CYCLOLOGIC

Let ABC be a triangle

Denote

1. Oa, Ob, Oc = the circumcenters of HBC, HCA, HAB, resp.

ABC, OaObOc are cyclologic, since Oa, Ob, Oc are the reflections of O in BC,CA,AB, resp.
Cyclologic center (OaObOc, ABC) = antigonal conjugate of O = X(265)

2. Sa, Sb, Sc = the X(54) of HBC, HCA, HAB, resp.

ABC,SaSbSc are cyclologic

Cyclologic centers?

ETC

X(72679) = X(2)X(3)∩X(146)X(1614)

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

Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 9773.

X(72679) lies on these lines: {2, 3}, {74, 59493}, {146, 1614}, {567, 34798}, {1154, 18442}, {2777, 10274}, {2888, 17702}, {3410, 12278}, {3448, 11440}, {3521, 10610}, {4316, 66593}, {4324, 66610}, {5000, 5001}, {5663, 12254}, {5876, 12383}, {5890, 43838}, {5895, 6800}, {5944, 7728}, {6102, 43818}, {6241, 64102}, {6288, 64180}, {6689, 66751}, {7592, 34796}, {7691, 35240}, {11381, 41482}, {11454, 67903}, {11559, 13418}, {11565, 51522}, {11591, 12121}, {12111, 14683}, {12203, 14676}, {12226, 22948}, {12244, 13491}, {12317, 45731}, {12606, 43846}, {13394, 68058}, {13445, 44829}, {13568, 34545}, {14644, 20191}, {15062, 18400}, {15108, 68018}, {15305, 34785}, {18128, 43806}, {18474, 40242}, {19467, 64025}, {19924, 66731}, {20127, 38898}, {22647, 41673}, {25738, 45736}, {26206, 59411}, {32196, 66739}, {43577, 61134}, {43613, 45286}, {43808, 68424}, {43831, 51033}, {44882, 63069}, {48880, 66736}, {48892, 66730}, {48898, 66750}, {52058, 63548}, {64179, 66737}

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X(72680) = X(2)X(3)∩X(54)X(5448)

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

Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 9773.

X(72680) lies on these lines: {2, 3}, {49, 68424}, {54, 5448}, {76, 54705}, {110, 21659}, {113, 1614}, {125, 11440}, {146, 6241}, {156, 11597}, {195, 43835}, {265, 5876}, {389, 1531}, {394, 22555}, {399, 45731}, {567, 67861}, {1154, 43865}, {1204, 26913}, {1352, 40317}, {1495, 41482}, {1503, 63069}, {1568, 13403}, {1899, 64025}, {1975, 65518}, {1994, 12241}, {2888, 9927}, {3060, 32352}, {3410, 5907}, {3448, 12111}, {3583, 4354}, {3585, 4351}, {3818, 66750}, {5000, 5001}, {5012, 43831}, {5254, 52058}, {5449, 14644}, {5475, 26216}, {5562, 50435}, {5654, 9545}, {5889, 18390}, {5890, 43816}, {5895, 13203}, {5944, 61574}, {6102, 43821}, {6146, 43605}, {6288, 15060}, {6800, 64024}, {7592, 43838}, {7689, 26917}, {7706, 15024}, {7728, 11561}, {9143, 61751}, {9306, 12278}, {9544, 19467}, {9729, 68550}, {9934, 19506}, {10113, 11591}, {10539, 12289}, {11425, 59771}, {11439, 11550}, {11441, 14683}, {11456, 40640}, {11538, 13599}, {11572, 44870}, {11750, 14157}, {12022, 22660}, {12134, 15052}, {12162, 25739}, {12163, 61701}, {12233, 34545}, {12281, 59493}, {12293, 12319}, {12370, 56292}, {12383, 61753}, {12824, 41589}, {12897, 51392}, {13219, 41009}, {13418, 21400}, {13434, 18388}, {13445, 22816}, {13579, 60618}, {13582, 60122}, {13585, 40448}, {14249, 58736}, {14639, 39818}, {14643, 32171}, {14674, 47065}, {14675, 14880}, {15030, 18383}, {15032, 58885}, {15056, 18392}, {15058, 18394}, {15061, 32210}, {15062, 20299}, {15068, 64183}, {15072, 16223}, {15087, 43575}, {15305, 18381}, {15800, 66739}, {17500, 51032}, {17845, 35264}, {17860, 52367}, {18296, 67624}, {18350, 30522}, {18376, 66756}, {18405, 32346}, {18418, 26881}, {18427, 32365}, {18504, 61747}, {20191, 23515}, {22793, 66741}, {26156, 29181}, {26166, 70210}, {26206, 36990}, {26883, 64718}, {29012, 66730}, {30529, 51254}, {31363, 62938}, {34782, 35265}, {34783, 43808}, {34944, 68015}, {35820, 66612}, {35821, 66611}, {36518, 64063}, {36982, 41738}, {37495, 51391}, {37638, 68009}, {41257, 66755}, {41365, 46924}, {41590, 63740}, {41724, 45187}, {43681, 54640}, {44829, 51403}, {46261, 64032}, {48901, 66736}, {51756, 68017}, {54704, 60285}, {54932, 60258}, {59494, 64256}, {60121, 60191}, {63392, 63839}, {63683, 67915}

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