711}, {4927, 32739}, {5151, 32659}, {8752, 22082}, {9456, 17460}, {21129, 32719}
X(36805) = cevapoint of X(2) and X(4358)
X(36805) = trilinear pole of line {8, 513}
X(36805) = barycentric product X(i)*X(j) for these {i,j}: {75, 1120}, {264, 1811}, {668, 23836}, {693, 6079}, {3596, 8686}
X(36805) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1149}, {2, 16610}, {3, 23205}, {4, 1878}, {8, 3880}, {10, 4695}, {44, 20972}, {75, 1266}, {100, 23832}, {106, 17109}, {274, 16711}, {513, 6085}, {519, 17460}, {667, 8660}, {693, 4927}, {1120, 1}, {1811, 3}, {3264, 20900}, {3699, 23705}, {3762, 21129}, {3880, 6018}, {3992, 21041}, {4358, 16594}, {5440, 22082}, {6079, 100}, {8686, 56}, {23836, 513}, {30939, 17195}

X(36806) =  BARYCENTRIC QUOTIENT X(8)/X(4155)

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

X(36806) lies on these lines: {190, 4584}, {645, 3287}, {646, 4140}, {661, 799}, {880, 2395}, {3699, 4631}, {4583, 8707}, {4610, 4621}

X(36806) = X(i)-isoconjugate of X(j) for these (i,j): {181, 8632}, {604, 4155}, {669, 16609}, {798, 1284}, {1356, 3570}, {1400, 4455}, {1402, 21832}, {1428, 4079}, {1874, 3049}, {1918, 7212}, {1919, 7235}, {2086, 29055}, {3747, 7180}
X(36806) = cevapoint of X(i) and X(j) for these (i,j): {799, 2396}, {3709, 35104}, {3907, 3985}
X(36806) = trilinear pole of line {8, 7257}
X(36806) = barycentric product X(i)*X(j) for these {i,j}: {261, 4583}, {314, 4589}, {333, 4639}, {335, 4631}, {660, 18021}, {2311, 4602}, {3596, 36066}, {4518, 4623}, {4584, 28660}, {4612, 18895}, {7257, 18827}
X(36806) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4155}, {21, 4455}, {99, 1284}, {261, 659}, {274, 7212}, {314, 4010}, {333, 21832}, {643, 3747}, {645, 2238}, {646, 4037}, {660, 181}, {668, 7235}, {799, 16609}, {811, 1874}, {874, 3027}, {875, 1356}, {1808, 810}, {2185, 8632}, {2311, 798}, {2396, 16591}, {3287, 2086}, {4518, 4705}, {4562, 2171}, {4583, 12}, {4584, 1400}, {4589, 65}, {4610, 1429}, {4612, 1914}, {4623, 1447}, {4631, 239}, {4636, 2210}, {4639, 226}, {4876, 4079}, {6064, 3573}, {7058, 4435}, {7256, 4433}, {7257, 740}, {7258, 3985}, {18021, 3766}, {18827, 4017}, {36066, 56}

preamble

X(36807) =  BARYCENTRIC QUOTIENT X(8)/X(5853)

Barycentrics    (a^2 - a*b + 2*b^2 - 2*a*c - b*c + c^2)*(a^2 - 2*a*b + b^2 - a*c - b*c + 2*c^2) : :
X(36807) = 4 X[4859] - 5 X[27191]

X(36807) lies on the conic {{A,B,C,X(2),X(7) and these lines: {2, 1280}, {7, 190}, {27, 1810}, {75, 646}, {86, 645}, {273, 6335}, {310, 30821}, {335, 17266}, {346, 1086}, {673, 3912}, {675, 6078}, {903, 4582}, {1088, 4554}, {1268, 29604}, {1440, 28753}, {1477, 8707}, {1997, 36620}, {3717, 24841}, {4366, 29572}, {4422, 4648}, {4440, 36606}, {4779, 20533}, {4869, 5845}, {4904, 6558}, {4997, 6548}, {5205, 14201}, {6384, 30822}, {6557, 16078}, {6650, 28530}, {7249, 27805}, {7318, 28738}, {9055, 17265}, {14621, 17244}, {17292, 27483}, {17310, 32096}, {17381, 28626}, {17755, 27475}, {17780, 31226}, {20131, 29599}, {21453, 33116}, {26582, 29579}, {29607, 32108}

X(36807) = reflection of X(i) in X(j) for these {i,j}: {190, 3161}, {4373, 1086}
X(36807) = isotomic conjugate of X(3008)
X(36807) = antitomic image of X(4373)
X(36807) = isotomic conjugate of the complement of X(3912)
X(36807) = polar conjugate of the isogonal conjugate of X(1810)
X(36807) = X(i)-cross conjugate of X(j) for these (i,j): {918, 190}, {1280, 35160}, {3717, 75}, {14942, 18025}, {24841, 903}
X(36807) = X(i)-isoconjugate of X(j) for these (i,j): {6, 1279}, {19, 20780}, {31, 3008}, {56, 2348}, {57, 8647}, {100, 8659}, {105, 20662}, {604, 5853}, {692, 6084}, {2976, 34080}, {8751, 20749}
X(36807) = cevapoint of X(2) and X(3912)
X(36807) = trilinear pole of line {8, 514}
X(36807) = barycentric product X(i)*X(j) for these {i,j}: {8, 35160}, {75, 1280}, {264, 1810}, {668, 35355}, {1477, 3596}, {3261, 6078}
X(36807) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1279}, {2, 3008}, {3, 20780}, {8, 5853}, {9, 2348}, {55, 8647}, {514, 6084}, {644, 23704}, {649, 8659}, {672, 20662}, {1280, 1}, {1477, 56}, {1810, 3}, {1818, 20749}, {3667, 2976}, {3912, 16593}, {3930, 20680}, {5853, 3021}, {6078, 101}, {35160, 7}, {35355, 513}

X(36808) =  X(6)X(31)∩X(9)X(32771)

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

Let E9 be the circumellipse centered at X(9). Dan Reznik discovered a triangle indicated by arrows in the following video: Thomson cubic of family of 3-periodics is E9. Peter Moses found (February 17, 2020) that its vertices, A', B', C' can by found by solving the following system for x,y,z:

a^2 z y^2 - a^2 y z^2 + b^2 x z^2 - b^2 z x^2 + c^2 y x^2 - c^2 x y^2 = 0
a y z + b z x + c x y = 0
x + y + z = 1

The vertices A', B', C' lie on these cubics: K002, K101, K317. The tangents to E9 at A', B', C' form a triangle A'', B'', C'' whose vertices lie on the cubic K002 and on the following conic:

bc(b + c)x^2 + ca(c + a)y^2 + ab(a + b)z^2 + 2(bc + ca + ab)(ayz + bzx + cxy) = 0,

which passes through X(i) for i = 5540, 9359, 16554, 24578 and the vertices of the excentral triangle. The perspector of this conic is X(6), and the center is X(36808).

X(36808) lies on these lines: {6, 31}, {9, 32771}, {38, 5283}, {39, 1185}, {63, 169}, {81, 4253}, {573, 7411}, {579, 5276}, {748, 20459}, {896, 20665}, {940, 2350}, {1180, 4283}, {1621, 3730}, {2249, 30257}, {2979, 24484}, {3219, 31317}, {3501, 32945}, {4712, 10477}, {7075, 32919}, {7193, 32664}, {19734, 28274}, {20229, 22060}, {21369, 32933}, {21384, 32864}, {21387, 32845}, {24578, 32860}, {24727, 32915}, {26035, 33163}

X(36809) =  (name pending)

Barycentrics    (S^2+SB SC)/(SA(3S^2+5 SB SC)) : :
Barycentrics    (a^2 (b^2+c^2)-(b^2-c^2)^2)/((b^2+c^2-a^2) (a^4+3 a^2 (b^2+c^2)-4 (b^2-c^2)^2)) : :

Let H be the orthocenter and M the midpoint of AH. Let Ba and Ca be the orthogonal projections of B and C on CM and BM, respectively. Define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'B'C' be the triangle having sidelines BaCa, CbAb, AcBc. Then A'B'C' is perspective to ABC, and the perspector is X(36809). (Angel Montesdeoca, February 21,2020)

See Angel Montesdeoca, Euclid 641 .

X(36809) lies on this line: {3855, 8797}

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Perspectors associated with mid-trace triangles: X(36810)-X(36413)

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This preamble and centers X(36810)-X(36413) were contributed by Clark Kimberling and Peter Moses, February 21, 2020

Let P = p : q : r and U = u : v : w be points not on the sidelines BC, CA, AB of a triangle ABC
Let A' = AP∩BC, A' = AU∩BC', and A* = midpoint of A' and A''
Define B* and C* cyclically
The triangle A*B*C* is here named the mid-trace triangle of P and U, denoted by M(P,U)

A* = 0 : 2 q v + r v + q w : 2 r w + r v + q w
B* = 2 p u + p w + r u : 0 : 2 r w + p w + r u
C* = 2 p u + q u + p v : 2 q v + q u + p v

For given P, the locus of a point X = x : y : z such that M(P,X) is a cevian triangle is given by the cubic

p (q r + r^2 + q p) y^2 z - p (q r + q^2 + r p) y z^2 + (cyclic) + (p - q) (p - r) (q - r) x y z = 0,

here named the mid-cevian cubic of P, denoted by MC(P)
If P is on the line at infinity, then MC(P) is the union of the line at infiniity (x + y + z = 0) and the circumconic

p2(q - r) y z + q2(r - p) z x + r2(p - q) x y = 0

The following points lie on MC(P): A, B, C, P, 1/p : : , p - 2q - 2r : : 1/r, and p(- p + q + r) : :

For further developments, see Bernard Gibert's page, CL069 Mid-Cevian Cubics.

X(36810) =  PERSPECTOR OF THESE TRIANGLES: M(X(1),X(2)) AND 2nd SHARYGIN

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

X(36810) lies on these lines: {3722,3745}, {4974,8299}

X(36811) =  PERSPECTOR OF THESE TRIANGLES: M(X(2),X(6)) AND ANTI-1st-BROCARD

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

X(36811) lies on these lines: {98,20190}, {99,5007}, {147,7931}, {385,732}, {1281,4974}, {1916,5182}, {3329,5989}, {3589,11606}, {5085,5984}, {5149,17128}, {7840,12830}, {7923,32528}, {8289,10352}, {8782,32449}, {10997,13196}, {14778,34482}, {19910,22521}

X(36811) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4027, 5026, 8290}, {4027, 8290, 385}, {5989, 10353, 3329}

X(36812) =  PERSPECTOR OF THESE TRIANGLES: M(X(2),X(274)) AND GEMINI 110

Barycentrics    a^2*b^2 + 4*a^2*b*c + 4*a*b^2*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 : :
X(36812) = 3 X[2] + X[274], 9 X[2] - X[1655], 3 X[274] + X[1655], X[1655] - 3 X[16589], 7 X[3624] - 3 X[30571]

X(36812) lies on these lines: {2, 39}, {10, 24656}, {21, 32456}, {32, 33035}, {83, 16911}, {86, 20970}, {99, 16912}, {115, 33034}, {187, 16917}, {377, 7842}, {386, 15668}, {405, 7816}, {442, 625}, {443, 7761}, {519, 25130}, {574, 33036}, {620, 2795}, {626, 8728}, {740, 1125}, {1015, 16819}, {1078, 16994}, {1213, 16887}, {1500, 31996}, {1506, 33033}, {1509, 20142}, {1573, 31997}, {1574, 27255}, {2092, 25457}, {2140, 34824}, {2238, 17175}, {2548, 33026}, {3008, 6703}, {3624, 30571}, {3634, 25109}, {3734, 11108}, {3828, 25102}, {4253, 17259}, {4698, 25092}, {4751, 17030}, {5007, 17000}, {5021, 20156}, {5051, 25468}, {5275, 7805}, {5475, 33028}, {6692, 6706}, {7603, 33045}, {7737, 33039}, {7738, 33027}, {7748, 33029}, {7760, 16993}, {7780, 16992}, {7800, 17582}, {7804, 11321}, {7815, 16408}, {7825, 17528}, {7849, 17529}, {8589, 33063}, {10436, 28252}, {15271, 16863}, {15489, 24220}, {15513, 33062}, {16454, 25497}, {16604, 31238}, {17303, 30110}, {17759, 32009}, {18424, 33057}, {19862, 21264}, {19878, 20530}, {22036, 27478}, {23447, 27164}, {27147, 29985}, {27274, 28653}, {31415, 33037}

X(36812) = midpoint of X(274) and X (16589)
X(36812) = complement of X (16589)
X(36812) = X(i)-complementary conjugate of X(j) for these (i,j): {8708, 4129}, {32009, 3454}
X(36812) = crosssum of X(6) and X (21753)
X(36812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 274, 16589}, {17175, 29460, 2238}, {25457, 25508, 2092}

X(36813) =  PERSPECTOR OF THESE TRIANGLES: M(X(2),X(75)) AND 2ND SHARYGIN

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

X(36813) lies on no line X(i)X(j) for 0 < i < j < 36812.

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Points on mid-cevian cubics: X(36814)-X(36431)

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This preamble and centers X(36814)-X(36431) were contributed by Clark Kimberling and Peter Moses, February 21, 2020

The family of mid-cevian cubics is introduced just before X(36810); specifically, if P is not on BC or CA or AB, then the cubic MC(P) is given by

p (q r + r^2 + q p) y^2 z - p (q r + q^2 + r p) y z^2 + (cyclic) + (p - q) (p - r) (q - r) x y z = 0.

MC(X(1)) passes through A, B, C and X(i) for these i: 1, 9, 75, 87, 993, 3679, 24806, 36814, 36815, 36816, 36817, 36818, 36819, 36871, 36872, 36873
MC(X(4)) = K616 passes through A, B, C and X(i) for these i: 4, 69, 376, 1249, 3421, 5485, 6601, 9214, 34208, 36874, 36875, 36876, 36877, 36878, 41325, 51830, 51831, 51832, 51833, 51834, 51835, 53133
MC(X(6)) passes through the vertices of the Brocard triangle and X(i) for these i: 3, 6, 76, 599, 3224, 9462, 14608, 19127, 36820, 36821, 36822, 36823, 36879, 36880, 36881, 46023, 46024
MC(X(7)) passes through A, B, C and X(i) for these i: 7, 8, 3160, 6172, 27818, 36588, 36887, 36888
MC(X(67)) passes through the vertices of the crcular points at infinity and X(i) for these i: 67, 265, 316, 524, 8724, 11646, 15900, 34319, 36824, 36825, 36826, 36833, 36882, 36883, 36884
MC(X(69)) passes through A, B, C and X(i) for these i: 4, 69, 1992, 6337, 6604, 34403, 36889, 36890, 36891, 36892, 36893, 36894, 36895
MC(X(74)) passes through A, B, C and X(i) for these i: 74, 477, 895, 3260, 5627, 10706, 36896, 43574
MC(X(76)) passes through A, B, C and X(i) for these i: 6, 76, 264, 598, 6374, 7757, 52756
MC(X(98)) passes through A, B, C and X(i) for these i: 98, 325, 842, 5503, 6054, 16092, 36897, 36898, 36899
MC(X(100)) passes through A, B, C and X(i) for these i: 100, {100, 693, 1290, 4767, 5375, 10707, 37143, 51562
MC(X(110)) passes through A, B, C and X(i) for these i: 110, 476, 850, 9140, 9146, 17708, 27867, 36827, 36828, 36829, 36830, 36831, 36885, 36886
MC(X(476)) passes through A, B, C and X(i) for these i: 110, 476, 3268, 9140, 17708, 34312
MC(X(850)) passes through A, B, C and X(i) for these i: 110, 850, 3268, 8599, 9141, 9979, 36900, 36901

X(36814) =  X(1)X(513)∩X(75)X(537)

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

X(36814) lies on the cubic MC(X(1)) and these lines: {1, 513}, {75, 537}, {87, 106}, {88, 36263}, {726, 21140}, {901, 3550}, {1320, 3551}, {3257, 16468}, {3662, 4013}, {4080, 30942}, {4386, 17969}, {6548, 24427}, {9456, 16779}, {17109, 21214}, {19634, 31164}, {20347, 23822}, {24325, 24517}, {25034, 31139}

X(36814) = X(i)-isoconjugate of X(j) for these (i,j): {44, 20332}, {519, 727}, {902, 3226}, {1319, 8851}, {1404, 36799}, {1960, 8709}, {2251, 32020}, {3285, 27809}, {4358, 34077}, {17780, 23355}
X(36814) = crossdifference of every pair of points on line {44, 14408}
X(36814) = barycentric product X(i)*X(j) for these {i,j}: {88, 726}, {901, 20908}, {903, 1575}, {1022, 23354}, {1463, 4997}, {3009, 20568}, {3257, 3837}, {4080, 18792}, {4622, 21053}, {5376, 21140}, {9456, 35538}
X(36814) = barycentric quotient X(i)/X(j) for these {i,j}: {88, 3226}, {106, 20332}, {726, 4358}, {903, 32020}, {1320, 36799}, {1463, 3911}, {1575, 519}, {2316, 8851}, {3009, 44}, {3257, 8709}, {3837, 3762}, {4674, 27809}, {6373, 1635}, {9456, 727}, {17475, 4432}, {18792, 16704}, {20777, 22356}, {20785, 5440}, {21760, 902}, {21830, 21805}, {23354, 24004}

X(36815) =  X(1)X(523)∩X(9)X(80)

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

X(36815) lies on the cubic MC(X(1)) and these lines: {1, 523}, {9, 80}, {75, 99}, {87, 1411}, {238, 4124}, {341, 4076}, {513, 24402}, {655, 7672}, {740, 3573}, {874, 35544}, {2222, 2726}, {2783, 24436}, {3684, 4037}, {3797, 27941}, {3877, 3903}, {3923, 24482}, {4448, 24428}, {4613, 6187}, {9282, 14584}, {16067, 29857}, {17278, 24918}, {17279, 25683}, {24461, 34230}

X(36815) = X(15507)-cross conjugate of X(238)
X(36815) = X(i)-isoconjugate of X(j) for these (i,j): {36, 291}, {292, 3218}, {295, 1870}, {320, 1911}, {335, 7113}, {741, 758}, {813, 3960}, {1443, 7077}, {1808, 1835}, {1922, 20924}, {1983, 4444}, {2196, 17923}, {2311, 18593}, {2361, 7233}, {3572, 4585}, {3724, 18827}, {3936, 18268}, {4453, 34067}, {4562, 21758}, {4584, 21828}
X(36815) = cevapoint of X(i) and X(j) for these (i,j): {740, 4432}, {4124, 4448}
X(36815) = trilinear pole of line {2238, 4435}
X(36815) = barycentric product X(i)*X(j) for these {i,j}: {80, 239}, {238, 18359}, {350, 2161}, {655, 3716}, {659, 36804}, {740, 24624}, {759, 3948}, {1411, 3975}, {1914, 20566}, {1921, 6187}, {2006, 3685}, {2238, 14616}, {3684, 18815}, {4435, 35174}, {6740, 16609}, {30940, 34857}, {34079, 35544}
X(36815) = barycentric quotient X(i)/X(j) for these {i,j}: {80, 335}, {238, 3218}, {239, 320}, {242, 17923}, {350, 20924}, {659, 3960}, {740, 3936}, {812, 4453}, {1284, 18593}, {1429, 1443}, {1447, 17078}, {1914, 36}, {2006, 7233}, {2161, 291}, {2201, 1870}, {2210, 7113}, {2238, 758}, {3573, 4585}, {3684, 4511}, {3685, 32851}, {3716, 3904}, {3747, 2245}, {3948, 35550}, {4010, 4707}, {4155, 2610}, {4435, 3738}, {4455, 21828}, {4693, 27757}, {4800, 23884}, {6187, 292}, {6740, 36800}, {7193, 22128}, {8300, 27950}, {15507, 16586}, {16514, 3792}, {18359, 334}, {20566, 18895}, {24624, 18827}, {27918, 4089}, {34079, 741}, {36804, 4583}

X(36816) =  X(1)X(514)∩X(9)X(75)

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

X(36816) lies on the cubic MC(X(1)) and these lines: {1, 514}, {9, 75}, {10, 14267}, {105, 993}, {536, 23343}, {666, 6654}, {3679, 13576}, {4363, 16482}, {4670, 16494}, {6381, 23891}, {7962, 14942}, {9315, 14727}, {16831, 27922}

X(36816) = X(14433)-cross conjugate of X(23891)
X(36816) = X(i)-isoconjugate of X(j) for these (i,j): {518, 739}, {665, 898}, {918, 32718}, {1026, 23892}, {2223, 3227}, {2254, 34075}, {9454, 31002}
X(36816) = cevapoint of X(536) and X(4465)
X(36816) = trilinear pole of line {899, 4728}
X(36816) = barycentric product X(i)*X(j) for these {i,j}: {105, 6381}, {536, 673}, {666, 4728}, {899, 2481}, {927, 14430}, {1438, 35543}, {3230, 18031}, {3768, 36803}, {4526, 34085}
X(36816) = barycentric quotient X(i)/X(j) for these {i,j}: {536, 3912}, {666, 4607}, {673, 3227}, {891, 2254}, {899, 518}, {919, 34075}, {1438, 739}, {2481, 31002}, {3230, 672}, {3768, 665}, {3994, 3932}, {4009, 3717}, {4465, 17755}, {4706, 4684}, {4728, 918}, {6381, 3263}, {14431, 4088}, {14942, 36798}, {19945, 3675}, {23343, 1026}, {32666, 32718}, {36086, 898}

X(36817) =  X(1)X(512)∩X(9)X(87)

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

X(36817) lies on the cubic MC(X(1)) and these lines: {1, 512}, {9, 87}, {31, 4584}, {75, 670}, {291, 3679}, {741, 993}

X(36817) = X(i)-isoconjugate of X(j) for these (i,j): {715, 740}, {3747, 18826}
X(36817) = barycentric product X(i)*X(j) for these {i,j}: {2229, 18827}, {18268, 35532}
X(36817) = barycentric quotient X(i)/X(j) for these {i,j}: {714, 3948}, {2229, 740}, {18268, 715}

X(36818) =  X(1)X(4777)∩X(75)X(4597)

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

X(36818) lies on the cubic MC(X(1)) and these lines: {1, 4777}, {75, 4597}, {100, 993}, {1023, 4908}, {4432, 25690}, {4618, 36594}

X(36818) = X(901)-isoconjugate of X(14315)
X(36818) = barycentric quotient X(1635)/X(14315)

X(36819) =  X(1)X(522)∩X(9)X(48)

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

X(36819) lies on the cubic MC(X(1)) and these lines: {1, 522}, {9, 48}, {10, 14266}, {33, 1309}, {75, 77}, {78, 765}, {518, 2283}, {609, 32641}, {1026, 1818}, {1376, 15635}, {1458, 1861}, {2191, 36123}, {2284, 3693}, {2720, 2751}, {3870, 36037}

X(36819) = X(i)-isoconjugate of X(j) for these (i,j): {105, 517}, {294, 1465}, {666, 3310}, {673, 2183}, {859, 13576}, {885, 23981}, {908, 1438}, {919, 10015}, {1024, 24029}, {1416, 6735}, {1457, 14942}, {1769, 36086}, {1785, 36057}, {1814, 14571}, {2195, 22464}, {2804, 32735}, {4246, 10099}, {22350, 36124}, {23696, 23706}, {32666, 36038}
X(36819) = cevapoint of X(2340) and X(14439)
X(36819) = crosssum of X(517) and X(15507)
X(36819) = crossdifference of every pair of points on line {1769, 2183}
X(36819) = barycentric product X(i)*X(j) for these {i,j}: {104, 3912}, {518, 34234}, {672, 18816}, {909, 3263}, {918, 36037}, {1026, 2401}, {1458, 36795}, {1809, 5236}, {1818, 16082}, {2250, 30941}, {2254, 13136}, {3717, 34051}, {25083, 36123}
X(36819) = barycentric quotient X(i)/X(j) for these {i,j}: {104, 673}, {241, 22464}, {518, 908}, {665, 1769}, {672, 517}, {909, 105}, {918, 36038}, {1026, 2397}, {1458, 1465}, {1795, 1814}, {2223, 2183}, {2250, 13576}, {2254, 10015}, {2283, 24029}, {2342, 294}, {2356, 14571}, {2423, 1027}, {2720, 36146}, {3693, 6735}, {3912, 3262}, {3930, 17757}, {5089, 1785}, {14439, 1145}, {14578, 36057}, {17435, 35015}, {18206, 17139}, {18816, 18031}, {20683, 21801}, {20752, 22350}, {32641, 36086}, {32669, 32735}, {34234, 2481}, {34858, 1438}, {36037, 666}

X(36820) =  X(3)X(67)∩X(6)X(826)

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

X(36820) lies on the cubic MC(X(6)) and these lines: {3, 67}, {6, 826}, {76, 4577}, {935, 2698}, {9076, 15080}, {20975, 32242}

X(36820) = X(i)-isoconjugate of X(j) for these (i,j): {23, 1581}, {316, 1967}, {694, 16568}, {733, 18715}, {1934, 18374}, {9468, 20944}
X(36820) = cevapoint of X(732) and X(5026)
X(36820) = crossdifference of every pair of points on line {2492, 9019}
X(36820) = barycentric product X(i)*X(j) for these {i,j}: {67, 385}, {419, 34897}, {732, 9076}, {804, 17708}, {935, 24284}, {1691, 18019}, {1966, 2157}, {3455, 3978}, {5026, 10415}, {8791, 12215}
X(36820) = barycentric quotient X(i)/X(j) for these {i,j}: {67, 1916}, {385, 316}, {804, 9979}, {1580, 16568}, {1691, 23}, {1966, 20944}, {2157, 1581}, {2236, 18715}, {3455, 694}, {4039, 21094}, {4107, 21205}, {5026, 7664}, {5027, 2492}, {8623, 9019}, {9076, 14970}, {11183, 18311}, {14602, 18374}, {17708, 18829}, {18019, 18896}

X(36821) =  X(6)X(512)∩X(76)X(338)

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

X(36821) lies on the cubic MC(X(6)) and these lines: {6, 512}, {76, 338}, {111, 12149}, {895, 30496}, {3224, 19127}, {5077, 9462}

X(36821) = crosssum of X(187) and X(5026)
X(36821) = X(i)-isoconjugate of X(j) for these (i,j): {699, 14210}, {896, 3225}
X(36821) = barycentric product X(i)*X(j) for these {i,j}: {111, 698}, {671, 3229}, {897, 2227}, {18023, 32748}, {32740, 35524}
X(36821) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 3225}, {698, 3266}, {895, 8858}, {2227, 14210}, {3229, 524}, {9429, 351}, {32540, 5967}, {32740, 699}, {32748, 187}

X(36822) =  X(3)X(76)∩X(6)X(523)

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

X(36822) lies on the cubics K297 and MC(X(6)) and these lines: {3, 76}, {6, 523}, {287, 10602}, {381, 34175}, {385, 4226}, {538, 5118}, {599, 20021}, {1976, 9462}, {2854, 34227}, {2966, 14614}, {3111, 3734}, {5968, 9154}, {11286, 14608}, {13137, 35930}, {15048, 36157}, {23342, 30736}

X(36822) = crosssum of X(511) and X(6786)
X(36822) = trilinear pole of line {3231, 9148}
X(36822) = crossdifference of every pair of points on line {511, 2491}
X(36822) = X(i)-isoconjugate of X(j) for these (i,j): {729, 1959}, {1755, 3228}, {3569, 36133}, {9417, 34087}
X(36822) = barycentric product X(i)*X(j) for these {i,j}: {98, 538}, {290, 3231}, {1821, 2234}, {1976, 30736}, {2395, 23342}, {2966, 9148}, {6786, 34536}, {18024, 33875}
X(36822) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 3228}, {290, 34087}, {538, 325}, {887, 2491}, {888, 3569}, {1976, 729}, {2234, 1959}, {2715, 32717}, {2966, 9150}, {3231, 511}, {5118, 2421}, {5967, 14608}, {6786, 36790}, {9148, 2799}, {14609, 5968}, {23342, 2396}, {33875, 237}, {36084, 36133}
X(36822) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5652, 35606, 1316}, {8870, 14382, 32540}

X(36823) =  X(6)X(525)∩X(76)X(648)

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

X(36823) lies on the cubics K527 and MC(X(6)) and these lines: {3, 1177}, {6, 525}, {76, 648}, {249, 20806}, {2211, 34138}, {2373, 15066}, {2421, 6393}, {2710, 10423}, {4235, 22151}, {5182, 9289}, {14355, 15407}, {14966, 36212}
on K527

X(36823) = cevapoint of X(3289) and X(9155)
X(36823) = trilinear pole of line {237, 684}
X(36823) = X(i)-isoconjugate of X(j) for these (i,j): {98, 18669}, {293, 5523}, {336, 14580}, {858, 1910}, {1821, 2393}, {1976, 20884}, {14961, 36120}
X(36823) = barycentric product X(i)*X(j) for these {i,j}: {297, 18876}, {325, 1177}, {511, 2373}, {6333, 10423}
X(36823) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 5523}, {237, 2393}, {325, 1236}, {511, 858}, {1177, 98}, {1755, 18669}, {1959, 20884}, {2211, 14580}, {2373, 290}, {3289, 14961}, {9155, 5181}, {10422, 9154}, {10423, 685}, {17209, 17172}, {18876, 287}

X(36824) =  X(67)X(512)∩X(110)X(524)

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

X(36824) lies on the cubic MC(X(67)) and these lines: {67, 512}, {110, 524}, {316, 670}, {1634, 7813}, {4576, 9019}

X(36825) = X(82)-isoconjugate of X(2854)
X(36825) = trilinear pole of line {39, 14424}
X(36825) = barycentric product X(i)*X(j) for these {i,j}: {141, 2770}, {1930, 36150}, {8024, 32741}
X(36825) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 2854}, {2770, 83}, {32741, 251}, {35325, 7482}, {36150, 82}

X(36825) =  X(30)X(99)∩X(67)X(523)

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

X(36825) lies on the cubics K091 and MC(X(67)) and these lines: {30, 99}, {67, 523}, {5649, 11007}, {14995, 36194}

X(36826) =  X(67)X(526)∩X(98)X(20126)

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

X(36826) lies on the cubic MC(X(67)) and these lines: {67, 526}, {98, 20126}, {265, 290}, {524, 9186}, {2715, 15900}, {5967, 34319}

X(36827) =  X(67)X(524)∩X(110)X(249)

Barycentrics    a^2*(a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(b^2 + c^2) : :
X(36827) = 4 X[3231] - X[15107]

X(36827) lies on the cubic MC(X(110)) and these lines: {67, 524}, {110, 249}, {111, 694}, {182, 10560}, {297, 8753}, {352, 17964}, {660, 36085}, {670, 850}, {671, 14957}, {826, 4576}, {1291, 35191}, {1634, 3005}, {2421, 9178}, {3565, 35188}, {5012, 10559}, {5380, 8050}, {5468, 34290}, {5968, 7998}, {7883, 14246}, {9463, 14609}, {10097, 11634}, {10562, 27867}, {20021, 31125}

X(36827) = isogonal conjugate of X(22105)
X(36827) = X(i)-isoconjugate of X(j) for these (i,j): {1, 22105}, {82, 690}, {83, 2642}, {187, 18070}, {351, 3112}, {1648, 4599}, {4062, 18108}, {4593, 21906}, {4750, 18098}, {10566, 21839}, {14210, 18105}, {14273, 34055}, {14419, 18082}, {23889, 34294}
X(36827) = crosspoint of X(691) and X(892)
X(36827) = crosssum of X(351) and X(690)
X(36827) = trilinear pole of line {39, 1634}
X(36827) = crossdifference of every pair of points on line {1648, 11183}
X(36827) = barycentric product X(i)*X(j) for these {i,j}: {38, 36085}, {39, 892}, {110, 31125}, {111, 4576}, {141, 691}, {671, 1634}, {1930, 36142}, {5380, 16696}, {7813, 34574}, {8024, 32729}, {14424, 34539}, {23297, 32583}, {30786, 35325}
X(36827) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 22105}, {39, 690}, {141, 35522}, {688, 21906}, {691, 83}, {892, 308}, {895, 4580}, {897, 18070}, {1634, 524}, {1843, 14273}, {1964, 2642}, {3005, 1648}, {3051, 351}, {3917, 14417}, {4576, 3266}, {8041, 14424}, {8623, 11183}, {9019, 18311}, {9178, 34294}, {17187, 4750}, {30489, 23287}, {31125, 850}, {32583, 10130}, {32729, 251}, {32740, 18105}, {35325, 468}, {35359, 26235}, {36085, 3112}, {36142, 82}
X(36827) = {X(691),X(32583)}-harmonic conjugate of X(110)

X(36828) =  X(6)X(6032)∩X(110)X(112)

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

X((36828) lies on the cubic MC(X(110)) and these lines: {6, 6032}, {110, 112}, {476, 26714}, {648, 850}, {805, 9087}, {1576, 3005}, {1613, 2502}, {2421, 9146}, {3016, 7737}, {3051, 11646}, {7998, 22240}, {10562, 27867}

X(36828) = crosspoint of X(648) and X(11636)
X(36828) = crosssum of X(647) and X(3906)
X(36828) = X(1577)-isoconjugate of X(19151)
X(36828) = barycentric product X(i)*X(j) for these {i,j}: {99, 9971}, {110, 5169}, {691, 8262}
X(36828) = barycentric quotient X(i)/X(j) for these {i,j}: {1576, 19151}, {5169, 850}, {8262, 35522}, {9971, 523}

X(36829) =  X(3)X(9140)∩X(99)X(476)

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

X((36829) lies on the cubic MC(X(110)) and these lines: {3, 9140}, {23, 6054}, {99, 476}, {107, 20189}, {110, 351}, {237, 8724}, {925, 930}, {1995, 11184}, {3448, 23217}, {5012, 23158}, {5640, 9155}, {7608, 16042}, {12273, 34333}, {14984, 34834}, {17434, 32661}, {18316, 35921}, {19911, 34013}

X(36829) = X(18117)-cross conjugate of X(566)
X(36829) = X(661)-isoconjugate of X(7578)
X(36829) = cevapoint of X(566) and X(18117)
X(36829) = trilinear pole of line {566, 23039}
X(36829) = barycentric product X(i)*X(j) for these {i,j}: {99, 566}, {648, 23039}, {4558, 7577}, {4590, 18117}
X(36829) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 7578}, {566, 523}, {7577, 14618}, {18117, 115}, {23039, 525}
X(36829) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1634, 15329, 110}, {1634, 23181, 15329}

X(36830) =  X(110)X(647)∩X(112)X(476)

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

X((36830) lies on the cubic MC(X(110)) and these lines: {2, 9514}, {32, 6792}, {50, 230}, {110, 647}, {112, 476}, {187, 9218}, {248, 9140}, {441, 3580}, {691, 10561}, {850, 2966}, {1576, 3005}, {1914, 19622}, {3265, 4558}, {3284, 23061}, {6587, 7471}, {7480, 23964}, {8574, 14366}, {8651, 32729}, {11610, 30789}, {17434, 32661}, {22391, 23293}, {23584, 27866}

X(36830) = complement of X(13485)
X(36830) = complement of the isogonal conjugate of X(7669)
X(36830) = complement of the isotomic conjugate of X(3448)
X(36830) = isogonal conjugate of the polar conjugate of X(30716)
X(36830) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 110}, {2643, 33967}, {3448, 2887}, {7669, 10}, {8574, 8287}, {14366, 21254}, {16562, 141}, {20941, 626}, {21092, 21245}, {21203, 21252}, {22146, 18589}, {30716, 21259}
X(36830) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 110}, {14366, 7669}
X(36830) = X(i)-cross conjugate of X(j) for these (i,j): {7669, 14366}, {8574, 7669}
X(36830) = X(i)-isoconjugate of X(j) for these (i,j): {661, 13485}, {662, 6328}, {1577, 3447}
X(36830) = cevapoint of X(7669) and X(8574)
X(36830) = crosspoint of X(2) and X(3448)
X(36830) = crosssum of X(6) and X(3447)
X(36830) = trilinear pole of line {7669, 22146}
X(36830) = crossdifference of every pair of points on line {868, 6328}
X(36830) = center of hyperbola {{A,B,C,X(249),X(250),PU(2)}} (the isogonal conjugate of line X(115)X(125))
X(36830) = complementary conjugate of complement of X(7669)
X(36830) = crosssum of circumcircle intercepts of line PU(40) (line X(115)X(125))
X(36830) = barycentric product X(i)*X(j) for these {i,j}: {3, 30716}, {99, 7669}, {110, 3448}, {163, 20941}, {523, 14366}, {648, 22146}, {662, 16562}, {2966, 34349}, {4556, 21092}, {4570, 21203}, {4590, 8574}
X(36830) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 13485}, {512, 6328}, {1576, 3447}, {3448, 850}, {7669, 523}, {8574, 115}, {14366, 99}, {16562, 1577}, {20941, 20948}, {21203, 21207}, {22146, 525}, {30716, 264}, {34349, 2799}
X(36830) = {X(647),X(23357)}-harmonic conjugate of X(110)

X(36831) =  X(30)X(74)∩X(110)X(250)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) : :
X(36831) = 4 X[852] - X[23061]

X((36831) lies on the Johnson circumconic [see K714]), the cubic MC(X(110)), and these lines: {30, 74}, {110, 250}, {850, 6528}, {852, 14919}, {1625, 17434}, {1993, 9717}, {3060, 35908}, {3470, 15801}, {5640, 35910}, {5889, 14264}, {6368, 35360}, {10313, 18877}, {14380, 15329}, {14385, 34148}, {16080, 35098}

X(36831) = reflection of X(110) in X(7480)
X(36831) = X(i)-isoconjugate of X(j) for these (i,j): {30, 2616}, {54, 36035}, {275, 2631}, {1637, 2167}, {1784, 23286}, {2173, 15412}, {2190, 9033}, {2623, 14206}
X(36831) = cevapoint of X(51) and X(2081)
X(36831) = crosssum of X(1637) and X(9409)
X(36831) = trilinear pole of line {216, 1625}
X(36831) = barycentric product X(i)*X(j) for these {i,j}: {74, 14570}, {216, 16077}, {311, 32640}, {343, 1304}, {1494, 1625}, {1568, 34568}, {2349, 2617}, {5562, 15459}, {14213, 36034}, {14919, 35360}, {16080, 23181}, {18695, 36131}, {28706, 32715}
X(36831) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 1637}, {74, 15412}, {216, 9033}, {217, 9409}, {418, 1636}, {1154, 5664}, {1304, 275}, {1625, 30}, {1953, 36035}, {2081, 3258}, {2159, 2616}, {2433, 8901}, {2617, 14206}, {14570, 3260}, {15459, 8795}, {16077, 276}, {17434, 1650}, {18877, 23286}, {23181, 11064}, {32640, 54}, {32695, 8884}, {32715, 8882}, {36034, 2167}, {36131, 2190}

X(36832) =  SINGULAR FOCUS OF THE CUBIC MC(X(67))

Barycentrics    a^18 - 4*a^16*b^2 + 2*a^14*b^4 + 4*a^12*b^6 - 3*a^10*b^8 + 3*a^8*b^10 - 4*a^6*b^12 - 2*a^4*b^14 + 4*a^2*b^16 - b^18 - 4*a^16*c^2 + 20*a^14*b^2*c^2 - 26*a^12*b^4*c^2 + 12*a^10*b^6*c^2 - 9*a^6*b^10*c^2 + 25*a^4*b^12*c^2 - 23*a^2*b^14*c^2 + 5*b^16*c^2 + 2*a^14*c^4 - 26*a^12*b^2*c^4 + 43*a^10*b^4*c^4 - 33*a^8*b^6*c^4 - 7*a^6*b^8*c^4 - 8*a^4*b^10*c^4 + 19*a^2*b^12*c^4 - 2*b^14*c^4 + 4*a^12*c^6 + 12*a^10*b^2*c^6 - 33*a^8*b^4*c^6 + 74*a^6*b^6*c^6 - 19*a^4*b^8*c^6 + 23*a^2*b^10*c^6 - 14*b^12*c^6 - 3*a^10*c^8 - 7*a^6*b^4*c^8 - 19*a^4*b^6*c^8 - 46*a^2*b^8*c^8 + 12*b^10*c^8 + 3*a^8*c^10 - 9*a^6*b^2*c^10 - 8*a^4*b^4*c^10 + 23*a^2*b^6*c^10 + 12*b^8*c^10 - 4*a^6*c^12 + 25*a^4*b^2*c^12 + 19*a^2*b^4*c^12 - 14*b^6*c^12 - 2*a^4*c^14 - 23*a^2*b^2*c^14 - 2*b^4*c^14 + 4*a^2*c^16 + 5*b^2*c^16 - c^18 : :

X(36832) lies on these lines: {30,16339), (67,10748), (126,542), (2780,3818), (2854,15067), (6698,14650), (7761,32424), (9129,24206), (20304,28662), (32274,33962}

X(36833) =  X(67)X(691)∩X(110)X(14357)

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

X(36833) lies on the circumcircle and these lines: {67, 691}, {110, 14357}, {827, 34319}, {2715, 15900}, {3455, 20404}, {11635, 32244}

X(36834) =  X(1)X(4688)∩X(2)X(3707)

Barycentrics    5*a^2 + 7*a*b + 7*a*c + 8*b*c : :

X(36834) is mentioned at K317.

Let E9, A'B'C', A"B"C" be as at X(36808). Then X(36834) is the perspector of E9 wrt A'B'C', and the perspector of triangles A'B'C' and A"B"C". (Randy Hutson, March 29, 2020)

X(36834) lies on these lines: {1, 4688}, {2, 3707}, {6, 31312}, {9, 4670}, {57, 5333}, {86, 1449}, {142, 24604}, {190, 16676}, {192, 3247}, {1086, 25055}, {1125, 5698}, {1698, 17392}, {1730, 5437}, {2345, 29606}, {3243, 36480}, {3616, 4779}, {3624, 4675}, {3663, 28641}, {3664, 4748}, {4393, 31313}, {4470, 4873}, {4472, 29573}, {4648, 29604}, {4726, 17318}, {4798, 17284}, {5436, 25526}, {5750, 29627}, {6707, 17272}, {7290, 36554}, {16667, 31238}, {17237, 34595}, {17274, 29612}, {17286, 29589}, {17335, 31311}, {17374, 19875}, {20195, 31191}, {26039, 29600}, {28604, 29618}

X(36834) = barycentric product X(75)*X(14969)
X(36834) = barycentric quotient X(14969)/X(1)
X(36834) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4659, 16826, 3247}, {10436, 16826, 4659}

X(36835) =  X(1)X(3711)∩X(2)X(5223)

Barycentrics    a*(a^2 + 2*a*b - 3*b^2 + 2*a*c - 26*b*c - 3*c^2) : :
X(36835) = 6 X[2] - X[11034], 7 X[3624] - 2 X[18490]

X(36835) lies one these lines: {1, 3711}, {2, 5223}, {10, 12541}, {11, 9819}, {40, 11379}, {165, 3683}, {210, 30350}, {936, 35016}, {946, 1698}, {1001, 3158}, {2093, 19876}, {3062, 18230}, {3243, 3740}, {3305, 9352}, {3339, 3649}, {3624, 4866}, {3634, 5328}, {3731, 17756}, {3925, 30308}, {5251, 7987}, {5437, 15481}, {5531, 11715}, {5658, 30326}, {5659, 20196}, {5660, 11407}, {10398, 33993}, {10857, 30291}, {11531, 25917}, {15298, 33995}, {16487, 17125}, {16832, 24003}, {17718, 34595}, {30283, 30389}

X(36835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 30393, 10980}, {1698, 5316, 7988}, {3624, 4866, 30343}

X(36836) =  X(3)X(6)∩X(20)X(396)

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

See Kadir Altintas and Peter Moses, Euclid 645 .

The two circles mentioned in Euclid 645 are in the Schoutte coaxal system. As such, the squared radius can be nicely written as 7 a^2 b^2 c^2 / (Sqrt[3] (a^2 + b^2 + c^2) + 8 S)^2. A point associated with X(36836) is X(36843), for which the analogous circle has squared radius 7 a^2 b^2 c^2 / (Sqrt[3] (a^2 + b^2 + c^2) - 8. (Peter Moses, February 26, 2020).

X(36836) lies on these lines: {2, 5339}, {3, 6}, {4, 16644}, {13, 1657}, {14, 3526}, {17, 382}, {18, 5054}, {20, 396}, {22, 34424}, {64, 11243}, {140, 10654}, {376, 397}, {394, 11146}, {395, 3523}, {398, 631}, {548, 10653}, {617, 11307}, {619, 11312}, {623, 11309}, {628, 11299}, {630, 11306}, {632, 18581}, {635, 11301}, {1080, 22532}, {1092, 11137}, {1656, 16241}, {2307, 5217}, {3090, 5321}, {3091, 23302}, {3146, 11488}, {3303, 7051}, {3304, 10638}, {3412, 15696}, {3516, 8740}, {3524, 16773}, {3525, 5334}, {3529, 5318}, {3532, 36296}, {3534, 16962}, {3545, 5349}, {3627, 18582}, {5067, 5343}, {5071, 5365}, {5072, 16966}, {5076, 16808}, {5079, 16809}, {5198, 11475}, {5335, 17538}, {5344, 11001}, {5350, 33703}, {6694, 11298}, {8739, 15750}, {9761, 22114}, {10303, 23303}, {10601, 11145}, {10606, 35469}, {10632, 35502}, {10640, 22333}, {10641, 11403}, {10658, 15039}, {10984, 11134}, {11542, 15704}, {11543, 14869}, {15040, 36209}, {15041, 36208}, {15533, 35304}, {15681, 16267}, {15683, 22235}, {15700, 16963}, {15701, 16268}, {20416, 23013}, {21359, 33387}, {22334, 32585}, {23261, 35732}, {30471, 32821}

X(36836) = Brocard-circle-inverse of X(36843)

X(36837) =  EULER LINE INTERCEPT OF X(1141)X(6343)

Barycentrics    2*a^16-5*a^14*b^2-5*a^12*b^4+31*a^10*b^6-45*a^8*b^8+33*a^6*b^10-15*a^4*b^12+5*a^2*b^14-b^16-5*a^14*c^2-2*a^12*b^2*c^2+31*a^10*b^4*c^2-24*a^8*b^6*c^2-27*a^6*b^8*c^2+50*a^4*b^10*c^2-31*a^2*b^12*c^2+8*b^14*c^2-5*a^12*c^4+31*a^10*b^2*c^4-12*a^8*b^4*c^4-15*a^6*b^6*c^4-34*a^4*b^8*c^4+63*a^2*b^10*c^4-28*b^12*c^4+31*a^10*c^6-24*a^8*b^2*c^6-15*a^6*b^4*c^6-2*a^4*b^6*c^6-37*a^2*b^8*c^6+56*b^10*c^6-45*a^8*c^8-27*a^6*b^2*c^8-34*a^4*b^4*c^8-37*a^2*b^6*c^8-70*b^8*c^8+33*a^6*c^10+50*a^4*b^2*c^10+63*a^2*b^4*c^10+56*b^6*c^10-15*a^4*c^12-31*a^2*b^2*c^12-28*b^4*c^12+5*a^2*c^14+8*b^2*c^14-c^16 : :
Barycentrics    4 S^4+S^2 (57 R^4+4 SB SC-44 R^2 SW+8 SW^2)-SB SC (43 R^4-36 R^2 SW+8 SW^2) : :
X(36837) = 3*X(3)+X(28237), 3*X(5)-4*X(13469), 3*X(5)-2*X(15335), 3*X(547)-4*X(12056), 3*X(549)-X(10205), 3*X(549)-2*X(15334), 5*X(632)-4*X(12057),7*X(3090)-8*X(34420), 3*X(8703)-2*X(15336), X(10126)-4*X(15327), 3*X(10285)-X(28237), 3*X(11539)-2*X(15333), 4*X(13469)+3*X(14142),3*X(14142)+2*X(15335), 4*X(16239)-3*X(34479)

As a point on the Euler line, X(36837) has Shinagawa coefficients (57 R^4 - 44 R^2 SW + 4 (S^2 + 2 SW^2),-43 R^4 + 36 R^2 SW + 4 (S^2 - 2 SW^2)).

See Tran Quang Hung and Ercole Suppa, Euclid 655 .

X(36837) lies on these lines: {2,3}, {54,24385}, {1141,6343}, {1263,25042}, {8254,16337}, {10610,12026}, {14140,34804}, {31879,34598}, {32423,32551}, {32744,33545}

X(36837) = midpoint of X(i) and X(j) for these {i,j}: {3,10285}, {5,14142}, {550,20120}
X(36837) = reflection of X(i) in X(j) for these (i,j): (4,19940), (140,15327), (546,15957), (3853,25404), (5066,25403), (10126,140), (10205,15334), (15335,13469), (20030,5501), (27868,10289), (31879,34598)
X(36837) = anticomplement of X(10289)
X(36837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,27868,10289), (549,10205,15334), (13469,15335,5)

X(36838) =  ISOTOMIC CONJUGATE OF X(4130)

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

See Tran Quang Hung and César Lozada, Euclid 659 .

X(36838) lies on the conics {{A, B, C, X(85), X(18026)}}, {{A, B, C, X(277), X(1783)}} and these lines: {7, 3022}, {85, 23058}, {279, 34018}, {658, 3732}, {664, 4569}, {927, 934}, {1088, 1111}, {4573, 4617}, {7056, 7215}, {9442, 10481}, {17079, 30682}, {24002, 24011}

X(36838) = isotomic conjugate of X(4130)
X(36838) = cevapoint of X(i) and X(j) for these {i,j}: {7, 650}, {514, 10481}, {658, 4626}, {1088, 24002}
X(36838) = X(i)-cross conjugate of-X(j) for these (i,j): (650, 7), (658, 4569), (1088, 24011)
X(36838) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 4105}, {9, 8641}, {32, 4163}, {55, 657}
X(36838) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 4105), (2, 4130), (7, 3900), (56, 8641)
X(36838) = X(1742)-Zayin conjugate of-X(657)
X(36838) = trilinear pole of the tangent to the Feuerbach hyperbola at X(7)
X(36838) = cevapoint of Feuerbach hyperbola intercepts of Gergonne line
X(36838) = trilinear pole of the line {7, 354 } (line X(2)X(6) of the intouch triangle)
X(36838) = barycentric product X(i)*X(j) for these {i, j}: {7, 4569}, {75, 4626}, {76, 4617}, {85, 658}, {190, 23062}, {226, 4635}
X(36838) = barycentric quotient X(i)/X(j) for these (i, j): (1, 4105), (7, 3900), (56, 8641), (57, 657), (65, 4524), (75, 4163)
X(36838) = trilinear product X(i)*X(j) for these {i, j}: {2, 4626}, {7, 658}, {57, 4569}, {65, 4635}, {75, 4617}, {76, 6614}
X(36838) = trilinear quotient X(i)/X(j) for these (i, j): (2, 4105), (7, 657), (57, 8641), (75, 4130), (76, 4163), (85, 3900)

X(36839) = X(13)X(5916)∩X(14)X(8014)

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

See Tran Quang Hung and César Lozada, Euclid 659 .

X(36839) lies on the conics {{A, B, C, X(2), X(648)}}, {{A, B, C, X(14), X(99)}} and these lines: {2, 10217}, {13, 5916}, {14, 8014}, {110, 5618}, {476, 5995}, {1640, 23588}, {2407, 17402}, {3457, 14181}, {4240, 36306}, {9214, 11080}, {16963, 36211}

X(36839) = cevapoint of X(13) and X(523)
X(36839) = X(523)-cross conjugate of-X(13)
X(36839) = X(i)-isoconjugate-of-X(j) for these {i,j}: {523, 1094}, {661, 11131}, {798, 11129}
X(36839) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (13, 23870), (99, 11129), (110, 11131), (163, 1094)
X(36839) = trilinear pole of the tangent to the Kiepert hyperbola at X(13)
X(36839) = trilinear pole of line X(13)X(15) (the line through X(13) perpendicular to its trilinear polar)
X(36839) = X(1576)-vertex conjugate of X(36840)
X(36839) = barycentric product X(i)*X(j) for these {i, j}: {13, 23895}, {99, 11080}, {300, 5995}, {476, 11078}
X(36839) = barycentric quotient X(i)/X(j) for these (i, j): (13, 23870), (99, 11129), (110, 11131), (163, 1094), (476, 11092), (1989, 23284), (23588, 36840)
X(36839) = trilinear product X(i)*X(j) for these {i, j}: {662, 11080}, {2153, 23895}
X(36839) = trilinear quotient X(i)/X(j) for these (i, j): (110, 1094), (662, 11131), (799, 11129), (2153, 6137), (2166, 23284)

X(36840) = X(13)X(8015)∩X(14)X(5917)

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

See Tran Quang Hung and César Lozada, Euclid 659 .

X(36840) lies on the conics {{A, B, C, X(2), X(648)}}, {{A, B, C, X(13), X(99)}} and these lines: {2, 10218}, {13, 8015}, {14, 5917}, {110, 5619}, {476, 5994}, {1640, 23588}, {2407, 17403}, {3458, 14177}, {4240, 36309}, {9214, 11085}, {16962, 36210}

X(36840) = cevapoint of X(14) and X(523)
X(36840) = X(523)-cross conjugate of-X(14)
X(36840) = X(i)-isoconjugate-of-X(j) for these {i,j}: {523, 1095}, {661, 11130}, {798, 11128}, {2152, 23871}
X(36840) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (14, 23871), (99, 11128), (110, 11130), (163, 1095)
X(36840) = trilinear pole of the tangent to the Kiepert hyperbola at X(14)
X(36840) = trilinear pole of the line {14, 16}
X(36840) = trilinear pole of line X(14)X(16) (the line through X(14) perpendicular to its trilinear polar)
X(36840) = X(1576)-vertex conjugate of X(36839)
X(36840) = barycentric product X(i)*X(j) for these {i, j}: {14, 23896}, {99, 11085}, {299, 5619}, {301, 5994}, {476, 11092}, {648, 10218}
X(36840) = barycentric quotient X(i)/X(j) for these (i, j): (14, 23871), (99, 11128), (110, 11130), (163, 1095), (476, 11078), (1989, 23283), (23588, 36839)
X(36840) = trilinear product X(i)*X(j) for these {i, j}: {162, 10218}, {662, 11085}, {2154, 23896}
X(36840) = trilinear quotient X(i)/X(j) for these (i, j): (110, 1095), (662, 11130), (799, 11128), (2154, 6138), (2166, 23283)

X(36841) = X(2)X(34570)∩X(3)X(11596)

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

See Tran Quang Hung and César Lozada, Euclid 659 .

X(36841) lies on the conics {{A, B, C, X(20), X(4235)}}, {{A, B, C, X(112), X(1461)}} and these lines: {2, 34570}, {3, 11596}, {98, 22143}, {99, 112}, {645, 4592}, {658, 662}, {691, 20187}, {1632, 5467}, {1992, 7763}, {2452, 22085}, {2966, 31998}, {3053, 14772}, {4563, 34211}, {8754, 10723}, {13479, 14060}, {14615, 15905}, {18879, 30528}, {20975, 34473}

X(36841) = cevapoint of X(20) and X(6587)
X(36841) = X(i)-isoconjugate-of-X(j) for these {i,j}: {64, 661}, {253, 798}, {459, 810}, {512, 2184}
X(36841) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (20, 523), (99, 253), (107, 6526), (110, 64)
X(36841) = trilinear pole of the line {20, 154} (the tangent to hyperbola {{A,B,C,X(4),X(20)}} at X(20))
X(36841) = barycentric product X(i)*X(j) for these {i, j}: {20, 99}, {110, 14615}, {154, 670}, {610, 799}, {643, 33673}, {645, 18623}
X(36841) = barycentric quotient X(i)/X(j) for these (i, j): (20, 523), (99, 253), (107, 6526), (110, 64), (122, 5489), (154, 512)
X(36841) = trilinear product X(i)*X(j) for these {i, j}: {20, 662}, {99, 610}, {110, 18750}, {154, 799}, {163, 14615}, {204, 4563}
X(36841) = trilinear quotient X(i)/X(j) for these (i, j): (20, 661), (99, 2184), (110, 2155), (154, 798), (163, 33581), (204, 2489)
X(36841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (648, 4558, 99), (2407, 4558, 648)

X(36842) = X(30)X(54)∩X(140)X(1157)

Barycentrics    (S^2+SA*SB)*(S^2+SA*SC)*(4*S^2+(SB+SC)*(2*R^2-4*SA+SB+SC)) : :

See Tran Quang Hung and César Lozada, Euclid 659 .

X(36842) lies on these lines: {4, 15619}, {5, 23338}, {30, 54}, {97, 34002}, {137, 20414}, {140, 1157}, {252, 3628}, {275, 15559}, {389, 36161}, {523, 30490}, {546, 1141}, {547, 7604}, {548, 25042}, {933, 1166}, {1493, 25150}, {3574, 27196}, {5501, 24385}, {6150, 13856}, {6689, 12060}, {7745, 14586}, {8254, 32744}, {8901, 33332}, {9820, 15958}, {10615, 31376}, {12026, 31879}, {13160, 19179}, {13564, 16030}, {14130, 16035}, {14865, 19172}, {15425, 23280}, {16337, 33545}, {19552, 23337}, {24147, 32551}
X(36842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{3, 6, P2}, {3, 15, 22236}, {3, 61, 22238}, {3, 62, 11481}, {3, 5238, 11480}, {3, 5611, 5864}, {3, 11485, 62}, {3, 11486, 5351}, {3, 22236, 6}, {4, 16772, 16644}, {15, 5238, 3}, {15, 5352, 61}, {15, 10645, 11485}, {15, 11480, 6}, {15, 21158, 10613}, {20, 396, 5340}, {61, 5238, 5352}, {61, 5352, 3}, {61, 22238, 6}, {62, 10645, 3}, {398, 631, 16645}, {1151, 1152, 15}, {6396, 17852, P2}, {6409, 6425, P2}, {6410, 6426, P2}, {6433, 10147, P2}, {6434, 10148, P2}, {10541, 15815, P2}, {10613, 21158, 19780}, {10645, 11485, 11481}, {11480, 11481, 10645}, {11480, 22236, 3}, {11481, 11485, 6}, {14540, 21158, 3}, {16241, 16964, 1656}, {22236, 22238, 61}, {22331, 31884, 36843}

X(36843) =  X(3)X(6)∩X(20)X(395)

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

See X(36836).

X(36843) lies on these lines: {2, 5340}, {3, 6}, {4, 16645}, {13, 3526}, {14, 1657}, {17, 5054}, {18, 382}, {20, 395}, {22, 34425}, {64, 11244}, {140, 10653}, {376, 398}, {383, 22531}, {394, 11145}, {396, 3523}, {397, 631}, {548, 10654}, {616, 11308}, {618, 11311}, {624, 11310}, {627, 11300}, {629, 11305}, {632, 18582}, {636, 11302}, {1092, 11134}, {1250, 3304}, {1656, 16242}, {3090, 5318}, {3091, 23303}, {3146, 11489}, {3303, 19373}, {3411, 15696}, {3516, 8739}, {3524, 16772}, {3525, 5335}, {3529, 5321}, {3532, 36297}, {3534, 16963}, {3545, 5350}, {3627, 18581}, {5067, 5344}, {5071, 5366}, {5072, 16967}, {5076, 16809}, {5079, 16808}, {5198, 11476}, {5204, 7127}, {5334, 17538}, {5343, 11001}, {5349, 33703}, {6695, 11297}, {8740, 15750}, {9763, 22113}, {10303, 23302}, {10601, 11146}, {10606, 35470}, {10633, 35502}, {10639, 22333}, {10642, 11403}, {10657, 15039}, {10984, 11137}, {11542, 14869}, {11543, 15704}, {15040, 36208}, {15041, 36209}, {15533, 35303}, {15681, 16268}, {15683, 22237}, {15700, 16962}, {15701, 16267}, {20415, 23006}, {21360, 33386}, {22334, 32586}, {23251, 35732}, {30472, 32821}

X(36843) = Brocard-circle-inverse of X(36836)
X(36843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 36836}, {3, 16, 22238}, {3, 61, 11480}, {3, 62, 22236}, {3, 5237, 11481}, {3, 5615, 5865}, {3, 11485, 5352}, {3, 11486, 61}, {3, 22238, 6}, {4, 16773, 16645}, {16, 5237, 3}, {16, 5351, 62}, {16, 10646, 11486}, {16, 11481, 6}, {16, 21159, 10614}, {20, 395, 5339}, {61, 10646, 3}, {62, 5237, 5351}, {62, 5351, 3}, {62, 22236, 6}, {397, 631, 16644}, {1151, 1152, 16}, {6396, 17852, 36836}, {6409, 6425, 36836}, {6410, 6426, 36836}, {6433, 10147, 36836}, {6434, 10148, 36836}, {10541, 15815, 36836}, {10614, 21159, 19781}, {10646, 11486, 11480}, {11480, 11481, 10646}, {11480, 11486, 6}, {11481, 22238, 3}, {14541, 21159, 3}, {16242, 16965, 1656}, {22236, 22238, 62}, {22331, 31884, 36836}

X(36844) =  PERSPECTOR OF THESE TRIANGLES: TANGENTIAL-OF-ANTICOMPLEMENTARY AND GEMINI 29

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

X(36844) lies on these lines: {1,4}, {2,197}, {55,26118}, {69,35614}, {75,1370}, {149,7391}, {222,29207}, {312,3436}, {377,32773}, {406,9798}, {429,8192}, {518,5928}, {1352,35645}, {1376,30778}, {1617,19542}, {2385,20223}, {2478,32942}, {2550,7386}, {2886,26052}, {3421,3974}, {3474,26929}, {6601,15314}, {6818,33171}, {6822,32783}, {7169,10431}, {7381,27491}, {7392,26105}, {8270,21621}, {12588,21334}, {16063,33110}, {17135,21270}, {20539,33088}, {21293,32064}, {22654,27505}, {23843,27379}

X(36845) =  PERSPECTOR OF THESE TRIANGLES: SODDY AND GEMINI 29

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

For the Soddy triangle, see X(31528).

X(36845) lies on these lines: {1,2}, {4,3555}, {7,3434}, {11,5748}, {40,1208}, {55,5744}, {56,3189}, {57,5853}, {63,390}, {69,4514}, {72,1058}, {75,14548}, {81,4344}, {100,1617}, {149,152}, {189,1814}, {193,10025}, {210,26105}, {219,30619}, {226,3243}, {278,1280}, {321,5807}, {329,497}, {346,2257}, {347,3875}, {354,2550}, {355,5804}, {377,3889}, {388,5175}, {443,5045}, {479,35312}, {480,25893}, {496,26129}, {515,15239}, {516,9965}, {517,5768}, {524,24352}, {527,9580}, {528,2094}, {664,17093}, {668,18153}, {758,30305}, {908,5274}, {942,5082}, {944,7580}, {950,6762}, {952,19541}, {956,3488}, {962,3868}, {1004,4308}, {1005,3486}, {1056,3419}, {1088,6604}, {1108,3693}, {1155,34607}, {1214,3896}, {1320,3427}, {1331,17127}, {1420,12437}, {1445,7674}, {1468,4339}, {1482,8727}, {1621,5273}, {1697,24391}, {1723,3161}, {1788,3913}, {1997,3699}, {2136,4848}, {2256,5839}, {2328,16704}, {2478,5815}, {2886,3475}, {2975,4313}, {3058,5698}, {3158,3911}, {3174,8732}, {3218,9778}, {3304,35985}, {3305,5686}, {3333,6904}, {3340,21627}, {3421,5722}, {3428,5731}, {3476,35990}, {3485,3813}, {3487,24390}, {3668,4452}, {3677,3755}, {3681,18228}, {3689,17728}, {3697,17559}, {3742,26040}, {3772,4864}, {3869,9785}, {3874,4295}, {3883,14552}, {3885,14110}, {3914,4310}, {3927,15172}, {3952,8055}, {4000,17597}, {4305,8666}, {4309,6763}, {4314,17576}, {4318,18623}, {4323,34195}, {4358,5423}, {4442,15590}, {4512,30331}, {4640,10385}, {4648,4883}, {4661,31018}, {4712,32915}, {4855,5265}, {4860,34612}, {4878,28778}, {4880,34719}, {4899,30568}, {4971,25355}, {4981,5296}, {5084,34790}, {5177,21620}, {5219,24386}, {5226,11680}, {5249,11038}, {5284,18230}, {5534,6848}, {5603,8226}, {5687,26062}, {5691,18452}, {5730,14022}, {5734,10883}, {5745,10389}, {5749,24552}, {5758,12116}, {5761,10943}, {5770,10679}, {5811,10531}, {5850,20214}, {5932,20221}, {6327,17145}, {6919,21075}, {6939,18908}, {7270,19790}, {7308,24393}, {8232,24389}, {8271,34036}, {9052,35645}, {9440,25885}, {9779,31053}, {9804,11024}, {10106,12625}, {10167,35514}, {10591,21077}, {11523,12053}, {12245,31786}, {12247,25416}, {12526,12575}, {12541,14923}, {12573,21454}, {13407,31418}, {15299,20588}, {16496,24210}, {16572,21096}, {16750,30941}, {17140,21283}, {17163,18698}, {17766,24283}, {18141,32850}, {20095,23958}, {21183,21302}, {25080,27804}, {28610,30332}, {31527,32003}, {32099,33075}, {32943,33163}, {33141,33144}, {34699,34744}

X(36845) = isotomic conjugate of isogonal conjugate of X(21002)
X(36845) = complement of X(20015)
X(36845) = anticomplement of X(200)
X(36845) = polar conjugate of isogonal conjugate of X(22153)
X(36845) = barycentric quotient X(21096)/X(10)

X(36846) =  PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND GEMINI 29

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

For the Bevan-antipodal triangle, see X(34488).

X(36846) lies on these lines: {1,2}, {3,3895}, {9,3890}, {11,11256}, {21,31393}, {34,1120}, {40,3885}, {46,2802}, {55,11260}, {56,3880}, {57,1476}, {63,3057}, {65,10912}, {77,3875}, {84,1320}, {100,1420}, {149,5691}, {224,1317}, {269,4452}, {312,1222}, {405,31792}, {516,20076}, {518,2098}, {528,8544}, {529,12701}, {644,16572}, {664,4350}, {728,26690}, {758,30323}, {912,1482}, {956,5250}, {958,5919}, {962,12650}, {999,10914}, {1012,13600}, {1100,5782}, {1106,35281}, {1108,3692}, {1319,3913}, {1331,3915}, {1376,3893}, {1388,4917}, {1442,4460}, {1467,12632}, {1470,8668}, {1697,2975}, {1699,20060}, {1706,5253}, {1836,13463}, {2099,11520}, {3174,12630}, {3218,7991}, {3243,16133}, {3304,3306}, {3340,3873}, {3434,10106}, {3436,12053}, {3445,16610}, {3554,17314}, {3576,3871}, {3612,25439}, {3681,15829}, {3753,7373}, {3813,5252}, {3869,6762}, {3874,25415}, {3878,3951}, {3879,7190}, {3889,11529}, {3891,21147}, {3894,11280}, {3911,12640}, {3984,5289}, {4018,8148}, {4188,13462}, {4190,4315}, {4297,20075}, {4301,5905}, {4308,12541}, {4318,34039}, {4320,17480}, {4342,12527}, {4373,7271}, {4430,7995}, {4512,30337}, {4652,5119}, {4695,11512}, {4860,10107}, {4900,17572}, {5048,12635}, {5080,9614}, {5086,24392}, {5141,5726}, {5176,9581}, {5273,7320}, {5288,12514}, {5303,35445}, {5450,12703}, {5687,24928}, {5748,18220}, {5777,10222}, {5854,12832}, {5882,10884}, {6872,12575}, {7274,32093}, {8256,17728}, {9578,11680}, {9579,34605}, {9845,11220}, {9846,25722}, {9850,17616}, {10247,18908}, {10475,35634}, {10827,24387}, {11009,12559}, {11373,17757}, {113