leftri rightri

This is PART 12: Centers X(22001) - X(24000)

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(22001) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22001) lies on these lines: {9, 5307}, {10, 407}, {37, 226}, {40, 17860}, {63, 321}, {71, 6358}, {72, 515}, {92, 573}, {190, 2064}, {329, 21078}, {516, 1824}, {527, 3175}, {758, 2901}, {993, 13733}, {1868, 12572}, {2321, 8896}, {2328, 7009}, {3029, 6044}, {3191, 18446}, {3869, 10454}, {3970, 3995}, {3998, 22003}, {17862, 20367}, {22004, 22019}, {22009, 22033}

X(22002) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22002) lies on these lines: {37, 226}, {63, 22022}, {72, 5882}, {228, 22027}, {321, 20879}, {516, 21807}, {572, 2167}, {894, 18646}, {3218, 3995}, {18662, 21363}, {22013, 22033}

X(22003) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22003) lies on these lines: {10, 2652}, {37, 142}, {72, 2801}, {99, 101}, {100, 6011}, {307, 21069}, {320, 22047}, {321, 20879}, {514, 3882}, {522, 4436}, {527, 4053}, {1018, 1020}, {2295, 14750}, {3159, 8720}, {3998, 22001}, {4033, 4169}, {4791, 18740}, {5773, 21061}, {18698, 21811}

X(22004) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22004) lies on these lines: {10, 6058}, {37, 3452}, {72, 3244}, {321, 20879}, {1999, 3219}, {22001, 22019}

X(22005) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22005) lies on these lines: {3, 37}, {72, 4513}, {101, 1819}, {169, 1824}, {321, 857}, {346, 21078}, {1334, 4456}, {3159, 3950}, {3970, 3995}, {4043, 20926}, {4222, 5011}, {5074, 22011}, {21017, 21029}, {21070, 22022}

X(22006) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22006) lies on these lines: {37, 226}, {321, 1848}, {908, 20336}, {946, 12618}, {5279, 17171}

X(22007) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22007) lies on these lines: {37, 226}, {321, 20884}, {824, 22043}

X(22008) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22008) lies on these lines: {10, 872}, {37, 141}, {69, 21061}, {72, 3717}, {76, 4043}, {213, 17353}, {226, 306}, {312, 22000}, {344, 3294}, {495, 5295}, {1423, 17296}, {2901, 13161}, {3159, 6541}, {3588, 3882}, {3596, 4417}, {3687, 4967}, {3695, 10381}, {3932, 7064}, {3995, 17242}, {4150, 4153}, {15523, 21803}, {16574, 17137}, {20336, 21078}, {20496, 21076}, {21099, 22046}, {22009, 22015}, {22016, 22031}

X(22009) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22009) lies on these lines: {10, 7109}, {37, 744}, {306, 1230}, {321, 4766}, {2205, 17766}, {4109, 21085}, {4150, 4177}, {21093, 22039}, {22001, 22033}, {22008, 22015}

X(22010) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22010) lies on these lines: {37, 3782}, {72, 22791}, {190, 17167}, {226, 3995}, {306, 4043}, {321, 908}, {3159, 12047}, {17781, 21061}

X(22011) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22011) lies on these lines: {2, 22013}, {10, 762}, {37, 39}, {75, 2140}, {226, 4153}, {274, 4568}, {321, 1930}, {335, 16887}, {514, 1909}, {894, 17200}, {1018, 17164}, {1086, 6292}, {1089, 21808}, {1215, 16600}, {2321, 12609}, {3239, 21193}, {3294, 4115}, {3754, 4095}, {3822, 4136}, {3930, 4647}, {3934, 21208}, {3963, 17867}, {3992, 21921}, {3995, 16826}, {4043, 18157}, {4054, 21073}, {4066, 21071}, {4071, 11263}, {4075, 16589}, {4692, 17451}, {5074, 22005}, {5279, 10461}, {5299, 18098}, {16552, 17165}, {16720, 17205}, {21194, 22042}

X(22012) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22012) lies on these lines: {37, 3589}, {72, 3883}, {75, 2140}, {83, 4360}, {86, 4568}, {226, 306}, {313, 21067}, {350, 22013}, {732, 3879}, {2667, 3159}, {3663, 22035}, {3954, 4357}, {3970, 20336}, {3995, 17011}

X(22013) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22013) lies on these lines: {2, 22011}, {37, 714}, {42, 21067}, {306, 1230}, {310, 4568}, {321, 20433}, {350, 22012}, {726, 21814}, {3294, 3757}, {3741, 3954}, {3840, 22035}, {4103, 4651}, {18152, 18833}, {21093, 22026}, {21877, 22036}, {22002, 22033}

X(22014) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22014) lies on these lines: {37, 57}, {72, 4853}, {226, 21801}, {228, 5537}, {321, 908}, {517, 21361}, {2171, 4656}, {3175, 22021}, {3970, 3995}, {4043, 20928}, {4053, 22034}, {5850, 21328}

X(22015) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^4 b + a^3 b^2 - a^2 b^3 + a b^4 - a^4 c + 2 a^3 b c - 2 a b^3 c + b^4 c + a^3 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(22015) lies on these lines: {37, 2886}, {226, 3970}, {312, 21070}, {321, 20431}, {497, 3294}, {22008, 22009}

X(22016) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22016) lies on these lines: {2, 37}, {76, 17242}, {213, 17121}, {313, 3943}, {314, 17261}, {726, 3953}, {740, 3214}, {872, 3896}, {984, 3702}, {1089, 3993}, {1269, 17243}, {2321, 3948}, {3121, 21895}, {3728, 3971}, {3760, 20435}, {3765, 17314}, {3770, 17315}, {3912, 22019}, {3932, 21927}, {3950, 3963}, {3970, 14210}, {3992, 4709}, {3994, 21080}, {6378, 7230}, {6381, 21070}, {17143, 17260}, {17144, 17349}, {17229, 18133}, {17240, 18144}, {17269, 18044}, {20706, 21071}, {21435, 21830}, {22008, 22031}

X(22017) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22017) lies on these lines: {37, 537}, {321, 1930}, {3753, 4169}, {3930, 4714}, {3992, 21067}, {4125, 21101}

X(22018) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22018) lies on these lines: {5, 37}, {29, 101}, {72, 5179}, {312, 21070}, {321, 857}, {469, 22000}, {1737, 2198}, {1826, 21077}, {2478, 3294}, {3159, 21090}, {3970, 22032}, {4043, 21579}, {4150, 4153}

X(22019) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22019) lies on these lines: {10, 7064}, {37, 142}, {72, 4301}, {76, 4043}, {144, 10446}, {226, 3175}, {321, 908}, {344, 2140}, {2321, 4377}, {2486, 21865}, {3294, 18230}, {3674, 3970}, {3912, 22016}, {4133, 21077}, {4924, 21627}, {5074, 22005}, {17197, 17351}, {17353, 17761}, {18698, 21809}, {20683, 21927}, {21065, 21091}, {21069, 21073}, {22001, 22004}

X(22020) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22020) lies on these lines: {2, 10468}, {8, 10478}, {10, 12}, {37, 3452}, {200, 10888}, {228, 6745}, {306, 21069}, {312, 21070}, {321, 908}, {329, 573}, {386, 3191}, {946, 5295}, {956, 19701}, {1764, 3588}, {1999, 17182}, {2064, 4568}, {2092, 4415}, {2901, 21616}, {3159, 17748}, {3294, 18228}, {3421, 5712}, {3596, 4417}, {3912, 22028}, {3998, 22001}, {5815, 19853}, {14973, 15281}, {22031, 22034}

X(22021) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22021) lies on these lines: {1, 6}, {4, 2901}, {10, 2294}, {19, 3811}, {35, 1761}, {42, 7237}, {48, 22836}, {57, 3998}, {65, 3694}, {69, 18726}, {71, 758}, {101, 1474}, {145, 5802}, {199, 228}, {226, 306}, {284, 5279}, {319, 18714}, {329, 3995}, {346, 5746}, {442, 594}, {519, 1953}, {527, 18650}, {579, 2198}, {912, 1765}, {950, 17452}, {965, 3940}, {1018, 21853}, {1400, 15556}, {1500, 10381}, {1751, 3187}, {1766, 18446}, {1824, 2900}, {1826, 21077}, {1848, 22000}, {1880, 4551}, {1897, 8748}, {1901, 3943}, {1959, 3879}, {2092, 3721}, {2178, 11517}, {2260, 3874}, {2345, 3487}, {2893, 6542}, {3125, 21857}, {3158, 3198}, {3159, 3950}, {3175, 22014}, {3219, 4877}, {3419, 17299}, {3586, 4898}, {3670, 4261}, {3684, 16547}, {3686, 17451}, {3726, 17053}, {3822, 21675}, {3870, 4463}, {3875, 19791}, {3912, 20336}, {3958, 4067}, {3962, 4047}, {3987, 21858}, {3991, 21871}, {4007, 5295}, {4016, 4424}, {4018, 21866}, {4029, 21809}, {4037, 22039}, {4043, 20444}, {4069, 20702}, {4086, 22041}, {4158, 10974}, {4659, 7201}, {4851, 18733}, {4876, 15314}, {5257, 21033}, {6356, 18642}, {7146, 17296}, {10445, 22035}, {16548, 18598}, {17315, 18720}, {17362, 17443}, {17377, 18041}, {17388, 17444}, {17757, 21933}, {21039, 22312}, {21068, 21096}, {22031, 22040}

X(22022) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22022) lies on these lines: {37, 5745}, {63, 22002}, {72, 519}, {321, 908}, {1999, 3219}, {4066, 21075}, {4133, 4135}, {4847, 21807}, {21062, 21069}, {21070, 22005}, {21273, 21363}

X(22023) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^9 - a^8 b + a^5 b^4 + a^4 b^5 - a^8 c + b^8 c + a^5 c^4 - b^5 c^4 + a^4 c^5 - b^4 c^5 + b c^8) : :

X(22023) lies on these lines: {37, 16582}, {321, 2172}, {3995, 17492}, {21079, 22000}

X(22024) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22024) lies on these lines: {1, 3159}, {10, 20966}, {37, 714}, {38, 321}, {537, 3175}, {596, 19863}, {740, 22275}, {758, 2901}, {835, 2206}, {4003, 6682}, {4362, 5282}, {10453, 20068}, {20671, 21877}, {21070, 22026}, {21093, 22000}

X(22025) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22025) lies on these lines: {37, 6292}, {321, 17873}, {3159, 6541}, {3912, 3995}, {4109, 4129}

X(22026) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22026) lies on these lines: {37, 744}, {321, 20898}, {672, 3741}, {3840, 22032}, {3912, 3995}, {17766, 18098}, {21070, 22024}, {21093, 22013}

X(22027) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22027) lies on these lines: {10, 201}, {37, 800}, {72, 519}, {228, 22002}, {321, 4712}, {516, 1824}, {522, 4640}, {756, 17874}, {1867, 19925}, {1897, 2328}, {3870, 3995}, {3930, 3950}, {4075, 21075}, {4362, 5282}, {5223, 17156}, {7211, 21804}, {21807, 22000}

X(22028) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22028) lies on these lines: {10, 3728}, {37, 2998}, {72, 19222}, {76, 321}, {194, 17149}, {213, 668}, {306, 3948}, {313, 21024}, {1575, 18148}, {3178, 20710}, {3264, 20255}, {3721, 21435}, {3912, 22020}, {4033, 20691}, {4043, 20943}, {6374, 18837}, {6381, 21070}, {9229, 9239}, {16589, 21827}, {20892, 21240}, {21257, 22189}

X(22029) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22029) lies on these lines: {37, 537}, {190, 18645}, {321, 3452}, {2321, 4103}, {3125, 5257}, {3971, 4029}, {3992, 21801}, {4035, 21062}, {21070, 22030}

X(22030) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22030) lies on these lines: {37, 519}, {321, 3262}, {2321, 21088}, {3943, 3971}, {4042, 16672}, {21070, 22029}

X(22031) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22031) lies on these lines: {10, 2486}, {37, 142}, {72, 21627}, {190, 17197}, {226, 3995}, {313, 2321}, {321, 3452}, {918, 22035}, {946, 3159}, {3175, 22000}, {3191, 12437}, {3294, 20257}, {3912, 18150}, {4010, 21093}, {4069, 13576}, {4422, 17761}, {4728, 22032}, {7064, 21927}, {21090, 21091}, {22008, 22016}, {22020, 22034}, {22021, 22040}

X(22032) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22032) lies on these lines: {11, 37}, {306, 20496}, {321, 20431}, {3840, 22026}, {3970, 22018}, {4043, 21580}, {4054, 21073}, {4120, 21090}, {4728, 22031}

X(22033) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

Barycentrics    (a - b) (a - c) (b + c) (2 a^3 + 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(22033) lies on these lines: {37, 16592}, {321, 20903}, {1023, 4115}, {2796, 21833}, {4024, 4427}, {22001, 22009}, {22002, 22013}

X(22034) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22034) lies on these lines: {1, 19747}, {2, 37}, {38, 4519}, {44, 19750}, {72, 3586}, {210, 3994}, {226, 3943}, {329, 17299}, {594, 4656}, {726, 21342}, {740, 3967}, {1089, 4646}, {1100, 19739}, {1279, 4387}, {1999, 17351}, {2321, 4415}, {2901, 3244}, {3159, 3626}, {3187, 16669}, {3198, 6154}, {3696, 3971}, {3701, 21896}, {3723, 19746}, {3751, 4942}, {3782, 17231}, {3914, 6057}, {3931, 4066}, {3932, 21949}, {3948, 21868}, {3950, 17056}, {4035, 4052}, {4044, 20691}, {4053, 22014}, {4096, 4709}, {4431, 5743}, {4654, 17311}, {5271, 16814}, {5905, 17374}, {7230, 16583}, {7308, 17119}, {11679, 17262}, {16676, 19744}, {17022, 17118}, {22020, 22031}

X(22035) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22035) lies on these lines: {10, 762}, {37, 537}, {321, 1111}, {335, 4568}, {918, 22031}, {3159, 3970}, {3663, 22012}, {3840, 22013}, {4013, 21044}, {4075, 21808}, {4120, 21090}, {4169, 21888}, {4958, 22045}, {9055, 17761}, {10445, 22021}, {21070, 22036}

X(22036) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22036) lies on these lines: {37, 39}, {72, 14839}, {76, 321}, {115, 4136}, {187, 8669}, {194, 3995}, {313, 21412}, {519, 14537}, {538, 3175}, {730, 2901}, {1089, 3721}, {1500, 21101}, {3125, 3701}, {3700, 3906}, {3727, 4692}, {3734, 3905}, {3735, 4385}, {3967, 16583}, {3970, 4037}, {3971, 16589}, {3992, 21951}, {3994, 21808}, {4066, 21024}, {4103, 21868}, {4109, 21093}, {4125, 21025}, {4135, 7230}, {4424, 21021}, {4721, 17489}, {4920, 7794}, {12699, 17299}, {15810, 17132}, {17165, 20963}, {20691, 21067}, {21070, 22035}, {21877, 22013}

X(22037) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22037) lies on these lines: {10, 690}, {37, 3960}, {72, 3887}, {74, 2372}, {99, 101}, {321, 3762}, {514, 4024}, {525, 4129}, {918, 22031}, {2785, 13181}, {3566, 4807}, {3667, 4064}, {3906, 4806}, {3947, 18006}, {3995, 21222}, {4049, 4080}, {4066, 18003}

X(22038) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22038) lies on these lines: {37, 4892}, {306, 1230}, {321, 20904}, {3006, 4115}, {3261, 3835}

X(22039) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22039) lies on these lines: {37, 714}, {76, 321}, {716, 3175}, {718, 2901}, {726, 21877}, {3701, 22171}, {3948, 22200}, {3995, 17486}, {4037, 22021}, {4135, 21070}, {21093, 22009}

X(22040) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22040) lies on these lines: {2, 37}, {72, 5809}, {726, 21346}, {1089, 4356}, {1441, 3950}, {1446, 21096}, {2901, 6765}, {3674, 3970}, {3701, 3755}, {3702, 7174}, {3896, 4878}, {3932, 21955}, {3971, 21039}, {4098, 18698}, {10889, 21061}, {22021, 22031}

X(22041) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22041) lies on these lines: {37, 4529}, {321, 4171}, {3239, 4064}, {3261, 3835}, {4024, 20294}, {4086, 22021}, {8045, 22044}

X(22042) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22042) lies on these lines: {10, 21960}, {37, 522}, {321, 20907}, {514, 4079}, {657, 3294}, {1577, 4171}, {2321, 4036}, {3239, 4024}, {3261, 4043}, {3686, 8702}, {3700, 7180}, {3709, 4151}, {3950, 4140}, {4791, 21070}, {8714, 21348}, {17233, 18158}, {21194, 22011}

X(22043) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22043) lies on these lines: {10, 4155}, {37, 812}, {321, 4728}, {335, 2786}, {514, 4079}, {523, 4129}, {804, 3993}, {824, 22007}, {918, 22031}, {1577, 21834}, {3835, 4024}, {3995, 21297}, {4033, 4103}, {4043, 20950}

X(22044) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22044) lies on these lines: {37, 523}, {321, 4374}, {514, 4079}, {522, 649}, {661, 4815}, {784, 21348}, {798, 4151}, {802, 4500}, {1577, 21099}, {3700, 8672}, {4043, 7199}, {4705, 21960}, {5214, 21061}, {6367, 17990}, {8045, 22041}

X(22045) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22045) lies on these lines: {37, 6377}, {42, 3952}, {244, 321}, {537, 3175}, {726, 17154}, {740, 22313}, {2802, 2901}, {3159, 3244}, {4010, 21093}, {4958, 22035}

X(22046) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22046) lies on these lines: {321, 20910}, {824, 22007}, {3261, 3835}, {21099, 22008}

X(22047) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22046) lies on these lines: {37, 524}, {226, 306}, {320, 22003}, {514, 4079}, {3912, 4053}, {4043, 20956}, {4062, 21829}, {4115, 17264}, {16704, 17019}

X(22048) =  (A,B,C,X(2); A',B',C',X(37)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(10)

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

X(22048) lies on these lines: {1, 3159}, {10, 20703}, {37, 3589}, {321, 1930}, {538, 3175}, {3948, 21067}, {4044, 21101}, {4568, 16826}

X(22049) =  (name pending)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (35 a^12-82 a^10 b^2-3 a^8 b^4+132 a^6 b^6-83 a^4 b^8-18 a^2 b^10+19 b^12-82 a^10 c^2+262 a^8 b^2 c^2-196 a^6 b^4 c^2-100 a^4 b^6 c^2+150 a^2 b^8 c^2-34 b^10 c^2-3 a^8 c^4-196 a^6 b^2 c^4+366 a^4 b^4 c^4-132 a^2 b^6 c^4-35 b^8 c^4+132 a^6 c^6-100 a^4 b^2 c^6-132 a^2 b^4 c^6+100 b^6 c^6-83 a^4 c^8+150 a^2 b^2 c^8-35 b^4 c^8-18 a^2 c^10-34 b^2 c^10+19 c^12) : :

See Kadir Altintas, Antreas Hatzipolakis, and Peter Moses, Hyacinthos 28137.

X(22049) lies on this line: {2,3}

X(22050) =  (name pending)

Barycentrics    90 a^16-689 a^14 b^2+2139 a^12 b^4-3497 a^10 b^6+3215 a^8 b^8-1595 a^6 b^10+329 a^4 b^12+21 a^2 b^14-13 b^16-689 a^14 c^2+3214 a^12 b^2 c^2-4761 a^10 b^4 c^2+1684 a^8 b^6 c^2+1585 a^6 b^8 c^2-990 a^4 b^10 c^2-167 a^2 b^12 c^2+124 b^14 c^2+2139 a^12 c^4-4761 a^10 b^2 c^4+1212 a^8 b^4 c^4+685 a^6 b^6 c^4+834 a^4 b^8 c^4+375 a^2 b^10 c^4-484 b^12 c^4-3497 a^10 c^6+1684 a^8 b^2 c^6+685 a^6 b^4 c^6-346 a^4 b^6 c^6-229 a^2 b^8 c^6+1028 b^10 c^6+3215 a^8 c^8+1585 a^6 b^2 c^8+834 a^4 b^4 c^8-229 a^2 b^6 c^8-1310 b^8 c^8-1595 a^6 c^10-990 a^4 b^2 c^10+375 a^2 b^4 c^10+1028 b^6 c^10+329 a^4 c^12-167 a^2 b^2 c^12-484 b^4 c^12+21 a^2 c^14+124 b^2 c^14-13 c^16 : :

See Kadir Altintas, Antreas Hatzipolakis, and Peter Moses, Hyacinthos 28137.

X(22050) lies on this line: {2,3}

X(22051) =  X(2)X(12316)∩X(5)X(195)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28138.

X(22051) lies on these lines: {2, 12316}, {5, 195}, {30, 54}, {51, 13368}, {110, 13163}, {113, 137}, {140, 389}, {143, 10096}, {252, 10289}, {468, 6242}, {539, 5066}, {547, 1209}, {548, 10610}, {549, 12307}, {568, 10125}, {1157, 10126}, {1173, 14643}, {1263, 20030}, {1594, 2914}, {1656, 12325}, {3530, 7691}, {3542, 12175}, {3564, 19150}, {3580, 3628}, {3627, 12254}, {3850, 6288}, {3853, 5893}, {3881, 5901}, {4994, 15557}, {5056, 13432}, {5898, 18369}, {5965, 12812}, {6152, 21841}, {6153, 10095}, {6676, 12606}, {6696, 10628}, {7356, 15325}, {7583, 12971}, {7584, 12965}, {10066, 15172}, {10203, 13353}, {10224, 18912}, {10619, 20585}, {10677, 11543}, {10678, 11542}, {11805, 15089}, {11808, 13451}, {12161, 18356}, {12363, 16197}, {14216, 17824}, {18946, 19347}

X(22051) = X(22051) = midpoint of X(i) and X(j) for these {i,j}: {2914, 11804}, {10113, 14049}, {15801, 21230}
X(22051) = reflection of X(i) in X(j) for these {i,j}: {140, 8254}, {546, 3574}, {548, 10610}, {6153, 10095}, {6288, 3850}, {7691, 3530}, {8254, 12242}, {10619, 20585}, {21230, 3628}
X(22051) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {143, 15806, 10096}, {1656, 12325, 21357}, {11803, 12242, 140}

X(22052) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

Let A'B'C' be the cevian triangle of X(3). X(22052) is the perspector, wrt A'B'C', of the circumconic of A'B'C' centered at X(140). (Randy Hutson, November 30, 2018)

X(22052) lies on these lines: {2, 10985}, {3, 6}, {53, 550}, {71, 22055}, {95, 401}, {97, 323}, {140, 233}, {230, 10691}, {232, 6636}, {393, 3522}, {418, 1495}, {631, 10986}, {940, 21503}, {1040, 10987}, {1196, 9609}, {1216, 14533}, {1249, 21735}, {1368, 3054}, {1971, 3819}, {2165, 7748}, {3055, 6676}, {3087, 3523}, {3131, 10642}, {3132, 10641}, {3289, 22352}, {3357, 17849}, {3481, 21354}, {3620, 6389}, {3631, 15526}, {6509, 15066}, {6640, 11614}, {6749, 15712}, {7484, 10314}, {7485, 10311}, {7492, 15355}, {7502, 14576}, {7749, 9722}, {8703, 18487}, {8908, 10133}, {9220, 18564}, {10313, 15246}, {10319, 10988}, {17277, 22359}, {18424, 18531}, {22062, 22085}

X(22052) = isogonal conjugate of polar conjugate of X(140)
X(22052) = complement of X(32002)
X(22052) = complement of polar conjugate of X(2963)
X(22052) = isotomic conjugate of polar conjugate of X(13366)
X(22052) = crosspoint of X(32585) and X(32586)
X(22052) = X(92)-isoconjugate of X(1173)
X(22052) = crosssum of X(472) and X(473)
X(22052) = inverse-in-Brocard-circle of X(10979)
X(22052) = Schoute-circle-inverse of X(389)
X(22052) = {X(15),X(16)}-harmonic conjugate of X(389)

X(22053) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22053) lies on these lines: {1, 376}, {2, 2635}, {3, 73}, {20, 2654}, {33, 5732}, {34, 8726}, {35, 1066}, {36, 1064}, {42, 1155}, {48, 1473}, {55, 1407}, {56, 4300}, {57, 955}, {63, 1818}, {71, 3917}, {77, 1040}, {109, 15931}, {142, 17194}, {184, 20780}, {201, 1071}, {216, 22410}, {221, 8273}, {223, 10857}, {228, 3937}, {241, 10391}, {269, 10383}, {278, 21151}, {354, 1418}, {497, 1742}, {577, 22054}, {581, 15803}, {601, 7742}, {631, 1745}, {971, 7069}, {1038, 10884}, {1042, 2646}, {1044, 3485}, {1193, 4252}, {1214, 7004}, {1333, 17187}, {1393, 9940}, {1401, 2223}, {1409, 22400}, {1427, 17603}, {1457, 3576}, {1465, 11227}, {1790, 4575}, {1836, 3000}, {1935, 6986}, {1936, 7411}, {2003, 13329}, {2183, 4191}, {2197, 22418}, {2267, 7484}, {3057, 4322}, {3075, 3651}, {3190, 3928}, {3475, 4334}, {3601, 4306}, {3682, 3916}, {3920, 18450}, {3942, 17441}, {4551, 10164}, {5122, 5396}, {5165, 20973}, {5287, 8544}, {7987, 10571}, {9371, 10178}, {11020, 17092}, {18591, 22064}, {20755, 20783}, {22066, 22449}, {22069, 22084}, {22070, 22088}, {22341, 22347}

X(22053) = isogonal conjugate of polar conjugate of X(142)
X(22053) = isotomic conjugate of polar conjugate of X(1475)
X(22053) = crosssum of X(4) and X(33)
X(22053) = crosspoint of X(3) and X(77)
X(22053) = {X(3),X(73)}-harmonic conjugate of X(22072)
X(22053) = X(19)-isoconjugate of X(32008)
X(22053) = X(92)-isoconjugate of X(1174)

X(22054) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22054) lies on these lines: {2, 20289}, {3, 48}, {6, 5204}, {9, 5267}, {19, 3576}, {36, 284}, {37, 1055}, {42, 4497}, {55, 37519}, {172, 5110}, {187, 2300}, {198, 2267}, {199, 14547}, {228, 20775}, {501, 2150}, {515, 21011}, {517, 17438}, {572, 2183}, {573, 2317}, {574, 2273}, {577, 22053}, {579, 7280}, {584, 1475}, {604, 1470}, {609, 5105}, {610, 7987}, {672, 2174}, {902, 16685}, {1030, 2269}, {1100, 17454}, {1125, 1839}, {1193, 1333}, {1201, 5301}, {1385, 1953}, {1400, 2278}, {1404, 4271}, {1436, 8273}, {1444, 20769}, {1449, 4262}, {1457, 1950}, {1630, 2272}, {1631, 2293}, {1761, 4511}, {1790, 4288}, {1826, 4297}, {1901, 15326}, {2173, 13624}, {2178, 2268}, {2193, 4303}, {2197, 22059}, {2245, 21748}, {2256, 5217}, {2287, 5303}, {2294, 2646}, {2302, 11012}, {2347, 4268}, {3916, 3958}, {3949, 5440}, {4299, 5747}, {4466, 18650}, {4471, 20978}, {4855, 5227}, {4860, 16884}, {5011, 16553}, {6511, 10607}, {6684, 21012}, {7117, 18591}, {11573, 22162}, {14597, 22056}, {15586, 17443}, {17647, 21675}, {20729, 22077}, {20750, 22096}, {20752, 22352}, {20756, 20784}, {20757, 22062}, {22073, 22447}, {22118, 22350}

X(22054) = isogonal conjugate of isotomic conjugate of X(4001)
X(22054) = isogonal conjugate of polar conjugate of X(1125)
X(22054) = isotomic conjugate of polar conjugate of X(2308)
X(22054) = X(19)-isoconjugate of X(1268)
X(22054) = X(92)-isoconjugate of X(1126)

X(22055) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22055) lies on these lines: {3, 906}, {71, 22052}, {187, 672}, {577, 22071}, {647, 22375}, {1951, 2077}, {1983, 13006}, {3284, 22059}, {5546, 17100}

X(22055) = isogonal conjugate of polar conjugate of X(3035)
X(22055) = isotomic conjugate of polar conjugate of X(20958)

X(22056) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22056) lies on these lines: on lines {3, 2197}, {71, 22052}, {187, 2269}, {577, 22070}, {1950, 11012}, {2193, 7117}, {3284, 22058}, {14597, 22054}, {22079, 22378}

X(22056) = isogonal conjugate of polar conjugate of X(4999)
X(22056) = isotomic conjugate of polar conjugate of X(20959)
X(22056) = X(92)-isoconjugate of X(18772)

X(22057) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22057) lies on these lines: {1, 6349}, {3, 31}, {42, 1214}, {43, 6350}, {71, 22069}, {73, 228}, {326, 4176}, {426, 22421}, {497, 614}, {577, 22053}, {1066, 20764}, {1458, 7011}, {1473, 7124}, {2193, 17187}, {3120, 18588}, {3682, 3998}, {3720, 17073}, {3917, 22074}, {21530, 21935}, {22060, 22070}, {22064, 22400}, {22399, 22418}, {22404, 22434}

X(22057) = isogonal conjugate of polar conjugate of X(18589)
X(22057) = isotomic conjugate of polar conjugate of X(23620)

X(22058) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22058) lies on these lines: {3, 22122}, {71, 216}, {570, 672}, {1100, 21798}, {2197, 22356}, {2260, 3002}, {2269, 3003}, {3284, 22056}, {4466, 18606}, {7117, 18591}, {20819, 22449}, {20821, 22062}, {20975, 22389}, {22065, 22073}

X(22058) = isogonal conjugate of polar conjugate of X(25639)
X(22058) = isotomic conjugate of polar conjugate of X(20961)
X(22058) = {X(71),X(216)}-harmonic conjugate of X(22059)

X(22059) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22059) lies on these lines: {3, 22123}, {71, 216}, {570, 2269}, {672, 3003}, {2183, 13006}, {2197, 22054}, {2252, 22350}, {3284, 22055}, {3917, 22429}, {7117, 22356}, {8607, 21801}, {20729, 22095}, {20731, 22084}, {20775, 22169}, {20777, 20975}, {20821, 22087}, {22410, 22435}, {22414, 22428}

X(22059) = isogonal conjugate of polar conjugate of X(3814)
X(22059) = isotomic conjugate of polar conjugate of X(20962)
X(22059) = {X(71),X(216)}-harmonic conjugate of X(22058)

X(22060) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22060) lies on these lines: {3, 63}, {9, 4191}, {36, 846}, {38, 2223}, {56, 968}, {57, 1011}, {71, 3917}, {199, 3220}, {295, 1796}, {354, 8053}, {527, 21319}, {614, 20992}, {851, 5745}, {896, 20967}, {942, 17524}, {993, 3980}, {1040, 18606}, {1402, 4414}, {1444, 22389}, {1790, 7193}, {1818, 3690}, {2300, 17187}, {3218, 4184}, {3219, 4210}, {3286, 3666}, {3305, 16059}, {3306, 16058}, {3677, 16688}, {3683, 20470}, {3706, 4436}, {3928, 19346}, {3937, 20730}, {3941, 17599}, {4303, 22076}, {4640, 16678}, {4641, 5132}, {5122, 16374}, {5249, 8731}, {5285, 16064}, {5303, 11688}, {5437, 16373}, {10436, 16343}, {16574, 19339}, {17194, 20367}, {18210, 18607}, {18591, 22420}, {20731, 22061}, {20735, 20756}, {20736, 22409}, {20780, 22352}, {20785, 22062}, {20821, 22077}, {22057, 22070}, {22074, 22400}, {22084, 22405}, {22128, 22139}

X(22060) = isogonal conjugate of polar conjugate of X(3739)
X(22060) = isotomic conjugate of polar conjugate of X(20963)
X(22060) = {X(3),X(63)}-harmonic conjugate of X(228)
X(22060) = X(19)-isoconjugate of X(32009)

X(22061) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22061) lies on these lines: {3, 295}, {48, 78}, {71, 73}, {72, 2200}, {101, 3678}, {172, 1691}, {228, 22364}, {419, 1215}, {756, 9310}, {813, 2698}, {1237, 14382}, {2295, 18905}, {2304, 3811}, {3690, 15377}, {4019, 12215}, {4303, 20729}, {9016, 16689}, {20731, 22060}, {20752, 22065}, {20785, 22345}, {22069, 22422}, {22342, 22375}, {22350, 22447}, {22367, 22373}

X(22061) = isogonal conjugate of polar conjugate of X(1215)
X(22061) = isotomic conjugate of polar conjugate of X(20964)
X(22061) = {X(71),X(73)}-harmonic conjugate of X(20727)
X(22061) = X(19)-isoconjugate of X(32010)
X(22061) = X(92)-isoconjugate of X(1178)

X(22062) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22062) lies on these lines: {3, 69}, {6, 11175}, {22, 16990}, {71, 20730}, {141, 237}, {160, 599}, {183, 7467}, {216, 3289}, {264, 22712}, {418, 6389}, {1078, 9230}, {1232, 2782}, {1634, 3631}, {1843, 5188}, {3231, 8265}, {3589, 5201}, {3619, 11328}, {7484, 7736}, {7485, 7774}, {7779, 15246}, {9407, 19121}, {9917, 16043}, {10790, 16045}, {11574, 20975}, {14575, 19126}, {20731, 22412}, {20757, 22054}, {20785, 22060}, {20821, 22058}, {20823, 22065}, {22052, 22085}, {22138, 22151}

X(22062) = isogonal conjugate of polar conjugate of X(3934)
X(22062) = isotomic conjugate of polar conjugate of X(20965)

X(22063) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22063) lies on these lines: {1, 281}, {3, 22124}, {6, 41}, {19, 1457}, {42, 21860}, {71, 216}, {102, 112}, {204, 3192}, {219, 22350}, {221, 1436}, {393, 2654}, {577, 22053}, {610, 10571}, {614, 3554}, {800, 2300}, {820, 836}, {995, 2257}, {1033, 21148}, {1108, 1201}, {1386, 8766}, {1409, 7117}, {1953, 14571}, {2272, 21767}, {2289, 22131}, {2293, 2638}, {2635, 3087}, {3284, 22357}, {3553, 7221}, {3666, 6508}, {4303, 15905}, {5105, 14482}, {5158, 22356}, {8608, 16685}, {14597, 22088}, {15851, 20818}

X(22063) = isogonal conjugate of polar conjugate of X(946)
X(22063) = X(92)-isoconjugate of X(947)

X(22064) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22064) lies on these lines: {3, 22125}, {71, 20728}, {216, 20731}, {3917, 22069}, {7004, 18589}, {18591, 22053}, {20727, 20819}, {20734, 20826}, {20821, 22413}, {22057, 22400}, {22065, 22401}, {22070, 22440}

X(22064) = isogonal conjugate of polar conjugate of X(17046)
X(22064) = isotomic conjugate of polar conjugate of X(23636)

X(22065) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22065) lies on these lines: {3, 48}, {19, 10476}, {72, 4020}, {172, 672}, {220, 20471}, {255, 20812}, {392, 2179}, {960, 1755}, {1125, 14964}, {1610, 2272}, {1613, 2275}, {1791, 2196}, {1812, 7116}, {2260, 2303}, {2269, 18755}, {3688, 18758}, {3690, 20777}, {3730, 5267}, {3917, 20727}, {4426, 20460}, {6626, 17209}, {7117, 20750}, {14547, 16372}, {16604, 20459}, {20735, 20827}, {20752, 22061}, {20757, 22409}, {20823, 22062}, {22058, 22073}, {22064, 22401}

X(22065) = isogonal conjugate of polar conjugate of X(3741)
X(22065) = isotomic conjugate of polar conjugate of X(2309)

X(22066) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22066) lies on these lines: {3, 48}, {78, 20785}, {1575, 20460}, {2179, 17614}, {2318, 20777}, {3056, 20996}, {3917, 20755}, {4020, 5440}, {7117, 20727}, {20729, 22070}, {20731, 22435}, {20732, 20824}, {20750, 22072}, {22053, 22449}, {22096, 22381}

X(22066) = isogonal conjugate of polar conjugate of X(3840)
X(22066) = isotomic conjugate of polar conjugate of X(22343)
X(22066) = X(19)-isoconjugate of X(32011)

X(22067) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22067) lies on these lines: {3, 22083}, {71, 3917}, {228, 3784}, {1473, 20818}, {1818, 3937}, {3292, 20780}, {17616, 21807}, {20731, 20757}, {20733, 22094}, {22082, 22350}, {22084, 22406}

X(22067) = isogonal conjugate of polar conjugate of X(3834)
X(22067) = isotomic conjugate of polar conjugate of isogonal conjugate of X(32012)
X(22067) = X(19)-isoconjugate of X(32012)

X(22068) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22068) lies on these lines: {3, 1331}, {71, 3917}, {3784, 22080}

X(22068) = isogonal conjugate of polar conjugate of X(34824)
X(22068) = isotomic conjugate of polar conjugate of isogonal conjugate of X(32013)
X(22068) = X(19)-isoconjugate of X(32013)

X(22069) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22069) lies on these lines: {3, 22130}, {31, 1779}, {43, 63}, {71, 22057}, {75, 18022}, {228, 22094}, {307, 3778}, {656, 21912}, {1737, 21935}, {3917, 22064}, {20727, 22404}, {20823, 22411}, {22053, 22084}, {22061, 22422}

X(22069) = isogonal conjugate of polar conjugate of X(20305)
X(22069) = isotomic conjugate of polar conjugate of X(23619)
X(22069) = crosssum of X(4) and X(31)
X(22069) = crosspoint of X(3) and X(75)

X(22070) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22070) lies on these lines: {1, 3002}, {3, 906}, {39, 213}, {48, 18591}, {71, 216}, {73, 20752}, {219, 2197}, {577, 22056}, {607, 3428}, {614, 1194}, {800, 2269}, {1107, 9284}, {1951, 11012}, {2193, 22122}, {2886, 16699}, {2968, 21915}, {3057, 8608}, {3730, 13006}, {3917, 20727}, {6467, 22389}, {8735, 15908}, {16588, 17451}, {18210, 18671}, {18589, 18606}, {20729, 22066}, {20734, 20755}, {20822, 22427}, {22053, 22088}, {22057, 22060}, {22064, 22440}, {22416, 22432}

X(22070) = isogonal conjugate of polar conjugate of X(2886)
X(22070) = isotomic conjugate of polar conjugate of X(21746)
X(22070) = {X(71),X(216)}-harmonic conjugate of X(22071)
X(22070) = X(92)-isoconjugate of X(3449)

X(22071) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22071) lies on these lines: {3, 2197}, {37, 5432}, {39, 2269}, {55, 4261}, {71, 216}, {212, 8606}, {219, 4587}, {573, 13006}, {577, 22055}, {608, 10310}, {672, 800}, {1409, 22350}, {1950, 2077}, {2092, 2268}, {2252, 3990}, {3270, 20753}, {3917, 22064}, {3949, 7004}, {6467, 20777}, {8607, 21871}, {10950, 21858}, {11998, 17362}, {14749, 17398}, {17053, 17452}, {20729, 20732}, {20730, 22413}, {20731, 22440}, {20819, 20820}

X(22071) = isogonal conjugate of polar conjugate of X(1329)
X(22071) = isotomic conjugate of polar conjugate of X(23638)
X(22071) = {X(71),X(216)}-harmonic conjugate of X(22070)
X(22071) = X(92)-isoconjugate of X(3450)

X(22072) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22072) lies on these lines: {1, 631}, {2, 2654}, {3, 73}, {20, 2635}, {33, 936}, {34, 6282}, {35, 1064}, {36, 1066}, {40, 1457}, {42, 2646}, {43, 3486}, {55, 1191}, {56, 7074}, {71, 216}, {72, 7004}, {78, 345}, {165, 10571}, {201, 17102}, {228, 22347}, {376, 1745}, {386, 1453}, {404, 1936}, {497, 978}, {517, 1393}, {602, 8069}, {899, 1837}, {950, 3216}, {960, 9371}, {995, 1697}, {1038, 20277}, {1042, 1155}, {1149, 2098}, {1201, 3057}, {1364, 22082}, {1458, 5204}, {1468, 22768}, {1470, 1496}, {1802, 7124}, {1818, 4855}, {1935, 6909}, {2269, 4261}, {3074, 6906}, {3075, 6940}, {3100, 17280}, {3214, 10950}, {3682, 5440}, {3937, 22376}, {4297, 4551}, {4300, 5217}, {4324, 6127}, {5044, 7069}, {5399, 13624}, {5438, 7070}, {9581, 17749}, {11376, 17278}, {20727, 20728}, {20750, 22066}, {20752, 22088}, {20781, 20786}, {22076, 22418}, {22079, 22369}, {22341, 22346}

X(22072) = isogonal conjugate of polar conjugate of X(3452)
X(22072) = isotomic conjugate of polar conjugate of X(2347)
X(22072) = crosssum of X(4) and X(34)
X(22072) = crosspoint of X(3) and X(78)
X(22072) = {X(3),X(73)}-harmonic conjugate of X(22053)
X(22072) = X(92)-isoconjugate of X(3451)

X(22073) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22073) lies on these lines: {3, 22133}, {9, 14963}, {71, 73}, {216, 3289}, {442, 1953}, {604, 2245}, {1474, 3430}, {1901, 21801}, {2092, 20228}, {2260, 10974}, {2294, 17056}, {3142, 21011}, {3269, 22428}, {20729, 22080}, {20730, 22084}, {20759, 20830}, {20820, 22433}, {22054, 22447}, {22058, 22065}

X(22073) = isogonal conjugate of polar conjugate of X(3454)
X(22073) = isotomic conjugate of polar conjugate of X(20966)
X(22073) = X(92)-isoconjugate of X(3453)

X(22074) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22074) lies on these lines: {3, 1409}, {6, 1732}, {48, 184}, {71, 216}, {78, 219}, {213, 2268}, {612, 2256}, {614, 21334}, {869, 1253}, {1193, 1682}, {1333, 2361}, {1880, 14110}, {2197, 22350}, {2286, 7078}, {3057, 16685}, {3100, 3786}, {3230, 17452}, {3917, 22057}, {3958, 7004}, {14597, 22054}, {17440, 20963}, {20732, 22099}, {22060, 22400}

X(22074) = isogonal conjugate of polar conjugate of X(960)
X(22074) = isotomic conjugate of polar conjugate of X(20967)
X(22074) = X(19)-isoconjugate of X(31643)
X(22074) = X(92)-isoconjugate of X(961)

X(22075) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22075) lies on these lines: {3, 22135}, {6, 21213}, {22, 11610}, {32, 184}, {154, 3162}, {206, 17409}, {216, 8779}, {394, 18876}, {418, 22391}, {1691, 1899}, {2351, 14600}, {14597, 22362}

X(22075) = isogonal conjugate of polar conjugate of X(206)
X(22075) = isotomic conjugate of polar conjugate of X(20968)

X(22076) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22076) lies on these lines: {1, 10974}, {2, 970}, {3, 49}, {10, 3142}, {12, 22299}, {21, 511}, {40, 851}, {42, 10822}, {51, 405}, {56, 2245}, {65, 17056}, {71, 73}, {72, 306}, {78, 3781}, {125, 21530}, {181, 10474}, {199, 2360}, {201, 18592}, {228, 3682}, {373, 11108}, {389, 1006}, {392, 4205}, {404, 3819}, {407, 14110}, {408, 2972}, {411, 5907}, {429, 960}, {442, 517}, {474, 5650}, {500, 17524}, {573, 13738}, {581, 1011}, {758, 3178}, {857, 3661}, {946, 3136}, {958, 16980}, {976, 3688}, {1154, 5428}, {1193, 1682}, {1201, 20966}, {1213, 2262}, {1214, 1425}, {1332, 1791}, {1364, 22361}, {1495, 2915}, {1818, 22369}, {1834, 3057}, {1901, 21871}, {1993, 13323}, {2082, 2238}, {2280, 20970}, {2328, 3145}, {2392, 3647}, {2476, 15488}, {2979, 4189}, {3060, 16865}, {3191, 21319}, {3269, 20728}, {3454, 3878}, {3649, 20718}, {3651, 6000}, {3730, 14963}, {3784, 4652}, {3869, 3936}, {3877, 5051}, {3916, 3937}, {3925, 22300}, {3948, 19582}, {3954, 21799}, {4188, 7998}, {4199, 5250}, {4259, 19765}, {4260, 19767}, {4303, 22060}, {5047, 5943}, {5164, 16589}, {5230, 10480}, {5320, 16471}, {5396, 16287}, {5446, 7489}, {5640, 16859}, {5754, 16286}, {5972, 12826}, {6044, 6737}, {6101, 7508}, {6675, 18180}, {6688, 17536}, {6875, 11412}, {6876, 11459}, {6905, 11793}, {6906, 15644}, {6909, 13348}, {6912, 13598}, {6914, 10625}, {6920, 10110}, {6942, 7999}, {6985, 15030}, {6986, 9729}, {7078, 7085}, {7117, 20750}, {7580, 11381}, {8582, 10440}, {9306, 11337}, {10219, 17546}, {11451, 17570}, {14915, 16117}, {15082, 17535}, {16418, 21969}, {16858, 21849}, {20738, 20787}, {20821, 22350}, {22072, 22418}, {22082, 22094}, {22097, 22345}

X(22076) = isogonal conjugate of polar conjugate of X(1211)
X(22076) = isotomic conjugate of polar conjugate of X(2092)
X(22076) = crosssum of X(4) and X(28)
X(22076) = crosspoint of X(3) and X(72)
X(22076) = X(19)-isoconjugate of X(14534)
X(22076) = X(92)-isoconjugate of X(1169)

X(22077) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22077) lies on these lines: {3, 22137}, {71, 22348}, {228, 20727}, {2525, 8611}, {20729, 22054}, {20821, 22060}, {22094, 22409}

X(22077) = isogonal conjugate of polar conjugate of X(21249)
X(22077) = isotomic conjugate of polar conjugate of X(20969)

X(22078) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22078) lies on these lines: {3, 1176}, {1818, 22345}, {3313, 14096}, {3618, 9821}, {3917, 20775}, {11574, 20975}, {20729, 22054}

X(22078) = isogonal conjugate of polar conjugate of X(6292)
X(22078) = isotomic conjugate of polar conjugate of X(11205)

X(22079) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22079) lies on these lines: {3, 77}, {31, 5065}, {48, 184}, {55, 1100}, {71, 3270}, {604, 1253}, {861, 20262}, {1011, 7070}, {1040, 18606}, {1212, 1827}, {1398, 5584}, {1475, 2293}, {4319, 20992}, {14557, 20853}, {15837, 20990}, {20775, 20780}, {22056, 22378}, {22072, 22369}

X(22079) = isogonal conjugate of polar conjugate of X(1212)
X(22079) = isotomic conjugate of polar conjugate of X(20229)
X(22079) = X(19)-isoconjugate of X(31618)
X(22079) = X(92)-isoconjugate of X(1170)

X(22080) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22080) lies on these lines: {3, 49}, {9, 15496}, {31, 2092}, {35, 10974}, {51, 573}, {55, 2245}, {71, 228}, {125, 440}, {165, 851}, {187, 2206}, {199, 1495}, {212, 8606}, {373, 16058}, {430, 1213}, {442, 3579}, {464, 1899}, {511, 4184}, {516, 3136}, {572, 13366}, {661, 11124}, {902, 20966}, {926, 2624}, {970, 16452}, {991, 19346}, {1030, 2194}, {1155, 17056}, {1195, 20967}, {1211, 4640}, {1230, 4427}, {1331, 1796}, {2308, 20970}, {2610, 6139}, {3142, 6684}, {3784, 22068}, {3819, 4210}, {3916, 4001}, {3937, 20730}, {3955, 20733}, {4191, 5650}, {4204, 4512}, {5651, 11350}, {6000, 7430}, {9306, 11340}, {20666, 21838}, {20729, 22073}, {20749, 20820}, {20975, 22371}, {22372, 22429}

X(22080) = crosssum of X(4) and X(27)
X(22080) = isogonal conjugate of polar conjugate of X(1213)
X(22080) = isotomic conjugate of polar conjugate of X(20970)
X(22080) = crosspoint of X(3) and X(71)
X(22080) = X(19)-isoconjugate of X(32014)
X(22080) = X(92)-isoconjugate of X(1171)

X(22081) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22081) lies on these lines: {3, 15373}, {63, 69}, {228, 20759}, {3784, 20736}, {3917, 20755}, {20729, 20732}, {20730, 22053}, {20820, 20826}

X(22081) = isogonal conjugate of polar conjugate of X(34832)
X(22081) = isotomic conjugate of polar conjugate of X(20971)

X(22082) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22082) lies on these lines: {3, 1331}, {71, 7117}, {1149, 6018}, {1332, 1811}, {1364, 22072}, {3917, 22083}, {3977, 5440}, {4587, 20818}, {5151, 16594}, {22067, 22350}, {22076, 22094}, {22369, 22373}

X(22082) = isogonal conjugate of polar conjugate of X(16594)
X(22082) = isotomic conjugate of polar conjugate of X(20972)

X(22083) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22083) lies on these lines: {3, 22067}, {71, 22134}, {3917, 22082}, {5440, 22370}

X(22083) = isogonal conjugate of polar conjugate of complement of X(89)
X(22083) = isogonal conjugate of polar conjugate of complementary conjugate of X(34824)
X(22083) = isotomic conjugate of polar conjugate of X(20973)

X(22084) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22084) lies on these lines: {3, 22145}, {11, 244}, {71, 20728}, {103, 8750}, {216, 22440}, {603, 2594}, {1459, 3270}, {1473, 4286}, {3269, 22433}, {3917, 22428}, {3937, 20975}, {7117, 22437}, {20730, 22073}, {20731, 22059}, {20819, 20830}, {22053, 22069}, {22060, 22405}, {22067, 22406}, {22418, 22435}

X(22084) = isogonal conjugate of polar conjugate of X(116)
X(22084) = isotomic conjugate of polar conjugate of X(20974)

X(22085) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22085) lies on these lines: {3, 895}, {99, 9512}, {577, 20819}, {648, 21166}, {1576, 9155}, {1634, 5191}, {3284, 22087}, {7669, 9145}, {9723, 14575}, {20756, 20784}, {22052, 22062}, {22093, 22399}

X(22085) = isogonal conjugate of polar conjugate of X(620)
X(22085) = isotomic conjugate of polar conjugate of X(20976)

X(22086) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22086) lies on these lines: {6, 654}, {31, 926}, {42, 6139}, {44, 1639}, {520, 647}, {649, 6363}, {665, 21742}, {906, 1331}, {918, 4641}, {1635, 20972}, {1769, 14399}, {2092, 2624}, {3937, 7117}, {21786, 22108}, {22144, 22148}

X(22086) = isogonal conjugate of polar conjugate of X(900)
X(22086) = isotomic conjugate of polar conjugate of X(1960)
X(22086) = X(19)-isoconjugate of X(4555)
X(22086) = X(92)-isoconjugate of X(901)

X(22087) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22087) lies on these lines: {3, 22151}, {216, 3289}, {237, 3001}, {566, 14096}, {2393, 9155}, {2524, 3049}, {3284, 22085}, {5024, 5166}, {8681, 20975}, {14570, 21531}, {20821, 22059}

X(22087) = isogonal conjugate of polar conjugate of X(625)
X(22087) = isotomic conjugate of polar conjugate of X(20977)

X(22088) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22088) lies on these lines: {1, 2272}, {3, 48}, {6, 1106}, {19, 3333}, {41, 1470}, {73, 7117}, {198, 5022}, {603, 7124}, {610, 2260}, {672, 3207}, {910, 1475}, {1202, 1615}, {1466, 2266}, {1953, 5045}, {2183, 4253}, {2253, 4020}, {2275, 20995}, {2317, 4251}, {4322, 8608}, {4860, 17474}, {7177, 7289}, {9310, 22768}, {14597, 22063}, {15656, 17558}, {20727, 22435}, {20752, 22072}, {22053, 22070}

X(22088) = isogonal conjugate of polar conjugate of X(11019)
X(22088) = isotomic conjugate of polar conjugate of X(20978)

X(22089) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22089) lies on these lines: {3, 525}, {74, 2706}, {99, 22456}, {512, 684}, {520, 11589}, {523, 2071}, {647, 22091}, {669, 3265}, {804, 3267}, {2524, 3049}, {2797, 14618}, {3357, 23103}, {4558, 9218}, {5664, 18570}, {7484, 9209}, {8673, 9409}, {9210, 14096}, {15143, 16229}, {15411, 16695}

X(22089) = isogonal conjugate of polar conjugate of X(30476)
X(22089) = isotomic conjugate of polar conjugate of X(2451)

X(22090) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22090) lies on these lines: {1, 17072}, {3, 22154}, {42, 2533}, {43, 4147}, {386, 514}, {521, 656}, {663, 1193}, {1946, 22384}, {2524, 3049}, {3835, 17921}, {4040, 5313}, {4885, 17478}, {16695, 20979}, {20731, 20757}, {20821, 22406}

X(22090) = isogonal conjugate of polar conjugate of X(3835)
X(22090) = isotomic conjugate of polar conjugate of X(20979)
X(22090) = X(19)-isoconjugate of X(4598)

X(22091) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22091) lies on these lines: {3, 905}, {36, 1734}, {56, 3900}, {404, 4391}, {521, 23087}, {647, 22089}, {663, 2821}, {667, 2254}, {3733, 7655}, {4188, 17496}, {4367, 9511}, {20731, 20757}

X(22091) = isogonal conjugate of polar conjugate of X(4885)
X(22091) = isotomic conjugate of polar conjugate of X(20980)
X(22091) = X(19)-isoconjugate of X(30610)

X(22092) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22092) lies on these lines: {3, 22155}, {39, 665}, {441, 525}, {1459, 22095}, {2275, 4435}, {3937, 7117}, {4526, 17053}, {5069, 22108}, {6373, 20663}, {20731, 20757}

X(22092) = isogonal conjugate of polar conjugate of X(3837)
X(22092) = isotomic conjugate of polar conjugate of X(6373)
X(22092) = X(19)-isoconjugate of X(8709)

X(22093) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22093) lies on these lines: {3, 810}, {58, 14838}, {171, 3907}, {419, 4369}, {741, 2698}, {750, 21052}, {905, 22384}, {940, 17478}, {1010, 21259}, {1459, 1946}, {1468, 4041}, {1691, 20981}, {3406, 4444}, {4252, 21789}, {14382, 17103}, {20731, 20757}, {22085, 22399}, {22403, 22444}, {22441, 22443}

X(22093) = isogonal conjugate of polar conjugate of X(4369)
X(22093) = isotomic conjugate of polar conjugate of X(20981)
X(22093) = X(19)-isoconjugate of X(27805)

X(22094) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22094) lies on these lines: {3, 4575}, {71, 22434}, {125, 656}, {228, 22069}, {1818, 22406}, {1834, 12832}, {2088, 2624}, {2605, 3024}, {2972, 3270}, {3269, 7117}, {3937, 20975}, {20729, 20825}, {20733, 22067}, {20738, 22420}, {20749, 20820}, {22076, 22082}, {22077, 22409}, {22097, 22405}, {22363, 22402}, {22404, 22439}

X(22094) = isogonal conjugate of polar conjugate of X(8287)
X(22094) = isotomic conjugate of polar conjugate of X(20982)

X(22095) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22095) lies on these lines: {3, 22157}, {39, 3063}, {513, 4261}, {1459, 22092}, {2092, 20980}, {2276, 21348}, {2524, 3049}, {17072, 21347}, {20729, 22059}, {20828, 22387}

X(22095) = isogonal conjugate of polar conjugate of X(21260)
X(22095) = isotomic conjugate of polar conjugate of X(20983)

X(22096) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22096) lies on these lines: {3, 1332}, {48, 2196}, {69, 23086}, {71, 20759}, {228, 22357}, {237, 7113}, {667, 3271}, {854, 5137}, {1086, 3733}, {1437, 17971}, {2643, 8639}, {3248, 8660}, {3937, 22379}, {7117, 20975}, {20750, 22054}, {20777, 22356}, {22066, 22381}

X(22096) = isogonal conjugate of polar conjugate of X(1015)
X(22096) = isotomic conjugate of polar conjugate of X(1977)
X(22096) = X(19)-isoconjugate of X(31625)
X(22096) = X(92)-isoconjugate of X(1016)

X(22097) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22097) lies on these lines: {1, 19262}, {2, 2183}, {3, 73}, {9, 20205}, {36, 1412}, {40, 388}, {42, 4259}, {48, 394}, {55, 1350}, {57, 573}, {63, 69}, {81, 2260}, {142, 1730}, {184, 22390}, {198, 17811}, {223, 10856}, {226, 1764}, {228, 1818}, {238, 5324}, {497, 6210}, {511, 14547}, {553, 20367}, {572, 2003}, {672, 4641}, {940, 1400}, {1193, 4267}, {1211, 19608}, {1284, 21334}, {1331, 5314}, {1368, 21912}, {1458, 16678}, {1762, 7291}, {1788, 9548}, {1790, 4288}, {1796, 1797}, {1804, 7099}, {1812, 7116}, {1848, 2354}, {1936, 4220}, {1993, 2317}, {2185, 17209}, {2269, 3666}, {2318, 3781}, {2328, 3220}, {2347, 4383}, {2635, 4192}, {2654, 9840}, {2999, 4266}, {3198, 5784}, {3218, 17778}, {3219, 17280}, {3452, 21361}, {3485, 10476}, {3687, 3882}, {3720, 18165}, {3730, 3929}, {3752, 4271}, {3911, 21363}, {3937, 20730}, {3942, 18607}, {4466, 18651}, {4643, 5928}, {5218, 20368}, {5307, 10444}, {7193, 22139}, {7293, 20780}, {10571, 10882}, {17147, 21271}, {18141, 21371}, {22076, 22345}, {22094, 22405}, {22369, 22412}

X(22097) = isogonal conjugate of polar conjugate of X(4357)
X(22097) = isotomic conjugate of polar conjugate of X(1193)
X(22097) = X(19)-isoconjugate of X(1220)

X(22098) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22098) lies on these lines: {3, 22162}, {71, 73}, {4303, 22447}, {8540, 8586}, {20729, 22350}, {20731, 20757}, {20752, 22414}

X(22098) = isogonal conjugate of polar conjugate of X(4892)
X(22098) = isotomic conjugate of polar conjugate of X(20984)

X(22099) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22099) lies on these lines: {3, 295}, {48, 63}, {71, 3917}, {73, 22447}, {228, 20775}, {572, 3509}, {1755, 16678}, {2200, 3916}, {4303, 20727}, {20732, 22074}

X(22099) = isogonal conjugate of polar conjugate of X(24325)
X(22099) = isotomic conjugate of polar conjugate of X(20985)

X(22100) = X(5)X(524)∩X(7812)X(9487)

Barycentrics    5 a^10-21 a^8 (b^2+c^2) +a^6 (34 b^4+28 b^2 c^2+34 c^4)+a^4 (-31 b^6+15 b^4 c^2+15 b^2 c^4-31 c^6) +15 a^2 (b^2-c^2)^2 (b^4-b^2 c^2+c^4) - (b^2-c^2)^2 (2 b^6-3 b^4 c^2-3 b^2 c^4+2 c^6) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28146.

X(22100) lies on these lines: {5,524}, {7812,9487}, {8787,9966}, {11317,14262}

X(22101) = X(2)X(195)∩X(6)X(3459)

Barycentrics    a^2 (a^20-10 a^18 (b^2+c^2)+(b^2-c^2)^8 (b^4-b^2 c^2+c^4)+a^16 (45 b^4+69 b^2 c^2+45 c^4)-2 a^2 (b^2-c^2)^6 (5 b^6-b^4 c^2-b^2 c^4+5 c^6)-4 a^14 (30 b^6+49 b^4 c^2+49 b^2 c^4+30 c^6)+a^12 (210 b^8+278 b^6 c^2+303 b^4 c^4+278 b^2 c^6+210 c^8)-6 a^10 (42 b^10+26 b^8 c^2+27 b^6 c^4+27 b^4 c^6+26 b^2 c^8+42 c^10)-2 a^6 (b^2-c^2)^2 (60 b^10+2 b^8 c^2-3 b^6 c^4-3 b^4 c^6+2 b^2 c^8+60 c^10)+a^4 (b^2-c^2)^2 (45 b^12-84 b^10 c^2+24 b^8 c^4+29 b^6 c^6+24 b^4 c^8-84 b^2 c^10+45 c^12)+a^8 (210 b^12-100 b^10 c^2+2 b^8 c^4+b^6 c^6+2 b^4 c^8-100 b^2 c^10+210 c^12)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28146.

X(22101) lies on these lines: {2,195}, {6,3459}, {288,1157}, {1263,14627}, {1994,15345}

X(22102) = COMPLEMENT OF X(3259)

Barycentrics    2 a^6-4 a^5 b-2 a^4 b^2+6 a^3 b^3-a^2 b^4-2 a b^5+b^6-4 a^5 c+16 a^4 b c-10 a^3 b^2 c-8 a^2 b^3 c+8 a b^4 c-2 b^5 c-2 a^4 c^2-10 a^3 b c^2+ 20 a^2 b^2 c^2-6 a b^3 c^2-b^4 c^2+6 a^3 c^3-8 a^2 b c^3-6 a b^2 c^3+4 b^3 c^3-a^2 c^4+8 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6 : :
X(22102) = 3 X[2] + X[901], 5 X[631] - X[953]

X(22102) lies on the nine-point circle of the medial triangle and on these lines: {1,14026}, {2,901}, {3,2222}, {36,19335}, {100,6075}, {104,6073}, {150,4998}, {165,3322}, {513,3035}, {517,1387}, {620,4369}, {631,953}, {1054,7336}, {1155,5988}, {2810,15632}, {3025,5432}, {5433,13756}, {6681,6715}

X(22102) = complement of X(3259)
X(22102) = midpoint of X(i) and X(j) for these {i,j}: {100, 6075}, {104, 6073}, {901, 3259}, {15632, 15635}
X(22102) = X(9268)-complementary conjugate of X(119)
X(22102) = {X(2),X(901)}-harmonic conjugate of X(3259)
X(22102) = center of rectangular bicevian hyperbola of X(2) and X(901)
X(22102) = centroid of ABCX(901)

X(22103) = COMPLEMENT OF X(2679)

Barycentrics    a^2 (-a^6 b^6+a^4 b^8+2 a^8 b^2 c^2-3 a^6 b^4 c^2+4 a^4 b^6 c^2-3 a^2 b^8 c^2+b^10 c^2-3 a^6 b^2 c^4+2 a^4 b^4 c^4-a^2 b^6 c^4-2 b^8 c^4-a^6 c^6+4 a^4 b^2 c^6-a^2 b^4 c^6+ 4 b^6 c^6+a^4 c^8-3 a^2 b^2 c^8-2 b^4 c^8+b^2 c^10) : :
X(22103) = 3 X[2] + X[805], 5 X[631] - X[2698], 3 X[51] - X[16979]

X(22103) lies on the on the nine-point circle of the medial triangle and one these lines: on lines {2,805}, {51,16979}, {98,6072}, {99,6071}, {230,511}, {512,620}, {631,2698}, {5976,16068}, {6787,7835}, {15630,15631}

X(22103) = complement X(2679)
X(22103) = midpoint of X(i) and X(j) for these {i,j}: {98, 6072}, {99, 6071}, {805, 2679}, {5976, 16068}, {15630, 15631}
X(22103) = center of rectangular bicevian hyperbola of X(2) and X(805)
X(22103) = centroid of ABCX(805)
X(22103) = intersection of axes of parabolas {{A,B,C,X(511),X(805)}} and {{A,B,C,X(512),X(669)}}
X(22103) = {X(2),X(805)}-harmonic conjugate of X(2679)

X(22104) = COMPLEMENT OF X(3258)

Barycentrics    2 a^12-4 a^10 b^2+3 a^6 b^6+a^4 b^8-3 a^2 b^10+b^12-4 a^10 c^2+12 a^8 b^2 c^2-7 a^6 b^4 c^2-10 a^4 b^6 c^2+10 a^2 b^8 c^2-b^10 c^2-7 a^6 b^2 c^4+20 a^4 b^4 c^4-7 a^2 b^6 c^4 -5 b^8 c^4+3 a^6 c^6-10 a^4 b^2 c^6-7 a^2 b^4 c^6+10 b^6 c^6+a^4 c^8+10 a^2 b^2 c^8-5 b^4 c^8-3 a^2 c^10-b^2 c^10+c^12 : :
X(22104) = 3 X[2] + X[476], X[477] - 5 X[631], 9 X[11539] - X[11749], 3 X[5943] - 2 X[12052], 3 X[5627] + X[12383], 3 X[5642] - X[14611], 9 X[2] - X[14731], 3 X[3258] - X[14731], 3 X[476] + X[14731], 3 X[376] + X[14989], 3 X[51] - X[16978], 5 X[15059] - X[17511], 3 X[549] + X[18319], 5 X[1656] - X[20957], X[10113] - 3 X[21315]

X(22104) lies on these lines: {2,476}, {3,16177}, {30,6699}, {51,16978}, {74,1553}, {110,6070}, {125,7471}, {126,9179}, {140,16168}, {376,14989}, {468,6036}, {477,631}, {511,11657}, {523,5972}, {542,3233}, {549,18319}, {1656,20957}, {3154,6723}, {5627,12383}, {5642,14611}, {5943,12052}, {6720,14341}, {10113,21315}, {11539,11749}, {15059,17511}

X(22104) = midpoint of X(i) and X(j) for these {i,j}: {74, 1553}, {110, 6070}, {125, 7471}, {126, 9179}, {476, 3258}, {3233, 12079}
X(22104) = reflection of X(i) in X(j) for these {i,j}: {3154, 6723}, {5972, 12068}
X(22104) = reflection X(5972) in the Euler line
X(22104) = reflection of X(32223) in the orthic axis
X(22104) = complement X(3258)
X(22104) = X(15395)-complementary conjugate of X(10)
X(22104) = {X(2),X(476)}-harmonic conjugate of X(3258)
X(22104) = centroid of ABCX(476)
X(22104) = intersection of axes of parabolas {{A,B,C,X(30),X(476)}} and {{A,B,C,X(476),X(523)}}

X(22105) = MIDPOINT OF X(4580) AND X(18105)

Barycentrics    (a^2+b^2) (b^2-c^2) (2 a^2-b^2-c^2) (a^2+c^2) : :
X(22105) = 3 X[9185] - X[14277], 3 X[9189] - X[14278], 2 X[3589] - 3 X[14428]

X(22105) lies on these lines: {5,11620}, {23,385}, {83,9180}, {99,827}, {111,9076}, {115,804}, {141,5113}, {308,14606}, {351,7664}, {690,5026}, {3228,14970}, {3589,14428}, {9185,14277}, {9189,14278}, {9293,17997}

X(22105) = midpoint of X(4580) and X(18105)
X(22105) = reflection of X(i) in X(j) for these {i,j}: {5, 11620}, {141, 5113}
X(22105) = isogonal conjugate of X(36827)
X(22105) = X(i)-cross conjugate of X(j) for these (i,j): {18311, 523}, {21906, 524}
X(22105) = X(i)-isoconjugate of X(j) for these (i,j): {38, 691}, {892, 1964}, {897, 1634}, {923, 4576}, {5380, 17187}
X(22105) = cevapoint of X(351) and X(690)
X(22105) = trilinear pole of line {1648, 11183}
X(22105) = crossdifference of every pair of points on line {39, 1634}
X(22105) = barycentric product X(i)*X(j) for these {i,j}: {83, 690}, {308, 351}, {468, 4580}, {689, 21906}, {896, 18070}, {1648, 4577}, {1799, 14273}, {2642, 3112}, {3266, 18105}, {4062, 10566}, {4750, 18082}, {9076, 18311}, {11183, 14970}, {14432, 18097}, {19326, 20483}
X(22105) = barycentric quotient X(i)/X(j) for these {i,j}: {83, 892}, {187, 1634}, {251, 691}, {351, 39}, {524, 4576}, {690, 141}, {1648, 826}, {1649, 7813}, {2642, 38}, {4062, 4568}, {4750, 16887}, {11183, 732}, {14273, 427}, {14417, 3933}, {14419, 16696}, {14424, 7794}, {18098, 5380}, {18105, 111}, {21839, 4553}, {21906, 3005}
X(22105) = trilinear product X(i)*X(j) for these {i,j}: {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(22106) = X(13436)-CEVA CONJUGATE OF X(6365)

Barycentrics    (b - c)^2 (b c - S) : :

X(22106) lies on the incircle and these lines: {482,1360}, {918,1086}, {1335,13436}, {1361,8243}, {3321,5393}

X(22106) = X(13436)-Ceva conjugate of X(6365)
X(22106) = X(i)-isoconjugate of X(j) for these (i,j): {59, 13427}, {101, 6136}, {1110, 1336}, {2149, 13426}, {6065, 13460}
X(22106) = barycentric product X(i)*X(j) for these {i,j}: {11, 13436}, {693, 6365}, {1086, 5391}, {1111, 3084}, {1358, 13458}, {1565, 13387}
X(22106) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 13426}, {513, 6136}, {606, 1110}, {1086, 1336}, {1335, 1252}, {1358, 13459}, {1565, 13386}, {2170, 13427}, {3084, 765}, {3942, 6212}, {5391, 1016}, {6365, 100}, {13387, 15742}, {13436, 4998}, {13458, 4076}
X(22106) = {X(1086),X(1111)}-harmonic conjugate of X(22107)

X(22107) = X(13453)-CEVA CONJUGATE OF X(6364)

Barycentrics    (b - c)^2 (b c + S) : :

X(22107) lies on the incircle and these lines: {481,1360}, {918,1086}, {1124,13453}, {3321,5405}

X(22107) = X(13453)-Ceva conjugate of X(6364)
X(22107) = X(i)-isoconjugate of X(j) for these (i,j): {59, 13456}, {101, 6135}, {1110, 1123}, {2149, 13454}, {6065, 13438}
X(22107) = barycentric product X(i)*X(j) for these {i,j}: {11, 13453}, {693, 6364}, {1086, 1267}, {1111, 3083}, {1358, 13425}, {1565, 13386}
X(22107) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 13454}, {513, 6135}, {605, 1110}, {1086, 1123}, {1124, 1252}, {1267, 1016}, {1358, 13437}, {1565, 13387}, {2170, 13456}, {3083, 765}, {3942, 6213}, {6364, 100}, {13386, 15742}, {13425, 4076}, {13453, 4998}
{X(1086),X(1111)}-harmonic conjugate of X(22106)

X(22108) = MIDPOINT OF X(2590) and X(2591)

Barycentrics    a^2 (b-c) (a^2-2 a b+b^2-2 a c+b c+c^2) : :
X(22108) = X[649] + 3 X[657]

X(22108) lies on these lines: {6,665}, {9,900}, {37,4435}, {44,513}, {45,4526}, {101,692}, {523,21390}, {667,9029}, {909,911}, {1024,2161}, {2170,17463}, {2291,6139}, {2605,3063}, {2820,4869}, {3196,8658}, {3709,21007}, {3766,17277}, {3887,6594}, {4491,8659}, {5069,22092}, {5540,6084}, {8632,14407}, {8638,15624}, {21131,21832}, {21786,22086}

X(22108) = midpoint of X(2590) and X(2591)
X(22108) = isogonal conjugate of X(37143)
X(22108) = X(i)-zayin conjugate of X(j) for these (i,j): {9, 1308}, {650, 3254}, {1308, 1638}, {1638, 9}, {2826, 57}, {5527, 658}, {5536, 651}
X(22108) = X(2742)-Ceva conjugate of X(55)
X(22108) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1308}, {651, 3254}
X(22108) = crosspoint of X(i) and X(j) for these (i,j): {1, 37143}, {57, 14733}, {101, 2291}
X(22108) = crossdifference of every pair of points on line {1, 528}
X(22108) = crosssum of X(i) and X(j) for these (i,j): {1, 22108}, {9, 6366}, {514, 527}, {522, 5199}, {650, 18839}
X(22108) = barycentric product X(i)*X(j) for these {i,j}: {1, 3887}, {75, 8645}, {513, 3935}, {514, 5526}, {522, 2078}, {649, 17264}, {693, 19624}
X(22108) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1308}, {663, 3254}, {2078, 664}, {3887, 75}, {3935, 668}, {5526, 190}, {8645, 1}, {17264, 1978}, {19624, 100}
X(22108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1635, 2246, 659), (3063, 6586, 2605)

X(22109) = X(3)X(125)∩X(26)X(113)

Barycentrics    a^2 (a^2-b^2-c^2) (a^12-2 a^10 b^2-a^8 b^4+4 a^6 b^6-a^4 b^8-2 a^2 b^10+b^12-2 a^10 c^2+3 a^8 b^2 c^2-a^6 b^4 c^2-2 a^4 b^6 c^2+5 a^2 b^8 c^2-3 b^10 c^2-a^8 c^4-a^6 b^2 c^4+ 2 a^4 b^4 c^4-3 a^2 b^6 c^4+3 b^8 c^4+4 a^6 c^6-2 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(22109) = 2 X[11430] - 5 X[15051], 3 X[15035] - X[15463]

X(22109) lies on these lines: {3,125}, {20,13293}, {22,2777}, {23,1531}, {24,5972}, {26,113}, {52,12228}, {69,12584}, {74,7512}, {110,5562}, {143,9826}, {186,249}, {569,12236}, {974,10984}, {1092,1511}, {1350,15141}, {1539,17714}, {1568,2070}, {1594,18428}, {2935,11414}, {2937,7728}, {3043,11412}, {3047,12273}, {3917,17701}, {5181,15577}, {5504,13367}, {5642,14070}, {5663,7502}, {5889,12227}, {6636,15055}, {6723,7509}, {6759,12825}, {7387,13202}, {7503,7687}, {7506,12900}, {7525,12041}, {7526,12295}, {7575,11064}, {9626,12368}, {9715,10117}, {9967,19138}, {10024,19479}, {10272,12107}, {10539,20773}, {10634,10664}, {10635,10663}, {10721,12088}, {10733,14118}, {10897,12892}, {10898,12891}, {11430,13434}, {12225,19506}, {13198,21649}, {13564,20127}, {14984,19131}, {15085,19456}, {17834,19504}

X(22109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 2931, 125), (3, 12121, 12901), (3, 12310, 19457), (110, 7488, 13289), (110, 7691, 12219), (9715, 12168, 10117), (10117, 12168, 15063)

X(22110) = X(2)X(6)∩X(30)X(114)

Barycentrics    2 a^4-5 a^2 b^2+5 b^4-5 a^2 c^2-2 b^2 c^2+5 c^4 : :
X(22110) = X[230] + 2 X[325], 5 X[230] - 2 X[385], 5 X[2] - X[385], 5 X[325] + X[385], 2 X[625] + X[6390], 7 X[325] - X[7779], 7 X[2] + X[7779], 7 X[230] + 2 X[7779], 7 X[385] + 5 X[7779], X[671] + 3 X[7799], 3 X[7779] - 7 X[7840], 3 X[325] - X[7840], 3 X[2] + X[7840], 3 X[230] + 2 X[7840], 3 X[385] + 5 X[7840], 3 X[5215] + X[7845], X[230] - 10 X[7925], X[2] - 5 X[7925], X[325] + 5 X[7925], X[7840] + 15 X[7925], 7 X[385] - 15 X[8859], 7 X[230] - 6 X[8859], 7 X[2] - 3 X[8859], 7 X[325] + 3 X[8859], X[7779] + 3 X[8859], 7 X[7840] + 9 X[8859], X[187] - 3 X[9167], 2 X[6722] - 3 X[10150], 2 X[549] - 3 X[10256], X[8591] + 3 X[14041], X[11054] - 5 X[14061], X[7813] + 3 X[14971], 5 X[6390] - 2 X[15301], 5 X[625] + X[15301], 8 X[385] - 5 X[15480], 8 X[2] - X[15480], 4 X[230] - X[15480], 8 X[325] + X[15480], 8 X[7840] + 3 X[15480], 8 X[7779] + 7 X[15480], X[15993] - 3 X[21358], 7 X[15702] - 3 X[21445]

X(22110) lies on these lines: {2,6}, {5,7801}, {30,114}, {39,8360}, {98,12151}, {99,8352}, {115,8355}, {126,9193}, {140,7810}, {147,10488}, {187,9167}, {316,8598}, {338,3266}, {468,9164}, {523,7625}, {538,2023}, {543,625}, {546,7863}, {547,9466}, {549,7818}, {574,12040}, {598,7835}, {620,3849}, {626,8359}, {632,7854}, {671,7799}, {858,10717}, {1153,7848}, {1503,6054}, {1506,8367}, {2549,11165}, {3291,9165}, {3363,3734}, {3530,7873}, {3564,6055}, {3628,7794}, {3788,7745}, {3933,7862}, {5077,7618}, {5159,15526}, {5215,7845}, {5254,11318}, {5648,9759}, {5976,15814}, {6722,10150}, {7181,21057}, {7495,9829}, {7617,7908}, {7622,7761}, {7752,7789}, {7753,8368}, {7762,7940}, {7763,7841}, {7764,7817}, {7769,7883}, {7807,7812}, {7813,14971}, {7827,7899}, {7833,7912}, {7866,9606}, {7874,8365}, {7907,9939}, {8290,8786}, {8364,9698}, {8591,14041}, {8705,12093}, {8716,16041}, {8724,15980}, {8787,12830}, {9607,14064}, {9756,11180}, {11054,14061}, {11057,11149}, {11185,20112}, {12036,13857}, {15702,21445}

X(22110) = midpoint of X(i) and X(j) for these {i,j}: {2, 325}, {99, 8352}, {316, 8598}, {8724, 15980}
X(22110) = reflection of X(i) in X(j) for these {i,j}: {115, 8355}, {230, 2}
X(22110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 7610), (2, 183, 15597), (2, 599, 11168), (2, 1007, 11184), (2, 7610, 3054), (2, 7779, 8859), (2, 7788, 13468), (2, 9766, 5306), (2, 9770, 6), (2, 9771, 3055), (2, 11163, 597), (2, 11184, 3815), (2, 21356, 15271), (141, 9771, 2), (597, 11163, 9300), (1007, 7778, 3815), (3734, 8176, 3363), (3788, 7775, 8369), (6189, 6190, 11160), (7752, 7870, 8370), (7775, 8369, 7745), (7778, 11184, 2), (7870, 8370, 7789)
X(22110) = orthoptic circle of the Steiner inellipe inverse of X(14916)
X(22110) = complement of X(22329)
X(22110) = isotomic of the isogonal of X(5107)
X(22110) = X(2)-daleth conjugate of X(599)
X(22110) = X(i)-complementary conjugate of X(j) for these (i,j): {2709, 4369}, {5503, 2887}
X(22110) = X(2)-Hirst inverse of X(11160)
X(22110) = crosspoint of X(2) and X(5503)
X(22110) = crossdifference of every pair of points on line {512, 1384}
X(22110) = crosssum of X(6) and X(2030)
X(22110) = barycentric product X(76)*X(5107)
X(22110) = barycentric quotient X(5107)/X(6)

X(22111) = CROSSSUM OF X(2) AND X(9741)

Barycentrics    a^2 (a^4-a^2 b^2+4 b^4-a^2 c^2-16 b^2 c^2+4 c^4) : :

X(22111) lies on these lines: {2,5503}, {6,373}, {51,11173}, {111,182}, {184,2502}, {187,3148}, {352,576}, {511,20481}, {574,3124}, {597,16317}, {647,9171}, {1383,10545}, {1384,3066}, {1995,2030}, {3098,13192}, {5166,9813}, {5354,11451}, {5476,5913}, {5640,11580}, {6032,11647}, {6792,11178}, {7606,10160}, {7617,9169}, {8288,18362}, {10485,20998}

X(22111) = crossdifference of every pair of points on line {1499, 8598}
X(22111) = crosssum of X(2) and X(9741)
X(22111) = psi-transform of X(13492)
X(22111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 8585, 5651), (111, 7708, 182)

X(22112) = X(2)X(98)∩X(3)X(373)

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

X(22112) is the perspector of the Thomson-Gibert-Moses hyperbola wrt triangle X(2)X(3)X(6). (Randy Hutson, October 15, 2018)

X(22112) lies on these lines: {2,98}, {3,373}, {5,16654}, {6,5646}, {22,6688}, {23,17508}, {51,1350}, {52,13154}, {140,13142}, {381,8717}, {468,19124}, {511,21766}, {567,1092}, {569,632}, {574,3124}, {575,15066}, {576,7998}, {578,3525}, {582,16296}, {631,11424}, {868,7913}, {1204,7395}, {1351,3917}, {1397,17124}, {1437,16863}, {1495,5085}, {1656,10984}, {1790,16409}, {1974,5094}, {1993,15516}, {1995,5092}, {2175,17125}, {2972,5158}, {3091,13347}, {3098,5640}, {3231,5034}, {3292,5050}, {3628,13336}, {3819,5097}, {4550,10620}, {5056,15431}, {5067,6759}, {5118,15482}, {5159,19131}, {5398,19249}, {5562,15805}, {5643,11002}, {5645,5888}, {5943,7485}, {6784,9145}, {6800,12045}, {6803,21659}, {7392,14927}, {7492,10545}, {7509,11695}, {7550,11438}, {7570,7703}, {7889,14003}, {8371,8723}, {8541,12039}, {8585,20998}, {8722,14096}, {9275,17749}, {10303,13346}, {11451,15246}, {11935,15723}, {12100,20192}, {13323,17531}, {13329,16373}, {13366,17811}, {14805,15040}, {14926,18435}, {15033,15702}, {15080,16042}, {15720,21970}, {16051,19126}, {16063,19130}

X(22112) = crossdifference of every pair of points on line {3569, 9123}
X(22112) = trilinear product of vertices of Stammler triangle
X(22112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 110, 16187), (2, 182, 5651), (3, 5544, 3066), (110, 16187, 5651), (182, 5651, 184), (182, 9306, 11003), (182, 16187, 110), (575, 15082, 15066), (3066, 5544, 373), (3917, 10601, 15004), (5085, 11284, 1495), (5640, 7496, 3098), (7484, 17825, 51), (7998, 15018, 576), (10601, 16419, 3917)

X(22113) = ANTICOMPLEMENT OF X(627)

Barycentrics    a^4+2 a^2 b^2-3 b^4+2 a^2 c^2+6 b^2 c^2-3 c^4+2 Sqrt[3] (3 a^2-b^2-c^2) S : :
X(22113) = 3 X[2] - 4 X[17], 9 X[2] - 8 X[629], 3 X[627] - 4 X[629], 3 X[17] - 2 X[629], 15 X[2] - 16 X[6673], 5 X[627] - 8 X[6673], 5 X[629] - 6 X[6673], 5 X[17] - 4 X[6673], 7 X[3622] - 8 X[11739], 5 X[3091] - 4 X[16626]

X(22113) lies on the curves Q088 and K906, and on these lines: {2,17}, {4,3180}, {5,3181}, {13,633}, {20,6770}, {61,622}, {148,16001}, {193,576}, {299,397}, {530,5238}, {617,16965}, {628,10653}, {2896,16941}, {3105,5335}, {3600,18973}, {3622,11739}, {3926,11132}, {5340,5859}, {5613,13571}

X(22113) = anticomplement X(627)
X(22113) = reflection of X(i) in X(j) for these {i,j}: {4, 16629}, {627, 17}
X(22113) = anticomplement of the isotomic of X(19712)
X(22113) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3489, 8}, {19712, 6327}
X(22113) = X(19712)-Ceva conjugate of X(2)

X(22113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 627, 2), (193,3091,22114), (5873, 20425, 4)

X(22114) = ANTICOMPLEMENT OF X(628)

Barycentrics    a^4+2 a^2 b^2-3 b^4+2 a^2 c^2+6 b^2 c^2-3 c^4-2 Sqrt[3] (3 a^2-b^2-c^2) S : :
X(22114) = 3 X[2] - 4 X[18], 9 X[2] - 8 X[630], 3 X[628] - 4 X[630], 3 X[18] - 2 X[630], 15 X[2] - 16 X[6674], 5 X[628] - 8 X[6674], 5 X[630] - 6 X[6674], 5 X[18] - 4 X[6674], 7 X[3622] - 8 X[11740], 5 X[3091] - 4 X[16627]

X(22114) lies on the curves Q088 and K906, and on these lines: {2,18}, {4,3181}, {5,3180}, {14,634}, {20,6773}, {62,621}, {148,16002}, {193,576}, {298,398}, {531,5237}, {616,16964}, {627,10654}, {2896,16940}, {3104,5334}, {3600,18972}, {3622,11740}, {3926,11133}, {5339,5858}, {5617,13571}

X(22114) = anticomplement X(628)
X(22114) = reflection of X(i) in X(j) for these {i,j}: {4, 16628}, {628, 18}
X(22114) = anticomplement of the isotomic of X(19713)
X(22114) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3490, 8}, {19713, 6327}
X(22114) = X(19713)-Ceva conjugate of X(2)
X(22114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 628, 2), (193,3091,22113), (5872, 20426, 4)

X(22115) = MIDPOINT OF X(186) AND X(323)

Barycentrics    a^4 (a^2-b^2-c^2) (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) : :
X(22115) = X[323] + 2 X[1511], X[3] + 2 X[3292], 4 X[1511] - X[3581], 2 X[323] + X[3581], 2 X[5609] + X[7464], X[399] + 2 X[10564], X[265] - 4 X[11064], X[10510] + 2 X[12584], 3 X[10540] - 2 X[14157], 3 X[110] - X[14157], 2 X[403] - 3 X[14643], 8 X[5159] - 5 X[15027], 2 X[7575] - 5 X[15034], 4 X[10257] - 3 X[15061], X[12429] - 4 X[15123], 2 X[1147] + X[15136], 4 X[11694] - X[15360], 3 X[15035] - 2 X[15646], 4 X[16534] - X[18325], 7 X[15020] - 4 X[18571], 3 X[3] - 2 X[21663], 3 X[3292] + X[21663]

X(22115) lies on these lines: {2,567}, {3,49}, {4,18350}, {5,15033}, {6,9676}, {15,3200}, {16,3201}, {20,156}, {22,13340}, {23,13391}, {24,6243}, {30,110}, {32,9603}, {35,2477}, {36,215}, {50,18334}, {54,140}, {60,5453}, {68,6640}, {69,19129}, {125,539}, {154,12083}, {165,9621}, {182,599}, {186,323}, {187,9696}, {195,389}, {265,2072}, {376,9544}, {378,15068}, {381,9306}, {382,10539}, {399,2935}, {403,14643}, {498,9653}, {499,9666}, {500,17104}, {511,2070}, {520,6760}, {524,15462}, {526,15470}, {548,9705}, {549,5012}, {550,1614}, {568,1993}, {569,3526}, {574,9604}, {576,13321}, {578,1656}, {631,9545}, {858,15132}, {974,10816}, {993,9702}, {1199,1493}, {1351,19136}, {1495,5899}, {1568,17702}, {1594,6288}, {1657,5895}, {1658,11412}, {1994,5946}, {2063,18466}, {2071,5663}, {2888,6143}, {2931,18127}, {2937,10282}, {2979,7502}, {3044,12042}, {3047,12041}, {3060,12106}, {3153,12383}, {3202,9821}, {3203,12054}, {3205,5238}, {3206,5237}, {3289,10317}, {3431,18882}, {3518,10263}, {3520,5876}, {3524,11003}, {3530,9706}, {3548,6193}, {3564,5622}, {3575,15800}, {3576,9586}, {3580,12228}, {3628,13434}, {3851,11424}, {4299,9652}, {4302,9667}, {5050,9027}, {5055,5651}, {5066,13482}, {5159,15027}, {5446,13621}, {5462,14627}, {5609,7464}, {5891,11430}, {5892,13366}, {5907,14130}, {5943,15038}, {5944,7512}, {6090,9818}, {6101,7488}, {6409,9677}, {6445,9687}, {6642,9777}, {7506,17810}, {7514,14805}, {7527,15060}, {7574,15139}, {7575,15034}, {7666,10274}, {7799,10411}, {7987,9622}, {8717,15689}, {8718,12103}, {8780,18534}, {9145,15365}, {9301,9418}, {9730,15087}, {9820,10024}, {9927,10255}, {10091,10149}, {10096,13392}, {10110,18369}, {10151,15472}, {10226,11440}, {10272,11563}, {10510,11649}, {11250,12111}, {11422,15045}, {11441,12084}, {11442,18281}, {11459,18570}, {11591,14118}, {11693,15303}, {11694,15360}, {11695,15047}, {12118,18404}, {12161,17928}, {12254,13470}, {12278,18377}, {12429,15123}, {12902,13851}, {13336,15720}, {13371,14516}, {13445,14094}, {13564,15644}, {13596,15052}, {14106,19552}, {14984,18449}, {15020,18571}, {15035,15646}, {15316,15317}, {16089,18831}, {16386,20127}, {16534,18325}, {18438,20806}

X(22115) = midpoint of X(i) and X(j) for these {i,j}: {186, 323}, {399, 18859}, {3153, 12383}, {13445, 14094}
X(22115) = reflection of X(i) in X(j) for these {i,j}: {125, 14156}, {186, 1511}, {265, 2072}, {2072, 11064}, {3581, 186}, {5899, 1495}, {10096, 13392}, {10540, 110}, {11563, 10272}, {12902, 13851}, {18403, 1568}, {18859, 10564}, {20127, 16386}
X(22115) = isogonal conjugate of X(6344)
X(22115) = isotomic conjugate of X(18817)
X(22115) = X(i)-Ceva conjugate of X(j) for these (i,j): {323, 50}, {5504, 3}, {10411, 8552}, {14919, 577}
X(22115) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6344}, {4, 2166}, {19, 94}, {31, 18817}, {75, 18384}, {92, 1989}, {158, 265}, {162, 10412}, {328, 1096}, {811, 15475}, {823, 14582}, {1784, 5627}, {1969, 11060}, {1973, 20573}
X(22115) = crosspoint of X(i) and X(j) for these (i,j): {95, 2986}, {328, 11140}
X(22115) = crossdifference of every pair of points on line {53, 2501}
X(22115) = crosssum of X(i) and X(j) for these (i,j): {51, 3003}, {1989, 18384}, {1990, 14583}
X(22115) = barycentric product X(i)*X(j) for these {i,j}: {3, 323}, {50, 69}, {63, 6149}, {97, 1154}, {110, 8552}, {184, 7799}, {186, 394}, {249, 16186}, {305, 19627}, {340, 577}, {520, 14590}, {526, 4558}, {647, 10411}, {1092, 14165}, {1273, 14533}, {1511, 14919}, {2624, 4592}, {3265, 14591}, {4563, 14270}, {6148, 18877}, {11064, 14385}, {14918, 19210}
X(22115) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18817}, {3, 94}, {6, 6344}, {32, 18384}, {48, 2166}, {50, 4}, {69, 20573}, {184, 1989}, {186, 2052}, {323, 264}, {340, 18027}, {394, 328}, {520, 14592}, {526, 14618}, {577, 265}, {647, 10412}, {1147, 18883}, {1154, 324}, {2088, 2970}, {3043, 14165}, {3049, 15475}, {3284, 14254}, {3289, 14356}, {6149, 92}, {7799, 18022}, {8552, 850}, {10411, 6331}, {11062, 13450}, {11077, 14859}, {14270, 2501}, {14355, 16081}, {14385, 16080}, {14533, 1141}, {14575, 11060}, {14590, 6528}, {14591, 107}, {16186, 338}, {18877, 5627}, {19627, 25}
X(22115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1147, 49), (3, 3167, 18445), (3, 9703, 184), (15, 3200, 11137), (16, 3201, 11134), (24, 16266, 6243), (54, 140, 13353), (184, 1147, 9703), (184, 9703, 49), (323, 1511, 3581), (378, 15068, 18435), (549, 5012, 13339), (1092, 1147, 3), (1092, 3292, 15136), (1216, 13367, 3), (1493, 12006, 1199), (1511, 3043, 11597), (1993, 6644, 568), (2979, 11464, 7502), (3917, 18475, 3), (5562, 12038, 3), (5892, 13366, 15037), (5944, 10627, 7512), (9306, 13352, 381), (10282, 10625, 2937), (10539, 13346, 382), (11412, 11449, 1658), (11441, 12084, 18439)

X(22116) = X(1)X(3()∩X(10)X(514)

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

X(22116) lies on the cubics K1038, K1068, K1069) and these lines: {1,39}, {2,19897}, {8,4562}, {9,9470}, {10,514}, {12,85}, {56,4564}, {76,4583}, {335,16593}, {518,3252}, {660,5220}, {813,1083}, {1026,4447}, {1376,9503}, {3573,17798}, {3675,3912}, {3932,18157}, {4075,17758}, {8256,9311}, {17169,18827}

X(22116) = X(i)-complementary conjugate of X(j) for these (i,j): {518, 20551}, {672, 20343}, {727, 518}, {2223, 20532}, {3226, 20544}, {20332, 20335}
X(22116) = X(i)-Ceva conjugate of X(j) for these (i,j): {291, 518}, {4583, 918}
X(22116) = X(i)-cross conjugate of X(j) for these (i,j): {3675, 876}, {4712, 518}
X(22116) = X(i)-Hirst inverse of X(j) for these (i,j): {291, 4876}, {518, 3252}
X(22116) = cevapoint of X(i) and X(j) for these (i,j): {3126, 3675}, {6184, 20683}
X(22116) = trilinear pole of line {2254, 3930}
X(22116) = crossdifference of every pair of points on line {659, 1914}
X(22116) = crosssum of X(238) and X(8300)
X(22116) = X(2254)-zayin conjugate of X(659)
X(22116) = barycentric product X(i)X(j) for these {i,j}: {75, 3252}, {241, 4518}, {291, 3912}, {292, 3263}, {334, 672}, {335, 518}, {337, 5089}, {660, 918}, {665, 4583}, {1026, 4444}, {1916, 4447}, {2223, 18895}, {2254, 4562}, {3693, 7233}, {3930, 18827}, {4088, 4584}, {4876, 9436}
X(22116) = X(i)-isoconjugate of X(j) for these (i,j): {6, 6654}, {105, 238}, {239, 1438}, {294, 1429}, {666, 8632}, {673, 1914}, {812, 919}, {1027, 3573}, {1416, 3685}, {1428, 14942}, {1447, 2195}, {1462, 3684}, {1814, 2201}, {2210, 2481}, {5009, 13576}, {8751, 20769}, {14599, 18031}
X(22116) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6654}, {241, 1447}, {291, 673}, {292, 105}, {295, 1814}, {334, 18031}, {335, 2481}, {518, 239}, {660, 666}, {665, 659}, {672, 238}, {918, 3766}, {926, 4435}, {1026, 3570}, {1458, 1429}, {1818, 20769}, {1911, 1438}, {2223, 1914}, {2254, 812}, {2284, 3573}, {2340, 3684}, {2356, 2201}, {3252, 1}, {3263, 1921}, {3572, 1027}, {3693, 3685}, {3717, 3975}, {3912, 350}, {3930, 740}, {3932, 3948}, {4447, 385}, {4712, 17755}, {4876, 14942}, {5089, 242}, {6184, 8299}, {7077, 294}, {8299, 4366}, {9436, 10030}, {9454, 2210}, {9455, 14599}, {14439, 4432}, {17435, 4124}, {20683, 2238}, {20752, 7193}

X(22117) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22117) lies on these lines: {1, 3683}, {3, 73}, {6, 13404}, {33, 5779}, {55, 2003}, {81, 954}, {109, 6244}, {165, 1419}, {278, 5762}, {329, 15252}, {394, 1260}, {405, 3562}, {582, 1167}, {651, 7580}, {971, 7070}, {999, 5398}, {1074, 18541}, {1103, 3579}, {1407, 13329}, {1496, 16466}, {1617, 2361}, {1754, 6180}, {1771, 9709}, {1795, 22141}, {1936, 19541}, {3074, 11108}, {3075, 16408}, {3167, 20752}, {3745, 15298}, {3990, 15905}, {4667, 13405}, {5759, 18623}, {6056, 7011}, {6149, 8069}, {7193, 23089}, {7290, 12915}, {9654, 13408}, {20796, 20799}, {20797, 22149}, {22132, 22139} X(22117) = isogonal conjugate of polar conjugate of X(144)
X(22117) = isotomic conjugate of polar conjugate of X(3207)
X(22117) = X(19)-isoconjugate of X(10405)
X(22117) = X(92)-isoconjugate of X(11051)

X(22118) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22118) lies on these lines: {1, 1333}, {3, 2197}, {6, 8071}, {48, 255}, {63, 18604}, {160, 692}, {216, 22123}, {218, 5065}, {219, 577}, {517, 1950}, {573, 1415}, {608, 11249}, {2169, 3990}, {2327, 22126}, {2911, 5063}, {3284, 22122}, {3562, 7054}, {4261, 14793}, {5124, 13006}, {5841, 8736}, {15905, 22131}, {20793, 23086}, {22054, 22350}

X(22118) = isogonal conjugate of polar conjugate of X(2975)
X(22118) = isotomic conjugate of polar conjugate of X(20986)
X(22118) = X(92)-isoconjugate of X(34434)

X(22119) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22119) lies on these lines: {3, 31}, {6, 1214}, {48, 222}, {63, 22131}, {81, 6349}, {219, 22130}, {380, 2999}, {394, 22134}, {608, 11347}, {857, 17902}, {940, 17073}, {997, 17811}, {1040, 7290}, {1073, 1260}, {3101, 8743}, {3157, 7016}, {3195, 7580}, {3772, 18588}, {4329, 17903}, {5230, 21530}, {5711, 18641}, {20967, 22341}

X(22119) = isogonal conjugate of polar conjugate of X(4329)
X(22119) = isotomic conjugate of polar conjugate of X(3556)
X(22119) = X(19)-isoconjugate of X(7219)
X(22119) = X(92)-isoconjugate of X(7169)

X(22120) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22120) lies on these lines: {3, 6}, {26, 10313}, {30, 8743}, {112, 12084}, {127, 7759}, {155, 22146}, {194, 15013}, {230, 6640}, {232, 7517}, {248, 15317}, {339, 7754}, {382, 2207}, {441, 1993}, {1060, 5280}, {1062, 5299}, {1147, 8779}, {1180, 15818}, {1368, 5359}, {1576, 2353}, {2072, 3767}, {2548, 10024}, {2549, 18563}, {3087, 7528}, {3146, 8744}, {3172, 12085}, {3546, 5304}, {3548, 7735}, {3549, 7736}, {3815, 6639}, {3926, 22151}, {3927, 22131}, {3933, 20806}, {5254, 18404}, {5286, 18531}, {5305, 11585}, {5354, 16051}, {5523, 18569}, {5938, 20993}, {6644, 10312}, {7400, 14930}, {7506, 10311}, {7553, 8745}, {7737, 15075}, {7758, 14376}, {7890, 15526}, {10255, 13881}, {12605, 15048}, {13861, 15355}, {15341, 22660}, {16502, 18455}, {19597, 22143}

X(22120) = isogonal conjugate of polar conjugate of X(7391)
X(22120) = isotomic conjugate of polar conjugate of X(20987)
X(22120) = X(92)-isoconjugate of X(34436)

X(22121) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^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(22121) lies on these lines: {3, 6}, {30, 8744}, {112, 18859}, {232, 5899}, {323, 441}, {399, 13509}, {647, 22155}, {1368, 5354}, {1576, 5938}, {1657, 8743}, {2070, 10313}, {2207, 5073}, {2549, 18564}, {3289, 22146}, {5159, 11580}, {5523, 7574}, {6390, 22151}, {7545, 15355}, {8779, 22115}, {10985, 13621}, {15075, 18565}, {16784, 18455}, {16785, 18447}

X(22121) = isogonal conjugate of polar conjugate of X(5189)
X(22121) = isotomic conjugate of polar conjugate of X(19596)
X(22121) = X(92)-isoconjugate of X(34437)

X(22122) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22122) lies on these lines: {1, 6}, {3, 22058}, {48, 22144}, {69, 20808}, {216, 906}, {2193, 22070}, {2259, 5396}, {2286, 23073}, {3284, 22118}, {22126, 22133}, {22143, 23094}

X(22122) = isogonal conjugate of polar conjugate of isogonal conjugate of X(34441)
X(22122) = isogonal conjugate of polar conjugate of complement of X(20066)
X(22122) = isogonal conjugate of polar conjugate of anticomplement of X(35)
X(22122) = isotomic conjugate of polar conjugate of X(20988)
X(22122) = X(92)-isoconjugate of X(34441)

X(22123) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22123) lies on these lines: {1, 6}, {3, 22059}, {59, 7115}, {216, 22118}, {284, 2594}, {521, 2522}, {692, 2393}, {906, 3284}, {1332, 20808}, {1783, 7359}, {2193, 2197}, {2302, 5399}, {3157, 19350}, {4282, 5172}, {7124, 23073}, {20744, 22145}, {20796, 22143}, {22144, 22356}

X(22123) = isogonal conjugate of polar conjugate of X(5080)
X(22123) = isotomic conjugate of polar conjugate of X(20989)
X(22123) = X(92)-isoconjugate of X(34442)

X(22124) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22124) lies on these lines: {1, 6}, {3, 22063}, {48, 222}, {109, 1436}, {198, 10571}, {221, 610}, {517, 2331}, {602, 1622}, {1064, 4254}, {1409, 7124}, {1604, 2199}, {1783, 7003}, {3157, 20818}, {3211, 22144}, {3284, 23073}, {22147, 23071}

X(22124) = isotomic conjugate of polar conjugate of X(20991)
X(22124) = isogonal conjugate of polar conjugate of X(962)
X(22124) = X(92)-isoconjugate of X(963)

X(22125) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22125) lies on these lines: {3, 22064}, {6, 142}, {219, 20740}, {222, 3211}, {306, 394}, {20739, 20806}

X(22125) = isogonal conjugate of polar conjugate of X(21285)
X(22125) = isotomic conjugate of polar conjugate of X(1626)

X(22126) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22126) lies on these lines: {1, 16699}, {3, 48}, {6, 1125}, {69, 20811}, {72, 20752}, {78, 4574}, {218, 1468}, {220, 993}, {394, 4001}, {2274, 9605}, {2327, 22118}, {2911, 5021}, {3927, 22163}, {4020, 22458}, {4047, 14597}, {17135, 17911}, {20762, 20809}, {20796, 23077}, {22122, 22133}

X(22127) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22127) lies on these lines: {3, 48}, {6, 978}, {72, 22163}, {78, 20752}, {101, 15654}, {172, 218}, {222, 348}, {394, 7124}, {610, 10476}, {1613, 16502}, {3496, 20995}, {3940, 22164}, {4020, 20760}, {5776, 15486}, {7078, 20762}, {10453, 17920}, {20739, 22144}, {20741, 22131}, {20745, 20812}, {22158, 23088}

X(22127) = isogonal conjugate of polar conjugate of X(10453)
X(22127) = isotomic conjugate of polar conjugate of X(20992)
X(22127) = X(92)-isoconjugate of X(34445)

X(22128) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22128) lies on these lines: {2, 2003}, {3, 22067}, {6, 3306}, {9, 15066}, {57, 1993}, {63, 77}, {72, 23070}, {78, 3157}, {81, 3664}, {84, 11441}, {110, 3220}, {184, 3784}, {221, 11682}, {228, 22161}, {283, 4303}, {320, 17923}, {323, 1443}, {651, 908}, {758, 4351}, {905, 4131}, {960, 8614}, {1203, 5253}, {1259, 23072}, {1331, 1818}, {1332, 3977}, {1437, 11573}, {1473, 3167}, {1790, 4288}, {1795, 22350}, {1797, 22356}, {1812, 4001}, {1943, 14213}, {1944, 14206}, {2979, 5285}, {3193, 4292}, {3292, 3937}, {3305, 17811}, {3916, 22136}, {3917, 3955}, {4511, 11700}, {4855, 7078}, {4867, 6126}, {5310, 7186}, {5422, 5437}, {5440, 22141}, {6507, 7099}, {6515, 20266}, {7171, 11456}, {7289, 20806}, {9037, 20989}, {14597, 22133}, {17976, 22148}, {20746, 22156}, {22060, 22139}

X(22128) = isogonal conjugate of polar conjugate of X(320)
X(22128) = isotomic conjugate of polar conjugate of X(36)
X(22128) = X(19)-isoconjugate of X(80)

X(22129) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22129) lies on these lines: {2, 1407}, {3, 1331}, {6, 2243}, {57, 10601}, {63, 77}, {81, 2255}, {220, 15066}, {221, 2975}, {283, 23072}, {329, 17074}, {651, 5744}, {940, 4415}, {958, 1406}, {971, 2000}, {1191, 16948}, {1259, 4303}, {1413, 4296}, {1473, 3796}, {1977, 16781}, {2003, 3928}, {3157, 3916}, {3219, 17811}, {3784, 7085}, {4652, 7078}, {5710, 20076}, {6360, 20477}, {6511, 10607}, {14996, 20059}

X(22129) = isogonal conjugate of polar conjugate of anticomplement of X(45)
X(22129) = isotomic conjugate of polar conjugate of X(999)
X(22129) = X(19)-isoconjugate of X(1000)

X(22130) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22130) lies on these lines: {3, 22069}, {6, 226}, {31, 916}, {219, 22119}, {222, 3942}, {306, 394}, {323, 20017}, {1993, 3187}, {2650, 3157}, {14543, 18676}, {17811, 20106}, {17902, 21270}, {20760, 22156}

X(22130) = isogonal conjugate of polar conjugate of X(21270)
X(22130) = isotomic conjugate of polar conjugate of X(23843)

X(22131) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22131) lies on these lines: {1, 6}, {3, 906}, {8, 1783}, {41, 1064}, {48, 4303}, {63, 22119}, {101, 10571}, {169, 5452}, {222, 22153}, {277, 1462}, {394, 4001}, {517, 607}, {602, 672}, {692, 19153}, {1409, 3211}, {1802, 22350}, {1814, 17170}, {1951, 11249}, {2172, 3556}, {2178, 21744}, {2207, 3419}, {2286, 20818}, {2289, 22063}, {3157, 20752}, {3434, 17905}, {3827, 18596}, {3927, 22120}, {8608, 11508}, {8735, 10525}, {15905, 22118}, {20741, 22127}

X(22132) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^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(22132) lies on these lines: {1, 6}, {3, 2197}, {48, 22350}, {71, 255}, {159, 692}, {181, 6056}, {222, 10319}, {306, 394}, {478, 1766}, {517, 608}, {604, 1066}, {610, 1103}, {651, 4329}, {906, 15905}, {1264, 1332}, {1333, 8069}, {1409, 3157}, {1950, 11248}, {2303, 3085}, {3197, 18598}, {3211, 20752}, {4261, 8071}, {5285, 7074}, {5301, 11508}, {5776, 9370}, {7124, 20818}, {8736, 10526}, {18650, 20744}, {20741, 20745}, {20765, 20770}, {22117, 22139}, {22144, 22147}

X(22133) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22133) lies on these lines: {2, 6}, {3, 22073}, {71, 3955}, {219, 3157}, {283, 18591}, {511, 1474}, {572, 5562}, {573, 1092}, {651, 18631}, {2327, 3284}, {7078, 8766}, {14597, 22128}, {20742, 22145}, {22122, 22126}

X(22134) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22134) lies on these lines: {1, 6}, {3, 1409}, {47, 1333}, {48, 255}, {63, 16697}, {71, 22083}, {77, 20744}, {394, 22119}, {517, 1880}, {573, 10571}, {577, 828}, {602, 604}, {651, 17134}, {692, 18611}, {906, 2289}, {1064, 2269}, {1332, 3718}, {1397, 2352}, {1682, 7066}, {1766, 4559}, {2268, 21741}, {2280, 21743}, {2286, 3157}, {2288, 4254}, {2327, 3561}, {3167, 20752}, {3211, 7124}, {3692, 4574}, {4047, 17102}, {7352, 15945}, {20745, 22163}

X(22135) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22135) lies on these lines: {3, 22075}, {6, 25}, {394, 10316}, {1503, 13854}, {5596, 8879}, {17409, 19149}

X(22135) = isogonal conjugate of polar conjugate of X(5596)
X(22135) = isotomic conjugate of polar conjugate of X(20993)

X(22136) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22136) lies on these lines: {1, 15910}, {3, 49}, {6, 4658}, {21, 323}, {63, 23070}, {72, 18447}, {78, 22141}, {81, 6675}, {110, 2915}, {191, 8614}, {219, 3157}, {399, 16117}, {405, 1993}, {442, 3193}, {451, 2895}, {474, 15066}, {500, 2328}, {501, 1030}, {511, 20831}, {942, 2323}, {1330, 4585}, {1332, 3695}, {1994, 5047}, {2979, 20833}, {3560, 16266}, {3564, 21530}, {3916, 22128}, {3940, 7078}, {4205, 15988}, {4423, 16472}, {5422, 16842}, {5706, 17528}, {5752, 9306}, {6883, 12161}, {6985, 15068}, {7193, 11573}, {7580, 11441}, {10601, 16853}, {11004, 16859}, {15018, 17534}, {16408, 17811}, {16855, 17825}, {17814, 19541}, {17971, 22158}, {17976, 23079}, {20740, 22146}, {20762, 20809}, {22161, 22458}

X(22135) = X(92)-isoconjugate of X(34427)

X(22137) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22137) lies on these lines: {3, 22077}, {6, 16587}, {48, 3784}, {63, 20808}, {219, 23068}, {20739, 20760}

X(22137) = isogonal conjugate of polar conjugate of X(21289)
X(22137) = isotomic conjugate of polar conjugate of X(20994)

X(22138) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22138) lies on these lines: {3, 1176}, {6, 8623}, {48, 3784}, {69, 22143}, {206, 21512}, {255, 3781}, {394, 20794}, {1974, 9821}, {3313, 6660}, {9969, 21513}, {13111, 17500}, {17976, 22458}, {22062, 22151}

X(22139) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22139) lies on these lines: {3, 49}, {6, 16058}, {58, 20849}, {63, 17972}, {71, 3955}, {81, 8731}, {110, 199}, {212, 3781}, {219, 7015}, {228, 17976}, {238, 21334}, {323, 4184}, {440, 3564}, {511, 2328}, {573, 9306}, {582, 16422}, {1011, 1993}, {1214, 17975}, {1331, 3690}, {1350, 20841}, {1351, 13615}, {1654, 2905}, {2651, 17778}, {2979, 16064}, {3219, 21318}, {3819, 13329}, {4191, 15066}, {4199, 15988}, {5422, 16373}, {6090, 11350}, {6822, 17349}, {7193, 22097}, {16059, 17811}, {22060, 22128}, {22117, 22132}, {22143, 23081}

X(22139) = isogonal conjugate of polar conjugate of X(1654)
X(22139) = isotomic conjugate of polar conjugate of X(18755)
X(22139) = X(19)-isoconjugate of X(6625)

X(22140) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22140) lies on these lines: {3, 15373}, {219, 20785}, {222, 20742}, {394, 7124}, {20741, 20745}, {20807, 20814}

X(22141) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22141) lies on these lines: {3, 1331}, {78, 22136}, {219, 1807}, {394, 22142}, {651, 9945}, {1616, 10700}, {1795, 22117}, {3722, 16466}, {3927, 7004}, {4855, 23070}, {5315, 21870}, {5440, 22128}, {7074, 10703}, {16483, 17460}, {23079, 23083}

X(22142) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22142) lies on these lines: {3, 22067}, {219, 23071}, {394, 22141}

X(22143) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22143) lies on these lines: {3, 895}, {6, 694}, {32, 2936}, {69, 22138}, {71, 20802}, {99, 11596}, {248, 6391}, {648, 2782}, {1576, 2854}, {1942, 15316}, {2055, 21651}, {2393, 6660}, {2407, 9512}, {2452, 13188}, {2510, 22146}, {3095, 8541}, {3284, 8681}, {5467, 7669}, {6321, 8754}, {9214, 12355}, {9976, 15919}, {10765, 21309}, {15143, 15262}, {17976, 20746}, {19597, 22120}, {20740, 20795}, {20766, 22356}, {20785, 20813}, {20796, 22123}, {20806, 22152}, {22122, 23094}, {22139, 23081}, {22144, 22158}, {22145, 22148}

X(22143) = isogonal conjugate of polar conjugate of X(148)
X(22143) = isotomic conjugate of polar conjugate of X(20998)
X(22143) = crosssum of polar conjugates of PU(40)
X(22143) = X(19)-isoconjugate of X(35511)
X(22143) = X(92)-isoconjugate of X(9217)

X(22144) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22144) lies on these lines: {3, 906}, {6, 101}, {48, 22122}, {219, 1807}, {239, 21602}, {294, 15251}, {607, 1482}, {952, 1783}, {1421, 5540}, {1565, 1814}, {1951, 22765}, {3157, 22153}, {3211, 22124}, {4361, 21429}, {5299, 16550}, {8735, 10738}, {14578, 15905}, {17976, 20811}, {20739, 22127}, {20752, 23071}, {20762, 20809}, {20769, 20808}, {22086, 22148}, {22123, 22356}, {22132, 22147}, {22143, 22158}, {22146, 22156}

X(22145) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22145) lies on these lines: {3, 22084}, {6, 7}, {219, 20740}, {222, 3942}, {345, 394}, {692, 20871}, {1993, 3210}, {2003, 2288}, {2808, 8750}, {20742, 22133}, {20744, 22123}, {22143, 22148}

X(22145) = isogonal conjugate of polar conjugate of X(150)
X(22145) = isotomic conjugate of polar conjugate of X(20999)
X(22145) = X(92)-isoconjugate of X(34179)

X(22146) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^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(22146) lies on these lines: {3, 248}, {6, 13}, {30, 13509}, {39, 49}, {112, 5663}, {155, 22120}, {195, 15093}, {232, 10540}, {287, 339}, {394, 4175}, {511, 13115}, {568, 10311}, {577, 23039}, {1154, 10313}, {1562, 17702}, {1968, 18439}, {1970, 14130}, {1971, 2070}, {1993, 22253}, {2079, 2088}, {2420, 10620}, {2510, 22143}, {2871, 11641}, {3289, 22121}, {5938, 14917}, {6102, 10312}, {7735, 18917}, {8779, 10317}, {10316, 18436}, {10766, 14984}, {14961, 22115}, {15905, 18877}, {20740, 22136}, {22144, 22156}

X(22147) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22147) lies on these lines: {3, 48}, {6, 7373}, {9, 10246}, {19, 8148}, {281, 12645}, {394, 23089}, {517, 18594}, {610, 12702}, {1375, 20110}, {2256, 6767}, {2323, 10680}, {5049, 16667}, {5120, 17796}, {20752, 22149}, {22124, 23071}, {22132, 22144}

X(22147) = isogonal conjugate of polar conjugate of X(2516)
X(22147) = isotomic conjugate of polar conjugate of X(21000)
X(22147) = X(19)-isoconjugate of X(36606)

X(22148) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22148) lies on these lines: {3, 1331}, {6, 6377}, {63, 17972}, {109, 2810}, {222, 295}, {394, 22149}, {1054, 14122}, {1407, 16059}, {3157, 20805}, {3167, 23089}, {3784, 20786}, {3955, 22390}, {4641, 20601}, {7078, 23085}, {10756, 14936}, {17976, 22128}, {20741, 20785}, {20744, 20796}, {22086, 22144}, {22143, 22145}, {22158, 22384}, {22458, 23070}, {23083, 23091}

X(22148) = isogonal conjugate of polar conjugate of X(4440)
X(22148) = isotomic conjugate of polar conjugate of X(9259)
X(22148) = X(19)-isoconjugate of X(6630)

X(22149) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22149) lies on these lines: {3, 63}, {57, 16409}, {144, 4192}, {219, 20785}, {222, 17976}, {329, 19540}, {394, 22148}, {846, 3295}, {851, 20078}, {956, 11688}, {968, 6767}, {1282, 6244}, {1376, 4090}, {1403, 1757}, {2223, 16570}, {2318, 3784}, {3218, 16059}, {3219, 16058}, {3504, 23091}, {3684, 16557}, {3955, 23095}, {3980, 9709}, {4067, 15654}, {9965, 16056}, {10025, 19541}, {16574, 19342}, {20745, 20765}, {20752, 22147}, {20797, 22117}, {20818, 22163}

X(22149) = isogonal conjugate of polar conjugate of X(1278)
X(22149) = isotomic conjugate of polar conjugate of X(16969)
X(22149) = X(19)-isoconjugate of X(38247)
X(22149) = X(92)-isoconjugate of X(36614)

X(22150) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(3001)

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 - 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 - 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 - 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(22150) lies on these lines: {2, 3}, {86, 3001}

X(22151) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^4 - b^4 + b^2 c^2 - c^4) : :
Barycentrics    (cot A)(2 sin 2A - 3 tan ω) : :

Barycentrics    4 cos^2 A - 3 cot A tan ω : :

X(22151) lies on these lines: {2, 6}, {3, 22087}, {22, 19153}, {23, 6593}, {49, 15074}, {110, 2393}, {182, 5890}, {184, 11511}, {186, 249}, {206, 12220}, {287, 328}, {316, 8744}, {401, 14570}, {525, 3049}, {542, 1568}, {575, 1199}, {576, 1092}, {648, 3260}, {858, 2892}, {895, 3292}, {1147, 8538}, {1176, 11574}, {1332, 20808}, {1350, 10298}, {1351, 6644}, {1352, 7577}, {1503, 3153}, {1531, 10706}, {1570, 15560}, {1576, 3001}, {2071, 2781}, {2072, 3564}, {2930, 15826}, {2987, 14910}, {3060, 19136}, {3167, 10602}, {3266, 17708}, {3284, 4558}, {3313, 19121}, {3926, 22120}, {5038, 22416}, {5050, 7514}, {5622, 13754}, {5651, 9813}, {5866, 10766}, {6090, 11405}, {6390, 22121}, {6636, 19127}, {6660, 9407}, {6776, 18445}, {7464, 9970}, {8541, 9306}, {8549, 11441}, {8705, 19596}, {9512, 21531}, {9723, 15905}, {9967, 18475}, {9968, 12279}, {9971, 13595}, {9977, 14763}, {10564, 10752}, {10989, 19379}, {11179, 15032}, {11180, 15068}, {11470, 13346}, {14649, 18860}, {14853, 18420}, {14927, 19149}, {14984, 18449}, {15038, 18583}, {15053, 19161}, {15516, 19150}, {15818, 19125}, {16163, 19924}, {17206, 22366}, {19118, 21213}, {22062, 22138}

X(22151) = reflection of X(186) in X(15462)
X(22151) = isogonal conjugate of X(8791)
X(22151) = isotomic conjugate of polar conjugate of X(23)
X(22151) = inverse-in-MacBeath-circumconic of X(69)
X(22151) = X(19)-isoconjugate of X(67)
X(22151) = X(92)-isoconjugate of X(3455)
X(22151) = crossdifference of every pair of points on line X(512)X(1843)

X(22152) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22152) lies on these lines: {3, 69}, {6, 3229}, {25, 7779}, {160, 3630}, {193, 11328}, {219, 20785}, {237, 20080}, {264, 13108}, {2782, 14615}, {3095, 14913}, {3157, 17976}, {3289, 20233}, {5020, 7774}, {7467, 10513}, {7855, 9917}, {7877, 10790}, {8266, 15533}, {13188, 20477}, {16419, 16990}, {20769, 23086}, {20806, 22143}

X(22152) = isogonal conjugate of polar conjugate of X(20081)
X(22152) = isotomic conjugate of polar conjugate of X(21001)
X(22152) = X(19)-isoconjugate of X(38262)
X(22152) = X(92)-isoconjugate of X(36615)

X(22153) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22153) lies on these lines: {3, 48}, {6, 3333}, {9, 12675}, {56, 101}, {169, 354}, {220, 3576}, {222, 22131}, {910, 12704}, {946, 5781}, {2272, 10306}, {2911, 5022}, {3157, 22144}, {3197, 6769}, {3555, 7719}, {3730, 8273}, {7078, 7124}

X(22153) = isogonal conjugate of polar conjugate of anticomplement of X(200)
X(22153) = isogonal conjugate of polar conjugate of X(36845)
X(22153) = isotomic conjugate of polar conjugate of X(21002)

X(22154) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22154) lies on these lines: {3, 22090}, {6, 514}, {525, 3049}, {663, 16466}, {810, 22160}, {838, 3733}, {905, 4131}, {1203, 4040}, {1459, 17976}, {5711, 17072}, {6332, 20808}, {7252, 14349}, {17922, 20295}

X(22154) = isogonal conjugate of polar conjugate of anticomplement of X(649)
X(22154) = isotomic conjugate of polar conjugate of X(4057)
X(22154) = X(19)-isoconjugate of X(8050)

X(22155) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22155) lies on these lines: {3, 22092}, {6, 665}, {647, 22121}, {905, 4131}, {1459, 22157}, {2196, 22384}, {2530, 7252}, {4435, 16502}, {22086, 22144}

X(22155) = isogonal conjugate of polar conjugate of anticomplement of X(659)
X(22155) = isotomic conjugate of polar conjugate of X(21003)

X(22156) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22156) lies on these lines: {3, 4575}, {6, 16592}, {43, 5348}, {78, 22136}, {212, 3781}, {20741, 20813}, {20746, 22128}, {20760, 22130}, {22143, 22145}, {22144, 22146}

X(22156) = isogonal conjugate of polar conjugate of X(21221)
X(22156) = isotomic conjugate of polar conjugate of X(21004)

X(22157) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22157) lies on these lines: {3, 22095}, {6, 513}, {521, 2522}, {525, 3049}, {832, 7252}, {1459, 22155}, {5280, 21390}, {20816, 23092}

X(22157) = isogonal conjugate of polar conjugate of X(21301)
X(22157) = isotomic conjugate of polar conjugate of X(21005)

X(22158) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22158) lies on these lines: {3, 1332}, {48, 20762}, {219, 2196}, {17971, 22136}, {20760, 23073}, {20794, 20795}, {20796, 22356}, {22127, 23088}, {22143, 22144}, {22148, 22384}

X(22158) = isogonal conjugate of polar conjugate of X(9263)
X(22158) = isotomic conjugate of polar conjugate of X(1979)
X(22158) = X(19)-isoconjugate of X(9295)
X(22158) = X(92)-isoconjugate of X(9265)

X(22159) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22159) lies on these lines: {3, 2524}, {6, 512}, {525, 3049}, {647, 22121}, {826, 3050}, {2451, 3800}, {2510, 15451}, {5359, 5996}, {8711, 21006}

X(22159) = isogonal conjugate of polar conjugate of anticomplement of X(669)
X(22159) = isotomic conjugate of polar conjugate of X(21006)

X(22160) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22160) lies on these lines: {3, 905}, {21, 17496}, {55, 1734}, {405, 4391}, {514, 21789}, {647, 8673}, {810, 22154}, {1459, 4091}, {2401, 6914}, {3295, 3900}, {3309, 8641}, {3733, 8637}, {3803, 8642}, {5248, 8714}, {6002, 13245}, {16158, 21301}, {20796, 22383}

X(22160) = isogonal conjugate of polar conjugate of anticomplement of X(693)
X(22160) = isotomic conjugate of polar conjugate of X(21007)
X(22160) = isogonal conjugate of polar conjugate of X(17494)

X(22161) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22161) lies on these lines: {1, 9959}, {3, 73}, {55, 7186}, {63, 17972}, {219, 20785}, {228, 22128}, {333, 20256}, {394, 17976}, {651, 4192}, {1331, 3917}, {3167, 23095}, {3562, 9840}, {3781, 20804}, {4020, 7116}, {22136, 22458}

X(22161) = isogonal conjugate of polar conjugate of X(6646)
X(22161) = isotomic conjugate of polar conjugate of X(21008)

X(22162) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22162) lies on these lines: {3, 22098}, {6, 5883}, {219, 3157}, {905, 4131}, {4574, 20741}, {11573, 22054}

X(22162) = isogonal conjugate of polar conjugate of X(17491)
X(22162) = isotomic conjugate of polar conjugate of X(21009)

X(22163) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22163) lies on these lines: {3, 295}, {6, 982}, {48, 3955}, {63, 77}, {71, 3784}, {72, 22127}, {579, 1401}, {846, 2256}, {2200, 20805}, {3684, 20995}, {3927, 22126}, {4334, 17754}, {5120, 5364}, {5227, 14597}, {20739, 23070}, {20745, 22134}, {20818, 22149}

X(22163) = isogonal conjugate of polar conjugate of anticomplement of X(984)
X(22163) = isotomic conjugate of polar conjugate of X(21010)
X(22163) = isogonal conjugate of polar conjugate of X(24349)

X(22164) =  (A,B,C,X(2); A',B',C',X(3)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22164) lies on these lines: {3, 295}, {6, 3874}, {63, 20744}, {71, 11573}, {72, 20752}, {219, 3157}, {2200, 20785}, {2284, 3730}, {3694, 14597}, {3940, 22127}, {4456, 8679}, {17165, 17915}, {20741, 23070}, {20760, 23076}, {22457, 23084}, {23077, 23083}

X(22164) = isogonal conjugate of polar conjugate of X(17165)
X(22164) = isotomic conjugate of polar conjugate of X(20990)
X(22164) = X(92)-isoconjugate of X(34443)

X(22165) =  X(2)X(6)∩X(7)X(4478)

Barycentrics    5(b^2+c^2)-4a^2 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28154.

X(22165) lies on these lines: {2, 6}, {7, 4478}, {8, 7238}, {76, 8352}, {182, 11812}, {315, 11317}, {319, 7263}, {320, 4665}, {376, 15069}, {511, 3845}, {518, 3919}, {519, 4743}, {542, 8703}, {543, 7848}, {545, 17294}, {547, 576}, {549, 8550}, {575, 11539}, {620, 8787}, {625, 16509}, {633, 5349}, {634, 5350}, {1078, 12151}, {1086, 4405}, {1350, 11001}, {1352, 3830}, {1353, 10168}, {1503, 3534}, {2321, 4912}, {2854, 3917}, {2930, 6636}, {2979, 8705}, {3081, 12583}, {3094, 11055}, {3098, 15690}, {3363, 9466}, {3416, 4677}, {3545, 11477}, {3564, 12100}, {3818, 12101}, {3819, 9027}, {3828, 4663}, {3849, 14929}, {3860, 18358}, {3933, 7810}, {3943, 4741}, {4364, 17374}, {4399, 7232}, {4643, 16676}, {4851, 16673}, {4969, 17227}, {4971, 17274}, {5066, 5480}, {5085, 15719}, {5092, 19711}, {5206,7767}, {5254, 7883}, {5476, 10109}, {5569, 7908}, {5585, 11147}, {5648, 6030}, {5921, 15697}, {5965, 15713}, {5969, 14711}, {6101, 12061}, {6776, 15698}, {7277, 17228}, {7485, 8546}, {7750, 9855}, {7751, 8360}, {7759, 8367}, {7768, 8370}, {7794, 8369}, {7811, 8598}, {7813, 15810}, {7820, 19661}, {7821, 12815}, {7854, 8359}, {7869, 8365}, {7896, 8355}, {8353, 11161}, {9830, 15300}, {10519, 19708}, {10541, 15708}, {11179, 11898}, {11645, 19710}, {14645, 19662}, {15685, 18440}, {15687, 18553}, {17132, 17345}, {17133, 17372}, {17243, 17344}, {17246, 17373}, {17272, 17390}, {17273, 17388}, {17287, 17365}, {17288, 17362}, {17295, 17334}, {17296, 17332}

X(22165) = midpoint of X(i) and X(j) for these {i,j}: {2, 15533}, {6, 11160}, {69, 599}, {376, 15069}, {1350, 11180}, {5648, 13169}, {11179, 11898}
X(22165) = reflection of X(i) in X(j) for these {i,j}: {141, 599}, {576, 547}, {597, 141}, {599, 3631}, {1353, 10168}, {1992, 3589}, {3629, 597}, {4663, 3828}, {5480, 11178}, {8550, 549}, {8584, 2}, {8787, 620}, {15687, 18553}
X(22165) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2, 69, 15533}, {2, 8584, 597}, {69, 141, 3630}, {141, 8584, 2}, {3589, 3620, 141}, {9771, 15598, 11168}

X(22166) =  X(1)X(2)∩X(4902)X(7988)

Barycentrics    -7a^3 + 6a^2 (b + c) - 12(b - c)^2(b + c) + a(17 b^2 - 30 b c + 17 c^2) : :

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(22166) and X(22266) are equal.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28177.

X(22166) lies on these lines: {1, 2}, {4902, 7988}

X(22167) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22167) lies on these lines: {10, 22172}, {37, 42}, {38, 192}, {75, 244}, {141, 3123}, {145, 984}, {256, 6542}, {536, 4022}, {594, 3122}, {678, 15624}, {688, 21834}, {714, 4043}, {726, 3702}, {740, 4642}, {982, 1278}, {1221, 18059}, {2170, 20864}, {2228, 17229}, {2292, 3993}, {2310, 3056}, {2321, 3778}, {3120, 21927}, {3121, 6378}, {3747, 21061}, {3764, 17299}, {3840, 20892}, {3877, 17460}, {3943, 21035}, {3954, 20686}, {3963, 21100}, {3994, 21080}, {4033, 21238}, {4046, 21936}, {4392, 4788}, {4443, 17233}, {4492, 17311}, {4516, 7237}, {4695, 4709}, {4772, 17063}, {4941, 17236}, {7148, 21024}, {17355, 20456}, {20681, 21809}, {20703, 21804}, {20707, 22168}, {21827, 22206}, {22170, 22175}, {22171, 22173}, {22177, 22193}, {22180, 22185}, {22188, 22211}, {22207, 22210}

X(22168) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22168) lies on these lines: {10, 22169}, {1441, 18210}, {2171, 21807}, {3778, 4516}, {20707, 22167}, {20975, 21011}, {22171, 22181}, {22172, 22210}, {22201, 22209}

X(22169) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22169) lies on these lines: {10, 22168}, {42, 181}, {71, 20975}, {216, 22389}, {307, 18210}, {6467, 20777}, {20775, 22059}, {20821, 22370}, {22173, 22174}, {22175, 22195}, {22176, 22194}, {22181, 22213}, {22200, 22201}

X(22170) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22170) lies on these lines: {10, 22168}, {20975, 21012}, {22167, 22175}, {22186, 22213}

X(22171) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22171) lies on these lines: {10, 22200}, {37, 22325}, {321, 3125}, {756, 3954}, {982, 21902}, {3124, 15523}, {3452, 17435}, {3701, 22039}, {3721, 3971}, {3773, 21954}, {3930, 21796}, {7237, 20709}, {20255, 21416}, {21345, 22215}, {21827, 22220}, {22167, 22173}, {22168, 22181}, {22177, 22198}, {22180, 22189}, {22188, 22193}, {22201, 22204}, {22207, 22208}

X(22172) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22172) lies on these lines: {9, 20456}, {10, 22167}, {37, 3122}, {42, 4890}, {71, 20984}, {142, 3123}, {192, 17065}, {244, 3663}, {256, 16826}, {291, 17261}, {756, 3986}, {982, 17247}, {1400, 3747}, {1964, 8610}, {2228, 17243}, {2309, 17053}, {3009, 21746}, {3728, 5257}, {3764, 16777}, {3948, 21095}, {3963, 21257}, {4022, 4364}, {4356, 4642}, {4357, 21330}, {4443, 4687}, {4446, 4664}, {4484, 16675}, {4499, 7240}, {4695, 4780}, {4704, 12782}, {4941, 17063}, {20683, 21826}, {20684, 21827}, {20686, 21808}, {20711, 21101}, {21345, 22232}, {22168, 22210}, {22187, 22197}, {22201, 22227}

X(22173) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22173) lies on these lines: {2, 2170}, {10, 20684}, {42, 16583}, {43, 17451}, {210, 20706}, {756, 20681}, {2171, 2238}, {2294, 21904}, {3740, 20593}, {3930, 4685}, {4642, 21838}, {4695, 21877}, {16606, 21951}, {20686, 20709}, {21044, 21925}, {21345, 22220}, {21827, 22215}, {22167, 22171}, {22169, 22174}, {22193, 22211}, {22194, 22219}

X(22174) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22174) lies on these lines: {2, 256}, {9, 20984}, {10, 22167}, {38, 17065}, {42, 21892}, {238, 19318}, {244, 4357}, {750, 1716}, {756, 3778}, {1213, 3122}, {1962, 2092}, {2228, 4698}, {3123, 3739}, {4022, 4708}, {4772, 4941}, {5224, 21330}, {8040, 20966}, {14815, 17514}, {17063, 17236}, {18904, 21921}, {21827, 22201}, {22169, 22173}, {22176, 22198}, {22182, 22210}, {22203, 22204}

X(22175) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22175) lies on these lines: {10, 22194}, {22167, 22170}, {22169, 22195}

X(22176) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22176) lies on these lines: {10, 22195}, {22167, 22170}, {22169, 22194}, {22174, 22198}

X(22177) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22177) lies on these lines: {228, 1962}, {18671, 20760}, {21827, 22197}, {22167, 22193}, {22169, 22173}, {22171, 22198}, {22184, 22194}

X(22178) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22178) lies on these lines: {10, 22168}, {20975, 21016}

X(22179) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22179) lies on these lines: {10, 22168}, {20975, 21017}, {22223, 22228}

X(22180) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22180) lies on these lines: {42, 3970}, {1930, 4475}, {3125, 7148}, {3728, 3954}, {21827, 22203}, {22167, 22185}, {22171, 22189}, {22181, 22188}, {22190, 22210}

X(22181) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22181) lies on these lines: {10, 22204}, {22168, 22171}, {22169, 22213}, {22180, 22188}, {22209, 22218}

X(22182) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22182) lies on these lines: {20707, 22167}, {22174, 22210}, {22189, 22196}

X(22183) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22183) lies on these lines: {20707, 22167}

X(22184) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22184) lies on these lines: {10, 21345}, {37, 4685}, {75, 6379}, {740, 21838}, {756, 20688}, {1015, 4359}, {1107, 4970}, {1500, 3896}, {1962, 6155}, {2229, 17163}, {3121, 21020}, {3210, 16975}, {3696, 16584}, {3741, 6377}, {4093, 4111}, {4457, 21897}, {4651, 21327}, {4709, 21877}, {18904, 21085}, {22167, 22171}, {22177, 22194}, {22206, 22215}

X(22185) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22185) lies on these lines: {10, 22232}, {3294, 20681}, {21802, 21803}, {22167, 22180}

X(22186) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22186) lies on these lines: {10, 22201}, {1084, 21022}, {3121, 3963}, {21238, 21835}, {22168, 22171}, {22170, 22213}, {22204, 22209}

X(22187) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22187) lies on these lines: {1423, 21328}, {20707, 22167}, {22169, 22173}, {22172, 22197}

X(22188) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22188) lies on these lines: {22167, 22211}, {22171, 22193}, {22180, 22181}

X(22189) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22189) lies on these lines: {10, 22167}, {37, 22293}, {76, 21330}, {244, 20888}, {756, 3970}, {3122, 21024}, {3123, 21240}, {3501, 20984}, {3728, 4890}, {3778, 21071}, {3954, 20711}, {21257, 22028}, {22171, 22180}, {22182, 22196}, {22202, 22210}

X(22190) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22190) lies on these lines: {10, 22167}, {76, 244}, {756, 3954}, {3122, 7148}, {3123, 20255}, {3721, 20711}, {6376, 21330}, {22180, 22210}

X(22191) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22191) lies on these lines: {10, 22208}, {22167, 22171}, {22215, 22220}, {22222, 22223}

X(22192) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22192) lies on these lines: {10, 22207}, {22167, 22171}

X(22193) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22193) lies on these lines: {21827, 22225}, {22167, 22177}, {22171, 22188}, {22173, 22211}

X(22194) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22194) lies on these lines: {10, 22175}, {3675, 20880}, {3721, 4516}, {3954, 20704}, {4890, 21804}, {20707, 22167}, {22169, 22176}, {22171, 22180}, {22173, 22219}, {22177, 22184}

X(22195) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22195) lies on these lines: {10, 22176}, {756, 21804}, {2321, 4516}, {20545, 20633}, {20594, 20864}, {20684, 22206}, {20707, 22167}, {22169, 22175}, {22171, 22188}, {22210, 22214}

X(22196) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22196) lies on these lines: {37, 2209}, {181, 756}, {321, 17891}, {2643, 21713}, {22167, 22180}, {22168, 22171}, {22182, 22189}

X(22197) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22197) lies on these lines: {1, 41}, {10, 20684}, {28, 19554}, {72, 20706}, {213, 2171}, {960, 20593}, {1953, 2176}, {1959, 16827}, {2218, 9447}, {3294, 21809}, {16524, 18671}, {21044, 21930}, {21827, 22177}, {22167, 22180}, {22172, 22187}, {22210, 22219}, {22218, 22220}

X(22198) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22198) lies on these lines: {1962, 4890}, {3061, 20864}, {20684, 21827}, {20707, 22167}, {22171, 22177}, {22174, 22176}

X(22199) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(22199) lies on these lines: {1, 21838}, {2, 668}, {6, 23374}, {38, 3121}, {39, 42}, {43, 2275}, {75, 6379}, {76, 21223}, {244, 22173}, {292, 3961}, {518, 16584}, {519, 21877}, {672, 20228}, {726, 21345}, {893, 32913}, {982, 6377}, {984, 21827}, {1011, 2241}, {1278, 36645}, {1449, 2276}, {1500, 17018}, {1574, 4651}, {1575, 4541}, {1757, 30646}, {2092, 16778}, {2229, 17135}, {2238, 17053}, {2886, 16592}, {3510, 24575}, {3681, 21830}, {3741, 16606}, {3778, 20462}, {3971, 20363}, {4022, 6375}, {8624, 21750}, {9284, 29655}, {9336, 25502}, {16058, 16781}, {16345, 31490}, {16746, 16887}, {17165, 21327}, {18152, 26815}, {18172, 20255}, {18904, 33064}, {18905, 29673}, {20457, 23638}, {20688, 32925}, {20859, 20870}, {20861, 23636}, {20864, 23362}, {21080, 23488}, {21226, 31008}, {21330, 22171}, {21757, 23579}, {21796, 37657}, {22343, 23415}, {23413, 23418}, {23414, 23416}, {23420, 23436}, {23423, 23428}, {23431, 23452}, {23448, 23451}, {23470, 23538}, {23546, 23548}, {23574, 24534}, {24519, 34063}, {25286, 27035}, {25287, 27091}, {26973, 30955}, {30647, 32912}

X(22200) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22200) lies on these lines: {1, 3981}, {2, 20861}, {10, 22171}, {42, 2054}, {51, 21760}, {71, 20461}, {213, 21813}, {740, 21954}, {1196, 1197}, {3051, 20961}, {3122, 16584}, {3125, 3914}, {3271, 21757}, {3720, 20859}, {3721, 4425}, {3774, 21936}, {3778, 21838}, {3948, 22039}, {14599, 20988}, {17889, 20271}, {20684, 21827}, {22169, 22201}

X(22201) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22201) lies on these lines: {10, 22186}, {37, 4033}, {142, 1646}, {1084, 21035}, {2092, 21814}, {3121, 3778}, {21798, 21819}, {21827, 22174}, {22168, 22209}, {22169, 22200}, {22171, 22204}, {22172, 22227}

X(22202) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22202) lies on these lines: {1, 6}, {10, 22171}, {986, 21883}, {2292, 21820}, {3124, 20653}, {3125, 4647}, {3721, 4037}, {22167, 22180}, {22189, 22210}, {22207, 22225}

X(22203) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22203) lies on these lines: {1962, 20703}, {21827, 22180}, {22174, 22204}

X(22204) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22204) lies on these lines: {10, 22181}, {2092, 3930}, {22171, 22201}, {22174, 22203}, {22186, 22209}

X(22205) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22205) lies on these lines: {37, 16609}, {1334, 21830}, {3709, 7064}, {20684, 21827}, {21795, 21796}

X(22206) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22206) lies on these lines: {10, 22171}, {37, 43}, {76, 321}, {141, 21416}, {756, 762}, {984, 21883}, {2321, 3971}, {3097, 21877}, {3124, 8013}, {3125, 21020}, {3208, 21879}, {3681, 21839}, {3728, 21838}, {20684, 22195}, {20690, 21833}, {21827, 22167}, {22184, 22215}

X(22207) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22207) lies on these lines: {10, 22192}, {37, 1018}, {2087, 4738}, {22167, 22210}, {22171, 22208}, {22202, 22225}

X(22208) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22208) lies on these lines: {10, 22191}, {37, 758}, {22171, 22207}

X(22209) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22209) lies on these lines: {10, 22213}, {3121, 4516}, {21043, 21906}, {22168, 22201}, {22181, 22218}, {22186, 22204}, {22210, 22227}, {22211, 22215}

X(22210) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22210) lies on these lines: {10, 22175}, {11, 1111}, {2642, 2643}, {17463, 21138}, {18210, 21144}, {22167, 22207}, {22168, 22172}, {22174, 22182}, {22180, 22190}, {22189, 22202}, {22195, 22214}, {22197, 22219}, {22209, 22227}, {22212, 22225}, {22215, 22216}

X(22211) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22211) lies on these lines: {22167, 22188}, {22173, 22193}, {22209, 22215}

X(22212) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22212) lies on these lines: {4516, 21824}, {22210, 22225}

X(22213) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22213) lies on these lines: {10, 22209}, {21047, 21906}, {22169, 22181}, {22170, 22186}

X(22214) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22214) lies on these lines: {10, 22167}, {37, 4890}, {210, 21826}, {2321, 3122}, {3123, 21255}, {3663, 21330}, {3728, 3986}, {3778, 3950}, {4029, 21035}, {4431, 17065}, {5257, 21699}, {21100, 21257}, {22195, 22210}

X(22215) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22215) lies on these lines: {10, 22192}, {244, 665}, {21345, 22171}, {21827, 22173}, {22184, 22206}, {22191, 22220}, {22209, 22211}, {22210, 22216}

X(22216) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22216) lies on these lines: {1962, 14404}, {4132, 7234}, {4453, 14421}, {4730, 21828}, {22210, 22215}

X(22217) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22217) lies on these lines: {22168, 22171}, {22221, 22226}

X(22218) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22218) lies on these lines: {10, 22186}, {37, 42}, {313, 1084}, {321, 6378}, {561, 2998}, {6375, 20891}, {6377, 20892}, {17451, 20363}, {21257, 21835}, {22181, 22209}, {22197, 22220}

X(22219) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

Barycentrics    a (b + c) (a^3 b^2 - 2 a^2 b^3 + a b^4 - 2 a^3 b c + 3 a^2 b^2 c - 6 a b^3 c + b^4 c + a^3 c^2 + 3 a^2 b c^2 + 10 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - 6 a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(22219) lies on these lines: {10, 22167}, {244, 3673}, {2310, 20271}, {3122, 21049}, {3123, 21258}, {22173, 22194}, {22197, 22210}

X(22220) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22220) lies on these lines: {10, 22167}, {37, 65}, {244, 726}, {518, 1149}, {740, 4695}, {756, 4090}, {984, 3616}, {986, 4704}, {2170, 20363}, {3122, 3932}, {3123, 3836}, {3125, 20688}, {3728, 3842}, {3778, 4078}, {3790, 17065}, {3930, 21830}, {3993, 4642}, {20366, 20598}, {21345, 22173}, {21827, 22171}, {22191, 22215}, {22197, 22218}

X(22221) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22221) lies on these lines: {2491, 21050}, {22217, 22226}

X(22222) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22222) lies on these lines: {10, 22229}, {512, 16589}, {665, 4391}, {2524, 21838}, {2533, 3709}, {4147, 21348}, {22191, 22223}

X(22223) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22223) lies on these lines: {3766, 14431}, {4455, 22319}, {17990, 21832}, {22179, 22228}, {22191, 22222}, {22210, 22215}

X(22224) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22224) lies on these lines: {10, 21056}, {667, 22381}, {798, 21051}, {1577, 21832}, {4079, 22320}, {4705, 9279}, {14407, 21901}, {16589, 21836}, {17072, 20979}, {22191, 22222}

X(22225) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22225) lies on these lines: {3125, 21824}, {21827, 22193}, {22202, 22207}, {22209, 22211}, {22210, 22212}

X(22226) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22226) lies on these lines: {75, 21351}, {325, 523}, {3221, 3728}, {4516, 22227}, {20711, 21834}, {22217, 22221}

X(22226) = isotomic conjugate of X(35573)

X(22227) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22227) lies on these lines: {351, 865}, {1646, 3123}, {4516, 22226}, {22172, 22201}, {22209, 22210}

X(22228) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22228) lies on these lines: {9402, 21836}, {22179, 22223}, {22217, 22221}

X(22229) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22229) lies on these lines: {10, 22222}, {37, 21051}, {512, 1500}, {523, 21901}, {665, 1734}, {1577, 4140}, {2276, 4367}, {2489, 3709}, {7180, 8611}, {17072, 21348}

X(22230) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22230) lies on these lines: {2, 257}, {9, 8238}, {10, 20684}, {37, 42}, {38, 20284}, {1011, 2312}, {1215, 20706}, {1926, 18152}, {2170, 3741}, {2227, 3863}, {2280, 2287}, {2292, 21838}, {3496, 4203}, {3721, 16606}, {3938, 16969}, {4095, 4651}, {4642, 21877}, {15523, 21025}, {16569, 16611}, {21827, 22171}, {22184, 22206}

X(22231) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22231) lies on these lines: {21829, 21839}, {22167, 22180}, {22191, 22222}

X(22232) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(37)

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

X(22232) lies on these lines: {10, 22185}, {1962, 21796}, {21345, 22172}, {21827, 22174}, {22167, 22171}

X(22233) =  X(6)X(1173)∩X(3567)X(15750)

Barycentrics    a^2 (5 a^8-23 a^6 (b^2+c^2)+ 39 a^4 (b^4+c^4)+29 a^2 (-b^6+a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6)+b^2 c^2 (54 b^2 c^2-35 (b^4+c^4))+8 (b^8+ c^8)) : :

See Kadir Altintas and Peter Moses, Hyacinthos 28156.

X(22233) lies on these lines: {6,1173}, {3567,15750}, {5447,15018}, {9777,11464}, {10545,14627}, {11438,15004}, {13363,15019}

X(22234) =  X(3)X(6)∩X(524)X(632)

Barycentrics    a^2*(5*a^4-9*(b^2+c^2)*a^2+2*(2*b^2-c^2)*(b^2-2*c^2)) : :
X(22234) = X(3)+9*X(6), 4*X(3)-9*X(182), X(3)-6*X(575), 2*X(3)+3*X(576), 19*X(3)-9*X(1350), 11*X(3)+9*X(1351), 14*X(3)-9*X(3098), 13*X(3)-18*X(5092), 7*X(3)+3*X(11477), X(3)+3*X(11482), 5*X(3)-9*X(12017), 23*X(3)-18*X(14810), 7*X(3)-12*X(20190), 4*X(6)+X(182), 3*X(6)+2*X(575), 6*X(6)-X(576), 19*X(6)+X(1350), 11*X(6)-X(1351), 14*X(6)+X(3098), 7*X(6)+3*X(5050)

Let ABC be a triangle, G its centroid and A'B'C' its McCay triangle. Let Ka be the symmedian point of GB'C' and Ka' the reflection of Ka in B'C'. Define Kb' and Kc' cyclically. The lines AKa', BKb', CKc' concur in X(22234).

See César Lozada, Hyacinthos 28167.

X(22234) lies on these lines: {2, 10185}, {3, 6}, {23, 15004}, {140, 8584}, {184, 14002}, {524, 632}, {542, 3091}, {546, 5476}, {597, 3628}, {895, 13472}, {1199, 15058}, {1352, 15022}, {1353, 6329}, {1493, 8542}, {1595, 15471}, {1992, 3525}, {1995, 13366}, {2548, 20398}, {2854, 15026}, {3090, 7856}, {3146, 11179}, {3292, 5422}, {3518, 8541}, {3526, 15534}, {3529, 20423}, {3544, 14561}, {3564, 12812}, {3567, 11649}, {3618, 5965}, {3818, 3857}, {4663, 15178}, {5032, 10168}, {5072, 18553}, {5079, 15069}, {5480, 12102}, {5609, 9976}, {5643, 5651}, {9306, 11422}, {10169, 18381}, {10282, 11216}, {11004, 22112}, {11255, 12107}, {11470, 14865}, {12105, 15826}, {12151, 17130}, {12811, 18583}, {14035, 18800}, {14869, 20583}, {14912, 19130}, {15018, 16187}, {17538, 19924}

X(22234) = inverse-in-Brocard-circle of X(22330)
X(22234) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(8589)
X(22234) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 22330), (6, 182, 15520), (6, 575, 576), (6, 5050, 5097), (61, 62, 574), (371, 372, 8589), (575, 576, 182), (575, 5097, 20190), (575, 20190, 5050), (576, 3098, 11477), (3098, 5050, 182), (3592, 3594, 15815), (5050, 5097, 3098), (5050, 11477, 20190), (5097, 20190, 11477), (6419, 6420, 39), (11477, 20190, 3098)

X(22235) =  X(2)X(397)∩X(13)X(20)

Barycentrics    (2*SB+sqrt(3)*S)*(2*SC+sqrt(3)*S) : :

Let ABC be a triangle, G its centroid and A'B'C' its inner Napoleon triangle. Let Ka be the symmedian point of GB'C' and Ka' the reflection of Ka in B'C'. Define Kb' and Kc' cyclically. The lines AKa', BKb', CKc' concur in X(22235).

See César Lozada, Hyacinthos 28167.

X(22235) lies on the Kiepert hyperbola and these lines: {2, 397}, {4, 11408}, {6, 5068}, {13, 20}, {14, 3091}, {15, 5366}, {16, 10188}, {17, 3523}, {18, 5056}, {61, 3839}, {62, 7486}, {396, 3146}, {398, 3854}, {459, 470}, {2041, 9693}, {2043, 14241}, {2044, 14226}, {2045, 3316}, {2046, 3317}, {3424, 5869}, {3522, 5340}, {3543, 12816}, {3832, 5339}, {5059, 5318}, {5237, 15721}, {5343, 16808}, {5352, 15697}, {5485, 11303}, {10303, 10653}, {10304, 16965}, {10611, 22113}, {10654, 12821}, {11289, 18840}, {11290, 18841}, {11304, 18842}, {15640, 16962}, {15717, 16644}

X(22235) = isogonal conjugate of X(22236)
X(22235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5068, 22237), (17, 5335, 3523), (5340, 11488, 3522)

X(22236) =  ISOGONAL CONJUGATE OF X(22235)

Trilinears    2 cos A + sqrt(3) sin A : :
Barycentrics    a^2*(2*SA+sqrt(S)) : :

See César Lozada, Hyacinthos 28167.

X(22236) lies on these lines: {2, 398}, {3, 6}, {4, 396}, {5, 5339}, {13, 382}, {14, 1656}, {17, 381}, {18, 3526}, {20, 397}, {30, 5340}, {55, 2307}, {140, 16645}, {154, 3129}, {203, 3295}, {302, 22114}, {303, 7773}, {394, 11127}, {395, 631}, {546, 18582}, {550, 10653}, {617, 11289}, {627, 5858}, {628, 11290}, {632, 11543}, {633, 11307}, {634, 5859}, {636, 11297}, {999, 7005}, {1080, 5868}, {1147, 11137}, {1498, 11243}, {1593, 8740}, {1657, 16965}, {1993, 11146}, {2981, 14170}, {3090, 5334}, {3091, 5321}, {3130, 17810}, {3146, 5318}, {3205, 9703}, {3303, 10638}, {3304, 7051}, {3515, 8739}, {3523, 16773}, {3529, 5335}, {3543, 5350}, {3545, 5343}, {3627, 11542}, {3628, 18581}, {3642, 6694}, {3830, 16267}, {3832, 5349}, {3855, 5365}, {5072, 16809}, {5076, 16960}, {5079, 16966}, {5198, 10641}, {5217, 7127}, {5362, 16865}, {5366, 15682}, {5367, 17572}, {5422, 11145}, {5869, 6770}, {6671, 11309}, {6695, 11302}, {9763, 11304}, {10303, 11489}, {10594, 10632}, {10676, 17826}, {11244, 17821}, {11403, 11408}, {11555, 15441}, {13846, 18585}, {13847, 15765}, {14138, 16626}, {15668, 21903}, {15693, 16963}, {15694, 16268}, {15720, 16242}, {17259, 21932}

X(22236) = reflection of X(22238) in X(22331)
X(22236) = isogonal conjugate of X(22235)
X(22236) = inverse-in-Brocard-circle of X(22238)
X(22236) = X(22333)-cross conjugate of X(22238)
X(22236) = X(22334)-Ceva conjugate of X(22238)
X(22236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 22238), (3, 61, 6), (3, 5611, 5865), (3, 5865, 1350), (3, 11485, 61), (3, 11486, 5237), (6, 11480, 11481), (15, 61, 3), (16, 5352, 3), (39, 10541, 22238), (61, 3389, 6420), (61, 5238, 62), (61, 14539, 7772), (62, 5238, 3), (398, 16772, 2), (576, 21401, 3), (3311, 3365, 6), (3389, 3390, 10646), (5237, 10645, 3)

X(22237) =  X(2)X(398)∩X(14)X(20)

Barycentrics    (2*SB-sqrt(3)*S)*(2*SC-sqrt(3)*S) : :

Let ABC be a triangle, G its centroid and A'B'C' its outer Napoleon triangle. Let Ka be the symmedian point of GB'C' and Ka' the reflection of Ka in B'C'. Define Kb' and Kc' cyclically. Then the lines AKa', BKb', CKc' concur in X(22237).

See César Lozada, Hyacinthos 28167.

X(22237) lies on the Kiepert hyperbola and these lines: {2, 398}, {4, 11409}, {6, 5068}, {13, 3091}, {14, 20}, {15, 10187}, {16, 5365}, {17, 5056}, {18, 3523}, {61, 7486}, {62, 3839}, {395, 3146}, {397, 3854}, {459, 471}, {2042, 9693}, {2043, 14226}, {2044, 14241}, {2045, 3317}, {2046, 3316}, {3424, 5868}, {3522, 5339}, {3543, 12817}, {3832, 5340}, {5059, 5321}, {5238, 15721}, {5344, 16809}, {5351, 15697}, {5485, 11304}, {10303, 10654}, {10304, 16964}, {10653, 12820}, {11289, 18841}, {11290, 18840}, {11303, 18842}, {15640, 16963}, {15717, 16645}

X(22237) = isogonal conjugate of X(22238)
X(22237) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5068, 22235), (18, 5334, 3523), (5339, 11489, 3522)

X(22238) =  ISOGONAL CONJUGATE OF X(22237)

Trilinears    2 cos A - sqrt(3) sin A : :
Barycentrics    a^2*(2*SA-sqrt(S)) : :
X(22238) = 2*X(15)-5*X(19780)

See César Lozada, Hyacinthos 28167.

X(22238) lies on these lines: {2, 397}, {3, 6}, {4, 395}, {5, 5340}, {13, 1656}, {14, 382}, {17, 3526}, {18, 381}, {20, 398}, {30, 5339}, {56, 7127}, {140, 16644}, {154, 3130}, {202, 3295}, {302, 7773}, {303, 22113}, {383, 5869}, {394, 11126}, {396, 631}, {532, 11302}, {546, 18581}, {550, 10654}, {616, 11290}, {627, 11289}, {628, 5859}, {632, 11542}, {633, 5858}, {634, 11308}, {635, 11298}, {999, 7006}, {1147, 11134}, {1250, 3303}, {1498, 11244}, {1593, 8739}, {1657, 16964}, {1993, 11145}, {2307, 5204}, {3090, 5335}, {3091,f 5318}, {3129, 17810}, {3146, 5321}, {3206, 9703}, {3304, 19373}, {3515, 8740}, {3523, 16772}, {3529, 5334}, {3543, 5349}, {3545, 5344}, {3627, 11543}, {3628, 18582}, {3643, 6695}, {3830, 16268}, {3832, 5350}, {3855, 5366}, {5072, 16808}, {5076, 16961}, {5079, 16967}, {5198, 10642}, {5362, 17572}, {5365, 15682}, {5367, 16865}, {5422, 11146}, {5868, 6773}, {6151, 14169}, {6672, 11310}, {6694, 11301}, {9761, 11303}, {10303, 11488}, {10594, 10633}, {10675, 17827}, {11243, 17821}, {11403, 11409}, {11556, 15442}, {13846, 15765}, {13847, 18585}, {14139, 16627}, {15668, 21932}, {15693, 16962}, {15694, 16267}, {15720, 16241}, {17259, 21903}

X(22238) = reflection of X(22236) in X(22331)
X(22238) = isogonal conjugate of X(22237)
X(22238) = inverse-in-Brocard-circle of X(22236)
X(22238) = X(22333)-cross conjugate of X(22236)
X(22238) = X(22334)-Ceva conjugate of X(22236)
X(22238) = {{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 22236), (3, 62, 6), (3, 5615, 5864), (3, 5864, 1350), (3, 11485, 5238), (3, 11486, 62), (6, 11481, 11480), (15, 5351, 3), (16, 61, 5237), (16, 62, 3), (39, 10541, 22236), (62, 3365, 6419), (62, 14538, 7772), (371, 372, 11486), (397, 16773, 2), (1151, 1152, 11481), (3312, 3389, 6), (3364, 3365, 10645), (5238, 10646, 3)

X(22239) =  X(4)X(2693)∩X(24,477)

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

X(22239) lies on the circumcircle and these lines: {4,2693}, {24,477}, {25,2697}, {28,2694}, {30,5897}, {74,403}, {110,8057}, {111,16318}, {112,6587}, {186,1294}, {468,1297}, {523,1301}, {691,2409}, {841,18533}, {842,6353}, {925,7480}, {1290,7435}, {1295,2074}, {2691,4244}, {3565,7473}, {4240,10420}, {5878,18809}, {5896,10151}, {7471,13398}, {7482,20187}

X(22239) = reflection of X(1301) in the Euler line
X(22239) = polar circle inverse of X(16177)
X(22239) = Collings transform of X(10151)
X(22239) = X(9033)-cross conjugate of X(4)
X(22239) = X(656)-isoconjugate of X(2071)
X(22239) = cevapoint of X(i) and X(j) for these (i,j): {25, 1637}, {523, 10151}
X(22239) = trilinear pole of line {6, 1562}
X(22239) = Λ;(X(1636), X(2433))
X(22239) = barycentric product X(648)*X(11744)
X(22239) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 2071}, {1637, 16177}, {11744, 525}

X(22240) =  X(2)X(216)∩X(3,112)

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

X(22240) lies on these lines: {2,216}, {3,112}, {6,22}, {20,39}, {23,5158}, {26,10312}, {32,7488}, {53,5133}, {187,10298}, {217,5889}, {376,14961}, {401,3329}, {566,858}, {570,1370}, {574,2071}, {577,6636}, {800,1194}, {1249,7494}, {1500,9538}, {1609,1627}, {1625,11459}, {1968,14118}, {1990,7495}, {1995,11062}, {2070,10986}, {2207,7503}, {2275,4296}, {2276,3100}, {2373,9087}, {2493,18573}, {2697,6795}, {2979,3289}, {3003,7493}, {3087,7500}, {3091,3199}, {3153,5475}, {3269,15072}, {3284,7492}, {3314,11672}, {3331,15305}, {5013,11413}, {5024,21312}, {5359,8573}, {5523,15760}, {6676,16318}, {6997,14576}, {7426,16328}, {7502,10317}, {7512,10316}, {7539,11197}, {7745,12225}, {7761,13219}, {9157,19153}, {9300,13351}, {9605,11414}, {9909,15851}, {10314,13595}, {10979,15246}, {11174,20477}, {15340,18474}, {15574,20806}, {16303,16387}, {19149,19158}

X(22240) = crosssum of X(125) and X(3288)
X(22240) = X(3402)-anticomplementary conjugate of X(8878)
X(22240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 232, 15355), (6, 22, 10313), (216, 232, 2), (800, 1194, 5304), (11417, 11418, 19121)

X(22241) =  X(3)X(69)∩X(25,8024)

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

X(22241) lies on these lines: {3,69}, {25,8024}, {76,6642}, {99,15574}, {315,12085}, {325,9818}, {394,14961}, {1593,7776}, {1975,7387}, {2071,10513}, {5024,15066}, {7393,7763}, {8369,8573}

X(22242) =  (name pending)

Barycentrics    a^2 (2 a^2-b^2-c^2-3 Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : :

X(22242) lies on this line: {3,6}

X(22242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 574, 14630), (6, 1380, 1379), (6, 1384, 3557), (6, 2028, 1380)

X(22243) =  (name pending)

Barycentrics    a^2 (2 a^2-b^2-c^2+3 Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : :

X(22243) lies on this line: {3,6}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 574, 14631), (6, 1379, 1380), (6, 1384, 3558), (6, 2029, 1379)

X(22244) =  X(597)X(14632)∩X(599)X(6177)

Barycentrics    (a^2+b^2-2 c^2+3 Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) (a^2-2 b^2+c^2+3 Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : :

X(22244) lies on the Kiepert hyperbola and these lines: {597,14632}, {599,6177}, {2482,3413}, {3414,5461}

X(22245) =  X(597)X(14633)∩X(599)X(6178)

Barycentrics    (a^2+b^2-2 c^2-3 Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) (a^2-2 b^2+c^2-3 Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : :

X(22245) lies on the Kiepert hyperbola and these lines: {597,14633}, {599,6178}, {2482,3414}, {3413,5461}

X(22246) =  MIDPOINT OF X(14482) AND X(14930)

Barycentrics    a^2 (5 a^2+11 b^2+11 c^2) : :

X(22246) lies on these lines: {3,6}, {30,14482}, {538,14535}, {1180,20850}, {1285,15688}, {1383,9909}, {1597,8744}, {2549,15684}, {3054,5319}, {3108,21448}, {3793,5032}, {3815,15703}, {3830,15048}, {3851,5286}, {5054,5304}, {5055,7736}, {5070,5305}, {5354,16419}, {6767,9331}, {7373,9336}, {7735,15694}, {7737,15685}, {7738,17800}, {7739,14269}, {7770,20105}, {8148,9575}, {8362,20080}, {9300,19709}, {13192,20854}, {14996,21526}, {14997,21514}, {15681,18907}, {15722,21843}

X(22246) = midpoint of X(i) and X(j) for these {i,j}: {14482, 14930}
X(22246) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 39, 1384), (6, 3053, 14075), (6, 5013, 5008), (6, 5024, 21309), (6, 5210, 5007), (574, 5041, 6), (5008, 5013, 15655), (5024, 21309, 3), (6199, 6395, 12017), (11485, 11486, 5085), (15603, 21309, 1384)

X(22247) =  MIDPOINT OF X(2) AND X(620)

Barycentrics    10 a^4-10 a^2 b^2+7 b^4-10 a^2 c^2-4 b^2 c^2+7 c^4 : :
X(22247) = 7 X[2] + X[99], 5 X[2] - X[115], 5 X[99] + 7 X[115], 17 X[115] - 5 X[148], 17 X[2] - X[148], 17 X[99] + 7 X[148], X[99] - 7 X[620], X[115] + 5 X[620], X[148] + 17 X[620], 9 X[148] - 17 X[671], 9 X[115] - 5 X[671], 9 X[2] - X[671], 9 X[620] + X[671], 9 X[99] + 7 X[671], 3 X[99] - 7 X[2482], 3 X[620] - X[2482], 3 X[2] + X[2482], X[671] + 3 X[2482], 3 X[115] + 5 X[2482], 3 X[148] + 17 X[2482], X[114] + 3 X[5054], X[325] + 3 X[5215], 3 X[148] - 17 X[5461], 3 X[115] - 5 X[5461], X[671] - 3 X[5461], 3 X[620] + X[5461], 3 X[99] + 7 X[5461], 2 X[148] - 17 X[6722], 2 X[671] - 9 X[6722], 2 X[115] - 5 X[6722], 2 X[5461] - 3 X[6722], 2 X[620] + X[6722], 2 X[2482] + 3 X[6722], 2 X[99] + 7 X[6722], 15 X[99] - 7 X[8591], 15 X[620] - X[8591], 5 X[2482] - X[8591], 15 X[2] + X[8591], 3 X[115] + X[8591], 5 X[5461] + X[8591], 15 X[6722] + 2 X[8591], 5 X[671] + 3 X[8591], 15 X[148] + 17 X[8591], 7 X[3619] + X[8593], 11 X[671] - 3 X[8596], 11 X[5461] - X[8596], 11 X[2482] + X[8596], 11 X[8591] + 5 X[8596], 7 X[3526] + X[8724], 3 X[141] + X[8787], X[7813] + 3 X[8859], 11 X[115] - 15 X[9166], 11 X[5461] - 9 X[9166], X[8596] - 9 X[9166], 11 X[6722] - 6 X[9166], 11 X[2] - 3 X[9166], 11 X[620] + 3 X[9166], 11 X[2482] + 9 X[9166], X[2482] - 9 X[9167], X[620] - 3 X[9167], X[2] + 3 X[9167], X[6722] + 6 X[9167], X[5461] + 9 X[9167], X[9166] + 11 X[9167], X[115] + 15 X[9167], 5 X[1656] - X[9880], 7 X[3624] + X[9881], 7 X[9780] + X[9884], 15 X[3763] + X[10488], 13 X[10303] - X[10991], 11 X[5070] + X[10992], X[6036] - 3 X[11539], 7 X[3526] - X[11623], 7 X[3090] + X[12117], 17 X[3533] - X[12243], 13 X[5461] - 15 X[14061], 13 X[6722] - 10 X[14061], 13 X[2] - 5 X[14061], 13 X[620] + 5 X[14061], 13 X[2482] + 15 X[14061], 9 X[5054] - X[14830], 3 X[114] + X[14830], 7 X[115] - 15 X[14971], 7 X[9166] - 11 X[14971], 7 X[5461] - 9 X[14971], 7 X[6722] - 6 X[14971], 7 X[2] - 3 X[14971], 7 X[9167] + X[14971], X[99] + 3 X[14971], 7 X[620] + 3 X[14971], 7 X[2482] + 9 X[14971], 11 X[3525] + X[14981], 11 X[8591] - 15 X[15300], 11 X[99] - 7 X[15300], 11 X[2482] - 3 X[15300], 11 X[620] - X[15300], 11 X[2] + X[15300], 3 X[9166] + X[15300], 11 X[6722] + 2 X[15300], 11 X[5461] + 3 X[15300], X[8596] + 3 X[15300], 11 X[115] + 5 X[15300], 11 X[671] + 9 X[15300], 11 X[148] + 17 X[15300], X[6055] + 3 X[15561], X[6055] - 5 X[15694], 3 X[15561] + 5 X[15694], X[10722] + 7 X[15698], X[6033] + 7 X[15701], X[6054] + 7 X[15702], X[98] - 9 X[15709], X[12042] - 5 X[15713], X[12258] - 5 X[19862], X[9875] - 17 X[19872], X[11725] - 3 X[19883], 4 X[16239] - X[20398], 2 X[140] + X[20399], 5 X[5071] + 3 X[21166], X[5477] + 3 X[21356], X[18800] + 3 X[21358]

X(22247) lies on these lines: {2,99}, {30,6721}, {98,15709}, {114,5054}, {140,542}, {141,1153}, {325,5215}, {549,2794}, {597,6680}, {599,3788}, {754,22110}, {1656,9880}, {1992,7764}, {2782,10124}, {2796,19878}, {3055,14762}, {3090,12117}, {3525,14981}, {3526,8724}, {3533,12243}, {3619,8593}, {3624,9881}, {3763,10488}, {5026,19662}, {5070,10992}, {5071,21166}, {5182,7815}, {5477,21356}, {5569,7778}, {5969,6683}, {6033,15701}, {6036,11539}, {6054,15702}, {6055,15561}, {6292,8786}, {7749,7870}, {7801,8860}, {7804,9771}, {7810,7907}, {7813,8859}, {7817,9607}, {7830,7940}, {7833,11149}, {7880,11168}, {8252,13968}, {8253,13908}, {8997,13847}, {9780,9884}, {9875,19872}, {9876,16419}, {10303,10991}, {10722,15698}, {11164,18424}, {11725,19883}, {12042,15713}, {12258,19862}, {13846,13989}, {15810,15814}, {16239,20398}, {18800,21358}

X(22247) = midpoint of X(i) and X(j) for these {i,j}: {2, 620}, {2482, 5461}, {5026, 19662}, {8724, 11623}
X(22247) = reflection of X(i) in X(j) for these {i,j}: {6722, 2}
X(22247) = complement X(5461)
X(22247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 99, 14971), (2, 2482, 5461), (2, 7618, 7844), (2, 7622, 4045), (2, 9167, 620), (115, 2482, 8591), (620, 5461, 2482), (15561, 15694, 6055)
X(22247) = X(2)-daleth conjugate of X(8591)

X(22248) =  (name pending)

Barycentrics    18 a^10-37 a^8 b^2+2 a^6 b^4+36 a^4 b^6-20 a^2 b^8+b^10-37 a^8 c^2+42 a^6 b^2 c^2-31 a^4 b^4 c^2+29 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-31 a^4 b^2 c^4-18 a^2 b^4 c^4+2 b^6 c^4+36 a^4 c^6 +29 a^2 b^2 c^6+2 b^4 c^6-20 a^2 c^8-3 b^2 c^8+c^10 : :
X(22248) = 6 X[186] - X[548], 9 X[140] - 4 X[858], 3 X[2070] + 2 X[3530], 8 X[468] - 3 X[5066], X[140] + 4 X[7575], X[858] + 9 X[7575], X[3853] - 6 X[10096], X[5189] - 6 X[11812], 2 X[23] + 3 X[12100]

X(22248) lies on this line: {2,3}

X(22248) = {X(140),X(15690)}-harmonic conjugate of X(10300)

X(22249) =  MIDPOINT OF X(140) AND X(7575)

Barycentrics    6 a^10-13 a^8 b^2+2 a^6 b^4+12 a^4 b^6-8 a^2 b^8+b^10-13 a^8 c^2+18 a^6 b^2 c^2-13 a^4 b^4 c^2+11 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-13 a^4 b^2 c^4-6 a^2 b^4 c^4+2 b^6 c^4+12 a^4 c^6+ 11 a^2 b^2 c^6+2 b^4 c^6-8 a^2 c^8-3 b^2 c^8+c^10 : :
X(22249) = X[548] - 8 X[1511], 9 X[140] - 2 X[3448], 9 X[547] - 16 X[5972], 11 X[546] - 4 X[10733], 2 X[5972] - 9 X[11694], X[547] - 8 X[11694], 15 X[547] - 8 X[11801], 10 X[5972] - 3 X[11801], 15 X[11694] - X[11801], 4 X[110] + 3 X[12100], 8 X[5642] - X[12101], 2 X[12383] + 5 X[12812], X[546] - 8 X[13392], 5 X[546] - 12 X[14643], 10 X[13392] - 3 X[14643], 6 X[11812] + X[14683], X[12308] + 6 X[14891], 2 X[13392] + 5 X[15034], 13 X[1511] + X[15063], 13 X[548] + 8 X[15063], 13 X[5972] - 6 X[15088], 15 X[548] - 8 X[16111], 15 X[1511] - X[16111], 15 X[15063] + 13 X[16111]

X(22249) lies on these lines: {2,3}, {3292,11694}, {11561,13392}, {11695,13365}, {18400,20396}

X(22249) = midpoint of X(i) and X(j) for these {i,j}: {140,7575}, {468,18571}, {546,10295}, {548,11799}, {7426,12100}, {10096,15646}, {11694,15361}, {12105,15122}
X(22249) = reflection of X(15122) in X(12108)
X(22249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 12100, 7496), (186, 14940, 10295), (468, 18579, 18571), (548, 10096, 11799), (7575, 15646, 7488), (11799, 15646, 548)

X(22250) =  X(110)X(12100)∩X(140)X(3448)

Barycentrics    26 a^10-69 a^8 b^2+50 a^6 b^4+4 a^4 b^6-12 a^2 b^8+b^10-69 a^8 c^2+98 a^6 b^2 c^2-47 a^4 b^4 c^2+21 a^2 b^6 c^2-3 b^8 c^2+50 a^6 c^4-47 a^4 b^2 c^4-18 a^2 b^4 c^4+ 2 b^6 c^4+4 a^4 c^6+21 a^2 b^2 c^6+2 b^4 c^6-12 a^2 c^8-3 b^2 c^8+c^10 : :

X(22250) lies on these lines: {110,12100}, {140,3448}, {546,10733}, {547,5972}, {548,1511}, {5642,12101}, {11812,14683}, {12308,14891}, {12383,12812}

X(22251) =  MIDPOINT OF X(3) AND X(20125)

Barycentrics    8 a^10-21 a^8 b^2+14 a^6 b^4+4 a^4 b^6-6 a^2 b^8+b^10-21 a^8 c^2+32 a^6 b^2 c^2-17 a^4 b^4 c^2+9 a^2 b^6 c^2-3 b^8 c^2+14 a^6 c^4-17 a^4 b^2 c^4-6 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+ 9 a^2 b^2 c^6+2 b^4 c^6-6 a^2 c^8-3 b^2 c^8+c^10 : :
X(22251) = 2 X[110] + 3 X[549], X[5] + 4 X[1511], 6 X[140] - X[3448], X[399] + 4 X[3530], 3 X[5] - 8 X[5972], 3 X[1511] + 2 X[5972], 13 X[5] - 8 X[7687], 13 X[5972] - 3 X[7687], 13 X[1511] + 2 X[7687], 3 X[550] + 2 X[7728], 4 X[5642] + X[8703], 18 X[7687] - 13 X[10113], 9 X[5] - 4 X[10113], 6 X[5972] - X[10113], 9 X[1511] + X[10113], X[7728] - 6 X[10272], X[550] + 4 X[10272], 7 X[3857] - 2 X[10733], 4 X[125] - 9 X[11539], X[11539] + 4 X[11693], X[125] + 9 X[11693], X[110] - 6 X[11694], X[549] + 4 X[11694], 4 X[7471] + X[11749], X[9143] + 4 X[11812], X[10620] - 6 X[12100], 3 X[3845] + 2 X[12121], 9 X[3524] + X[12308], 4 X[3628] + X[12383], 3 X[10283] + 2 X[12778], 11 X[5] - 16 X[12900], 11 X[5972] - 6 X[12900], 11 X[1511] + 4 X[12900], 6 X[547] - X[12902], X[3] + 4 X[13392], X[3627] - 6 X[14643], 9 X[5054] + X[14683], 2 X[10264] - 7 X[14869], 3 X[14643] + 7 X[15020], X[3627] + 14 X[15020], 5 X[632] - 2 X[15027], X[632] + 2 X[15034], X[15027] + 5 X[15034], X[550] - 6 X[15035], 2 X[10272] + 3 X[15035], X[7728] + 9 X[15035], 3 X[5655] + 7 X[15036], 8 X[12108] + 7 X[15039], 4 X[3861] - 9 X[15046], 3 X[15027] - 5 X[15059], 3 X[632] - 2 X[15059], 3 X[15034] + X[15059], 6 X[3819] - X[15101], X[1353] - 6 X[15462], 4 X[11801] - 9 X[15699], 4 X[113] + X[15704], X[12317] - 11 X[15720], 9 X[8703] - 4 X[16111], 9 X[5642] + X[16111], X[14677] + 4 X[16534], 4 X[12041] - 9 X[17504], 2 X[10721] + 3 X[19710], 4 X[13392] - X[20125], 7 X[549] - 2 X[20126], 14 X[11694] + X[20126], 7 X[110] + 3 X[20126], 3 X[16532] + 2 X[22115]

X(22251) lies on these lines: {3,13392}, {5,1511}, {30,15040}, {110,549}, {113,15704}, {125,11539}, {140,3448}, {399,3530}, {541,15714}, {542,15713}, {547,12902}, {550,7728}, {632,15027}, {1353,15462}, {3524,12308}, {3627,14643}, {3628,12383}, {3819,15101}, {3845,12121}, {3857,10733}, {3861,15046}, {5054,14683}, {5642,8703}, {5655,15036}, {5663,15712}, {7471,11749}, {9143,11812}, {10264,14869}, {10283,12778}, {10620,12100}, {10721,19710}, {10819,19116}, {10820,19117}, {11801,15699}, {12041,17504}, {12108,15039}, {12317,15720}, {14677,16534}, {16532,22115}

X(22251) = midpoint of X(3) and X(20125)
X(22251) = {X(10272),X(15035)}-harmonic conjugate of X(550)

X(22252) =  X(32)X(8790)∩X(3186)X(3511)

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

X(22252) lies on these lines: {32,8790}, {3186,3511}

X(22252) = antitomic conjugate of X(9230)
X(22252) = X(i)-isoconjugate of X(j) for these (i,j): {695, 18272}, {9236, 19573}, {9288, 19566}, {14946, 18270}
X(22252) = barycentric quotient X(i)/X(j) for these {i,j}: {384, 19566}, {1582, 18272}, {1925, 18276}, {1965, 18271}, {9230, 19573}, {16985, 19585}

X(22253) =  X(3)X(194)∩X(6)X(538)

Barycentrics    a^4+3 a^2 b^2+3 a^2 c^2-4 b^2 c^2 : :
X(22253) = 3 X[6] - 2 X[3734], 3 X[599] - 4 X[4045], 4 X[2549] - 3 X[5077], 2 X[141] - 3 X[7739], X[3734] - 3 X[7798], 5 X[3734] - 6 X[7804], 5 X[6] - 4 X[7804], 5 X[7798] - 2 X[7804], 16 X[7804] - 15 X[11286], 8 X[3734] - 9 X[11286], 4 X[6] - 3 X[11286], 8 X[7798] - 3 X[11286], 2 X[69] - 3 X[11287], 4 X[10796] - 5 X[11482], 3 X[11287] - 4 X[15048], 4 X[7848] - 3 X[15533], 3 X[3060] - 2 X[16983], 3 X[11159] - 4 X[18907], 3 X[1992] - 2 X[18907]

X(22253) lies on these lines: {2,14482}, {3,194}, {5,6392}, {6,538}, {25,8267}, {30,193}, {39,15271}, {69,11287}, {76,9605}, {99,1384}, {115,9766}, {141,7739}, {148,3830}, {183,5024}, {187,8716}, {192,6767}, {325,11318}, {330,7373}, {376,3793}, {381,7774}, {382,7762}, {384,20105}, {511,14532}, {524,2549}, {543,10488}, {546,2996}, {574,8667}, {599,4045}, {671,7926}, {754,6144}, {1003,7766}, {1184,19568}, {1351,2782}, {1597,9308}, {1654,11359}, {1655,11108}, {1657,20065}, {1975,3972}, {1992,11159}, {1993,22146}, {1995,9870}, {3053,7781}, {3060,16983}, {3180,11296}, {3181,11295}, {3210,3732}, {3363,5485}, {3629,7737}, {3843,7785}, {3851,13571}, {3926,5305}, {3933,5286}, {5013,7751}, {5041,17130}, {5054,17008}, {5055,7777}, {5073,7823}, {5254,7758}, {5304,8369}, {5309,7778}, {5319,7789}, {5346,7863}, {5355,7801}, {5858,6772}, {5859,6775}, {5969,11173}, {6390,7735}, {7738,7767}, {7748,7890}, {7764,13881}, {7765,7784}, {7770,7839}, {7773,7905}, {7779,7841}, {7780,15815}, {7788,7790}, {7796,7851}, {7797,7881}, {7800,9607}, {7817,7908}, {7827,7868}, {7845,11648}, {7848,15533}, {7861,7916}, {7864,7879}, {7872,7882}, {7887,7906}, {7895,7902}, {8359,15589}, {8556,15482}, {10796,11482}, {11054,11163}, {11185,15484}, {11285,17129}, {11354,17379}, {14712,15681}, {14929,20080}, {15694,17004}, {15703,17005}, {16370,17002}, {16371,17001}, {16417,16997}, {16418,16998}, {16857,17000}, {20794,21177}

X(22253) = reflection of X(i) in X(j) for these {i,j}: {6, 7798}, {69, 15048}, {7737, 3629}, {11159, 1992}, {20080, 14929}
X(22253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 17131, 15271), (69, 15048, 11287), (99, 14614, 1384), (183, 7757, 5024), (194, 7754, 3), (1003, 7766, 21309), (3933, 5286, 7866), (5254, 7758, 7776), (5309, 7813, 7778), (6390, 7735, 11288), (7765, 7855, 7784), (7781, 7805, 3053), (7839, 20081, 7770)

X(22254) =  X(2)X(39)∩X(30)X(17941)

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

X(22254) lies on these lines: {2,39}, {30,17941}, {316,5468}, {671,1641}, {868,7809}, {1989,18896}, {5108,7790}

X(22254) = X(798)-isoconjugate of X(20404)
X(22254) = crosssum of X(6041) and X(21906)
X(22254) = barycentric quotient X(99)/X(20404)
X(22254) = {X(2),X(2396)}-harmonic conjugate of X(7799)

X(22255) =  (name pending)

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

X(22255) lies on the cubic K091 and this line: {523,10510}

X(22256) =  X(67)X(316)∩X(99)X(523)

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

X(22256) lies on the cubic K091 and these lines: {67,316}, {99,523}

X(22257) =  X(5)X(8798)∩X(52)X(382)

Barycentrics    (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-3 a^12+7 a^10 b^2+2 a^8 b^4-18 a^6 b^6+17 a^4 b^8-5 a^2 b^10+7 a^10 c^2-16 a^8 b^2 c^2+18 a^6 b^4 c^2-12 a^4 b^6 c^2-a^2 b^8 c^2+4 b^10 c^2+2 a^8 c^4+18 a^6 b^2 c^4-10 a^4 b^4 c^4+6 a^2 b^6 c^4-16 b^8 c^4-18 a^6 c^6-12 a^4 b^2 c^6+6 a^2 b^4 c^6+24 b^6 c^6+17 a^4 c^8-a^2 b^2 c^8-16 b^4 c^8-5 a^2 c^10+4 b^2 c^10) : :
X(22257) = 5 X[631] - 9 X[1075], 4 X[5] - 3 X[8798], 11 X[5070] - 9 X[14059], 7 X[3832] - 9 X[14249], 2 X[5] - 3 X[14363], 5 X[631] - 3 X[15318], 3 X[1075] - X[15318]

X(22257) lies on the cubic K096 and these lines: {5,8798}, {52,382}, {216,631}, {324,3832}, {548,15912}, {5070,14059}

X(22257) = reflection of X(8798) in X(14363)
X(22257) = X(20)-Ceva conjugate of X(5)

X(22258) =  ISOGONAL CONJUGATE OF X(11061)

Barycentrics    a^2 (a^8-2 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^4-3 a^2 b^2 c^4+2 b^4 c^4+2 a^2 c^6+2 b^2 c^6-3 c^8) (a^8-2 a^6 b^2+2 a^4 b^4+2 a^2 b^6-3 b^8+a^4 b^2 c^2- 3 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(22258) lies on the cubic K108 and these lines:{3,15899}, {187,8428}, {858,6390}, {2393,2930}, {5094,14357}

X(22258) = isogonal conjugate of X(11061)
X(22258) = X(i)-cross conjugate of X(j) for these (i,j): {1205, 4}, {3455, 6}
X(22258) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11061}, {896, 10416}, {15900, 16568}
X(22258) = X(23)-vertex conjugate of X(23)
X(22258) = X(25)-vertex conjugate of X(3447)
X(22258) = crosssum of X(2930) and X(15141)
X(22258) = barycentric product X(i)*X(j) for these {i,j}: {6, 14364}, {671, 10417}
X(22258) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11061}, {111, 10416}, {3455, 15900}, {10417, 524}, {14364, 76}

X(22259) =  ISOGONAL CONJUGATE OF X(14360)

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

X(22259) lies on the cubic K108 and these lines: {2,13140}, {23,524}, {187,18374}, {1499,5621}, {2393,10355}

X(22259) = isogonal conjugate of X(14360)
X(22259) = anticomplement X(13140)
X(22259) = X(3455)-cross conjugate of X(25)
X(22259) = X(524)-vertex conjugate of X(524)
X(22259) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14360}, {2, 16563}, {75, 2930}, {662, 18310}, {14210, 15899}
X(22259) = barycentric product X(6)X(13574)
X(22259) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14360}, {31, 16563}, {32, 2930}, {512, 18310}, {13574, 76}

X(22260) =  X(6)X(512)∩X(141)X(523)

Barycentrics    a^2 (b^2-c^2)^3 : :
X(22260) = 5 X[3618] - 3 X[5652], 2 X[6] - 3 X[9171], 2 X[5092] - 3 X[9175], X[6] - 3 X[9178], 2 X[141] - 3 X[11182], 4 X[3589] - 3 X[11183], 4 X[2492] - 3 X[14428], 2 X[5027] - 3 X[14428]

X(22260) lies on the cubic K153 and these lines: on lines {6,512}, {141,523}, {520,6391}, {669,6041}, {850,1502}, {888,3569}, {1499,21850}, {1648,8029}, {1974,2422}, {2492,2872}, {3221,9973}, {3566,15583}, {3589,11183}, {3618,5652}, {4024,21713}, {4108,7806}, {4705,21810}, {5092,9175}, {5996,7777}, {6071,21906}, {6088,9208}, {7927,18311}, {9137,10546}

X(22260) = reflection of X(i) in X(j) for these {i,j}: {5027, 2492}, {9171, 9178}, {9426, 2489}
X(22260) = isogonal conjugate of X(31614)
X(22260) = X(i)-Ceva conjugate of X(j) for these (i,j): {512, 3124}, {850, 115}, {2489, 1084}, {9178, 21906}
X(22260) = X(i)-isoconjugate of X(j) for these (i,j): {249, 799}, {643, 7340}, {662, 4590}, {670, 1101}, {763, 6632}, {1414, 6064}, {4556, 4601}, {4567, 4610}, {4570, 4623}, {4592, 18020}, {4612, 4620}
X(22260) = crosspoint of X(i) and X(j) for these (i,j): {115, 850}, {512, 3124}, {2489, 8754}
X(22260) = crossdifference of every pair of points on line {249, 524}
X(22260) = crosssum of X(i) and X(j) for these (i,j): {99, 4590}, {249, 1576}, {523, 14061}, {524, 14443}
X(22260) = barycentric product X(i)*X(j) for these {i,j}: {6, 8029}, {42, 21131}, {115, 512}, {125, 2489}, {338, 669}, {513, 21833}, {523, 3124}, {525, 2971}, {594, 8034}, {647, 8754}, {649, 21043}, {661, 2643}, {762, 764}, {798, 1109}, {850, 1084}, {868, 2422}, {1365, 3709}, {1648, 9178}, {2088, 15475}, {2207, 5489}, {2333, 21134}, {2501, 20975}, {2799, 15630}, {2970, 3049}, {3120, 4079}, {3121, 4036}, {3122, 4024}, {3125, 4705}, {4092, 7180}, {4117, 20948}, {5466, 21906}, {6328, 8574}, {6535, 21143}, {10278, 19610}, {10630, 14443}, {12079, 14398}
X(22260) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 670}, {338, 4609}, {512, 4590}, {669, 249}, {1084, 110}, {1109, 4602}, {1356, 4565}, {1645, 5118}, {1924, 1101}, {2086, 17941}, {2489, 18020}, {2643, 799}, {2971, 648}, {3122, 4610}, {3124, 99}, {3125, 4623}, {3709, 6064}, {4079, 4600}, {4117, 163}, {4516, 4631}, {4705, 4601}, {7063, 5546}, {7180, 7340}, {8027, 763}, {8029, 76}, {8034, 1509}, {8754, 6331}, {9427, 1576}, {15630, 2966}, {20975, 4563}, {21043, 1978}, {21131, 310}, {21143, 6628}, {21833, 668}, {21906, 5468}
{X(2492),X(5027)}-harmonic conjugate of X(14428)

X(22261) =  ISOGONAL CONJUGATE OF X(5889)

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

X(22261) lies on the conic {{A,B,C,X(4),X(5)}}, the cubic K158, and on these lines: {4,8154}, {5,578}, {24,13450}, {53,571}, {311,1975}, {1141,12289}, {1658,5961}, {2165,14533}, {3071,8911}, {3613,11424}, {6293,9512}, {7544,10548}, {8800,12605}, {14674,18436}, {14889,18377}, {15033,16837}

X(22261) = isogonal conjugate of X(5889)
X(22261) = X(i)-cross conjugate of X(j) for these (i,j): {136, 523}, {418, 6}, {21659, 4}
X(22261) = cevapoint of X(i) and X(j) for these (i,j): {3, 12429}
X(22261) = trilinear pole of line {2451, 12077}
X(22261) = barycentric quotient X(6)/X(5889)

X(22262) =  X(159)X(394)∩X(206)X(19615)

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

X(22262) lies on the cubic K161 and these lines: {159,394}, {206,19615}, {315,5596}

X(22262) = X(19615)-Ceva conjugate of X(32)
X(22262) = X(i)-isoconjugate of X(j) for these (i,j): {2, 20931}, {75, 5596}, {76, 16544}, {274, 21079}, {304, 8879}, {561, 20993}, {668, 21190}, {1969, 22135}
X(22262) = barycentric product X(i)*X(j) for these {i,j}: {66, 19615}, {2156, 19616}, {2353, 19613}
X(22262) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 20931}, {32, 5596}, {560, 16544}, {1501, 20993}, {1918, 21079}, {1919, 21190}, {1974, 8879}, {14575, 22135}, {19615, 315}, {19616, 20641}

X(22263) =  ISOGONAL CONJUGATE OF X(14826)

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

X(22263) = lies on the cubic K173 and these lines: {232,1184}, {325,7386}, {511,1181}, {3089,6530}

X(22263) = isogonal conjugate of X(14826)

X(22264) =  MIDPOINT OF X(125) AND X(647)

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

X(22264) = X[850] - 5 X[15059]

X(22264) lies on the cubic K869 and these lines: {2,879}, {125,647}, {468,512}, {520,11064}, {525,5159}, {690,9209}, {850,15059}, {974,9242}, {1942,14220}, {2433,5094}, {3049,3231}, {3154,15359}, {6698,9030}, {8675,15118}, {11176,14271}

X(22264) = midpoint of X(125) and X(647)
X(22264) = crossdifference of every pair of points on line {4230, 6787}
X(22264) = barycentric product X(525)X(2452)
X(22264) = barycentric quotient X(2452)/X(648)

X(22265) =  MIDPOINT OF X(146) AND X(5984)

Barycentrics    a^14-2 a^12 b^2+3 a^10 b^4-7 a^8 b^6+8 a^6 b^8-3 a^4 b^10-2 a^12 c^2+5 a^8 b^4 c^2-3 a^6 b^6 c^2-3 a^4 b^8 c^2+5 a^2 b^10 c^2-2 b^12 c^2+3 a^10 c^4+5 a^8 b^2 c^4-9 a^6 b^4 c^4+6 a^4 b^6 c^4-13 a^2 b^8 c^4+6 b^10 c^4-7 a^8 c^6-3 a^6 b^2 c^6+6 a^4 b^4 c^6+16 a^2 b^6 c^6-4 b^8 c^6+8 a^6 c^8-3 a^4 b^2 c^8-13 a^2 b^4 c^8-4 b^6 c^8-3 a^4 c^10+5 a^2 b^2 c^10+6 b^4 c^10-2 b^2 c^12 : :
X(22265) = 4 X[115] - 3 X[14644], 2 X[11005] - 3 X[14644], 2 X[125] - 3 X[14651], 2 X[10264] - 3 X[14849], 4 X[140] - 3 X[14850], 2 X[99] - 3 X[15035], 4 X[12042] - 3 X[15055], 3 X[9140] - 4 X[15535], 3 X[11632] - 2 X[15535], 3 X[9140] - 2 X[15545], 3 X[11632] - X[15545], 3 X[14651] - X[18331], 7 X[15036] - 6 X[21166]

X(22265) lies on the cubic K873 and these lines: {2,11656}, {4,542}, {74,98}, {99,15035}, {110,1316}, {111,1640}, {113,147}, {115,6794}, {125,14651}, {140,14850}, {146,5984}, {148,17702}, {247,3448}, {541,11177}, {842,3906}, {868,9140}, {1511,13188}, {1648,14834}, {2777,9862}, {2784,12368}, {2794,10721}, {2966,11676}, {5465,6054}, {5622,18338}, {5663,12188}, {6055,11006}, {6321,10733}, {8724,15000}, {10264,14849}, {10766,11646}, {11623,15357}, {12042,15055}, {13169,19905}, {13172,16163}, {15036,21166}, {15928,16261}

X(22265) = midpoint of X(146) and X(5984)
X(22265) = reflection of X(i) in X(j) for these {i,j}: {2, 11656}, {4, 16278}, {74, 98}, {110, 18332}, {147, 113}, {6054, 5465}, {9140, 11632}, {10706, 9144}, {10733, 6321}, {11005, 115}, {11006, 6055}, {13169, 19905}, {13172, 16163}, {13188, 1511}, {14094, 15342}, {15357, 11623}, {15545, 15535}, {18331, 125}
X(22265) = anticomplement of the anticomplement of X(33511)
X(22265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 11005, 14644), (11632, 15545, 15535), (14651, 18331, 125), (15535, 15545, 9140)

X(22266) = X(1)X(2)∩X(548)X(11231)

Barycentrics    8*a+15*b+15*c : :
X(22266) = 4*X(1)+15*X(10), 34*X(1)-15*X(3244), 14*X(1)+5*X(3625), 9*X(1)+10*X(3626), X(1)-20*X(3634), X(1)+18*X(3828), 10*X(1)+9*X(4669), 7*X(1)+12*X(4691), 13*X(1)+6*X(4701), 12*X(2)+7*X(10), 26*X(2)-7*X(551), 33*X(2)-14*X(1125), 18*X(2)+X(3625), 5*X(2)+14*X(3828), 15*X(2)+4*X(4691), 13*X(10)+6*X(551), 11*X(10)+8*X(1125), X(10)-20*X(1698), 17*X(10)+2*X(3244), 39*X(10)-20*X(3617)

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(22166) and X(22266) are equal.

See César Lozada, Hyacinthos 28173.

X(22266) lies on these lines: {1, 2}, {548, 11231}, {1657, 10164}, {3579, 3850}, {3627, 10175}, {3740, 4537}, {3812, 4525}, {3817, 12812}, {3843, 6684}, {3947, 4114}, {4072, 16674}, {4744, 5044}, {5072, 18483}, {5217, 19538}, {9956, 15712}, {10172, 12702}, {12108, 17502}, {14891, 18480}, {15828, 17303}, {17538, 19925}

X(22266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3617, 4701), (1, 4691, 3625), (1, 19877, 3634), (1, 19883, 15808), (2, 4668, 1125), (1125, 20057, 551), (1698, 19877, 3828), (3244, 19875, 10), (3626, 3634, 19872), (3626, 19872, 19862), (3634, 3828, 1), (3634, 9780, 19862), (4678, 5550, 1), (4691, 20053, 4669), (9780, 19862, 10), (9780, 19872, 3626)

X(22267) = (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(3053)

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

X(22267) lies on these lines:

X(22268) = X(2)X(14978)∩X(3)X(233)

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

See Kadir Altintas and Ercole Suppa, Hyacinthos 28175.

X(22268) lies on these lines: {2,14978}, {3,233}, {5,12013}, {97,140}, {195,394}, {216,14938}, {632,14919}

X(22268) = crosssum of X(i) and X(j) for these (i,j): {195,15805}

X(22269) = (name pending)

Barycentrics    (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (2 a^12-14 a^10 b^2+39 a^8 b^4-56 a^6 b^6+44 a^4 b^8-18 a^2 b^10+3 b^12-14 a^10 c^2+48 a^8 b^2 c^2-34 a^6 b^4 c^2-38 a^4 b^6 c^2+56 a^2 b^8 c^2-18 b^10 c^2+39 a^8 c^4-34 a^6 b^2 c^4-12 a^4 b^4 c^4-38 a^2 b^6 c^4+45 b^8 c^4-56 a^6 c^6-38 a^4 b^2 c^6-38 a^2 b^4 c^6-60 b^6 c^6+44 a^4 c^8+56 a^2 b^2 c^8+45 b^4 c^8-18 a^2 c^10-18 b^2 c^10+3 c^12) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28176.

X(22268) lies on this line: {4, 12013}

X(22270) = X(3)X(6748)∩X(97)X(631)

Barycentrics    1/(a^8-4a^6(b^2+c^2)+6a^4(b^4+b^2c^2+c^4) -4a^2(b^2-c^2)^2(b^2+c^2)+(b^2-c^2)^2(b^4-4b^2c^2+c^4)) : :

See Antreas Hatzipolakis and Angel Montesdeoca Hyacinthos Hyacinthos 28174 and HG060918.

X(22270) lies on these lines: :{3,6748}, {97,631}, {140,394}, {1073,3526}, {1214,6958}, {1232,3926}, {3525,14919}, {3682,21012}, {13336,17974}

X(22271) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22271) lies on these lines: {8, 22272}, {10, 141}, {37, 42}, {44, 1918}, {71, 3059}, {72, 3696}, {75, 3681}, {192, 19998}, {200, 15624}, {313, 22289}, {319, 4553}, {513, 4416}, {517, 4793}, {536, 4685}, {594, 4111}, {668, 6385}, {674, 3686}, {692, 2287}, {740, 3159}, {758, 4732}, {899, 4022}, {984, 3293}, {2874, 4148}, {3294, 4068}, {3555, 16828}, {3664, 9038}, {3688, 17362}, {3690, 4046}, {3740, 4698}, {3779, 17275}, {3789, 4657}, {3842, 4015}, {3873, 4751}, {3941, 21384}, {3943, 7064}, {3952, 4043}, {4053, 21804}, {4061, 8804}, {4097, 15733}, {4134, 4709}, {4399, 14839}, {4517, 17299}, {4557, 21061}, {4661, 4699}, {4686, 22313}, {4690, 17792}, {4735, 21857}, {4738, 22306}, {5044, 15569}, {5739, 11677}, {5814, 22283}, {6007, 17332}, {8053, 16552}, {9054, 17049}, {17135, 18137}, {17330, 21746}, {17751, 20923}, {20694, 21873}, {21035, 21858}, {21083, 21085}, {21881, 21897}, {22274, 22280}, {22282, 22296}, {22291, 22309}

X(22272) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22272) lies on these lines: {8, 22271}, {10, 4523}, {42, 1953}, {209, 1824}, {692, 1172}, {1234, 4463}, {4651, 21271}, {21858, 21889}, {22277, 22308}

X(22273) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22273) lies on these lines: {10, 4523}, {42, 48}, {46, 3293}, {200, 22276}, {209, 3198}, {916, 11500}, {1486, 2333}, {2801, 22312}, {4651, 21270}, {18747, 20243}, {19998, 20074}, {22278, 22279}, {22280, 22298}, {22281, 22297}, {22286, 22311}

X(22274) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22274) lies on these lines: {10, 4523}, {42, 17438}, {228, 21855}, {15624, 21860}, {22271, 22280}, {22289, 22311}

X(22275) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22275) lies on these lines: {8, 22300}, {10, 12}, {37, 7109}, {38, 42}, {40, 12548}, {55, 10477}, {69, 22282}, {71, 3693}, {75, 3681}, {306, 4437}, {312, 3869}, {313, 22291}, {321, 14973}, {354, 6682}, {517, 3706}, {537, 4685}, {714, 22316}, {740, 22024}, {986, 3293}, {1233, 22285}, {1234, 4463}, {1918, 3744}, {3210, 4661}, {3690, 3932}, {3876, 19874}, {3909, 20290}, {4001, 8679}, {4030, 9052}, {4061, 17658}, {4113, 4692}, {5044, 16828}, {5718, 9564}, {9020, 22277}, {16574, 16678}, {17137, 18138}, {18057, 22293}, {20693, 21858}, {20716, 22321}, {22306, 22307}

X(22276) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22276) lies on these lines: {1, 16455}, {5, 10}, {6, 31}, {8, 15232}, {9, 375}, {37, 181}, {51, 3683}, {63, 8679}, {65, 17056}, {72, 3704}, {100, 1812}, {197, 219}, {200, 22273}, {210, 430}, {220, 2333}, {226, 15282}, {227, 7066}, {306, 4437}, {511, 4640}, {518, 4028}, {528, 4685}, {573, 3185}, {692, 5285}, {756, 21801}, {1155, 3917}, {1376, 3781}, {1402, 2245}, {1631, 2187}, {1869, 7957}, {2258, 4277}, {2318, 4557}, {2321, 14973}, {2323, 20986}, {2389, 2900}, {3190, 15624}, {3293, 5119}, {3428, 3682}, {3434, 4651}, {3579, 13754}, {3681, 20243}, {3792, 17596}, {3827, 8896}, {3869, 4417}, {3870, 9049}, {3931, 10974}, {4061, 8804}, {4259, 17594}, {4271, 20967}, {4531, 4849}, {4646, 10822}, {5752, 12514}, {5943, 15254}, {6690, 6703}, {7998, 9352}, {8013, 21011}, {8568, 22279}, {15733, 22312}, {19998, 20075}

X(22277) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

In the plane of a triangle ABC, let
I = incenter;
DEF = intouch triangle;
(O) = circumcircle;
P = perpendicular bisector of segment ID;
Ai = points of intersection of P and (O), and define Bi and Ci cyclically, for i = 1, 2;
Di = reflection of D in IAi, for i = 1, 2;
A' = E1E2∩F1F2, and define B' and C' cyclically;
The lines DA', EB', FC' concur in X(22277). See X(22277). (Angel Montesdeoca, August 6, 2022)

X(22277) lies on these lines: {1, 9049}, {6, 31}, {10, 141}, {37, 4890}, {41, 1631}, {43, 4446}, {44, 21746}, {46, 3293}, {48, 4497}, {65, 21867}, {69, 4651}, {72, 4026}, {181, 4849}, {193, 19998}, {210, 1213}, {218, 1486}, {354, 17245}, {511, 3579}, {524, 4685}, {579, 15624}, {583, 2223}, {742, 22316}, {758, 4085}, {872, 3778}, {1002, 4648}, {1100, 3688}, {1155, 22440}, {1269, 17165}, {1334, 4068}, {1362, 1418}, {1386, 9052}, {1400, 4557}, {1469, 3214}, {1475, 16679}, {1826, 1827}, {1843, 2355}, {1964, 20456}, {2092, 4735}, {2160, 7077}, {2174, 17798}, {2260, 2340}, {2277, 4484}, {2294, 21039}, {2321, 21865}, {2333, 7716}, {2388, 3997}, {2876, 9969}, {3271, 16669}, {3555, 4966}, {3589, 9054}, {3629, 9025}, {3681, 5224}, {3755, 20718}, {3789, 17327}, {3799, 17315}, {3827, 22290}, {3868, 4429}, {3873, 17234}, {3879, 4553}, {3941, 4253}, {4090, 20723}, {4430, 17232}, {4517, 16777}, {4524, 8675}, {4661, 17238}, {4705, 9029}, {4848, 20617}, {4852, 14839}, {4946, 9024}, {5800, 12587}, {5846, 22328}, {6007, 17351}, {9004, 22278}, {9016, 20455}, {9020, 22275}, {9021, 22285}, {9040, 22320}, {9055, 21080}, {13576, 15320}, {15185, 16593}, {17049, 17348}, {17366, 20358}, {19586, 21699}, {21863, 21889}, {22272, 22308}

X(22278) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22278) lies on these lines: {1, 16297}, {5, 10}, {42, 244}, {71, 374}, {72, 4714}, {75, 3681}, {165, 7416}, {210, 20718}, {373, 3058}, {375, 516}, {392, 19870}, {518, 4685}, {528, 5943}, {553, 9026}, {1730, 15621}, {3212, 22297}, {3293, 5902}, {3696, 14973}, {3880, 4891}, {4430, 17490}, {9004, 22277}, {14923, 18743}, {18142, 20244}, {21867, 22291}, {21888, 21902}, {22273, 22279}, {22296, 22309}

X(22279) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22279) lies on these lines: {10, 141}, {37, 3122}, {42, 1100}, {65, 20713}, {75, 22289}, {86, 4553}, {210, 21014}, {244, 17457}, {291, 16696}, {319, 4651}, {354, 15523}, {513, 894}, {674, 5750}, {1018, 4068}, {1213, 20683}, {1215, 20723}, {1631, 16788}, {3293, 4649}, {3589, 17049}, {3688, 17398}, {3753, 21867}, {3779, 17303}, {3873, 17228}, {3941, 17754}, {3943, 4890}, {4026, 20718}, {4670, 17792}, {4685, 4725}, {5285, 8021}, {6007, 7227}, {8053, 16549}, {8568, 22276}, {9049, 19868}, {14839, 17045}, {16606, 21878}, {16732, 21922}, {17140, 18143}, {17142, 18046}, {17165, 18133}, {17369, 21746}, {17384, 20358}, {21860, 21891}, {22273, 22278}, {22281, 22301}, {22303, 22304}

X(22280) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22280) lies on these lines: {10, 116}, {42, 17439}, {100, 3565}, {210, 21711}, {1824, 5139}, {3699, 3799}, {4557, 21859}, {22271, 22274}, {22273, 22298}

X(22281) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22281) lies on these lines: {10, 8230}, {42, 17440}, {22271, 22274}, {22273, 22297}, {22279, 22301}

X(22282) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22282) lies on these lines: {10, 1368}, {42, 65}, {69, 22275}, {197, 940}, {306, 22299}, {322, 22298}, {517, 4028}, {1824, 17874}, {3827, 8896}, {4651, 22297}, {10441, 11500}, {22271, 22296}, {22273, 22278}

X(22283) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22283) lies on these lines: {10, 4523}, {42, 17442}, {55, 5283}, {1228, 4463}, {5814, 22271}

X(22284) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22284) lies on these lines: {10, 4523}, {42, 18669}, {514, 22319}

X(22285) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22285) lies on these lines: {10, 626}, {42, 2240}, {72, 3696}, {1233, 22275}, {4463, 22296}, {9021, 22277}, {22286, 22291}, {22293, 22308}

X(22286) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics    a (b + c) (a^5 b - a b^5 + a^5 c - b^5 c - a c^5 - b c^5) : :

X(22286) lies on these lines: {10, 16580}, {42, 4118}, {313, 22288}, {1234, 4463}, {21889, 22316}, {22273, 22311}, {22285, 22291}

X(22287) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22287) lies on these lines: {8, 22271}, {10, 21236}, {42, 17443}, {3753, 21867}, {20713, 22292}, {21035, 21889}

X(22288) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22288) lies on these lines: {8, 22271}, {10, 21237}, {42, 17444}, {313, 22286}, {3697, 21670}, {4010, 4036}, {21022, 22304}

X(22289) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22289) lies on these lines: {10, 37}, {42, 4852}, {75, 22279}, {76, 22292}, {239, 18082}, {308, 17143}, {313, 22271}, {314, 4553}, {321, 21865}, {350, 4651}, {536, 21035}, {1234, 4463}, {3293, 4716}, {3706, 15523}, {4686, 22323}, {5178, 17751}, {13476, 20913}, {17135, 18040}, {20716, 21873}, {21889, 22304}, {22274, 22311}

X(22290) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22290) lies on these lines: {8, 22271}, {10, 21239}, {42, 2262}, {517, 22312}, {2357, 21866}, {3827, 22277}, {22273, 22278}

X(22291) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22291) lies on these lines: {10, 17047}, {42, 17447}, {313, 22275}, {21867, 22278}, {22271, 22309}, {22285, 22286}

X(22292) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22292) lies on these lines: {8, 22328}, {10, 141}, {42, 1107}, {72, 20716}, {76, 22289}, {1233, 22275}, {1909, 4651}, {3678, 20723}, {3681, 6376}, {14973, 20683}, {19998, 21226}, {20691, 21035}, {20713, 22287}, {21868, 22323}

X(22293) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22293) lies on these lines: {8, 22327}, {10, 141}, {37, 22189}, {42, 17448}, {72, 20723}, {4735, 20691}, {7148, 21868}, {18057, 22275}, {20683, 21025}, {21024, 21865}, {22285, 22308}

X(22294) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22294) lies on these lines: {2, 22325}, {10, 908}, {42, 982}, {51, 4450}, {75, 3681}, {181, 4972}, {517, 4358}, {518, 4706}, {693, 2533}, {752, 20962}, {758, 4674}, {3218, 16506}, {3293, 3868}, {3909, 4645}, {3952, 20718}, {4673, 14923}, {14973, 17163}

X(22295) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22295) lies on these lines: {10, 11}, {42, 3742}, {75, 3681}, {210, 4732}, {3753, 21870}

X(22296) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22296) lies on these lines: {10, 21243}, {42, 2611}, {313, 22275}, {1824, 1882}, {4463, 22285}, {22271, 22282}, {22278, 22309}

X(22297) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

Barycentrics    a (b + c) (a^4 b - a^3 b^2 + a^2 b^3 - a b^4 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c - a^3 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(22297) lies on these lines: {8, 22271}, {10, 116}, {42, 17451}, {65, 21867}, {210, 21024}, {1233, 22275}, {3212, 22278}, {4059, 9004}, {4651, 22282}, {22273, 22281}

X(22298) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22298) lies on these lines: {8, 22271}, {10, 8230}, {42, 17452}, {210, 430}, {313, 22275}, {322, 22282}, {612, 2352}, {872, 21801}, {22273, 22280}, {22308, 22312}

X(22299) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22299) lies on these lines: {1, 16287}, {5, 10}, {6, 10480}, {8, 22271}, {9, 12435}, {12, 22076}, {37, 65}, {40, 3185}, {42, 3057}, {72, 1089}, {306, 22282}, {312, 3869}, {375, 18250}, {518, 21080}, {674, 950}, {758, 3159}, {958, 10441}, {1214, 20617}, {1216, 5841}, {1826, 1829}, {1834, 10822}, {1869, 1902}, {2200, 6603}, {2829, 15644}, {3035, 15489}, {3293, 5697}, {3682, 14110}, {3690, 21677}, {3725, 4642}, {3753, 16828}, {3781, 5794}, {3827, 8804}, {3917, 7354}, {3962, 3994}, {4553, 7270}, {4651, 14923}, {5247, 18178}, {5251, 18180}, {5267, 5482}, {5562, 11827}, {8679, 12527}, {17747, 21024}, {20245, 21596}

X(22300) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22300) lies on these lines: {1, 5132}, {5, 10}, {8, 22275}, {28, 692}, {35, 18180}, {42, 65}, {51, 6284}, {71, 1212}, {72, 3696}, {171, 18178}, {181, 1834}, {185, 6253}, {197, 5706}, {209, 1829}, {375, 12572}, {389, 5842}, {392, 16828}, {513, 1770}, {518, 22316}, {910, 2200}, {1104, 1918}, {1376, 10441}, {1706, 12435}, {1715, 15622}, {1824, 1882}, {1826, 1902}, {2550, 22301}, {2807, 20420}, {3057, 21321}, {3191, 4557}, {3214, 22313}, {3293, 5903}, {3579, 6097}, {3827, 22277}, {3869, 4651}, {3877, 19874}, {3925, 22076}, {4255, 10473}, {4292, 8679}, {4673, 14923}, {4999, 15489}, {5295, 14973}, {5438, 10439}, {5446, 5840}, {7354, 16980}, {11553, 21319}, {21853, 21874}, {22308, 22317}

X(22301) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22301) lies on these lines: {6, 31}, {8, 22271}, {10, 3781}, {69, 22275}, {1155, 22412}, {2550, 22300}, {3588, 4557}, {3728, 21801}, {3869, 17788}, {5853, 22312}, {14624, 21865}, {17792, 22325}, {20694, 21871}, {20697, 21882}, {21011, 21728}, {22279, 22281}

X(22302) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22302) lies on these lines: {10, 21247}, {42, 17453}, {209, 21875}

X(22303) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22303) lies on these lines: {10, 21248}, {42, 2240}, {4651, 20911}, {22279, 22304}

X(22304) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22304) lies on these lines: {10, 16580}, {42, 17457}, {518, 3293}, {3961, 22325}, {20693, 21858}, {21022, 22288}, {21889, 22289}, {22279, 22303}

X(22305) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22305) lies on these lines: {10, 21250}, {42, 17459}, {210, 21868}, {536, 4685}, {18057, 22275}, {20713, 20721}

X(22306) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22306) lies on these lines: {10, 11}, {37, 1018}, {42, 17460}, {65, 3159}, {72, 3701}, {80, 4553}, {244, 5439}, {517, 4358}, {537, 21080}, {942, 17154}, {2835, 8804}, {3762, 14288}, {3931, 14752}, {4002, 19874}, {4738, 22271}, {20722, 22326}, {22275, 22307}

X(22307) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22307) lies on these lines: {10, 908}, {37, 758}, {42, 3899}, {72, 4066}, {517, 4793}, {519, 21080}, {3159, 4067}, {4135, 4525}, {22275, 22306}

X(22308) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22308) lies on these lines: {10, 116}, {42, 2170}, {72, 20716}, {891, 20507}, {3753, 21867}, {4651, 21272}, {4730, 21888}, {4738, 22271}, {10914, 22328}, {22272, 22277}, {22285, 22293}, {22298, 22312}, {22300, 22317}, {22310, 22321}

X(22309) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22309) lies on these lines: {10, 21252}, {42, 17463}, {4145, 21889}, {22271, 22291}, {22278, 22296}

X(22310) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22310) lies on these lines: {10, 21253}, {42, 3708}, {101, 2870}, {4155, 21889}, {21293, 21602}, {22308, 22321}

X(22311) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22311) lies on these lines: {10, 8287}, {42, 17467}, {100, 21891}, {2805, 21043}, {4436, 4705}, {4553, 17934}, {22273, 22286}, {22274, 22289}

X(22312) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22312) lies on these lines: {7, 4651}, {9, 42}, {10, 141}, {71, 3174}, {72, 3755}, {144, 19998}, {200, 579}, {209, 3059}, {210, 5257}, {516, 5752}, {517, 22290}, {527, 4685}, {758, 21867}, {1738, 5904}, {2092, 4849}, {2321, 20683}, {2801, 22273}, {3056, 4700}, {3293, 5223}, {3662, 4661}, {3681, 4357}, {3686, 3779}, {3707, 21746}, {3778, 21805}, {4029, 7064}, {4058, 21865}, {4067, 20713}, {4878, 21061}, {5853, 22301}, {9054, 17348}, {11038, 19874}, {15733, 22276}, {21039, 22021}, {22298, 22308}

X(22313) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22313) lies on these lines: {10, 11}, {37, 14752}, {42, 244}, {65, 3293}, {100, 18191}, {209, 2835}, {210, 321}, {517, 5400}, {518, 4706}, {537, 4685}, {740, 22045}, {891, 20507}, {900, 15914}, {2254, 22323}, {2262, 21858}, {3214, 22300}, {3271, 6154}, {3880, 4742}, {3893, 17751}, {4145, 21889}, {4686, 22271}, {20718, 21805}, {21832, 21888}

X(22314) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22314) lies on these lines: {10, 4928}, {42, 1635}, {210, 4155}, {513, 4380}, {812, 4685}, {891, 20507}, {3699, 3799}, {4139, 4524}, {4651, 21297}, {4705, 4825}, {4730, 21894}, {4773, 9032}, {4849, 17989}, {4893, 21727}

X(22315) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22315) lies on these lines: {10, 16581}, {42, 17472}, {1234, 4463}, {4083, 4408}

X(22316) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22316) lies on these lines: {1, 20150}, {10, 37}, {42, 75}, {192, 4651}, {209, 744}, {239, 1918}, {321, 872}, {518, 22300}, {536, 4685}, {714, 22275}, {730, 17362}, {742, 22277}, {899, 18137}, {1278, 19998}, {2667, 3896}, {4022, 17135}, {4043, 4365}, {4362, 15624}, {4726, 4946}, {21889, 22286}

X(22317) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22317) lies on these lines: {10, 141}, {210, 21921}, {1254, 21896}, {3212, 22278}, {4651, 16284}, {20683, 21049}, {22300, 22308}

X(22318) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22318) lies on these lines: {10, 3907}, {42, 17478}, {3900, 4036}, {4083, 4408}

X(22319) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22319) lies on these lines: {10, 21261}, {512, 3700}, {513, 22322}, {514, 22284}, {693, 2533}, {891, 20507}, {4455, 22223}

X(22320) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22320) lies on these lines: {10, 512}, {42, 4367}, {484, 513}, {693, 2533}, {798, 21901}, {814, 4507}, {834, 17072}, {1019, 3293}, {1577, 4132}, {3214, 4784}, {4079, 22224}, {4490, 21727}, {9040, 22277}

X(22321) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22321) lies on these lines: {10, 125}, {42, 2611}, {65, 3120}, {72, 3701}, {149, 517}, {150, 20940}, {210, 15523}, {518, 17763}, {526, 18004}, {756, 21319}, {758, 21093}, {1824, 1893}, {2610, 21888}, {2771, 15343}, {2801, 3937}, {2809, 14740}, {2818, 12691}, {3681, 3781}, {3869, 17777}, {4018, 4080}, {4145, 21889}, {4551, 18210}, {12019, 15906}, {16560, 20999}, {20716, 22275}, {22308, 22310}

X(22322) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22322) lies on these lines: {10, 21262}, {42, 17458}, {513, 22319}, {4010, 4036}, {4083, 4408}, {5283, 16692}

X(22323) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22323) lies on these lines: {10, 537}, {37, 3122}, {42, 678}, {209, 2877}, {291, 4553}, {2254, 22313}, {3293, 4663}, {4651, 4690}, {4686, 22289}, {4730, 21888}, {9016, 20455}, {14404, 21893}, {15523, 21342}, {17154, 18150}, {17351, 18082}, {17448, 22328}, {21868, 22292}

X(22324) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22324) lies on these lines: {10, 21263}, {514, 22284}, {4083, 4408}

X(22325) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22325) lies on these lines: {1, 16302}, {2, 22294}, {5, 10}, {37, 22171}, {38, 42}, {72, 3293}, {181, 4026}, {210, 321}, {536, 4685}, {740, 14973}, {1215, 20718}, {1376, 1764}, {3057, 17751}, {3681, 17147}, {3877, 18743}, {3961, 22304}, {4891, 9957}, {17792, 22301}

X(22326) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22326) lies on these lines: {10, 625}, {72, 3696}, {693, 2533}, {20722, 22306}

X(22327) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22327) lies on these lines: {8, 22293}, {10, 3934}, {42, 1100}, {75, 3681}, {321, 20723}, {524, 4685}, {2238, 21865}

X(22328) =  (A,B,C,X(2); A',B',C',X(10)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(37)

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

X(22328) lies on these lines: {1, 22279}, {8, 22292}, {10, 3934}, {42, 2229}, {72, 3696}, {213, 21865}, {239, 18087}, {274, 4553}, {291, 18172}, {308, 17143}, {732, 17792}, {1089, 20723}, {1107, 21035}, {4651, 17152}, {5846, 22277}, {10914, 22308}, {15523, 20358}, {17448, 22323}, {20694, 21879}

X(22329) =  X(2)X(6)∩X(30)X(98)

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

Let P be a point on the circumcircle. Let T be the trilinear pole of the polar of P wrt the Brocard circle. Let T' be the isogonal conjugate of T. The locus of T' as P varies is a hyperbola centered at X(22329). (Randy Hutson, September 8, 2018)

X(22329) lies on these lines: {2, 6}, {4, 11172}, {5, 6179}, {23, 7669}, {30, 98}, {32, 8370}, {76, 8369}, {83, 8367}, {99, 9136}, {111, 6094}, {115, 3849}, {140, 7760}, {187, 543}, {237, 9149}, {297, 6103}, {315, 11318}, {316, 3793}, {351, 523}, {381, 9753}, {468, 648}, {511, 6055}, {530, 6109}, {531, 6108}, {538, 1569}, {542, 1513}, {549, 7757}, {574, 5569}, {598, 3363}, {620, 5215}, {625, 14971}, {736, 6661}, {754, 5461}, {858, 7668}, {892, 16317}, {1078, 5305}

X(22329) = midpoint of X(37785) and X(37786)
X(22329) = isotomic conjugate of X(5503)
X(22329) = complement of X(7840)
X(22329) = anticomplement of X(22110)

X(22330) =  MIDPOINT OF X(575) AND X(576)

Barycentrics    a^2(4 a^4 - 9 a^2 (b^2 + c^2) + 5 b^4 + 5 c^4 - 8 b^2 c^2) : :
X(22330) = X(3) - 9 X(6)

X(22330) lies on these lines: {3, 6}, {4, 17503}, {5, 8584}, {23, 13366}, {51, 9544}, {143, 11649}, {323, 5643}, {373, 11004}, {394, 10219}, {397, 16002}, {398, 16001}, {524, 3628}, {542, 546}, {597, 632}, {895, 1173}, {1199, 8718}

X(22330) = midpoint of X(575) and X(576)
X(22330) = isogonal conjugate of X(10185)
X(22330) = inverse-in-Brocard-circle of X(22234)
X(22330) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(8588)
X(22330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 22234), (61, 62 187), (371, 372, 8588)

X(22331) =  MIDPOINT OF X(22236) AND X(22238)

Trilinears    3 sin A - 4 cos A tan ω : :
Trilinears    4 cos A - 3 sin A cot ω : :
Barycentrics    a^2 (7 a^2 - b^2 - c^2) : :

X(22331) lies on these lines: {3,6}, {20,5306}, {23,1184}, {112,10594}, {115,5076}, {172,3303}, {230,3091}, {382,7755}, {384,8667}, {439,1992}, {546,7737}, {548,7739}, {550,5319}, {599,14001}, {609,3746}, {632,2548}, {980,21517}, {999,9341}, {1003,6179}, {1285,3090}, {1572,15178}, {1611,1627}, {1657,5309}, {1914,3304}, {1968,5198}, {2207,3518}, {2549,12103}, {3146,7735}, {3517,14581}, {3522,9607}, {3523,9300}, {3524,9606}, {3526,7753}, {3529,5254}, {3534,7765}, {3552,14614}, {3627,3767}, {3628,18907}, {3629,6337}, {3763,3785}, {3793,7795}, {3815,10303}, {3851,14537}, {3926,6144}, {5072,7746}, {5077,7902}, {5079,5475}, {5204,5332}, {5217,7296}, {5266,16672}, {5275,16865}, {5277,16842}, {5286,17538}, {5305,15704}, {5337,21496}, {5346,6781}, {5359,7492}, {5563,7031}, {6103,12173}, {7610,16924}, {7749,15484}, {7759,11288}, {7760,8716}, {7770,8556}, {7778,20065}, {7780,11286}, {7784,8363}, {7793,15271}, {7819,19661}, {7851,14712}, {7907,11184}, {7922,8366}, {8369,14023}, {8778,10311}, {9698,15720}, {9756,12110}, {9766,16925}, {9939,14043}, {11285,12150}, {11291,13847}, {11292,13846}, {11648,17800}, {11672,22333}, {12812,18584}, {14045,19569}, {14869,21843}

X(22331) = midpoint of X(22236) and X(22238)
X(22331) = X(5020)-Ceva conjugate of X(19132)
X(22331) = inverse-in-Brocard-circle of X(22332)
X(22331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 22332), (3, 7772, 5013), (6, 3053, 5023), (6, 5023, 15815), (6, 9601, 6422), (32, 39, 21309), (32, 1384, 3053), (32, 3053, 6), (32, 5171, 12212), (32, 5206, 5008), (61, 62, 5093), (187, 7772, 3), (1151, 1152, 3098), (3053, 5013, 187), (3592, 3594, 576), (5008, 5206, 9605), (5023, 15815, 5585), (5085, 12212, 6), (5210, 21309, 6)

X(22332) =  INVERSE-IN-BROCARD-CIRCLE OF X(22331)

Trilinears    3 sin A + 4 cos A tan ω : :
Trilinears    4 cos A + 3 sin A cot ω : :
Barycentrics    a^2 (a^2 - 7 b^2 - 7 c^2) : :

Let X be a point on the 2nd Brocard circle. The locus of the symmedian point of triangle XPU(1) as X varies is an ellipse with center X(22332). (Randy Hutson, September 8, 2018)

X(22332) lies on these lines: {2, 9607}, {3, 6}, {4, 9606}, {20, 9300}, {45, 988}, {115, 5079}, {140, 7739}, {194, 15271}, {230, 10303}, {232, 11403}, {546, 2549}, {549, 5319}, {599, 16043}, {632, 3767}, {1180, 1611}

X(22332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 22331), (22236, 22238, 182)

X(22333) =  CROSSPOINT OF X(22236) AND X(22238)

Barycentrics    a^2 (7 a^4 - 6 a^2 (b^2 + c^2) - (b^2 - c^2)^2) (7 a^4 + 7 b^4 + 7 c^4 - 14 a^2 b^2 - 14 a^2 c^2 + 2 b^2 c^2) : :

X(22333) lies on these lines: {113, 5076}, {141, 10303}, {1192, 1511}, {11672, 22331}

X(22333) = crosspoint of X(22236) and X(22238)

X(22334) =  CEVAPOINT OF X(22236) AND X(22238)

Trilinears    1/(3 cos A - cos B cos C) : :
Trilinears    1/(sec A - 4 sec B sec C) : :
Barycentrics    a^2 / (7 a^4 - b^4 - c^4 - 6 a^2 b^2 - 6 a^2 c^2 + 2 b^2 c^2) : :

X(22334) lies on the Jerabek hyperbola and these lines: {3, 13474}, {6, 9968}, {25, 3532}, {54, 1498}, {64, 5198}, {66, 5895}, {67, 12173}, {68, 3627}, {69, 3146}, {72, 1750}, {73, 3303}, {74, 1192}, {265, 5076}, {381, 14861}, {382, 3519}, {389, 3531}, {546, 4846}, {1173, 12290}, {1176, 10541}, {1181, 13472}

X(22334) isogonal conjugate of X(3522)
X(22334) cevapoint of X(22236) and X(22238)

X(22335) = X(4)X(8254)∩X(381)X(3459)

Barycentrics    (S^2+SB*SC)*(2*SB-13*R^2+4*SW)*(2*SC-13*R^2+4*SW) : :
Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^6-3 a^4 b^2-3 a^2 b^4+3 b^6-5 a^4 c^2-a^2 b^2 c^2-5 b^4 c^2+a^2 c^4+b^2 c^4+c^6) (3 a^6-5 a^4 b^2+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^4-5 b^2 c^4+3 c^6) : : (Peter Moses)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28188.

X(22335) lies on the conic {{A, B, C, X(4), X(5)}} and these lines: {4, 8254}, {381, 3459}, {546, 1141}, {1263, 3574}, {1487, 3850}, {3845, 15619}

X(22335) = isogonal conjugate of X(25042)

X(22336) = ISOGONAL CONJUGATE OF X(7496)

Barycentrics    (a^4+5*c^2*a^2+c^4-b^4)*(a^4+5*b^2*a^2+b^4-c^4) : :
X(22336) = X(69)-7*X(7693)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28188.

X(22336) lies on the Jerabek hyperbola and these lines: {3, 5476}, {51, 67}, {54, 18374}, {69, 7693}, {74, 5480}, {248, 13338}, {895, 8584}, {1173, 8550}, {1176, 6329}, {1177, 10169}, {1503, 14483}, {3431, 14853}, {5486, 9971}, {6776, 14491}, {9969, 13622}, {9973, 17040}, {15360, 20582}, {19136, 19151}

X(22336) = isogonal conjugate of X(7496)

X(22337) = REFLECTION OF X(3) IN X(133)

Barycentrics    S^4-(8*R^2*(6*R^2+SA-2*SW)-2*SA^2-3*SB*SC+SW^2)*S^2+(4*R^2-SW)*(108*R^2-11*SW)*SB*SC : :
X(22337) = 3*X(3)-4*X(6716), 2*X(122)-3*X(381), 3*X(133)-2*X(6716), 4*X(11732)-5*X(18493)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28188.

X(22337) lies on these lines: {3, 133}, {4, 2972}, {5, 1294}, {30, 107}, {64, 265}, {122, 381}, {1478, 7158}, {1479, 3324}, {1559, 6760}, {1657, 3184}, {2790, 6321}, {2797, 6033}, {2803, 10742}, {2811, 10741}, {2822, 10739}, {2828, 10738}, {2833, 15521}, {2839, 15522}, {2846, 10740}, {2848, 12918}, {3146, 5667}, {3627, 10152}, {3845, 10714}, {7517, 14703}, {7728, 9033}, {9520, 10743}, {9524, 10744}, {9528, 10746}, {9529, 10748}, {10762, 21850}, {11718, 18481}, {11732, 18493}, {14673, 18534}

X(22337) = midpoint of X(3146) and X(5667)
X(22337) = reflection of X(i) in X(j) for these (i,j): (3, 133), (1294,5), (1657, 3184), (10762, 21850)
X(22337) = X(133)-of-X3-ABC reflections-triangle
X(22337) = X(1294)-of-Johnson-triangle

X(22338) = REFLECTION OF X(3) IN X(5512)

Barycentrics    6*(12*R^2-SW)*S^4+(81*(SA-SW)*R^2-(6*SA-7*SW)*SW)*SA*S^2-3*SB*SC*SW^3 : :
X(22338) = 3*X(3)-4*X(6719), 3*X(4)-X(14360), 4*X(111)-3*X(14666), 2*X(126)-3*X(381), 3*X(3543)+X(20099), 3*X(5512)-2*X(6719), 3*X(10748)-2*X(14360), 3*X(14561)-2*X(14688)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28188.

X(22338) lies on these lines: {3, 5512}, {4, 10748}, {5, 1296}, {20, 14650}, {30, 111}, {126, 381}, {265, 2780}, {382, 11258}, {543, 3830}, {590, 11835}, {615, 11836}, {1478, 6019}, {1479, 3325}, {2793, 6321}, {2805, 10742}, {2813, 10741}, {2819, 10747}, {2824, 10739}, {2830, 10738}, {2837, 15521}, {2843, 15522}, {2852, 10740}, {2854, 7728}, {3146, 14654}, {3534, 9172}, {3543, 20099}, {3627, 10734}, {3845, 10717}, {7517, 14657}, {7665, 14653}, {9129, 12121}, {9522, 10743}, {9526, 10744}, {9529, 10745}, {9531, 10746}, {10765, 21850}, {11721, 18481}, {14561, 14688}, {14645, 18346}

X(22338) = midpoint of X(i) and X(j) for these {i,j}: {382, 11258}, {3146, 14654}
X(22338) = reflection of X(i) in X(j) for these (i,j): (3, 5512), (20, 14650), (3534, 9172), (10765, 21850), (1296,5)
X(22338) = X(1296)-of-Johnson-triangle
X(22338) = X(5512)-of-X3-ABC-reflections-triangle

X(22339) = ISOTOMIC CONJUGATE OF X(1113)

Barycentrics    1/(2 a^4-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) J) : :

X(22339) lies on the cubics K242, K606, K1070 and these lines: {2,2592}, {69,2574}, {99,1113}, {264,1347}, {287,8116}, {306,2582}, {325,523}, {339,1313}, {1114,2373}, {1494,10719}, {2593,2799}, {13219,14807}, {14360,14808}

X(22339) = isotomic conjugate of X(1113)
X(22339) = anticomplement X(8105)
X(22339) = X(i)-Ceva conjugate of X(j) for these (i,j): {6331, 2593}, {15164, 69}
X(22339) = X(i)-cross conjugate of X(j) for these (i,j): {125, 2593}, {1313, 2}, {2574, 2592}
X(22339) = cevapoint of X(2) and X(14807)
X(22339) = crosspoint of X(264) and X(15164)
X(22339) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {63, 14807}, {162, 2593}, {662, 2575}, {1113, 5905}, {1822, 2}, {2575, 21221}, {2576, 193}, {2579, 148}, {2580, 4}, {2583, 3448}, {2586, 6515}, {8115, 8}, {15164, 21270}
X(22339) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2576}, {25, 1822}, {31, 1113}, {32, 2580}, {112, 2579}, {163, 8106}, {184, 2586}, {560, 15164}, {1576, 2589}, {1973, 8115}
X(22339) = barycentric product X(i)*X(j) for these {i,j}: {69, 2592}, {75, 2582}, {76, 2574}, {304, 2588}, {305, 8105}, {525, 15165}, {561, 2578}, {850, 8116}, {1114, 3267}, {1823, 20948}, {1969, 2584}, {2581, 14208}
X(22339) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2576}, {2, 1113}, {63, 1822}, {69, 8115}, {75, 2580}, {76, 15164}, {92, 2586}, {523, 8106}, {525, 2575}, {656, 2579}, {850, 2593}, {1114, 112}, {1313, 8105}, {1577, 2589}, {1823, 163}, {2574, 6}, {2578, 31}, {2581, 162}, {2582, 1}, {2584, 48}, {2588, 19}, {2592, 4}, {8105, 25}, {8115, 15461}, {8116, 110}, {14208, 2583}, {15165, 648}
X(22339) = {X(850),X(3268)}-harmonic conjugate of X(22340)
X(22339) = {P",U"}-harmonic conjugate of X(2), where P" and U" are the isotomic conjugates of the imaginary foci of the orthic inconic

X(22340) = ISOTOMIC CONJUGATE OF X(1114)

Barycentrics    1/(2 a^4-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) J) : :

X(22340) lies on the cubics K242, K606, K1070 and these lines: {2,2593}, {69,2575}, {99,1114}, {264,1346}, {287,8115}, {306,2583}, {325,523}, {339,1312}, {1113,2373}, {1494,10720}, {2592,2799}, {13219,14808}, {14360,14807}

X(22340) = isotomic conjugate of X(1114)
X(22340) = anticomplement X(8106)
X(22340) = X(i)-Ceva conjugate of X(j) for these (i,j): {6331, 2592}, {15165, 69}
X(22340) = X(i)-cross conjugate of X(j) for these (i,j): {125, 2592}, {1312, 2}, {2575, 2593}
X(22340) = cevapoint of X(2) and X(14808)
X(22340) = crosspoint of X(264) and X(15165)
X(22340) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {63, 14808}, {162, 2592}, {662, 2574}, {1114, 5905}, {1823, 2}, {2574, 21221}, {2577, 193}, {2578, 148}, {2581, 4}, {2582, 3448}, {2587, 6515}, {8116, 8}, {15165, 21270}
X(22340) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2577}, {25, 1823}, {31, 1114}, {32, 2581}, {112, 2578}, {163, 8105}, {184, 2587}, {560, 15165}, {1576, 2588}, {1973, 8116}
X(22340) = barycentric product X(i)*X(j) for these {i,j}: {69, 2593}, {75, 2583}, {76, 2575}, {304, 2589}, {305, 8106}, {525, 15164}, {561, 2579}, {850, 8115}, {1113, 3267}, {1822, 20948}, {1969, 2585}, {2580, 14208}
X(22340) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2577}, {2, 1114}, {63, 1823}, {69, 8116}, {75, 2581}, {76, 15165}, {92, 2587}, {523, 8105}, {525, 2574}, {656, 2578}, {850, 2592}, {1113, 112}, {1312, 8106}, {1577, 2588}, {1822, 163}, {2575, 6}, {2579, 31}, {2580, 162}, {2583, 1}, {2585, 48}, {2589, 19}, {2593, 4}, {8106, 25}, {8115, 110}, {8116, 15460}, {14208, 2582}, {15164, 648}
X(22340) = {X(850),X(3268)}-harmonic conjugate of X(22339)
X(22340) = {P",U"}-harmonic conjugate of X(2), where P" and U" are the isotomic conjugates of the real foci of the orthic inconic

X(22341) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22341) lies on these lines: {1, 3}, {10, 856}, {12, 18641}, {34, 13738}, {48, 577}, {72, 23067}, {73, 228}, {95, 404}, {108, 1294}, {109, 2360}, {184, 3215}, {198, 1035}, {212, 7114}, {216, 2260}, {221, 3185}, {225, 851}, {227, 11214}, {243, 411}, {255, 1092}, {283, 296}, {326, 1259}, {336, 1231}, {417, 8763}, {418, 22344}, {500, 20122}, {580, 19365}, {581, 19366}, {828, 3990}, {912, 22457}, {1042, 3724}, {1071, 20803}, {1075, 8762}, {1248, 1935}, {1284, 18589}, {1399, 2194}, {1400, 18591}, {1408, 2193}, {1435, 4191}, {1465, 16453}, {1474, 1950}, {1708, 19762}, {1745, 13855}, {1788, 6350}, {1816, 1896}, {1825, 21318}, {1875, 7420}, {1877, 13724}, {1882, 3149}, {2169, 19210}, {2720, 2744}, {3485, 6349}, {3682, 7066}, {4055, 7138}, {4225, 4296}, {5433, 7515}, {6198, 7421}, {10090, 14679}, {11375, 17073}, {16451, 17080}, {20727, 22375}, {20967, 22119}, {22053, 22347}, {22072, 22346}, {22363, 22364}

X(22341) = isogonal conjugate of X(1896)
X(22341) = isotomic conjugate of polar conjugate of X(1409)
X(22341) = X(19)-isoconjugate of X(31623)
X(22341) = X(92)-isoconjugate of X(1172)

X(22342) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22342) lies on these lines: {3, 201}, {35, 186}, {36, 12005}, {48, 3215}, {50, 1399}, {55, 20838}, {65, 3724}, {73, 228}, {252, 2962}, {477, 2222}, {1155, 15443}, {1393, 16453}, {1451, 2352}, {2171, 2178}, {3465, 7421}, {20277, 20764}, {22061, 22375}

X(22342) = isogonal conjugate of polar conjugate of X(16577)
X(22342) = isotomic conjugate of polar conjugate of X(21741)
X(22342) = {X(22346),X(22347)}-harmonic conjugate of X(3)

X(22343) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(2), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(22343) lies on these lines: {1,4704}, {2,87}, {6,31}, {9,3009}, {37,3248}, {39,20667}, {44,1964}, {86,799}, {190,18170}, {192,18194}, {238,1201}, {256,8843}, {560,4268}, {572,2210}, {869,1743}, {872,16669}, {894,9359}, {899,1740}, {1015,22172}, {1045,17121}, {1149,15485}, {1178,1931}, {1193,5145}, {1334,21760}, {1475,14758}, {1977,21759}, {2053,2275}, {2234,17348}, {2347,20663}, {2667,16666}, {3271,3778}, {3720,17379}, {3736,16477}, {3747,20228}, {3764,5069}, {3840,17178}, {4003,17477}, {4128,21332}, {5053,7122}, {7189,17333}, {16571,16816}, {16604,22174}, {17351,17445}, {17448,22167}, {20456,21746}, {20460,20864}, {21757,21838}

X(22343) = isogonal conjugate of X(32011)
X(22343) = crosssum of X(2) and X(43)
X(22343) = polar conjugate of isotomic conjugate of X(22066)
X(22343) = crosspoint of X(3) and X(87)

X(22344) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22344) lies on these lines: {3, 63}, {25, 1466}, {35, 20999}, {46, 15654}, {56, 15854}, {73, 3937}, {100, 9369}, {184, 603}, {418, 22341}, {855, 1210}, {908, 19514}, {942, 7428}, {1106, 2187}, {1437, 4575}, {1470, 3556}, {1818, 22413}, {1828, 3752}, {3185, 5204}, {3689, 15625}, {3911, 13724}, {4188, 17350}, {5122, 16453}, {6705, 13734}, {8192, 10310}, {11509, 22654}, {13738, 15803}, {17102, 18210}, {20775, 20780}, {22364, 22386}, {22378, 22390}

X(22344) = isogonal conjugate of polar conjugate of X(3752)
X(22344) = isotomic conjugate of polar conjugate of X(20228)
X(22344) = X(19)-isoconjugate of X(32017)

X(22345) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22345) lies on these lines: {1, 15654}, {3, 63}, {35, 3961}, {36, 1046}, {39, 21744}, {42, 16980}, {48, 577}, {55, 8192}, {56, 3185}, {57, 13738}, {58, 4215}, {100, 4696}, {184, 255}, {197, 10834}, {198, 1466}, {222, 1410}, {283, 7015}, {404, 894}, {851, 4292}, {855, 950}, {859, 942}, {908, 19513}, {1193, 20967}, {1210, 13724}, {1399, 20986}, {1402, 1468}, {1408, 7113}, {1437, 18604}, {1486, 10835}, {1496, 2187}, {1763, 19763}, {1798, 4558}, {1818, 22078}, {1829, 3666}, {1894, 15844}, {2200, 4020}, {2352, 4252}, {2594, 8679}, {3145, 3220}, {3216, 21361}, {3218, 4225}, {3682, 3917}, {3868, 4216}, {3937, 4303}, {4185, 15509}, {4191, 15803}, {4245, 5439}, {5044, 16374}, {5217, 15624}, {6245, 13734}, {6734, 9840}, {7004, 18673}, {7289, 18606}, {9798, 11507}, {10882, 12526}, {10902, 20999}, {12680, 15622}, {13411, 21319}, {17102, 17441}, {17609, 18613}, {18210, 18732}, {20775, 22364}, {20778, 22386}, {20784, 22375}, {20785, 22061}, {22076, 22097}, {22347, 22361}

X(22345) = isogonal conjugate of polar conjugate of X(3666)
X(22345) = isotomic conjugate of polar conjugate of X(2300)
X(22345) = X(19)-isoconjugate of X(30710)
X(22345) = X(92)-isoconjugate of X(2298)

X(22346) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22346) lies on these lines: {3, 201}, {36, 5083}, {212, 7125}, {228, 3937}, {1155, 3724}, {1830, 16578}, {7069, 7416}, {8677, 22399}, {15906, 16453}, {22072, 22341}

X(22346) = isogonal conjugate of polar conjugate of X(16578)
X(22346) = isotomic conjugate of polar conjugate of X(21742)
X(22346) = {X(3),X(22342)}-harmonic conjugate of X(22347)

X(22347) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22347) lies on these lines: {3, 201}, {35, 7512}, {228, 22072}, {1393, 7420}, {1831, 16579}, {2646, 3724}, {7069, 16287}, {22053, 22341}, {22345, 22361}

X(22347) = isogonal conjugate of polar conjugate of X(16579)
X(22347) = isotomic conjugate of polar conjugate of X(21743)
X(22347) = {X(3),X(22342)}-harmonic conjugate of X(22346)

X(22348) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22348) lies on these lines: {3, 23068}, {42, 18210}, {71, 22077}, {73, 228}, {1193, 5320}, {7117, 20229}, {11393, 16580}, {22364, 22422}

X(22348) = isogonal conjugate of polar conjugate of X(16580)
X(22348) = isotomic conjugate of polar conjugate of X(21744)

X(22349) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22349) lies on these lines: {3, 23069}, {71, 22438}, {73, 228}, {22384, 22387}

X(22349) = isogonal conjugate of polar conjugate of X(16581)
X(22349) = isotomic conjugate of polar conjugate of X(21745)

X(22350) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22350) lies on these lines: {1, 2}, {3, 73}, {5, 2654}, {6, 2289}, {20, 1745}, {21, 3074}, {30, 2635}, {31, 8069}, {33, 5720}, {35, 4300}, {36, 59}, {40, 10571}, {46, 1042}, {48, 22132}, {55, 1064}, {56, 1066}, {58, 1167}, {71, 22083}, {72, 17102}, {109, 2077}, {216, 3990}, {219, 22063}, {221, 10310}, {223, 6282}, {226, 1074}, {227, 14110}, {244, 5570}, {404, 3075}, {515, 4551}, {517, 1457}, {521, 656}, {581, 3601}, {651, 6909}, {672, 3002}, {758, 1735}, {859, 2183}, {908, 1785}, {912, 7004}, {999, 1450}, {1038, 10360}, {1040, 18446}, {1060, 20277}, {1076, 5930}, {1155, 1464}, {1385, 5399}, {1409, 22071}, {1468, 22766}, {1496, 8071}, {1497, 16466}, {1739, 12736}, {1795, 22128}, {1801, 17187}, {1802, 22131}, {1807, 18455}, {1935, 6906}, {1936, 6905}, {2197, 22074}, {2252, 22059}, {2318, 3940}, {2361, 5172}, {2594, 2646}, {2650, 13750}, {2933, 14529}, {3100, 3465}, {3428, 7074}, {3468, 4296}, {3583, 6127}, {3915, 11508}, {4306, 15803}, {4337, 5010}, {5396, 14547}, {6001, 9371}, {6198, 7551}, {7117, 20752}, {9370, 12114}, {10523, 21935}, {20729, 22098}, {20821, 22076}, {22054, 22118}, {22061, 22447}, {22067, 22082}

X(22350) = isogonal conjugate of X(36123)
X(22350) = isogonal conjugate of polar conjugate of X(908)
X(22350) = isotomic conjugate of polar conjugate of X(2183)
X(22350) = crossdifference of every pair of points on line X(19)X(649)
X(22350) = X(19)-isoconjugate of X(34234)
X(22350) = X(92)-isoconjugate of X(909)

X(22351) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(8588)

Barycentrics    9 a^4 - a^3 b - 9 a^2 b^2 - a b^3 - a^3 c - a^2 b c - a b^2 c - b^3 c - 9 a^2 c^2 - a b c^2 - a c^3 - b c^3 : :

X(22351) lies on these lines: {2, 3}, {86, 8588}, {2482, 17271}, {7618, 17346}, {8182, 17378}, {8584, 18755}, {8589, 17277}, {15533, 17206}, {15655, 17379}

X(22351) = {X(2),X(3)}-harmonic conjugate of X(22355)

X(22352) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22352) lies on these lines: {2, 1495}, {3, 49}, {6, 21969}, {22, 51}, {23, 5643}, {25, 373}, {26, 13336}, {39, 1501}, {52, 7525}, {54, 15644}, {110, 3819}, {125, 6676}, {154, 5646}, {186, 16836}, {187, 3051}, {199, 13329}, {216, 8779}, {228, 20778}, {376, 11427}, {389, 7512}, {428, 3589}, {511, 1994}, {548, 10610}, {572, 16064}, {575, 3060}, {578, 10323}, {631, 10282}, {1176, 11574}, {1194, 1691}, {1340, 21032}, {1341, 21036}, {1350, 11402}, {1368, 13394}, {1428, 5310}, {1503, 7499}, {1511, 12100}, {1614, 11793}, {1619, 23041}, {1692, 20859}, {1799, 12215}, {1843, 5157}, {1899, 7494}, {1915, 5116}, {1993, 3098}, {2070, 5892}, {2076, 14153}, {2194, 5096}, {2330, 5322}, {2916, 9969}, {2937, 5462}, {2979, 11003}, {3066, 20850}, {3289, 22052}, {3398, 21512}, {3431, 19708}, {3518, 11695}, {3523, 14826}, {3524, 11464}, {3530, 5944}, {3534, 14805}, {3690, 5314}, {3787, 15513}, {3934, 10328}, {3937, 3955}, {4048, 8891}, {4175, 6390}, {5007, 11205}, {5020, 22112}, {5026, 15822}, {5050, 15004}, {5135, 5347}, {5446, 13353}, {5650, 6800}, {5946, 7555}, {6146, 16197}, {6467, 19126}, {6515, 11179}, {6660, 12054}, {6688, 13595}, {6689, 17712}, {6759, 7509}, {6823, 21659}, {7383, 9833}, {7400, 19467}, {7488, 9729}, {7495, 21243}, {7500, 14561}, {7502, 9730}, {7503, 11381}, {7514, 15030}, {7516, 10539}, {7550, 14157}, {7556, 15045}, {7558, 18381}, {7592, 14531}, {7998, 9544}, {8041, 14567}, {8627, 20965}, {8703, 10564}, {8718, 13474}, {9714, 15805}, {9738, 13616}, {9739, 13617}, {9909, 10601}, {10110, 12088}, {10170, 10540}, {10219, 16042}, {10298, 20791}, {10541, 17810}, {10691, 11064}, {11414, 11424}, {11449, 15717}, {11513, 21641}, {11514, 21640}, {11515, 21648}, {11516, 21647}, {11572, 13160}, {13347, 17928}, {13419, 14788}, {13434, 13598}, {13445, 14118}, {13851, 15760}, {14128, 23060}, {14130, 14641}, {14855, 18570}, {15107, 21849}, {17704, 22467}, {20752, 22054}, {20780, 22060}

X(22352) = isogonal conjugate of polar conjugate of X(3589)
X(22352) = isotomic conjugate of polar conjugate of X(5007)
X(22352) = isogonal conjugate of isotomic conjugate of X(7767)
X(22352) = X(92)-isoconjugate of X(3108)
X(22352) = {X(3),X(49)}-harmonic conjugate of X(5447)

X(22353) = X(5)X(568)∩X(186)X(476)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28208.

X(22353) lies on these lines: {5, 568}, {186, 476}, {230, 15355}

X(22354) = X(41)X(2361)∩X(672)X(5036)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28208.

X(22354) lies on these lines: {41, 2361}, {672, 5036}

X(22355) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(8589)

Barycentrics    9 a^4 + a^3 b - 9 a^2 b^2 + a b^3 + a^3 c + a^2 b c + a b^2 c + b^3 c - 9 a^2 c^2 + a b c^2 + a c^3 + b c^3 : :

X(22355) lies on these lines: {2,3}, {86,8589}, {2482,17297}, {4487,7354}, {7618,17378}, {8182,17346}, {8588,17277}, {15655,17349}, {17206,22165}

X(22355) = {X(2),X(3)}-harmonic conjugate of X(22351)

X(22356) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22356) lies on these lines: {1, 1731}, {3, 48}, {6, 1201}, {9, 2317}, {19, 7982}, {37, 21748}, {44, 1319}, {101, 953}, {112, 2755}, {184, 2318}, {220, 2267}, {228, 22372}, {284, 3746}, {374, 16666}, {517, 2173}, {519, 8756}, {520, 647}, {604, 2911}, {610, 7991}, {672, 3446}, {692, 2340}, {899, 5137}, {902, 3285}, {952, 7359}, {1023, 2325}, {1055, 2245}, {1100, 21808}, {1332, 20769}, {1334, 2278}, {1420, 1732}, {1473, 22435}, {1618, 2272}, {1797, 22128}, {1953, 10222}, {2174, 2269}, {2182, 6603}, {2197, 22058}, {2246, 18839}, {2256, 3303}, {2260, 5563}, {2261, 2324}, {2273, 7772}, {2300, 5007}, {2347, 3204}, {2364, 16676}, {3009, 16795}, {3942, 6510}, {3984, 5227}, {4466, 9028}, {5053, 5526}, {5158, 22063}, {7117, 22059}, {8609, 17439}, {15178, 17438}, {20754, 20975}, {20760, 23082}, {20766, 22143}, {20777, 22096}, {20796, 22158}, {22123, 22144}

X(22356) = isogonal conjugate of X(6336)
X(22356) = isotomic conjugate of polar conjugate of X(902)
X(22356) = X(19)-isoconjugate of X(903)

X(22357) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22357) lies on these lines: {3, 48}, {228, 22096}, {284, 5563}, {1055, 2278}, {1201, 3285}, {1385, 2173}, {1790, 1797}, {1953, 15178}, {2267, 3207}, {3284, 22063}, {3304, 37519}, {3955, 23081}, {4289, 17474}, {10222, 17438}

X(22357) = isogonal conjugate of polar conjugate of X(551)
X(22357) = isotomic conjugate of polar conjugate of X(21747)

X(22358) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(10485)

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

X(22358) lies on these lines: {2, 3}, {86, 10485}, {8586, 17277}

X(22359) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(10979)

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

X(22359) lies on these lines: {2, 3}, {86, 10979}, {17277, 22052}

X(22360) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(12212)

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

X(22360) lies on these lines: {2, 3}, {86, 12212}, {2271, 16990}, {3329, 17206}, {13331, 17277}

X(22361) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22361) lies on these lines: {1, 6875}, {3, 73}, {21, 1936}, {47, 1064}, {55, 1468}, {58, 14547}, {71, 577}, {165, 21147}, {283, 6514}, {411, 1935}, {417, 8763}, {602, 1450}, {896, 1858}, {902, 3057}, {1006, 3075}, {1040, 4652}, {1155, 1254}, {1193, 2361}, {1259, 2318}, {1333, 2269}, {1364, 22076}, {1399, 4300}, {1407, 5204}, {1457, 11012}, {1745, 6876}, {1794, 1795}, {2646, 2650}, {3074, 6905}, {3601, 4257}, {3915, 10966}, {3916, 7004}, {5217, 7074}, {6149, 14794}, {14597, 22054}, {20753, 22390}, {20775, 20780}, {22345, 22347}

X(22361) = isogonal conjugate of polar conjugate of X(5745)
X(22361) = isotomic conjugate of polar conjugate of X(21748)

X(22362) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22362) lies on these lines: {3, 23074}, {73, 22422}, {228, 22402}, {14597, 22075}

X(22362) = isogonal conjugate of polar conjugate of X(16582)
X(22362) = isotomic conjugate of polar conjugate of X(21749)

X(22363) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22363) lies on these lines: {3, 326}, {25, 18615}, {31, 1974}, {71, 228}, {184, 14597}, {1040, 1473}, {1402, 1918}, {2178, 18611}, {7083, 16583}, {20775, 20780}, {22094, 22402}, {22341, 22364}

X(22363) = isogonal conjugate of polar conjugate of X(16583)
X(22363) = isotomic conjugate of polar conjugate of X(21750)
X(22363) = crosssum of X(4) and X(75)
X(22363) = crosspoint of X(3) and X(31)

X(22364) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22364) lies on these lines: {3, 304}, {71, 22367}, {73, 22373}, {228, 22061}, {682, 22368}, {863, 16583}, {20775, 22345}, {22341, 22363}, {22344, 22386}, {22348, 22422}

X(22364) = isogonal conjugate of polar conjugate of X(16584)
X(22364) = isotomic conjugate of polar conjugate of X(21751)

X(22365) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(10316)

Barycentrics    (a^2 - b^2 - c^2) (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 + a^2 b^5 c + a b^6 c + b^7 c - a^6 c^2 - a^5 b c^2 - 2 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 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 - 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 + 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(22365) lies on these lines: {2, 3}, {86, 10316}, {5224, 14376}, {7767, 18643}, {17206, 20806}

X(22366) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(10317)

Barycentrics    (a^2 - b^2 - c^2) (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 + a^2 b^5 c + a b^6 c + b^7 c - a^6 c^2 - a^5 b c^2 - 2 a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 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 - 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 + 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(22366) lies on these lines: {2, 3}, {86, 10317}, {17206, 22151}

X(22367) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22367) lies on these lines: {3, 23077}, {71, 22364}, {228, 20727}, {1818, 22078}, {3690, 20777}, {22061, 22373}

X(22367) = isogonal conjugate of polar conjugate of X(16587)
X(22367) = isotomic conjugate of polar conjugate of X(21752)

X(22368) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22368) lies on these lines: {3, 348}, {55, 2295}, {212, 7116}, {408, 22369}, {682, 22364}, {20775, 20780}

X(22368) = isogonal conjugate of polar conjugate of X(16588)
X(22368) = isotomic conjugate of polar conjugate of X(9449)

X(22369) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22369) lies on these lines: {3, 69}, {71, 228}, {237, 1030}, {408, 22368}, {851, 5218}, {966, 1011}, {1213, 8053}, {1654, 4184}, {1818, 22076}, {2092, 2223}, {2238, 20992}, {2245, 3779}, {2642, 8638}, {3941, 4272}, {4191, 4648}, {4210, 17300}, {18591, 20728}, {20750, 22054}, {22072, 22079}, {22082, 22373}, {22097, 22412}

X(22369) = isogonal conjugate of polar conjugate of X(16589)
X(22369) = isotomic conjugate of polar conjugate of X(21753)

X(22370) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22370) lies on these lines: {1, 3778}, {2, 2269}, {3, 22378}, {9, 1654}, {40, 4645}, {43, 2209}, {46, 3178}, {48, 1332}, {55, 17792}, {57, 17300}, {63, 69}, {75, 21231}, {77, 2197}, {78, 3781}, {87, 2108}, {100, 1253}, {190, 17786}, {192, 1423}, {193, 672}, {219, 20769}, {228, 3504}, {239, 3169}, {304, 4019}, {344, 2183}, {385, 7075}, {573, 3912}, {579, 3879}, {604, 21495}, {894, 3501}, {966, 3305}, {1018, 3729}, {1025, 1419}, {1040, 22418}, {1334, 17257}, {1400, 5933}, {1424, 7783}, {1716, 3747}, {1742, 3888}, {1818, 4855}, {2245, 4851}, {2268, 15988}, {3056, 8299}, {3218, 17375}, {3219, 17343}, {3306, 4648}, {3685, 6210}, {3730, 4416}, {3779, 3870}, {3784, 22413}, {4110, 4595}, {4266, 17353}, {4271, 17279}, {4553, 15624}, {4660, 5119}, {5036, 17311}, {5440, 22083}, {8680, 20930}, {9025, 20992}, {14923, 17868}, {16574, 17296}, {16609, 20171}, {17294, 21061}, {17298, 20367}, {17363, 21384}, {17379, 17754}, {20775, 20787}, {20777, 20794}, {20821, 22169}

X(22370) = isogonal conjugate of polar conjugate of X(6376)
X(22370) = isotomic conjugate of polar conjugate of X(43)
X(22370) = X(3)-Ceva conjugate of X(63)
X(22370) = X(19)-isoconjugate of X(87)

X(22371) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22371) lies on these lines: {3, 1331}, {71, 22372}, {184, 23073}, {212, 3270}, {228, 22096}, {1623, 2810}, {20975, 22080}

X(22371) = isogonal conjugate of polar conjugate of X(4370)
X(22371) = isotomic conjugate of polar conjugate of X(1017)
X(22371) = X(92)-isoconjugate of X(2226)

X(22372) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22372) lies on these lines: {3, 22067}, {71, 22371}, {228, 22356}, {22080, 22429}

X(22372) = isogonal conjugate of polar conjugate of X(16590)
X(22372) = isotomic conjugate of polar conjugate of X(21754)

X(22373) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22373) lies on these lines: {3, 4592}, {73, 22364}, {228, 22375}, {667, 20982}, {3023, 4367}, {3937, 22386}, {7117, 20975}, {20727, 22381}, {20738, 20787}, {20754, 20777}, {22061, 22367}, {22082, 22369}

X(22373) = isogonal conjugate of polar conjugate of X(16592)
X(22373) = isotomic conjugate of polar conjugate of X(21755)

X(22374) =  (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22380)

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

X(22374) lies on these lines: {2, 3}, {4383, 22380}, {25083, 36504}

X(22375) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22375) lies on these lines: {3, 3708}, {228, 22373}, {647, 22055}, {20727, 22341}, {20784, 22345}, {22061, 22342}

X(22375) = isogonal conjugate of polar conjugate of X(16598)
X(22375) = isotomic conjugate of polar conjugate of X(21756)

X(22376) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22376) lies on these lines: {3, 63}, {36, 20843}, {3893, 8683}, {3937, 22072}, {5122, 7428}, {20780, 22378}

X(22376) = isogonal conjugate of polar conjugate of X(16602)

X(22377) =  (X(3),X(6),X(75),X(1); X(3),X(2),X(75),X(1)) COLLINEATION IMAGE OF X(50)

Barycentrics    a^8 - a^7 b - 3 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 - 3 a^6 c^2 + a^5 b c^2 + 2 a^4 b^2 c^2 + a^2 b^4 c^2 + a b^5 c^2 + a^5 c^3 + a^4 b c^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 + a b^3 c^4 + a^3 c^5 + 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(22377) lies on these lines: {2, 3}, {50, 86}, {323, 17206}, {566, 17277}, {3580, 18755}, {17271, 18375}

X(22378) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22378) lies on these lines: {3, 22370}, {48, 20777}, {69, 20757}, {71, 20759}, {228, 20775}, {1444, 22449}, {20753, 20781}, {20780, 22376}, {22056, 22079}, {22344, 22390}

X(22378) = isogonal conjugate of polar conjugate of X(16604)
X(22378) = isotomic conjugate of polar conjugate of X(21757)

X(22379) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22379) lies on these lines: {3, 23087}, {36, 3738}, {56, 1769}, {526, 3724}, {667, 6085}, {905, 2850}, {1459, 1946}, {3937, 22096}, {4768, 8666}

X(22379) = isogonal conjugate of polar conjugate of X(3960)
X(22379) = isotomic conjugate of polar conjugate of X(21758)
X(22379) = X(19)-isoconjugate of X(36804)

X(22380) =  (pending)

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

X(22380) lies on these lines: {3, 6}, {980, 18134}, {986, 2276}, {1575, 16583}, {1759, 17596}, {2275, 16787}, {5283, 16062}

X(22381) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22381) lies on these lines: {3, 22370}, {25, 2053}, {32, 2209}, {63, 3504}, {87, 19762}, {98, 932}, {184, 15373}, {228, 22061}, {667, 22224}, {1402, 3747}, {1799, 22449}, {2196, 17970}, {2319, 5285}, {14199, 17797}, {20727, 22373}, {20996, 21857}, {22066, 22096}

X(22381) = isogonal conjugate of polar conjugate of X(16606)
X(22381) = isotomic conjugate of polar conjugate of X(21759)
X(22381) = X(19)-isoconjugate of X(31008)
X(22381) = X(92)-isoconjugate of X(27644)

X(22382) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22382) lies on these lines: {3, 822}, {48, 8611}, {284, 16612}, {450, 8062}, {662, 18020}, {1813, 9358}, {2249, 2706}, {6332, 8632}, {20731, 20757}

X(22382) = isogonal conjugate of polar conjugate of X(8062)
X(22382) = isotomic conjugate of polar conjugate of X(21761)

X(22383) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22383) lies on these lines: {6, 650}, {31, 8641}, {81, 693}, {112, 2719}, {513, 1430}, {514, 21117}, {520, 647}, {521, 2522}, {649, 854}, {654, 6589}, {656, 2523}, {661, 20980}, {667, 838}, {788, 8646}, {810, 822}, {894, 21438}, {905, 4131}, {940, 4885}, {1021, 21173}, {1364, 22432}, {2451, 17418}, {2504, 17094}, {2720, 7115}, {3287, 6590}, {3288, 4449}, {3738, 16612}, {3758, 21611}, {4394, 21786}, {4790, 21007}, {5040, 20983}, {6373, 8633}, {6586, 9404}, {9010, 21005}, {11269, 15280}, {20729, 22059}, {20731, 20757}, {20796, 22160}, {22444, 22445}

X(22383) = isogonal conjugate of X(6335)
X(22383) = isotomic conjugate of polar conjugate of X(667)
X(22383) = X(2)-Ceva conjugate of X(34467)
X(22383) = crossdifference of every pair of points on line X(4)X(8)
X(22383) = polar conjugate of isotomic conjugate of X(23224)
X(22383) = X(19)-isoconjugate of X(668)

X(22384) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22384) lies on these lines: {3, 22437}, {31, 2254}, {58, 3960}, {238, 3716}, {514, 21761}, {520, 647}, {580, 2814}, {595, 3887}, {659, 3808}, {810, 22154}, {905, 22093}, {928, 8578}, {1193, 8648}, {1331, 1332}, {1468, 14413}, {1724, 3762}, {1946, 22090}, {2196, 22155}, {3915, 4895}, {3937, 22096}, {22148, 22158}, {22349, 22387}, {23069, 23092}

X(22384) = isogonal conjugate of polar conjugate of X(812)
X(22384) = isotomic conjugate of polar conjugate of X(8632)
X(22384) = X(19)-isoconjugate of X(4562)
X(22384) = X(92)-isoconjugate of X(813)

X(22385) =  (pending)

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

X(22385) lies on these lines: {3,6}

X(22386) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22386) lies on these lines: {3, 4561}, {669, 4128}, {1015, 8637}, {3937, 22373}, {16695, 21138}, {20778, 22345}, {22344, 22364}

X(22386) = isogonal conjugate of polar conjugate of X(6377)
X(22386) = isotomic conjugate of polar conjugate of X(21762)

X(22387) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22387) lies on these lines: {3, 23092}, {3736, 4369}, {20731, 20757}, {20828, 22095}, {22349, 22384}

X(22387) = isotomic conjugate of polar conjugate of X(21763)

X(22388) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22388) lies on these lines: {3, 4025}, {32, 21122}, {187, 237}, {228, 652}, {1011, 3239}, {2352, 6589}, {4191, 7658}

X(22388) = isogonal conjugate of polar conjugate of X(6586)

X(22389) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22389) lies on these lines: {3, 22370}, {48, 184}, {63, 20794}, {69, 22449}, {71, 20775}, {216, 22169}, {237, 2269}, {283, 7015}, {1444, 22060}, {2223, 7122}, {6467, 22070}, {11574, 20821}, {18210, 18606}, {20750, 22054}, {20769, 23079}, {20975, 22058}

X(22389) = isogonal conjugate of polar conjugate of X(1107)
X(22389) = isotomic conjugate of polar conjugate of X(1197)
X(22389) = X(19)-isoconjugate of X(1221)
X(22389) = X(92)-isoconjugate of X(1258)

X(22390) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22390) lies on these lines: {3, 48}, {22, 14547}, {42, 5347}, {56, 19133}, {73, 1176}, {182, 2183}, {184, 22097}, {198, 5085}, {212, 3796}, {228, 20778}, {326, 4652}, {511, 2317}, {560, 1193}, {572, 3220}, {603, 1804}, {1400, 5135}, {1790, 4575}, {1890, 17023}, {2174, 5096}, {2260, 5138}, {2293, 20872}, {2318, 5314}, {3955, 22148}, {4259, 21748}, {4265, 7113}, {5132, 9454}, {8766, 17102}, {20753, 22361}, {20775, 22345}, {22344, 22378}

X(22390) = isogonal conjugate of polar conjugate of X(17023)
X(22390) = isotomic conjugate of polar conjugate of X(21764)

X(22391) =  X(2)-CEVA CONJUGATE OF X(184)

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

Let P be a point on the circumcircle. Let T be the trilinear pole of the polar of P wrt the polar circle (i.e., T is the polar conjugate of P). Let T' be the isogonal conjugate of T. (T' is also the barycentric product X(3)*P.) The locus of T' as P varies is the circumconic {{A,B,C,X(112),X(248)}}, the isogonal conjugate of line X(297)X(525), which is the polar conjugate of the circumcircle. The center of the conic is X(22391). This conic is an ellipse if ABC is acute, and a hyperbola if ABC is obtuse. The conic passes through X(112), X(248), X(1415), X(1576), X(4558), X(14578), X(14908), and X(18877). The perspector of the conic is X(184). (Randy Hutson, September 9, 2018)

The conic {{A,B,C,X(112),X(248)}} is also the locus of barycentric product of circumcircle antipodes. (Randy Hutson, January 15, 2019)

X(22391) lies on these lines: {2, 11610}, {32, 51}, {184, 14600}, {230, 427}, {248, 1899}, {343, 441}, {426, 577}, {578, 14773}, {647, 9306}, {1627, 9753}

X(22391) = isogonal conjugate of polar conjugate of X(157)
X(22391) = isotomic conjugate of polar conjugate of X(2909)
X(22391) = complement of isogonal conjugate of X(157)
X(22391) = X(2)-Ceva conjugate of X(184)
X(22391) = perspector of circumconic centered at X(184)
X(22391) = barycentric product X(i)*X(j) for these {i,j}: {3, 157}, {48, 21374}, {184, 11442}

X(22392) =  X(1)X(5)∩X(20)X(386)

Barycentrics    a (3 a^5 b-6 a^3 b^3+3 a b^5+3 a^5 c-3 a^3 b^2 c+a^2 b^3 c-b^5 c-3 a^3 b c^2-6 a^2 b^2 c^2-3 a b^3 c^2-6 a^3 c^3+a^2 b c^3-3 a b^2 c^3+2 b^3 c^3+3 a c^5-b c^5) : :
X(22392) = 3 (b + c) (c + a) (a + b)X[1] - 8 a b c X[5]

X(22392) lies on the cubic K1071 these lines: {1,5}, {20,386}, {40,5754}, {42,4301}, {43,9568}, {165,970}, {500,3530}, {581,631}, {991,15717}, {1064,3293}, {2051,5691}, {3017,6960}, {3832,19767}, {4192,7991}, {4658,6915}, {5312,9589}, {5453,16239}, {7982,19648}, {7987,21363}, {9275,13434}, {9706,17104}, {11224,15488}, {16189,19646}

X(22393) =  (pending)

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

X(22393) lies on these lines: {3,6} et al

X(22394) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22394) lies on these lines: {71, 22057}, {255, 6505}, {1214, 4055}, {3747, 3914}, {3915, 4642}, {3917, 22399}, {22409, 22434}

X(22394) = isogonal conjugate of polar conjugate of X(21231)
X(22394) = isotomic conjugate of polar conjugate of X(23621)

X(22395) =  (pending)

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

X(22395) lies on these lines: {3, 6}

X(22396) =  (pending)

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

X(22396) lies on these lines: {3, 6}l

X(22397) =  (pending)

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

X(22397) lies on these lines: {3, 6}

X(22398) =  (pending)

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

X(22398) lies on these lines: {3, 6}, {982, 2275}, {1107, 4438}

X(22399) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22399) lies on these lines: {71, 20728}, {1332, 20778}, {3270, 20749}, {3917, 22394}, {8677, 22346}, {22057, 22418}, {22085, 22093}

X(22399) = isogonal conjugate of polar conjugate of X(21232)
X(22399) = isotomic conjugate of polar conjugate of X(23622)

X(22400) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22400) lies on these lines: {63, 212}, {71, 22418}, {1409, 22053}, {3747, 21334}, {3917, 22394}, {22057, 22064}, {22060, 22074}, {22345, 22347}

X(22400) = isogonal conjugate of polar conjugate of X(21233)
X(22400) = isotomic conjugate of polar conjugate of X(23623)

X(22401) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

Barycentrics    a^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) : :

X(22401) lies on these lines: {3, 6}, {20, 232}, {30, 3199}, {115, 11585}, {127, 7821}, {185, 3289}, {230, 16196}, {441, 7789}, {682, 6467}, {980, 18592}, {1015, 1062}, {1038, 2276}, {1040, 2275}, {1060, 1500}, {1194, 7386}, {1196, 1368}, {1506, 15760}, {1589, 8962}, {1625, 10575}, {1843, 11326}, {1968, 11413}, {2207, 21312}, {2549, 6643}, {2883, 11672}, {3146, 15355}, {3269, 5562}, {3291, 16051}, {3522, 22240}, {3546, 3767}, {3548, 7746}, {3815, 6823}, {3917, 22416}, {3926, 6338}, {3933, 15526}, {3964, 6461}, {6337, 6509}, {6748, 9825}, {7603, 10024}, {7736, 10996}, {7748, 15075}, {7749, 10257}, {7756, 12605}, {7801, 14376}, {7816, 15013}, {10311, 17928}, {10313, 22467}, {20727, 22421}, {22057, 22060}, {22064, 22065}

X(22401) = isogonal conjugate of polar conjugate of X(1368)
X(22401) = isotomic conjugate of polar conjugate of X(6467)

X(22402) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22402) lies on these lines: {71, 22057}, {228, 22362}, {3778, 4466}, {22094, 22363}

X(22402) = isogonal conjugate of polar conjugate of X(16607)
X(22402) = isotomic conjugate of polar conjugate of X(23624)

X(22403) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22403) lies on these lines: {71, 22057}, {22093, 22444}

X(22403) = isogonal conjugate of polar conjugate of X(21234)
X(22403) = isotomic conjugate of polar conjugate of X(23625)

X(22404) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22404) lies on these lines: {71, 22077}, {20727, 22069}, {20819, 22411}, {22057, 22434}, {22094, 22439}

X(22404) = isogonal conjugate of polar conjugate of X(21235)
X(22404) = isotomic conjugate of polar conjugate of X(23626)

X(22405) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22405) lies on these lines: {3917, 22064}, {22060, 22084}, {22094, 22097}, {22412, 22420}

X(22405) = isogonal conjugate of polar conjugate of X(21236)
X(22405) = isotomic conjugate of polar conjugate of X(23627)

X(22406) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22406) lies on these lines: {1818, 22094}, {3917, 22064}, {20821, 22090}, {20823, 22432}, {22067, 22084}

X(22406) = isogonal conjugate of polar conjugate of X(21237)
X(22406) = isotomic conjugate of polar conjugate of X(23628)

X(22407) =  (pending)

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

X(22407) lies on these lines: {3,6}

X(22408) =  (pending)

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

X(22408) lies on these lines: {3,6}

X(22409) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22409) lies on these lines: {71, 228}, {3917, 20730}, {7116, 17977}, {20727, 22069}, {20736, 22060}, {20757, 22065}, {22077, 22094}, {22394, 22434}

X(22409) = isogonal conjugate of polar conjugate of X(21238)
X(22409) = isotomic conjugate of polar conjugate of X(23629)

X(22410) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22410) lies on these lines: {39, 20229}, {216, 22053}, {1473, 7117}, {3917, 22064}, {17102, 18652}, {22057, 22060}, {22059, 22435}

X(22410) = isogonal conjugate of polar conjugate of X(21239)
X(22410) = isotomic conjugate of polar conjugate of X(23630)

X(22411) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22411) lies on these lines: {3917, 20820}, {20727, 22416}, {20819, 22404}, {20823, 22069}

X(22411) = isogonal conjugate of polar conjugate of X(17047)
X(22411) = isotomic conjugate of polar conjugate of X(23631)

X(22412) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22412) lies on these lines: {69, 3784}, {71, 3917}, {1155, 22301}, {1818, 22078}, {3781, 3916}, {20727, 20819}, {20730, 22073}, {20731, 22062}, {22097, 22369}, {22405, 22420}

X(22412) = isogonal conjugate of polar conjugate of X(21240)
X(22412) = isotomic conjugate of polar conjugate of X(23632)

X(22413) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22413) lies on these lines: {39, 21751}, {69, 3937}, {71, 3917}, {1818, 22344}, {3781, 4652}, {3784, 22370}, {20727, 20734}, {20730, 22071}, {20819, 20830}, {20821, 22064}

X(22413) = isogonal conjugate of polar conjugate of X(20255)
X(22413) = isotomic conjugate of polar conjugate of X(22199)

X(22414) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22414) lies on these lines: {71, 22083}, {2524, 3049}, {3269, 20825}, {3917, 20727}, {7117, 20729}, {20752, 22098}, {22059, 22428}

X(22414) = isogonal conjugate of polar conjugate of X(21241)
X(22414) = isotomic conjugate of polar conjugate of X(23633)

X(22415) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22415) lies on these lines: {71, 7117}, {3917, 20727}

X(22415) = isogonal conjugate of polar conjugate of X(21242)
X(22415) = isotomic conjugate of polar conjugate of X(23634)

X(22416) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22416) lies on these lines: {2, 9290}, {3, 248}, {6, 5889}, {39, 3289}, {69, 194}, {185, 216}, {217, 13754}, {232, 5907}, {343, 5254}, {394, 5013}, {574, 1092}, {577, 8565}, {1216, 14961}, {1506, 1568}, {1625, 5876}, {1970, 14118}, {1971, 7488}, {2088, 7749}, {3124, 13881}, {3199, 15030}, {3331, 12162}, {3917, 22401}, {5038, 22151}, {7512, 13509}, {7691, 10313}, {12111, 22240}, {14901, 22109}, {15056, 15355}, {20727, 22411}, {22070, 22432}

X(22416) = isogonal conjugate of polar conjugate of X(21243)
X(22416) = isotomic conjugate of polar conjugate of X(23635)
X(22416) = crosssum of X(4) and X(32)
X(22416) = crosspoint of X(3) and X(76)

X(22417) =  (pending)

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

X(22417) lies on these lines: {3,6}

X(22418) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22418) lies on these lines: {71, 22400}, {212, 5314}, {306, 7004}, {1040, 22370}, {1364, 20732}, {2197, 22053}, {3778, 21334}, {3917, 22064}, {20727, 22411}, {20821, 20824}, {22057, 22399}, {22072, 22076}, {22084, 22435}

X(22418) = isogonal conjugate of polar conjugate of X(21244)
X(22418) = isotomic conjugate of polar conjugate of X(23637)

X(22419) =  (pending)

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

X(22419) lies on these lines: {3,6}

X(22420) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22420) lies on these lines: {216, 3289}, {17052, 21318}, {18591, 22060}, {20727, 22069}, {20738, 22094}, {20821, 22076}, {20822, 22432}

X(22420) = isogonal conjugate of polar conjugate of X(21245)
X(22420) = isotomic conjugate of polar conjugate of X(23639)

X(22421) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22421) lies on these lines: {3, 73}, {55, 2274}, {326, 1040}, {426, 22057}, {497, 1740}, {1010, 2654}, {1936, 13588}, {2269, 17187}, {2309, 21321}, {3009, 21333}, {3736, 14547}, {3917, 22064}, {20727, 22401}, {20824, 22449}, {22060, 22074}

X(22421) = isogonal conjugate of polar conjugate of X(21246)
X(22421) = isotomic conjugate of polar conjugate of X(23640)

X(22422) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22422) lies on these lines: {73, 22362}, {22061, 22069}, {22348, 22364}

X(22422) = isogonal conjugate of polar conjugate of X(21247)
X(22422) = isotomic conjugate of polar conjugate of X(23641)

X(22423) =  (pending)

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

X(22423) lies on these lines: {3,6}

X(22424) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22424) lies on these lines: {3, 1176}, {39, 3051}, {2525, 5489}, {2979, 9917}, {3095, 10519}, {7767, 20975}, {7795, 14003}, {20821, 22060}

X(22424) = isogonal conjugate of polar conjugate of X(21248)
X(22424) = isotomic conjugate of polar conjugate of X(23642)

X(22425) =  (pending)

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

X(22425) lies on these lines: {3, 6}, {1500, 3721}, {1759, 2276}, {2223, 20966}, {2240, 21838}, {5051, 16589}, {5283, 17676}

X(22426) =  (pending)

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

X(22426) lies on these lines: {3, 6}, {37, 4660}, {941, 17300}, {980, 17378}, {1908, 2243}, {2223, 3764}

X(22427) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22427) lies on these lines: {3, 20738}, {78, 3781}, {3917, 20755}, {20727, 20734}, {20821, 20824}, {20822, 22070}

X(22427) = isogonal conjugate of polar conjugate of X(21250)
X(22427) = isotomic conjugate of polar conjugate of X(23643)

X(22428) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22428) lies on these lines: {55, 4286}, {71, 7117}, {900, 1635}, {1293, 8752}, {2267, 4271}, {3269, 22073}, {3917, 22084}, {20727, 22429}, {20975, 22080}, {22059, 22414}

X(22428) = isogonal conjugate of polar conjugate of X(121)
X(22428) = isotomic conjugate of polar conjugate of X(23644)

X(22429) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22429) lies on these lines: {71, 22083}, {3917, 22059}, {20727, 22428}, {22080, 22372}

X(22429) = isogonal conjugate of polar conjugate of X(21251)
X(22429) = isotomic conjugate of polar conjugate of X(23645)

X(22430) =  (pending)

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

X(22430) lies on these lines: {3, 6}, {1500, 1759}

X(22431) =  (pending)

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

X(22431) lies on these lines: {3, 6}, {1759, 9331}

X(22432) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22432) lies on these lines: {1364, 22383}, {3269, 7117}, {3917, 20820}, {20822, 22420}, {20823, 22406}, {22070, 22416}

X(22432) = isogonal conjugate of polar conjugate of X(21252)
X(22432) = isotomic conjugate of polar conjugate of X(23646)

X(22433) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22433) lies on these lines: {3269, 22084}, {20820, 22073}

X(22433) = isogonal conjugate of polar conjugate of X(21253)
X(22433) = isotomic conjugate of polar conjugate of X(23647)

X(22434) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22434) lies on these lines: {71, 22094}, {20735, 20756}, {22057, 22404}, {22394, 22409}

X(22434) = isogonal conjugate of polar conjugate of X(21254)
X(22434) = isotomic conjugate of polar conjugate of X(23648)

X(22435) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22435) lies on these lines: {71, 3917}, {394, 20780}, {1473, 22356}, {1818, 3784}, {2318, 3937}, {20727, 22088}, {20731, 22066}, {22059, 22410}, {22084, 22418}

X(22435) = isogonal conjugate of polar conjugate of X(21255)
X(22435) = isotomic conjugate of polar conjugate of X(23649)

X(22436) =  (pending)

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

X(22436) lies on these lines: {3, 6}, {1500, 2243}, {14020, 16589}

X(22437) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22437) lies on these lines: {3, 22384}, {3960, 4256}, {7117, 22084}, {20731, 20757}

X(22437) = isogonal conjugate of polar conjugate of X(21255)
X(22437) = isotomic conjugate of polar conjugate of X(23650)

X(22438) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22438) lies on these lines: {71, 22349}, {20727, 22069}, {20828, 22095}

X(22438) = isogonal conjugate of polar conjugate of X(21256)
X(22438) = isotomic conjugate of polar conjugate of X(23651)

X(22439) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22439) lies on these lines: {63, 20736}, {71, 228}, {216, 20729}, {3917, 20755}, {22094, 22404}

X(22439) = isogonal conjugate of polar conjugate of X(21257)
X(22439) = isotomic conjugate of polar conjugate of X(23652)

X(22440) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22440) lies on these lines: {3, 1803}, {71, 3917}, {77, 3270}, {185, 4303}, {216, 22084}, {373, 2635}, {1155, 22277}, {1253, 1362}, {1425, 10884}, {1439, 10167}, {3000, 21746}, {3937, 6467}, {20731, 22071}, {22064, 22070}

X(22440) = isogonal conjugate of polar conjugate of X(21258)
X(22440) = isotomic conjugate of polar conjugate of X(23653)

X(22441) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22441) lies on these lines: {4269, 8062}, {20828, 22095}, {22093, 22443}

X(22441) = isogonal conjugate of polar conjugate of X(21259)
X(22441) = isotomic conjugate of polar conjugate of X(23654)

X(22442) =  (pending)

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

X(22442) lies on these lines: {3, 6}, {1015, 3721}, {1759, 2275}

X(22443) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22443) lies on these lines: {44, 513}, {71, 1459}, {522, 579}, {1400, 21960}, {2524, 3049}, {17072, 21388}, {22093, 22441}

X(22443) = isogonal conjugate of polar conjugate of X(21255)
X(22443) = isotomic conjugate of polar conjugate of X(23655)

X(22444) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22444) lies on these lines: {812, 4283}, {2524, 3049}, {7117, 22084}, {22093, 22403}, {22383, 22445}

X(22444) = isogonal conjugate of polar conjugate of X(21261)
X(22444) = isotomic conjugate of polar conjugate of X(23656)

X(22445) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22445) lies on these lines: {20821, 22090}, {20828, 22095}, {22383, 22444}

X(22445) = isogonal conjugate of polar conjugate of X(21262)
X(22445) = isotomic conjugate of polar conjugate of X(23657)

X(22446) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22446) lies on these lines: {20828, 22095}, {22093, 22403}

X(22446) = isogonal conjugate of polar conjugate of X(21263)
X(22446) = isotomic conjugate of polar conjugate of X(23658)

X(22447) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22447) lies on these lines: {3, 9247}, {71, 216}, {73, 22099}, {3917, 20755}, {4303, 22098}, {5267, 14963}, {7117, 20750}, {22054, 22073}, {22061, 22350}

X(22447) = isogonal conjugate of polar conjugate of X(3846)
X(22447) = isotomic conjugate of polar conjugate of X(23659)

X(22448) =  (pending)

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

X(22448) lies on these lines: {3, 6}, {2243, 2275}, {3291, 16048}, {3721, 3976}, {9465, 17522}

X(22449) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(22449) lies on these lines: {3, 63}, {38, 18758}, {69, 22389}, {71, 20730}, {672, 14096}, {1444, 22378}, {1799, 22381}, {3917, 20727}, {4640, 20878}, {5322, 17798}, {20819, 22058}, {20824, 22421}, {22053, 22066}

X(22449) = isogonal conjugate of polar conjugate of X(21264)
X(22449) = isotomic conjugate of polar conjugate of X(23660)

X(22450) =  (pending)

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

X(22450) lies on these lines: {3, 6}, {1015, 1759}, {3721, 4694}

X(22451) = X(4)X(1511)∩X(1989)X(3284)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28200.

X(22451) lies on these lines: {4,1511}, {1989,3284}

X(22451) = barycentric quotient X(18487)/X(1539)

X(22452) = (name pending)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28200.

X(22452) lies on this line: {3,10113}

X(22453) = ISOGONAL CONJUGATE OF X(4680)

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

See César E. Lozada, Hyacinthos 28201.

X(22453) lies on these lines: {35, 976}, {2174, 2273}

X(22453) = isogonal conjugate of X(4680)

X(22454) = X(95)X(2070)∩X(252)X(3518)

Barycentrics    SB*SC*(S^2+SA*SB)*(S^2+SA*SC)*(S^2-(3*R^2-SB-2*SW)*SB)*(S^2-(3*R^2-SC-2*SW)*SC) : :

See César E. Lozada, Hyacinthos 28201.

X(22454) lies on these lines: {95, 2070}, {252, 3518}

X(22455) = ISOGONAL CONJUGATE OF X(1531)

Barycentrics    SC*SB*(S^2+3*SA*SC)*(S^2+3*SA*SB)*(S^2-3*SA*SC)*(S^2-3*SA*SB)*(SB+SC) : :

See César E. Lozada, Hyacinthos 28201.

X(22455) lies on these lines: {3, 1494}, {25, 16263}, {32, 8749}, {74, 184}, {186, 5627}, {9139, 14908}, {10151, 16243}, {10152, 13596}

X(22455) = isogonal conjugate of X(1531)
X(22455) = trilinear pole of the line {2433, 3049}

X(22456) = ISOTOMIC CONJUGATE OF X(684)

Barycentrics    (SB^2-SA*SC)*(SC^2-SA*SB)*(SA^2-SC^2)*(SA^2-SB^2)*SB*SC : :
Barycentrics    (csc A)/(sec B sin^3 C - sec C sin^3 B) : :

Let A', B', C' be the intersections of line X(4)X(69) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(22456). (Randy Hutson, October 15, 2018)

Let A"B"C" be the circumsymmedial triangle. Let A* be the pole, wrt the polar circle, of line B"C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(264). The lines A"A*, B"B*, C"C* concur in X(22456). (Randy Hutson, October 15, 2018)

See César E. Lozada, Hyacinthos 28201.

X(22456) lies on the circumcircle and these lines: {4, 2679}, {69, 2706}, {74, 290}, {76, 2710}, {98, 16083}, {99, 22089}, {110, 685}, {111, 16081}, {112, 2966}, {264, 842}, {286, 2699}, {314, 2707}, {340, 9161}, {729, 6531}, {805, 877}, {879, 2713}, {1294, 6394}, {1297, 5999}, {1821, 2249}, {2373, 18024}, {2395, 9091}, {2697, 15915}, {2857, 18022}, {4230, 6037}

X(22456) = isotomic conjugate of X(684)
X(22456) = anticomplement of X(38974)
X(22456) = polar conjugate of X(3569)
X(22456) = polar circle-inverse of X(2679)
X(22456) = trilinear pole of the line {6, 264}
X(22456) = X(63)-isoconjugate of X(2491)
X(22456) = Ψ(X(3), X(76))
X(22456) = Ψ(X(6), X(264))
X(22456) = Ψ(X(32), X(4))
X(22456) = Λ(X(3269), X(9409))
X(22456) = Λ(trilinear polar of X(184))
X(22456) = Λ(trilinear polar of X(237))
X(22456) = Λ(PU(89))
X(22456) = Λ(PU(109))
X(22456) = perspector, wrt circumsymmedial triangle, of polar circle

X(22457) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22457) lies on these lines: {3, 201}, {912, 22341}, {3157, 7016}, {9645, 11248}, {22164, 23084} X(22457) = isogonal conjugate of polar conjugate of X(17479)
X(22457) = isotomic conjugate of polar conjugate of X(21768)

X(22458) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(22458) lies on these lines: {1, 18174}, {3, 63}, {7, 16415}, {9, 16286}, {57, 16414}, {255, 20803}, {329, 19543}, {496, 15507}, {603, 23067}, {859, 3868}, {942, 4245}, {1437, 15409}, {1634, 17104}, {2200, 20785}, {2801, 15622}, {2810, 5399}, {3218, 16453}, {3219, 16287}, {3305, 16291}, {3647, 8053}, {3682, 11573}, {3876, 16374}, {3881, 18613}, {4020, 22126}, {4696, 5687}, {5044, 19261}, {5273, 16290}, {5439, 19250}, {6763, 16678}, {7483, 21319}, {10436, 16408}, {12635, 15654}, {15905, 20764}, {17976, 22138}, {19513, 20245}, {20794, 23076}, {20797, 23091}, {20802, 23084}, {22136, 22161}, {22148, 23070}

X(22459) =  (pending)

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

X(22459) lies on these lines: {3, 6}, {1759, 9336}

X(22460) =  (pending)

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

X(22460) lies on these lines: {3, 6}, {1015, 2243}

X(22461) = X(35)X(37)∩X(3746)X(8143)

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

See Antreas Hatzipolakis, Paul Yiu, and Peter Moses, Hyacinthos 28215.

X(22461) lies on these lines: {35,37}, {3746,8143}

X(22461) = X(1)-waw conjugate of X(3746)

X(22462) = X(2)X(3)∩X(49)X(373)

Barycentrics    a^2 (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+5 a^4 b^2 c^2-11 a^2 b^4 c^2+8 b^6 c^2-11 a^2 b^2 c^4-14 b^4 c^4+2 a^2 c^6+8 b^2 c^6-c^8) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28218.

X(22462) lies on these lines: {2,3}, {49,373}, {74,11017}, {110,15047}, {156,11465}, {195,15026}, {399,12006}, {1493,12834}, {1511,12046}, {3567,12316}, {5643,9705}, {5898,8254}, {5943,14627}, {6688,13353}, {9704,10601}, {9706,15039}, {10263,10545}{10540,11695}, {12308,13630}, {15024,15087}, {15037,18350}

X(22462) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,11484,19709}, {7506,11284,5070}, {11414,15701,3}
X(22462) = crossdifference of every pair of points on line X(647)X(13152)

X(22463) = MIDPOINT OF X(3) AND X(50)

Barycentrics    (SB+SC)*((3*R^2+2*SA-SW)*S^2+(3*R^2-SW)*SA*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28213.

X(22463) lies on these lines: {3, 6}, {858, 6036}, {924, 14270}

X(22463) = midpoint of X(3) and X(50)

X(22464) = ISOGONAL CONJUGATE OF X(2342)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28213.

X(22464) lies on the cubic K660 and these lines: {1, 7}, {2, 20223}, {37, 21617}, {44, 5723}, {57, 16548}, {63, 278}, {75, 225}, {85, 4389}, {88, 655}, {109, 2861}, {223, 5905}, {226, 17080}, {239, 17950}, {241, 1086}, {283, 8822}, {320, 664}, {522, 693}, {527, 651}, {553, 17074}, {653, 8755}, {894, 17086}, {903, 17078}, {908, 1465}, {934, 2716}, {948, 4419}, {1020, 20367}, {1068, 1119}, {1072, 3673}, {1214, 5249}, {1254, 13161}, {1358, 13756}, {1427, 3782}, {1440, 4373}, {1441, 4357}, {1445, 4000}, {1447, 1758}, {1456, 17768}, {1457, 17139}, {1737, 18815}, {1804, 11249}, {1936, 7012}, {1937, 2481}, {1943, 4001}, {2302, 18162}, {3262, 6735}, {3666, 6354}, {3755, 7672}, {3868, 5930}, {3912, 4552}, {4572, 18891}, {4656, 5226}, {5219, 16676}, {5222, 12848}, {5228, 17301}, {5236, 8680}, {6180, 17276}, {7053, 10680}, {7279, 14794}, {9312, 17274}, {9965, 18623}, {10404, 15832}, {14564, 16666}

X(22464) = isogonal conjugate of X(2342)
X(22464) = X(50)-of-intouch triangle
X(22464) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 347, 77), (7, 1442, 3664), (7, 3672, 7190), (175, 176, 5731), (269, 4862, 7), (279, 4346, 7), (948, 4419, 8545), (3638, 3639, 21578), (3663, 3668, 7)

X(22465) = MIDPOINT OF X(1) AND X(22464)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28213.

X(22465) lies on these lines: {1, 7}, {522, 3960}, {1068, 1861}, {2323, 5850}, {3333, 16548}, {6745, 16586}, {16272, 18839}

leftri

Miscellaneous centers: X(22466)-X(23049)

rightri

Centers X(22466)-X(23049) were contributed by César Eliud Lozada, September 11, 2018.

X(22466) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC TO 3rd HATZIPOLAKIS

Barycentrics    (S^2+(16*R^2+SB-4*SW)*SB)*(S^2+(16*R^2+SC-4*SW)*SC) : :

The reciporcal orthologic center of these triangles is X(12241).

X(22466) lies on the Jerabek hyperbola and these lines: {3,2929}, {4,18936}, {5,5504}, {6,17837}, {54,403}, {56,18978}, {64,13399}, {68,5876}, {73,19472}, {74,5894}, {185,11744}, {265,12162}, {381,15317}, {389,3521}, {578,16867}, {895,15044}, {1173,12233}, {1176,19142}, {3426,18381}, {3431,7505}, {4846,13630}, {6145,13851}, {6288,7687}, {6391,15069}, {9927,15316}, {11559,16003}, {11572,15321}, {15002,18388}, {18396,18532}

X(22466) = isogonal conjugate of X(22467)
X(22466) = {X(19083), X(19084)}-harmonic conjugate of X(6)
X(22466) = perspector of 2nd Droz-Farny circle

X(22467) = ISOGONAL CONJUGATE OF X(22466)

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6+5*b^2*c^2*a^4+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^6-c^6)*(b^2-c^2)) : :
X(22467) = 2*(5*R^2-SW)*X(3)+R^2*X(4)

As a point on the Euler line, X(22467) has Shinagawa coefficients (E-4*F, 4*F).

X(22467) lies on these lines: {2,3}, {49,1511}, {54,5504}, {74,12162}, {107,1105}, {110,185}, {182,11443}, {184,10574}, {323,1092}, {389,1994}, {394,1192}, {567,12006}, {569,15045}, {578,15043}, {974,3047}, {1078,1236}, {1147,5890}, {1181,9544}, {1204,9306}, {1209,20191}, {1587,9682}, {1620,17811}, {1968,15355}, {1975,5866}, {1993,9786}, {2079,5254}, {2888,12359}, {2929,13567}, {2931,12022}, {3043,14708}, {3060,13346}, {3357,15305}, {3431,15317}, {3448,14516}, {3521,14643}, {3567,13352}, {3581,6101}, {3917,7691}, {4297,9590}, {5012,9729}, {5218,9659}, {5265,10832}, {5281,10831}, {5422,11425}, {5446,10564}, {5462,15033}, {5640,11424}, {5643,15023}, {5651,11454}, {5663,18350}, {5877,7891}, {5894,10117}, {5907,11440}, {6241,10539}, {6288,13561}, {6759,15072}, {6800,17821}, {6801,18284}, {7288,9672}, {7689,11459}, {8718,14855}, {8780,12174}, {8907,18910}, {9539,11399}, {9591,12512}, {9637,19366}, {9705,15034}, {9706,15020}, {9932,18916}, {10312,14961}, {10516,15578}, {10540,13491}, {10546,11439}, {10575,14157}, {10605,11441}, {10984,11202}, {11003,19357}, {11064,13568}, {11381,13445}, {11430,13434}, {11468,15058}, {12118,18912}, {12901,14644}, {13289,15030}, {13366,15012}, {13403,16163}, {14249,21396}, {14831,15801}, {14927,20987}, {17854,20771}, {18911,19467}

X(22467) = isogonal conjugate of X(22466)
X(22467) = crosspoint, wrt excentral or tangential triangle, of X(3) and X(2929)
X(22467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 2071), (3, 22, 3522), (3, 2915, 7411), (3, 2937, 548), (3, 14130, 10226), (4, 631, 3548), (22, 3522, 16661), (22, 4232, 23), (24, 12082, 9714), (1593, 1995, 3832), (5004, 5005, 1368), (7387, 7516, 6643), (7387, 14130, 18560), (10226, 14130, 3520), (14002, 17578, 1598), (14709, 14710, 2)

X(22468) = ISOTOMIC CONJUGATE OF X(22466)

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

X(22468) lies on these lines: {4,69}, {325,6677}, {801,13567}, {1078,16196}, {3964,7799}

X(22468) = isotomic conjugate of X(22466)
X(22468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69, 317, 14615), (317, 14615, 316)

X(22469) = CYCLOLOGIC CENTER OF THESE TRIANGLES: AAOA TO ARA

Barycentrics    (SB+SC)*((R^2-SW)*(3*R^2-SW)^3*S^4+(3*R^2-SW)*(18*(9*SA+7*SW)*R^8+(27*SA^2-237*SA*SW-157*SW^2)*R^6-(45*SA^2-127*SA*SW-27*SW^2)*SW*R^4+(21*SA^2-31*SA*SW+5*SW^2)*SW^2*R^2-(3*SA^2-3*SA*SW+SW^2)*SW^3)*S^2-((357*SA-101*SW)*R^8-2*(85*SA-21*SW)*SW*R^6-4*(SA-5*SW)*SW^2*R^4+10*(SA-SW)*SW^3*R^2-(SA-SW)*SW^4)*SA*SW^2) : :

X(22469) lies on these lines: {}

X(22470) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ARA TO AOAA

Barycentrics    (SB+SC)*((R^2-SW)*(9*R^2-2*SW)*(3*R^2-SW)^2*S^4+(-27*(9*SA+14*SW)*R^10+(423*SW*SA+121*SW^2+81*SA^2)*R^8-SW*(51*SW^2+153*SA^2+190*SW*SA)*R^6+SW^2*(93*SA^2-3*SW*SA+23*SW^2)*R^4-SW^3*(23*SA^2+3*SW^2-15*SW*SA)*R^2+2*SA*SW^4*(SA-SW))*S^2-((231*SA-331*SW)*R^6-13*(SA-6*SW)*SW*R^4-(31*SA-17*SW)*SW^2*R^2+(5*SA-4*SW)*SW^3)*R^2*SA*SW^2) : :

X(22470) lies on these lines: {}

X(22471) = CYCLOLOGIC CENTER OF THESE TRIANGLES: AAOA TO JOHNSON

Barycentrics    a^2*(a^34-5*(b^2+c^2)*a^32+(b^4+29*b^2*c^2+c^4)*a^30+(b^2+c^2)*(43*b^4-81*b^2*c^2+43*c^4)*a^28-(107*b^8+107*c^8+b^2*c^2*(115*b^4-192*b^2*c^2+115*c^4))*a^26+(b^2+c^2)*(55*b^8+55*c^8+b^2*c^2*(387*b^4-674*b^2*c^2+387*c^4))*a^24+(165*b^12+165*c^12-(467*b^8+467*c^8+b^2*c^2*(317*b^4-1490*b^2*c^2+317*c^4))*b^2*c^2)*a^22-(b^2+c^2)*(297*b^12+297*c^12-(79*b^8+79*c^8+13*b^2*c^2*(113*b^4-242*b^2*c^2+113*c^4))*b^2*c^2)*a^20+(99*b^16+99*c^16+(965*b^12+965*c^12-(1614*b^8+1614*c^8+b^2*c^2*(1269*b^4-4498*b^2*c^2+1269*c^4))*b^2*c^2)*b^2*c^2)*a^18+(b^2+c^2)*(209*b^16+209*c^16-(965*b^12+965*c^12-(512*b^8+512*c^8+b^2*c^2*(4073*b^4-7470*b^2*c^2+4073*c^4))*b^2*c^2)*b^2*c^2)*a^16-(253*b^20+253*c^20+(145*b^16+145*c^16-(2295*b^12+2295*c^12-2*(1648*b^8+1648*c^8+b^2*c^2*(1413*b^4-4073*b^2*c^2+1413*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^14+(b^2+c^2)*(65*b^20+65*c^20+(533*b^16+533*c^16-(2139*b^12+2139*c^12-2*(468*b^8+468*c^8+7*b^2*c^2*(427*b^4-771*b^2*c^2+427*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12+(b^2-c^2)^2*(71*b^20+71*c^20-3*(65*b^16+65*c^16+(219*b^12+219*c^12-2*(332*b^8+332*c^8+b^2*c^2*(145*b^4-587*b^2*c^2+145*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10-(b^4-c^4)*(b^2-c^2)*(67*b^20+67*c^20+(69*b^16+69*c^16-(671*b^12+671*c^12-2*(410*b^8+410*c^8+7*b^2*c^2*(88*b^4-211*b^2*c^2+88*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(b^2-c^2)^4*(23*b^20+23*c^20+(163*b^16+163*c^16+(135*b^12+135*c^12-2*(238*b^8+238*c^8+b^2*c^2*(11*b^4-457*b^2*c^2+11*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^6*(b^2+c^2)*(3*b^16+3*c^16+(37*b^12+37*c^12+(126*b^8+126*c^8+b^2*c^2*(19*b^4-170*b^2*c^2+19*c^4))*b^2*c^2)*b^2*c^2)*a^4-(b^2-c^2)^8*(b^2+c^2)^2*(b^8+c^8-2*b^2*c^2*(6*b^4+19*b^2*c^2+6*c^4))*b^2*c^2*a^2+(b^2-c^2)^10*b^2*c^2*(b^2+c^2)^3*(b^4+4*b^2*c^2+c^4)) : :

X(22471) lies on the line {378,15136}

X(22472) = CYCLOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO AAOA

Barycentrics    4*(9*R^2-2*SW)*(3*R^2-SW)^2*S^4+(324*R^8*(27*R^2-26*SW)+3*(81*SA^2+9*SA*SW+1027*SW^2)*R^6-4*(63*SA^2-14*SA*SW+127*SW^2)*SW*R^4+16*(5*SA^2-2*SA*SW+2*SW^2)*SW^2*R^2-4*(2*SA-SW)*SA*SW^3)*S^2+(972*R^8*(27*R^2-32*SW)+21*SW^2*R^2*(705*R^4+20*SW^2-168*SW*R^2)-20*SW^5)*SB*SC : :

X(22472) lies on these lines: {}

X(22473) = CYCLOLOGIC CENTER OF THESE TRIANGLES: AAOA TO MEDIAL

Barycentrics    (SB+SC)*(2*(3*R^2-SW)^2*S^4-(3*R^2-SW)*(2*R^4*(18*SA-11*SW)-R^2*(6*SA^2+18*SA*SW-13*SW^2)+2*SA^2*SW+2*SA*SW^2-2*SW^3)*S^2+(4*R^2-SW)*R^2*SB*SC*SW^2)*((3*R^2-SW)*(6*R^2+2*SA-SW)*S^2-(4*R^2-SW)*SA^2*SW) : :

X(22473) lies on these lines: {}

X(22474) = CYCLOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO AAOA

Barycentrics    ((18*R^4-11*R^2*SW+2*SW^2)*S^2-2*R^2*SB*SC*SW)*(2*(3*R^2-SW)^2*S^2+R^2*SA^2*SW)*(4*S^2+(3*SA+SW)*(SA-SW)) : :

X(22474) lies on the nine-point circle and these lines: {}

X(22475) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^6-3*(b+c)*(b^2+c^2)*a^5-(5*b^4+5*c^4-2*(3*b^2-4*b*c+3*c^2)*b*c)*a^4+3*(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3+(2*b^4+2*c^4+(4*b^2+b*c+4*c^2)*b*c)*(b-c)^2*a^2+3*(b^2-c^2)*(b-c)*b^2*c^2*a+(b^2-c^2)^2*b^2*c^2 : :
X(22475) = 3*X(1)+X(22650) = 3*X(1)-X(22713) = X(39)+2*X(13464) = X(76)-7*X(9624) = 3*X(262)-X(22650) = 3*X(262)+X(22713) = 3*X(3576)-X(22676) = 5*X(3616)-X(6194) = X(4301)+2*X(13334) = 3*X(5603)+X(7709) = 4*X(5901)-X(12263) = 4*X(6683)-X(11362) = 5*X(7786)+X(7982) = X(7976)+5*X(8227) = 3*X(10246)+X(22728)

The reciprocal orthologic center of these triangles is X(3).

X(22475) lies on these lines: {1,262}, {2,22697}, {39,13464}, {76,9624}, {511,551}, {515,22682}, {999,22680}, {1125,15819}, {1319,18971}, {1385,12264}, {1386,11710}, {2646,22711}, {2782,12258}, {3295,22556}, {3576,22676}, {3616,6194}, {3656,11171}, {4301,13334}, {5603,7709}, {5901,12263}, {6683,11362}, {7786,7982}, {7976,8227}, {9955,22681}, {10246,22728}, {10595,12782}, {11257,11522}, {11363,22480}, {11364,22521}, {11365,22655}, {11366,22668}, {11367,22672}, {11368,22678}, {11370,22699}, {11371,22700}, {11373,22703}, {11374,22704}, {11375,22705}, {11376,22706}, {11377,22709}, {11378,22710}, {11831,22698}, {12194,21445}, {13883,22720}, {13936,22721}, {14881,15178}, {18991,19063}, {18992,19064}

X(22475) = midpoint of X(i) and X(j) for these {i,j}: {1, 262}, {3656, 11171}
X(22475) = complement of X(22697)
X(22475) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22650, 22713), (262, 22713, 22650)

X(22476) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO 3rd HATZIPOLAKIS

Barycentrics    (b+c)*a^15-2*b*c*a^14-(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^12+(b+c)*(b^4+c^4-(4*b^2-15*b*c+4*c^2)*b*c)*a^11-2*(2*b^6+2*c^6-(b^4+c^4+(3*b^2-11*b*c+3*c^2)*b*c)*b*c)*a^10+(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+(19*b^2-22*b*c+19*c^2)*b*c)*b*c)*a^9+(5*b^8+5*c^8-2*(4*b^6+4*c^6+(b^4+c^4-9*(b^2-b*c+c^2)*b*c)*b*c)*b*c)*a^8-(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6-(2*b^4+2*c^4+(9*b^2-23*b*c+9*c^2)*b*c)*b*c)*b*c)*a^7+2*(b^4+c^4-3*(b^2+3*b*c+c^2)*b*c)*(b-c)^4*b*c*a^6-(b^2-c^2)*(b-c)*(b^8+c^8+2*(2*b^6+2*c^6-7*(b^4+c^4+(b^2-b*c+c^2)*b*c)*b*c)*b*c)*a^5-(b^2-c^2)^2*(b-c)^2*(5*b^6+5*c^6+(6*b^4+6*c^4-(7*b^2+6*b*c+7*c^2)*b*c)*b*c)*a^4+(b^4-c^4)*(b^2-c^2)^2*(b-c)*(3*b^4+3*c^4+(2*b^2-7*b*c+2*c^2)*b*c)*a^3+2*(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(2*b^2+3*b*c+2*c^2)*a^2-(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22476) lies on these lines: {1,22466}, {2,22941}, {515,22833}, {999,22776}, {1125,22966}, {1319,18978}, {2646,22965}, {3295,22559}, {3576,22951}, {3616,22647}, {5603,22533}, {5886,22955}, {9955,22800}, {10246,22979}, {11363,22483}, {11364,22524}, {11365,22658}, {11368,22747}, {11370,22945}, {11371,22947}, {11373,22956}, {11374,22957}, {11375,22958}, {11376,22959}, {11377,22963}, {11831,22943}, {13883,22976}, {13936,22977}, {18991,19083}, {18992,19084}

X(22476) = midpoint of X(1) and X(22466)
X(22476) = complement of X(22941)
X(22476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22653, 22969), (22466, 22969, 22653)

X(22477) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO EHRMANN-MID

Barycentrics    8*a^13-8*(b+c)*a^12-10*(b^2-3*b*c+c^2)*a^11+(b+c)*(28*b^2-51*b*c+28*c^2)*a^10-3*(10*b^4+10*c^4+b*c*(13*b^2-38*b*c+13*c^2))*a^9-2*(b+c)*(13*b^4+13*c^4-b*c*(60*b^2-91*b*c+60*c^2))*a^8+(66*b^6+66*c^6-(57*b^4+57*c^4+b*c*(115*b^2-216*b*c+115*c^2))*b*c)*a^7-(b+c)*(10*b^6+10*c^6+(54*b^4+54*c^4-b*c*(251*b^2-378*b*c+251*c^2))*b*c)*a^6-(38*b^8+38*c^8-(105*b^6+105*c^6-(79*b^4+79*c^4+b*c*(141*b^2-308*b*c+141*c^2))*b*c)*b*c)*a^5+(b+c)*(26*b^8+26*c^8-(48*b^6+48*c^6+(61*b^4+61*c^4-b*c*(246*b^2-325*b*c+246*c^2))*b*c)*b*c)*a^4-(b^2-c^2)^2*(33*b^4+33*c^4-20*b*c*(5*b^2-6*b*c+5*c^2))*b*c*a^3-(b^2-c^2)^2*(b+c)*(10*b^6+10*c^6-3*(11*b^4+11*c^4-2*b*c*(10*b^2-13*b*c+10*c^2))*b*c)*a^2+2*(b^2-c^2)^4*(2*b^4+2*c^4-3*(b^2-b*c+c^2)*b*c)*a+4*(b^2-c^2)^4*(b+c)*b^2*c^2 : :

The reciprocal cyclologic center of these triangles is X(22478).

X(22477) lies on the line {9955,22478}

X(22478) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO ANTI-AQUILA

Barycentrics    8*a^16-12*(b+c)*a^15-2*(4*b^2-21*b*c+4*c^2)*a^14+36*(b^2-c^2)*(b-c)*a^13-2*(22*b^4+22*c^4+27*b*c*(b^2-3*b*c+c^2))*a^12-6*(b+c)*(2*b^4+2*c^4-b*c*(25*b^2-43*b*c+25*c^2))*a^11+2*(34*b^6+34*c^6-9*(6*b^4+6*c^4+b*c*(7*b^2-18*b*c+7*c^2))*b*c)*a^10-6*(b+c)*(10*b^6+10*c^6-(11*b^4+11*c^4+b*c*(40*b^2-83*b*c+40*c^2))*b*c)*a^9+(40*b^8+40*c^8+(144*b^6+144*c^6-(407*b^4+407*c^4+24*b*c*(5*b^2-29*b*c+5*c^2))*b*c)*b*c)*a^8+6*(b^2-c^2)*(b-c)*(10*b^6+10*c^6-(34*b^4+34*c^4+b*c*(11*b^2-40*b*c+11*c^2))*b*c)*a^7-(128*b^10+128*c^10-(114*b^8+114*c^8+(453*b^6+453*c^6-2*(315*b^4+315*c^4+b*c*(127*b^2-444*b*c+127*c^2))*b*c)*b*c)*b*c)*a^6+6*(b^2-c^2)*(b-c)*(2*b^8+2*c^8+(30*b^6+30*c^6-(43*b^4+43*c^4+b*c*(11*b^2-60*b*c+11*c^2))*b*c)*b*c)*a^5+(b^2-c^2)^2*(68*b^8+68*c^8-(198*b^6+198*c^6-(217*b^4+217*c^4+3*b*c*(46*b^2-145*b*c+46*c^2))*b*c)*b*c)*a^4-6*(b^2-c^2)^3*(b-c)*(6*b^6+6*c^6-(b^4+c^4+2*b*c*(8*b^2-15*b*c+8*c^2))*b*c)*a^3+(b^2-c^2)^4*(4*b^6+4*c^6+(48*b^4+48*c^4-b*c*(163*b^2-198*b*c+163*c^2))*b*c)*a^2+6*(b^2-c^2)^5*(b-c)*(b^2-b*c+c^2)*(2*b^2-3*b*c+2*c^2)*a-4*(b^2-c^2)^6*(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2) : :

The reciprocal cyclologic center of these triangles is X(22477).

X(22478) lies on the line {9955,22477}

X(22479) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 2nd CIRCUMPERP TANGENTIAL

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

X(22479) lies on these lines: {3,1829}, {4,2975}, {24,10269}, {25,34}, {28,1851}, {33,10966}, {36,7713}, {39,607}, {55,11396}, {58,1473}, {104,7487}, {184,3556}, {235,22753}, {427,958}, {428,11194}, {604,2354}, {608,5019}, {956,5090}, {988,1039}, {999,11363}, {1112,22586}, {1191,14975}, {1201,2212}, {1452,1470}, {1475,1973}, {1593,1753}, {1597,1900}, {1598,1878}, {1838,4185}, {1843,22769}, {1862,22560}, {1870,14017}, {1902,22770}, {1905,8071}, {3516,5584}, {3575,11390}, {4186,11399}, {5186,22514}, {5253,6353}, {5260,8889}, {5410,19014}, {5411,19013}, {7677,7717}, {11380,22520}, {11381,22778}, {11384,11493}, {11385,11492}, {11386,22744}, {11388,22756}, {11389,22757}, {11392,22759}, {11393,22760}, {11394,22761}, {11395,22762}, {11398,20832}, {11400,22768}, {11576,22781}, {11832,22755}, {12131,22504}, {12132,22565}, {12133,22583}, {12134,22659}, {12135,12513}, {12136,18237}, {12137,12773}, {12138,22775}, {12139,22777}, {12140,19478}, {12141,22774}, {12142,22773}, {12143,22779}, {12144,22780}, {12145,19159}, {12146,22782}, {12147,22595}, {12148,22624}, {13166,19162}, {13668,22783}, {13743,16114}, {13788,22784}, {13884,22763}, {13937,22764}, {22480,22680}, {22481,22771}, {22482,22772}, {22483,22776}

X(22479) = {X(3), X(1829)}-harmonic conjugate of X(11383)

X(22480) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 1st BROCARD-REFLECTED

Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)*(3*(b^2+c^2)*a^8-4*(b^4+b^2*c^2+c^4)*a^6-(b^2+c^2)*(b^4+13*b^2*c^2+c^4)*a^4+2*(b^2-c^2)^2*(b^4+3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(22480) = 4*X(6756)-X(12143)

The reciprocal orthologic center of these triangles is X(3).

X(22480) lies on these lines: {4,2896}, {25,262}, {33,22711}, {34,18971}, {235,22682}, {427,15819}, {428,511}, {1593,22676}, {1598,22728}, {1843,12131}, {1907,5188}, {2023,10985}, {2782,7576}, {3518,11272}, {5064,22712}, {5090,22697}, {5410,19064}, {5411,19063}, {6756,12143}, {7487,7709}, {7713,22650}, {10594,14881}, {11363,22475}, {11380,22521}, {11383,22556}, {11385,22672}, {11388,22699}, {11389,22700}, {11390,22703}, {11391,22704}, {11392,22705}, {11393,22706}, {11394,22709}, {11396,22713}, {11398,22729}, {11399,22730}, {11400,22731}, {11401,22732}, {11832,22698}, {13884,22720}, {13937,22721}, {22479,22680}

X(22481) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22481) lies on these lines: {4,617}, {18,25}, {33,22865}, {34,18972}, {235,22831}, {427,630}, {468,6674}, {1593,22843}, {1598,16628}, {1843,5965}, {5090,22851}, {5410,19072}, {5411,19069}, {6756,12142}, {6995,22114}, {7487,22531}, {7713,22651}, {11363,11740}, {11380,22522}, {11383,22557}, {11386,22745}, {11388,22853}, {11389,22854}, {11390,22857}, {11391,22858}, {11392,22859}, {11393,22860}, {11394,22863}, {11396,22867}, {11398,22884}, {11399,22885}, {11400,22886}, {11401,22887}, {11832,22852}, {13884,22876}, {13937,22877}, {22479,22771}

X(22482) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22482) lies on these lines: {4,616}, {17,25}, {33,22910}, {34,18973}, {235,22832}, {427,629}, {428,532}, {468,6673}, {1593,22890}, {1598,16629}, {1843,5965}, {5090,22896}, {5410,19070}, {5411,19071}, {6756,12141}, {6995,22113}, {7487,22532}, {7713,22652}, {11363,11739}, {11380,22523}, {11383,22558}, {11386,22746}, {11388,22898}, {11389,22899}, {11390,22902}, {11391,22903}, {11392,22904}, {11393,22905}, {11394,22908}, {11395,22909}, {11396,22912}, {11398,22929}, {11399,22930}, {11400,22931}, {11401,22932}, {11832,22897}, {13884,22921}, {13937,22922}, {22479,22772}

X(22483) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 3rd HATZIPOLAKIS

Barycentrics    SB*SC*((5*R^2-SW)*SA^2+2*(4*R^2-SW)*(5*R^2-SW)*SA+R^2*(61*SW^2+256*R^4-232*SW*R^2)-5*SW^3) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22483) lies on these lines: {4,801}, {25,22466}, {33,22965}, {34,18978}, {235,22833}, {427,22966}, {1593,22951}, {1598,22979}, {1843,21652}, {1853,2929}, {5090,22941}, {5410,19084}, {5411,19083}, {6644,22834}, {7487,22533}, {7506,22808}, {7713,22653}, {11363,22476}, {11380,22524}, {11383,22559}, {11386,22747}, {11388,22945}, {11389,22947}, {11390,22956}, {11391,22957}, {11392,22958}, {11393,22959}, {11394,22963}, {11395,22964}, {11396,22969}, {11398,22980}, {11399,22981}, {11400,22982}, {11401,22983}, {11832,22943}, {13884,22976}, {13937,22977}, {19460,22662}, {22467,22581}, {22479,22776}, {22530,22953}

X(22483) = {X(4), X(22750)}-harmonic conjugate of X(22800)

X(22484) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO ANTI-ARTZT

Barycentrics    S^2+(9*SA-5*SW)*S+9*SB*SC : :
X(22484) = 2*X(2)-3*X(486) = 5*X(2)-3*X(487) = 7*X(2)-6*X(642) = X(2)+3*X(12221) = 5*X(486)-2*X(487) = 7*X(486)-4*X(642) = X(486)+2*X(12221) = 7*X(487)-10*X(642) = X(487)+5*X(12221) = 2*X(642)+7*X(12221) = 2*X(3830)+3*X(6280) = X(3830)-3*X(12601) = X(4677)-3*X(9906) = 4*X(4745)-3*X(12787) = X(6280)+2*X(12601)

The reciprocal orthologic center of these triangles is X(12158).

X(22484) lies on these lines: {2,371}, {381,6281}, {524,1328}, {542,13810}, {591,6561}, {1327,1992}, {3564,3845}, {3830,6280}, {4677,9906}, {4745,12787}, {5066,6290}, {6319,22562}, {8703,12123}, {11001,12256}, {12296,15640}, {12509,15698}, {13650,13846}, {13711,13932}, {15685,22809}

X(22484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (591, 6561, 13712), (3845, 15534, 22485)

X(22485) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO ANTI-ARTZT

Barycentrics    (80*a^8-146*(b^2+c^2)*a^6-2*(33*b^4+182*b^2*c^2+33*c^4)*a^4+2*(b^2+c^2)*(89*b^4-118*b^2*c^2+89*c^4)*a^2-2*(23*b^4-34*b^2*c^2+23*c^4)*(b^2-c^2)^2)*S+4*S^2*(5*a^6-96*(b^2+c^2)*a^4+3*(19*b^4-26*b^2*c^2+19*c^4)*a^2+2*(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4)) : :
X(22485) = 2*X(2)-3*X(485) = 5*X(2)-3*X(488) = 7*X(2)-6*X(641) = 11*X(2)-12*X(6118) = X(2)+3*X(12222) = 5*X(485)-2*X(488) = 7*X(485)-4*X(641) = 11*X(485)-8*X(6118) = X(485)+2*X(12222) = 7*X(488)-10*X(641) = 11*X(488)-20*X(6118) = X(488)+5*X(12222) = 11*X(641)-14*X(6118) = 2*X(641)+7*X(12222) = 4*X(6118)+11*X(12222)

The reciprocal orthologic center of these triangles is X(12159).

X(22485) lies on these lines: {2,372}, {381,6278}, {524,1327}, {542,13691}, {1328,1992}, {1991,6560}, {3564,3845}, {3830,6279}, {4677,9907}, {4745,12788}, {5066,6289}, {5860,6564}, {6320,22563}, {8703,12124}, {11001,12257}, {12297,15640}, {12510,15698}, {13662,13712}, {13771,13847}, {13834,13850}, {15685,22810}

X(22485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1991, 6560, 13835), (3845, 15534, 22484)

X(22486) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO 1st BROCARD-REFLECTED

Barycentrics    4*(b^2+c^2)*a^4-(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^2+c^2)*b^2*c^2 : :
X(22486) = X(76)+2*X(13330) = X(194)-3*X(5032) = 3*X(262)-4*X(5476) = 4*X(3934)-3*X(21356) = 2*X(5052)+X(18906) = 4*X(8584)-X(11055)

The reciprocal orthologic center of these triangles is X(6).

X(22486) lies on these lines: {2,51}, {3,11155}, {6,99}, {32,13085}, {39,7618}, {69,5475}, {76,524}, {182,13586}, {183,11173}, {194,5032}, {376,13354}, {384,576}, {538,1992}, {542,11361}, {575,3552}, {597,3094}, {599,7809}, {698,8584}, {732,12156}, {1351,10796}, {1916,11150}, {2782,8593}, {3095,8369}, {3102,11157}, {3103,11158}, {3104,16646}, {3105,16647}, {3106,11153}, {3107,11154}, {3363,7697}, {3407,5039}, {3934,8176}, {4048,12151}, {5038,7782}, {5104,7771}, {5107,7804}, {5976,11163}, {6248,11180}, {7753,14645}, {7770,11477}, {7775,18806}, {7833,19924}, {8359,9821}, {8541,15014}, {8550,19687}, {11059,13410}, {11151,11171}, {11152,18800}, {11160,14994}, {11161,11317}, {11179,11257}, {11288,14848}, {11318,14881}, {13637,22722}, {13757,22723}, {15004,16951}, {22493,22702}, {22494,22701}

X(22486) = reflection of X(i) in X(j) for these (i,j): (69, 9466), (376, 13354), (11152, 18800), (11160, 14994)
X(22486) = {X(6), X(1003)}-harmonic conjugate of X(5182)

X(22487) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO INNER-FERMAT

Barycentrics    -2*S*(23*a^2-4*b^2-4*c^2)+(7*a^4-(b^2+c^2)*a^2-2*b^4+8*b^2*c^2-2*c^4)*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(5858).

X(22487) lies on these lines: {2,18}, {5858,7813}, {8584,12155}, {11159,22488}, {12154,15534}, {13637,22878}

X(22488) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO OUTER-FERMAT

Barycentrics    2*S*(23*a^2-4*b^2-4*c^2)+(7*a^4-(b^2+c^2)*a^2-2*b^4+8*b^2*c^2-2*c^4)*sqrt(3) : :

The reciprocal orthologic center of these triangles is X(5859).

X(22488) lies on these lines: {2,17}, {5859,7813}, {8584,12154}, {11159,22487}, {12155,15534}, {13637,22923}, {13757,22924}

X(22489) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO ANTI-ARTZT

Barycentrics    -2*S*(5*a^2+2*b^2+2*c^2)+sqrt(3)*(a^4-5*(b^2+c^2)*a^2+4*(b^2-c^2)^2) : :
X(22489) = 2*X(2)+X(13) = 7*X(2)-X(616) = 5*X(2)-2*X(618) = X(2)+2*X(5459) = 4*X(2)-X(5463) = X(2)-4*X(6669) = 7*X(13)+2*X(616) = 5*X(13)+4*X(618) = X(13)-4*X(5459) = 2*X(13)+X(5463) = X(13)+8*X(6669) = 5*X(616)-14*X(618) = X(616)+14*X(5459) = 4*X(616)-7*X(5463) = X(618)+5*X(5459) = 8*X(618)-5*X(5463) = X(618)-10*X(6669) = 8*X(5459)+X(5463) = X(5459)+2*X(6669) = X(5463)-16*X(6669)

The reciprocal orthologic center of these triangles is X(12155).

X(22489) lies on these lines: {2,13}, {14,5461}, {17,9763}, {30,21156}, {61,22491}, {99,22577}, {115,5464}, {141,22580}, {376,5478}, {381,6771}, {395,9112}, {396,22572}, {524,16267}, {542,5050}, {543,5470}, {547,5617}, {549,5473}, {551,7975}, {619,671}, {620,9116}, {623,16960}, {633,20394}, {1656,20415}, {3526,16001}, {3582,10062}, {3584,10078}, {3679,11705}, {3828,12781}, {5071,6770}, {5460,6777}, {5472,16645}, {5474,9880}, {6671,19106}, {6722,6778}, {9204,11625}, {11284,13859}, {11295,16808}, {11296,16241}, {11303,13083}, {11542,22495}, {12258,12780}, {13103,15694}, {13908,19076}, {13917,19054}, {13968,19075}, {13982,19053}, {14830,22797}, {19709,22796}, {21358,21360}

X(22489) = inner-Napoleon-circle-inverse of X(35752)
X(22489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13, 5463), (2, 5459, 13), (13, 16242, 23006), (619, 671, 9114), (5459, 6669, 2), (6108, 18582, 13)

X(22490) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO ANTI-ARTZT

Barycentrics    15*S^2+sqrt(3)*(3*SA-7*SW)*S+9*SB*SC : :
X(22490) = 2*X(2)+X(14) = 7*X(2)-X(617) = 5*X(2)-2*X(619) = X(2)+2*X(5460) = 4*X(2)-X(5464) = X(2)-4*X(6670) = 7*X(14)+2*X(617) = 5*X(14)+4*X(619) = X(14)-4*X(5460) = 2*X(14)+X(5464) = X(14)+8*X(6670) = 5*X(617)-14*X(619) = X(617)+14*X(5460) = 4*X(617)-7*X(5464) = X(619)+5*X(5460) = 8*X(619)-5*X(5464) = X(619)-10*X(6670) = 8*X(5460)+X(5464) = X(5460)+2*X(6670) = X(5464)-16*X(6670)

The reciprocal orthologic center of these triangles is X(12154).

X(22490) lies on these lines: {2,14}, {13,5461}, {18,9761}, {30,21157}, {62,22492}, {99,22578}, {115,5463}, {141,22579}, {376,5479}, {381,6774}, {395,22571}, {396,9113}, {524,16268}, {542,5050}, {543,5469}, {547,5613}, {549,5474}, {551,7974}, {618,671}, {620,9114}, {624,16961}, {634,20395}, {1656,20416}, {3526,16002}, {3582,10061}, {3584,10077}, {3679,11706}, {3828,12780}, {5071,6773}, {5459,6778}, {5471,16644}, {5473,9880}, {6672,19107}, {6722,6777}, {9205,11627}, {11284,13858}, {11295,16242}, {11296,16809}, {11304,13084}, {11543,22496}, {12258,12781}, {13102,15694}, {13908,19074}, {13916,19054}, {13968,19073}, {13981,19053}, {14830,22796}, {19709,22797}, {21358,21359}

X(22490) = outer-Napoleon-circle-inverse of X(36330)
X(22490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14, 5464), (2, 5460, 14), (14, 16241, 23013), (618, 671, 9116), (5460, 6670, 2), (6109, 18581, 14)

X(22491) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO ANTI-ARTZT

Barycentrics    3*S^2+sqrt(3)*(3*SA-SW)*S+9*SB*SC : :
X(22491) = 2*X(3845)+X(5858) = 4*X(5066)-X(5859)

The reciprocal orthologic center of these triangles is X(12155).

X(22491) lies on these lines: {2,14}, {5,9763}, {13,1992}, {30,9761}, {61,22489}, {69,16809}, {115,22579}, {193,16808}, {298,11185}, {376,13084}, {381,524}, {395,7737}, {398,11305}, {532,3839}, {543,5617}, {3845,5858}, {3849,20428}, {5066,5859}, {5210,16645}, {5321,11295}, {5459,13705}, {5479,7620}, {5613,7617}, {6695,22237}, {6782,12155}, {7618,13102}, {7775,16626}, {9734,9886}, {9762,9770}, {9885,16002}, {11160,22494}, {16630,22572}

X(22491) = reflection of X(376) in X(13084)
X(22491) = reflection of X(22492) in X(381)
X(22491) = anticomplement of X(13083)
X(22491) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 22496, 1992), (381, 20426, 20423), (1352, 7615, 22492), (3642, 5460, 2)

X(22492) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO ANTI-ARTZT

Barycentrics    3*S^2-sqrt(3)*(3*SA-SW)*S+9*SB*SC : :
X(22492) = 2*X(3845)+X(5859) = 4*X(5066)-X(5858)

The reciprocal orthologic center of these triangles is X(12154).

X(22492) lies on these lines: {2,13}, {5,9761}, {14,1992}, {30,9763}, {62,22490}, {69,16808}, {115,22580}, {193,16809}, {299,11185}, {376,13083}, {381,524}, {396,7737}, {397,11306}, {532,3545}, {543,5613}, {3845,5859}, {3849,20429}, {5066,5858}, {5210,16644}, {5318,11296}, {5460,13703}, {5478,7620}, {5617,7617}, {6694,22235}, {6783,12154}, {7618,13103}, {7775,16627}, {9734,9885}, {9760,9770}, {9886,16001}, {11160,22493}, {16631,22571}

X(22492) = reflection of X(376) in X(13083)
X(22492) = reflection of X(22491) in X(381)
X(22492) = anticomplement of X(13084)
X(22492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 22495, 1992), (381, 20425, 20423), (1352, 7615, 22491), (3643, 5459, 2)

X(22493) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO ANTI-ARTZT

Barycentrics    3*S^2+sqrt(3)*(9*SA-SW)*S+9*SB*SC : :
X(22493) = 4*X(2)-3*X(16962) = 2*X(2)-3*X(21359) = 3*X(13)-2*X(22495) = 8*X(623)-5*X(16960) = 4*X(623)-3*X(22489) = 5*X(16960)-6*X(22489)

The reciprocal orthologic center of these triangles is X(12155).

X(22493) lies on these lines: {2,18}, {13,524}, {14,599}, {69,16809}, {76,12817}, {99,298}, {183,9760}, {302,13083}, {316,22576}, {381,11477}, {617,13084}, {623,16960}, {754,5858}, {2482,6780}, {3180,5459}, {3534,22890}, {3849,6779}, {5464,5569}, {7840,9762}, {9113,9117}, {9763,16966}, {11160,22492}, {11178,20426}, {11295,16964}, {22486,22702}, {22577,23004}

X(22493) = reflection of X(i) in X(j) for these (i,j): (3180, 5459), (22577, 23004)
X(22493) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 15533, 22494), (5464, 9761, 16242)

X(22494) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO ANTI-ARTZT

Barycentrics    3*S^2-sqrt(3)*(9*SA-SW)*S+9*SB*SC : :
X(22494) = 4*X(2)-3*X(16963) = 2*X(2)-3*X(21360) = 3*X(14)-2*X(22496) = 8*X(624)-5*X(16961) = 4*X(624)-3*X(22490) = 5*X(16961)-6*X(22490)

The reciprocal orthologic center of these triangles is X(12154).

X(22494) lies on these lines: {2,17}, {13,599}, {14,524}, {69,16808}, {76,12816}, {99,299}, {183,9762}, {303,13084}, {316,22575}, {381,11477}, {616,13083}, {624,16961}, {754,5859}, {2482,6779}, {3181,5460}, {3534,22843}, {3849,6780}, {5463,5569}, {7840,9760}, {9112,9115}, {9761,16967}, {11160,22491}, {11178,20425}, {11296,16965}, {22486,22701}, {22578,23005}

X(22494) = reflection of X(i) in X(j) for these (i,j): (3181, 5460), (22578, 23005)
X(22494) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 15533, 22493), (5463, 9763, 16241)

X(22495) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO ANTI-ARTZT

Barycentrics    3*S^2-sqrt(3)*(9*SA-5*SW)*S+9*SB*SC : :
X(22495) = 2*X(2)-3*X(16267) = 3*X(13)-X(22493) = 2*X(115)-3*X(22571) = X(9116)-3*X(16529) = 4*X(11542)-3*X(22489)

The reciprocal orthologic center of these triangles is X(12155).

X(22495) lies on these lines: {2,17}, {13,524}, {14,1992}, {16,7622}, {61,11295}, {115,22571}, {193,16808}, {298,5459}, {381,576}, {396,5463}, {538,3105}, {542,20425}, {543,22997}, {2996,12816}, {3629,16809}, {5066,16627}, {5464,6783}, {5858,7775}, {6779,9885}, {7774,9760}, {9116,16529}, {9762,22998}, {10646,13083}, {11542,22489}

X(22495) = reflection of X(i) in X(j) for these (i,j): (14, 22573), (298, 5459), (5464, 6783)
X(22495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 15534, 22496), (1992, 22492, 14)

X(22496) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO ANTI-ARTZT

Barycentrics    3*S^2+sqrt(3)*(9*SA-5*SW)*S+9*SB*SC : :
X(22496) = 2*X(2)-3*X(16268) = 3*X(14)-X(22494) = 2*X(115)-3*X(22572) = X(9114)-3*X(16530) = 4*X(11543)-3*X(22490)

The reciprocal orthologic center of these triangles is X(12154).

X(22496) lies on these lines: {2,18}, {13,1992}, {14,524}, {15,7622}, {62,11296}, {115,22572}, {193,16809}, {299,5460}, {381,576}, {395,5464}, {532,11054}, {538,3104}, {542,20426}, {543,22998}, {2996,12817}, {3629,16808}, {5066,16626}, {5463,6782}, {5859,7775}, {6780,9886}, {7774,9762}, {9114,16530}, {9760,22997}, {10645,13084}, {11543,22490}

X(22496) = reflection of X(i) in X(j) for these (i,j): (13, 22574), (299, 5460), (5463, 6782)
X(22496) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 15534, 22495), (1992, 22491, 13)

X(22497) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA TO 3rd HATZIPOLAKIS

Barycentrics    SB*SC*(SB+SC)*((24*R^2+2*SA-5*SW)*R^2*S^2+2*(4*R^2-SW)*(8*R^2-SW)*SA^2) : :
X(22497) = 3*X(11402)-2*X(19460) = 3*X(11402)-4*X(22529)

The reciprocal orthologic center of these triangles is X(9729).

X(22497) lies on these lines: {3,22528}, {4,22550}, {6,21652}, {25,2929}, {154,22658}, {184,17837}, {427,22555}, {1974,9968}, {1993,22534}, {3515,22962}, {3516,22549}, {5410,22960}, {5411,22961}, {7395,22834}, {7484,22581}, {7592,22535}, {9777,22530}, {9818,22808}, {11245,18936}, {11284,22973}, {11402,19460}, {11403,22538}, {11405,22830}, {11406,22840}, {11408,22974}, {11409,22975}, {11410,22978}, {16030,19198}, {18386,22816}, {19118,19142}, {19404,19488}

X(22497) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2929, 22970, 25), (19460, 22529, 11402)

X(22498) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND 1st BROCARD-REFLECTED

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

Let A'B'C' be the 1st anti-Brocard triangle. X(22498) is the radical center of the circumcircles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, June 7, 2019)

X(22498) lies on these lines: {2,4159}, {3,9772}, {6,1916}, {99,736}, {114,8295}, {147,7897}, {5026,10334}, {5999,6033}, {7840,12215}, {7931,8290}, {12177,22503}, {13586,16508}

X(22498) = X(3407)-of-1st-anti-Brocard-triangle
X(22498) = 1st-anti-Brocard-isogonal conjugate of X(76)
X(22498) = {X(1916), X(8289)}-harmonic conjugate of X(19120)

X(22499) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS REFLECTION

Barycentrics    (b^2*c^2*a^24+3*(b^2+c^2)*b^2*c^2*a^22-(b^8+c^8+(21*b^4+29*b^2*c^2+21*c^4)*b^2*c^2)*a^20+3*(b^2+c^2)*(6*b^4-19*b^2*c^2+6*c^4)*b^2*c^2*a^18+(20*b^12+20*c^12+(106*b^8+106*c^8+(365*b^4+412*b^2*c^2+365*c^4)*b^2*c^2)*b^2*c^2)*a^16-(b^2+c^2)*(64*b^12+64*c^12+(229*b^8+229*c^8+(387*b^4+125*b^2*c^2+387*c^4)*b^2*c^2)*b^2*c^2)*a^14+2*(45*b^16+45*c^16+(116*b^12+116*c^12+(52*b^8+52*c^8-(109*b^4-32*b^2*c^2+109*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12-(b^2+c^2)*(64*b^16+64*c^16-(127*b^12+127*c^12+7*(50*b^8+50*c^8+(25*b^4+26*b^2*c^2+25*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10+(20*b^20+20*c^20-(183*b^16+183*c^16+(151*b^12+151*c^12-(281*b^8+281*c^8-(121*b^4+948*b^2*c^2+121*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(b^2+c^2)*(82*b^16+82*c^16-(275*b^12+275*c^12-(313*b^8+313*c^8+(323*b^4-1014*b^2*c^2+323*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2*a^6-(b^2-c^2)^2*(b^20+c^20+(9*b^16+9*c^16-(62*b^12+62*c^12-(122*b^8+122*c^8+(5*b^4-414*b^2*c^2+5*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3*(b^12+c^12+(3*b^8+3*c^8-(21*b^4-76*b^2*c^2+21*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2*a^2+(b^2-c^2)^6*(b^4+6*b^2*c^2+c^4)*b^6*c^6)*S+4*S^2*(b^2*c^2*a^22-3*(b^2+c^2)*b^2*c^2*a^20-(b^8+c^8+2*(b^4+8*b^2*c^2+c^4)*b^2*c^2)*a^18+6*(b^2+c^2)*(b^8+c^8+3*(b^2+c^2)^2*b^2*c^2)*a^16-(12*b^12+12*c^12+(29*b^8+29*c^8+(20*b^4-33*b^2*c^2+20*c^4)*b^2*c^2)*b^2*c^2)*a^14+(b^2+c^2)*(7*b^12+7*c^12-(44*b^8+44*c^8+(101*b^4+75*b^2*c^2+101*c^4)*b^2*c^2)*b^2*c^2)*a^12+(7*b^16+7*c^16+(95*b^12+95*c^12+(157*b^8+157*c^8+8*(16*b^4+17*b^2*c^2+16*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10-(b^2+c^2)*(12*b^16+12*c^16+(38*b^12+38*c^12-(91*b^8+91*c^8+(b^4-104*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(6*b^20+6*c^20-(8*b^16+8*c^16+(67*b^12+67*c^12-(103*b^8+103*c^8+8*(3*b^4-41*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2+c^2)*(b^20+c^20-(11*b^16+11*c^16-(b^4+4*b^2*c^2+c^4)*(22*b^8+22*c^8-(77*b^4-114*b^2*c^2+77*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^16+c^16-(4*b^12+4*c^12-(11*b^8+11*c^8+3*(3*b^4-14*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)^3*(b^4-3*b^2*c^2+c^4)*b^6*c^6) : :

X(22499) lies on these lines: {511,13029}, {1916,6401}, {4027,11984}, {5989,11986}, {8289,22785}, {8302,11937}, {8303,11938}, {8304,11941}, {8305,11942}, {8306,11959}, {8307,11960}, {8308,11963}, {8309,11964}, {8310,11967}, {8311,11969}, {8312,11971}, {8313,11973}, {8314,11975}, {8315,11977}, {8316,11979}, {8317,11981}, {9772,14167}, {11983,22500}, {19375,19390}

X(22500) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) REFLECTION

Barycentrics    (b^2*c^2*a^24+3*(b^2+c^2)*b^2*c^2*a^22-(b^8+c^8+(21*b^4+29*b^2*c^2+21*c^4)*b^2*c^2)*a^20+3*(b^2+c^2)*(6*b^4-19*b^2*c^2+6*c^4)*b^2*c^2*a^18+(20*b^12+20*c^12+(106*b^8+106*c^8+(365*b^4+412*b^2*c^2+365*c^4)*b^2*c^2)*b^2*c^2)*a^16-(b^2+c^2)*(64*b^12+64*c^12+(229*b^8+229*c^8+(387*b^4+125*b^2*c^2+387*c^4)*b^2*c^2)*b^2*c^2)*a^14+2*(45*b^16+45*c^16+(116*b^12+116*c^12+(52*b^8+52*c^8-(109*b^4-32*b^2*c^2+109*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12-(b^2+c^2)*(64*b^16+64*c^16-(127*b^12+127*c^12+7*(50*b^8+50*c^8+(25*b^4+26*b^2*c^2+25*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10+(20*b^20+20*c^20-(183*b^16+183*c^16+(151*b^12+151*c^12-(281*b^8+281*c^8-(121*b^4+948*b^2*c^2+121*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(b^2+c^2)*(82*b^16+82*c^16-(275*b^12+275*c^12-(313*b^8+313*c^8+(323*b^4-1014*b^2*c^2+323*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2*a^6-(b^2-c^2)^2*(b^20+c^20+(9*b^16+9*c^16-(62*b^12+62*c^12-(122*b^8+122*c^8+(5*b^4-414*b^2*c^2+5*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3*(b^12+c^12+(3*b^8+3*c^8-(21*b^4-76*b^2*c^2+21*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2*a^2+(b^2-c^2)^6*(b^4+6*b^2*c^2+c^4)*b^6*c^6)*S-4*S^2*(b^2*c^2*a^22-3*(b^2+c^2)*b^2*c^2*a^20-(b^8+c^8+2*(b^4+8*b^2*c^2+c^4)*b^2*c^2)*a^18+6*(b^2+c^2)*(b^8+c^8+3*(b^2+c^2)^2*b^2*c^2)*a^16-(12*b^12+12*c^12+(29*b^8+29*c^8+(20*b^4-33*b^2*c^2+20*c^4)*b^2*c^2)*b^2*c^2)*a^14+(b^2+c^2)*(7*b^12+7*c^12-(44*b^8+44*c^8+(101*b^4+75*b^2*c^2+101*c^4)*b^2*c^2)*b^2*c^2)*a^12+(7*b^16+7*c^16+(95*b^12+95*c^12+(157*b^8+157*c^8+8*(16*b^4+17*b^2*c^2+16*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10-(b^2+c^2)*(12*b^16+12*c^16+(38*b^12+38*c^12-(91*b^8+91*c^8+(b^4-104*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(6*b^20+6*c^20-(8*b^16+8*c^16+(67*b^12+67*c^12-(103*b^8+103*c^8+8*(3*b^4-41*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2+c^2)*(b^20+c^20-(11*b^16+11*c^16-(b^4+4*b^2*c^2+c^4)*(22*b^8+22*c^8-(77*b^4-114*b^2*c^2+77*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^16+c^16-(4*b^12+4*c^12-(11*b^8+11*c^8+3*(3*b^4-14*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)^3*(b^4-3*b^2*c^2+c^4)*b^6*c^6) : :

X(22500) lies on these lines: {511,13031}, {1916,6402}, {4027,11985}, {8289,22786}, {8302,11939}, {8304,11943}, {8306,11961}, {8308,11965}, {8310,11970}, {8311,11968}, {8312,11974}, {8313,11972}, {8314,11978}, {8316,11982}, {9772,14168}, {11983,22499}

X(22501) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 1st ANTI-BROCARD

Barycentrics    S^4-(-2*SA^2+11*SB*SC+SW^2)*S^2-S*(SW*(6*SA-SW)*(SA-SW)+(9*SA-SW)*S^2)+5*SB*SC*SW^2 : :
X(22501) = 3*X(486)-2*X(13926)

The reciprocal orthologic center of these triangles is X(9867).

X(22501) lies on these lines: {98,486}, {115,19105}, {542,1328}, {6231,6561}, {6280,22617}, {7840,22562}, {12221,22613}, {22502,22505}

X(22502) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 1st ANTI-BROCARD

Barycentrics    S^4-(-2*SA^2+11*SB*SC+SW^2)*S^2+S*(SW*(6*SA-SW)*(SA-SW)+(9*SA-SW)*S^2)+5*SB*SC*SW^2 : :
X(22502) = 3*X(485)-2*X(13873)

The reciprocal orthologic center of these triangles is X(9868).

X(22502) lies on these lines: {98,485}, {115,19102}, {542,1327}, {6230,6560}, {6279,22646}, {7840,22563}, {12222,22642}, {22501,22505}

X(22503) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO 1st BROCARD-REFLECTED

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

The reciprocal orthologic center of these triangles is X(12177).

X(22503) lies on these lines: {2,51}, {3,22679}, {30,8592}, {147,316}, {325,9772}, {850,8704}, {1916,15980}, {2023,8586}, {2080,3329}, {2782,7840}, {3095,7864}, {3314,7697}, {4027,22525}, {5939,5999}, {7709,7774}, {7900,11257}, {8290,11676}, {8291,14538}, {8292,14539}, {8350,18860}, {12177,22498}, {13334,20088}

X(22503) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(34095)
X(22503) = {X(262), X(22677)}-harmonic conjugate of X(2)

X(22504) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999).

X(22504) lies on these lines: {3,11711}, {30,22565}, {36,9860}, {55,7970}, {56,98}, {99,3428}, {104,9862}, {114,958}, {115,22753}, {147,2975}, {517,13173}, {542,11194}, {690,22583}, {956,9864}, {993,2792}, {999,11710}, {1001,11724}, {2782,11249}, {2783,22560}, {2784,8666}, {2787,22775}, {2794,12114}, {2799,19159}, {3027,10966}, {3149,13178}, {5584,21166}, {6033,22758}, {6226,22757}, {6227,22756}, {8980,22763}, {9861,22654}, {10053,22766}, {10069,22767}, {10269,12042}, {11492,12180}, {11493,12179}, {12131,22479}, {12176,22520}, {12181,22755}, {12184,22759}, {12185,22760}, {12186,22761}, {12187,22762}, {12188,22765}, {12189,22768}, {13967,22764}, {18761,22505}, {19013,19055}, {19014,19056}, {22680,22769}

X(22505) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 1st ANTI-BROCARD

Barycentrics    2*a^8-(b^2+c^2)*a^6-(b^4+c^4)*a^4+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(2*b^4-b^2*c^2+2*c^4)*(b^2-c^2)^2 : :
X(22505) = 3*X(4)+X(147) = 5*X(4)-X(148) = 3*X(4)-X(6321) = 9*X(4)+X(14692) = 5*X(147)+3*X(148) = X(147)-3*X(6033) = 3*X(147)-X(14692) = 2*X(147)+3*X(22515) = X(148)+5*X(6033) = 3*X(148)-5*X(6321) = 9*X(148)+5*X(14692) = 2*X(148)-5*X(22515) = 3*X(6033)+X(6321) = 9*X(6033)-X(14692) = 2*X(6033)+X(22515) = 3*X(6321)+X(14692) = 2*X(6321)-3*X(22515)

The reciprocal orthologic center of these triangles is X(5999).

X(22505) lies on these lines: {3,7899}, {4,147}, {5,2794}, {20,15561}, {30,114}, {98,381}, {99,382}, {115,546}, {146,15545}, {316,5976}, {542,1353}, {543,15687}, {549,6721}, {550,620}, {671,14269}, {690,1539}, {1478,12185}, {1479,12184}, {1657,21166}, {2023,5475}, {2783,22938}, {2784,18483}, {2787,22799}, {2799,19160}, {3023,3585}, {3027,3583}, {3091,9862}, {3543,8724}, {3545,14830}, {3818,22681}, {3830,6054}, {3832,14651}, {3839,5984}, {3843,12188}, {3850,10991}, {3851,14061}, {3853,14981}, {3857,20398}, {3858,11623}, {4027,14041}, {5026,5103}, {5066,6055}, {5149,7825}, {5985,17577}, {6226,18511}, {6227,18509}, {6287,16044}, {7687,15535}, {7728,11005}, {7841,10352}, {7970,18525}, {8980,18538}, {9772,19910}, {9818,9861}, {9860,18492}, {9864,12699}, {9880,14893}, {9955,11710}, {10053,10895}, {10069,10896}, {10086,12953}, {10089,12943}, {10742,10768}, {10753,18440}, {11737,14971}, {12117,15684}, {12176,18502}, {12178,18491}, {12179,18495}, {12180,18497}, {12181,18507}, {12182,18516}, {12183,18517}, {12186,18520}, {12187,18522}, {12189,18542}, {12190,18544}, {13665,19056}, {13785,19055}, {13967,18762}, {14230,22625}, {14233,22596}, {15704,20399}, {17504,22247}, {18761,22504}, {22501,22502}

X(22505) = midpoint of X(i) and X(j) for these {i,j}: {3, 10722}, {4, 6033}, {99, 382}, {146, 15545}, {3543, 8724}, {3830, 6054}, {7728, 11005}, {7970, 18525}, {9864, 12699}, {10742, 10768}, {10753, 18440}, {12117, 15684}, {12181, 18507}
X(22505) = reflection of X(i) in X(j) for these (i,j): (115, 546), (550, 620), (9880, 14893)
X(22505) = complement of X(38741)
X(22505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 147, 6321), (3830, 13188, 10723), (3843, 12188, 14639), (6033, 6321, 147), (6054, 10723, 13188)

X(22506) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO INNER-FERMAT

Barycentrics    a^8-(8*b^4+17*b^2*c^2+8*c^4)*a^4+2*(b^2+c^2)*(5*b^4-2*b^2*c^2+5*c^4)*a^2+2*sqrt(3)*(a^4-b^4+b^2*c^2-c^4)*(a^2+b^2+c^2)*S-(3*b^4+b^2*c^2+3*c^4)*(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(22507).

X(22506) lies on these lines: {2,18}, {3,22748}, {316,22508}, {325,5983}, {1916,11603}, {4027,22526}, {5965,5982}, {8291,11133}, {16648,22507}

X(22507) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 1st ANTI-BROCARD

Barycentrics    2*(2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3)*S-((b^2+c^2)*a^2-b^4-c^4)*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2) : :
X(22507) = 4*X(114)-3*X(5613) = 2*X(114)-3*X(5617) = 3*X(616)-X(13172) = 3*X(5470)-4*X(20253) = 3*X(5613)-2*X(22509) = 3*X(5617)-X(22509) = X(5984)-3*X(6773) = 5*X(14061)-4*X(20415) = 3*X(14639)-2*X(16001) = 3*X(14651)-4*X(20416)

The reciprocal orthologic center of these triangles is X(22506).

X(22507) lies on these lines: {2,98}, {3,22736}, {5,6778}, {14,11603}, {99,633}, {148,16002}, {299,5983}, {616,13172}, {2782,3104}, {5470,20253}, {5479,13103}, {5611,6298}, {5858,13102}, {6033,16626}, {6775,22998}, {6782,11646}, {13349,14905}, {14061,20415}, {14639,16001}, {14651,16627}, {16648,22506}

X(22507) = reflection of X(148) in X(16002)
X(22507) = reflection of X(22509) in X(114)
X(22507) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (114, 22509, 5613), (1352, 5984, 22509), (5617, 22509, 114), (6230, 6231, 5617)

X(22508) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-BROCARD TO OUTER-FERMAT

Barycentrics    a^8-(8*b^4+17*b^2*c^2+8*c^4)*a^4+2*(b^2+c^2)*(5*b^4-2*b^2*c^2+5*c^4)*a^2-2*sqrt(3)*(a^4-b^4+b^2*c^2-c^4)*(a^2+b^2+c^2)*S-(3*b^4+b^2*c^2+3*c^4)*(b^2-c^2)^2 : :

The reciprocal orthologic center of these triangles is X(22509).

X(22508) lies on these lines: {2,17}, {316,22506}, {325,5982}, {1916,11602}, {4027,22527}, {5965,5983}, {8292,11132}, {16649,22509}

X(22509) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 1st ANTI-BROCARD

Barycentrics    2*(2*a^6-2*(b^2+c^2)*a^4+(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3)*S+((b^2+c^2)*a^2-b^4-c^4)*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2) : :
X(22509) = 2*X(114)-3*X(5613) = 4*X(114)-3*X(5617) = 3*X(617)-X(13172) = 3*X(5469)-4*X(20252) = 3*X(5613)-X(22507) = 3*X(5617)-2*X(22507) = X(5984)-3*X(6770) = 5*X(14061)-4*X(20416) = 3*X(14639)-2*X(16002) = 3*X(14651)-4*X(20415)

The reciprocal orthologic center of these triangles is X(22508).

X(22509) lies on these lines: {2,98}, {3,22737}, {5,6777}, {13,11602}, {99,634}, {148,16001}, {298,5982}, {532,22570}, {617,13172}, {2782,3105}, {5469,20252}, {5478,13102}, {5615,6299}, {5859,13103}, {6033,16627}, {6772,22997}, {6783,11646}, {13350,14904}, {14061,20416}, {14639,16002}, {14651,16626}, {16649,22508}

X(22509) = reflection of X(148) in X(16001)
X(22509) = reflection of X(22507) in X(114)
X(22509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (114, 22507, 5617), (1352, 5984, 22507), (5613, 22507, 114), (6230, 6231, 5613)

X(22510) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 1st ANTI-BROCARD

Barycentrics    sqrt(3)*SA*S^2-S*(8*S^2+3*SA^2-3*SW^2)-sqrt(3)*SB*SC*SW : :
X(22510) = X(13)+2*X(6109) = X(14)+2*X(396) = 2*X(14)+X(22997) = X(15)+2*X(115) = 2*X(15)+X(23004) = X(16)-4*X(230) = X(98)+2*X(7684) = X(99)-4*X(6671) = 4*X(115)-X(23004) = 2*X(187)+X(23005) = X(298)-4*X(6670) = X(385)+2*X(624) = 4*X(396)-X(22997) = 2*X(623)-5*X(14061) = 2*X(6774)+X(20425)

The reciprocal orthologic center of these triangles is X(5979).

X(22510) lies on these lines: {2,3106}, {4,16631}, {5,14}, {13,98}, {15,115}, {16,230}, {18,298}, {30,5470}, {62,6774}, {99,6671}, {187,23005}, {203,10061}, {385,624}, {511,6034}, {524,16268}, {532,16530}, {542,16267}, {618,5983}, {619,11289}, {623,14061}, {635,7901}, {3104,7746}, {3105,3767}, {3107,5309}, {5238,5474}, {5463,22573}, {5479,16964}, {5978,6669}, {5999,6108}, {6036,14538}, {6114,16966}, {6295,11303}, {6321,13350}, {6775,16241}, {6777,6783}, {6778,11542}, {7005,10077}, {9735,11648}, {9753,22694}, {11304,12204}, {11543,22850}, {11602,14138}, {11707,13178}, {13102,22236}, {13103,19780}, {14651,22688}, {16808,22512}

X(22510) = centroid of X(13)X(14)X(15)
X(22510) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 17, 5613), (14, 396, 22997), (15, 115, 23004), (61, 22891, 14), (398, 20253, 14), (6670, 14137, 18), (6777, 16960, 6783)

X(22511) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 1st ANTI-BROCARD

Barycentrics    sqrt(3)*SA*S^2+S*(8*S^2+3*SA^2-3*SW^2)-sqrt(3)*SB*SC*SW : :
X(22511) = X(13)+2*X(395) = 2*X(13)+X(22998) = X(14)+2*X(6108) = X(15)-4*X(230) = X(16)+2*X(115) = 2*X(16)+X(23005) = X(98)+2*X(7685) = X(99)-4*X(6672) = 4*X(115)-X(23005) = 2*X(187)+X(23004) = X(299)-4*X(6669) = X(385)+2*X(623) = 4*X(395)-X(22998) = 2*X(624)-5*X(14061) = 2*X(6771)+X(20426)

The reciprocal orthologic center of these triangles is X(5978).

X(22511) lies on these lines: {2,3107}, {4,16630}, {5,13}, {14,98}, {15,230}, {16,115}, {17,299}, {30,5469}, {61,6771}, {99,6672}, {187,23004}, {202,10062}, {385,623}, {511,6034}, {524,16267}, {542,16268}, {618,11290}, {619,5982}, {624,14061}, {636,7901}, {3104,3767}, {3105,7746}, {3106,5309}, {5237,5473}, {5464,22574}, {5478,16965}, {5979,6670}, {5999,6109}, {6036,14539}, {6115,16967}, {6321,13349}, {6582,11304}, {6772,16242}, {6777,11543}, {6778,6782}, {7006,10078}, {9736,11648}, {9753,22693}, {11303,12205}, {11542,22894}, {11603,14139}, {11708,13178}, {13102,19781}, {13103,22238}, {14651,22690}, {16809,22513}

X(22511) = centroid of X(13)X(14)X(16)
X(22511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 18, 5617), (13, 395, 22998), (16, 115, 23005), (62, 22846, 13), (397, 20252, 13), (6669, 14136, 17), (6778, 16961, 6782)

X(22512) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 1st ANTI-BROCARD

Barycentrics    -S*((SA-SW)*(3*SA-SW)+2*S^2)+2*sqrt(3)*SB*SC*SW : :
X(22512) = 2*X(619)-3*X(11297) = 4*X(6670)-3*X(11298)

The reciprocal orthologic center of these triangles is X(5979).

X(22512) lies on these lines: {3,6114}, {4,32}, {14,16}, {15,5613}, {61,6778}, {187,383}, {381,6109}, {398,9113}, {542,6772}, {543,616}, {617,7865}, {619,11297}, {621,3734}, {622,754}, {624,6295}, {1080,5475}, {2549,5334}, {2782,3104}, {3094,22707}, {3098,22861}, {3815,9750}, {5318,18907}, {5343,22531}, {5460,11296}, {5479,16942}, {5978,7880}, {5979,22568}, {5981,7761}, {6033,6115}, {6108,14830}, {6670,11298}, {6774,18581}, {6782,20426}, {6783,11485}, {7051,12951}, {7804,11303}, {9140,21467}, {10638,12941}, {11486,14137}, {16808,22510}, {18582,22797}, {22689,23019}, {22693,22708}, {22906,23013}

X(22512) = reflection of X(22513) in X(115)
X(22512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 7737, 22513), (14, 19107, 23004)

X(22513) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 1st ANTI-BROCARD

Barycentrics    S*((SA-SW)*(3*SA-SW)+2*S^2)+2*sqrt(3)*SB*SC*SW : :
X(22513) = 2*X(618)-3*X(11298) = 4*X(6669)-3*X(11297)

The reciprocal orthologic center of these triangles is X(5978).

X(22513) lies on these lines: {3,6115}, {4,32}, {13,15}, {16,5617}, {62,6777}, {187,1080}, {381,6108}, {383,5475}, {397,9112}, {542,6775}, {543,617}, {616,7865}, {618,11298}, {621,754}, {622,3734}, {623,6582}, {1250,12942}, {2549,5335}, {2782,3105}, {3094,22708}, {3098,22907}, {3815,9749}, {5321,18907}, {5344,22532}, {5459,11295}, {5478,16943}, {5978,22570}, {5979,7880}, {5980,7761}, {6033,6114}, {6109,14830}, {6669,11297}, {6771,18582}, {6782,11486}, {6783,20425}, {7804,11304}, {9140,21466}, {11485,14136}, {12952,19373}, {16809,22511}, {18581,22796}, {22687,23025}, {22694,22707}, {22862,23006}

X(22513) = reflection of X(22512) in X(115)
X(22513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 7737, 22512), (13, 19106, 23005), (5335, 6770, 5472)

X(22514) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st ANTI-BROCARD

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

The reciprocal parallelogic center of these triangles is X(385).

X(22514) lies on these lines: {3,11710}, {36,13174}, {55,7983}, {56,99}, {98,3428}, {104,13172}, {114,22753}, {115,958}, {148,2975}, {517,12178}, {519,12326}, {542,22583}, {543,11194}, {690,22586}, {956,13178}, {993,11599}, {999,11711}, {1001,11725}, {2782,11249}, {2783,22775}, {2785,8301}, {2787,22560}, {2794,19159}, {2799,19162}, {3023,10966}, {3149,9864}, {4027,22520}, {5186,22479}, {5969,22769}, {6319,22756}, {6320,22757}, {6321,22758}, {8782,22744}, {8997,22763}, {9881,16371}, {10086,22766}, {10089,22767}, {11492,13177}, {11493,13176}, {12114,13180}, {12258,16418}, {13175,22654}, {13179,22755}, {13182,22759}, {13183,22760}, {13184,22761}, {13185,22762}, {13188,22765}, {13189,15452}, {13989,22764}, {18761,22515}, {19013,19108}, {19014,19109}

X(22515) = PARALLELOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 1st ANTI-BROCARD

Barycentrics    2*a^8-3*(b^2+c^2)*a^6+(b^4+4*b^2*c^2+c^4)*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(2*b^4-3*b^2*c^2+2*c^4)*(b^2-c^2)^2 : :
X(22515) = 3*X(2)-4*X(15092) = 3*X(3)-5*X(14061) = X(3)-3*X(14639) = 5*X(4)-X(147) = 3*X(4)+X(148) = 3*X(4)-X(6033) = 11*X(4)-X(14692) = 3*X(147)+5*X(148) = 3*X(147)-5*X(6033) = X(147)+5*X(6321) = 11*X(147)-5*X(14692) = 2*X(147)-5*X(22505) = X(148)-3*X(6321) = 11*X(148)+3*X(14692) = 2*X(148)+3*X(22505) = X(6033)+3*X(6321) = 11*X(6033)-3*X(14692) = 2*X(6033)-3*X(22505) = 11*X(6321)+X(14692) = 2*X(6321)+X(22505) = 3*X(10723)+5*X(14061) = X(10723)+3*X(14639) = 5*X(14061)-9*X(14639)

The reciprocal parallelogic center of these triangles is X(385).

X(22515) lies on these lines: {2,15092}, {3,10723}, {4,147}, {5,620}, {30,115}, {76,18547}, {98,382}, {99,381}, {114,546}, {542,1539}, {543,3845}, {549,6722}, {550,6036}, {671,3830}, {690,10113}, {1478,13183}, {1479,13182}, {1656,21166}, {2023,7748}, {2482,5066}, {2777,15535}, {2783,22799}, {2787,22938}, {2794,3627}, {2799,19163}, {3023,3583}, {3027,3585}, {3044,10540}, {3091,13172}, {3146,14651}, {3534,9166}, {3543,9862}, {3818,5969}, {3839,8724}, {3843,13188}, {3850,10992}, {3857,20399}, {3860,15300}, {3861,14981}, {4027,14042}, {5026,19130}, {5055,12117}, {5461,8703}, {5939,9993}, {6054,12355}, {6319,18509}, {6320,18511}, {7747,12829}, {7845,13449}, {7951,15452}, {7983,18525}, {8782,18500}, {8997,18538}, {9167,10109}, {9818,13175}, {9955,11711}, {9996,11185}, {10053,12953}, {10069,12943}, {10086,10895}, {10089,10896}, {10733,18332}, {10742,10769}, {10754,18440}, {11801,15357}, {12041,15359}, {12100,14971}, {12295,16278}, {12699,13178}, {12902,15342}, {13173,18491}, {13174,18492}, {13176,18495}, {13177,18497}, {13179,18507}, {13180,18516}, {13181,18517}, {13184,18520}, {13185,18522}, {13189,18542}, {13190,18544}, {13665,19109}, {13785,19108}, {13989,18762}, {14830,15682}, {15704,20398}, {18761,22514}

X(22515) = midpoint of X(i) and X(j) for these {i,j}: {3, 10723}, {4, 6321}, {98, 382}, {671, 3830}, {3543, 11632}, {6054, 12355}, {7983, 18525}, {10733, 18332}, {10742, 10769}, {10754, 18440}, {12295, 16278}, {12699, 13178}, {12902, 15342}, {13179, 18507}, {14830, 15682}
X(22515) = reflection of X(i) in X(j) for these (i,j): (114, 546), (550, 6036), (2482, 5066), (5026, 19130), (12041, 15359)
X(22515) = complement of X(38730)
X(22515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 148, 6033), (671, 10722, 12188), (3091, 13172, 15561), (3830, 12188, 10722), (6033, 6321, 148), (10723, 14639, 3), (12355, 14269, 6054)

X(22516) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-BROCARD TO ANTI-INNER-GREBE

Barycentrics    a^2*(16*(a^2+b^2+c^2)*(a^12-16*(b^2+c^2)*a^10+(161*b^4-243*b^2*c^2+161*c^4)*a^8+10*(b^2+c^2)*(8*b^4+21*b^2*c^2+8*c^4)*a^6+(35*b^8+35*c^8-2*b^2*c^2*(14*b^4+1377*b^2*c^2+14*c^4))*a^4+2*(b^2+c^2)*(64*b^8+64*c^8-5*b^2*c^2*(83*b^4-190*b^2*c^2+83*c^4))*a^2+59*b^10*c^2-727*b^8*c^4-5*b^12+59*b^2*c^10-5*c^12+2306*b^6*c^6-727*b^4*c^8)*S-15*a^16+2*(b^2+c^2)*a^14-(542*b^4-2575*b^2*c^2+542*c^4)*a^12-2*(b^2+c^2)*(763*b^4-710*b^2*c^2+763*c^4)*a^10-(236*b^8+236*c^8+b^2*c^2*(18693*b^4-9674*b^2*c^2+18693*c^4))*a^8-2*(b^2+c^2)*(637*b^8+637*c^8-10*b^2*c^2*(130*b^4+1953*b^2*c^2+130*c^4))*a^6-(1586*b^12+1586*c^12-(12269*b^8+12269*c^8-2*b^2*c^2*(8309*b^4-14599*b^2*c^2+8309*c^4))*b^2*c^2)*a^4+2*(b^2+c^2)*(247*b^12+247*c^12-(4106*b^8+4106*c^8-39*b^2*c^2*(495*b^4-1172*b^2*c^2+495*c^4))*b^2*c^2)*a^2-26834*b^8*c^8+4519*b^6*c^10+3982*b^4*c^12-919*b^2*c^14+75*c^16+75*b^16-919*b^14*c^2+3982*b^12*c^4+4519*b^10*c^6) : :

The reciprocal cyclologic center of these triangles is X(22517).

X(22516) lies on these lines: {}

X(22517) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO 4th ANTI-BROCARD

Barycentrics    a^2*((168*a^18-1084*(b^2+c^2)*a^16-40*(27*b^4+71*b^2*c^2+27*c^4)*a^14+4*(b^2+c^2)*(2798*b^4+7175*b^2*c^2+2798*c^4)*a^12+8*(351*b^8+351*c^8-38*b^2*c^2*(71*b^4+559*b^2*c^2+71*c^4))*a^10-4*(b^2+c^2)*(2928*b^8+2928*c^8+b^2*c^2*(13947*b^4-5582*b^2*c^2+13947*c^4))*a^8-8*(141*b^12+141*c^12+(3011*b^8+3011*c^8-b^2*c^2*(17909*b^4+32934*b^2*c^2+17909*c^4))*b^2*c^2)*a^6+4*(b^2+c^2)*(370*b^12+370*c^12+(4657*b^8+4657*c^8-2*b^2*c^2*(12655*b^4-21619*b^2*c^2+12655*c^4))*b^2*c^2)*a^4-32*(24*b^16+24*c^16-(16*b^12+16*c^12+(715*b^8+715*c^8-2*b^2*c^2*(754*b^4+1775*b^2*c^2+754*c^4))*b^2*c^2)*b^2*c^2)*a^2+4*(b^2+c^2)*(31*b^16+31*c^16-(477*b^12+477*c^12-(1410*b^8+1410*c^8+b^2*c^2*(1293*b^4-6818*b^2*c^2+1293*c^4))*b^2*c^2)*b^2*c^2))*S-13*a^20-573*(b^2+c^2)*a^18+(3464*b^4+7779*b^2*c^2+3464*c^4)*a^16-2*(b^2+c^2)*(518*b^4+9477*b^2*c^2+518*c^4)*a^14-2*(3991*b^8+3991*c^8+5*b^2*c^2*(3147*b^4+224*b^2*c^2+3147*c^4))*a^12+2*(b^2+c^2)*(2465*b^8+2465*c^8+b^2*c^2*(18371*b^4+66416*b^2*c^2+18371*c^4))*a^10+4*(1059*b^12+1059*c^12+(9344*b^8+9344*c^8-b^2*c^2*(25031*b^4+19948*b^2*c^2+25031*c^4))*b^2*c^2)*a^8-2*(b^2+c^2)*(1558*b^12+1558*c^12+(3159*b^8+3159*c^8-2*b^2*c^2*(9733*b^4-61237*b^2*c^2+9733*c^4))*b^2*c^2)*a^6+(339*b^16+339*c^16-2*(7137*b^12+7137*c^12-(22100*b^8+22100*c^8+b^2*c^2*(37593*b^4-20843*b^2*c^2+37593*c^4))*b^2*c^2)*b^2*c^2)*a^4-(b^2+c^2)*(205*b^16+205*c^16-2*(3865*b^12+3865*c^12-(15796*b^8+15796*c^8-b^2*c^2*(7351*b^4+28565*b^2*c^2+7351*c^4))*b^2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^2*(44*b^16+44*c^16+(267*b^12+267*c^12-(2370*b^8+2370*c^8-b^2*c^2*(3237*b^4+12236*b^2*c^2+3237*c^4))*b^2*c^2)*b^2*c^2)) : :

The reciprocal cyclologic center of these triangles is X(22516).

X(22517) lies on these lines: {}

X(22518) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-BROCARD TO ANTI-OUTER-GREBE

Barycentrics    a^2*(16*(a^2+b^2+c^2)*(a^12-16*(b^2+c^2)*a^10+(161*b^4-243*b^2*c^2+161*c^4)*a^8+10*(b^2+c^2)*(8*b^4+21*b^2*c^2+8*c^4)*a^6+(35*b^8+35*c^8-2*b^2*c^2*(14*b^4+1377*b^2*c^2+14*c^4))*a^4+2*(b^2+c^2)*(64*b^8+64*c^8-5*b^2*c^2*(83*b^4-190*b^2*c^2+83*c^4))*a^2+59*b^10*c^2-727*b^8*c^4-5*b^12+59*b^2*c^10-5*c^12+2306*b^6*c^6-727*b^4*c^8)*S+15*a^16-2*(b^2+c^2)*a^14+(542*b^4-2575*b^2*c^2+542*c^4)*a^12+2*(b^2+c^2)*(763*b^4-710*b^2*c^2+763*c^4)*a^10+(236*b^8+236*c^8+b^2*c^2*(18693*b^4-9674*b^2*c^2+18693*c^4))*a^8+2*(b^2+c^2)*(637*b^8+637*c^8-10*b^2*c^2*(130*b^4+1953*b^2*c^2+130*c^4))*a^6+(1586*b^12+1586*c^12-(12269*b^8+12269*c^8-2*b^2*c^2*(8309*b^4-14599*b^2*c^2+8309*c^4))*b^2*c^2)*a^4-2*(b^2+c^2)*(247*b^12+247*c^12-(4106*b^8+4106*c^8-39*b^2*c^2*(495*b^4-1172*b^2*c^2+495*c^4))*b^2*c^2)*a^2+26834*b^8*c^8-4519*b^6*c^10-3982*b^4*c^12+919*b^2*c^14-75*c^16-75*b^16+919*b^14*c^2-3982*b^12*c^4-4519*b^10*c^6) : :

The reciprocal cyclologic center of these triangles is X(22519).

X(22518) lies on these lines: {}

X(22519) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 4th ANTI-BROCARD

Barycentrics    a^2*((168*a^18-1084*(b^2+c^2)*a^16-40*(27*b^4+71*b^2*c^2+27*c^4)*a^14+4*(b^2+c^2)*(2798*b^4+7175*b^2*c^2+2798*c^4)*a^12+8*(351*b^8+351*c^8-38*b^2*c^2*(71*b^4+559*b^2*c^2+71*c^4))*a^10-4*(b^2+c^2)*(2928*b^8+2928*c^8+b^2*c^2*(13947*b^4-5582*b^2*c^2+13947*c^4))*a^8-8*(141*b^12+141*c^12+(3011*b^8+3011*c^8-b^2*c^2*(17909*b^4+32934*b^2*c^2+17909*c^4))*b^2*c^2)*a^6+4*(b^2+c^2)*(370*b^12+370*c^12+(4657*b^8+4657*c^8-2*b^2*c^2*(12655*b^4-21619*b^2*c^2+12655*c^4))*b^2*c^2)*a^4-32*(24*b^16+24*c^16-(16*b^12+16*c^12+(715*b^8+715*c^8-2*b^2*c^2*(754*b^4+1775*b^2*c^2+754*c^4))*b^2*c^2)*b^2*c^2)*a^2+4*(b^2+c^2)*(31*b^16+31*c^16-(477*b^12+477*c^12-(1410*b^8+1410*c^8+b^2*c^2*(1293*b^4-6818*b^2*c^2+1293*c^4))*b^2*c^2)*b^2*c^2))*S+13*a^20+573*(b^2+c^2)*a^18-(3464*b^4+7779*b^2*c^2+3464*c^4)*a^16+2*(b^2+c^2)*(518*b^4+9477*b^2*c^2+518*c^4)*a^14+2*(3991*b^8+3991*c^8+5*b^2*c^2*(3147*b^4+224*b^2*c^2+3147*c^4))*a^12-2*(b^2+c^2)*(2465*b^8+2465*c^8+b^2*c^2*(18371*b^4+66416*b^2*c^2+18371*c^4))*a^10-4*(1059*b^12+1059*c^12+(9344*b^8+9344*c^8-b^2*c^2*(25031*b^4+19948*b^2*c^2+25031*c^4))*b^2*c^2)*a^8+2*(b^2+c^2)*(1558*b^12+1558*c^12+(3159*b^8+3159*c^8-2*b^2*c^2*(9733*b^4-61237*b^2*c^2+9733*c^4))*b^2*c^2)*a^6-(339*b^16+339*c^16-2*(7137*b^12+7137*c^12-(22100*b^8+22100*c^8+b^2*c^2*(37593*b^4-20843*b^2*c^2+37593*c^4))*b^2*c^2)*b^2*c^2)*a^4+(b^2+c^2)*(205*b^16+205*c^16-2*(3865*b^12+3865*c^12-(15796*b^8+15796*c^8-b^2*c^2*(7351*b^4+28565*b^2*c^2+7351*c^4))*b^2*c^2)*b^2*c^2)*a^2+(b^2-c^2)^2*(44*b^16+44*c^16+(267*b^12+267*c^12-(2370*b^8+2370*c^8-b^2*c^2*(3237*b^4+12236*b^2*c^2+3237*c^4))*b^2*c^2)*b^2*c^2)) : :

The reciprocal cyclologic center of these triangles is X(22518).

X(22519) lies on these lines: {}

X(22520) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 2nd CIRCUMPERP TANGENTIAL

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

X(22520) lies on these lines: {3,11490}, {32,56}, {36,10789}, {55,10800}, {83,958}, {98,22753}, {104,10788}, {182,3428}, {384,22779}, {727,4257}, {956,10791}, {995,4279}, {999,11364}, {2080,10269}, {2975,7787}, {3398,11249}, {4027,22514}, {5253,7793}, {6196,16476}, {7976,17034}, {10790,22654}, {10793,22757}, {10794,12110}, {10796,22758}, {10797,22759}, {10798,22760}, {10799,10804}, {10801,22766}, {10802,22767}, {10803,22768}, {11194,12150}, {11380,22479}, {11492,11838}, {11493,11837}, {11839,22755}, {11840,22761}, {11841,22762}, {11842,22765}, {12176,22504}, {12191,22565}, {12192,22583}, {12193,22659}, {12195,12513}, {12196,18237}, {12197,22770}, {12198,12773}, {12199,22775}, {12200,22777}, {12201,19478}, {12202,22778}, {12204,22774}, {12205,22773}, {12206,22780}, {12207,19159}, {12208,22781}, {12209,22782}, {12210,22595}, {12211,22624}, {12212,22769}, {13193,22586}, {13194,22560}, {13195,19162}, {13672,22783}, {13743,16115}, {13792,22784}, {13885,22763}, {13938,22764}, {17023,21010}, {18502,18761}, {18993,19013}, {18994,19014}, {22521,22680}, {22522,22771}, {22523,22772}, {22524,22776}

X(22520) = {X(3), X(12194)}-harmonic conjugate of X(11490)

X(22521) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 1st BROCARD-REFLECTED

Barycentrics    3*a^8-6*(b^2+c^2)*a^6+(b^2-3*b*c-c^2)*(b^2+3*b*c-c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(22521) = 4*X(3398)-X(7470) = 2*X(5007)+X(12110)

The reciprocal orthologic center of these triangles is X(3).

X(22521) lies on these lines: {4,3172}, {5,20088}, {6,7709}, {32,262}, {61,22523}, {62,22522}, {83,15819}, {98,5008}, {99,5097}, {182,22676}, {376,5050}, {385,7697}, {511,12150}, {576,3972}, {1003,5093}, {1656,7900}, {2080,3329}, {2548,9754}, {2782,12191}, {3398,7470}, {3524,19661}, {3533,10155}, {5007,12110}, {5171,7878}, {5188,9751}, {5306,14651}, {5480,9862}, {5999,11842}, {6179,10358}, {6194,7787}, {7694,9753}, {7757,15520}, {7785,20576}, {9166,14160}, {10789,22650}, {10790,22655}, {10791,22697}, {10792,22699}, {10793,22700}, {10794,22703}, {10795,22704}, {10797,22705}, {10798,22706}, {10799,22711}, {10800,22713}, {10801,22729}, {10802,22730}, {10803,22731}, {10804,22732}, {11364,22475}, {11380,22480}, {11490,22556}, {11837,22668}, {11838,22672}, {11839,22698}, {11840,22709}, {11841,22710}, {12176,12212}, {12835,18971}, {13860,21309}, {13885,22720}, {13938,22721}, {14537,14639}, {14693,17005}, {14912,15428}, {18502,22681}, {18993,19063}, {18994,19064}, {22520,22680}

X(22521) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 10788, 11676), (32, 262, 21445)

X(22522) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO INNER-FERMAT

Barycentrics    5*a^8-9*(b^2+c^2)*a^6+(2*b^4-15*b^2*c^2+2*c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2+2*(a^4+2*(b^2+c^2)*a^2+b^2*c^2)*sqrt(3)*S*a^2+(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22522) lies on these lines: {18,32}, {62,22521}, {83,630}, {98,22831}, {182,22843}, {628,7787}, {1078,6674}, {3398,12204}, {5965,12212}, {10788,22531}, {10789,22651}, {10790,22656}, {10791,22851}, {10792,22853}, {10793,22854}, {10794,22857}, {10795,22858}, {10796,16627}, {10797,22859}, {10798,22860}, {10799,22865}, {10800,22867}, {10801,22884}, {10802,22885}, {10803,22886}, {10804,22887}, {11364,11740}, {11380,22481}, {11490,22557}, {11837,22669}, {11838,22673}, {11839,22852}, {11840,22863}, {11841,22864}, {11842,16628}, {12110,16965}, {12835,18972}, {13885,22876}, {13938,22877}, {18502,22794}, {18993,19069}, {18994,19072}, {22520,22771}

X(22523) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO OUTER-FERMAT

Barycentrics    5*a^8-9*(b^2+c^2)*a^6+(2*b^4-15*b^2*c^2+2*c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-2*(a^4+2*(b^2+c^2)*a^2+b^2*c^2)*sqrt(3)*S*a^2+(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22523) lies on these lines: {17,32}, {61,22521}, {83,629}, {98,22832}, {182,22890}, {532,12150}, {627,7787}, {1078,6673}, {3398,12205}, {5965,12212}, {10788,22532}, {10789,22652}, {10790,22657}, {10791,22896}, {10792,22898}, {10793,22899}, {10794,22902}, {10795,22903}, {10796,16626}, {10797,22904}, {10798,22905}, {10799,22910}, {10800,22912}, {10801,22929}, {10802,22930}, {10803,22931}, {10804,22932}, {11364,11739}, {11380,22482}, {11490,22558}, {11837,22670}, {11838,22674}, {11839,22897}, {11840,22908}, {11841,22909}, {11842,16629}, {12110,16964}, {12835,18973}, {13885,22921}, {13938,22922}, {18502,22795}, {18993,19071}, {18994,19070}, {22520,22772}

X(22524) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 3rd HATZIPOLAKIS

Barycentrics    (5*R^2-SW)*(16*R^2-SA-3*SW)*S^4+(16*(11*SA^2-18*SA*SW+8*SW^2)*R^4-(70*SA^2-107*SA*SW+45*SW^2)*SW*R^2+(7*SA^2-10*SA*SW+4*SW^2)*SW^2)*S^2+(4*R^2-SW)*(16*R^2-3*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22524) lies on these lines: {32,22466}, {83,22966}, {98,22833}, {182,22951}, {7787,22647}, {10788,22533}, {10789,22653}, {10790,22658}, {10791,22941}, {10792,22945}, {10793,22947}, {10794,22956}, {10795,22957}, {10796,22955}, {10797,22958}, {10798,22959}, {10799,22965}, {10800,22969}, {10801,22980}, {10802,22981}, {10803,22982}, {10804,22983}, {11364,22476}, {11380,22483}, {11490,22559}, {11839,22943}, {11840,22963}, {11841,22964}, {11842,22979}, {12835,18978}, {13885,22976}, {13938,22977}, {18502,22800}, {18993,19083}, {18994,19084}, {22520,22776}

X(22525) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO 1st BROCARD-REFLECTED

Barycentrics    3*a^10-9*(b^2+c^2)*a^8+(13*b^4+7*b^2*c^2+13*c^4)*a^6-3*(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^4+(2*b^8+2*c^8-(11*b^4-6*b^2*c^2+11*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(22525) = X(385)-4*X(575)

The reciprocal orthologic center of these triangles is X(12177).

X(22525) lies on these lines: {182,7771}, {325,9755}, {385,575}, {511,5182}, {524,5050}, {576,3972}, {2782,12151}, {4027,22503}, {7894,22234}, {10131,22679}, {11159,20423}

X(22526) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO INNER-FERMAT

Barycentrics    5*a^10-16*(b^2+c^2)*a^8+(21*b^4+11*b^2*c^2+21*c^4)*a^6-13*(b^4-c^4)*(b^2-c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*b^2*c^2+(3*b^8+3*c^8-2*(9*b^4-2*b^2*c^2+9*c^4)*b^2*c^2)*a^2+2*(a^8-(b^2+c^2)*a^6-3*b^2*c^2*a^4+(b^6+c^6)*a^2+(b^4+c^4)*b^2*c^2)*sqrt(3)*S : :

The reciprocal orthologic center of these triangles is X(22507).

X(22526) lies on these lines: {6,22683}, {182,22736}, {4027,22506}, {10131,22748}

X(22527) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-BROCARD TO OUTER-FERMAT

Barycentrics    -5*a^10+16*(b^2+c^2)*a^8-(21*b^4+11*b^2*c^2+21*c^4)*a^6+13*(b^4-c^4)*(b^2-c^2)*a^4-3*(b^4-c^4)*(b^2-c^2)*b^2*c^2-(3*b^8+3*c^8-2*(9*b^4-2*b^2*c^2+9*c^4)*b^2*c^2)*a^2+2*(a^8-(b^2+c^2)*a^6-3*b^2*c^2*a^4+(b^6+c^6)*a^2+(b^4+c^4)*b^2*c^2)*sqrt(3)*S : :

The reciprocal orthologic center of these triangles is X(22509).

X(22527) lies on these lines: {6,22685}, {182,22737}, {532,5182}, {4027,22508}, {10131,22749}

X(22528) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((-16*R^2+5*SA+3*SW)*S^2+(8*R^2*(32*R^2-3*SA-10*SW)+5*SA^2-5*SB*SC+6*SW^2)*SA) : :
X(22528) = 3*X(2)-4*X(22581) = 9*X(2)-8*X(22973) = 3*X(2979)-X(22534) = 3*X(3060)-4*X(22530) = 3*X(22581)-2*X(22973) = 3*X(22970)-4*X(22973)

The reciprocal orthologic center of these triangles is X(9729).

X(22528) lies on these lines: {2,22581}, {3,22497}, {4,22834}, {20,1204}, {22,1620}, {30,22808}, {64,394}, {69,11440}, {97,19198}, {511,21652}, {1370,22555}, {1619,12279}, {1993,19460}, {2071,5907}, {2979,22534}, {3060,22530}, {3100,22954}, {3101,22840}, {3146,22538}, {3153,22816}, {4296,19472}, {5012,22529}, {6515,18936}, {7488,22962}, {7691,16386}, {11412,21312}, {11414,22550}, {11416,22830}, {11417,22960}, {11418,22961}, {11420,22974}, {11421,22975}, {12086,12294}, {12219,15054}, {13567,22466}, {17811,22966}, {19121,19142}, {19406,19488}, {19407,19489}

X(22528) = midpoint of X(11412) and X(22535)
X(22528) = reflection of X(i) in X(j) for these (i,j): (4, 22834), (3146, 22538)
X(22528) = anticomplement of X(22970)
X(22528) = {X(22581), X(22970)}-harmonic conjugate of X(2)

X(22529) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-CONWAY TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((6*R^2-SW)^2*S^2-(4*R^2-SW)^2*(8*R^2+SA-2*SW)*SA) : :
X(22529) = 3*X(11402)-X(19460) = 3*X(11402)+X(22497)

The reciprocal orthologic center of these triangles is X(9729).

X(22529) lies on these lines: {6,2929}, {54,403}, {182,22581}, {184,22970}, {389,22962}, {567,22808}, {569,22834}, {1147,22955}, {2904,5890}, {5012,22528}, {9306,22973}, {11402,19460}, {11422,22534}, {11423,22535}, {11424,22538}, {11425,22549}, {11426,22550}, {11427,22555}, {11428,22840}, {11429,22954}, {11430,22978}, {11536,22952}, {12233,15472}, {13366,21652}, {14912,18936}, {17809,17837}, {18388,22816}, {19153,22658}, {19365,19472}, {19408,19488}, {19409,19489}

X(22529) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2929, 22530), (11402, 22497, 19460)

X(22530) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((76*R^4-30*R^2*SW+3*SW^2)*S^2+(4*R^2-SW)^2*(8*R^2+SA-2*SW)*SA) : :
X(22530) = 3*X(51)+X(21652) = 3*X(51)-X(22970) = 3*X(568)+X(22808) = 3*X(3060)+X(22528) = 5*X(3567)-X(22750) = 9*X(5640)-X(22534) = 3*X(5943)-2*X(22973) = 7*X(9781)+X(22535)

The reciprocal orthologic center of these triangles is X(9729).

X(22530) lies on these lines: {4,18936}, {6,2929}, {25,19460}, {51,21652}, {52,22834}, {185,22538}, {378,19360}, {511,22581}, {568,22808}, {578,22962}, {974,12241}, {3060,22528}, {3567,22750}, {5640,22534}, {5943,22973}, {6217,22947}, {6218,22945}, {6642,22955}, {6746,9969}, {9777,22497}, {9781,22535}, {9786,22549}, {9792,19198}, {10151,22968}, {11432,22550}, {11433,22555}, {11435,22840}, {11436,22954}, {11438,22978}, {17810,17837}, {18390,22816}, {19039,19083}, {19040,19084}, {19366,19472}, {19410,19488}, {19411,19489}, {22483,22953}

X(22530) = midpoint of X(i) and X(j) for these {i,j}: {52, 22834}, {185, 22538}, {22483, 22953}
X(22530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2929, 22529), (51, 21652, 22970)

X(22531) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO INNER-FERMAT

Barycentrics    -2*S*(7*a^4-6*(b^2+c^2)*a^2-(b^2-c^2)^2)+sqrt(3)*(a^2-b^2-c^2)*(3*a^4+(b^2-c^2)^2) : :
X(22531) = 3*X(4)-4*X(22831) = 3*X(18)-2*X(22831) = 4*X(630)-5*X(631) = 7*X(3090)-8*X(6674) = 5*X(3091)-4*X(22794) = 7*X(3528)-2*X(22845) = 3*X(5603)-4*X(11740) = 3*X(5657)-2*X(22851) = 3*X(7967)-2*X(22867) = 2*X(11603)-3*X(14651) = 3*X(11845)-2*X(22852)

The reciprocal orthologic center of these triangles is X(3).

X(22531) lies on these lines: {2,16627}, {3,299}, {4,16}, {20,6773}, {24,22656}, {30,16628}, {32,16941}, {98,5488}, {104,22771}, {315,11133}, {388,22884}, {397,19780}, {398,16940}, {497,22885}, {515,22651}, {630,631}, {1204,3098}, {3085,22859}, {3086,22860}, {3090,6674}, {3091,22794}, {3104,7709}, {3528,22845}, {4293,18972}, {4294,22865}, {5334,7756}, {5343,22512}, {5344,12815}, {5603,11740}, {5657,22851}, {5869,11481}, {7487,22481}, {7581,19072}, {7582,19069}, {7967,22867}, {8260,22238}, {9862,22745}, {10645,22855}, {10646,22850}, {10783,22853}, {10784,22854}, {10785,22857}, {10786,22858}, {10788,22522}, {10805,22886}, {10806,22887}, {11491,22557}, {11603,14651}, {11843,22669}, {11844,22673}, {11845,22852}, {11846,22863}, {11847,22864}, {12252,14538}, {13886,22876}, {13939,22877}, {16965,22846}

X(22531) = midpoint of X(20) and X(22114)
X(22531) = reflection of X(4) in X(18)
X(22531) = anticomplement of X(16627)
X(22531) = {X(3522), X(6776)}-harmonic conjugate of X(22532)

X(22532) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO OUTER-FERMAT

Barycentrics    2*S*(7*a^4-6*(b^2+c^2)*a^2-(b^2-c^2)^2)+sqrt(3)*(a^2-b^2-c^2)*(3*a^4+(b^2-c^2)^2) : :
X(22532) = 3*X(4)-4*X(22832) = 3*X(17)-2*X(22832) = 3*X(376)-2*X(22890) = 4*X(629)-5*X(631) = 7*X(3090)-8*X(6673) = 5*X(3091)-4*X(22795) = 7*X(3528)-2*X(22844) = 3*X(5603)-4*X(11739) = 3*X(5657)-2*X(22896) = 3*X(7967)-2*X(22912) = 2*X(11602)-3*X(14651) = 3*X(11845)-2*X(22897)

The reciprocal orthologic center of these triangles is X(3).

X(22532) lies on these lines: {2,16626}, {3,298}, {4,15}, {20,6770}, {24,22657}, {30,16629}, {32,16940}, {98,5487}, {104,22772}, {315,11132}, {376,532}, {388,22929}, {397,16941}, {398,19781}, {497,22930}, {515,22652}, {629,631}, {1204,3098}, {3085,22904}, {3086,22905}, {3090,6673}, {3091,22795}, {3105,7709}, {3528,22844}, {4293,18973}, {4294,22910}, {5335,7756}, {5343,12815}, {5344,22513}, {5603,11739}, {5657,22896}, {5868,11480}, {7487,22482}, {7581,19070}, {7582,19071}, {7967,22912}, {8259,22236}, {9862,22746}, {10645,22894}, {10646,22901}, {10783,22898}, {10784,22899}, {10785,22902}, {10786,22903}, {10788,22523}, {10805,22931}, {10806,22932}, {11491,22558}, {11602,14651}, {11843,22670}, {11844,22674}, {11845,22897}, {11846,22908}, {11847,22909}, {12252,14539}, {13886,22921}, {13939,22922}, {16964,22891}

X(22532) = midpoint of X(20) and X(22113)
X(22532) = reflection of X(4) in X(17)
X(22532) = anticomplement of X(16626)
X(22532) = {X(3522), X(6776)}-harmonic conjugate of X(22531)

X(22533) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 3rd HATZIPOLAKIS

Barycentrics    SA*((11*R^2-SA-SW)*S^2-(SA-SW)*(R^2*(8*R^2-6*SA-7*SW)+SA^2-SB*SC+SW^2)) : :
X(22533) = 3*X(4)-4*X(22833) = 3*X(376)-2*X(22951) = 5*X(631)-4*X(22966) = 5*X(3091)-4*X(22800) = 3*X(5603)-4*X(22476) = 3*X(5657)-2*X(22941) = 3*X(7967)-2*X(22969) = 3*X(11845)-2*X(22943) = 3*X(22466)-2*X(22833)

The reciprocal orthologic center of these triangles is X(12241).

X(22533) lies on these lines: {2,22953}, {3,22647}, {4,18936}, {5,19361}, {20,1204}, {24,22658}, {26,22550}, {30,22979}, {68,3546}, {104,22776}, {125,2888}, {186,2917}, {206,1614}, {376,22951}, {388,22980}, {497,22981}, {515,22653}, {631,22966}, {1181,22972}, {3085,22958}, {3086,22959}, {3091,22800}, {3448,12111}, {4293,18978}, {4294,22965}, {5603,22476}, {5657,22941}, {5876,18933}, {5925,12244}, {6353,22662}, {6623,22970}, {7487,22483}, {7581,19084}, {7582,19083}, {7967,22969}, {9862,22747}, {10783,22945}, {10784,22947}, {10785,22956}, {10786,22957}, {10788,22524}, {10805,22982}, {10806,22983}, {10938,13491}, {11411,18436}, {11431,22968}, {11457,12250}, {11491,22559}, {11799,18914}, {11845,22943}, {11846,22963}, {11847,22964}, {13886,22976}, {13939,22977}, {14216,22538}, {15532,18946}, {18925,18952}, {19467,22962}

X(22533) = reflection of X(4) in X(22466)
X(22533) = anticomplement of X(22955)

X(22534) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((3*R^2*(8*R^2-5*SW)+2*SW^2)*S^2+(512*R^6-8*R^4*(13*SA+34*SW)+R^2*SW*(54*SW+31*SA)-2*SA*SW^2-4*SW^3)*SA) : :
X(22534) = 3*X(2979)-2*X(22528) = 3*X(3060)-4*X(22970) = 9*X(5640)-8*X(22530) = 9*X(7998)-8*X(22581) = 5*X(11439)-4*X(22538) = 5*X(11444)-4*X(22834) = 15*X(11451)-16*X(22973) = 3*X(11459)-2*X(22808)

The reciprocal orthologic center of these triangles is X(9729).

X(22534) lies on these lines: {2,21652}, {3,22535}, {22,17837}, {25,5889}, {110,2929}, {1993,22497}, {2979,22528}, {3060,22970}, {5012,19460}, {5640,22530}, {7998,22581}, {11422,22529}, {11439,22538}, {11440,22549}, {11442,22555}, {11443,22830}, {11444,22834}, {11445,22840}, {11446,22954}, {11447,22960}, {11448,22961}, {11449,22962}, {11451,22973}, {11452,22974}, {11453,22975}, {11454,22978}, {11459,22808}, {12270,14683}, {12279,17845}, {12280,13598}, {12825,22979}, {18392,22816}, {18911,18936}, {19122,19142}, {19167,19198}, {19367,19472}, {19412,19488}, {19413,19489}

X(22534) = anticomplement of X(21652)

X(22535) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((3*R^2*(8*R^2-5*SW)+2*SW^2)*S^2+(736*R^6-4*R^4*(26*SA+89*SW)+31*R^2*SW*(2*SW+SA)-2*SA*SW^2-4*SW^3)*SA) : :
X(22535) = 5*X(3567)-4*X(22970) = 3*X(5890)-2*X(22750) = 7*X(7999)-8*X(22581) = 7*X(9781)-8*X(22530) = 3*X(11455)-4*X(22538) = 3*X(11459)-4*X(22834) = 17*X(11465)-16*X(22973)

The reciprocal orthologic center of these triangles is X(9729).

X(22535) lies on these lines: {3,22534}, {4,21652}, {24,17837}, {54,19460}, {74,22549}, {1614,2929}, {3567,22970}, {5890,22750}, {7592,22497}, {7999,22581}, {9781,22530}, {11412,21312}, {11423,22529}, {11455,22538}, {11456,22550}, {11457,12281}, {11458,22830}, {11459,22834}, {11460,22840}, {11461,22954}, {11462,22960}, {11463,22961}, {11464,22962}, {11465,22973}, {11466,22974}, {11467,22975}, {11468,22978}, {12111,22808}, {18394,22816}, {18912,18936}, {19123,19142}, {19168,19198}, {19368,19472}, {19414,19488}, {19415,19489}

X(22535) = reflection of X(i) in X(j) for these (i,j): (4, 21652), (11412, 22528), (12111, 22808)

X(22536) = PERSPECTOR OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 3rd ANTI-TRI-SQUARES

Barycentrics    S^4-(2*R^2*(5*SA-9*SW)-2*SA^2+3*SB*SC+4*SW^2)*S^2+S*(2*(SA-SW)*(17*R^2*SA+4*R^2*SW-SW^2)+(20*R^2+SA+3*SW)*S^2)-2*(4*R^2+SW)*SB*SC*SW : :

X(22536) lies on these lines: {372,22553}, {22588,22644}

X(22537) = PERSPECTOR OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 4th ANTI-TRI-SQUARES

Barycentrics    S^4-(2*R^2*(5*SA-9*SW)-2*SA^2+3*SB*SC+4*SW^2)*S^2-S*(2*(SA-SW)*(17*R^2*SA+4*R^2*SW-SW^2)+(20*R^2+SA+3*SW)*S^2)-2*(4*R^2+SW)*SB*SC*SW : :

X(22537) lies on these lines: {371,22554}, {22615,22619}

X(22538) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS TO 3rd HATZIPOLAKIS

Barycentrics    SB*SC*(2*S^2-8*R^2*(40*R^2+SA-16*SW)+SA^2-2*SB*SC-13*SW^2) : :
X(22538) = 3*X(4)-X(22750) = 5*X(3091)-4*X(22973) = 5*X(11439)-X(22534) = 3*X(11455)+X(22535) = 2*X(22750)-3*X(22970)

The reciprocal orthologic center of these triangles is X(9729).

X(22538) lies on these lines: {4,801}, {20,22581}, {24,22978}, {25,22549}, {30,22834}, {33,19472}, {34,22954}, {64,13399}, {125,1885}, {185,22530}, {378,22962}, {382,22808}, {1498,19460}, {1593,2929}, {1595,15432}, {1597,22550}, {3091,22973}, {3146,22528}, {7507,22971}, {9927,11472}, {10151,22966}, {11381,21652}, {11403,22497}, {11424,22529}, {11439,22534}, {11442,11469}, {11455,22535}, {11470,22830}, {11471,22840}, {11473,22960}, {11474,22961}, {11475,22974}, {11476,22975}, {12134,12295}, {12162,21651}, {12293,22979}, {12324,18936}, {13473,16656}, {13488,18488}, {14216,22533}, {15811,17837}, {19124,19142}, {19169,19198}, {19416,19488}, {19417,19489}

X(22538) = midpoint of X(i) and X(j) for these {i,j}: {382, 22808}, {3146, 22528}, {11381, 21652}
X(22538) = reflection of X(i) in X(j) for these (i,j): (20, 22581), (185, 22530)

X(22539) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO 4th BROCARD

Barycentrics    (SB+SC)*(11*S^4+(84*R^2-20*SW)*S^3+(4*R^2*(27*R^2-75*SA-13*SW)+11*SA^2+56*SA*SW-10*SB*SC+6*SW^2)*S^2+(-8*R^2*(1458*R^4-1080*R^2*SW+269*SW^2)+180*SW^3)*S+4*(9*R^2-2*SW)*(9*R^2*(-2*SW+3*SA)-7*SA^2+7*SB*SC+5*SW^2)*SA) : :

The reciprocal cyclologic center of these triangles is X(22540).

X(22539) lies on the line {6,22542}

X(22540) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 4th BROCARD TO ANTI-INNER-GREBE

Barycentrics    15*S^4+4*(3*R^2*(27*R^2-6*SA-7*SW)+4*SA^2-5*SB*SC+SW^2)*S^2+(SA-SW)*(18*R^2+SA-4*SW)*SW^2 : :

The reciprocal cyclologic center of these triangles is X(22539).

X(22540) lies on the orthocentroidal circle and these lines: {}

X(22541) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 1st TRI-SQUARES-CENTRAL

Barycentrics    10*S^2-15*(SA-SW)*S+3*(3*SA-2*SW)*(SA-SW) : :

The reciprocal orthologic center of these triangles is X(13665).

X(22541) lies on these lines: {2,13662}, {6,1327}, {30,19103}, {371,13666}, {1384,13712}, {1588,13687}, {3068,13701}, {3299,13715}, {3301,13714}, {5410,13668}, {6417,13713}, {7581,13674}, {7583,13692}, {7585,13678}, {7586,13988}, {7969,13702}, {11055,13669}, {11147,12159}, {13651,13846}, {13665,22806}, {13667,18992}, {13672,18994}, {13675,19000}, {13679,19004}, {13680,19006}, {13682,19008}, {13683,19010}, {13685,19012}, {13688,13883}, {13689,19018}, {13693,19024}, {13694,19026}, {13695,19028}, {13696,19030}, {13697,19032}, {13698,19034}, {13699,19038}, {13711,22616}, {13716,19048}, {13717,19050}, {13770,13932}, {18986,18996}, {19014,22783}

X(22541) = {X(6), X(1327)}-harmonic conjugate of X(19099)

X(22542) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 4th BROCARD

Barycentrics    (SB+SC)*(11*S^4-(84*R^2-20*SW)*S^3+(4*R^2*(27*R^2-75*SA-13*SW)+11*SA^2+56*SA*SW-10*SB*SC+6*SW^2)*S^2-(-8*R^2*(1458*R^4-1080*R^2*SW+269*SW^2)+180*SW^3)*S+4*(9*R^2-2*SW)*(9*R^2*(3*SA-2*SW)-7*SA^2+7*SB*SC+5*SW^2)*SA) : :

The reciprocal cyclologic center of these triangles is X(22543).

X(22542) lies on the line {6,22539}

X(22543) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO EHRMANN-VERTEX

Barycentrics    a^2*(a^38-3*(b^2+c^2)*a^36-(b^4-3*b^2*c^2+c^4)*a^34+(b^2+c^2)*(10*b^4+b^2*c^2+10*c^4)*a^32-2*(2*b^8+2*c^8+b^2*c^2*(3*b^4-4*b^2*c^2+3*c^4))*a^30-(b^2+c^2)*(10*b^8+10*c^8+b^2*c^2*(13*b^4+33*b^2*c^2+13*c^4))*a^28+(4*b^12+4*c^12+(7*b^8+7*c^8-b^2*c^2*(17*b^4+11*b^2*c^2+17*c^4))*b^2*c^2)*a^26+(b^2+c^2)*(2*b^12+2*c^12+(28*b^8+28*c^8+b^2*c^2*(39*b^4+31*b^2*c^2+39*c^4))*b^2*c^2)*a^24+(6*b^16+6*c^16-(24*b^12+24*c^12-(15*b^8+15*c^8+b^2*c^2*(57*b^4+85*b^2*c^2+57*c^4))*b^2*c^2)*b^2*c^2)*a^22-(b^2+c^2)*(11*b^12+11*c^12+(42*b^8+42*c^8+b^2*c^2*(17*b^4+20*b^2*c^2+17*c^4))*b^2*c^2)*b^2*c^2*a^20-(6*b^20+6*c^20-(47*b^16+47*c^16-(2*b^12+2*c^12+(65*b^8+65*c^8+4*b^2*c^2*(21*b^4+19*b^2*c^2+21*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^18-2*(b^2+c^2)*(b^20+c^20+2*(3*b^16+3*c^16-(19*b^12+19*c^12-(4*b^8+4*c^8+b^2*c^2*(b^4+9*b^2*c^2+c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^16-2*(2*b^20+2*c^20+(19*b^16+19*c^16-5*(7*b^12+7*c^12-(3*b^8+3*c^8-2*b^2*c^2*(3*b^4-4*b^2*c^2+3*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*(b^2+c^2)^2*a^14+(b^4-c^4)*(b^2-c^2)*(10*b^20+10*c^20+(17*b^16+17*c^16-(61*b^12+61*c^12+(102*b^8+102*c^8+b^2*c^2*(111*b^4+82*b^2*c^2+111*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12+(b^4-c^4)^2*(4*b^20+4*c^20+3*(11*b^16+11*c^16+(5*b^12+5*c^12-(13*b^8+13*c^8+b^2*c^2*(13*b^4+20*b^2*c^2+13*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10-(b^4-c^4)^2*(b^2+c^2)*(10*b^20+10*c^20-(16*b^16+16*c^16+(27*b^12+27*c^12-(5*b^8+5*c^8+b^2*c^2*(7*b^4+34*b^2*c^2+7*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(b^4-c^4)^2*(b^24+c^24-(20*b^20+20*c^20-(5*b^16+5*c^16+(37*b^12+37*c^12+3*(6*b^8+6*c^8+b^2*c^2*(b^4-24*b^2*c^2+c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^6+(b^2-c^2)^4*(b^2+c^2)^5*(3*b^16+3*c^16-(5*b^12+5*c^12+(4*b^8+4*c^8+3*(b^4-4*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-(b^2-c^2)^6*(b^2+c^2)^4*(b^16+c^16-(4*b^12+4*c^12+b^2*c^2*(b^4+4*b^2*c^2+c^4)*(2*b^4-3*b^2*c^2+2*c^4))*b^2*c^2)*a^2-(b^2-c^2)^8*b^2*c^2*(b^2+c^2)^5*(b^8+c^8)) : :

The reciprocal cyclologic center of these triangles is X(22544).

X(22543) lies on the line {19130,22544}

X(22544) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO ANTI-HONSBERGER

Barycentrics    a^42-4*(b^2+c^2)*a^40+(b^4+9*b^2*c^2+c^4)*a^38+(b^2+c^2)*(15*b^4-4*b^2*c^2+15*c^4)*a^36-2*(8*b^8+8*c^8+3*b^2*c^2*(5*b^4+b^2*c^2+5*c^4))*a^34-(b^2+c^2)*(17*b^8+17*c^8+b^2*c^2*(b^4+73*b^2*c^2+c^4))*a^32+(28*b^12+28*c^12+(36*b^8+36*c^8+b^2*c^2*(44*b^4+101*b^2*c^2+44*c^4))*b^2*c^2)*a^30+(b^2+c^2)*(4*b^12+4*c^12+(50*b^8+50*c^8+b^2*c^2*(91*b^4+60*b^2*c^2+91*c^4))*b^2*c^2)*a^28-(10*b^16+10*c^16+(50*b^12+50*c^12+(55*b^8+55*c^8+b^2*c^2*(83*b^4+41*b^2*c^2+83*c^4))*b^2*c^2)*b^2*c^2)*a^26-(b^2+c^2)*(4*b^16+4*c^16+(88*b^12+88*c^12+(69*b^8+69*c^8+b^2*c^2*(127*b^4+115*b^2*c^2+127*c^4))*b^2*c^2)*b^2*c^2)*a^24-(10*b^20+10*c^20-(114*b^16+114*c^16-(10*b^12+10*c^12+b^2*c^2*(18*b^4-13*b^2*c^2+18*c^4)*(3*b^4+4*b^2*c^2+3*c^4))*b^2*c^2)*b^2*c^2)*a^22+(b^2+c^2)*(10*b^20+10*c^20+(30*b^16+30*c^16+(163*b^12+163*c^12+(53*b^8+53*c^8+4*b^2*c^2*(5*b^4+28*b^2*c^2+5*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^20+(4*b^24+4*c^24-(146*b^20+146*c^20-(17*b^16+17*c^16+(122*b^12+122*c^12+(139*b^8+139*c^8+2*b^2*c^2*(53*b^4+30*b^2*c^2+53*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^18+(b^2+c^2)*(10*b^24+10*c^24+(26*b^20+26*c^20-(261*b^16+261*c^16-(92*b^12+92*c^12+(21*b^8+21*c^8-2*b^2*c^2*(b^4+6*b^2*c^2+c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^16-(4*b^24+4*c^24-(116*b^20+116*c^20-(176*b^16+176*c^16-(69*b^12+69*c^12-(135*b^8+135*c^8-b^2*c^2*(233*b^4-278*b^2*c^2+233*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*(b^2+c^2)^2*a^14-(b^4-c^4)*(b^2-c^2)*(28*b^24+28*c^24+(58*b^20+58*c^20-(109*b^16+109*c^16+(154*b^12+154*c^12+(123*b^8+123*c^8+4*b^2*c^2*(19*b^4+32*b^2*c^2+19*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12+(b^4-c^4)^2*(17*b^24+17*c^24-(78*b^20+78*c^20+(27*b^16+27*c^16-(85*b^12+85*c^12+(3*b^8+3*c^8+b^2*c^2*(51*b^4-62*b^2*c^2+51*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10+(b^4-c^4)^3*(b^2-c^2)*(16*b^20+16*c^20+(24*b^16+24*c^16-(27*b^12+27*c^12+(41*b^8+41*c^8+b^2*c^2*(13*b^4+30*b^2*c^2+13*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8-(b^2-c^2)^6*(b^2+c^2)^2*(15*b^20+15*c^20+(7*b^16+7*c^16-(41*b^12+41*c^12+2*(42*b^8+42*c^8+b^2*c^2*(48*b^4+41*b^2*c^2+48*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^8*(b^2+c^2)^3*(b^16+c^16+(b^4+b^2*c^2+c^4)*(16*b^8+16*c^8+b^2*c^2*(21*b^4+8*b^2*c^2+21*c^4))*b^2*c^2)*a^4+(b^2-c^2)^10*(b^2+c^2)^4*(4*b^12+4*c^12+(8*b^8+8*c^8+b^2*c^2*(13*b^4+12*b^2*c^2+13*c^4))*b^2*c^2)*a^2-(b^4+b^2*c^2+c^4)*(b^2-c^2)^12*(b^2+c^2)^5*(b^4+c^4) : :

The reciprocal cyclologic center of these triangles is X(22543).

X(22544) lies on the line {19130,22543}

X(22545) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO 2nd EHRMANN

Barycentrics    a^2*(a^18-4*(b^2+c^2)*a^16-(b^4-20*b^2*c^2+c^4)*a^14+(b^2+c^2)*(13*b^4-28*b^2*c^2+13*c^4)*a^12-(2*b^8+2*c^8+b^2*c^2*(28*b^4-45*b^2*c^2+28*c^4))*a^10-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*(7*b^4-12*b^2*c^2+7*c^4)*a^8+(3*b^12+3*c^12-b^4*c^4*(29*b^4-56*b^2*c^2+29*c^4))*a^6+(b^4-c^4)*(b^2-c^2)*(5*b^8+5*c^8-2*b^2*c^2*(10*b^4-11*b^2*c^2+10*c^4))*a^4-(b^4-c^4)^2*(b^8+c^8-b^2*c^2*(8*b^4-21*b^2*c^2+8*c^4))*a^2-(b^4-c^4)^2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*b^2*c^2) : :

The reciprocal cyclologic center of these triangles is X(22546).

X(22545) lies on the line {6,22546}

X(22546) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO ANTI-HONSBERGER

Barycentrics    a^2*(a^18-6*(b^2+c^2)*a^16+(7*b^4+36*b^2*c^2+7*c^4)*a^14+(b^2+c^2)*(17*b^4-72*b^2*c^2+17*c^4)*a^12-(26*b^8+26*c^8+b^2*c^2*(28*b^4-109*b^2*c^2+28*c^4))*a^10-(b^2+c^2)*(14*b^8+14*c^8-3*b^2*c^2*(43*b^4-66*b^2*c^2+43*c^4))*a^8+(27*b^12+27*c^12-(56*b^8+56*c^8+23*b^2*c^2*(3*b^4-8*b^2*c^2+3*c^4))*b^2*c^2)*a^6+(b^2+c^2)*(b^12+c^12-(46*b^8+46*c^8-b^2*c^2*(163*b^4-244*b^2*c^2+163*c^4))*b^2*c^2)*a^4-3*(b^4-c^4)^2*(3*b^8+3*c^8-b^2*c^2*(16*b^4-23*b^2*c^2+16*c^4))*a^2+(b^4-c^4)^3*(b^2-c^2)*(2*b^4-7*b^2*c^2+2*c^4)) : :

The reciprocal cyclologic center of these triangles is X(22545).

X(22546) lies on the line {6,22545}

X(22547) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO TRINH

Barycentrics    a^2*(4*a^26-2*(b^2+c^2)*a^24-(12*b^4-53*b^2*c^2+12*c^4)*a^22-(b^2+c^2)*(2*b^4+11*b^2*c^2+2*c^4)*a^20+(20*b^8+20*c^8-b^2*c^2*(8*b^2-b*c-8*c^2)*(8*b^2+b*c-8*c^2))*a^18+4*(b^2+c^2)*(3*b^8+3*c^8-2*b^2*c^2*(7*b^4-8*b^2*c^2+7*c^4))*a^16-2*(12*b^12+12*c^12+(b^8+c^8+6*(b^4-3*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*a^14-2*(b^2+c^2)*(2*b^12+2*c^12-(55*b^8+55*c^8-b^2*c^2*(115*b^4-132*b^2*c^2+115*c^4))*b^2*c^2)*a^12+2*(6*b^16+6*c^16-(6*b^12+6*c^12+(33*b^8+33*c^8-2*b^2*c^2*(24*b^4-11*b^2*c^2+24*c^4))*b^2*c^2)*b^2*c^2)*a^10-2*(b^4-c^4)*(b^2-c^2)^3*(5*b^8+5*c^8+13*b^2*c^2*(2*b^4+3*b^2*c^2+2*c^4))*a^8+(b^4-c^4)^2*(4*b^12+4*c^12+(21*b^8+21*c^8+2*b^2*c^2*(12*b^4-b^2*c^2+12*c^4))*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)*(6*b^16+6*c^16-(15*b^12+15*c^12-(60*b^8+60*c^8-b^2*c^2*(73*b^4-76*b^2*c^2+73*c^4))*b^2*c^2)*b^2*c^2)*a^4-(b^4-c^4)^2*(4*b^16+4*c^16-(4*b^12+4*c^12-(63*b^8+63*c^8+2*b^2*c^2*(8*b^4+41*b^2*c^2+8*c^4))*b^2*c^2)*b^2*c^2)*a^2-2*(b^2-c^2)^4*b^2*c^2*(b^2+c^2)^5*(2*b^4-b^2*c^2+2*c^4)) : :

The reciprocal cyclologic center of these triangles is X(22548).

X(22547) lies on the line {5092,22548}

X(22548) = CYCLOLOGIC CENTER OF THESE TRIANGLES: TRINH TO ANTI-HONSBERGER

Barycentrics    a^2*(4*a^28-10*(b^2+c^2)*a^26-(4*b^4-77*b^2*c^2+4*c^4)*a^24+(b^2+c^2)*(28*b^4-141*b^2*c^2+28*c^4)*a^22-(24*b^8+24*c^8-b^2*c^2*(15*b^4+353*b^2*c^2+15*c^4))*a^20+(b^2+c^2)*(14*b^8+14*c^8-b^2*c^2*(25*b^4+174*b^2*c^2+25*c^4))*a^18+(4*b^12+4*c^12-(24*b^8+24*c^8-b^2*c^2*(95*b^4+248*b^2*c^2+95*c^4))*b^2*c^2)*a^16-(b^2+c^2)*(56*b^12+56*c^12-(318*b^8+318*c^8-b^2*c^2*(631*b^4-690*b^2*c^2+631*c^4))*b^2*c^2)*a^14+(52*b^16+52*c^16-(178*b^12+178*c^12+(101*b^8+101*c^8-2*b^2*c^2*(62*b^4-73*b^2*c^2+62*c^4))*b^2*c^2)*b^2*c^2)*a^12+(b^2+c^2)*(18*b^16+18*c^16-(66*b^12+66*c^12-(401*b^8+401*c^8-2*b^2*c^2*(269*b^4-217*b^2*c^2+269*c^4))*b^2*c^2)*b^2*c^2)*a^10-(28*b^20+28*c^20+(19*b^16+19*c^16+(169*b^12+169*c^12+(164*b^8+164*c^8-b^2*c^2*(157*b^4+542*b^2*c^2+157*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8-(b^4-c^4)*(b^2-c^2)*(4*b^16+4*c^16+(25*b^12+25*c^12-(321*b^8+321*c^8+b^2*c^2*(349*b^4+782*b^2*c^2+349*c^4))*b^2*c^2)*b^2*c^2)*a^6+(b^4-c^4)^2*(131*b^12+131*c^12-(200*b^8+200*c^8-b^2*c^2*(47*b^4-460*b^2*c^2+47*c^4))*b^2*c^2)*b^2*c^2*a^4+(b^4-c^4)*(b^2-c^2)^3*(10*b^16+10*c^16-(29*b^12+29*c^12+(167*b^8+167*c^8+b^2*c^2*(263*b^4+398*b^2*c^2+263*c^4))*b^2*c^2)*b^2*c^2)*a^2-2*(b^4+b^2*c^2+c^4)*(b^2-c^2)^6*(b^2+c^2)^2*(2*b^8+2*c^8+b^2*c^2*(7*b^4+6*b^2*c^2+7*c^4))) : :

The reciprocal cyclologic center of these triangles is X(22547).

X(22548) lies on the line {5092,22547}

X(22549) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((-5*R^2+SA+SW)*S^2+(2*R^2*(40*R^2-2*SA-13*SW)+SA^2-SB*SC+2*SW^2)*SA) : :
X(22549) = 3*X(3)-X(22550) = 3*X(3)-2*X(22962) = 4*X(5)-3*X(22971) = 3*X(2929)-2*X(22550) = 3*X(2929)-4*X(22962) = X(2929)-4*X(22978) = 3*X(5085)-2*X(19142) = X(22550)-6*X(22978)

The reciprocal orthologic center of these triangles is X(9729).

X(22549) lies on these lines: {3,2929}, {5,22971}, {20,10117}, {25,22538}, {55,19472}, {56,22954}, {64,394}, {68,10264}, {74,22535}, {141,3520}, {185,19460}, {378,22750}, {382,22816}, {1092,2935}, {1151,22960}, {1152,22961}, {1204,21652}, {1350,15073}, {1593,22970}, {2071,2888}, {2917,16163}, {3516,22497}, {3964,15874}, {5085,19142}, {5584,22840}, {5646,7503}, {5925,9914}, {6101,12163}, {9786,22530}, {10620,18436}, {11425,22529}, {11440,22534}, {11472,12084}, {11477,22830}, {11479,22973}, {11480,22974}, {11481,22975}, {12162,18859}, {12307,15644}, {13021,19488}, {13022,19489}, {14130,14926}, {15068,22585}, {15622,22559}, {17928,22833}, {18913,18936}, {19172,19198}

X(22549) = midpoint of X(i) and X(j) for these {i,j}: {20, 22555}, {64, 17837}
X(22549) = reflection of X(i) in X(j) for these (i,j): (3, 22978), (382, 22816), (11477, 22830)
X(22549) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22550, 22962), (22550, 22962, 2929)

X(22550) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((10*R^2-2*SA-2*SW)*S^2+(R^2*(8*R^2+8*SA-11*SW)-2*SA^2+2*SB*SC+2*SW^2)*SA) : :
X(22550) = 3*X(3)-2*X(22549) = 3*X(3)-4*X(22962) = 5*X(3)-4*X(22978) = 3*X(2929)-X(22549) = 3*X(2929)-2*X(22962) = 5*X(2929)-2*X(22978) = 5*X(3843)-4*X(22816) = 5*X(3843)-6*X(22971) = 3*X(5050)-4*X(19142) = 5*X(11482)-4*X(22830) = 5*X(22549)-6*X(22978) = 2*X(22816)-3*X(22971) = 5*X(22962)-3*X(22978)

The reciprocal orthologic center of these triangles is X(9729).

X(22550) lies on these lines: {3,2929}, {4,22497}, {5,22555}, {24,12310}, {25,5889}, {26,22533}, {52,12316}, {155,11557}, {999,19472}, {1181,21652}, {1351,7506}, {1593,7703}, {1597,22538}, {1598,22970}, {2070,9920}, {2904,3167}, {3295,22954}, {3311,22960}, {3312,22961}, {3517,12309}, {3527,6642}, {3843,22816}, {5050,19142}, {5446,15136}, {6644,22647}, {6759,17837}, {7387,11820}, {7517,12315}, {8780,9937}, {10306,22840}, {11414,22528}, {11426,22529}, {11432,22530}, {11456,22535}, {11482,22830}, {11484,22973}, {11485,22974}, {11486,22975}, {11801,12084}, {12308,18378}, {13346,13376}, {18914,18936}, {19173,19198}, {19347,19460}, {19418,19488}, {19419,19489}

X(22550) = reflection of X(3) in X(2929)
X(22550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2929, 22549, 22962), (22549, 22962, 3), (22816, 22971, 3843)

X(22551) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO 1st EXCOSINE

Barycentrics    (SB+SC)*(3*S^6+(-4*R^2*(56*R^2+5*SA-30*SW)+5*SA^2-SB*SC-16*SW^2)*S^4+2*(4*R^2-SW)*(4*R^2*(4*R^2*SW-4*SA^2-2*SW^2+7*SA*SW)+SW*(SW^2-9*SA*SW+6*SA^2))*S^2-(4*R^2-SW)^3*SA*SW^2) : :

The reciprocal cyclologic center of these triangles is X(22552).

X(22551) lies on these lines: {3,129}, {25,1298}, {130,1598}, {154,14673}, {1181,21661}, {1303,11414}, {6759,22552}, {9920,11641}, {13175,17834}

X(22552) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st EXCOSINE TO ANTI-INCIRCLE-CIRCLES

Barycentrics    (SB+SC)*(3*S^6+2*(8*R^2*(2*R^2-SA-SW)+2*SA^2-SB*SC+SW^2)*S^4-(4*R^2-SW)^2*(8*R^2*(6*SA-SW)-20*SA^2+4*SB*SC+SW^2)*S^2-2*(4*R^2-SW)^2*SB*SC*SW^2) : :

The reciprocal cyclologic center of these triangles is X(22551).

X(22552) lies on these lines: {6,130}, {25,21661}, {129,17814}, {394,1303}, {1181,1298}, {6759,22551}

X(22553) = PERSPECTOR OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND 3rd ANTI-TRI-SQUARES

Barycentrics    S^4-(6*R^2*(SA-SW)-SB*SC-SW^2)*S^2-S*((SA-SW)*(2*SA*(9*R^2+SA)-2*SB*SC-SW^2)+(20*R^2+3*SA+SW)*S^2)+3*SB*SC*SW^2 : :

X(22553) lies on these lines: {372,22536}, {1328,5491}, {6565,22589}

X(22554) = PERSPECTOR OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND 4th ANTI-TRI-SQUARES

Barycentrics    S^4-(6*R^2*(SA-SW)-SB*SC-SW^2)*S^2+S*((SA-SW)*(2*SA*(9*R^2+SA)-2*SB*SC-SW^2)+(20*R^2+3*SA+SW)*S^2)+3*SB*SC*SW^2 : :

X(22554) lies on these lines: {371,22537}, {1327,5490}, {6564,22620}

X(22555) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO 3rd HATZIPOLAKIS

Barycentrics    SA*((9*R^2-2*SA)*S^2-2*(SA-SW)*(R^2*(16*R^2-5*SA-8*SW)+SA^2-SB*SC+SW^2)) : :
X(22555) = 3*X(4)-4*X(22816) = 3*X(376)-4*X(22978) = 5*X(631)-4*X(22962) = 3*X(1992)-4*X(22830) = 5*X(3618)-4*X(19142) = 7*X(3832)-6*X(22971)

The reciprocal orthologic center of these triangles is X(9729).

X(22555) lies on these lines: {2,2929}, {4,801}, {5,22550}, {20,10117}, {68,18933}, {69,22466}, {376,22978}, {381,15436}, {388,19472}, {427,22497}, {497,22954}, {631,22962}, {1370,22528}, {1503,17837}, {1899,18936}, {1992,22830}, {2550,22840}, {3068,22960}, {3069,22961}, {3153,14516}, {3618,19142}, {3832,22971}, {4549,6643}, {5562,12325}, {6776,19460}, {7386,22581}, {7392,22973}, {11411,18436}, {11427,22529}, {11433,22530}, {11442,22534}, {11457,12281}, {11487,14128}, {11488,22974}, {11489,22975}, {12225,22658}, {12319,12902}, {15435,22968}, {19174,19198}, {19420,19488}, {19421,19489}, {20806,22972}

X(22555) = reflection of X(20) in X(22549)
X(22555) = anticomplement of X(2929)
X(22555) = {X(1899), X(21652)}-harmonic conjugate of X(18936)

X(22556) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 1st BROCARD-REFLECTED

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

The reciprocal orthologic center of these triangles is X(3).

X(22556) lies on these lines: {3,22680}, {35,22650}, {55,262}, {56,22713}, {100,6194}, {197,22655}, {511,4421}, {1376,15819}, {2782,12326}, {3295,22475}, {5687,22697}, {7697,11499}, {7709,11491}, {10310,22676}, {11248,12339}, {11383,22480}, {11490,22521}, {11492,22668}, {11493,22672}, {11494,22678}, {11496,22682}, {11497,22699}, {11498,22700}, {11500,22704}, {11501,22705}, {11502,22706}, {11503,22709}, {11504,22710}, {11507,22729}, {11508,22730}, {11509,18971}, {11510,22732}, {11848,22698}, {11849,22728}, {12178,12329}, {13887,22720}, {13940,22721}, {18491,22681}, {18999,19063}, {19000,19064}

X(22557) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO INNER-FERMAT

Barycentrics    a*(2*sqrt(3)*(a^3-(b+c)*a^2+(b+c)^2*a-(b+c)*(b^2+c^2))*S*a+5*a^6+10*a^3*b^3+10*a^3*b^2*c+10*a^3*b*c^2+10*c^3*a^3+4*b^5*c-8*c^3*b^3+4*b*c^5-5*(b+c)*a^5-2*(5*b^2-b*c+5*c^2)*a^4+(5*b^4-6*b^3*c-10*b^2*c^2-6*b*c^3+5*c^4)*a^2+(-5*b^5-5*b^4*c+10*b^3*c^2+10*b^2*c^3-5*b*c^4-5*c^5)*a) : :

The reciprocal orthologic center of these triangles is X(3).

X(22557) lies on these lines: {3,22771}, {18,55}, {35,22651}, {56,22867}, {100,628}, {197,22656}, {630,1376}, {1001,6674}, {3295,11740}, {5687,22851}, {5965,12329}, {10310,22843}, {11248,12336}, {11383,22481}, {11490,22522}, {11491,22531}, {11492,22669}, {11493,22673}, {11494,22745}, {11496,22831}, {11497,22853}, {11498,22854}, {11499,16627}, {11500,22858}, {11501,22859}, {11502,22860}, {11503,22863}, {11504,22864}, {11507,22884}, {11508,22885}, {11509,18972}, {11510,22887}, {11848,22852}, {11849,16628}, {13887,22876}, {13940,22877}, {18491,22794}, {18999,19069}, {19000,19072}

X(22558) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO OUTER-FERMAT

Barycentrics    a*(-2*sqrt(3)*(a^3-(b+c)*a^2+(b+c)^2*a-(b+c)*(b^2+c^2))*S*a+5*a^6+10*a^3*b^3+10*a^3*b^2*c+10*a^3*b*c^2+10*c^3*a^3+4*b^5*c-8*c^3*b^3+4*b*c^5-5*(b+c)*a^5-2*(5*b^2-b*c+5*c^2)*a^4+(5*b^4-6*b^3*c-10*b^2*c^2-6*b*c^3+5*c^4)*a^2+(-5*b^5-5*b^4*c+10*b^3*c^2+10*b^2*c^3-5*b*c^4-5*c^5)*a) : :

The reciprocal orthologic center of these triangles is X(3).

X(22558) lies on these lines: {3,22772}, {17,55}, {35,22652}, {56,22912}, {100,627}, {197,22657}, {532,4421}, {629,1376}, {1001,6673}, {3295,11739}, {5687,22896}, {5965,12329}, {10310,22890}, {11248,12337}, {11383,22482}, {11490,22523}, {11491,22532}, {11492,22670}, {11493,22674}, {11494,22746}, {11496,22832}, {11497,22898}, {11498,22899}, {11499,16626}, {11500,22903}, {11501,22904}, {11502,22905}, {11503,22908}, {11504,22909}, {11507,22929}, {11508,22930}, {11509,18973}, {11510,22932}, {11848,22897}, {11849,16629}, {13887,22921}, {13940,22922}, {18491,22795}, {18999,19071}, {19000,19070}

X(22559) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 3rd HATZIPOLAKIS

Barycentrics    a*((7*(8*R^2-3*SW)*R^2+2*SW^2)*(b*c+(-a+b+c)*a)*S^2-((24*(3*R^2-SW)*R^2+2*SW^2-(5*R^2-SW)*SA)*S^2+(4*R^2-SW)*(16*R^2-3*SW)*SB*SC)*b*c) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22559) lies on these lines: {3,22776}, {35,22653}, {55,22466}, {56,22969}, {100,22647}, {197,22658}, {1376,22956}, {3295,22476}, {5687,22941}, {7074,22972}, {10310,22951}, {11383,22483}, {11490,22524}, {11491,22533}, {11494,22747}, {11496,22833}, {11497,22945}, {11498,22947}, {11499,22955}, {11500,22957}, {11501,22958}, {11502,22959}, {11503,22963}, {11504,22964}, {11507,22980}, {11508,22981}, {11509,18978}, {11510,22983}, {11848,22943}, {11849,22979}, {13887,22976}, {13940,22977}, {15622,22549}, {18491,22800}, {18999,19083}, {19000,19084}

X(22560) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ANTI-MANDART-INCIRCLE

Barycentrics    a^2*(-a+b+c)*(a^4-(2*b^2-3*b*c+2*c^2)*a^2-b*c*(b+c)*a+b^4+c^4-4*b^3*c+8*b^2*c^2-4*b*c^3) : :
X(22560) = 2*X(104)-3*X(11194) = 4*X(119)-3*X(11236) = 3*X(3928)-X(12767) = 2*X(10738)-3*X(11235)

The reciprocal cyclologic center of these triangles is X(13025).

X(22560) lies on these lines: {1,6596}, {3,2802}, {11,958}, {35,12653}, {36,2932}, {55,1320}, {56,100}, {63,17638}, {80,956}, {104,376}, {106,3939}, {119,11236}, {149,2975}, {153,529}, {214,999}, {405,16173}, {517,12332}, {518,6326}, {519,12331}, {952,11249}, {993,21630}, {1001,1387}, {1012,14217}, {1145,1376}, {1862,22479}, {2136,13144}, {2771,22583}, {2783,22504}, {2787,22514}, {2800,12330}, {2806,19162}, {2831,19159}, {3035,3085}, {3149,12751}, {3169,21773}, {3680,7280}, {3738,4491}, {3811,22935}, {3928,12767}, {4421,10269}, {5119,17652}, {5204,8668}, {5220,18254}, {5288,9897}, {5289,12740}, {5563,15015}, {5840,12114}, {6174,11239}, {6264,11012}, {6265,10680}, {6366,8301}, {6702,9708}, {6906,13463}, {6913,16174}, {8666,12773}, {8674,22586}, {8730,9945}, {9024,22769}, {10074,10609}, {10087,22766}, {10310,18861}, {10530,18962}, {10738,11235}, {11492,13230}, {11493,13228}, {12641,15180}, {13194,22520}, {13222,22654}, {13235,22744}, {13268,22755}, {13269,22756}, {13270,22757}, {13273,22759}, {13274,22760}, {13275,22761}, {13276,22762}, {13278,22768}, {13922,22763}, {13991,22764}, {18761,22938}, {19013,19112}, {19014,19113}

X(22560) = reflection of X(i) in X(j) for these (i,j): (149, 3813), (3811, 22935), (6264, 11260)
X(22560) = circumperp conjugate of X(14664)
X(22560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (36, 5541, 2932), (1145, 10090, 1376), (1320, 4996, 55)

X(22561) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND 1st BROCARD-REFLECTED

Barycentrics    27*S^6+9*(6*SA^2-5*SW^2)*S^4+(24*SA^2+48*SB*SC-7*SW^2)*SW^2*S^2+(2*SA^2+16*SB*SC+SW^2)*SW^4 : :

X(22561) lies on these lines: {3,8289}, {6,11152}, {99,22564}, {385,16508}, {2482,10810}, {5652,9485}, {8592,11317}

X(22562) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO ANTI-MCCAY

Barycentrics    3*S^4+(-6*SA^2+15*SB*SC+SW^2)*S^2+S*(SW*(3*SA^2+15*SB*SC+SW^2)+3*(3*SA-4*SW)*S^2)-9*SB*SC*SW^2 : :
X(22562) = 3*X(486)-2*X(13927)

The reciprocal orthologic center of these triangles is X(9891).

X(22562) lies on these lines: {486,490}, {542,6281}, {543,1328}, {642,8786}, {2482,12123}, {6054,9758}, {6290,10488}, {6319,22484}, {6561,9892}, {7840,22501}, {14645,22591}, {22563,22566}

X(22563) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO ANTI-MCCAY

Barycentrics    3*S^4+(-6*SA^2+15*SB*SC+SW^2)*S^2-S*(SW*(3*SA^2+15*SB*SC+SW^2)+3*(3*SA-4*SW)*S^2)-9*SB*SC*SW^2 : :
X(22563) = 3*X(485)-2*X(13874)

The reciprocal orthologic center of these triangles is X(9893).

X(22563) lies on these lines: {485,489}, {542,6278}, {543,1327}, {641,8786}, {2482,12124}, {6054,9757}, {6289,10488}, {6320,22485}, {6560,9894}, {7840,22502}, {14645,22592}, {22562,22566}

X(22564) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO 1st BROCARD-REFLECTED

Barycentrics    4*(b^2+c^2)*a^6-(b^4-5*b^2*c^2+c^4)*a^4-2*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2+(b^4-b^2*c^2+c^4)*b^2*c^2 : :
X(22564) = 6*X(5215)-5*X(7786)

The reciprocal orthologic center of these triangles is X(99).

X(22564) lies on these lines: {2,51}, {76,3849}, {98,10810}, {99,22561}, {187,7757}, {316,7848}, {385,5104}, {524,8592}, {538,8591}, {1003,9301}, {1916,8587}, {2076,14614}, {2080,4027}, {2782,9855}, {3329,8586}, {5162,9888}, {5215,7786}, {5976,7840}, {7793,13085}, {7833,9821}, {7883,18806}, {8290,12151}, {8598,11152}, {8704,9485}, {9772,9773}, {9889,11054}

X(22564) = reflection of X(316) in X(9466)

X(22565) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ANTI-MCCAY

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

The reciprocal orthologic center of these triangles is X(9855).

X(22565) lies on these lines: {3,12326}, {30,22504}, {36,9875}, {55,9884}, {56,671}, {104,12243}, {519,13173}, {542,12114}, {543,11194}, {956,9881}, {958,2482}, {993,8301}, {999,12258}, {2782,22680}, {2796,8666}, {2975,8591}, {3428,12117}, {5969,22779}, {8724,22758}, {9830,22769}, {9876,22654}, {9878,22744}, {9880,22753}, {9882,22756}, {9883,22757}, {10054,22766}, {10070,22767}, {10966,12354}, {11492,12346}, {11493,12345}, {11711,16418}, {12132,22479}, {12191,22520}, {12347,22755}, {12350,22759}, {12351,22760}, {12352,22761}, {12353,22762}, {12355,22765}, {12356,22768}, {13178,16371}, {13908,22763}, {13968,22764}, {18761,22566}, {19013,19057}, {19014,19058}

X(22566) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO ANTI-MCCAY

Barycentrics    3*S^4-(6*SA^2-15*SB*SC-SW^2)*S^2-9*SB*SC*SW^2 : :
X(22566) = 5*X(2)-X(9862) = 3*X(4)+X(8591) = 3*X(5)-2*X(5461) = 5*X(5)-2*X(11623) = 7*X(5)-4*X(20398) = 5*X(5461)-3*X(11623) = 7*X(5461)-6*X(20398) = 5*X(6033)+X(9862) = 2*X(6033)+X(12042) = 3*X(6033)+X(14830) = X(8591)-3*X(8724) = 2*X(9862)-5*X(12042) = 3*X(9862)-5*X(14830) = 7*X(11623)-10*X(20398) = 3*X(12042)-2*X(14830)

The reciprocal orthologic center of these triangles is X(9855).

X(22566) lies on these lines: {2,5191}, {3,11149}, {4,8591}, {5,542}, {30,114}, {98,5055}, {99,3830}, {115,5066}, {147,3545}, {262,381}, {376,15561}, {382,12117}, {543,3845}, {546,9880}, {547,6055}, {549,2794}, {550,20399}, {620,8703}, {625,5026}, {804,18309}, {1478,12351}, {1479,12350}, {2080,10487}, {2548,6034}, {2796,18483}, {3091,12243}, {3534,10722}, {3583,12354}, {3585,18969}, {3628,10991}, {3656,9864}, {3818,8176}, {3839,6321}, {3843,12355}, {3853,10992}, {5071,11177}, {5182,11318}, {5655,11005}, {5939,10033}, {5969,7775}, {5976,7809}, {6036,15699}, {6721,11539}, {7728,11006}, {7934,9774}, {8355,18800}, {8593,18440}, {8860,10104}, {9166,12188}, {9167,12100}, {9760,22570}, {9762,22568}, {9818,9876}, {9875,18492}, {9878,18500}, {9881,12699}, {9882,18509}, {9883,18511}, {9884,18525}, {9888,11184}, {9955,12258}, {10054,10895}, {10056,12185}, {10070,10896}, {10072,12184}, {10109,14971}, {10488,12177}, {10516,19905}, {12101,15300}, {12191,18502}, {12326,18491}, {12345,18495}, {12346,18497}, {12347,18507}, {12348,18516}, {12349,18517}, {12352,18520}, {12353,18522}, {12356,18542}, {12357,18544}, {13188,14269}, {13665,19058}, {13785,19057}, {13908,18538}, {13968,18762}, {15681,21166}, {18761,22565}, {22562,22563}

X(22566) = midpoint of X(i) and X(j) for these {i,j}: {2, 6033}, {4, 8724}, {99, 3830}, {147, 11632}, {382, 12117}, {3534, 10722}, {3656, 9864}, {5655, 11005}, {7728, 11006}, {8593, 18440}, {9881, 12699}, {9884, 18525}, {12347, 18507}
X(22566) = reflection of X(115) in X(5066)
X(22566) = complement of X(14830)
X(22566) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (147, 3545, 11632), (9166, 19709, 15092), (12188, 19709, 9166)

X(22567) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO INNER-FERMAT

Barycentrics    -2*(17*a^4-2*(b^2+c^2)*a^2-b^4-11*b^2*c^2-c^4)*S+sqrt(3)*(a^6+9*(b^2+c^2)*a^4-3*(3*b^4+b^2*c^2+3*c^4)*a^2+c^6+b^6) : :

The reciprocal orthologic center of these triangles is X(22568).

X(22567) lies on these lines: {2,18}, {8594,11054}, {16650,22568}

X(22568) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO ANTI-MCCAY

Barycentrics    3*(3*SA^2+6*SB*SC-SW^2)*SW-sqrt(3)*(3*SA-SW)^2*S : :
X(22568) = 2*X(2482)-3*X(9885) = 4*X(2482)-3*X(9886) = 3*X(9762)-2*X(22566) = 3*X(9885)-X(22570) = 3*X(9886)-2*X(22570)

The reciprocal orthologic center of these triangles is X(22567).

X(22568) lies on these lines: {2,99}, {3,22866}, {2936,3129}, {3104,5463}, {5464,6296}, {5979,22512}, {8724,16626}, {9115,10754}, {9762,22566}, {16650,22567}

X(22568) = anticomplement of X(33460)
X(22568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (671, 9888, 22570), (2482, 22570, 9886), (7618, 8596, 22570), (8591, 9890, 22570), (9885, 22570, 2482), (9892, 9894, 9885)

X(22569) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO OUTER-FERMAT

Barycentrics    9*(3*SA+SW)*S^2+sqrt(3)*S*(45*S^2+27*SA^2-36*SB*SC-14*SW^2)-3*(9*SB*SC+SW^2)*SW : :

The reciprocal orthologic center of these triangles is X(22570).

X(22569) lies on these lines: {2,17}, {8595,11054}, {16651,22570}

X(22570) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO ANTI-MCCAY

Barycentrics    3*(3*SA^2+6*SB*SC-SW^2)*SW+sqrt(3)*(3*SA-SW)^2*S : :
X(22570) = 4*X(2482)-3*X(9885) = 2*X(2482)-3*X(9886) = 3*X(9760)-2*X(22566) = 3*X(9885)-2*X(22568) = 3*X(9886)-X(22568)

The reciprocal orthologic center of these triangles is X(22569).

X(22570) lies on these lines: {2,99}, {3,22911}, {532,22509}, {2936,3130}, {3105,5464}, {5463,6297}, {5978,22513}, {8724,16627}, {9117,10754}, {9760,22566}, {16651,22569}

X(22570) = anticomplement of X(33461)
X(22570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (671, 9888, 22568), (2482, 22568, 9885), (7618, 8596, 22568), (8591, 9890, 22568), (9886, 22568, 2482), (9892, 9894, 9886)

X(22571) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO ANTI-MCCAY

Barycentrics    sqrt(3)*(9*SA-4*SW)*S^2-S*(24*S^2+12*SB*SC-11*SW^2+15*SA^2)+3*sqrt(3)*SB*SC*SW : :
X(22571) = 2*X(115)+X(22495) = X(5464)-4*X(11542) = 5*X(16960)+X(22577)

The reciprocal orthologic center of these triangles is X(8595).

X(22571) lies on these lines: {14,8584}, {17,9886}, {61,22575}, {115,22495}, {395,22490}, {524,5470}, {532,9166}, {543,16267}, {5464,11542}, {10991,16001}, {16631,22492}, {16808,22579}, {16960,22577}

X(22572) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    sqrt(3)*(9*SA-4*SW)*S^2+S*(24*S^2+12*SB*SC-11*SW^2+15*SA^2)+3*sqrt(3)*SB*SC*SW : :
X(22572) = 2*X(115)+X(22496) = X(5463)-4*X(11543) = 5*X(16961)+X(22578)

The reciprocal orthologic center of these triangles is X(8594).

X(22572) lies on these lines: {13,8584}, {18,9885}, {62,22576}, {115,22496}, {396,22489}, {524,5469}, {543,16268}, {5463,11543}, {10991,16002}, {16630,22491}, {16809,22580}, {16961,22578}

X(22573) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    9*S^2-sqrt(3)*(3*SA-SW)*S+6*SA^2+3*SB*SC-4*SW^2 : :
X(22573) = X(298)-3*X(9166) = X(5463)-3*X(22510) = X(5464)-3*X(16267) = X(8595)-3*X(8859) = X(9114)-5*X(16960) = 3*X(16529)+X(22578) = 3*X(16962)+X(22577)

The reciprocal orthologic center of these triangles is X(8595).

X(22573) lies on these lines: {14,1992}, {115,524}, {148,8594}, {298,9166}, {299,11054}, {381,22579}, {396,543}, {532,5460}, {2482,22892}, {3839,5479}, {5032,9113}, {5459,22691}, {5461,22847}, {5463,22510}, {5464,16267}, {5471,8584}, {5472,9830}, {6114,11163}, {6303,13757}, {6307,13637}, {6775,9763}, {8593,9112}, {8595,8859}, {8860,13084}, {9114,16960}, {9201,9979}, {9760,18582}, {9886,16644}, {10654,22575}, {11632,20425}, {13874,13876}, {13927,13929}, {16529,22578}, {16962,22577}

X(22573) = midpoint of X(i) and X(j) for these {i,j}: {14, 22495}, {148, 8594}, {299, 11054}, {11632, 20425}
X(22573) = X(16)-pedal-to-X(15)-pedal similarity image of X(2)

X(22574) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    9*S^2+sqrt(3)*(3*SA-SW)*S+6*SA^2+3*SB*SC-4*SW^2 : :
X(22574) = X(299)-3*X(9166) = X(5463)-3*X(16268) = X(5464)-3*X(22511) = X(8594)-3*X(8859) = X(9116)-5*X(16961) = 3*X(16530)+X(22577) = 3*X(16963)+X(22578)

The reciprocal orthologic center of these triangles is X(8594).

X(22574) lies on these lines: {13,1992}, {115,524}, {148,8595}, {298,11054}, {299,9166}, {381,22580}, {395,543}, {2482,22848}, {3839,5478}, {5032,9112}, {5460,22692}, {5461,22893}, {5463,16268}, {5464,22511}, {5471,9830}, {5472,8584}, {6115,11163}, {6302,13757}, {6306,13637}, {6772,9761}, {8593,9113}, {8594,8859}, {8860,13083}, {9116,16961}, {9200,9979}, {9762,18581}, {9885,16645}, {10653,22576}, {11632,20426}, {13874,13875}, {13927,13928}, {16530,22577}, {16963,22578}

X(22574) = midpoint of X(i) and X(j) for these {i,j}: {13, 22496}, {148, 8595}, {298, 11054}, {11632, 20426}
X(22574) = X(15)-pedal-to-X(16)-pedal similarity image of X(2)

X(22575) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    3*SW*S^2-sqrt(3)*S*(3*S^2-SW^2+3*SA^2+15*SB*SC)+9*SW*SB*SC : :
X(22575) = 2*X(13084)-3*X(22490)

The reciprocal orthologic center of these triangles is X(8595).

X(22575) lies on these lines: {2,11154}, {5,9886}, {13,11317}, {61,22571}, {114,381}, {115,11295}, {303,5464}, {316,22494}, {524,20429}, {598,11603}, {630,11303}, {5321,22579}, {5460,11489}, {5474,13083}, {5479,7620}, {8352,23004}, {9166,11299}, {10654,22573}, {11301,14971}, {11304,13084}, {16809,22577}

X(22575) = reflection of X(5474) in X(13083)

X(22576) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    3*SW*S^2+sqrt(3)*S*(3*S^2-SW^2+3*SA^2+15*SB*SC)+9*SW*SB*SC : :
X(22576) = 2*X(13083)-3*X(22489)

The reciprocal orthologic center of these triangles is X(8594).

X(22576) lies on these lines: {2,11153}, {5,9885}, {14,11317}, {62,22572}, {114,381}, {115,11296}, {302,5463}, {316,22493}, {524,20428}, {598,11602}, {629,11304}, {5318,22580}, {5459,11488}, {5473,13084}, {5478,7620}, {8352,23005}, {9166,11300}, {10653,22574}, {11302,14971}, {11303,13083}, {16808,22578}

X(22576) = reflection of X(5473) in X(13084)

X(22577) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    3*(3*SA-2*SW)*S^2-sqrt(3)*S*(6*S^2+30*SB*SC-7*SW^2+15*SA^2)+9*SW*SB*SC : :
X(22577) = 3*X(13)-2*X(5464) = 2*X(99)-3*X(22489) = 3*X(671)-2*X(5460) = 4*X(671)-3*X(5469) = 2*X(2482)-3*X(5470) = 4*X(5460)-3*X(5463) = 8*X(5460)-9*X(5469) = 2*X(5463)-3*X(5469) = 4*X(5464)-3*X(9114) = 3*X(16530)-4*X(22574) = 5*X(16960)-6*X(22571) = 3*X(16962)-4*X(22573)

The reciprocal orthologic center of these triangles is X(8595).

X(22577) lies on these lines: {13,543}, {18,671}, {99,22489}, {115,9116}, {382,542}, {2482,5470}, {5459,8591}, {5461,11312}, {5473,11632}, {8596,22113}, {9886,16966}, {16530,22574}, {16809,22575}, {16960,22571}, {16962,22573}, {16964,22579}, {22493,23004}

X(22577) = reflection of X(i) in X(j) for these (i,j): (5473, 11632), (22493, 23004)
X(22577) = {X(671), X(5463)}-harmonic conjugate of X(5469)

X(22578) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO ANTI-MCCAY

Barycentrics    3*(3*SA-2*SW)*S^2+sqrt(3)*S*(6*S^2+30*SB*SC-7*SW^2+15*SA^2)+9*SW*SB*SC : :
X(22578) = 3*X(14)-2*X(5463) = 2*X(99)-3*X(22490) = 3*X(671)-2*X(5459) = 4*X(671)-3*X(5470) = 2*X(2482)-3*X(5469) = 4*X(5459)-3*X(5464) = 8*X(5459)-9*X(5470) = 4*X(5463)-3*X(9116) = 2*X(5464)-3*X(5470) = 3*X(16529)-4*X(22573) = 5*X(16961)-6*X(22572) = 3*X(16963)-4*X(22574)

The reciprocal orthologic center of these triangles is X(8594).

X(22578) lies on these lines: {14,543}, {17,671}, {99,22490}, {115,9114}, {382,542}, {2482,5469}, {5460,8591}, {5461,11311}, {5474,11632}, {8596,22114}, {9885,16967}, {16529,22573}, {16808,22576}, {16961,22572}, {16963,22574}, {16965,22580}, {22494,23005}

X(22578) = reflection of X(i) in X(j) for these (i,j): (5474, 11632), (22494, 23005)
X(22578) = {X(671), X(5464)}-harmonic conjugate of X(5470)

X(22579) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO ANTI-MCCAY

Barycentrics    3*(3*SA-2*SW)*S^2-2*sqrt(3)*(3*SA-2*SW)*SW*S-9*SB*SC*SW : :
X(22579) = 2*X(141)-3*X(22490) = 2*X(5459)-3*X(6034) = 4*X(6670)-3*X(21358)

The reciprocal orthologic center of these triangles is X(8595).

X(22579) lies on these lines: {2,16940}, {4,542}, {14,524}, {15,9886}, {61,597}, {115,22491}, {141,22490}, {381,22573}, {396,9760}, {543,10654}, {599,636}, {3104,5463}, {3181,10754}, {5026,9114}, {5321,22575}, {5459,6034}, {5476,5613}, {6109,9761}, {6114,9763}, {6670,21358}, {8584,23004}, {11160,22114}, {11632,20426}, {15534,16942}, {16808,22571}, {16964,22577}

X(22579) = reflection of X(599) in X(5460)
X(22579) = {X(1992), X(20423)}-harmonic conjugate of X(22580)

X(22580) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO ANTI-MCCAY

Barycentrics    3*(3*SA-2*SW)*S^2+2*sqrt(3)*(3*SA-2*SW)*SW*S-9*SB*SC*SW : :
X(22580) = 2*X(141)-3*X(22489) = 2*X(5460)-3*X(6034) = 4*X(6669)-3*X(21358)

The reciprocal orthologic center of these triangles is X(8594).

X(22580) lies on these lines: {2,16941}, {4,542}, {13,524}, {16,9885}, {62,597}, {115,22492}, {141,22489}, {381,22574}, {395,9762}, {543,10653}, {599,635}, {3105,5464}, {3180,10754}, {5026,9116}, {5318,22576}, {5460,6034}, {5476,5617}, {6108,9763}, {6115,9761}, {6669,21358}, {8584,23005}, {11160,22113}, {11632,20425}, {15534,16943}, {16809,22572}, {16965,22578}

X(22580) = reflection of X(599) in X(5459)
X(22580) = {X(1992), X(20423)}-harmonic conjugate of X(22579)

X(22581) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO 3rd HATZIPOLAKIS

Barycentrics    (4*(5*R^2-SW)*S^2+(SA-SW)*(16*R^2-3*SW)*(8*R^2+SA-SW))*SA : :
X(22581) = 3*X(2)+X(22528) = 3*X(3)+X(22808) = 5*X(631)-X(22750) = 3*X(3917)+X(21652) = 9*X(7998)-X(22534) = 7*X(7999)+X(22535) = X(22528)+2*X(22973) = X(22808)-3*X(22834)

The reciprocal orthologic center of these triangles is X(9729).

X(22581) lies on these lines: {2,22528}, {3,2929}, {20,22538}, {69,18936}, {95,19198}, {141,22966}, {182,22529}, {394,19460}, {511,22530}, {631,22750}, {1038,19472}, {1040,22954}, {1368,5894}, {3357,3546}, {3548,22800}, {3917,21652}, {5907,6696}, {7386,22555}, {7484,22497}, {7998,22534}, {7999,22535}, {10319,22840}, {11487,22955}, {11511,22830}, {11513,22960}, {11514,22961}, {11515,22974}, {11516,22975}, {12358,20417}, {17811,17837}, {18531,22816}, {19126,19142}, {19422,19488}, {19423,19489}, {22467,22483}

X(22581) = midpoint of X(i) and X(j) for these {i,j}: {3, 22834}, {20, 22538}
X(22581) = anticomplement of X(22973)
X(22581) = complement of X(22970)
X(22581) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22528, 22970), (2, 22970, 22973)

X(22582) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((R^2*(551*R^2-66*SA-216*SW)+4*SW*(5*SW+3*SA))*S^2+3*(11*R^2-2*SW)*(R^2*(132*R^2-42*SW+SA)-2*SA^2+2*SB*SC+4*SW^2)*SA) : :

X(22582) lies on these lines: {399,13630}, {974,22585}

X(22583) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ANTI-ORTHOCENTROIDAL

Barycentrics    a^2*(a^11-(b+c)*a^10-(3*b^2-2*b*c+3*c^2)*a^9+(b+c)*(3*b^2-4*b*c+3*c^2)*a^8+(2*b^4+2*c^4-(3*b-2*c)*(2*b-3*c)*b*c)*a^7-(b+c)*(2*b^4+2*c^4-(8*b^2-13*b*c+8*c^2)*b*c)*a^6+2*(b^6+c^6+3*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*b*c)*a^5-2*(b^2-c^2)*(b-c)*(b^4+c^4+(2*b^2-3*b*c+2*c^2)*b*c)*a^4-(3*b^6+3*c^6+2*(4*b^4+4*c^4+(8*b^2+13*b*c+8*c^2)*b*c)*b*c)*(b-c)^2*a^3+(b^2-c^2)*(b-c)^3*(3*b^4+3*c^4+(4*b^2+b*c+4*c^2)*b*c)*a^2+(b^2-c^2)^2*(b^6+c^6+(7*b^2+2*b*c+7*c^2)*b^2*c^2)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c)^3) : :

The reciprocal orthologic center of these triangles is X(12112).

X(22583) lies on these lines: {1,2778}, {3,11720}, {30,19478}, {36,9904}, {55,7978}, {56,74}, {104,12244}, {110,3428}, {113,958}, {125,22753}, {146,2975}, {399,2779}, {517,13204}, {541,11194}, {542,22514}, {690,22504}, {956,12368}, {999,11709}, {1001,11723}, {1539,18761}, {2771,22560}, {2777,12114}, {2781,19162}, {3028,10966}, {3149,13211}, {5584,15035}, {5663,11249}, {7725,22756}, {7726,22757}, {7728,22758}, {8674,22775}, {8994,22763}, {9517,19159}, {9919,22654}, {9984,22744}, {10065,22766}, {10081,22767}, {10269,12041}, {10620,22765}, {10628,22781}, {11492,12366}, {11493,12365}, {12133,22479}, {12192,22520}, {12369,22755}, {12373,22759}, {12374,22760}, {12377,22761}, {12378,22762}, {12381,22768}, {13969,22764}, {17702,22659}, {19013,19059}, {19014,19060}

X(22584) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO ANTI-ORTHOCENTROIDAL

Barycentrics    (S^2-SB*SC)*(3*S^2+3*(21*R^2-4*SA-8*SW)*R^2+3*SA^2-2*SB*SC+2*SW^2) : :
X(22584) = 3*X(3)-4*X(12358) = 7*X(3)-8*X(13416) = 2*X(113)-3*X(18435) = 2*X(185)-3*X(15061) = 3*X(265)-2*X(21649) = 3*X(381)-2*X(1986) = 2*X(1511)-3*X(11459) = 3*X(7723)-2*X(12358) = 7*X(7723)-4*X(13416) = 3*X(11459)-X(12270) = 4*X(11591)-3*X(15035) = 3*X(12111)+X(15100) = 3*X(12281)-X(15100) = 7*X(12358)-6*X(13416) = X(21649)-3*X(21650)

The reciprocal orthologic center of these triangles is X(3581).

Let triangle A*B*C* be as described at X(7723). Then X(22584) = X(3)-of A*B*C*. (Randy Hutson, October 15, 2018)

X(22584) lies on these lines: {3,74}, {5,7722}, {30,12219}, {113,10254}, {125,5448}, {146,3410}, {185,15061}, {265,1531}, {381,1986}, {382,12292}, {542,18438}, {567,12227}, {568,7687}, {1112,3843}, {1154,10733}, {1539,7731}, {1656,14708}, {2072,10264}, {2777,18439}, {2781,18440}, {2914,7527}, {3043,18570}, {3448,18404}, {3627,6242}, {3830,12133}, {3851,13148}, {5055,9826}, {5504,11559}, {5562,12121}, {5889,10113}, {5890,20304}, {5907,11562}, {6000,20127}, {6102,14644}, {6243,12295}, {6288,7728}, {7574,18441}, {7724,18453}, {7727,18455}, {9818,12165}, {9976,18449}, {10317,14901}, {10540,13289}, {10657,18468}, {10658,18470}, {11557,15030}, {11561,15060}, {11806,15027}, {11807,16194}, {12273,18561}, {12290,13201}, {12317,18531}, {12375,18457}, {12376,18459}, {12429,12902}, {12901,22115}, {13630,15059}, {14130,15463}, {15043,15088}, {15760,21357}, {17702,18436}, {17835,18451}, {18445,19457}, {18447,19470}, {18462,19484}, {18463,19485}, {18563,22815}, {18917,18933}, {19129,19140}, {19176,19195}

X(22584) = midpoint of X(12290) and X(13201)
X(22584) = reflection of X(i) in X(j) for these (i,j): (3, 7723), (265, 21650), (382, 12292), (5889, 10113), (6243, 12295)
X(22584) = inverse of X(12412) in the Stammler circle
X(22584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (399, 10620, 12412), (5907, 11562, 14643), (7731, 15305, 1539), (11459, 12270, 1511)

X(22585) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((-R^2*(71*R^2-36*SA-24*SW)-2*SW*(3*SA+SW))*S^2+6*(6*R^2-SW)*(R^2*(16*R^2-5*SA-3*SW)+SA^2-SB*SC)*SA) : :

The reciprocal orthologic center of these triangles is X(974).

X(22585) lies on these lines: {6,17837}, {974,22582}, {1514,2914}, {2929,11456}, {15068,22549}

X(22586) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ANTI-ORTHOCENTROIDAL

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

The reciprocal parallelogic center of these triangles is X(323).

X(22586) lies on these lines: {3,11709}, {36,2948}, {55,7984}, {56,110}, {63,10693}, {74,3428}, {104,12383}, {113,22753}, {125,958}, {265,22758}, {399,22765}, {517,12327}, {542,11194}, {690,22514}, {952,12334}, {956,13211}, {993,13605}, {999,11720}, {1001,11735}, {1112,22479}, {1511,10269}, {2163,6126}, {2771,6261}, {2778,5709}, {2781,19159}, {2836,3576}, {2854,22769}, {2975,3448}, {3149,12368}, {3556,15647}, {3560,12261}, {5584,15055}, {5663,11249}, {7732,22756}, {7733,22757}, {8674,22560}, {8998,22763}, {9517,19162}, {10088,22766}, {10091,22767}, {10113,18761}, {11492,13209}, {11493,13208}, {12114,13213}, {12310,22654}, {12903,22759}, {12904,22760}, {13193,22520}, {13210,22744}, {13212,22755}, {13215,22761}, {13216,22762}, {13217,22768}, {13990,22764}, {19013,19110}, {19014,19111}, {19478,22781}

X(22587) = PERSPECTOR OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES AND CIRCUMORTHIC

Barycentrics    5*S^4-(2*R^2*(7*SA-9*SW)-4*SA^2+3*SB*SC+5*SW^2)*S^2-S*((SA-SW)*(2*R^2*(17*SA-2*SW)-6*SA^2+6*SB*SC+SW^2)+(44*R^2-5*SA-3*SW)*S^2)+(8*R^2-SW)*SB*SC*SW : :

X(22587) lies on these lines: {}

X(22588) = PERSPECTOR OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES AND EHRMANN-VERTEX

Barycentrics    S^4-(3*R^2*SW-SB*SC-SW^2)*S^2+S*((SA-SW)*(3*R^2*(12*SA-SW)-14*SA^2+14*SB*SC+SW^2)+(40*R^2-15*SA-SW)*S^2)-9*(3*R^2-SW)*SB*SC*SW : :

X(22588) lies on these lines: {1328,1989}, {22536,22644}

X(22589) = PERSPECTOR OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES AND ORTHIC

Barycentrics    (2*R^2+SA-SW)*S^2+S*((SA-SW)*(6*R^2-SA-2*SW)-2*S^2)+2*(9*R^2-2*SW)*SB*SC : :

X(22589) lies on these lines: {381,485}, {486,12240}, {6565,22553}, {16310,22620}

X(22590) = PERSPECTOR OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES AND INNER-SQUARES

Barycentrics    (4*R^2+8*SA-5*SW)*S^2+(SA-SW)*(12*R^2+8*SA-7*SW)*S+(36*R^2-13*SW)*SB*SC : :

X(22590) lies on the line {371,1328}

X(22591) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 4th ANTI-TRI-SQUARES

Barycentrics    (6*SA-3*SW)*S^2+S*((2*SA+SW)*(SA-SW)+4*S^2)+3*SB*SC*SW : :
X(22591) = 3*X(485)-4*X(13881) = 3*X(485)-2*X(22592) = 3*X(486)-2*X(13881) = 3*X(486)-X(22592)

The reciprocal orthologic center of these triangles is X(22592).

X(22591) lies on these lines: {5,6}, {487,641}, {488,6561}, {1328,2996}, {5420,12257}, {6337,13701}, {6565,12222}, {7612,10194}, {12601,22615}, {14645,22562}, {22625,22646}

X(22591) = midpoint of X(488) and X(12221)
X(22591) = reflection of X(487) in X(641)
X(22591) = reflection of X(22592) in X(13881)
X(22591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (486, 22592, 13881), (13881, 22592, 485)

X(22592) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 3rd ANTI-TRI-SQUARES

Barycentrics    3*(2*SA-SW)*S^2-(4*S^2+(2*SA+SW)*(SA-SW))*S+3*SB*SC*SW : :
X(22592) = 3*X(485)-2*X(13881) = 3*X(485)-X(22591) = 3*X(486)-4*X(13881) = 3*X(486)-2*X(22591)

The reciprocal orthologic center of these triangles is X(22591).

X(22592) lies on these lines: {5,6}, {487,6560}, {488,642}, {1327,2996}, {5418,12256}, {6337,13821}, {6564,12221}, {7612,10195}, {12602,22644}, {14645,22563}, {22596,22617}

X(22592) = midpoint of X(487) and X(12222)
X(22592) = reflection of X(488) in X(642)
X(22592) = reflection of X(22591) in X(13881)
X(22592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (485, 22591, 13881), (13881, 22591, 486)

X(22593) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4+(2*SA^2+7*SB*SC+SW^2)*S^2+3*(SA-SW)*(3*S^2-SW*(2*SA-SW))*S+7*SB*SC*SW^2 : :
X(22593) = 5*X(486)-2*X(3103) = 3*X(486)-2*X(22725) = 3*X(3103)-5*X(22725)

The reciprocal orthologic center of these triangles is X(22594).

X(22593) lies on these lines: {262,486}, {511,1328}, {6561,22726}, {12221,22614}, {13330,22622}

X(22593) = {X(13330), X(22681)}-harmonic conjugate of X(22622)

X(22594) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 3rd ANTI-TRI-SQUARES

Barycentrics    (3*SA-4*SW)*S^2+(SA^2+2*SB*SC+SW^2)*S+SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22593).

X(22594) lies on these lines: {2,371}, {3,22726}, {6,12217}, {83,6419}, {99,372}, {182,22716}, {194,6420}, {511,22718}, {575,3734}, {3311,14535}, {3564,6228}, {6033,6230}, {11174,22725}

X(22594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (486, 487, 6315), (575, 3734, 22623)

X(22595) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 3rd ANTI-TRI-SQUARES

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

The reciprocal orthologic center of these triangles is X(486).

X(22595) lies on these lines: {3,12343}, {30,22784}, {36,9906}, {55,7980}, {56,486}, {104,12256}, {487,2975}, {642,958}, {956,12787}, {993,8225}, {999,12268}, {3428,12123}, {3564,22624}, {6251,22753}, {6280,22757}, {6281,22756}, {6290,22758}, {9921,22654}, {9986,22744}, {10067,22766}, {10083,22767}, {10966,13081}, {11492,12485}, {11493,12484}, {12114,12928}, {12147,22479}, {12210,22520}, {12601,22765}, {12799,22755}, {12948,22759}, {12958,22760}, {13002,22761}, {13003,22762}, {13132,22768}, {13921,22763}, {13933,22764}, {18761,22596}, {19013,19104}, {19014,19105}

X(22596) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 3rd ANTI-TRI-SQUARES

Barycentrics    SA*S^2-(S^2+5*SB*SC)*S+3*SB*SC*SW : :
X(22596) = 3*X(4)+X(487) = 5*X(4)-X(12296) = 7*X(4)+X(12509) = 3*X(381)-X(486) = X(487)-3*X(6290) = 5*X(487)+3*X(12296) = 7*X(487)-3*X(12509) = 5*X(3091)-X(12256) = 9*X(3839)-X(12221) = 5*X(3843)+X(6281) = 5*X(3843)-X(12601) = 5*X(6290)+X(12296) = 7*X(6290)-X(12509) = 7*X(12296)+5*X(12509) = 3*X(22807)+2*X(22820)

The reciprocal orthologic center of these triangles is X(486).

X(22596) lies on these lines: {4,487}, {5,6119}, {30,642}, {381,486}, {382,12123}, {546,576}, {1478,12958}, {1479,12948}, {1597,12984}, {1598,12972}, {3091,12256}, {3583,13081}, {3585,18989}, {3839,12221}, {3843,6281}, {3861,22819}, {5395,14244}, {6280,18511}, {6560,13934}, {6564,7745}, {7980,18525}, {9758,15294}, {9818,9921}, {9906,18492}, {9955,12268}, {9986,18500}, {10067,10895}, {10083,10896}, {12210,18502}, {12343,18491}, {12484,18495}, {12485,18497}, {12699,12787}, {12799,18507}, {12928,18516}, {12938,18517}, {12978,18535}, {13002,18520}, {13003,18522}, {13132,18542}, {13133,18544}, {13665,19105}, {13711,21309}, {13785,19104}, {13921,18538}, {13933,18762}, {14233,22505}, {14269,22809}, {15765,22606}, {18585,22605}, {18761,22595}, {22592,22617}

X(22596) = midpoint of X(i) and X(j) for these {i,j}: {4, 6290}, {382, 12123}, {7980, 18525}, {12699, 12787}, {12799, 18507}
X(22596) = {X(546), X(3818)}-harmonic conjugate of X(22625)

X(22597) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO INNER-FERMAT

Barycentrics    218*(3*a^4+4*(b^2+c^2)*a^2+(-2*(b^2+c^2)*a^2+6*(b^2-c^2)^2)*sqrt(3)-13*(b^2-c^2)^2)*S+(28-15*sqrt(3))*(109*a^6-275*(b^2+c^2)*a^4+(187*b^4-134*b^2*c^2+187*c^4)*a^2+(-50*(b^2+c^2)*a^4+2*(17*b^4-32*b^2*c^2+17*c^4)*a^2+16*(b^4-c^4)*(b^2-c^2))*sqrt(3)-21*(b^4-c^4)*(b^2-c^2)) : :
X(22597) = 3*X(486)-2*X(22881)

The reciprocal orthologic center of these triangles is X(22598).

X(22597) lies on these lines: {18,486}, {6561,22882}, {12221,22603}, {22626,22794}

X(22598) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 3rd ANTI-TRI-SQUARES

Barycentrics    39*S^2+(4*sqrt(3)+3)*(13*SB*SC+(13*SA-5*SW-2*sqrt(3)*SW)*S) : :
X(22598) = 4*X(642)-3*X(6300) = 2*X(642)-3*X(6301) = 3*X(6300)-2*X(22600) = 3*X(6301)-X(22600)

The reciprocal orthologic center of these triangles is X(22597).

X(22598) lies on these lines: {2,371}, {3,22882}, {3104,22610}, {3564,22629}, {5339,22627}, {5615,12601}, {6290,16626}, {16645,22881}

X(22598) = anticomplement of X(33449)
X(22598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (486, 487, 6301), (486, 12221, 22600), (642, 22600, 6300), (6301, 22600, 642)

X(22599) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO OUTER-FERMAT

Barycentrics    218*(3*a^4+4*a^2*(b^2+c^2)-sqrt(3)*(6*(b^2-c^2)^2-2*a^2*(b^2+c^2))-13*(b^2-c^2)^2)*S+(28+15*sqrt(3))*(109*a^6-275*(b^2+c^2)*a^4+a^2*(187*b^4-134*b^2*c^2+187*c^4)-sqrt(3)*(-50*(b^2+c^2)*a^4+16*(b^4-c^4)*(b^2-c^2)+2*a^2*(17*b^4-32*b^2*c^2+17*c^4))-21*(b^4-c^4)*(b^2-c^2)) : :
X(22599) = 3*X(486)-2*X(22926)

The reciprocal orthologic center of these triangles is X(22600).

X(22599) lies on these lines: {17,486}, {532,1328}, {6561,22927}, {12221,22601}, {22628,22795}

X(22600) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 3rd ANTI-TRI-SQUARES

Barycentrics    39*S^2+(3-4*sqrt(3))*((13*SA-5*SW+2*sqrt(3)*SW)*S+13*SB*SC) : :
X(22600) = 2*X(642)-3*X(6300) = 4*X(642)-3*X(6301) = 3*X(6300)-X(22598) = 3*X(6301)-2*X(22598)

The reciprocal orthologic center of these triangles is X(22599).

X(22600) lies on these lines: {2,371}, {3,22927}, {532,22919}, {3105,22609}, {3564,22627}, {5340,22629}, {5611,12601}, {6290,16627}, {16644,22926}

X(22600) = anticomplement of X(33451)
X(22600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (486, 487, 6300), (486, 12221, 22598), (642, 22598, 6301), (6300, 22598, 642)

X(22601) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 3rd FERMAT-DAO

Barycentrics    (10*sqrt(3)+27)*(33*SA-24*SW+4*sqrt(3)*SW)*S^2+99*(4*sqrt(3)+3)*SW*SB*SC-33*(-26*S^2+(2*sqrt(3)-5)*(3*SA+3*SW+2*sqrt(3)*SW)*(SA-SW))*S : :
X(22601) = 3*X(486)-2*X(13929)

The reciprocal orthologic center of these triangles is X(22602).

X(22601) lies on these lines: {13,486}, {148,22603}, {6302,6561}, {12221,22599}, {22605,22998}, {22630,22796}

X(22602) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    13*(1-sqrt(3))*(sqrt(3)*(2*SA-3*SW)+SW)*S^2-26*sqrt(3)*SB*SC*SW+(3*sqrt(3)+1)*(52*S^2+3*(4+sqrt(3))*(2*SA-SW+sqrt(3)*SW)*(SA-SW))*S : :
X(22602) = X(13)+2*X(13929) = 4*X(11542)-X(22609) = 5*X(16960)+X(22607)

The reciprocal orthologic center of these triangles is X(22601).

X(22602) lies on these lines: {13,486}, {17,6300}, {61,22605}, {5459,22631}, {11542,22609}, {16808,22611}, {16960,22607}

X(22603) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 4th FERMAT-DAO

Barycentrics    (27-10*sqrt(3))*(33*SA-24*SW-4*sqrt(3)*SW)*S^2-33*(-26*S^2+(-5-2*sqrt(3))*(SA-SW)*(3*SA+3*SW-2*sqrt(3)*SW))*S+99*(3-4*sqrt(3))*SB*SC*SW : :
X(22603) = 3*X(486)-2*X(13928)

The reciprocal orthologic center of these triangles is X(22604).

X(22603) lies on these lines: {14,486}, {148,22601}, {6303,6561}, {12221,22597}, {22606,22997}, {22632,22797}

X(22604) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    13*(1+sqrt(3))*(-sqrt(3)*(2*SA-3*SW)+SW)*S^2+(1-3*sqrt(3))*(52*S^2+3*(4-sqrt(3))*(SA-SW)*(2*SA-SW-sqrt(3)*SW))*S+26*sqrt(3)*SB*SC*SW : :
X(22604) = X(14)+2*X(13928) = 4*X(11543)-X(22610) = 5*X(16961)+X(22608)

The reciprocal orthologic center of these triangles is X(22603).

X(22604) lies on these lines: {14,486}, {18,6301}, {62,22606}, {5460,22633}, {11543,22610}, {16809,22612}, {16961,22608}

X(22605) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    (2+sqrt(3))*(-SA*sqrt(3)+SW)*S^2+sqrt(3)*(1+sqrt(3))*SW*SB*SC-(S^2+(9+4*sqrt(3))*SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(22601).

X(22605) lies on these lines: {5,6300}, {14,486}, {61,22602}, {381,1991}, {642,18586}, {3564,22635}, {5321,22611}, {6301,15765}, {10654,13929}, {16808,22609}, {16809,22607}, {18585,22596}, {22601,22998}

X(22605) = {X(381), X(6290)}-harmonic conjugate of X(22606)

X(22606) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    (2-sqrt(3))*(sqrt(3)*SA+SW)*S^2-(S^2+(9-4*sqrt(3))*SB*SC)*S-sqrt(3)*(1-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22603).

X(22606) lies on these lines: {5,6301}, {13,486}, {62,22604}, {381,1991}, {642,18587}, {3564,22634}, {5318,22612}, {6300,18585}, {10653,13928}, {15765,22596}, {16808,22608}, {16809,22610}, {22603,22997}

X(22606) = {X(381), X(6290)}-harmonic conjugate of X(22605)

X(22607) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    (sqrt(3)+3)*(2*SA-(-3+2*sqrt(3))*SW)*S^2+3*(-sqrt(3)+1)*SW*SB*SC+(8*S^2*sqrt(3)+(1+sqrt(3))*(SA-SW)*(6*SA+(-3+2*sqrt(3))*SW))*S : :
X(22607) = 3*X(13)-2*X(22609) = 4*X(13929)-3*X(16962) = 5*X(16960)-6*X(22602)

The reciprocal orthologic center of these triangles is X(22601).

X(22607) lies on these lines: {13,22609}, {18,486}, {621,6115}, {6300,16966}, {6565,9732}, {13929,16962}, {16809,22605}, {16960,22602}, {16964,22611}

X(22608) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    (3-sqrt(3))*(2*SA-(-3-2*sqrt(3))*SW)*S^2+(-8*sqrt(3)*S^2+(1-sqrt(3))*(SA-SW)*(6*SA+(-3-2*sqrt(3))*SW))*S+3*(1+sqrt(3))*SB*SC*SW : :
X(22608) = 3*X(14)-2*X(22610) = 4*X(13928)-3*X(16963) = 5*X(16961)-6*X(22604)

The reciprocal orthologic center of these triangles is X(22603).

X(22608) lies on these lines: {14,22610}, {17,486}, {622,6114}, {6301,16967}, {6565,9732}, {13928,16963}, {16808,22606}, {16961,22604}, {16965,22612}

X(22609) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    3*(3*sqrt(3)+4)*(22*SA+3*(-5+sqrt(3))*SW)*S^2+33*(sqrt(3)+3)*SW*SB*SC-(132*S^2+11*sqrt(3)*(SA-SW)*(-6*SA+SW*(sqrt(3)+3)))*S : :
X(22609) = 3*X(13)-X(22607) = 4*X(11542)-3*X(22602) = 2*X(13929)-3*X(16267)

The reciprocal orthologic center of these triangles is X(22601).

X(22609) lies on these lines: {13,22607}, {16,6300}, {17,486}, {61,22611}, {3105,22600}, {6290,6565}, {11542,22602}, {13929,16267}, {16808,22605}

X(22610) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    3*(4-3*sqrt(3))*(22*SA+3*(-5-sqrt(3))*SW)*S^2-(132*S^2-11*sqrt(3)*(SA-SW)*(-6*SA+(3-sqrt(3))*SW))*S+33*(3-sqrt(3))*SB*SC*SW : :
X(22610) = 3*X(14)-X(22608) = 4*X(11543)-3*X(22604) = 2*X(13928)-3*X(16268)

The reciprocal orthologic center of these triangles is X(22603).

X(22610) lies on these lines: {14,22608}, {15,6301}, {18,486}, {62,22612}, {3104,22598}, {6290,6565}, {11543,22604}, {13928,16268}, {16809,22606}

X(22611) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    (SA+sqrt(3)*SW)*S^2+(sqrt(3)+3)*SW*SB*SC-(2*S^2-(SA-SW)*(-3*SA+(1+sqrt(3))*SW))*S : :

The reciprocal orthologic center of these triangles is X(22601).

X(22611) lies on these lines: {4,372}, {15,6300}, {61,22609}, {381,13929}, {3104,22598}, {5321,22605}, {16808,22602}, {16964,22607}

X(22612) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 3rd ANTI-TRI-SQUARES

Barycentrics    (SA-sqrt(3)*SW)*S^2-(2*S^2-(SA-SW)*(-3*SA+(1-sqrt(3))*SW))*S+(3-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22603).

X(22612) lies on these lines: {4,372}, {16,6301}, {62,22610}, {381,13928}, {3105,22600}, {5318,22606}, {16809,22604}, {16965,22608}

X(22613) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 1st NEUBERG

Barycentrics    S^4-(SA+4*SW)*S^3+(2*SA^2-3*SB*SC-SW^2)*S^2-(7*SA^2-SB*SC-3*SW^2)*SW*S-3*SB*SC*SW^2 : :
X(22613) = 3*X(486)-2*X(13930)

The reciprocal orthologic center of these triangles is X(6316).

X(22613) lies on these lines: {76,486}, {511,6280}, {538,1328}, {639,7864}, {6318,6561}, {12221,22501}, {14881,22642}

X(22614) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 2nd NEUBERG

Barycentrics    3*S^4+(SA-5*SW)*S^3+(2*SA^2-SB*SC+SW^2)*S^2-(2*SA-SW)*(3*SA-7*SW)*SW*S+15*SB*SC*SW^2 : :
X(22614) = 3*X(486)-2*X(13931)

The reciprocal orthologic center of these triangles is X(6315).

X(22614) lies on these lines: {83,486}, {754,1328}, {6317,6561}, {12221,22593}, {22643,22803}

X(22615) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO OUTER-SQUARES

Barycentrics    S^2-(SA-SW)*S-7*SB*SC : :
X(22615) = 3*X(486)-2*X(1152) = 2*X(486)-3*X(1328) = 7*X(486)-6*X(13847) = 5*X(486)-4*X(13966) = 9*X(486)-8*X(13993) = 4*X(642)-3*X(13835) = 4*X(1152)-9*X(1328) = 7*X(1152)-9*X(13847) = 5*X(1152)-6*X(13966) = 3*X(1152)-4*X(13993) = 7*X(1328)-4*X(13847) = 15*X(1328)-8*X(13966) = 27*X(1328)-16*X(13993) = 15*X(13847)-14*X(13966) = 9*X(13966)-10*X(13993)

The reciprocal orthologic center of these triangles is X(486).

X(22615) lies on these lines: {4,371}, {5,6409}, {6,3627}, {20,5420}, {30,486}, {372,3146}, {376,10577}, {381,5418}, {382,3071}, {546,1151}, {548,8252}, {550,10194}, {590,3843}, {615,1657}, {642,13835}, {1327,6470}, {1503,9975}, {1587,17578}, {1588,3543}, {1598,9682}, {1656,6496}, {2043,16242}, {2044,16241}, {3053,13834}, {3070,3830}, {3091,6200}, {3311,5076}, {3365,19107}, {3390,19106}, {3529,6396}, {3592,12102}, {3628,6411}, {3832,6484}, {3839,9540}, {3845,6429}, {3850,8253}, {3853,6431}, {3858,10195}, {3861,8981}, {5059,6487}, {5064,18289}, {5072,6455}, {5073,13785}, {5079,6451}, {6410,15704}, {6412,12103}, {6425,18538}, {6437,13925}, {6454,11541}, {6460,15682}, {7408,8854}, {7409,8280}, {7500,18290}, {7692,12123}, {8276,18535}, {8976,14269}, {9647,10896}, {9660,10895}, {9677,15033}, {9683,9818}, {12240,14915}, {12601,22591}, {12963,13711}, {12969,13770}, {13836,22809}, {13846,14893}, {13951,17800}, {22537,22619}

X(22615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6459, 6564), (4, 6561, 485), (6, 3627, 22644), (20, 6565, 5420), (371, 6564, 13886), (382, 3071, 6560), (3832, 9541, 10576), (6459, 13886, 371), (9541, 10576, 9680), (15704, 18762, 6410)

X(22616) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 1st TRI-SQUARES-CENTRAL

Barycentrics    8*S^3+(12*SA-7*SW)*S^2+3*(SA-SW)*SW*S+9*SB*SC*SW : :
X(22616) = 3*X(486)-X(1327) = 3*X(486)-2*X(13932)

The reciprocal orthologic center of these triangles is X(13711).

X(22616) lies on these lines: {381,486}, {591,6561}, {3564,13846}, {6337,13701}, {6463,13678}, {13691,13769}, {13711,22541}, {13770,19099}

X(22616) = {X(486), X(1327)}-harmonic conjugate of X(13932)

X(22617) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES TO 4th TRI-SQUARES

Barycentrics    8*S^3+(4*SA-3*SW)*S^2+(8*SA-SW)*(SA-SW)*S-3*SB*SC*SW : :
X(22617) = 3*X(486)-2*X(13934)

The reciprocal orthologic center of these triangles is X(13934).

X(22617) lies on these lines: {3,486}, {372,12296}, {485,6251}, {487,6565}, {638,12221}, {642,12322}, {1328,5491}, {6280,22501}, {6463,13678}, {14537,19104}, {22592,22596}

X(22617) = 3rd-anti-tri-squares-isogonal conjugate of X(32498)
X(22617) = {X(3071), X(12601)}-harmonic conjugate of X(486)

X(22618) = PERSPECTOR OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES AND CIRCUMORTHIC

Barycentrics    5*S^4-(2*R^2*(7*SA-9*SW)-4*SA^2+3*SB*SC+5*SW^2)*S^2+S*((SA-SW)*(2*R^2*(17*SA-2*SW)-6*SA^2+6*SB*SC+SW^2)+(44*R^2-5*SA-3*SW)*S^2)+(8*R^2-SW)*SB*SC*SW : :

X(22618) lies on these lines: {}

X(22619) = PERSPECTOR OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES AND EHRMANN-VERTEX

Barycentrics    S^4-(3*SW*R^2-SB*SC-SW^2)*S^2-S*((SA-SW)*(3*R^2*(12*SA-SW)-14*SA^2+14*SB*SC+SW^2)+(40*R^2-15*SA-SW)*S^2)-9*(3*R^2-SW)*SB*SC*SW : :

X(22619) lies on these lines: {1327,1989}, {22537,22615}

X(22620) = PERSPECTOR OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES AND ORTHIC

Barycentrics    (2*R^2+SA-SW)*S^2-S*((SA-SW)*(6*R^2-SA-2*SW)-2*S^2)+2*(9*R^2-2*SW)*SB*SC : :

X(22620) lies on these lines: {381,486}, {485,12239}, {5406,13712}, {6560,10133}, {6564,22554}, {16310,22589}

X(22621) = PERSPECTOR OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES AND OUTER-SQUARES

Barycentrics    (4*R^2+8*SA-5*SW)*S^2-(SA-SW)*(12*R^2+8*SA-7*SW)*S+(36*R^2-13*SW)*SB*SC : :

X(22621) lies on the line {372,1327}

X(22622) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4+(2*SA^2+7*SB*SC+SW^2)*S^2+7*SW^2*SB*SC-3*(SA-SW)*(3*S^2-SW*(2*SA-SW))*S : :
X(22622) = 5*X(485)-2*X(3102) = 3*X(485)-2*X(22724) = 3*X(3102)-5*X(22724)

The reciprocal orthologic center of these triangles is X(22623).

X(22622) lies on these lines: {262,485}, {511,1327}, {6560,22727}, {12222,22643}, {13330,22593}

X(22622) = {X(13330), X(22681)}-harmonic conjugate of X(22593)

X(22623) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 4th ANTI-TRI-SQUARES

Barycentrics    (3*SA-4*SW)*S^2-(SA^2+2*SB*SC+SW^2)*S+SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22622).

X(22623) lies on these lines: {6,12218}, {83,6420}, {99,371}, {182,22718}, {194,6419}, {511,22716}, {575,3734}, {3312,14535}, {3564,6229}, {6033,6231}, {11174,22724}

X(22623) = {X(575), X(3734)}-harmonic conjugate of X(22594)

X(22624) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 4th ANTI-TRI-SQUARES

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

The reciprocal orthologic center of these triangles is X(485).

X(22624) lies on these lines: {3,12344}, {30,22783}, {36,9907}, {55,7981}, {56,485}, {104,12257}, {488,2975}, {641,958}, {956,12788}, {999,12269}, {3428,12124}, {3564,22595}, {6250,22753}, {6278,22757}, {6279,22756}, {6289,22758}, {9922,22654}, {9987,22744}, {10068,22766}, {10084,22767}, {10966,13082}, {11492,12487}, {11493,12486}, {12114,12929}, {12148,22479}, {12211,22520}, {12602,22765}, {12800,22755}, {12949,22759}, {12959,22760}, {13004,22761}, {13005,22762}, {13134,22768}, {13879,22763}, {13880,22764}, {18761,22625}, {19013,19102}, {19014,19103}

X(22625) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 4th ANTI-TRI-SQUARES

Barycentrics    S^2*SA+(S^2+5*SB*SC)*S+3*SB*SC*SW : :
X(22625) = 3*X(4)+X(488) = 5*X(4)-X(12297) = 7*X(4)+X(12510) = 3*X(5)-2*X(6118) = 3*X(381)-X(485) = X(488)-3*X(6289) = 5*X(488)+3*X(12297) = 7*X(488)-3*X(12510) = 5*X(3091)-X(12257) = 9*X(3839)-X(12222) = 5*X(3843)+X(6278) = 5*X(3843)-X(12602) = 5*X(6289)+X(12297) = 7*X(6289)-X(12510) = 7*X(12297)+5*X(12510) = 3*X(22806)+2*X(22819)

The reciprocal orthologic center of these triangles is X(485).

X(22625) lies on these lines: {4,488}, {5,6118}, {30,641}, {371,18539}, {381,485}, {382,12124}, {546,576}, {1479,12949}, {1597,12985}, {1598,12973}, {3091,12257}, {3583,13082}, {3585,18988}, {3839,12222}, {3843,6278}, {3861,22820}, {5395,14229}, {6279,18509}, {6561,13882}, {6565,7745}, {7981,18525}, {9757,15293}, {9818,9922}, {9907,18492}, {9955,12269}, {9987,18500}, {10068,10895}, {10084,10896}, {12211,18502}, {12344,18491}, {12486,18495}, {12487,18497}, {12699,12788}, {12800,18507}, {12929,18516}, {12939,18517}, {12979,18535}, {13004,18520}, {13005,18522}, {13134,18542}, {13135,18544}, {13665,19103}, {13785,19102}, {13834,21309}, {13879,18538}, {13880,18762}, {14230,22505}, {14269,22810}, {15765,22634}, {18585,22635}, {18761,22624}, {22591,22646}

X(22625) = midpoint of X(i) and X(j) for these {i,j}: {4, 6289}, {382, 12124}, {7981, 18525}, {12699, 12788}, {12800, 18507}
X(22625) = {X(546), X(3818)}-harmonic conjugate of X(22596)

X(22626) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO INNER-FERMAT

Barycentrics    218*(3*a^4+4*a^2*(b^2+c^2)-sqrt(3)*(6*(b^2-c^2)^2-2*a^2*(b^2+c^2))-13*(b^2-c^2)^2)*S-(28+15*sqrt(3))*(109*a^6-275*(b^2+c^2)*a^4+a^2*(187*b^4-134*b^2*c^2+187*c^4)-sqrt(3)*(-50*(b^2+c^2)*a^4+16*(b^4-c^4)*(b^2-c^2)+2*a^2*(17*b^4-32*b^2*c^2+17*c^4))-21*(b^4-c^4)*(b^2-c^2)) : :
X(22626) = 3*X(485)-2*X(22880)

The reciprocal orthologic center of these triangles is X(22627).

X(22626) lies on these lines: {18,485}, {6560,22883}, {12222,22632}, {22597,22794}

X(22627) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 4th ANTI-TRI-SQUARES

Barycentrics    39*S^2+(3-4*sqrt(3))*(-(13*SA-5*SW+2*sqrt(3)*SW)*S+13*SB*SC) : :
X(22627) = 4*X(641)-3*X(6304) = 2*X(641)-3*X(6305) = 3*X(6304)-2*X(22629) = 3*X(6305)-X(22629)

The reciprocal orthologic center of these triangles is X(22626).

X(22627) lies on these lines: {2,372}, {3,22883}, {3104,22639}, {3564,22600}, {5339,22598}, {5615,12602}, {6289,16626}, {16645,22880}

X(22627) = anticomplement of X(33448)
X(22627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (485, 488, 6305), (485, 12222, 22629), (641, 22629, 6304), (6305, 22629, 641)

X(22628) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO OUTER-FERMAT

Barycentrics    -218*(3*a^4+4*(b^2+c^2)*a^2+(-2*(b^2+c^2)*a^2+6*(b^2-c^2)^2)*sqrt(3)-13*(b^2-c^2)^2)*S+(28-15*sqrt(3))*(109*a^6-275*(b^2+c^2)*a^4+(187*b^4-134*b^2*c^2+187*c^4)*a^2+(-50*(b^2+c^2)*a^4+2*(17*b^4-32*b^2*c^2+17*c^4)*a^2+16*(b^4-c^4)*(b^2-c^2))*sqrt(3)-21*(b^4-c^4)*(b^2-c^2)) : :
X(22628) = 3*X(485)-2*X(22925)

The reciprocal orthologic center of these triangles is X(22629).

X(22628) lies on these lines: {17,485}, {532,1327}, {6560,22928}, {12222,22630}, {22599,22795}

X(22629) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 4th ANTI-TRI-SQUARES

Barycentrics    39*S^2+(4*sqrt(3)+3)*(13*SB*SC-(13*SA-5*SW-2*sqrt(3)*SW)*S) : :
X(22629) = 2*X(641)-3*X(6304) = 4*X(641)-3*X(6305) = 3*X(6304)-X(22627) = 3*X(6305)-2*X(22627)

The reciprocal orthologic center of these triangles is X(22628).

X(22629) lies on these lines: {2,372}, {3,22928}, {532,22917}, {3105,22638}, {3564,22598}, {5340,22600}, {5611,12602}, {6289,16627}, {16644,22925}

X(22629) = anticomplement of X(33450)
X(22629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (485, 488, 6304), (485, 12222, 22627), (641, 22627, 6305), (6304, 22627, 641)

X(22630) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 3rd FERMAT-DAO

Barycentrics    (27-10*sqrt(3))*(33*SA-24*SW-4*sqrt(3)*SW)*S^2+33*(-26*S^2+(-2*sqrt(3)-5)*(SA-SW)*(3*SA+3*SW-2*sqrt(3)*SW))*S+99*(3-4*sqrt(3))*SB*SC*SW : :
X(22630) = 3*X(485)-2*X(13876)

The reciprocal orthologic center of these triangles is X(22631).

X(22630) lies on these lines: {13,485}, {148,22632}, {6306,6560}, {12222,22628}, {22601,22796}, {22634,22998}

X(22631) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    13*(1+sqrt(3))*(-sqrt(3)*(2*SA-3*SW)+SW)*S^2-(1-3*sqrt(3))*(52*S^2+3*(4-sqrt(3))*(SA-SW)*(2*SA-SW-sqrt(3)*SW))*S+26*sqrt(3)*SB*SC*SW : :
X(22631) = X(13)+2*X(13876) = 4*X(11542)-X(22638) = 5*X(16960)+X(22636)

The reciprocal orthologic center of these triangles is X(22630).

X(22631) lies on these lines: {13,485}, {17,6304}, {61,22634}, {5459,22602}, {11542,22638}, {16808,22640}, {16960,22636}

X(22632) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 4th FERMAT-DAO

Barycentrics    (10*sqrt(3)+27)*(33*SA-24*SW+4*sqrt(3)*SW)*S^2+99*(4*sqrt(3)+3)*SW*SB*SC+33*(-26*S^2+(2*sqrt(3)-5)*(3*SA+3*SW+2*sqrt(3)*SW)*(SA-SW))*S : :
X(22632) = 3*X(485)-2*X(13875)

The reciprocal orthologic center of these triangles is X(22633).

X(22632) lies on these lines: {14,485}, {148,22630}, {6307,6560}, {12222,22626}, {22603,22797}, {22635,22997}

X(22633) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    13*(1-sqrt(3))*(sqrt(3)*(2*SA-3*SW)+SW)*S^2-26*sqrt(3)*SB*SC*SW-(3*sqrt(3)+1)*(52*S^2+3*(4+sqrt(3))*(2*SA-SW+sqrt(3)*SW)*(SA-SW))*S : :
X(22633) = X(14)+2*X(13875) = 4*X(11543)-X(22639) = 5*X(16961)+X(22637)

The reciprocal orthologic center of these triangles is X(22632).

X(22633) lies on these lines: {14,485}, {18,6305}, {62,22635}, {5460,22604}, {11543,22639}, {16809,22641}, {16961,22637}

X(22634) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    (2-sqrt(3))*(sqrt(3)*SA+SW)*S^2+(S^2+(9-4*sqrt(3))*SB*SC)*S-sqrt(3)*(1-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22630).

X(22634) lies on these lines: {5,6304}, {14,485}, {61,22631}, {381,591}, {641,18587}, {3564,22606}, {5321,22640}, {6305,18585}, {10654,13876}, {15765,22625}, {16808,22638}, {16809,22636}, {22630,22998}

X(22634) = {X(381), X(6289)}-harmonic conjugate of X(22635)

X(22635) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    (2+sqrt(3))*(-SA*sqrt(3)+SW)*S^2+sqrt(3)*(1+sqrt(3))*SW*SB*SC+(S^2+(9+4*sqrt(3))*SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(22632).

X(22635) lies on these lines: {5,6305}, {13,485}, {62,22633}, {381,591}, {641,18586}, {3564,22605}, {5318,22641}, {6304,15765}, {10653,13875}, {16808,22637}, {16809,22639}, {18585,22625}, {22632,22997}

X(22635) = {X(381), X(6289)}-harmonic conjugate of X(22634)

X(22636) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    (3-sqrt(3))*(2*SA-(-3-2*sqrt(3))*SW)*S^2-(-8*sqrt(3)*S^2+(1-sqrt(3))*(SA-SW)*(6*SA+(-3-2*sqrt(3))*SW))*S+3*(1+sqrt(3))*SB*SC*SW : :
X(22636) = 3*X(13)-2*X(22638) = 4*X(13876)-3*X(16962) = 5*X(16960)-6*X(22631)

The reciprocal orthologic center of these triangles is X(22630).

X(22636) lies on these lines: {13,22638}, {18,485}, {621,6115}, {6304,16966}, {6564,9733}, {13876,16962}, {16809,22634}, {16960,22631}, {16964,22640}

X(22637) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    (sqrt(3)+3)*(2*SA-(-3+2*sqrt(3))*SW)*S^2+3*(1-sqrt(3))*SW*SB*SC-(8*S^2*sqrt(3)+(1+sqrt(3))*(SA-SW)*(6*SA+(-3+2*sqrt(3))*SW))*S : :
X(22637) = 3*X(14)-2*X(22639) = 4*X(13875)-3*X(16963) = 5*X(16961)-6*X(22633)

The reciprocal orthologic center of these triangles is X(22632).

X(22637) lies on these lines: {14,22639}, {17,485}, {622,6114}, {6305,16967}, {6564,9733}, {13875,16963}, {16808,22635}, {16961,22633}, {16965,22641}

X(22638) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    3*(4-3*sqrt(3))*(22*SA+3*(-5-sqrt(3))*SW)*S^2+(132*S^2-11*sqrt(3)*(SA-SW)*(-6*SA+(3-sqrt(3))*SW))*S+33*(3-sqrt(3))*SB*SC*SW : :
X(22638) = 3*X(13)-X(22636) = 4*X(11542)-3*X(22631) = 2*X(13876)-3*X(16267)

The reciprocal orthologic center of these triangles is X(22630).

X(22638) lies on these lines: {13,22636}, {16,6304}, {17,485}, {61,22640}, {3105,22629}, {6289,6564}, {11542,22631}, {13876,16267}, {16808,22634}

X(22639) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    3*(3*sqrt(3)+4)*(22*SA+3*(-5+sqrt(3))*SW)*S^2+33*(sqrt(3)+3)*SW*SB*SC+(132*S^2+11*sqrt(3)*(SA-SW)*(-6*SA+SW*(sqrt(3)+3)))*S : :
X(22639) = 3*X(14)-X(22637) = 4*X(11543)-3*X(22633) = 2*X(13875)-3*X(16268)

The reciprocal orthologic center of these triangles is X(22632).

X(22639) lies on these lines: {14,22637}, {15,6305}, {18,485}, {62,22641}, {3104,22627}, {6289,6564}, {11543,22633}, {13875,16268}, {16809,22635}

X(22640) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    (SA-sqrt(3)*SW)*S^2+(2*S^2-(SA-SW)*(-3*SA+(1-sqrt(3))*SW))*S+(3-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22630).

X(22640) lies on these lines: {4,371}, {15,6304}, {61,22638}, {381,13876}, {3104,22627}, {5321,22634}, {16808,22631}, {16964,22636}

X(22641) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 4th ANTI-TRI-SQUARES

Barycentrics    (SA+sqrt(3)*SW)*S^2+(sqrt(3)+3)*SW*SB*SC+(2*S^2-(SA-SW)*(-3*SA+(1+sqrt(3))*SW))*S : :

The reciprocal orthologic center of these triangles is X(22632).

X(22641) lies on these lines: {4,371}, {16,6305}, {62,22639}, {381,13875}, {3105,22629}, {5318,22635}, {16809,22633}, {16965,22637}

X(22642) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 1st NEUBERG

Barycentrics    S^4+(SA+4*SW)*S^3+(2*SA^2-3*SB*SC-SW^2)*S^2+(7*SA^2-SB*SC-3*SW^2)*SW*S-3*SB*SC*SW^2 : :
X(22642) = 3*X(485)-2*X(13877)

The reciprocal orthologic center of these triangles is X(6312).

X(22642) lies on these lines: {76,485}, {511,6279}, {538,1327}, {640,7864}, {6314,6560}, {12222,22502}, {14881,22613}

X(22643) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 2nd NEUBERG

Barycentrics    3*S^4-(SA-5*SW)*S^3+(2*SA^2-SB*SC+SW^2)*S^2+(2*SA-SW)*(3*SA-7*SW)*SW*S+15*SB*SC*SW^2 : :
X(22643) = 3*X(485)-2*X(13878)

The reciprocal orthologic center of these triangles is X(6311).

X(22643) lies on these lines: {83,485}, {754,1327}, {6313,6560}, {12222,22622}, {22614,22803}

X(22644) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO INNER-SQUARES

Barycentrics    S^2+(SA-SW)*S-7*SB*SC : :
X(22644) = 3*X(485)-2*X(1151) = 2*X(485)-3*X(1327) = 5*X(485)-4*X(8981) = 7*X(485)-6*X(13846) = 9*X(485)-8*X(13925) = 4*X(641)-3*X(13712) = 4*X(1151)-9*X(1327) = 5*X(1151)-6*X(8981) = 7*X(1151)-9*X(13846) = 3*X(1151)-4*X(13925) = 15*X(1327)-8*X(8981) = 7*X(1327)-4*X(13846) = 27*X(1327)-16*X(13925) = 14*X(8981)-15*X(13846) = 9*X(8981)-10*X(13925)

The reciprocal orthologic center of these triangles is X(485).

X(22644) lies on these lines: {4,372}, {5,6410}, {6,3627}, {20,5418}, {30,485}, {371,3146}, {376,10576}, {381,5420}, {382,3070}, {546,1152}, {548,8253}, {550,10195}, {590,1657}, {615,3843}, {641,13712}, {1131,8960}, {1328,6471}, {1503,9974}, {1587,3543}, {1588,17578}, {1656,6497}, {2043,16241}, {2044,16242}, {3053,13711}, {3068,9681}, {3071,3830}, {3091,6396}, {3312,5076}, {3364,19107}, {3389,19106}, {3529,6200}, {3594,12102}, {3628,6412}, {3832,6485}, {3839,13935}, {3845,6430}, {3850,8252}, {3853,6432}, {3858,10194}, {3861,13966}, {5059,6486}, {5064,18290}, {5072,6456}, {5073,13665}, {5079,6452}, {6250,21736}, {6409,15704}, {6411,12103}, {6426,18762}, {6438,13993}, {6453,11541}, {6459,15682}, {7408,8855}, {7409,8281}, {7500,18289}, {7690,12124}, {8277,18535}, {8976,9680}, {9682,12085}, {12239,14915}, {12602,22592}, {12962,13651}, {12968,13834}, {13713,22810}, {13847,14893}, {13951,14269}, {22536,22588}

X(22644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6460, 6565), (4, 6560, 486), (6, 3627, 22615), (20, 6564, 5418), (372, 6565, 13939), (382, 3070, 6561), (1131, 9541, 8960), (6460, 13939, 372), (15704, 18538, 6409)

X(22645) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    -8*S^3+(12*SA-7*SW)*S^2-3*(SA-SW)*SW*S+9*SB*SC*SW : :
X(22645) = 3*X(485)-X(1328) = 3*X(485)-2*X(13850)

The reciprocal orthologic center of these triangles is X(13834).

X(22645) lies on these lines: {381,485}, {1991,6560}, {3564,13847}, {6337,13821}, {6462,13798}, {13651,19100}, {13810,13833}, {13834,19101}

X(22645) = {X(485), X(1328)}-harmonic conjugate of X(13850)

X(22646) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES TO 3rd TRI-SQUARES

Barycentrics    -8*S^3+(4*SA-3*SW)*S^2-(8*SA-SW)*(SA-SW)*S-3*SB*SC*SW : :
X(22646) = 3*X(485)-2*X(13882)

The reciprocal orthologic center of these triangles is X(13882).

X(22646) lies on these lines: {3,485}, {371,12297}, {486,6250}, {488,6564}, {637,12222}, {641,12323}, {1327,5490}, {6279,22502}, {6462,13798}, {14537,19103}, {22591,22625}

X(22646) = 4th-anti-tri-squares-isogonal conjugate of X(32499)
X(22646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (485, 12124, 5418), (3070, 12602, 485)

X(22647) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO 3rd HATZIPOLAKIS

Barycentrics    SA*((11*R^2-SA-SW)*S^2-(SA-SW)*(2*R^2*(32*R^2-3*SA-14*SW)+SA^2-SB*SC+3*SW^2)) : :
X(22647) = 3*X(2)-4*X(22966) = 3*X(4)-4*X(22800) = 5*X(3091)-4*X(22833) = 5*X(3616)-4*X(22476) = 2*X(22800)-3*X(22955)

The reciprocal orthologic center of these triangles is X(12241).

X(22647) lies on these lines: {2,22466}, {3,22533}, {4,801}, {5,22979}, {8,22941}, {10,22653}, {20,22662}, {22,22658}, {69,11440}, {100,22559}, {145,22969}, {146,5895}, {388,18978}, {394,22972}, {497,22959}, {1270,22947}, {1271,22945}, {2071,2888}, {2896,22747}, {2929,13567}, {2975,22776}, {3085,22980}, {3086,22981}, {3091,22833}, {3434,22956}, {3436,22957}, {3548,22808}, {3616,22476}, {4240,22943}, {5449,22834}, {5562,15103}, {6241,12383}, {6462,22963}, {6463,22964}, {6644,22550}, {7585,19084}, {7586,19083}, {7787,22524}, {8972,22976}, {10116,12118}, {10528,22982}, {10529,22983}, {11064,22971}, {12254,16163}, {13941,22977}, {16013,22978}, {18912,22962}, {18936,22953}

X(22647) = reflection of X(i) in X(j) for these (i,j): (4, 22955), (8, 22941), (20, 22951), (145, 22969), (4240, 22943)
X(22647) = anticomplement of X(22466)
X(22647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18978, 22958, 388), (22466, 22966, 2), (22959, 22965, 497)

X(22648) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO ANTICOMPLEMENTARY

Barycentrics    5*S^4-(6*(2*SA-SW)*R^2-6*SA^2+7*SB*SC+3*SW^2)*S^2+3*SW^2*SB*SC-2*sqrt(3)*((R^2+SA-SW)*S^2+6*R^2*SB*SC)*S : :

The reciprocal cyclologic center of these triangles is X(616).

X(22648) lies on these lines: {2,3480}, {4,3181}, {14,11601}, {617,2926}, {6106,18581}

X(22648) = anticomplement of X(3480)

X(22649) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO ANTICOMPLEMENTARY

Barycentrics    5*S^4-(6*(2*SA-SW)*R^2-6*SA^2+7*SB*SC+3*SW^2)*S^2+3*SW^2*SB*SC+2*sqrt(3)*((R^2+SA-SW)*S^2+6*R^2*SB*SC)*S : :

The reciprocal cyclologic center of these triangles is X(617).

X(22649) lies on these lines: {2,3479}, {4,3180}, {13,11600}, {532,1337}, {616,2925}, {6107,18582}

X(22649) = anticomplement of X(3479)

X(22650) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^6-3*(b+c)*(b^2+c^2)*a^5+(b^2+b*c+c^2)*(b^2+5*b*c+c^2)*a^4+3*(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3-(2*b^2+b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*(b-c)^2*a^2+3*(b^2-c^2)*(b-c)*b^2*c^2*a-2*(b^2-c^2)^2*b^2*c^2 : :
X(22650) = 3*X(1)-4*X(22475) = 3*X(1)-2*X(22713) = 3*X(165)-2*X(22676) = 3*X(262)-2*X(22475) = 3*X(262)-X(22713) = 4*X(355)-X(9902) = 5*X(1698)-4*X(15819) = 3*X(1699)-4*X(22682) = 2*X(3095)+X(5881) = 3*X(3097)-2*X(7709) = 3*X(3679)-2*X(22697) = 4*X(5188)-7*X(9588) = 3*X(5587)-2*X(7697) = X(5691)+2*X(12782) = X(7982)-4*X(14881)

The reciprocal orthologic center of these triangles is X(3).

X(22650) lies on these lines: {1,262}, {8,7985}, {10,6194}, {35,22556}, {36,22680}, {40,9903}, {57,18971}, {165,22676}, {355,9902}, {511,3679}, {515,3097}, {517,22728}, {1697,22711}, {1698,15819}, {1699,14839}, {2782,9875}, {3095,5881}, {3099,22678}, {3751,9860}, {5188,9588}, {5587,7697}, {5588,22700}, {5589,22699}, {5691,12782}, {5727,12837}, {7713,22480}, {7982,14881}, {7989,12263}, {8185,22655}, {8186,22668}, {8187,22672}, {8188,22709}, {8189,22710}, {8931,9746}, {9578,22705}, {9581,22706}, {10789,22521}, {10826,22703}, {10827,22704}, {11852,22698}, {13888,22720}, {13942,22721}, {18492,22681}, {19003,19063}, {19004,19064}, {19875,22712}

X(22650) = reflection of X(1) in X(262)
X(22650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (262, 22713, 22475), (22475, 22713, 1)

X(22651) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO INNER-FERMAT

Barycentrics    2*(a^2+2*(b+c)*a-b^2-c^2)*sqrt(3)*S*a-(a+b+c)*(3*a^4-5*(b+c)*a^3+(b^2+10*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-4*(b^2-c^2)^2) : :
X(22651) = 3*X(1)-4*X(11740) = 3*X(1)-2*X(22867) = 3*X(18)-2*X(11740) = 3*X(18)-X(22867) = 3*X(165)-2*X(22843) = 4*X(630)-5*X(1698) = 3*X(1699)-4*X(22831) = 7*X(3624)-8*X(6674) = 3*X(5587)-2*X(16627) = 3*X(11852)-2*X(22852) = 5*X(18492)-4*X(22794)

The reciprocal orthologic center of these triangles is X(3).

X(22651) lies on these lines: {1,18}, {8,22114}, {10,628}, {35,22557}, {36,22771}, {40,9900}, {57,18972}, {165,22843}, {355,9901}, {515,22531}, {517,16628}, {630,1698}, {1697,22865}, {1699,22831}, {3099,22745}, {3624,6674}, {3751,5965}, {5587,16627}, {5588,22854}, {5589,22853}, {7713,22481}, {8185,22656}, {8186,22669}, {8187,22673}, {8188,22863}, {8189,22864}, {9578,22859}, {9581,22860}, {10789,22522}, {10826,22857}, {10827,22858}, {11852,22852}, {13888,22876}, {13942,22877}, {18492,22794}, {19003,19069}, {19004,19072}

X(22651) = midpoint of X(8) and X(22114)
X(22651) = reflection of X(1) in X(18)
X(22651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 22867, 11740), (11740, 22867, 1)

X(22652) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO OUTER-FERMAT

Barycentrics    -2*(a^2+2*(b+c)*a-b^2-c^2)*sqrt(3)*S*a-(a+b+c)*(3*a^4-5*(b+c)*a^3+(b^2+10*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-4*(b^2-c^2)^2) : :
X(22652) = 3*X(1)-4*X(11739) = 3*X(1)-2*X(22912) = 3*X(17)-2*X(11739) = 3*X(17)-X(22912) = 3*X(165)-2*X(22890) = 4*X(629)-5*X(1698) = 3*X(1699)-4*X(22832) = 7*X(3624)-8*X(6673) = 3*X(3679)-2*X(22896) = 3*X(5587)-2*X(16626) = 3*X(11852)-2*X(22897) = 5*X(18492)-4*X(22795)

The reciprocal orthologic center of these triangles is X(3).

X(22652) lies on these lines: {1,17}, {8,22113}, {10,627}, {35,22558}, {36,22772}, {40,9901}, {57,18973}, {165,22890}, {355,9900}, {515,22532}, {517,16629}, {532,3679}, {629,1698}, {1697,22910}, {1699,22832}, {3099,22746}, {3624,6673}, {3751,5965}, {5587,16626}, {5588,22899}, {5589,22898}, {7713,22482}, {8185,22657}, {8186,22670}, {8187,22674}, {8188,22908}, {8189,22909}, {9578,22904}, {9581,22905}, {10789,22523}, {10826,22902}, {10827,22903}, {11852,22897}, {13888,22921}, {13942,22922}, {18492,22795}, {19003,19071}, {19004,19070}

X(22652) = midpoint of X(8) and X(22113)
X(22652) = reflection of X(1) in X(17)
X(22652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 22912, 11739), (11739, 22912, 1)

X(22653) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 3rd HATZIPOLAKIS

Barycentrics    3*a^16-(b+c)*a^15-(9*b^2-2*b*c+9*c^2)*a^14+(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(5*b^4+5*c^4-(4*b^2-41*b*c+4*c^2)*b*c)*a^12-(b+c)*(b^4+c^4-(4*b^2-15*b*c+4*c^2)*b*c)*a^11+(7*b^6+7*c^6-(2*b^4+2*c^4+(45*b^2-22*b*c+45*c^2)*b*c)*b*c)*a^10-(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+(19*b^2-22*b*c+19*c^2)*b*c)*b*c)*a^9-(5*b^8+5*c^8-2*(4*b^6+4*c^6-(8*b^4+8*c^4+3*(3*b^2-17*b*c+3*c^2)*b*c)*b*c)*b*c)*a^8+(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6-(2*b^4+2*c^4+(9*b^2-23*b*c+9*c^2)*b*c)*b*c)*b*c)*a^7-(3*b^8+3*c^8+2*(4*b^6+4*c^6-(11*b^4+11*c^4+(17*b^2+3*b*c+17*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^6+(b^2-c^2)*(b-c)*(b^8+c^8+2*(2*b^6+2*c^6-7*(b^4+c^4+(b^2-b*c+c^2)*b*c)*b*c)*b*c)*a^5-(b^2-c^2)^2*(b-c)^2*(b^6+c^6+2*(3*b^4+3*c^4-(b^2+12*b*c+c^2)*b*c)*b*c)*a^4-(b^4-c^4)*(b^2-c^2)^2*(b-c)*(3*b^4+3*c^4+(2*b^2-7*b*c+2*c^2)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(5*b^2+12*b*c+5*c^2)*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-2*(b^2-c^2)^6*(b^2+c^2)^2 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22653) lies on these lines: {1,22466}, {10,22647}, {35,22559}, {36,22776}, {40,22840}, {57,18978}, {165,22951}, {515,22533}, {517,22979}, {1697,22965}, {1698,22966}, {1699,22833}, {3099,22747}, {3679,22941}, {5587,22955}, {5588,22947}, {5589,22945}, {7713,22483}, {8185,22658}, {8188,22963}, {8189,22964}, {9578,22958}, {9581,22959}, {10789,22524}, {10826,22956}, {10827,22957}, {11852,22943}, {13888,22976}, {13942,22977}, {18492,22800}, {19003,19083}, {19004,19084}

X(22653) = reflection of X(1) in X(22466)
X(22653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22466, 22969, 22476), (22476, 22969, 1)

X(22654) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 2nd CIRCUMPERP TANGENTIAL

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

X(22654) lies on these lines: {1,159}, {3,10}, {4,17111}, {11,4186}, {20,11677}, {22,2975}, {24,104}, {25,34}, {28,3433}, {36,8185}, {55,8192}, {65,1473}, {119,21479}, {198,1212}, {222,14529}, {388,4224}, {405,19836}, {517,12517}, {859,7742}, {956,8193}, {960,20876}, {962,1633}, {963,1436}, {999,11365}, {1043,16876}, {1191,7083}, {1329,16434}, {1406,3937}, {1460,4252}, {1598,22753}, {1602,1610}, {1616,16686}, {1617,1661}, {1995,5253}, {2178,16968}, {2182,12680}, {2551,19649}, {3086,4222}, {3145,8240}, {3189,20871}, {3304,20988}, {3428,11414}, {4185,7354}, {4214,12943}, {4999,19544}, {5204,20989}, {5260,7485}, {5594,22757}, {5595,22756}, {6642,10269}, {7078,8679}, {7293,19860}, {7387,11249}, {7428,8069}, {7517,22765}, {8071,11334}, {8190,11493}, {8191,11492}, {8194,22761}, {8195,22762}, {9630,11396}, {9861,22504}, {9876,22565}, {9908,22659}, {9909,11194}, {9910,18237}, {9911,22770}, {9912,12773}, {9913,22775}, {9914,22778}, {9915,22774}, {9916,22773}, {9917,22779}, {9918,22780}, {9919,22583}, {9920,22781}, {9921,22595}, {9922,22624}, {10037,22766}, {10046,22767}, {10790,22520}, {10831,22759}, {10832,22760}, {10833,10835}, {10834,22768}, {10896,17516}, {11641,19162}, {11853,22755}, {12310,22586}, {12410,12513}, {12411,22777}, {12412,19478}, {12413,19159}, {12414,22782}, {13175,22514}, {13222,22560}, {13680,22783}, {13737,20470}, {13743,16119}, {13800,22784}, {13889,22763}, {13943,22764}, {16828,19286}, {18242,21484}, {19005,19013}, {19006,19014}, {22655,22680}, {22656,22771}, {22657,22772}, {22658,22776}

X(22654) = isogonal conjugate of isotomic conjugate of X(21279)
X(22654) = complement of X(20211)
X(22654) = polar conjugate of isotomic conjugate of X(23122)
X(22654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3220, 3556), (1, 13730, 1486), (3, 9798, 197), (159, 18610, 1486), (956, 20833, 8193), (999, 20831, 11365), (12513, 20872, 12410)

X(22655) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 1st BROCARD-REFLECTED

Barycentrics    (SB+SC)*(S^4+(4*R^2*SW+SA^2-3*SB*SC-2*SW^2)*S^2-(4*R^2+2*SA-3*SW)*SA*SW^2) : :
X(22655) = 4*X(26)-X(9917)

The reciprocal orthologic center of these triangles is X(3).

X(22655) lies on these lines: {3,3734}, {22,6194}, {24,7709}, {25,262}, {26,9917}, {39,3517}, {76,9715}, {154,511}, {159,9861}, {197,22556}, {237,9752}, {538,10245}, {1598,22682}, {2782,9876}, {3095,9714}, {3515,11257}, {5594,22700}, {5595,22699}, {7387,9918}, {7517,22728}, {8185,22650}, {8190,22668}, {8191,22672}, {8192,22713}, {8193,22697}, {8194,22709}, {8195,22710}, {9754,20885}, {9818,22681}, {10037,22729}, {10046,22730}, {10790,22521}, {10828,22678}, {10829,22703}, {10830,22704}, {10831,22705}, {10832,22706}, {10833,22711}, {10834,22731}, {10835,22732}, {11365,22475}, {11414,22676}, {11853,22698}, {13889,22720}, {13943,22721}, {18954,18971}, {19005,19063}, {19006,19064}, {22654,22680}

X(22656) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO INNER-FERMAT

Barycentrics    (SB+SC)*(sqrt(3)*(2*S^2*R^2-SW*SB*SC)+(3*S^2+SA*(2*R^2+3*SA-5*SW))*S) : :

The reciprocal orthologic center of these triangles is X(3).

X(22656) lies on these lines: {3,624}, {18,25}, {22,628}, {23,22114}, {24,22531}, {26,9916}, {159,5965}, {197,22557}, {1598,22831}, {5020,6674}, {5594,22854}, {5595,22853}, {7387,9915}, {7517,16628}, {8185,22651}, {8190,22669}, {8191,22673}, {8192,22867}, {8193,22851}, {8194,22863}, {8195,22864}, {9818,22794}, {10037,22884}, {10046,22885}, {10790,22522}, {10828,22745}, {10829,22857}, {10830,22858}, {10831,22859}, {10832,22860}, {10833,22865}, {10834,22886}, {10835,22887}, {11365,11740}, {11414,22843}, {11853,22852}, {13889,22876}, {13943,22877}, {18954,18972}, {19005,19069}, {19006,19072}, {22654,22771}

X(22657) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO OUTER-FERMAT

Barycentrics    (SB+SC)*(sqrt(3)*(2*S^2*R^2-SW*SB*SC)-(3*S^2+SA*(2*R^2+3*SA-5*SW))*S) : :

The reciprocal orthologic center of these triangles is X(3).

X(22657) lies on these lines: {3,623}, {17,25}, {22,627}, {23,22113}, {24,22532}, {26,9915}, {159,5965}, {197,22558}, {532,9909}, {1598,22832}, {5020,6673}, {5594,22899}, {5595,22898}, {7387,9916}, {7517,16629}, {8185,22652}, {8190,22670}, {8191,22674}, {8192,22912}, {8193,22896}, {8194,22908}, {8195,22909}, {9818,22795}, {10037,22929}, {10046,22930}, {10790,22523}, {10828,22746}, {10829,22902}, {10830,22903}, {10831,22904}, {10832,22905}, {10833,22910}, {10834,22931}, {10835,22932}, {11365,11739}, {11414,22890}, {11853,22897}, {13889,22921}, {13943,22922}, {18954,18973}, {19005,19071}, {19006,19070}, {22654,22772}

X(22658) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 3rd HATZIPOLAKIS

Barycentrics    ((6*R^2-SW)*R^2*S^2+(16*(8*R^2+SA-7*SW)*R^4-3*(SA-9*SW)*R^2*SW-2*SW^3)*SA)*(SB+SC) : :
X(22658) = 3*X(154)-X(22972)

The reciprocal orthologic center of these triangles is X(12241).

X(22658) lies on these lines: {3,22955}, {22,22647}, {24,22533}, {25,22466}, {154,22497}, {159,2929}, {197,22559}, {1204,1660}, {1598,22833}, {1619,12279}, {1657,9919}, {2070,9920}, {3532,13171}, {3556,15071}, {5594,22947}, {5595,22945}, {5925,9914}, {7517,22979}, {8185,22653}, {8192,22969}, {8193,22941}, {8194,22963}, {8195,22964}, {9818,22800}, {10037,22980}, {10046,22981}, {10790,22524}, {10828,22747}, {10829,22956}, {10830,22957}, {10831,22958}, {10832,22959}, {10833,22965}, {10834,22982}, {10835,22983}, {11365,22476}, {11414,22951}, {11853,22943}, {12163,12412}, {12225,22555}, {13889,22976}, {13943,22977}, {18954,18978}, {19005,19083}, {19006,19084}, {19153,22529}, {22654,22776}

X(22659) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ARIES

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

The reciprocal orthologic center of these triangles is X(9833).

X(22659) lies on these lines: {1,90}, {3,914}, {30,22778}, {36,9896}, {55,9933}, {56,68}, {104,11411}, {539,11194}, {956,9928}, {958,1147}, {999,12259}, {2975,6193}, {3428,12118}, {3564,22595}, {9908,22654}, {9923,22744}, {9927,22753}, {9929,22756}, {9930,22757}, {10055,22766}, {10071,22767}, {10269,12359}, {10966,12428}, {11493,12415}, {12114,12422}, {12134,22479}, {12193,22520}, {12418,22755}, {12426,22761}, {12427,22762}, {12429,22765}, {12430,22768}, {13909,22763}, {13970,22764}, {17702,22583}, {18761,22660}, {19013,19061}, {19014,19062}

X(22660) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO ARIES

Barycentrics    (-a^2+b^2+c^2)*(3*(b^2+c^2)*a^6-(5*b^4-6*b^2*c^2+5*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
X(22660) = X(3)-3*X(5654) = 3*X(4)+X(6193) = 3*X(4)-X(12293) = 3*X(5)-2*X(5449) = X(68)-3*X(381) = 3*X(155)-X(6193) = 3*X(155)+X(12293) = 3*X(381)+X(12164) = X(382)+3*X(3167) = 2*X(546)+X(15083) = 3*X(3167)-X(12118) = 3*X(5448)-X(5449) = 4*X(5448)-X(12359) = 4*X(5449)-3*X(12359) = 3*X(5654)-2*X(9820)

The reciprocal orthologic center of these triangles is X(9833).

X(22660) lies on these lines: {2,12163}, {3,4549}, {4,155}, {5,389}, {11,7352}, {12,6238}, {20,9707}, {25,9932}, {26,16252}, {30,156}, {49,18563}, {52,113}, {68,381}, {110,6240}, {125,21971}, {140,7689}, {141,11591}, {143,15873}, {146,12086}, {184,12605}, {185,1568}, {343,10024}, {382,3167}, {403,5889}, {427,12162}, {539,3845}, {541,15115}, {542,18383}, {546,576}, {550,5944}, {632,20191}, {858,6241}, {912,946}, {1069,1478}, {1154,15761}, {1181,18531}, {1204,10257}, {1216,6823}, {1479,3157}, {1498,14790}, {1503,18569}, {1531,21659}, {1539,3627}, {1593,9938}, {1594,7703}, {1596,5446}, {1597,12301}, {1598,9937}, {1614,12225}, {1619,5878}, {1658,10192}, {1885,13352}, {1906,11576}, {1907,16194}, {2931,3518}, {3070,10666}, {3071,10665}, {3088,11469}, {3091,11411}, {3548,10605}, {3574,7403}, {3575,10539}, {3580,16868}, {3583,12428}, {3585,18970}, {3843,9936}, {5133,15058}, {5198,12166}, {5318,10662}, {5321,10661}, {5504,7728}, {5562,15760}, {5576,18435}, {5655,7540}, {5663,6247}, {5891,7399}, {5893,16266}, {5894,11250}, {6146,18404}, {6243,11799}, {6561,8909}, {6644,13568}, {6696,18281}, {6804,15805}, {7526,19908}, {7547,11442}, {7706,9825}, {9703,18562}, {9704,18564}, {9818,9908}, {9896,18492}, {9923,18500}, {9928,12699}, {9929,18509}, {9930,18511}, {9933,18525}, {9955,12259}, {10055,10895}, {10071,10896}, {10110,12235}, {10272,12893}, {10295,11449}, {10982,19458}, {11381,15063}, {11438,16238}, {11459,13160}, {11745,13861}, {12061,14984}, {12161,12241}, {12193,18502}, {12309,18535}, {12319,16658}, {12328,18491}, {12415,18495}, {12418,18507}, {12422,18516}, {12423,18517}, {12426,18520}, {12427,18522}, {12430,18542}, {12431,18544}, {13292,18390}, {13665,19062}, {13785,19061}, {13909,18538}, {13970,18762}, {14094,15133}, {14788,15056}, {15305,15559}, {15341,22120}, {15738,20303}, {16534,20772}, {17814,18420}, {18567,19479}, {18761,22659}

X(22660) = midpoint of X(i) and X(j) for these {i,j}: {4, 155}, {68, 12164}, {146, 12302}, {382, 12118}, {1498, 14790}, {5504, 7728}, {9928, 12699}, {9933, 18525}, {12418, 18507}, {14094, 15133}
X(22660) = reflection of X(i) in X(j) for these (i,j): (3, 9820), (5, 5448), (26, 16252), (550, 12038), (5894, 11250)
X(22660) = complement of X(12163)
X(22660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5654, 9820), (4, 6193, 12293), (4, 11441, 12134), (5, 6102, 13567), (52, 113, 235), (155, 12293, 6193), (185, 1568, 11585), (381, 12164, 68), (382, 3167, 12118), (3091, 11411, 14852), (3574, 15030, 7403), (5907, 18388, 5), (10024, 18436, 343), (18404, 18445, 6146)

X(22661) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO ARIES

Barycentrics    SA*(2*(3*R^2-SA)*S^2-(SA-SW)*(26*R^4-7*R^2*(SA+2*SW)+2*SA^2-2*SB*SC+2*SW^2)) : :
X(22661) = 3*X(4)+X(12318) = 3*X(381)-X(9937) = 5*X(3843)-X(12309) = X(17836)+3*X(18405)

The reciprocal orthologic center of these triangles is X(7387).

X(22661) lies on these lines: {3,20302}, {4,155}, {5,9932}, {20,12302}, {30,9938}, {49,12118}, {64,14790}, {68,265}, {381,9937}, {382,12301}, {539,18568}, {542,9926}, {1147,18388}, {2931,7505}, {3153,11411}, {3564,18377}, {3583,9931}, {3585,19471}, {3832,18427}, {3843,12309}, {5448,11818}, {5654,18350}, {5907,9927}, {6288,15739}, {6564,12424}, {6565,12425}, {6759,17702}, {7689,14791}, {9820,18420}, {10660,16809}, {11457,15133}, {12166,18386}, {12235,18390}, {12271,18392}, {12282,18394}, {12359,18531}, {12417,18406}, {12429,18403}, {13754,18381}, {13851,21651}, {17836,18405}, {18396,19458}, {18414,19486}, {18415,19487}, {18918,18934}, {19130,19141}, {19177,19196}

X(22661) = midpoint of X(382) and X(12301)
X(22661) = reflection of X(3) in X(20302)
X(22661) = {X(155), X(12293)}-harmonic conjugate of X(14516)

X(22662) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARIES TO 3rd HATZIPOLAKIS

Barycentrics    (192*R^6+8*R^4*(10*SA-23*SW)-(31*SA-49*SW)*R^2*SW+(3*SA-4*SW)*SW^2)*S^2-(2*R^2*(64*R^4-96*R^2*SW+29*SW^2)-5*SW^3)*SB*SC : :

The reciprocal orthologic center of these triangles is X(22663).

X(22662) lies on these lines: {20,22647}, {25,22953}, {159,2929}, {235,22466}, {1368,22955}, {1498,17837}, {6146,22750}, {6353,22533}, {10539,22808}, {19460,22483}

X(22663) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd HATZIPOLAKIS TO ARIES

Barycentrics    SA*(2*(-SW+7*R^2)*S^2-(SA-SW)*(R^2*(8*R^2-4*SA-7*SW)+SW^2)) : :

The reciprocal orthologic center of these triangles is X(22662).

X(22663) lies on these lines: {5,6}, {974,22953}, {1885,5889}, {11245,17928}, {11264,13630}, {11585,15317}, {12420,13861}, {15316,18952}

X(22663) = midpoint of X(5) and X(12421)

X(22664) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND 1st BROCARD-REFLECTED

Barycentrics    3*S^6-3*SW^2*S^4-(SA^2-4*SB*SC-SW^2)*SW^2*S^2-2*SB*SC*SW^4 : :
X(22664) = 2*X(8719)-3*X(21166)

X(22664) lies on these lines: {2,2794}, {3,9743}, {6,98}, {30,9877}, {99,5999}, {114,7710}, {147,3424}, {542,9770}, {1503,6054}, {1513,10722}, {2548,10991}, {2782,9764}, {5652,9775}, {6055,14561}, {8719,21166}, {12042,14535}

X(22664) = midpoint of X(147) and X(3424)

X(22665) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO INNER-FERMAT

Barycentrics    3*SW*(S^2+3*SB*SC)-sqrt(3)*(10*S^2-3*SW*(3*SA-SW))*S : :
X(22665) = X(18)+2*X(22871)

The reciprocal orthologic center of these triangles is X(5858).

X(22665) lies on these lines: {2,18}, {381,7764}, {5858,5965}, {6115,11121}, {7788,22850}, {7837,22855}, {9760,9766}, {13638,22878}, {13758,22879}

X(22666) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO OUTER-FERMAT

Barycentrics    3*SW*(S^2+3*SB*SC)+sqrt(3)*(10*S^2-3*SW*(3*SA-SW))*S : :
X(22666) = X(17)+2*X(22916)

The reciprocal orthologic center of these triangles is X(5859).

X(22666) lies on these lines: {2,17}, {381,7764}, {5859,5965}, {6114,11122}, {7788,22894}, {7837,22901}, {9762,9766}, {13638,22923}, {13758,22924}

X(22667) = PERSPECTOR OF THESE TRIANGLES: ASCELLA AND 2nd CIRCUMPERP TANGENTIAL

Barycentrics    a*(a^9-2*(b+c)*a^8-2*(b^2-b*c+c^2)*a^7+6*(b^3+c^3)*a^6+2*(b^2+6*b*c+c^2)*b*c*a^5-2*(b+c)*(3*b^4+3*c^4-(5*b^2-12*b*c+5*c^2)*b*c)*a^4+2*(b^6+c^6-(5*b^4+5*c^4+(9*b^2+22*b*c+9*c^2)*b*c)*b*c)*a^3+2*(b+c)*(b^6+c^6-(b^4+c^4-(7*b^2-30*b*c+7*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^4+c^4-6*(b^2+b*c+c^2)*b*c)*a-2*(b^2-c^2)^3*(b-c)*b*c) : :

X(22667) lies on these lines: {142,958}, {942,12513}, {1001,9856}, {1125,18237}, {1467,7091}, {6892,22775}, {9942,12114}, {9945,13205}, {9946,12773}, {12436,22777}

X(22668) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 1st BROCARD-REFLECTED

Barycentrics    (3*(b^2+c^2)*a^6-3*(b+c)*(b^2+c^2)*a^5-(b^4+c^4-2*b*c*(3*b^2+b*c+3*c^2))*a^4+3*(b+c)*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a^3-(2*b^4+2*c^4+b*c*(4*b^2+7*b*c+4*c^2))*(b-c)^2*a^2+3*(b^2-c^2)*(b-c)*b^2*c^2*a-(b^2-c^2)^2*b^2*c^2)*D+12*S^2*(-a+b+c)*a^2*((b^2+c^2)*a^2+b^2*c^2) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(22668) lies on these lines: {262,5597}, {511,11207}, {2782,12345}, {5598,22713}, {5599,15819}, {5601,6194}, {7697,8200}, {7709,11843}, {8186,22650}, {8190,22655}, {8196,22682}, {8197,22697}, {8198,22699}, {11366,22475}, {11384,22480}, {11492,22556}, {11493,22680}, {11822,22676}, {11837,22521}, {11861,22678}, {11865,22703}, {11867,22704}, {11869,22705}, {11871,22706}, {11873,22711}, {11875,22728}, {11877,22729}, {11879,22730}, {11881,22731}, {11883,22732}, {13890,22720}, {13944,22721}, {18495,22681}, {18955,18971}, {19007,19063}, {19008,19064}

X(22669) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO INNER-FERMAT

Barycentrics    (6*a*((b+c)*a-b^2-c^2)*S-sqrt(3)*(a+b+c)*(4*a^4-2*(b^2-5*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-(b+c)*(5*a^3+2*b^3+2*c^3-2*(b+c)*b*c)))*D+2*(3*(a^2+b^2+c^2)*S-10*S^2*sqrt(3))*(-a+b+c)*(a+b+c)*a^2 : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(22669) lies on these lines: {18,5597}, {628,5601}, {630,5599}, {5598,22867}, {5965,12452}, {8186,22651}, {8190,22656}, {8196,22831}, {8197,22851}, {8198,22853}, {8199,22854}, {8200,16627}, {11366,11740}, {11384,22481}, {11492,22557}, {11493,22771}, {11822,22843}, {11837,22522}, {11843,22531}, {11861,22745}, {11865,22857}, {11867,22858}, {11869,22859}, {11871,22860}, {11873,22865}, {11875,16628}, {11877,22884}, {11879,22885}, {11881,22886}, {11883,22887}, {13890,22876}, {13944,22877}, {18495,22794}, {18955,18972}, {19007,19069}, {19008,19072}

X(22670) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO OUTER-FERMAT

Barycentrics    (-6*a*((b+c)*a-b^2-c^2)*S-sqrt(3)*(a+b+c)*(4*a^4-2*(b^2-5*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-(b+c)*(5*a^3+2*b^3+2*c^3-2*(b+c)*b*c)))*D+2*(-3*(a^2+b^2+c^2)*S-10*S^2*sqrt(3))*(-a+b+c)*(a+b+c)*a^2 : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(22670) lies on these lines: {17,5597}, {532,11207}, {627,5601}, {629,5599}, {5598,22912}, {5965,12452}, {8186,22652}, {8190,22657}, {8196,22832}, {8197,22896}, {8198,22898}, {8199,22899}, {8200,16626}, {11366,11739}, {11384,22482}, {11492,22558}, {11493,22772}, {11822,22890}, {11837,22523}, {11843,22532}, {11861,22746}, {11865,22902}, {11867,22903}, {11869,22904}, {11871,22905}, {11873,22910}, {11875,16629}, {11877,22929}, {11879,22930}, {11881,22931}, {11883,22932}, {13890,22921}, {13944,22922}, {18495,22795}, {18955,18973}, {19007,19071}, {19008,19070}

X(22671) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 3rd HATZIPOLAKIS

Barycentrics    (2*a^16-(b+c)*a^15-2*(3*b^2-b*c+3*c^2)*a^14+(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(3*b^4+3*c^4-4*b*c*(b^2-7*b*c+c^2))*a^12-(b+c)*(b^4+c^4-b*c*(4*b^2-15*b*c+4*c^2))*a^11+2*(3*b^6+3*c^6-(b^4+c^4+b*c*(16*b^2-11*b*c+16*c^2))*b*c)*a^10-(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+b*c*(19*b^2-22*b*c+19*c^2))*b*c)*a^9-(5*b^8+5*c^8-2*(4*b^6+4*c^6-(5*b^4+5*c^4+b*c*(9*b^2-37*b*c+9*c^2))*b*c)*b*c)*a^8+(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6-(2*b^4+2*c^4+b*c*(9*b^2-23*b*c+9*c^2))*b*c)*b*c)*a^7-2*(b^8+c^8+(3*b^6+3*c^6-(9*b^4+9*c^4+2*b*c*(6*b^2-b*c+6*c^2))*b*c)*b*c)*(b-c)^2*a^6+(b^2-c^2)*(b-c)*(b^8+c^8+2*(2*b^6+2*c^6-7*(b^4+c^4+b*c*(b^2-b*c+c^2))*b*c)*b*c)*a^5+(b^2-c^2)^2*(b-c)^2*(b^6+c^6-(2*b^4+2*c^4+b*c*(b^2-14*b*c+c^2))*b*c)*a^4-(b^4-c^4)*(b^2-c^2)^2*(b-c)*(3*b^4+3*c^4+b*c*(2*b^2-7*b*c+2*c^2))*a^3+2*(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6)*D+4*S^2*(-a+b+c)*a^2*(a^12-2*(b^2+c^2)*a^10-(b^2-3*b*c-c^2)*(b^2+3*b*c-c^2)*a^8+(b^2+c^2)*(4*b^4-13*b^2*c^2+4*c^4)*a^6-(b^8+c^8+b^2*c^2*(9*b^4-28*b^2*c^2+9*c^4))*a^4-(b^4-c^4)*(b^2-c^2)*(2*b^4-9*b^2*c^2+2*c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(12241).

X(22671) lies on these lines: {55,22675}, {5597,22466}, {5598,22969}, {5599,22966}, {5601,22647}, {8186,22653}, {8190,22658}, {8196,22833}, {8197,22941}, {8198,22945}, {8199,22947}, {8200,22955}, {8201,22963}, {8202,22964}, {11366,22476}, {11384,22483}, {11492,22559}, {11493,22776}, {11822,22951}, {11837,22524}, {11843,22533}, {11861,22747}, {11863,22943}, {11865,22956}, {11867,22957}, {11869,22958}, {11871,22959}, {11873,22965}, {11875,22979}, {11877,22980}, {11879,22981}, {11881,22982}, {11883,22983}, {13890,22976}, {13944,22977}, {18495,22800}, {18955,18978}, {19007,19083}, {19008,19084}

X(22672) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 1st BROCARD-REFLECTED

Barycentrics    (-3*(b^2+c^2)*a^6+3*(b+c)*(b^2+c^2)*a^5+(b^4+c^4-2*b*c*(3*b^2+b*c+3*c^2))*a^4-3*(b+c)*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*a^3+(2*b^4+2*c^4+b*c*(4*b^2+7*b*c+4*c^2))*(b-c)^2*a^2-3*(b^2-c^2)*(b-c)*b^2*c^2*a+(b^2-c^2)^2*b^2*c^2)*D+12*S^2*(-a+b+c)*a^2*((b^2+c^2)*a^2+b^2*c^2) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(22672) lies on these lines: {262,5598}, {511,11208}, {2782,12346}, {5597,22713}, {5600,15819}, {5602,6194}, {7697,8207}, {7709,11844}, {8187,22650}, {8191,22655}, {8203,22682}, {8204,22697}, {8205,22699}, {8206,22700}, {11253,12477}, {11367,22475}, {11385,22480}, {11492,22680}, {11493,22556}, {11823,22676}, {11838,22521}, {11862,22678}, {11866,22703}, {11868,22704}, {11870,22705}, {11872,22706}, {11874,22711}, {11876,22728}, {11878,22729}, {11880,22730}, {11882,22731}, {11884,22732}, {13891,22720}, {13945,22721}, {18497,22681}, {18956,18971}, {19009,19063}, {19010,19064}

X(22673) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO INNER-FERMAT

Barycentrics    (6*a*((b+c)*a-b^2-c^2)*S+sqrt(3)*(a+b+c)*(-4*a^4+2*(b^2-5*b*c+c^2)*a^2+(5*b^2*c+5*b*c^2-5*c^3-5*b^3)*a+(b+c)*(5*a^3+2*(b^2-c^2)*(b-c))))*D-2*(3*(a^2+b^2+c^2)*S-10*S^2*sqrt(3))*(-a+b+c)*(a+b+c)*a^2 : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(3).

X(22673) lies on these lines: {18,5598}, {628,5602}, {630,5600}, {5597,22867}, {5965,12453}, {8187,22651}, {8191,22656}, {8203,22831}, {8204,22851}, {8205,22853}, {8206,22854}, {8207,16627}, {11367,11740}, {11385,22481}, {11492,22771}, {11493,22557}, {11823,22843}, {11838,22522}, {11844,22531}, {11862,22745}, {11866,22857}, {11868,22858}, {11870,22859}, {11872,22860}, {11874,22865}, {11876,16628}, {11878,22884}, {11880,22885}, {11882,22886}, {11884,22887}, {13891,22876}, {13945,22877}, {18497,22794}, {18956,18972}, {19009,19069}, {19010,19072}

X(22674) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO OUTER-FERMAT

Barycentrics    (-6*a*((b+c)*a-b^2-c^2)*S+sqrt(3)*(a+b+c)*(-4*a^4+2*(b^2-5*b*c+c^2)*a^2+(5*b^2*c+5*b*c^2-5*c^3-5*b^3)*a+(b+c)*(5*a^3+2*(b^2-c^2)*(b-c))))*D-2*(-3*(a^2+b^2+c^2)*S-10*S^2*sqrt(3))*(-a+b+c)*(a+b+c)*a^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22674) lies on these lines: {17,5598}, {532,11208}, {627,5602}, {629,5600}, {5597,22912}, {5965,12453}, {8187,22652}, {8191,22657}, {8203,22832}, {8204,22896}, {8205,22898}, {8206,22899}, {8207,16626}, {11367,11739}, {11385,22482}, {11492,22772}, {11493,22558}, {11823,22890}, {11838,22523}, {11844,22532}, {11862,22746}, {11866,22902}, {11868,22903}, {11870,22904}, {11872,22905}, {11874,22910}, {11876,16629}, {11878,22929}, {11880,22930}, {11882,22931}, {11884,22932}, {13891,22921}, {13945,22922}, {18497,22795}, {18956,18973}, {19009,19071}, {19010,19070}

X(22675) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 3rd HATZIPOLAKIS

Barycentrics    -(2*a^16-(b+c)*a^15-2*(3*b^2-b*c+3*c^2)*a^14+(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(3*b^4+3*c^4-4*b*c*(b^2-7*b*c+c^2))*a^12-(b+c)*(b^4+c^4-b*c*(4*b^2-15*b*c+4*c^2))*a^11+2*(3*b^6+3*c^6-(b^4+c^4+b*c*(16*b^2-11*b*c+16*c^2))*b*c)*a^10-(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+b*c*(19*b^2-22*b*c+19*c^2))*b*c)*a^9-(5*b^8+5*c^8-2*(4*b^6+4*c^6-(5*b^4+5*c^4+b*c*(9*b^2-37*b*c+9*c^2))*b*c)*b*c)*a^8+(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6-(2*b^4+2*c^4+b*c*(9*b^2-23*b*c+9*c^2))*b*c)*b*c)*a^7-2*(b^8+c^8+(3*b^6+3*c^6-(9*b^4+9*c^4+2*b*c*(6*b^2-b*c+6*c^2))*b*c)*b*c)*(b-c)^2*a^6+(b^2-c^2)*(b-c)*(b^8+c^8+2*(2*b^6+2*c^6-7*(b^4+c^4+b*c*(b^2-b*c+c^2))*b*c)*b*c)*a^5+(b^2-c^2)^2*(b-c)^2*(b^6+c^6-(2*b^4+2*c^4+b*c*(b^2-14*b*c+c^2))*b*c)*a^4-(b^4-c^4)*(b^2-c^2)^2*(b-c)*(3*b^4+3*c^4+b*c*(2*b^2-7*b*c+2*c^2))*a^3+2*(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6)*D+4*S^2*(-a+b+c)*a^2*(a^12-2*(b^2+c^2)*a^10-(b^2-3*b*c-c^2)*(b^2+3*b*c-c^2)*a^8+(b^2+c^2)*(4*b^4-13*b^2*c^2+4*c^4)*a^6-(b^8+c^8+b^2*c^2*(9*b^4-28*b^2*c^2+9*c^4))*a^4-(b^4-c^4)*(b^2-c^2)*(2*b^4-9*b^2*c^2+2*c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2) : : , where D=4*S*sqrt(R*(4*R+r))

The reciprocal orthologic center of these triangles is X(12241).

X(22675) lies on these lines: {55,22671}, {5597,22969}, {5598,22466}, {5600,22966}, {5602,22647}, {8187,22653}, {8191,22658}, {8203,22833}, {8204,22941}, {8205,22945}, {8206,22947}, {8207,22955}, {8208,22963}, {8209,22964}, {11367,22476}, {11385,22483}, {11492,22776}, {11493,22559}, {11823,22951}, {11838,22524}, {11844,22533}, {11862,22747}, {11864,22943}, {11866,22956}, {11868,22957}, {11870,22958}, {11872,22959}, {11874,22965}, {11876,22979}, {11878,22980}, {11880,22981}, {11882,22982}, {11884,22983}, {13891,22976}, {13945,22977}, {18497,22800}, {18956,18978}, {19009,19083}, {19010,19084}

X(22676) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 1st BROCARD-REFLECTED

Barycentrics    6*(b^2+c^2)*a^6-(4*b^4+b^2*c^2+4*c^4)*a^4-2*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(22676) = 8*X(3)-5*X(7786) = 7*X(3)-4*X(11272) = 5*X(3)-2*X(14881) = 3*X(3)-X(22728) = 2*X(20)+X(76) = X(20)+2*X(5188) = X(76)-4*X(5188) = 4*X(262)-5*X(7786) = 7*X(262)-8*X(11272) = 5*X(262)-4*X(14881) = 3*X(262)-2*X(22728) = 25*X(7786)-16*X(14881) = 15*X(7786)-8*X(22728) = 10*X(11272)-7*X(14881) = 12*X(11272)-7*X(22728) = 6*X(14881)-5*X(22728)

The reciprocal orthologic center of these triangles is X(3).

X(22676) lies on these lines: {2,22682}, {3,83}, {4,7831}, {20,76}, {30,7697}, {35,22729}, {36,22730}, {39,3522}, {55,18971}, {56,22711}, {69,15428}, {99,1350}, {147,7850}, {165,22650}, {182,22521}, {183,14532}, {315,7710}, {316,7694}, {371,19064}, {372,19063}, {376,511}, {382,22681}, {515,22697}, {517,22713}, {548,3095}, {550,9821}, {1078,9756}, {1503,7811}, {1513,7934}, {1593,22480}, {2023,5210}, {2782,3534}, {2794,9772}, {3098,10000}, {3146,3934}, {3428,22680}, {3528,13334}, {3529,6248}, {3576,22475}, {4297,7976}, {4316,10063}, {4324,10079}, {5085,12150}, {5092,10788}, {5171,7470}, {5999,7771}, {6179,9755}, {6284,22706}, {6661,21167}, {6683,15717}, {7354,22705}, {7768,8721}, {7782,22679}, {7803,9748}, {7828,9752}, {7884,9753}, {7926,9744}, {7937,13862}, {8350,18860}, {8703,11171}, {9466,15683}, {9540,22720}, {9778,14839}, {10304,21163}, {10310,22556}, {11055,15697}, {11248,22731}, {11249,22732}, {11261,14810}, {11299,22694}, {11300,22693}, {11414,22655}, {11822,22668}, {11823,22672}, {11824,22699}, {11825,22700}, {11826,22703}, {11827,22704}, {11828,22709}, {11829,22710}, {12251,17538}, {12512,12782}, {13935,22721}, {14927,14994}

X(22676) = midpoint of X(20) and X(6194)
X(22676) = reflection of X(i) in X(j) for these (i,j): (4, 15819), (382, 22681), (11261, 14810)
X(22676) = anticomplement of X(22682)
X(22676) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 5188, 76), (550, 9821, 11257), (5999, 8722, 7771)

X(22677) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO 1st BROCARD-REFLECTED

Barycentrics    ((2*b^2+2*c^2)^2-b^2*c^2)*a^6-3*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^4+(2*b^8+2*c^8-(5*b^4+6*b^2*c^2+5*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(22677) = 4*X(140)-X(13330) = 2*X(576)-5*X(7786) = X(1351)-4*X(10007) = X(6194)-3*X(10519) = 3*X(10516)-2*X(22681) = 4*X(11272)-X(11477)

The reciprocal orthologic center of these triangles is X(12177).

X(22677) lies on these lines: {2,51}, {3,5026}, {69,7709}, {114,9743}, {140,13330}, {141,7697}, {182,7771}, {384,22679}, {524,11171}, {575,7793}, {576,7786}, {599,2782}, {1351,10007}, {1352,7761}, {1469,22729}, {2896,11257}, {3056,22730}, {3094,15048}, {3098,10000}, {3314,9772}, {3785,13334}, {8179,8586}, {9751,12216}, {10008,14994}, {10516,22681}, {11179,21163}, {11272,11477}

X(22677) = midpoint of X(69) and X(7709)
X(22677) = reflection of X(11179) in X(21163)
X(22677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22503, 262), (22714, 22715, 22712), (22726, 22727, 6194)

X(22678) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 1st BROCARD-REFLECTED

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

The reciprocal orthologic center of these triangles is X(3).

X(22678) lies on these lines: {3,10333}, {4,2896}, {32,262}, {511,7811}, {2782,9878}, {3094,3269}, {3096,15819}, {3098,10000}, {3099,22650}, {5188,7849}, {7793,14881}, {7865,22712}, {9301,13860}, {9857,22697}, {9863,10335}, {9993,22682}, {9994,22699}, {9995,22700}, {9997,22713}, {10038,22729}, {10047,22730}, {10347,11261}, {10828,22655}, {10871,22703}, {10872,22704}, {10873,22705}, {10874,22706}, {10875,22709}, {10876,22710}, {10877,22711}, {10878,22731}, {10879,22732}, {11368,22475}, {11494,22556}, {11861,22668}, {11862,22672}, {11885,22698}, {13892,22720}, {13946,22721}, {18957,18971}, {19011,19063}, {19012,19064}, {22680,22744}

X(22678) = midpoint of X(9863) and X(10335)

X(22679) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^10-2*(5*b^4+8*b^2*c^2+5*c^4)*a^8+(b^2+c^2)*(5*b^4-13*b^2*c^2+5*c^4)*a^6+(4*b^8+4*c^8+(17*b^4+9*b^2*c^2+17*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(b^4+c^4)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*b^2*c^2 : :

The reciprocal orthologic center of these triangles is X(12177).

X(22679) lies on these lines: {3,22503}, {4,2896}, {262,5171}, {315,9772}, {384,22677}, {511,7833}, {2782,9939}, {5188,7752}, {7709,20065}, {7782,22676}, {7802,9863}, {10131,22525}, {11261,12110}, {15819,16921}

X(22680) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st BROCARD-REFLECTED

Barycentrics    a*(3*(b^2+c^2)*a^8-3*(b+c)*(b^2+c^2)*a^7-3*(2*b^4+2*c^4-(4*b^2-3*b*c+4*c^2)*b*c)*a^6+3*(b+c)*(2*b^4+2*c^4-(4*b^2-3*b*c+4*c^2)*b*c)*a^5+(3*b^6+3*c^6-(8*b^4+8*c^4-(15*b^2-2*b*c+15*c^2)*b*c)*b*c)*a^4-3*(b^2-c^2)*(b-c)*(b^4+c^4-2*(b^2+c^2)*b*c)*a^3-(4*b^4+4*c^4+(5*b^2+14*b*c+5*c^2)*b*c)*(b-c)^2*b*c*a^2-3*(b^2-c^2)*(b-c)^3*b^2*c^2*a-2*(b^2-c^2)^2*b^3*c^3) : :

The reciprocal orthologic center of these triangles is X(3).

X(22680) lies on these lines: {3,22556}, {36,22650}, {55,22713}, {56,262}, {104,7709}, {511,11194}, {956,22697}, {958,15819}, {999,22475}, {2782,22565}, {2975,6194}, {3428,22676}, {7697,22758}, {10966,22711}, {11249,22780}, {11492,22672}, {11493,22668}, {12114,22703}, {18761,22681}, {19013,19063}, {19014,19064}, {22479,22480}, {22504,22769}, {22520,22521}, {22654,22655}, {22678,22744}, {22682,22753}, {22698,22755}, {22699,22756}, {22700,22757}, {22705,22759}, {22706,22760}, {22709,22761}, {22710,22762}, {22720,22763}, {22721,22764}, {22728,22765}, {22729,22766}, {22730,22767}, {22731,22768}

X(22681) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^6+(b^4+4*b^2*c^2+c^4)*a^4-(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^2-5*(b^2-c^2)^2*b^2*c^2 : :
X(22681) = 3*X(4)+X(6194) = 5*X(4)+X(9821) = 7*X(5)-4*X(6683) = 5*X(5)-2*X(13334) = X(39)-4*X(3850) = X(76)+5*X(3843) = X(262)-3*X(381) = 7*X(262)-3*X(7757) = 7*X(381)-X(7757) = 2*X(546)+X(6248) = 4*X(546)-X(14881) = 5*X(3843)-X(22728) = X(6194)-3*X(7697) = 5*X(6194)-3*X(9821) = 2*X(6248)+X(14881) = 10*X(6683)-7*X(13334) = 5*X(7697)-X(9821)

The reciprocal orthologic center of these triangles is X(3).

X(22681) lies on these lines: {4,2896}, {5,4045}, {30,15810}, {39,3850}, {76,3843}, {262,381}, {382,22676}, {511,3845}, {546,6248}, {547,21163}, {732,18546}, {1478,22706}, {1479,22705}, {2023,18424}, {3091,7709}, {3095,3832}, {3545,11171}, {3583,22711}, {3585,18971}, {3627,3934}, {3818,22505}, {3830,22712}, {3851,11257}, {3853,5188}, {5072,7786}, {6321,9772}, {9466,14893}, {9755,10796}, {9756,12042}, {9818,22655}, {9955,22475}, {10516,22677}, {10895,22729}, {10896,22730}, {12699,22697}, {13330,22593}, {13665,19064}, {13785,19063}, {18491,22556}, {18492,22650}, {18495,22668}, {18497,22672}, {18502,22521}, {18507,22698}, {18509,22699}, {18511,22700}, {18516,22703}, {18517,22704}, {18520,22709}, {18522,22710}, {18525,22713}, {18538,22720}, {18542,22731}, {18544,22732}, {18761,22680}, {18762,22721}

X(22681) = midpoint of X(i) and X(j) for these {i,j}: {4, 7697}, {76, 22728}, {382, 22676}, {3830, 22712}, {6321, 9772}, {12699, 22697}, {18507, 22698}, {18525, 22713}
X(22681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (546, 6248, 14881), (22593, 22622, 13330)

X(22682) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^6+2*(b^4+4*b^2*c^2+c^4)*a^4-5*(b^4-c^4)*(b^2-c^2)*a^2-4*(b^2-c^2)^2*b^2*c^2 : :
X(22682) = 2*X(4)+X(39) = 3*X(4)+X(7709) = 5*X(4)+X(11257) = 4*X(5)-X(5188) = X(20)-4*X(6683) = 3*X(39)-2*X(7709) = 5*X(39)-2*X(11257) = X(76)-7*X(3832) = 3*X(262)-X(7709) = 5*X(262)-X(11257) = 3*X(381)-X(7697) = 4*X(381)-X(9466) = 3*X(381)+X(22728) = X(5052)-4*X(5480) = 4*X(7697)-3*X(9466) = 5*X(7709)-3*X(11257) = 3*X(9466)+4*X(22728)

The reciprocal orthologic center of these triangles is X(3).

X(22682) lies on these lines: {2,22676}, {4,39}, {5,5188}, {11,18971}, {12,22711}, {20,6683}, {30,21163}, {32,9756}, {76,3832}, {98,5008}, {115,5052}, {187,13860}, {235,22480}, {371,22720}, {372,22721}, {381,511}, {382,13334}, {515,22475}, {538,3839}, {546,6248}, {574,8719}, {625,13862}, {1352,7845}, {1478,22730}, {1479,22729}, {1503,7753}, {1513,7603}, {1587,19063}, {1588,19064}, {1598,22655}, {1699,14839}, {2782,3845}, {2794,14537}, {3091,3934}, {3095,3843}, {3146,7786}, {3202,11424}, {3545,22712}, {3627,11272}, {3714,19925}, {3767,9748}, {3830,11171}, {3851,9821}, {5007,9755}, {5097,12188}, {5309,14853}, {5587,22697}, {5603,22713}, {5999,7804}, {6201,22700}, {6202,22699}, {7470,9751}, {7746,9752}, {8196,22668}, {8203,22672}, {8212,22709}, {8213,22710}, {8589,11676}, {9765,9772}, {9993,22678}, {10531,22731}, {10532,22732}, {10893,22703}, {10894,22704}, {10895,22705}, {10896,22706}, {10991,18907}, {11477,17131}, {11496,22556}, {11897,22698}, {12110,21445}, {12263,12571}, {13335,18502}, {13354,15980}, {14492,14639}, {22680,22753}

X(22682) = midpoint of X(i) and X(j) for these {i,j}: {4, 262}, {3830, 11171}
X(22682) = complement of X(22676)
X(22682) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 22728, 7697), (546, 14881, 6248)

X(22683) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO INNER-FERMAT

Barycentrics    (5*a^4-9*(b^2+c^2)*a^2+2*(2*b^2-c^2)*(b^2-2*c^2))*a^2+2*sqrt(3)*(a^4+2*b^2*c^2)*S : :

The reciprocal orthologic center of these triangles is X(22684).

X(22683) lies on these lines: {2,18}, {3,22684}, {6,22526}, {62,99}, {575,22687}, {576,22689}, {3734,22234}, {5965,22737}, {6033,16627}

X(22683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3734, 22234, 22685), (22882, 22883, 22871)

X(22684) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 1st BROCARD-REFLECTED

Barycentrics    6*sqrt(3)*a^2*((b^2+c^2)*a^2-b^4-c^4)*S+(a^4-(b^2+c^2)*a^2-2*b^2*c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(22684) = 2*X(15819)-3*X(22714) = 4*X(15819)-3*X(22715) = X(22686)-3*X(22714) = 2*X(22686)-3*X(22715)

The reciprocal orthologic center of these triangles is X(22683).

X(22684) lies on these lines: {2,51}, {3,22683}, {398,3104}, {7697,16626}

X(22684) = anticomplement of X(33462)
X(22684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3104, 22702, 22707), (15819, 22686, 22715), (22686, 22714, 15819), (22726, 22727, 22714)

X(22685) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO OUTER-FERMAT

Barycentrics    (5*a^4-9*(b^2+c^2)*a^2+2*(2*b^2-c^2)*(b^2-2*c^2))*a^2-2*sqrt(3)*(a^4+2*b^2*c^2)*S : :

The reciprocal orthologic center of these triangles is X(22686).

X(22685) lies on these lines: {2,17}, {3,22686}, {6,22527}, {61,99}, {575,22689}, {576,22687}, {3734,22234}, {5965,22736}, {6033,16626}

X(22685) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3734, 22234, 22683), (22927, 22928, 22916)

X(22686) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 1st BROCARD-REFLECTED

Barycentrics    -6*sqrt(3)*a^2*((b^2+c^2)*a^2-b^4-c^4)*S+(a^4-(b^2+c^2)*a^2-2*b^2*c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(22686) = 4*X(15819)-3*X(22714) = 2*X(15819)-3*X(22715) = 2*X(22684)-3*X(22714) = X(22684)-3*X(22715)

The reciprocal orthologic center of these triangles is X(22685).

X(22686) lies on these lines: {2,51}, {3,22685}, {397,3105}, {7697,16627}

X(22686) = anticomplement of X(33463)
X(22686) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3105, 22701, 22708), (15819, 22684, 22714), (22684, 22715, 15819), (22726, 22727, 22715)

X(22687) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 3rd FERMAT-DAO

Barycentrics    3*(SA-2*SW)*S^2+3*SW*SB*SC-sqrt(3)*(SA^2+2*SB*SC+SW^2)*S : :
X(22687) = 3*X(5463)+X(23006) = 3*X(12155)-X(23006)

The reciprocal orthologic center of these triangles is X(22688).

X(22687) lies on these lines: {2,13}, {3,22715}, {6,12214}, {15,99}, {61,194}, {62,83}, {182,2782}, {298,22998}, {542,3642}, {575,22683}, {576,22685}, {619,8724}, {620,6771}, {621,6777}, {623,5617}, {627,7785}, {629,16627}, {1916,3106}, {2482,13083}, {5981,8289}, {6034,6772}, {7753,9115}, {9885,16508}, {11174,22691}, {11304,23005}, {11486,14535}, {14061,22846}, {22513,23025}

X(22687) = 1st-Brocard-isogonal conjugate of X(3642)
X(22687) = inverse of X(22689) in the Brocard circle
X(22687) = inverse of X(5979) in the inner-Napoleon circle
X(22687) = X(15)-of-1st-Brocard-triangle
X(22687) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 5463, 5979), (16, 5980, 6582), (182, 3734, 22689), (5463, 6779, 616), (6302, 6306, 6298)

X(22688) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4-(2*SA^2-5*SB*SC-5*SW^2)*S^2+5*SW^2*SB*SC-sqrt(3)*(S^2+SW*(2*SA+3*SW))*(SA-SW)*S : :
X(22688) = 2*X(13)+X(3106) = X(13)+2*X(22691) = X(3106)-4*X(22691) = 4*X(11542)-X(22701) = 5*X(16960)+X(22695)

The reciprocal orthologic center of these triangles is X(22687).

X(22688) lies on these lines: {13,262}, {17,3105}, {61,22693}, {511,16267}, {2782,5470}, {3107,7697}, {11272,16627}, {11542,22701}, {13331,22690}, {14651,22510}, {16808,22707}, {16960,22695}

X(22688) = {X(13), X(22691)}-harmonic conjugate of X(3106)

X(22689) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 4th FERMAT-DAO

Barycentrics    3*(SA-2*SW)*S^2+3*SW*SB*SC+sqrt(3)*(SA^2+2*SB*SC+SW^2)*S : :
X(22689) = 3*X(5464)+X(23013) = 3*X(12154)-X(23013)

The reciprocal orthologic center of these triangles is X(22690).

X(22689) lies on these lines: {2,14}, {3,22714}, {6,12213}, {16,99}, {61,83}, {62,194}, {182,2782}, {299,22997}, {542,3643}, {575,22685}, {576,22683}, {618,8724}, {620,6774}, {622,6778}, {624,5613}, {628,7785}, {630,16626}, {1916,3107}, {2482,13084}, {5980,8289}, {6034,6775}, {7753,9117}, {9886,16508}, {11174,22692}, {11303,23004}, {11485,14535}, {14061,22891}, {22512,23019}

X(22689) = inverse of X(22687) in the Brocard circle
X(22689) = inverse of X(5978) in the outer-Napoleon circle
X(22689) = X(16)-of-1st-Brocard-triangle
X(22689) = 1st-Brocard-isogonal conjugate of X(3643)
X(22689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 5464, 5978), (15, 5981, 6295), (182, 3734, 22687), (5464, 6780, 617), (6303, 6307, 6299)

X(22690) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4-(2*SA^2-5*SB*SC-5*SW^2)*S^2+5*SW^2*SB*SC+sqrt(3)*(S^2+SW*(2*SA+3*SW))*(SA-SW)*S : :
X(22690) = 2*X(14)+X(3107) = X(14)+2*X(22692) = X(3107)-4*X(22692) = 4*X(11543)-X(22702) = 5*X(16961)+X(22696)

The reciprocal orthologic center of these triangles is X(22689).

X(22690) lies on these lines: {14,262}, {18,3104}, {62,22694}, {511,16268}, {2782,5469}, {3106,7697}, {11272,16626}, {11543,22702}, {13331,22688}, {14651,22511}, {16809,22708}, {16961,22696}

X(22690) = {X(14), X(22692)}-harmonic conjugate of X(3107)

X(22691) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    2*S^4-(SA-2*SW)*SW*S^2+SW^2*SB*SC-sqrt(3)*(SA+SW)*(SA-SW)*SW*S : :
X(22691) = X(13)-3*X(22688) = X(3106)+3*X(22688) = 3*X(16267)-X(22701) = 3*X(16962)+X(22695)

The reciprocal orthologic center of these triangles is X(22687).

X(22691) lies on these lines: {2,3107}, {5,39}, {13,262}, {15,5999}, {17,1916}, {61,98}, {62,3329}, {381,22707}, {396,511}, {630,5976}, {3105,22712}, {5459,22573}, {5470,9760}, {5617,13331}, {6581,11305}, {6694,7792}, {6772,11171}, {7709,16635}, {7786,11290}, {10654,22693}, {11174,22687}, {13876,22724}, {13929,22725}, {15819,22892}, {16267,22701}, {16644,22715}, {16962,22695}

X(22691) = midpoint of X(i) and X(j) for these {i,j}: {13, 3106}, {6772, 22708}
X(22691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2009, 2010, 6115), (3106, 22688, 13)

X(22692) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    2*S^4-(SA-2*SW)*SW*S^2+SW^2*SB*SC+sqrt(3)*(SA+SW)*(SA-SW)*SW*S : :
X(22692) = X(14)-3*X(22690) = X(3107)+3*X(22690) = 3*X(16268)-X(22702) = 3*X(16963)+X(22696)

The reciprocal orthologic center of these triangles is X(22689).

X(22692) lies on these lines: {2,3106}, {5,39}, {14,262}, {16,5999}, {18,1916}, {61,3329}, {62,98}, {381,22708}, {395,511}, {629,5976}, {3104,22712}, {5460,22574}, {5469,9762}, {5613,13331}, {6294,11306}, {6695,7792}, {6775,11171}, {7709,16634}, {7786,11289}, {10653,22694}, {11174,22689}, {13875,22724}, {13928,22725}, {15819,22848}, {16268,22702}, {16645,22714}, {16963,22696}

X(22692) = midpoint of X(i) and X(j) for these {i,j}: {14, 3107}, {6775, 22707}
X(22692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2009, 2010, 6114), (3107, 22690, 14)

X(22693) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 11th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    (SA^2-4*SB*SC-SW^2)*S^2-4*SW^2*SB*SC-sqrt(3)*(S^2*SA-SW*SB*SC)*S : :
X(22693) = 2*X(4)+X(3104)

The reciprocal orthologic center of these triangles is X(22687).

X(22693) lies on these lines: {4,3104}, {5,22715}, {14,262}, {15,13860}, {39,5339}, {61,22688}, {381,511}, {623,13862}, {3095,16628}, {5321,22707}, {5480,22702}, {7685,9993}, {9753,22511}, {10654,22691}, {11300,22676}, {14881,16626}, {16808,22701}, {16809,22695}, {22512,22708}

X(22694) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 12th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    (SA^2-4*SB*SC-SW^2)*S^2-4*SW^2*SB*SC+sqrt(3)*(S^2*SA-SW*SB*SC)*S : :
X(22694) = 2*X(4)+X(3105)

The reciprocal orthologic center of these triangles is X(22689).

X(22694) lies on these lines: {4,3105}, {5,22714}, {13,262}, {16,13860}, {39,5340}, {62,22690}, {381,511}, {624,13862}, {3095,16629}, {5318,22708}, {5480,22701}, {7684,9993}, {9753,22510}, {10653,22692}, {11299,22676}, {14881,16627}, {16808,22696}, {16809,22702}, {22513,22707}

X(22695) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4+(4*SA^2-7*SB*SC-SW^2)*S^2-7*SW^2*SB*SC-sqrt(3)*(3*S^2-SW*(4*SA+SW))*(SA-SW)*S : :
X(22695) = 3*X(13)-2*X(22701) = 5*X(16960)-6*X(22688) = 3*X(16962)-4*X(22691)

The reciprocal orthologic center of these triangles is X(22687).

X(22695) lies on these lines: {13,511}, {16,10788}, {18,262}, {3104,22708}, {5475,22696}, {9762,21359}, {16809,22693}, {16960,22688}, {16962,22691}, {16964,22707}, {16966,22715}

X(22696) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4+(4*SA^2-7*SB*SC-SW^2)*S^2-7*SW^2*SB*SC+sqrt(3)*(3*S^2-SW*(4*SA+SW))*(SA-SW)*S : :
X(22696) = 3*X(14)-2*X(22702) = 5*X(16961)-6*X(22690) = 3*X(16963)-4*X(22692)

The reciprocal orthologic center of these triangles is X(22689).

X(22696) lies on these lines: {14,511}, {15,10788}, {17,262}, {3105,22707}, {5475,22695}, {9760,21360}, {16808,22694}, {16961,22690}, {16963,22692}, {16965,22708}, {16967,22714}

X(22697) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 1st BROCARD-REFLECTED

Barycentrics    3*(b+c)*(b^2+c^2)*a^5-(b^4+c^4+(6*b^2+b*c+6*c^2)*b*c)*a^4-3*(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3+(b^2+b*c+c^2)*(b^2-3*b*c+c^2)*(b+c)^2*a^2-3*(b^2-c^2)*(b-c)*b^2*c^2*a+2*(b^2-c^2)^2*b^2*c^2 : :
X(22697) = X(76)+2*X(11362) = 3*X(3679)-X(22650) = 4*X(3934)-X(7982) = 2*X(5188)+X(5881) = 3*X(5587)-2*X(22682) = 4*X(5690)-X(12782) = 3*X(5790)-X(22728) = 2*X(6248)+X(7991) = 4*X(6684)-X(7976) = 7*X(9588)-4*X(13334) = X(12245)+2*X(12263) = 3*X(22712)-X(22713)

The reciprocal orthologic center of these triangles is X(3).

X(22697) lies on these lines: {1,15819}, {2,22475}, {8,6194}, {10,262}, {65,22705}, {72,22704}, {76,11362}, {355,12783}, {511,3679}, {515,22676}, {517,7697}, {519,22712}, {956,22680}, {1018,6210}, {1737,22730}, {1837,22711}, {2782,3654}, {3057,22706}, {3416,9864}, {3934,7982}, {4424,7235}, {4737,4899}, {5090,22480}, {5188,5881}, {5252,18971}, {5587,22682}, {5687,22556}, {5688,22700}, {5689,22699}, {5690,12782}, {5790,22728}, {6248,7991}, {6684,7976}, {8193,22655}, {8197,22668}, {8204,22672}, {8214,22709}, {8215,22710}, {9588,13334}, {9755,12197}, {9857,22678}, {10039,22729}, {10791,22521}, {10914,22703}, {10915,22731}, {10916,22732}, {12245,12263}, {12699,22681}, {13883,19064}, {13893,22720}, {13936,19063}, {13947,22721}

X(22697) = midpoint of X(8) and X(6194)
X(22697) = reflection of X(i) in X(j) for these (i,j): (1, 15819), (12699, 22681)
X(22697) = anticomplement of X(22475)

X(22698) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 1st BROCARD-REFLECTED

Barycentrics    (S^2-3*SB*SC)*(6*S^4-(36*R^2*SA-7*SA^2+4*SB*SC-SW^2)*S^2-(12*R^2*(12*SW*R^2+3*SA^2+2*SW*SA-6*SW^2)-SW*(11*SA^2+4*SW*SA-9*SW^2))*SW) : :
X(22698) = X(76)+2*X(15774) = X(7709)-3*X(11845) = 3*X(11831)-2*X(22475) = 3*X(11852)-X(22650) = 3*X(11897)-2*X(22682) = 3*X(11911)-X(22728)

The reciprocal orthologic center of these triangles is X(3).

X(22698) lies on these lines: {30,7697}, {76,15774}, {262,402}, {511,1651}, {1650,15819}, {2782,12347}, {4240,6194}, {7709,11845}, {11251,12795}, {11831,22475}, {11832,22480}, {11839,22521}, {11848,22556}, {11852,22650}, {11853,22655}, {11885,22678}, {11897,22682}, {11901,22699}, {11902,22700}, {11903,22703}, {11904,22704}, {11905,22705}, {11906,22706}, {11907,22709}, {11908,22710}, {11909,22711}, {11910,22713}, {11911,22728}, {11912,22729}, {11913,22730}, {11914,22731}, {11915,22732}, {12181,12583}, {13894,22720}, {13948,22721}, {18507,22681}, {18958,18971}, {19017,19063}, {19018,19064}, {22680,22755}

X(22698) = midpoint of X(4240) and X(6194)
X(22698) = reflection of X(i) in X(j) for these (i,j): (262, 402), (1650, 15819), (18507, 22681)

X(22699) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 1st BROCARD-REFLECTED

Barycentrics    (3*SA-SW)*S^4+(SA-SW)*SW^2*S^2+2*SW^3*SB*SC-(S^4-SW*(SA-SW)*S^2+SW^2*SB*SC)*S : :
X(22699) = 4*X(5875)-X(6273)

The reciprocal orthologic center of these triangles is X(3).

X(22699) lies on these lines: {6,98}, {76,6281}, {511,5861}, {1161,6275}, {1271,6194}, {1352,22727}, {2782,9882}, {5589,22650}, {5591,15819}, {5595,22655}, {5605,22713}, {5689,22697}, {5875,6273}, {6202,22682}, {6215,7697}, {7709,10783}, {8198,22668}, {8205,22672}, {8216,22709}, {8217,22710}, {8974,22720}, {9994,22678}, {10040,22729}, {10048,22730}, {10792,22521}, {10919,22703}, {10921,22704}, {10923,22705}, {10925,22706}, {10927,22711}, {10929,22731}, {10931,22732}, {11370,22475}, {11388,22480}, {11497,22556}, {11824,22676}, {11901,22698}, {11916,22728}, {13949,22721}, {18509,22681}, {18959,18971}, {22680,22756}

X(22700) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 1st BROCARD-REFLECTED

Barycentrics    (3*SA-SW)*S^4+(SA-SW)*SW^2*S^2+2*SW^3*SB*SC+(S^4-SW*(SA-SW)*S^2+SW^2*SB*SC)*S : :
X(22700) = 4*X(5874)-X(6272)

The reciprocal orthologic center of these triangles is X(3).

X(22700) lies on these lines: {6,98}, {76,6278}, {511,5860}, {1160,6274}, {1270,6194}, {1352,22726}, {2782,9883}, {5588,22650}, {5590,15819}, {5594,22655}, {5604,22713}, {5688,22697}, {5874,6272}, {6201,22682}, {6214,7697}, {7709,10784}, {8199,22668}, {8206,22672}, {8218,22709}, {8219,22710}, {8975,22720}, {9995,22678}, {10041,22729}, {10049,22730}, {10793,22521}, {10920,22703}, {10922,22704}, {10924,22705}, {10926,22706}, {10928,22711}, {10930,22731}, {10932,22732}, {11371,22475}, {11389,22480}, {11498,22556}, {11825,22676}, {11902,22698}, {11917,22728}, {13950,22721}, {18511,22681}, {18960,18971}, {22680,22757}

X(22701) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4+(2*SA^2+SB*SC+SW^2)*S^2+SW^2*SB*SC-sqrt(3)*(3*S^2-SW*(2*SA-SW))*(SA-SW)*S : :
X(22701) = 3*X(13)-X(22695) = 4*X(11542)-3*X(22688) = 3*X(16267)-2*X(22691)

The reciprocal orthologic center of these triangles is X(22687).

X(22701) lies on these lines: {13,511}, {15,11676}, {16,22715}, {17,262}, {61,10796}, {62,385}, {396,3106}, {397,3105}, {2782,22997}, {3107,5464}, {5475,7697}, {5480,22694}, {11542,22688}, {16267,22691}, {16808,22693}, {22486,22494}

X(22701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7697, 13330, 22702), (22686, 22708, 3105)

X(22702) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 1st BROCARD-REFLECTED

Barycentrics    3*S^4+(2*SA^2+SB*SC+SW^2)*S^2+SW^2*SB*SC+sqrt(3)*(3*S^2-SW*(2*SA-SW))*(SA-SW)*S : :
X(22702) = 3*X(14)-X(22696) = 4*X(11543)-3*X(22690) = 3*X(16268)-2*X(22692)

The reciprocal orthologic center of these triangles is X(22689).

X(22702) lies on these lines: {14,511}, {15,22714}, {16,11676}, {18,262}, {61,385}, {62,10796}, {395,3107}, {398,3104}, {2782,22998}, {3106,5463}, {5475,7697}, {5480,22693}, {11543,22690}, {16268,22692}, {16809,22694}, {22486,22493}

X(22702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7697, 13330, 22701), (22684, 22707, 3104)

X(22703) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 1st BROCARD-REFLECTED

Barycentrics    (b^4+c^4-(6*b^2-b*c+6*c^2)*b*c)*a^7-(b+c)*(b^4+c^4-(6*b^2-b*c+6*c^2)*b*c)*a^6-2*(b^2+b*c+c^2)*(b^4+c^4-(3*b^2-7*b*c+3*c^2)*b*c)*a^5+2*(b^2-c^2)*(b-c)*(b^4+c^4-2*(b^2+c^2)*b*c)*a^4+(b^6+c^6+2*(2*b^4+2*c^4+(b^2+5*b*c+c^2)*b*c)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(b^6+c^6-2*(3*b^2-b*c+3*c^2)*b^2*c^2)*a^2+2*(b^2-c^2)*(b-c)*(b^3+c^3)*b^2*c^2*a-2*(b^2-c^2)^3*(b-c)*b^2*c^2 : :
X(22703) = 4*X(10943)-X(12923)

The reciprocal orthologic center of these triangles is X(3).

X(22703) lies on these lines: {11,262}, {12,22731}, {355,7697}, {511,11235}, {1376,15819}, {2782,12348}, {3434,6194}, {7709,10785}, {10523,22729}, {10525,12924}, {10794,22521}, {10826,22650}, {10829,22655}, {10871,22678}, {10893,22682}, {10914,22697}, {10919,22699}, {10920,22700}, {10943,12923}, {10944,22705}, {10945,22709}, {10946,22710}, {10947,22711}, {10948,22730}, {10949,22732}, {11373,22475}, {11390,22480}, {11826,22676}, {11865,22668}, {11866,22672}, {11903,22698}, {11928,22728}, {12114,22680}, {12182,12586}, {13895,22720}, {13952,22721}, {18516,22681}, {18961,18971}, {19023,19063}, {19024,19064}

X(22704) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 1st BROCARD-REFLECTED

Barycentrics    (b^4+c^4+(6*b^2+b*c+6*c^2)*b*c)*a^7-(b+c)*(b^4+c^4+(6*b^2+b*c+6*c^2)*b*c)*a^6-2*(b^2+3*b*c+c^2)*(b^4+c^4-(3*b^2-b*c+3*c^2)*b*c)*a^5+2*(b+c)*(b^6+c^6+(2*b^4+2*c^4-(7*b^2-2*b*c+7*c^2)*b*c)*b*c)*a^4+(b^2-c^2)^2*(b^4+c^4-3*(2*b^2+b*c+2*c^2)*b*c)*a^3-(b^2-c^2)^2*(b+c)*(b^4+c^4-(2*b^2+3*b*c+2*c^2)*b*c)*a^2+2*(b^2-c^2)^2*(b^2-3*b*c+c^2)*b^2*c^2*a-2*(b^2-c^2)^3*(b-c)*b^2*c^2 : :
X(22704) = 4*X(10942)-X(12933)

The reciprocal orthologic center of these triangles is X(3).

X(22704) lies on these lines: {11,22732}, {12,262}, {72,22697}, {355,7697}, {511,11236}, {958,15819}, {2782,12349}, {3436,6194}, {7709,10786}, {10523,22730}, {10526,12934}, {10795,22521}, {10827,22650}, {10830,22655}, {10872,22678}, {10894,22682}, {10921,22699}, {10922,22700}, {10942,12933}, {10950,22706}, {10951,22709}, {10952,22710}, {10953,22711}, {10954,22729}, {10955,22731}, {11374,22475}, {11391,22480}, {11500,22556}, {11827,22676}, {11867,22668}, {11868,22672}, {11904,22698}, {11929,22728}, {12183,12587}, {13896,22720}, {13953,22721}, {18517,22681}, {18962,18971}, {19025,19063}, {19026,19064}

X(22705) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 1st BROCARD-REFLECTED

Barycentrics    (b^4+c^4+(6*b^2+b*c+6*c^2)*b*c)*a^4-(b^2+b*c+c^2)*(b^2-3*b*c+c^2)*(b+c)^2*a^2-2*(b^2-c^2)^2*b^2*c^2 : :
X(22705) = 2*X(495)+X(10063) = 4*X(495)-X(12837) = 2*X(10063)+X(12837)

The reciprocal orthologic center of these triangles is X(3).

X(22705) lies on these lines: {1,7697}, {4,22711}, {5,22730}, {12,262}, {56,15819}, {65,22697}, {76,15888}, {388,6194}, {495,10063}, {511,11237}, {1478,12944}, {1479,22681}, {2782,10056}, {3023,9772}, {3085,7709}, {3303,6248}, {3304,3934}, {3584,11171}, {5188,9657}, {5270,9821}, {5434,22712}, {7354,22676}, {9578,22650}, {9654,22728}, {9755,10799}, {10797,22521}, {10831,22655}, {10873,22678}, {10895,22682}, {10944,22703}, {10956,22731}, {10957,22732}, {11375,22475}, {11392,22480}, {11501,22556}, {11869,22668}, {11870,22672}, {11905,22698}, {11930,22709}, {11931,22710}, {12184,12588}, {13897,22720}, {13954,22721}, {19027,19063}, {19028,19064}, {22680,22759}

X(22705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7697, 22706), (388, 6194, 18971), (495, 10063, 12837)

X(22706) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 1st BROCARD-REFLECTED

Barycentrics    (b^4+c^4-(6*b^2-b*c+6*c^2)*b*c)*a^4-(b^2+3*b*c+c^2)*(b^2-b*c+c^2)*(b-c)^2*a^2-2*(b^2-c^2)^2*b^2*c^2 : :
X(22706) = 2*X(496)+X(10079) = 4*X(496)-X(12836) = 2*X(10079)+X(12836)

The reciprocal orthologic center of these triangles is X(3).

X(22706) lies on these lines: {1,7697}, {4,18971}, {5,22729}, {11,262}, {55,15819}, {496,10079}, {497,6194}, {511,11238}, {1478,22681}, {1479,12954}, {2782,10072}, {3027,9772}, {3057,22697}, {3058,22712}, {3086,7709}, {3303,3934}, {3304,6248}, {3582,11171}, {4857,9821}, {5188,9670}, {6284,22676}, {9581,22650}, {9669,22728}, {9755,12835}, {10798,22521}, {10832,22655}, {10874,22678}, {10896,22682}, {10925,22699}, {10926,22700}, {10950,22704}, {10958,22731}, {10959,22732}, {11376,22475}, {11393,22480}, {11502,22556}, {11871,22668}, {11872,22672}, {11906,22698}, {11932,22709}, {11933,22710}, {12185,12589}, {13898,22720}, {13955,22721}, {14986,18982}, {19029,19063}, {19030,19064}, {22680,22760}

X(22706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7697, 22705), (496, 10079, 12836), (497, 6194, 22711)

X(22707) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 1st BROCARD-REFLECTED

Barycentrics    (SA^2+2*SB*SC-SW^2)*S^2+2*SW^2*SB*SC+sqrt(3)*(SA-SW)*(SA+SW)*SW*S : :

The reciprocal orthologic center of these triangles is X(22687).

X(22707) lies on these lines: {4,39}, {14,2782}, {15,22715}, {30,3107}, {61,10796}, {381,22691}, {398,3104}, {511,10654}, {736,6581}, {3094,22512}, {3105,22696}, {5321,22693}, {6775,11171}, {7804,10613}, {16808,22688}, {16964,22695}, {22513,22694}

X(22707) = reflection of X(6775) in X(22692)
X(22707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2549, 7709, 22708), (3104, 22702, 22684)

X(22708) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 1st BROCARD-REFLECTED

Barycentrics    (SA^2+2*SB*SC-SW^2)*S^2+2*SW^2*SB*SC-sqrt(3)*(SA-SW)*(SA+SW)*SW*S : :

The reciprocal orthologic center of these triangles is X(22689).

X(22708) lies on these lines: {4,39}, {13,2782}, {16,22714}, {30,3106}, {62,10796}, {381,22692}, {397,3105}, {511,10653}, {736,6294}, {3094,22513}, {3104,22695}, {5318,22694}, {6772,11171}, {7804,10614}, {16809,22690}, {16965,22696}, {22512,22693}

X(22708) = reflection of X(6772) in X(22691)
X(22708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2549, 7709, 22707), (3105, 22701, 22686)

X(22709) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st BROCARD-REFLECTED

Barycentrics    3*(SA^2-SW^2)*S^4+(2*SA^2+SB*SC-2*SW^2)*SW^2*S^2+SW^4*SB*SC-((4*R^2-3*SA)*S^4-8*SW^2*R^2*SB*SC+SW*(4*R^2*(2*SA+SW)-3*SA^2+3*SB*SC+3*SW^2)*S^2)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22709) lies on these lines: {262,493}, {511,12152}, {2782,12352}, {6194,6462}, {6461,22710}, {7697,8220}, {7709,11846}, {8188,22650}, {8194,22655}, {8210,22713}, {8212,22682}, {8214,22697}, {8216,22699}, {8218,22700}, {8222,15819}, {10669,12994}, {10875,22678}, {10945,22703}, {10951,22704}, {11377,22475}, {11394,22480}, {11503,22556}, {11828,22676}, {11840,22521}, {11907,22698}, {11930,22705}, {11932,22706}, {11947,22711}, {11949,22728}, {11951,22729}, {11953,22730}, {11955,22731}, {11957,22732}, {12186,12590}, {13899,22720}, {13956,22721}, {18520,22681}, {18963,18971}, {19031,19063}, {19032,19064}, {22680,22761}

X(22710) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st BROCARD-REFLECTED

Barycentrics    3*(SA^2-SW^2)*S^4+(2*SA^2+SB*SC-2*SW^2)*SW^2*S^2+SW^4*SB*SC+((4*R^2-3*SA)*S^4-8*SW^2*R^2*SB*SC+SW*(4*R^2*(2*SA+SW)-3*SA^2+3*SB*SC+3*SW^2)*S^2)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22710) lies on these lines: {262,494}, {511,12153}, {2782,12353}, {6194,6463}, {6461,22709}, {7697,8221}, {7709,11847}, {8189,22650}, {8195,22655}, {8211,22713}, {8213,22682}, {8215,22697}, {8217,22699}, {8219,22700}, {8223,15819}, {10673,12995}, {10876,22678}, {10946,22703}, {10952,22704}, {11378,22475}, {11395,22480}, {11504,22556}, {11829,22676}, {11841,22521}, {11908,22698}, {11931,22705}, {11933,22706}, {11948,22711}, {11950,22728}, {11952,22729}, {11954,22730}, {11956,22731}, {11958,22732}, {12187,12591}, {13900,22720}, {13957,22721}, {18522,22681}, {18964,18971}, {19033,19063}, {19034,19064}, {22680,22762}

X(22711) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^6-(b^4+c^4+2*(3*b^2-b*c+3*c^2)*b*c)*a^4-(2*b^4+2*c^4-(4*b^2-7*b*c+4*c^2)*b*c)*(b+c)^2*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(22711) = 2*X(6284)+X(18982) = X(13077)-4*X(15171)

The reciprocal orthologic center of these triangles is X(3).

X(22711) lies on these lines: {1,13078}, {3,22730}, {4,22705}, {11,15819}, {12,22682}, {33,22480}, {55,262}, {56,22676}, {76,9670}, {390,12837}, {497,6194}, {511,3058}, {1479,7697}, {1697,22650}, {1837,22697}, {2023,10987}, {2098,22713}, {2646,22475}, {2782,12354}, {3027,3056}, {3095,4309}, {3295,22728}, {3583,22681}, {3746,14881}, {4294,7709}, {6284,18982}, {9668,10063}, {9772,12185}, {10799,22521}, {10833,22655}, {10877,22678}, {10927,22699}, {10928,22700}, {10947,22703}, {10953,22704}, {10965,22731}, {10966,22680}, {11238,22712}, {11873,22668}, {11874,22672}, {11909,22698}, {11947,22709}, {11948,22710}, {13077,15171}, {13901,22720}, {13958,22721}, {19037,19063}, {19038,19064}

X(22711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 6194, 22706), (3295, 22728, 22729)

X(22712) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO 1st BROCARD-REFLECTED

Barycentrics    2*(b^2+c^2)*a^6-(2*b^4+b^2*c^2+2*c^4)*a^4-4*(b^2+c^2)*b^2*c^2*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(22712) = 2*X(3)+X(76) = 4*X(3)-X(11257) = 5*X(3)+X(13108) = X(4)-4*X(3934) = X(4)+2*X(5188) = 2*X(5)+X(9821) = X(20)+2*X(6248) = 2*X(76)+X(11257) = 5*X(76)-2*X(13108) = X(98)+2*X(5976) = X(262)+2*X(6194) = X(262)-4*X(15819) = 2*X(3934)+X(5188) = X(6194)+2*X(15819) = 5*X(11257)+4*X(13108)

The reciprocal orthologic center of these triangles is X(99).

X(22712) lies on these lines: {2,51}, {3,76}, {4,3934}, {5,3096}, {20,6248}, {30,7697}, {35,10079}, {36,10063}, {39,631}, {40,12263}, {69,9744}, {114,3314}, {140,3095}, {141,1513}, {182,385}, {194,3523}, {230,3094}, {264,22062}, {371,19089}, {372,19090}, {376,9466}, {383,3642}, {384,5171}, {519,22697}, {538,3524}, {542,8592}, {543,11167}, {549,7757}, {575,7766}, {576,3329}, {599,6054}, {698,13468}, {726,10164}, {732,5085}, {736,21445}, {842,9832}, {1080,3643}, {1350,13860}, {1351,11174}, {1352,16990}, {1385,7976}, {1587,8992}, {1588,13983}, {1656,7944}, {1799,3425}, {1916,6036}, {2021,21843}, {2080,3972}, {2709,5108}, {2794,7810}, {3058,22706}, {3098,5999}, {3102,5420}, {3103,5418}, {3104,22692}, {3105,22691}, {3106,16242}, {3107,16241}, {3398,6179}, {3399,6680}, {3406,8150}, {3515,12143}, {3525,6683}, {3526,11272}, {3545,22682}, {3582,22730}, {3584,22729}, {3734,8722}, {3815,13330}, {3830,22681}, {4108,8704}, {5007,10359}, {5050,14614}, {5052,7736}, {5055,22728}, {5064,22480}, {5092,8350}, {5204,18982}, {5217,13077}, {5306,13331}, {5432,12837}, {5433,12836}, {5434,22705}, {5657,14839}, {5969,7610}, {5987,12584}, {6309,7751}, {6684,12782}, {6776,14994}, {7422,18304}, {7616,8782}, {7746,10357}, {7770,12110}, {7780,8149}, {7793,13335}, {7804,10788}, {7815,18806}, {7824,9737}, {7841,14639}, {7846,20576}, {7865,22678}, {7870,15561}, {7898,13449}, {7987,9902}, {8556,9756}, {9301,10347}, {9769,15035}, {10267,13110}, {10269,13109}, {11055,15693}, {11151,11152}, {11237,18971}, {11238,22711}, {11672,14252}, {12007,15480}, {13083,21156}, {13084,21157}, {13086,14651}, {13862,16986}, {14711,15698}, {15717,20081}, {19875,22650}

X(22712) = midpoint of X(2) and X(6194)
X(22712) = reflection of X(i) in X(j) for these (i,j): (2, 15819), (3830, 22681)
X(22712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 76, 11257), (3, 183, 98), (140, 3095, 7786), (183, 5976, 76), (194, 3523, 13334), (631, 12251, 39), (1350, 15271, 13860), (3524, 7709, 21163), (3734, 8722, 11676), (3934, 5188, 4), (5085, 8667, 9755), (5980, 5981, 7771), (6194, 15819, 262), (22714, 22715, 22677)

X(22713) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 1st BROCARD-REFLECTED

Barycentrics    6*(b^2+c^2)*a^6-6*(b+c)*(b^2+c^2)*a^5-(2*b^2-7*b*c+2*c^2)*(2*b^2+b*c+2*c^2)*a^4+6*(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3-2*(b^4+c^4+(b+2*c)*(2*b+c)*b*c)*(b-c)^2*a^2+6*(b^2-c^2)*(b-c)*b^2*c^2*a-(b^2-c^2)^2*b^2*c^2 : :
X(22713) = 3*X(1)-2*X(22475) = 3*X(1)-X(22650) = 3*X(262)-4*X(22475) = 3*X(262)-2*X(22650) = 4*X(1483)-X(7976) = 3*X(5603)-2*X(22682) = 4*X(5882)-X(11257) = X(7709)-3*X(7967) = 5*X(7786)-8*X(15178) = 3*X(10247)-X(22728) = X(12782)-4*X(13607) = 2*X(22697)-3*X(22712)

The reciprocal orthologic center of these triangles is X(3).

X(22713) lies on these lines: {1,262}, {8,15819}, {55,22680}, {56,22556}, {145,6194}, {511,3241}, {517,22676}, {519,22697}, {952,7697}, {1482,7977}, {1483,7976}, {2098,22711}, {2099,18971}, {2782,9884}, {3242,7970}, {5597,22672}, {5598,22668}, {5603,22682}, {5604,22700}, {5605,22699}, {5882,11257}, {7709,7967}, {7786,15178}, {7968,19063}, {7969,19064}, {7972,10063}, {8192,22655}, {8210,22709}, {8211,22710}, {9997,22678}, {10247,22728}, {10800,22521}, {10944,22703}, {10950,22704}, {11396,22480}, {11910,22698}, {12782,13607}, {13902,22720}, {13959,22721}, {18525,22681}

X(22713) = midpoint of X(145) and X(6194)
X(22713) = reflection of X(i) in X(j) for these (i,j): (8, 15819), (18525, 22681)
X(22713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22650, 22475), (22475, 22650, 262), (22731, 22732, 262)

X(22714) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO 1st BROCARD-REFLECTED

Barycentrics    2*sqrt(3)*S*((b^2+c^2)*a^2-b^4-c^4)*a^2+(a^4-(b^2+c^2)*a^2-2*b^2*c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(22714) = 4*X(140)-X(3105) = 2*X(15819)+X(22684) = 4*X(15819)-X(22686) = 2*X(22684)+X(22686)

The reciprocal orthologic center of these triangles is X(22689).

X(22714) lies on these lines: {2,51}, {3,22689}, {5,22694}, {15,22702}, {16,22708}, {18,3104}, {76,627}, {140,3105}, {182,5980}, {302,23024}, {395,3106}, {2782,5463}, {3107,16242}, {3643,5617}, {5613,9749}, {7709,14145}, {7761,20428}, {7771,13350}, {9885,11171}, {16645,22692}, {16967,22696}

X(22714) = anticomplement of X(33479)
X(22714) = outer-Napoleon-circle-inverse of X(33873)
X(22714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15819, 22684, 22686), (22677, 22712, 22715), (22726, 22727, 22684)

X(22715) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-NAPOLEON TO 1st BROCARD-REFLECTED

Barycentrics    -2*sqrt(3)*S*((b^2+c^2)*a^2-b^4-c^4)*a^2+(a^4-(b^2+c^2)*a^2-2*b^2*c^2)*(3*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(22715) = 4*X(140)-X(3104) = 4*X(15819)-X(22684) = 2*X(15819)+X(22686) = X(22684)+2*X(22686)

The reciprocal orthologic center of these triangles is X(22687).

X(22715) lies on these lines: {2,51}, {3,22687}, {5,22693}, {15,22707}, {16,22701}, {17,3105}, {76,628}, {140,3104}, {182,5981}, {303,23018}, {396,3107}, {2782,5464}, {3106,16241}, {3642,5613}, {5617,9750}, {7709,14144}, {7761,20429}, {7771,13349}, {9886,11171}, {16644,22691}, {16966,22695}

X(22715) = anticomplement of X(33478)
X(22715) = inner-Napoleon-circle-inverse of X(33873)
X(22715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15819, 22686, 22684), (22677, 22712, 22714), (22726, 22727, 22686)

X(22716) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (SA-4*SW)*S^2+3*SW*SB*SC-(SA^2+2*SB*SC+SW^2)*S : :
X(22716) = X(6560)-3*X(13669) = X(6560)+3*X(13712)

The reciprocal orthologic center of these triangles is X(22717).

X(22716) lies on these lines: {2,1327}, {6,13673}, {83,372}, {99,6200}, {182,22594}, {194,371}, {511,22623}, {639,21737}, {642,14244}, {3734,5092}, {5591,13674}, {6033,13692}, {6312,12968}, {6398,14535}, {6411,13828}, {9892,16508}, {12124,21736}

X(22716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13712, 13710), (3734, 5092, 22718)

X(22717) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO 1st BROCARD-REFLECTED

Barycentrics    2*S^4-(SA-2*SW)*SW*S^2+SW^2*SB*SC-(SA-SW)*(S^2+SW*(SA+2*SW))*S : :

The reciprocal orthologic center of these triangles is X(22716).

X(22717) lies on these lines: {39,485}, {76,3316}, {511,3068}, {590,3103}, {638,7786}, {1352,7736}, {2023,6230}, {3102,7583}, {3312,6312}, {6314,8976}, {19064,22723}, {22720,22727}, {22724,22726}

X(22718) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    (SA-4*SW)*S^2+3*SW*SB*SC+(SA^2+2*SB*SC+SW^2)*S : :
X(22718) = X(6561)-3*X(13789) = X(6561)+3*X(13835)

The reciprocal orthologic center of these triangles is X(22719).

X(22718) lies on these lines: {2,1328}, {6,13793}, {83,371}, {99,6396}, {182,22623}, {194,372}, {511,22594}, {641,14229}, {3734,5092}, {5590,13794}, {6033,13812}, {6221,14535}, {6316,12963}, {6412,13708}, {9894,16508}

X(22718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13835, 13830), (3734, 5092, 22716)

X(22719) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO 1st BROCARD-REFLECTED

Barycentrics    2*S^4-(SA-2*SW)*SW*S^2+SW^2*SB*SC+(SA-SW)*(S^2+SW*(SA+2*SW))*S : :

The reciprocal orthologic center of these triangles is X(22718).

X(22719) lies on these lines: {39,486}, {76,3317}, {511,3069}, {615,3102}, {637,7786}, {1352,7736}, {2023,6231}, {3103,7584}, {3311,6316}, {6318,13951}, {19063,22722}, {22721,22726}, {22725,22727}

X(22720) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 1st BROCARD-REFLECTED

Barycentrics    4*S^4-(SA-4*SW)*SW*S^2+SW^2*SB*SC-3*(S^2+SW^2)*(SA-SW)*S : :
X(22720) = X(8992)-4*X(13925)

The reciprocal orthologic center of these triangles is X(3).

X(22720) lies on these lines: {2,19064}, {39,8960}, {262,3068}, {371,22682}, {511,13846}, {590,15819}, {2782,13908}, {3316,19089}, {6194,8972}, {7585,19063}, {7697,8976}, {7709,13886}, {8974,22699}, {8975,22700}, {8980,13910}, {8981,8993}, {8992,13925}, {9540,22676}, {13883,22475}, {13884,22480}, {13885,22521}, {13887,22556}, {13888,22650}, {13889,22655}, {13891,22672}, {13892,22678}, {13893,22697}, {13894,22698}, {13895,22703}, {13896,22704}, {13897,22705}, {13898,22706}, {13899,22709}, {13900,22710}, {13901,22711}, {13902,22713}, {13903,22728}, {13904,22729}, {13905,22730}, {13906,22731}, {13907,22732}, {18538,22681}, {18965,18971}, {22680,22763}, {22717,22727}

X(22721) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 1st BROCARD-REFLECTED

Barycentrics    4*S^4-(SA-4*SW)*SW*S^2+SW^2*SB*SC+3*(S^2+SW^2)*(SA-SW)*S : :
X(22721) = X(13983)-4*X(13993)

The reciprocal orthologic center of these triangles is X(3).

X(22721) lies on these lines: {2,19063}, {6,22720}, {262,3069}, {372,22682}, {511,13847}, {615,15819}, {2782,13968}, {3317,19090}, {6194,13941}, {7586,19064}, {7697,13951}, {7709,13939}, {13935,22676}, {13936,22475}, {13937,22480}, {13938,22521}, {13940,22556}, {13942,22650}, {13943,22655}, {13944,22668}, {13945,22672}, {13946,22678}, {13947,22697}, {13948,22698}, {13949,22699}, {13950,22700}, {13952,22703}, {13953,22704}, {13954,22705}, {13955,22706}, {13956,22709}, {13957,22710}, {13958,22711}, {13959,22713}, {13961,22728}, {13962,22729}, {13963,22730}, {13964,22731}, {13965,22732}, {13966,13984}, {13967,13972}, {13983,13993}, {18762,22681}, {18966,18971}, {22680,22764}, {22719,22726}

X(22722) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO 1st BROCARD-REFLECTED

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

The reciprocal orthologic center of these triangles is X(6).

X(22722) lies on these lines: {2,3787}, {39,7585}, {76,13707}, {262,13638}, {371,12110}, {372,13885}, {511,3068}, {538,13639}, {590,13330}, {698,13647}, {732,13648}, {1271,3934}, {2782,13640}, {3103,19090}, {5058,7878}, {5062,6179}, {5861,14994}, {5969,13642}, {6272,13877}, {13637,22486}, {19063,22719}, {19064,22727}

X(22722) = reflection of X(76) in X(13707)
X(22722) = {X(2), X(5052)}-harmonic conjugate of X(22723)

X(22723) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO 1st BROCARD-REFLECTED

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

The reciprocal orthologic center of these triangles is X(6).

X(22723) lies on these lines: {2,3787}, {39,7586}, {76,13827}, {262,13758}, {371,13938}, {372,12110}, {511,3069}, {538,13759}, {615,13330}, {698,13766}, {732,13767}, {1270,3934}, {2782,13760}, {3102,19089}, {5058,6179}, {5062,7878}, {5860,14994}, {5969,13761}, {6273,13930}, {13757,22486}, {19063,22726}, {19064,22717}

X(22723) = reflection of X(76) in X(13827)
X(22723) = {X(2), X(5052)}-harmonic conjugate of X(22722)

X(22724) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO 1st BROCARD-REFLECTED

Barycentrics    4*S^4-(SA-4*SW)*SW*S^2+SW^2*SB*SC-3*(SA^2-SW^2)*SW*S : :
X(22724) = 2*X(485)+X(3102) = 3*X(485)-X(22622) = 3*X(3102)+2*X(22622) = 4*X(6118)-X(13877)

The reciprocal orthologic center of these triangles is X(22623).

X(22724) lies on these lines: {39,1656}, {262,485}, {371,6222}, {511,13846}, {590,13926}, {641,13878}, {3103,13882}, {6118,13877}, {13875,22692}, {13876,22691}, {22717,22726}

X(22725) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO 1st BROCARD-REFLECTED

Barycentrics    4*S^4-(SA-4*SW)*SW*S^2+SW^2*SB*SC+3*(SA^2-SW^2)*SW*S : :
X(22725) = 2*X(486)+X(3103) = 3*X(486)-X(22593) = 3*X(3103)+2*X(22593)

The reciprocal orthologic center of these triangles is X(22594).

X(22725) lies on these lines: {39,1656}, {262,486}, {372,6399}, {511,13847}, {615,13873}, {642,13931}, {3102,13934}, {11174,22594}, {13928,22692}, {13929,22691}, {22719,22727}

X(22726) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-VECTEN TO 1st BROCARD-REFLECTED

Barycentrics    2*S^4+(SA+2*SW)*SW*S^2-SW^2*SB*SC+3*(S^2*SA-SW*SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(22594).

X(22726) lies on these lines: {2,51}, {3,22594}, {39,19102}, {182,10852}, {615,13873}, {1352,22700}, {3102,7584}, {6228,6289}, {6561,22593}, {11171,13700}, {19063,22723}, {22717,22724}, {22719,22721}

X(22726) = complement of X(33435)
X(22726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6194, 22677, 22727), (22684, 22714, 22727), (22686, 22715, 22727)

X(22727) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-VECTEN TO 1st BROCARD-REFLECTED

Barycentrics    2*S^4+(SA+2*SW)*SW*S^2-SW^2*SB*SC-3*(S^2*SA-SW*SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(22623).

X(22727) lies on these lines: {2,51}, {39,19105}, {182,10851}, {590,13926}, {1352,22699}, {3103,7583}, {6229,6290}, {6560,22622}, {11171,13820}, {19064,22722}, {22717,22720}, {22719,22725}

X(22727) = complement of X(33434)
X(22727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6194, 22677, 22726), (22684, 22714, 22726), (22686, 22715, 22726)

X(22728) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 1st BROCARD-REFLECTED

Barycentrics    3*(b^2+c^2)*a^6+(b^4+7*b^2*c^2+c^4)*a^4-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :
X(22728) = 7*X(3)-10*X(7786) = 5*X(3)-8*X(11272) = X(3)-4*X(14881) = 3*X(3)-2*X(22676) = 4*X(4)-X(13108) = 7*X(4)-X(20081) = 7*X(262)-5*X(7786) = 5*X(262)-4*X(11272) = 3*X(262)-X(22676) = 5*X(7786)-14*X(14881) = 15*X(7786)-7*X(22676) = 2*X(11272)-5*X(14881) = 12*X(11272)-5*X(22676) = 7*X(13108)-4*X(20081) = 6*X(14881)-X(22676)

The reciprocal orthologic center of these triangles is X(3).

X(22728) lies on these lines: {3,83}, {4,7779}, {5,6194}, {30,7709}, {39,1657}, {76,3843}, {194,3627}, {381,511}, {382,3095}, {517,22650}, {546,12251}, {999,18971}, {1350,11261}, {1351,12188}, {1384,2023}, {1598,22480}, {1656,7914}, {2782,3830}, {3094,15484}, {3104,5340}, {3105,5339}, {3295,22711}, {3526,5188}, {3534,11171}, {3934,5072}, {5055,22712}, {5073,11257}, {5475,22695}, {5790,22697}, {5999,11842}, {6417,19064}, {6418,19063}, {7517,22655}, {7757,15684}, {7785,10335}, {9301,13860}, {9654,22705}, {9655,12836}, {9668,12837}, {9669,22706}, {10007,14535}, {10246,22475}, {10247,22713}, {11849,22556}, {11875,22668}, {11876,22672}, {11911,22698}, {11916,22699}, {11917,22700}, {11928,22703}, {11929,22704}, {11949,22709}, {11950,22710}, {12000,22731}, {12001,22732}, {13334,15696}, {13903,22720}, {13961,22721}, {15688,21163}, {15980,21850}, {22680,22765}

X(22728) = reflection of X(i) in X(j) for these (i,j): (3, 262), (76, 22681), (1350, 11261), (3534, 11171)
X(22728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13111, 18501), (7697, 22682, 381), (18971, 22730, 999), (22711, 22729, 3295)

X(22729) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1st BROCARD-REFLECTED

Barycentrics    (2*b^4+2*c^4+(6*b^2+5*b*c+6*c^2)*b*c)*a^4-2*(b^2+b*c+c^2)*(b^4+c^4-(b^2+b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(22729) = 4*X(495)-X(10063) = 2*X(495)+X(12837) = X(10063)+2*X(12837)

The reciprocal orthologic center of these triangles is X(3).

X(22729) lies on these lines: {1,262}, {3,18971}, {5,22706}, {12,7697}, {35,22676}, {55,10064}, {388,7709}, {495,10063}, {498,15819}, {511,10056}, {611,10053}, {1469,22677}, {1479,22682}, {2782,10054}, {3085,6194}, {3095,15888}, {3295,22711}, {3299,19063}, {3301,19064}, {3303,14881}, {3304,11272}, {3584,22712}, {4317,13334}, {5270,11257}, {5434,11171}, {5563,7786}, {9654,13077}, {10037,22655}, {10038,22678}, {10039,22697}, {10040,22699}, {10041,22700}, {10523,22703}, {10801,22521}, {10802,21445}, {10895,22681}, {10954,22704}, {11398,22480}, {11507,22556}, {11877,22668}, {11878,22672}, {11912,22698}, {11951,22709}, {11952,22710}, {12782,13407}, {13904,22720}, {13962,22721}, {22680,22766}

X(22729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 262, 22730), (495, 12837, 10063), (3295, 22728, 22711)

X(22730) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1st BROCARD-REFLECTED

Barycentrics    (2*b^4+2*c^4-(6*b^2-5*b*c+6*c^2)*b*c)*a^4-2*(b^2-b*c+c^2)*(b^4+c^4+(b^2-b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(22730) = 4*X(496)-X(10079) = 2*X(496)+X(12836) = X(10079)+2*X(12836)

The reciprocal orthologic center of these triangles is X(3).

X(22730) lies on these lines: {1,262}, {3,22711}, {5,22705}, {11,7697}, {36,22676}, {56,10080}, {496,10079}, {497,7709}, {499,15819}, {511,10072}, {613,10069}, {999,18971}, {1478,22682}, {1737,22697}, {2782,10070}, {3056,22677}, {3058,11171}, {3086,6194}, {3299,19064}, {3301,19063}, {3303,11272}, {3304,14881}, {3582,22712}, {3746,7786}, {4309,13334}, {4857,11257}, {9669,18982}, {10046,22655}, {10047,22678}, {10048,22699}, {10049,22700}, {10523,22704}, {10801,21445}, {10802,22521}, {10896,22681}, {10948,22703}, {11399,22480}, {11508,22556}, {11879,22668}, {11880,22672}, {11913,22698}, {11953,22709}, {11954,22710}, {13905,22720}, {13963,22721}, {22680,22767}

X(22730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 262, 22729), (496, 12836, 10079), (999, 22728, 18971)

X(22731) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 1st BROCARD-REFLECTED

Barycentrics    (2*b^4+2*c^4+(12*b^2+5*b*c+12*c^2)*b*c)*a^7-(b+c)*(2*b^4+2*c^4+(12*b^2+5*b*c+12*c^2)*b*c)*a^6-(2*b^2+b*c+2*c^2)*(2*b^4+2*c^4+(b^2-12*b*c+c^2)*b*c)*a^5+(b+c)*(4*b^6+4*c^6+(8*b^4+8*c^4-(19*b^2-2*b*c+19*c^2)*b*c)*b*c)*a^4+(2*b^6+2*c^6-(4*b^4+4*c^4+11*(b+c)^2*b*c)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(2*b^6+2*c^6-(3*b^2+14*b*c+3*c^2)*b^2*c^2)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*b^2*c^2*a-(b^2-c^2)^3*(b-c)*b^2*c^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22731) lies on these lines: {1,262}, {12,22703}, {511,11239}, {2782,12356}, {5552,15819}, {6194,10528}, {7697,10942}, {7709,10805}, {10531,22682}, {10679,13112}, {10803,22521}, {10834,22655}, {10878,22678}, {10915,22697}, {10929,22699}, {10930,22700}, {10955,22704}, {10956,22705}, {10958,22706}, {10965,22711}, {11248,22676}, {11400,22480}, {11509,18971}, {11881,22668}, {11882,22672}, {11914,22698}, {11955,22709}, {11956,22710}, {12000,22728}, {12189,12594}, {13906,22720}, {13964,22721}, {18542,22681}, {19047,19063}, {19048,19064}, {22680,22768}

X(22731) = {X(262), X(22713)}-harmonic conjugate of X(22732)

X(22732) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 1st BROCARD-REFLECTED

Barycentrics    (2*b^4+2*c^4-(12*b^2-5*b*c+12*c^2)*b*c)*a^7-(b+c)*(2*b^4+2*c^4-(12*b^2-5*b*c+12*c^2)*b*c)*a^6-(2*b^2+b*c+2*c^2)*(2*b^4+2*c^4-(7*b^2-16*b*c+7*c^2)*b*c)*a^5+(b+c)*(4*b^6+4*c^6-(16*b^4+16*c^4-(29*b^2-22*b*c+29*c^2)*b*c)*b*c)*a^4+(2*b^6+2*c^6+(4*b^2+b*c+c^2)*(b^2+b*c+4*c^2)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(2*b^6+2*c^6-(3*b-c)*(b-3*c)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2*a-(b^2-c^2)^3*(b-c)*b^2*c^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22732) lies on these lines: {1,262}, {11,22704}, {511,11240}, {2782,12357}, {6194,10529}, {7697,10943}, {7709,10806}, {10527,15819}, {10532,22682}, {10680,13113}, {10804,22521}, {10835,22655}, {10879,22678}, {10916,22697}, {10931,22699}, {10932,22700}, {10949,22703}, {10957,22705}, {10959,22706}, {10966,22680}, {11249,22676}, {11401,22480}, {11510,22556}, {11883,22668}, {11884,22672}, {11915,22698}, {11957,22709}, {11958,22710}, {12001,22728}, {12190,12595}, {13907,22720}, {13965,22721}, {18544,22681}, {18967,18971}, {19049,19063}, {19050,19064}

X(22732) = {X(262), X(22713)}-harmonic conjugate of X(22731)

X(22733) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 1st BROCARD-REFLECTED

Barycentrics    (a^8+7*(b^2+c^2)*a^6-4*(b^4+b^2*c^2+c^4)*a^4+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2+(b^4-b^2*c^2+c^4)*b^2*c^2)*(b^2-c^2) : :
X(22733) = 2*X(9131)+X(13307) = X(13306)-4*X(14610)

The reciprocal parallelogic center of these triangles is X(3).

X(22733) lies on these lines: {351,13308}, {512,9123}, {804,8592}, {5466,14327}, {9131,13307}, {9135,9147}, {13306,14610}

X(22734) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 1st BROCARD-REFLECTED

Barycentrics    (a^8-5*(b^2+c^2)*a^6+2*(b^2-c^2)^2*a^4+2*(b^2+c^2)*(b^4+c^4)*a^2+(b^4-b^2*c^2+c^4)*b^2*c^2)*(b^2-c^2) : :
X(22734) = 2*X(9979)+X(13306)

The reciprocal parallelogic center of these triangles is X(3).

X(22734) lies on these lines: {2,3569}, {23,9420}, {351,13308}, {512,9185}, {804,5466}, {4108,9208}, {9979,13306}

X(22735) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD-REFLECTED TO 2nd BROCARD

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

The reciprocal cyclologic center of these triangles is X(3).

X(22735) lies on the cubic K509 and these lines: {2,2782}, {30,11673}, {98,237}, {99,14096}, {115,3117}, {694,804}, {2396,8842}, {2450,8569}, {5149,10328}, {6321,14957}, {11328,12188}

X(22735) = centroid of (degenerate) reflection triangle of ABC wrt 1st Brocard triangle

X(22736) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO INNER-FERMAT

Barycentrics    (9*SA+4*SW)*S^2-SB*SC*SW-sqrt(3)*(2*S^2+SA^2-4*SB*SC+SW^2)*S : :

The reciprocal orthologic center of these triangles is X(22507).

X(22736) lies on these lines: {2,18}, {3,22507}, {76,11603}, {182,22526}, {299,22846}, {384,22748}, {636,16628}, {3314,5983}, {5464,11149}, {5965,22685}, {7697,16627}, {7761,22737}, {10000,22745}, {11306,20378}

X(22736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 628, 22866), (22882, 22883, 22869)

X(22737) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st BROCARD TO OUTER-FERMAT

Barycentrics    (9*SA+4*SW)*S^2-SB*SC*SW+sqrt(3)*(2*S^2+SA^2-4*SB*SC+SW^2)*S : :

The reciprocal orthologic center of these triangles is X(22509).

X(22737) lies on these lines: {2,17}, {3,22509}, {76,11602}, {182,22527}, {298,22891}, {384,22749}, {635,16629}, {3314,5982}, {5463,11149}, {5965,22683}, {7697,16626}, {7761,22736}, {10000,22746}, {11305,20377}

X(22737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 627, 22911), (22927, 22928, 22914)

X(22738) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 2nd BROCARD

Barycentrics    9*S^2*(S^2+2*SA^2+3*SB*SC-SW^2)+3*SW^2*SB*SC-sqrt(3)*((48*R^2+5*SA+3*SW)*S^2+SW*(14*SA+3*SW)*(SA-SW))*S : :

The reciprocal cyclologic center of these triangles is X(3).

X(22738) lies on the cubic K509 and these lines: {2,5470}, {4,8450}, {17,930}, {6778,14447}

X(22738) = inverse of X(22846) in the inner-Napoleon circle

X(22739) = CYCLOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 2nd BROCARD

Barycentrics    9*S^2*(S^2+2*SA^2+3*SB*SC-SW^2)+3*SW^2*SB*SC+sqrt(3)*((48*R^2+5*SA+3*SW)*S^2+SW*(14*SA+3*SW)*(SA-SW))*S : :

The reciprocal cyclologic center of these triangles is X(3).

X(22739) lies on the cubic K509 and these lines: {2,5469}, {4,8451}, {18,930}, {6777,14446}

X(22739) = inverse of X(22891) in the outer-Napoleon circle
X(22739) = inner-Napoleon-circle-inverse of X(36782)

X(22740) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd BROCARD TO 1st LEMOINE-DAO

Barycentrics    a^2*(2*sqrt(3)*(3*a^14-9*(b^2+c^2)*a^12+3*(7*b^4-12*b^2*c^2+7*c^4)*a^10-(b^2+c^2)*(13*b^4-74*b^2*c^2+13*c^4)*a^8-(23*b^8+23*c^8-2*b^2*c^2*(23*b^4-59*b^2*c^2+23*c^4))*a^6+(b^2+c^2)*(25*b^8+25*c^8-2*b^2*c^2*(53*b^4-57*b^2*c^2+53*c^4))*a^4-(b^2-c^2)^2*(b^8+c^8-2*b^2*c^2*(2*b^4+25*b^2*c^2+2*c^4))*a^2-(b^8-c^8)*(b^2-c^2)*(b^2-3*c^2)*(3*b^2-c^2))*S+3*a^16-2*(13*b^4+16*b^2*c^2+13*c^4)*a^12+6*(b^2+c^2)*(3*b^4+5*b^2*c^2+3*c^4)*a^10+2*(2*b^4-b^2*c^2+2*c^4)*(10*b^4+31*b^2*c^2+10*c^4)*a^8-4*(b^2+c^2)*(9*b^8+9*c^8+16*b^2*c^2*(4*b^4-5*b^2*c^2+4*c^4))*a^6-2*(7*b^12+7*c^12-(120*b^8+120*c^8-b^2*c^2*(153*b^4-44*b^2*c^2+153*c^4))*b^2*c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*(9*b^8+9*c^8-b^2*c^2*(13*b^4-32*b^2*c^2+13*c^4))*a^2-(b^2-c^2)^2*(3*b^12+3*c^12+(30*b^8+30*c^8-b^2*c^2*(73*b^4-88*b^2*c^2+73*c^4))*b^2*c^2)) : :

The reciprocal cyclologic center of these triangles is X(22741).

X(22740) lies on the Brocard circle and these lines: {}

X(22741) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 2nd BROCARD

Barycentrics    2*(4*a^18-7*(b^2+c^2)*a^16-(26*b^4-35*b^2*c^2+26*c^4)*a^14+2*(b^2+c^2)*(55*b^4-87*b^2*c^2+55*c^4)*a^12-(173*b^8+173*c^8-b^2*c^2*(115*b^4+103*b^2*c^2+115*c^4))*a^10+2*(b^2+c^2)*(76*b^8+76*c^8-3*b^2*c^2*(51*b^4-40*b^2*c^2+51*c^4))*a^8-(88*b^12+88*c^12-(121*b^8+121*c^8+6*b^2*c^2*(13*b^4-23*b^2*c^2+13*c^4))*b^2*c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*(17*b^8+17*c^8-b^2*c^2*(11*b^4+16*b^2*c^2+11*c^4))*a^4-(b^2-c^2)^2*(5*b^12+5*c^12+(9*b^8+9*c^8-4*b^2*c^2*(13*b^4-16*b^2*c^2+13*c^4))*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)^3*(-b^8-c^8+2*b^2*c^2*(3*b^4-7*b^2*c^2+3*c^4)))*sqrt(3)*S-6*a^20+27*(b^2+c^2)*a^18-(59*b^4+35*b^2*c^2+59*c^4)*a^16+(b^2+c^2)*(116*b^4-235*b^2*c^2+116*c^4)*a^14-3*(75*b^8+75*c^8-b^2*c^2*(97*b^4+175*b^2*c^2+97*c^4))*a^12+(b^2+c^2)*(309*b^8+309*c^8-b^2*c^2*(595*b^4-149*b^2*c^2+595*c^4))*a^10-(258*b^12+258*c^12-(173*b^8+173*c^8+2*b^2*c^2*(43*b^4+113*b^2*c^2+43*c^4))*b^2*c^2)*a^8+(b^2+c^2)*(126*b^12+126*c^12-(169*b^8+169*c^8-2*b^2*c^2*(38*b^4-15*b^2*c^2+38*c^4))*b^2*c^2)*a^6-(b^2-c^2)^2*(31*b^12+31*c^12+(85*b^8+85*c^8+4*b^2*c^2*(4*b^2+c^2)*(b^2+4*c^2))*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(2*b^12+2*c^12-(b^2-c^2)^2*b^2*c^2*(35*b^4+29*b^2*c^2+35*c^4))*a^2+(b^4-c^4)^2*(b^2-c^2)^2*(3*b^8+3*c^8-4*b^2*c^2*(4*b^4-7*b^2*c^2+4*c^4)) : :

The reciprocal cyclologic center of these triangles is X(22740).

X(22741) lies on these lines: {}

X(22742) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd BROCARD TO 2nd LEMOINE-DAO

Barycentrics    a^2*(-2*sqrt(3)*(3*a^14-9*(b^2+c^2)*a^12+3*(7*b^4-12*b^2*c^2+7*c^4)*a^10-(b^2+c^2)*(13*b^4-74*b^2*c^2+13*c^4)*a^8-(23*b^8+23*c^8-2*b^2*c^2*(23*b^4-59*b^2*c^2+23*c^4))*a^6+(b^2+c^2)*(25*b^8+25*c^8-2*b^2*c^2*(53*b^4-57*b^2*c^2+53*c^4))*a^4-(b^2-c^2)^2*(b^8+c^8-2*b^2*c^2*(2*b^4+25*b^2*c^2+2*c^4))*a^2-(b^8-c^8)*(b^2-c^2)*(b^2-3*c^2)*(3*b^2-c^2))*S+3*a^16-2*(13*b^4+16*b^2*c^2+13*c^4)*a^12+6*(b^2+c^2)*(3*b^4+5*b^2*c^2+3*c^4)*a^10+2*(2*b^4-b^2*c^2+2*c^4)*(10*b^4+31*b^2*c^2+10*c^4)*a^8-4*(b^2+c^2)*(9*b^8+9*c^8+16*b^2*c^2*(4*b^4-5*b^2*c^2+4*c^4))*a^6-2*(7*b^12+7*c^12-(120*b^8+120*c^8-b^2*c^2*(153*b^4-44*b^2*c^2+153*c^4))*b^2*c^2)*a^4+2*(b^4-c^4)*(b^2-c^2)*(9*b^8+9*c^8-b^2*c^2*(13*b^4-32*b^2*c^2+13*c^4))*a^2-(b^2-c^2)^2*(3*b^12+3*c^12+(30*b^8+30*c^8-b^2*c^2*(73*b^4-88*b^2*c^2+73*c^4))*b^2*c^2)) : :

The reciprocal cyclologic center of these triangles is X(22743).

X(22742) lies on the Brocard circle and these lines: {}

X(22743) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 2nd BROCARD

Barycentrics    -2*(4*a^18-7*(b^2+c^2)*a^16-(26*b^4-35*b^2*c^2+26*c^4)*a^14+2*(b^2+c^2)*(55*b^4-87*b^2*c^2+55*c^4)*a^12-(173*b^8+173*c^8-b^2*c^2*(115*b^4+103*b^2*c^2+115*c^4))*a^10+2*(b^2+c^2)*(76*b^8+76*c^8-3*b^2*c^2*(51*b^4-40*b^2*c^2+51*c^4))*a^8-(88*b^12+88*c^12-(121*b^8+121*c^8+6*b^2*c^2*(13*b^4-23*b^2*c^2+13*c^4))*b^2*c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*(17*b^8+17*c^8-b^2*c^2*(11*b^4+16*b^2*c^2+11*c^4))*a^4-(b^2-c^2)^2*(5*b^12+5*c^12+(9*b^8+9*c^8-4*b^2*c^2*(13*b^4-16*b^2*c^2+13*c^4))*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)^3*(-b^8-c^8+2*b^2*c^2*(3*b^4-7*b^2*c^2+3*c^4)))*sqrt(3)*S-6*a^20+27*(b^2+c^2)*a^18-(59*b^4+35*b^2*c^2+59*c^4)*a^16+(b^2+c^2)*(116*b^4-235*b^2*c^2+116*c^4)*a^14-3*(75*b^8+75*c^8-b^2*c^2*(97*b^4+175*b^2*c^2+97*c^4))*a^12+(b^2+c^2)*(309*b^8+309*c^8-b^2*c^2*(595*b^4-149*b^2*c^2+595*c^4))*a^10-(258*b^12+258*c^12-(173*b^8+173*c^8+2*b^2*c^2*(43*b^4+113*b^2*c^2+43*c^4))*b^2*c^2)*a^8+(b^2+c^2)*(126*b^12+126*c^12-(169*b^8+169*c^8-2*b^2*c^2*(38*b^4-15*b^2*c^2+38*c^4))*b^2*c^2)*a^6-(b^2-c^2)^2*(31*b^12+31*c^12+(85*b^8+85*c^8+4*b^2*c^2*(4*b^2+c^2)*(b^2+4*c^2))*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(2*b^12+2*c^12-(b^2-c^2)^2*b^2*c^2*(35*b^4+29*b^2*c^2+35*c^4))*a^2+(b^4-c^4)^2*(b^2-c^2)^2*(3*b^8+3*c^8-4*b^2*c^2*(4*b^4-7*b^2*c^2+4*c^4)) : :

The reciprocal cyclologic center of these triangles is X(22742).

X(22743) lies on these lines: {}

X(22744) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 2nd CIRCUMPERP TANGENTIAL

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

X(22744) lies on these lines: {3,9941}, {32,56}, {36,3099}, {55,9997}, {104,9862}, {956,9857}, {958,3096}, {999,11368}, {2896,2975}, {3094,22769}, {3098,3428}, {4475,20999}, {5253,10583}, {8782,22514}, {9301,22765}, {9821,11249}, {9873,10871}, {9878,22565}, {9923,22659}, {9981,22774}, {9982,22773}, {9983,22779}, {9984,22583}, {9985,22781}, {9986,22595}, {9987,22624}, {9993,22753}, {9994,22756}, {9995,22757}, {9996,22758}, {10038,22766}, {10047,22767}, {10873,22759}, {10874,22760}, {10875,22761}, {10876,22762}, {10877,10879}, {10878,22768}, {11386,22479}, {11492,11862}, {11493,11861}, {11885,22755}, {12495,12513}, {12496,18237}, {12497,22770}, {12498,12773}, {12499,22775}, {12500,22777}, {12501,19478}, {12502,22778}, {12503,19159}, {12504,22782}, {13210,22586}, {13235,22560}, {13236,19162}, {13685,22783}, {13743,16123}, {13805,22784}, {13892,22763}, {13946,22764}, {18500,18761}, {19011,19013}, {19012,19014}, {22678,22680}, {22745,22771}, {22746,22772}, {22747,22776}

X(22744) = {X(3), X(9941)}-harmonic conjugate of X(11494)

X(22745) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO INNER-FERMAT

Barycentrics    4*S^4+(11*SA^2-21*SW*SA+SW^2)*S^2-3*SW^2*SB*SC+sqrt(3)*(S^2-SW*(2*SA+SW))*(SA-SW)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22745) lies on these lines: {15,628}, {16,5872}, {18,32}, {630,3096}, {3094,5965}, {3098,22843}, {3099,22651}, {3105,9982}, {6674,7846}, {9301,16628}, {9821,9981}, {9857,22851}, {9862,22531}, {9878,9989}, {9993,22831}, {9994,22853}, {9995,22854}, {9996,16627}, {9997,22867}, {10000,22736}, {10038,22884}, {10047,22885}, {10828,22656}, {10871,22857}, {10872,22858}, {10873,22859}, {10874,22860}, {10875,22863}, {10876,22864}, {10877,22865}, {10878,22886}, {10879,22887}, {11368,11740}, {11386,22481}, {11494,22557}, {11861,22669}, {11862,22673}, {11885,22852}, {13892,22876}, {13946,22877}, {18500,22794}, {18957,18972}, {19011,19069}, {19012,19072}, {22744,22771}

X(22746) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO OUTER-FERMAT

Barycentrics    4*S^4+(11*SA^2-21*SW*SA+SW^2)*S^2-3*SW^2*SB*SC-sqrt(3)*(S^2-SW*(2*SA+SW))*(SA-SW)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22746) lies on these lines: {15,5873}, {16,627}, {17,32}, {532,3105}, {629,3096}, {3094,5965}, {3098,22890}, {3099,22652}, {3104,9981}, {6673,7846}, {9301,16629}, {9821,9982}, {9857,22896}, {9862,22532}, {9878,9988}, {9993,22832}, {9994,22898}, {9995,22899}, {9996,16626}, {9997,22912}, {10000,22737}, {10038,22929}, {10047,22930}, {10828,22657}, {10871,22902}, {10872,22903}, {10873,22904}, {10874,22905}, {10875,22908}, {10876,22909}, {10877,22910}, {10878,22931}, {10879,22932}, {11368,11739}, {11386,22482}, {11494,22558}, {11861,22670}, {11862,22674}, {11885,22897}, {13892,22921}, {13946,22922}, {18500,22795}, {18957,18973}, {19011,19071}, {19012,19070}, {22744,22772}

X(22747) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 3rd HATZIPOLAKIS

Barycentrics    (5*R^2-SW)*(16*R^2-SA-3*SW)*S^4+(16*(11*SA^2-18*SA*SW+4*SW^2)*R^4-(35*SA-11*SW)*(2*SA-3*SW)*SW*R^2+(7*SA^2-14*SA*SW+4*SW^2)*SW^2)*S^2-3*(4*R^2-SW)*(16*R^2-3*SW)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22747) lies on these lines: {32,22466}, {2896,22647}, {3096,22966}, {3098,22951}, {3099,22653}, {9301,22979}, {9857,22941}, {9862,22533}, {9993,22833}, {9994,22945}, {9995,22947}, {9996,22955}, {9997,22969}, {10038,22980}, {10047,22981}, {10828,22658}, {10871,22956}, {10872,22957}, {10873,22958}, {10874,22959}, {10875,22963}, {10876,22964}, {10877,22965}, {10878,22982}, {10879,22983}, {11368,22476}, {11386,22483}, {11494,22559}, {11885,22943}, {13892,22976}, {13946,22977}, {18500,22800}, {18957,18978}, {19011,19083}, {19012,19084}, {22744,22776}

X(22748) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO INNER-FERMAT

Barycentrics    14*S^4+(9*SA^2-19*SB*SC-SW^2)*S^2-SW^2*SB*SC+sqrt(3)*(3*S^2+(5*SA-6*SW)*SA)*SW*S : :

The reciprocal orthologic center of these triangles is X(22507).

X(22748) lies on these lines: {3,22506}, {15,628}, {315,5983}, {384,22736}, {7802,22749}, {10131,22526}

X(22749) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th BROCARD TO OUTER-FERMAT

Barycentrics    14*S^4+(9*SA^2-19*SB*SC-SW^2)*S^2-SW^2*SB*SC-sqrt(3)*(3*S^2+(5*SA-6*SW)*SA)*SW*S : :

The reciprocal orthologic center of these triangles is X(22509).

X(22749) lies on these lines: {16,627}, {315,5982}, {384,22737}, {532,7833}, {7802,22748}, {10131,22527}

X(22750) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC TO 3rd HATZIPOLAKIS

Barycentrics    SC*SB*(SB+SC)*((24*R^2+2*SA-5*SW)*S^2+(8*R^2-3*SW)*SA^2) : :
X(22750) = 3*X(4)-2*X(22538) = 5*X(631)-4*X(22581) = 7*X(3090)-8*X(22973) = 5*X(3567)-4*X(22530) = 3*X(5890)-X(22535) = X(22538)-3*X(22970) = 2*X(22833)-3*X(22971)

The reciprocal orthologic center of these triangles is X(9729).

X(22750) lies on these lines: {2,22834}, {3,22497}, {4,801}, {5,22808}, {24,1192}, {25,5889}, {54,403}, {110,235}, {184,6622}, {186,8718}, {206,1614}, {378,22549}, {389,21652}, {631,22581}, {1112,15801}, {1147,6623}, {1181,17837}, {1596,18350}, {1598,11387}, {1660,18945}, {1870,19472}, {3089,6193}, {3090,22973}, {3518,7722}, {3520,22978}, {3567,22530}, {5890,22535}, {6146,22662}, {6197,22840}, {6198,22954}, {6240,10721}, {6353,6759}, {6403,7716}, {7592,19460}, {7699,22833}, {8537,9781}, {10540,21841}, {10632,22974}, {10633,22975}, {10880,22960}, {10881,22961}, {12292,16835}, {14644,20303}, {15033,22968}, {18504,22979}, {18560,22951}, {18916,18936}, {19424,19488}, {19425,19489}

X(22750) = reflection of X(4) in X(22970)
X(22750) = anticomplement of X(22834)
X(22750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1614, 3542, 19128), (22483, 22800, 4)

X(22751) = CYCLOLOGIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC TO EHRMANN-SIDE

Barycentrics    (SB+SC)*((5*R^2+SC-2*SW)*S^2-5*(3*R^2-SW)*SA*SB)*((-2*SW+5*R^2+SB)*S^2-5*SC*SA*(3*R^2-SW)) : :

The reciprocal cyclologic center of these triangles is X(22752).

X(22751) lies on the circumcircle and these lines: {3,12092}, {4,14103}, {5,22752}, {107,16868}, {110,1658}, {476,18403}

X(22751) = reflection of X(4) in X(14103)
X(22751) = circumperp conjugate of X(12092)
X(22751) = antipode of X(12092) in the circumcircle

X(22752) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO CIRCUMORTHIC

Barycentrics    a^22-3*(b^2+c^2)*a^20-(b^4-10*b^2*c^2+c^4)*a^18+(b^2+c^2)*(11*b^4-15*b^2*c^2+11*c^4)*a^16-2*(3*b^8+3*c^8+2*b^2*c^2*(7*b^4-6*b^2*c^2+7*c^4))*a^14-(b^2+c^2)*(14*b^8+14*c^8-3*b^2*c^2*(21*b^4-26*b^2*c^2+21*c^4))*a^12+(14*b^12+14*c^12-(13*b^8+13*c^8+b^2*c^2*(52*b^4-87*b^2*c^2+52*c^4))*b^2*c^2)*a^10+(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*(2*b^4-4*b^2*c^2+c^4)*(b^4-4*b^2*c^2+2*c^4)*a^8-(b^2-c^2)^2*(11*b^12+11*c^12-2*(6*b^8+6*c^8+b^2*c^2*(7*b^4-20*b^2*c^2+7*c^4))*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)^3*(b^8+c^8+b^2*c^2*(11*b^4-15*b^2*c^2+11*c^4))*a^4+(b^2-c^2)^6*(3*b^8+3*c^8-b^2*c^2*(b^4+5*b^2*c^2+c^4))*a^2-(b^4+b^2*c^2+c^4)*(b^2-c^2)^8*(b^2+c^2) : :
X(22752) = 3*X(381)-2*X(14103)

The reciprocal cyclologic center of these triangles is X(22751).

X(22752) lies on these lines: {5,22751}, {30,12092}, {186,20957}, {265,5889}, {381,14103}, {10255,10745}

X(22753) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND EULER

Barycentrics    a*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3+(b-c)^4*a^2-(b^2-c^2)*(b-c)^3*a+2*(b^2-c^2)^2*b*c) : :
X(22753) = X(6244)-3*X(16417) = X(10860)-3*X(21164)

X(22753) lies on these lines: {1,227}, {2,3428}, {3,142}, {4,11}, {5,958}, {8,6915}, {10,6918}, {12,6834}, {20,5253}, {30,7956}, {35,11522}, {36,1012}, {40,392}, {46,12672}, {55,5603}, {57,6001}, {72,12704}, {84,3062}, {98,22520}, {113,22586}, {114,22514}, {115,22504}, {119,11236}, {125,22583}, {127,19159}, {132,19162}, {221,3075}, {235,22479}, {354,18446}, {355,10680}, {371,22763}, {372,22764}, {377,15908}, {381,535}, {388,6848}, {404,962}, {405,5715}, {411,3616}, {496,20420}, {499,6831}, {515,999}, {517,997}, {518,5720}, {546,18761}, {631,5584}, {758,2095}, {859,17188}, {940,1064}, {942,6261}, {944,3304}, {952,18491}, {956,5231}, {960,5709}, {993,3817}, {1006,4423}, {1158,9856}, {1191,3072}, {1193,5706}, {1329,6944}, {1385,5806}, {1389,14497}, {1410,1893}, {1420,16616}, {1466,4295}, {1470,1519}, {1478,1532}, {1479,22766}, {1482,2802}, {1490,3333}, {1512,5252}, {1537,10090}, {1587,19013}, {1588,19014}, {1598,22654}, {2077,16371}, {2098,11501}, {2099,11502}, {2260,5776}, {2346,5703}, {2478,11827}, {2551,6964}, {2717,2728}, {2886,6826}, {2932,14217}, {2975,3091}, {3035,6970}, {3058,10596}, {3073,4252}, {3085,6927}, {3295,4342}, {3303,10595}, {3336,12767}, {3337,15071}, {3339,7971}, {3436,6953}, {3486,5804}, {3555,17857}, {3556,16252}, {3560,9955}, {3574,22781}, {3576,7580}, {3600,12667}, {3614,10599}, {3651,8273}, {3656,4421}, {3678,5780}, {3683,21165}, {3742,18443}, {3816,6827}, {3820,8169}, {3871,5734}, {3925,6854}, {3927,20117}, {4190,11826}, {4298,6260}, {4301,10306}, {4413,5657}, {4999,6824}, {5056,5260}, {5080,6945}, {5085,16792}, {5204,6906}, {5217,6942}, {5251,7988}, {5258,7989}, {5298,7965}, {5427,21669}, {5432,6880}, {5433,6833}, {5434,12115}, {5435,14647}, {5438,6769}, {5443,15175}, {5450,18483}, {5478,22773}, {5479,22774}, {5480,22769}, {5536,5692}, {5550,6986}, {5563,5691}, {5687,7982}, {5708,5884}, {5719,20330}, {5721,11269}, {5732,10177}, {5812,21616}, {5844,8168}, {5882,6744}, {5901,10267}, {5903,13253}, {6201,22757}, {6202,22756}, {6244,16417}, {6245,18237}, {6247,22778}, {6248,22779}, {6249,22780}, {6250,22624}, {6251,22595}, {6253,12116}, {6256,18990}, {6284,6934}, {6361,6940}, {6667,6978}, {6684,16408}, {6690,6954}, {6691,6891}, {6705,21628}, {6832,7958}, {6835,10527}, {6839,11680}, {6844,10589}, {6847,7288}, {6864,19843}, {6883,8167}, {6909,9812}, {6912,9779}, {6922,10200}, {6924,11248}, {6938,15326}, {6941,10895}, {6966,21154}, {6969,10590}, {6979,11681}, {7171,15726}, {7420,10478}, {8071,12047}, {8158,9709}, {8196,11493}, {8203,11492}, {8212,22761}, {8213,22762}, {8666,19925}, {8668,13463}, {8727,15325}, {9624,10902}, {9708,10175}, {9880,22565}, {9927,22659}, {9940,12520}, {9993,22744}, {10113,19478}, {10247,18524}, {10393,16193}, {10597,10786}, {10860,21164}, {10943,18517}, {10950,18967}, {10958,18962}, {11897,22755}, {12001,18518}, {12599,22777}, {12600,22782}, {12705,15803}, {13687,22783}, {13743,16125}, {13807,22784}, {14110,19861}, {14793,18393}, {16203,18481}, {17572,20070}, {22680,22682}, {22771,22831}, {22772,22832}, {22776,22833}

X(22753) = midpoint of X(4) and X(4293)
X(22753) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3149, 11500), (3, 946, 11496), (3, 5886, 1001), (4, 56, 12114), (4, 7681, 10893), (5, 11249, 958), (36, 1699, 1012), (56, 10896, 22760), (355, 10680, 12513), (381, 22765, 22758), (5603, 6905, 55), (5805, 5886, 946), (6834, 10532, 12), (6918, 22770, 10), (22758, 22765, 11194)

X(22754) = PERSPECTOR OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND EXCENTERS-MIDPOINTS

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

X(22754) lies on these lines: {1,11505}, {2,1476}, {9,56}, {10,999}, {100,7320}, {119,1656}, {214,3295}, {442,3086}, {474,1145}, {1125,6260}, {1376,12640}, {3304,3698}, {3616,10427}, {3812,7373}, {5013,6184}, {5253,5435}, {5836,15347}, {6691,9708}, {10269,18237}, {11249,22777}, {11517,17614}, {12709,19861}, {16410,22767}

X(22755) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND GOSSARD

Barycentrics    a*(a^11-2*(b^2+c^2)*a^9-2*(b+c)*b*c*a^8-(2*b^4-9*b^2*c^2+2*c^4)*a^7+2*(b+c)*(b^2+c^2)*b*c*a^6+(8*b^6+8*c^6-(2*b^4+2*c^4+(9*b^2-4*b*c+9*c^2)*b*c)*b*c)*a^5+2*(b+c)*(b^2-2*c^2)*(2*b^2-c^2)*b*c*a^4-(b^2-c^2)^2*(7*b^4+7*c^4-(4*b^2-15*b*c+4*c^2)*b*c)*a^3-6*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4+2*c^4-(2*b-c)*(b-2*c)*b*c)*a+2*(b^2-c^2)^2*(b+c)*(b^4+3*b^2*c^2+c^4)*b*c)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(22755) lies on these lines: {3,11848}, {30,3428}, {36,11852}, {55,11910}, {56,402}, {104,11845}, {958,1650}, {999,11831}, {1376,16210}, {1651,11194}, {2975,4240}

X(22756) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND INNER-GREBE

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

X(22756) lies on these lines: {1,8903}, {3,3641}, {6,41}, {36,5589}, {55,5605}, {104,10783}, {956,5689}, {958,5591}, {999,11370}, {1161,11249}, {1271,2975}, {3428,11824}, {5595,22654}, {5861,11194}, {5871,10919}, {6202,22753}, {6215,22758}, {6227,22504}, {6258,18237}, {6263,12773}, {6267,22778}, {6270,22773}, {6271,22774}, {6273,22779}, {6275,22780}, {6277,22781}, {6279,22624}, {6281,22595}, {6319,22514}, {7725,22583}, {7732,22586}, {8198,11493}, {8205,11492}, {8216,22761}, {8217,22762}, {8974,22763}, {9882,22565}, {9929,22659}, {9994,22744}, {10040,22766}, {10048,22767}, {10792,22520}, {10923,22759}, {10925,22760}, {10927,10931}, {10929,22768}, {11388,22479}, {11901,22755}, {11916,22765}, {12513,12627}, {12697,22770}, {12753,22775}, {12801,22777}, {12803,19478}, {12805,19159}, {12807,22782}, {13269,22560}, {13282,19162}, {13690,22783}, {13743,16130}, {13810,22784}, {13949,22764}, {18509,18761}, {22680,22699}, {22771,22853}, {22772,22898}, {22776,22945}

X(22756) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3641, 11497), (56, 22769, 22757)

X(22757) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND OUTER-GREBE

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

X(22757) lies on these lines: {1,8904}, {3,3640}, {6,41}, {36,5588}, {55,5604}, {104,10784}, {956,5688}, {958,5590}, {999,11371}, {1160,11249}, {1270,2975}, {3428,11825}, {5594,22654}, {5860,11194}, {5870,10920}, {6201,22753}, {6214,22758}, {6226,22504}, {6257,18237}, {6262,12773}, {6266,22778}, {6268,22773}, {6269,22774}, {6272,22779}, {6274,22780}, {6276,22781}, {6278,22624}, {6280,22595}, {6320,22514}, {7726,22583}, {7733,22586}, {8199,11493}, {8206,11492}, {8218,22761}, {8219,22762}, {8975,22763}, {9883,22565}, {9930,22659}, {9995,22744}, {10041,22766}, {10049,22767}, {10793,22520}, {10924,22759}, {10926,22760}, {10928,10932}, {10930,22768}, {11389,22479}, {11902,22755}, {11917,22765}, {12513,12628}, {12698,22770}, {12754,22775}, {12802,22777}, {12804,19478}, {12806,19159}, {12808,22782}, {13270,22560}, {13283,19162}, {13691,22783}, {13743,16131}, {13811,22784}, {13950,22764}, {18511,18761}, {22680,22700}, {22771,22854}, {22772,22899}, {22776,22947}

X(22757) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3640, 11498), (56, 22769, 22756)

X(22758) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND JOHNSON

Barycentrics    a*(a^6-(b+c)*a^5-2*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)*a^3+(b^4+c^4-2*(b^2-3*b*c+c^2)*b*c)*a^2-(b^2-c^2)*(b-c)^3*a-2*(b^2-c^2)^2*b*c) : :
X(22758) = 2*X(226)-3*X(5886) = 2*X(993)+X(18519) = 5*X(1656)-4*X(3822) = 2*X(3579)-3*X(21165) = 3*X(5603)-X(5905)

X(22758) lies on these lines: {1,90}, {2,104}, {3,10}, {4,2975}, {5,56}, {8,6906}, {11,6929}, {12,6862}, {21,944}, {30,3428}, {35,5881}, {36,5587}, {40,5258}, {48,5778}, {55,952}, {63,517}, {80,11502}, {100,6950}, {110,19478}, {214,15064}, {226,999}, {265,22586}, {381,535}, {388,6824}, {404,5818}, {405,1385}, {474,9956}, {497,6930}, {498,10942}, {519,10679}, {527,3656}, {529,7680}, {550,5584}, {601,10459}, {631,5260}, {758,1482}, {946,8666}, {962,3648}, {1001,2801}, {1006,5731}, {1060,1455}, {1125,16203}, {1329,6958}, {1352,22769}, {1468,5707}, {1470,1737}, {1479,10943}, {1483,3303}, {1484,11238}, {1532,18516}, {1621,7967}, {1656,3822}, {1837,8071}, {2077,3679}, {2099,14988}, {2178,5816}, {2478,10785}, {2550,6948}, {2551,6891}, {2646,14872}, {2829,2886}, {3036,15813}, {3085,6892}, {3086,6893}, {3090,5253}, {3095,22779}, {3149,18480}, {3244,12000}, {3304,5901}, {3359,9623}, {3421,6935}, {3434,5840}, {3436,6833}, {3534,11495}, {3556,9833}, {3576,5251}, {3577,3928}, {3579,21165}, {3600,6846}, {3601,5534}, {3612,17857}, {3616,5811}, {3653,16857}, {3654,6244}, {3655,16418}, {3816,20418}, {3897,12528}, {3913,11849}, {4189,11491}, {4293,6826}, {4421,12331}, {4999,6863}, {5080,6830}, {5126,10157}, {5204,6924}, {5229,6867}, {5248,5882}, {5252,8069}, {5265,6964}, {5288,7982}, {5303,6942}, {5307,7497}, {5433,6959}, {5440,18908}, {5444,5660}, {5552,6977}, {5563,8227}, {5603,5905}, {5613,22774}, {5617,22773}, {5657,6909}, {5690,10310}, {5691,6985}, {5694,5730}, {5770,18391}, {5817,7677}, {5878,22778}, {6033,22504}, {6214,22757}, {6215,22756}, {6256,6842}, {6259,18237}, {6287,22780}, {6288,22781}, {6289,22624}, {6290,22595}, {6321,22514}, {6825,12667}, {6831,10526}, {6837,10532}, {6839,20067}, {6850,19843}, {6859,10590}, {6860,10599}, {6872,12116}, {6888,20060}, {6905,18491}, {6910,10786}, {6917,7354}, {6940,9780}, {6944,7288}, {6951,12248}, {6952,11681}, {6973,10589}, {6980,10742}, {7428,15623}, {7583,19014}, {7584,19013}, {7697,22680}, {7701,11014}, {7728,22583}, {8200,11493}, {8207,11492}, {8220,22761}, {8221,22762}, {8724,22565}, {8757,10571}, {8976,22763}, {9940,19520}, {9947,13624}, {9996,22744}, {10074,11729}, {10085,13369}, {10529,10531}, {10573,11509}, {10596,11240}, {10738,11235}, {10749,19162}, {10796,22520}, {10944,11508}, {10950,11507}, {10993,17784}, {11501,15446}, {12001,13464}, {12332,19914}, {12699,22770}, {12737,15558}, {12856,22777}, {12918,19159}, {12919,22782}, {13188,14663}, {13692,22783}, {13812,22784}, {13951,22764}, {16626,22772}, {16627,22771}, {22776,22955}

X(22758) = midpoint of X(i) and X(j) for these {i,j}: {3, 18519}, {3434, 6938}
X(22758) = reflection of X(i) in X(j) for these (i,j): (3, 993), (55, 6914)
X(22758) = complement of X(12115)
X(22758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7330, 5887), (2, 104, 10269), (3, 355, 11499), (3, 5790, 1376), (3, 18518, 6796), (3, 18525, 11500), (4, 2975, 11249), (8, 6906, 11248), (10, 5450, 3), (21, 944, 10267), (958, 12114, 3), (5267, 6796, 3), (5790, 18515, 3), (11249, 18761, 4), (22759, 22760, 1)

X(22759) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 1st JOHNSON-YFF

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

X(22759) lies on these lines: {1,90}, {2,12}, {3,5252}, {4,10957}, {5,22767}, {8,11509}, {10,1470}, {11,6893}, {21,3476}, {36,9578}, {55,944}, {57,5258}, {65,956}, {104,3085}, {226,8666}, {355,8071}, {405,1319}, {495,22766}, {497,10949}, {498,10269}, {603,10459}, {952,11507}, {993,10106}, {999,11375}, {1001,1388}, {1012,3057}, {1191,7299}, {1317,3303}, {1399,5710}, {1408,19259}, {1420,3646}, {1476,5047}, {1478,6842}, {1479,18761}, {1836,22770}, {2098,11496}, {2099,3868}, {2217,8192}, {2886,18961}, {3086,6898}, {3256,3632}, {3304,3487}, {3340,5288}, {3428,6850}, {3435,10834}, {3485,18967}, {3601,9845}, {3614,6981}, {3624,5193}, {3913,14882}, {4293,6897}, {5204,6940}, {5219,5563}, {5432,6961}, {5584,15326}, {6892,15888}, {6911,10827}, {6913,11376}, {6914,11508}, {6937,9657}, {6941,10895}, {8668,12648}, {9613,11012}, {9654,22765}, {10088,19478}, {10572,18519}, {10680,12047}, {10797,22520}, {10831,22654}, {10873,22744}, {10896,13729}, {10923,22756}, {10924,22757}, {11011,12559}, {11248,12647}, {11392,22479}, {11492,11870}, {11493,11869}, {11499,14793}, {11905,22755}, {11930,22761}, {11931,22762}, {12184,22504}, {12350,22565}, {12373,22583}, {12678,18237}, {12739,12773}, {12763,22775}, {12837,22779}, {12859,22777}, {12903,22586}, {12940,22778}, {12941,22774}, {12942,22773}, {12944,22780}, {12945,19159}, {12946,22781}, {12947,22782}, {12948,22595}, {12949,22624}, {13182,22514}, {13273,22560}, {13296,19162}, {13695,22783}, {13743,16140}, {13815,22784}, {13897,22763}, {13954,22764}, {18838,19860}, {19013,19027}, {19014,19028}, {22680,22705}, {22771,22859}, {22772,22904}, {22776,22958}

X(22759) = anticomplement of X(15843)
X(22759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22758, 22760), (3, 5252, 11501), (21, 3476, 11510), (104, 3085, 22768), (355, 8071, 11502), (388, 2975, 56)

X(22760) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 2nd JOHNSON-YFF

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

X(22760) lies on these lines: {1,90}, {2,10958}, {3,1737}, {4,11}, {5,22766}, {8,21}, {12,6824}, {25,2217}, {28,1857}, {35,5727}, {36,6985}, {65,1012}, {80,11499}, {119,10320}, {145,10965}, {355,8069}, {388,6837}, {405,997}, {411,5204}, {474,17606}, {495,16617}, {496,22767}, {497,2975}, {499,6842}, {517,920}, {855,3556}, {944,11510}, {950,993}, {952,11508}, {956,3057}, {960,7082}, {999,10404}, {1001,10394}, {1006,4305}, {1210,1470}, {1319,1898}, {1376,5086}, {1388,21740}, {1420,10864}, {1454,7686}, {1468,2654}, {1478,6841}, {1479,7491}, {1482,12758}, {1603,20989}, {1610,15494}, {1697,5258}, {1776,2098}, {1788,6909}, {2099,11496}, {3058,11111}, {3085,10955}, {3304,3485}, {3428,6284}, {3586,11012}, {3601,5251}, {3612,6883}, {3614,6855}, {3813,10947}, {3924,7004}, {4252,5348}, {4295,21669}, {5172,11500}, {5217,6875}, {5218,5260}, {5229,6870}, {5253,6871}, {5259,13384}, {5288,7962}, {5432,6857}, {5433,6825}, {5584,15338}, {5603,18967}, {5722,8071}, {6796,17010}, {6828,10895}, {6838,7288}, {6867,7173}, {6869,15326}, {6906,11509}, {6911,10826}, {6913,11375}, {6914,11507}, {6924,12019}, {7680,18962}, {7742,18481}, {8581,20323}, {8666,12053}, {8758,21147}, {9657,10883}, {9669,22765}, {10058,10573}, {10091,19478}, {10106,12617}, {10395,17647}, {10448,14547}, {10798,22520}, {10832,22654}, {10874,22744}, {10925,22756}, {10926,22757}, {11114,11194}, {11393,22479}, {11492,11872}, {11493,11871}, {11906,22755}, {11932,22761}, {11933,22762}, {12185,22504}, {12351,22565}, {12374,22583}, {12589,22769}, {12665,12739}, {12679,18237}, {12701,22770}, {12740,12773}, {12836,22779}, {12860,22777}, {12904,22586}, {12950,22778}, {12951,22774}, {12952,22773}, {12954,22780}, {12955,19159}, {12956,22781}, {12957,22782}, {12958,22595}, {12959,22624}, {13183,22514}, {13274,22560}, {13297,19162}, {13696,22783}, {13743,16141}, {13816,22784}, {13898,22763}, {13955,22764}, {14793,15446}, {14800,15079}, {17603,19520}, {18254,22836}, {19013,19029}, {19014,19030}, {22680,22706}, {22771,22860}, {22772,22905}

X(22760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 90, 5887), (1, 22758, 22759), (3, 1837, 11502), (21, 3486, 55), (56, 10896, 22753), (104, 3086, 56), (355, 8069, 11501), (497, 2975, 10966), (1210, 5450, 1470), (1319, 1898, 6261), (6906, 18391, 11509), (10058, 10573, 11248)

X(22761) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND LUCAS HOMOTHETIC

Barycentrics    a^2*(-8*(a^8+(b+c)*a^7+2*(b^2+3*b*c+c^2)*a^6+(b+c)*(b^2+5*b*c+c^2)*a^5-7*b*c*(b-c)^2*a^4-(b+c)*(b^2+c^2)*(b^2+18*b*c+c^2)*a^3-2*(b^2+c^2)*(b^4+c^4+2*(2*b^2+9*b*c+2*c^2)*b*c)*a^2-(b+c)*(b^2+3*b*c+c^2)*(b^4+10*b^2*c^2+c^4)*a-(b^3-c^3)*(b-c)*(b^2+c^2)^2)*S+3*a^10-(b+3*c)*(3*b+c)*a^8-32*b*c*(b+c)*a^7-2*(3*b^4+3*c^4+4*b*c*(3*b^2+17*b*c+3*c^2))*a^6+8*b*c*(b-3*c)*(3*b-c)*(b+c)*a^5+2*(3*b^6+3*c^6+(42*b^4+42*c^4+b*c*(51*b^2+40*b*c+51*c^2))*b*c)*a^4+16*(b+c)*(3*b^4+3*c^4+2*b*c*(3*b^2+7*b*c+3*c^2))*b*c*a^3+(3*b^8+3*c^8-2*(28*b^6+28*c^6-(20*b^4+20*c^4+3*b*c*(12*b^2+55*b*c+12*c^2))*b*c)*b*c)*a^2-8*(b+c)*(5*b^6+5*c^6+(2*b^4+2*c^4-b*c*(13*b^2+12*b*c+13*c^2))*b*c)*b*c*a-3*(b^4-c^4)^2*(b-c)^2) : :

X(22761) lies on these lines: {3,11503}, {36,8188}, {55,8210}, {56,493}, {104,11846}, {956,8214}, {958,8222}, {999,11377}, {2975,6462}, {3428,11828}, {6461,22762}, {8194,22654}, {8201,11493}, {8208,11492}, {8212,22753}, {8216,22756}, {8218,22757}, {8220,22758}, {9838,10945}, {10669,11249}, {10875,22744}, {10966,11947}, {11194,12152}, {11394,22479}, {11840,22520}, {11907,22755}, {11930,22759}, {11932,22760}, {11949,22765}, {11951,22766}, {11953,22767}, {11955,22768}, {12186,22504}, {12352,22565}, {12377,22583}, {12426,22659}, {12513,12636}, {12590,22769}, {12741,12773}, {12765,22775}, {12861,22777}, {12894,19478}, {12986,22778}, {12988,22774}, {12990,22773}, {12992,22779}, {12994,22780}, {12996,19159}, {12998,22781}, {13000,22782}, {13002,22595}, {13004,22624}, {13184,22514}, {13215,22586}, {13275,22560}, {13298,19162}, {13697,22783}, {13743,16161}, {13817,22784}, {13899,22763}, {13956,22764}, {18237,18245}, {18520,18761}, {19013,19031}, {19014,19032}, {22680,22709}, {22770,22841}, {22771,22863}, {22772,22908}, {22776,22963}

X(22761) = {X(3), X(12440)}-harmonic conjugate of X(11503)

X(22762) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*(8*(a^8+(b+c)*a^7+2*(b^2+3*b*c+c^2)*a^6+(b+c)*(b^2+5*b*c+c^2)*a^5-7*b*c*(b-c)^2*a^4-(b+c)*(b^2+c^2)*(b^2+18*b*c+c^2)*a^3-2*(b^2+c^2)*(b^4+c^4+2*(2*b^2+9*b*c+2*c^2)*b*c)*a^2-(b+c)*(b^2+3*b*c+c^2)*(b^4+10*b^2*c^2+c^4)*a-(b^3-c^3)*(b-c)*(b^2+c^2)^2)*S+3*a^10-(b+3*c)*(3*b+c)*a^8-32*b*c*(b+c)*a^7-2*(3*b^4+3*c^4+4*b*c*(3*b^2+17*b*c+3*c^2))*a^6+8*b*c*(b-3*c)*(3*b-c)*(b+c)*a^5+2*(3*b^6+3*c^6+(42*b^4+42*c^4+b*c*(51*b^2+40*b*c+51*c^2))*b*c)*a^4+16*(b+c)*(3*b^4+3*c^4+2*b*c*(3*b^2+7*b*c+3*c^2))*b*c*a^3+(3*b^8+3*c^8-2*(28*b^6+28*c^6-(20*b^4+20*c^4+3*b*c*(12*b^2+55*b*c+12*c^2))*b*c)*b*c)*a^2-8*(b+c)*(5*b^6+5*c^6+(2*b^4+2*c^4-b*c*(13*b^2+12*b*c+13*c^2))*b*c)*b*c*a-3*(b^4-c^4)^2*(b-c)^2) : :

X(22762) lies on these lines: {3,11504}, {36,8189}, {55,8211}, {56,494}, {104,11847}, {956,8215}, {958,8223}, {999,11378}, {2975,6463}, {3428,11829}, {6461,22761}, {8195,22654}, {8202,11493}, {8209,11492}, {8213,22753}, {8217,22756}, {8219,22757}, {8221,22758}, {9839,10946}, {10673,11249}, {10876,22744}, {10966,11948}, {11194,12153}, {11395,22479}, {11841,22520}, {11908,22755}, {11931,22759}, {11933,22760}, {11950,22765}, {11952,22766}, {11954,22767}, {11956,22768}, {12187,22504}, {12353,22565}, {12378,22583}, {12427,22659}, {12513,12637}, {12591,22769}, {12742,12773}, {12766,22775}, {12862,22777}, {12895,19478}, {12987,22778}, {12989,22774}, {12991,22773}, {12993,22779}, {12995,22780}, {12997,19159}, {12999,22781}, {13001,22782}, {13003,22595}, {13005,22624}, {13185,22514}, {13216,22586}, {13276,22560}, {13299,19162}, {13698,22783}, {13743,16162}, {13818,22784}, {13900,22763}, {13957,22764}, {18237,18246}, {18522,18761}, {19013,19033}, {19014,19034}, {22680,22710}, {22770,22842}, {22771,22864}, {22772,22909}, {22776,22964}

X(22762) = {X(3), X(12441)}-harmonic conjugate of X(11504)

X(22763) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 3rd TRI-SQUARES-CENTRAL

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

X(22763) lies on these lines: {2,19014}, {3,8983}, {6,978}, {36,13888}, {55,13902}, {56,3068}, {104,13886}, {371,22753}, {404,19000}, {474,18991}, {485,12114}, {590,958}, {956,13893}, {999,13883}, {1125,13940}, {1376,7969}, {2975,8972}, {3149,9583}, {3304,19066}, {3428,9540}, {3616,18999}, {4413,19065}, {5253,7585}, {6409,11495}, {7580,9615}, {7583,10269}, {8974,22756}, {8975,22757}, {8976,22758}, {8980,22504}, {8981,11249}, {8987,18237}, {8988,12773}, {8991,22778}, {8992,22779}, {8993,22780}, {8995,22781}, {8997,22514}, {8998,22586}, {10966,13901}, {11194,13846}, {11492,13891}, {11493,13890}, {12513,13911}, {13743,16148}, {13848,22784}, {13884,22479}, {13885,22520}, {13889,22654}, {13894,22755}, {13897,22759}, {13898,22760}, {13899,22761}, {13900,22762}, {13903,22765}, {13904,22766}, {13905,22767}, {13906,19030}, {13908,22565}, {13909,22659}, {13910,22769}, {13912,22770}, {13913,22775}, {13914,22777}, {13916,22774}, {13917,22773}, {13918,19159}, {13919,22782}, {13921,22595}, {13922,22560}, {13923,19162}, {13936,16408}, {13947,16862}, {18538,18761}, {22680,22720}, {22771,22876}, {22772,22921}, {22776,22976}

X(22763) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 8983, 13887), (5253, 7585, 19013)

X(22764) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 4th TRI-SQUARES-CENTRAL

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

X(22764) lies on these lines: {2,19013}, {3,13940}, {6,978}, {36,13942}, {55,13959}, {56,3069}, {104,13939}, {372,22753}, {404,18999}, {474,18992}, {486,12114}, {615,958}, {956,13947}, {999,13936}, {1125,13887}, {1376,7968}, {2975,13941}, {3304,19065}, {3428,13935}, {3616,19000}, {4413,19066}, {5253,7586}, {6410,11495}, {7584,10269}, {10966,13958}, {11194,13847}, {11249,13966}, {11492,13945}, {11493,13944}, {12513,13973}, {12773,13976}, {13743,16149}, {13849,22784}, {13880,22624}, {13883,16408}, {13893,16862}, {13933,22595}, {13937,22479}, {13938,22520}, {13943,22654}, {13946,22744}, {13948,22755}, {13949,22756}, {13950,22757}, {13951,22758}, {13954,22759}, {13955,22760}, {13956,22761}, {13957,22762}, {13961,22765}, {13962,22766}, {13963,22767}, {13964,19029}, {13967,22504}, {13968,22565}, {13969,22583}, {13970,22659}, {13972,22769}, {13974,18237}, {13975,22770}, {13977,22775}, {13978,22777}, {13979,19478}, {13980,22778}, {13981,22774}, {13982,22773}, {13983,22779}, {13984,22780}, {13985,19159}, {13986,22781}, {13987,22782}, {13988,22783}, {13989,22514}, {13990,22586}, {13991,22560}, {13992,19162}, {18761,18762}, {22680,22721}, {22771,22877}, {22772,22922}, {22776,22977}

X(22764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13971, 13940), (5253, 7586, 19014)

X(22765) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND X3-ABC REFLECTIONS

Barycentrics    a^2*(a^5-(b+c)*a^4-(2*b^2-3*b*c+2*c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^4+c^4-3*(b-c)^2*b*c)*a-(b^2-c^2)*(b-c)^3) : :
X(22765) = 3*X(3)-2*X(2077) = 5*X(3)-2*X(5537) = 3*X(36)-X(2077) = 5*X(36)-X(5537) = X(40)-3*X(5131) = 5*X(1656)-4*X(3814) = 7*X(3526)-8*X(6681) = 2*X(5057)-5*X(18493) = 2*X(5176)-3*X(5790) = X(5180)-3*X(5603) = 3*X(5298)-2*X(6713) = 3*X(5886)-2*X(11813)

X(22765) lies on the cubic K725 and these lines: {1,3}, {4,20067}, {5,2975}, {8,6924}, {21,5901}, {30,104}, {74,6584}, {100,5844}, {110,859}, {119,529}, {140,5253}, {145,6942}, {195,22781}, {355,8666}, {381,535}, {382,11928}, {388,6863}, {399,22586}, {404,5690}, {496,7491}, {499,6971}, {515,12747}, {519,12331}, {573,21773}, {758,6265}, {946,12600}, {952,6905}, {956,5176}, {958,1656}, {993,5886}, {995,5398}, {1012,18515}, {1056,6954}, {1351,9037}, {1478,6980}, {1483,11491}, {1532,10742}, {1598,1878}, {1621,7508}, {1727,17638}, {1951,22144}, {2687,5606}, {2718,6011}, {2800,4973}, {3086,6928}, {3149,18525}, {3218,14988}, {3421,6970}, {3436,6959}, {3526,6681}, {3556,14530}, {3560,5057}, {3600,6825}, {3622,6875}, {3628,5260}, {3843,18761}, {3877,19525}, {4188,12245}, {4189,10595}, {4293,6923}, {4297,16117}, {4299,10525}, {4996,19907}, {5054,10197}, {5123,9708}, {5146,7497}, {5180,5603}, {5251,11230}, {5258,9956}, {5265,6891}, {5267,13464}, {5298,6713}, {5450,12699}, {5694,6763}, {5762,7677}, {5780,17615}, {5840,15326}, {5887,13465}, {6417,19014}, {6418,19013}, {6597,12919}, {6834,20076}, {6842,18990}, {6862,10532}, {6868,14986}, {6882,15325}, {6906,22791}, {6910,10597}, {6915,18357}, {6917,10527}, {6934,10529}, {6936,10586}, {6949,20060}, {6958,7288}, {6962,10805}, {7517,22654}, {9301,22744}, {9654,22759}, {9669,22760}, {10090,19914}, {10620,22583}, {11495,15688}, {11499,12513}, {11500,18526}, {11842,22520}, {11911,22755}, {11916,22756}, {11917,22757}, {11949,22761}, {11950,22762}, {12188,22504}, {12355,22565}, {12429,22659}, {12601,22595}, {12602,22624}, {12684,18237}, {12872,22777}, {12902,19478}, {13093,22778}, {13102,22774}, {13103,22773}, {13108,22779}, {13111,22780}, {13115,19159}, {13126,22782}, {13188,22514}, {13310,19162}, {13713,22783}, {13836,22784}, {13903,22763}, {13961,22764}, {14217,15228}, {15611,17734}, {16628,22771}, {16629,22772}, {17455,19297}, {18519,19541}, {22680,22728}, {22776,22979}

X(22765) = midpoint of X(i) and X(j) for these {i,j}: {1, 5535}, {4, 20067}, {14217, 15228}
X(22765) = reflection of X(6882) in X(15325)
X(22765) = circumperp conjugate of X(3579)
X(22765) = inverse of X(1385) in the circumcircle
X(22765) = inverse of X(13750) in the incircle
X(22765) = inverse of X(1482) in the Stammler circle
X(22765) = isogonal conjugate of antigonal conjugate of X(1389)
X(22765) = X(2070)-of-2nd-circumperp-triangle
X(22765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,37621), (1, 36, 5172), (1, 7280, 14795), (3, 1482, 11849), (3, 8148, 11248), (3, 10247, 55), (3, 10680, 1482), (3, 12001, 3295), (36, 5193, 5126), (40, 5131, 10225), (56, 22767, 999), (1381, 1382, 1385), (1385, 6583, 1), (2095, 10246, 1482), (2446, 2447, 13750), (5204, 11248, 3), (11009, 14792, 14882)

X(22766) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND INNER-YFF

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

X(22766) lies on these lines: {1,3}, {5,22760}, {11,6917}, {12,6862}, {47,4252}, {104,388}, {149,4190}, {226,5450}, {377,3086}, {404,18391}, {411,4305}, {442,499}, {474,1737}, {495,22759}, {497,6934}, {498,958}, {601,1457}, {611,5135}, {613,4259}, {920,960}, {939,2163}, {952,11501}, {956,10039}, {993,12527}, {1004,11019}, {1012,12047}, {1056,6977}, {1201,1497}, {1210,17647}, {1259,18389}, {1376,10573}, {1436,1781}, {1478,6831}, {1479,22753}, {1537,10044}, {1709,18237}, {1785,4185}, {1788,6940}, {1804,3664}, {1837,6911}, {2164,17443}, {2178,2278}, {2975,3085}, {3149,10572}, {3299,19013}, {3301,19014}, {3485,6906}, {3486,6905}, {3556,7428}, {3560,11375}, {3582,17528}, {3600,6890}, {4293,6836}, {4295,6909}, {4413,18395}, {5248,17010}, {5730,19525}, {5784,15299}, {5880,10094}, {6860,10590}, {6889,7288}, {6918,10826}, {6924,11502}, {6959,10958}, {6984,10589}, {7177,14878}, {8068,10742}, {8581,15518}, {10037,22654}, {10038,22744}, {10040,22756}, {10041,22757}, {10053,22504}, {10054,22565}, {10055,22659}, {10056,11194}, {10057,12773}, {10059,22777}, {10060,22778}, {10061,22774}, {10062,22773}, {10063,22779}, {10064,22780}, {10065,22583}, {10066,22781}, {10067,22595}, {10072,10948}, {10085,12664}, {10086,22514}, {10087,22560}, {10088,22586}, {10090,10609}, {10106,12616}, {10801,22520}, {10950,11499}, {11398,20832}, {11912,22755}, {11951,22761}, {11952,22762}, {12513,12647}, {12903,19478}, {13116,19159}, {13128,22782}, {13311,19162}, {13714,22783}, {13743,16152}, {13837,22784}, {13904,22763}, {13962,22764}, {22680,22729}, {22771,22884}, {22772,22929}, {22776,22980}

X(22766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 8071), (1, 56, 22767), (1, 3338, 5570), (3, 999, 65), (36, 3612, 3), (36, 5563, 3361), (46, 14803, 3), (55, 56, 11249), (55, 18967, 1482), (56, 10966, 22765), (56, 22768, 3), (999, 1482, 18967), (999, 3295, 12001), (3295, 12001, 5048), (3295, 22765, 10966), (5563, 14803, 46)

X(22767) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND OUTER-YFF

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

X(22767) lies on these lines: {1,3}, {5,22759}, {11,6929}, {12,6959}, {47,1191}, {104,497}, {255,1201}, {378,15500}, {388,6834}, {411,1476}, {474,10039}, {496,22760}, {498,13747}, {499,958}, {613,22769}, {952,11502}, {956,1737}, {997,20588}, {1056,6880}, {1145,1376}, {1210,8666}, {1387,6914}, {1473,1795}, {1478,1532}, {1479,10948}, {1603,15617}, {1745,9363}, {1804,3663}, {2178,4271}, {2478,2975}, {2829,15845}, {3074,21214}, {3085,5253}, {3299,19014}, {3301,19013}, {3434,13279}, {3476,6905}, {3554,15817}, {3560,11376}, {3600,6838}, {3825,15866}, {4186,11399}, {4293,6925}, {4421,10087}, {5252,6911}, {5433,10320}, {5450,12053}, {5533,11238}, {5687,10094}, {5790,10057}, {5840,10947}, {5854,15813}, {6872,14986}, {6917,10957}, {6918,10827}, {6924,11501}, {6967,7288}, {7580,21578}, {8070,10895}, {10046,22654}, {10047,22744}, {10048,22756}, {10049,22757}, {10051,10074}, {10069,22504}, {10070,22565}, {10071,22659}, {10072,11113}, {10073,12773}, {10075,22777}, {10076,22778}, {10077,22774}, {10079,22779}, {10080,22780}, {10081,22583}, {10082,22781}, {10083,22595}, {10084,22624}, {10085,18237}, {10089,22514}, {10091,22586}, {10573,12513}, {10802,22520}, {10896,18761}, {10944,11499}, {11913,22755}, {11953,22761}, {11954,22762}, {12904,19478}, {13117,19159}, {13129,22782}, {13312,19162}, {13715,22783}, {13743,16153}, {13838,22784}, {13905,22763}, {13963,22764}, {16410,22754}, {22680,22730}, {22771,22885}, {22772,22930}, {22776,22981}

X(22767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 11508), (1, 36, 8069), (1, 56, 22766), (1, 3338, 13750), (1, 8071, 11507), (1, 14793, 55), (1, 17437, 65), (3, 999, 1319), (36, 5119, 3), (36, 5563, 13462), (55, 56, 10269), (56, 3428, 36), (56, 10966, 3), (999, 15934, 3304), (999, 22765, 56), (1420, 11012, 7742)

X(22768) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND INNER-YFF TANGENTS

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

X(22768) lies on these lines: {1,3}, {2,10958}, {11,377}, {12,6833}, {104,3085}, {119,6862}, {198,2278}, {224,10391}, {388,6890}, {404,3486}, {442,10200}, {474,1837}, {497,4190}, {498,10942}, {939,19349}, {944,11501}, {956,10915}, {958,5432}, {993,21075}, {997,1858}, {1012,11375}, {1058,13199}, {1376,5554}, {1468,22072}, {1696,2182}, {2057,3711}, {2178,2268}, {2252,2256}, {2330,12594}, {2361,4252}, {2975,5218}, {3086,6897}, {3434,10959}, {3485,6909}, {3614,6860}, {4293,6899}, {4294,10596}, {4305,6905}, {4413,5794}, {4861,8668}, {4995,11194}, {5252,12616}, {5433,6889}, {5450,13411}, {6256,6831}, {6284,6934}, {6836,7354}, {6911,10572}, {6917,10896}, {6940,18391}, {6955,10947}, {6966,15888}, {6984,7173}, {7951,18542}, {9310,22088}, {10058,11729}, {10785,10957}, {10803,22520}, {10827,18519}, {10834,22654}, {10878,22744}, {10929,22756}, {10930,22757}, {11112,11238}, {11376,12609}, {11400,22479}, {11496,15950}, {11914,22755}, {11955,22761}, {11956,22762}, {12189,22504}, {12356,22565}, {12381,22583}, {12430,22659}, {12513,12648}, {12686,18237}, {12711,17614}, {12739,15528}, {12749,12773}, {12775,22775}, {12874,22777}, {12905,19478}, {13094,22778}, {13104,22774}, {13105,22773}, {13109,22779}, {13112,22780}, {13118,19159}, {13121,22781}, {13130,22782}, {13132,22595}, {13189,15452}, {13217,22586}, {13278,22560}, {13313,19162}, {13716,22783}, {13743,16154}, {13839,22784}, {13906,19030}, {13964,19029}, {16408,17606}, {17611,20849}, {19013,19038}, {19014,19037}, {22680,22731}, {22771,22886}, {22772,22931}, {22776,22982}

X(22768) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 11509), (1, 35, 10679), (1, 55, 10965), (1, 10269, 56), (1, 14803, 3), (3, 999, 46), (3, 2646, 55), (3, 22766, 56), (55, 56, 10966), (55, 3304, 2098), (56, 5217, 3428), (104, 3085, 22759), (404, 3486, 11502), (1385, 8069, 11510), (3085, 10805, 10956)

X(22769) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st EHRMANN

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

The reciprocal orthologic center of these triangles is X(3).

X(22769) lies on these lines: {1,159}, {3,518}, {6,41}, {19,4327}, {22,3873}, {25,354}, {36,3751}, {38,55}, {57,197}, {65,8192}, {69,2975}, {104,5848}, {141,958}, {182,2810}, {210,7484}, {222,20986}, {375,17825}, {390,1633}, {511,11249}, {524,11194}, {542,19478}, {610,4321}, {611,5135}, {613,22767}, {674,1350}, {732,22779}, {942,9798}, {956,3416}, {984,16560}, {991,2876}, {993,9028}, {999,1386}, {1001,4364}, {1253,20780}, {1279,7083}, {1351,9037}, {1352,22758}, {1466,2933}, {1480,2841}, {1503,12114}, {1593,12680}, {1598,13374}, {1610,3600}, {1617,3185}, {1843,22479}, {2182,8581}, {2330,12594}, {2385,3663}, {2646,19459}, {2781,19162}, {2836,10246}, {2854,22586}, {3056,10966}, {3094,22744}, {3098,9052}, {3296,17562}, {3475,4224}, {3555,8193}, {3564,22595}, {3576,9004}, {3618,5253}, {3619,5260}, {3681,7485}, {3740,16419}, {3742,5020}, {3818,18761}, {3844,9708}, {3870,7293}, {3913,9053}, {3916,15592}, {4185,10404}, {4293,5800}, {4421,9041}, {4430,6636}, {4661,15246}, {4860,20989}, {5045,11365}, {5085,9026}, {5096,5204}, {5138,15654}, {5480,22753}, {5563,16475}, {5584,9049}, {5846,12513}, {5847,8666}, {5965,22771}, {5969,22514}, {6642,13373}, {7395,14872}, {8185,18398}, {9021,12635}, {9024,22560}, {9830,22565}, {10391,16541}, {10829,17625}, {10833,18839}, {11492,12453}, {11493,12452}, {12212,22520}, {12583,22755}, {12588,22759}, {12589,22760}, {12590,22761}, {12591,22762}, {13910,22763}, {13972,22764}, {18613,37519}, {22504,22680}, {22783,22784}

X(22769) = midpoint of X(1) and X(7289)
X(22769) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3220, 1486), (56, 198, 20470), (3242, 4265, 55), (22756, 22757, 56)

X(22770) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 3rd EXTOUCH

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2-b*c+c^2)*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4+c^4-2*(b^2-5*b*c+c^2)*b*c)*a-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)) : :
X(22770) = 3*X(3)-2*X(11248) = 3*X(381)-2*X(10526) = 2*X(5450)-3*X(11194) = 3*X(5603)-X(5758)

The reciprocal orthologic center of these triangles is X(4).

X(22770) lies on these lines: {1,3}, {4,956}, {5,2551}, {8,3149}, {10,6918}, {63,12672}, {104,6361}, {106,15663}, {145,411}, {210,5780}, {219,945}, {347,7053}, {355,4847}, {381,10526}, {387,19543}, {388,6907}, {405,5603}, {442,10532}, {474,5657}, {495,6825}, {496,6827}, {515,12513}, {516,8666}, {518,6261}, {519,11500}, {529,6256}, {573,2256}, {580,1191}, {944,7580}, {946,958}, {952,6985}, {962,1012}, {993,4301}, {1001,13464}, {1006,10595}, {1044,9363}, {1056,6908}, {1058,6987}, {1108,1766}, {1259,11682}, {1260,5730}, {1376,11362}, {1457,7078}, {1478,15908}, {1479,11827}, {1498,2818}, {1532,3436}, {1537,11415}, {1597,1872}, {1621,5734}, {1630,20818}, {1656,19854}, {1657,5840}, {1699,5258}, {1702,19014}, {1703,19013}, {1836,22759}, {1902,22479}, {2800,12330}, {2802,22775}, {2817,9798}, {3086,6922}, {3421,6848}, {3427,5770}, {3555,18446}, {3560,5698}, {3600,6916}, {3617,6915}, {3622,6986}, {3651,7967}, {3654,16417}, {3656,16418}, {3820,6944}, {3889,18444}, {3897,20835}, {3913,6796}, {3927,5887}, {4299,11826}, {4423,9624}, {4679,18493}, {5220,20117}, {5251,11522}, {5288,5691}, {5450,11194}, {5687,6905}, {5690,6911}, {5721,19648}, {5763,5901}, {5795,7682}, {5886,11108}, {6001,22778}, {6600,22836}, {6601,6869}, {6737,11499}, {6831,10527}, {6834,17757}, {6836,10529}, {6842,9654}, {6850,18990}, {6865,14986}, {6868,15171}, {6889,10597}, {6891,15325}, {6893,7956}, {6909,20070}, {6923,9655}, {6925,20076}, {6927,7080}, {6928,9669}, {6932,20060}, {6936,10596}, {6962,10528}, {6980,11929}, {7330,9856}, {7491,9668}, {9911,22654}, {10531,11113}, {10599,17530}, {10609,12776}, {11230,16853}, {11231,16863}, {12197,22520}, {12331,13996}, {12497,22744}, {12520,12675}, {12645,18518}, {12670,12842}, {12671,12687}, {12696,22755}, {12697,22756}, {12698,22757}, {12699,22758}, {12701,22760}, {13329,15287}, {13912,22763}, {13975,22764}, {16293,19860}, {16410,19861}, {18761,22793}, {22761,22841}, {22762,22842}

X(22770) = reflection of X(i) in X(j) for these (i,j): (382, 10525), (3913, 6796), (5763, 5901)
X(22770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1482, 3295), (3, 8148, 10679), (3, 10247, 16202), (3, 10680, 999), (3, 12001, 10246), (36, 7991, 10310), (36, 10310, 3), (40, 56, 3), (65, 12704, 2095), (2077, 5204, 3), (3304, 5584, 3576), (7982, 11011, 1482), (7982, 11012, 55), (10902, 16200, 3303), (11012, 16204, 10267), (12702, 22765, 3)

X(22771) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO INNER-FERMAT

Barycentrics    a*(2*sqrt(3)*(a^4-2*(b+c)*b*c*a-(b^2+c^2)*(b-c)^2)*S*a+(a+b+c)*(5*a^6+10*a^3*b^3-10*a^3*b^2*c-10*a^3*b*c^2+10*c^3*a^3-4*b^5*c+8*c^3*b^3-4*b*c^5-5*(b+c)*a^5-2*(5*b^2-9*b*c+5*c^2)*a^4+(5*b^4-14*b^3*c+30*b^2*c^2-14*b*c^3+5*c^4)*a^2+(-5*b^5+15*b^4*c-10*b^3*c^2-10*b^2*c^3+15*b*c^4-5*c^5)*a)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22771) lies on these lines: {3,22557}, {18,56}, {36,22651}, {55,22867}, {104,22531}, {628,2975}, {630,958}, {956,22851}, {999,11740}, {3428,22843}, {5965,22769}, {10966,22865}, {11249,22774}, {11492,22673}, {11493,22669}, {12114,22857}, {16627,22758}, {16628,22765}, {18761,22794}, {19013,19069}, {19014,19072}, {22479,22481}, {22520,22522}, {22654,22656}, {22744,22745}, {22753,22831}, {22755,22852}, {22756,22853}, {22757,22854}, {22759,22859}, {22760,22860}, {22761,22863}, {22762,22864}, {22763,22876}, {22764,22877}, {22766,22884}, {22767,22885}, {22768,22886}

X(22772) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO OUTER-FERMAT

Barycentrics    a*(-2*sqrt(3)*(a^4-2*(b+c)*b*c*a-(b^2+c^2)*(b-c)^2)*S*a+(a+b+c)*(5*a^6+10*a^3*b^3-10*a^3*b^2*c-10*a^3*b*c^2+10*c^3*a^3-4*b^5*c+8*c^3*b^3-4*b*c^5-5*(b+c)*a^5-2*(5*b^2-9*b*c+5*c^2)*a^4+(5*b^4-14*b^3*c+30*b^2*c^2-14*b*c^3+5*c^4)*a^2+(-5*b^5+15*b^4*c-10*b^3*c^2-10*b^2*c^3+15*b*c^4-5*c^5)*a)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22772) lies on these lines: {3,22558}, {17,56}, {36,22652}, {55,22912}, {104,22532}, {532,11194}, {627,2975}, {629,958}, {956,22896}, {999,11739}, {3428,22890}, {5965,22769}, {10966,22910}, {11249,22773}, {11492,22674}, {11493,22670}, {12114,22902}, {16626,22758}, {16629,22765}, {18761,22795}, {19013,19071}, {19014,19070}, {22479,22482}, {22520,22523}, {22654,22657}, {22744,22746}, {22753,22832}, {22755,22897}, {22756,22898}, {22757,22899}, {22759,22904}, {22760,22905}, {22761,22908}, {22762,22909}, {22763,22921}, {22764,22922}, {22766,22929}, {22767,22930}, {22768,22931}

X(22773) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 3rd FERMAT-DAO

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

The reciprocal orthologic center of these triangles is X(13).

X(22773) lies on these lines: {3,12337}, {13,56}, {36,9901}, {55,7975}, {104,6770}, {542,19478}, {616,2975}, {618,958}, {956,12781}, {999,11705}, {3428,5473}, {5478,22753}, {5617,22758}, {6268,22757}, {6270,22756}, {6771,10269}, {9916,22654}, {9982,22744}, {10062,22766}, {10078,22767}, {10966,13076}, {11249,22772}, {11492,12473}, {11493,12472}, {12114,12922}, {12142,22479}, {12205,22520}, {12793,22755}, {12942,22759}, {12952,22760}, {12990,22761}, {12991,22762}, {13103,22765}, {13105,22768}, {13917,22763}, {13982,22764}, {19013,19073}, {19014,19074}

X(22774) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 4th FERMAT-DAO

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

The reciprocal orthologic center of these triangles is X(14).

X(22774) lies on these lines: {3,12336}, {14,56}, {36,9900}, {55,7974}, {542,19478}, {617,2975}, {619,958}, {956,12780}, {999,11706}, {3428,5474}, {5479,22753}, {5613,22758}, {6269,22757}, {6271,22756}, {6774,10269}, {9915,22654}, {9981,22744}, {10061,22766}, {10077,22767}, {10966,13075}, {11249,22771}, {11492,12471}, {11493,12470}, {12114,12921}, {12141,22479}, {12204,22520}, {12792,22755}, {12941,22759}, {12951,22760}, {12988,22761}, {12989,22762}, {13102,22765}, {13104,22768}, {13916,22763}, {13981,22764}, {18761,22797}, {19013,19075}, {19014,19076}

X(22775) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40).

X(22775) lies on these lines: {3,214}, {4,11}, {35,13253}, {36,1727}, {40,2932}, {46,17654}, {55,10698}, {72,2949}, {80,3149}, {84,3065}, {100,3428}, {119,958}, {153,2975}, {392,2950}, {411,6224}, {515,12747}, {517,13205}, {952,11249}, {956,12751}, {993,21635}, {999,11715}, {1001,6914}, {1012,18393}, {1317,10966}, {1537,10044}, {2771,6261}, {2783,22514}, {2787,22504}, {2802,22770}, {2806,19159}, {2831,19162}, {3035,6954}, {3560,12611}, {5180,6909}, {5204,18861}, {5251,15017}, {5253,6888}, {5450,5886}, {6667,6859}, {6702,6918}, {6713,6862}, {6892,22667}, {6905,12247}, {6906,15950}, {6910,21154}, {6911,12619}, {6980,10742}, {7280,7971}, {7580,12119}, {8068,10894}, {8069,12758}, {8071,11570}, {8674,22583}, {9913,22654}, {10051,10074}, {10267,19907}, {10310,17100}, {10680,12737}, {11492,12463}, {11493,12462}, {11499,19914}, {11571,14793}, {12138,22479}, {12199,22520}, {12499,22744}, {12608,16128}, {12752,22755}, {12753,22756}, {12754,22757}, {12763,22759}, {12765,22761}, {12766,22762}, {12775,22768}, {13370,18238}, {13913,22763}, {13977,22764}, {17660,18446}, {18761,22799}, {19013,19081}, {19014,19082}

X(22775) = reflection of X(153) in X(18242)
X(22775) = circumperp conjugate of X(14690)
X(22775) = inverse of X(11713) in the circumcircle
X(22775) = {X(1537), X(10058)}-harmonic conjugate of X(11496)

X(22776) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 3rd HATZIPOLAKIS

Barycentrics    a*(a^18-(b+c)*a^17-2*(2*b^2-3*b*c+2*c^2)*a^16+4*(b^3+c^3)*a^15+(4*b^4+4*c^4-(18*b^2-25*b*c+18*c^2)*b*c)*a^14-(b+c)*(4*b^4+4*c^4-(4*b-3*c)*(3*b-4*c)*b*c)*a^13+(4*b^6+4*c^6+(8*b^4+8*c^4-43*(b-c)^2*b*c)*b*c)*a^12-(b+c)*(4*b^6+4*c^6+(4*b^4+4*c^4-(43*b^2-60*b*c+43*c^2)*b*c)*b*c)*a^11-2*(5*b^8+5*c^8-(11*b^6+11*c^6+(3*b^4+3*c^4-(51*b^2-64*b*c+51*c^2)*b*c)*b*c)*b*c)*a^10+2*(b+c)*(5*b^8+5*c^8-(10*b^6+10*c^6+(3*b^4+3*c^4-2*(19*b^2-32*b*c+19*c^2)*b*c)*b*c)*b*c)*a^9+2*(2*b^10+2*c^10-(10*b^8+10*c^8-(23*b^6+23*c^6-(14*b^4+14*c^4+(53*b^2-120*b*c+53*c^2)*b*c)*b*c)*b*c)*b*c)*a^8-2*(b^2-c^2)*(b-c)*(2*b^8+2*c^8-(6*b^6+6*c^6-(9*b^4+9*c^4+2*(8*b^2-15*b*c+8*c^2)*b*c)*b*c)*b*c)*a^7+(4*b^10+4*c^10+(2*b^8+2*c^8-(35*b^6+35*c^6-(26*b^4+26*c^4+5*(11*b^2-8*b*c+11*c^2)*b*c)*b*c)*b*c)*b*c)*(b-c)^2*a^6-(b^2-c^2)*(b-c)^3*(4*b^8+4*c^8+(3*b^2-2*b*c+3*c^2)*(4*b^2-9*b*c+4*c^2)*(b+c)^2*b*c)*a^5-(b^2-c^2)^2*(b-c)^2*(4*b^8+4*c^8+(3*b^4+3*c^4+2*(4*b^2-19*b*c+4*c^2)*b*c)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)^2*(b-c)^3*(4*b^4+4*c^4+(4*b^2-5*b*c+4*c^2)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(b^4+c^4+2*(2*b^2+7*b*c+2*c^2)*b*c)*a^2-(b^2-c^2)^5*(b-c)^3*(b^2+c^2)^2*a-2*(b^2-c^2)^6*(b^2+c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22776) lies on these lines: {3,22559}, {36,22653}, {55,22969}, {56,18978}, {104,22533}, {956,22941}, {958,22957}, {999,22476}, {2929,20838}, {2975,22647}, {3428,22951}, {10966,22965}, {12114,22956}, {18761,22800}, {19013,19083}, {19014,19084}, {22479,22483}, {22520,22524}, {22654,22658}, {22744,22747}, {22753,22833}, {22755,22943}, {22756,22945}, {22757,22947}, {22758,22955}, {22759,22958}, {22760,22959}, {22761,22963}, {22762,22964}, {22763,22976}, {22764,22977}, {22765,22979}, {22766,22980}, {22767,22981}, {22768,22982}

X(22777) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO HUTSON EXTOUCH

Barycentrics    a^2*(a^8-2*(b+c)*a^7-2*(b^2+5*b*c+c^2)*a^6+2*(b+c)*(3*b^2+2*b*c+3*c^2)*a^5+2*(13*b^2+6*b*c+13*c^2)*b*c*a^4-2*(b^2-c^2)*(b-c)*(b+3*c)*(3*b+c)*a^3+2*(b^6+c^6-(11*b^4+11*c^4+(5*b^2+34*b*c+5*c^2)*b*c)*b*c)*a^2+2*(b^2-c^2)*(b-c)^3*(b^2+6*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)*(b^2-6*b*c+c^2)) : :
X(22777) = 3*X(3)-X(12631) = 3*X(376)+X(15998) = 3*X(12333)-2*X(12631)

The reciprocal orthologic center of these triangles is X(40).

X(22777) lies on these lines: {3,12333}, {36,9898}, {55,8000}, {56,7160}, {104,12249}, {376,15998}, {956,12777}, {958,12858}, {999,12260}, {1001,13464}, {1490,3428}, {2975,9874}, {5584,12842}, {5920,17624}, {10059,22766}, {10075,22767}, {10966,12863}, {10993,12773}, {11249,22754}, {11492,12465}, {11493,12464}, {12114,12857}, {12139,22479}, {12200,22520}, {12411,22654}, {12436,22667}, {12500,22744}, {12599,22753}, {12789,22755}, {12801,22756}, {12802,22757}, {12856,22758}, {12859,22759}, {12860,22760}, {12861,22761}, {12862,22762}, {12872,22765}, {12874,22768}, {13914,22763}, {13978,22764}, {18761,22801}, {19013,19085}, {19014,19086}

X(22778) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO MIDHEIGHT

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

The reciprocal orthologic center of these triangles is X(4).

X(22778) lies on these lines: {1,7169}, {3,12335}, {30,22659}, {36,9899}, {55,7973}, {56,64}, {104,12250}, {154,5584}, {197,9121}, {221,1496}, {956,12779}, {958,2883}, {999,12262}, {1498,3428}, {2777,19478}, {2975,6225}, {3357,10269}, {5878,22758}, {6000,11249}, {6001,22770}, {6247,22753}, {6266,22757}, {6267,22756}, {7355,10966}, {8991,22763}, {9914,22654}, {10060,22766}, {10076,22767}, {11381,22479}, {11492,12469}, {11493,12468}, {12114,12920}, {12202,22520}, {12502,22744}, {12791,22755}, {12940,22759}, {12950,22760}, {12986,22761}, {12987,22762}, {13093,22765}, {13094,22768}, {13980,22764}, {18761,22802}, {19013,19087}, {19014,19088}

X(22778) = {X(1498), X(3428)}-harmonic conjugate of X(3556)

X(22779) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(22779) lies on these lines: {36,9902}, {39,958}, {55,7976}, {56,76}, {58,10800}, {104,12251}, {194,2975}, {384,22520}, {511,12114}, {538,11194}, {726,8666}, {732,22769}, {956,12782}, {999,12263}, {1001,5145}, {2782,11249}, {3095,22758}, {3097,5258}, {3428,11257}, {5969,22565}, {6248,22753}, {6272,22757}, {6273,22756}, {8301,15654}, {8992,22763}, {9917,22654}, {9983,22744}, {10063,22766}, {10079,22767}, {10966,13077}, {11492,12475}, {11493,12474}, {12143,22479}, {12513,14839}, {12794,22755}, {12836,22760}, {12837,22759}, {12992,22761}, {12993,22762}, {13108,22765}, {13109,22768}, {13983,22764}, {14881,18761}, {19013,19089}, {19014,19090}

X(22780) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3).

X(22780) lies on these lines: {3,12339}, {36,9903}, {55,7977}, {56,83}, {104,12252}, {732,22769}, {754,11194}, {956,12783}, {958,6292}, {999,12264}, {2896,2975}, {3428,12122}, {6249,22753}, {6274,22757}, {6275,22756}, {6287,22758}, {8666,17766}, {8993,22763}, {9918,22654}, {10064,22766}, {10080,22767}, {10966,13078}, {11249,22680}, {11492,12477}, {11493,12476}, {12114,12924}, {12144,22479}, {12206,22520}, {12795,22755}, {12944,22759}, {12954,22760}, {12994,22761}, {12995,22762}, {13111,22765}, {13112,22768}, {13984,22764}, {18761,22803}, {19013,19091}, {19014,19092}

X(22781) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO REFLECTION

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

The reciprocal orthologic center of these triangles is X(4).

X(22781) lies on these lines: {1,2917}, {3,12341}, {36,9905}, {54,56}, {55,7979}, {104,12254}, {195,22765}, {539,11194}, {956,12785}, {958,1209}, {999,12266}, {1154,11249}, {2888,2975}, {3428,7691}, {3574,22753}, {6276,22757}, {6277,22756}, {6288,22758}, {8995,22763}, {9920,22654}, {9985,22744}, {10066,22766}, {10082,22767}, {10269,10610}, {10628,22583}, {10966,13079}, {11492,12481}, {11493,12480}, {11576,22479}, {12114,12926}, {12208,22520}, {12797,22755}, {12946,22759}, {12956,22760}, {12998,22761}, {12999,22762}, {13121,22768}, {13986,22764}, {18761,22804}, {19013,19095}, {19014,19096}, {19478,22586}

X(22782) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st SCHIFFLER

Barycentrics    a*(a^9-2*(b+c)*a^8-2*(b-c)^2*a^7+2*(3*b^2-2*b*c+3*c^2)*(b+c)*a^6-(6*b^2-17*b*c+6*c^2)*b*c*a^5-2*(b+c)*(3*b^4+3*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*a^4+(2*b^6+2*c^6-(11*b^2-4*b*c+11*c^2)*b^2*c^2)*a^3+2*(b+c)*(b^6+c^6-(b^2+b*c+c^2)*b^2*c^2)*a^2-(b^2-c^2)^2*(b^4+c^4-2*(b^2-3*b*c+c^2)*b*c)*a-2*(b^2-c^2)^3*(b-c)*b*c) : :

The reciprocal orthologic center of these triangles is X(79).

X(22782) lies on these lines: {3,12342}, {36,12409}, {55,13100}, {56,10266}, {104,12255}, {956,12786}, {958,12937}, {999,12267}, {1621,7354}, {2771,12745}, {2975,12849}, {3428,12556}, {5046,12615}, {6597,15071}, {6949,12623}, {10966,13080}, {11014,12513}, {11491,11826}, {11492,12483}, {11493,12482}, {12114,12927}, {12146,22479}, {12209,22520}, {12414,22654}, {12504,22744}, {12600,22753}, {12798,22755}, {12807,22756}, {12808,22757}, {12919,22758}, {12947,22759}, {12957,22760}, {13000,22761}, {13001,22762}, {13126,22765}, {13128,22766}, {13129,22767}, {13130,22768}, {13465,23016}, {13919,22763}, {13987,22764}, {18761,22805}, {19013,19097}, {19014,19098}

X(22782) = reflection of X(13465) in X(23016)

X(22783) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st TRI-SQUARES-CENTRAL

Barycentrics    a*(3*a*(a^4-2*b*c*(b+c)*a-(b^2+c^2)*(b-c)^2)*S-(a+b+c)*(3*a^6-3*(b+c)*a^5-2*(3*b^2-8*b*c+3*c^2)*a^4+6*(b^2-c^2)*(b-c)*a^3+(3*b^4+3*c^4-b*c*(11*b^2-18*b*c+11*c^2))*a^2-3*(b^2-c^2)*(b-c)^3*a-5*(b^2-c^2)^2*b*c)) : :

The reciprocal orthologic center of these triangles is X(13665).

X(22783) lies on these lines: {3,13675}, {30,22624}, {36,13679}, {55,13702}, {56,1327}, {104,13674}, {956,13688}, {958,13694}, {999,13667}, {2975,13678}, {3428,13666}, {10966,13699}, {11492,13683}, {11493,13682}, {12114,13693}, {13668,22479}, {13672,22520}, {13680,22654}, {13685,22744}, {13687,22753}, {13689,22755}, {13690,22756}, {13691,22757}, {13692,22758}, {13695,22759}, {13696,22760}, {13697,22761}, {13698,22762}, {13713,22765}, {13714,22766}, {13715,22767}, {13716,22768}, {13920,22763}, {13988,22764}, {18761,22806}, {19013,19099}, {19014,22541}, {22769,22784}

X(22784) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    a*(-3*a*(a^4-2*b*c*(b+c)*a-(b^2+c^2)*(b-c)^2)*S-(a+b+c)*(3*a^6-3*(b+c)*a^5-2*(3*b^2-8*b*c+3*c^2)*a^4+6*(b^2-c^2)*(b-c)*a^3+(3*b^4+3*c^4-b*c*(11*b^2-18*b*c+11*c^2))*a^2-3*(b^2-c^2)*(b-c)^3*a-5*(b^2-c^2)^2*b*c)) : :

The reciprocal orthologic center of these triangles is X(13785).

X(22784) lies on these lines: {3,13795}, {30,22595}, {36,13799}, {55,13822}, {56,1328}, {104,13794}, {956,13808}, {958,13814}, {999,13787}, {2975,13798}, {3428,13786}, {10966,13819}, {11492,13803}, {11493,13802}, {12114,13813}, {13788,22479}, {13792,22520}, {13800,22654}, {13805,22744}, {13807,22753}, {13809,22755}, {13810,22756}, {13811,22757}, {13812,22758}, {13815,22759}, {13816,22760}, {13817,22761}, {13818,22762}, {13836,22765}, {13837,22766}, {13838,22767}, {13839,22768}, {13848,22763}, {13849,22764}, {18761,22807}, {19013,19101}, {19014,19100}, {22769,22783}

X(22785) = PERSPECTOR OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND LUCAS REFLECTION

Barycentrics    (SB+SC)*(5*(4*R^2+SW)*S^4-3*(2*R^2-SW)*(2*R^2-SA)*SA*SW^2-(2*R^2*(24*R^2*SA+20*R^2*SW-6*SA^2+3*SA*SW-5*SW^2)+SW*(3*SA^2-5*SW^2))*S^2+4*(S^4+(R^2*(4*R^2-3*SA+SW)+2*SW^2)*S^2+(4*R^4*(3*SA+SW)-R^2*(3*SA^2+3*SA*SW+4*SW^2)+SW^3)*SW)*S) : :

X(22785) lies on these lines: {3,11967}, {6,6401}, {98,14167}, {6199,11941}, {6200,11973}, {6221,11959}, {6395,11942}, {6396,11971}, {6398,11960}, {6433,11963}, {6434,11964}, {6435,11975}, {6436,11977}, {8289,22499}, {8375,11937}, {8376,11938}, {11983,22786}, {19379,19390}

X(22786) = PERSPECTOR OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND LUCAS(-1) REFLECTION

Barycentrics    (SB+SC)*(5*(4*R^2+SW)*S^4-3*(2*R^2-SW)*(2*R^2-SA)*SA*SW^2-(2*R^2*(24*R^2*SA+20*R^2*SW-6*SA^2+3*SA*SW-5*SW^2)+SW*(3*SA^2-5*SW^2))*S^2-4*(S^4+(R^2*(4*R^2-3*SA+SW)+2*SW^2)*S^2+(4*R^4*(3*SA+SW)-R^2*(3*SA^2+3*SA*SW+4*SW^2)+SW^3)*SW)*S) : :

X(22786) lies on these lines: {3,11968}, {6,6402}, {98,14168}, {6199,11943}, {6200,11972}, {6221,11961}, {6395,11944}, {6396,11974}, {6398,11962}, {6433,11965}, {6434,11966}, {6435,11978}, {6436,11976}, {8289,22500}, {8375,11939}, {8376,11940}, {11983,22785}, {19379,19391}

X(22787) = PERSPECTOR OF THESE TRIANGLES: 3rd CONWAY AND INNER-YFF

Barycentrics    (b+2*c)*(2*b+c)*a^11+3*(b+c)*(2*b^2+b*c+2*c^2)*a^10-2*(3*b^2-2*b*c+3*c^2)*b*c*a^9-8*(b+c)*(b^2+c^2)*(2*b^2+b*c+2*c^2)*a^8-2*(6*b^6+6*c^6+(15*b^4+15*c^4+2*(24*b^2+25*b*c+24*c^2)*b*c)*b*c)*a^7+2*(b+c)*(6*b^6+6*c^6-(17*b^4+17*c^4+2*(14*b^2+27*b*c+14*c^2)*b*c)*b*c)*a^6+4*(4*b^8+4*c^8+(5*b^6+5*c^6-(3*b^4+3*c^4+(39*b^2+58*b*c+39*c^2)*b*c)*b*c)*b*c)*a^5+4*(b+c)*(12*b^6+12*c^6+(5*b^4+5*c^4-2*(8*b^2+17*b*c+8*c^2)*b*c)*b*c)*b*c*a^4-(6*b^8+6*c^8-(29*b^6+29*c^6+(26*b^4+26*c^4-(5*b^2+104*b*c+5*c^2)*b*c)*b*c)*b*c)*(b+c)^2*a^3-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6+(9*b^4+9*c^4-2*(15*b^2+17*b*c+15*c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)^2*(b+c)^2*(3*b^4+3*c^4-2*(b^2+3*b*c+c^2)*b*c)*b*c*a-4*(b^2-c^2)^4*(b+c)*b^2*c^2 : :

X(22787) lies on these lines: {10057,12551}, {10442,15298}, {10478,22788}, {11021,18223}, {12435,12647}

X(22788) = PERSPECTOR OF THESE TRIANGLES: 3rd CONWAY AND OUTER-YFF

Barycentrics    (b+2*c)*(2*b+c)*a^11+3*(b+c)*(2*b^2+b*c+2*c^2)*a^10-2*(3*b^2-2*b*c+3*c^2)*b*c*a^9-8*(b+c)*(2*b^4+2*c^4-(b-c)^2*b*c)*a^8-6*(2*b^6+2*c^6-(3*b^4+2*b^2*c^2+3*c^4)*b*c)*a^7+2*(b+c)*(6*b^6+6*c^6-(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*b*c)*a^6+4*(b^2-b*c+c^2)*(4*b^6+4*c^6+(b^4+c^4-2*(3*b^2+5*b*c+3*c^2)*b*c)*b*c)*a^5-4*(7*b^2-10*b*c+7*c^2)*(b+c)^3*b^2*c^2*a^4-(6*b^8+6*c^8-(13*b^6+13*c^6-(38*b^4+38*c^4-(27*b^2-8*b*c+27*c^2)*b*c)*b*c)*b*c)*(b+c)^2*a^3-(b^2-c^2)^2*(b+c)*(2*b^2+b*c+2*c^2)*(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2-2*(b^2-c^2)^2*(b+c)^2*(3*b^4+3*c^4-2*(b^2-b*c+c^2)*b*c)*b*c*a-4*(b^2-c^2)^4*(b+c)*b^2*c^2 : :

X(22788) lies on these lines: {1479,12547}, {10073,12551}, {10442,15299}, {10478,22787}, {10573,12435}, {11021,18224}

X(22789) = PERSPECTOR OF THESE TRIANGLES: 3rd CONWAY AND INNER-YFF TANGENTS

Barycentrics    (b+2*c)*(2*b+c)*a^14+2*(b+c)*(2*b^2-b*c+2*c^2)*a^13-2*(4*b^4+4*c^4+(7*b^2-6*b*c+7*c^2)*b*c)*a^12-4*(b^3+c^3)*(5*b^2-b*c+5*c^2)*a^11+(b^2+4*b*c+c^2)*(10*b^4+10*c^4-(37*b^2+4*b*c+37*c^2)*b*c)*a^10+2*(b+c)*(20*b^6+20*c^6-(41*b^4+41*c^4-2*(12*b^2-43*b*c+12*c^2)*b*c)*b*c)*a^9+4*(7*b^6+7*c^6+(55*b^4+55*c^4-4*(3*b^2+b*c+3*c^2)*b*c)*b*c)*b*c*a^8-8*(b+c)*(5*b^8+5*c^8-(16*b^6+16*c^6-(7*b^4+7*c^4-5*(5*b^2+2*b*c+5*c^2)*b*c)*b*c)*b*c)*a^7-(10*b^10+10*c^10+(37*b^8+37*c^8+2*(57*b^6+57*c^6-(128*b^4+128*c^4+(110*b^2+37*b*c+110*c^2)*b*c)*b*c)*b*c)*b*c)*a^6+2*(b+c)*(10*b^10+10*c^10-(51*b^8+51*c^8-2*(17*b^6+17*c^6-(22*b^4+22*c^4-(10*b^2+63*b*c+10*c^2)*b*c)*b*c)*b*c)*b*c)*a^5+2*(b^2-c^2)^2*(4*b^8+4*c^8+(b+3*c)*(3*b+c)*(3*b^4+3*c^4-8*(b^2-b*c+c^2)*b*c)*b*c)*a^4-4*(b^2-c^2)^2*(b+c)*(b^8+c^8-(10*b^6+10*c^6-(9*b^4+9*c^4-8*(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*b*c)*a^3-(b^2-c^2)^2*(b+c)^2*(2*b^8+2*c^8-(b^6+c^6+(26*b^4+26*c^4-(33*b^2-32*b*c+33*c^2)*b*c)*b*c)*b*c)*a^2-2*(b^2-c^2)^5*(b-c)*(3*b^2+2*b*c+3*c^2)*b*c*a-4*(b^2-c^2)^6*b^2*c^2 : :

X(22789) lies on these lines: {11021,18225}, {12115,12547}, {12435,12648}, {12551,12749}

X(22790) = PERSPECTOR OF THESE TRIANGLES: 3rd CONWAY AND OUTER-YFF TANGENTS

Barycentrics    (b+2*c)*(2*b+c)*a^14+2*(b+c)*(2*b^2-b*c+2*c^2)*a^13-2*(4*b^4+4*c^4+(15*b^2+14*b*c+15*c^2)*b*c)*a^12-4*(b+c)*(5*b^4+5*c^4-(2*b^2-9*b*c+2*c^2)*b*c)*a^11+(10*b^6+10*c^6+(67*b^4+67*c^4+2*(b^2-13*b*c+c^2)*b*c)*b*c)*a^10+2*(b+c)*(20*b^6+20*c^6-(9*b^4+9*c^4-2*(4*b^2+5*b*c+4*c^2)*b*c)*b*c)*a^9-4*(17*b^6+17*c^6-(11*b^4+11*c^4-4*(b^2+11*b*c+c^2)*b*c)*b*c)*b*c*a^8-8*(b+c)*(5*b^8+5*c^8-(4*b^6+4*c^6-(b^4+c^4+(19*b^2-30*b*c+19*c^2)*b*c)*b*c)*b*c)*a^7-(10*b^10+10*c^10-(27*b^8+27*c^8-2*(25*b^6+25*c^6-(32*b^4+32*c^4+13*(6*b^2-7*b*c+6*c^2)*b*c)*b*c)*b*c)*b*c)*a^6+2*(b+c)*(10*b^10+10*c^10-(19*b^8+19*c^8-2*(9*b^6+9*c^6+(26*b^4+26*c^4-(46*b^2-63*b*c+46*c^2)*b*c)*b*c)*b*c)*b*c)*a^5+2*(b^2-c^2)^2*(4*b^8+4*c^8+(b^6+c^6+(18*b^4+18*c^4-(7*b^2-72*b*c+7*c^2)*b*c)*b*c)*b*c)*a^4-4*(b^2-c^2)^2*(b+c)*(b^8+c^8-(6*b^6+6*c^6-(7*b^4+7*c^4-4*(3*b^2-b*c+3*c^2)*b*c)*b*c)*b*c)*a^3-(b^2-c^2)^2*(b+c)^2*(2*b^8+2*c^8-(b^6+c^6+(10*b^4+10*c^4-(b^2+c^2)*b*c)*b*c)*b*c)*a^2-2*(b^2-c^2)^5*(b-c)*(3*b^2+2*b*c+3*c^2)*b*c*a-4*(b^2-c^2)^6*b^2*c^2 : :

X(22790) lies on these lines: {11021,18226}, {12116,12547}, {12435,12649}, {12551,12750}

X(22791) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO EXCENTERS-MIDPOINTS

Barycentrics    2*(b+c)*a^3+(b^2-4*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(22791) = 5*X(1)-3*X(3655) = X(1)-3*X(3656) = 3*X(1)-X(18481) = 3*X(2)-5*X(18493) = 3*X(3)-5*X(3616) = X(3)-3*X(5603) = 3*X(3)-X(6361) = 5*X(3)-3*X(9778) = 3*X(962)+5*X(3616) = X(962)+3*X(5603) = X(962)+2*X(5901) = 3*X(962)+X(6361) = 5*X(962)+3*X(9778) = 5*X(3616)-9*X(5603) = 5*X(3616)-6*X(5901) = 5*X(3616)-X(6361) = 25*X(3616)-9*X(9778) = X(3655)-5*X(3656) = 3*X(3655)+5*X(12699) = 9*X(3655)-5*X(18481) = 3*X(3656)+X(12699) = 9*X(3656)-X(18481) = 3*X(5603)-2*X(5901) = 9*X(5603)-X(6361) = 5*X(5603)-X(9778) = 6*X(5901)-X(6361) = 10*X(5901)-3*X(9778) = 3*X(12699)+X(18481) = X(12702)-5*X(18493) = X(16118)-3*X(16159)

The reciprocal orthologic center of these triangles is X(10).

X(22791) lies on these lines: {1,30}, {2,12702}, {3,962}, {4,145}, {5,10}, {7,7373}, {8,381}, {11,5903}, {12,5697}, {20,10246}, {35,15950}, {38,5492}, {40,140}, {46,11376}, {56,1387}, {57,11373}, {63,19919}, {65,496}, {72,22010}, {80,11280}, {119,13996}, {165,3530}, {226,9957}, {265,7978}, {355,546}, {376,3622}, {382,944}, {390,6869}, {392,8728}, {442,3877}, {484,5433}, {495,3057}, {497,12433}, {515,1483}, {516,550}, {518,21850}, {519,3845}, {528,22836}, {529,22837}, {547,1698}, {548,3576}, {549,1125}, {551,8703}, {632,6684}, {758,3813}, {908,10914}, {912,9856}, {938,1159}, {942,12053}, {956,11415}, {999,4295}, {1000,5261}, {1012,10680}, {1058,6851}, {1145,11681}, {1210,7743}, {1319,1770}, {1388,4299}, {1478,2098}, {1479,2099}, {1480,5711}, {1484,2800}, {1511,11723}, {1519,10942}, {1565,17753}, {1572,5305}, {1595,1902}, {1596,1829}, {1656,5657}, {1657,5731}, {1697,11374}, {1737,10593}, {2095,6847}, {2102,10751}, {2103,10750}, {2140,20328}, {2475,5330}, {2771,3874}, {2802,11698}, {2807,6102}, {2975,3648}, {3091,4678}, {3146,7967}, {3149,10679}, {3241,3830}, {3244,15687}, {3295,3485}, {3333,5586}, {3336,12515}, {3340,5722}, {3416,18358}, {3419,11682}, {3434,5730}, {3476,9655}, {3486,9668}, {3487,6767}, {3488,4323}, {3543,3623}, {3545,3617}, {3560,5698}, {3583,10950}, {3585,10944}, {3600,18541}, {3621,3839}, {3628,7991}, {3633,14893}, {3634,15699}, {3652,6763}, {3671,5045}, {3679,5066}, {3753,17527}, {3754,3816}, {3818,5846}, {3832,20052}, {3834,12610}, {3843,12645}, {3850,4668}, {3851,5818}, {3853,5691}, {3857,4746}, {3858,4701}, {3860,4677}, {3861,5881}, {3869,6841}, {3871,18524}, {3880,16616}, {3884,5499}, {3898,11263}, {3899,21677}, {3913,18491}, {3940,5082}, {3962,5887}, {3988,20117}, {4004,6922}, {4018,8727}, {4029,10445}, {4127,5694}, {4297,15178}, {4318,18447}, {4342,21620}, {4389,10446}, {4861,5057}, {4867,18406}, {5054,5550}, {5055,9780}, {5074,21258}, {5076,10248}, {5119,11375}, {5221,10072}, {5248,5428}, {5250,6675}, {5432,5443}, {5493,10165}, {5554,17556}, {5563,11246}, {5693,7965}, {5708,14986}, {5754,19998}, {5758,6913}, {5761,19541}, {5771,6824}, {5787,7971}, {5805,12700}, {5840,19907}, {5843,11372}, {5853,18482}, {5883,13145}, {5884,6583}, {5905,18519}, {5919,13407}, {6033,7983}, {6097,16678}, {6221,13902}, {6259,12650}, {6264,16128}, {6265,14217}, {6321,7970}, {6398,13959}, {6644,11365}, {6738,18527}, {6762,18540}, {6836,10596}, {6842,10129}, {6905,11849}, {6906,22765}, {6911,10306}, {6914,11249}, {6923,10532}, {6924,11248}, {6925,10597}, {6928,10531}, {6981,8166}, {7173,18395}, {7377,17230}, {7508,11012}, {7514,8193}, {7530,9798}, {7555,9591}, {7561,18453}, {7580,16202}, {7718,18494}, {7728,7984}, {7962,9612}, {7973,14216}, {7989,12811}, {8192,18534}, {8196,11253}, {8203,11252}, {8666,17768}, {9566,19853}, {9625,12107}, {9669,18391}, {9818,12410}, {9905,22051}, {9933,12293}, {9943,13373}, {10021,16139}, {10039,10592}, {10109,19875}, {10164,14869}, {10264,12261}, {10272,12778}, {10284,18242}, {10386,10624}, {10431,10806}, {10572,11011}, {10573,10896}, {10695,10741}, {10696,10747}, {10697,10739}, {10699,15521}, {10700,15522}, {10703,10740}, {10705,12918}, {10733,12898}, {10749,13099}, {10785,13226}, {10800,14880}, {10826,11545}, {10895,12647}, {10912,18516}, {11014,11827}, {11024,16863}, {11539,19862}, {11551,17609}, {11709,14677}, {11725,12042}, {11735,12041}, {11801,13211}, {12102,16189}, {12195,18502}, {12436,16004}, {12454,18495}, {12455,18497}, {12495,18500}, {12512,17502}, {12513,18761}, {12514,16617}, {12619,16174}, {12626,18507}, {12627,18509}, {12628,18511}, {12635,18517}, {12636,18520}, {12637,18522}, {12648,18542}, {12649,18544}, {12735,12943}, {12773,13126}, {13665,19066}, {13785,19065}, {13911,18538}, {13973,18762}, {14269,20050}, {14377,17044}, {14839,14881}, {14923,17757}, {15326,21842}, {15713,19883}, {15808,17504}, {16150,20067}, {16212,18508}, {17563,17614}

X(22791) = midpoint of X(i) and X(j) for these {i,j}: {1, 12699}, {3, 962}, {4, 1482}, {8, 8148}, {265, 7978}, {355, 7982}, {382, 944}, {2102, 10751}, {2103, 10750}, {3241, 3830}, {5758, 8158}, {5787, 7971}, {5905, 18519}, {6033, 7983}, {6259, 12650}, {6264, 16128}, {6265, 14217}, {6321, 7970}, {7728, 7984}, {7973, 14216}, {9933, 12293}, {10695, 10741}, {10696, 10747}, {10697, 10739}, {10699, 15521}, {10700, 15522}, {10703, 10740}, {10705, 12918}, {10733, 12898}, {10749, 13099}, {12626, 18507}
X(22791) = reflection of X(i) in X(j) for these (i,j): (3, 5901), (5, 946), (8, 18357), (40, 140), (355, 546), (1511, 11723), (3416, 18358), (3679, 5066), (4297, 15178), (5884, 6583), (9905, 22051), (9943, 13373), (10264, 12261), (12619, 16174)
X(22791) = complement of X(12702)
X(22791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 79, 5434), (1, 1836, 18990), (1, 12701, 15171), (3, 5603, 5901), (4, 145, 18525), (4, 20060, 10742), (10, 946, 9955), (946, 7686, 7956), (962, 3616, 6361), (962, 5603, 3), (1482, 18525, 145), (3616, 6361, 3), (3656, 12699, 1), (5603, 6361, 3616), (12702, 18493, 2), (15170, 16137, 1)

X(22792) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO EXTOUCH

Barycentrics    2*a^7-(b+c)*a^6-(3*b^2-8*b*c+3*c^2)*a^5-2*(b+c)*b*c*a^4-2*(b-c)^2*b*c*a^3+(b^2-c^2)*(b-c)*(3*b^2+4*b*c+3*c^2)*a^2+(b^2-c^2)^2*(b^2-6*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(22792) = 3*X(4)-X(5787) = 3*X(4)+X(6223) = 5*X(4)-X(9799) = 3*X(5)-2*X(6705) = X(84)-3*X(381) = 5*X(3091)-X(12246) = X(3146)+3*X(5658) = 5*X(3843)-X(12684) = 2*X(5450)-3*X(11230) = X(5787)+3*X(6259) = 5*X(5787)-3*X(9799) = X(6223)-3*X(6259) = 5*X(6223)+3*X(9799) = 5*X(6259)+X(9799) = X(7992)-5*X(18492)

The reciprocal orthologic center of these triangles is X(40).

X(22792) lies on these lines: {4,7}, {5,6692}, {30,6260}, {56,1538}, {84,381}, {153,10914}, {377,10157}, {382,1490}, {388,17622}, {515,1483}, {516,12607}, {517,6256}, {546,6245}, {1158,9956}, {1385,2829}, {1466,9579}, {1478,9856}, {1479,12678}, {1699,3304}, {1709,10895}, {1768,17606}, {1836,13601}, {1898,13273}, {2099,5691}, {2475,5927}, {2478,11227}, {2771,12761}, {3057,12763}, {3091,12246}, {3146,5658}, {3579,18242}, {3583,12680}, {3585,12688}, {3824,6913}, {3843,12684}, {3916,6932}, {4298,7956}, {5044,6850}, {5046,10167}, {5049,10531}, {5084,10156}, {5086,9809}, {5122,6834}, {5253,17618}, {5439,13729}, {5450,11230}, {5499,11231}, {5777,6923}, {5887,16128}, {6001,10107}, {6257,18511}, {6258,18509}, {6929,9940}, {7330,15239}, {7971,18525}, {7992,18492}, {8987,18538}, {9654,12705}, {9780,14646}, {9818,9910}, {9955,12114}, {9957,12115}, {10085,10896}, {10728,21740}, {11681,17613}, {12196,18502}, {12330,18491}, {12456,18495}, {12457,18497}, {12496,18500}, {12616,22798}, {12667,12699}, {12668,18507}, {12675,18527}, {12676,18516}, {12677,18517}, {12686,18542}, {12687,18544}, {13665,19068}, {13785,19067}, {13974,18762}, {18237,18761}, {18245,18520}, {18246,18522}

X(22792) = midpoint of X(i) and X(j) for these {i,j}: {4, 6259}, {382, 1490}, {7971, 18525}, {12667, 12699}, {12668, 18507}
X(22792) = reflection of X(i) in X(j) for these (i,j): (1158, 9956), (1385, 12608), (3579, 18242)
X(22792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6223, 5787), (1478, 12679, 9856), (5787, 6259, 6223)

X(22793) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 3rd EXTOUCH

Barycentrics    2*a^4+(b+c)*a^3-2*b*c*a^2-(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(22793) = X(3)-3*X(1699) = 5*X(3)-7*X(3624) = 3*X(3)-5*X(8227) = 2*X(3)-3*X(11230) = 5*X(4)-X(8) = 3*X(4)-X(355) = 3*X(4)+X(962) = X(4)+3*X(9812) = 3*X(4)-7*X(10248) = 9*X(4)-X(12245) = 3*X(8)-5*X(355) = 3*X(8)+5*X(962) = X(8)+15*X(9812) = 9*X(8)-5*X(12245) = X(8)+5*X(12699) = 2*X(8)-5*X(18480) = X(355)+9*X(9812) = X(355)-7*X(10248) = 3*X(355)-X(12245) = X(355)+3*X(12699) = 2*X(355)-3*X(18480) = 15*X(1699)-7*X(3624) = 9*X(1699)-5*X(8227) = 3*X(1699)-2*X(9955) = 7*X(3624)-10*X(9955) = 14*X(3624)-15*X(11230) = 5*X(8227)-6*X(9955) = 10*X(8227)-9*X(11230) = 4*X(9955)-3*X(11230)

The reciprocal orthologic center of these triangles is X(4).

X(22793) lies on these lines: {1,382}, {3,1699}, {4,8}, {5,516}, {7,15008}, {10,546}, {11,1770}, {20,5886}, {30,551}, {35,17605}, {40,381}, {46,10896}, {56,7743}, {57,9669}, {65,3521}, {79,354}, {80,17501}, {140,3817}, {149,3555}, {165,1656}, {226,15171}, {390,5714}, {392,2475}, {484,17606}, {495,10624}, {496,4292}, {497,5045}, {499,5122}, {515,1483}, {519,15687}, {528,21077}, {535,11260}, {548,10165}, {549,12512}, {550,1125}, {631,9779}, {632,10171}, {942,1479}, {944,3543}, {952,3853}, {971,16127}, {999,9579}, {1001,3824}, {1155,7741}, {1156,11662}, {1212,5134}, {1319,10483}, {1387,4311}, {1478,9957}, {1482,3830}, {1538,3149}, {1657,3576}, {1697,9654}, {1698,3851}, {1702,13665}, {1703,13785}, {1709,11928}, {1717,9630}, {2635,5399}, {2646,18393}, {2771,7728}, {2777,12261}, {2778,19506}, {2800,22938}, {2802,22799}, {2807,5446}, {3057,3585}, {3058,13407}, {3090,9778}, {3091,6361}, {3146,5603}, {3295,9580}, {3333,18541}, {3338,11238}, {3474,10591}, {3526,7988}, {3528,5550}, {3529,3616}, {3530,19862}, {3534,7987}, {3544,19877}, {3615,5196}, {3628,10164}, {3652,10032}, {3653,11001}, {3654,3839}, {3655,10595}, {3660,7702}, {3671,12433}, {3679,14269}, {3701,21282}, {3753,5046}, {3832,5657}, {3838,5248}, {3843,5587}, {3845,4745}, {3850,5493}, {3855,9780}, {3861,11362}, {3911,10593}, {3916,11680}, {3944,5266}, {4293,11373}, {4294,11374}, {4295,5225}, {4299,5126}, {4302,11375}, {4309,17718}, {4312,5708}, {4314,5719}, {4324,5443}, {4325,16173}, {4333,5204}, {4338,5221}, {4848,12019}, {5049,10404}, {5054,16192}, {5073,10246}, {5076,7982}, {5119,10895}, {5183,18395}, {5250,17532}, {5270,5919}, {5290,6767}, {5439,9782}, {5536,7701}, {5556,11037}, {5563,16118}, {5697,18513}, {5698,5791}, {5709,5789}, {5715,10267}, {5720,12651}, {5790,7991}, {5804,9800}, {5805,6851}, {5840,9945}, {5844,12102}, {5881,8148}, {5885,6895}, {5899,9626}, {5903,18514}, {6001,22802}, {6240,11363}, {6265,10724}, {6284,12047}, {6797,12764}, {6840,13145}, {6841,7965}, {6915,17618}, {6943,17613}, {6985,11496}, {7686,7706}, {7957,18406}, {7973,18405}, {8976,9616}, {9590,18378}, {9593,15484}, {9624,17800}, {9818,9911}, {10306,18491}, {10308,13243}, {10386,13405}, {10431,10531}, {10446,17361}, {10728,12737}, {10742,14217}, {10916,17768}, {11012,13743}, {11248,19541}, {11365,12085}, {11531,12645}, {11699,17702}, {12053,18990}, {12197,18502}, {12458,18495}, {12459,18497}, {12497,18500}, {12563,15935}, {12696,18507}, {12697,18509}, {12698,18511}, {12703,18542}, {12704,18544}, {12747,13253}, {13369,13374}, {13912,18538}, {13975,18762}, {14869,19878}, {15172,21620}, {15931,16117}, {16160,22936}, {16200,18526}, {17579,17614}, {17748,17764}, {18520,22841}, {18522,22842}, {18761,22770}

X(22793) = midpoint of X(i) and X(j) for these {i,j}: {1, 382}, {4, 12699}, {149, 16128}, {1482, 5691}, {3146, 18481}, {3655, 15682}, {5881, 8148}, {6265, 10724}, {10728, 12737}, {10742, 14217}, {11531, 12645}, {12696, 18507}, {12747, 13253}
X(22793) = reflection of X(i) in X(j) for these (i,j): (3, 9955), (5, 18483), (10, 546), (20, 13624), (40, 9956), (550, 1125), (1071, 6583), (13369, 13374)
X(22793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1699, 9955), (3, 9955, 11230), (4, 962, 355), (4, 9812, 12699), (5, 3579, 11231), (20, 5886, 13624), (40, 381, 9956), (79, 4857, 354), (355, 12699, 962), (946, 4297, 5901), (962, 10248, 4), (4297, 5901, 1385), (6684, 12571, 5), (6684, 18483, 12571), (9812, 10248, 962)

X(22794) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO INNER-FERMAT

Barycentrics    -2*S*(3*a^4+(b^2+c^2)*a^2-4*(b^2-c^2)^2)+(a^6+2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3) : :
X(22794) = 3*X(4)+X(628) = 3*X(5)-2*X(6674) = X(18)-3*X(381) = X(628)-3*X(16627) = 5*X(3091)-X(22531) = 9*X(3839)-X(22114) = 5*X(3843)-X(16628) = 9*X(14269)+X(22845) = 5*X(18492)-X(22651)

The reciprocal orthologic center of these triangles is X(3).

X(22794) lies on these lines: {4,617}, {5,6672}, {18,381}, {30,630}, {62,10612}, {382,22843}, {546,5478}, {1351,3818}, {1478,22860}, {1479,22859}, {3091,22531}, {3583,22865}, {3585,18972}, {3839,22114}, {5318,8260}, {5873,16002}, {6033,11603}, {7747,16808}, {9818,22656}, {9955,11740}, {10895,22884}, {10896,22885}, {12699,22851}, {13665,19072}, {13785,19069}, {14269,22845}, {18491,22557}, {18492,22651}, {18495,22669}, {18497,22673}, {18500,22745}, {18502,22522}, {18507,22852}, {18509,22853}, {18511,22854}, {18516,22857}, {18517,22858}, {18520,22863}, {18522,22864}, {18525,22867}, {18538,22876}, {18542,22886}, {18544,22887}, {18761,22771}, {18762,22877}, {22597,22626}

X(22794) = midpoint of X(i) and X(j) for these {i,j}: {4, 16627}, {382, 22843}, {6033, 11603}, {12699, 22851}, {18507, 22852}, {18525, 22867}
X(22794) = {X(3818), X(3843)}-harmonic conjugate of X(22795)

X(22795) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO OUTER-FERMAT

Barycentrics    2*S*(3*a^4+(b^2+c^2)*a^2-4*(b^2-c^2)^2)+(a^6+2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2))*sqrt(3) : :
X(22795) = 3*X(4)+X(627) = 3*X(5)-2*X(6673) = X(17)-3*X(381) = X(627)-3*X(16626) = 5*X(3091)-X(22532) = 9*X(3839)-X(22113) = 5*X(3843)-X(16629) = 9*X(14269)+X(22844) = 5*X(18492)-X(22652)

The reciprocal orthologic center of these triangles is X(3).

X(22795) lies on these lines: {4,616}, {5,6671}, {17,381}, {30,629}, {61,10611}, {382,22890}, {532,3845}, {546,5479}, {1351,3818}, {1478,22905}, {1479,22904}, {3091,22532}, {3583,22910}, {3585,18973}, {3839,22113}, {5321,8259}, {5872,16001}, {6033,11602}, {7747,16809}, {9818,22657}, {9955,11739}, {10895,22929}, {10896,22930}, {12699,22896}, {13665,19070}, {13785,19071}, {14269,22844}, {18491,22558}, {18492,22652}, {18495,22670}, {18497,22674}, {18500,22746}, {18502,22523}, {18507,22897}, {18509,22898}, {18511,22899}, {18516,22902}, {18517,22903}, {18520,22908}, {18522,22909}, {18525,22912}, {18538,22921}, {18542,22931}, {18544,22932}, {18761,22772}, {18762,22922}, {22599,22628}

X(22795) = midpoint of X(i) and X(j) for these {i,j}: {4, 16626}, {382, 22890}, {6033, 11602}, {12699, 22896}, {18507, 22897}, {18525, 22912}
X(22795) = {X(3818), X(3843)}-harmonic conjugate of X(22794)

X(22796) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 3rd FERMAT-DAO

Barycentrics    sqrt(3)*(S^2*SA+3*SW*SB*SC)+2*(S^2+3*SB*SC)*S : :
X(22796) = 3*X(4)+X(616) = 3*X(5)-2*X(6669) = X(13)-3*X(381) = 4*X(546)-X(16001) = X(616)-3*X(5617) = 5*X(1656)-3*X(21156) = 5*X(3091)-X(6770) = 5*X(3091)-2*X(20415) = 5*X(3843)-X(13103) = 3*X(5469)-X(12188) = X(5474)-3*X(15561) = 4*X(6669)-3*X(6771) = X(9901)-5*X(18492) = X(14830)-3*X(22490) = 5*X(19709)-3*X(22489)

The reciprocal orthologic center of these triangles is X(13).

X(22796) lies on these lines: {4,616}, {5,6669}, {6,13}, {15,22892}, {30,618}, {114,1080}, {382,5473}, {398,14136}, {543,6298}, {546,5478}, {621,7809}, {626,3642}, {1478,12952}, {1479,12942}, {1656,21156}, {2782,5479}, {2794,6774}, {3091,6770}, {3583,13076}, {3585,18974}, {3830,5463}, {3843,13103}, {3850,20252}, {5066,5459}, {5318,6782}, {5321,6115}, {5474,15561}, {5965,20425}, {6108,22847}, {6670,12042}, {7975,18525}, {9818,9916}, {9901,18492}, {9955,11705}, {9982,18500}, {10061,12185}, {10062,10895}, {10077,12184}, {10078,10896}, {11121,14492}, {12205,18502}, {12337,18491}, {12472,18495}, {12473,18497}, {12699,12781}, {12793,18507}, {12922,18516}, {12932,18517}, {12990,18520}, {12991,18522}, {13105,18542}, {13107,18544}, {13917,18538}, {13982,18762}, {14830,22490}, {16530,16965}, {18581,22513}, {18761,22773}, {18764,22998}, {19709,22489}, {22601,22630}

X(22796) = midpoint of X(i) and X(j) for these {i,j}: {4, 5617}, {382, 5473}, {3830, 5463}, {7975, 18525}, {12699, 12781}, {12793, 18507}
X(22796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 381, 19130), (381, 3818, 22797), (6777, 16808, 5472)

X(22797) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 4th FERMAT-DAO

Barycentrics    sqrt(3)*(S^2*SA+3*SW*SB*SC)-2*(S^2+3*SB*SC)*S : :
X(22797) = 3*X(4)+X(617) = 3*X(5)-2*X(6670) = X(14)-3*X(381) = 4*X(546)-X(16002) = X(617)-3*X(5613) = 5*X(1656)-3*X(21157) = 5*X(3091)-X(6773) = 5*X(3091)-2*X(20416) = 5*X(3843)-X(13102) = 3*X(5470)-X(12188) = X(5473)-3*X(15561) = 4*X(6670)-3*X(6774) = X(9900)-5*X(18492) = X(14830)-3*X(22489) = 5*X(19709)-3*X(22490)

The reciprocal orthologic center of these triangles is X(14).

X(22797) lies on these lines: {4,617}, {5,6670}, {6,13}, {16,22848}, {30,619}, {114,383}, {382,5474}, {397,14137}, {543,6299}, {546,5479}, {622,7809}, {626,3643}, {1478,12951}, {1479,12941}, {1656,21157}, {2782,5478}, {2794,6771}, {3091,6773}, {3583,13075}, {3585,18975}, {3830,5464}, {3843,13102}, {3850,20253}, {5066,5460}, {5318,6114}, {5321,6783}, {5473,15561}, {5965,20426}, {6109,22893}, {6669,12042}, {7974,18525}, {9818,9915}, {9900,18492}, {9955,11706}, {9981,18500}, {10061,10895}, {10062,12185}, {10077,10896}, {10078,12184}, {11122,14492}, {12204,18502}, {12336,18491}, {12470,18495}, {12471,18497}, {12699,12780}, {12792,18507}, {12921,18516}, {12931,18517}, {12988,18520}, {12989,18522}, {13104,18542}, {13106,18544}, {13916,18538}, {13981,18762}, {14830,22489}, {16529,16964}, {18582,22512}, {18761,22774}, {18765,22997}, {19709,22490}, {22603,22632}

X(22797) = midpoint of X(i) and X(j) for these {i,j}: {4, 5613}, {382, 5474}, {3830, 5464}, {7974, 18525}, {12699, 12780}, {12792, 18507}
X(22797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 381, 19130), (381, 3818, 22796)

X(22798) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 2nd FUHRMANN

Barycentrics    (b+c)*a^6-2*(b^2-b*c+c^2)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+2*(2*b^4-b^2*c^2+2*c^4)*a^3-(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2-2*(b+c)*(b^2-c^2)*(b^3-c^3)*a+(b^2-c^2)^3*(b-c) : :
X(22798) = 3*X(4)+X(3648) = 3*X(5)-2*X(6701) = 3*X(21)-X(18481) = X(79)-3*X(381) = 5*X(3091)-X(16116) = 7*X(3624)-3*X(16132) = X(3648)-3*X(3652) = X(3649)-3*X(6841) = X(5441)-3*X(13743) = 3*X(6841)-2*X(9955) = 3*X(7701)+X(16118) = 3*X(7701)+5*X(18492) = 3*X(13743)+X(18525) = X(16118)-5*X(18492) = 3*X(16160)-X(22791)

The reciprocal orthologic center of these triangles is X(3).

X(22798) lies on these lines: {4,3648}, {5,3833}, {10,30}, {11,113}, {21,18481}, {46,1749}, {55,5441}, {79,381}, {355,21669}, {382,16113}, {546,16125}, {758,3813}, {1210,11544}, {1385,12617}, {1478,16141}, {1479,16140}, {2475,16138}, {3065,10742}, {3091,16116}, {3583,16142}, {3585,18977}, {3624,16132}, {3650,6734}, {3826,5499}, {3839,20084}, {3843,16150}, {3850,10265}, {4297,12104}, {5694,8727}, {5885,8226}, {6175,10308}, {6245,11230}, {6361,16139}, {7548,16128}, {9818,16119}, {10895,16152}, {10896,16153}, {11684,12699}, {12616,22792}, {12619,19925}, {12620,22801}, {12623,22805}, {13624,15670}, {13665,19080}, {13785,19079}, {16115,18502}, {16117,18491}, {16121,18495}, {16122,18497}, {16123,18500}, {16129,18507}, {16130,18509}, {16131,18511}, {16148,18538}, {16149,18762}, {16154,18542}, {16155,18544}, {16161,18520}, {16162,18522}, {17768,18482}

X(22798) = midpoint of X(i) and X(j) for these {i,j}: {4, 3652}, {355, 21669}, {382, 16113}, {2475, 16138}, {3065, 10742}, {11684, 12699}, {16129, 18507}
X(22798) = reflection of X(i) in X(j) for these (i,j): (1385, 16617), (4297, 12104)
X(22798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3649, 6841, 9955), (7701, 18492, 16118), (13743, 18525, 5441)

X(22799) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO INNER-GARCIA

Barycentrics    2*a^7-2*(b+c)*a^6-2*(b^2-4*b*c+c^2)*a^5+2*(b+c)*(b^2-3*b*c+c^2)*a^4-(2*b^4+2*c^4+(b^2-8*b*c+c^2)*b*c)*a^3+2*(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)^2*(2*b^2-7*b*c+2*c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(22799) = 5*X(4)-X(149) = 3*X(4)+X(153) = 3*X(4)-X(10738) = 5*X(5)-4*X(6667) = 3*X(5)-2*X(6713) = 5*X(119)-3*X(6174) = 3*X(149)+5*X(153) = 3*X(149)-5*X(10738) = X(149)+5*X(10742) = 2*X(149)-5*X(22938) = X(153)-3*X(10742) = 2*X(153)+3*X(22938) = 6*X(6667)-5*X(6713) = X(10738)+3*X(10742) = 2*X(10738)-3*X(22938)

The reciprocal orthologic center of these triangles is X(40).

X(22799) lies on these lines: {3,10728}, {4,145}, {5,2829}, {11,546}, {30,119}, {80,1836}, {100,382}, {104,381}, {214,5087}, {515,12611}, {528,15687}, {550,3035}, {1145,5080}, {1317,3583}, {1387,1478}, {1479,12735}, {1484,3845}, {1539,8674}, {1699,12737}, {1768,18492}, {2783,22515}, {2787,22505}, {2800,18480}, {2801,18482}, {2802,22793}, {2806,19160}, {2831,19163}, {3091,12248}, {3543,13199}, {3627,5840}, {3628,21154}, {3830,10711}, {3843,12773}, {3858,20418}, {5221,12019}, {5434,15180}, {5587,12515}, {5691,6265}, {6256,11729}, {7972,18514}, {9818,9913}, {9955,11715}, {10058,10895}, {10074,10896}, {10087,12953}, {10090,12943}, {10707,14269}, {10759,18440}, {10956,15171}, {11604,14496}, {12199,18502}, {12332,18491}, {12462,18495}, {12463,18497}, {12499,18500}, {12619,19925}, {12699,12751}, {12701,12749}, {12752,18507}, {12753,18509}, {12754,18511}, {12761,16112}, {12762,18517}, {12765,18520}, {12766,18522}, {12775,18542}, {12776,18544}, {13665,19082}, {13785,19081}, {13913,18538}, {13977,18762}, {15704,20400}, {18761,22775}

X(22799) = midpoint of X(i) and X(j) for these {i,j}: {3, 10728}, {4, 10742}, {80, 16128}, {100, 382}, {3627, 11698}, {3830, 10711}, {5691, 6265}, {10759, 18440}, {12699, 12751}, {12752, 18507}
X(22799) = reflection of X(i) in X(j) for these (i,j): (11, 546), (550, 3035), (12619, 19925)
X(22799) = complement of X(38753)
X(22799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 153, 10738), (1478, 12764, 1387), (1479, 12763, 12735), (3830, 12331, 10724), (10711, 10724, 12331), (10738, 10742, 153)

X(22800) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 3rd HATZIPOLAKIS

Barycentrics    S^4-(R^2*(40*R^2+5*SA-18*SW)-SA^2+SB*SC+2*SW^2)*S^2-(R^2*(104*R^2-35*SW)+3*SW^2)*SB*SC : :
X(22800) = 3*X(4)+X(22647) = 3*X(381)-X(22466) = 5*X(3091)-X(22533) = 5*X(3843)-X(22979) = 5*X(18492)-X(22653) = X(22647)-3*X(22955)

The reciprocal orthologic center of these triangles is X(12241).

X(22800) lies on these lines: {4,801}, {30,22966}, {113,389}, {143,15873}, {381,15317}, {382,22951}, {546,22833}, {1478,22959}, {1479,22958}, {2072,18488}, {2929,7506}, {3091,22533}, {3548,22581}, {3583,22965}, {3585,18978}, {3843,22979}, {4846,22973}, {5448,18418}, {6644,22802}, {9818,22658}, {9955,22476}, {10895,22980}, {10896,22981}, {12084,22978}, {12699,22941}, {13665,19084}, {13785,19083}, {15043,18504}, {18491,22559}, {18492,22653}, {18500,22747}, {18502,22524}, {18507,22943}, {18509,22945}, {18511,22947}, {18516,22956}, {18517,22957}, {18520,22963}, {18522,22964}, {18525,22969}, {18538,22976}, {18542,22982}, {18544,22983}, {18761,22776}, {18762,22977}, {22808,22972}

X(22800) = midpoint of X(i) and X(j) for these {i,j}: {4, 22955}, {382, 22951}, {12699, 22941}, {18507, 22943}, {18525, 22969}, {22808, 22972}
X(22800) = {X(4), X(22750)}-harmonic conjugate of X(22483)

X(22801) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO HUTSON EXTOUCH

Barycentrics    2*a^10-3*(b+c)*a^9-2*(2*b^2+9*b*c+2*c^2)*a^8+2*(b+c)*(3*b^2+8*b*c+3*c^2)*a^7+2*(b^2+7*b*c+c^2)*(b+c)^2*a^6-16*(b+c)*(b^2+c^2)*b*c*a^5-2*(b^6+c^6-(11*b^4+11*c^4-3*(b^2-6*b*c+c^2)*b*c)*b*c)*a^4-2*(b^4-c^4)*(b-c)*(3*b^2+14*b*c+3*c^2)*a^3+2*(b^2-c^2)^2*(2*b^4+2*c^4-(13*b^2+10*b*c+13*c^2)*b*c)*a^2+(b^2-c^2)^3*(b-c)*(3*b^2+22*b*c+3*c^2)*a-2*(b^2-c^2)^4*(b-c)^2 : :
X(22801) = 3*X(4)+X(9874) = 3*X(381)-X(7160) = 5*X(3091)-X(12249) = 5*X(3843)-X(12872) = X(9874)-3*X(12856) = X(9898)-5*X(18492)

The reciprocal orthologic center of these triangles is X(40).

X(22801) lies on these lines: {4,9874}, {30,12864}, {381,7160}, {382,12120}, {546,12599}, {1478,12860}, {1479,12859}, {3091,12249}, {3583,12863}, {3585,18979}, {3843,12872}, {8000,18525}, {9818,12411}, {9898,18492}, {9955,12260}, {10059,10895}, {10075,10896}, {12200,18502}, {12333,18491}, {12464,18495}, {12465,18497}, {12500,18500}, {12611,12612}, {12620,22798}, {12699,12777}, {12789,18507}, {12801,18509}, {12802,18511}, {12855,15172}, {12857,18516}, {12858,18482}, {12861,18520}, {12862,18522}, {12874,18542}, {12875,18544}, {13665,19086}, {13785,19085}, {13914,18538}, {13978,18762}, {18761,22777}

X(22801) = midpoint of X(i) and X(j) for these {i,j}: {4, 12856}, {382, 12120}, {8000, 18525}, {12699, 12777}, {12789, 18507}

X(22802) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO MIDHEIGHT

Barycentrics    a^10-(5*b^4-8*b^2*c^2+5*c^4)*a^6+5*(b^4-c^4)*(b^2-c^2)*a^4-6*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(22802) = 3*X(3)-X(5925) = 5*X(3)-6*X(10182) = 3*X(4)+X(6225) = 5*X(4)-X(12324) = 3*X(4)-X(14216) = 5*X(4)-2*X(14864) = 4*X(4)-3*X(18376) = 3*X(4)-2*X(18383) = 3*X(5878)-X(6225) = 5*X(5878)+X(12324) = 3*X(5878)+X(14216) = 5*X(5878)+2*X(14864) = 4*X(5878)+3*X(18376) = 2*X(5878)+X(18381) = 3*X(5878)+2*X(18383) = 3*X(5895)+X(5925) = 5*X(5895)+6*X(10182) = 5*X(5925)-18*X(10182) = 5*X(6225)+3*X(12324) = 5*X(6225)+6*X(14864) = 4*X(6225)+9*X(18376) = 2*X(6225)+3*X(18381)

The reciprocal orthologic center of these triangles is X(4).

X(22802) lies on these lines: {2,18504}, {3,113}, {4,51}, {5,3357}, {20,10282}, {30,156}, {64,381}, {74,16868}, {140,5894}, {146,2888}, {154,1657}, {184,18560}, {195,382}, {221,9668}, {235,1514}, {403,1204}, {541,5449}, {542,12293}, {546,6247}, {548,10192}, {550,11202}, {576,1353}, {578,1885}, {1181,13403}, {1478,12950}, {1479,12940}, {1539,13491}, {1562,8743}, {1568,11413}, {1593,18388}, {1596,13568}, {1614,9934}, {1656,10606}, {1853,3843}, {2192,9655}, {2778,5694}, {2781,5876}, {2818,10525}, {2904,11456}, {3091,7703}, {3146,5656}, {3153,12279}, {3526,8567}, {3534,17821}, {3583,7355}, {3585,6285}, {3830,12315}, {3850,15105}, {4846,9729}, {5073,17845}, {5076,18405}, {5270,11189}, {5448,12084}, {5663,9927}, {6001,22793}, {6145,18550}, {6243,18325}, {6266,18511}, {6267,18509}, {6293,18439}, {6624,15005}, {6644,22800}, {6689,7526}, {6816,16836}, {7401,18489}, {7505,21663}, {7689,15761}, {7973,18525}, {8991,18538}, {9786,22971}, {9818,9914}, {9899,18492}, {9955,12262}, {10060,10895}, {10076,10896}, {10111,12295}, {10274,11805}, {10483,10535}, {10540,18565}, {10575,18404}, {10675,19107}, {10676,19106}, {10990,16219}, {11441,15063}, {11468,12244}, {11695,18537}, {12161,12897}, {12173,13419}, {12174,18396}, {12202,18502}, {12233,13488}, {12335,18491}, {12468,18495}, {12469,18497}, {12502,18500}, {12699,12779}, {12791,18507}, {12920,18516}, {12930,18517}, {12986,18520}, {12987,18522}, {13094,18542}, {13095,18544}, {13665,19088}, {13785,19087}, {13980,18762}, {13997,18809}, {14530,17800}, {14641,14791}, {14915,18569}, {15072,16223}, {15125,18281}, {15811,18494}, {18761,22778}

X(22802) = midpoint of X(i) and X(j) for these {i,j}: {3, 5895}, {4, 5878}, {3146, 9833}, {5073, 17845}, {6293, 18439}, {7973, 18525}, {12699, 12779}, {12791, 18507}
X(22802) = reflection of X(i) in X(j) for these (i,j): (5, 5893), (20, 10282), (64, 20299), (550, 16252), (7689, 15761), (13997, 18809)
X(22802) = complement of X(20427)
X(22802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 185, 18390), (4, 6225, 14216), (4, 11457, 13851), (4, 12290, 11550), (4, 14216, 18383), (4, 18381, 18376), (64, 381, 20299), (140, 5894, 11204), (550, 16252, 11202), (3146, 5656, 9833), (3526, 8567, 10193), (3843, 13093, 1853), (5878, 14216, 6225), (12244, 14940, 11468), (14216, 18383, 18381)

X(22803) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 2nd NEUBERG

Barycentrics    2*a^8+3*(b^2+c^2)*a^6+(b^4+8*b^2*c^2+c^4)*a^4-(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^2-(b^2-c^2)^2*(2*b^2+c^2)*(b^2+2*c^2) : :
X(22803) = 3*X(4)+X(2896) = 3*X(5)-2*X(6704) = X(83)-3*X(381) = 5*X(1656)-3*X(9751) = X(2896)-3*X(6287) = 5*X(3091)-X(12252) = 9*X(3839)-X(20088) = 5*X(3843)-X(13111) = 9*X(7617)-5*X(8150) = X(9903)-5*X(18492)

The reciprocal orthologic center of these triangles is X(3).

X(22803) lies on these lines: {2,8725}, {4,2896}, {5,5092}, {30,6292}, {83,381}, {115,546}, {382,7910}, {732,3818}, {754,3845}, {1478,12954}, {1479,12944}, {1656,9751}, {2548,13331}, {3091,12252}, {3583,13078}, {3585,18983}, {3839,20088}, {3843,13111}, {3851,7919}, {3861,13449}, {6033,11606}, {6274,18511}, {6275,18509}, {7617,8150}, {7842,15687}, {7882,18553}, {7977,18525}, {8993,18538}, {9478,12042}, {9818,9918}, {9903,18492}, {9955,12264}, {10064,10895}, {10080,10896}, {12206,18502}, {12339,18491}, {12476,18495}, {12477,18497}, {12699,12783}, {12795,18507}, {12924,18516}, {12934,18517}, {12994,18520}, {12995,18522}, {13112,18542}, {13113,18544}, {13665,19092}, {13785,19091}, {13984,18762}, {16630,16809}, {16631,16808}, {17766,18483}, {18761,22780}, {22614,22643}

X(22803) = midpoint of X(i) and X(j) for these {i,j}: {4, 6287}, {382, 12122}, {6033, 11606}, {7977, 18525}, {12699, 12783}, {12795, 18507}
X(22803) = complement of X(8725)

X(22804) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO REFLECTION

Barycentrics    (R^2-SA)*S^2+(17*R^2-7*SW)*SB*SC : :
X(22804) = 3*X(4)+X(2888) = 5*X(4)+X(3519) = 7*X(4)+X(12325) = 3*X(5)-2*X(6689) = X(54)-3*X(381) = 4*X(546)-X(1493) = 3*X(546)-X(22051) = 3*X(1493)-4*X(22051) = 5*X(2888)-3*X(3519) = X(2888)-3*X(6288) = 7*X(2888)-3*X(12325) = X(3519)-5*X(6288) = 7*X(3519)-5*X(12325) = 3*X(3574)-2*X(22051) = X(5876)+2*X(11576) = 7*X(6288)-X(12325) = 4*X(6689)-3*X(10610)

The reciprocal orthologic center of these triangles is X(4).

X(22804) lies on these lines: {3,7703}, {4,93}, {5,5944}, {30,1209}, {54,156}, {113,137}, {140,11572}, {195,3843}, {265,10095}, {382,7691}, {539,3845}, {973,6102}, {1478,12956}, {1479,12946}, {1511,1594}, {1539,6153}, {2917,7526}, {3091,12254}, {3153,14128}, {3583,13079}, {3585,18984}, {3627,21230}, {3830,12307}, {3850,8254}, {3851,9707}, {3858,12242}, {3861,13142}, {4846,6145}, {5946,18912}, {5965,21850}, {6000,11802}, {6276,18511}, {6277,18509}, {6286,18514}, {6696,12041}, {7356,18513}, {7579,11449}, {7687,11804}, {7730,12111}, {7979,18525}, {8995,18538}, {9818,9920}, {9905,18492}, {9927,11743}, {9955,12266}, {9985,18500}, {10066,10895}, {10082,10896}, {10110,10115}, {11565,13353}, {11801,13163}, {11808,13754}, {12061,18553}, {12208,18502}, {12316,14269}, {12341,18491}, {12363,18404}, {12480,18495}, {12481,18497}, {12606,18403}, {12699,12785}, {12797,18507}, {12926,18516}, {12936,18517}, {12998,18520}, {12999,18522}, {13121,18542}, {13122,18544}, {13365,13630}, {13367,13413}, {13423,22815}, {13665,19096}, {13785,19095}, {13986,18762}, {15030,18567}, {15052,15091}, {15060,18377}, {15067,18569}, {18761,22781}

X(22804) = midpoint of X(i) and X(j) for these {i,j}: {4, 6288}, {382, 7691}, {3627, 21230}, {7979, 18525}, {12699, 12785}, {12797, 18507}, {13423, 22815}
X(22804) = reflection of X(i) in X(j) for these (i,j): (3, 13565), (140, 20584)

X(22805) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 1st SCHIFFLER

Barycentrics    2*(b+c)*a^9-(5*b^2-4*b*c+5*c^2)*a^8-2*(b+c)*(b^2+c^2)*a^7+2*(7*b^4+7*c^4-6*(b^2+c^2)*b*c)*a^6-2*(b+c)*(3*b^4+3*c^4-(b^2+b*c+c^2)*b*c)*a^5-(12*b^6+12*c^6-(10*b^4+10*c^4-(b^2+8*b*c+c^2)*b*c)*b*c)*a^4+2*(b^2-c^2)*(b-c)*(5*b^4+5*c^4+(8*b^2+5*b*c+8*c^2)*b*c)*a^3+(b^2-c^2)^2*(2*b^4+13*b^2*c^2+2*c^4)*a^2-2*(b^2-c^2)^3*(b-c)*(2*b^2+3*b*c+2*c^2)*a+(b^2-c^2)^4*(b-c)^2 : :
X(22805) = 3*X(4)+X(12849) = 3*X(381)-X(10266) = 5*X(3091)-X(12255) = 5*X(3843)-X(13126) = X(12409)-5*X(18492) = X(12849)-3*X(12919)

The reciprocal orthologic center of these triangles is X(79).

X(22805) lies on these lines: {4,12146}, {30,13089}, {381,10266}, {382,12556}, {546,12600}, {1478,12957}, {1479,12947}, {3091,12255}, {3583,13080}, {3585,18985}, {3843,13126}, {6595,10742}, {9818,12414}, {9955,12267}, {10895,13128}, {10896,13129}, {12209,18502}, {12342,18491}, {12409,18492}, {12482,18495}, {12483,18497}, {12504,18500}, {12611,12615}, {12623,22798}, {12699,12786}, {12798,18507}, {12807,18509}, {12808,18511}, {12927,18516}, {12937,18517}, {13000,18520}, {13001,18522}, {13100,18525}, {13130,18542}, {13131,18544}, {13665,19098}, {13785,19097}, {13919,18538}, {13987,18762}, {18761,22782}

X(22805) = midpoint of X(i) and X(j) for these {i,j}: {4, 12919}, {382, 12556}, {6595, 10742}, {12699, 12786}, {12798, 18507}, {13100, 18525}

X(22806) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 1st TRI-SQUARES-CENTRAL

Barycentrics    3*S^2*SA+9*SW*SB*SC+(5*S^2+9*SB*SC)*S : :
X(22806) = 3*X(4)+X(13678) = 3*X(381)-X(1327) = 5*X(3091)-X(13674) = 5*X(3843)-X(13713) = X(13678)-3*X(13692) = X(13679)-5*X(18492) = X(13691)+5*X(19709) = 5*X(22625)-2*X(22819)

The reciprocal orthologic center of these triangles is X(13665).

X(22806) lies on these lines: {4,13668}, {30,641}, {381,486}, {382,13666}, {546,13687}, {597,3818}, {1478,13696}, {1479,13695}, {3091,13674}, {3583,13699}, {3585,18986}, {3830,13712}, {3843,13713}, {6565,9300}, {9818,13680}, {9955,13667}, {10895,13714}, {10896,13715}, {12699,13688}, {13665,22541}, {13672,18502}, {13675,18491}, {13679,18492}, {13682,18495}, {13683,18497}, {13685,18500}, {13689,18507}, {13690,18509}, {13691,18511}, {13693,18516}, {13694,18517}, {13697,18520}, {13698,18522}, {13702,18525}, {13716,18542}, {13717,18544}, {13785,19099}, {13920,18538}, {13988,18762}, {18761,22783}

X(22806) = midpoint of X(i) and X(j) for these {i,j}: {4, 13692}, {382, 13666}, {3830, 13712}, {12699, 13688}, {13689, 18507}, {13702, 18525}
X(22806) = {X(3818), X(5066)}-harmonic conjugate of X(22807)

X(22807) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    3*S^2*SA+9*SW*SB*SC-(5*S^2+9*SB*SC)*S : :
X(22807) = 3*X(4)+X(13798) = 3*X(381)-X(1328) = 5*X(3091)-X(13794) = 5*X(3843)-X(13836) = X(13798)-3*X(13812) = X(13799)-5*X(18492) = X(13810)+5*X(19709) = 5*X(22596)-2*X(22820)

The reciprocal orthologic center of these triangles is X(13785).

X(22807) lies on these lines: {4,13788}, {30,642}, {381,485}, {382,13786}, {546,13807}, {597,3818}, {1478,13816}, {1479,13815}, {3091,13794}, {3583,13819}, {3585,18987}, {3830,13835}, {3843,13836}, {6564,9300}, {9818,13800}, {9955,13787}, {10895,13837}, {10896,13838}, {12699,13808}, {13665,19100}, {13785,19101}, {13792,18502}, {13795,18491}, {13799,18492}, {13802,18495}, {13803,18497}, {13805,18500}, {13809,18507}, {13810,18509}, {13811,18511}, {13813,18516}, {13814,18517}, {13817,18520}, {13818,18522}, {13822,18525}, {13839,18542}, {13840,18544}, {13848,18538}, {13849,18762}, {18761,22784}

X(22807) = midpoint of X(i) and X(j) for these {i,j}: {4, 13812}, {382, 13786}, {3830, 13835}, {12699, 13808}, {13809, 18507}, {13822, 18525}
X(22807) = {X(3818), X(5066)}-harmonic conjugate of X(22806)

X(22808) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO 3rd HATZIPOLAKIS

Barycentrics    SA*((SA-SW)*(3*SA^2+R^2*(-16*SA-23*SW+40*R^2)-3*SB*SC+3*SW^2)+(-20*R^2+3*SA+SW)*S^2) : :
X(22808) = 3*X(3)-4*X(22581) = 3*X(381)-2*X(22970) = 3*X(568)-4*X(22530) = 9*X(5055)-8*X(22973) = 3*X(11459)-X(22534) = 2*X(22581)-3*X(22834)

The reciprocal orthologic center of these triangles is X(9729).

X(22808) lies on these lines: {3,2929}, {5,22750}, {30,22528}, {155,22953}, {265,11585}, {381,22970}, {382,22538}, {394,12429}, {567,22529}, {568,22530}, {2072,6288}, {3519,12358}, {3548,22647}, {3580,7691}, {5055,22973}, {6640,18466}, {6643,10627}, {7506,22483}, {8549,18440}, {9815,22833}, {9818,22497}, {10539,22662}, {11411,18436}, {11459,22534}, {12111,22535}, {12605,18442}, {13474,18403}, {13754,21652}, {14216,18404}, {17837,18451}, {18445,19460}, {18447,19472}, {18449,22830}, {18453,22840}, {18455,22954}, {18457,22960}, {18459,22961}, {18462,19488}, {18463,19489}, {18468,22974}, {18470,22975}, {18563,20127}, {18917,18936}, {19129,19142}, {19176,19198}, {22800,22972}

X(22808) = midpoint of X(12111) and X(22535)
X(22808) = reflection of X(i) in X(j) for these (i,j): (3, 22834), (382, 22538)

X(22809) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO LUCAS ANTIPODAL

Barycentrics    (4*SA-SW)*S^2-3*SW*SB*SC-2*(S^2-5*SB*SC)*S : :
X(22809) = 3*X(3)-4*X(486) = 5*X(3)-4*X(12123) = 3*X(381)-2*X(487) = 5*X(486)-3*X(12123) = 2*X(486)-3*X(12601) = 3*X(568)-4*X(12237) = 8*X(642)-9*X(5055) = 5*X(3843)-4*X(6290) = 7*X(3851)-8*X(6251) = 3*X(11459)-X(12274) = 2*X(12123)-5*X(12601) = 3*X(13836)-2*X(22615) = 9*X(14269)-8*X(22596) = X(15685)-4*X(22484)

The reciprocal orthologic center of these triangles is X(3).

X(22809) lies on these lines: {3,486}, {5,12509}, {30,12221}, {381,487}, {382,3564}, {567,12229}, {568,12237}, {642,5055}, {1657,12256}, {3843,6290}, {3851,6251}, {5899,9921}, {6221,13881}, {6767,13081}, {7373,18989}, {9818,12169}, {9906,12702}, {11459,12274}, {12111,12285}, {12147,18535}, {12320,18531}, {12597,18449}, {12662,18453}, {12910,18455}, {12960,18457}, {12966,18459}, {12980,18468}, {12981,18470}, {13754,21653}, {13836,22615}, {14269,22596}, {15685,22484}, {17839,18451}, {18403,22817}, {18445,19461}, {18447,19473}, {18462,19490}, {18917,18937}, {19129,19143}, {19176,19199}

X(22809) = midpoint of X(12111) and X(12285)
X(22809) = reflection of X(i) in X(j) for these (i,j): (3, 12601), (382, 12296), (1657, 12256)

X(22810) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO LUCAS(-1) ANTIPODAL

Barycentrics    (4*SA-SW)*S^2-3*SW*SB*SC+2*(S^2-5*SB*SC)*S : :
X(22810) = 3*X(3)-4*X(485) = 5*X(3)-4*X(12124) = 3*X(381)-2*X(488) = 5*X(485)-3*X(12124) = 2*X(485)-3*X(12602) = 3*X(568)-4*X(12238) = 8*X(641)-9*X(5055) = 5*X(3843)-4*X(6289) = 7*X(3851)-8*X(6250) = 16*X(6118)-15*X(15694) = 3*X(11459)-X(12275) = 2*X(12124)-5*X(12602) = 3*X(13713)-2*X(22644) = 9*X(14269)-8*X(22625) = X(15685)-4*X(22485)

The reciprocal orthologic center of these triangles is X(3).

X(22810) lies on these lines: {3,485}, {5,12510}, {30,12222}, {381,488}, {382,3564}, {567,12230}, {568,12238}, {641,5055}, {1657,12257}, {3843,6289}, {3851,6250}, {5899,9922}, {6118,15694}, {6398,13881}, {6767,13082}, {7373,18988}, {9818,12170}, {9907,12702}, {11459,12275}, {12111,12286}, {12148,18535}, {12321,18531}, {12598,18449}, {12663,18453}, {12911,18455}, {12961,18457}, {12967,18459}, {12982,18468}, {12983,18470}, {13713,22644}, {13754,21654}, {14269,22625}, {15685,22485}, {17842,18451}, {18403,22818}, {18445,19462}, {18447,19474}, {18463,19491}, {18917,18938}, {19129,19144}, {19176,19200}

X(22810) = midpoint of X(12111) and X(12286)
X(22810) = reflection of X(i) in X(j) for these (i,j): (3, 12602), (382, 12297), (1657, 12257)

X(22811) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO LUCAS CENTRAL

Barycentrics    a^2*(-a^2+b^2+c^2)*(-(b^2+c^2)*a^6+4*S*c^2*a^2*b^2+(b^4+b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-(b^4+c^4)*(b^2-c^2)^2) : :
X(22811) = 3*X(3)-4*X(12360) = 3*X(381)-2*X(6291) = 9*X(5055)-8*X(9823) = 3*X(11459)-X(12276)

The reciprocal orthologic center of these triangles is X(3).

X(22811) lies on these lines: {3,6}, {5,6239}, {30,12223}, {381,6291}, {382,12298}, {488,14984}, {5055,9823}, {6252,18453}, {7362,18447}, {9818,12171}, {11459,12276}, {12111,12287}, {12256,15074}, {12322,18531}, {13754,21655}, {17840,18451}, {18403,22819}, {18445,19463}, {18462,19492}, {18463,19494}, {18917,18941}, {19176,19201}

X(22811) = midpoint of X(12111) and X(12287)
X(22811) = reflection of X(i) in X(j) for these (i,j): (3, 12603), (382, 12298)
X(22811) = {X(3), X(18438)}-harmonic conjugate of X(22812)

X(22812) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO LUCAS(-1) CENTRAL

Barycentrics    a^2*(-a^2+b^2+c^2)*(-(b^2+c^2)*a^6-4*S*c^2*a^2*b^2+(b^4+b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-(b^4+c^4)*(b^2-c^2)^2) : :
X(22812) = 3*X(3)-4*X(12361) = 3*X(381)-2*X(6406) = 9*X(5055)-8*X(9824) = 3*X(11459)-X(12277)

The reciprocal orthologic center of these triangles is X(3).

X(22812) lies on these lines: {3,6}, {5,6400}, {30,12224}, {381,6406}, {382,12299}, {487,14984}, {3060,8964}, {5055,9824}, {6404,18453}, {6405,18455}, {7353,18447}, {9818,12172}, {11459,12277}, {12111,12288}, {12257,15074}, {12323,18531}, {13754,21656}, {17843,18451}, {18403,22820}, {18445,19464}, {18462,19495}, {18463,19493}, {18917,18942}, {19176,19202}

X(22812) = midpoint of X(12111) and X(12288)
X(22812) = reflection of X(i) in X(j) for these (i,j): (3, 12604), (382, 12299)
X(22812) = {X(3), X(18438)}-harmonic conjugate of X(22811)

X(22813) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO LUCAS REFLECTION

Barycentrics    SA*((8*R^2-SA+SW)*S^2+2*(2*S^2-(SA-SW)*(11*R^2-2*SA-2*SW))*S-SW*(SA-SW)*(6*R^2-2*SA-SW)) : :
X(22813) = 3*X(3)-4*X(13027) = 3*X(381)-2*X(13051) = 3*X(568)-4*X(13013) = 9*X(5055)-8*X(13053) = 3*X(11459)-X(13015)

The reciprocal orthologic center of these triangles is X(10670).

X(22813) lies on these lines: {3,485}, {5,13035}, {30,13009}, {381,13051}, {382,13019}, {567,13011}, {568,13013}, {5055,13053}, {9818,13007}, {11459,13015}, {12111,13017}, {13025,18531}, {13037,18449}, {13041,18453}, {13043,18455}, {13045,18457}, {13047,18459}, {13057,18468}, {13059,18470}, {13754,21657}, {17841,18451}, {18403,22821}, {18445,19465}, {18447,19475}, {18463,19497}, {18917,18943}, {19129,19147}, {19176,19203}

X(22813) = midpoint of X(12111) and X(13017)
X(22813) = reflection of X(i) in X(j) for these (i,j): (3, 13039), (382, 13019)

X(22814) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO LUCAS(-1) REFLECTION

Barycentrics    SA*((8*R^2-SA+SW)*S^2-2*(2*S^2-(SA-SW)*(11*R^2-2*SA-2*SW))*S-SW*(SA-SW)*(6*R^2-2*SA-SW)) : :
X(22814) = 3*X(3)-4*X(13028) = 3*X(381)-2*X(13052) = 3*X(568)-4*X(13014) = 9*X(5055)-8*X(13054) = 3*X(11459)-X(13016)

The reciprocal orthologic center of these triangles is X(10674).

X(22814) lies on these lines: {3,486}, {5,13036}, {30,13010}, {381,13052}, {382,13020}, {567,13012}, {568,13014}, {5055,13054}, {9818,13008}, {11459,13016}, {12111,13018}, {13026,18531}, {13038,18449}, {13042,18453}, {13044,18455}, {13046,18457}, {13048,18459}, {13058,18468}, {13060,18470}, {13754,21658}, {17844,18451}, {18403,22822}, {18445,19466}, {18447,19476}, {18462,19496}, {18917,18944}, {19129,19148}, {19176,19204}

X(22814) = midpoint of X(12111) and X(13018)
X(22814) = reflection of X(i) in X(j) for these (i,j): (3, 13040), (382, 13020)

X(22815) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO REFLECTION

Barycentrics    (S^2-SB*SC)*(S^2+R^2*(19*R^2-4*SA-12*SW)+SA^2-2*SB*SC+2*SW^2) : :
X(22815) = 3*X(3)-4*X(12363) = 3*X(54)-2*X(6102) = 3*X(381)-2*X(6152) = 3*X(568)-4*X(12242) = 5*X(3843)-4*X(11576) = 9*X(5055)-8*X(9827) = 4*X(5907)-3*X(6288) = 3*X(7691)-4*X(10627) = 5*X(10574)-6*X(10610) = 3*X(11459)-X(12280) = 3*X(12022)-2*X(12899) = 2*X(12363)-3*X(12606) = 2*X(14449)-3*X(20424)

The reciprocal orthologic center of these triangles is X(6243).

X(22815) lies on these lines: {3,54}, {5,6242}, {30,12226}, {265,3519}, {381,6152}, {382,12300}, {539,18436}, {550,7722}, {567,12234}, {568,12242}, {1147,15091}, {1209,10255}, {2072,21230}, {2888,18404}, {2914,7488}, {3574,10254}, {3843,11576}, {5055,9827}, {5907,6288}, {5965,18438}, {6243,18388}, {6255,18453}, {6286,18455}, {7356,18447}, {7542,22051}, {9818,12175}, {9977,18449}, {10024,14449}, {10575,10628}, {10677,18468}, {10678,18470}, {11459,12280}, {11585,11804}, {12022,12899}, {12111,12291}, {12325,18531}, {12965,18457}, {12971,18459}, {13423,22804}, {13754,18442}, {14978,19177}, {17846,18451}, {18400,18439}, {18445,19468}, {18462,19502}, {18463,19503}, {18563,22584}, {18917,18946}, {19129,19150}, {19176,19207}

X(22815) = midpoint of X(12111) and X(12291)
X(22815) = reflection of X(i) in X(j) for these (i,j): (3, 12606), (382, 12300), (13423, 22804)

X(22816) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO 3rd HATZIPOLAKIS

Barycentrics    (8*R^2+SA-2*SW)*R^2*S^2+(200*R^4-79*R^2*SW+8*SW^2)*SB*SC : :
X(22816) = 3*X(4)+X(22555) = 3*X(381)-X(2929) = 5*X(3843)-X(22550) = 5*X(3843)-3*X(22971) = X(17837)+3*X(18405) = X(22550)-3*X(22971)

The reciprocal orthologic center of these triangles is X(9729).

X(22816) lies on these lines: {4,801}, {5,13293}, {30,22978}, {125,15062}, {265,12162}, {381,2929}, {382,22549}, {542,22830}, {3153,22528}, {3583,22954}, {3585,19472}, {3843,22550}, {3845,18428}, {6564,22960}, {6565,22961}, {9927,22833}, {13474,18403}, {13851,21652}, {16808,22974}, {16809,22975}, {17837,18405}, {18386,22497}, {18388,22529}, {18390,22530}, {18392,22534}, {18394,22535}, {18396,19460}, {18404,22834}, {18406,22840}, {18414,19488}, {18415,19489}, {18418,22966}, {18420,22973}, {18531,22581}, {18918,18936}, {19130,19142}, {19177,19198}

X(22816) = midpoint of X(382) and X(22549)
X(22816) = {X(3843), X(22550)}-harmonic conjugate of X(22971)

X(22817) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO LUCAS ANTIPODAL

Barycentrics    S^2*R^2*SA+(3*R^2-SW)*SW*SB*SC-(S^2*R^2+(5*R^2-2*SW)*SB*SC)*S : :
X(22817) = 3*X(4)+X(12320) = 3*X(381)-X(12978) = 5*X(3843)-X(12311) = X(17839)+3*X(18405)

The reciprocal orthologic center of these triangles is X(3).

X(22817) lies on these lines: {3,18415}, {4,487}, {5,12972}, {30,9921}, {381,12978}, {382,12303}, {486,10898}, {542,12597}, {642,18420}, {3153,12221}, {3564,18569}, {3583,12910}, {3585,19473}, {3843,12311}, {6564,12960}, {6565,12966}, {12169,18386}, {12229,18388}, {12237,18390}, {12274,18392}, {12285,18394}, {12601,18404}, {12662,18406}, {12980,16808}, {12981,16809}, {13851,21653}, {17839,18405}, {18396,19461}, {18403,22809}, {18414,19490}, {18918,18937}, {19130,19143}, {19177,19199}

X(22817) = midpoint of X(382) and X(12303)

X(22818) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO LUCAS(-1) ANTIPODAL

Barycentrics    S^2*R^2*SA+(3*R^2-SW)*SW*SB*SC+(S^2*R^2+(5*R^2-2*SW)*SB*SC)*S : :
X(22818) = 3*X(4)+X(12321) = 3*X(381)-X(12979) = 5*X(3843)-X(12312) = X(17842)+3*X(18405)

The reciprocal orthologic center of these triangles is X(3).

X(22818) lies on these lines: {3,18414}, {4,488}, {5,12973}, {30,9922}, {381,12979}, {382,12304}, {485,10897}, {542,12598}, {641,18420}, {3153,12222}, {3564,18569}, {3583,12911}, {3585,19474}, {3843,12312}, {6564,12961}, {6565,12967}, {12170,18386}, {12230,18388}, {12238,18390}, {12275,18392}, {12286,18394}, {12602,18404}, {12663,18406}, {12982,16808}, {12983,16809}, {13851,21654}, {17842,18405}, {18396,19462}, {18403,22810}, {18415,19491}, {18918,18938}, {19130,19144}, {19177,19200}

X(22818) = midpoint of X(382) and X(12304)

X(22819) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO LUCAS CENTRAL

Barycentrics    a^6+2*b^2*c^2*a^2+4*(a^2-b^2+c^2)*(a^2+b^2-c^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(22819) = 3*X(4)+X(12322) = 3*X(381)-X(1151) = 5*X(3843)-X(12313) = X(17840)+3*X(18405) = 5*X(22625)-3*X(22806)

The reciprocal orthologic center of these triangles is X(3).

X(22819) lies on these lines: {4,69}, {5,12974}, {30,641}, {182,14233}, {381,1151}, {382,12305}, {542,9974}, {543,6311}, {3070,18539}, {3071,5476}, {3153,12223}, {3583,6283}, {3585,7362}, {3843,12313}, {3861,22596}, {5076,18511}, {5965,12602}, {6252,18406}, {6564,12962}, {6565,7747}, {9823,18420}, {10667,16808}, {10668,16809}, {12171,18386}, {12231,18388}, {12239,18390}, {12276,18392}, {12287,18394}, {12360,18531}, {12603,18404}, {13851,21655}, {17840,18405}, {18396,19463}, {18403,22811}, {18414,19492}, {18415,19494}, {18918,18941}, {19130,19145}, {19177,19201}

X(22819) = midpoint of X(382) and X(12305)
X(22819) = {X(4), X(3818)}-harmonic conjugate of X(22820)

X(22820) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO LUCAS(-1) CENTRAL

Barycentrics    a^6+2*b^2*c^2*a^2-4*(a^2-b^2+c^2)*(a^2+b^2-c^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(22820) = 3*X(4)+X(12323) = 3*X(381)-X(1152) = 5*X(3843)-X(12314) = X(17843)+3*X(18405) = 5*X(22596)-3*X(22807)

The reciprocal orthologic center of these triangles is X(3).

X(22820) lies on these lines: {4,69}, {5,12975}, {30,642}, {182,14230}, {381,1152}, {382,12306}, {542,9975}, {543,6315}, {3070,5476}, {3153,12224}, {3583,6405}, {3585,7353}, {3843,12314}, {3861,22625}, {5076,18509}, {5965,12601}, {6404,18406}, {6564,7747}, {6565,12969}, {9824,18420}, {10671,16808}, {10672,16809}, {12172,18386}, {12232,18388}, {12240,18390}, {12277,18392}, {12288,18394}, {12361,18531}, {12604,18404}, {13851,21656}, {17843,18405}, {18396,19464}, {18403,22812}, {18414,19495}, {18415,19493}, {18918,18942}, {19130,19146}, {19177,19202}

X(22820) = midpoint of X(382) and X(12306)
X(22820) = {X(4), X(3818)}-harmonic conjugate of X(22819)

X(22821) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO LUCAS REFLECTION

Barycentrics    (R^2*SA-2*SB*SC)*S^2+(3*R^2-2*SW)*SW*SB*SC+(S^2*R^2+(R^2-4*SW)*SB*SC)*S : :
X(22821) = 3*X(4)+X(13025) = 3*X(381)-X(13055)

The reciprocal orthologic center of these triangles is X(10670).

X(22821) lies on these lines: {4,488}, {5,13049}, {30,13061}, {381,13055}, {382,13021}, {542,13037}, {3153,13009}, {3583,13043}, {3585,19475}, {3843,13023}, {6564,13045}, {6565,13047}, {13007,18386}, {13011,18388}, {13013,18390}, {13015,18392}, {13017,18394}, {13027,18531}, {13039,18404}, {13041,18406}, {13053,18420}, {13057,16808}, {13059,16809}, {13851,21657}, {17841,18405}, {18396,19465}, {18403,22813}, {18415,19497}, {18918,18943}, {19130,19147}, {19177,19203}

X(22821) = midpoint of X(382) and X(13021)

X(22822) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO LUCAS(-1) REFLECTION

Barycentrics    (R^2*SA-2*SB*SC)*S^2+(3*R^2-2*SW)*SW*SB*SC-(S^2*R^2+(R^2-4*SW)*SB*SC)*S : :
X(22822) = 3*X(4)+X(13026) = 3*X(381)-X(13056)

The reciprocal orthologic center of these triangles is X(10674).

X(22822) lies on these lines: {4,487}, {5,13050}, {30,13062}, {381,13056}, {382,13022}, {542,13038}, {3153,13010}, {3583,13044}, {3585,19476}, {3843,13024}, {6564,13046}, {6565,13048}, {13008,18386}, {13012,18388}, {13014,18390}, {13016,18392}, {13018,18394}, {13028,18531}, {13040,18404}, {13042,18406}, {13054,18420}, {13058,16808}, {13060,16809}, {13851,21658}, {17844,18405}, {18396,19466}, {18403,22814}, {18414,19496}, {18918,18944}, {19130,19148}, {19177,19204}

X(22822) = midpoint of X(382) and X(13022)

X(22823) = CYCLOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO JOHNSON

Barycentrics    (R^2*(6*R^2+SA-2*SW)-2*SB*SC)*S^2+(R^2*(54*R^2-35*SW)+6*SW^2)*SB*SC : :
X(22823) = 3*X(381)-X(13558) = 4*X(11801)-3*X(14854)

The reciprocal cyclologic center of these triangles is X(265).

X(22823) lies on these lines: {4,110}, {5,5961}, {30,13496}, {131,18404}, {381,13558}, {925,3153}, {11801,14854}, {18403,20957}

X(22824) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO TO 1st EHRMANN

Barycentrics    (SB+SC)*(sqrt(3)*(3*R^2*(18*R^2-5*SW)+2*SW^2)*S^2+(27*R^2*S^2-SW*(3*R^2*(-SW+6*SA)-4*SA^2+4*SB*SC))*S-sqrt(3)*SW*(9*R^2-2*SW)*SA^2) : :

The reciprocal orthologic center of these triangles is X(14174).

X(22824) lies on these lines: {6,2981}, {511,16247}, {2854,16259}, {14173,16642}, {16638,22826}

X(22825) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO TO 1st EHRMANN

Barycentrics    (SB+SC)*(sqrt(3)*(3*R^2*(18*R^2-5*SW)+2*SW^2)*S^2-(27*R^2*S^2-SW*(3*R^2*(-SW+6*SA)-4*SA^2+4*SB*SC))*S-sqrt(3)*SW*(9*R^2-2*SW)*SA^2) : :

The reciprocal orthologic center of these triangles is X(14180).

X(22825) lies on these lines: {6,6151}, {511,16248}, {2854,16260}, {14179,16643}, {16639,22827}

X(22826) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th FERMAT-DAO TO 1st EHRMANN

Barycentrics    (S+sqrt(3)*SB)*(S+sqrt(3)*SC)*((3*SA-7*SW)*S+sqrt(3)*(SA-SW)*(3*SA+2*SW)) : :

The reciprocal orthologic center of these triangles is X(14174).

X(22826) lies on these lines: {6,8014}, {13,524}, {69,11119}, {2854,16461}, {10217,16459}, {14173,16463}, {16638,22824}

X(22827) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th FERMAT-DAO TO 1st EHRMANN

Barycentrics    (-S+sqrt(3)*SB)*(-S+sqrt(3)*SC)*(-(3*SA-7*SW)*S+sqrt(3)*(SA-SW)*(3*SA+2*SW)) : :

The reciprocal orthologic center of these triangles is X(14180).

X(22827) lies on these lines: {6,8015}, {14,524}, {69,11120}, {2854,16462}, {10218,16460}, {14179,16464}, {16639,22825}

X(22828) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(3*(2*R^2*(7*R^2-4*SW)+SW*(SW+SA))*S^2-(R^2*(64*R^2+15*SA-33*SW)-3*SA^2+3*SB*SC+4*SW^2)*SA*SW) : :

The reciprocal orthologic center of these triangles is X(22829).

X(22828) lies on these lines: {8542,22966}, {9970,22955}, {12584,22962}

X(22829) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd HATZIPOLAKIS TO 1st EHRMANN

Barycentrics    a^2*((b^2+c^2)*a^4-10*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)) : :
X(22829) = 7*X(6)-3*X(51) = 5*X(6)-X(1843) = 3*X(6)+X(6467) = 3*X(6)-X(9969) = 11*X(6)-3*X(9971) = 9*X(6)-X(9973) = 15*X(51)-7*X(1843) = 9*X(51)+7*X(6467) = 9*X(51)-7*X(9969) = 11*X(51)-7*X(9971) = 27*X(51)-7*X(9973) = 3*X(1843)+5*X(6467) = 3*X(1843)-5*X(9969) = 11*X(1843)-15*X(9971) = 9*X(1843)-5*X(9973) = 11*X(6467)+9*X(9971) = 3*X(6467)+X(9973) = 11*X(9969)-9*X(9971) = 3*X(9969)-X(9973)

The reciprocal orthologic center of these triangles is X(22828).

X(22829) lies on these lines: {6,25}, {54,19142}, {141,9027}, {511,548}, {524,7734}, {597,14913}, {1992,3313}, {2854,6329}, {3564,14128}, {3589,8681}, {3618,15531}, {3629,11574}, {3630,3819}, {3917,6144}, {5097,11255}, {5421,20975}, {5446,15520}, {5462,15516}, {5486,17040}, {6391,8542}, {6776,12290}, {8550,15105}, {8584,17710}, {11649,21852}

X(22829) = midpoint of X(3629) and X(11574)
X(22829) = reflection of X(5462) in X(15516)
X(22829) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 6467, 9969), (6, 19459, 19136)

X(22830) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*((4*R^2*(42*R^2-17*SW)+7*SW^2)*S^2-(4*R^2-SW)*(8*R^2+SA-2*SW)*SA*SW) : :
X(22830) = 3*X(6)-X(2929) = 3*X(1992)+X(22555) = 2*X(2929)-3*X(19142) = 5*X(11482)-X(22550) = 3*X(17813)+X(17837)

The reciprocal orthologic center of these triangles is X(9729).

X(22830) lies on these lines: {6,2929}, {511,22978}, {542,22816}, {575,22962}, {895,15044}, {1992,22555}, {3090,8542}, {3520,5622}, {8537,9781}, {8538,22834}, {8539,22840}, {8540,22954}, {8541,22970}, {8548,9818}, {9813,22973}, {10602,19460}, {11405,22497}, {11416,22528}, {11443,22534}, {11458,22535}, {11470,22538}, {11477,22549}, {11482,22550}, {11511,22581}, {17813,17837}, {18449,22808}, {18919,18936}, {19178,19198}, {19369,19472}, {19426,19488}, {19427,19489}, {21639,21652}

X(22830) = midpoint of X(11477) and X(22549)

X(22831) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO INNER-FERMAT

Barycentrics    -2*S*(4*a^4+3*(b^2+c^2)*a^2-7*(b^2-c^2)^2)+sqrt(3)*(3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)) : :
X(22831) = 3*X(4)+X(22531) = 3*X(18)-X(22531) = X(628)-5*X(3091) = 3*X(1699)+X(22651) = 7*X(3832)+X(22114) = 11*X(3855)-X(22845) = 3*X(5587)-X(22851) = 3*X(5603)-X(22867) = X(11603)-3*X(14639) = 3*X(11897)-X(22852)

The reciprocal orthologic center of these triangles is X(3).

X(22831) lies on these lines: {2,22843}, {3,6674}, {4,16}, {5,619}, {11,18972}, {12,22865}, {17,23013}, {98,22522}, {115,398}, {235,22481}, {371,22876}, {372,22877}, {515,11740}, {546,5478}, {628,3091}, {1478,22885}, {1479,22884}, {1587,19069}, {1588,19072}, {1598,22656}, {1699,22651}, {3832,22114}, {3850,7684}, {3855,22845}, {3858,5480}, {5340,16943}, {5349,12815}, {5587,22851}, {5603,22867}, {6201,22854}, {6202,22853}, {6695,20378}, {8196,22669}, {8203,22673}, {8212,22863}, {8213,22864}, {8260,10612}, {9993,22745}, {10531,22886}, {10532,22887}, {10893,22857}, {10894,22858}, {10895,22859}, {10896,22860}, {11496,22557}, {11603,14639}, {11897,22852}, {13687,18585}, {13807,15765}, {16808,22856}, {22753,22771}

X(22831) = midpoint of X(4) and X(18)
X(22831) = reflection of X(3) in X(6674)
X(22831) = complement of X(22843)
X(22831) = {X(3858), X(5480)}-harmonic conjugate of X(22832)

X(22832) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO OUTER-FERMAT

Barycentrics    2*S*(4*a^4+3*(b^2+c^2)*a^2-7*(b^2-c^2)^2)+sqrt(3)*(3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)) : :
X(22832) = 3*X(4)+X(22532) = 3*X(17)-X(22532) = 3*X(381)-X(16626) = 3*X(381)+X(16629) = X(627)-5*X(3091) = 3*X(1699)+X(22652) = 7*X(3832)+X(22113) = 11*X(3855)-X(22844) = 3*X(5587)-X(22896) = 3*X(5603)-X(22912) = X(11602)-3*X(14639) = 3*X(11897)-X(22897)

The reciprocal orthologic center of these triangles is X(3).

X(22832) lies on these lines: {2,22890}, {3,6673}, {4,15}, {5,618}, {11,18973}, {12,22910}, {18,23006}, {98,22523}, {115,397}, {235,22482}, {371,22921}, {372,22922}, {381,532}, {515,11739}, {546,5479}, {627,3091}, {1478,22930}, {1479,22929}, {1587,19071}, {1588,19070}, {1598,22657}, {1699,22652}, {3832,22113}, {3850,7685}, {3855,22844}, {3858,5480}, {5339,16942}, {5350,12815}, {5587,22896}, {5603,22912}, {6201,22899}, {6202,22898}, {6694,20377}, {8196,22670}, {8203,22674}, {8212,22908}, {8213,22909}, {8259,10611}, {9993,22746}, {10531,22931}, {10532,22932}, {10893,22902}, {10894,22903}, {10895,22904}, {10896,22905}, {11496,22558}, {11602,14639}, {11897,22897}, {13687,15765}, {13807,18585}, {16809,22900}, {22753,22772}

X(22832) = midpoint of X(4) and X(17)
X(22832) = reflection of X(3) in X(6673)
X(22832) = complement of X(22890)
X(22832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 16629, 16626), (3858, 5480, 22831)

X(22833) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO 3rd HATZIPOLAKIS

Barycentrics    S^4+(R^2*(16*R^2-5*SA-3*SW)+SA^2-SB*SC)*S^2+(2*R^2*(88*R^2-35*SW)+7*SW^2)*SB*SC : :
X(22833) = 3*X(4)+X(22533) = 3*X(381)-X(22955) = 3*X(381)+X(22979) = 3*X(1699)+X(22653) = 5*X(3091)-X(22647) = 3*X(5587)-X(22941) = 3*X(5603)-X(22969) = 3*X(11897)-X(22943) = 3*X(22466)-X(22533) = X(22750)-3*X(22971)

The reciprocal orthologic center of these triangles is X(12241).

X(22833) lies on these lines: {2,22951}, {4,18936}, {5,12897}, {11,18978}, {12,22965}, {98,22524}, {125,1885}, {235,22483}, {371,22976}, {372,22977}, {378,2929}, {381,22955}, {515,22476}, {546,22800}, {974,22948}, {1478,22981}, {1479,22980}, {1587,19083}, {1588,19084}, {1598,22658}, {1699,22653}, {3091,22647}, {3574,10151}, {5587,22941}, {5603,22969}, {6201,22947}, {6202,22945}, {7699,22750}, {7706,10095}, {8212,22963}, {8213,22964}, {9815,22808}, {9927,22816}, {9993,22747}, {10531,22982}, {10532,22983}, {10893,22956}, {10894,22957}, {10895,22958}, {10896,22959}, {11250,22962}, {11496,22559}, {11897,22943}, {17928,22549}, {22753,22776}

X(22833) = midpoint of X(4) and X(22466)
X(22833) = complement of X(22951)
X(22833) = {X(381), X(22979)}-harmonic conjugate of X(22955)

X(22834) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO 3rd HATZIPOLAKIS

Barycentrics    SA*((20*R^2-3*SA-SW)*S^2+(SA-SW)*(2*R^2*(8*R^2+8*SA+SW)-3*SA^2+3*SB*SC-SW^2)) : :
X(22834) = 5*X(1656)-4*X(22973) = 5*X(11444)-X(22534) = 3*X(11459)+X(22535) = 2*X(22581)+X(22808)

The reciprocal orthologic center of these triangles is X(9729).

X(22834) lies on these lines: {2,22750}, {3,2929}, {4,22528}, {5,22970}, {30,22538}, {52,22530}, {68,3546}, {125,16196}, {155,19460}, {569,22529}, {1060,19472}, {1062,22954}, {1092,22953}, {1209,10257}, {1352,3548}, {1368,5562}, {1656,22973}, {2072,18488}, {4549,6643}, {5449,22647}, {6247,11585}, {6644,22483}, {7395,22497}, {7723,16003}, {8251,22840}, {8538,22830}, {10634,22974}, {10635,22975}, {10897,22960}, {10898,22961}, {11411,18936}, {11444,22534}, {11459,22535}, {12362,21663}, {12605,16111}, {17814,17822}, {18404,22816}, {18531,20427}, {19131,19142}, {19179,19198}, {19428,19488}, {19429,19489}

X(22834) = midpoint of X(i) and X(j) for these {i,j}: {3, 22808}, {4, 22528}
X(22834) = reflection of X(i) in X(j) for these (i,j): (3, 22581), (52, 22530)
X(22834) = complement of X(22750)

X(22835) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st ZANIAH TO 3rd EULER

Barycentrics    (b+c)*a^6+(b^2-4*b*c+c^2)*a^5-4*(b^2-c^2)*(b-c)*a^4-2*(b-c)^4*a^3+(b^2-c^2)*(b-c)*(5*b^2-2*b*c+5*c^2)*a^2+(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(22835) = X(36)+3*X(1699) = X(1538)+2*X(16174) = X(2077)-5*X(8227) = 5*X(3091)-X(5176) = X(3814)-3*X(3817) = 3*X(4881)+X(10724) = X(5048)-3*X(5603) = X(5080)-9*X(9779) = X(5087)-4*X(9955) = X(5126)+2*X(18483)

The reciprocal cyclologic center of these triangles is X(946).

X(22835) lies on these lines: {1,10893}, {2,13528}, {4,1319}, {5,10}, {11,1519}, {30,18857}, {36,1012}, {119,3880}, {474,2077}, {496,12608}, {515,1387}, {516,6681}, {912,12611}, {962,6931}, {1155,6833}, {1537,1737}, {1837,10598}, {1878,3259}, {2096,3086}, {3057,6941}, {3091,5176}, {3660,18238}, {3698,6975}, {3838,5886}, {4881,10724}, {5048,5252}, {5057,6837}, {5080,6957}, {5126,18483}, {5180,6860}, {5183,6879}, {5193,12114}, {5570,12047}, {5587,17618}, {5720,11235}, {6256,11373}, {6261,9669}, {6834,12701}, {6841,20288}, {6958,12699}, {6966,9812}, {7741,12672}, {9614,11500}, {9943,18856}, {10531,11375}, {10593,12616}, {10596,17718}, {10785,12679}, {10827,11522}, {10957,18839}, {11238,18446}, {12053,18242}

X(22835) = midpoint of X(i) and X(j) for these {i,j}: {4, 1319}, {11, 1519}, {1537, 1737}
X(22835) = reflection of X(i) in X(j) for these (i,j): (5570, 13374), (9943, 18856)
X(22835) = complement of X(13528)
X(22835) = inverse of X(7681) in the nine-point circle
X(22835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (496, 12608, 12675), (946, 3817, 7680), (946, 7681, 7686), (5603, 6968, 5252)

X(22836) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798E TO EXCENTERS-MIDPOINTS

Barycentrics    a*(a^3-(b+c)*a^2-(b^2+c^2)*a+b^3+c^3) : :
X(22836) = 3*X(1)+X(6765) = 9*X(1)-X(11519) = 5*X(1)-X(12629) = X(2136)+3*X(16200) = 3*X(3158)+X(7982) = X(3189)+3*X(5603) = 3*X(3576)+X(11523) = 3*X(4421)-X(12702) = 3*X(4930)+X(12702) = 5*X(8227)-X(12625) = 3*X(10246)-X(12513) = 3*X(10247)-X(10912) = 3*X(11235)-5*X(18493) = 3*X(11236)-X(18525)

The reciprocal orthologic center of these triangles is X(442).

X(22836) lies on these lines: {1,2}, {3,758}, {4,6326}, {20,5538}, {21,5692}, {30,18243}, {35,3869}, {36,3868}, {40,6876}, {46,4084}, {48,22021}, {55,3878}, {56,214}, {57,12559}, {63,3612}, {65,5440}, {72,993}, {79,17579}, {80,11681}, {100,5903}, {182,518}, {191,4189}, {224,4292}, {226,17647}, {326,3664}, {329,4305}, {354,17614}, {376,16132}, {377,11263}, {381,18549}, {404,5902}, {405,10176}, {474,5883}, {500,540}, {515,10526}, {516,6261}, {517,6796}, {524,5453}, {528,22791}, {535,18481}, {550,17768}, {908,10572}, {912,5450}, {920,17010}, {944,6903}, {946,12437}, {950,21616}, {952,12607}, {958,3678}, {960,5248}, {986,4256}, {991,17770}, {999,3881}, {1046,4257}, {1055,17736}, {1155,4018}, {1259,18389}, {1319,3555}, {1320,21398}, {1376,3754}, {1392,13143}, {1479,11813}, {1482,2802}, {1807,10570}, {1837,3814}, {2099,5687}, {2136,16200}, {2268,21078}, {2278,4053}, {2320,7161}, {2800,11248}, {2801,12114}, {2900,3817}, {2975,5904}, {3061,4251}, {3157,11700}, {3158,7982}, {3159,3191}, {3189,5603}, {3218,3901}, {3295,3884}, {3303,3898}, {3304,3892}, {3336,4188}, {3338,11520}, {3419,11375}, {3496,4262}, {3553,17355}, {3554,4856}, {3560,20117}, {3576,11523}, {3601,12514}, {3647,16370}, {3649,11112}, {3680,14497}, {3681,3897}, {3689,10914}, {3735,18755}, {3743,19765}, {3746,3877}, {3813,5901}, {3816,12433}, {3822,5794}, {3825,5722}, {3833,16408}, {3871,5697}, {3873,5563}, {3876,5251}, {3880,13374}, {3894,4881}, {3916,3962}, {3918,9709}, {3927,4127}, {3951,4525}, {3970,9310}, {3984,4134}, {3988,5220}, {3991,6603}, {4006,4390}, {4015,9708}, {4297,18446}, {4299,5905}, {4302,11415}, {4347,10571}, {4421,4930}, {4658,18465}, {4851,17073}, {4852,18261}, {4973,5204}, {5057,11015}, {5086,7951}, {5119,11682}, {5180,20066}, {5221,16371}, {5239,7006}, {5240,7005}, {5253,18398}, {5426,16865}, {5438,11529}, {5441,11114}, {5443,11680}, {5497,19582}, {5506,16859}, {5535,6942}, {5541,11280}, {5587,6873}, {5693,6906}, {5694,6914}, {5696,8543}, {5720,6866}, {5736,18698}, {5853,13464}, {5854,19907}, {6224,20060}, {6282,12512}, {6598,6829}, {6600,22770}, {6692,17706}, {6701,17528}, {6909,15071}, {6924,22935}, {6940,15016}, {6958,10265}, {6972,9803}, {7269,17151}, {7354,10609}, {7373,20116}, {7483,21677}, {7987,18444}, {8227,12625}, {8728,11281}, {9619,16973}, {10246,12513}, {10247,10912}, {10269,12005}, {10393,12572}, {10543,11113}, {10950,17757}, {10965,15558}, {11009,14923}, {11014,12245}, {11235,18493}, {11236,12738}, {11260,15178}, {11571,17100}, {11684,17549}, {12436,12563}, {13746,17188}, {15654,20760}, {15792,17512}, {18254,22760}

X(22836) = midpoint of X(i) and X(j) for these {i,j}: {1, 3811}, {3, 12635}, {946, 12437}, {1482, 3913}, {4421, 4930}
X(22836) = reflection of X(i) in X(j) for these (i,j): (3813, 5901), (11260, 15178)
X(22836) = inverse of X(5529) in the hexyl circle
X(22836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2, 30143), (1, 78, 10), (1, 997, 1125), (1, 3216, 3924), (1, 3632, 4861), (1, 3870, 3244), (1, 5312, 17016), (1, 5313, 5262), (1, 19861, 551), (8, 498, 10), (35, 4867, 3869), (55, 5730, 3878), (145, 5552, 10573), (3935, 4861, 3632), (5552, 10573, 10), (6737, 13411, 10)

X(22837) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798I TO EXCENTERS-MIDPOINTS

Barycentrics    a*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-3*b*c+c^2)*(b+c)) : :
X(22837) = 3*X(1)-X(3811) = 5*X(1)-X(6765) = 7*X(1)+X(11519) = 3*X(1)+X(12629) = 3*X(3576)+X(3680) = 3*X(3829)-2*X(18357) = X(3913)-3*X(10246) = X(6762)+3*X(16200) = 3*X(10165)-X(12640) = 3*X(10247)-X(12635) = 3*X(11194)-X(12702) = 3*X(11235)-X(18525) = 3*X(11236)-5*X(18493)

The reciprocal orthologic center of these triangles is X(1145).

X(22837) lies on these lines: {1,2}, {3,2802}, {30,13463}, {35,3885}, {36,14923}, {72,5048}, {100,21842}, {101,4051}, {141,18261}, {214,1388}, {405,3898}, {515,10525}, {517,5450}, {518,576}, {529,22791}, {535,12699}, {758,1482}, {944,6264}, {952,3813}, {956,2098}, {958,3884}, {962,12543}, {993,3057}, {999,3754}, {1145,5433}, {1319,10914}, {1320,2975}, {1329,1387}, {1385,3880}, {1392,3467}, {1442,17151}, {1479,21630}, {1483,5499}, {2099,3874}, {3304,5883}, {3338,3919}, {3436,11813}, {3445,6095}, {3553,4856}, {3554,17355}, {3555,11011}, {3576,3680}, {3612,3895}, {3678,5289}, {3730,4919}, {3746,3897}, {3753,20323}, {3814,11376}, {3817,10599}, {3825,11373}, {3829,18357}, {3868,11009}, {3869,5288}, {3877,5258}, {3889,5425}, {3890,5251}, {3893,5440}, {3913,10246}, {3968,16408}, {4067,11682}, {4193,16173}, {4430,16126}, {5119,5267}, {5176,7741}, {5248,9957}, {5330,5692}, {5438,11525}, {5690,5854}, {5693,10698}, {5696,14151}, {5853,13607}, {5882,21627}, {5901,12607}, {6265,11256}, {6597,14497}, {6647,14377}, {6762,16200}, {6914,10284}, {6941,12751}, {7962,12514}, {8256,15325}, {8668,10269}, {9802,20066}, {10165,12640}, {10247,12635}, {10953,12053}, {11010,12653}, {11194,12702}, {11235,18525}, {11236,18493}, {11524,15015}, {12740,15863}, {13464,21077}, {18393,20060}, {19907,20400}

X(22837) = midpoint of X(i) and X(j) for these {i,j}: {3, 10912}, {5882, 21627}, {6265, 11256}
X(22837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3632, 4511), (1, 4853, 997), (1, 12629, 3811), (1, 19860, 551), (8, 499, 10), (145, 10527, 12647), (956, 2098, 3878), (997, 4853, 3626), (1320, 2975, 5697), (1388, 5687, 214), (6264, 11014, 944), (10527, 12647, 10)

X(22838) = PERSPECTOR OF THESE TRIANGLES: 2nd EXCOSINE AND INNER-SQUARES

Barycentrics    3*S^4+(8*R^2*(16*R^2-SA-5*SW)+2*SA^2-SB*SC+2*SW^2)*S^2-4*(4*R^2-SW)^2*SB*SC-(S^2+16*(4*R^2-SW)*R^2)*(SA-SW)*S : :

X(22838) lies on these lines: {6,22839}, {64,485}, {371,6525}, {3068,3183}, {18288,18289}

X(22839) = PERSPECTOR OF THESE TRIANGLES: 2nd EXCOSINE AND OUTER-SQUARES

Barycentrics    3*S^4+(8*R^2*(16*R^2-SA-5*SW)+2*SA^2-SB*SC+2*SW^2)*S^2-4*(4*R^2-SW)^2*SB*SC+(S^2+16*(4*R^2-SW)*R^2)*(SA-SW)*S : :

X(22839) lies on these lines: {6,22838}, {64,486}, {372,6525}, {3069,3183}, {8281,17830}, {18288,18290}

X(22840) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTANGENTS TO 3rd HATZIPOLAKIS

Barycentrics    a*(a^15+(b+c)*a^14-3*(b^2+c^2)*a^13-3*(b+c)*(b^2+c^2)*a^12+(b^4+c^4-(2*b^2-11*b*c+2*c^2)*b*c)*a^11+(b+c)*(b^4+c^4+(2*b^2+11*b*c+2*c^2)*b*c)*a^10+(5*b^4+5*c^4+(17*b^2+18*b*c+17*c^2)*b*c)*(b-c)^2*a^9+(5*b^4+5*c^4-(17*b^2-18*b*c+17*c^2)*b*c)*(b+c)^3*a^8-(5*b^8+5*c^8+2*(4*b^6+4*c^6+(3*b^4-11*b^2*c^2+3*c^4)*b*c)*b*c)*a^7-(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6-(3*b^4-11*b^2*c^2+3*c^4)*b*c)*b*c)*a^6-(b^2-c^2)^2*(b^6+c^6-(2*b^4+2*c^4+(17*b^2+20*b*c+17*c^2)*b*c)*b*c)*a^5-(b^2-c^2)^2*(b+c)*(b^6+c^6+(2*b^4+2*c^4-(17*b^2-20*b*c+17*c^2)*b*c)*b*c)*a^4+(b^2-c^2)^2*(b+c)^2*(3*b^6+3*c^6-2*(2*b^2+3*b*c+2*c^2)*(b^2-b*c+c^2)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(3*b^6+3*c^6+2*(b^2+b*c+c^2)*(2*b^2-3*b*c+2*c^2)*b*c)*a^2-(b^3+c^3)*(b+c)*(b^2-c^2)^3*(b^4-c^4)*a-(b^3-c^3)*(b^2-c^2)^5*(b^2+c^2)) : :

The reciprocal orthologic center of these triangles is X(9729).

X(22840) lies on these lines: {19,22970}, {40,22653}, {55,2929}, {65,775}, {71,22466}, {2550,22555}, {3101,22528}, {3197,17837}, {3611,21652}, {5415,22960}, {5416,22961}, {5584,22549}, {6197,22750}, {7688,22978}, {8251,22834}, {8539,22830}, {9816,22973}, {10306,22550}, {10319,22581}, {10636,22974}, {10637,22975}, {10902,22962}, {11406,22497}, {11428,22529}, {11435,22530}, {11445,22534}, {11460,22535}, {11471,22538}, {18406,22816}, {18453,22808}, {18921,18936}, {19133,19142}, {19181,19198}, {19350,19460}, {19432,19488}, {19433,19489}

X(22841) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(4).

X(22841) lies on these lines: {1,11828}, {3,11377}, {4,8214}, {10,8212}, {40,493}, {46,11953}, {65,11947}, {515,12636}, {516,9838}, {517,10669}, {946,8222}, {962,6462}, {1702,19032}, {1703,19031}, {1836,11930}, {1902,11394}, {2800,13275}, {2802,12765}, {3057,18963}, {5119,11951}, {5812,10951}, {5840,12741}, {6001,12986}, {6361,11846}, {6461,22842}, {7982,8210}, {7991,8188}, {8194,9911}, {8201,12458}, {8208,12459}, {8216,12697}, {8218,12698}, {8220,12699}, {10306,11503}, {10875,12497}, {10945,12700}, {10981,12441}, {11840,12197}, {11907,12696}, {11932,12701}, {11949,12702}, {11955,12703}, {11957,12704}, {13899,13912}, {13956,13975}, {18520,22793}, {22761,22770}

X(22842) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(4).

X(22842) lies on these lines: {1,11829}, {3,11378}, {4,8215}, {10,8213}, {40,494}, {46,11954}, {65,11948}, {515,12637}, {516,9839}, {517,10673}, {946,8223}, {962,6463}, {1702,19034}, {1703,19033}, {1836,11931}, {1902,11395}, {2800,13276}, {2802,12766}, {3057,18964}, {5119,11952}, {5812,10952}, {5840,12742}, {6001,12987}, {6361,11847}, {6461,22841}, {7982,8211}, {7991,8189}, {8195,9911}, {8202,12458}, {8209,12459}, {8217,12697}, {8219,12698}, {8221,12699}, {10306,11504}, {10876,12497}, {10946,12700}, {10981,12440}, {11841,12197}, {11908,12696}, {11933,12701}, {11950,12702}, {11956,12703}, {11958,12704}, {13900,13912}, {13957,13975}, {18522,22793}, {22762,22770}

X(22843) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO INNER-FERMAT

Barycentrics    -2*(9*a^4-7*(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3)*S+3*a^2*(a^4+2*(b^2+c^2)*a^2-3*b^4-2*b^2*c^2-3*c^4) : :
X(22843) = 3*X(3)-X(16628) = 3*X(18)-2*X(16628) = 3*X(165)-X(22651) = 4*X(550)+X(22845) = 5*X(631)-4*X(6674) = 5*X(3522)-X(22114) = 3*X(3576)-2*X(11740) = 2*X(10612)-3*X(21157) = 2*X(14139)-3*X(21159) = 3*X(21156)-2*X(22846)

The reciprocal orthologic center of these triangles is X(3).

X(22843) lies on these lines: {2,22831}, {3,14}, {4,630}, {15,22862}, {20,622}, {30,16627}, {35,22884}, {36,22885}, {55,18972}, {56,22865}, {165,22651}, {182,22522}, {371,19072}, {372,19069}, {382,22794}, {515,22851}, {517,22867}, {548,14538}, {550,5473}, {631,6674}, {1350,5965}, {1593,22481}, {2043,13666}, {2044,13786}, {3098,22745}, {3411,13349}, {3428,22771}, {3522,22114}, {3534,22494}, {3576,11740}, {5352,21156}, {5983,9749}, {6284,22860}, {6772,16772}, {7354,22859}, {7748,11480}, {7782,11133}, {9540,22876}, {10310,22557}, {10646,22856}, {11248,22886}, {11249,22887}, {11414,22656}, {11822,22669}, {11823,22673}, {11824,22853}, {11825,22854}, {11826,22857}, {11827,22858}, {11828,22863}, {11829,22864}, {13935,22877}, {14139,21159}

X(22843) = midpoint of X(20) and X(628)
X(22843) = reflection of X(i) in X(j) for these (i,j): (4, 630), (382, 22794)
X(22843) = anticomplement of X(22831)
X(22843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5339, 21157), (550, 14541, 5473), (1350, 15696, 22890)

X(22844) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO OUTER-FERMAT

Barycentrics    5*a^2*(-a^2+b^2+c^2)-2*sqrt(3)*(-2*c^2-2*b^2+3*a^2)*S : :
X(22844) = 6*X(2)-5*X(17) = 3*X(2)-5*X(627) = 9*X(2)-10*X(629) = 21*X(2)-20*X(6673) = 9*X(2)-5*X(22113) = 3*X(17)-4*X(629) = 7*X(17)-8*X(6673) = 3*X(17)-2*X(22113) = 4*X(546)-5*X(16626) = 4*X(550)-5*X(22890) = 3*X(627)-2*X(629) = 7*X(627)-4*X(6673) = 3*X(627)-X(22113) = 7*X(629)-6*X(6673) = 12*X(6673)-7*X(22113)

The reciprocal orthologic center of these triangles is X(22845).

X(22844) lies on these lines: {2,17}, {3,5965}, {15,11008}, {16,3631}, {61,3629}, {69,5237}, {298,7860}, {382,5864}, {524,5238}, {546,16626}, {550,5474}, {618,3412}, {3104,22901}, {3244,22912}, {3528,22532}, {3626,22896}, {3851,16629}, {3855,22832}, {5340,21359}, {5351,5464}, {5463,22236}, {5487,12821}, {5858,16964}, {5982,6778}, {11309,16960}, {12815,16645}, {14269,22795}

X(22844) = anticomplement of X(33465)
X(22844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (627, 22113, 629), (629, 22113, 17), (22927, 22928, 627)

X(22845) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO INNER-FERMAT

Barycentrics    5*a^2*(-a^2+b^2+c^2)+2*sqrt(3)*(-2*c^2-2*b^2+3*a^2)*S : :
X(22845) = 6*X(2)-5*X(18) = 3*X(2)-5*X(628) = 9*X(2)-10*X(630) = 21*X(2)-20*X(6674) = 9*X(2)-5*X(22114) = 3*X(18)-4*X(630) = 7*X(18)-8*X(6674) = 3*X(18)-2*X(22114) = 4*X(546)-5*X(16627) = 4*X(550)-5*X(22843) = 3*X(628)-2*X(630) = 7*X(628)-4*X(6674) = 3*X(628)-X(22114) = 7*X(630)-6*X(6674) = 12*X(6674)-7*X(22114)

The reciprocal orthologic center of these triangles is X(22844).

X(22845) lies on these lines: {2,18}, {3,5965}, {15,3631}, {16,11008}, {62,3629}, {69,5238}, {299,7860}, {382,5865}, {524,5237}, {546,16627}, {550,5473}, {619,3411}, {3105,22855}, {3244,22867}, {3528,22531}, {3626,22851}, {3851,16628}, {3855,22831}, {5339,21360}, {5352,5463}, {5464,22238}, {5488,12820}, {5859,16965}, {11310,16961}, {12815,16644}, {14269,22794}

X(22845) = anticomplement of X(33464)
X(22845) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (628, 22114, 630), (630, 22114, 18), (22882, 22883, 628)

X(22846) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd FERMAT-DAO TO INNER-FERMAT

Barycentrics    3*(SA-2*SW)*S^2+3*SW*SB*SC-sqrt(3)*(16*S^2+3*(SA-SW)*(3*SA+SW))*S : :
X(22846) = X(13)+2*X(22847) = 3*X(5470)-X(11603) = 4*X(11542)-X(22855) = 5*X(16960)+X(22849) = 3*X(21156)-X(22843)

The reciprocal orthologic center of these triangles is X(616).

X(22846) lies on these lines: {2,5470}, {3,16631}, {5,13}, {14,6770}, {15,115}, {16,13103}, {17,628}, {61,16628}, {182,18362}, {299,22736}, {542,10612}, {618,6674}, {621,16529}, {630,6669}, {1080,5478}, {3054,5473}, {3107,7697}, {5237,12815}, {5352,21156}, {5965,10611}, {6772,22893}, {6778,18581}, {8859,12205}, {9982,16808}, {10062,22885}, {10078,22884}, {11303,22866}, {11542,22855}, {14061,22687}, {16941,18582}, {16960,22849}, {16965,22531}, {19069,19074}, {19072,19073}

X(22846) = reflection of X(i) in X(j) for these (i,j): (618, 6674), (630, 6669)
X(22846) = complement of X(14145)
X(22846) = inverse of X(22738) in the inner-Napoleon circle
X(22846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 22511, 62), (182, 18362, 22891)

X(22847) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO INNER-FERMAT

Barycentrics    14*S^2+sqrt(3)*(SA+SW)*S+3*(2*SA+SW)*(SA-SW) : :
X(22847) = 3*X(2)+X(11121) = X(13)+3*X(18) = X(13)-3*X(22846) = 3*X(12815)-X(22892) = 3*X(16267)-X(22855) = 3*X(16962)+X(22849)

The reciprocal orthologic center of these triangles is X(616).

X(22847) lies on these lines: {2,11121}, {5,13}, {30,10617}, {115,618}, {141,6034}, {381,22861}, {383,9756}, {398,6771}, {549,5469}, {616,16645}, {628,16644}, {630,6673}, {3642,7746}, {3643,18362}, {5461,22573}, {6036,6109}, {6108,22796}, {6118,13876}, {6775,22891}, {9166,14904}, {10654,16628}, {11290,14145}, {16267,22855}, {16962,22849}

X(22847) = midpoint of X(115) and X(22848)
X(22847) = {X(6034), X(14061)}-harmonic conjugate of X(22893)

X(22848) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO INNER-FERMAT

Barycentrics    (2*S+(SA-SW)*sqrt(3))*(-4*S+(SA+SW)*sqrt(3)) : :
X(22848) = X(14)-3*X(18) = 2*X(14)-3*X(10612) = 3*X(12815)-2*X(22893) = 3*X(16268)-X(22856) = 3*X(16963)+X(22850)

The reciprocal orthologic center of these triangles is X(14).

X(22848) lies on these lines: {2,6151}, {3,14}, {16,22797}, {99,11121}, {114,6108}, {115,618}, {140,14137}, {381,22862}, {396,630}, {617,5471}, {629,6674}, {641,13875}, {642,13928}, {1649,9200}, {2482,22574}, {3411,16529}, {3589,22892}, {6303,13701}, {6307,13821}, {6772,14145}, {6780,22849}, {7749,22866}, {9886,11147}, {10653,16627}, {15819,22692}, {16268,22856}, {16963,22850}

X(22848) = midpoint of X(i) and X(j) for these {i,j}: {99, 11121}, {6780, 22849}
X(22848) = reflection of X(115) in X(22847)

X(22849) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO INNER-FERMAT

Barycentrics    (19*SA-4*SW)*S^2-3*SW*SB*SC+sqrt(3)*(12*S^2+(SA-SW)*(13*SA+SW))*S : :
X(22849) = 2*X(15)-3*X(18) = 5*X(16960)-6*X(22846) = 3*X(16962)-4*X(22847)

The reciprocal orthologic center of these triangles is X(616).

X(22849) lies on these lines: {15,18}, {16,22114}, {628,16966}, {3411,19780}, {6780,22848}, {16628,16809}, {16960,22846}, {16962,22847}, {16964,22861}

X(22849) = reflection of X(6780) in X(22848)
X(22849) = {X(16628), X(22850)}-harmonic conjugate of X(16809)

X(22850) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO INNER-FERMAT

Barycentrics    (7*SA+2*SW)*S^2+3*SW*SB*SC-sqrt(3)*(4*S^2+(SA-SW)^2)*S : :
X(22850) = 3*X(14)-2*X(22856) = 6*X(18)-5*X(16961) = 3*X(16963)-4*X(22848)

The reciprocal orthologic center of these triangles is X(14).

X(22850) lies on these lines: {2,3170}, {6,17}, {14,299}, {15,628}, {16,5613}, {303,22866}, {2381,11601}, {3104,22871}, {3105,16627}, {3643,11132}, {6114,7779}, {6672,22998}, {7788,22665}, {10646,22531}, {11301,16241}, {11543,22510}, {16628,16809}, {16941,18582}, {16963,22848}, {16965,22862}, {18581,22114}

X(22850) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 22855, 6), (628, 22861, 15), (16809, 22849, 16628), (16967, 22901, 16961)

X(22851) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO INNER-FERMAT

Barycentrics    -2*(-a^3+b^3+b^2*c+b*c^2+c^3)*sqrt(3)*S+(a+b+c)*(a^4-5*(b+c)*a^3+2*(b^2+5*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-3*(b^2-c^2)^2) : :
X(22851) = 5*X(1698)-4*X(6674) = 5*X(3617)-X(22114) = 4*X(3626)+X(22845) = 3*X(5587)-2*X(22831) = 3*X(5657)-X(22531) = 3*X(5790)-X(16628)

The reciprocal orthologic center of these triangles is X(3).

X(22851) lies on these lines: {1,630}, {2,11740}, {8,628}, {10,18}, {65,22859}, {72,22858}, {355,12780}, {515,22843}, {517,16627}, {519,22867}, {956,22771}, {1018,6191}, {1698,6674}, {1737,22885}, {1837,22865}, {3057,22860}, {3416,5965}, {3617,22114}, {3626,22845}, {5090,22481}, {5252,18972}, {5587,22831}, {5657,22531}, {5687,22557}, {5688,22854}, {5689,22853}, {5690,12781}, {5790,16628}, {8193,22656}, {8197,22669}, {8204,22673}, {8214,22863}, {8215,22864}, {9857,22745}, {10039,22884}, {10791,22522}, {10914,22857}, {10915,22886}, {10916,22887}, {12699,22794}, {13883,19072}, {13893,22876}, {13936,19069}, {13947,22877}

X(22851) = midpoint of X(8) and X(628)
X(22851) = reflection of X(i) in X(j) for these (i,j): (1, 630), (12699, 22794)
X(22851) = anticomplement of X(11740)

X(22852) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO INNER-FERMAT

Barycentrics    (4*a^12-11*(b^2+c^2)*a^10+(3*b^4+26*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(18*b^4-43*b^2*c^2+18*c^4)*a^6-(b^2-c^2)^2*(22*b^4+29*b^2*c^2+22*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(9*b^4+5*b^2*c^2+9*c^4)*a^2+2*(-4*b^2*c^2*a^6+(b^2+c^2)*a^8+3*(b^2+c^2)*b^4*c^4-(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^4+4*(b^4-c^4)^2*a^2-(b^2+c^2)*(b^8+c^8+(b^4-b^2*c^2+c^4)*b^2*c^2))*sqrt(3)*S-(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^4)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3).

X(22852) lies on these lines: {18,402}, {30,16627}, {628,4240}, {630,1650}, {5965,12583}, {6674,15183}, {11251,12792}, {11740,11831}, {11832,22481}, {11839,22522}, {11845,22531}, {11848,22557}, {11852,22651}, {11853,22656}, {11885,22745}, {11897,22831}, {11901,22853}, {11902,22854}, {11903,22857}, {11904,22858}, {11905,22859}, {11906,22860}, {11907,22863}, {11908,22864}, {11909,22865}, {11910,22867}, {11911,16628}, {11912,22884}, {11913,22885}, {11914,22886}, {11915,22887}, {13894,22876}, {13948,22877}, {18507,22794}, {18958,18972}, {19017,19069}, {19018,19072}, {22755,22771}

X(22852) = midpoint of X(628) and X(4240)
X(22852) = reflection of X(i) in X(j) for these (i,j): (18, 402), (18507, 22794)

X(22853) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO INNER-FERMAT

Barycentrics    61*(10-sqrt(3))*(97*SA+(-37+6*sqrt(3))*SW)*S^2+11834*SW*SB*SC-5917*(3*S^2+SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22853) lies on these lines: {6,17}, {628,1271}, {630,5591}, {1161,6271}, {5589,22651}, {5595,22656}, {5605,22867}, {5689,22851}, {5875,6270}, {6202,22831}, {6215,16627}, {8198,22669}, {8205,22673}, {8216,22863}, {8217,22864}, {8974,22876}, {9994,22745}, {10040,22884}, {10048,22885}, {10783,22531}, {10792,22522}, {10919,22857}, {10921,22858}, {10923,22859}, {10925,22860}, {10927,22865}, {10929,22886}, {10931,22887}, {11370,11740}, {11388,22481}, {11497,22557}, {11824,22843}, {11901,22852}, {11916,16628}, {13949,22877}, {18509,22794}, {18959,18972}, {22756,22771}

X(22854) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO INNER-FERMAT

Barycentrics    61*(10+sqrt(3))*(97*SA+(-37-6*sqrt(3))*SW)*S^2+5917*(3*S^2+SB*SC)*S+11834*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(3).

X(22854) lies on these lines: {6,17}, {628,1270}, {630,5590}, {1160,6269}, {5588,22651}, {5594,22656}, {5604,22867}, {5688,22851}, {5874,6268}, {6201,22831}, {6214,16627}, {8199,22669}, {8206,22673}, {8218,22863}, {8219,22864}, {8975,22876}, {9995,22745}, {10041,22884}, {10049,22885}, {10784,22531}, {10793,22522}, {10920,22857}, {10922,22858}, {10924,22859}, {10926,22860}, {10928,22865}, {10930,22886}, {10932,22887}, {11371,11740}, {11389,22481}, {11498,22557}, {11825,22843}, {11902,22852}, {11917,16628}, {13950,22877}, {18511,22794}, {18960,18972}, {22757,22771}

X(22855) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO INNER-FERMAT

Barycentrics    (17*SA-8*SW)*S^2+3*SW*SB*SC-sqrt(3)*(4*S^2+SA^2-SW^2)*S : :
X(22855) = 3*X(18)-5*X(16960) = 4*X(11542)-3*X(22846) = 3*X(16267)-2*X(22847)

The reciprocal orthologic center of these triangles is X(616).

X(22855) lies on these lines: {6,17}, {16,628}, {61,22861}, {299,11133}, {3105,22845}, {5464,5859}, {5873,16964}, {6778,13103}, {7837,22665}, {10645,22531}, {11542,22846}, {16267,22847}, {16529,19780}, {16627,16809}, {16628,16808}, {18582,22114}

X(22855) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 22850, 18), (16627, 22856, 16809), (16960, 22894, 16966)

X(22856) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO INNER-FERMAT

Barycentrics    (11*SA-8*SW)*S^2-3*SW*SB*SC+sqrt(3)*(8*S^2+(SA-SW)*(7*SA-SW))*S : :
X(22856) = 3*X(14)-X(22850) = 3*X(18)-4*X(11543) = 3*X(16268)-2*X(22848)

The reciprocal orthologic center of these triangles is X(14).

X(22856) lies on these lines: {6,16628}, {14,299}, {15,18}, {16,5471}, {62,22862}, {628,18581}, {630,16967}, {3104,5334}, {5321,6777}, {6672,6780}, {10646,22843}, {16268,22848}, {16627,16809}, {16808,22831}, {16961,23013}

X(22856) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5334, 22114, 22861), (16809, 22855, 16627)

X(22857) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22857) lies on these lines: {11,18}, {12,22886}, {355,16627}, {628,3434}, {630,1376}, {5965,12586}, {10523,22884}, {10525,12921}, {10785,22531}, {10794,22522}, {10826,22651}, {10829,22656}, {10871,22745}, {10893,22831}, {10914,22851}, {10919,22853}, {10920,22854}, {10943,12922}, {10944,22859}, {10945,22863}, {10946,22864}, {10947,22865}, {10948,22885}, {10949,22887}, {11373,11740}, {11390,22481}, {11826,22843}, {11865,22669}, {11866,22673}, {11903,22852}, {11928,16628}, {12114,22771}, {13895,22876}, {13952,22877}, {18516,22794}, {18961,18972}, {19023,19069}, {19024,19072}

X(22858) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO INNER-FERMAT

Barycentrics    2*(a^6-(b^2+c^2)*a^4+(2*b*c^3+b^4+c^4+2*b^3*c+2*b^2*c^2)*a^2-(b+c)*(2*a^3*b*c+b^5-b^4*c-b*c^4+c^5))*sqrt(3)*S+(a+b+c)*(a^7-(b+c)*a^6+(b^2+10*b*c+c^2)*a^5-(b+c)*(b^2+12*b*c+c^2)*a^4-(5*b^4-22*b^2*c^2+5*c^4)*a^3+(b^2-c^2)*(b-c)*(5*b^2+16*b*c+5*c^2)*a^2+(b^2-c^2)^2*(b-3*c)*(3*b-c)*a-3*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22858) lies on these lines: {11,22887}, {12,18}, {72,22851}, {355,16627}, {628,3436}, {630,958}, {5965,12587}, {10523,22885}, {10526,12931}, {10786,22531}, {10795,22522}, {10827,22651}, {10830,22656}, {10872,22745}, {10894,22831}, {10921,22853}, {10922,22854}, {10942,12932}, {10950,22860}, {10951,22863}, {10952,22864}, {10953,22865}, {10954,22884}, {10955,22886}, {11374,11740}, {11391,22481}, {11500,22557}, {11827,22843}, {11867,22669}, {11868,22673}, {11904,22852}, {11929,16628}, {13896,22876}, {13953,22877}, {18517,22794}, {18962,18972}, {19025,19069}, {19026,19072}

X(22859) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22859) lies on these lines: {1,16627}, {4,22865}, {5,22885}, {12,18}, {56,630}, {65,22851}, {388,628}, {495,10062}, {1478,12941}, {1479,22794}, {3027,11603}, {3085,22531}, {5261,22114}, {5965,12588}, {7354,22843}, {9578,22651}, {9654,16628}, {10797,22522}, {10831,22656}, {10873,22745}, {10895,22831}, {10923,22853}, {10924,22854}, {10944,22857}, {10956,22886}, {10957,22887}, {11375,11740}, {11392,22481}, {11501,22557}, {11869,22669}, {11870,22673}, {11905,22852}, {11930,22863}, {11931,22864}, {13897,22876}, {13954,22877}, {14145,18974}, {19027,19069}, {19028,19072}, {22759,22771}

X(22859) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16627, 22860), (388, 628, 18972)

X(22860) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22860) lies on these lines: {1,16627}, {4,18972}, {5,22884}, {11,18}, {55,630}, {496,10078}, {497,628}, {1478,22794}, {1479,12951}, {3023,11603}, {3057,22851}, {3086,22531}, {5274,22114}, {5965,12589}, {6284,22843}, {9581,22651}, {9669,16628}, {10798,22522}, {10832,22656}, {10874,22745}, {10896,22831}, {10925,22853}, {10926,22854}, {10950,22858}, {10958,22886}, {10959,22887}, {11376,11740}, {11393,22481}, {11502,22557}, {11871,22669}, {11872,22673}, {11906,22852}, {11932,22863}, {11933,22864}, {13076,14145}, {13898,22876}, {13955,22877}, {19029,19069}, {19030,19072}, {22760,22771}

X(22860) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16627, 22859), (497, 628, 22865)

X(22861) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO INNER-FERMAT

Barycentrics    (SA+2*SW)*S^2+3*SW*SB*SC-sqrt(3)*(2*S^2+(SA-SW)*(3*SA-SW))*S : :
X(22861) = 3*X(18)-X(22862)

The reciprocal orthologic center of these triangles is X(616).

X(22861) lies on these lines: {4,16}, {5,19780}, {14,148}, {15,628}, {32,16627}, {61,22855}, {381,22847}, {624,22866}, {3098,22512}, {3104,5334}, {5321,16628}, {6114,10646}, {6782,16940}, {7693,21466}, {7737,9996}, {8260,11486}, {9982,16808}, {10653,19130}, {16964,22849}

X(22861) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15, 22850, 628), (5334, 22114, 22856)

X(22862) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO INNER-FERMAT

Barycentrics    (2*SA-5*SW)*S^2+3*SW*SB*SC-sqrt(3)*(4*S^2+(SA-SW)*(6*SA+SW))*S : :
X(22862) = 3*X(18)-2*X(22861)

The reciprocal orthologic center of these triangles is X(14).

X(22862) lies on these lines: {4,16}, {6,23013}, {13,99}, {15,22843}, {62,22856}, {381,22848}, {628,5335}, {630,18582}, {1250,22884}, {3054,5473}, {3105,22845}, {5318,16627}, {5321,8260}, {6115,14145}, {7747,11486}, {11133,11303}, {11308,16966}, {12017,22906}, {14137,23004}, {16808,16943}, {16965,22850}, {19373,22885}, {22513,23006}

X(22863) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO INNER-FERMAT

Barycentrics    18*sqrt(3)*S^4-3*sqrt(3)*((-6+10*sqrt(3))*SA^2+24*R^2*SA+6*SB*SC-(-6+7*sqrt(3))*SW^2)*S^2-9*SW^2*SB*SC+9*((8*R^2-10*SA+10*SW)*S^2+(-8*R^2*SA+sqrt(3)*(SW+2*SA)*SW)*(SA-SW))*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22863) lies on these lines: {18,493}, {628,6462}, {630,8222}, {5965,12590}, {6461,22864}, {8188,22651}, {8194,22656}, {8210,22867}, {8212,22831}, {8214,22851}, {8216,22853}, {8218,22854}, {8220,16627}, {10669,12988}, {10875,22745}, {10945,22857}, {10951,22858}, {11377,11740}, {11394,22481}, {11503,22557}, {11828,22843}, {11840,22522}, {11846,22531}, {11907,22852}, {11930,22859}, {11932,22860}, {11947,22865}, {11949,16628}, {11951,22884}, {11953,22885}, {11955,22886}, {11957,22887}, {13899,22876}, {13956,22877}, {18520,22794}, {18963,18972}, {19031,19069}, {19032,19072}, {22761,22771}

X(22864) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO INNER-FERMAT

Barycentrics    18*sqrt(3)*S^4-3*sqrt(3)*(24*R^2*SA+(-6-10*sqrt(3))*SA^2+6*SB*SC-(-6-7*sqrt(3))*SW^2)*S^2+9*((8*R^2-10*SA+10*SW)*S^2+(SA-SW)*(-8*R^2*SA-sqrt(3)*SW*(2*SA+SW)))*S+9*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22864) lies on these lines: {18,494}, {628,6463}, {630,8223}, {5965,12591}, {6461,22863}, {8189,22651}, {8195,22656}, {8211,22867}, {8213,22831}, {8215,22851}, {8217,22853}, {8219,22854}, {8221,16627}, {10673,12989}, {10876,22745}, {10946,22857}, {10952,22858}, {11378,11740}, {11395,22481}, {11504,22557}, {11829,22843}, {11841,22522}, {11847,22531}, {11908,22852}, {11931,22859}, {11933,22860}, {11948,22865}, {11950,16628}, {11952,22884}, {11954,22885}, {11956,22886}, {11958,22887}, {13900,22876}, {13957,22877}, {18522,22794}, {18964,18972}, {19033,19069}, {19034,19072}, {22762,22771}

X(22865) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22865) lies on these lines: {1,13075}, {3,22885}, {4,22859}, {11,630}, {12,22831}, {18,55}, {33,22481}, {56,22843}, {390,22114}, {497,628}, {1479,16627}, {1697,22651}, {1837,22851}, {2098,22867}, {2646,11740}, {3056,5965}, {3295,16628}, {3583,22794}, {4294,22531}, {5432,6674}, {10799,22522}, {10833,22656}, {10877,22745}, {10927,22853}, {10928,22854}, {10947,22857}, {10953,22858}, {10965,22886}, {10966,22771}, {11603,13183}, {11873,22669}, {11874,22673}, {11909,22852}, {11947,22863}, {11948,22864}, {12952,14145}, {13076,15171}, {13901,22876}, {13958,22877}, {19037,19069}, {19038,19072}

X(22865) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 628, 22860), (3295, 16628, 22884)

X(22866) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO INNER-FERMAT

Barycentrics    (3*SA+4*SW)*S^2-3*SW*SB*SC-sqrt(3)*(10*S^2+3*SA^2-4*SB*SC-SW^2)*S : :
X(22866) = 2*X(630)+X(22869) = 4*X(630)-X(22871) = 2*X(22869)+X(22871)

The reciprocal orthologic center of these triangles is X(22568).

X(22866) lies on these lines: {2,18}, {3,22568}, {76,16241}, {303,22850}, {624,22861}, {1078,3643}, {3642,7746}, {6294,11171}, {6298,10104}, {7749,22848}, {11303,22846}

X(22866) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 628, 22736), (630, 22869, 22871)

X(22867) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO INNER-FERMAT

Barycentrics    2*(a^2-(b+c)*a+2*b^2+2*c^2)*sqrt(3)*S*a+(a+b+c)*(9*a^4-10*(b+c)*a^3-(7*b^2-20*b*c+7*c^2)*a^2+10*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2) : :
X(22867) = 3*X(1)-2*X(11740) = 3*X(1)-X(22651) = 3*X(18)-4*X(11740) = 3*X(18)-2*X(22651) = 4*X(3244)+X(22845) = 5*X(3616)-4*X(6674) = 5*X(3623)-X(22114) = 3*X(5603)-2*X(22831) = 3*X(7967)-X(22531) = 3*X(10247)-X(16628)

The reciprocal orthologic center of these triangles is X(3).

X(22867) lies on these lines: {1,18}, {8,630}, {55,22771}, {56,22557}, {145,628}, {517,22843}, {519,22851}, {952,16627}, {1482,7974}, {1483,7975}, {2098,22865}, {2099,18972}, {3242,5965}, {3244,22845}, {3616,6674}, {3623,22114}, {5597,22673}, {5598,22669}, {5603,22831}, {5604,22854}, {5605,22853}, {7967,22531}, {7968,19069}, {7969,19072}, {8192,22656}, {8210,22863}, {8211,22864}, {9997,22745}, {10247,16628}, {10800,22522}, {10944,22857}, {10950,22858}, {11396,22481}, {11910,22852}, {13902,22876}, {13959,22877}, {18525,22794}

X(22867) = midpoint of X(145) and X(628)
X(22867) = reflection of X(i) in X(j) for these (i,j): (8, 630), (18525, 22794)
X(22867) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22651, 11740), (11740, 22651, 18), (22886, 22887, 18)

X(22868) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 1st NEUBERG

Barycentrics    6*SW*S^2-(SA^2-SW^2)*sqrt(3)*S+3*(3*SA^2-SW^2)*SW : :
X(22868) = 2*X(39)-3*X(6294) = 4*X(39)-3*X(6581) = 3*X(6294)-X(22913) = 3*X(6581)-2*X(22913)

The reciprocal orthologic center of these triangles is X(22869).

X(22868) lies on these lines: {2,39}, {3,22869}, {621,7758}, {698,3104}, {732,3105}, {3095,16626}, {5981,7751}

X(22868) = anticomplement of X(33466)
X(22868) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 22913, 6581), (6294, 22913, 39), (6314, 6318, 6294)

X(22869) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st NEUBERG TO INNER-FERMAT

Barycentrics    3*(SA+2*SW)*S^2-9*SW*SB*SC-sqrt(3)*(16*S^2+9*SA^2-10*SB*SC-3*SW^2)*S : :
X(22869) = 2*X(630)-3*X(22866) = 3*X(22866)-X(22871)

The reciprocal orthologic center of these triangles is X(22868).

X(22869) lies on these lines: {2,18}, {3,22868}, {98,14541}, {3098,22914}, {3642,7755}, {5865,9756}, {5965,22916}, {6287,7684}, {6295,7751}, {6582,7780}, {10645,20081}

X(22869) = circumtangential isogonal conjugate of X(62)
X(22869) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22866, 22871, 630), (22882, 22883, 22736)

X(22870) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 2nd NEUBERG

Barycentrics    12*SW*S^2-sqrt(3)*(SA-3*SW)*(SA+SW)*S+3*(3*SA^2-6*SB*SC-SW^2)*SW : :
X(22870) = 4*X(6292)-3*X(6296) = 2*X(6292)-3*X(6297) = 3*X(6296)-2*X(22915) = 3*X(6297)-X(22915)

The reciprocal orthologic center of these triangles is X(22871).

X(22870) lies on these lines: {2,32}, {3,22871}, {732,3104}, {6287,7685}

X(22870) = anticomplement of X(33468)
X(22870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6292, 22915, 6296), (6297, 22915, 6292), (6313, 6317, 6297)

X(22871) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd NEUBERG TO INNER-FERMAT

Barycentrics    sqrt(3)*(S^2*SA+3*SB*SC*SW)+(2*S^2+9*SA^2-8*SB*SC-3*SW^2)*S : :
X(22871) = X(18)-3*X(22665) = 4*X(630)-3*X(22866) = 3*X(22866)-2*X(22869)

The reciprocal orthologic center of these triangles is X(22870).

X(22871) lies on these lines: {2,18}, {3,22870}, {13,13571}, {14,7814}, {3095,16627}, {3104,22850}, {3643,7796}, {3818,22916}, {5965,22914}, {6298,7764}, {6299,7759}, {16626,16628}

X(22871) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (628, 22114, 61), (630, 22869, 22866), (22882, 22883, 22683)

X(22872) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 1st TRI-SQUARES-CENTRAL

Barycentrics    47*S^2+(1+4*sqrt(3))*((3*SA+(2*sqrt(3)-3)*SW)*S-9*SB*SC) : :
X(22872) = 2*X(13701)-3*X(13704) = 4*X(13701)-3*X(13706) = 3*X(13704)-X(22917) = 3*X(13706)-2*X(22917)

The reciprocal orthologic center of these triangles is X(22873).

X(22872) lies on these lines: {2,1327}, {30,6305}, {2044,13687}, {3104,23011}, {5460,13928}, {13692,16626}, {16645,22874}

X(22872) = anticomplement of X(33470)
X(22872) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1327, 22917), (2, 13712, 13704), (13678, 13712, 22917), (13701, 22917, 13706), (13704, 22917, 13701)

X(22873) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO INNER-FERMAT

Barycentrics    78*((b^2+c^2)*sqrt(3)-74*a^2-15*b^2-15*c^2)*S+3*(34+15*sqrt(3))*((39*(b^2+c^2)*a^2-7*(b^2-c^2)^2)*sqrt(3)+26*a^4-91*(b^2+c^2)*a^2+41*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(22872).

X(22873) lies on these lines: {11489,22875}, {19072,22879}, {22876,22883}, {22880,22882}

X(22874) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-FERMAT TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    47*S^2+(1-4*sqrt(3))*((-3*SA-(-3-2*sqrt(3))*SW)*S-9*SB*SC) : :
X(22874) = 2*X(13821)-3*X(13824) = 4*X(13821)-3*X(13826) = 3*X(13824)-X(22919) = 3*X(13826)-2*X(22919)

The reciprocal orthologic center of these triangles is X(22875).

X(22874) lies on these lines: {2,1328}, {30,6301}, {2043,13807}, {3104,23012}, {5460,13850}, {13812,16626}, {16645,22872}

X(22874) = anticomplement of X(33472)
X(22874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1328, 22919), (2, 13835, 13824), (13798, 13835, 22919), (13821, 22919, 13826), (13824, 22919, 13821)

X(22875) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO INNER-FERMAT

Barycentrics    78*(-sqrt(3)*(b^2+c^2)-74*a^2-15*b^2-15*c^2)*S-3*(34-15*sqrt(3))*(-sqrt(3)*(-7*(b^2-c^2)^2+39*a^2*(b^2+c^2))+26*a^4-91*a^2*(b^2+c^2)+41*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(22874).

X(22875) lies on these lines: {11489,22873}, {19069,22878}, {22877,22882}, {22881,22883}

X(22876) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO INNER-FERMAT

Barycentrics    1358*S^2+(-10+sqrt(3))*(97*SA+20*sqrt(3)*SW-91*SW)*S+97*(SA-SW)*(SA+2*sqrt(3)*SW) : :

The reciprocal orthologic center of these triangles is X(3).

X(22876) lies on these lines: {2,19072}, {6,6674}, {18,3068}, {371,22831}, {590,630}, {628,8972}, {5965,13910}, {7585,19069}, {8974,22853}, {8975,22854}, {8976,16627}, {8981,13916}, {9540,22843}, {11740,13883}, {13884,22481}, {13885,22522}, {13886,22531}, {13887,22557}, {13888,22651}, {13889,22656}, {13890,22669}, {13891,22673}, {13892,22745}, {13893,22851}, {13894,22852}, {13895,22857}, {13896,22858}, {13897,22859}, {13898,22860}, {13899,22863}, {13900,22864}, {13901,22865}, {13902,22867}, {13903,16628}, {13904,22884}, {13905,22885}, {13906,22886}, {13907,22887}, {13917,13925}, {18538,22794}, {18965,18972}, {22763,22771}, {22873,22883}

X(22876) = {X(6), X(6674)}-harmonic conjugate of X(22877)

X(22877) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO INNER-FERMAT

Barycentrics    1358*S^2-(10+sqrt(3))*(-97*SA+(91+20*sqrt(3))*SW)*S-97*(SA-SW)*(-SA+2*sqrt(3)*SW) : :

The reciprocal orthologic center of these triangles is X(3).

X(22877) lies on these lines: {2,19069}, {6,6674}, {18,3069}, {372,22831}, {615,630}, {628,13941}, {5965,13972}, {7586,19072}, {11740,13936}, {13935,22843}, {13937,22481}, {13938,22522}, {13939,22531}, {13940,22557}, {13942,22651}, {13943,22656}, {13944,22669}, {13945,22673}, {13946,22745}, {13947,22851}, {13948,22852}, {13949,22853}, {13950,22854}, {13951,16627}, {13952,22857}, {13953,22858}, {13954,22859}, {13955,22860}, {13956,22863}, {13957,22864}, {13958,22865}, {13959,22867}, {13961,16628}, {13962,22884}, {13963,22885}, {13964,22886}, {13965,22887}, {13966,13981}, {13982,13993}, {18762,22794}, {18966,18972}, {22764,22771}, {22875,22882}

X(22877) = {X(6), X(6674)}-harmonic conjugate of X(22876)

X(22878) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO INNER-FERMAT

Barycentrics    44*S^2-(5+sqrt(3))*(19*SA+2*sqrt(3)*SW-13*SW)*S-(5*sqrt(3)+3)*(SA-SW)*(3*SA-2*SW) : :

The reciprocal orthologic center of these triangles is X(5858).

X(22878) lies on these lines: {2,22879}, {13637,22487}, {13638,22665}, {13644,22923}, {19069,22875}, {19072,22883}

X(22879) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO INNER-FERMAT

Barycentrics    44*S^2+(5-sqrt(3))*((19*SA-(2*sqrt(3)+13)*SW)*S+sqrt(3)*(3*SA-2*SW)*(SA-SW)) : :

The reciprocal orthologic center of these triangles is X(5858).

X(22879) lies on these lines: {2,22878}, {13757,22487}, {13758,22665}, {13763,22924}, {19069,22882}, {19072,22873}

X(22880) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO INNER-FERMAT

Barycentrics    598*S^2+13*(4*sqrt(3)-5)*(SA-SW+2*sqrt(3)*SW)*S-(-1+10*sqrt(3))*(SA-SW)*(-13*SA+sqrt(3)*SW-9*SW) : :
X(22880) = 3*X(485)-X(22626)

The reciprocal orthologic center of these triangles is X(22627).

X(22880) lies on these lines: {18,485}, {590,22883}, {630,13882}, {641,13875}, {6118,13876}, {6305,13850}, {6674,11312}, {12815,22925}, {16645,22627}, {22873,22882}

X(22881) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO INNER-FERMAT

Barycentrics    598*S^2-13*(5+4*sqrt(3))*(-SA+SW+2*sqrt(3)*SW)*S-(10*sqrt(3)+1)*(SA-SW)*(13*SA+sqrt(3)*SW+9*SW) : :
X(22881) = 3*X(486)-X(22597)

The reciprocal orthologic center of these triangles is X(22598).

X(22881) lies on these lines: {18,486}, {615,22882}, {630,13934}, {642,13928}, {6301,13932}, {6674,11312}, {12815,22926}, {16645,22598}, {22875,22883}

X(22882) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-VECTEN TO INNER-FERMAT

Barycentrics    299*S^2+13*(10*sqrt(3)+1)*SB*SC-(sqrt(3)+30)*(-13*SA+5*SW+2*sqrt(3)*SW)*S : :

The reciprocal orthologic center of these triangles is X(22598).

X(22882) lies on these lines: {2,18}, {3,22598}, {615,22881}, {5965,22928}, {6289,16627}, {6561,22597}, {19069,22879}, {22873,22880}, {22875,22877}

X(22882) = complement of X(33437)
X(22882) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 22114, 22883), (628, 22845, 22883), (22683, 22871, 22883), (22736, 22869, 22883)

X(22883) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-VECTEN TO INNER-FERMAT

Barycentrics    299*S^2-13*(-1+10*sqrt(3))*SB*SC+(sqrt(3)-30)*(13*SA-5*SW+2*sqrt(3)*SW)*S : :

The reciprocal orthologic center of these triangles is X(22627).

X(22883) lies on these lines: {2,18}, {3,22627}, {590,22880}, {3642,8960}, {5965,22927}, {6290,16627}, {6560,22626}, {19072,22878}, {22873,22876}, {22875,22881}

X(22883) = complement of X(33436)
X(22883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18, 22114, 22882), (628, 22845, 22882), (22683, 22871, 22882), (22736, 22869, 22882)

X(22884) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22884) lies on these lines: {1,18}, {3,18972}, {5,22860}, {12,16627}, {35,22843}, {55,10061}, {388,22531}, {495,10062}, {498,630}, {499,6674}, {611,5965}, {628,3085}, {1250,22862}, {1479,22831}, {3295,16628}, {3299,19069}, {3301,19072}, {10037,22656}, {10038,22745}, {10039,22851}, {10040,22853}, {10041,22854}, {10077,10612}, {10078,22846}, {10523,22857}, {10801,22522}, {10895,22794}, {10954,22858}, {11398,22481}, {11507,22557}, {11877,22669}, {11878,22673}, {11912,22852}, {11951,22863}, {11952,22864}, {12815,22930}, {13904,22876}, {13962,22877}, {22766,22771}

X(22884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18, 22885), (3295, 16628, 22865)

X(22885) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO INNER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22885) lies on these lines: {1,18}, {3,22865}, {5,22859}, {11,16627}, {36,22843}, {56,10077}, {496,10078}, {497,22531}, {498,6674}, {499,630}, {613,5965}, {628,3086}, {999,16628}, {1478,22831}, {1737,22851}, {3299,19072}, {3301,19069}, {10046,22656}, {10047,22745}, {10048,22853}, {10049,22854}, {10061,10612}, {10062,22846}, {10523,22858}, {10802,22522}, {10896,22794}, {10948,22857}, {11399,22481}, {11508,22557}, {11879,22669}, {11880,22673}, {11913,22852}, {11953,22863}, {11954,22864}, {12815,22929}, {13905,22876}, {13963,22877}, {14986,22114}, {19373,22862}, {22767,22771}

X(22885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18, 22884), (999, 16628, 18972)

X(22886) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INNER-FERMAT

Barycentrics    2*(a^4-2*(b^2+b*c+c^2)*a^2+2*(b+c)*b*c*a+b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*sqrt(3)*S*a^2+(a+b+c)*(a^7-(b+c)*a^6-4*(b^2+4*b*c+c^2)*a^5+2*(b+c)*(2*b^2+9*b*c+2*c^2)*a^4+(5*b^2-12*b*c+5*c^2)*(b^2+4*b*c+c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+24*b*c+5*c^2)*a^2-2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+2*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22886) lies on these lines: {1,18}, {12,22857}, {628,10528}, {630,5552}, {5965,12594}, {10531,22831}, {10679,13104}, {10803,22522}, {10805,22531}, {10834,22656}, {10878,22745}, {10915,22851}, {10929,22853}, {10930,22854}, {10942,16627}, {10955,22858}, {10956,22859}, {10958,22860}, {10965,22865}, {11248,22843}, {11400,22481}, {11509,18972}, {11881,22669}, {11882,22673}, {11914,22852}, {11955,22863}, {11956,22864}, {12000,16628}, {13906,22876}, {13964,22877}, {18542,22794}, {19047,19069}, {19048,19072}, {22768,22771}

X(22886) = {X(18), X(22867)}-harmonic conjugate of X(22887)

X(22887) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INNER-FERMAT

Barycentrics    2*(a^4+b^4+c*(4*b^3+c^3-2*b*c*(b-2*c))-2*(b+c)*b*c*a-2*(b^2-b*c+c^2)*a^2)*sqrt(3)*S*a^2+(a+b+c)*(a^7-(b+c)*a^6-4*(b^2-5*b*c+c^2)*a^5+2*(b+c)*(2*b^2-11*b*c+2*c^2)*a^4+(5*b^4+5*c^4-2*(10*b^2-21*b*c+10*c^2)*b*c)*a^3-(b^2-c^2)*(b-c)*(5*b^2-16*b*c+5*c^2)*a^2-2*(b^4-c^4)*(b^2-c^2)*a+2*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22887) lies on these lines: {1,18}, {11,22858}, {628,10529}, {630,10527}, {5965,12595}, {10532,22831}, {10680,13106}, {10804,22522}, {10806,22531}, {10835,22656}, {10879,22745}, {10916,22851}, {10931,22853}, {10932,22854}, {10943,16627}, {10949,22857}, {10957,22859}, {10959,22860}, {10966,22771}, {11249,22843}, {11401,22481}, {11510,22557}, {11883,22669}, {11884,22673}, {11915,22852}, {11957,22863}, {11958,22864}, {12001,16628}, {13907,22876}, {13965,22877}, {18544,22794}, {18967,18972}, {19049,19069}, {19050,19072}

X(22887) = {X(18), X(22867)}-harmonic conjugate of X(22886)

X(22888) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO INNER-FERMAT

Barycentrics    -(sqrt(3)*(4*a^4-(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*a^2-2*S*(12*a^4-7*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4))*(b^2-c^2) : :

The reciprocal parallelogic center of these triangles is X(3).

X(22888) lies on these lines: {351,9201}, {9135,22933}, {13305,14610}

X(22889) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO INNER-FERMAT

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

The reciprocal parallelogic center of these triangles is X(3).

X(22889) lies on these lines: {2,14447}, {351,9201}, {3569,22934}, {6137,9979}

X(22890) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO OUTER-FERMAT

Barycentrics    2*(9*a^4-7*(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3)*S+3*a^2*(a^4+2*(b^2+c^2)*a^2-3*b^4-2*b^2*c^2-3*c^4) : :
X(22890) = 3*X(3)-X(16629) = 3*X(17)-2*X(16629) = 3*X(165)-X(22652) = 3*X(376)-X(22532) = 4*X(550)+X(22844) = 5*X(631)-4*X(6673) = 5*X(3522)-X(22113) = 2*X(10611)-3*X(21156) = 2*X(14138)-3*X(21158) = 3*X(21157)-2*X(22891)

The reciprocal orthologic center of these triangles is X(3).

X(22890) lies on these lines: {2,22832}, {3,13}, {4,629}, {16,22906}, {20,621}, {30,16626}, {35,22929}, {36,22930}, {55,18973}, {56,22910}, {165,22652}, {182,22523}, {371,19070}, {372,19071}, {376,532}, {382,22795}, {515,22896}, {517,22912}, {548,14539}, {550,5474}, {631,6673}, {1350,5965}, {1593,22482}, {2043,13786}, {2044,13666}, {3098,22746}, {3412,13350}, {3522,22113}, {3534,22493}, {5351,21157}, {5982,9750}, {6284,22905}, {6775,16773}, {7354,22904}, {7748,11481}, {7782,11132}, {9540,22921}, {10310,22558}, {10645,22900}, {11248,22931}, {11249,22932}, {11414,22657}, {11822,22670}, {11823,22674}, {11824,22898}, {11825,22899}, {11826,22902}, {11827,22903}, {11828,22908}, {11829,22909}, {13935,22922}, {14138,21158}

X(22890) = midpoint of X(20) and X(627)
X(22890) = reflection of X(i) in X(j) for these (i,j): (4, 629), (382, 22795)
X(22890) = anticomplement of X(22832)
X(22890) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5340, 21156), (550, 14540, 5474), (1350, 15696, 22843)

X(22891) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th FERMAT-DAO TO OUTER-FERMAT

Barycentrics    3*(SA-2*SW)*S^2+3*SW*SB*SC+sqrt(3)*(16*S^2+3*(SA-SW)*(SW+3*SA))*S : :
X(22891) = X(14)+2*X(22893) = 3*X(5469)-X(11602) = 4*X(11543)-X(22901) = 5*X(16961)+X(22895) = 3*X(21157)-X(22890)

The reciprocal orthologic center of these triangles is X(617).

X(22891) lies on these lines: {2,5469}, {3,16630}, {5,14}, {13,6773}, {15,13102}, {16,115}, {18,627}, {62,16629}, {182,18362}, {298,22737}, {383,5479}, {532,5460}, {542,10611}, {619,6673}, {622,16530}, {629,6670}, {3054,5474}, {3106,7697}, {5238,12815}, {5351,21157}, {5965,10612}, {6775,22847}, {6777,18582}, {8859,12204}, {9981,16809}, {10061,22930}, {10077,22929}, {11304,22911}, {11543,22901}, {14061,22689}, {16940,18581}, {16961,22895}, {16964,22532}, {19070,19075}, {19071,19076}

X(22891) = reflection of X(i) in X(j) for these (i,j): (619, 6673), (629, 6670)
X(22891) = complement of X(14144)
X(22891) = inverse of X(22739) in the outer-Napoleon circle
X(22891) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 22510, 61), (182, 18362, 22846)

X(22892) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO TO OUTER-FERMAT

Barycentrics    (-2*S+(SA-SW)*sqrt(3))*(4*S+(SA+SW)*sqrt(3)) : :
X(22892) = X(13)-3*X(17) = 2*X(13)-3*X(10611) = 3*X(12815)-2*X(22847) = 3*X(16267)-X(22900) = 3*X(16962)+X(22894)

The reciprocal orthologic center of these triangles is X(13).

X(22892) lies on these lines: {2,2981}, {3,13}, {15,22796}, {99,11122}, {114,6109}, {115,619}, {140,14136}, {381,22906}, {395,629}, {396,532}, {616,5472}, {630,6673}, {641,13876}, {642,13929}, {1649,9201}, {2482,22573}, {3412,16530}, {3589,22848}, {6302,13701}, {6306,13821}, {6775,14144}, {6779,22895}, {7749,22911}, {9885,11147}, {10654,16626}, {15819,22691}, {16267,22900}, {16962,22894}

X(22892) = midpoint of X(i) and X(j) for these {i,j}: {99, 11122}, {6779, 22895}
X(22892) = reflection of X(115) in X(22893)

X(22893) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO TO OUTER-FERMAT

Barycentrics    14*S^2-sqrt(3)*(SA+SW)*S+3*(2*SA+SW)*(SA-SW) : :
X(22893) = 3*X(2)+X(11122) = X(14)+3*X(17) = X(14)-3*X(22891) = 3*X(12815)-X(22848) = 3*X(16268)-X(22901) = 3*X(16963)+X(22895)

The reciprocal orthologic center of these triangles is X(617).

X(22893) lies on these lines: {2,11122}, {5,14}, {30,10616}, {115,619}, {141,6034}, {381,22907}, {395,532}, {397,6774}, {549,5470}, {617,16644}, {627,16645}, {629,6674}, {1080,9756}, {3642,18362}, {3643,7746}, {5461,22574}, {6036,6108}, {6109,22797}, {6118,13875}, {6772,22846}, {9166,14905}, {10653,16629}, {11289,14144}, {16268,22901}, {16963,22895}

X(22893) = midpoint of X(115) and X(22892)
X(22893) = {X(6034), X(14061)}-harmonic conjugate of X(22847)

X(22894) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO TO OUTER-FERMAT

Barycentrics    (7*SA+2*SW)*S^2+3*SW*SB*SC+sqrt(3)*(4*S^2+(SA-SW)^2)*S : :
X(22894) = 3*X(13)-2*X(22900) = 6*X(17)-5*X(16960) = 3*X(16962)-4*X(22892)

The reciprocal orthologic center of these triangles is X(13).

X(22894) lies on these lines: {2,3171}, {6,17}, {13,298}, {15,5617}, {16,627}, {302,22911}, {2380,11600}, {3104,16626}, {3105,22916}, {3642,11133}, {6115,7779}, {6671,22997}, {7788,22666}, {10645,22532}, {11302,16242}, {11542,22511}, {16629,16808}, {16940,18581}, {16962,22892}, {16964,22906}, {18582,22113}

X(22894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 22901, 6), (627, 22907, 16), (16808, 22895, 16629), (16966, 22855, 16960)

X(22895) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO TO OUTER-FERMAT

Barycentrics    (19*SA-4*SW)*S^2-3*SW*SB*SC-sqrt(3)*(12*S^2+(SA-SW)*(13*SA+SW))*S : :
X(22895) = 3*X(14)-2*X(22901) = 2*X(16)-3*X(17) = 5*X(16961)-6*X(22891) = 3*X(16963)-4*X(22893)

The reciprocal orthologic center of these triangles is X(617).

X(22895) lies on these lines: {14,532}, {15,22113}, {16,17}, {627,16967}, {3412,19781}, {5474,6778}, {6779,22892}, {16629,16808}, {16961,22891}, {16963,22893}, {16965,22907}

X(22895) = reflection of X(6779) in X(22892)
X(22895) = {X(16629), X(22894)}-harmonic conjugate of X(16808)

X(22896) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO OUTER-FERMAT

Barycentrics    2*(-a^3+b^3+c*b^2+b*c^2+c^3)*sqrt(3)*S+(a+b+c)*(a^4-5*(b+c)*a^3+2*(b^2+5*b*c+c^2)*a^2+5*(b^2-c^2)*(b-c)*a-3*(b^2-c^2)^2) : :
X(22896) = 5*X(1698)-4*X(6673) = 5*X(3617)-X(22113) = 4*X(3626)+X(22844) = 3*X(3679)-X(22652) = 3*X(5587)-2*X(22832) = 3*X(5657)-X(22532) = 3*X(5790)-X(16629)

The reciprocal orthologic center of these triangles is X(3).

X(22896) lies on these lines: {1,629}, {2,11739}, {8,627}, {10,17}, {65,22904}, {72,22903}, {355,12781}, {515,22890}, {517,16626}, {519,22912}, {532,3679}, {956,22772}, {1018,6192}, {1698,6673}, {1737,22930}, {1837,22910}, {3057,22905}, {3416,5965}, {3617,22113}, {3626,22844}, {5090,22482}, {5252,18973}, {5587,22832}, {5657,22532}, {5687,22558}, {5688,22899}, {5689,22898}, {5690,12780}, {5790,16629}, {8193,22657}, {8197,22670}, {8204,22674}, {8214,22908}, {8215,22909}, {9857,22746}, {10039,22929}, {10791,22523}, {10914,22902}, {10915,22931}, {10916,22932}, {12699,22795}, {13883,19070}, {13893,22921}, {13936,19071}, {13947,22922}

X(22896) = midpoint of X(8) and X(627)
X(22896) = reflection of X(i) in X(j) for these (i,j): (1, 629), (12699, 22795)
X(22896) = anticomplement of X(11739)

X(22897) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO OUTER-FERMAT

Barycentrics    (4*a^12-11*(b^2+c^2)*a^10+(3*b^4+26*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(18*b^4-43*b^2*c^2+18*c^4)*a^6-(b^2-c^2)^2*(22*b^4+29*b^2*c^2+22*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*(9*b^4+5*b^2*c^2+9*c^4)*a^2-2*(-4*a^6*b^2*c^2+(b^2+c^2)*a^8+3*(b^2+c^2)*b^4*c^4-(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^4+4*(b^4-c^4)^2*a^2-(b^2+c^2)*(b^8+c^8+(b^4-b^2*c^2+c^4)*b^2*c^2))*sqrt(3)*S-(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^4)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(3).

X(22897) lies on these lines: {17,402}, {30,16626}, {532,1651}, {627,4240}, {629,1650}, {5965,12583}, {6673,15183}, {11251,12793}, {11739,11831}, {11832,22482}, {11839,22523}, {11845,22532}, {11848,22558}, {11852,22652}, {11853,22657}, {11885,22746}, {11897,22832}, {11901,22898}, {11902,22899}, {11903,22902}, {11904,22903}, {11905,22904}, {11906,22905}, {11907,22908}, {11908,22909}, {11909,22910}, {11910,22912}, {11911,16629}, {11912,22929}, {11913,22930}, {11914,22931}, {11915,22932}, {13894,22921}, {13948,22922}, {18507,22795}, {18958,18973}, {19017,19071}, {19018,19070}, {22755,22772}

X(22897) = midpoint of X(627) and X(4240)
X(22897) = reflection of X(i) in X(j) for these (i,j): (17, 402), (18507, 22795)

X(22898) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO OUTER-FERMAT

Barycentrics    61*(10+sqrt(3))*(97*SA+(-37-6*sqrt(3))*SW)*S^2-5917*(3*S^2+SB*SC)*S+11834*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(3).

X(22898) lies on these lines: {6,17}, {532,5861}, {627,1271}, {629,5591}, {1161,6270}, {5589,22652}, {5595,22657}, {5689,22896}, {5875,6271}, {6202,22832}, {6215,16626}, {8198,22670}, {8205,22674}, {8216,22908}, {8217,22909}, {8974,22921}, {9994,22746}, {10040,22929}, {10048,22930}, {10783,22532}, {10792,22523}, {10919,22902}, {10921,22903}, {10923,22904}, {10925,22905}, {10927,22910}, {10929,22931}, {10931,22932}, {11370,11739}, {11388,22482}, {11497,22558}, {11824,22890}, {11901,22897}, {11916,16629}, {13949,22922}, {18509,22795}, {18959,18973}, {22756,22772}

X(22899) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO OUTER-FERMAT

Barycentrics    61*(10-sqrt(3))*(97*SA+(-37+6*sqrt(3))*SW)*S^2+11834*SW*SB*SC+5917*(3*S^2+SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22899) lies on these lines: {6,17}, {532,5860}, {627,1270}, {629,5590}, {1160,6268}, {5588,22652}, {5594,22657}, {5604,22912}, {5688,22896}, {5874,6269}, {6201,22832}, {6214,16626}, {8199,22670}, {8206,22674}, {8218,22908}, {8219,22909}, {8975,22921}, {9995,22746}, {10041,22929}, {10049,22930}, {10784,22532}, {10793,22523}, {10920,22902}, {10922,22903}, {10924,22904}, {10926,22905}, {10928,22910}, {10930,22931}, {10932,22932}, {11371,11739}, {11389,22482}, {11498,22558}, {11825,22890}, {11902,22897}, {11917,16629}, {13950,22922}, {18511,22795}, {18960,18973}, {22757,22772}

X(22900) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO OUTER-FERMAT

Barycentrics    (11*SA-8*SW)*S^2-3*SW*SB*SC-sqrt(3)*(8*S^2+(SA-SW)*(7*SA-SW))*S : :
X(22900) = 3*X(13)-X(22894) = 3*X(17)-4*X(11542) = 3*X(16267)-2*X(22892)

The reciprocal orthologic center of these triangles is X(13).

X(22900) lies on these lines: {6,16629}, {13,298}, {15,5472}, {16,17}, {61,22906}, {627,18582}, {629,16966}, {3105,5335}, {5318,6778}, {6671,6779}, {10645,22890}, {16267,22892}, {16626,16808}, {16809,22832}, {16960,23006}

X(22900) = isogonal conjugate of X(37747)
X(22900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5335, 22113, 22907), (16808, 22901, 16626)

X(22901) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO OUTER-FERMAT

Barycentrics    (17*SA-8*SW)*S^2+3*SW*SB*SC+sqrt(3)*(4*S^2+SA^2-SW^2)*S : :
X(22901) = 3*X(14)-X(22895) = 3*X(17)-5*X(16961) = 4*X(11543)-3*X(22891) = 3*X(16268)-2*X(22893)

The reciprocal orthologic center of these triangles is X(617).

X(22901) lies on these lines: {6,17}, {14,532}, {15,627}, {62,22907}, {298,11132}, {3104,22844}, {5463,5858}, {5872,16965}, {6777,13102}, {7837,22666}, {10646,22532}, {11543,22891}, {16268,22893}, {16530,19781}, {16626,16808}, {16629,16809}, {18581,22113}

X(22901) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 22894, 17), (16626, 22900, 16808), (16961, 22850, 16967)

X(22902) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22902) lies on these lines: {11,17}, {12,22931}, {355,16626}, {532,11235}, {627,3434}, {629,1376}, {5965,12586}, {10523,22929}, {10525,12922}, {10785,22532}, {10794,22523}, {10826,22652}, {10829,22657}, {10871,22746}, {10893,22832}, {10914,22896}, {10919,22898}, {10920,22899}, {10943,12921}, {10944,22904}, {10945,22908}, {10946,22909}, {10947,22910}, {10948,22930}, {10949,22932}, {11373,11739}, {11390,22482}, {11826,22890}, {11865,22670}, {11866,22674}, {11903,22897}, {11928,16629}, {12114,22772}, {13895,22921}, {13952,22922}, {18516,22795}, {18961,18973}, {19023,19071}, {19024,19070}

X(22903) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO OUTER-FERMAT

Barycentrics    -2*(a^6-(b^2+c^2)*a^4+(2*b*c^3+b^4+c^4+2*b^3*c+2*b^2*c^2)*a^2-(b+c)*(2*a^3*b*c+b^5-c*b^4-b*c^4+c^5))*sqrt(3)*S+(a+b+c)*(a^7-(b+c)*a^6+(b^2+10*b*c+c^2)*a^5-(b+c)*(b^2+12*b*c+c^2)*a^4-(5*b^4-22*b^2*c^2+5*c^4)*a^3+(b^2-c^2)*(b-c)*(5*b^2+16*b*c+5*c^2)*a^2+(b^2-c^2)^2*(b-3*c)*(3*b-c)*a-3*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22903) lies on these lines: {11,22932}, {12,17}, {72,22896}, {355,16626}, {532,11236}, {627,3436}, {629,958}, {5965,12587}, {10523,22930}, {10526,12932}, {10786,22532}, {10795,22523}, {10827,22652}, {10830,22657}, {10872,22746}, {10894,22832}, {10921,22898}, {10922,22899}, {10942,12931}, {10950,22905}, {10951,22908}, {10952,22909}, {10953,22910}, {10954,22929}, {10955,22931}, {11374,11739}, {11391,22482}, {11500,22558}, {11827,22890}, {11867,22670}, {11868,22674}, {11904,22897}, {11929,16629}, {13896,22921}, {13953,22922}, {18517,22795}, {18962,18973}, {19025,19071}, {19026,19070}

X(22904) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22904) lies on these lines: {1,16626}, {4,22910}, {5,22930}, {12,17}, {56,629}, {65,22896}, {388,627}, {495,10061}, {532,11237}, {1478,12942}, {1479,22795}, {3027,11602}, {3085,22532}, {5261,22113}, {5965,12588}, {7354,22890}, {9654,16629}, {10797,22523}, {10831,22657}, {10873,22746}, {10895,22832}, {10923,22898}, {10924,22899}, {10944,22902}, {10956,22931}, {10957,22932}, {11375,11739}, {11392,22482}, {11501,22558}, {11869,22670}, {11905,22897}, {11930,22908}, {11931,22909}, {13897,22921}, {13954,22922}, {14144,18975}, {19027,19071}, {19028,19070}, {22759,22772}

X(22904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16626, 22905), (388, 627, 18973)

X(22905) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22905) lies on these lines: {1,16626}, {4,18973}, {5,22929}, {11,17}, {55,629}, {496,10077}, {497,627}, {532,11238}, {1478,22795}, {1479,12952}, {3023,11602}, {3057,22896}, {3086,22532}, {5274,22113}, {5965,12589}, {6284,22890}, {9581,22652}, {9669,16629}, {10798,22523}, {10832,22657}, {10874,22746}, {10896,22832}, {10925,22898}, {10926,22899}, {10950,22903}, {10958,22931}, {10959,22932}, {11376,11739}, {11393,22482}, {11502,22558}, {11871,22670}, {11872,22674}, {11906,22897}, {11932,22908}, {11933,22909}, {13075,14144}, {13898,22921}, {13955,22922}, {19029,19071}, {19030,19070}, {22760,22772}

X(22905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16626, 22904), (497, 627, 22910)

X(22906) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO OUTER-FERMAT

Barycentrics    (2*SA-5*SW)*S^2+3*SW*SB*SC+sqrt(3)*(4*S^2+(SA-SW)*(6*SA+SW))*S : :
X(22906) = 3*X(17)-2*X(22907)

The reciprocal orthologic center of these triangles is X(13).

X(22906) lies on these lines: {4,15}, {6,23006}, {14,99}, {16,22890}, {61,22900}, {193,532}, {381,22892}, {627,5334}, {629,18581}, {3054,5474}, {3104,22844}, {5318,8259}, {5321,16626}, {6114,14144}, {7051,22930}, {7747,11485}, {10638,22929}, {11132,11304}, {11307,16967}, {12017,22862}, {14136,23005}, {16809,16942}, {16964,22894}, {22512,23013}

X(22907) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO OUTER-FERMAT

Barycentrics    (SA+2*SW)*S^2+3*SW*SB*SC+sqrt(3)*(2*S^2+(SA-SW)*(3*SA-SW))*S : :
X(22907) = 3*X(17)-X(22906)

The reciprocal orthologic center of these triangles is X(617).

X(22907) lies on these lines: {4,15}, {5,19781}, {13,148}, {16,627}, {32,16626}, {62,22901}, {69,532}, {381,22893}, {623,22911}, {3098,22513}, {3105,5335}, {5318,16629}, {6115,10645}, {6783,16941}, {7693,21467}, {7737,9996}, {8259,11485}, {9981,16809}, {10654,19130}, {16965,22895}

X(22907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16, 22894, 627), (5335, 22113, 22900)

X(22908) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO OUTER-FERMAT

Barycentrics    -18*sqrt(3)*S^4+3*sqrt(3)*(24*R^2*SA+(-6-10*sqrt(3))*SA^2+6*SB*SC-(-6-7*sqrt(3))*SW^2)*S^2+9*((8*R^2-10*SA+10*SW)*S^2+(SA-SW)*(-8*R^2*SA-sqrt(3)*SW*(2*SA+SW)))*S-9*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(3).

X(22908) lies on these lines: {17,493}, {532,12152}, {627,6462}, {629,8222}, {5965,12590}, {6461,22909}, {8188,22652}, {8194,22657}, {8210,22912}, {8212,22832}, {8214,22896}, {8216,22898}, {8218,22899}, {8220,16626}, {10669,12990}, {10875,22746}, {10945,22902}, {10951,22903}, {11377,11739}, {11394,22482}, {11503,22558}, {11828,22890}, {11840,22523}, {11846,22532}, {11907,22897}, {11930,22904}, {11932,22905}, {11947,22910}, {11949,16629}, {11951,22929}, {11953,22930}, {11955,22931}, {11957,22932}, {13899,22921}, {13956,22922}, {18520,22795}, {18963,18973}, {19031,19071}, {19032,19070}, {22761,22772}

X(22909) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO OUTER-FERMAT

Barycentrics    18*sqrt(3)*S^4-3*sqrt(3)*((-6+10*sqrt(3))*SA^2+24*R^2*SA+6*SB*SC-(-6+7*sqrt(3))*SW^2)*S^2-9*SW^2*SB*SC-9*((8*R^2-10*SA+10*SW)*S^2+(-8*R^2*SA+sqrt(3)*(SW+2*SA)*SW)*(SA-SW))*S : :

The reciprocal orthologic center of these triangles is X(3).

X(22909) lies on these lines: {17,494}, {532,12153}, {627,6463}, {629,8223}, {5965,12591}, {6461,22908}, {8189,22652}, {8195,22657}, {8211,22912}, {8213,22832}, {8215,22896}, {8217,22898}, {8219,22899}, {8221,16626}, {10673,12991}, {10876,22746}, {10946,22902}, {10952,22903}, {11378,11739}, {11395,22482}, {11504,22558}, {11829,22890}, {11841,22523}, {11847,22532}, {11908,22897}, {11931,22904}, {11933,22905}, {11948,22910}, {11950,16629}, {11952,22929}, {11954,22930}, {11956,22931}, {11958,22932}, {13900,22921}, {13957,22922}, {18522,22795}, {18964,18973}, {19033,19071}, {19034,19070}, {22762,22772}

X(22910) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22910) lies on these lines: {1,13076}, {3,22930}, {4,22904}, {11,629}, {12,22832}, {17,55}, {33,22482}, {56,22890}, {390,22113}, {497,627}, {532,3058}, {1479,16626}, {1697,22652}, {1837,22896}, {2098,22912}, {2646,11739}, {3056,5965}, {3295,16629}, {3583,22795}, {4294,22532}, {5432,6673}, {10799,22523}, {10833,22657}, {10927,22898}, {10928,22899}, {10947,22902}, {10953,22903}, {10965,22931}, {10966,22772}, {11602,13183}, {11873,22670}, {11874,22674}, {11909,22897}, {11947,22908}, {11948,22909}, {12951,14144}, {13075,15171}, {13901,22921}, {13958,22922}, {19037,19071}, {19038,19070}

X(22910) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 627, 22905), (3295, 16629, 22929)

X(22911) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MCCAY TO OUTER-FERMAT

Barycentrics    (3*SA+4*SW)*S^2-3*SW*SB*SC+sqrt(3)*(10*S^2+3*SA^2-4*SB*SC-SW^2)*S : :
X(22911) = 2*X(629)+X(22914) = 4*X(629)-X(22916) = 2*X(22914)+X(22916)

The reciprocal orthologic center of these triangles is X(22570).

X(22911) lies on these lines: {2,17}, {3,22570}, {76,16242}, {302,22894}, {623,22907}, {1078,3642}, {3643,7746}, {6299,10104}, {6581,11171}, {7749,22892}, {11304,22891}

X(22911) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 627, 22737), (629, 22914, 22916)

X(22912) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO OUTER-FERMAT

Barycentrics    -2*(a^2-(b+c)*a+2*b^2+2*c^2)*sqrt(3)*S*a+(a+b+c)*(9*a^4-10*(b+c)*a^3-(7*b^2-20*b*c+7*c^2)*a^2+10*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2) : :
X(22912) = 3*X(1)-2*X(11739) = 3*X(1)-X(22652) = 3*X(17)-4*X(11739) = 3*X(17)-2*X(22652) = 4*X(3244)+X(22844) = 5*X(3616)-4*X(6673) = 5*X(3623)-X(22113) = 3*X(5603)-2*X(22832) = 3*X(7967)-X(22532) = 3*X(10247)-X(16629)

The reciprocal orthologic center of these triangles is X(3).

X(22912) lies on these lines: {1,17}, {8,629}, {55,22772}, {56,22558}, {145,627}, {517,22890}, {519,22896}, {532,3241}, {952,16626}, {1482,7975}, {1483,7974}, {2098,22910}, {2099,18973}, {3242,5965}, {3244,22844}, {3616,6673}, {3623,22113}, {5597,22674}, {5598,22670}, {5603,22832}, {5604,22899}, {7967,22532}, {7968,19071}, {7969,19070}, {8192,22657}, {8210,22908}, {8211,22909}, {9997,22746}, {10247,16629}, {10800,22523}, {10944,22902}, {10950,22903}, {11396,22482}, {11910,22897}, {13902,22921}, {13959,22922}, {18525,22795}

X(22912) = midpoint of X(145) and X(627)
X(22912) = reflection of X(i) in X(j) for these (i,j): (8, 629), (18525, 22795)
X(22912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22652, 11739), (11739, 22652, 17), (22931, 22932, 17)

X(22913) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 1st NEUBERG

Barycentrics    6*S^2*SW+(SA^2-SW^2)*sqrt(3)*S+3*(3*SA^2-SW^2)*SW : :
X(22913) = 4*X(39)-3*X(6294) = 2*X(39)-3*X(6581) = 3*X(6294)-2*X(22868) = 3*X(6581)-X(22868)

The reciprocal orthologic center of these triangles is X(22914).

X(22913) lies on these lines: {2,39}, {3,22914}, {622,7758}, {698,3105}, {732,3104}, {3095,16627}, {5980,7751}

X(22913) = anticomplement of X(33467)
X(22913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 22868, 6294), (6314, 6318, 6581), (6581, 22868, 39)

X(22914) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st NEUBERG TO OUTER-FERMAT

Barycentrics    3*(SA+2*SW)*S^2-9*SW*SB*SC+sqrt(3)*(16*S^2+9*SA^2-10*SB*SC-3*SW^2)*S : :
X(22914) = 2*X(629)-3*X(22911) = 3*X(22911)-X(22916)

The reciprocal orthologic center of these triangles is X(22913).

X(22914) lies on these lines: {2,17}, {3,22913}, {98,14540}, {3098,22869}, {3643,7755}, {5864,9756}, {5965,22871}, {6287,7685}, {6295,7780}, {6582,7751}, {10646,20081}

X(22914) = circumtangential isogonal conjugate of X(61)
X(22914) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22911, 22916, 629), (22927, 22928, 22737)

X(22915) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 2nd NEUBERG

Barycentrics    12*S^2*SW+sqrt(3)*(SA-3*SW)*(SA+SW)*S+3*(3*SA^2-6*SB*SC-SW^2)*SW : :
X(22915) = 2*X(6292)-3*X(6296) = 4*X(6292)-3*X(6297) = 3*X(6296)-X(22870) = 3*X(6297)-2*X(22870)

The reciprocal orthologic center of these triangles is X(22916).

X(22915) lies on these lines: {2,32}, {3,22916}, {732,3105}, {6287,7684}

X(22915) = anticomplement of X(33469)
X(22915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6292, 22870, 6297), (6296, 22870, 6292), (6313, 6317, 6296)

X(22916) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd NEUBERG TO OUTER-FERMAT

Barycentrics    sqrt(3)*(S^2*SA+3*SW*SB*SC)-(2*S^2+9*SA^2-8*SB*SC-3*SW^2)*S : :
X(22916) = X(17)-3*X(22666) = 4*X(629)-3*X(22911) = 3*X(22911)-2*X(22914)

The reciprocal orthologic center of these triangles is X(22915).

X(22916) lies on these lines: {2,17}, {3,22915}, {13,7814}, {14,13571}, {3095,16626}, {3105,22894}, {3642,7796}, {3818,22871}, {5965,22869}, {6298,7759}, {6299,7764}, {16627,16629}

X(22916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (627, 22113, 62), (629, 22914, 22911), (22927, 22928, 22685)

X(22917) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 1st TRI-SQUARES-CENTRAL

Barycentrics    47*S^2+(1-4*sqrt(3))*((3*SA+(-3-2*sqrt(3))*SW)*S-9*SB*SC) : :
X(22917) = 4*X(13701)-3*X(13704) = 2*X(13701)-3*X(13706) = 3*X(13704)-2*X(22872) = 3*X(13706)-X(22872)

The reciprocal orthologic center of these triangles is X(22918).

X(22917) lies on these lines: {2,1327}, {30,6304}, {532,22629}, {2043,13687}, {3105,23002}, {5459,13929}, {13692,16627}, {16644,22919}

X(22917) = anticomplement of X(33471)
X(22917) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1327, 22872), (2, 13712, 13706), (13678, 13712, 22872), (13701, 22872, 13704), (13706, 22872, 13701)

X(22918) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES-CENTRAL TO OUTER-FERMAT

Barycentrics    78*(-sqrt(3)*(b^2+c^2)-74*a^2-15*b^2-15*c^2)*S+3*(34-15*sqrt(3))*(-sqrt(3)*(-7*(b^2-c^2)^2+39*a^2*(b^2+c^2))+26*a^4-91*a^2*(b^2+c^2)+41*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(22917).

X(22918) lies on these lines: {532,3068}, {11488,22920}, {19070,22924}, {22921,22928}, {22925,22927}

X(22919) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-FERMAT TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    47*S^2+(1+4*sqrt(3))*(-(3*SA+(2*sqrt(3)-3)*SW)*S-9*SB*SC) : :
X(22919) = 4*X(13821)-3*X(13824) = 2*X(13821)-3*X(13826) = 3*X(13824)-2*X(22874) = 3*X(13826)-X(22874)

The reciprocal orthologic center of these triangles is X(22920).

X(22919) lies on these lines: {2,1328}, {30,6300}, {532,22600}, {2044,13807}, {3105,23003}, {5459,13850}, {13812,16627}, {16644,22917}

X(22919) = anticomplement of X(33473)
X(22919) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1328, 22874), (2, 13835, 13826), (13798, 13835, 22874), (13821, 22874, 13824), (13826, 22874, 13821)

X(22920) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES-CENTRAL TO OUTER-FERMAT

Barycentrics    -78*((b^2+c^2)*sqrt(3)-74*a^2-15*b^2-15*c^2)*S+3*(34+15*sqrt(3))*((39*(b^2+c^2)*a^2-7*(b^2-c^2)^2)*sqrt(3)+26*a^4-91*(b^2+c^2)*a^2+41*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(22919).

X(22920) lies on these lines: {532,3069}, {11488,22918}, {19071,22923}, {22922,22927}, {22926,22928}

X(22921) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO OUTER-FERMAT

Barycentrics    1358*S^2+(10+sqrt(3))*(-97*SA+(91+20*sqrt(3))*SW)*S-97*(SA-SW)*(-SA+2*sqrt(3)*SW) : :

The reciprocal orthologic center of these triangles is X(3).

X(22921) lies on these lines: {2,19070}, {6,6673}, {17,3068}, {371,22832}, {532,6305}, {590,629}, {627,8972}, {5965,13910}, {7585,19071}, {8974,22898}, {8975,22899}, {8976,16626}, {8981,13917}, {9540,22890}, {11739,13883}, {13884,22482}, {13885,22523}, {13886,22532}, {13887,22558}, {13889,22657}, {13891,22674}, {13893,22896}, {13894,22897}, {13895,22902}, {13896,22903}, {13897,22904}, {13898,22905}, {13899,22908}, {13900,22909}, {13901,22910}, {13902,22912}, {13903,16629}, {13904,22929}, {13905,22930}, {13906,22931}, {13907,22932}, {18538,22795}, {18965,18973}, {22763,22772}, {22918,22928}

X(22921) = {X(6), X(6673)}-harmonic conjugate of X(22922)

X(22922) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO OUTER-FERMAT

Barycentrics    1358*S^2-(-10+sqrt(3))*(97*SA+20*sqrt(3)*SW-91*SW)*S+97*(SA-SW)*(SA+2*sqrt(3)*SW) : :

The reciprocal orthologic center of these triangles is X(3).

X(22922) lies on these lines: {2,19071}, {6,6673}, {17,3069}, {372,22832}, {532,6301}, {615,629}, {627,13941}, {5965,13972}, {7586,19070}, {11739,13936}, {13935,22890}, {13937,22482}, {13938,22523}, {13939,22532}, {13940,22558}, {13942,22652}, {13943,22657}, {13944,22670}, {13945,22674}, {13946,22746}, {13947,22896}, {13948,22897}, {13949,22898}, {13950,22899}, {13951,16626}, {13952,22902}, {13953,22903}, {13954,22904}, {13955,22905}, {13956,22908}, {13957,22909}, {13958,22910}, {13959,22912}, {13961,16629}, {13962,22929}, {13963,22930}, {13964,22931}, {13965,22932}, {13966,13982}, {13981,13993}, {18762,22795}, {18966,18973}, {22764,22772}, {22920,22927}

X(22922) = {X(6), X(6673)}-harmonic conjugate of X(22921)

X(22923) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO OUTER-FERMAT

Barycentrics    44*S^2+(5-sqrt(3))*(-(19*SA-(2*sqrt(3)+13)*SW)*S+sqrt(3)*(3*SA-2*SW)*(SA-SW)) : :

The reciprocal orthologic center of these triangles is X(5859).

X(22923) lies on these lines: {2,22924}, {532,3068}, {13637,22488}, {13638,22666}, {13644,22878}, {19070,22928}, {19071,22920}

X(22924) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO OUTER-FERMAT

Barycentrics    44*S^2+(5+sqrt(3))*(19*SA+2*sqrt(3)*SW-13*SW)*S-(5*sqrt(3)+3)*(SA-SW)*(3*SA-2*SW) : :

The reciprocal orthologic center of these triangles is X(5859).

X(22924) lies on these lines: {2,22923}, {532,3069}, {13757,22488}, {13758,22666}, {13763,22879}, {19070,22918}, {19071,22927}

X(22925) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES TO OUTER-FERMAT

Barycentrics    598*S^2+13*(5+4*sqrt(3))*(-SA+SW+2*sqrt(3)*SW)*S-(10*sqrt(3)+1)*(SA-SW)*(13*SA+sqrt(3)*SW+9*SW) : :
X(22925) = 3*X(485)-X(22628)

The reciprocal orthologic center of these triangles is X(22629).

X(22925) lies on these lines: {17,485}, {532,6305}, {590,22928}, {629,13882}, {641,13876}, {6118,13875}, {6304,13850}, {6673,11311}, {12815,22880}, {16644,22629}, {22918,22927}

X(22926) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES TO OUTER-FERMAT

Barycentrics    598*S^2-13*(4*sqrt(3)-5)*(SA-SW+2*sqrt(3)*SW)*S-(-1+10*sqrt(3))*(SA-SW)*(-13*SA+sqrt(3)*SW-9*SW) : :
X(22926) = 3*X(486)-X(22599)

The reciprocal orthologic center of these triangles is X(22600).

X(22926) lies on these lines: {17,486}, {532,6301}, {615,22927}, {629,13934}, {642,13929}, {6300,13932}, {6673,11311}, {12815,22881}, {16644,22600}, {22920,22928}

X(22927) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-VECTEN TO OUTER-FERMAT

Barycentrics    299*S^2-13*(-1+10*sqrt(3))*SB*SC-(sqrt(3)-30)*(13*SA-5*SW+2*sqrt(3)*SW)*S : :

The reciprocal orthologic center of these triangles is X(22600).

X(22927) lies on these lines: {2,17}, {3,22600}, {615,22926}, {5965,22883}, {6289,16626}, {6561,22599}, {19071,22924}, {22918,22925}, {22920,22922}

X(22927) = complement of X(33439)
X(22927) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 22113, 22928), (627, 22844, 22928), (22685, 22916, 22928), (22737, 22914, 22928)

X(22928) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-VECTEN TO OUTER-FERMAT

Barycentrics    299*S^2+13*(10*sqrt(3)+1)*SB*SC+(sqrt(3)+30)*(-13*SA+5*SW+2*sqrt(3)*SW)*S : :

The reciprocal orthologic center of these triangles is X(22629).

X(22928) lies on these lines: {2,17}, {3,22629}, {590,22925}, {3643,8960}, {5965,22882}, {6290,16626}, {6560,22628}, {19070,22923}, {22918,22921}, {22920,22926}

X(22928) = complement of X(33438)
X(22928) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17, 22113, 22927), (627, 22844, 22927), (22685, 22916, 22927), (22737, 22914, 22927)

X(22929) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22929) lies on these lines: {1,17}, {3,18973}, {5,22905}, {12,16626}, {35,22890}, {388,22532}, {495,10061}, {498,629}, {499,6673}, {532,10056}, {611,5965}, {627,3085}, {1479,22832}, {3295,16629}, {3299,19071}, {3301,19070}, {10037,22657}, {10038,22746}, {10039,22896}, {10040,22898}, {10077,22891}, {10078,10611}, {10523,22902}, {10801,22523}, {10895,22795}, {10954,22903}, {11398,22482}, {11507,22558}, {11877,22670}, {11878,22674}, {11912,22897}, {11951,22908}, {11952,22909}, {12815,22885}, {13904,22921}, {13962,22922}, {22766,22772}

X(22929) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17, 22930), (3295, 16629, 22910)

X(22930) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO OUTER-FERMAT

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

The reciprocal orthologic center of these triangles is X(3).

X(22930) lies on these lines: {1,17}, {3,22910}, {5,22904}, {11,16626}, {36,22890}, {56,10078}, {496,10077}, {497,22532}, {498,6673}, {499,629}, {532,10072}, {613,5965}, {627,3086}, {1478,22832}, {1737,22896}, {3299,19070}, {3301,19071}, {7051,22906}, {10046,22657}, {10048,22898}, {10049,22899}, {10061,22891}, {10062,10611}, {10523,22903}, {10802,22523}, {10896,22795}, {10948,22902}, {11399,22482}, {11508,22558}, {11880,22674}, {11913,22897}, {11953,22908}, {11954,22909}, {12815,22884}, {13905,22921}, {13963,22922}, {14986,22113}

X(22930) = {X(1), X(17)}-harmonic conjugate of X(22929)

X(22931) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO OUTER-FERMAT

Barycentrics    -2*(a^4-2*(b^2+b*c+c^2)*a^2+2*(b+c)*b*c*a+b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*sqrt(3)*S*a^2+(a+b+c)*(a^7-(b+c)*a^6-4*(b^2+4*b*c+c^2)*a^5+2*(b+c)*(2*b^2+9*b*c+2*c^2)*a^4+(5*b^2-12*b*c+5*c^2)*(b^2+4*b*c+c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+24*b*c+5*c^2)*a^2-2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+2*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22931) lies on these lines: {1,17}, {12,22902}, {532,11239}, {627,10528}, {629,5552}, {5965,12594}, {10531,22832}, {10679,13105}, {10803,22523}, {10805,22532}, {10834,22657}, {10878,22746}, {10915,22896}, {10929,22898}, {10930,22899}, {10942,16626}, {10955,22903}, {10956,22904}, {10958,22905}, {10965,22910}, {11248,22890}, {11400,22482}, {11509,18973}, {11881,22670}, {11882,22674}, {11914,22897}, {11955,22908}, {11956,22909}, {12000,16629}, {13906,22921}, {13964,22922}, {18542,22795}, {19047,19071}, {19048,19070}, {22768,22772}

X(22931) = {X(17), X(22912)}-harmonic conjugate of X(22932)

X(22932) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO OUTER-FERMAT

Barycentrics    -2*(a^4+b^4+c*(4*b^3+c^3-2*b*c*(b-2*c))-2*(b+c)*b*c*a-2*(b^2-b*c+c^2)*a^2)*sqrt(3)*S*a^2+(a+b+c)*(a^7-(b+c)*a^6-4*(b^2-5*b*c+c^2)*a^5+2*(b+c)*(2*b^2-11*b*c+2*c^2)*a^4+(5*b^4+5*c^4-2*(10*b^2-21*b*c+10*c^2)*b*c)*a^3-(b^2-c^2)*(b-c)*(5*b^2-16*b*c+5*c^2)*a^2-2*(b^4-c^4)*(b^2-c^2)*a+2*(b^2-c^2)^3*(b-c)) : :

The reciprocal orthologic center of these triangles is X(3).

X(22932) lies on these lines: {1,17}, {11,22903}, {532,11240}, {627,10529}, {629,10527}, {5965,12595}, {10532,22832}, {10680,13107}, {10804,22523}, {10806,22532}, {10835,22657}, {10879,22746}, {10916,22896}, {10931,22898}, {10932,22899}, {10943,16626}, {10949,22902}, {10957,22904}, {10959,22905}, {10966,22772}, {11249,22890}, {11401,22482}, {11510,22558}, {11883,22670}, {11884,22674}, {11915,22897}, {11957,22908}, {11958,22909}, {12001,16629}, {13907,22921}, {13965,22922}, {18544,22795}, {18967,18973}, {19049,19071}, {19050,19070}

X(22932) = {X(17), X(22912)}-harmonic conjugate of X(22931)

X(22933) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO OUTER-FERMAT

Barycentrics    -(sqrt(3)*(4*a^4-(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*a^2+2*S*(12*a^4-7*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4))*(b^2-c^2) : :

The reciprocal parallelogic center of these triangles is X(3).

X(22933) lies on these lines: {351,9200}, {9135,22888}, {13304,14610}

X(22934) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO OUTER-FERMAT

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

The reciprocal parallelogic center of these triangles is X(3).

X(22934) lies on these lines: {2,14446}, {351,9200}, {3569,22889}, {6138,9979}

X(22935) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798E TO FUHRMANN

Barycentrics    a*(-2*a^6+3*(b+c)*a^5+3*(b-c)^2*a^4-(b+c)*(6*b^2-7*b*c+6*c^2)*a^3+6*(b^2-b*c+c^2)*b*c*a^2+(b^2-c^2)*(b-c)*(3*b^2-b*c+3*c^2)*a-(b^4-c^4)*(b^2-c^2)) : :
X(22935) = 3*X(3)-X(1768) = X(3)-3*X(15015) = 2*X(11)-3*X(11230) = 3*X(100)+X(10698) = X(149)-3*X(5886) = 3*X(214)-X(11715) = 3*X(1385)-2*X(11715) = X(1768)+3*X(6326) = X(1768)-9*X(15015) = 4*X(3035)-3*X(11231) = 3*X(6265)-X(10698) = X(6326)+3*X(15015) = 2*X(10609)+X(18480) = 3*X(11231)-2*X(12619) = X(12738)+2*X(13624)

The reciprocal orthologic center of these triangles is X(191).

X(22935) lies on the cubic K798 and these lines: {1,6797}, {3,191}, {10,140}, {11,6881}, {20,16128}, {30,21635}, {35,17638}, {72,4996}, {80,2646}, {100,517}, {104,6986}, {119,6831}, {149,5886}, {153,18481}, {355,6224}, {376,9809}, {381,15017}, {404,5885}, {515,11698}, {528,11729}, {550,18243}, {631,9803}, {942,10090}, {1125,1484}, {1155,11571}, {1319,7972}, {1482,5541}, {1537,10993}, {2800,3579}, {2801,15481}, {2802,19907}, {2932,4855}, {3576,5531}, {3811,22560}, {3871,17652}, {3916,12532}, {4413,6264}, {5126,10074}, {5587,12747}, {5603,20095}, {5660,10742}, {5790,9897}, {5818,20085}, {5840,9945}, {5887,17100}, {5901,21630}, {6261,12332}, {6702,20104}, {6901,11604}, {6924,22836}, {7508,10176}, {7743,13274}, {8674,11699}, {8715,10284}, {9802,10595}, {9856,12775}, {9955,10738}, {9957,10087}, {10225,14988}, {10247,12653}, {11219,13151}, {12645,21842}, {12699,13199}, {12702,13253}, {12737,15178}, {13145,17654}, {13146,13743}, {20117,22936}

X(22935) = midpoint of X(i) and X(j) for these {i,j}: {1, 12331}, {3, 6326}, {20, 16128}, {100, 6265}, {104, 12738}, {119, 10609}, {153, 18481}, {355, 6224}, {1482, 5541}, {1537, 10993}, {3811, 22560}, {6261, 12332}, {12699, 13199}, {12702, 13253}, {13146, 13743}
X(22935) = reflection of X(i) in X(j) for these (i,j): (80, 9956), (104, 13624), (12737, 15178)
X(22935) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3035, 12619, 11231), (3576, 5531, 12773), (5660, 12119, 10742), (6326, 15015, 3), (10087, 12740, 9957), (10090, 12739, 942)

X(22936) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798E TO 2nd FUHRMANN

Barycentrics    a*(-2*a^6+(b+c)*a^5+(5*b^2-2*b*c+5*c^2)*a^4-(b+c)*(2*b^2-b*c+2*c^2)*a^3-2*(2*b^4+b^2*c^2+2*c^4)*a^2+(b^3-c^3)*(b^2-c^2)*a+(b^2-c^2)^2*(b+c)^2) : :
X(22936) = 3*X(3)-X(16143) = X(1385)+2*X(3652) = X(3579)-4*X(3647) = X(3648)+2*X(9955) = 3*X(5426)-2*X(15178) = 4*X(5428)-3*X(17502) = 2*X(5499)-3*X(11231) = 3*X(5886)-X(14450) = 3*X(7701)+X(16143) = 4*X(10021)-3*X(11230) = 3*X(11230)-2*X(11263) = X(11278)+2*X(11684) = 5*X(15674)-X(16116)

The reciprocal orthologic center of these triangles is X(7701).

X(22936) lies on the cubic K798 and these lines: {1,13465}, {3,7701}, {5,12615}, {10,30}, {21,104}, {35,3065}, {58,8143}, {65,1749}, {79,17605}, {140,21635}, {191,517}, {355,15680}, {549,18243}, {758,11260}, {1155,16118}, {1770,13852}, {2475,9956}, {3648,3916}, {3651,5927}, {3683,13624}, {4861,11278}, {4999,12611}, {5251,13145}, {5260,12515}, {5426,15178}, {5428,17502}, {5499,11231}, {5694,6914}, {5885,7489}, {5886,14450}, {6701,20107}, {6853,16128}, {6888,12600}, {6906,18259}, {10021,11230}, {10058,14883}, {12769,13126}, {15674,16116}, {16139,21669}, {16160,22793}, {16617,17768}, {20117,22935}

X(22936) = midpoint of X(i) and X(j) for these {i,j}: {1, 13465}, {3, 7701}, {21, 3652}, {191, 13743}, {355, 15680}, {3648, 16159}, {3651, 16138}, {12769, 13126}, {16139, 21669}
X(22936) = reflection of X(2475) in X(9956)
X(22936) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4640, 18480, 3579), (10021, 11263, 11230)

X(22937) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798I TO 2nd FUHRMANN

Barycentrics    a*(2*a^6-(b+c)*a^5-(5*b^2+2*b*c+5*c^2)*a^4+(b+c)*(2*b^2-b*c+2*c^2)*a^3+2*(2*b^2+3*b*c+2*c^2)*(b^2-b*c+c^2)*a^2-(b^3-c^3)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)) : :
X(22937) = 3*X(3)+X(13465) = 3*X(3)-X(16132) = 3*X(165)+X(7701) = 3*X(165)-X(16117) = 3*X(191)-X(13465) = 3*X(191)+X(16132) = 2*X(442)-3*X(11231) = 5*X(631)-X(14450) = X(1482)-3*X(5426) = X(3579)+2*X(3647) = X(3656)-3*X(15672) = X(11684)+2*X(13624) = X(11684)+3*X(21161) = 2*X(13624)-3*X(21161) = 4*X(18253)-X(18480)

The reciprocal orthologic center of these triangles is X(6326).

X(22937) lies on the cubic K798 and these lines: {2,16159}, {3,191}, {10,30}, {21,517}, {35,1749}, {40,13743}, {71,2290}, {79,1155}, {140,11263}, {165,7701}, {442,11231}, {500,896}, {516,16160}, {582,4414}, {631,14450}, {758,1385}, {846,8143}, {946,10021}, {1006,5885}, {1482,5426}, {3651,3652}, {3654,15677}, {3656,15672}, {3683,9955}, {3916,4511}, {3925,13852}, {4995,13995}, {5122,16140}, {5273,18517}, {5432,14526}, {5499,6684}, {5603,15676}, {5657,15680}, {5886,15674}, {6675,11230}, {6701,20104}, {6841,7965}, {6902,11604}, {7743,16155}, {9956,16113}, {10164,11277}, {10246,16126}, {10895,16118}, {11010,17636}, {11259,21376}, {11699,16164}, {16138,17613}, {19861,21165}

X(22937) = midpoint of X(i) and X(j) for these {i,j}: {3, 191}, {21, 16139}, {40, 13743}, {3651, 3652}, {3654, 15677}
X(22937) = reflection of X(i) in X(j) for these (i,j): (946, 10021), (5499, 6684), (11699, 16164)
X(22937) = complement of X(16159)
X(22937) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 13465, 16132), (35, 1749, 17637), (165, 7701, 16117), (191, 16132, 13465)

X(22938) = PARALLELOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO INNER-GARCIA

Barycentrics    2*a^7-2*(b+c)*a^6-2*(b-c)^2*a^5+2*(b^3+c^3)*a^4-(2*b^4+2*c^4-(b^2+c^2)*b*c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-2*c)*(2*b-c)*a-2*(b^2-c^2)^3*(b-c) : :
X(22938) = 3*X(4)+X(149) = 5*X(4)-X(153) = 3*X(4)-X(10742) = 3*X(5)-2*X(3035) = X(100)-3*X(381) = 3*X(119)-X(6154) = 5*X(149)+3*X(153) = X(149)-3*X(10738) = 2*X(149)+3*X(22799) = X(153)+5*X(10738) = 3*X(153)-5*X(10742) = 2*X(153)-5*X(22799) = 6*X(546)-X(6154) = 3*X(10738)+X(10742) = 2*X(10738)+X(22799)

The reciprocal parallelogic center of these triangles is X(1).

X(22938) lies on these lines: {3,10724}, {4,145}, {5,3035}, {11,30}, {80,5560}, {100,381}, {104,382}, {119,546}, {214,9955}, {355,14217}, {516,12619}, {528,3845}, {548,21154}, {549,6667}, {550,6713}, {946,11567}, {962,19914}, {1145,18357}, {1317,3585}, {1385,16174}, {1387,1388}, {1478,12735}, {1484,2829}, {1539,2771}, {1699,6265}, {1770,20118}, {1836,10073}, {2550,6929}, {2783,22505}, {2787,22515}, {2800,22793}, {2802,18480}, {2806,19163}, {2831,19160}, {3045,10540}, {3091,13199}, {3543,12248}, {3579,6702}, {3656,7972}, {3818,9024}, {3830,10707}, {3839,20095}, {3843,12331}, {3850,10993}, {3857,20400}, {4996,13743}, {5066,6174}, {5533,7354}, {5541,18492}, {5690,10525}, {5691,12737}, {5848,21850}, {5886,12119}, {6033,10769}, {6284,8068}, {6321,10768}, {6734,19919}, {6924,10893}, {7728,10778}, {8148,12531}, {8674,10113}, {9803,10248}, {9812,12247}, {9818,13222}, {10057,12701}, {10058,12953}, {10074,12943}, {10087,10895}, {10090,10896}, {10308,11604}, {10711,14269}, {10739,10772}, {10740,10777}, {10741,10770}, {10747,10771}, {10750,10782}, {10751,10781}, {10755,18440}, {10773,15521}, {10774,15522}, {10780,12918}, {12019,12764}, {12047,12743}, {12611,18483}, {13194,18502}, {13205,18491}, {13228,18495}, {13230,18497}, {13235,18500}, {13268,18507}, {13269,18509}, {13270,18511}, {13271,18516}, {13272,18517}, {13275,18520}, {13276,18522}, {13278,18542}, {13279,18544}, {13665,19113}, {13785,19112}, {13922,18538}, {13991,18762}, {16173,18481}, {18240,18527}, {18761,22560}

X(22938) = midpoint of X(i) and X(j) for these {i,j}: {3, 10724}, {4, 10738}, {80, 12699}, {104, 382}, {355, 14217}, {962, 19914}, {1484, 3627}, {3830, 10707}, {5691, 12737}, {6033, 10769}, {6321, 10768}, {7728, 10778}, {8148, 12531}, {10739, 10772}, {10740, 10777}, {10741, 10770}, {10747, 10771}, {10750, 10782}, {10751, 10781}, {10755, 18440}, {10773, 15521}, {10774, 15522}, {10780, 12918}, {13268, 18507}
X(22938) = reflection of X(i) in X(j) for these (i,j): (119, 546), (214, 9955), (550, 6713), (1145, 18357), (1385, 16174), (3579, 6702), (12611, 18483)
X(22938) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 149, 10742), (1478, 13274, 12735), (1479, 13273, 1387), (3830, 12773, 10728), (10707, 10728, 12773), (10738, 10742, 149)

X(22939) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INNER-GARCIA TO ATIK

Barycentrics    a*(9*a^12+12*(b+c)*a^11-2*(15*b^2-38*b*c+15*c^2)*a^10-12*(b+c)*(7*b^2-8*b*c+7*c^2)*a^9+(b+3*c)*(3*b+c)*(13*b^2-50*b*c+13*c^2)*a^8+8*(b+c)*(27*b^4+27*c^4-2*b*c*(18*b^2-b*c+18*c^2))*a^7-4*(9*b^6+9*c^6+(114*b^4+114*c^4-b*c*(77*b^2+236*b*c+77*c^2))*b*c)*a^6-24*(b+c)*(11*b^6+11*c^6-(12*b^4+12*c^4-b*c*(b^2-32*b*c+c^2))*b*c)*a^5+(39*b^6+39*c^6+(650*b^4+650*c^4-3*b*c*(333*b^2-292*b*c+333*c^2))*b*c)*(b+c)^2*a^4+4*(b^2-c^2)*(b-c)*(39*b^6+39*c^6+(54*b^4+54*c^4+b*c*(25*b^2-108*b*c+25*c^2))*b*c)*a^3-2*(b^2-c^2)^2*(15*b^6+15*c^6+(194*b^4+194*c^4+b*c*(169*b^2+524*b*c+169*c^2))*b*c)*a^2-4*(b^2-c^2)^2*(b+c)*(9*b^6+9*c^6-b^2*c^2*(49*b^2-16*b*c+49*c^2))*a+(b^2-c^2)^4*(b+3*c)^2*(3*b+c)^2) : :

The reciprocal cyclologic center of these triangles is X(22940).

X(22939) lies on these lines: {}

X(22940) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ATIK TO INNER-GARCIA

Barycentrics    a*(9*(b+c)*a^16-6*(b+c)^2*a^15-(b+c)*(102*b^2-205*b*c+102*c^2)*a^14+2*(69*b^4+69*c^4-(9*b^2+202*b*c+9*c^2)*b*c)*a^13+3*(b+c)*(106*b^4+106*c^4-7*(61*b^2-98*b*c+61*c^2)*b*c)*a^12-2*(303*b^6+303*c^6-(292*b^4+292*c^4+(871*b^2-1800*b*c+871*c^2)*b*c)*b*c)*a^11-(b+c)*(366*b^6+366*c^6-(3173*b^4+3173*c^4-2*(4179*b^2-5375*b*c+4179*c^2)*b*c)*b*c)*a^10+2*(585*b^8+585*c^8-(1035*b^6+1035*c^6+(1348*b^4+1348*c^4-(4307*b^2-4826*b*c+4307*c^2)*b*c)*b*c)*b*c)*a^9-(b+c)*(3985*b^6+3985*c^6-(14842*b^4+14842*c^4-(21727*b^2-22828*b*c+21727*c^2)*b*c)*b*c)*b*c*a^8-2*(585*b^10+585*c^10-(1610*b^8+1610*c^8+3*(297*b^6+297*c^6-2*(828*b^4+828*c^4-(605*b^2+298*b*c+605*c^2)*b*c)*b*c)*b*c)*b*c)*a^7+(b+c)*(366*b^10+366*c^10+(2655*b^8+2655*c^8-2*(6753*b^6+6753*c^6-(9234*b^4+9234*c^4-(9206*b^2-8509*b*c+9206*c^2)*b*c)*b*c)*b*c)*b*c)*a^6+2*(303*b^12+303*c^12-(1279*b^10+1279*c^10+(202*b^8+202*c^8-(4701*b^6+4701*c^6-(575*b^4+575*c^4+2*(1967*b^2+1062*b*c+1967*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^5-(b+c)*(318*b^12+318*c^12+(875*b^10+875*c^10-(6318*b^8+6318*c^8-(8335*b^6+8335*c^6-2*(1739*b^4+1739*c^4-(2563*b^2-6906*b*c+2563*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^4-2*(b^2-c^2)^2*(69*b^10+69*c^10-(504*b^8+504*c^8-(137*b^6+137*c^6+2*(832*b^4+832*c^4-(375*b^2-152*b*c+375*c^2)*b*c)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(102*b^8+102*c^8+(315*b^6+315*c^6-(598*b^4+598*c^4-b*c*(821*b^2+320*b*c+821*c^2))*b*c)*b*c)*a^2+2*(b^2-c^2)^4*(b-c)^2*(3*b^6+3*c^6-(71*b^4+71*c^4+3*b*c*(47*b^2+26*b*c+47*c^2))*b*c)*a+(b^2-c^2)^7*(b-c)*(-9*c^2-21*b*c-9*b^2)) : :

The reciprocal cyclologic center of these triangles is X(22939).

X(22940) lies on these lines: {}

X(22941) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO 3rd HATZIPOLAKIS

Barycentrics    2*a*(7*R^2*(8*R^2-3*SW)+2*SW^2)*(2*R*S-SB*b-SC*c)+(2*R^2*(3*S^2+(8*R^2+8*SA-7*SW)*SA)-SW*(S^2+3*SA^2-2*SA*SW))*(SB+SC) : :
X(22941) = 3*X(3679)-X(22653) = 3*X(5587)-2*X(22833) = 3*X(5657)-X(22533) = 3*X(5790)-X(22979)

The reciprocal orthologic center of these triangles is X(12241).

X(22941) lies on these lines: {1,22966}, {2,22476}, {8,22647}, {10,22466}, {65,775}, {72,22957}, {515,22951}, {517,22955}, {519,22969}, {956,22776}, {1737,22981}, {1837,22965}, {3057,22959}, {3679,22653}, {5090,22483}, {5252,18978}, {5587,22833}, {5657,22533}, {5687,22559}, {5688,22947}, {5689,22945}, {5790,22979}, {8193,22658}, {8214,22963}, {8215,22964}, {9857,22747}, {10039,22980}, {10791,22524}, {10914,22956}, {10915,22982}, {10916,22983}, {12699,22800}, {13883,19084}, {13893,22976}, {13936,19083}, {13947,22977}

X(22941) = midpoint of X(8) and X(22647)
X(22941) = reflection of X(i) in X(j) for these (i,j): (1, 22966), (12699, 22800)
X(22941) = anticomplement of X(22476)

X(22942) = CYCLOLOGIC CENTER OF THESE TRIANGLES: GARCIA-REFLECTION TO 5th MIXTILINEAR

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

The reciprocal cyclologic center of these triangles is X(1320).

X(22942) lies on these lines: {1,1145}, {57,3021}, {1280,11019}, {5563,6011}

X(22942) = inverse of X(3035) in the incircle

X(22943) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 3rd HATZIPOLAKIS

Barycentrics    (S^2-3*SB*SC)*(S^4-(R^2*(14*R^2+5*SA-8*SW)-SA^2+SB*SC+SW^2)*S^2-4608*R^8-32*(21*SA-130*SW)*R^6-2*(17*SA^2-197*SA*SW+694*SW^2)*R^4+(17*SA^2-77*SA*SW+203*SW^2)*SW*R^2-(2*SA^2-5*SA*SW+11*SW^2)*SW^2) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22943) lies on these lines: {30,22951}, {402,22466}, {1650,22966}, {4240,22647}, {11831,22476}, {11832,22483}, {11839,22524}, {11845,22533}, {11848,22559}, {11852,22653}, {11853,22658}, {11885,22747}, {11897,22833}, {11901,22945}, {11902,22947}, {11903,22956}, {11904,22957}, {11905,22958}, {11906,22959}, {11907,22963}, {11908,22964}, {11909,22965}, {11910,22969}, {11911,22979}, {11912,22980}, {11913,22981}, {11914,22982}, {11915,22983}, {13894,22976}, {13948,22977}, {18507,22800}, {18958,18978}, {19017,19083}, {19018,19084}, {22755,22776}

X(22943) = midpoint of X(4240) and X(22647)
X(22943) = reflection of X(i) in X(j) for these (i,j): (1650, 22966), (18507, 22800)

X(22944) = PERSPECTOR OF THESE TRIANGLES: INNER-GREBE AND 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(3*S^4-2*(26*R^2-SA-3*SW)*S^3+(4*R^2*(-8*SW-SA+112*R^2)+SA^2-2*SB*SC)*S^2+2*(4*R^2*(-SW+2*SA)+SA^2-SB*SC-2*SW^2)*SA*S-32*R^2*SB*SC*SW) : :

X(22944) lies on these lines: {1161,13630}, {12241,22945}

X(22945) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 3rd HATZIPOLAKIS

Barycentrics    6*(4*R^2-SW)*S^4+(R^2*(16*R^2-5*SA-3*SW)+SA*SW)*S^3-2*(4*R^2-SW)*(2*R^2*(7*SA-5*SW)-3*SA^2+3*SB*SC+2*SW^2)*S^2+(4*R^2-SW)*(16*R^2-3*SW)*SB*SC*S-2*(4*R^2-SW)*(16*R^2-3*SW)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(12241).

X(22945) lies on these lines: {6,17837}, {1271,22647}, {2929,8903}, {5589,22653}, {5591,22966}, {5595,22658}, {5605,22969}, {5689,22941}, {6202,22833}, {6215,22955}, {6218,22530}, {8216,22963}, {8217,22964}, {8974,22976}, {9994,22747}, {10040,22980}, {10048,22981}, {10783,22533}, {10792,22524}, {10919,22956}, {10921,22957}, {10923,22958}, {10925,22959}, {10927,22965}, {10929,22982}, {10931,22983}, {11370,22476}, {11388,22483}, {11497,22559}, {11824,22951}, {11901,22943}, {11916,22979}, {12241,22944}, {13949,22977}, {18509,22800}, {18959,18978}, {22756,22776}

X(22946) = PERSPECTOR OF THESE TRIANGLES: OUTER-GREBE AND 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(3*S^4+2*(26*R^2-SA-3*SW)*S^3+(4*R^2*(-8*SW-SA+112*R^2)+SA^2-2*SB*SC)*S^2-2*(4*R^2*(-SW+2*SA)+SA^2-SB*SC-2*SW^2)*SA*S-32*R^2*SB*SC*SW) : :

X(22946) lies on these lines: {1160,13630}, {12241,22947}

X(22947) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 3rd HATZIPOLAKIS

Barycentrics    6*(4*R^2-SW)*S^4-(R^2*(16*R^2-5*SA-3*SW)+SA*SW)*S^3-2*(4*R^2-SW)*(2*R^2*(7*SA-5*SW)-3*SA^2+3*SB*SC+2*SW^2)*S^2-(4*R^2-SW)*(16*R^2-3*SW)*SB*SC*S-2*(4*R^2-SW)*(16*R^2-3*SW)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(12241).

X(22947) lies on these lines: {6,17837}, {1270,22647}, {2929,8904}, {5588,22653}, {5590,22966}, {5594,22658}, {5604,22969}, {5688,22941}, {6201,22833}, {6214,22955}, {6217,22530}, {8218,22963}, {8219,22964}, {8975,22976}, {9995,22747}, {10041,22980}, {10049,22981}, {10784,22533}, {10793,22524}, {10920,22956}, {10922,22957}, {10924,22958}, {10926,22959}, {10928,22965}, {10930,22982}, {10932,22983}, {11371,22476}, {11389,22483}, {11498,22559}, {11825,22951}, {11902,22943}, {11917,22979}, {12241,22946}, {13950,22977}, {18511,22800}, {18960,18978}, {22757,22776}

X(22948) = PERSPECTOR OF THESE TRIANGLES: 3rd HATZIPOLAKIS AND ORTHIC

Barycentrics    SB*SC*(SB+SC)*(5*SA^2+S^2)*(SA+SW-10*R^2) : :
X(22948) = 3*X(6030)-X(15086) = 7*X(7999)-3*X(15103) = X(12111)-3*X(15062)

X(22948) lies on these lines: {4,3521}, {6,6241}, {24,8718}, {30,6152}, {52,12897}, {113,1594}, {155,378}, {389,13202}, {403,9729}, {974,22833}, {1154,13420}, {1493,2914}, {1593,2904}, {1843,6240}, {1885,1986}, {3520,11591}, {3574,6000}, {3575,11817}, {5890,5895}, {6030,15086}, {6102,16880}, {7576,10575}, {7729,18912}, {7999,15103}, {10594,15072}, {13431,13754}

X(22949) = PERSPECTOR OF THESE TRIANGLES: 3rd HATZIPOLAKIS AND ORTHOCENTROIDAL

Barycentrics    (SB+SC)*((3*R^2-2*SW)*S^2+(11*R^2-2*SW)*(44*R^2-5*SA-6*SW)*SA) : :

X(22949) lies on these lines: {74,195}, {381,6241}, {974,22971}, {1205,1992}, {2452,13489}, {5654,12281}, {7699,20299}, {11468,12163}, {17505,18394}

X(22950) = PERSPECTOR OF THESE TRIANGLES: 3rd HATZIPOLAKIS AND REFLECTION

Barycentrics    (SB+SC)*((61*R^2-10*SW)*S^2-(9*R^2-2*SW)*(36*R^2-7*SA-2*SW)*SA) : :
X(22950) = 3*X(11455)-4*X(13603)

X(22950) lies on these lines: {6,11455}, {54,22972}, {382,3567}, {399,11702}, {1173,15084}, {5888,7514}, {5890,11807}, {12281,16176}, {13403,13423}

X(22951) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 3rd HATZIPOLAKIS

Barycentrics    (R^2*(96*R^2+5*SA-39*SW)-SA*SW+4*SW^2)*S^2-(2*R^2*(88*R^2-35*SW)+7*SW^2)*SB*SC : :
X(22951) = 3*X(3)-X(22979) = 3*X(165)-X(22653) = 3*X(376)-X(22533) = 3*X(3576)-2*X(22476) = 3*X(22466)-2*X(22979)

The reciprocal orthologic center of these triangles is X(12241).

X(22951) lies on these lines: {2,22833}, {3,2929}, {4,22966}, {20,22647}, {30,22943}, {35,22980}, {36,22981}, {55,18978}, {56,22965}, {110,2883}, {165,22653}, {182,22524}, {371,19084}, {372,19083}, {376,22533}, {382,22800}, {515,22941}, {517,22969}, {1593,22483}, {3098,22747}, {3428,22776}, {3576,22476}, {4549,10627}, {6284,22959}, {7354,22958}, {7691,16386}, {9540,22976}, {9627,19472}, {10310,22559}, {10575,12121}, {11248,22982}, {11249,22983}, {11414,22658}, {11824,22945}, {11825,22947}, {11826,22956}, {11827,22957}, {11828,22963}, {11829,22964}, {12118,13491}, {13935,22977}, {15644,18442}, {17818,22953}, {18560,22750}

X(22951) = midpoint of X(20) and X(22647)
X(22951) = reflection of X(i) in X(j) for these (i,j): (4, 22966), (382, 22800)
X(22951) = anticomplement of X(22833)

X(22952) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HYACINTH TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(2*(5*R^2-SW)*(6*R^2-SW)*S^2+(2*R^4*(136*R^2-4*SA-79*SW)+R^2*SW*(31*SW+SA)-2*SW^3)*SA) : :

The reciprocal orthologic center of these triangles is X(19481).

X(22952) lies on these lines: {389,6677}, {1147,2929}, {1181,21652}, {1493,11802}, {6102,11557}, {11536,22529}, {11806,12897}, {15120,19511}, {15134,19480}

X(22953) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd HYACINTH TO 3rd HATZIPOLAKIS

Barycentrics    (4*(5*R^2-SW)*(6*R^2-SW)*S^2+(SB+SC)*(8*R^4*(16*R^2-5*SA-17*SW)+R^2*SW*(37*SW+13*SA)-SW^2*(3*SW+SA)))*SA : :

The reciprocal orthologic center of these triangles is X(22663).

X(22953) lies on these lines: {2,22533}, {6,17837}, {25,22662}, {155,22808}, {185,10112}, {974,22663}, {1092,22834}, {10116,10938}, {17818,22951}, {18936,22647}, {22483,22530}

X(22953) = reflection of X(22483) in X(22530)

X(22954) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTANGENTS TO 3rd HATZIPOLAKIS

Barycentrics    a*(4*(7*(8*R^2-3*SW)*R^2+2*SW^2)*S^2+((8*R^2+SA-2*SW)*S^2-(24*R^2-5*SW)*SB*SC)*b*c) : :

The reciprocal orthologic center of these triangles is X(9729).

X(22954) lies on these lines: {1,18978}, {33,22970}, {34,22538}, {35,22962}, {36,22978}, {55,2929}, {56,22549}, {497,22555}, {1040,22581}, {1062,22834}, {1250,22975}, {2066,22960}, {2192,17837}, {2330,19142}, {3100,22528}, {3270,21652}, {3583,22816}, {5414,22961}, {6198,22750}, {6284,19505}, {7071,22497}, {8540,22830}, {9627,22466}, {9817,22973}, {10638,22974}, {10895,22971}, {11429,22529}, {11436,22530}, {11446,22534}, {11461,22535}, {12888,18970}, {18455,22808}, {18922,18936}, {19182,19198}, {19354,19460}, {19434,19488}, {19435,19489}

X(22955) = ORTHOLOGIC CENTER OF THESE TRIANGLES: JOHNSON TO 3rd HATZIPOLAKIS

Barycentrics    ((6*R^2-3*SA-SW)*S^2+(2*R^2*(8*R^2+8*SA-7*SW)-3*SA^2+3*SB*SC+2*SW^2)*SA)*(SB+SC) : :
X(22955) = 3*X(381)-2*X(22833) = 3*X(381)-X(22979) = 3*X(5587)-X(22653) = 3*X(5886)-2*X(22476) = X(22647)+2*X(22800)

The reciprocal orthologic center of these triangles is X(12241).

X(22955) lies on these lines: {1,22958}, {2,22533}, {3,22658}, {4,801}, {5,5504}, {11,22981}, {12,22980}, {30,22943}, {49,8550}, {54,5972}, {110,185}, {155,2929}, {355,22956}, {381,22833}, {517,22941}, {578,22973}, {952,22969}, {1147,22529}, {1351,7506}, {1352,3548}, {1368,22662}, {1478,18978}, {1479,22965}, {2071,5907}, {2072,6288}, {3292,15801}, {5587,22653}, {5878,10539}, {5886,22476}, {6214,22947}, {6215,22945}, {6642,22530}, {7583,19084}, {7584,19083}, {7728,18350}, {8220,22963}, {8221,22964}, {8976,22976}, {9970,22828}, {9996,22747}, {10796,22524}, {10942,22982}, {10943,22983}, {11472,12084}, {11487,22581}, {11499,22559}, {12421,16238}, {13352,22968}, {13951,22977}, {22758,22776}

X(22955) = midpoint of X(4) and X(22647)
X(22955) = reflection of X(i) in X(j) for these (i,j): (3, 22966), (4, 22800)
X(22955) = complement of X(22533)
X(22955) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 22979, 22833), (22958, 22959, 1)

X(22956) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 3rd HATZIPOLAKIS

Barycentrics    a*(a^18-(b+c)*a^17-(5*b^2-6*b*c+5*c^2)*a^16+(b+c)*(5*b^2-4*b*c+5*c^2)*a^15+(9*b^4+9*c^4-(20*b^2-27*b*c+20*c^2)*b*c)*a^14-(b+c)*(9*b^4+9*c^4-(14*b^2-27*b*c+14*c^2)*b*c)*a^13-(5*b^6+5*c^6-(16*b^4+16*c^4-(49*b^2-86*b*c+49*c^2)*b*c)*b*c)*a^12+(b+c)*(5*b^6+5*c^6-(12*b^4+12*c^4-(49*b^2-60*b*c+49*c^2)*b*c)*b*c)*a^11-(5*b^8+5*c^8-(12*b^6+12*c^6+(29*b^4+29*c^4-2*(53*b^2-54*b*c+53*c^2)*b*c)*b*c)*b*c)*a^10+(b+c)*(5*b^8+5*c^8-(10*b^6+10*c^6+(29*b^4+29*c^4-4*(20*b^2-27*b*c+20*c^2)*b*c)*b*c)*b*c)*a^9+(9*b^10+9*c^10-(20*b^8+20*c^8-(9*b^6+9*c^6+2*(2*b^4+2*c^4-(41*b^2-96*b*c+41*c^2)*b*c)*b*c)*b*c)*b*c)*a^8-(b^2-c^2)*(b-c)*(9*b^8+9*c^8-2*(b^6+c^6+(2*b^4+2*c^4-(5*b^2-29*b*c+5*c^2)*b*c)*b*c)*b*c)*a^7-(5*b^10+5*c^10+(6*b^8+6*c^8+(18*b^6+18*c^6-(26*b^4+26*c^4+(55*b^2-8*b*c+55*c^2)*b*c)*b*c)*b*c)*b*c)*(b-c)^2*a^6+(b^2-c^2)*(b-c)*(5*b^10+5*c^10+(4*b^8+4*c^8+(14*b^6+14*c^6-(18*b^4+18*c^4+7*(5*b^2-4*b*c+5*c^2)*b*c)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)^2*(b-c)^2*(b^8+c^8+2*(b^6+c^6+(b^4+15*b^2*c^2+c^4)*b*c)*b*c)*a^4-(b^2-c^2)^3*(b-c)*(b^8+c^8+2*(3*b^6+3*c^6+(5*b^4+5*c^4-(3*b^2-5*b*c+3*c^2)*b*c)*b*c)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(4*b^2+11*b*c+4*c^2)*b*c*a^2+(b^2-c^2)^5*(b-c)*(b^2+c^2)*(2*b^2+b*c+2*c^2)*b*c*a-2*(b^2-c^2)^6*(b^2+c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22956) lies on these lines: {11,22466}, {12,22982}, {355,22955}, {1376,22559}, {3434,22647}, {10523,22980}, {10785,22533}, {10794,22524}, {10826,22653}, {10829,22658}, {10871,22747}, {10893,22833}, {10914,22941}, {10919,22945}, {10920,22947}, {10944,22958}, {10945,22963}, {10946,22964}, {10947,22965}, {10948,22981}, {10949,22983}, {11373,22476}, {11390,22483}, {11826,22951}, {11903,22943}, {11928,22979}, {12114,22776}, {13895,22976}, {13952,22977}, {18516,22800}, {18961,18978}, {19023,19083}, {19024,19084}

X(22957) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 3rd HATZIPOLAKIS

Barycentrics    a*(a^18-(b+c)*a^17-(5*b^2+2*b*c+5*c^2)*a^16+5*(b+c)*(b^2+c^2)*a^15+(9*b^4+9*c^4+(4*b^2+19*b*c+4*c^2)*b*c)*a^14-(b+c)*(9*b^4+9*c^4-(2*b^2-19*b*c+2*c^2)*b*c)*a^13-(5*b^6+5*c^6-(4*b^4+4*c^4-(33*b^2+26*b*c+33*c^2)*b*c)*b*c)*a^12+(b+c)*(5*b^6+5*c^6-(8*b^4+8*c^4-33*(b^2+c^2)*b*c)*b*c)*a^11-(5*b^8+5*c^8+(12*b^6+12*c^6-(37*b^4+37*c^4+2*(11*b^2+10*b*c+11*c^2)*b*c)*b*c)*b*c)*a^10+(b+c)*(5*b^8+5*c^8+(10*b^6+10*c^6-(37*b^4+37*c^4-4*(b^2-5*b*c+c^2)*b*c)*b*c)*b*c)*a^9+(9*b^10+9*c^10-(23*b^6+23*c^6-2*(22*b^4+22*c^4-(5*b^2+52*b*c+5*c^2)*b*c)*b*c)*b^2*c^2)*a^8-(b+c)*(9*b^10+9*c^10-(23*b^6+23*c^6-2*(16*b^4+16*c^4-(5*b^2+24*b*c+5*c^2)*b*c)*b*c)*b^2*c^2)*a^7-(b^2-c^2)^2*(5*b^8+5*c^8-(12*b^6+12*c^6-(13*b^4+13*c^4+4*(8*b^2-9*b*c+8*c^2)*b*c)*b*c)*b*c)*a^6+(b^2-c^2)^2*(b+c)*(5*b^8+5*c^8-(10*b^6+10*c^6-(13*b^4+13*c^4+2*(11*b^2-18*b*c+11*c^2)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)^2*(b-c)^4*(b^6+c^6+(9*b^2+22*b*c+9*c^2)*b^2*c^2)*a^4-(b^2-c^2)^3*(b-c)*(b^8+c^8-2*(3*b^6+3*c^6-(b^4+c^4+(5*b^2-3*b*c+5*c^2)*b*c)*b*c)*b*c)*a^3-(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(4*b^2+13*b*c+4*c^2)*b*c*a^2-(b^2-c^2)^5*(b-c)*(b^2+c^2)*(2*b^2-b*c+2*c^2)*b*c*a+2*(b^2-c^2)^6*(b^2+c^2)^2*b*c) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22957) lies on these lines: {11,22983}, {12,22466}, {72,22941}, {355,22955}, {958,22776}, {3436,22647}, {10523,22981}, {10786,22533}, {10795,22524}, {10827,22653}, {10830,22658}, {10872,22747}, {10894,22833}, {10921,22945}, {10922,22947}, {10950,22959}, {10951,22963}, {10952,22964}, {10953,22965}, {10954,22980}, {10955,22982}, {11374,22476}, {11391,22483}, {11500,22559}, {11827,22951}, {11904,22943}, {11929,22979}, {13896,22976}, {13953,22977}, {18517,22800}, {18962,18978}, {19025,19083}, {19026,19084}

X(22958) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(2*(8*R^2+8*SA-7*SW)*R^2*SA+(6*R^2-SW)*S^2-(3*SA-2*SW)*SA*SW+2*(7*R^2*(8*R^2-3*SW)+2*SW^2)*b*c) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22958) lies on these lines: {1,22955}, {4,22965}, {12,22466}, {56,22966}, {65,775}, {388,18978}, {495,22980}, {1479,22800}, {3085,22533}, {6198,22750}, {7354,22951}, {9578,22653}, {9654,22979}, {10797,22524}, {10831,22658}, {10873,22747}, {10895,22833}, {10923,22945}, {10924,22947}, {10944,22956}, {10956,22982}, {10957,22983}, {11375,22476}, {11392,22483}, {11501,22559}, {11905,22943}, {11930,22963}, {11931,22964}, {13897,22976}, {13954,22977}, {19027,19083}, {19028,19084}, {22759,22776}

X(22958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22955, 22959), (388, 22647, 18978)

X(22959) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 3rd HATZIPOLAKIS

Barycentrics    ((6*R^2-SW)*S^2-2*b*c*(2*SW^2+56*R^4-21*R^2*SW)+2*R^2*SA*(8*R^2+8*SA-7*SW)-SA*SW*(-2*SW+3*SA))*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22959) lies on these lines: {1,22955}, {4,18978}, {5,22980}, {11,22466}, {55,22966}, {496,22981}, {497,22647}, {1478,22800}, {1870,19472}, {3057,22941}, {6284,22951}, {9581,22653}, {9669,22979}, {10798,22524}, {10832,22658}, {10874,22747}, {10896,22833}, {10925,22945}, {10926,22947}, {10950,22957}, {10958,22982}, {10959,22983}, {11376,22476}, {11393,22483}, {11502,22559}, {11906,22943}, {11932,22963}, {11933,22964}, {13898,22976}, {13955,22977}, {19029,19083}, {19030,19084}

X(22959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22955, 22958), (497, 22647, 22965)

X(22960) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS TO 3rd HATZIPOLAKIS

Barycentrics    ((5*R^2-SW)*S^2+(7*R^2*(8*R^2-3*SW)+2*SW^2)*S+(4*R^2-SW)*(8*R^2+SA-2*SW)*SA)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(9729).

X(22960) lies on these lines: {6,2929}, {372,22962}, {1151,22549}, {2066,22954}, {2067,19472}, {3068,22555}, {3311,22550}, {5410,22497}, {5412,22970}, {5415,22840}, {6200,22978}, {6413,22466}, {6564,22816}, {10880,22750}, {10897,22834}, {10961,22973}, {11417,22528}, {11447,22534}, {11462,22535}, {11473,22538}, {11513,22581}, {17819,17837}, {18457,22808}, {18923,18936}, {19183,19198}, {19355,19460}, {19436,19488}, {19438,19489}, {21640,21652}

X(22960) = {X(6), X(2929)}-harmonic conjugate of X(22961)

X(22961) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS TO 3rd HATZIPOLAKIS

Barycentrics    ((5*R^2-SW)*S^2-(7*R^2*(8*R^2-3*SW)+2*SW^2)*S+(4*R^2-SW)*(8*R^2+SA-2*SW)*SA)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(9729).

X(22961) lies on these lines: {6,2929}, {371,22962}, {1152,22549}, {3069,22555}, {3312,22550}, {5411,22497}, {5413,22970}, {5414,22954}, {5416,22840}, {6396,22978}, {6414,22466}, {6502,19472}, {6565,22816}, {10881,22750}, {10898,22834}, {10963,22973}, {11418,22528}, {11448,22534}, {11463,22535}, {11474,22538}, {11514,22581}, {17820,17837}, {18459,22808}, {18924,18936}, {19184,19198}, {19356,19460}, {19437,19489}, {19439,19488}, {21641,21652}

X(22961) = {X(6), X(2929)}-harmonic conjugate of X(22960)

X(22962) = ORTHOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 3rd HATZIPOLAKIS

Barycentrics    ((5*R^2-SA-SW)*S^2+(R^2*(88*R^2+4*SA-37*SW)-SA^2+SB*SC+4*SW^2)*SA)*(SB+SC) : :
X(22962) = 3*X(3)-X(22549) = 3*X(3)+X(22550) = X(382)-3*X(22971) = 5*X(631)-X(22555) = 3*X(2929)+X(22549) = 3*X(2929)-X(22550) = 2*X(2929)+X(22978) = 5*X(17821)-X(17837) = 2*X(22549)-3*X(22978) = 2*X(22550)+3*X(22978)

The reciprocal orthologic center of these triangles is X(9729).

X(22962) lies on these lines: {3,2929}, {5,13293}, {15,22975}, {16,22974}, {24,1533}, {35,22954}, {36,19472}, {54,9729}, {110,185}, {113,3521}, {186,8718}, {371,22961}, {372,22960}, {378,22538}, {382,22971}, {389,22529}, {511,19142}, {575,22830}, {578,22530}, {631,22555}, {1147,13630}, {1658,8717}, {3515,22497}, {6642,18418}, {6644,22800}, {6723,14130}, {6759,13491}, {7488,22528}, {8907,15078}, {10902,22840}, {11250,22833}, {11449,22534}, {11464,22535}, {11702,14708}, {12084,22968}, {12584,22828}, {13367,21652}, {17821,17837}, {18912,22647}, {18925,18936}, {19185,19198}, {19357,19460}, {19440,19488}, {19441,19489}, {19467,22533}

X(22962) = midpoint of X(3) and X(2929)
X(22962) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22550, 22549), (2929, 22549, 22550)

X(22963) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 3rd HATZIPOLAKIS

Barycentrics    (16*R^4*(7*SA^2-6*SW^2)-R^2*SW*(42*SA^2+5*SA*SW-39*SW^2)+SW^2*(4*SA^2+SA*SW-4*SW^2))*S^2+(4*R^2-SW)*(16*R^2-3*SW)*SW^2*SB*SC-2*(-4*(4*R^2-SW)*(16*R^2-3*SW)*R^2*SB*SC+(4*R^4*(40*R^2-9*SA-4*SW)+R^2*SW*(-13*SW+17*SA)-2*(SA-SW)*SW^2)*S^2)*S : :

The reciprocal orthologic center of these triangles is X(12241).

X(22963) lies on these lines: {493,22466}, {6461,22964}, {6462,22647}, {8188,22653}, {8194,22658}, {8210,22969}, {8212,22833}, {8214,22941}, {8216,22945}, {8218,22947}, {8220,22955}, {8222,22966}, {10875,22747}, {10945,22956}, {10951,22957}, {11377,22476}, {11394,22483}, {11503,22559}, {11828,22951}, {11840,22524}, {11846,22533}, {11907,22943}, {11930,22958}, {11932,22959}, {11947,22965}, {11949,22979}, {11951,22980}, {11953,22981}, {11955,22982}, {11957,22983}, {13899,22976}, {13956,22977}, {18520,22800}, {18963,18978}, {19031,19083}, {19032,19084}, {22761,22776}

X(22964) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 3rd HATZIPOLAKIS

Barycentrics    (16*R^4*(7*SA^2-6*SW^2)-R^2*SW*(42*SA^2+5*SA*SW-39*SW^2)+SW^2*(4*SA^2+SA*SW-4*SW^2))*S^2+(4*R^2-SW)*(16*R^2-3*SW)*SW^2*SB*SC+2*(-4*(4*R^2-SW)*(16*R^2-3*SW)*R^2*SB*SC+(4*R^4*(40*R^2-9*SA-4*SW)+R^2*SW*(-13*SW+17*SA)-2*(SA-SW)*SW^2)*S^2)*S : :

The reciprocal orthologic center of these triangles is X(12241).

X(22964) lies on these lines: {494,22466}, {6461,22963}, {6463,22647}, {8189,22653}, {8195,22658}, {8211,22969}, {8213,22833}, {8215,22941}, {8217,22945}, {8219,22947}, {8221,22955}, {8223,22966}, {10876,22747}, {10946,22956}, {10952,22957}, {11378,22476}, {11395,22483}, {11504,22559}, {11829,22951}, {11841,22524}, {11847,22533}, {11908,22943}, {11931,22958}, {11933,22959}, {11948,22965}, {11950,22979}, {11952,22980}, {11954,22981}, {11956,22982}, {11958,22983}, {13900,22976}, {13957,22977}, {18522,22800}, {18964,18978}, {19033,19083}, {19034,19084}, {22762,22776}

X(22965) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 3rd HATZIPOLAKIS

Barycentrics    (5*R^2-SW)*(16*R^2-SA-3*SW)*S^2-(SB+SC)*((56*R^4-21*R^2*SW+2*SW^2)*b*c+(24*R^2-5*SW)*(5*R^2-SW)*SA) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22965) lies on these lines: {1,18978}, {3,22981}, {4,22958}, {11,22966}, {12,22833}, {33,22483}, {56,22951}, {497,22647}, {1697,22653}, {1837,22941}, {2098,22969}, {2646,22476}, {3295,22979}, {3583,22800}, {4294,22533}, {10799,22524}, {10833,22658}, {10877,22747}, {10927,22945}, {10928,22947}, {10947,22956}, {10953,22957}, {10965,22982}, {10966,22776}, {11909,22943}, {11947,22963}, {11948,22964}, {13901,22976}, {13958,22977}, {19037,19083}, {19038,19084}

X(22965) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 22647, 22959), (3295, 22979, 22980)

X(22966) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO 3rd HATZIPOLAKIS

Barycentrics    (SA*(16*R^2+SA-4*SW)+S^2)*(SA-3*SW+16*R^2)*(SB+SC) : :
X(22966) = 3*X(2)+X(22647) = 5*X(631)-X(22533) = 5*X(1656)-X(22979) = 5*X(1698)-X(22653)

The reciprocal orthologic center of these triangles is X(12241).

X(22966) lies on these lines: {1,22941}, {2,22466}, {3,22658}, {4,22951}, {5,12897}, {6,2929}, {8,22969}, {11,22965}, {12,18978}, {30,22800}, {55,22959}, {56,22958}, {83,22524}, {113,5893}, {141,22581}, {378,22549}, {427,22483}, {498,22980}, {499,22981}, {590,22976}, {615,22977}, {631,22533}, {958,22776}, {1125,22476}, {1147,13630}, {1181,9705}, {1209,10257}, {1376,22559}, {1493,11802}, {1650,22943}, {1656,22979}, {1698,22653}, {1885,22970}, {3068,19084}, {3069,19083}, {3096,22747}, {4550,11250}, {5181,13367}, {5552,22982}, {5590,22947}, {5591,22945}, {5907,11598}, {6640,18466}, {8222,22963}, {8223,22964}, {8542,22828}, {10151,22538}, {10527,22983}, {11449,15748}, {17811,22528}, {18418,22816}

X(22966) = midpoint of X(i) and X(j) for these {i,j}: {1, 22941}, {3, 22955}, {4, 22951}, {8, 22969}, {1650, 22943}
X(22966) = complement of X(22466)
X(22966) = {X(2), X(22647)}-harmonic conjugate of X(22466)

X(22967) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd HATZIPOLAKIS TO MIDHEIGHT

Barycentrics    (SA*(-3*SW+16*R^2)*(8*R^2-SW-SA)-2*S^2*R^2)*(SB+SC) : :
X(22967) = X(64)+3*X(185) = 5*X(64)+3*X(6293) = X(64)-9*X(7729) = 5*X(185)-X(6293) = X(185)+3*X(7729) = X(5925)+3*X(14831) = X(6293)+15*X(7729)

The reciprocal orthologic center of these triangles is X(22968).

X(22967) lies on these lines: {4,15887}, {6,64}, {546,5462}, {974,22968}, {3357,12161}, {3629,5894}, {5925,14831}, {6241,15011}, {6247,18388}, {6644,6759}, {7529,12315}, {9729,22973}, {11250,13754}, {11381,15010}

X(22968) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MIDHEIGHT TO 3rd HATZIPOLAKIS

Barycentrics    S^4+(2*R^2*(32*R^2-3*SA-11*SW)+SA^2-SB*SC+2*SW^2)*S^2+(16*R^2-3*SW)^2*SB*SC : :
X(22968) = X(22466)+3*X(22971) = 3*X(22466)+X(22972) = X(22970)-3*X(22971) = 3*X(22970)-X(22972) = 9*X(22971)-X(22972)

The reciprocal orthologic center of these triangles is X(22967).

X(22968) lies on these lines: {5,12897}, {6,17837}, {389,5893}, {546,12235}, {974,22967}, {1514,15887}, {1593,2929}, {5448,18418}, {7687,13488}, {10110,12236}, {10151,22530}, {11431,22533}, {12084,22962}, {13352,22955}, {15033,22750}, {15435,22555}

X(22968) = {X(22466), X(22971)}-harmonic conjugate of X(22970)

X(22969) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO 3rd HATZIPOLAKIS

Barycentrics    3*a^16-2*(b+c)*a^15-(9*b^2-4*b*c+9*c^2)*a^14+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^13+(4*b^4+4*c^4-(8*b^2-43*b*c+8*c^2)*b*c)*a^12-2*(b+c)*(b^4+c^4-(4*b^2-15*b*c+4*c^2)*b*c)*a^11+(11*b^6+11*c^6-(4*b^4+4*c^4+(51*b^2-44*b*c+51*c^2)*b*c)*b*c)*a^10-2*(b+c)*(5*b^6+5*c^6-(2*b^4+2*c^4+(19*b^2-22*b*c+19*c^2)*b*c)*b*c)*a^9-2*(5*b^8+5*c^8-(8*b^6+8*c^6-(7*b^4+7*c^4+6*(3*b-c)*(b-3*c)*b*c)*b*c)*b*c)*a^8+2*(b+c)*(5*b^8+5*c^8-2*(4*b^6+4*c^6-(2*b^4+2*c^4+(9*b^2-23*b*c+9*c^2)*b*c)*b*c)*b*c)*a^7-(3*b^8+3*c^8+2*(5*b^6+5*c^6-(16*b^4+16*c^4+(19*b^2-9*b*c+19*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^6+2*(b^2-c^2)*(b-c)*(b^8+c^8+2*(2*b^6+2*c^6-7*(b^4+c^4+(b^2-b*c+c^2)*b*c)*b*c)*b*c)*a^5+(b^2-c^2)^2*(b-c)^2*(4*b^6+4*c^6-(5*b^2-18*b*c+5*c^2)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)^2*(b-c)*(3*b^4+3*c^4+(2*b^2-7*b*c+2*c^2)*b*c)*a^3+(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(b^2+6*b*c+c^2)*a^2+2*(b^2-c^2)^5*(b-c)*(b^2+c^2)^2*a-(b^2+c^2)^2*(b^2-c^2)^6 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22969) lies on these lines: {1,22466}, {8,22966}, {55,22776}, {56,22559}, {145,22647}, {517,22951}, {519,22941}, {952,22955}, {2098,22965}, {2099,18978}, {5603,22833}, {5604,22947}, {5605,22945}, {7967,22533}, {7968,19083}, {7969,19084}, {8192,22658}, {8210,22963}, {8211,22964}, {9997,22747}, {10247,22979}, {10800,22524}, {10944,22956}, {10950,22957}, {11396,22483}, {11910,22943}, {13902,22976}, {13959,22977}, {18525,22800}

X(22969) = midpoint of X(145) and X(22647)
X(22969) = reflection of X(i) in X(j) for these (i,j): (8, 22966), (18525, 22800)
X(22969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22653, 22476), (22476, 22653, 22466), (22982, 22983, 22466)

X(22970) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHIC TO 3rd HATZIPOLAKIS

Barycentrics    SC*SB*(SA+SW-8*R^2)*(SA-3*SW+16*R^2) : :
X(22970) = 3*X(2)-4*X(22973) = 3*X(51)-X(21652) = 3*X(51)-2*X(22530) = 3*X(381)-X(22808) = 3*X(3060)+X(22534) = 5*X(3567)-X(22535) = X(22466)-3*X(22971) = X(22528)-4*X(22973) = X(22538)+2*X(22750) = 2*X(22968)-3*X(22971) = 2*X(22968)+X(22972) = 3*X(22971)+X(22972)

The reciprocal orthologic center of these triangles is X(9729).

X(22970) lies on these lines: {2,22528}, {4,801}, {5,22834}, {6,17837}, {19,22840}, {24,1533}, {25,2929}, {33,22954}, {34,19472}, {51,21652}, {52,1596}, {184,22529}, {185,235}, {193,11470}, {275,19198}, {378,22978}, {381,22808}, {403,9729}, {1425,1858}, {1593,22549}, {1598,22550}, {1660,21659}, {1843,1906}, {1885,22966}, {1974,19142}, {1986,15063}, {3060,22534}, {3089,5878}, {3567,22535}, {3574,10151}, {3575,13202}, {5412,22960}, {5413,22961}, {5448,22979}, {6622,15740}, {6623,22533}, {8541,22830}, {10019,16622}, {10564,13488}, {10641,22974}, {10642,22975}, {11433,18936}, {19446,19488}, {19447,19489}

X(22970) = midpoint of X(4) and X(22750)
X(22970) = anticomplement of X(22581)
X(22970) = complement of X(22528)
X(22970) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22528, 22581), (6, 17837, 19460), (6, 22972, 17837), (25, 22497, 2929), (51, 21652, 22530), (22466, 22971, 22968), (22581, 22973, 2), (22971, 22972, 22466)

X(22971) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHOCENTROIDAL TO 3rd HATZIPOLAKIS

Barycentrics    (R^2*(48*R^2-SA-19*SW)+2*SW^2)*S^2+2*(16*R^2-3*SW)*(6*R^2-SW)*SB*SC : :
X(22971) = 2*X(4)+X(2929) = X(22466)-4*X(22968) = X(22466)+2*X(22970) = 2*X(22466)+X(22972) = X(22750)+2*X(22833) = 2*X(22968)+X(22970) = 8*X(22968)+X(22972) = 4*X(22970)-X(22972)

The reciprocal orthologic center of these triangles is X(974).

X(22971) lies on these lines: {4,2929}, {6,17837}, {113,195}, {974,22949}, {7699,22750}, {9786,22802}, {10895,22954}, {10896,19472}, {11064,22647}, {13352,22979}

X(22971) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22466, 22970, 22972), (22968, 22970, 22466)

X(22972) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION TO 3rd HATZIPOLAKIS

Barycentrics    ((11*R^2-4*SA-2*SW)*S^2-4*(R^2*(16*R^2-3*SW-5*SA)+SA^2-SB*SC)*SA)*(SB+SC) : :
X(22972) = 3*X(154)-2*X(22658) = 3*X(22466)-4*X(22968) = 2*X(22466)-3*X(22971) = 2*X(22968)-3*X(22970) = 8*X(22968)-9*X(22971) = 4*X(22970)-3*X(22971)

The reciprocal orthologic center of these triangles is X(54).

X(22972) lies on these lines: {6,17837}, {24,1192}, {54,22950}, {64,21650}, {154,22497}, {382,13419}, {394,22647}, {1181,22533}, {6759,10938}, {7074,22559}, {9512,17703}, {11472,12084}, {11807,12316}, {12308,18378}, {17810,21652}, {17811,22528}, {20806,22555}, {22800,22808}

X(22972) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17837, 22970, 6), (22466, 22970, 22971)

X(22973) = ORTHOLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO 3rd HATZIPOLAKIS

Barycentrics    S^4-(4*R^2*(64*R^2+SA-25*SW)-SA^2+SB*SC+10*SW^2)*S^2-(16*R^2-3*SW)*(8*R^2-SW)*SB*SC : :
X(22973) = 9*X(2)-X(22528) = 3*X(2)+X(22970) = 9*X(373)-X(21652) = 5*X(1656)-X(22834) = 3*X(5943)-X(22530) = 15*X(11451)+X(22534) = 17*X(11465)-X(22535) = X(22528)-3*X(22581) = X(22528)+3*X(22970)

The reciprocal orthologic center of these triangles is X(9729).

X(22973) lies on these lines: {2,22528}, {5,12897}, {373,21652}, {542,15119}, {578,22955}, {1656,22834}, {2929,5020}, {4846,22800}, {5943,22530}, {9306,22529}, {9729,22967}, {9813,22830}, {9815,18388}, {9817,22954}, {9826,15012}, {10601,19460}, {10643,22974}, {10644,22975}, {10961,22960}, {10963,22961}, {11451,22534}, {11465,22535}, {17825,17837}, {18928,18936}, {19137,19142}, {19188,19198}, {19372,19472}, {19448,19488}, {19449,19489}

X(22973) = complement of X(22581)
X(22973) = {X(2), X(22970)}-harmonic conjugate of X(22581)

X(22974) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(3*(5*R^2-SW)*S^2+sqrt(3)*(7*R^2*(8*R^2-3*SW)+2*SW^2)*S+3*(8*R^2+SA-2*SW)*(4*R^2-SW)*SA) : :

The reciprocal orthologic center of these triangles is X(9729).

X(22974) lies on these lines: {6,2929}, {16,22962}, {7051,19472}, {10632,22750}, {10634,22834}, {10636,22840}, {10638,22954}, {10641,22970}, {10643,22973}, {10645,22978}, {11408,22497}, {11420,22528}, {11452,22534}, {11466,22535}, {11475,22538}, {11480,22549}, {11485,22550}, {11488,22555}, {11515,22581}, {16808,22816}, {17826,17837}, {18468,22808}, {18929,18936}, {19190,19198}, {19363,19460}, {19450,19488}, {19451,19489}, {21647,21652}

X(22974) = {X(6), X(2929)}-harmonic conjugate of X(22975)

X(22975) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO 3rd HATZIPOLAKIS

Barycentrics    (SB+SC)*(3*(5*R^2-SW)*S^2-sqrt(3)*(7*R^2*(8*R^2-3*SW)+2*SW^2)*S+3*(8*R^2+SA-2*SW)*(4*R^2-SW)*SA) : :

The reciprocal orthologic center of these triangles is X(9729).

X(22975) lies on these lines: {6,2929}, {15,22962}, {1250,22954}, {10633,22750}, {10635,22834}, {10637,22840}, {10642,22970}, {10644,22973}, {10646,22978}, {11409,22497}, {11421,22528}, {11453,22534}, {11467,22535}, {11476,22538}, {11481,22549}, {11486,22550}, {11489,22555}, {11516,22581}, {16809,22816}, {17827,17837}, {18470,22808}, {18930,18936}, {19191,19198}, {19364,19460}, {19373,19472}, {19452,19488}, {19453,19489}, {21648,21652}

X(22975) = {X(6), X(2929)}-harmonic conjugate of X(22974)

X(22976) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 3rd HATZIPOLAKIS

Barycentrics    S^4+(R^2*(128*R^2-5*SA-45*SW)+SA^2-SB*SC+4*SW^2)*S^2+2*(7*R^2*(8*R^2-3*SW)+2*SW^2)*(SB+SC)*S+(4*R^2-SW)*(16*R^2-3*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(12241).

X(22976) lies on these lines: {2,19084}, {6,22977}, {371,22833}, {590,22966}, {7585,19083}, {8972,22647}, {8974,22945}, {8975,22947}, {8976,22955}, {9540,22951}, {13883,22476}, {13884,22483}, {13885,22524}, {13886,22533}, {13887,22559}, {13888,22653}, {13889,22658}, {13892,22747}, {13893,22941}, {13894,22943}, {13895,22956}, {13896,22957}, {13898,22959}, {13899,22963}, {13900,22964}, {13901,22965}, {13902,22969}, {13903,22979}, {13904,22980}, {13905,22981}, {13906,22982}, {13907,22983}, {18538,22800}, {18965,18978}, {22763,22776}

X(22977) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 3rd HATZIPOLAKIS

Barycentrics    S^4+(R^2*(128*R^2-5*SA-45*SW)+SA^2-SB*SC+4*SW^2)*S^2-2*(7*R^2*(8*R^2-3*SW)+2*SW^2)*(SB+SC)*S+(4*R^2-SW)*(16*R^2-3*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(12241).

X(22977) lies on these lines: {2,19083}, {6,22976}, {372,22833}, {615,22966}, {3069,22466}, {7586,19084}, {13935,22951}, {13936,22476}, {13937,22483}, {13938,22524}, {13939,22533}, {13940,22559}, {13941,22647}, {13942,22653}, {13943,22658}, {13946,22747}, {13947,22941}, {13948,22943}, {13949,22945}, {13950,22947}, {13951,22955}, {13952,22956}, {13953,22957}, {13954,22958}, {13955,22959}, {13956,22963}, {13957,22964}, {13958,22965}, {13959,22969}, {13961,22979}, {13962,22980}, {13963,22981}, {13964,22982}, {13965,22983}, {18762,22800}, {18966,18978}, {22764,22776}

X(22978) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TRINH TO 3rd HATZIPOLAKIS

Barycentrics    ((5*R^2-SA-SW)*S^2-(R^2*(136*R^2-4*SA-47*SW)+SA^2-SB*SC+4*SW^2)*SA)*(SB+SC) : :
X(22978) = 3*X(3)-X(2929) = 5*X(3)-X(22550) = 3*X(376)+X(22555) = 5*X(1656)-3*X(22971) = X(2929)+3*X(22549) = 5*X(2929)-3*X(22550) = 2*X(2929)-3*X(22962) = 3*X(10606)+X(17837) = 5*X(22549)+X(22550) = 2*X(22549)+X(22962) = 2*X(22550)-5*X(22962)

The reciprocal orthologic center of these triangles is X(9729).

X(22978) lies on these lines: {3,2929}, {24,22538}, {30,22816}, {35,19472}, {36,22954}, {74,5562}, {376,22555}, {378,22970}, {511,22830}, {550,13289}, {1656,22971}, {2071,5907}, {3098,15074}, {3357,5876}, {3519,20417}, {3520,22750}, {4550,11250}, {5092,19142}, {5448,19511}, {6200,22960}, {6396,22961}, {7688,22840}, {7689,10627}, {7691,13348}, {9818,22973}, {10605,19460}, {10606,17837}, {10645,22974}, {10646,22975}, {11410,22497}, {11430,22529}, {11438,22530}, {11454,22534}, {11468,22535}, {12084,22800}, {12359,12901}, {16013,22647}, {16111,18442}, {18931,18936}, {19192,19198}, {19454,19488}, {19455,19489}

X(22978) = midpoint of X(3) and X(22549)
X(22978) = circumtangential isogonal conjugate of X(22467)

X(22979) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 3rd HATZIPOLAKIS

Barycentrics    (R^2*(24*R^2+10*SA-15*SW)-2*SA*SW+2*SW^2)*S^2-(R^2*(184*R^2-77*SW)+8*SW^2)*SB*SC : :
X(22979) = 3*X(3)-2*X(22951) = 3*X(381)-4*X(22833) = 3*X(381)-2*X(22955) = 5*X(1656)-4*X(22966) = 5*X(3843)-4*X(22800) = 3*X(5790)-2*X(22941) = 3*X(10246)-4*X(22476) = 3*X(10247)-2*X(22969) = 3*X(11911)-2*X(22943) = 3*X(22466)-X(22951)

The reciprocal orthologic center of these triangles is X(12241).

X(22979) lies on these lines: {3,2929}, {5,22647}, {30,22533}, {195,5893}, {381,22833}, {517,22653}, {999,18978}, {1598,22483}, {1656,22966}, {3295,22965}, {3843,22800}, {5448,22970}, {5790,22941}, {6417,19084}, {6418,19083}, {7517,22658}, {9301,22747}, {9654,22958}, {9669,22959}, {10112,11744}, {10246,22476}, {10247,22969}, {10620,20427}, {11842,22524}, {11849,22559}, {11911,22943}, {11916,22945}, {11917,22947}, {11928,22956}, {11929,22957}, {11949,22963}, {11950,22964}, {12000,22982}, {12001,22983}, {12111,12282}, {12293,22538}, {12825,22534}, {12902,18439}, {13352,22971}, {13903,22976}, {13961,22977}, {18504,22750}, {19362,19460}, {22765,22776}

X(22979) = reflection of X(3) in X(22466)
X(22979) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18978, 22981, 999), (22833, 22955, 381), (22965, 22980, 3295)

X(22980) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 3rd HATZIPOLAKIS

Barycentrics    (5*R^2-SW)*(16*R^2-SA-3*SW)*S^2+(SB+SC)*((56*R^4+2*SW^2-21*R^2*SW)*b*c-(4*R^2-SW)*(16*R^2-3*SW)*SA) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22980) lies on these lines: {1,22466}, {3,18978}, {5,22959}, {12,22955}, {35,22951}, {388,22533}, {495,22958}, {498,22966}, {1479,22833}, {3085,22647}, {3295,22965}, {3299,19083}, {3301,19084}, {10037,22658}, {10038,22747}, {10039,22941}, {10040,22945}, {10041,22947}, {10523,22956}, {10801,22524}, {10895,22800}, {10954,22957}, {11398,22483}, {11507,22559}, {11912,22943}, {11951,22963}, {11952,22964}, {13904,22976}, {13962,22977}, {18447,19472}, {22766,22776}

X(22980) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22466, 22981), (3295, 22979, 22965)

X(22981) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 3rd HATZIPOLAKIS

Barycentrics    (5*R^2-SW)*(16*R^2-SA-3*SW)*S^2+(SB+SC)*(-(56*R^4+2*SW^2-21*R^2*SW)*b*c-(4*R^2-SW)*(16*R^2-3*SW)*SA) : :

The reciprocal orthologic center of these triangles is X(12241).

X(22981) lies on these lines: {1,22466}, {3,22965}, {11,22955}, {36,22951}, {496,22959}, {497,22533}, {499,22966}, {999,18978}, {1478,22833}, {1737,22941}, {3086,22647}, {3299,19084}, {3301,19083}, {10046,22658}, {10047,22747}, {10048,22945}, {10049,22947}, {10523,22957}, {10802,22524}, {10896,22800}, {10948,22956}, {11399,22483}, {11508,22559}, {11913,22943}, {11953,22963}, {11954,22964}, {13905,22976}, {13963,22977}, {18455,22808}, {22767,22776}

X(22981) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 22466, 22980), (999, 22979, 18978)

X(22982) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 3rd HATZIPOLAKIS

Barycentrics    a^19-(b+c)*a^18-4*(b-c)^2*a^17+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^16+(5*b^4+5*c^4-3*(8*b^2-9*b*c+8*c^2)*b*c)*a^15-(b+c)*(5*b^4+5*c^4-9*(2*b^2-3*b*c+2*c^2)*b*c)*a^14-(b^6+c^6-4*(3*b^4+3*c^4-(11*b^2-28*b*c+11*c^2)*b*c)*b*c)*a^13+(b+c)*(b^6+c^6-2*(4*b^4+4*c^4-(22*b^2-43*b*c+22*c^2)*b*c)*b*c)*a^12-(b^8+c^8-2*(12*b^6+12*c^6-(b^4+c^4+(64*b^2-71*b*c+64*c^2)*b*c)*b*c)*b*c)*a^11+(b+c)*(b^8+c^8-2*(11*b^6+11*c^6-(b^4+c^4+(51*b^2-71*b*c+51*c^2)*b*c)*b*c)*b*c)*a^10-(b^10+c^10+(20*b^8+20*c^8-(45*b^6+45*c^6-4*(10*b^4+10*c^4+(25*b^2-74*b*c+25*c^2)*b*c)*b*c)*b*c)*b*c)*a^9+(b+c)*(b^10+c^10+(20*b^8+20*c^8-(45*b^6+45*c^6-4*(7*b^4+7*c^4+5*(5*b^2-12*b*c+5*c^2)*b*c)*b*c)*b*c)*b*c)*a^8-(b^10+c^10+(10*b^8+10*c^8+(20*b^6+20*c^6-(82*b^4+82*c^4+(77*b^2-64*b*c+77*c^2)*b*c)*b*c)*b*c)*b*c)*(b-c)^2*a^7+(b^2-c^2)*(b-c)*(b^10+c^10+(8*b^8+8*c^8+(16*b^6+16*c^6-(74*b^4+74*c^4+3*(19*b^2-28*b*c+19*c^2)*b*c)*b*c)*b*c)*b*c)*a^6+(b^2-c^2)^2*(b-c)^2*(5*b^8+5*c^8+(14*b^6+14*c^6-(9*b^4+9*c^4+4*(4*b^2-11*b*c+4*c^2)*b*c)*b*c)*b*c)*a^5-(b^2-c^2)^3*(b-c)*(5*b^8+5*c^8+(18*b^6+18*c^6-(b^4+c^4+2*(11*b^2-12*b*c+11*c^2)*b*c)*b*c)*b*c)*a^4-4*(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(b^4-4*b^2*c^2+c^4)*a^3+2*(b^2-c^2)^5*(b-c)*(b^2+c^2)*(2*b^4+2*c^4+(3*b^2-2*b*c+3*c^2)*b*c)*a^2+(b^2-c^2)^6*(b^2+c^2)^2*(b^2-4*b*c+c^2)*a-(b^2-c^2)^7*(b-c)*(b^2+c^2)^2 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22982) lies on these lines: {1,22466}, {12,22956}, {5552,22966}, {10528,22647}, {10531,22833}, {10803,22524}, {10805,22533}, {10834,22658}, {10878,22747}, {10915,22941}, {10929,22945}, {10930,22947}, {10942,22955}, {10955,22957}, {10956,22958}, {10958,22959}, {10965,22965}, {11248,22951}, {11400,22483}, {11509,18978}, {11914,22943}, {11955,22963}, {11956,22964}, {12000,22979}, {13906,22976}, {13964,22977}, {18542,22800}, {19047,19083}, {19048,19084}, {22768,22776}

X(22982) = {X(22466), X(22969)}-harmonic conjugate of X(22983)

X(22983) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 3rd HATZIPOLAKIS

Barycentrics    a^19-(b+c)*a^18-4*(b^2+b*c+c^2)*a^17+2*(b+c)*(2*b^2+b*c+2*c^2)*a^16+(5*b^4+5*c^4+(12*b^2+11*b*c+12*c^2)*b*c)*a^15-(b+c)*(b^2+b*c+c^2)*(5*b^2+b*c+5*c^2)*a^14-(b^6+c^6+4*(b^4+c^4+3*(b^2+5*b*c+c^2)*b*c)*b*c)*a^13+(b+c)*(b^6+c^6+2*(6*b^2+17*b*c+6*c^2)*b^2*c^2)*a^12-(b^8+c^8+2*(10*b^6+10*c^6-(7*b^4+7*c^4+(38*b^2-17*b*c+38*c^2)*b*c)*b*c)*b*c)*a^11+(b+c)*(b^8+c^8+2*(9*b^6+9*c^6-(7*b^4+7*c^4+(25*b^2-17*b*c+25*c^2)*b*c)*b*c)*b*c)*a^10-(b^10+c^10-(20*b^8+20*c^8-(19*b^6+19*c^6-4*(4*b^4+4*c^4+(11*b^2-46*b*c+11*c^2)*b*c)*b*c)*b*c)*b*c)*a^9+(b+c)*(b^10+c^10-(20*b^8+20*c^8-(19*b^6+19*c^6-4*(b^4+c^4+(11*b^2-32*b*c+11*c^2)*b*c)*b*c)*b*c)*b*c)*a^8-(b^10+c^10-(2*b^8+2*c^8+(20*b^6+20*c^6-(b+2*c)*(2*b+c)*(23*b^2-20*b*c+23*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^7+(b^2-c^2)*(b-c)*(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*(b^6+c^6-2*(2*b^4+2*c^4-(b^2+9*b*c+c^2)*b*c)*b*c)*a^6+(b^4-c^4)*(b^2-c^2)*(b-c)^2*(5*b^6+5*c^6-2*(b^4+c^4+(7*b^2-3*b*c+7*c^2)*b*c)*b*c)*a^5-(b^2-c^2)^3*(b-c)*(5*b^8+5*c^8-(6*b^6+6*c^6+(17*b^4+17*c^4-2*(5*b^2-4*b*c+5*c^2)*b*c)*b*c)*b*c)*a^4-4*(b^4-c^4)^2*(b^2-c^2)^2*(b-c)*(b^3-c^3)*a^3+2*(b^2-c^2)^5*(b-c)^3*(b^2+c^2)*(2*b^2+3*b*c+2*c^2)*a^2+(b^2-c^2)^6*(b^2+c^2)^3*a-(b^2-c^2)^7*(b-c)*(b^2+c^2)^2 : :

The reciprocal orthologic center of these triangles is X(12241).

X(22983) lies on these lines: {1,22466}, {11,22957}, {10527,22966}, {10529,22647}, {10532,22833}, {10804,22524}, {10806,22533}, {10835,22658}, {10879,22747}, {10916,22941}, {10931,22945}, {10932,22947}, {10943,22955}, {10949,22956}, {10957,22958}, {10959,22959}, {10966,22776}, {11249,22951}, {11401,22483}, {11510,22559}, {11915,22943}, {11957,22963}, {11958,22964}, {12001,22979}, {13907,22976}, {13965,22977}, {18544,22800}, {18967,18978}, {19049,19083}, {19050,19084}

X(22983) = {X(22466), X(22969)}-harmonic conjugate of X(22982)

X(22984) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 3rd HATZIPOLAKIS

Barycentrics    (SB-SC)*(3*(5*R^2-SW)*(8*R^2-SA-SW)*S^2-8*R^4*(12*SW^2+21*SA^2-28*SA*SW)+R^2*SW*(63*SA^2-79*SA*SW+34*SW^2)-SW^2*(6*SA^2-7*SA*SW+3*SW^2)) : :

The reciprocal parallelogic center of these triangles is X(12241).

X(22984) lies on the line {351,22985}

X(22985) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 3rd HATZIPOLAKIS

Barycentrics    (SB-SC)*(3*(5*R^2-SW)*(8*R^2-SA-SW)*S^2+8*(21*SA^2-28*SA*SW+2*SW^2)*R^4-(63*SA^2-89*SA*SW+8*SW^2)*SW*R^2+(6*SA^2-9*SA*SW+SW^2)*SW^2) : :

The reciprocal parallelogic center of these triangles is X(12241).

X(22985) lies on the line {351,22984}

X(22986) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 3rd HATZIPOLAKIS TO MIDHEIGHT

Barycentrics    (5*R^2-SW)*(R^2*(112*R^2+SA-43*SW)+4*SW^2)*S^4+(-30720*R^10+64*(SA+445*SW)*R^8+4*(215*SA^2-217*SA*SW-2618*SW^2)*R^6-2*(246*SA^2-243*SA*SW-953*SW^2)*SW*R^4+(94*SA^2-93*SA*SW-171*SW^2)*SW^2*R^2-6*(SA^2-SA*SW-SW^2)*SW^3)*S^2+2*(4*R^2-SW)*(16*R^2-3*SW)^2*(6*R^2-SW)^2*SB*SC : :

The reciprocal cyclologic center of these triangles is X(974).

X(22986) lies on the line {113,5893}

X(22987) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st HATZIPOLAKIS TO 2nd HATZIPOLAKIS

Barycentrics    a*(2*(b+c)*a^14-(3*b^2+16*b*c+3*c^2)*a^13-2*(b+c)*(2*b^2-15*b*c+2*c^2)*a^12+(12*b^4+12*c^4+(22*b^2-57*b*c+22*c^2)*b*c)*a^11-2*(b+c)*(4*b^4+4*c^4+3*(13*b^2-10*b*c+13*c^2)*b*c)*a^10-(15*b^6+15*c^6-(54*b^4+54*c^4+(115*b^2+22*b*c+115*c^2)*b*c)*b*c)*a^9+2*(b+c)*(15*b^6+15*c^6+4*(3*b^2-5*b*c+3*c^2)*(b^2-3*b*c+c^2)*b*c)*a^8-2*(58*b^6+58*c^6+(14*b^4+14*c^4+(19*b^2+150*b*c+19*c^2)*b*c)*b*c)*b*c*a^7-2*(b+c)*(15*b^8+15*c^8-(42*b^6+42*c^6-(5*b^4+5*c^4+2*(b^2-36*b*c+c^2)*b*c)*b*c)*b*c)*a^6+(15*b^8+15*c^8+2*(7*b^6+7*c^6-(49*b^4+49*c^4-(112*b^2-127*b*c+112*c^2)*b*c)*b*c)*b*c)*(b+c)^2*a^5+2*(b^3+c^3)*(b+c)^2*(4*b^6+4*c^6-(37*b^4+37*c^4-6*(13*b^2-19*b*c+13*c^2)*b*c)*b*c)*a^4-(12*b^8+12*c^8-(78*b^6+78*c^6-(235*b^4+235*c^4-2*(224*b^2-275*b*c+224*c^2)*b*c)*b*c)*b*c)*(b+c)^4*a^3+2*(b^2-c^2)^3*(b-c)*(2*b^6+2*c^6+(b^4+c^4+7*(b^2+c^2)*b*c)*b*c)*a^2+(b^2-c^2)^4*(3*b^6+3*c^6-(2*b^2-3*b*c+2*c^2)*(9*b^2-4*b*c+9*c^2)*b*c)*a-2*(b^2-c^2)^5*(b-c)^5) : :

The reciprocal orthologic center of these triangles is X(22988).

X(22987) lies on these lines: {}

X(22988) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd HATZIPOLAKIS TO 1st HATZIPOLAKIS

Barycentrics    a*(2*(b+c)*a^7+(b^2-4*b*c+c^2)*a^6-2*(b+c)*(2*b^2-b*c+2*c^2)*a^5-(b^4+c^4-2*(5*b^2-4*b*c+5*c^2)*b*c)*a^4+2*(b+c)*(b^4+c^4)*a^3-(b^6+c^6+(4*b^4+4*c^4-(11*b^2-20*b*c+11*c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)*(b-c)^3*b*c*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2)*(a^9-2*(b+c)*a^8+6*b*c*a^7+4*(b^2-c^2)*(b-c)*a^6-(6*b^4+6*c^4+(2*b^2+3*b*c+2*c^2)*b*c)*a^5+4*(b+c)*(2*b^2-b*c+2*c^2)*b*c*a^4+2*(4*b^6+4*c^6-(7*b^2-13*b*c+7*c^2)*(b+c)^2*b*c)*a^3-4*(b+c)*(b^6+c^6-2*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*b*c)*a^2-(3*b-c)*(b-3*c)*(b^2-b*c+c^2)^2*(b+c)^2*a+2*(b^2-c^2)^3*(b-c)^3)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

The reciprocal orthologic center of these triangles is X(22987).

X(22988) lies on these lines: {}

X(22989) = CYCLOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO URSA MINOR

Barycentrics    a*((b^2-3*b*c+c^2)*a^14-(b+c)*(13*b^2-36*b*c+13*c^2)*a^13+(78*b^4+78*c^4-b*c*(59*b^2+192*b*c+59*c^2))*a^12-2*(b+c)*(143*b^4+143*c^4-5*b*c*(29*b^2+28*b*c+29*c^2))*a^11+(715*b^6+715*c^6+(299*b^4+299*c^4-3*b*c*(278*b^2+483*b*c+278*c^2))*b*c)*a^10-(b+c)*(1287*b^6+1287*c^6-2*(313*b^4+313*c^4+b*c*(27*b^2+850*b*c+27*c^2))*b*c)*a^9+(1716*b^8+1716*c^8+(447*b^6+447*c^6-(476*b^4+476*c^4+b*c*(1109*b^2+1480*b*c+1109*c^2))*b*c)*b*c)*a^8-6*(b^2-c^2)*(b-c)*(286*b^6+286*c^6+(190*b^4+190*c^4+b*c*(410*b^2+271*b*c+410*c^2))*b*c)*a^7+(1287*b^8+1287*c^8+(969*b^6+969*c^6+(824*b^4+824*c^4+b*c*(851*b^2+950*b*c+851*c^2))*b*c)*b*c)*(b-c)^2*a^6-(b^2-c^2)*(b-c)^3*(715*b^6+715*c^6+(420*b^4+420*c^4+b*c*(919*b^2+466*b*c+919*c^2))*b*c)*a^5+(286*b^8+286*c^8+(83*b^6+83*c^6-(120*b^4+120*c^4+b*c*(233*b^2+300*b*c+233*c^2))*b*c)*b*c)*(b-c)^4*a^4-2*(b^2-c^2)*(b-c)^5*(39*b^6+39*c^6+(5*b^4+5*c^4-b*c*(15*b^2+31*b*c+15*c^2))*b*c)*a^3+(13*b^8+13*c^8+(11*b^6+11*c^6-(53*b^4+53*c^4+2*b*c*(b^2-17*b*c+c^2))*b*c)*b*c)*(b-c)^6*a^2-(b^2-c^2)*(b-c)^7*(b^6+c^6+2*(3*b^4+3*c^4-5*(2*b^2-b*c+2*c^2)*b*c)*b*c)*a+(b^4+c^4-2*b*c*(b^2+b*c+c^2))*(b-c)^10*b*c) : :

The reciprocal cyclologic center of these triangles is X(22990).

X(22989) lies on the line {5572,22990}

X(22990) = CYCLOLOGIC CENTER OF THESE TRIANGLES: URSA MINOR TO HONSBERGER

Barycentrics    a*((b+c)*a^13-(9*b^2+22*b*c+9*c^2)*a^12+2*(b+c)*(15*b^2+47*b*c+15*c^2)*a^11-2*(15*b^4+15*c^4+b*c*(151*b^2+244*b*c+151*c^2))*a^10-(b+c)*(89*b^4+89*c^4-7*b*c*(58*b^2+65*b*c+58*c^2))*a^9+(369*b^6+369*c^6-2*(12*b^4+12*c^4+b*c*(383*b^2+551*b*c+383*c^2))*b*c)*a^8-(b+c)*(636*b^6+636*c^6-(484*b^4+484*c^4-b*c*(113*b^2-854*b*c+113*c^2))*b*c)*a^7+(636*b^6+636*c^6+(1124*b^4+1124*c^4+b*c*(1539*b^2+1678*b*c+1539*c^2))*b*c)*(b-c)^2*a^6-(b^2-c^2)*(b-c)*(369*b^6+369*c^6-2*(33*b^4+33*c^4-b*c*(241*b^2-9*b*c+241*c^2))*b*c)*a^5+(89*b^6+89*c^6+2*(67*b^4+67*c^4+b*c*(78*b^2+85*b*c+78*c^2))*b*c)*(b-c)^4*a^4+(b^2-c^2)*(b-c)^3*(30*b^6+30*c^6-(42*b^4+42*c^4-b*c*(115*b^2+6*b*c+115*c^2))*b*c)*a^3-(30*b^6+30*c^6-(14*b^4+14*c^4-b*c*(13*b^2+34*b*c+13*c^2))*b*c)*(b-c)^6*a^2+(b^2-c^2)*(b-c)^5*(9*b^6+9*c^6-(36*b^4+36*c^4-b*c*(53*b^2-60*b*c+53*c^2))*b*c)*a-(b^4+c^4-2*b*c*(b^2+b*c+c^2))*(b-c)^10) : :

The reciprocal cyclologic center of these triangles is X(22989).

X(22990) lies on the line {5572,22989}

X(22991) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798E TO HUTSON EXTOUCH

Barycentrics    2*(b+c)*a^9-3*(b^2+c^2)*a^8-2*(b+c)*(3*b^2+5*b*c+3*c^2)*a^7+2*(5*b^4+5*c^4-(b^2-16*b*c+c^2)*b*c)*a^6+6*(b^3+c^3)*(b^2+6*b*c+c^2)*a^5-2*(6*b^4+6*c^4+(9*b^2+14*b*c+9*c^2)*b*c)*(b-c)^2*a^4-2*(b^2-c^2)*(b-c)*(b^4+c^4+(17*b^2+8*b*c+17*c^2)*b*c)*a^3+2*(b^2-c^2)^2*(3*b^4+3*c^4-(3*b^2+8*b*c+3*c^2)*b*c)*a^2+10*(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(442).

X(22991) lies on these lines: {3,11281}, {10,12864}, {1020,3333}, {1125,5763}, {2346,5703}, {6265,12521}, {9957,12855}, {10582,12120}, {11019,12599}, {12777,20007}

X(22991) = midpoint of X(3) and X(16134)

X(22992) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798I TO HUTSON EXTOUCH

Barycentrics    (b^2+4*b*c+c^2)*a^8-2*(b+c)*(b^2-5*b*c+c^2)*a^7-2*(b^4+c^4+(15*b^2+8*b*c+15*c^2)*b*c)*a^6+2*(b+c)*(3*b^4+3*c^4-(7*b^2+16*b*c+7*c^2)*b*c)*a^5+2*(25*b^4+25*c^4+2*(4*b^2+31*b*c+4*c^2)*b*c)*b*c*a^4-2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+(7*b^2-16*b*c+7*c^2)*b*c)*a^3+2*(b^2-c^2)^2*(b^4+c^4-13*(b^2+c^2)*b*c)*a^2+2*(b^2-c^2)^3*(b-c)*(b^2+5*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(12732).

X(22992) lies on these lines: {3,12731}, {10,12864}, {3428,12777}, {3652,12516}, {5049,12855}, {6943,9874}, {8273,15998}, {12260,14986}

X(22992) = midpoint of X(3) and X(12731)

X(22993) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND 2nd ZANIAH

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

X(22993) lies on these lines: {2,178}, {8,8390}, {9,363}, {10,9836}, {210,17607}, {236,11923}, {518,11026}, {936,8111}, {958,8109}, {960,9805}, {1125,11039}, {1329,8380}, {1376,8107}, {2886,8377}, {3035,13260}, {3036,12733}, {3740,11222}, {5044,12488}, {5745,11854}, {5777,12673}, {6732,7028}, {8140,8580}, {8385,18230}, {9783,18228}, {9847,18247}, {11527,15829}, {11530,12879}, {11856,18227}, {11892,18229}, {11922,18234}, {12561,18249}, {12574,18250}, {12707,18251}, {12719,18252}, {12759,18254}, {12851,18255}, {12878,18257}, {12882,18259}, {12886,18248}, {16135,18253}, {17621,18236}

X(22993) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11685, 8113), (8, 8390, 12633)

X(22994) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND 2nd ZANIAH

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

X(22994) lies on these lines: {2,8114}, {8,178}, {9,164}, {10,9837}, {188,8135}, {210,17608}, {518,11027}, {958,8110}, {960,9806}, {1125,11040}, {1329,8381}, {1376,8108}, {2886,8378}, {3035,13261}, {3036,12734}, {3452,12885}, {3740,11223}, {5044,12489}, {5273,11887}, {5745,11855}, {5777,12674}, {7028,8138}, {8140,8580}, {8242,10494}, {8386,18230}, {9787,18228}, {9849,18247}, {10233,16016}, {11528,15829}, {11857,18227}, {11893,18229}, {11925,18234}, {11926,18235}, {12562,18249}, {12576,18250}, {12708,18251}, {12720,18252}, {12760,18254}, {12852,18255}, {12881,18248}, {12883,18257}, {12887,18259}, {16136,18253}, {17623,18236}

X(22994) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11686, 8114), (8, 8392, 12634)

X(22995) = CYCLOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 2nd ZANIAH

Barycentrics    2*a^10-2*(b+c)*a^9-(13*b^2-4*b*c+13*c^2)*a^8-2*(b+c)*(b^2-18*b*c+c^2)*a^7+2*(7*b^2-13*b*c+7*c^2)*(b^2+6*b*c+c^2)*a^6+2*(b+c)*(b^4+c^4-2*b*c*(3*b^2+23*b*c+3*c^2))*a^5-2*(2*b^6+2*c^6+(19*b^4+19*c^4+2*b*c*(26*b^2-147*b*c+26*c^2))*b*c)*a^4+2*(b+c)*(b^6+c^6+(14*b^4+14*c^4-5*b*c*(9*b^2-4*b*c+9*c^2))*b*c)*a^3+2*(3*b^4+3*c^4+2*b*c*(5*b-4*c)*(4*b-5*c))*(b+c)^2*b*c*a^2-20*(b^4-c^4)*(b^2-c^2)*b*c*(b+c)*a+(b^4-c^4)^2*(b+c)^2 : :

The reciprocal cyclologic center of these triangles is X(22996).

X(22995) lies on the line {1125,22996}

X(22996) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO INCIRCLE-CIRCLES

Barycentrics    2*a^9-2*(b+c)*a^8+(5*b^2-32*b*c+5*c^2)*a^7+(b+c)*(b^2+30*b*c+c^2)*a^6-(11*b^4+11*c^4+2*b*c*(7*b^2-33*b*c+7*c^2))*a^5+(b+c)*(9*b^4+9*c^4-2*b*c*(12*b^2+41*b*c+12*c^2))*a^4+(3*b^6+3*c^6+(52*b^4+52*c^4-3*b*c*(13*b^2-88*b*c+13*c^2))*b*c)*a^3-(b+c)*(9*b^6+9*c^6-(6*b^4+6*c^4-b*c*(67*b^2-60*b*c+67*c^2))*b*c)*a^2+(b^6+c^6+3*b^2*c^2*(9*b^2-16*b*c+9*c^2))*(b+c)^2*a+(b^4-c^4)*(b^2-c^2)*(b+c)*(b^2-4*b*c+c^2) : :

The reciprocal cyclologic center of these triangles is X(22995).

X(22996) lies on the Spieker circle and these lines: {1125,22995}, {3755,4906}

X(22997) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 2nd ISODYNAMIC-DAO

Barycentrics    (9*SA-4*SW)*S^2-sqrt(3)*(3*SA-SW)*(SB+SC)*S+3*SB*SC*SW : :
X(22997) = X(14)-3*X(16529) = 2*X(14)-3*X(22510) = 2*X(115)-3*X(16267) = 4*X(230)-3*X(16268) = 2*X(396)-3*X(16529) = 4*X(396)-3*X(22510) = 3*X(5469)-5*X(16960) = 3*X(5470)-4*X(11542) = 2*X(6109)-3*X(16962) = X(6777)-3*X(16962)

The reciprocal orthologic center of these triangles is X(22998).

X(22997) lies on these lines: {5,14}, {15,542}, {16,524}, {30,6778}, {99,532}, {115,16267}, {187,8724}, {194,617}, {230,9113}, {298,619}, {299,22689}, {511,23007}, {512,22999}, {523,15743}, {543,22495}, {2782,22701}, {5460,16966}, {5463,8593}, {5469,16960}, {5470,11542}, {5965,9115}, {5969,23000}, {6054,6109}, {6671,22894}, {6772,22509}, {9114,23006}, {9760,22496}, {16001,20425}, {18765,22797}, {22603,22606}, {22632,22635}

X(22997) = reflection of X(i) in X(j) for these (i,j): (13, 6783), (15, 9117), (298, 619), (22998, 187)
X(22997) = isogonal conjugate of X(32908)
X(22997) = X(13)-antipedal-to-X(14)-antipedal similarity image of X(14)
X(22997) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 396, 22510), (14, 16529, 396), (6777, 16962, 6109)

X(22998) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 1st ISODYNAMIC-DAO

Barycentrics    (9*SA-4*SW)*S^2+sqrt(3)*(3*SA-SW)*(SB+SC)*S+3*SB*SC*SW : :
X(22998) = X(13)-3*X(16530) = 2*X(13)-3*X(22511) = 2*X(115)-3*X(16268) = 4*X(230)-3*X(16267) = 2*X(395)-3*X(16530) = 4*X(395)-3*X(22511) = 3*X(5469)-4*X(11543) = 3*X(5470)-5*X(16961) = 2*X(6108)-3*X(16963) = X(6778)-3*X(16963)

The reciprocal orthologic center of these triangles is X(22997).

X(22998) lies on these lines: {5,13}, {15,524}, {16,542}, {30,6777}, {115,16268}, {187,8724}, {194,616}, {230,9112}, {298,22687}, {299,618}, {385,532}, {511,23014}, {512,23008}, {523,11586}, {543,22496}, {2782,22702}, {5459,16967}, {5464,8593}, {5469,11543}, {5470,16961}, {5965,9117}, {5969,23009}, {6054,6108}, {6672,22850}, {6775,22507}, {9116,23013}, {9762,22495}, {16002,20426}, {18764,22796}, {22601,22605}, {22630,22634}

X(22998) = isogonal conjugate of X(32906)
X(22998) = reflection of X(i) in X(j) for these (i,j): (14, 6782), (16, 9115), (299, 618), (22997, 187)
X(22998) = X(14)-antipedal-to-X(13)-antipedal similarity image of X(13)
X(22998) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 395, 22511), (13, 16530, 395), (6778, 16963, 6108)

X(22999) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO INNER-LE VIET AN

Barycentrics    (SB+SC)*(6*S^4-(3*R^2-2*SW)*SA*SW^2-(27*R^2*SA-8*SA^2+2*SB*SC+2*SW^2)*S^2+sqrt(3)*((9*R^2-2*SW)*S^2+(R^2*SW-2*SA^2)*SW)*S) : :

The reciprocal orthologic center of these triangles is X(14181).

X(22999) lies on these lines: {13,23007}, {16,3231}, {61,23017}, {511,6321}, {512,22997}, {622,11582}

X(23000) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 1st NEUBERG

Barycentrics    3*(SA^2+SW*SA-SW^2)*S^2-3*SW^2*SB*SC+sqrt(3)*((SA+4*SW)*S^2+SW*(7*SA^2-SB*SC-3*SW^2))*S : :

The reciprocal orthologic center of these triangles is X(6582).

X(23000) lies on these lines: {13,538}, {15,385}, {16,6581}, {17,76}, {61,23018}, {62,12215}, {194,622}, {698,3105}, {3095,3818}, {3643,7757}, {5969,22997}, {14539,14880}

X(23001) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 2nd NEUBERG

Barycentrics    9*S^4+3*(2*SA^2-3*SB*SC+SW^2)*S^2+15*SW^2*SB*SC-sqrt(3)*((SA-5*SW)*S^2-SW*(2*SA-SW)*(3*SA-7*SW))*S : :

The reciprocal orthologic center of these triangles is X(6298).

X(23001) lies on these lines: {13,754}, {16,6296}, {17,83}, {61,23019}, {623,9866}, {732,3105}, {6287,19130}

X(23002) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (2-sqrt(3))*(18*SA-7*(sqrt(3)+3)*SW)*S^2-9*(sqrt(3)-1)*SW*SB*SC-3*(4*S^2-(-2+sqrt(3))*(SB+SC)*(-6*SA+(sqrt(3)+3)*SW))*S : :

The reciprocal orthologic center of these triangles is X(13705).

X(23002) lies on these lines: {16,13706}, {17,1327}, {61,23020}, {3105,22917}, {13692,23011}

X(23003) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    (2+sqrt(3))*(18*SA-7*(3-sqrt(3))*SW)*S^2+3*(4*S^2-(-2-sqrt(3))*(SB+SC)*(-6*SA+(3-sqrt(3))*SW))*S-9*(-1-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13825).

X(23003) lies on these lines: {16,13826}, {17,1328}, {61,23021}, {3105,22919}, {13812,23012}

X(23004) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO 2nd ISODYNAMIC-DAO

Barycentrics    3*S^2*SA-3*SW*SB*SC+sqrt(3)*(8*S^2-(SB+SC)*(9*SA+SW))*S : :
X(23004) = 2*X(15)-3*X(22510) = 4*X(115)-3*X(22510) = 2*X(187)-3*X(22511) = 2*X(396)-3*X(5470) = 2*X(624)-3*X(14041) = 3*X(5469)-2*X(6109) = 3*X(5470)-X(6780) = 4*X(6036)-3*X(21158) = 4*X(6671)-5*X(14061) = 4*X(6672)-3*X(13586) = 2*X(7684)-3*X(14639) = 2*X(9117)-3*X(16267) = 4*X(11542)-3*X(16529)

The reciprocal parallelogic center of these triangles is X(23005).

X(23004) lies on these lines: {4,3105}, {6,13102}, {14,16}, {15,115}, {61,5254}, {62,5471}, {98,11602}, {99,623}, {148,621}, {187,22511}, {396,5470}, {511,6321}, {512,23007}, {617,18582}, {619,16966}, {624,14041}, {635,17128}, {636,7911}, {1080,5479}, {2549,3106}, {2782,20428}, {3054,5474}, {3104,7748}, {3107,5475}, {5237,20416}, {5318,6778}, {5460,16242}, {5464,20112}, {5469,6109}, {5613,16808}, {5978,6115}, {5983,7925}, {6036,21158}, {6108,8859}, {6671,14061}, {6672,13586}, {6694,7923}, {6774,10646}, {6777,10722}, {7684,14639}, {7685,11676}, {7746,16630}, {8352,22575}, {8584,22579}, {9117,16267}, {11303,22689}, {11542,16529}, {14137,22862}, {14539,15980}, {22493,22577}

X(23004) = midpoint of X(i) and X(j) for these {i,j}: {148, 621}, {6777, 19106}, {22493, 22577}
X(23004) = reflection of X(i) in X(j) for these (i,j): (15, 115), (99, 623), (1080, 5479), (14539, 15980)
X(23004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 19107, 22512), (15, 115, 22510), (5470, 6780, 396), (5479, 6114, 16809), (6321, 11646, 23005)

X(23005) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 1st ISODYNAMIC-DAO

Barycentrics    3*S^2*SA-3*SW*SB*SC-sqrt(3)*(8*S^2-(SB+SC)*(9*SA+SW))*S : :
X(23005) = 2*X(16)-3*X(22511) = 4*X(115)-3*X(22511) = 2*X(187)-3*X(22510) = 2*X(395)-3*X(5469) = 2*X(623)-3*X(14041) = 3*X(5469)-X(6779) = 3*X(5470)-2*X(6108) = 4*X(6036)-3*X(21159) = 4*X(6671)-3*X(13586) = 4*X(6672)-5*X(14061) = 2*X(7685)-3*X(14639) = 2*X(9115)-3*X(16268) = 4*X(11543)-3*X(16530)

The reciprocal parallelogic center of these triangles is X(23004).

X(23005) lies on these lines: {4,3104}, {6,13103}, {13,15}, {16,115}, {61,5472}, {62,5254}, {98,11603}, {99,624}, {148,622}, {187,22510}, {383,5478}, {395,5469}, {511,6321}, {512,23014}, {616,18581}, {618,16967}, {623,14041}, {635,7911}, {636,17128}, {2549,3107}, {2782,20429}, {3054,5473}, {3105,7748}, {3106,5475}, {5238,20415}, {5321,6777}, {5459,16241}, {5460,8595}, {5463,20112}, {5470,6108}, {5617,16809}, {5979,6114}, {5982,7925}, {6036,21159}, {6109,8859}, {6671,13586}, {6672,14061}, {6695,7923}, {6771,10645}, {6778,10722}, {7684,11676}, {7685,14639}, {7746,16631}, {8352,22576}, {8584,22580}, {9115,16268}, {11304,22687}, {11543,16530}, {14136,22906}, {14538,15980}, {22494,22578}

X(23005) = midpoint of X(i) and X(j) for these {i,j}: {148, 622}, {6778, 19107}, {22494, 22578}
X(23005) = reflection of X(i) in X(j) for these (i,j): (16, 115), (99, 624), (383, 5478), (14538, 15980)
X(23005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 19106, 22513), (16, 115, 22511), (5469, 6779, 395), (5478, 6115, 16808), (6321, 11646, 23004)

X(23006) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 1st ISODYNAMIC-DAO

Barycentrics    3*(2*SA-SW)*S^2-3*SW*SB*SC-(4*S^2-(SB+SC)*(6*SA-SW))*S*sqrt(3) : :
X(23006) = 3*X(5463)-4*X(22687) = 3*X(12155)-2*X(22687)

The reciprocal parallelogic center of these triangles is X(23004).

X(23006) lies on these lines: {2,13}, {3,5472}, {4,6782}, {6,22906}, {14,9880}, {15,5473}, {18,22832}, {62,5254}, {99,6783}, {115,11486}, {187,20425}, {381,9115}, {511,23023}, {512,23028}, {1250,10062}, {1351,5477}, {3105,11257}, {5318,5617}, {5471,6321}, {5475,5615}, {5478,18581}, {5611,6781}, {6771,11481}, {6777,10722}, {7746,16629}, {9114,22997}, {10078,19373}, {10646,21156}, {16001,22238}, {16530,16809}, {16960,22900}, {22513,22862}

X(23006) = homothetic center of antipedal triangle of X(13) and 1st isodynamic-Dao triangle
X(23006) = antipedal-circle-of-X(13)-inverse of X(16)
X(23006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 6779, 5463), (13, 16242, 22489), (616, 5335, 6115), (5335, 6115, 13), (5473, 9112, 15), (8595, 9762, 5463), (11486, 13103, 115)

X(23007) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO TO INNER-LE VIET AN

Barycentrics    (SB+SC)*(-(3*R^2-2*SA+2*SW)*SA*SW^2+(3*(3*SA-4*SW)*R^2-2*SA^2+2*SB*SC+4*SW^2)*S^2-sqrt(3)*((3*R^2+2*SW)*S^2-((12*SA+SW)*R^2-2*SA^2)*SW)*S) : :

The reciprocal parallelogic center of these triangles is X(14187).

X(23007) lies on these lines: {13,22999}, {16,237}, {61,23022}, {511,22997}, {512,23004}, {10653,11002}

X(23008) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO OUTER-LE VIET AN

Barycentrics    (SB+SC)*(6*S^4-(3*R^2-2*SW)*SA*SW^2-(27*R^2*SA-8*SA^2+2*SB*SC+2*SW^2)*S^2-sqrt(3)*((9*R^2-2*SW)*S^2+(SW*R^2-2*SA^2)*SW)*S) : :

The reciprocal orthologic center of these triangles is X(14177).

X(23008) lies on these lines: {14,23014}, {15,3231}, {62,23023}, {511,6321}, {512,22998}, {621,11581}

X(23009) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 1st NEUBERG

Barycentrics    3*(SA^2+SA*SW-SW^2)*S^2-3*SW^2*SB*SC-sqrt(3)*((SA+4*SW)*S^2+SW*(7*SA^2-SB*SC-3*SW^2))*S : :

The reciprocal orthologic center of these triangles is X(6295).

X(23009) lies on these lines: {14,538}, {15,6294}, {16,385}, {18,76}, {61,12215}, {62,23024}, {194,621}, {698,3104}, {3095,3818}, {3642,7757}, {5969,22998}, {14538,14880}

X(23010) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 2nd NEUBERG

Barycentrics    9*S^4+3*(2*SA^2-3*SB*SC+SW^2)*S^2+15*SW^2*SB*SC+sqrt(3)*((SA-5*SW)*S^2-SW*(2*SA-SW)*(3*SA-7*SW))*S : :

The reciprocal orthologic center of these triangles is X(6299).

X(23010) lies on these lines: {14,754}, {15,6297}, {18,83}, {62,23025}, {624,9866}, {732,3104}, {6287,19130}

X(23011) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (2+sqrt(3))*(18*SA-7*(3-sqrt(3))*SW)*S^2-3*(4*S^2-(-2-sqrt(3))*(SB+SC)*(-6*SA+(3-sqrt(3))*SW))*S-9*(-1-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13703).

X(23011) lies on these lines: {15,13704}, {18,1327}, {62,23026}, {3104,22872}, {13692,23002}

X(23012) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    (2-sqrt(3))*(18*SA-7*(sqrt(3)+3)*SW)*S^2-9*(sqrt(3)-1)*SW*SB*SC+3*(4*S^2-(-2+sqrt(3))*(SB+SC)*(-6*SA+(sqrt(3)+3)*SW))*S : :

The reciprocal orthologic center of these triangles is X(13823).

X(23012) lies on these lines: {15,13824}, {18,1328}, {62,23027}, {3104,22874}, {13812,23003}

X(23013) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 2nd ISODYNAMIC-DAO

Barycentrics    3*(2*SA-SW)*S^2-3*SW*SB*SC+(4*S^2-(SB+SC)*(6*SA-SW))*S*sqrt(3) : :
X(23013) = 3*X(5464)-4*X(22689) = 3*X(12154)-2*X(22689)

The reciprocal parallelogic center of these triangles is X(23005).

X(23013) lies on these lines: {2,14}, {3,5471}, {4,6783}, {6,22862}, {13,9880}, {16,5474}, {17,22831}, {61,5254}, {99,6782}, {115,11485}, {187,20426}, {381,9117}, {511,23017}, {512,23022}, {1351,5477}, {3104,11257}, {5321,5613}, {5472,6321}, {5475,5611}, {5479,18582}, {5615,6781}, {6774,11480}, {6778,10722}, {7051,10077}, {7746,16628}, {9116,22998}, {10061,10638}, {10645,21157}, {16002,22236}, {16529,16808}, {16961,22856}, {22512,22906}

X(23013) = antipedal-circle-of-X(14)-inverse of X(15)
X(23013) = homothetic center of antipedal triangle of X(14) and 2nd isodynamic-Dao triangle
X(23013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 6780, 5464), (14, 16241, 22490), (617, 5334, 6114), (5334, 6114, 14), (5474, 9113, 16), (8594, 9760, 5464), (11485, 13102, 115)

X(23014) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO TO OUTER-LE VIET AN

Barycentrics    (SB+SC)*(-(3*R^2-2*SA+2*SW)*SA*SW^2+(3*(3*SA-4*SW)*R^2-2*SA^2+2*SB*SC+4*SW^2)*S^2+sqrt(3)*((3*R^2+2*SW)*S^2-((12*SA+SW)*R^2-2*SA^2)*SW)*S) : :

The reciprocal parallelogic center of these triangles is X(14185).

X(23014) lies on these lines: {14,23008}, {15,237}, {62,23028}, {511,22998}, {512,23005}, {10654,11002}

X(23015) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798E TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(10266).

X(23015) lies on these lines: {3,16145}, {10,191}, {758,12745}, {6265,12524}, {6952,7701}

X(23015) = midpoint of X(3) and X(16145)

X(23016) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798I TO 1st SCHIFFLER

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

The reciprocal orthologic center of these triangles is X(11604).

X(23016) lies on these lines: {3,12745}, {10,191}, {758,12524}, {1749,10266}, {3652,12519}, {6853,12660}, {13465,22782}

X(23016) = midpoint of X(i) and X(j) for these {i,j}: {3, 12745}, {13465, 22782}

X(23017) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO INNER-LE VIET AN

Barycentrics    (SB+SC)*(3*S^4-(3*R^2-SW)*SA*SW^2+(-9*R^2*(-SW+2*SA)+4*SA^2-SB*SC-SW^2)*S^2+sqrt(3)*(R^2*(-SW+3*SA)-S^2-SA^2)*S*SW) : :

The reciprocal orthologic center of these triangles is X(14181).

X(23017) lies on these lines: {14,512}, {15,14182}, {61,22999}, {511,23013}, {5475,23023}, {10654,23022}

X(23217) = isogonal conjugate of polar conjugate of X(34990)

X(23018) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 1st NEUBERG

Barycentrics    3*S^2*SA*SW-3*SW^2*SB*SC-sqrt(3)*((SA-3*SW)*S^2-SW*(3*SA-SW)*SA)*S : :
X(23018) = 3*X(3107)-4*X(3589)

The reciprocal orthologic center of these triangles is X(6582).

X(23018) lies on these lines: {4,69}, {6,12214}, {14,5969}, {15,6581}, {61,23000}, {303,22715}, {538,10654}, {698,3104}, {3098,5981}, {3107,3589}

X(23019) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 2nd NEUBERG

Barycentrics    15*SB*SC*SW^2+3*(SA-2*SW)*S^2*SW+(SA+SW)*(S^2+(3*SA-4*SW)*SW)*sqrt(3)*S : :

The reciprocal orthologic center of these triangles is X(6298).

X(23019) lies on these lines: {4,83}, {15,6296}, {61,23001}, {732,3104}, {754,10654}, {22512,22689}

X(23020) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (9*SA-12*SW+sqrt(3)*SW)*S^2+(6*S^2+3*(SB+SC)*(-3*SA+sqrt(3)*SW-3*SW))*S+9*(sqrt(3)-1)*SW*SB*SC : :

The reciprocal orthologic center of these triangles is X(13705).

X(23020) lies on these lines: {4,1327}, {15,13706}, {61,23002}, {3104,22872}

X(23021) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    (9*SA-12*SW-sqrt(3)*SW)*S^2-(6*S^2+3*(SB+SC)*(-3*SA-3*SW-sqrt(3)*SW))*S+9*(-1-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13825).

X(23021) lies on these lines: {4,1328}, {15,13826}, {61,23003}, {3104,22874}

X(23022) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st LEMOINE-DAO TO INNER-LE VIET AN

Barycentrics    (SB+SC)*(S^4+(3*R^2-SA+2*SW)*S^2*SW-(3*R^2-SA+SW)*SA*SW^2-sqrt(3)*((6*R^2+SW)*S^2-(R^2*(9*SA-SW)-SA^2)*SW)*S) : :

The reciprocal parallelogic center of these triangles is X(14187).

X(23022) lies on these lines: {14,511}, {15,14188}, {61,23007}, {512,23013}, {10654,23017}

X(23023) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO OUTER-LE VIET AN

Barycentrics    (SB+SC)*(3*S^4+(4*SA^2-SB*SC-9*R^2*(2*SA-SW)-SW^2)*S^2-sqrt(3)*(-S^2-SA^2+R^2*(3*SA-SW))*SW*S-SA*(3*R^2-SW)*SW^2) : :

The reciprocal orthologic center of these triangles is X(14177).

X(23023) lies on these lines: {13,512}, {16,14178}, {62,23008}, {511,23006}, {5475,23017}, {10653,23028}

X(23024) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 1st NEUBERG

Barycentrics    3*SA*SW*S^2+sqrt(3)*((SA-3*SW)*S^2-SA*(3*SA-SW)*SW)*S-3*SB*SC*SW^2 : :
X(23024) = 3*X(3106)-4*X(3589)

The reciprocal orthologic center of these triangles is X(6295).

X(23024) lies on these lines: {4,69}, {6,12213}, {13,5969}, {16,6294}, {62,23009}, {302,22714}, {538,10653}, {698,3105}, {3098,5980}, {3106,3589}

X(23025) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 2nd NEUBERG

Barycentrics    3*(SA-2*SW)*SW*S^2-sqrt(3)*(SA+SW)*(S^2+(3*SA-4*SW)*SW)*S+15*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6299).

X(23025) lies on these lines: {4,83}, {16,6297}, {62,23010}, {732,3105}, {754,10653}, {22513,22687}

X(23026) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 1st TRI-SQUARES-CENTRAL

Barycentrics    (9*SA-12*SW-sqrt(3)*SW)*S^2+(6*S^2+3*(SB+SC)*(-3*SA-3*SW-sqrt(3)*SW))*S+9*(-1-sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13703).

X(23026) lies on these lines: {4,1327}, {16,13704}, {62,23011}, {3105,22917}

X(23027) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO 2nd TRI-SQUARES-CENTRAL

Barycentrics    (9*SA-12*SW+sqrt(3)*SW)*S^2-(6*S^2+3*(SB+SC)*(-3*SA+sqrt(3)*SW-3*SW))*S+9*(sqrt(3)-1)*SW*SB*SC : :

The reciprocal orthologic center of these triangles is X(13823).

X(23027) lies on these lines: {4,1328}, {16,13824}, {62,23012}, {3105,22919}

X(23028) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd LEMOINE-DAO TO OUTER-LE VIET AN

Barycentrics    (SB+SC)*(sqrt(3)*S^4+3*(6*R^2+SW)*S^3+sqrt(3)*(3*R^2-SA+2*SW)*SW*S^2-3*(-SA^2+R^2*(9*SA-SW))*SW*S-sqrt(3)*(3*R^2-SA+SW)*SA*SW^2) : :

The reciprocal parallelogic center of these triangles is X(14185).

X(23028) lies on these lines: {13,511}, {16,14186}, {62,23014}, {512,23006}, {10653,23023}

X(23029) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 1st MORLEY-MIDPOINT

Barycentrics    24*b*c*a*(2*a*z*y+x*(c*y+b*z))+4*(3*a^2+2*sqrt(3)*S)*b*c*x+2*(3*a^2+9*b^2-3*c^2+2*sqrt(3)*S)*a*c*y+2*(3*a^2-3*b^2+9*c^2+2*sqrt(3)*S)*a*b*z+3*(b^2+c^2)*a^2-3*(b^2-c^2)^2+2*sqrt(3)*(b^2+c^2)*S : : , where x = cos(A/3)
Barycentrics    Csc[B/3] Sin[B]+Csc[C/3] Sin[C] : :

X(23029) lies on the line {2,3603}

X(23029) = complement of X(3603)

X(23030) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 2nd MORLEY-MIDPOINT

Barycentrics    24*b*c*a*(2*a*z*y+x*(c*y+b*z))+4*(3*a^2+2*sqrt(3)*S)*b*c*x+2*(3*a^2+9*b^2-3*c^2+2*sqrt(3)*S)*a*c*y+2*(3*a^2-3*b^2+9*c^2+2*sqrt(3)*S)*a*b*z+3*(b^2+c^2)*a^2-3*(b^2-c^2)^2+2*sqrt(3)*(b^2+c^2)*S : : , where x=cos(A/3 -2*Pi/3)
Barycentrics    Sec[B/3 - Pi/6]*Sin[B] + Sec[C/3 - Pi/6]*Sin[C] : :

X(23030) lies on the line {2,3604}

X(23030) = complement of X(3604)

X(23031) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 3rd MORLEY-MIDPOINT

Barycentrics    24*b*c*a*(2*a*z*y+x*(c*y+b*z))+4*(3*a^2+2*sqrt(3)*S)*b*c*x+2*(3*a^2+9*b^2-3*c^2+2*sqrt(3)*S)*a*c*y+2*(3*a^2-3*b^2+9*c^2+2*sqrt(3)*S)*a*b*z+3*(b^2+c^2)*a^2-3*(b^2-c^2)^2+2*sqrt(3)*(b^2+c^2)*S : : , where x=cos(A/3 -4*Pi/3)
Barycentrics    Sec[B/3 + Pi/6]*Sin[B] + Sec[C/3 + Pi/6]*Sin[C] : :

X(23031) lies on the line {2,3602}

X(23031) = complement of X(3602)

X(23032) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 1st MORLEY-ADJUNCT MIDPOINT

Barycentrics    b*c*(3*(c*y+b*z)*a+4*sqrt(3)*S+6*a^2)*x+6*a^2*b*c*y*z+(3*a^2+9*b^2-3*c^2+2*sqrt(3)*S)*a*c*y+(3*a^2-3*b^2+9*c^2+2*sqrt(3)*S)*a*b*z+4*sqrt(3)*(b^2+c^2)*S+6*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : : , where x=sec(A/3)

X(23032) lies on the line {2,16840}

X(23033) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 2nd MORLEY-ADJUNCT MIDPOINT

Barycentrics    b*c*(3*(c*y+b*z)*a+4*sqrt(3)*S+6*a^2)*x+6*a^2*b*c*y*z+(3*a^2+9*b^2-3*c^2+2*sqrt(3)*S)*a*c*y+(3*a^2-3*b^2+9*c^2+2*sqrt(3)*S)*a*b*z+4*sqrt(3)*(b^2+c^2)*S+6*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : : , where x=sec((A-2*Pi)/3)

X(23033) lies on these lines: {}

X(23034) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 3rd MORLEY-ADJUNCT MIDPOINT

Barycentrics    b*c*(3*(c*y+b*z)*a+4*sqrt(3)*S+6*a^2)*x+6*a^2*b*c*y*z+(3*a^2+9*b^2-3*c^2+2*sqrt(3)*S)*a*c*y+(3*a^2-3*b^2+9*c^2+2*sqrt(3)*S)*a*b*z+4*sqrt(3)*(b^2+c^2)*S+6*(b^2+c^2)*a^2-6*(b^2-c^2)^2 : : , where x=sec((A-4*Pi)/3)

X(23034) lies on these lines: {}

X(23035) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 1st SCHIFFLER

Barycentrics    (2*a^11-(b+c)*a^10-2*(5*b^2+4*b*c+5*c^2)*a^9+7*(b+c)*(b^2+c^2)*a^8+2*(9*b^4+9*c^4+(7*b^2+8*b*c+7*c^2)*b*c)*a^7-2*(b+c)*(8*b^4+8*c^4-3*(b+c)^2*b*c)*a^6-2*(7*b^6+7*c^6+(3*b^4+3*c^4+2*(b^2+b*c+c^2)*b*c)*b*c)*a^5+(b+c)*(16*b^6+16*c^6-(10*b^4+10*c^4+(11*b^2-4*b*c+11*c^2)*b*c)*b*c)*a^4+2*(2*b^8+2*c^8-(b^6+c^6+(2*b^4-b^2*c^2+2*c^4)*b*c)*b*c)*a^3-(b+c)*(7*b^8+7*c^8-(6*b^6+6*c^6+(12*b^4+12*c^4-(4*b^2+13*b*c+4*c^2)*b*c)*b*c)*b*c)*a^2+2*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*b*c*a+(b^2-c^2)^3*(b-c)*(b^4-b^2*c^2+c^4))*(b-c) : :

The reciprocal parallelogic center of these triangles is X(79).

X(23035) lies on these lines: {351,23036}, {8702,9131}

X(23036) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 1st SCHIFFLER

Barycentrics    (b-c)*(2*a^11-5*(b+c)*a^10-2*(b^2-4*b*c+c^2)*a^9+13*(b+c)*(b^2+c^2)*a^8-2*(b^4+c^4+(9*b^2+4*b*c+9*c^2)*b*c)*a^7-2*(b+c)*(7*b^4+7*c^4+(b^2+10*b*c+c^2)*b*c)*a^6+2*(b^6+c^6+(7*b^4+7*c^4+2*(b^2+b*c+c^2)*b*c)*b*c)*a^5+(b+c)*(10*b^6+10*c^6+(2*b^4+2*c^4+(5*b^2-4*b*c+5*c^2)*b*c)*b*c)*a^4-2*(3*b^6+3*c^6+(2*b^4+2*c^4-(b+2*c)*(2*b+c)*b*c)*b*c)*b*c*a^3-(b+c)*(5*b^4+5*c^4-(2*b^2+7*b*c+2*c^2)*b*c)*(b^4-b^2*c^2+c^4)*a^2+2*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*b*c*a+(b^2-c^2)^3*(b-c)*(b^4-b^2*c^2+c^4)) : :

The reciprocal parallelogic center of these triangles is X(79).

X(23036) lies on these lines: {351,23035}, {8702,9979}

X(23037) = CYCLOLOGIC CENTER OF THESE TRIANGLES: REFLECTION TO YIU

Barycentrics    a^36-13*(b^2+c^2)*a^34+(79*b^4+141*b^2*c^2+79*c^4)*a^32-13*(b^2+c^2)*(23*b^4+31*b^2*c^2+23*c^4)*a^30+(794*b^8+794*c^8+b^2*c^2*(2128*b^4+2799*b^2*c^2+2128*c^4))*a^28-(b^2+c^2)*(1585*b^8+1585*c^8+2*b^2*c^2*(1404*b^4+1897*b^2*c^2+1404*c^4))*a^26+2*(1248*b^12+1248*c^12+(3266*b^8+3266*c^8+b^2*c^2*(5102*b^4+5841*b^2*c^2+5102*c^4))*b^2*c^2)*a^24-(b^2+c^2)*(3211*b^12+3211*c^12+2*(1971*b^8+1971*c^8+b^2*c^2*(3401*b^4+3049*b^2*c^2+3401*c^4))*b^2*c^2)*a^22+(3432*b^16+3432*c^16+(5643*b^12+5643*c^12+(7630*b^8+7630*c^8+b^2*c^2*(9051*b^4+9563*b^2*c^2+9051*c^4))*b^2*c^2)*b^2*c^2)*a^20-(b^2+c^2)*(3003*b^16+3003*c^16-(187*b^12+187*c^12-(3543*b^8+3543*c^8+b^2*c^2*(320*b^4+3793*b^2*c^2+320*c^4))*b^2*c^2)*b^2*c^2)*a^18+(2002*b^20+2002*c^20+(451*b^16+451*c^16+(630*b^12+630*c^12+(934*b^8+934*c^8+b^2*c^2*(925*b^4+943*b^2*c^2+925*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^16-(b^2+c^2)*(793*b^20+793*c^20-(793*b^16+793*c^16-(195*b^12+195*c^12+(278*b^8+278*c^8-b^2*c^2*(254*b^4-345*b^2*c^2+254*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^14-(130*b^24+130*c^24-(1278*b^20+1278*c^20-(2477*b^16+2477*c^16-(1913*b^12+1913*c^12-(616*b^8+616*c^8-5*b^2*c^2*(15*b^4-b^2*c^2+15*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12+(b^4-c^4)*(b^2-c^2)*(493*b^20+493*c^20-(2112*b^16+2112*c^16-(3871*b^12+3871*c^12-(4593*b^8+4593*c^8-b^2*c^2*(4636*b^4-4671*b^2*c^2+4636*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10-(b^2-c^2)^4*(416*b^20+416*c^20-(974*b^16+974*c^16-(756*b^12+756*c^12-(118*b^8+118*c^8+b^2*c^2*(88*b^4+107*b^2*c^2+88*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+(b^2-c^2)^6*(b^2+c^2)*(207*b^16+207*c^16-(598*b^12+598*c^12-(859*b^8+859*c^8-5*b^2*c^2*(170*b^4-149*b^2*c^2+170*c^4))*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^8*(65*b^16+65*c^16-(111*b^12+111*c^12-2*(42*b^8+42*c^8-b^2*c^2*(3*b^4+34*b^2*c^2+3*c^4))*b^2*c^2)*b^2*c^2)*a^4+(b^2-c^2)^12*(b^2+c^2)*(12*b^8+12*c^8-b^2*c^2*(9*b^4-25*b^2*c^2+9*c^4))*a^2-(b^4+c^4)^2*(b^2-c^2)^14 : :

The reciprocal cyclologic center of these triangles is X(23038).

X(23037) lies on the reflection circle and these lines: {}

X(23038) = CYCLOLOGIC CENTER OF THESE TRIANGLES: YIU TO REFLECTION

Barycentrics    (7*S^6+(-R^2*(6*R^2+23*SA-3*SW)+5*SA^2-8*SB*SC)*S^4+(-9*R^8+(-68*SA-16*SW)*R^6+(107*SA^2-11*SA*SW+24*SW^2)*R^4-(71*SA^2-36*SA*SW+9*SW^2)*SW*R^2+(6*SA-SW)*(2*SA-SW)*SW^2)*S^2+(R^2*(R^2+3*SW)-SW^2)*(3*R^6-2*R^2*(-3*R^2*SW+R^2*SA+SA*SW)-3*R^2*SW^2+SA*SW^2)*SA)*(SB+SC) : :

The reciprocal cyclologic center of these triangles is X(23037).

X(23038) lies on the Yiu circle and these lines: {}

X(23039) = EULEROLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO 1st ANTI-CIRCUMPERP

Barycentrics    a^2*((b^2+c^2)*a^4-(2*b^4+b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(-a^2+b^2+c^2) : :
X(23039) = 5*X(2)-4*X(13363) = 5*X(3)-2*X(185) = X(3)-4*X(1216) = 5*X(3)-8*X(5447) = X(3)+2*X(5562) = 2*X(3)+X(18436) = X(185)-10*X(1216) = X(185)-5*X(3917) = X(185)-4*X(5447) = X(185)+5*X(5562) = 4*X(185)+5*X(18436) = 3*X(568)-4*X(5946) = 5*X(568)-8*X(13363) = X(568)-4*X(15067) = 5*X(1216)-2*X(5447) = 2*X(1216)+X(5562) = 8*X(1216)+X(18436) = 5*X(3917)-4*X(5447) = 4*X(3917)+X(18436) = 4*X(5447)+5*X(5562) = 4*X(5562)-X(18436) = 5*X(5946)-6*X(13363) = X(5946)-3*X(15067) = 2*X(13363)-5*X(15067)

The reciprocal eulerologic center of these triangles is X(11459).

X(23039) lies on these lines: {2,568}, {3,49}, {4,2889}, {5,3060}, {20,5876}, {22,10540}, {26,18350}, {30,2979}, {51,5055}, {52,1656}, {68,3519}, {69,265}, {110,7502}, {140,5889}, {143,3090}, {156,7512}, {182,15087}, {183,18322}, {195,569}, {323,14805}, {343,2072}, {373,15703}, {376,5663}, {381,511}, {382,5907}, {389,3526}, {399,2916}, {546,15056}, {547,5640}, {548,6241}, {549,5890}, {550,12111}, {567,1993}, {577,22146}, {631,6102}, {632,15043}, {1350,12083}, {1351,12039}, {1352,9019}, {1511,10298}, {1568,10254}, {1614,6030}, {1657,12162}, {1658,7691}, {1994,7550}, {2070,9306}, {2781,5655}, {2937,10539}, {3091,10263}, {3153,15108}, {3313,18440}, {3419,18330}, {3522,13491}, {3523,13630}, {3525,12006}, {3530,10574}, {3534,6000}, {3544,16982}, {3567,3628}, {3581,6644}, {3627,15058}, {3819,5054}, {3830,15030}, {3851,5446}, {4549,7723}, {5056,10095}, {5067,15026}, {5068,13421}, {5070,5462}, {5071,11002}, {5072,10110}, {5449,22815}, {5609,7492}, {5650,5892}, {6090,14070}, {6193,11821}, {6288,18569}, {6293,10282}, {6592,13505}, {6759,13564}, {7386,18917}, {7393,12160}, {7485,13339}, {7503,16266}, {7506,17834}, {7509,12161}, {7516,7592}, {7517,17814}, {7528,11487}, {7574,11649}, {7577,15110}, {7729,11204}, {7731,10272}, {8681,9967}, {8703,15072}, {8717,12308}, {9729,15720}, {10201,12824}, {10219,16625}, {10224,21230}, {10264,12273}, {10299,11592}, {10575,13348}, {10628,11202}, {11381,17800}, {11442,14791}, {11451,15699}, {11562,15040}, {11660,13160}, {12100,20791}, {12103,12279}, {12290,15704}, {12294,18535}, {12825,20127}, {13346,14130}, {13504,14072}, {14683,15101}, {14845,21849}, {14855,15688}, {14915,15681}, {15022,18874}, {15024,16881}, {15028,16239}, {15032,15246}, {15082,15723}, {15687,16261}, {15693,16836}, {16163,22584}, {17702,18564}, {18392,18572}, {19129,20806}, {19709,21969}

X(23039) = reflection of X(i) in X(j) for these (i,j): (2, 15067), (3, 3917), (4, 15060), (51, 10170), (52, 5943), (381, 5891), (382, 16194), (3830, 15030), (7729, 11204)
X(23039) = anticomplement of X(5946)
X(23039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 394, 22115), (3, 5562, 18436), (3, 9703, 18475), (5, 11412, 6243), (5, 14449, 9781), (20, 5876, 18439), (22, 15068, 10540), (51, 10170, 5055), (52, 5943, 13321), (52, 11793, 1656), (185, 5447, 3), (1216, 5562, 3), (3292, 18475, 9703), (5876, 10627, 20), (6101, 11591, 4), (11412, 11444, 5)

X(23040) = EULEROLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO ANTI-EULER

Barycentrics    a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(4*a^4-8*(b^2+c^2)*a^2+4*b^4+9*b^2*c^2+4*c^4) : :
X(23040) = 8*(4*R^2-SW)*X(3)-R^2*X(4)

The reciprocal eulerologic center of these triangles does not exist
As a point on the Euler line, X(23040) has Shinagawa coefficients (16*F, E-16*F).

X(23040) lies on these lines: {2,3}, {54,20421}, {112,15515}, {185,3431}, {1249,15109}, {1614,11204}, {3043,15055}, {3098,8537}, {8567,9707}, {8588,10312}, {11454,12038}, {11468,13367}, {11470,17508}, {12112,17821}

X(23040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 378, 17506), (3, 3520, 21844), (3, 10226, 20), (3, 11250, 10298), (20, 6143, 4), (186, 3520, 1593), (1593, 15750, 3517), (3431, 11270, 185), (3520, 21844, 4), (3524, 3528, 10996), (10298, 11250, 3529), (15750, 17506, 21844)

X(23041) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO KOSNITA

Barycentrics    a^2*(3*a^10-5*(b^2+c^2)*a^8-2*(b^2+c^2)^2*a^6+6*(b^2+c^2)*(b^4+c^4)*a^4-(b^2-c^2)^4*a^2-(b^4-c^4)^2*(b^2+c^2)) : :
X(23041) = X(3)+2*X(206) = 7*X(3)+2*X(9968) = 2*X(3)+X(19149) = X(6)+2*X(15577) = X(6)+5*X(17821) = X(64)-4*X(15578) = X(66)-4*X(140) = 2*X(154)+X(10249) = X(159)+2*X(182) = 2*X(159)+X(8549) = X(159)-4*X(10282) = 4*X(182)-X(8549) = X(182)+2*X(10282) = 7*X(206)-X(9968) = 4*X(206)-X(19149) = X(8549)+8*X(10282) = 4*X(9968)-7*X(19149) = 2*X(15577)-5*X(17821)

The reciprocal eulerologic center of these triangles is X(23042).

X(23041) lies on these lines: {2,154}, {3,206}, {6,24}, {64,15578}, {66,140}, {159,182}, {161,10601}, {511,11202}, {631,5596}, {1176,17928}, {1177,1511}, {1350,7488}, {1352,7542}, {1495,19124}, {1498,7509}, {1658,19139}, {1974,13367}, {2393,5050}, {2781,15035}, {3147,6776}, {3313,9715}, {3515,19125}, {3517,9969}, {3526,6697}, {3589,7401}, {3827,10202}, {5092,6759}, {5480,7487}, {6000,17508}, {6643,16252}, {7405,9833}, {7544,17845}, {8550,15585}, {9924,15582}, {10303,20079}, {10541,15581}, {11449,19121}, {13289,15141}, {14561,23049}, {14788,20300}

X(23041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 206, 19149), (6, 17821, 15577), (159, 182, 8549), (182, 10282, 159), (3515, 19125, 19161), (7488, 20806, 1350)

X(23042) = EULEROLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO ANTI-HONSBERGER

Barycentrics    a^2*(3*a^10-6*(b^2+c^2)*a^8-2*b^2*c^2*a^6+2*(b^2+c^2)*(3*b^4-b^2*c^2+3*c^4)*a^4-3*(b^2-c^2)^2*(b^4+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(23042) = X(3)+5*X(19132) = X(6)+2*X(10282) = 2*X(154)+X(10250) = X(159)+2*X(575) = X(182)+2*X(206) = 2*X(182)+X(6759) = 4*X(206)-X(6759) = X(576)+2*X(15577) = X(1147)+2*X(19154) = X(1351)+5*X(17821) = X(1353)+2*X(15585) = X(1498)+5*X(12017) = X(3357)-4*X(5092) = X(3357)+2*X(19149) = 4*X(3589)-X(18381) = 2*X(5092)+X(19149) = X(11202)+2*X(19153)

The reciprocal eulerologic center of these triangles is X(23041).

X(23042) lies on these lines: {3,19132}, {5,182}, {6,3517}, {154,5050}, {159,575}, {184,11433}, {389,19125}, {511,11202}, {576,15577}, {578,1974}, {1092,19121}, {1147,19154}, {1351,17821}, {1353,15585}, {1498,12017}, {1971,5034}, {3357,5092}, {3564,10192}, {3618,9833}, {5085,6000}, {5171,15257}, {5596,20299}, {5656,10984}, {5965,10274}, {6467,9707}, {6593,13289}, {9306,19131}, {9968,15578}, {10182,19126}, {10539,19129}, {11204,17508}, {14561,18400}, {15139,16187}, {15582,22234}, {19118,19357}, {19127,21167}

X(23042) = midpoint of X(154) and X(5050)
X(23042) = reflection of X(11204) in X(17508)
X(23042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (182, 206, 6759), (5092, 19149, 3357)

X(23043) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO EHRMANN-VERTEX

Barycentrics    4*S^4+(R^2*(36*R^2-15*SA-10*SW)+4*SA^2-4*SB*SC)*S^2-(15*R^2-4*SW)*(36*R^2-7*SW)*SB*SC : :
X(23043) = 2*X(7728)+X(19506)

The reciprocal eulerologic center of these triangles is X(4).

X(23043) lies on these lines: {2,2777}, {4,11564}, {5,16219}, {113,11202}, {154,18561}, {1539,2781}, {5663,18376}, {6000,7728}, {10628,16194}, {10706,18400}, {10721,13619}, {11744,13623}, {13293,15646}

X(23044) = EULEROLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO ARA

Barycentrics    (SB+SC)*(2*R^2*S^2-(2*R^2*(-2*SW+6*R^2-SA)+SW^2)*SA) : :
X(23044) = (2*R^2-SW)^2*X(3)-4*R^4*X(4)

There is not eulerologic center (Ara, Ehrmann-side).

X(23044) lies on these lines: {2,3}, {6759,9908}, {9645,10831}, {13754,19141}

X(23044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10243, 26), (22, 7503, 376), (22, 15078, 7512), (376, 6353, 6803), (7387, 14070, 22), (9909, 18534, 7387)

X(23045) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-GARCIA TO ATIK

Barycentrics    a*((b+c)*a^8-2*(b^2-b*c+c^2)*a^7-2*(b^2-c^2)*(b-c)*a^6+2*(3*b^4+3*c^4-(3*b^2-20*b*c+3*c^2)*b*c)*a^5-4*(b+c)*(3*b^2+2*b*c+3*c^2)*b*c*a^4-2*(3*b^6+3*c^6-(3*b^4+3*c^4+(7*b^2+18*b*c+7*c^2)*b*c)*b*c)*a^3+2*(b^2-c^2)^2*(b+c)*(b^2+6*b*c+c^2)*a^2+2*(b^2-c^2)^2*(b^4+c^4-(b^2+24*b*c+c^2)*b*c)*a-(b^2-c^2)^3*(b-c)*(b^2+6*b*c+c^2)) : :

There is not eulerologic center (Atik, outer-Garcia).

X(23045) lies on these lines: {4,8}, {496,10863}, {971,8582}, {3091,10569}, {3304,17604}, {6245,10855}, {8581,9581}, {8583,10157}, {9709,10860}, {9711,15587}, {9842,11019}, {12019,13227}

X(23045) = {X(9947), X(10241)}-harmonic conjugate of X(8)

X(23046) = EULEROLOGIC CENTER OF THESE TRIANGLES: SUBMEDIAL TO EHRMANN-MID

Barycentrics    8*a^4+5*(b^2+c^2)*a^2-13*(b^2-c^2)^2 : :
X(23046) = 13*X(2)-7*X(3) = 5*X(2)+7*X(4) = 4*X(2)-7*X(5) = 17*X(2)-14*X(140) = 19*X(2)-7*X(376) = X(2)-7*X(381) = X(2)+14*X(546) = 11*X(2)-14*X(547) = 5*X(2)-2*X(548) = 10*X(2)-7*X(549) = 22*X(2)-7*X(550) = 7*X(2)-X(1657) = 11*X(2)-7*X(3524) = 25*X(2)-7*X(3534) = 3*X(2)-7*X(3545) = 2*X(2)+X(3627) = 11*X(2)+7*X(3830) = X(2)+7*X(3839) = X(2)+5*X(3843) = 2*X(2)+7*X(3845)

There is not eulerologic center (Ehrmann-mid, submedial)
As a point on the Euler line, X(23046) has Shinagawa coefficients (5, 21).

X(23046) lies on these lines: {2,3}, {538,22681}, {590,6476}, {615,6477}, {671,14692}, {754,20112}, {1327,7584}, {1328,7583}, {3625,22791}, {3630,21850}, {3633,3656}, {4668,18357}, {4691,18483}, {4995,18514}, {5298,18513}, {5318,16268}, {5321,16267}, {5476,12007}, {5876,21849}, {6144,20423}, {6417,14241}, {6418,14226}, {10706,11801}, {10733,11694}, {10895,15170}, {11381,18874}, {12295,22251}, {12571,13607}, {13364,16194}, {13451,18435}, {13482,18350}, {18424,18907}

X(23046) = reflection of X(2) in X(14892)
X(23046) = complement of X(15689)
X(23046) = inverse of X(15684) in the orthocentroidal circle
X(23046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3543, 17538), (2, 15684, 548), (140, 3543, 19710), (382, 3854, 12811), (382, 5071, 12100), (549, 15684, 15686), (3146, 15694, 15690), (3543, 3855, 19709), (3543, 15022, 15698), (3543, 15698, 17800), (3543, 19709, 140), (3544, 5073, 16239), (5055, 15684, 15706), (5072, 15684, 2), (14093, 15684, 15683), (18586, 18587, 3522)

X(23047) = EULEROLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO EULER

Barycentrics    (2*a^6-(b^2+c^2)*a^4-4*(b^4+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2))*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :
X(23047) = (4*R^2-SW)*X(3)+(10*R^2-3*SW)*X(4)

There is not eulerologic center (Euler, Ehrmann-vertex)
As a point on the Euler line, X(23047) has Shinagawa coefficients (F, E+5*F).

X(23047) lies on these lines: {2,3}, {125,13568}, {265,13292}, {578,18376}, {946,12135}, {973,1112}, {1398,5229}, {1503,11572}, {1514,13474}, {1699,5090}, {1829,19925}, {1879,6748}, {1902,18483}, {1986,11801}, {2883,11550}, {3564,8537}, {3574,12241}, {3817,11363}, {5448,12134}, {5480,11470}, {5893,11381}, {6146,18383}, {6403,15056}, {6746,13754}, {7699,12289}, {7718,9779}, {7745,16318}, {10880,18538}, {10881,18762}, {10895,11393}, {10896,11392}, {11245,12233}, {11402,18945}, {12022,18394}, {12162,15432}, {12370,18379}, {18405,19467}, {18474,22660}

X(23047) = inverse of X(12173) in the orthocentroidal circle
X(23047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 235, 428), (4, 381, 235), (4, 403, 6756), (4, 546, 10151), (4, 1885, 13473), (4, 3855, 3089), (4, 6623, 5198), (4, 7378, 11403), (4, 7541, 407), (4, 7563, 430), (4, 15559, 13488), (4, 18560, 3853), (5, 3627, 1658), (1595, 3845, 4), (3146, 8889, 3516), (3861, 13488, 4)

X(23048) = EULEROLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO 2nd EHRMANN

Barycentrics    5*S^4-(4*R^2*(6*SA-5*SW)-5*SA^2+5*SB*SC+4*SW^2)*S^2+(12*R^2-5*SW)*SB*SC*SW : :
X(23048) = X(154)-3*X(14848) = 2*X(576)+X(18381) = 2*X(8549)+X(22802) = X(9927)+2*X(11255) = 2*X(11216)+X(18376) = X(11477)+2*X(20299)

The reciprocal eulerologic center of these triangles is X(23049).

X(23048) lies on these lines: {4,11458}, {6,18400}, {30,10250}, {154,14848}, {182,10169}, {381,17813}, {542,11216}, {576,13292}, {597,11202}, {1350,10193}, {1351,1853}, {1503,15520}, {2393,5476}, {3153,11443}, {6000,20423}, {8541,18390}, {8549,22802}, {9927,11255}, {10192,18583}, {10249,19924}, {10602,18388}, {11206,13366}, {11405,18396}, {11477,20299}, {18449,18474}

X(23048) = midpoint of X(i) and X(j) for these {i,j}: {381, 17813}, {1351, 1853}
X(23048) = reflection of X(i) in X(j) for these (i,j): (182, 10169), (1350, 10193), (10192, 18583)

X(23049) = EULEROLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO EHRMANN-VERTEX

Barycentrics    S^4-(6*R^2*(SA-SW)-SA^2+SB*SC+SW^2)*S^2+4*(3*R^2-SW)*SB*SC*SW : :
X(23049) = 2*X(4)+X(8549) = X(6)+2*X(18382) = X(66)+2*X(21850) = X(159)-4*X(19130) = X(576)+2*X(18383) = X(1350)-4*X(20300) = 4*X(5480)-X(19149) = X(5925)-4*X(15579) = X(8548)+2*X(18377) = 2*X(10113)+X(13248) = X(11216)+2*X(18376) = X(11255)+2*X(18379)

The reciprocal eulerologic center of these triangles is X(23048).

X(23049) lies on these lines: {4,6}, {30,10249}, {66,21850}, {159,19130}, {206,567}, {265,1351}, {381,2393}, {511,14852}, {542,11216}, {576,18383}, {858,1350}, {895,18434}, {1352,10297}, {1853,2781}, {1995,15577}, {3546,21167}, {5476,18400}, {5925,15579}, {6403,14644}, {6642,10182}, {7464,15578}, {7529,9920}, {7547,15073}, {8537,18394}, {8548,18377}, {10113,13248}, {10169,11179}, {10516,11188}, {10602,18386}, {11255,18379}, {11416,18392}, {11470,11572}, {13434,17845}, {14561,23041}, {18430,18440}, {18494,19136}

X(23049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12022, 14853, 6), (18430, 18449, 18440)

X(23050) = X(1)X(475)∩X(9)X(8750)

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

See Angel Montesdeoca, HG300818.

X(23050) lies on these lines: {1,475}, {9,8750}, {19,25}, {34,1883}, {75,1897}, {200,3192}, {210,3195}, {318,5263}, {474,17102}, {594,2331}, {975,6198}, {1041,5236}, {1249,2345}, {1876,3242}, {2207,7079}, {2550,7952}, {8270,11677}, {12329,20613} {98,6530}, {107,685}, {112,2966}, {648,17932}, {2422,2442}, {14273,15459}

X(23051) = X(10)X(400)∩X(19)X(38)

Barycentrics    a/(3a^2+b^2+c^2) : :

See Angel Montesdeoca, HG300818.

X(23051) lies on these lines: {10,4000}, {19,38}, {37,614}, {63,82}, {65,3242}, {75,16750}, {158,20883}, {225,8801}, {759,907}, {969,3873}, {1910,17467}, {2186,17445}, {2345,3677}, {3668,11677}, {8769,17446}, {16517,18785}

X(23052) = X(1)X(19)∩X(4)X(3663)

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

See Angel Montesdeoca, HG300818.

X(23052) lies on these lines: {1,19}, {4,3663}, {33,3666}, {34,6180}, {38,1096}, {63,162}, {75,1895}, {158,20883}, {278,3677}, {281,7174}, {474,17102}, {518,2331}, {811,3403}, {982,1435}, {984,7079}, {986,11471}, {1040,19649}, {1210,1861}, {1767,8270}, {1783,5223}, {1859,17599}, {3242,14571}, {4310,5236}, {4847,17903}, {5573,17917}

X(23053) = X(2)X(6)∩X(671)X(3524)

Barycentrics    17a^4-20a^2(b^2+c^2)+11b^4-26b^2c^2+11c^4 : :

See Angel Montesdeoca, HG300818.

X(23053) lies on these lines: {2,6}, {671,3524}, {1153,2549}, {3545,13449}, {5067,7812}, {5210,20112}, {7607,11172}, {7612,11179}, {11147,16509}, {14568,15709}

X(23054) = X(1992)X(16509)∩X(4232)X(8860)

Barycentrics    1/(19a^4-40a^2b^2+13b^4-40a^2c^2-10b^2c^2+13c^4) : :

See Angel Montesdeoca, HG300818.

X(23054) lies on these lines: {1992,16509}, {4232,8860}

X(23055) = X(2)X(6)∩X(98)X(11172)

Barycentrics    11a^4-8a^2(b^2+c^2)+5b^4-14b^2c^2+5c^4 : :

See Angel Montesdeoca, HG300818.

X(23055) lies on these lines: {2,6}, {98,11172}, {99,5485}, {115,8182}, {187,7615}, {376,671}, {381,9752}, {598,1285}, {1384,3363}, {2549,5569}, {3090,7812}, {3524,14568}, {3533,7760}, {3543,9756}, {3545,10788}, {3785,11318}, {5067,6179}, {5461,16041}, {6054,9754}, {7617,7737}, {7619,7798}, {7620,8598}, {7710,11177}, {7757,15702}, {7810,14064}, {7817,16043}, {9166,14907}, {9167,17131}, {9209,14977}, {9741,11054}, {9759,11061}, {11159,16509}, {12150,18842} : :

X(23055) = reflection of X(1007) in X(2)

X(23056) = X(926)X(2170)∩X(2246)X(4845)

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

See Emmanuel José García and Angel Montesdeoca, AdGeom 4943 and HG100918.

X(23056) lies on these lines: {926,2170}, {2246,4845}, {3119,3900}, {4162,7004} : :

X(23057) = X(1)X(22254)∩X(145)X(3716)

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

See Emmanuel José García and Angel Montesdeoca, AdGeom 4943 and HG200918.

X(23056) lies on these lines: {1,2254}, {145,3716}, {513,4162}, {519,14430}, {663,14077}, {891,3251}, {905,4959}, {1635,3722}, {2814,16200}, {2832,10699}, {3244,3762}, {3295,8648}, {3900,14414}, {8572,20315}

X(23057) = reflection of X(i) in X(j) for thiese (i,j): (2254,14413), (14413,1)

X(23058) = X(1)X(1146)∩X(4)X(9)

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

See Kadir Altintas and Angel Montesdeoca, HG110918.

X(23058) lies on these lines: {1,1146}, {2,3160}, {4,9}, {5,5514}, {41,5727}, {101,5881}, {142,10004}, {219,4034}, {220,3679}, {282,7100}, {610,5787}, {728,6735}, {910,5691}, {938,1449}, {1212,1698}, {1375,16832}, {1419,5942}, {1446,18634}, {1737,16572}, {2082,9581}, {2262,5806}, {2324,4007}, {2886,5574}, {3061,17284}, {3119,5219}, {3632,6603}, {3673,4858}, {3684,12625}, {4136,4901}, {4515,4873}, {4534,11376}, {4875,5231}, {5437,20205}, {5540,10826}, {5880,15725}, {6506,7741}, {6706,20195}, {6913,7367}, {7988,13609}, {7991,17747}, {8558,9579}, {9367,16975}, {17435,20271}

X(23058) = midpoint of X(i) and X(j) for these {i,j}: {3160, 10405}, {7090, 14121}
X(23058) = reflection of X10004) in X(142)

X(23059) = X(3)X(74)∩X(60)X(65)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28223.

X(23059) lies on these lines: {3,74}, {21,6001}, {46,17104}, {60,65}, {229,13750}, {517,1437}, {1768,1789}, {1790,10902}, {3615,6831}

X(23060) = X(3)X(74)∩X(30)X(12242)

Barycentrics    a^2 (4 a^8-11 a^6 b^2+9 a^4 b^4-a^2 b^6-b^8-11 a^6 c^2-2 a^4 b^2 c^2+10 a^2 b^4 c^2+3 b^6 c^2+9 a^4 c^4+10 a^2 b^2 c^4-4 b^4 c^4-a^2 c^6+3 b^2 c^6-c^8) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28223.

X(23060) lies on these lines: {3,74}, {30,12242}, {143,17714}, {546,13470}, {1173,5899}, {1493,13391}, {1503,13565}, {3518,12006}, {3857,15432}, {7525,15083}, {10263,11423}, {10594,13364}, {10610,14865}, {11017,14157}, {11264,16618}, {12010,20304}, {12088,16982}, {12105,15012}, {12107,13630}, {12812,20190}, {15806,17712}

X(23061) = X(3)X(54)∩X(23)X(110)

Barycentrics    a^2 (a^4-3 a^2 b^2+2 b^4-3 a^2 c^2+b^2 c^2+2 c^4) : :
X(23061) = 2 X[23] - 3 X[110], X[23] - 3 X[323], 5 X[23] - 6 X[1495], 5 X[110] - 4 X[1495], 5 X[323] - 2 X[1495], 6 X[1568] - 5 X[3091], 3 X[1495] - 5 X[3292], 3 X[110] - 4 X[3292], 3 X[323] - 2 X[3292], 3 X[3580] - 4 X[5159], 4 X[858] - 3 X[9140], 2 X[895] - 3 X[11416], 4 X[10510] - 3 X[11416], 4 X[7464] - 3 X[13445], 11 X[3525] - 12 X[14156], 4 X[5609] - 3 X[14157], 6 X[186] - 7 X[15020], 6 X[2071] - 5 X[15021], 12 X[2072] - 11 X[15025], 12 X[403] - 13 X[15029], 4 X[7575] - 5 X[15034], 2 X[3581] - 3 X[15035], 8 X[10297] - 7 X[15044], 3 X[13445] - 2 X[15054], 4 X[10564] - 3 X[15055], 16 X[5159] - 15 X[15059], 4 X[3580] - 5 X[15059], 8 X[1495] - 5 X[15107], 4 X[23] - 3 X[15107], 8 X[3292] - 3 X[15107], 4 X[323] - X[15107], 7 X[15057] - 8 X[15122], 4 X[468] - 3 X[15360], 9 X[11416] - 8 X[15826], 3 X[895] - 4 X[15826], 3 X[10510] - 2 X[15826], 3 X[10733] - 4 X[18323], 3 X[10706] - 2 X[18325], 9 X[15035] - 8 X[18571], 3 X[3581] - 4 X[18571], 3 X[9143] - X[20063], 5 X[15034] - 6 X[22115], 2 X[7575] - 3 X[22115]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28226.

X(23061) lies on these lines: {2,576}, {3,54}, {6,5888}, {23,110}, {30,14094}, {49,7555}, {51,16042}, {67,524}, {69,8542}, {111,8586}, {155,12082}, {182,11004}, {184,6030}, {186,15020}, {193,11511}, {352,3291}, {394,1995}, {403,15029}, {468,15360}, {542,5189}, {575,1994}, {852,14919}, {1147,7556}, {1173,3628}, {1199,5447}, {1216,7550}, {1350,15080}, {1351,5640}, {1568,3091}, {2071,15021}, {2072,15025}, {2076,20976}, {2452,9159}, {2889,12242}, {2930,9019}, {2937,9705}, {2987,14510}, {3098,11003}, {3124,15514}, {3146,12278}, {3231,5111}, {3266,9146}, {3448,5965}, {3525,14156}, {3529,12118}, {3580,5159}, {3581,15035}, {4232,11470}, {5028,9463}, {5050,21766}, {5094,8537}, {5097,5650}, {5104,14567}, {5297,19369}, {5422,11482}, {5562,7527}, {5609,13391}, {5611,11131}, {5615,11130}, {5651,10545}, {6090,10546}, {6243,12106}, {6515,8538}, {7292,8540}, {7464,13445}, {7512,9706}, {7545,10263}, {7575,15034}, {7772,8623}, {8541,11160}, {9143,19924}, {9155,9301}, {9306,14002}, {9968,11206}, {9972,21243}, {10116,11271}, {10297,15044}, {10300,11245}, {10552,14712}, {10564,15055}, {10706,18325}, {10733,18323}, {11440,13346}, {11449,17834}, {11451,17811}, {12164,12279}, {13248,15126}, {13366,15246}, {13421,13621}, {13431,18128}, {13595,21969}, {15057,15122}, {15062,18436}, {15520,22112}, {16982,18369}, {19128,19504}

X(23061) = reflection of X(i) in X(j) for these {i,j}: {23, 3292}, {54, 15137}, {110, 323}, {895, 10510}, {15054, 7464}, {15107, 110}
X(23061) = isotomic conjugate of isogonal conjugate of X(39231)
X(23061) = crossdifference of every pair of points on line {1640, 12073}
X(23061) = crosspoint of X(i) and X(j) for these (i,j): {249, 892}
X(23061) = crosssum of X(i) and X(j) for these (i,j): {115, 351}, {187, 13366}
X(23061) = barycentric product X(i)*X(j) for these {i,j}: {3266, 10558}, {5468, 10562}
X(23061) = trilinear product of X(10558) and X(14210)
X(23061) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {897, 2888}, {923, 17035}, {2148, 8591}, {2167, 14360}
X(23061) = barycentric quotient X(i)/X(j) for these {i,j}: {10558, 111}, {10562, 5466}
X(23061) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 576, 15019), (2, 15019, 5643), (3, 1993, 11422), (3, 11422, 5012), (23, 323, 3292), (23, 3292, 110), (394, 11477, 1995), (575, 3917, 7496), (895, 10510, 11416), (1351, 15066, 5640), (1993, 2979, 5012), (1994, 7496, 575), (1995, 11477, 3060), (2979, 11422, 3), (5097, 5650, 15018), (5643, 15019, 12834), (5651, 11002, 10545), (7464, 15054, 13445), (7492, 9716, 184), (7575, 22115, 15034), (8586, 9225, 20977), (9225, 20977, 111)

X(23062) = ISOGONAL CONJUGATE OF X(6602)

Barycentrics    tan^2 A/2 sec^2 A/2 : :

See Randy Hutson, Hyacinthos 28227.

Let A38B38C38 be Gemini triangle 38. Let A' be the center of conic {{A,B,C,B38,C38}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(23062). (Randy Hutson, January 15, 2019)

X(23062) lies on these lines: {7,354}, {57,10509}, {85,142}, {269,4626}, {279,1418}, {658,1445}, {664,3174}, {738,1434}, {934,2369}, {1446,10004}, {1996,8232}, {2191,4350}, {6046,7233}, {8732,17093}, {10481,15841}, {11495,14189}

X(23062) = isogonal conjugate of X(6602)
X(23062) = isotomic conjugate of X(728)
X(23062) = barycentric square of X(555)
X(23062) = X(i)-beth conjugate of X(j) for these (i,j): {85, 10004}, {4616, 269}
X(23062) = X(i)-cross conjugate of X(j) for these (i,j): {279, 1088}, {2170, 3676}
XX(23062) = cevapoint of X(i) and X(j) for these (i,j): {7, 8732}, {57, 4350}, {279, 479}, {2170, 3676}
X(23062) = crosssum of X(3022) and X(6607)
(23062) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6602}, {6, 480}, {8, 14827}, {9, 1253}, {31, 728}, {32, 5423}, {33, 1802}, {41, 200}, {55, 220}, {101, 4105}, {212, 7079}, {219, 7071}, {341, 9447}, {346, 2175}, {607, 1260}, {644, 8641}, {657, 3939}, {692, 4130}, {1110, 3119}, {1146, 6066}, {1252, 3022}, {1334, 2328}, {1500, 6061}, {2192, 7368}, {2194, 4515}, {2212, 3692}, {2318, 2332}, {2346, 8551}, {3063, 4578}, {4524, 5546}, {6065, 14936}, {7054, 7064}, {7074, 7367}, {8012, 10482}
X(23062) = barycentric product X(i)*X(j) for these {i,j}: {7, 1088}, {75, 479}, {76, 738}, {85, 279}, {269, 6063}, {273, 7056}, {331, 7177}, {348, 1847}, {555, 555}, {561, 7023}, {693, 4626}, {873, 6046}, {1119, 7182}, {1407, 20567}, {1434, 1446}, {1502, 7366}, {3261, 4617}, {3676, 4569}, {4077, 4616}, {4635, 7178}, {18810, 21314}
X(23062) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 480}, {2, 728}, {6, 6602}, {7, 200}, {34, 7071}, {56, 1253}, {57, 220}, {75, 5423}, {77, 1260}, {85, 346}, {222, 1802}, {223, 7368}, {226, 4515}, {244, 3022}, {269, 55}, {273, 7046}, {278, 7079}, {279, 9}, {331, 7101}, {348, 3692}, {479, 1}, {513, 4105}, {514, 4130}, {552, 1098}, {555, 6731}, {604, 14827}, {658, 644}, {664, 4578}, {693, 4163}, {738, 6}, {757, 6061}, {934, 3939}, {1014, 2328}, {1086, 3119}, {1088, 8}, {1106, 2175}, {1111, 4081}, {1119, 33}, {1254, 7064}, {1358, 2310}, {1396, 2332}, {1398, 2212}, {1407, 41}, {1418, 8012}, {1422, 7367}, {1427, 1334}, {1434, 2287}, {1435, 607}, {1439, 2318}, {1441, 4082}, {1446, 2321}, {1475, 8551}, {1847, 281}, {3668, 210}, {3669, 657}, {3673, 4012}, {3674, 3965}, {3676, 3900}, {4017, 4524}, {4350, 6600}, {4554, 6558}, {4566, 4069}, {4569, 3699}, {4573, 7259}, {4616, 643}, {4617, 101}, {4625, 7256}, {4626, 100}, {4635, 645}, {4637, 5546}, {5435, 4936}, {6046, 756}, {6063, 341}, {6612, 7118}, {6614, 692}, {7023, 31}, {7045, 6065}, {7053, 212}, {7056, 78}, {7143, 872}, {7147, 1500}, {7177, 219}, {7178, 4171}, {7182, 1265}, {7185, 4073}, {7195, 4319}, {7197, 612}, {7203, 21789}, {7204, 4517}, {7216, 3709}, {7339, 1110}, {7366, 32}, {7371, 6726}, {10481, 3059}, {10509, 6605}, {14256, 2324}, {17096, 1021}, {20618, 3949}

X(23063) = X(115)X(244)∩X(678)X(1283)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28238.

X(23063) is the 4th intersection, other than the vertices of the incentral triangle, of the incentral inellipse and the incentral circle. (Randy Hutson, October 15, 2018)

X(23063) lies on the incentral inellipse, the incentral circle, and on these lines: {115,244}, {678,1283}, {756,8701}, {2310,3024}, {7004,14101}

X(23064) = X(30)X(511)∩X(6578)X(8701)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28238.

X(23064) lies on these lines: {30, 511}, {6578, 8701}

X(23065) = (name pending)

Barycentrics    a (a^7 b+4 a^6 b^2+2 a^5 b^3-7 a^4 b^4-7 a^3 b^5+2 a^2 b^6+4 a b^7+b^8+a^7 c+10 a^6 b c+12 a^5 b^2 c-14 a^4 b^3 c-23 a^3 b^4 c +a^2 b^5 c+10 a b^6 c+3 b^7 c+4 a^6 c^2+12 a^5 b c^2-10 a^4 b^2 c^2-33 a^3 b^3 c^2-18 a^2 b^4 c^2+3 a b^5 c^2+2 b^6 c^2+2 a^5 c^3 -14 a^4 b c^3-33 a^3 b^2 c^3-32 a^2 b^3 c^3-17 a b^4 c^3-3 b^5 c^3-7 a^4 c^4-23 a^3 b c^4-18 a^2 b^2 c^4-17 a b^3 c^4-6 b^4 c^4-7 a^3 c^5 +a^2 b c^5+3 a b^2 c^5-3 b^3 c^5+2 a^2 c^6+10 a b c^6+2 b^2 c^6+4 a c^7+3 b c^7+c^8) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28238.

X(23065) lies on these lines: {517, 3723}

X(23066) = (name pending)

Barycentrics    a (b-c) (a^8+4 a^7 b+4 a^6 b^2-4 a^5 b^3-10 a^4 b^4-4 a^3 b^5+4 a^2 b^6+4 a b^7+b^8+4 a^7 c+10 a^6 b c+2 a^5 b^2 c-16 a^4 b^3 c-16 a^3 b^4 c +2 a^2 b^5 c+10 a b^6 c+4 b^7 c+4 a^6 c^2+2 a^5 b c^2-11 a^4 b^2 c^2-16 a^3 b^3 c^2-17 a^2 b^4 c^2-2 a b^5 c^2+4 b^6 c^2-4 a^5 c^3-16 a^4 b c^3 -16 a^3 b^2 c^3-30 a^2 b^3 c^3-26 a b^4 c^3-4 b^5 c^3-10 a^4 c^4-16 a^3 b c^4-17 a^2 b^2 c^4-26 a b^3 c^4-10 b^4 c^4-4 a^3 c^5+2 a^2 b c^5-2 a b^2 c^5 -4 b^3 c^5+4 a^2 c^6+10 a b c^6+4 b^2 c^6+4 a c^7+4 b c^7+c^8) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28238.

X(23066) lies on these lines: {30, 511}

X(23067) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23067) lies on these lines: {3, 201}, {46, 15443}, {55, 11028}, {56, 214}, {72, 22341}, {100, 108}, {101, 109}, {110, 15439}, {219, 296}, {222, 295}, {228, 1214}, {603, 22458}, {643, 4564}, {1020, 4551}, {1260, 7011}, {1331, 1813}, {1376, 6358}, {1393, 16414}, {1708, 2352}, {1825, 11248}, {2222, 6011}, {2599, 11849}, {3185, 8270}, {3428, 11713}, {4561, 4571}, {4574, 23084}, {7078, 20764}, {17975, 17976}

X(23067) = isogonal conjugate of polar conjugate of X(4552)
X(23067) = isotomic conjugate of polar conjugate of X(4559)
X(23067) = X(19)-isoconjugate of X(4560)
X(23067) = X(92)-isoconjugate of X(7252)

X(23068) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23068) lies on these lines: {3, 22348}, {219, 22137}, {1260, 23071}, {3157, 7016}

X(23068) = isogonal conjugate of polar conjugate of X(17481)
X(23068) = isotomic conjugate of polar conjugate of X(21771)

X(23069) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23069) lies on these lines: {3, 22349}, {3157, 7016}, {22384, 23092}

X(23069) = isogonal conjugate of polar conjugate of X(17482)
X(23069) = isotomic conjugate of polar conjugate of X(21772)

X(23070) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23070) lies on these lines: {1, 399}, {3, 73}, {5, 651}, {6, 5708}, {30, 3562}, {63, 22136}, {72, 22128}, {81, 6147}, {109, 5399}, {140, 17074}, {195, 3461}, {221, 1482}, {381, 8757}, {394, 3927}, {912, 18447}, {942, 2003}, {1071, 18455}, {1079, 1454}, {1393, 14627}, {1419, 5709}, {1498, 12684}, {1771, 18524}, {1935, 7489}, {4306, 5398}, {4855, 22141}, {4860, 16472}, {5221, 16473}, {5706, 18541}, {5707, 6180}, {5779, 17814}, {5790, 9370}, {10571, 22765}, {15066, 15650}, {20739, 22163}, {20741, 22164}, {22148, 22458}

X(23070) = isogonal conjugate of polar conjugate of X(17483)
X(23070) = isotomic conjugate of polar conjugate of X(21773)

X(23071) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23071) lies on these lines: {1, 195}, {3, 73}, {5, 3562}, {6, 15934}, {30, 651}, {35, 8614}, {72, 18447}, {81, 5719}, {201, 7100}, {219, 22142}, {221, 12702}, {382, 8757}, {394, 3940}, {399, 3465}, {484, 6126}, {517, 1456}, {549, 17074}, {582, 4306}, {912, 18455}, {943, 5453}, {1149, 7373}, {1203, 5045}, {1260, 23068}, {1419, 3587}, {1459, 17976}, {1935, 13743}, {2392, 20872}, {3173, 18445}, {4551, 18524}, {4574, 20741}, {4585, 16086}, {5172, 6149}, {5440, 22128}, {5758, 18624}, {6180, 18541}, {8144, 12528}, {9370, 18525}, {20752, 22144}, {22124, 22147}

X(23071) = isogonal conjugate of polar conjugate of X(17484)
X(23071) = isotomic conjugate of polar conjugate of X(19297)
X(23071) = X(19)-isoconjugate of X(21739)
X(23071) = X(92)-isoconjugate of X(19302)

X(23072) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23072) lies on these lines: {3, 73}, {34, 2095}, {47, 1617}, {58, 999}, {109, 1413}, {221, 22770}, {283, 22129}, {495, 4340}, {517, 1394}, {580, 1407}, {651, 3149}, {942, 1453}, {991, 5399}, {1012, 3562}, {1060, 3927}, {1092, 7053}, {1259, 22128}, {1456, 12704}, {1771, 9370}, {1935, 6913}, {3075, 6918}, {3167, 20805}, {3664, 11374}, {6149, 7742}, {7011, 7335}, {8757, 19541}, {9538, 13243}, {11700, 12635}, {15905, 20764}

X(23072) = isogonal conjugate of polar conjugate of X(9965)
X(23072) = isotomic conjugate of polar conjugate of X(37519)

X(23073) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23073) lies on these lines: {3, 48}, {6, 5563}, {19, 10222}, {45, 16554}, {184, 22371}, {218, 7113}, {222, 1797}, {284, 3303}, {610, 7982}, {944, 7359}, {1388, 1731}, {1482, 2173}, {1732, 5126}, {2256, 3746}, {2286, 22122}, {2323, 3207}, {3284, 22124}, {7124, 22123}, {13462, 16670}, {16189, 18594}, {16547, 16884}, {17455, 22767}, {20760, 22158}

X(23073) = isogonal conjugate of polar conjugate of X(3241)
X(23073) = X(19)-isoconjugate of X(36588)

X(23074) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^9 b - 2 a^5 b^5 + a b^9 + a^9 c + a^8 b c - a b^8 c - b^9 c - 2 a^5 c^5 + 2 b^5 c^5 - a b c^8 + a c^9 - b c^9) : :

X(23074) lies on these lines: {1, 159}, {3, 22362}

X(23074) = isogonal conjugate of polar conjugate of X(21215)
X(23074) = isotomic conjugate of polar conjugate of X(21774)

X(23075) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23075) lies on these lines: {1, 7083}, {3, 326}, {31, 15370}, {219, 7015}, {255, 7193}, {2300, 3167}, {3186, 3732}, {3564, 15976}, {20764, 23076}

X(23075) = isogonal conjugate of polar conjugate of X(21216)
X(23075) = isotomic conjugate of polar conjugate of X(21775)

X(23076) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23076) lies on these lines: {3, 304}, {219, 23077}, {863, 21216}, {3157, 23083}, {19597, 23078}, {20760, 22164}, {20764, 23075}, {20794, 22458}, {20805, 23091}

X(23076) = isogonal conjugate of polar conjugate of X(17486)
X(23076) = isotomic conjugate of polar conjugate of X(21776)

X(23077) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23077) lies on these lines: {3, 22367}, {219, 23076}, {17976, 22138}, {20739, 20760}, {20796, 22126}, {22164, 23083}

X(23077) = isogonal conjugate of polar conjugate of X(21217)
X(23077) = isotomic conjugate of polar conjugate of X(21777)

X(23078) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23078) lies on these lines: {3, 348}, {255, 7193}, {19597, 23076}

X(23078) = isogonal conjugate of polar conjugate of X(21218)
X(23078) = isotomic conjugate of polar conjugate of X(21778)

X(23079) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23079) lies on these lines: {3, 69}, {48, 20762}, {71, 20796}, {219, 7015}, {1030, 1634}, {1654, 16372}, {2895, 20848}, {3511, 3882}, {4254, 11328}, {4648, 16420}, {7078, 20793}, {8681, 18591}, {11343, 20139}, {17778, 20845}, {17976, 22136}, {20740, 20795}, {20769, 22389}, {22141, 23083}

X(23079) = isogonal conjugate of polar conjugate of X(1655)
X(23079) = isotomic conjugate of polar conjugate of X(21779)

X(23080) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23080) lies on these lines: {3, 22370}, {219, 20785}, {1332, 20794}, {7078, 17976}, {20762, 20818}

X(23080) = isogonal conjugate of polar conjugate of X(21219)
X(23080) = isotomic conjugate of polar conjugate of X(21780)

X(23081) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (5 a^2 - 5 a b - b^2 - 5 a c + 7 b c - c^2) : :

X(23081) lies on these lines: {3, 1331}, {219, 23082}, {3955, 22357}, {20760, 22158}, {22139, 22143}

X(23081) = isogonal conjugate of polar conjugate of X(17487)
X(23081) = isotomic conjugate of polar conjugate of X(21781)

X(23082) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23082) lies on these lines: {3, 22067}, {219, 23081}, {20760, 22356}

X(23082) = isogonal conjugate of polar conjugate of X(17488)
X(23082) = isotomic conjugate of polar conjugate of X(21782)

X(23083) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23083) lies on these lines: {3, 4592}, {3157, 23076}, {20739, 23088}, {20760, 23084}, {20766, 20796}, {22141, 23079}, {22143, 22144}, {22148, 23091}, {22164, 23077}

X(23083) = isogonal conjugate of polar conjugate of X(21220)
X(23083) = isotomic conjugate of polar conjugate of X(21783)

X(23084) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23084) lies on these lines: {3, 3708}, {525, 6516}, {647, 906}, {4574, 23067}, {20739, 20764}, {20760, 23083}, {20802, 22458}, {22164, 22457}

X(23084) = isogonal conjugate of polar conjugate of X(6758)
X(23084) = isotomic conjugate of polar conjugate of X(21784)
X(23084) = X(19)-isoconjugate of X(7372)

X(23085) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23085) lies on these lines: {3, 63}, {56, 4650}, {329, 19514}, {404, 17350}, {603, 7193}, {5687, 9369}, {5744, 9840}, {7078, 22148}, {7288, 15507}, {15803, 16059}

X(23085) = isogonal conjugate of polar conjugate of X(17490)
X(23085) = isotomic conjugate of polar conjugate of X(21785)

X(23086) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23086) lies on these lines: {3, 22370}, {28, 330}, {48, 3955}, {56, 87}, {69, 22096}, {104, 932}, {219, 2196}, {295, 20753}, {603, 7193}, {604, 11328}, {982, 18194}, {1332, 20787}, {1333, 2162}, {1436, 2319}, {1437, 20805}, {1472, 7121}, {3733, 4361}, {6384, 18749}, {8843, 20992}, {20765, 20799}, {20769, 22152}, {20793, 22118}, {20796, 20818}

X(23086) = isogonal conjugate of polar conjugate of X(330)
X(23086) = isotomic conjugate of polar conjugate of X(2162)
X(23086) = X(63)-cross conjugate of X(3)
X(23086) = X(19)-isoconjugate of X(192)
X(23086) = X(92)-isoconjugate of X(2176)

X(23087) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23087) lies on these lines: {3, 22379}, {56, 3738}, {521, 22091}, {667, 9048}, {900, 10074}, {905, 9051}, {999, 1769}, {1331, 1813}, {1459, 4091}, {1795, 8677}, {3733, 4063}, {4491, 14812}, {4768, 12513}, {22148, 22158}

X(23087) = isogonal conjugate of polar conjugate of X(21222)
X(23087) = isotomic conjugate of polar conjugate of X(21786)

X(23088) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^3 b^3 - a^3 b^2 c + a^2 b^3 c - 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(23088) lies on these lines: {3, 22370}, {2200, 20794}, {3167, 20796}, {20739, 23083}, {20760, 22164}, {22127, 22158}

X(23088) = isogonal conjugate of polar conjugate of X(21223)
X(23088) = isotomic conjugate of polar conjugate of X(21787)

X(23089) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23089) lies on these lines: {3, 63}, {81, 7373}, {101, 1407}, {144, 16435}, {189, 952}, {198, 3928}, {222, 20818}, {329, 19517}, {394, 22147}, {527, 15509}, {3167, 22148}, {3210, 3732}, {4383, 21362}, {5294, 21542}, {7193, 22117}, {9965, 11347}

X(23089) = isogonal conjugate of polar conjugate of X(4452)
X(23089) = isotomic conjugate of polar conjugate of X(1616)
X(23089) = X(19)-isoconjugate of X(6553)

X(23090) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23090) lies on these lines: {3, 822}, {6, 16612}, {110, 677}, {112, 6081}, {219, 8611}, {284, 2432}, {425, 2501}, {448, 525}, {520, 3733}, {521, 650}, {662, 7045}, {905, 4131}, {1172, 2431}, {3676, 18199}, {4765, 21007}, {7253, 15146}, {10015, 17925}

X(23090) = isogonal conjugate of polar conjugate of X(7253)
X(23090) = isotomic conjugate of polar conjugate of X(21789)
X(23090) = X(19)-isoconjugate of X(4566)

X(23091) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23091) lies on these lines: {3, 4561}, {3504, 22149}, {4574, 20760}, {20797, 22458}, {20805, 23076}, {22148, 23083}

X(23091) = isogonal conjugate of polar conjugate of X(21224)
X(23091) = isotomic conjugate of polar conjugate of X(21790)

X(23092) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23092) lies on these lines: {3, 22387}, {6, 4369}, {905, 4131}, {3049, 15419}, {4481, 7252}, {17217, 17921}, {20816, 22157}, {22384, 23069}

X(23092) = isogonal conjugate of polar conjugate of X(17217)
X(23092) = isotomic conjugate of polar conjugate of X(16695)

X(23093) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23093) lies on these lines: {3, 4025}, {647, 8673}, {652, 20760}, {3239, 16058}, {7658, 16059}

X(23093) = isogonal conjugate of polar conjugate of X(21225)
X(23093) = isotomic conjugate of polar conjugate of X(21791)

X(23094) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23094) lies on these lines: {3, 22370}, {48, 20762}, {219, 20794}, {1332, 20775}, {3167, 20752}, {4020, 7015}, {22122, 22143}

X(23094) = isogonal conjugate of polar conjugate of X(21226)
X(23094) = isotomic conjugate of polar conjugate of X(21792)

X(23095) =  (A,B,C,X(1); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23095) lies on these lines: {3, 48}, {101, 182}, {184, 1331}, {218, 19554}, {222, 17972}, {255, 20765}, {284, 16516}, {613, 19561}, {1437, 20805}, {2182, 20430}, {3167, 22161}, {3955, 22149}, {5009, 21769}, {20794, 22458}

X(23095) = isogonal conjugate of polar conjugate of X(4393)
X(23095) = isotomic conjugate of polar conjugate of X(21793)
X(23095) = X(19)-isoconjugate of X(27494)

X(23096) =  X(23)X(20185)∩X(25)X(1291)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-4 a^8 b^2+4 a^6 b^4+2 a^4 b^6-5 a^2 b^8+2 b^10-3 a^8 c^2+11 a^6 b^2 c^2-7 a^4 b^4 c^2+10 a^2 b^6 c^2-5 b^8 c^2+2 a^6 c^4-14 a^4 b^2 c^4-7 a^2 b^4 c^4+2 b^6 c^4+2 a^4 c^6+11 a^2 b^2 c^6+4 b^4 c^6-3 a^2 c^8-4 b^2 c^8+c^10) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-4 a^8 c^2+11 a^6 b^2 c^2-14 a^4 b^4 c^2+11 a^2 b^6 c^2-4 b^8 c^2+4 a^6 c^4-7 a^4 b^2 c^4-7 a^2 b^4 c^4+4 b^6 c^4+2 a^4 c^6+10 a^2 b^2 c^6+2 b^4 c^6-5 a^2 c^8-5 b^2 c^8+2 c^10) : :

If you have GeoGebra, you can view X(23096).

See Telv Cohl and Peter Moses, Hyacinthos 28244.

X(23096) lies on the circumcircle and these lines: {23,20185}, {25,1291}, {468,930}, {691,3518}, {2070,3565}, {2696,7576}, {10420,13595}, {11635,13621}

leftri

K244 Moses images: X(23097)-X(23110)

rightri

If a point P on the circumcircle of a triangle ABC has barycentrics p : q : r, then then point a^2 q r (c^2 q + b^2 r) : : lies on the cubic K244. The following fourteen examples of K244 Moses images were contributed by Peter Moses, September 13, 2018. See also the preamble just before X(23342).

The Moses K244 image of P is the trilinear cube of the isogonal conjugate of P. (Randy Hutson, November 30, 2018)

X(23097) =  MOSES K244 IMAGE OF 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)^3 : :

X(23907) lies on the cubic K244 and these lines: {4, 69}, {30, 14254}, {94, 10733}, {858, 18279}, {1495, 15454}, {1568, 11251}, {9003, 15063}, {16163, 16240}

X(23097) = trilinear cube of X(30)
X(23097) = isotomic conjugate of isogonal conjugate of X(3081)
X(23097) = barycentric product X(i)*X(j) for these {i,j}: {76, 3081}, {1099, 14206}, {3163, 3260}
X(23097) = barycentric quotient X(i)/X(j) for these {i,j}: {1099, 2349}, {3081, 6}, {3163, 74}, {14401, 14380}, {16163, 14919}, {16240, 8749}

X(23098) =  MOSES K244 IMAGE OF X(98)

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

X(23908) lies on the cubic K244 and these lines: {3, 2421}, {5, 76}, {32, 1092}, {446, 511}, {684, 690}, {2080, 10411}, {6394, 14941}, {9419, 11672}

X(23098) = trilinear cube of X(511)
X(23098) = barycentric product X(i)*X(j) for these {i,j}: {325, 11672}, {3569, 15631}
X(23098) = barycentric quotient X(i)/X(j) for these {i,j}: {2967, 16081}, {9419, 1976}, {11672, 98}

X(23099) =  MOSES K244 IMAGE OF X(99)

Barycentrics    a^4 (b-c)^3 (b+c)^3 : :
X(23098) = 3 X[2531] - 4 X[17415]

X(23909) lies on the cubic K244 and these lines: {32, 669}, {39, 512}, {76, 523}, {887, 2491}, {1499, 3095}, {1649, 3005}, {2793, 14272}, {4079, 21700}, {6071, 21906}, {9009, 13330}, {9178, 14263}, {14443, 20975}

X(23099) = reflection of X(i) in X(j) for these {i,j}: {887, 2491}, {14824, 39}
X(23099) = reflection of X(14824) in the Brocard axis
X(23099) = isogonal conjugate of the isotomic conjugate of X(22260)
X(23099) = X(i)-Ceva conjugate of X(j) for these (i,j): {523, 3124}, {669, 1084}
X(23099) = crosspoint of X(i) and X(j) for these (i,j): {523, 3124}, {669, 1084}
X(23099) = crossdifference of every pair of points on line {385, 3266}
X(23099) = crosssum of X(110) and X(4590)
X(23099) = trilinear cube of X(512)
X(23099) = X(i)-isoconjugate of X(j) for these (i,j): {249, 4602}, {799, 4590}, {1101, 4609}, {4600, 4623}, {4601, 4610}, {4620, 4631}, {4625, 6064}, {7257, 7340}
X(23099) = barycentric product X(i)*X(j) for these {i,j}: {6, 22260}, {32, 8029}, {115, 669}, {338, 9426}, {512, 3124}, {523, 1084}, {647, 2971}, {667, 21833}, {762, 8027}, {798, 2643}, {850, 9427}, {882, 2086}, {1109, 1924}, {1356, 3700}, {1500, 8034}, {1577, 4117}, {1918, 21131}, {1919, 21043}, {2489, 20975}, {3049, 8754}, {3121, 4705}, {3122, 4079}, {3249, 6535}, {3569, 15630}, {7063, 7178}, {9178, 21906}
X(23099) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 4609}, {669, 4590}, {1084, 99}, {1356, 4573}, {2086, 880}, {2643, 4602}, {2971, 6331}, {3121, 4623}, {3124, 670}, {3249, 6628}, {4117, 662}, {7063, 645}, {8029, 1502}, {9426, 249}, {9427, 110}, {21833, 6386}, {22260, 76}

X(23100) =  MOSES K244 IMAGE OF X(101)

Barycentrics    b^2 (b-c)^3 c^2 : :

X(239100) lies on the cubic K244 and these lines: {76, 3261}, {85, 514}, {649, 14377}, {693, 20880}, {1111, 21132}, {3673, 21201}, {19594, 21118}

X(23100) = reflection of X(14825) in X(2140)
X(23100) = isotomic conjugate of the isogonal conjugate of X(6545)
X(23100) = trilinear cube of X(514)
X(23100) = X(i)-isoconjugate of X(j) for these (i,j): {101, 1110}, {560, 6632}, {651, 6066}, {692, 1252}, {1253, 4619}, {1415, 6065}, {2149, 3939}, {2251, 6551}
X(23100) = crossdifference of every pair of points on line {6066, 9459}
X(23100) = barycentric product X(i)*X(j) for these {i,j}: {76, 6545}, {561, 764}, {693, 1111}, {850, 17205}, {1086, 3261}, {1090, 4569}, {1502, 21143}, {1577, 16727}, {1928, 8027}, {2973, 4025}, {4572, 7336}, {6063, 21132}, {7192, 21207}, {7199, 16732}, {16726, 20948}
X(23100) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 3939}, {76, 6632}, {244, 692}, {279, 4619}, {338, 4103}, {513, 1110}, {514, 1252}, {522, 6065}, {663, 6066}, {693, 765}, {764, 31}, {903, 6551}, {1086, 101}, {1090, 3900}, {1111, 100}, {1358, 109}, {1565, 1331}, {2969, 8750}, {2973, 1897}, {3120, 4557}, {3249, 1501}, {3261, 1016}, {3669, 2149}, {3676, 59}, {3942, 906}, {4466, 4574}, {4858, 644}, {4957, 4752}, {5532, 4105}, {6545, 6}, {6548, 9268}, {6549, 901}, {6550, 902}, {7192, 4570}, {7199, 4567}, {7336, 663}, {8027, 560}, {8034, 1918}, {8042, 1333}, {8661, 9459}, {14442, 1017}, {15634, 677}, {16726, 163}, {16727, 662}, {16732, 1018}, {17197, 5546}, {17205, 110}, {17880, 4571}, {21131, 1500}, {21132, 55}, {21133, 3730}, {21134, 3690}, {21143, 32}, {21202, 14887}, {21207, 3952}

X(23101) =  MOSES K244 IMAGE OF X(104)

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

X(23101) lies on the cubic K244 and these lines: {76, 3262}, {78, 1482}, {517, 14260}

X(23101) = trilinear cube of X(517)
X(23101) = barycentric product X(10015)X(15632)
X(23101) = barycentric quotient X(i)/X(j) for these {i,j}: {15632, 13136}, {21664, 16082}

X(23102) =  MOSES K244 IMAGE OF X(105)

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

X(23102) lies on the cubic K244 and these lines: {1, 728}, {8, 14947}, {76, 3263}, {1259, 3423}, {2481, 4518}, {2826, 3762}, {3126, 14506}, {3675, 3912}, {4712, 6184}

X(23102) = X(3263)-Ceva conjugate of X(4437)
X(23102) = X(1438)-isoconjugate of X(6185)
X(23102) = crosspoint of X(i) and X(j) for these (i,j): {3263, 4437}
X(23102) = trilinear cube of X(518)
X(23102) = barycentric product X(i)*X(j) for these {i,j}: {518, 4437}, {3263, 6184}, {3912, 4712}, {3932, 16728}
X(23102) = barycentric quotient X(i)/X(j) for these {i,j}: {518, 6185}, {1362, 1462}, {4437, 2481}, {4712, 673}, {6184, 105}

X(23103) =  MOSES K244 IMAGE OF X(107)

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

X(23103) lies on the cubic K244 and these lines: {76, 3265}, {523, 15318}, {525, 14059}, {684, 2848}, {3357, 22089}

X(23103) = trilinear cube of X(520)
X(23103) = barycentric quotient X(2972)/X(15352)

X(23104) =  MOSES K244 IMAGE OF X(109)

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

X(23104) lies on the cubic K244 and these lines: {318, 522}, {341, 4163}, {663, 10570}, {3701, 4397}

X(23104) = trilinear cube of X(522)
X(23104) = X(i)-isoconjugate of X(j) for these (i,j): {604, 4619}, {692, 7339}, {1110, 6614}, {1262, 1415}, {1461, 2149}
X(23104) = barycentric product X(i)*X(j) for these {i,j}: {646, 1090}, {1978, 5532}, {3261, 4081}, {4397, 4858}, {6332, 21666}
X(23104) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4619}, {11, 1461}, {514, 7339}, {522, 1262}, {764, 7366}, {1086, 6614}, {1090, 3669}, {1111, 4617}, {1146, 109}, {2310, 1415}, {2968, 1813}, {3119, 692}, {3239, 59}, {4081, 101}, {4130, 1110}, {4163, 1252}, {4391, 7045}, {4397, 4564}, {4858, 934}, {5532, 649}, {6545, 7023}, {21131, 7143}, {21132, 1407}, {21666, 653}

X(23105) =  MOSES K244 IMAGE OF X(110)

Barycentrics    b^2 (b-c)^3 c^2 (b+c)^3 : :
Trilinears    sin^3(B - C) : :
X(23105) = X[5489] + 3 X[8029], X[8151] - 3 X[14566], 3 X[10412] - X[18039]

X(23105) lies on the cubics K244 and K589 and on these lines: {4, 512}, {5, 523}, {68, 520}, {76, 850}, {99, 14734}, {110, 14781}, {338, 15359}, {525, 10279}, {647, 7746}, {670, 14728}, {690, 16003}, {826, 1209}, {868, 5489}, {924, 18381}, {1093, 18808}, {1116, 5664}, {2395, 3767}, {3548, 15421}, {4108, 14002}, {6041, 7755}, {6130, 14270}, {6249, 12073}, {7253, 14777}, {10278, 11007}

X(23105) = reflection of X(i) in X(j) for these {i,j}: {5664, 1116}, {14270, 6130}
X(23105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2395, 3767, 8574)
X(23105) = isotomic conjugate of the isogonal conjugate of X(8029)
X(23105) = X(i)-Ceva conjugate of X(j) for these (i,j): {850, 338}, {14618, 115}
X(23105) = X(i)-isoconjugate of X(j) for these (i,j): {110, 1101}, {163, 249}, {250, 4575}, {1983, 9273}, {2617, 14587}
X(23105) = crosspoint of X(i) and X(j) for these (i,j): {338, 850}, {523, 8901}
X(23105) = crossdifference of every pair of points on line {50, 3289}
X(23105) = trilinear cube of X(523)
X(23105) = pole wrt polar circle of line X(249)X(250)
X(23105) = Kirikami-Euler image of X(115)
X(23105) = trilinear product of vertices of Schroeter triangle
X(23105) = barycentric product X(i)*X(j) for these {i,j}: {76, 8029}, {115, 850}, {125, 14618}, {313, 21131}, {338, 523}, {339, 2501}, {525, 2970}, {1109, 1577}, {1502, 22260}, {2052, 5489}, {2643, 20948}, {3261, 21043}, {3267, 8754}, {4024, 21207}, {4036, 16732}, {8901, 18314}
X(23105) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 110}, {125, 4558}, {338, 99}, {339, 4563}, {523, 249}, {661, 1101}, {850, 4590}, {868, 2421}, {1084, 14574}, {1109, 662}, {1365, 4565}, {1648, 5467}, {2501, 250}, {2623, 14587}, {2643, 163}, {2970, 648}, {3120, 4556}, {3124, 1576}, {3708, 4575}, {4024, 4570}, {4036, 4567}, {4092, 5546}, {5489, 394}, {8029, 6}, {8288, 9145}, {8754, 112}, {8901, 18315}, {14618, 18020}, {15328, 18879}, {20902, 4592}, {21043, 101}, {21044, 4636}, {21046, 1331}, {21131, 58}, {21134, 1790}, {21207, 4610}, {21833, 692}, {22260, 32}

X(23106) =  MOSES K244 IMAGE OF X(111)

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

X(23106) lies on the cubic K244 and these lines: {2, 39}, {1649, 6077}, {2482, 16733}, {2793, 14278}, {6390, 9177}, {7813, 21906}

X(23106) = isotomic conjugate of the isogonal conjugate of X(8030)
X(23106) = X(923)-isoconjugate of X(10630)
X(23106) = trilinear cube of X(524)
X(23106) = barycentric product X(i)*X(j) for these {i,j}: {76, 8030}, {2482, 3266}
X(23106) = barycentric quotient X(i)/X(j) for these {i,j}: {524, 10630}, {1366, 7316}, {1649, 9178}, {2482, 111}, {5095, 8753}, {6390, 15398}, {7067, 5547}, {8030, 6}, {14444, 3124}

X(23107) =  MOSES K244 IMAGE OF X(112)

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

X(23107) lies on the cubic K244 and these lines: {76, 2394}, {647, 14376}, {3265, 3933}

X(23107) = trilinear cube of X(525)
X(23107) = barycentric product X(i)*X(j) for these {i,j}: {305, 5489}, {338, 4143}, {339, 3265}, {3267, 15526}, {14208, 17879}
X(23107) = barycentric quotient X(i)/X(j) for these {i,j}: {338, 6529}, {339, 107}, {2972, 1576}, {3265, 250}, {4143, 249}, {5489, 25}, {15526, 112}, {17879, 162}

X(23108) =  MOSES K244 IMAGE OF X(476)

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

X(23108) lies on the cubic K244 and these lines: {76, 3268}, {690, 14670}, {14270, 14385}

X(23108) = trilinear cube of X(526)
X(23108) = barycentric product X(3268)*X(18334)
X(23108) = barycentric quotient X(18334)/X(476)

X(23109) =  MOSES K244 IMAGE OF X(1113)

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

X(23109) lies on the cubics K244 and K1065 and on these lines: {3, 2574}, {4, 16071}, {523, 20408}, {684, 690}, {5489, 14499}, {9173, 11638}

X(23109) = crosspoint of X(1313) and X(2574)
X(23109) = crosssum of X(1113) and X(15461)
X(23109) = trilinear cube of X(2574)
X(23109) = X(2586)-isoconjugate of X(15461)
X(23109) = barycentric quotient X(15166)/X(1113)

X(23110) =  MOSES K244 IMAGE OF X(1114)

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

X(23110) lies on the cubics K244 and K1065 and on these lines: {3, 2575}, {4, 16070}, {523, 20409}, {684, 690}, {5489, 14500}, {9174, 11638}

X(23110) = crosspoint of X(1312) and X(2575)
X(23110) = crosssum of X(1114) and X(15460)
X(23110) = trilinear cube of X(2575)
X(23110) = X(2587)-isoconjugate of X(15460)
X(23110) = barycentric quotient X(15167)/X(1114)

X(23111) =  (name pending)

Trilinears    (4*cos(A)*cos(6*A)*cos(B-C)-cos(2*(B-C))-4*(2*cos(A)+cos(3*A))*cos(4*A)*cos(3*(B-C))+(4*cos(3*A)*cos(A)+1)*cos(4*(B-C))-4*(2*cos(A)+cos(3*A))*cos(5*A))*sin(A) : :
Barycentrics    a^4 (-a^18 b^6+8 a^16 b^8-28 a^14 b^10+56 a^12 b^12-70 a^10 b^14+56 a^8 b^16-28 a^6 b^18+8 a^4 b^20-a^2 b^22+2 a^20 b^2 c^2-11 a^18 b^4 c^2+26 a^16 b^6 c^2-35 a^14 b^8 c^2+27 a^12 b^10 c^2-a^10 b^12 c^2-29 a^8 b^14 c^2+39 a^6 b^16 c^2-25 a^4 b^18 c^2+8 a^2 b^20 c^2-b^22 c^2-11 a^18 b^2 c^4 +48 a^16 b^4 c^4-82 a^14 b^6 c^4+64 a^12 b^8 c^4-13 a^10 b^10 c^4-7 a^8 b^12 c^4-16 a^6 b^14 c^4+34 a^4 b^16 c^4-22 a^2 b^18 c^4+5 b^20 c^4-a^18 c^6+26 a^16 b^2 c^6-82 a^14 b^4 c^6+104 a^12 b^6 c^6-61 a^10 b^8 c^6+13 a^8 b^10 c^6+11 a^6 b^12 c^6-29 a^4 b^14 c^6+29 a^2 b^16 c^6-10 b^18 c^6+8 a^16 c^8-35 a^14 b^2 c^8+64 a^12 b^4 c^8-61 a^10 b^6 c^8+30 a^8 b^8 c^8-10 a^6 b^10 c^8+15 a^4 b^12 c^8-20 a^2 b^14 c^8+10 b^16 c^8-28 a^14 c^10+27 a^12 b^2 c^10-13 a^10 b^4 c^10+13 a^8 b^6 c^10-10 a^6 b^8 c^10-6 a^4 b^10 c^10+6 a^2 b^12 c^10-5 b^14 c^10+56 a^12 c^12-a^10 b^2 c^12-7 a^8 b^4 c^12+11 a^6 b^6 c^12+15 a^4 b^8 c^12+6 a^2 b^10 c^12+2 b^12 c^12-70 a^10 c^14-29 a^8 b^2 c^14-16 a^6 b^4 c^14-29 a^4 b^6 c^14-20 a^2 b^8 c^14-5 b^10 c^14+56 a^8 c^16+39 a^6 b^2 c^16+34 a^4 b^4 c^16+29 a^2 b^6 c^16+10 b^8 c^16-28 a^6 c^18-25 a^4 b^2 c^18-22 a^2 b^4 c^18-10 b^6 c^18+8 a^4 c^20+8 a^2 b^2 c^20 +5 b^4 c^20-a^2 c^22-b^2 c^22) : :

See Antreas Hatzipolakis, César Lozada, and Ercole Suppa, , Hyacinthos 28213 and Hyacinthos 28248.

X(23111) lies on this line: {49,50}

X(23112) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23112) lies on these lines: {3, 22394}, {6, 16577}, {219, 22119}, {329, 2427}, {394, 23113}, {916, 22057}, {4055, 23171}, {6505, 20744}, {18676, 21271}, {23121, 23139}

X(23112) = isogonal conjugate of polar conjugate of X(21271)

X(23113) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23113) lies on these lines: {3, 22399}, {6, 16578}, {219, 20740}, {394, 23112}, {651, 2427}, {662, 1625}, {906, 1813}, {1332, 4561}, {4558, 7254}, {6510, 20744}, {17906, 21272}, {22119, 23129}

X(23113) = isogonal conjugate of polar conjugate of X(21272)
X(23113) = isotomic conjugate of polar conjugate of X(23845)

X(23114) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23114) lies on these lines: {3, 22400}, {6, 16579}, {63, 16697}, {219, 23129}, {255, 20803}, {345, 4574}, {394, 23112}, {1214, 20744}, {1764, 4559}, {3927, 7078}, {18163, 21770}, {18677, 21273}, {20797, 22117}, {20799, 23175}, {22119, 22125}

X(23114) = isogonal conjugate of polar conjugate of X(21273)
X(23114) = isotomic conjugate of polar conjugate of X(23846)

X(23115) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23115) lies on these lines: {3, 6}, {20, 8743}, {24, 10313}, {30, 2207}, {63, 22119}, {112, 2138}, {127, 7776}, {155, 3289}, {230, 3548}, {232, 7387}, {248, 15316}, {394, 441}, {1038, 5280}, {1040, 5299}, {1062, 16502}, {1092, 8779}, {1184, 1368}, {1370, 3162}, {1498, 1625}, {1576, 20993}, {1634, 23172}, {1968, 12085}, {1975, 15013}, {2072, 13881}, {2548, 15760}, {2549, 12605}, {3087, 7401}, {3148, 23606}, {3172, 21312}, {3199, 18534}, {3269, 12163}, {3529, 8744}, {3546, 7735}, {3547, 7736}, {3549, 3815}, {3692, 22132}, {3767, 11585}, {3926, 20806}, {5254, 18531}, {5286, 6643}, {5359, 7386}, {6337, 22151}, {6389, 7758}, {6390, 6461}, {6638, 23174}, {6642, 10311}, {6748, 7528}, {7855, 15526}, {8721, 19149}, {10312, 17928}, {10323, 22240}, {10594, 15355}, {11441, 13509}, {12362, 15048}, {16781, 18455}, {20739, 23131}, {22125, 22126}

X(23115) = isogonal conjugate of polar conjugate of X(1370)
X(23115) = isogonal conjugate of isotomic conjugate of X(28419)
X(23115) = isotomic conjugate of polar conjugate of X(159)
\ X(23115) = X(19)-isoconjugate of X(13575)
X(23115) = X(92)-isoconjugate of X(34207)
X(23115) = crossdifference of every pair of points on the radical axis of any pair of {1st, 2nd and 3rd pedal circles of X(4)}

X(23116) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23116) lies on these lines: {3, 22402}, {6, 16580}, {219, 22119}, {2273, 18734}, {3211, 22145}, {5280, 18730}, {17492, 18680}, {20336, 20806}, {20760, 23074}, {22156, 23075}

X(23116) = isogonal conjugate of polar conjugate of X(17492)
X(23116) = isotomic conjugate of polar conjugate of X(23847)

X(23117) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23117) lies on these lines: {3, 22403}, {6, 16581}, {219, 22119}, {7254, 23147}, {18681, 21274}, {22156, 23193}

X(23117) = isogonal conjugate of polar conjugate of X(21274)
X(23117) = isotomic conjugate of polar conjugate of X(23848)

X(23118) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23118) lies on these lines: {3, 22404}, {6, 2887}, {219, 22137}, {17910, 21275}, {20739, 22130}, {20806, 23123}, {22119, 23139}, {22156, 23143}

X(23118) = isogonal conjugate of polar conjugate of X(21275)
X(23118) = isotomic conjugate of polar conjugate of X(23849)

X(23119) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23119) lies on these lines: {3, 22405}, {6, 5249}, {63, 22145}, {239, 1993}, {306, 394}, {7193, 22348}, {22156, 22161}, {23124, 23130}

X(23119) = isogonal conjugate of polar conjugate of X(21276)
X(23119) = isotomic conjugate of polar conjugate of X(23850)

X(23120) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23120) lies on these lines: {3, 22406}, {6, 908}, {306, 394}, {323, 6542}, {1790, 2197}, {1993, 1999}, {6332, 20808}, {17976, 22156}, {20811, 23137}, {22128, 22145}

X(23120) = isogonal conjugate of polar conjugate of X(21277)
X(23120) = isotomic conjugate of polar conjugate of X(1324)

X(23121) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23121) lies on these lines: {3, 22409}, {6, 1215}, {63, 20747}, {219, 7015}, {394, 20742}, {17912, 21278}, {20739, 22130}, {20769, 22126}, {22137, 22156}, {23112, 23139}

X(23121) = isogonal conjugate of polar conjugate of X(21280)
X(23121) = isotomic conjugate of polar conjugate of X(23852)

X(23122) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23122) lies on these lines: {3, 22410}, {6, 57}, {63, 22119}, {81, 13577}, {189, 1783}, {306, 394}, {971, 3195}, {1071, 16466}, {1433, 6765}, {1473, 22348}, {1771, 3293}, {3173, 22145}, {17284, 17811}, {22123, 23140}, {22144, 23089}

X(23122) = isogonal conjugate of polar conjugate of X(21279)
X(23122) = isotomic conjugate of polar conjugate of X(22654)

X(23123) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23123) lies on these lines: {3, 22411}, {6, 2886}, {394, 20807}, {3173, 22131}, {20739, 23128}, {20806, 23118}, {20811, 22130}

X(23123) = isogonal conjugate of polar conjugate of X(21278)
X(23123) = isotomic conjugate of polar conjugate of X(23851)

X(23124) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23124) lies on these lines: {3, 22412}, {6, 3739}, {63, 77}, {69, 20744}, {326, 22134}, {332, 23131}, {651, 20245}, {2323, 7175}, {17137, 17913}, {17206, 22126}, {17976, 22138}, {20739, 20806}, {20742, 22133}, {22161, 23079}, {23119, 23130}, {23193, 23526}

X(23125) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23125) lies on these lines: {3, 22413}, {6, 75}, {63, 77}, {651, 20348}, {1760, 21767}, {17976, 20805}, {20739, 20747}, {20742, 22132}, {20806, 22145}, {20808, 22125}, {22148, 22152}, {23075, 23526}

X(23125) = isogonal conjugate of polar conjugate of X(21281)
X(23125) = isotomic conjugate of polar conjugate of X(23853)

X(23126) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23126) lies on these lines: {3, 22414}, {6, 519}, {219, 22142}, {394, 4001}, {525, 3049}, {20741, 22144}, {20752, 22162}, {20813, 22146}, {22123, 23135}

X(23126) = isogonal conjugate of polar conjugate of X(21282)
X(23126) = isotomic conjugate of polar conjugate of X(23854)

X(23127) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23127) lies on these lines: {3, 22415}, {6, 551}, {219, 1807}, {394, 4001}

X(23127) = isogonal conjugate of polar conjugate of X(21283)
X(23127) = isotomic conjugate of polar conjugate of X(23855)

X(23128) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23128) lies on these lines: {3, 248}, {5, 6}, {20, 13509}, {26, 1971}, {32, 13754}, {39, 1147}, {52, 10311}, {112, 12111}, {115, 9927}, {157, 2909}, {172, 7352}, {187, 7689}, {217, 18445}, {230, 12359}, {232, 10539}, {394, 441}, {458, 1235}, {520, 8574}, {525, 1975}, {539, 5309}, {574, 12038}, {577, 1216}, {1069, 16502}, {1092, 14961}, {1614, 22240}, {1625, 8743}, {1914, 6238}, {1968, 12162}, {1970, 7526}, {2207, 18451}, {2549, 12118}, {3053, 12163}, {3167, 9605}, {3289, 22120}, {3815, 9820}, {5007, 15083}, {5286, 6193}, {5448, 5475}, {5449, 7746}, {5462, 10314}, {5523, 14516}, {5562, 8779}, {5889, 10312}, {6237, 10315}, {6422, 8909}, {7592, 9755}, {7735, 11411}, {7745, 22660}, {7748, 14901}, {7881, 15066}, {8882, 19194}, {9620, 9928}, {10313, 11412}, {10317, 18436}, {14965, 20806}, {20739, 23123}, {22131, 23137}

X(23129) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23129) lies on these lines: {3, 22418}, {6, 3452}, {219, 23114}, {306, 394}, {3173, 20744}, {3940, 7078}, {20739, 23123}, {20745, 20748}, {20808, 20812}, {22119, 23113}, {22145, 23140}

X(23129) = isogonal conjugate of polar conjugate of X(21286)
X(23129) = isotomic conjugate of polar conjugate of X(2933)

X(23130) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23130) lies on these lines: {2, 6}, {3, 22420}, {72, 18447}, {306, 22123}, {511, 2203}, {651, 18632}, {1092, 5752}, {7085, 22139}, {20739, 22130}, {20809, 23137}, {23119, 23124}

X(23130) = isogonal conjugate of polar conjugate of X(21287)
X(23130) = isotomic conjugate of polar conjugate of X(2915)

X(23131) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23131) lies on these lines: {1, 19259}, {3, 73}, {6, 5745}, {63, 16697}, {306, 394}, {332, 23124}, {651, 23512}, {1010, 3562}, {1790, 22118}, {1812, 22126}, {3719, 4574}, {4559, 21375}, {5783, 17811}, {6617, 22119}, {20739, 23115}, {20812, 23151}

X(23131) = isogonal conjugate of polar conjugate of X(20245)
X(23131) = isotomic conjugate of polar conjugate of X(23361)

X(23132) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^9 + a^8 b - a b^8 - b^9 + a^8 c - 2 a^4 b^4 c + b^8 c - 2 a^4 b c^4 + 2 a b^4 c^4 - a c^8 + b c^8 - c^9) : :

X(23132) lies on these lines: {3, 22422}, {6, 16582}, {159, 14529}, {3157, 23074}, {22130, 22164}, {23068, 23076}

X(23132) = isogonal conjugate of polar conjugate of X(21288)
X(23132) = isotomic conjugate of polar conjugate of X(23856)

X(23133) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^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(23133) lies on these lines: {3, 1176}, {6, 6292}, {63, 20808}, {394, 22120}, {1369, 8792}, {3933, 22121}, {7767, 22151}, {15141, 15270}

X(23133) = isogonal conjugate of polar conjugate of X(1369)
X(23133) = isotomic conjugate of polar conjugate of X(2916)

X(23134) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23134) lies on these lines: {3, 20738}, {6, 6376}, {394, 7124}, {20739, 20747}, {20808, 20812}, {20809, 22131}, {22136, 23080}, {22370, 23519}

X(23134) = isogonal conjugate of polar conjugate of anticomplement of X(2162)
X(23134) = isotomic conjugate of polar conjugate of X(23857)

X(23135) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23135) lies on these lines: {3, 22428}, {6, 644}, {219, 1807}, {345, 394}, {20739, 23136}, {22123, 23126}, {22133, 22146}, {22139, 22143}

X(23135) = isogonal conjugate of polar conjugate of X(21290)
X(23135) = isotomic conjugate of polar conjugate of X(23858)
X(23135) = X(92)-isoconjugate of X(34184)

X(23136) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23136) lies on these lines: {3, 22429}, {6, 16590}, {219, 22142}, {394, 22123}, {20739, 23135}, {22139, 23082}

X(23136) = isogonal conjugate of polar conjugate of X(21291)
X(23136) = isotomic conjugate of polar conjugate of X(23859)

X(23137) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23137) lies on these lines: {3, 22432}, {6, 11}, {394, 20807}, {2504, 22145}, {20809, 23130}, {20811, 23120}, {22131, 23128}, {22144, 22146}

X(23137) = isogonal conjugate of polar conjugate of X(21293)
X(23137) = isotomic conjugate of polar conjugate of X(23402)

X(23138) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^7 - a^5 b^2 + a^2 b^5 - b^7 - 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 + a^2 c^5 + b^2 c^5 - c^7) : :

X(23138) lies on these lines: {3, 22433}, {6, 8287}, {20807, 22133}, {22145, 22146}

X(23138) = isogonal conjugate of polar conjugate of X(21294)
X(23138) = isotomic conjugate of polar conjugate of X(23860)

X(23139) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23139) lies on these lines: {3, 22434}, {6, 16598}, {63, 20768}, {219, 22156}, {647, 1331}, {17914, 21295}, {22119, 23118}, {23112, 23121}

X(23139) = isogonal conjugate of polar conjugate of X(21295)
X(23139) = isotomic conjugate of polar conjugate of X(23861)

X(23140) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23140) lies on these lines: {3, 22435}, {6, 5437}, {63, 77}, {189, 7359}, {524, 20266}, {1407, 2323}, {1473, 3292}, {1818, 22117}, {2003, 7308}, {2289, 7099}, {3157, 3940}, {3167, 3784}, {3682, 23072}, {4001, 17073}, {5440, 7078}, {7232, 20268}, {17814, 18540}, {17917, 21296}, {20739, 22153}, {20744, 22127}, {22123, 23122}, {22145, 23129}, {22356, 23089}

X(23140) = isogonal conjugate of polar conjugate of X(21296)
X(23140) = isotomic conjugate of polar conjugate of X(5204)
X(23140) = X(19)-isoconjugate of X(7319)

X(23141) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23141) lies on these lines: {3, 22384}, {6, 3960}, {905, 4131}, {1191, 3887}, {1332, 4561}, {1797, 22086}, {2254, 16466}, {3762, 4383}, {4895, 16483}, {22090, 22160}, {22144, 22145}

X(23141) = isogonal conjugate of polar conjugate of X(21297)
X(23141) = isotomic conjugate of polar conjugate of X(4491)

X(23142) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23142) lies on these lines: {3, 22438}, {6, 4892}, {219, 23069}, {20739, 22130}, {20816, 22157}

X(23142) = isogonal conjugate of polar conjugate of X(21298)
X(23142) = isotomic conjugate of polar conjugate of X(23862)

X(23143) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23143) lies on these lines: {3, 22439}, {6, 43}, {219, 7015}, {222, 7182}, {394, 7124}, {20747, 22149}, {22156, 23118}

X(23143) = isogonal conjugate of polar conjugate of X(21299)
X(23143) = isotomic conjugate of polar conjugate of X(23863)

X(23144) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23144) lies on these lines: {3, 1803}, {6, 7}, {48, 1804}, {57, 16438}, {63, 77}, {155, 23070}, {218, 1445}, {221, 3868}, {241, 2911}, {268, 1815}, {307, 23151}, {377, 9370}, {603, 1259}, {1004, 4551}, {1037, 1362}, {1062, 1071}, {1407, 17092}, {1419, 2323}, {1439, 3211}, {1442, 2256}, {1449, 2003}, {1498, 3562}, {1813, 7053}, {1993, 9965}, {2808, 7071}, {3100, 12669}, {3197, 7291}, {3561, 23072}, {4306, 16471}, {5249, 10601}, {5273, 17074}, {5776, 21279}, {7074, 7411}, {7078, 10884}, {7177, 22153}, {7289, 19350}, {8271, 16465}, {8757, 10982}, {18650, 20744}, {22125, 22131}

X(23144) = isogonal conjugate of polar conjugate of X(6604)
X(23144) = isotomic conjugate of polar conjugate of X(1617)
X(23144) = X(19)-isoconjugate of X(6601)

X(23145) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23145) lies on these lines: {3, 22441}, {6, 8062}, {219, 23189}, {521, 650}, {3049, 15411}, {7254, 23146}, {20816, 22157}, {21348, 21388}

X(23145) = isogonal conjugate of polar conjugate of X(21300)
X(23145) = isotomic conjugate of polar conjugate of X(23864)

X(23146) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23146) lies on these lines: {3, 22443}, {6, 522}, {218, 657}, {219, 1459}, {525, 3049}, {652, 17975}, {2522, 23090}, {2911, 6586}, {3063, 3309}, {7254, 23145}

X(23146) = isogonal conjugate of polar conjugate of X(21302)
X(23146) = isotomic conjugate of polar conjugate of X(23865)

X(23147) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23147) lies on these lines: {3, 22444}, {6, 812}, {525, 3049}, {7254, 23117}, {22144, 22145}, {22383, 23148}

X(23147) = isogonal conjugate of polar conjugate of X(21303)
X(23147) = isotomic conjugate of polar conjugate of X(23866)

X(23148) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23148) lies on these lines: {3, 22445}, {6, 3835}, {6332, 20808}, {20816, 22157}, {22383, 23147}

X(23148) = isogonal conjugate of polar conjugate of X(21304)
X(23148) = isotomic conjugate of polar conjugate of X(23867)

X(23149) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23149) lies on these lines: {3, 22446}, {7254, 23117}, {20816, 22157}

X(23149) = isogonal conjugate of polar conjugate of X(21305)
X(23149) = isotomic conjugate of polar conjugate of X(23403)

X(23150) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23150) lies on these lines: {1, 6}, {3, 9247}, {48, 22133}, {78, 20770}, {222, 7183}, {283, 22070}, {304, 20742}, {394, 7124}, {517, 7119}, {2083, 20254}, {3157, 22163}, {4020, 7116}, {6056, 12836}, {7066, 12835}, {7193, 23620}, {20762, 20809}, {20766, 23165}, {22162, 23070}, {22164, 23071}

X(23150) = isogonal conjugate of polar conjugate of X(4388)
X(23150) = isotomic conjugate of polar conjugate of X(23868)
X(23150) = X(19)-isoconjugate of X(7224)
X(23150) = X(92)-isoconjugate of X(34250)

X(23151) =  (A,B,C,X(75); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23151) lies on these lines: {2, 218}, {3, 63}, {6, 4357}, {7, 2287}, {8, 13727}, {41, 11343}, {56, 18206}, {57, 16412}, {69, 219}, {81, 3616}, {141, 2911}, {198, 16574}, {200, 9441}, {213, 940}, {220, 3912}, {222, 348}, {239, 3673}, {307, 23144}, {329, 6996}, {379, 20347}, {394, 4001}, {599, 17796}, {672, 21477}, {857, 21285}, {894, 5783}, {942, 19309}, {965, 10436}, {1376, 20683}, {1429, 21384}, {1804, 16731}, {1959, 5730}, {2256, 3879}, {2271, 3666}, {2318, 20731}, {3218, 11329}, {3219, 16367}, {3713, 3729}, {3868, 19310}, {3869, 7291}, {3876, 19314}, {4361, 16732}, {4384, 5228}, {4513, 17294}, {4641, 5021}, {5044, 19313}, {5282, 21981}, {5439, 19321}, {5526, 17284}, {5711, 16830}, {5776, 10444}, {16551, 21078}, {16844, 19716}, {18747, 21276}, {20806, 22122}, {20812, 23131}, {22097, 23620}, {22152, 23094}

X(23151) = isogonal conjugate of polar conjugate of X(4441)
X(23151) = isotomic conjugate of polar conjugate of X(1001)
X(23151) = X(19)-isoconjugate of X(1002)

X(23152) =  REFLECTION OF X(3057) IN X(13756)

Barycentrics    a*( (b+c)*a^7-(b^2+6*b*c+c^2)*a^6-(b+c)*(b^2-8*b*c+c^2)*a^5+(b^4-10*b^2*c^2+c^4)*a^4-(b^4-3*b^2*c^2+c^4)*(b+c)*a^3+(b^4+c^4+(4*b^2-b*c+4*c^2)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^4+c^4-2*(3*b^2-4*b*c+3*c^2)*b*c)*a-(b^2-c^2)^2*(b-c)^4) : :
X(23152) = 3*X(354)-2*X(3025)

See Antreas Hatzipolakis, César Lozada, and Ercole Suppa, Hyacinthos 28253 and Hyacinthos 28256.

X(23152) lies on these lines: {80, 517}, {88, 105}, {354, 3025}, {513, 17660}, {953, 2646}, {1319, 4351}, {3057, 3326}, {3259, 17605}, {5048, 10702}

X(23152) = reflection of X(3057) in X(13756)
X(23152) = X(476)-of-Ursa-minor triangle
X(23152) = X(14731)-of-intouch triangle

X(23153) =  REFLECTION OF X(1) IN X(13756)

Barycentrics    a^2*((b^2+b*c+c^2)*a^6-6*(b+c)*b*c*a^5-3*(b^4+c^4-(b^2+6*b*c+c^2)*b*c)*a^4+6*(2*b-c)*(b-2*c)*(b+c)*b*c*a^3+3*(b^6+c^6-(3*b^4+3*c^4+(4*b^2-13*b*c+4*c^2)*b*c)*b*c)*a^2-6*(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)*b*c*a-(b^2-c^2)^2*(b^4+c^4-(5*b^2-9*b*c+5*c^2)*b*c)) : :
X(23153) = 3*X(1)-2*X(3025), X(3025)-3*X(13756)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28254.

X(23153) lies on these lines: {1, 3025}, {35, 953}, {36, 106}, {80, 517}, {513, 7972}, {3259, 7951}, {5119, 15737}, {5697, 18340}, {5903, 6788}

X(23153) = reflection of X(1) in X(13756)
X(23153) = X(13756)-of-Aquila triangle
X(23153) = X(18319)-of-Ursa-minor triangle

X(23154) =  X(1)X(855)∩X(3)X(1331)

Barycentrics    a^2*(-a^2+b^2+c^2)*((b^2+c^2)*a+(b^2-c^2)*(b-c)) : :
X(23154) = 3*X(51)-4*X(942), 2*X(72)-3*X(3917), 9*X(373)-10*X(5439), 3*X(3060)-4*X(12109), 6*X(3819)-5*X(3876), 3*X(3917)-4*X(11573), 8*X(5044)-9*X(5650), 3*X(11246)-2*X(22300)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28255.

X(23154) lies on these lines: {1, 855}, {3, 1331}, {8, 2810}, {48, 3284}, {51, 942}, {63, 22076}, {65, 8679}, {72, 3917}, {73, 22345}, {78, 3784}, {184, 3157}, {197, 1406}, {201, 3942}, {221, 8192}, {222, 1425}, {228, 4303}, {283, 22161}, {373, 5439}, {511, 3868}, {513, 6284}, {912, 5562}, {944, 2818}, {970, 3218}, {971, 11381}, {1193, 1401}, {1437, 23070}, {1473, 7078}, {2390, 3057}, {2392, 3874}, {2807, 15071}, {2808, 12111}, {2841, 5697}, {2842, 3878}, {3060, 12109}, {3313, 9021}, {3690, 3927}, {3781, 3951}, {3782, 18178}, {3819, 3876}, {4020, 22070}, {4185, 6180}, {4306, 13738}, {5044, 5650}, {5905, 10441}, {5907, 12528}, {6147, 18180}, {8614, 20986}, {10364, 21279}, {11246, 22300}, {20727, 20785}, {22344, 22350}

X(23154) = reflection of X(i) in X(j) for these (i,j): (72, 11573), (185, 1071)
X(23154) = isogonal conjugate of polar conjugate of X(3782)
X(23154) = isotomic conjugate of isogonal conjugate of X(23196)
X(23154) = isotomic conjugate of polar conjugate of X(17053)
X(23154) = X(19)-isoconjugate of X(2985)
X(23154) = {X(72), X(11573)}-harmonic conjugate of X(3917)

X(23155) =  X(1)X(2392)∩X(2)X(375)

Barycentrics    a^2*((b^2+c^2)*a^2-(b+c)*b*c*a-b^4-c^4+b*c*(b^2-b*c+c^2)) : :
X(23155) = 5*X(2)-4*X(375), X(8)-4*X(11573), 2*X(210)-3*X(7998), 5*X(3567)-8*X(13373), 4*X(3742)-3*X(5640), X(5889)-4*X(12675), 5*X(11444)-2*X(14872), X(12111)+2*X(12680)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28255.

X(23155) lies on these lines: {1, 2392}, {2, 375}, {8, 11573}, {81, 1469}, {100, 3784}, {210, 7998}, {354, 3060}, {511, 3873}, {518, 2979}, {674, 4430}, {1401, 4850}, {1993, 22769}, {2807, 11220}, {2810, 3681}, {2842, 3899}, {3567, 13373}, {3705, 3909}, {3742, 5640}, {3888, 5014}, {3938, 7186}, {4661, 9026}, {5889, 12675}, {7391, 12586}, {11444, 14872}, {12111, 12680}, {17063, 20962}

X(23156) =  X(1)X(2392)∩X(10)X(8679)

Barycentrics    a^2*((b^2+c^2)*a^3+(b^3+c^3)*a^2-(b^2+c^2)^2*a-(b+c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)) : :
X(23156) = 4*X(1125)-3*X(15049), 3*X(2979)-X(5904), 3*X(3060)-5*X(18398), 2*X(3678)-3*X(3917), 4*X(11793)-3*X(15064)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28255.

X(23156) lies on these lines: {1, 2392}, {10, 8679}, {52, 12005}, {511, 3874}, {942, 9037}, {970, 4973}, {1125, 15049}, {2779, 18481}, {2801, 5562}, {2842, 3869}, {2979, 5904}, {3060, 18398}, {3678, 3917}, {3754, 16980}, {4020, 14963}, {6583, 10263}, {11793, 15064}

X(23157) =  X(1)X(2392)∩X(375)X(19878)

Barycentrics    a^2*((b^2+c^2)*a^3+(b^3+c^3)*a^2-(b^4+4*b^2*c^2+c^4)*a-(b+c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)) : :
X(23157) = 3*X(375)-4*X(19878), 5*X(3616)-3*X(15049), 3*X(3819)-2*X(4015), 3*X(5883)-X(16980)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28255.

X(23157) lies on these lines: {1, 2392}, {375, 19878}, {511, 3881}, {519, 11573}, {1125, 8679}, {2810, 3678}, {2842, 3878}, {3616, 15049}, {3784, 8715}, {3819, 4015}, {5045, 9037}, {5883, 16980}

X(23158) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23158) lies on these lines: {2, 8901}, {3, 68}, {184, 23181}, {418, 3564}, {1634, 5063}, {3133, 13292}, {3135, 6515}, {3167, 3289}, {3168, 4230}, {3815, 5020}, {5421, 5943}, {6676, 20775}, {9544, 15329}, {9605, 11328}, {13409, 14984}, {20760, 23161}

X(23158) = isogonal conjugate of polar conjugate of anticomplement of X(311)
X(23158) = {X(23161),X(23162)}-harmonic conjugate of X(20760)

X(23159) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23159) lies on these lines: {3, 345}, {48, 3955}, {172, 11328}, {603, 20805}, {1812, 22149}, {3167, 22158}, {19597, 23076}, {20796, 22127}, {23075, 23168}, {23173, 23190}, {23183, 23194}

X(23159) = isogonal conjugate of polar conjugate of anticomplement of X(3596)

X(23160) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23160) lies on these lines: {3, 306}, {48, 3955}, {993, 3840}, {1437, 15409}, {1468, 11328}, {6638, 20764}, {20796, 22126}, {22118, 23162}, {22139, 23094}, {22158, 23165}, {23076, 23173}

X(23160) = isogonal conjugate of polar conjugate of X(17148)

X(23161) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23161) lies on these lines: {3, 23198}, {72, 16731}, {101, 1634}, {651, 7460}, {1331, 23181}, {1813, 23187}, {6638, 22117}, {20760, 23158}, {20794, 20795}

X(23161) = isogonal conjugate of polar conjugate of anticomplement of X(34387)
X(23161) = {X(20760),X(23158)}-harmonic conjugate of X(23162)

X(23162) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23162) lies on these lines: {3, 23199}, {6638, 20818}, {20760, 23158}, {20794, 22117}, {22118, 23160}

X(23162) = isogonal conjugate of polar conjugate of anticomplement of X(34388)
X(23162) = {X(20760),X(23158)}-harmonic conjugate of X(23161)

X(23163) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23163) lies on these lines: {3, 1176}, {3167, 3289}, {3504, 18899}, {6660, 15651}, {9723, 23200}, {19588, 22143}, {22135, 23173}

X(23163) = isogonal conjugate of polar conjugate of X(20065)

X(23164) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23164) lies on these lines: {3, 22087}, {1576, 6660}, {1634, 18365}, {3049, 23188}, {3167, 3289}, {3284, 8681}, {3917, 22138}, {5166, 21309}, {5191, 11416}, {9407, 20854}, {11328, 18371}, {13754, 15781}

X(23164) = isogonal conjugate of polar conjugate of X(14712)

X(23165) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23165) lies on these lines: {3, 23201}, {48, 22139}, {184, 17976}, {394, 23095}, {3167, 20752}, {3292, 22161}, {3955, 22356}, {7193, 22053}, {20766, 23150}, {22158, 23160}

X(23165) = isogonal conjugate of polar conjugate of anticomplement of X(319)

X(23166) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23166) lies on these lines: {3, 22067}, {71, 3955}, {184, 22161}, {1331, 3292}, {2003, 20834}, {3167, 20752}, {7193, 22148}, {20796, 22160}, {22356, 23081}, {23072, 23085}

X(23166) = isogonal conjugate of polar conjugate of X(20072)

X(23167) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23167) lies on these lines: {3, 23203}, {219, 7015}, {760, 23381}, {3157, 23074}, {7193, 22458}, {20794, 23194}

X(23167) = isogonal conjugate of polar conjugate of X(17489)

X(23168) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23168) lies on these lines: {3, 23204}, {614, 999}, {1385, 16058}, {1472, 11328}, {3167, 20752}, {6638, 20764}, {23075, 23159}

X(23168) = isogonal conjugate of polar conjugate of anticomplement of X(322)

X(23169) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23169) lies on these lines: {3, 63}, {9, 19261}, {57, 4245}, {329, 19550}, {859, 3218}, {1459, 4091}, {1634, 5127}, {3191, 5482}, {3219, 16374}, {3220, 20918}, {3305, 19248}, {3306, 19250}, {3682, 22435}, {3868, 7428}, {3928, 19251}, {3929, 19252}, {4574, 20785}, {4694, 23404}, {4973, 20470}, {5437, 19253}, {14597, 20818}, {15325, 15507}, {20794, 23179}, {20800, 23083}, {22147, 23185}, {22148, 23071}, {23091, 23186}

X(23169) = isogonal conjugate of polar conjugate of X(17495)

X(23170) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23170) lies on these lines: {3, 63}, {20794, 23091}

X(23170) = isogonal conjugate of polar conjugate of anticomplement of X(4671)

X(23171) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23171) lies on these lines: {1, 3}, {212, 23067}, {255, 20803}, {577, 16685}, {1437, 3561}, {1870, 7420}, {2968, 17135}, {3100, 7416}, {3167, 20752}, {4055, 23112}, {4192, 15252}, {4640, 10181}, {5307, 19541}, {5753, 11435}, {7580, 17134}, {13405, 20990}, {19597, 23076}, {20799, 23194}, {22118, 23160}

X(23171) = isogonal conjugate of polar conjugate of anticomplement of X(1441)

X(23172) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^10 b^2 - 2 a^6 b^6 + a^2 b^10 + a^10 c^2 + a^8 b^2 c^2 - a^2 b^8 c^2 - b^10 c^2 - 2 a^6 c^6 + 2 b^6 c^6 - a^2 b^2 c^8 + a^2 c^10 - b^2 c^10) : :

X(23172) lies on these lines: {3, 66}, {1634, 23115}, {3167, 22135}, {10317, 19597}, {20794, 22120}

X(23172) = isogonal conjugate of polar conjugate of anticomplement of X(18018)

X(23173) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23173) lies on these lines: {2, 10342}, {3, 305}, {22, 9865}, {25, 3511}, {184, 3504}, {3167, 23180}, {5206, 14472}, {6638, 10316}, {22135, 23163}, {23076, 23160}, {23159, 23190}

X(23173) = isogonal conjugate of polar conjugate of X(8264)

X(23174) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23174) lies on these lines: {2, 3511}, {3, 305}, {63, 23186}, {184, 23180}, {394, 20794}, {3504, 3796}, {6638, 23115}, {20796, 22126}

X(23174) = isogonal conjugate of polar conjugate of anticomplement of X(3978)
X(23174) = isogonal conjugate of polar conjugate of Steiner-circumellipse-inverse of X(39)

X(23175) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23175) lies on these lines: {3, 7182}, {19597, 23076}, {20799, 23114}

X(23175) = isogonal conjugate of polar conjugate of anticomplement of X(20567)

X(23176) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23176) lies on these lines: {3, 69}, {1634, 18755}, {20760, 22164}, {20761, 23083}, {20796, 22139}, {20797, 22458}, {22117, 23078}, {22126, 23094}, {23081, 23180}

X(23176) = isogonal conjugate of polar conjugate of anticomplement of X(310)

X(23177) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23177) lies on these lines: {3, 22370}, {2200, 3504}, {4574, 23076}, {20761, 20767}, {20796, 20799}, {20797, 20805}, {22139, 22140}

X(23177) = isogonal conjugate of polar conjugate of anticomplement of X(6384)

X(23178) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23178) lies on these lines: {3, 23179}, {20760, 22158}, {20794, 23091}, {22141, 23079}

X(23178) = isogonal conjugate of polar conjugate of anticomplement of X(20568)

X(23179) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23179) lies on these lines: {3, 23178}, {20760, 22356}, {20793, 22350}, {20794, 23169}, {22142, 23079}

X(23179) = isogonal conjugate of polar conjugate of anticomplement of X(20569)

X(23180) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23180) lies on these lines: {3, 4563}, {110, 3511}, {184, 23174}, {394, 3504}, {3167, 23173}, {20794, 23181}, {20800, 23186}, {22148, 23083}, {22158, 23190}, {23081, 23176}

X(23180) = isogonal conjugate of polar conjugate of X(25054)

X(23181) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23181) lies on the Johnson circumconic and these lines: {3, 125}, {6, 23584}, {25, 114}, {26, 16391}, {99, 107}, {100, 7450}, {110, 351}, {160, 7493}, {184, 23158}, {343, 418}, {394, 6638}, {454, 9937}, {476, 930}, {689, 6037}, {852, 11064}, {1331, 23161}, {1625, 2081}, {1995, 7664}, {3066, 11328}, {3133, 8905}, {3233, 7480}, {3265, 4576}, {3952, 15411}, {4243, 6516}, {4575, 23067}, {5406, 23246}, {5407, 23256}, {5408, 23248}, {5409, 23258}, {6676, 23195}, {6786, 10607}, {7468, 22089}, {10112, 16035}, {11634, 19909}, {11800, 18114}, {12429, 15653}, {13394, 20775}, {15106, 15919}, {15958, 23286}, {20794, 23180}

X(23181) = isogonal conjugate of polar conjugate of X(14570)
X(23181) = X(92)-isoconjugate of X(2623)
X(23181) = barycentric product X(110)*X(343)

X(23182) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23182) lies on these lines: {3, 1265}, {48, 3955}, {22117, 22158}

X(23182) = isogonal conjugate of polar conjugate of anticomplement of isotomic conjugate of X(3445)

X(23183) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^3 b^3 - a^3 b^2 c - a^2 b^3 c - 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(23183) lies on these lines: {3, 4561}, {2200, 3504}, {20785, 23088}, {20794, 22458}, {20803, 23078}, {20805, 23186}, {23159, 23194}

X(23183) = isogonal conjugate of polar conjugate of anticomplement of X(6382)

X(23184) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23184) lies on these lines: {3, 23220}, {647, 8673}, {859, 3904}, {4245, 10015}, {22158, 23091}

X(23184) = isogonal conjugate of polar conjugate of anticomplement of isogonal conjugate of X(32719)
X(23184) = isogonal conjugate of polar conjugate of anticomplement of isotomic conjugate of X(901)
X(23184) = isogonal conjugate of polar conjugate of anticomplement of anticomplement of X(3310)

X(23185) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23185) lies on these lines: {3, 3692}, {48, 3955}, {219, 23085}, {1409, 20805}, {7053, 23089}, {22124, 22148}, {22147, 23169}, {22158, 23075}

X(23185) = isogonal conjugate of polar conjugate of X(17480)

X(23186) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23186) lies on these lines: {3, 304}, {48, 3955}, {63, 23174}, {228, 23079}, {1459, 23093}, {2196, 20796}, {3504, 23075}, {3509, 3511}, {20752, 22158}, {20800, 23180}, {20805, 23183}, {23071, 23083}, {23091, 23169}

X(23186) = isogonal conjugate of polar conjugate of X(19565)

X(23187) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23187) lies on these lines: {3, 521}, {56, 21189}, {513, 22765}, {514, 3733}, {520, 6760}, {656, 22379}, {928, 2605}, {956, 4397}, {999, 6129}, {1459, 4091}, {1813, 23161}, {2509, 5120}, {20794, 23191}

X(23187) = isogonal conjugate of polar conjugate of X(17496)

X(23188) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23188) lies on these lines: {3, 23225}, {295, 22384}, {1459, 4091}, {3049, 23164}, {20760, 22086}, {22158, 23091}, {22383, 23093}

X(23188) = isogonal conjugate of polar conjugate of anticomplement of X(3766)

X(23189) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23189) lies on these lines: {3, 656}, {21, 7253}, {36, 238}, {110, 2720}, {219, 23145}, {405, 8062}, {521, 1946}, {652, 23090}, {654, 4282}, {759, 2716}, {924, 2605}, {958, 4086}, {1444, 15419}, {1459, 4091}, {2169, 20803}, {4560, 14024}, {4575, 23067}, {7192, 17094}, {8648, 17420}, {22384, 23069}, {23093, 23191}

X(23189) = isogonal conjugate of anticomplement of X(34588)
X(23189) = isogonal conjugate of polar conjugate of X(4560)
X(23189) = crossdifference of every pair of points on line X(12)X(37)
X(23189) = X(92)-isoconjugate of X(4559)

X(23190) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23190) lies on these lines: {3, 23227}, {1331, 20794}, {22158, 23180}, {23159, 23173}

X(23190) = isogonal conjugate of polar conjugate of anticomplement of X(6386)

X(23191) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23191) lies on these lines: {3, 15413}, {521, 23079}, {20794, 23187}, {23093, 23189}

X(23191) = isogonal conjugate of polar conjugate of anticomplement of isotomic conjugate of X(692)

X(23192) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23192) lies on these lines: {1, 3511}, {3, 304}, {255, 7193}, {2200, 3504}, {4020, 7015}, {20797, 22458}, {22163, 23088}, {23070, 23083}

X(23192) = isogonal conjugate of polar conjugate of anticomplement of X(1920)
X(23192) = isotomic conjugate of polar conjugate of X(34251)

X(23193) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23193) lies on these lines: {3, 23230}, {219, 7015}, {1459, 4091}, {6007, 16680}, {22156, 23117}, {23124, 23526}

X(23193) = isogonal conjugate of polar conjugate of X(17497)

X(23194) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23194) lies on these lines: {3, 63}, {20794, 23167}, {20799, 23171}, {23075, 23094}, {23076, 23160}, {23159, 23183}

X(23194) = isogonal conjugate of polar conjugate of anticomplement of X(33931)

X(23195) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23195) lies on these lines: {3, 68}, {6, 3135}, {22, 9744}, {25, 160}, {39, 51}, {184, 418}, {228, 23198}, {569, 3133}, {3148, 23208}, {6676, 23181}, {7512, 17401}, {14585, 22391}, {15653, 18925}, {16030, 23292}, {22352, 23217}

X(23195) = isogonal conjugate of isotomic conjugate of X(1216)
X(23195) = isogonal conjugate of polar conjugate of X(570)

X(23196) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23196) lies on these lines: {3, 345}, {184, 22096}, {228, 20775}, {237, 2352}, {854, 3772}, {2318, 20777}, {22363, 23204}, {22364, 22368}, {23209, 23227}, {23219, 23231}

X(23196) = isogonal conjugate of isotomic conjugate of X(23154)
X(23196) = isogonal conjugate of polar conjugate of X(17053)
X(23196) = X(92)-isoconjugate of X(2985)

X(23197) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23197) lies on these lines: {3, 306}, {25, 1470}, {36, 4215}, {39, 23624}, {160, 2352}, {228, 20775}, {418, 22341}, {1011, 3840}, {3690, 20777}, {7280, 19343}, {22056, 23199}, {22080, 22389}, {22096, 23201}, {22364, 23209}

X(23197) = isogonal conjugate of isotomic conjugate of X(11573)
X(23197) = isogonal conjugate of polar conjugate of complement of X(313)
X(23197) = isogonal conjugate of polar conjugate of crosssum of X(6) and X(10)
X(23197) = isogonal conjugate of polar conjugate of crosspoint of X(2) and X(58)

X(23198) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23198) lies on these lines: {3, 23161}, {48, 2196}, {160, 198}, {212, 418}, {228, 23195}, {237, 2183}, {2252, 20777}

X(23198) = isogonal conjugate of polar conjugate of X(13006)

X(23199) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23199) lies on these lines: {3, 23162}, {48, 418}, {55, 160}, {212, 20775}, {228, 23195}, {237, 14547}, {22056, 23197}

X(23199) = isogonal conjugate of polar conjugate of complement of X(34388)

X(23200) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23200) lies on these lines: {3, 22087}, {50, 237}, {184, 418}, {401, 9512}, {524, 5467}, {571, 19136}, {1384, 5166}, {2193, 22369}, {2393, 5191}, {3049, 23225}, {3284, 20975}, {7669, 18365}, {9723, 23163}, {10317, 14908}, {10602, 15905}, {21637, 22052}

X(23200) = isogonal conjugate of isotomic conjugate of X(3292)
X(23200) = isogonal conjugate of polar conjugate of X(187)
X(23200) = isotomic conjugate of polar conjugate of X(14567)
X(23200) = X(19)-isoconjugate of X(18023)
X(23200) = X(92)-isoconjugate of X(671)

X(23201) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23201) lies on these lines: {3, 23165}, {41, 20230}, {48, 184}, {51, 2317}, {63, 23095}, {198, 11402}, {199, 2323}, {604, 5320}, {1100, 2355}, {1437, 18604}, {1495, 14547}, {1790, 7193}, {1818, 22352}, {2182, 21807}, {2183, 13366}, {2194, 7113}, {2206, 2300}, {2302, 10536}, {3292, 22097}, {3690, 22356}, {3917, 22390}, {5314, 17976}, {7085, 20818}, {7293, 22067}, {20754, 22447}, {22054, 22080}, {22096, 23197}

X(23201) = isogonal conjugate of polar conjugate of X(1100)
X(23201) = isogonal conjugate of isotomic conjugate of X(3916)
X(23201) = X(92)-isoconjugate of X(1255)

X(23202) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23202) lies on these lines: {3, 22067}, {31, 5042}, {48, 184}, {55, 11402}, {255, 22344}, {603, 22376}, {692, 2361}, {810, 822}, {902, 1404}, {1260, 22147}, {1331, 7193}, {1410, 3215}, {1473, 22117}, {1495, 2183}, {1818, 3292}, {2003, 16064}, {3689, 23344}, {3937, 20780}, {3955, 22060}, {5314, 22139}, {7293, 22161}, {13366, 14547}, {20729, 22073}, {22097, 22352}, {22356, 22371}

X(23202) = isogonal conjugate of isotomic conjugate of X(5440)
X(23202) = isogonal conjugate of polar conjugate of X(44)
X(23202) = isotomic conjugate of polar conjugate of X(2251)
X(23202) = X(19)-isoconjugate of X(20568)
X(23202) = X(88)-isoconjugate of X(92)
X(23202) = crossdifference of every pair of points on line X(92)X(4462)

X(23203) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23203) lies on these lines: {3, 23167}, {71, 228}, {73, 22362}, {1486, 21808}, {1918, 3724}, {3185, 21059}, {9310, 18611}, {17439, 18612}, {17451, 18610}, {20775, 23231}, {20780, 22345}

X(23203) = isogonal conjugate of polar conjugate of X(16600)

X(23204) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23204) lies on these lines: {3, 23168}, {25, 34}, {31, 2208}, {42, 7117}, {48, 184}, {51, 2260}, {104, 4183}, {154, 2352}, {199, 11012}, {354, 18210}, {418, 22341}, {909, 11429}, {1011, 3576}, {1385, 8021}, {1436, 11406}, {1473, 7011}, {1576, 2194}, {2187, 2223}, {2360, 4215}, {11323, 12114}, {22363, 23196}

X(23204) = isogonal conjugate of polar conjugate of X(1108)
X(23204) = isogonal conjugate of isotomic conjugate of X(1071)

X(23205) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23205) lies on these lines: {3, 63}, {100, 4487}, {219, 23222}, {855, 3911}, {859, 5122}, {908, 19335}, {1459, 1946}, {1878, 16610}, {2077, 20999}, {3937, 22350}, {20775, 23215}, {20782, 22373}, {22072, 23154}, {22386, 23223}

X(23205) = isogonal conjugate of polar conjugate of X(16610)

X(23206) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23206) lies on these lines: {1, 7428}, {3, 63}, {9, 16374}, {35, 1626}, {36, 3185}, {46, 23361}, {48, 906}, {57, 859}, {65, 15654}, {100, 4737}, {603, 1437}, {855, 5722}, {908, 19550}, {988, 23359}, {2352, 4257}, {3218, 4216}, {3220, 11334}, {3305, 19261}, {3306, 4245}, {3338, 23383}, {3556, 8071}, {3586, 13744}, {3928, 19254}, {4191, 5122}, {4247, 4850}, {5010, 15624}, {5435, 19256}, {5437, 19241}, {5744, 19262}, {7308, 19249}, {8069, 22769}, {8192, 11248}, {9798, 11509}, {10085, 15622}, {11507, 22654}, {15803, 16453}, {17102, 18732}, {20775, 22386}

X(23206) = isogonal conjugate of polar conjugate of X(4850)

X(23207) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23207) lies on these lines: {1, 3}, {11, 440}, {31, 577}, {33, 1011}, {42, 216}, {48, 184}, {154, 3185}, {284, 11428}, {464, 497}, {572, 11429}, {573, 11436}, {579, 11435}, {902, 22052}, {1001, 21482}, {1364, 22097}, {1474, 4215}, {1630, 10536}, {1713, 1864}, {1859, 8021}, {1870, 7430}, {1951, 2299}, {2150, 2193}, {2177, 10979}, {2252, 3611}, {2260, 14547}, {2308, 3284}, {2328, 10535}, {2968, 4046}, {3100, 4184}, {3270, 22080}, {3286, 10391}, {3720, 18592}, {4428, 21503}, {5432, 7536}, {7004, 22060}, {7074, 15624}, {9817, 16058}, {10393, 19762}, {20781, 23231}, {22056, 23197}, {22345, 22347}, {22364, 22368}

X(23207) = isogonal conjugate of polar conjugate of isogonal conjugate of X(2982)
X(23207) = isogonal conjugate of polar conjugate of complement of X(1441)

X(23208) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23208) lies on these lines: {3, 66}, {6, 20960}, {22, 315}, {23, 7785}, {24, 9744}, {25, 2548}, {32, 184}, {39, 1843}, {99, 8783}, {147, 7488}, {154, 20993}, {187, 682}, {206, 10316}, {230, 11360}, {574, 11326}, {577, 22262}, {626, 7467}, {1614, 11674}, {1634, 3933}, {2549, 11325}, {2896, 6636}, {3148, 23195}, {3202, 3289}, {3493, 17938}, {4173, 9418}, {5188, 5562}, {5207, 8788}, {5254, 21177}, {6292, 14096}, {6660, 9918}, {7492, 7929}, {7493, 15652}, {7694, 22655}, {7758, 9917}, {7767, 8266}, {7854, 22062}, {8618, 22362}, {10317, 15257}, {11641, 13564}, {14023, 19597}, {14917, 22416}

X(23208) = X(3)-Ceva conjugate of X(39)
X(23208) = isogonal conjugate of isotomic conjugate of X(3313)
X(23208) = isogonal conjugate of polar conjugate of crosssum of X(6) and X(66)
X(23208) = isogonal conjugate of polar conjugate of crosspoint of X(2) and X(22)
X(23208) = isogonal conjugate of polar conjugate of X(2)-Ceva conjugate of X(427)

X(23209) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23209) lies on these lines: {2, 21444}, {3, 305}, {184, 17970}, {418, 682}, {864, 1196}, {3511, 16276}, {6636, 9865}, {14575, 22075}, {20775, 22352}, {22364, 23197}, {23196, 23227}

X(23209) = isogonal conjugate of polar conjugate of X(8265)
X(23209) = isogonal conjugate of isotomic conjugate of X(4173)

X(23210) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23210) lies on these lines: {3, 305}, {418, 22401}, {3690, 20777}, {3917, 20775}, {8891, 14096}, {15246, 21444}, {22060, 23223}, {22352, 23216}

X(23210) = isogonal conjugate of polar conjugate of isotomic conjugate of X(31622)
X(23210) = isogonal conjugate of polar conjugate of complement of X(308)
X(23210) = isogonal conjugate of polar conjugate of crosspoint of X(2) and X(39)
X(23210) = isogonal conjugate of polar conjugate of crosssum of X(6) and X(83)
X(23210) = X(19)-isoconjugate of X(31622)

X(23211) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23211) lies on these lines: {3, 7182}, {1040, 23229}, {20781, 22400}, {22364, 22368}

X(23211) = isogonal conjugate of polar conjugate of complement of X(20567)

X(23212) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23212) lies on these lines: {3, 69}, {212, 7116}, {228, 22061}, {237, 18755}, {2092, 18758}, {8618, 20970}, {20749, 22373}, {20777, 22080}, {20778, 22345}, {22065, 22389}, {22371, 23216}

X(23212) = isogonal conjugate of polar conjugate of X(21838)

X(23213) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23213) lies on these lines: {3, 22370}, {212, 20777}, {228, 23219}, {20749, 20755}, {20778, 22344}, {22080, 22081}

X(23213) = isogonal conjugate of polar conjugate of complement of X(6384)

X(23214) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23214) lies on these lines: {3, 23178}, {228, 22096}, {20775, 22386}, {22082, 22369}

X(23214) = isogonal conjugate of polar conjugate of complement of X(20568)

X(23215) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23215) lies on these lines: {3, 23178}, {228, 22356}, {20775, 23205}, {22083, 22369}

X(23215) = isogonal conjugate of polar conjugate of complement of X(20569)

X(23216) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23216) lies on these lines: {3, 4563}, {110, 21444}, {184, 17970}, {305, 3504}, {669, 3124}, {864, 1692}, {865, 6388}, {2972, 17423}, {3917, 23221}, {3937, 22373}, {20759, 22080}, {20775, 23217}, {20782, 23223}, {22096, 23227}, {22352, 23210}, {22371, 23212}

X(23216) = isogonal conjugate of polar conjugate of X(1084)
X(23216) = isogonal conjugate of X(4)-cross conjugate of X(6331)
X(23216) = crossdifference of every pair of points on line X(2396)X(4609) (the tangent to conic {{A,B,C,X(107),X(648)}} at X(6331))
X(23216) = X(92)-isoconjugate of X(34537)

X(23217) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23217) lies on these lines: {3, 125}, {22, 114}, {23, 16760}, {110, 9161}, {160, 6786}, {237, 18860}, {418, 2972}, {620, 21525}, {684, 22085}, {1624, 5642}, {4558, 13198}, {5972, 15329}, {14966, 23584}, {20775, 23216}, {22352, 23195}

X(23218) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23218) lies on these lines: {3, 1265}, {212, 22096}, {228, 20775}

X(23219) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23219) lies on these lines: {3, 4561}, {228, 23213}, {8844, 23385}, {20775, 22345}, {20785, 22381}, {22344, 23223}, {22347, 22368}, {22373, 23154}, {23196, 23231}

X(23519) = isotomic conjugate of polar conjugate of X(23546)

X(23220) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23220) lies on these lines: {3, 23184}, {187, 237}, {676, 23383}, {859, 10015}, {3904, 4216}, {4528, 15621}, {6366, 23361}, {22096, 22386}

X(23220) = isogonal conjugate of isotomic conjugate of X(8677)
X(23220) = isogonal conjugate of polar conjugate of X(3310)
X(23220) = X(92)-isoconjugate of X(13136)

X(23221) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23221) lies on these lines: {3, 3504}, {63, 22386}, {228, 23213}, {1799, 8858}, {3917, 23216}, {6636, 8782}, {20775, 22352}, {22363, 23223}

X(23221) = isogonal conjugate of polar conjugate of X(6375)
X(23221) = isotomic conjugate of polar conjugate of X(9490)

X(23222) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23222) lies on these lines: {3, 3692}, {32, 604}, {44, 198}, {48, 22344}, {71, 22376}, {154, 2352}, {219, 23205}, {228, 20775}, {2347, 7117}, {3937, 22063}, {4003, 18210}, {22096, 22363}

X(23222) = isogonal conjugate of polar conjugate of complement of X(341)

X(23223) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23223) lies on these lines: {3, 304}, {228, 20775}, {1459, 22388}, {20752, 22096}, {20782, 23216}, {22060, 23210}, {22344, 23219}, {22350, 22373}, {22363, 23221}, {22386, 23205}

X(23223) = isogonal conjugate of polar conjugate of complement of X(1921)

X(23224) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23224) lies on these lines: {3, 521}, {36, 238}, {56, 6129}, {104, 2734}, {110, 2719}, {520, 4091}, {649, 17412}, {656, 22091}, {680, 822}, {1444, 15413}, {1459, 1946}, {2605, 8641}, {2975, 4397}, {3126, 3433}, {4560, 14294}, {20775, 23228}

X(23224) = isogonal conjugate of polar conjugate of X(905)
X(23224) = isotomic conjugate of polar conjugate of X(22383)
X(23224) = X(19)-isoconjugate of X(6335)
X(23224) = X(92)-isoconjugate of X(1783)

X(23225) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23225) lies on these lines: {3, 23188}, {228, 22086}, {654, 2352}, {918, 3286}, {926, 2223}, {1402, 6139}, {1459, 1946}, {3049, 23200}, {22096, 22386}, {22383, 22388}

X(23225) = isogonal conjugate of polar conjugate of X(665)
X(23225) = X(92)-isoconjugate of X(666)

X(23226) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23226) lies on these lines: {3, 656}, {21, 8062}, {50, 2624}, {71, 22441}, {186, 14838}, {477, 759}, {513, 8648}, {652, 22382}, {993, 4086}, {1459, 1946}, {2611, 15470}, {4189, 7253}, {7004, 23286}, {14385, 17104}, {22346, 23217}, {22349, 22384}, {22388, 23228}

X(23226) = isogonal conjugate of polar conjugate of X(14838)
X(23226) = isotomic conjugate of polar conjugate of isogonal conjugate of X(15455)
X(23226) = X(19)-isoconjugate of X(15455)

X(23227) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23227) lies on these lines: {3, 23190}, {22096, 23216}, {23196, 23209}

X(23227) = isogonal conjugate of polar conjugate of complement of X(6386)

X(23228) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23228) lies on these lines: {3, 15413}, {20775, 23224}, {22388, 23226}

X(23228) = isogonal conjugate of polar conjugate of complement of isotomic conjugate of X(692)

X(23229) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23229) lies on these lines: {3, 304}, {228, 23213}, {1040, 23211}, {4303, 22373}, {20775, 20780}, {20778, 22345}, {22099, 22381}

X(23230) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23230) lies on these lines: {3, 23193}, {71, 228}, {1459, 1946}, {1633, 16609}, {6093, 8691}, {22094, 22403}

X(23230) = isogonal conjugate of polar conjugate of X(16611)

X(23231) =  (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(3)

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

X(23231) lies on these lines: {3, 63}, {31, 18759}, {48, 23526}, {20775, 23203}, {20781, 23207}, {22363, 22389}, {22364, 23197}, {23196, 23219}

X(23231) = isogonal conjugate of polar conjugate of complement of X(33931)

X(23232) = REFLECTION OF X(4) IN X(138)

Barycentrics    SB^2*SC^2*(SA-SB)*(SA-SC)*(5*SA*SC*S^2+(SC+2*SA)*(2*SC+SA)*SB^2) *(5*S^2*SA*SB+(SA+2*SB)*(2*SA+SB)*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28260.

X(23232) lies on the circumcircle and these lines: {4, 138}, {1298, 5890}

X(23232) = reflection of X(4) in X(138)
X(23232) = Collings transform of X(138)

X(23233) = REFLECTION OF X(4) IN X(139)

Barycentrics    SB^2*SC^2*(S^2+SA*SB)*(S^2+SA*SC)*(2*S^4-(S^2-SB^2)*SW^2-(-SA*SC+SB^2)*(S^2+SB^2))*(2*S^4-(S^2-SC^2)*SW^2-(-SA*SB+SC^2)*(S^2+SC^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28260.

X(23233) lies on the circumcircle and these lines: {4, 139}, {97, 13398}, {99, 8884}, {110, 467}, {112, 8883}, {324, 925}, {933, 11547}, {1303, 5889}, {6570, 14149}

X(23233) = reflection of X(4) in X(139)
X(23233) = Collings transform of X(139)
X(23233) = Λ(X(52), X(53)) (line X(52)X(53) is the van Aubel line of the orthic triangle)

X(23234) = X(2)X(98)∩X(99)X(381)

Barycentrics    9*S^4-(3*SA+SW)*SW*S^2-3*SB*SC*SW^2 : :
X(23234) = 4*X(2)-X(98), X(2)+2*X(114), 5*X(2)+X(147), 13*X(2)-X(5984), 7*X(2)-4*X(6036), 2*X(2)+X(6054), 5*X(2)-2*X(6055), 5*X(2)-8*X(6721), 7*X(2)-X(11177), X(98)+8*X(114), 5*X(98)+4*X(147), 13*X(98)-4*X(5984), 7*X(98)-16*X(6036), X(98)+2*X(6054), 5*X(98)-8*X(6055), 7*X(98)-4*X(11177), 10*X(114)-X(147), 7*X(114)+2*X(6036), 4*X(114)-X(6054), 5*X(114)+X(6055)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23234) lies on these lines: {2, 98}, {3, 11149}, {4, 2482}, {5, 671}, {11, 12350}, {12, 12351}, {30, 10242}, {99, 381}, {113, 11006}, {115, 5071}, {119, 12356}, {140, 14830}, {262, 5503}, {355, 9884}, {376, 620}, {468, 20774}, {485, 19058}, {486, 19057}, {543, 3545}, {547, 11632}, {549, 6033}, {551, 9864}, {576, 10487}, {599, 10753}, {625, 15483}, {631, 22247}, {842, 1551}, {946, 9881}, {1007, 9993}, {1513, 22110}, {1569, 18362}, {2782, 5055}, {2784, 19883}, {2794, 3524}, {2796, 3817}, {3055, 11646}, {3090, 5461}, {3091, 8591}, {3241, 11724}, {3455, 7550}, {3525, 10991}, {3534, 22505}, {3679, 7970}, {3815, 6034}, {3828, 21636}, {3832, 10992}, {3845, 10723}, {3851, 12355}, {4995, 12185}, {5066, 6321}, {5067, 11623}, {5068, 8596}, {5215, 21445}, {5298, 12184}, {5469, 6114}, {5470, 6115}, {5476, 7777}, {5478, 9116}, {5479, 9114}, {6174, 10768}, {6248, 11152}, {6785, 6786}, {7395, 9876}, {7486, 20398}, {7507, 12132}, {7607, 8787}, {7741, 10070}, {7775, 12110}, {7951, 10054}, {7989, 9875}, {8176, 19911}, {8227, 12258}, {8587, 10185}, {8781, 14492}, {8786, 10486}, {9753, 9770}, {9774, 22664}, {9830, 10516}, {9855, 13449}, {9860, 19876}, {9862, 15702}, {9878, 10356}, {9882, 10514}, {9883, 10515}, {10011, 22329}, {10358, 12191}, {10895, 18969}, {10896, 12354}, {11049, 12181}, {11163, 14848}, {11257, 11318}, {11656, 12900}, {12042, 15694}, {12093, 14984}, {12188, 15703}, {13188, 19709}, {13846, 19056}, {13847, 19055}, {14651, 14971}

X(23234) = reflection of X(14651) in X(14971)
X(23234) = X(5055)-of-1st anti-Brocard-triangle
X(23234) = X(9166)-of-Artzt-triangle
X(23234) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 114, 6054), (2, 147, 6055), (2, 6054, 98), (2, 11177, 6036), (4, 2482, 12117), (5, 8724, 671), (114, 6721, 147), (547, 11632, 14061), (3090, 12243, 5461), (3091, 8591, 9880), (5461, 14981, 12243), (6055, 6721, 2), (9877, 11184, 5503), (11178, 12177, 11161)

X(23235) = X(3)X(76)∩X(20)X(542)

Barycentrics    3*S^4+(6*SA^2-3*SA*SW-SW^2)*S^2+SB*SC*SW^2 : :
X(23235) = 9*X(2)-8*X(20398), 4*X(3)-3*X(98), 2*X(3)-3*X(99), 7*X(3)-6*X(12042), 5*X(3)-3*X(12188), X(3)-3*X(13188), 8*X(3)-9*X(21166), 7*X(98)-8*X(12042), 5*X(98)-4*X(12188), X(98)-4*X(13188), 2*X(98)-3*X(21166), 7*X(99)-4*X(12042), 5*X(99)-2*X(12188), 4*X(99)-3*X(21166), 10*X(12042)-7*X(12188), 2*X(12042)-7*X(13188), X(12188)-5*X(13188), 8*X(12188)-15*X(21166), 8*X(13188)-3*X(21166)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23235) lies on these lines: {2, 20398}, {3, 76}, {4, 543}, {5, 671}, {20, 542}, {24, 2936}, {114, 148}, {115, 3090}, {140, 11632}, {147, 3146}, {194, 576}, {376, 10991}, {384, 575}, {401, 3292}, {511, 14509}, {538, 11676}, {546, 6321}, {548, 14830}, {620, 3525}, {631, 2482}, {690, 14094}, {895, 11596}, {1569, 7772}, {1632, 2930}, {1656, 9166}, {2023, 22332}, {2794, 3529}, {2796, 4301}, {3023, 3303}, {3027, 3304}, {3455, 7512}, {3522, 11177}, {3523, 6055}, {3533, 9167}, {3575, 20774}, {3592, 19056}, {3594, 19055}, {3627, 6033}, {3628, 14061}, {3734, 7709}, {3746, 10086}, {3832, 8596}, {3843, 12355}, {3934, 15483}, {5067, 5461}, {5076, 22505}, {5171, 20081}, {5186, 5198}, {5563, 10089}, {5609, 15342}, {5613, 16001}, {5617, 16002}, {5965, 14712}, {5969, 10753}, {5986, 7492}, {6034, 9607}, {6036, 10303}, {6248, 7783}, {6278, 9883}, {6281, 9882}, {6419, 19109}, {6420, 19108}, {6776, 14928}, {7550, 13233}, {7787, 22234}, {7798, 10788}, {7799, 15980}, {7833, 11161}, {7839, 22330}, {7970, 7982}, {7991, 13174}, {8289, 13335}, {8716, 13860}, {9144, 16534}, {9624, 12258}, {9657, 18969}, {9670, 12354}, {9737, 14931}, {9830, 15069}, {9862, 17538}, {9881, 11362}, {11006, 16003}, {11403, 12131}, {12347, 15774}, {12350, 15888}, {12829, 22331}, {13334, 17128}, {14692, 15704}, {14850, 15535}, {15034, 22265}, {15301, 18860}

X(23235) = midpoint of X(147) and X(20094)
X(23235) = reflection of X(i) in X(j) for these (i,j): (4, 14981), (20, 10992), (376, 15300), (6776, 14928), (38664, 3)
X(23235) = 2nd Brocard circle-inverse of X(22712)
X(23235) = circumcircle-inverse of X(21166)
X(23235) = circumperp conjugate of X(34473)
X(23235) = X(1351)-of-1st anti-Brocard-triangle
X(23235) = X(2080)-of-6th Brocard-triangle
X(23235) = X(14981)-of-anti-Euler-triangle
X(23235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 14981, 6054), (20, 8591, 10992), (20, 10992, 12117), (98, 99, 21166), (114, 148, 14639), (115, 20399, 3090), (631, 12243, 11623), (2482, 11623, 631)

X(23236) = X(3)X(67)∩X(5)X(49)

Barycentrics    (9*R^2+3*SA-4*SW)*S^2-(9*R^2-5*SW)*SB*SC : :
X(23236) = 6*X(2)-5*X(15027), 9*X(2)-8*X(20396), 3*X(3)-2*X(16003), 4*X(3)-3*X(20126), 5*X(3)-4*X(20417), 2*X(4)-3*X(5655), X(4)-3*X(9143), 3*X(2930)-X(15069), 4*X(5609)-3*X(5655), 2*X(5609)-3*X(9143), 3*X(14850)-2*X(15545), 15*X(15027)-16*X(20396), 8*X(16003)-9*X(20126), 5*X(16003)-6*X(20417), 15*X(20126)-16*X(20417)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23236) lies on these lines: {2, 15027}, {3, 67}, {4, 5609}, {5, 49}, {20, 5663}, {30, 14094}, {74, 548}, {113, 3843}, {125, 3526}, {140, 9140}, {155, 382}, {381, 16534}, {539, 2070}, {541, 1657}, {547, 15025}, {549, 15020}, {550, 15054}, {568, 2854}, {569, 20301}, {631, 1511}, {632, 11694}, {858, 15132}, {1147, 15133}, {1173, 13163}, {1352, 14805}, {1539, 17578}, {1656, 5642}, {1907, 15472}, {2072, 15153}, {2771, 12680}, {2777, 12308}, {2781, 9833}, {2888, 5944}, {2917, 5898}, {2931, 19908}, {2948, 5881}, {3028, 4317}, {3292, 7574}, {3517, 12828}, {3519, 7488}, {3528, 12041}, {3530, 10264}, {3534, 10990}, {3564, 3581}, {3627, 10706}, {3832, 10113}, {3850, 15044}, {3853, 10733}, {3855, 20125}, {4325, 19470}, {4330, 7727}, {5054, 20397}, {5066, 15029}, {5067, 20304}, {5070, 5972}, {5654, 18430}, {6053, 12295}, {6193, 6243}, {6278, 12804}, {6281, 12803}, {6699, 15040}, {6776, 16270}, {7471, 14993}, {7486, 15081}, {7540, 9970}, {7579, 9703}, {7765, 14901}, {8703, 15021}, {9624, 12261}, {9657, 18968}, {9670, 12896}, {9714, 12309}, {9715, 12412}, {9976, 15037}, {10088, 15888}, {10540, 11799}, {10620, 15696}, {10657, 16965}, {10658, 16964}, {11362, 12778}, {11442, 18580}, {11449, 18356}, {11482, 15303}, {11579, 13339}, {11591, 12254}, {11693, 15694}, {12100, 13393}, {12105, 15360}, {12118, 18439}, {12140, 19504}, {12273, 15102}, {12790, 15774}, {13148, 18533}, {13352, 14982}, {13353, 15462}, {13391, 20063}, {14611, 20957}, {14851, 14934}, {15023, 17504}, {15059, 16239}, {15131, 18381}, {15135, 18445}, {15137, 18400}, {15463, 15559}, {15738, 19467}

X(23236) = midpoint of X(12273) and X(15102)
X(23236) = reflection of X(i) in X(j) for these (i,j): (4, 5609), (10620, 16163)
X(23236) = X(5609)-of-anti-Euler triangle
X(23236) = X(23039)-of-anti-orthocentroidal triangle
X(23236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5609, 5655), (4, 9143, 5609), (49, 14516, 6288), (382, 399, 15063), (382, 15063, 7728), (631, 3448, 20379), (631, 20379, 15061), (1511, 3448, 15061), (1511, 20379, 631), (1656, 15039, 5642), (3530, 10264, 15057), (9140, 15034, 140), (15035, 15057, 3530)

X(23237) = X(5)X(128)∩X(546)X(930)

Barycentrics    (S^2+SB*SC)*(8*S^2-R^2*(45*R^2+6*SA-38*SW)-4*SB*SC-8*SW^2) : :
X(23237) = X(4)+2*X(6592), X(5)+2*X(128), 5*X(5)-2*X(137), 4*X(5)-X(1263), 2*X(5)+X(14072), 5*X(5)+X(14073), 5*X(128)+X(137), 8*X(128)+X(1263), 4*X(128)-X(14072), 2*X(546)+X(930), X(550)-4*X(13372), X(1141)-4*X(3628), 5*X(1656)-2*X(12026), 5*X(3091)+X(13512), 7*X(3851)-X(11671)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23237) lies on these lines: {2, 11694}, {4, 6592}, {5, 128}, {546, 930}, {550, 13372}, {1141, 3628}, {1656, 12026}, {3091, 13512}, {3327, 10592}, {3851, 11671}, {6288, 6343}, {7159, 10593}, {7516, 15960}, {10224, 14769}, {11016, 15327}

X(23237) = X(15041)-of-orthic triangle
X(23237) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 128, 14072), (5, 14072, 1263), (5, 14073, 137)

X(23238) = X(5)X(128)∩X(548)X(930)

Barycentrics    (S^2+SB*SC)*(8*S^2-R^2*(63*R^2+18*SA-50*SW)-12*SB*SC-8*SW^2) : :
X(23238) = 5*X(5)-6*X(128), 7*X(5)-6*X(137), 4*X(5)-3*X(1263), 2*X(5)-3*X(14072), X(5)-3*X(14073), X(20)-3*X(13512), 7*X(128)-5*X(137), 8*X(128)-5*X(1263), 4*X(128)-5*X(14072), 2*X(548)-3*X(930), 5*X(631)-6*X(6592), 3*X(1141)-4*X(3530), 7*X(3526)-6*X(12026), 5*X(3843)-3*X(11671)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23238) lies on these lines: {5, 128}, {20, 10620}, {548, 930}, {631, 6592}, {1141, 3530}, {3526, 12026}, {3843, 11671}

X(23239) = X(3)X(107)∩X(20)X(133)

Barycentrics    S^4+(4*R^2-SW)*((48*R^2-SA-7*SW)*S^2-(36*R^2-7*SW)*SB*SC) : :
X(23239) = 2*X(3)+X(107), 4*X(3)-X(1294), X(4)+2*X(3184), X(4)-4*X(6716), 4*X(5)-X(10152), X(20)+2*X(133), X(40)+2*X(11718), 2*X(107)+X(1294), 2*X(122)-5*X(631), 2*X(122)+X(5667), 4*X(140)-X(10745), 4*X(182)-X(10762), 4*X(549)-X(10714), 5*X(631)+X(5667), X(3184)+2*X(6716)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23239) lies on these lines: {2, 2777}, {3, 107}, {4, 3184}, {5, 10152}, {20, 133}, {40, 11718}, {122, 631}, {140, 10745}, {182, 10762}, {549, 10714}, {550, 22337}, {1385, 10701}, {2797, 21166}, {2822, 21162}, {3324, 5204}, {5217, 7158}, {6713, 10775}, {9033, 11845}, {9528, 21161}, {14652, 22467}, {14703, 17928}

X(23239) = anticomplement of X(36520)
X(23239) = X(107)-Gibert-Moses centroid
X(23239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (631, 5667, 122), (3184, 6716, 4)

X(23240) = X(3)X(113)∩X(20)X(1075)

Barycentrics    SA*((40*R^2-11*SA+3*SW)*S^2-(SA-SW)*(8*R^2*(54*R^2-6*SA-23*SW)+10*SA^2-10*SB*SC+19*SW^2)) : :
X(23240) = 3*X(3)-2*X(122), X(122)-3*X(3184), 4*X(122)-3*X(10745), 3*X(381)-4*X(6716), 4*X(3184)-X(10745)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23240) lies on the cubic K725 and these lines: {3, 113}, {5, 10152}, {20, 1075}, {30, 107}, {110, 12113}, {133, 382}, {381, 6716}, {550, 933}, {1650, 10721}, {3324, 4299}, {4302, 7158}, {8703, 10714}, {9033, 12121}, {10706, 16190}, {11718, 12699}, {13557, 14934}, {14059, 20427}

X(23240) = midpoint of X(20) and X(5667)
X(23240) = reflection of X(3) in X(3184)
X(23240) = circumnormal isogonal conjugate of X(6760)
X(23240) = circumperp conjugate of X(13289)
X(23240) = circumcircle-inverse of X(13293)
X(23240) = X(3184)-of-X3-ABC-reflections-triangle
X(23240) = X(10152)-of-Johnson-triangle
X(23240) = X(10745)-of-ABC-X3-reflections-triangle

X(23241) = X(20)X(110)∩X(107)X(382)

Barycentrics    4*S^4+(4*R^2*(72*R^2-3*SA-29*SW)+3*SA^2-5*SB*SC+11*SW^2)*S^2-(4*R^2-SW)*(180*R^2-31*SW)*SB*SC : :
X(23241) = 4*X(5)-3*X(10152), 4*X(20)-3*X(1294), 3*X(107)-2*X(382), 6*X(122)-7*X(3528), 6*X(133)-5*X(17578), 4*X(548)-3*X(10745), 4*X(550)-3*X(10714), 5*X(631)-6*X(3184), 11*X(3855)-12*X(6716)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28261.

X(23241) lies on these lines: {5, 10152}, {20, 110}, {107, 382}, {122, 3528}, {133, 17578}, {548, 10745}, {550, 10714}, {631, 3184}, {1632, 5925}, {3855, 6716}

X(23242) = (name pending)

Trilinears    128*p^8-192*q*p^7+32*(4*q^2-5)*p^6+192*q*p^5-8*(2*q^2-1)*(4*q^2+9)*p^4+4*(24*q^2-23)*q*p^3+2*(8*q^2-3)*p^2-2*(6*q^2-5)*q*p-1+q^2 : : , where p=sin A/2 , q = cos(B/2- C/2) : :
Barycentrics    a (2 a^11-5 a^10 b-a^9 b^2+13 a^8 b^3-10 a^7 b^4-8 a^6 b^5+16 a^5 b^6-4 a^4 b^7-8 a^3 b^8+5 a^2 b^9+a b^10-b^11-5 a^10 c+20 a^9 b c-19 a^8 b^2 c -23 a^7 b^3 c+57 a^6 b^4 c-26 a^5 b^5 c-32 a^4 b^6 c+41 a^3 b^7 c-6 a^2 b^8 c-12 a b^9 c+5 b^10 c-a^9 c^2-19 a^8 b c^2+64 a^7 b^2 c^2-46 a^6 b^3 c^2 -59 a^5 b^4 c^2+110 a^4 b^5 c^2-38 a^3 b^6 c^2-38 a^2 b^7 c^2+34 a b^8 c^2-7 b^9 c^2+13 a^8 c^3-23 a^7 b c^3-46 a^6 b^2 c^3+136 a^5 b^3 c^3-74 a^4 b^4 c^3-74 a^3 b^5 c^3+99 a^2 b^6 c^3-26 a b^7 c^3-5 b^8 c^3-10 a^7 c^4+57 a^6 b c^4-59 a^5 b^2 c^4-74 a^4 b^3 c^4+158 a^3 b^4 c^4-60 a^2 b^5 c^4-35 a b^6 c^4 +22 b^7 c^4-8 a^6 c^5-26 a^5 b c^5+110 a^4 b^2 c^5-74 a^3 b^3 c^5-60 a^2 b^4 c^5+76 a b^5 c^5-14 b^6 c^5+16 a^5 c^6-32 a^4 b c^6-38 a^3 b^2 c^6+99 a^2 b^3 c^6-35 a b^4 c^6-14 b^5 c^6-4 a^4 c^7+41 a^3 b c^7-38 a^2 b^2 c^7-26 a b^3 c^7+22 b^4 c^7-8 a^3 c^8-6 a^2 b c^8+34 a b^2 c^8-5 b^3 c^8 +5 a^2 c^9-12 a b c^9-7 b^2 c^9+a c^10+5 b c^10-c^11) : :
X(23242) = (32*R^3-8*(5*s^2-4*SW)*R-94*r^3-S*s+50*SW*r)*X(36)+8*(2*R^3+(2*s^2-3*SW)*R+2*r^3+S*s-4*SW*r)*X(1411)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28265 and Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28270. .

X(23242) lies on this line: {36, 1411}

X(23243) = (name pending)

Trilinears    (16*p^4-24*q*p^3+8*q^2*p^2+2*q*p-2*q^2+1)*(16*p^4-16*q*p^3+8*(2*q^2-1)*p^2-4*q*p+1) : : , where p=sin A/2 , q = cos(B/2- C/2) : :
Barycentrics    a (a^6-a^5 b-a^4 b^2+2 a^3 b^3-a^2 b^4-a b^5+b^6-a^5 c+4 a^4 b c-3 a^3 b^2 c-2 a^2 b^3 c+4 a b^4 c-2 b^5 c-a^4 c^2-3 a^3 b c^2+7 a^2 b^2 c^2-3 a b^3 c^2-b^4 c^2+2 a^3 c^3-2 a^2 b c^3-3 a b^2 c^3+4 b^3 c^3-a^2 c^4+4 a b c^4-b^2 c^4-a c^5-2 b c^5+c^6) (2 a^6-3 a^5 b-3 a^4 b^2+6 a^3 b^3-3 a b^5+b^6-3 a^5 c +10 a^4 b c-5 a^3 b^2 c-9 a^2 b^3 c+8 a b^4 c-b^5 c-3 a^4 c^2-5 a^3 b c^2+16 a^2 b^2 c^2-5 a b^3 c^2-b^4 c^2+6 a^3 c^3-9 a^2 b c^3-5 a b^2 c^3+2 b^3 c^3+8 a b c^4-b^2 c^4-3 a c^5-b c^5+c^6) : :
X(23243) = (R^2+4*r^2)*X(3025)+(5*R^2-2*SW)*X(17705)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28265 and Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28270.

X(23243) lies on this line: {3025, 17705}

X(23244) = ALTINTAS-MONTESDEOCA MIDPOINT

Barycentrics    a(a-b-c)^3(a^6-3a^4(b-c)^2+3a^2(b-c)^2(b^2+6b c+c^2)-16a b(b-c)^2c(b+c)-(b-c)^4(b^2+6b c+c^2)) : :

Let DEF be the extouch triangle of ABC. Let Ha be the hyperbola with foci E and F that passes through A, and define Hb and Hc cyclically. The three hyperbolas have in common two points, and their midpoint is X(23244).

See Kadir Altintas and Angel Montesdeoca, HG150918.

X(23244) lies on this line: {40,971}

X(23245) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22385)

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

X(23245) lies on these lines: {2, 3}, {4383, 22385}

X(23246) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(1587), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23246) lies on these lines: {2, 3}, {216, 8956}, {5406, 23181}, {19356, 20794}

X(23247) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22393)

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

X(23247) lies on these lines: {2, 3}, {4383, 22393}

X(23248) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(3070), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23248) lies on these lines: {2, 3}, {5408, 23181}

X(23249) = X(2)X(1327)∩X(4)X(6)

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

X(23249) lies on these lines: {2, 1327}, {3, 18538}, {4, 6}, {5, 6398}, {20, 485}, {30, 3068}, {140, 6452}, {148, 22630}, {187, 21736}, {371, 3146}, {372, 3091}, {376, 590}, {381, 3069}, {382, 6199}, {486, 3832}, {546, 3312}, {550, 6451}, {615, 3545}, {631, 6412}, {632, 6456}, {637, 12222}, {638, 3620}, {1124, 5225}, {1132, 6436}, {1151, 3529}, {1152, 3090}, {1335, 5229}, {1495, 5200}, {1539, 19052}, {1597, 19006}, {1656, 6446}, {1657, 6445}, {1703, 19925}, {2043, 11488}, {2044, 11489}, {2071, 9682}, {3098, 21737}, {3311, 3627}, {3316, 3528}, {3366, 18582}, {3391, 18581}, {3522, 5418}, {3523, 10576}, {3524, 8253}, {3525, 6410}, {3543, 6561}, {3594, 13939}, {3619, 7389}, {3628, 6450}, {3830, 18512}, {3839, 6565}, {3843, 7584}, {3845, 13785}, {3850, 13951}, {3851, 13966}, {3853, 19117}, {3857, 13993}, {3861, 19116}, {5056, 5420}, {5059, 6480}, {5067, 6434}, {5068, 10577}, {5071, 6469}, {5076, 6417}, {5079, 6408}, {5411, 10151}, {5413, 6623}, {5414, 10590}, {5925, 8991}, {6241, 12239}, {6250, 13834}, {6251, 13770}, {6361, 13911}, {6409, 17538}, {6419, 22615}, {6425, 11541}, {6427, 12102}, {6433, 11001}, {6448, 12811}, {6449, 13925}, {6454, 15022}, {6455, 12103}, {6476, 9543}, {6490, 9693}, {6497, 14869}, {6502, 10591}, {6522, 12812}, {7374, 13711}, {7396, 8854}, {7687, 19039}, {7989, 13975}, {8276, 11413}, {8280, 10565}, {8376, 12256}, {9683, 12087}, {9690, 13903}, {9781, 12240}, {9955, 13959}, {10483, 13904}, {11008, 12322}, {12295, 19111}, {12699, 19066}, {12943, 19030}, {12953, 19028}, {13202, 19042}, {13889, 21312}, {13897, 15338}, {13898, 15326}, {13902, 18481}, {13915, 20127}, {13936, 18492}, {14229, 14240}, {14244, 14245}, {14269, 18510}, {14927, 19145}, {18480, 19065}, {18483, 18992}, {18535, 19005}

X(23250) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23249), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23250) lies on this line: {2,3}

X(23251) = X(2)X(6410)∩X(4)X(6)

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

X(23251) lies on these lines:{2, 6410}, {3, 3366}, {4, 6}, {5, 1152}, {20, 590}, {30, 485}, {64, 1321}, {98, 14240}, {115, 6423}, {140, 6412}, {141, 12323}, {230, 7374}, {371, 382}, {372, 381}, {486, 546}, {488, 23311}, {489, 1991}, {511, 12602}, {516, 13911}, {524, 12222}, {530, 22627}, {531, 22629}, {542, 9974}, {550, 5418}, {576, 12601}, {599, 638}, {615, 3091}, {1124, 3583}, {1131, 3068}, {1132, 19053}, {1328, 14893}, {1335, 3585}, {1478, 3298}, {1479, 3297}, {1578, 18531}, {1656, 6396}, {1657, 6200}, {1699, 7968}, {1703, 18492}, {2043, 16644}, {2044, 16645}, {2066, 12953}, {2067, 12943}, {2393, 6291}, {3069, 3832}, {3299, 18514}, {3301, 18513}, {3311, 3830}, {3312, 3843}, {3316, 17538}, {3529, 9540}, {3543, 6459}, {3545, 6430}, {3564, 22646}, {3592, 3627}, {3629, 12221}, {3763, 7389}, {3815, 7000}, {3839, 6471}, {3845, 6432}, {3850, 6438}, {3851, 6398}, {3853, 6431}, {3854, 13941}, {3856, 13993}, {3858, 18762}, {4299, 9661}, {4302, 9646}, {5023, 21736}, {5055, 6450}, {5056, 6434}, {5059, 6433}, {5068, 6469}, {5070, 6456}, {5072, 6454}, {5073, 6221}, {5076, 6419}, {5341, 6213}, {5406, 15234}, {5412, 12173}, {5414, 10895}, {5475, 6421}, {5691, 7969}, {6000, 12239}, {6212, 7297}, {6250, 6399}, {6406, 9969}, {6408, 19709}, {6418, 14269}, {6420, 13785}, {6422, 7748}, {6424, 7747}, {6429, 9541}, {6449, 17800}, {6453, 13903}, {6455, 15681}, {6470, 7585}, {6489, 12812}, {6497, 15694}, {6502, 10896}, {7507, 11474}, {7684, 10671}, {7685, 10672}, {7690, 13713}, {7756, 9600}, {8276, 12085}, {8280, 9909}, {8376, 18424}, {8909, 17702}, {9542, 10141}, {9647, 13904}, {9660, 13905}, {9682, 12084}, {9779, 13959}, {9812, 19066}, {10019, 13937}, {10110, 12240}, {10148, 15022}, {10533, 17845}, {10665, 12293}, {11291, 23312}, {11403, 19006}, {11480, 14814}, {11481, 14813}, {12231, 18400}, {12375, 12902}, {12571, 13971}, {13973, 19925}, {14238, 14245}, {15687, 19117}, {15752, 19039}, {19044, 19088}, {19130, 19146}, {19355, 21659}, {22961, 22971}

X(23252) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23251), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23252) lies on this line: {2,3}

X(23253) = X(2)X(12818)∩X(4)X(6)

Barycentrics    5 a^4 - 5 b^4 + 10 b^2 c^2 - 5 c^4 + 4 a^2 S : :

X(23253) lies on these lines: {2, 12818}, {4, 6}, {5, 6450}, {20, 5418}, {30, 6449}, {371, 1131}, {372, 3832}, {381, 6460}, {382, 3068}, {485, 3146}, {486, 3839}, {546, 3069}, {547, 6456}, {590, 3529}, {615, 3855}, {637, 11160}, {1132, 6420}, {1152, 3545}, {1657, 18538}, {3091, 6454}, {3311, 3853}, {3312, 3845}, {3316, 6409}, {3317, 6426}, {3522, 10576}, {3528, 8253}, {3533, 6412}, {3627, 6459}, {3830, 7583}, {3850, 6398}, {3858, 13951}, {3861, 13785}, {5056, 6396}, {5059, 6200}, {5067, 6410}, {5068, 5420}, {5073, 8981}, {5206, 21736}, {6250, 7374}, {6472, 13903}, {6481, 10194}, {6496, 15686}, {6497, 16239}, {6561, 17578}, {9682, 12086}, {10147, 11541}, {12102, 19117}, {12239, 12290}, {12323, 21356}, {13886, 15682}, {13993, 23046}, {14240, 14244}, {14269, 19053}, {14810, 21737}, {14893, 19116}, {15687, 19054}, {19066, 22793}, {19087, 23324}

X(23254) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23253), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (5 a^6 b^2 - 10 a^4 b^4 + 5 a^2 b^6 + 5 a^6 c^2 + 5 a^4 b^2 c^2 - 5 a^2 b^4 c^2 - 5 b^6 c^2 - 10 a^4 c^4 - 5 a^2 b^2 c^4 + 10 b^4 c^4 + 5 a^2 c^6 - 5 b^2 c^6 - 4 a^2 b^2 c^2 S) : :

X(23254) lies on this line: {2,3}

X(23255) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22395)

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

X(23255) lies on these lines: {2, 3}, {4383, 22395}

X(23256) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(1588), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23256) lies on these lines: {2, 3}, {5407, 23181}, {19355, 20794}

X(23257) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22396)

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

X(23257) lies on these lines: {2, 3}, {4383, 22396}

X(23258) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(3071), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23258) lies on these lines: {2, 3}, {5409, 23181}, {5943, 8961}

X(23259) = X(2)X(1328)∩X(4)X(6)

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

X(23259) lies on these lines: {2, 1328}, {3, 18762}, {4, 6}, {5, 6221}, {20, 486}, {30, 3069}, {140, 6451}, {148, 22601}, {176, 8973}, {371, 3091}, {372, 3146}, {376, 615}, {381, 3068}, {382, 6395}, {485, 3832}, {546, 3311}, {550, 6452}, {574, 21736}, {590, 3545}, {631, 6411}, {632, 6455}, {637, 3620}, {638, 12221}, {1124, 5229}, {1131, 6435}, {1151, 3090}, {1152, 3529}, {1335, 5225}, {1539, 19051}, {1597, 19005}, {1656, 6445}, {1657, 6446}, {1702, 19925}, {2043, 11489}, {2044, 11488}, {2066, 10590}, {2067, 10591}, {3312, 3627}, {3317, 3528}, {3367, 18582}, {3392, 18581}, {3522, 5420}, {3523, 10577}, {3524, 8252}, {3525, 6409}, {3543, 6560}, {3592, 13886}, {3619, 7388}, {3628, 6449}, {3634, 9582}, {3817, 9583}, {3830, 18510}, {3839, 6564}, {3843, 7583}, {3845, 13665}, {3850, 8976}, {3851, 8981}, {3853, 19116}, {3857, 13925}, {3861, 19117}, {5055, 9690}, {5056, 5418}, {5059, 6481}, {5067, 6433}, {5068, 10576}, {5071, 6468}, {5076, 6418}, {5079, 6407}, {5092, 21737}, {5218, 9660}, {5410, 10151}, {5412, 6623}, {5925, 13980}, {6241, 12240}, {6250, 13651}, {6251, 13711}, {6361, 13973}, {6410, 17538}, {6420, 22644}, {6426, 11541}, {6428, 12102}, {6434, 11001}, {6447, 12811}, {6450, 13993}, {6453, 15022}, {6456, 12103}, {6476, 9542}, {6496, 14869}, {6519, 12812}, {6811, 9600}, {7000, 13834}, {7288, 9647}, {7396, 8855}, {7486, 9681}, {7687, 19040}, {7989, 13912}, {8277, 11413}, {8281, 10565}, {8375, 12257}, {9543, 9680}, {9616, 10175}, {9682, 13595}, {9781, 12239}, {9955, 13902}, {10483, 13962}, {11008, 12323}, {12295, 19110}, {12699, 19065}, {12943, 19029}, {12953, 19027}, {13202, 19041}, {13883, 18492}, {13943, 21312}, {13954, 15338}, {13955, 15326}, {13959, 18481}, {13961, 17800}, {13979, 20127}, {14229, 14231}, {14236, 14244}, {14269, 18512}, {14927, 19146}, {18480, 19066}, {18483, 18991}, {18535, 19006}

X(23260) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23259), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23260) lies on this line: {2,3}

X(23261) = (name pending)

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

X(23261) lies on these lines: {2, 6409}, {3, 3367}, {4, 6}, {5, 1151}, {20, 615}, {30, 486}, {64, 1322}, {98, 14236}, {115, 6424}, {140, 6411}, {141, 12322}, {230, 7000}, {371, 381}, {372, 382}, {485, 546}, {487, 23312}, {490, 591}, {498, 9660}, {499, 9647}, {511, 12601}, {516, 13973}, {524, 12221}, {530, 22598}, {531, 22600}, {542, 9975}, {550, 5420}, {567, 9677}, {576, 12602}, {590, 3091}, {599, 637}, {1124, 3585}, {1131, 19054}, {1132, 3069}, {1327, 14893}, {1335, 3583}, {1478, 3297}, {1479, 3298}, {1506, 9600}, {1579, 18531}, {1656, 6200}, {1657, 6396}, {1699, 7969}, {1702, 18492}, {2043, 16645}, {2044, 16644}, {2066, 10895}, {2067, 10896}, {2393, 6406}, {3068, 3832}, {3090, 9541}, {3299, 18513}, {3301, 18514}, {3311, 3843}, {3312, 3830}, {3317, 17538}, {3529, 13935}, {3543, 6460}, {3545, 6429}, {3564, 22617}, {3594, 3627}, {3629, 12222}, {3763, 7388}, {3814, 9679}, {3815, 7374}, {3839, 6470}, {3845, 6431}, {3850, 6437}, {3851, 6221}, {3853, 6432}, {3854, 8972}, {3856, 13925}, {3858, 18538}, {5055, 6449}, {5056, 6433}, {5059, 6434}, {5068, 6468}, {5070, 6455}, {5072, 6453}, {5073, 6398}, {5076, 6420}, {5341, 6212}, {5407, 15233}, {5413, 12173}, {5414, 12953}, {5448, 8909}, {5475, 6422}, {5691, 7968}, {6000, 12240}, {6199, 8960}, {6213, 7297}, {6222, 6251}, {6291, 9969}, {6407, 19709}, {6417, 14269}, {6419, 13665}, {6421, 7748}, {6423, 7747}, {6430, 13939}, {6450, 17800}, {6454, 13961}, {6456, 15681}, {6471, 7586}, {6488, 12812}, {6496, 15694}, {6502, 12943}, {7507, 11473}, {7514, 9683}, {7603, 9674}, {7684, 10667}, {7685, 10668}, {7692, 13836}, {7988, 9615}, {7989, 9616}, {8277, 12085}, {8281, 9909}, {8375, 18424}, {8967, 22589}, {8983, 12571}, {9543, 10141}, {9676, 18350}, {9682, 13861}, {9692, 10139}, {9779, 13902}, {9812, 19065}, {10019, 13884}, {10110, 12239}, {10147, 15022}, {10666, 12293}, {11292, 23311}, {11403, 19005}, {11480, 14813}, {11481, 14814}, {11541, 17852}, {12232, 18400}, {12376, 12902}, {13911, 19925}, {14231, 14234}, {15687, 19116}, {15752, 19040}, {15815, 21736}, {19043, 19087}, {19130, 19145}, {19356, 21659}, {22960, 22971}

X(23262) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23261), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23262) lies on this line: {2,3}

X(23263) = (name pending)

Barycentrics    -5 a^4 + 5 b^4 - 10 b^2 c^2 + 5 c^4 + 4 a^2 S : :

X(23263) lies on these lines: {2, 12819}, {4, 6}, {5, 6449}, {20, 5420}, {30, 6450}, {371, 3832}, {372, 1132}, {381, 6459}, {382, 3069}, {485, 3839}, {486, 3146}, {546, 3068}, {547, 6455}, {590, 3855}, {615, 3529}, {638, 11160}, {1131, 6419}, {1151, 3545}, {1657, 18762}, {3091, 6453}, {3311, 3845}, {3312, 3853}, {3316, 6425}, {3317, 6410}, {3522, 10577}, {3528, 8252}, {3533, 6411}, {3627, 6460}, {3830, 7584}, {3850, 6221}, {3851, 9690}, {3858, 8976}, {3861, 13665}, {5056, 6200}, {5067, 6409}, {5068, 5418}, {5073, 13966}, {6251, 7000}, {6473, 13961}, {6480, 10195}, {6496, 16239}, {6497, 15686}, {6560, 17578}, {9583, 12571}, {9647, 10589}, {9660, 10588}, {10148, 11541}, {12102, 19116}, {12240, 12290}, {12322, 21356}, {13925, 23046}, {13939, 15682}, {14229, 14236}, {14269, 19054}, {14893, 19117}, {15687, 19053}, {17508, 21737}, {19065, 22793}, {19088, 23324}

X(23264) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23263), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (5 a^6 b^2 - 10 a^4 b^4 + 5 a^2 b^6 + 5 a^6 c^2 + 5 a^4 b^2 c^2 - 5 a^2 b^4 c^2 - 5 b^6 c^2 - 10 a^4 c^4 - 5 a^2 b^2 c^4 + 10 b^4 c^4 + 5 a^2 c^6 - 5 b^2 c^6 + 4 a^2 b^2 c^2 S) : :

X(23264) lies on this line: {2,3}

X(23265) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22397)

Barycentrics    a (6 a^7 + 6 a^6 b - 6 a^5 b^2 + 6 a^3 b^4 - 6 a^2 b^5 - 6 a b^6 + 6 a^6 c - 13 a^5 b c - 7 a^4 b^2 c - 5 a^3 b^3 c - 11 a^2 b^4 c - 6 a b^5 c - 12 b^6 c - 6 a^5 c^2 - 7 a^4 b c^2 - 2 a^3 b^2 c^2 - 5 a^2 b^3 c^2 - 8 a b^4 c^2 - 5 a^3 b c^3 - 5 a^2 b^2 c^3 - 12 a b^3 c^3 - 12 b^4 c^3 + 6 a^3 c^4 - 11 a^2 b c^4 - 8 a b^2 c^4 - 12 b^3 c^4 - 6 a^2 c^5 - 6 a b c^5 - 6 a c^6 - 12 b c^6) : :

X(23265) lies on these lines: {2, 3}, {4383, 22397}

X(23266) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(7581), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23266) lies on this line: {2,3}

X(23267) = X(2)X(6398)∩X(4)X(6)

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

X(23267) lies on these lines: {2, 6398}, {3, 8972}, {4, 6}, {5, 1131}, {20, 6221}, {30, 6199}, {140, 6446}, {146, 19052}, {371, 3529}, {372, 3090}, {376, 3068}, {378, 19006}, {381, 7586}, {382, 19117}, {485, 631}, {486, 3855}, {546, 6418}, {548, 13903}, {550, 6445}, {590, 3524}, {615, 5071}, {632, 6408}, {637, 11008}, {638, 3619}, {1132, 3843}, {1151, 17538}, {1152, 3525}, {1327, 6436}, {1384, 21736}, {1703, 5818}, {3069, 3545}, {3091, 3312}, {3146, 3311}, {3299, 5225}, {3301, 5229}, {3522, 6451}, {3523, 6452}, {3528, 6411}, {3533, 6481}, {3534, 9690}, {3590, 15720}, {3592, 11541}, {3620, 7389}, {3627, 6417}, {3832, 7584}, {3839, 13785}, {3845, 18510}, {4293, 19030}, {4294, 19028}, {5056, 13966}, {5067, 6438}, {5068, 13951}, {5072, 13993}, {5076, 6500}, {5411, 6623}, {5414, 8164}, {5418, 10299}, {6361, 13883}, {6407, 12103}, {6419, 22644}, {6434, 8253}, {6437, 9541}, {6450, 10303}, {6561, 15682}, {6622, 10881}, {6805, 15066}, {7375, 12323}, {8960, 21735}, {10590, 19037}, {10591, 18995}, {12017, 12602}, {12256, 13711}, {12257, 13651}, {13674, 14482}, {13790, 14061}, {13846, 19708}, {15081, 19059}, {18483, 19003}, {19046, 19060}

X(23268) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23267), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23268) lies on this line: {2,3}

X(23269) = (name pending)

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

X(23269) lies on these lines: {2, 6450}, {3, 1131}, {4, 6}, {20, 6449}, {372, 1327}, {376, 485}, {381, 13939}, {382, 7585}, {546, 7586}, {547, 6408}, {550, 8972}, {590, 3528}, {631, 6560}, {638, 21356}, {1132, 3845}, {1151, 11001}, {1152, 5067}, {1657, 9690}, {3068, 3529}, {3069, 3855}, {3090, 5420}, {3091, 13951}, {3146, 7583}, {3311, 3543}, {3312, 3832}, {3522, 8976}, {3523, 18538}, {3533, 6396}, {3534, 13925}, {3627, 18512}, {3830, 19117}, {3839, 7584}, {3850, 6395}, {3851, 13941}, {3853, 6417}, {3854, 18762}, {3861, 18510}, {5055, 6473}, {5056, 6398}, {5059, 6221}, {5066, 13961}, {5068, 13966}, {5071, 13935}, {5418, 21735}, {6410, 15702}, {6459, 15682}, {6472, 9693}, {6497, 15708}, {7374, 10846}, {7376, 12323}, {9540, 17538}, {11160, 12222}, {12602, 21737}, {13903, 15704}

X(23270) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23269), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (5 a^6 b^2 - 10 a^4 b^4 + 5 a^2 b^6 + 5 a^6 c^2 + 5 a^4 b^2 c^2 - 5 a^2 b^4 c^2 - 5 b^6 c^2 - 10 a^4 c^4 - 5 a^2 b^2 c^4 + 10 b^4 c^4 + 5 a^2 c^6 - 5 b^2 c^6 - 8 a^2 b^2 c^2 S) : :

X(23270) lies on this line: {2,3}

X(23271) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22398)

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

X(23271) lies on these lines: {2, 3}, {4383, 22398}

X(23272) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(7582), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23272) lies on this line: {2,3}

X(23273) = X(2)X(6221)∩X(4)X(6)

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

X(23273) lies on these lines: {2, 6221}, {3, 13939}, {4, 6}, {5, 1132}, {20, 6398}, {30, 6395}, {140, 6445}, {146, 19051}, {371, 3090}, {372, 3529}, {376, 3069}, {378, 19005}, {381, 7585}, {382, 19116}, {485, 3855}, {486, 631}, {546, 6417}, {548, 13961}, {550, 6446}, {590, 5071}, {615, 3524}, {632, 6407}, {637, 3619}, {638, 11008}, {1131, 3843}, {1151, 3525}, {1152, 17538}, {1328, 6435}, {1495, 19219}, {1702, 5818}, {2066, 8164}, {3054, 9602}, {3068, 3545}, {3091, 3311}, {3146, 3312}, {3299, 5229}, {3301, 5225}, {3522, 6452}, {3523, 6451}, {3526, 9690}, {3528, 6412}, {3533, 6480}, {3591, 15720}, {3594, 11541}, {3620, 7388}, {3627, 6418}, {3832, 7583}, {3839, 13665}, {3845, 18512}, {4293, 19029}, {4294, 19027}, {5024, 21736}, {5056, 8981}, {5067, 6437}, {5068, 8976}, {5072, 13925}, {5076, 6501}, {5410, 6623}, {5420, 10299}, {6361, 13936}, {6408, 12103}, {6420, 22615}, {6433, 8252}, {6438, 11001}, {6449, 10303}, {6476, 9680}, {6560, 15682}, {6622, 10880}, {6806, 15066}, {7376, 12322}, {7484, 9695}, {9692, 16239}, {10590, 19038}, {10591, 18996}, {12017, 12601}, {12256, 13770}, {12257, 13834}, {13670, 14061}, {13794, 14482}, {13847, 19708}, {15081, 19060}, {18483, 19004}, {19045, 19059}

X(23274) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23273), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

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

X(23274) lies on this line: {2,3}

X(23275) = (name pending)

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

X(23275) lies on these lines: {2, 6449}, {3, 1132}, {4, 6}, {20, 6450}, {371, 1328}, {376, 486}, {381, 13886}, {382, 7586}, {546, 7585}, {547, 6407}, {550, 13941}, {615, 3528}, {631, 6561}, {637, 21356}, {1131, 3845}, {1151, 5067}, {1152, 11001}, {1656, 9690}, {3068, 3855}, {3069, 3529}, {3090, 5418}, {3091, 8976}, {3146, 7584}, {3311, 3832}, {3312, 3543}, {3522, 13951}, {3523, 18762}, {3525, 9541}, {3533, 6200}, {3534, 13993}, {3627, 18510}, {3830, 19116}, {3839, 7583}, {3850, 6199}, {3851, 8972}, {3853, 6418}, {3854, 18538}, {3861, 18512}, {5055, 6472}, {5056, 6221}, {5059, 6398}, {5066, 13903}, {5068, 8981}, {5071, 9540}, {5420, 21735}, {6409, 15702}, {6460, 15682}, {6473, 17800}, {6496, 15708}, {7000, 10845}, {7375, 12322}, {7486, 9693}, {11160, 12221}, {13935, 17538}, {13961, 15704}

X(23276) = (A,B,C,X(6); A',B',C',X(3)) COLLINEATION IMAGE OF X(23275), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(3)

Barycentrics    a^2 (a^2 - b^2 - c^2) (5 a^6 b^2 - 10 a^4 b^4 + 5 a^2 b^6 + 5 a^6 c^2 + 5 a^4 b^2 c^2 - 5 a^2 b^4 c^2 - 5 b^6 c^2 - 10 a^4 c^4 - 5 a^2 b^2 c^4 + 10 b^4 c^4 + 5 a^2 c^6 - 5 b^2 c^6 + 8 a^2 b^2 c^2 S) : :

X(23276) lies on this line: {2,3}

X(23277) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22407)

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

X(23277) lies on these lines: {2, 3}, {4383, 22407}

X(23278) = (X(3),X(6),X(1),X(2); X(3),X(2),X(1),X(6)) COLLINEATION IMAGE OF X(22408)

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

X(23278) lies on these lines: {2, 3}, {4383, 22408}

X(23279) =  (name pending)

Barycentrics    a (2 a^9-5 a^8 b-a^7 b^2+12 a^6 b^3-9 a^5 b^4-6 a^4 b^5+13 a^3 b^6-4 a^2 b^7-5 a b^8+3 b^9-5 a^8 c+22 a^7 b c-21 a^6 b^2 c-21 a^5 b^3 c+50 a^4 b^4 c-24 a^3 b^5 c-17 a^2 b^6 c+23 a b^7 c-7 b^8 c-a^7 c^2-21 a^6 b c^2+62 a^5 b^2 c^2-40 a^4 b^3 c^2-38 a^3 b^4 c^2+63 a^2 b^5 c^2-23 a b^6 c^2-2 b^7 c^2 +12 a^6 c^3-21 a^5 b c^3-40 a^4 b^2 c^3+94 a^3 b^3 c^3-42 a^2 b^4 c^3-23 a b^5 c^3+18 b^6 c^3-9 a^5 c^4+50 a^4 b c^4-38 a^3 b^2 c^4-42 a^2 b^3 c^4+56 a b^4 c^4-12 b^5 c^4-6 a^4 c^5-24 a^3 b c^5+63 a^2 b^2 c^5-23 a b^3 c^5-12 b^4 c^5+13 a^3 c^6-17 a^2 b c^6-23 a b^2 c^6+18 b^3 c^6-4 a^2 c^7 +23 a b c^7-2 b^2 c^7-5 a c^8-7 b c^8+3 c^9) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28270.

X(23279) lies on this line: {5048,5882}

X(23280) =  MIDPOINT OF X(5) AND X(252)

Barycentrics    2 a^16-13 a^14 b^2+39 a^12 b^4-71 a^10 b^6+85 a^8 b^8-67 a^6 b^10+33 a^4 b^12-9 a^2 b^14+b^16-13 a^14 c^2+54 a^12 b^2 c^2-83 a^10 b^4 c^2+36 a^8 b^6 c^2+55 a^6 b^8 c^2-88 a^4 b^10 c^2+49 a^2 b^12 c^2-10 b^14 c^2+39 a^12 c^4-83 a^10 b^2 c^4+40 a^8 b^4 c^4+3 a^6 b^6 c^4+54 a^4 b^8 c^4-93 a^2 b^10 c^4+40 b^12 c^4-71 a^10 c^6+36 a^8 b^2 c^6+3 a^6 b^4 c^6+2 a^4 b^6 c^6+53 a^2 b^8 c^6-86 b^10 c^6+85 a^8 c^8+55 a^6 b^2 c^8+54 a^4 b^4 c^8+53 a^2 b^6 c^8+110 b^8 c^8-67 a^6 c^10-88 a^4 b^2 c^10-93 a^2 b^4 c^10-86 b^6 c^10+33 a^4 c^12+49 a^2 b^2 c^12+40 b^4 c^12-9 a^2 c^14-10 b^2 c^14+c^16 : :
X(23280) = X(5) + X(252)

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28270.

X(23280) lies on these lines: {5,252}, {30,14051}, {137,10126}, {140,6592}, {1209,1493}, {5501,6150}, {10610,15327}

X(23280) = midpoint X(5) and X(252)

X(23281) =  X(5)X(128)∩X(140,20414)

Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^12-16 a^10 b^2+36 a^8 b^4-44 a^6 b^6+31 a^4 b^8-12 a^2 b^10+2 b^12-16 a^10 c^2+42 a^8 b^2 c^2-28 a^6 b^4 c^2-17 a^4 b^6 c^2+30 a^2 b^8 c^2-11 b^10 c^2+36 a^8 c^4-28 a^6 b^2 c^4-a^4 b^4 c^4-18 a^2 b^6 c^4+26 b^8 c^4-44 a^6 c^6-17 a^4 b^2 c^6-18 a^2 b^4 c^6-34 b^6 c^6+31 a^4 c^8+30 a^2 b^2 c^8+26 b^4 c^8-12 a^2 c^10-11 b^2 c^10+2 c^12) : :
X(23281) = X[252] - 5 X[1656].

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28273.

X(23281) lies on these lines: {5,128}, {140,20414}, {252,1656}, {3628,10615}, {5501,13372}, {12242,13565}

X(23282) =  MIDPOINT OF X(4024) AND X(4064)

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

X(23282) lies on the cubic K1072 and these lines: {321,2517}, {513,7265}, {522,1324}, {523,661}, {1577,21121}, {2786,4840}, {3733,8045}, {4778,22037}

X(23282) = midpoint of X(4024) and X(4064)
X(23282) = reflection of X(i) in X(j) for these {i,j}: {3733, 8045}, {21121, 1577}
X(23282) = X(835)-Ceva conjugate of X(10)
X(23282) = X(i)-isoconjugate of X(j) for these (i,j): {835, 849}, {2214, 4556}
X(23282) = crosspoint of X(10) and X(835)
X(23282) = crosssum of X(58) and X(834)
X(23282) = barycentric product X(i)*X(j) for these {i,j}: {469, 4064}, {1089, 14349}, {4024, 5224}
X(23282) = barycentric quotient X(i)/X(j) for these {i,j}: {386, 4556}, {594, 835}, {834, 593}, {3876, 4612}, {4705, 2214}, {5224, 4610}, {14349, 757}

X(23283) =  X(476)-CEVA CONJUGATE OF X(13)

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

X(23283) lies on the cubics K979 and K1072 and on these lines: {13,690}, {14,15475}, {300,14295}, {395,523}, {526,14446}, {619,5664}, {1637,1989}, {2407,17402}, {6104,14270}, {8014,15358}, {8737,16230}, {9979,21469}, {10412,20579}

X(23283) = reflection of X(23284) in X(1637)
X(23283) = X(476)-Ceva conjugate of X(13)
X(23283) = crosspoint of X(13) and X(476)
X(23283) = trilinear pole of line {3258, 15610}
X(23283) = crossdifference of every pair of points on line {15, 1511}
X(23283) = crosssum of X(i) and X(j) for these (i,j): {15, 526}, {523, 6106}
X(23283) = X(523)-Hirst inverse of X(20578)
X(23283) = X(i)-isoconjugate of X(j) for these (i,j): {163, 11092}, {476, 1094}, {662, 11086}, {2154, 17402}
X(23283) = barycentric product X(i)*X(j) for these {i,j}: {299, 20578}, {300, 6138}, {523, 11078}, {850, 11081}, {3268, 11080}, {10412, 11130}, {11119, 14446}, {11128, 15475}
X(23283) = barycentric quotient X(i)/X(j) for these {i,j}: {16, 17402}, {512, 11086}, {523, 11092}, {526, 11131}, {2624, 1094}, {3268, 11129}, {3457, 5994}, {6138, 15}, {11078, 99}, {11080, 476}, {11081, 110}, {11130, 10411}, {14446, 618}, {14582, 10218}, {15475, 11085}, {20578, 14}

X(23284) =  X(476)-CEVA CONJUGATE OF X(14)

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

X(23284) lies on the cubics K979 and K1072 and on these lines: {13,15475}, {14,690}, {301,14295}, {396,523}, {526,14447}, {618,5664}, {1637,1989}, {2407,17403}, {6105,14270}, {8015,15358}, {8738,16230}, {9979,21468}, {10412,20578}

X(23284) = reflection of X(23283) in X(1637)
X(23284) = X(476)-Ceva conjugate of X(14)
X(23284) = crosspoint of X(14) and X(476)
X(23284) = trilinear pole of line {3258, 15609}
X(23284) = crossdifference of every pair of points on line {16, 1511}
X(23284) = crosssum of X(i) and X(j) for these (i,j): {16, 526}, {523, 6107}
X(23284) = X(523)-Hirst inverse of X(20579)
X(23284) = X(i)-isoconjugate of X(j) for these (i,j): {163, 11078}, {476, 1095}, {662, 11081}, {2153, 17403}
X(23284) = barycentric product X(i)*X(j) for these {i,j}: {298, 20579}, {301, 6137}, {523, 11092}, {850, 11086}, {3268, 11085}, {10412, 11131}, {11120, 14447}, {11129, 15475}
X(23284) = barycentric quotient X(i)/X(j) for these {i,j}: {15, 17403}, {512, 11081}, {523, 11078}, {526, 11130}, {2624, 1095}, {3268, 11128}, {3458, 5995}, {6137, 16}, {11085, 476}, {11086, 110}, {11092, 99}, {11131, 10411}, {14447, 619}, {14582, 10217}, {15475, 11080}, {20579, 13}

X(23285) =  MIDPOINT OF X(850) AND X(3267)

Barycentrics    b^2 c^2 (b^4-c^4) : :

X(23285) lies on the cubic K1072 and these lines: {2,4580}, {99,1287}, {325,523}, {338,15449}, {804,5152}, {808,9426}, {1225,15415}, {2799,18314}, {3049,9030}, {4086,21249}, {5664,6292}

X(23285) = midpoint of X(850) and X(3267)
X(23285) = isogonal conjugate of X(4630)
X(23285) = isotomic conjugate of X(827)
X(23285) = complement X(4580)
X(23285) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3456, 21220}, {15321, 21221}
X(23285) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 7668}, {31, 339}, {38, 127}, {112, 1215}, {162, 3934}, {163, 6676}, {250, 8060}, {427, 21253}, {648, 21238}, {662, 11574}, {1634, 18589}, {1843, 8287}, {1964, 15526}, {1973, 3124}, {2203, 21208}, {3051, 16573}, {3404, 3150}, {4020, 122}, {17171, 21252}, {17442, 125}, {20775, 16595}
X(23285) = complementary conjugate of complement of X(35325)
X(23285) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 339}, {689, 76}, {850, 826}, {1502, 338}
X(23285) = X(2528)-cross conjugate of X(826)
X(23285) = cevapoint of X(826) and X(2525)
X(23285) = crosspoint of X(76) and X(689)
X(23285) = crosssum of X(i) and X(j) for these (i,j): {32, 688}, {512, 9969}, {826, 6697}, {1576, 14574}, {8673, 11574}
X(23285) = crossdifference of every pair of points on line {32, 206}
X(23285) = X(850)-waw conjugate of X(18314)
X(23285) = pole wrt polar circle of line X(25)X(251)
X(23285) = anticomplement of isotomic conjugate of isogonal conjugate of X(37085)
X(23285) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4630}, {31, 827}, {32, 4599}, {82, 1576}, {162, 10547}, {163, 251}, {560, 4577}, {689, 1917}, {1101, 18105}, {1333, 4628}, {1501, 4593}, {3112, 14574}
X(23285) = barycentric product X(i)*X(j) for these {i,j}: {38, 20948}, {76, 826}, {141, 850}, {264, 2525}, {308, 2528}, {313, 16892}, {338, 4576}, {427, 3267}, {523, 8024}, {525, 1235}, {561, 8061}, {689, 15449}, {1502, 3005}, {1577, 1930}, {1928, 2084}, {3261, 15523}, {3933, 14618}, {4036, 16703}, {4568, 21207}, {14208, 20883}, {14424, 18023}, {15415, 16030}
X(23285) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 827}, {6, 4630}, {10, 4628}, {38, 163}, {39, 1576}, {75, 4599}, {76, 4577}, {115, 18105}, {141, 110}, {339, 4580}, {427, 112}, {523, 251}, {525, 1176}, {561, 4593}, {647, 10547}, {688, 1501}, {826, 6}, {850, 83}, {1235, 648}, {1502, 689}, {1577, 82}, {1930, 662}, {2084, 560}, {2525, 3}, {2528, 39}, {2530, 1333}, {3005, 32}, {3051, 14574}, {3267, 1799}, {3665, 4565}, {3703, 5546}, {3933, 4558}, {3954, 692}, {4036, 18098}, {4568, 4570}, {4576, 249}, {7794, 1634}, {7813, 5467}, {8024, 99}, {8061, 31}, {9494, 9233}, {14424, 187}, {15449, 3005}, {15523, 101}, {16030, 14586}, {16732, 18108}, {16887, 4556}, {16892, 58}, {18314, 17500}, {20021, 2715}, {20883, 162}, {20948, 3112}, {21016, 8750}, {21108, 1474}, {21123, 2206}, {21125, 5299}, {21207, 10566}

X(23286) =  X(125)-CROSS CONJUGATE OF X(3)

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

The trilinear polar of X(23286) passes through X(3269).

Let A'B'C' be the anticevian triangle of X(4). Let A"B"C" be the anticevian triangle of X(6) (i.e., the tangential triangle). Let A*B*C* be the tangential triangle, wrt A'B'C', of the bianticevian conic of X(4) and X(6). The lines A"A*, B"B*, C"C* concur in X(23286). (Randy Hutson, January 15, 2019)

Let P1 and P2 be the intersections, other than X(3) and X(4), of the Jerebek hyperbola and circle O(3,4). X(23286) is the cevapoint of P1 and P2. (Randy Hutson, January 15, 2019)

Let L be the line through X(3) parallel to BC, and let A' = L∩AX(110). Define B' and C' cyclically. The triangle A'B'C' is parallelogic to ABC (by construction), at X(3) and X(14380), and X(23286) is the unique fixed point of the affine transformation that maps ABC onto A'B'C'. (Angel Monesdeoca, June 27, 2020)

X(23286) lies on the cubic K1072 and these lines: {3,6368}, {50,647}, {54,8562}, {97,14329}, {186,523}, {477,1141}, {520,6760}, {656,22342}, {933,1304}, {1157,1510}, {1634,14587}, {2169,20803}, {2929,16035}, {3154,8901}, {3265,15414}, {3447,13558}, {6587,15422}, {7004,23226}, {9409,20188}, {15958,23181}

X(23286) = isogonal conjugate of X(35360)
X(23286) = isogonal conjugate of the anticomplement X(2972)
X(23286) = isotomic conjugate of polar conjugate of X(2623)
X(23286) = pole wrt polar circle of line X(5)X(324)
X(23286) = X(i)-Ceva conjugate of X(j) for these (i,j): {96, 8901}, {252, 125}, {933, 54}, {14586, 16030}, {15412, 2623}, {15958, 3}, {16813, 6}, {18315, 14533}, {18831, 19180}
X(23286) = X(i)-cross conjugate of X(j) for these (i,j): {125, 3}, {526, 14380}, {647, 15412}
X(23286) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2617}, {5, 162}, {19, 14570}, {51, 811}, {53, 662}, {92, 1625}, {99, 2181}, {112, 14213}, {158, 23181}, {163, 324}, {216, 823}, {250, 2618}, {648, 1953}, {799, 3199}, {933, 1087}, {1783, 17167}, {1897, 18180}, {2179, 6331}, {4575, 13450}, {4592, 14569}, {5379, 21102}
X(23286) = X(i)-vertex conjugate of X(j) for these (i,j): {54, 14157}, {14157, 54}
X(23286) = cevapoint of X(54) and X(19208)
X(23286) = crosspoint of X(i) and X(j) for these (i,j): {54, 933}, {95, 18315}, {107, 1173}
X(23286) = crossdifference of every pair of points on line {5, 53}
X(23286) = crosssum of X(i) and X(j) for these (i,j): {5, 6368}, {51, 12077}, {140, 520}, {389, 523}, {525, 14767}
X(23286) = barycentric product X(i)*X(j) for these {i,j}: {3, 15412}, {25, 15414}, {54, 525}, {63, 2616}, {69, 2623}, {95, 647}, {97, 523}, {125, 18315}, {275, 520}, {338, 15958}, {339, 14586}, {656, 2167}, {850, 14533}, {933, 15526}, {1141, 8552}, {1577, 2169}, {2148, 14208}, {2972, 16813}, {3265, 8882}, {3268, 11077}, {3269, 18831}, {3964, 15422}, {4558, 8901}, {4580, 16030}, {14618, 19210}
X(23286) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 14570}, {48, 2617}, {54, 648}, {95, 6331}, {97, 99}, {125, 18314}, {184, 1625}, {275, 6528}, {339, 15415}, {512, 53}, {520, 343}, {523, 324}, {525, 311}, {526, 14918}, {577, 23181}, {647, 5}, {656, 14213}, {669, 3199}, {798, 2181}, {810, 1953}, {924, 467}, {1459, 17167}, {1510, 14129}, {1636, 1568}, {2148, 162}, {2167, 811}, {2169, 662}, {2190, 823}, {2489, 14569}, {2501, 13450}, {2616, 92}, {2623, 4}, {3049, 51}, {3269, 6368}, {3708, 2618}, {6137, 6117}, {6138, 6116}, {8552, 1273}, {8882, 107}, {8884, 15352}, {8901, 14618}, {11077, 476}, {14270, 11062}, {14533, 110}, {14586, 250}, {15412, 264}, {15414, 305}, {15422, 1093}, {15958, 249}, {18315, 18020}, {19210, 4558}, {20975, 12077}, {22383, 18180}, {23224, 16697}

X(23287) =  MIDPOINT OF X(1649) AND X(14272)

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

X(23287) lies on the cubic K1072 and these lines: {99,5467}, {115,2793}, {187,8704}, {351,523}, {598,804}, {690,15303}, {1383,9147}, {1649,14272}, {2408,6088}, {2492,5466}, {6593,22255}, {9178,18818}, {9293,14764}

X(23287) = midpoint of X(i) and X(j) for these {i,j}: {1649, 14272}, {2408, 9485}
X(23287) = reflection of X(5466) in X(2492)
X(23287) = isogonal conjugate of X(32583)
X(23287) = X(598)-Ceva conjugate of X(20382)
X(23287) = X(20382)-cross conjugate of X(598)
X(23287) = cevapoint of X(i) and X(j) for these (i,j): {690, 9125}, {2492, 6088}
X(23287) = trilinear pole of line {1648, 5099}
X(23287) = crossdifference of every pair of points on line {574, 8542}
X(23287) = crosssum of X(i) and X(j) for these (i,j): {690, 16511}, {3906, 19510}
X(23287) = X(i)-isoconjugate of X(j) for these (i,j): {897, 9145}, {923, 9146}
X(23287) = barycentric product X(i)*X(j) for these {i,j}: {524, 8599}, {598, 690}, {892, 20382}, {1649, 18818}, {5466, 20380}, {10511, 18311}
X(23287) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 9145}, {351, 574}, {524, 9146}, {598, 892}, {690, 599}, {1383, 691}, {1648, 3906}, {8599, 671}, {9125, 11165}, {14273, 5094}, {20380, 5468}, {20382, 690}, {21839, 3908}, {21906, 17414}, {22105, 10130}

X(23288) =  MIDPOINT OF X(5466) AND X(14977)

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

X(23288) lies on the cubic K1072 and these lines: {2,523}, {67,690}, {111,6325}, {381,8704}, {599,3906}, {892,17941}, {1995,8599}, {2444,12073}, {2453,15922}, {2793,14666}, {2799,18007}, {8430,18575}, {10130,17436}

X(23288) = midpoint of X(5466) and X(14977)
X(23288) = reflection of X(i) in X(j) for these {i,j}: {1649, 18310}, {9178, 5466}
X(23288) = X(896)-isoconjugate of X(11636)
X(23288) = trilinear pole of line {3906, 8288}
X(23288) = crossdifference of every pair of points on line {187, 6593}
X(23288) = crosssum of X(i) and X(j) for these (i,j): {690, 8262}, {3906, 6698}
X(23288) = barycentric product X(i)*X(j) for these {i,j}: {599, 5466}, {671, 3906}, {892, 8288}, {5094, 14977}, {9178, 9464}, {17414, 18023}
X(23288) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 11636}, {574, 5467}, {599, 5468}, {690, 20380}, {3906, 524}, {5094, 4235}, {5466, 598}, {8288, 690}, {9178, 1383}, {17414, 187}

X(23289) =  CROSSSUM OF X(579) AND X(8676)

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

X(23289) lies on the cubic K1072 and these lines: {513,4077}, {522,21789}, {523,663}, {657,3700}, {885,2997}, {1305,14733}, {2424,21172}, {3900,4086}, {15313,16228}

X(23289) = X(1305)-Ceva conjugate of X(1751)
X(23289) = X(23289) = X(i)-isoconjugate of X(j) for these (i,j): {100, 4306}, {109, 3868}, {209, 1414}, {579, 651}, {664, 2352}, {934, 3190}, {1415, 18134}, {2198, 4573}, {4565, 22021}, {7045, 8676}
X(23289) = crosspoint of X(1305) and X(1751)
X(23289) = trilinear pole of line {14936, 21044}
X(23289) = crossdifference of every pair of points on line {579, 4306}
X(23289) = crosssum of X(579) and X(8676)
X(23289) = barycentric product X(i)*X(j) for these {i,j}: {272, 3700}, {522, 1751}, {650, 2997}, {657, 15467}, {1146, 1305}, {2218, 4391}
X(23289) = barycentric quotient X(i)/X(j) for these {i,j}: {272, 4573}, {522, 18134}, {649, 4306}, {650, 3868}, {657, 3190}, {663, 579}, {1146, 20294}, {1305, 1275}, {1751, 664}, {2218, 651}, {2997, 4554}, {3063, 2352}, {3064, 5125}, {3709, 209}, {4041, 22021}, {6589, 19367}, {14936, 8676}

X(23290) =  ISOGONAL CONJUGATE OF X(15958)

Barycentrics    b^2 c^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) : :

X(23290 lies on the cubic K1072 and these lines: {4,1510}, {137,2970}, {403,523}, {686,2501}, {924,13851}, {6368,18314}, {6750,23105}, {8562,14940}, {13399,20184}, {15328,22466}, {15422,16040}, {15424,18808}

X(23290) = isogonal conjugate of X(15958)
X(23290) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 2970}, {14618, 12077}
X(23290) = X(i)-cross conjugate of X(j) for these (i,j): {137, 4}, {12077, 18314}
X(23290) = crosspoint of X(107) and X(1179)
X(23290) = crossdifference of every pair of points on line {577, 1147}
X(23290) = crosssum of X(i) and X(j) for these (i,j): {389, 924}, {520, 1216}, {5449, 6368}
X(23290) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15958}, {48, 18315}, {54, 4575}, {63, 14586}, {97, 163}, {110, 2169}, {162, 19210}, {255, 933}, {656, 14587}, {662, 14533}, {2148, 4558}, {4100, 16813}
X(23290) = barycentric product X(i)*X(j) for these {i,j}: {4, 18314}, {5, 14618}, {25, 15415}, {53, 850}, {92, 2618}, {93, 20577}, {264, 12077}, {311, 2501}, {324, 523}, {525, 13450}, {2052, 6368}, {2081, 18817}, {2181, 20948}, {2970, 14570}, {3267, 14569}, {10412, 14918}, {15451, 18027}
X(23290) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18315}, {5, 4558}, {6, 15958}, {25, 14586}, {53, 110}, {112, 14587}, {311, 4563}, {324, 99}, {339, 15414}, {393, 933}, {512, 14533}, {523, 97}, {647, 19210}, {661, 2169}, {1093, 16813}, {1953, 4575}, {2052, 18831}, {2081, 22115}, {2181, 163}, {2501, 54}, {2618, 63}, {2970, 15412}, {3199, 1576}, {6116, 17403}, {6117, 17402}, {6368, 394}, {8754, 2623}, {12077, 3}, {13450, 648}, {14213, 4592}, {14569, 112}, {14618, 95}, {14918, 10411}, {15415, 305}, {15451, 577}, {15475, 11077}, {17434, 1092}, {18314, 69}, {21011, 1331}, {21102, 1790}, {21807, 906}

leftri

anti-Ursa-minor triangle and related centers: X(23291)-X(23333)

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This preamble and centers X(23291)-X(23333) were contributed by César Eliud Lozada, September 17, 2018.

The anti-Ursa-minor triangle of an acute triangle ABC is the triangle A'B'C' whose Ursa-minor triangle is ABC. This triangle A'B'C' can be constructed as the anti-Hutson-intouch triangle of the Euler triangle of ABC, and A'B'C' has the following properties:

The appearance of (T, i) in the following list means that triangles A'B'C' and T are perspective with perspector X(i) (an asterisk * means that triangles are homothetic):

(anti-Ascella*, 5094), (anti-Atik*, 23291), (1st anti-circumperp*, 858), (anti-Conway*, 23292), (2nd anti-Conway*, 13567), (3rd anti-Euler*, 23293), (4th anti-Euler*, 23294), (anti-excenters-reflections*, 235), (2nd anti-extouch*, 1899), (anti-Honsberger*, 3589), (anti-Hutson intouch*, 4), (anti-incircle-circles*, 1656), (anti-inverse-in-incircle*, 2), (6th anti-mixtilinear*, 1368), (1st anti-Sharygin*, 23295), (anti-tangential-midarc*, 12), (anti-Wasat*, 125), (AOA, 23296), (circumorthic*, 1594), (Ehrmann-mid, 5), (Ehrmann-side*, 2072), (Ehrmann-vertex*, 30), (2nd Ehrmann*, 524), (Euler, 6247), (2nd Euler*, 11585), (3rd Euler, 5), (4th Euler, 5), (5th Euler, 427), (1st excosine*, 1853), (extangents*, 3925), (Feuerbach, 3925), (intangents*, 11), (1st Kenmotu diagonals*, 590), (2nd Kenmotu diagonals*, 615), (Kosnita*, 140), (Lemoine, 23297), (Lucas antipodal tangents*, 23298), (Lucas(-1) antipodal tangents*, 23299), (medial, 141), (midheight, 23300), (1st Neuberg, 1503), (orthic*, 427), (Schroeter, 23301), (submedial*, 5), (tangential*, 2), (inner tri-equilateral*, 23302), (outer tri-equilateral*, 23303), (Trinh*, 30), (1st Zaniah, 23304), (2nd Zaniah, 23305)

The appearance of (T, i, j) in the following list means that triangles A'B'C' and T are orthologic with orthologic centers X(i) and X(j):

(ABC, 5, 3), (AAOA, 23306, 15136), (ABC-X3 reflections, 5, 3), (anti-Aquila, 5, 1385), (anti-Ara, 5, 4), (anti-Ascella, 12359, 12160), (anti-Atik, 12359, 6643), (5th anti-Brocard, 5, 3398), (2nd anti-circumperp-tangential, 5, 1), (1st anti-circumperp, 12359, 11412), (anti-Conway, 12359, 12161), (2nd anti-Conway, 12359, 5), (anti-Euler, 5, 20), (3rd anti-Euler, 12359, 5889), (4th anti-Euler, 12359, 11412), (anti-excenters-reflections, 12359, 12162), (2nd anti-extouch, 12359, 3), (anti-inner-Grebe, 5, 3312), (anti-outer-Grebe, 5, 3311), (anti-Honsberger, 12359, 19139), (anti-Hutson intouch, 12359, 12163), (anti-incircle-circles, 12359, 12164), (anti-inverse-in-incircle, 12359, 11411), (anti-Mandart-incircle, 5, 11248), (6th anti-mixtilinear, 12359, 1216), (anti-orthocentroidal, 10264, 3581), (1st anti-Sharygin, 12359, 19194), (anti-tangential-midarc, 12359, 7352), (anti-Wasat, 12359, 5562), (anticomplementary, 5, 4), (AOA, 23306, 15123), (Aquila, 5, 40), (Ara, 5, 7387), (Aries, 23307, 7387), (1st Auriga, 5, 11252), (2nd Auriga, 5, 11253), (5th Brocard, 5, 9821), (circumorthic, 12359, 5889), (2nd circumperp tangential, 5, 11249), (Ehrmann-mid, 5, 4), (Ehrmann-side, 12359, 18436), (Ehrmann-vertex, 12359, 9927), (1st Ehrmann, 141, 576), (2nd Ehrmann, 12359, 8548), (Euler, 5, 5), (2nd Euler, 12359, 5562), (1st excosine, 12359, 17834), (extangents, 12359, 6237), (outer-Garcia, 5, 355), (Gossard, 5, 11251), (inner-Grebe, 5, 1161), (outer-Grebe, 5, 1160), (3rd Hatzipolakis, 23308, 9729), (1st Hyacinth, 23306, 10112), (2nd Hyacinth, 23307, 3), (intangents, 12359, 6238), (Johnson, 5, 4), (inner-Johnson, 5, 10525), (outer-Johnson, 5, 10526), (1st Johnson-Yff, 5, 1478), (2nd Johnson-Yff, 5, 1479), (1st Kenmotu diagonals, 12359, 10665), (2nd Kenmotu diagonals, 12359, 10666), (Kosnita, 12359, 1147), (Lucas antipodal, 23309, 3), (Lucas antipodal tangents, 12359, 18939), (Lucas central, 23311, 3), (Lucas homothetic, 5, 10669), (Lucas reflection, 23313, 10670), (Lucas(-1) antipodal, 23310, 3), (Lucas(-1) antipodal tangents, 12359, 18940), (Lucas(-1) central, 23312, 3), (Lucas(-1) homothetic, 5, 10673), (Lucas(-1) reflection, 23314, 10674), (Macbeath, 3, 4), (Mandart-incircle, 5, 1), (medial, 5, 5), (midheight, 6247, 389), (5th mixtilinear, 5, 1482), (orthic, 12359, 52), (orthocentroidal, 10264, 568), (reflection, 21230, 6243), (submedial, 12359, 5462), (tangential, 12359, 155), (inner tri-equilateral, 12359, 10661), (outer tri-equilateral, 12359, 10662), (3rd tri-squares-central, 5, 8981), (4th tri-squares-central, 5, 13966), (Trinh, 12359, 7689), (X3-ABC reflections, 5, 3), (inner-Yff, 5, 55), (outer-Yff, 5, 56), (inner-Yff tangents, 5, 10679), (outer-Yff tangents, 5, 10680)

The appearance of (T, i, j) in the following list means that triangles A'B'C' and T are parallelogic with parallelogic centers X(i) and X(j):

(AAOA, 23315, 15139), (AOA, 23315, 15126), (1st Hyacinth, 23315, 10116), (1st Parry, 5, 351), (2nd Parry, 5, 351)

The appearance of (T, i, j) in the following list means that triangles A'B'C' and T are cyclologic with cyclologic centers X(i) and X(j):

(anti-Honsberger, 23316, 23317), (circummedial, 23318, 9076), (Kosnita, 23319, 23320), (2nd orthosymmedial, 23321, 23322)

The appearance of (T, i, j) in the following list means that triangles A'B'C' and T are eulerologic with eulerologic centers X(i) and X(j) (two dashes -- mean that eulerologic center does not exists):

(Ehrmann-mid, 23323, --), (Ehrmann-vertex, 23324, 23325), (2nd Ehrmann, 23326, 23327), (Euler, 5, --), (medial, 5, --), (Trinh, 23328, 23329)

The appearance of (T, i) in the following list means that triangles A'B'C' and T are inversely similar with center of inverse similitude X(i):

(AAOA, 23330), (AOA, 5), (1st Hyacinth, 23331)

The appearance of (T, i) in the following list means that A', B', C' and the vertices of triangle T lie on a conic with center X(i):

(ABC, 15449), (anti-Honsberger, 3589), (Kosnita, 140), (Wasat, 125)

The appearance of (i, j) in the following list means that X(i)-of-A'B'C' = X(j) ( for 1 ≤ i ≤ 5000 ):

(2, 23332), (3, 13371), (4, 12359), (5, 13561), (6, 23333), (110, 23319)

X(23291) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND ANTI-ATIK

Barycentrics    (-a^2+b^2+c^2)*(a^4+3*(b^2-c^2)^2) : :
X(23291) = 5*X(3091)-4*X(18418)

X(23291) lies on these lines: {2,98}, {3,15077}, {4,64}, {5,18909}, {6,8889}, {11,18922}, {12,18915}, {20,21663}, {30,18918}, {51,7378}, {66,19136}, {68,3546}, {69,1368}, {140,18925}, {141,18935}, {161,186}, {185,3091}, {193,21639}, {226,10360}, {235,12324}, {253,9307}, {343,7386}, {376,18396}, {394,16051}, {403,5656}, {427,9777}, {468,11206}, {511,7396}, {524,18919}, {590,18923}, {615,18924}, {631,6146}, {858,6515}, {868,18347}, {974,15081}, {1181,3090}, {1204,3146}, {1370,3580}, {1425,5261}, {1498,6622}, {1503,6353}, {1594,18916}, {1656,18914}, {1974,20079}, {2072,18917}, {2996,9289}, {3088,20299}, {3089,14216}, {3167,5159}, {3168,10002}, {3269,3981}, {3270,5274}, {3332,4213}, {3357,22538}, {3522,21659}, {3523,19467}, {3525,19357}, {3541,15033}, {3542,11457}, {3543,13851}, {3547,5449}, {3548,6193}, {3589,19119}, {3619,16419}, {3620,3819}, {3628,19347}, {3818,7398}, {3917,15073}, {3925,18921}, {3926,19599}, {5094,11245}, {5117,5286}, {5596,18374}, {6000,6623}, {6143,15872}, {6393,19583}, {6530,6619}, {6640,9703}, {6643,11821}, {6677,18440}, {6995,11550}, {7487,18381}, {7592,19360}, {8972,21640}, {9909,14927}, {10264,18933}, {10303,13367}, {10588,19349}, {10589,19354}, {11061,15128}, {11411,11585}, {11431,20303}, {11548,19125}, {12429,16196}, {13371,18951}, {13561,18952}, {13941,21641}, {14912,23292}, {15027,16270}, {15134,18281}, {15153,18405}, {18926,23298}, {18927,23299}, {18929,23302}, {18930,23303}, {18932,23306}, {18934,23307}, {18936,23308}, {18937,23309}, {18938,23310}, {18941,23311}, {18942,23312}, {18943,23313}, {18944,23314}, {18946,21230}, {18947,23315}, {19166,23295}

X(23291) = isogonal conjugate of X(34233)
X(23291) = crosssum of X(i) and X(j) for these (i,j): {3, 8780}, {25, 8778}, {154, 3053}, {577, 1974}
X(23291) = X(7396)-of-1st Brocard triangle
X(23291) = crosspoint of X(i) and X(j) for these {i,j}: {253, 2996}, {305, 2052}
X(23291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1899, 6776), (2, 5921, 9306), (2, 11442, 14826), (125, 1899, 2), (343, 7386, 10519), (427, 11433, 14853), (1853, 13567, 4), (5094, 11245, 11427), (6619, 14361, 6530), (8889, 18950, 6), (11442, 14826, 11180), (18911, 23293, 2), (18912, 23294, 3541)

X(23292) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND ANTI-CONWAY

Barycentrics    2*a^6-3*(b^2+c^2)*a^4+(b^4-c^4)*(b^2-c^2) : :
X(23292) = 3*X(2)+X(1993) = X(22)-3*X(13394) = 3*X(184)+X(11550) = 3*X(427)-X(11550) = 3*X(6800)+X(7391)

X(23292) lies on these lines: {2,6}, {3,12233}, {4,154}, {5,578}, {11,11429}, {12,19365}, {15,466}, {16,465}, {22,13394}, {24,11745}, {25,5480}, {30,11430}, {39,441}, {49,5576}, {50,97}, {51,468}, {52,7542}, {53,11547}, {54,1594}, {83,801}, {110,5133}, {125,11245}, {140,389}, {143,10020}, {159,3867}, {182,1368}, {184,427}, {185,6696}, {206,15809}, {235,11424}, {241,18652}, {275,1971}, {287,14153}, {297,1915}, {338,14920}, {378,15311}, {381,8780}, {397,471}, {398,470}, {401,10329}, {403,15033}, {428,1495}, {436,6530}, {440,572}, {458,5254}, {472,5318}, {473,5321}, {475,5706}, {511,6676}, {546,13403}, {549,11438}, {567,2072}, {569,11585}, {573,7536}, {575,5159}, {580,18641}, {581,7515}, {631,9786}, {647,14773}, {858,5012}, {952,21072}, {1073,9605}, {1092,7399}, {1125,16193}, {1151,1590}, {1152,1589}, {1181,3541}, {1192,3523}, {1199,6143}, {1350,7494}, {1352,3167}, {1370,3796}, {1375,1730}, {1498,3088}, {1585,3071}, {1586,3070}, {1587,3536}, {1588,3535}, {1593,1619}, {1595,6759}, {1614,15559}, {1620,15717}, {1656,11426}, {1708,17073}, {1834,11109}, {1843,15585}, {1848,2182}, {1853,6776}, {1885,5893}, {1899,5094}, {1907,16656}, {1990,2052}, {2261,5928}, {2979,7495}, {3127,13748}, {3128,13749}, {3431,18559}, {3516,5894}, {3526,11432}, {3542,10982}, {3564,21243}, {3567,10018}, {3574,3575}, {3917,7499}, {3925,11428}, {4074,7789}, {4185,5799}, {5020,14561}, {5066,7687}, {5085,7386}, {5092,10691}, {5136,5721}, {5169,9544}, {5432,11436}, {5433,19366}, {5446,13383}, {5449,13292}, {5462,16238}, {5486,11216}, {5622,15131}, {5643,22830}, {5654,9818}, {5752,7561}, {5943,5972}, {5944,11819}, {6090,7539}, {6101,7568}, {6202,19219}, {6353,14853}, {6354,17923}, {6524,15274}, {6618,10002}, {6723,15516}, {6756,10282}, {6800,7391}, {6823,13346}, {7378,11206}, {7403,10539}, {7404,17814}, {7487,17821}, {7507,19467}, {7526,19908}, {7576,11464}, {7577,12022}, {7667,22352}, {8226,17188}, {8263,9813}, {8964,9738}, {8966,13960}, {9225,19188}, {9545,14516}, {9729,16196}, {9730,10257}, {10096,13451}, {10110,21841}, {10125,16881}, {10154,21850}, {10264,12227}, {10300,20190}, {10516,14826}, {10602,23326}, {10605,23328}, {10619,11572}, {11225,12585}, {11422,23293}, {11423,23294}, {12024,15153}, {12161,12359}, {12228,23306}, {12229,23309}, {12230,23310}, {12231,23311}, {12232,23312}, {12234,21230}, {13011,23313}, {13012,23314}, {13198,23315}, {13352,15760}, {13419,16198}, {14157,16654}, {14216,19347}, {14912,23291}, {15211,19006}, {15212,19005}, {15583,19459}, {15644,16197}, {16030,23195}, {16577,17043}, {18396,23324}, {18755,21940}, {18914,20299}, {19408,23298}, {19409,23299}, {21659,23047}, {22529,23308}

X(23292) = midpoint of X(i) and X(j) for these {i,j}: {184, 427}, {13352, 15760}
X(23292) = polar conjugate of X(14860)
X(23292) = complement of X(343)
X(23292) = barycentric product X(1)*X(17859)
X(23292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 13567), (2, 394, 141), (2, 1993, 343), (2, 1994, 3580), (2, 3618, 17825), (2, 11427, 6), (3, 12233, 13568), (5, 578, 12241), (25, 10192, 15448), (49, 5576, 12134), (54, 1594, 6146), (125, 13366, 11245), (5480, 10192, 25), (11064, 14389, 3589), (11245, 13366, 12007)

X(23293) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 3rd ANTI-EULER

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

X(23293) lies on these lines: {2,98}, {3,12278}, {4,5449}, {5,5890}, {11,11446}, {12,19367}, {20,11204}, {22,1853}, {23,11550}, {30,11454}, {49,18356}, {51,5169}, {66,19121}, {69,6697}, {140,11449}, {141,12272}, {186,18474}, {235,11439}, {265,18570}, {343,858}, {378,14852}, {403,15305}, {427,3060}, {524,11443}, {590,11447}, {615,11448}, {631,1209}, {850,18022}, {1092,2888}, {1147,6143}, {1368,7998}, {1594,5889}, {1614,6639}, {1656,11441}, {1993,5094}, {2071,23329}, {2072,11459}, {3153,23325}, {3520,9927}, {3549,11457}, {3567,5576}, {3589,19122}, {3818,13595}, {3925,11445}, {4121,9464}, {4550,15081}, {5133,5640}, {5663,10254}, {5876,10255}, {5892,14789}, {6241,10024}, {6247,12279}, {6515,8889}, {6676,15080}, {6997,10545}, {7391,15107}, {7488,18381}, {7516,8907}, {7527,18390}, {7547,12163}, {7571,17825}, {7577,13754}, {8288,20859}, {10018,12134}, {10193,16163}, {10201,14157}, {10224,18436}, {10264,12270}, {10296,18376}, {10298,18400}, {10413,18362}, {10539,14940}, {10574,13160}, {11245,14389}, {11412,13371}, {11422,23292}, {11433,15019}, {11444,11585}, {11452,23302}, {11453,23303}, {11455,11799}, {11680,21252}, {12162,16868}, {12220,16789}, {12271,23307}, {12273,23306}, {12274,23309}, {12275,23310}, {12276,23311}, {12277,23312}, {12280,21230}, {12290,15761}, {12825,15060}, {12834,14561}, {13015,23313}, {13016,23314}, {13201,23315}, {13383,16659}, {13406,18439}, {13434,18912}, {13504,23319}, {13506,20625}, {14788,15028}, {15053,18420}, {15072,15760}, {16386,23328}, {18379,18562}, {18394,18563}, {19167,23295}, {19412,23298}, {19413,23299}, {20191,21844}, {22534,23308}

X(23293) = complement of X(9544)
X(23293) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1899, 5012), (2, 3410, 9306), (2, 3448, 184), (2, 23291, 18911), (3, 13561, 23294), (69, 23327, 11416), (125, 21243, 2), (140, 14516, 11449), (343, 858, 2979), (427, 3580, 3060), (1594, 12359, 5889), (3060, 7703, 427), (5012, 9140, 1899), (9140, 15059, 5622), (11704, 15058, 5)

X(23294) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 4th ANTI-EULER

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

X(23294) lies on these lines: {2,1614}, {3,12278}, {4,74}, {5,6241}, {6,17711}, {11,11461}, {12,19368}, {20,5449}, {24,1853}, {30,11468}, {54,70}, {66,19128}, {110,6640}, {140,11464}, {141,12283}, {184,6143}, {185,7577}, {186,18381}, {195,15106}, {235,11455}, {265,11250}, {403,6247}, {427,3567}, {468,16659}, {524,11458}, {590,11462}, {615,11463}, {631,21243}, {858,11412}, {1141,14050}, {1147,3448}, {1173,11433}, {1209,3523}, {1368,7999}, {1495,14864}, {1503,10018}, {1594,5890}, {1650,14059}, {1656,11456}, {2071,9927}, {2072,12111}, {3153,7689}, {3431,10619}, {3518,11550}, {3520,23329}, {3526,9707}, {3541,15033}, {3548,11442}, {3589,19123}, {3832,18488}, {3925,11460}, {5055,12174}, {5094,7592}, {5133,15024}, {5169,5462}, {5448,16003}, {5576,7703}, {5663,10255}, {5889,13371}, {6000,16868}, {6102,20379}, {6403,23300}, {6696,18560}, {6697,6776}, {6759,14940}, {7492,17712}, {7505,14157}, {7547,10605}, {7691,14791}, {7699,10937}, {7731,23315}, {7746,13509}, {8537,23327}, {8889,18916}, {9140,18281}, {9545,10116}, {9781,13567}, {10024,15072}, {10193,23040}, {10224,10264}, {10254,13491}, {10257,14516}, {10298,11750}, {11003,18128}, {11413,14852}, {11423,23292}, {11440,18404}, {11454,18563}, {11459,11585}, {11466,23302}, {11467,23303}, {11572,18559}, {12041,18379}, {12254,14076}, {12279,15761}, {12282,23307}, {12284,23306}, {12285,23309}, {12286,23310}, {12287,23311}, {12288,23312}, {12291,21230}, {12827,15034}, {13017,23313}, {13018,23314}, {13434,18952}, {13505,23319}, {14865,18390}, {16658,21841}, {17854,20304}, {18356,22115}, {18383,21663}, {18400,21844}, {18474,22467}, {19168,23295}, {19361,19504}, {19414,23298}, {19415,23299}, {22535,23308}

X(23294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11457, 1614), (3, 13561, 23293), (4, 20427, 10721), (125, 20299, 4), (403, 6247, 12290), (858, 12359, 11412), (1204, 23325, 4), (3541, 18912, 15033), (3541, 23291, 18912), (7505, 14216, 14157), (7703, 15043, 5576), (11704, 12290, 403), (11750, 20191, 10298), (12041, 18379, 18565), (13567, 15559, 9781)

X(23295) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 1st ANTI-SHARYGIN

Barycentrics    SB*SC*(S^2+SA*SB)*(S^2+SA*SC)*((8*R^2-SA-2*SW)*S^2-SB*SC*SW) : :

X(23295) lies on these lines: {2,19174}, {3,19205}, {4,19172}, {5,8884}, {11,19182}, {12,19175}, {30,19177}, {54,1594}, {95,1368}, {97,858}, {98,275}, {125,21638}, {140,19185}, {141,19197}, {230,8882}, {235,19169}, {403,19651}, {524,19178}, {590,19183}, {615,19184}, {1656,19173}, {1853,19180}, {1899,19170}, {2072,19176}, {3589,19171}, {3925,19181}, {4993,5133}, {4994,15559}, {5094,16030}, {6247,19206}, {6530,8794}, {7507,16035}, {8795,16089}, {9792,13567}, {10264,19195}, {11585,19179}, {12359,19194}, {13371,19210}, {13561,19211}, {19166,23291}, {19167,23293}, {19168,23294}, {19186,23298}, {19187,23299}, {19190,23302}, {19191,23303}, {19193,23306}, {19196,23307}, {19198,23308}, {19199,23309}, {19200,23310}, {19201,23311}, {19202,23312}, {19203,23313}, {19204,23314}, {19207,21230}, {19208,23315}

X(23295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 19174, 19189), (427, 8901, 275)

X(23296) = PERSPECTOR OF THESE TRIANGLES: ANTI-URSA-MINOR AND AOA

Barycentrics    SA*(2*(3*R^2*(12*R^2-5*SW)+SA*SW+SW^2)*S^2+(SB+SC)*(R^2*(6*SA-SW)-2*SA^2+2*SB*SC)*SW) : :
X(23296) = 3*X(3167)-X(11061)

X(23296) lies on these lines: {5,5181}, {6,15119}, {67,3564}, {141,6723}, {511,15125}, {542,6247}, {599,8548}, {895,5159}, {2854,15116}, {2892,12084}, {2930,12118}, {3167,11061}, {3448,19588}, {5622,16196}, {8681,19510}, {12359,15115}, {15121,15531}, {15123,23306}

X(23296) = midpoint of X(3448) and X(19588)

X(23297) = PERSPECTOR OF THESE TRIANGLES: ANTI-URSA-MINOR AND LEMOINE

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

X(23297) lies on these lines: {2,187}, {66,3618}, {83,9076}, {126,1506}, {597,8288}, {850,18311}, {858,3613}, {1502,3266}, {5169,7790}, {5996,8371}, {7664,8370}, {7665,16044}, {7813,8024}, {8801,17907}

X(23297) = isotomic conjugate of X(10130)
X(23297) = polar conjugate of X(32581)

X(23298) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics    S^2-2*(2*R^2-SW)*S-2*R^2*(SA+SW)-SB*SC+SW^2 : :

X(23298) lies on these lines: {2,6503}, {3,19498}, {4,13021}, {5,18980}, {11,19434}, {12,19370}, {30,18414}, {125,21642}, {140,19440}, {141,8222}, {235,19416}, {427,19446}, {524,19426}, {590,19436}, {615,19439}, {858,19406}, {1368,19422}, {1594,19424}, {1656,19418}, {1853,19430}, {1899,19358}, {2072,18462}, {3589,19134}, {3925,19432}, {3926,5490}, {5094,19404}, {6247,19500}, {10264,19484}, {11585,19428}, {12321,13055}, {12359,18939}, {13061,22818}, {13567,19410}, {18926,23291}, {19186,23295}, {19408,23292}, {19412,23293}, {19414,23294}, {19450,23302}, {19452,23303}, {19482,23306}, {19486,23307}, {19488,23308}, {19490,23309}, {19492,23311}, {19495,23312}, {19496,23314}, {19502,21230}, {19507,23315}

X(23298) = {X(2), X(19420)}-harmonic conjugate of X(8939)

X(23299) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics    S^2+2*(2*R^2-SW)*S-2*R^2*(SA+SW)-SB*SC+SW^2 : :

X(23299) lies on these lines: {2,6503}, {3,19499}, {4,13022}, {5,18981}, {11,19435}, {12,19371}, {30,18415}, {125,21643}, {140,19441}, {141,8223}, {235,19417}, {427,19447}, {524,19427}, {590,19438}, {615,19437}, {858,19407}, {1368,19423}, {1594,19425}, {1656,19419}, {1853,19431}, {1899,19359}, {2072,18463}, {3589,19135}, {3925,19433}, {3926,5491}, {5094,19405}, {6247,19501}, {10264,19485}, {11585,19429}, {12320,13056}, {12359,18940}, {13062,22817}, {13567,19411}, {18927,23291}, {19187,23295}, {19409,23292}, {19413,23293}, {19415,23294}, {19451,23302}, {19453,23303}, {19483,23306}, {19487,23307}, {19489,23308}, {19491,23310}, {19493,23312}, {19494,23311}, {19497,23313}, {19503,21230}, {19508,23315}

X(23300) = PERSPECTOR OF THESE TRIANGLES: ANTI-URSA-MINOR AND MIDHEIGHT

Barycentrics    (b^2+c^2)*a^6+(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*a^2-(b^4-c^4)^2 : :
X(23300) = X(6)+3*X(1853) = 2*X(6)-3*X(10169) = X(6)-3*X(23327) = X(66)-3*X(1853) = 2*X(66)+3*X(10169) = X(66)+3*X(23327) = X(141)-3*X(23332) = X(193)-3*X(11216) = 5*X(632)-2*X(15582) = 2*X(1853)+X(10169) = X(3098)-3*X(23329) = X(3818)-3*X(23325) = 2*X(6697)+X(15583) = 2*X(6697)-3*X(23332) = X(15583)+3*X(23332)

X(23300) lies on these lines: {2,159}, {4,9914}, {5,182}, {6,66}, {10,3827}, {30,18382}, {64,1907}, {67,13248}, {69,858}, {125,1205}, {140,15577}, {141,1368}, {157,441}, {161,7499}, {193,11216}, {343,3313}, {389,1595}, {468,20987}, {511,12235}, {542,23306}, {550,15578}, {632,15582}, {973,19161}, {1177,15321}, {1352,8549}, {1594,6776}, {1596,7687}, {1619,6997}, {1906,15752}, {1974,11550}, {2072,18440}, {2450,9722}, {2777,20301}, {2781,10264}, {2854,15116}, {3098,23329}, {3448,15141}, {3556,19784}, {3564,13371}, {3618,5133}, {3627,15579}, {3628,15581}, {3629,23315}, {3763,5646}, {3867,9969}, {5050,5576}, {5085,7399}, {5092,18400}, {5094,19459}, {5142,5800}, {5169,15431}, {6000,19130}, {6403,23294}, {7403,14216}, {7405,9833}, {8177,21536}, {8362,15270}, {8550,12242}, {8889,18935}, {10117,10301}, {10249,15760}, {10859,11019}, {11431,14853}, {11442,20806}, {12220,16789}, {12294,22530}, {17442,21916}

X(23300) = midpoint of X(i) and X(j) for these {i,j}: {6, 66}, {67, 13248}, {141, 15583}, {1352, 8549}, {3448, 15141}
X(23300) = reflection of X(i) in X(j) for these (i,j): (5, 20300), (141, 6697), (550, 15578)
X(23300) = complementary conjugate of X(3162)
X(23300) = complement of X(159)
X(23300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1853, 66), (66, 23327, 6), (141, 23332, 6697), (3618, 5596, 19153), (3867, 13567, 9969), (11574, 21243, 141), (14216, 14561, 19149), (15583, 23332, 141)

X(23301) = PERSPECTOR OF THESE TRIANGLES: ANTI-URSA-MINOR AND SCHROETER

Barycentrics    ((b^2+c^2)*a^2-b^2*c^2)*(b^2-c^2) : :
X(23301) = X(850)+3*X(5996) = X(850)-3*X(9148) = 2*X(2501)-3*X(10278) = X(3005)-3*X(5996) = X(3005)+3*X(9148) = 3*X(4108)-X(8664) = 2*X(6562)-3*X(14610) = 2*X(8651)-3*X(11176) = 3*X(9134)-X(12077)

X(23301) lies on these lines: {2,669}, {5,1499}, {115,9151}, {125,2679}, {126,9152}, {140,5926}, {141,9009}, {325,523}, {427,2501}, {512,625}, {525,4486}, {647,804}, {656,4010}, {661,2533}, {688,21262}, {924,6130}, {1368,10190}, {1853,1988}, {2450,3566}, {2486,22227}, {2799,12075}, {3221,21191}, {4108,8664}, {4128,5518}, {5025,14824}, {5133,10189}, {5169,8371}, {6562,14610}, {7668,8288}, {7752,23099}, {8639,21301}, {8651,11176}, {8653,15283}, {9134,12077}, {15523,21726}, {15667,18312}, {20909,21350}, {20952,21349}

X(23301) = midpoint of X(i) and X(j) for these {i,j}: {8639, 21301}, {20909, 21350}
X(23301) = isotomic conjugate of X(3222)
X(23301) = complement of X(669)
X(23301) = complementary conjugate of X(1084)
X(23301) = pole wrt nine-point circle of line X(2)X(6)
X(23301) = nine-point-circle-inverse of X(32525)
X(23301) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32531)
X(23301) = radical center of nine-point circles of {ABC, 1st Brocard triangle, 1st anti-Brocard triangle}
X(23301) = radical center of de Longchamps circles of {ABC, 1st Brocard triangle, 1st anti-Brocard triangle}

X(23302) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND INNER TRI-EQUILATERAL

Barycentrics    a^2+2*sqrt(3)*S : :
X(23302) = X(10645)-3*X(16241) = 3*X(16241)+X(16808)

X(23302) lies on these lines: {2,6}, {3,5318}, {4,11480}, {5,15}, {11,10638}, {12,7051}, {13,549}, {14,547}, {16,17}, {18,10188}, {30,10645}, {37,5243}, {39,630}, {44,5242}, {53,470}, {61,3628}, {62,632}, {115,619}, {125,21647}, {187,624}, {233,19295}, {235,11475}, {398,1656}, {427,10641}, {465,22052}, {466,10979}, {468,10642}, {471,6748}, {546,5238}, {550,5350}, {618,5472}, {631,5335}, {635,6673}, {636,7749}, {858,11420}, {1250,5432}, {1368,11515}, {1506,6694}, {1594,10632}, {1853,17826}, {1899,19363}, {2045,3070}, {2046,3071}, {2072,18468}, {2548,11312}, {2549,11298}, {2963,2981}, {3090,5334}, {3412,16961}, {3523,5340}, {3525,22238}, {3526,11486}, {3530,16965}, {3627,5352}, {3767,11311}, {3851,5349}, {3925,10636}, {5054,10653}, {5055,10654}, {5056,5339}, {5092,6115}, {5094,11408}, {5237,14869}, {5254,11289}, {5326,7127}, {5344,10299}, {5351,12108}, {5366,21735}, {5433,19373}, {5459,8589}, {5460,9117}, {5471,6670}, {5613,18358}, {5650,11624}, {6200,18585}, {6247,10675}, {6396,15765}, {6676,11516}, {6677,10644}, {6772,22489}, {6774,6783}, {7542,10635}, {7737,11297}, {7745,11290}, {8254,10678}, {9820,10662}, {10018,10633}, {10020,11268}, {10264,10657}, {10272,10658}, {10576,14814}, {10577,14813}, {10634,11585}, {10659,23307}, {10661,12359}, {10663,23306}, {10667,23311}, {10671,23312}, {10676,16252}, {10677,21230}, {10681,23315}, {11145,15109}, {11243,23332}, {11267,13371}, {11301,21843}, {11307,19780}, {11452,23293}, {11466,23294}, {11539,16242}, {12816,15690}, {12980,23309}, {12982,23310}, {13057,23313}, {13058,23314}, {13083,18424}, {13349,20415}, {14061,14905}, {15699,16962}, {18755,21903}, {18929,23291}, {19190,23295}, {19450,23298}, {19451,23299}, {20252,23005}, {21156,22513}, {22974,23308}

X(23302) = complement of X(302)
X(23302) = crosssum of X(6) and X(61)
X(23302) = crosspoint of X(2) and X(17)
X(23302) = X(2)-Ceva conjugate of X(629)
X(23302) = perspector of circumconic centered at X(629)
X(23302) = center of circumconic that is locus of trilinear poles of lines passing through X(629)
X(23302) = intersection of tangents to Evans conic at X(13) and X(15)
X(23302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 23303), (2, 303, 141), (2, 396, 395), (2, 11488, 6), (2, 16644, 396), (3, 18582, 5318), (5, 15, 5321), (6, 11488, 396), (6, 16644, 11488), (6, 23303, 395), (15, 16966, 5), (141, 3054, 23303), (396, 23303, 6), (590, 615, 396), (3055, 3589, 23303), (5321, 16772, 15)

X(23303) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND OUTER TRI-EQUILATERAL

Barycentrics    a^2-2*sqrt(3)*S : :
X(23303) = X(10646)-3*X(16242) = 3*X(16242)+X(16809)

X(23303) lies on these lines: {2,6}, {3,5321}, {4,11481}, {5,16}, {11,1250}, {12,19373}, {13,547}, {14,549}, {15,18}, {17,10187}, {30,10646}, {37,5242}, {39,629}, {44,5243}, {53,471}, {61,632}, {62,3628}, {115,618}, {125,21648}, {187,623}, {233,19294}, {235,11476}, {397,1656}, {427,10642}, {465,10979}, {466,22052}, {468,10641}, {470,6748}, {546,5237}, {550,5349}, {619,5471}, {631,5334}, {635,7749}, {636,6674}, {858,11421}, {1368,11516}, {1506,6695}, {1594,10633}, {1853,17827}, {1899,19364}, {2045,3071}, {2046,3070}, {2072,18470}, {2307,7294}, {2548,11311}, {2549,11297}, {2963,6151}, {3090,5335}, {3411,16960}, {3523,5339}, {3525,22236}, {3526,11485}, {3530,16964}, {3627,5351}, {3767,11312}, {3851,5350}, {3925,10637}, {5054,10654}, {5055,10653}, {5056,5340}, {5092,6114}, {5094,11409}, {5238,14869}, {5254,11290}, {5343,10299}, {5352,12108}, {5365,21735}, {5432,10638}, {5433,7051}, {5459,9115}, {5460,8589}, {5472,6669}, {5617,18358}, {5650,11626}, {6200,15765}, {6247,10676}, {6396,18585}, {6676,11515}, {6677,10643}, {6771,6782}, {6775,22490}, {7542,10634}, {7737,11298}, {7745,11289}, {8254,10677}, {9820,10661}, {10018,10632}, {10020,11267}, {10264,10658}, {10272,10657}, {10576,14813}, {10577,14814}, {10635,11585}, {10660,23307}, {10662,12359}, {10664,23306}, {10668,23311}, {10672,23312}, {10675,16252}, {10678,21230}, {10682,23315}, {11146,15109}, {11244,23332}, {11268,13371}, {11302,21843}, {11308,19781}, {11453,23293}, {11467,23294}, {11539,16241}, {12817,15690}, {12981,23309}, {12983,23310}, {13059,23313}, {13060,23314}, {13084,18424}, {13350,20416}, {14061,14904}, {15699,16963}, {18755,21932}, {18930,23291}, {19191,23295}, {19452,23298}, {19453,23299}, {20253,23004}, {21157,22512}, {22975,23308}

X(23303) = complement of X(303)
X(23303) = crosssum of X(6) and X(62)
X(23303) = crosspoint of X(2) and X(18)
X(23303) = X(2)-Ceva conjugate of X(630)
X(23303) = perspector of circumconic centered at X(630)
X(23303) = center of circumconic that is locus of trilinear poles of lines passing through X(630)
X(23303) = intersection of tangents to Evans conic at X(14) and X(16)
X(23303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 23302), (2, 302, 141), (2, 395, 396), (2, 11489, 6), (2, 16645, 395), (3, 18581, 5321), (5, 16, 5318), (6, 11489, 395), (6, 16645, 11489), (6, 23302, 396), (16, 16967, 5), (141, 3054, 23302), (395, 23302, 6), (590, 615, 395), (3055, 3589, 23302), (5318, 16773, 16)

X(23304) = PERSPECTOR OF THESE TRIANGLES: ANTI-URSA-MINOR AND 1st ZANIAH

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

X(23304) lies on these lines: {2,197}, {5,515}, {11,33}, {56,429}, {406,22654}, {858,11680}, {1368,2886}, {1572,5517}, {1595,7681}, {1883,10896}, {1904,7354}, {1997,11681}, {2823,7956}, {3086,5142}, {3741,20305}, {3827,21621}, {4187,19836}, {10859,11019}, {19542,20470}, {21252,23332}

X(23304) = complementary conjugate of X(478)
X(23304) = complement of X(197)
X(23304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 427, 17111), (1368, 2886, 23305)

X(23305) = PERSPECTOR OF THESE TRIANGLES: ANTI-URSA-MINOR AND 2nd ZANIAH

Barycentrics    (b^2+c^2)*a^3-(b+c)*(b^2+c^2)*a^2+(b^2-c^2)^2*a-(b^4-c^4)*(b-c) : :
X(23305) = 3*X(2)+X(11677) = X(18674)+3*X(21020)

X(23305) lies on these lines: {2,1486}, {5,516}, {10,3827}, {11,4319}, {12,2263}, {19,427}, {120,17279}, {141,2876}, {674,16608}, {858,4329}, {1329,3823}, {1368,2886}, {1375,1631}, {1836,21015}, {1953,21931}, {2195,17337}, {2835,3820}, {3816,17356}, {7399,21160}, {17452,21945}, {18674,21020}, {20621,23050}

X(23305) = complementary conjugate of X(5452)
X(23305) = complement of X(1486)
X(23305) = X(1485)-of-2nd Zaniah triangle
X(23305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11677, 1486), (1368, 2886, 23304)

X(23306) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO AAOA

Barycentrics    SA*(2*(7*R^2-2*SA)*S^2+(SB+SC)*(3*R^2*(6*R^2-5*SA-4*SW)+4*SA^2-4*SB*SC+2*SW^2)) : :
X(23306) = 3*X(2)+X(12319) = X(399)-3*X(5654) = 5*X(1656)-X(12310) = 3*X(1853)+X(17838) = 2*X(9820)+X(15133) = 3*X(10192)-2*X(20773) = 3*X(14852)-5*X(15081)

The reciprocal orthologic center of these triangles is X(15136).

X(23306) lies on these lines: {2,2931}, {4,12302}, {5,1511}, {11,12888}, {12,19469}, {30,12901}, {74,858}, {110,1594}, {113,427}, {125,5562}, {140,12893}, {141,14984}, {155,3448}, {235,12295}, {265,2072}, {399,5654}, {403,10733}, {524,12596}, {542,23300}, {590,12891}, {615,12892}, {912,13605}, {1147,10224}, {1368,6699}, {1568,21650}, {1656,12310}, {1853,17838}, {1899,19456}, {2854,20300}, {3548,22661}, {3564,9976}, {3574,16223}, {3589,19138}, {3818,10272}, {3925,12661}, {5094,12168}, {5449,11804}, {5576,14643}, {5663,6247}, {7542,22109}, {7577,12383}, {9927,11801}, {10024,12121}, {10117,14790}, {10192,20773}, {10255,12118}, {10264,13754}, {10663,23302}, {10664,23303}, {12140,20771}, {12228,23292}, {12233,14708}, {12236,13567}, {12273,23293}, {12284,23294}, {13160,15035}, {13392,13413}, {14852,15066}, {15123,23296}, {15760,16163}, {18531,19457}, {18932,23291}, {19193,23295}, {19482,23298}, {19483,23299}

X(23306) = midpoint of X(i) and X(j) for these {i,j}: {4, 12302}, {110, 15133}, {155, 3448}, {265, 5504}, {10117, 14790}
X(23306) = reflection of X(i) in X(j) for these (i,j): (110, 9820), (9927, 11801)
X(23306) = complement of X(2931)
X(23306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12319, 2931), (11585, 20302, 12359)

X(23307) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO ARIES

Barycentrics    SA*(2*(2*R^2-SA)*S^2+(SB+SC)*(2*R^2*(2*R^2-3*SA-2*SW)+2*SA^2-2*SB*SC+SW^2)) : :
X(23307) = 3*X(2)+X(12318) = 5*X(1656)-X(12309)

The reciprocal orthologic center of these triangles is X(7387).

X(23307) lies on these lines: {2,9937}, {4,12301}, {5,578}, {11,9931}, {12,19471}, {30,9938}, {68,394}, {125,21651}, {136,8800}, {140,9932}, {141,5449}, {155,427}, {235,12293}, {524,9926}, {590,12424}, {615,12425}, {858,11411}, {1216,1368}, {1594,6193}, {1595,22660}, {1656,12309}, {1853,17836}, {1899,19458}, {2072,12429}, {3167,5576}, {3564,13371}, {3589,19141}, {3925,12417}, {5094,12166}, {5654,7403}, {6247,13754}, {6823,18475}, {8548,18952}, {9908,14790}, {10659,23302}, {10660,23303}, {11064,20303}, {12118,15760}, {12225,12319}, {12235,13567}, {12271,23293}, {12282,23294}, {13383,15577}, {15647,17702}, {18934,23291}, {19196,23295}, {19486,23298}, {19487,23299}

X(23307) = midpoint of X(i) and X(j) for these {i,j}: {4, 12301}, {68, 15316}, {9908, 14790}
X(23307) = reflection of X(5) in X(20302)
X(23307) = complement of X(9937)
X(23307) = {X(2), X(12318)}-harmonic conjugate of X(9937)

X(23308) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO 3rd HATZIPOLAKIS

Barycentrics    (R^2*(64*R^2+SA-23*SW)+2*SW^2)*S^2+2*(4*R^2-SW)^2*SB*SC : :
X(23308) = 3*X(2)+X(22555) = 3*X(1853)+X(17837)

The reciprocal orthologic center of these triangles is X(9729).

X(23308) lies on these lines: {2,2929}, {4,22549}, {5,12897}, {11,22954}, {12,19472}, {30,22816}, {125,21652}, {140,22962}, {235,22538}, {403,22951}, {427,22970}, {524,22830}, {590,22960}, {615,22961}, {858,22528}, {1368,5894}, {1594,22750}, {1656,22550}, {1853,17837}, {1899,19460}, {2072,6288}, {3091,22971}, {3589,19142}, {3925,22840}, {5094,22497}, {5876,10264}, {6247,11585}, {6247,12162}, {9927,11801}, {11591,12359}, {11793,21230}, {13371,22800}, {13567,22530}, {15052,22585}, {15069,22533}, {18936,23291}, {19198,23295}, {19488,23298}, {19489,23299}, {22529,23292}, {22534,23293}, {22535,23294}, {22974,23302}, {22975,23303}

X(23308) = midpoint of X(4) and X(22549)
X(23308) = complement of X(2929)
X(23308) = {X(2), X(22555)}-harmonic conjugate of X(2929)

X(23309) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO LUCAS ANTIPODAL

Barycentrics    (4*R^2*(SA+SW)-SW^2)*S^2-SW^2*SB*SC-2*((6*R^2-SW)*S^2-(2*R^2+SW)*SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(3).

X(23309) lies on these lines: {2,12320}, {3,19499}, {4,12303}, {5,642}, {11,12910}, {12,19473}, {30,9921}, {66,3564}, {125,21653}, {140,12972}, {235,12296}, {427,487}, {486,1368}, {524,12597}, {590,12960}, {615,12966}, {858,12221}, {1594,12509}, {1595,6290}, {1656,12311}, {1853,17839}, {1899,19461}, {2072,22809}, {3589,19143}, {3925,12662}, {5094,12169}, {6823,12123}, {11585,12601}, {12229,23292}, {12237,13567}, {12274,23293}, {12285,23294}, {12980,23302}, {12981,23303}, {18937,23291}, {19199,23295}, {19490,23298}

X(23309) = midpoint of X(4) and X(12303)
X(23309) = complement of X(12978)

X(23310) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO LUCAS(-1) ANTIPODAL

Barycentrics    (4*R^2*(SA+SW)-SW^2)*S^2-SW^2*SB*SC+2*((6*R^2-SW)*S^2-(2*R^2+SW)*SB*SC)*S : :
X(23310) = 3*X(2)+X(12321) = 5*X(1656)-X(12312) = 3*X(1853)+X(17842)

The reciprocal orthologic center of these triangles is X(3).

X(23310) lies on these lines: {2,12321}, {3,19498}, {4,12304}, {5,641}, {11,12911}, {12,19474}, {30,9922}, {66,3564}, {125,21654}, {140,12973}, {235,12297}, {427,488}, {485,1368}, {524,12598}, {590,12961}, {615,12967}, {858,12222}, {1594,12510}, {1595,6289}, {1656,12312}, {1853,17842}, {1899,19462}, {2072,22810}, {3589,19144}, {3925,12663}, {5094,12170}, {6823,12124}, {11585,12602}, {12238,13567}, {12275,23293}, {12286,23294}, {12982,23302}, {12983,23303}, {19200,23295}

X(23310) = midpoint of X(4) and X(12304)
X(23310) = complement of X(12979)
X(23310) = {X(2), X(12321)}-harmonic conjugate of X(12979)

X(23311) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO LUCAS CENTRAL

Barycentrics    S*(b^2+c^2)+(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(23311) = 3*X(2)+X(12322) = 5*X(641)-3*X(13701) = 5*X(1656)-X(12313) = 3*X(1853)+X(17840)

The reciprocal orthologic center of these triangles is X(3).

X(23311) lies on these lines: {2,489}, {3,14233}, {4,12305}, {5,141}, {11,6283}, {12,7362}, {30,641}, {125,21655}, {140,12974}, {235,12298}, {343,15234}, {427,6291}, {485,524}, {486,3589}, {487,8253}, {492,3070}, {590,637}, {591,1587}, {597,7584}, {615,7389}, {642,3628}, {858,12223}, {1368,12360}, {1503,6289}, {1591,12239}, {1594,6239}, {1656,12313}, {1853,17840}, {1899,19463}, {2072,22811}, {3091,5590}, {3317,13783}, {3592,12221}, {3593,6460}, {3629,7583}, {3630,18538}, {3734,6251}, {3925,6252}, {5023,8252}, {5056,5591}, {5094,12171}, {6118,8981}, {6201,15835}, {6561,11315}, {8584,19117}, {10124,13821}, {10667,23302}, {10668,23303}, {11585,12603}, {12276,23293}, {12287,23294}, {19201,23295}, {19492,23298}, {19494,23299}

X(23311) = midpoint of X(4) and X(12305)
X(23311) = complement of X(1151)
X(23311) = complementary conjugate of X(33364)
X(23311) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12322, 1151), (5, 141, 23312), (5, 639, 141), (486, 11313, 3589)

X(23312) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO LUCAS(-1) CENTRAL

Barycentrics    -S*(b^2+c^2)+(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(23312) = 3*X(2)+X(12323) = 5*X(642)-3*X(13821) = 5*X(1656)-X(12314)

The reciprocal orthologic center of these triangles is X(3).

X(23312) lies on these lines: {2,490}, {3,14230}, {4,12306}, {5,141}, {11,6405}, {12,7353}, {30,642}, {125,21656}, {140,12975}, {235,12299}, {343,15233}, {427,6406}, {485,3589}, {486,524}, {488,8252}, {491,3071}, {590,7388}, {597,7583}, {615,638}, {641,3628}, {858,12224}, {1368,12361}, {1503,6290}, {1588,1991}, {1592,12240}, {1594,6400}, {1656,12314}, {1853,17843}, {1899,19464}, {2072,22812}, {3091,5591}, {3316,13663}, {3594,12222}, {3595,6459}, {3629,7584}, {3630,18762}, {3734,6250}, {3925,6404}, {5023,8253}, {5056,5590}, {5094,12172}, {6119,13966}, {6202,15834}, {6560,11316}, {8584,19116}, {10124,13701}, {10671,23302}, {10672,23303}, {11585,12604}, {12277,23293}, {12288,23294}, {19202,23295}, {19493,23299}, {19495,23298}

X(23312) = midpoint of X(4) and X(12306)
X(23312) = complement of X(1152)
X(23312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12323, 1152), (5, 141, 23311), (5, 640, 141), (485, 11314, 3589)

X(23313) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO LUCAS REFLECTION

Barycentrics    S^4-(2*R^2*(SA+SW)-SB*SC-SW^2)*S^2+SW^2*SB*SC-2*((2*R^2-SW)*S^2-(2*R^2+SW)*SB*SC)*S : :
X(23313) = 3*X(2)+X(13025) = 5*X(1656)-X(13023) = 3*X(1853)+X(17841)

The reciprocal orthologic center of these triangles is X(10670).

X(23313) lies on these lines: {2,13025}, {4,13021}, {5,641}, {11,13043}, {12,19475}, {30,13061}, {125,21657}, {140,13049}, {235,13019}, {427,13051}, {524,13037}, {590,13045}, {615,13047}, {858,13009}, {1368,13027}, {1594,13035}, {1656,13023}, {1853,17841}, {1899,19465}, {2072,22813}, {3070,6458}, {3589,19147}, {3613,7403}, {3925,13041}, {5094,13007}, {11585,13039}, {13013,13567}, {13015,23293}, {13017,23294}, {13057,23302}, {13059,23303}, {19203,23295}, {19497,23299}

X(23313) = midpoint of X(4) and X(13021)
X(23313) = complement of X(13055)
X(23313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13025, 13055), (3613, 7403, 23314)

X(23314) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO LUCAS(-1) REFLECTION

Barycentrics    S^4-(2*R^2*(SA+SW)-SB*SC-SW^2)*S^2+SW^2*SB*SC+2*((2*R^2-SW)*S^2-(2*R^2+SW)*SB*SC)*S : :

The reciprocal orthologic center of these triangles is X(10674).

X(23314) lies on these lines: {2,13026}, {4,13022}, {5,642}, {11,13044}, {12,19476}, {30,13062}, {125,21658}, {140,13050}, {235,13020}, {427,13052}, {524,13038}, {590,13046}, {615,13048}, {858,13010}, {1368,13028}, {1594,13036}, {1656,13024}, {1853,17844}, {1899,19466}, {2072,22814}, {3071,6457}, {3589,19148}, {3613,7403}, {3925,13042}, {5094,13008}, {11585,13040}, {13014,13567}, {13016,23293}, {13018,23294}, {13058,23302}, {13060,23303}, {19204,23295}, {19496,23298}

X(23314) = midpoint of X(4) and X(13022)
X(23314) = complement of X(13056)
X(23314) = {X(3613), X(7403)}-harmonic conjugate of X(23313)

X(23315) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO AAOA

Barycentrics    (R^2*(48*R^2-SA-19*SW)+2*SW^2)*S^2-2*(5*R^2-SW)*SB*SC*SW : :
X(23315) = 3*X(2)+X(13203) = X(110)-3*X(15131) = 2*X(125)-3*X(23332) = 5*X(1656)-X(9919) = 3*X(1853)-X(3448) = 3*X(1853)+X(17847) = 4*X(5972)-3*X(10192) = 2*X(6723)-3*X(15113) = X(9934)-3*X(14643) = 2*X(10113)-3*X(23324) = 3*X(10192)-2*X(15647) = 3*X(10606)-X(12244) = 2*X(11801)-3*X(23325) = 2*X(12041)-3*X(23328) = 3*X(14643)-2*X(16252)

The reciprocal parallelogic center of these triangles is X(15139).

X(23315) lies on these lines: {2,10117}, {4,2929}, {5,1539}, {6,2892}, {11,10118}, {12,19505}, {30,12893}, {51,125}, {64,146}, {66,15141}, {74,1594}, {110,858}, {113,2883}, {140,13289}, {141,13416}, {235,13202}, {399,14216}, {403,10721}, {511,15126}, {524,13248}, {590,13287}, {615,13288}, {946,2778}, {974,12233}, {1177,3589}, {1368,5972}, {1595,7687}, {1624,1650}, {1656,9919}, {1853,1993}, {1899,19504}, {2072,7728}, {2854,15583}, {2931,14790}, {3066,5169}, {3088,18933}, {3357,10224}, {3541,19457}, {3629,23300}, {3925,10119}, {5094,13171}, {5133,15059}, {5576,15061}, {5621,8889}, {5663,6247}, {5893,11744}, {5925,16868}, {6102,10115}, {6146,15463}, {6697,16776}, {6759,10272}, {7577,10606}, {7706,23329}, {7731,23294}, {9934,14643}, {10024,20127}, {10113,23324}, {10255,20427}, {10681,23302}, {10682,23303}, {11801,23325}, {12227,18914}, {12241,15472}, {13160,15055}, {13201,23293}, {13413,16219}, {14644,15559}, {15760,16111}, {17855,18388}, {19208,23295}, {19507,23298}, {19508,23299}

X(23315) = midpoint of X(i) and X(j) for these {i,j}: {4, 2935}, {6, 2892}, {64, 146}, {66, 15141}, {399, 14216}, {2931, 14790}
X(23315) = reflection of X(i) in X(j) for these (i,j): (74, 6696), (141, 15116), (1177, 3589), (6759, 10272), (9934, 16252)
X(23315) = complement of X(10117)
X(23315) = inverse of X(6716) in the nine-point circle
X(23315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 13203, 10117), (125, 1112, 13567), (1853, 17847, 3448), (5972, 15647, 10192), (9934, 14643, 16252)

X(23316) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO ANTI-HONSBERGER

Barycentrics
(b^2+c^2)*a^28-2*(b^2+c^2)^2*a^26-3*(b^2+c^2)*(b^4+c^4)*a^24+4*(2*b^8+2*c^8-(b^2-c^2)^2*b^2*c^2)*a^22+(b^2+c^2)*(b^8+c^8+b^2*c^2*(6*b^4+43*b^2*c^2+6*c^4))*a^20-2*(5*b^12+5*c^12-(23*b^8+23*c^8+12*b^2*c^2*(b^4+c^4))*b^2*c^2)*a^18+(b^2+c^2)*(5*b^12+5*c^12-(28*b^8+28*c^8+b^2*c^2*(37*b^4-84*b^2*c^2+37*c^4))*b^2*c^2)*a^16-8*(9*b^12+9*c^12-2*b^2*c^2*(b^8-9*b^4*c^4+c^8))*b^2*c^2*a^14-(b^2+c^2)*(5*b^16+5*c^16-2*(22*b^12+22*c^12-(13*b^8+13*c^8+b^2*c^2*(72*b^4-91*b^2*c^2+72*c^4))*b^2*c^2)*b^2*c^2)*a^12+2*(5*b^16+5*c^16+2*(5*b^12+5*c^12-(27*b^8+27*c^8-b^2*c^2*(67*b^4-87*b^2*c^2+67*c^4))*b^2*c^2)*b^2*c^2)*(b^2+c^2)^2*a^10-(b^2+c^2)*(b^20+c^20+(24*b^16+24*c^16-(67*b^12+67*c^12+2*(2*b^8+2*c^8-b^2*c^2*(57*b^4-100*b^2*c^2+57*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8-4*(b^4-c^4)^2*(2*b^16+2*c^16+(b^12+c^12+(2*b^8+2*c^8+b^2*c^2*(17*b^4-20*b^2*c^2+17*c^4))*b^2*c^2)*b^2*c^2)*a^6+(b^4-c^4)^2*(b^2+c^2)*(3*b^16+3*c^16-(2*b^12+2*c^12+(35*b^8+35*c^8-2*b^2*c^2*(27*b^4-44*b^2*c^2+27*c^4))*b^2*c^2)*b^2*c^2)*a^4+2*(b^4-c^4)^4*(b^12+c^12-(b^8+c^8-2*b^2*c^2*(3*b^4-4*b^2*c^2+3*c^4))*b^2*c^2)*a^2-(b^4+c^4)*(b^2+c^2)^5*(b^2-c^2)^8 : :

The reciprocal cyclologic center of these triangles is X(23317).

X(23316) lies on the line {3589,23317}

X(23317) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO ANTI-URSA-MINOR

Barycentrics
a^2*(a^30-2*(b^2+c^2)*a^28-3*(b^4+c^4)*a^26+4*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^24+(b^8+c^8+b^2*c^2*(8*b^4+39*b^2*c^2+8*c^4))*a^22-(b^2+c^2)*(10*b^8+10*c^8-b^2*c^2*(37*b^4-24*b^2*c^2+37*c^4))*a^20+(5*b^12+5*c^12-(36*b^8+36*c^8+b^2*c^2*(23*b^4-72*b^2*c^2+23*c^4))*b^2*c^2)*a^18-(b^2+c^2)*(37*b^8+37*c^8-5*b^2*c^2*(15*b^4-28*b^2*c^2+15*c^4))*b^2*c^2*a^16-(5*b^16+5*c^16-2*(28*b^12+28*c^12-(19*b^8+19*c^8+b^2*c^2*(40*b^4-27*b^2*c^2+40*c^4))*b^2*c^2)*b^2*c^2)*a^14+2*(b^2+c^2)*(5*b^16+5*c^16+(5*b^12+5*c^12-(18*b^8+18*c^8-b^2*c^2*(81*b^4-130*b^2*c^2+81*c^4))*b^2*c^2)*b^2*c^2)*a^12-(b^16+c^16+2*(15*b^12+15*c^12-(64*b^8+64*c^8-b^2*c^2*(113*b^4-137*b^2*c^2+113*c^4))*b^2*c^2)*b^2*c^2)*(b^2+c^2)^2*a^10-2*(b^4-c^4)*(b^2-c^2)*(4*b^16+4*c^16+(7*b^12+7*c^12+(25*b^8+25*c^8+b^2*c^2*(51*b^4+10*b^2*c^2+51*c^4))*b^2*c^2)*b^2*c^2)*a^8+(b^4-c^4)^2*(3*b^16+3*c^16-(35*b^8+35*c^8-8*b^2*c^2*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2))*b^4*c^4)*a^6+(b^4-c^4)^2*(b^2+c^2)*(2*b^16+2*c^16+(b^12+c^12+(8*b^8+8*c^8-5*b^2*c^2*(9*b^4-4*b^2*c^2+9*c^4))*b^2*c^2)*b^2*c^2)*a^4-(b^4-c^4)^2*(b^2+c^2)^2*(b^16+c^16-2*(3*b^12+3*c^12-(7*b^8+7*c^8-b^2*c^2*(11*b^4-21*b^2*c^2+11*c^4))*b^2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^4*(b^2+c^2)^5*(b^8+c^8-3*b^2*c^2*(b^4+c^4))*b^2*c^2) : :

The reciprocal cyclologic center of these triangles is X(23316).

X(23317) lies on the line {3589,23316}

X(23318) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO CIRCUMMEDIAL

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

The reciprocal cyclologic center of these triangles is X(9076).

X(23318) lies on the line {3818,10272}

X(23319) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO KOSNITA

Barycentrics    2*(7*R^2-2*SW)*S^4-(48*R^6+R^4*(5*SA-53*SW)-R^2*(5*SA^2-3*SA*SW-19*SW^2)+2*SW*(SA^2-SA*SW-SW^2))*S^2+(2*R^2-SW)*(5*R^2-2*SW)*SB*SC*SW : :
X(23319) = 5*X(1656)-X(15960)

The reciprocal cyclologic center of these triangles is X(23320).

X(23319) lies on these lines: {2,14652}, {5,11701}, {128,11585}, {137,427}, {140,23320}, {858,930}, {1141,1594}, {1147,10224}, {1368,13372}, {1656,15960}, {13504,23293}, {13505,23294}

X(23319) = complement of X(15959)
X(23319) = X(110)-of-anti-Ursa minor triangle

X(23320) = CYCLOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO ANTI-URSA-MINOR

Barycentrics    (7*S^4+(2*R^2*(11*R^2-13*SA-8*SW)+7*SA^2-8*SB*SC+3*SW^2)*S^2+(72*R^6+6*R^4*(2*SA-17*SW)-2*R^2*SW*(-25*SW+6*SA)+(3*SA-8*SW)*SW^2)*SA)*(SB+SC) : :
X(23320) = 3*X(3)+X(15960) = 3*X(15959)-X(15960)

The reciprocal cyclologic center of these triangles is X(23319).

X(23320) lies on these lines: {3,128}, {24,137}, {140,23319}, {186,1141}, {549,23333}, {930,7488}, {1263,7575}, {7399,15366}, {11449,13504}, {11464,13505}, {12359,12893}, {14652,22467}

X(23320) = midpoint of X(3) and X(15959)
X(23320) = X(110)-of-Kosnita triangle

X(23321) = CYCLOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO 2nd ORTHOSYMMEDIAL

Barycentrics
(b^2+c^2)*a^28-2*(b^2+c^2)^2*a^26+(b^2+c^2)*(b^4+5*b^2*c^2+c^4)*a^24-b^2*c^2*(5*b^4+2*b^2*c^2+5*c^4)*a^22-(b^2+c^2)*(3*b^8+3*c^8-b^2*c^2*(7*b^4-13*b^2*c^2+7*c^4))*a^20+(6*b^8+6*c^8-b^2*c^2*(11*b^4-18*b^2*c^2+11*c^4))*(b^2+c^2)^2*a^18-(b^2+c^2)*(3*b^12+3*c^12+(6*b^8+6*c^8-b^2*c^2*(5*b^4-7*b^2*c^2+5*c^4))*b^2*c^2)*a^16+(10*b^12+10*c^12+b^2*c^2*(4*b^4+b^2*c^2-c^4)*(b^4-b^2*c^2-4*c^4))*b^2*c^2*a^14+(b^2+c^2)*(3*b^16+3*c^16-2*(7*b^12+7*c^12-(10*b^8+10*c^8-b^2*c^2*(17*b^4-31*b^2*c^2+17*c^4))*b^2*c^2)*b^2*c^2)*a^12-2*(3*b^20+3*c^20-(5*b^16+5*c^16-(3*b^12+3*c^12-(5*b^8+5*c^8+2*b^2*c^2*(3*b^4-14*b^2*c^2+3*c^4))*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10+(b^4-c^4)*(b^2-c^2)*(3*b^16+3*c^16+(3*b^12+3*c^12+2*(2*b^8+2*c^8-b^2*c^2*(b^4+14*b^2*c^2+c^4))*b^2*c^2)*b^2*c^2)*a^8-(b^4-c^4)^2*(5*b^12+5*c^12+2*(b^4+b^2*c^2+c^4)*(b^4-5*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2*a^6-(b^8-c^8)*a^4*(b^2-c^2)^3*(b^12+c^12-(3*b^8+3*c^8+4*b^2*c^2*(3*b^4+5*b^2*c^2+3*c^4))*b^2*c^2)+(b^8-c^8)^2*a^2*(b^2-c^2)^2*(2*b^8+2*c^8-b^2*c^2*(3*b^4+2*b^2*c^2+3*c^4))-(b^2-c^2)^6*(b^2+c^2)^3*(b^4+c^4)^3 : :

The reciprocal cyclologic center of these triangles is X(23322).

X(23321) lies on these lines: {}

X(23322) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 2nd ORTHOSYMMEDIAL TO ANTI-URSA-MINOR

Barycentrics
a^16+(b^4+c^4)*a^12-(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^10+b^2*c^2*(2*b^4-3*b^2*c^2+2*c^4)*a^8+2*(b^6+c^6)*b^2*c^2*a^6-(b^12+c^12-b^2*c^2*(b^4+4*b^2*c^2+c^4)*(b^4-b^2*c^2+c^4))*a^4+(b^12-c^12)*a^2*(b^2-c^2)-(b^4-c^4)^2*(b^4+c^4)*(b^4-b^2*c^2+c^4) : :

The reciprocal cyclologic center of these triangles is X(23321).

X(23322) lies on these lines: {}

X(23323) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO EHRMANN-MID

Barycentrics    (5*R^2-SW)*S^2+(21*R^2-5*SW)*SB*SC : :
X(23323) = (5*R^2-SW)*X(3)+(13*R^2-3*SW)*X(4) = X(13446)-3*X(13570)

The reciprocal eulerologic center of triangles Ehrmann-mid to anti-Ursa-minor does not exist
As a point on the Euler line, X(23323) has Shinagawa coefficients (E-4*F, E-20*F).

X(23323) lies on these lines: {2,3}, {113,13851}, {1514,9826}, {1539,15311}, {3818,23048}, {4993,19651}, {5448,12370}, {5893,13491}, {6000,14708}, {7687,12236}, {10540,12228}, {11441,15317}, {12028,18576}, {12134,18379}, {12289,18504}, {13364,16227}, {13446,13570}, {15241,21268}, {18376,18418}, {22800,22804}

X(23323) = midpoint of X(3) and X(32534)
X(23323) = midpoint of X(113) and X(13851)
X(23323) = inverse of X(20) in the 1st Droz-Farny circle
X(23323) = inverse of X(15761) in the nine-point circle
X(23323) = X(403)-of-Ehrmann-mid triangle
X(23323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3091, 7506), (4, 12084, 3627), (25, 18568, 3627), (235, 18377, 11819), (381, 3843, 7394), (381, 5133, 3850), (381, 18386, 11818), (381, 18403, 403), (1312, 1313, 15761), (2071, 6644, 15646), (3091, 7526, 5), (3850, 3861, 13163), (5159, 12084, 15122), (10019, 12605, 13406), (11818, 18386, 3845), (16868, 18563, 10020)

X(23324) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO EHRMANN-VERTEX

Barycentrics    (8*R^2-SA-SW)*S^2+8*(3*R^2-SW)*SB*SC : :
X(23324) = 5*X(4)+X(64) = 7*X(4)-X(5895) = 2*X(4)+X(6247) = 11*X(4)+X(12250) = 8*X(4)+X(15105) = X(64)-5*X(1853) = 7*X(64)+5*X(5895) = 2*X(64)-5*X(6247) = 11*X(64)-5*X(12250) = 8*X(64)-5*X(15105) = 7*X(1853)+X(5895) = 11*X(1853)-X(12250) = 8*X(1853)-X(15105) = 2*X(5895)+7*X(6247) = 11*X(5895)+7*X(12250) = 8*X(5895)+7*X(15105) = 11*X(6247)-2*X(12250) = 4*X(6247)-X(15105)

The reciprocal eulerologic center of these triangles is X(23325).

X(23324) lies on these lines: {2,18405}, {4,64}, {5,5944}, {6,18918}, {30,11204}, {53,6529}, {66,3531}, {141,18382}, {154,3545}, {235,11572}, {265,3629}, {343,3153}, {381,597}, {382,6696}, {427,13851}, {524,23049}, {542,23326}, {546,2883}, {547,11202}, {550,10193}, {858,18392}, {1173,6145}, {1498,3832}, {1594,18394}, {1596,7687}, {1899,15153}, {2072,18430}, {2777,15687}, {3090,17845}, {3091,11206}, {3357,3853}, {3431,7577}, {3543,10606}, {3564,23048}, {3627,5894}, {3818,15583}, {3843,5893}, {3845,5946}, {3850,6759}, {3851,9833}, {3861,22802}, {3919,6001}, {5056,17821}, {5076,20427}, {5480,18390}, {5925,17578}, {6146,7547}, {6293,9781}, {7507,12241}, {7576,14644}, {7729,11455}, {8538,10297}, {8547,18537}, {8550,10169}, {10113,23315}, {10151,11550}, {10182,15699}, {11180,17813}, {11245,12233}, {11402,12024}, {11744,13603}, {11801,19506}, {12083,15578}, {12099,17853}, {12359,18377}, {13371,18379}, {15033,15139}, {15053,15138}, {15752,17822}, {15760,20300}, {22165,23039}

X(23324) = midpoint of X(i) and X(j) for these {i,j}: {2, 18405}, {4, 1853}, {3543, 10606}, {7729, 11455}, {11180, 17813}
X(23324) = reflection of X(i) in X(j) for these (i,j): (550, 10193), (8550, 10169), (10192,5)
X(23324) = X(10192)-of-Johnson triangle
X(23324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (546, 18381, 2883), (3627, 20299, 5894), (3843, 14216, 5893)

X(23325) = EULEROLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO ANTI-URSA-MINOR

Barycentrics    (12*R^2-SA-2*SW)*S^2+(12*R^2-5*SW)*SB*SC : :
X(23325) = X(3)+2*X(18383) = 2*X(4)+X(3357) = X(4)+2*X(20299) = 5*X(4)+X(20427) = 4*X(5)-X(6759) = 5*X(5)-2*X(16252) = 2*X(5)+X(18381) = 2*X(125)+X(19506) = X(182)-4*X(20300) = X(3357)-4*X(20299) = 5*X(3357)-2*X(20427) = X(3818)+2*X(23300) = 5*X(6759)-8*X(16252) = X(6759)+2*X(18381) = 4*X(10182)-3*X(11202) = 4*X(16252)+5*X(18381) = 10*X(20299)-X(20427)

The reciprocal eulerologic center of these triangles is X(23324).

X(23325) lies on these lines: {2,10182}, {3,18383}, {4,74}, {5,182}, {24,11572}, {30,11204}, {64,3843}, {66,19130}, {68,5965}, {154,5055}, {184,7577}, {185,7547}, {235,16654}, {265,13352}, {376,10193}, {378,13851}, {381,1853}, {389,7507}, {403,11550}, {427,16657}, {511,14852}, {524,23048}, {541,23043}, {542,5654}, {546,6247}, {547,10192}, {567,7579}, {568,10628}, {569,6145}, {576,12585}, {578,1594}, {1147,10224}, {1352,11511}, {1498,3851}, {1568,11442}, {1656,10282}, {1899,18388}, {2071,18392}, {2072,9306}, {2393,10170}, {2818,15050}, {2883,3850}, {3090,9833}, {3091,5643}, {3098,6697}, {3153,23293}, {3518,11704}, {3520,18394}, {3526,17845}, {3541,13403}, {3542,13419}, {3564,23326}, {3574,18912}, {3627,6696}, {3830,10606}, {3832,5878}, {3845,15311}, {3853,5894}, {3855,12324}, {3858,5893}, {5068,14862}, {5070,17821}, {5071,11206}, {5073,8567}, {5076,5925}, {5094,11430}, {5449,18569}, {5462,7564}, {5640,7565}, {6143,12289}, {6639,11750}, {6971,14925}, {7527,7703}, {7545,10117}, {7689,13561}, {7699,15032}, {8549,18553}, {8889,18918}, {9927,13346}, {9967,11793}, {10110,19161}, {10113,13293}, {10249,11645}, {10255,10539}, {10296,11454}, {10605,18386}, {11232,12161}, {11250,12893}, {11470,14853}, {11562,18439}, {11801,23315}, {12024,15153}, {12106,13289}, {14002,15025}, {14789,22112}, {15083,18356}, {15088,15647}, {15113,17702}, {15583,18358}, {15873,16198}, {18531,21243}

X(23325) = midpoint of X(i) and X(j) for these {i,j}: {3, 18405}, {265, 15131}, {381, 1853}, {3830, 10606}
X(23325) = reflection of X(376) in X(10193)
X(23325) = anticomplement of X(10182)
X(23325) = X(13289)-of-orthocentroidal triangle
X(23325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 125, 11438), (4, 20299, 3357), (4, 23294, 1204), (5, 18381, 6759), (546, 6247, 22802), (2072, 18474, 9306), (5094, 18396, 11430), (6697, 18382, 3098), (9927, 13371, 13346), (13561, 18377, 7689), (18376, 23332, 11204)

X(23326) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO 2nd EHRMANN

Barycentrics    4*a^8-7*(b^2+c^2)*a^6-(5*b^4-14*b^2*c^2+5*c^4)*a^4+7*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-c^4)^2 : :
X(23326) = 7*X(6)-X(5596) = 2*X(6)+X(15583) = X(159)-4*X(6329) = 2*X(576)+X(6247) = X(2883)+2*X(8549) = 5*X(3618)-2*X(15585) = X(3629)+2*X(23300) = X(3630)-4*X(6697) = 2*X(5596)+7*X(15583) = 2*X(6696)+X(11477) = 4*X(10169)-X(10192) = 2*X(11216)+X(23332) = 2*X(11255)+X(12359) = 5*X(11482)+X(14216)

The reciprocal eulerologic center of these triangles is X(23327).

X(23326) lies on these lines: {2,17813}, {4,6}, {30,10250}, {141,5159}, {159,6329}, {343,11416}, {427,21639}, {511,23328}, {524,11216}, {542,23324}, {576,6247}, {597,2393}, {858,11443}, {895,15131}, {1353,11232}, {1594,11458}, {1843,11746}, {1853,1992}, {1899,11405}, {2777,21850}, {2781,14831}, {3066,9924}, {3564,23325}, {3566,9171}, {3589,5544}, {3618,15585}, {3629,23300}, {3630,6697}, {6696,11477}, {8541,13567}, {10182,11695}, {11255,12359}, {11482,14216}, {11574,21167}, {15311,20423}

X(23326) = midpoint of X(i) and X(j) for these {i,j}: {2, 17813}, {895, 15131}, {1853, 1992}
X(23326) = reflection of X(597) in X(10169)

X(23327) = EULEROLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO ANTI-URSA-MINOR

Barycentrics    a^8-2*(b^2+c^2)*a^6-2*(b^2-c^2)^2*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-c^4)^2 : :
X(23327) = 2*X(5)+X(8549) = 2*X(6)+X(66) = X(6)+2*X(23300) = X(66)+4*X(10169) = X(66)-4*X(23300) = X(69)-4*X(6697) = 2*X(125)+X(13248) = X(159)-4*X(3589) = X(159)+2*X(15583) = 2*X(206)-5*X(3618) = X(1177)-4*X(15118) = X(1853)+2*X(10169) = 2*X(3589)+X(15583) = 5*X(3618)-X(11206) = X(11216)+2*X(23332)

The reciprocal eulerologic center of these triangles is X(23326).

X(23327) lies on these lines: {2,2393}, {4,1177}, {5,8549}, {6,66}, {30,10249}, {69,6697}, {125,8541}, {159,3589}, {182,18400}, {195,15141}, {206,3618}, {381,597}, {511,23048}, {524,11216}, {542,5654}, {568,2781}, {575,18381}, {576,18951}, {599,17813}, {895,15116}, {1352,2072}, {1597,5480}, {1660,7392}, {1992,9140}, {3098,10193}, {3753,3827}, {3767,7668}, {5094,5486}, {5159,8263}, {5476,6000}, {5890,14853}, {6247,11432}, {6776,7699}, {8537,23294}, {8542,16051}, {8548,13371}, {8889,18919}, {9833,13353}, {9968,12324}, {9971,12099}, {10127,23041}, {10168,11202}, {10541,17845}, {11204,19924}, {11255,13561}, {11442,22151}, {11511,21243}, {11645,18376}, {14003,20975}, {14984,18281}, {15578,18859}, {15579,20427}, {18382,19127}, {18583,19149}

X(23327) = midpoint of X(i) and X(j) for these {i,j}: {6, 1853}, {599, 17813}
X(23327) = reflection of X(i) in X(j) for these (i,j): (6, 10169), (159, 10192), (3098, 10193)
X(23327) = X(1177)-of-orthocentroidal triangle
X(23327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 23300, 66), (3589, 15583, 159), (11416, 23293, 69)

X(23328) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR TO TRINH

Barycentrics    4*a^10-7*(b^2+c^2)*a^8-4*(-4*b^2*c^2+(b^2-c^2)^2)*a^6+14*(b^4-c^4)*(b^2-c^2)*a^4-8*(b^4-c^4)^2*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(23328) = 2*X(3)+X(6247) = X(3)+2*X(6696) = 7*X(3)-X(9833) = 5*X(3)+X(14216) = X(4)+5*X(8567) = 2*X(5)+X(5894) = X(6247)-4*X(6696) = 7*X(6247)+2*X(9833) = 5*X(6247)-2*X(14216) = 14*X(6696)+X(9833) = 10*X(6696)-X(14216) = 2*X(6699)+X(11598) = 5*X(9833)+7*X(14216) = 5*X(11204)+X(18376) = 4*X(11204)+X(23324) = 3*X(11204)+X(23325) = 2*X(11204)+X(23332) = 2*X(12041)+X(23315) = 4*X(18376)-5*X(23324) = 3*X(18376)-5*X(23325) = X(18376)-5*X(23329) = 2*X(18376)-5*X(23332) = 3*X(23324)-4*X(23325) = X(23324)-4*X(23329) = X(23325)-3*X(23329) = 2*X(23325)-3*X(23332)

The reciprocal eulerologic center of these triangles is X(23329).

X(23328) lies on these lines: {2,10606}, {3,66}, {4,8567}, {5,1539}, {6,18931}, {20,18405}, {24,16654}, {30,11204}, {64,631}, {74,15131}, {140,2883}, {154,3524}, {185,15151}, {186,16658}, {343,2071}, {371,13980}, {372,8991}, {376,1853}, {378,13567}, {427,21663}, {511,23326}, {524,10249}, {548,18381}, {549,6000}, {550,20299}, {597,2781}, {858,11454}, {993,20307}, {1192,3088}, {1204,12233}, {1498,3523}, {1593,15873}, {1594,11468}, {1620,7487}, {1656,5893}, {1899,11410}, {2778,5883}, {2935,7527}, {3090,5895}, {3091,5925}, {3098,15583}, {3515,16621}, {3516,12241}, {3517,16656}, {3520,12022}, {3525,12250}, {3526,5878}, {3528,17845}, {3530,6759}, {3541,13568}, {3628,22802}, {3629,13352}, {3630,10564}, {5298,11189}, {5480,11438}, {6001,10164}, {6225,10303}, {6684,12262}, {7495,15138}, {7729,11459}, {8550,11430}, {8703,18400}, {9306,16976}, {9540,19087}, {9786,14853}, {10226,12901}, {10282,15712}, {11202,12100}, {11206,15692}, {11250,12359}, {11425,14912}, {12324,15717}, {13093,15720}, {13347,23042}, {13394,15072}, {13935,19088}, {14389,17835}, {14644,18560}, {15032,17847}, {15704,18383}, {16386,23293}, {16659,17506}, {16775,18281}

X(23328) = midpoint of X(i) and X(j) for these {i,j}: {2, 10606}, {20, 18405}, {64, 5656}, {74, 15131}, {376, 1853}, {7729, 11459}
X(23328) = reflection of X(i) in X(j) for these (i,j): (549, 10193), (11202, 12100)
X(23328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6696, 6247), (64, 631, 16252), (140, 3357, 2883), (1192, 3088, 11745), (1656, 20427, 5893), (2883, 3357, 15105), (12324, 15717, 17821)

X(23329) = EULEROLOGIC CENTER OF THESE TRIANGLES: TRINH TO ANTI-URSA-MINOR

Barycentrics    a^10-2*(b^2+c^2)*a^8-(b^4-8*b^2*c^2+c^4)*a^6+5*(b^4-c^4)*(b^2-c^2)*a^4-2*(b^2-c^2)^2*(2*b^4+3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(23329) = 5*X(2)-X(5656) = 7*X(3)-X(17845) = 2*X(3)+X(18381) = X(3)+2*X(20299) = 2*X(5)+X(3357) = 5*X(5)-2*X(5893) = X(5)+2*X(6696) = 4*X(5)-X(22802) = X(1853)+2*X(10193) = 7*X(1853)+X(17845) = 5*X(3357)+4*X(5893) = X(3357)-4*X(6696) = 2*X(3357)+X(22802) = X(5893)+5*X(6696) = 8*X(5893)-5*X(22802) = 8*X(6696)+X(22802) = 14*X(10193)-X(17845) = 4*X(10193)+X(18381) = 2*X(17845)+7*X(18381) = X(17845)+14*X(20299) = X(18381)-4*X(20299)

The reciprocal eulerologic center of these triangles is X(23328).

X(23329) lies on these lines: {2,5656}, {3,161}, {4,11270}, {5,3357}, {6,19348}, {20,18383}, {26,20191}, {30,11204}, {54,17711}, {64,1656}, {66,5092}, {74,7577}, {125,378}, {140,6247}, {154,5054}, {156,5498}, {186,11550}, {381,2777}, {382,8567}, {389,3541}, {427,11438}, {511,23048}, {524,10250}, {541,15113}, {542,10249}, {546,5894}, {549,1503}, {576,10169}, {578,11245}, {631,10282}, {632,16252}, {1204,1594}, {1498,3526}, {1899,11430}, {2071,23293}, {2781,5476}, {2883,3628}, {3088,10110}, {3090,5878}, {3091,20427}, {3098,23300}, {3153,11454}, {3515,13419}, {3516,13403}, {3520,23294}, {3523,9833}, {3525,12324}, {3534,18405}, {3542,13474}, {3546,11793}, {3548,5907}, {3582,11189}, {3818,6644}, {3843,5925}, {3851,5895}, {5056,12250}, {5067,6225}, {5070,13093}, {5094,10605}, {5169,15053}, {5449,12084}, {5890,10628}, {5972,18451}, {6001,11231}, {6143,6241}, {6639,10575}, {6640,12162}, {6723,11472}, {7404,11695}, {7502,15578}, {7505,11381}, {7506,18488}, {7583,13980}, {7584,8991}, {7689,13371}, {7706,23315}, {7729,18435}, {8889,18931}, {8976,19087}, {9306,10257}, {9919,21308}, {9927,11250}, {9956,12262}, {10168,19153}, {10181,10199}, {10201,14915}, {10274,20376}, {11243,16241}, {11244,16242}, {11410,18396}, {11456,13399}, {11457,13367}, {11598,20304}, {12041,19506}, {12290,14940}, {12315,14862}, {12359,13346}, {13382,18913}, {13754,18281}, {13951,19088}, {13997,16177}, {14156,15068}, {15087,17847}, {15125,18431}, {15131,20126}, {15720,17821}, {16003,18445}, {18475,18580}, {19924,23049}

X(23329) = midpoint of X(i) and X(j) for these {i,j}: {3, 1853}, {381, 10606}, {3534, 18405}, {7729, 18435}, {15131, 20126}
X(23329) = reflection of X(i) in X(j) for these (i,j): (3, 10193), (154, 10182), (576, 10169)
X(23329) = X(6759)-of-orthocentroidal triangle
X(23329) = X(10193)-of-X3-ABC reflections triangle
X(23329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 20299, 18381), (5, 3357, 22802), (5, 6696, 3357), (125, 378, 18390), (140, 6247, 6759), (154, 5054, 10182), (631, 14216, 10282), (5094, 10605, 18388), (11204, 23332, 18376), (11250, 13561, 9927), (18388, 20417, 10605)

X(23330) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: ANTI-URSA-MINOR AND AAOA

Barycentrics    (R^2*(18*R^2-3*SA-13*SW)+SA*SW+2*SW^2)*S^2-(2*R^2-SW)*SB*SC*SW : :

X(23330) lies on these lines: {2,15139}, {3,18432}, {5,12824}, {52,1594}, {54,9140}, {69,10510}, {70,13353}, {125,575}, {850,1235}, {858,3917}, {1993,5094}, {5640,11704}, {6240,18488}, {6403,7703}, {7495,18553}, {7579,13321}, {12038,12827}, {13160,15030}, {13371,21230}

X(23330) = midpoint of X(3) and X(18432)

X(23331) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: ANTI-URSA-MINOR AND 1st HYACINTH

Barycentrics    2*(6*R^2-SW)*S^4+(196*R^6-4*R^4*(9*SA+32*SW)+3*R^2*(4*SA^2-4*SB*SC+SW*(9*SW+2*SA))-2*SW*(SA^2-SB*SC+SW^2))*S^2-(4*R^2*(7*R^2-SW)-SW^2)*R^2*SB*SC : :

X(23331) lies on these lines: {265,1885}, {3574,21649}, {12359,13403}, {15462,16238}

X(23332) = X(2) OF ANTI-URSA-MINOR TRIANGLE

Barycentrics    (b^2+c^2)*a^4+2*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(23332) = 5*X(2)-X(11206) = 2*X(4)+X(5894) = 5*X(4)+X(5925) = X(4)+2*X(6696) = 4*X(5)-X(2883) = 2*X(5)+X(6247) = X(5)+2*X(20299) = X(154)+3*X(1853) = 2*X(154)-3*X(10192) = 5*X(154)-3*X(11206) = 2*X(1853)+X(10192) = 5*X(1853)+X(11206) = X(2883)+2*X(6247) = X(2883)+8*X(20299) = 5*X(5894)-2*X(5925) = X(5894)-4*X(6696) = X(5925)-10*X(6696) = X(5925)-5*X(10606) = X(6247)-4*X(20299) = 5*X(10192)-2*X(11206)

X(23332) lies on these lines: {2,154}, {3,16254}, {4,1192}, {5,2883}, {6,8889}, {11,11189}, {22,15578}, {30,11204}, {51,125}, {64,3091}, {66,3589}, {69,17813}, {140,11202}, {141,1368}, {159,16419}, {161,7485}, {206,11548}, {221,10588}, {343,858}, {376,18405}, {381,15311}, {403,11455}, {459,10002}, {468,11550}, {524,11216}, {542,15113}, {546,3357}, {549,18400}, {550,18383}, {590,11241}, {615,11242}, {632,10282}, {1154,12359}, {1204,23047}, {1329,2390}, {1350,7396}, {1370,18382}, {1498,3090}, {1585,14230}, {1586,14233}, {1594,5890}, {1595,15873}, {1619,11284}, {1656,14216}, {1660,18358}, {1899,5094}, {1971,3054}, {1994,17847}, {1995,15579}, {2072,18435}, {2192,10589}, {2777,3845}, {2886,20305}, {2935,13596}, {3070,13980}, {3071,8991}, {3089,16656}, {3146,8567}, {3523,17845}, {3525,17821}, {3526,9833}, {3535,13749}, {3536,13748}, {3541,12241}, {3542,16621}, {3564,10250}, {3566,10189}, {3614,7355}, {3619,9924}, {3628,6759}, {3740,3827}, {3763,15585}, {3818,6677}, {3832,5895}, {3843,20427}, {3850,15105}, {3851,5878}, {3855,12250}, {3867,9971}, {3925,11190}, {5012,15139}, {5056,12324}, {5068,6225}, {5071,5656}, {5072,13093}, {5079,12315}, {5117,5254}, {5133,7703}, {5142,5799}, {5159,9306}, {5644,14561}, {5891,11585}, {6001,10157}, {6145,20376}, {6285,7173}, {6293,15043}, {6622,15811}, {6640,12134}, {6723,15647}, {7378,17810}, {7403,14845}, {7484,15577}, {7488,20391}, {7505,16655}, {7507,13568}, {7687,11598}, {7729,15305}, {7989,12779}, {8167,18621}, {8280,13910}, {8281,13972}, {8549,16051}, {8584,10169}, {8703,10193}, {9140,15131}, {10117,13595}, {10182,11539}, {10184,23333}, {10224,22660}, {10257,18474}, {10691,21167}, {11064,11442}, {11243,23302}, {11244,23303}, {11427,12007}, {11801,13293}, {12262,19925}, {13160,20791}, {14076,21357}, {14361,15274}, {14855,15760}, {14940,16659}, {15153,18396}, {16219,19506}, {16386,18392}, {17825,19149}, {19132,20079}, {19209,23295}, {20303,22530}, {21252,23304}

X(23332) = midpoint of X(i) and X(j) for these {i,j}: {2, 1853}, {4, 10606}, {66, 19153}, {69, 17813}, {376, 18405}, {7729, 15305}, {9140, 15131}, {16219, 19506}
X(23332) = reflection of X(i) in X(j) for these (i,j): (8584, 10169), (8703, 10193)
X(23332) = complementary conjugate of X(1249)
X(23332) = complement of X(154)
X(23332) = X(2)-of-anti-Ursa minor triangle
X(23332) = X(9909)-of-1st orthosymmedial triangle
X(23332) = X(15647)-of-orthocentroidal triangle
X(23332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6696, 5894), (5, 6247, 2883), (5, 20299, 6247), (64, 3091, 5893), (125, 427, 13567), (141, 23300, 15583), (427, 13567, 5480), (858, 23293, 343), (1368, 21243, 141), (1656, 14216, 16252), (6697, 23300, 141), (11204, 23325, 18376), (13371, 13561, 12359), (18376, 23329, 11204)

X(23333) = X(6) OF ANTI-URSA-MINOR TRIANGLE

Barycentrics    (b^2+c^2)*a^6-(b^2+c^2)^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :
X(23333) = 3*X(1853)+X(17849)

X(23333) lies on these lines: {2,157}, {3,1485}, {5,182}, {6,2450}, {30,18380}, {53,232}, {125,6751}, {141,2871}, {160,297}, {549,23320}, {590,6413}, {615,6414}, {858,20477}, {868,20775}, {1853,17849}, {1899,2165}, {2072,6795}, {5133,11174}, {6697,14767}, {7668,9722}, {10184,23332}, {13160,20792}, {19212,23295}

X(23333) = complementary conjugate of X(22391)
X(23333) = complement of X(157)
X(23333) = X(6)-of-anti-Ursa-minor-triangle
X(23333) = X(6)-of-A'B'C' as defined at X(11585)

X(23334) =  ANTICOMPLEMENT OF X(8182)

Barycentrics    13*a^4+2*(b^2+c^2)*a^2+14*b^2*c^2-11*c^4-11*b^4 : :
X(23334) = 9*X(2)-8*X(1153), 5*X(2)-4*X(5569), 3*X(2)-4*X(8176), 3*X(4)-X(5485), X(20)-4*X(7775), 3*X(381)-2*X(16509), 10*X(1153)-9*X(5569), 2*X(1153)-3*X(8176), 4*X(1153)-3*X(8182), 3*X(3543)+X(11148), 3*X(3839)-2*X(7615), 3*X(3839)-X(9740), 2*X(5485)-3*X(7620), 3*X(5569)-5*X(8176), 6*X(5569)-5*X(8182), 3*X(9770)-2*X(11165)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28279.

X(23334) lies on these lines: {2, 187}, {4, 524}, {20, 7618}, {30, 7710}, {69, 11317}, {147, 543}, {193, 671}, {325, 11164}, {376, 11184}, {381, 16509}, {754, 3839}, {1007, 8598}, {1384, 8355}, {1992, 8352}, {3091, 7617}, {3146, 7843}, {3523, 7619}, {3524, 9771}, {3534, 12040}, {3545, 7610}, {3589, 18842}, {3620, 10302}, {3854, 7780}, {5032, 5286}, {5071, 15597}, {5077, 7736}, {5140, 21969}, {5177, 7621}, {5395, 7911}, {5503, 10722}, {7622, 10304}, {7759, 17578}, {7774, 8597}, {7823, 8859}, {7879, 8370}, {8667, 20112}, {9741, 9766}, {11160, 11185}, {11318, 19661}, {13449, 20423}, {16508, 22566}

X(23334) = midpoint of X(i) and X(j) for these {i,j}: {5503, 10722}, {9741, 15682}
X(23334) = reflection of X(i) in X(j) for these (i,j): (20, 7618), (376, 11184), (3534, 12040), (8667, 20112), (9741, 9766), (16508, 22566)
X(23334) = anticomplement of X(8182)
X(23334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3839, 9740, 7615), (8176, 8182, 2)

X(23335) =  ANTICOMPLEMENT OF X(13383)

Barycentrics    (b^2+c^2)*a^8-2*(b^4+4*b^2*c^2+c^4)*a^6+4*b^2*c^2*(b^2+c^2)*a^4+2*(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(23335) = X(68)-3*X(1853), X(1498)-3*X(5654)

As a point on the Euler line, X(23335) has Shinagawa coefficients (-E+2*F, 3*E+2*F).

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28279.

X(23335) lies on these lines: {2, 3}, {66, 3564}, {68, 1853}, {110, 16659}, {141, 5447}, {143, 19161}, {155, 14216}, {343, 10625}, {496, 8144}, {499, 9645}, {511, 12235}, {541, 15105}, {542, 14864}, {1092, 11550}, {1147, 1503}, {1351, 18951}, {1498, 5654}, {1568, 11381}, {1899, 13292}, {1993, 11457}, {2883, 5448}, {3260, 6662}, {3313, 10627}, {3357, 20302}, {5157, 19154}, {5446, 13567}, {5462, 5480}, {5891, 18488}, {6000, 22660}, {6146, 13352}, {6247, 13754}, {6696, 7689}, {6759, 9820}, {8550, 18128}, {8883, 8901}, {8981, 11265}, {10539, 11064}, {11266, 13966}, {11695, 19130}, {12160, 18917}, {12161, 18914}, {12412, 13203}, {13346, 18381}, {13391, 13561}, {13421, 20379}, {14561, 15805}, {15644, 21243}, {16318, 22120}

X(23335) = midpoint of X(i) and X(j) for these {i,j}: {155, 14216}, {12412, 13203}, {13346, 18381}
X(23335) = reflection of X(i) in X(j) for these (i,j): (2883, 5448), (6759, 9820)
X(23335) = anticomplement of X(13383)
X(23335) = complement of X(7387)
X(23335) = orthocentroidal circle-inverse of X(7529)
X(23335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7387, 13383), (2, 15559, 7403), (3, 382, 18533), (3, 5576, 7399), (5, 3627, 1596), (382, 2072, 235), (382, 13473, 3627), (1368, 1595, 5), (1594, 15760, 5), (2041, 2042, 9714), (3147, 7500, 9714), (3523, 5169, 14788), (5073, 10255, 11799), (5576, 7399, 5), (7526, 14791, 12362), (9825, 16198, 11818)

X(23336) =  COMPLEMENT OF X(15761)

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

As a point on the Euler line, X(23336) has Shinagawa coefficients (E-12*F, -3*E+4*F)).)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28279.

X(23336) lies on these lines: {2, 3}, {125, 12370}, {156, 6247}, {389, 6699}, {511, 20191}, {1154, 20376}, {1511, 12134}, {5433, 8144}, {5663, 6696}, {5876, 11064}, {5907, 14156}, {6689, 16836}, {8254, 15739}, {11264, 20379}, {11425, 18952}, {12038, 20299}, {12359, 22663}, {12421, 15136}, {15120, 16270}, {18488, 20773}

X(23336) = midpoint of X(i) and X(j) for these {i,j}: {156, 6247}, {12038, 20299}
X(23336) = complement of X(15761)
X(23336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12084, 15761), (3, 18281, 13371), (5, 549, 17928), (5, 1885, 546), (5, 10257, 140), (140, 546, 16238), (140, 548, 6676), (140, 13383, 10125), (378, 6640, 5), (2071, 6143, 10024), (3523, 14790, 18324), (3526, 12085, 10201), (5094, 16977, 6677), (7568, 18569, 13383), (10125, 13383, 10020), (12086, 14940, 11799)

X(23337) =  X(140)X(6150)∩X(547)X(14143)

Barycentrics    9*S^4-(2*R^2*(11*R^2+10*SA-13*SW)-8*SA^2+19*SB*SC+7*SW^2)*S^2+(2*R^2*(R^2-5*SW)+5*SW^2)*SB*SC : :
X(23337) = 3*X(547)-2*X(14143)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28279.

X(23337) lies on these lines: {140, 6150}, {547, 14143}, {1539, 3853}, {5066, 20413}

X(23338) =  REFLECTION OF X(3628) IN X(15425)

Barycentrics    S^4-(R^2*(39*R^2+10*SA-34*SW)-4*SA^2+11*SB*SC+7*SW^2)*S^2-(R^2*(11*R^2-4*SW)-SW^2)*SB*SC : :
X(23338) = 3*X(547)-X(14143)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28279.

X(23338) lies on these lines: {30, 13856}, {547, 14143}, {3628, 10615}, {3850, 7687}, {7604, 19552}

X(23338) = reflection of X(3628) in X(15425)

X(23339) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(3), where A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23339) lies on these lines: {1, 18175}, {3, 3741}, {22, 16681}, {25, 225}, {31, 1469}, {48, 16721}, {75, 16678}, {159, 1626}, {1762, 3185}, {2933, 11350}, {5248, 12579}, {16683, 23382}, {18613, 23388}

X(23340) =  MIDPOINT OF X(4) AND X(3885)

Barycentrics    a*((b+c)*a^5-(b^2+6*b*c+c^2)*a^4-2*(b+c)*(b^2-5*b*c+c^2)*a^3+2*(b^4+c^4+2*(b^2-4*b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-8*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(23340) = 4*X(1)-3*X(10202), 5*X(1)-4*X(13373), 7*X(1)-5*X(15016), 3*X(65)-4*X(6583), 3*X(392)-2*X(5690), 2*X(942)-3*X(10247), 4*X(942)-3*X(10273), 3*X(944)-X(9961), 8*X(3628)-7*X(4002), 3*X(3655)-2*X(9943), 3*X(3656)-2*X(7686), 5*X(3698)-6*X(11230), 3*X(3753)-4*X(5901), 3*X(3877)-X(12245), 5*X(3890)-3*X(5657)

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28285.

X(23340) lies on these lines: {1, 3}, {4, 3885}, {5, 6735}, {8, 6893}, {72, 5844}, {119, 946}, {145, 912}, {355, 3880}, {392, 5690}, {496, 17622}, {519, 5887}, {944, 9961}, {952, 12672}, {962, 12115}, {971, 18526}, {1000, 5555}, {1210, 15558}, {1320, 12775}, {1519, 10942}, {1699, 11929}, {1872, 1877}, {2136, 5720}, {2800, 3244}, {2950, 12773}, {3555, 14988}, {3585, 12749}, {3625, 20117}, {3628, 4002}, {3633, 5693}, {3655, 9943}, {3656, 7686}, {3698, 11230}, {3753, 5901}, {3869, 6930}, {3877, 5084}, {3878, 5795}, {3881, 15528}, {3884, 11362}, {3890, 5657}, {3898, 6684}, {4301, 12608}, {5053, 21853}, {5252, 10525}, {5439, 10283}, {5552, 5603}, {5587, 11928}, {5734, 6970}, {5761, 6848}, {5777, 12625}, {5836, 5886}, {6256, 12699}, {6827, 9785}, {6882, 12053}, {6923, 12700}, {6958, 11373}, {6971, 7743}, {7330, 12629}, {7491, 10624}, {7680, 13463}, {7967, 13369}, {7970, 13189}, {7978, 13217}, {7983, 12189}, {7984, 12381}, {9856, 18525}, {10526, 12701}, {10595, 17567}, {10698, 13278}, {10705, 13118}, {10738, 12751}, {10866, 18527}, {10912, 11496}, {12650, 12686}, {12705, 18519}, {13099, 13313}

X(23340) = midpoint of X(i) and X(j) for these {i,j}: {4, 3885}, {3633, 5693}
X(23340) = reflection of X(i) in X(j) for these (i,j): (1071, 1483), (3625, 20117), (7491, 10624)
X(23340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 40, 10269), (1, 2077, 1385), (1, 3359, 16203), (1, 5903, 18838), (1, 11010, 14803), (1, 12703, 11248), (942, 9957, 20789), (946, 10915, 119), (1482, 10679, 1), (2098, 11509, 1), (2099, 10965, 1), (3746, 11014, 1385), (10596, 12245, 5554), (10942, 22791, 1519), (12702, 16203, 3359)

X(23341) =  X(35)X(60)∩X(758)X(3057)

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

See Antreas Hatzipolakis, César Lozada, Hyacinthos 28285.

X(23341) lies on these lines: {35, 60}, {758, 3057}, {946, 6003}, {5563, 6584}, {8674, 13604}, {11009, 23153}

leftri

K229 Moses images: X(23342)-X(23351)

rightri

Let P be a point of the circumcircle of a triangle ABC, and let Q be the trilinear pole of the line X(6)P. Let X be the Q-Ceva conjugate of P, so that also, X = crosspoint of P and Q. Then X lies on the cubic K229. The following ten examples of K229 Moses images were contributed by Peter Moses, September 18, 2018. See also the preamble just before X(23097).

X(23342) =  K229 MOSES IMAGE OF X(99)

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

X(23342) lies on the cubics K229 and K740 and on these lines: {2,6}, {99,670}, {110,9066}, {476,2858}, {523,2396}, {538,14609}, {892,9178}, {1975,2453}, {2930,5989}, {4226,14588}, {4590,5467}, {6082,9080}, {7754,14700}, {9023,9146}, {11052,21906}, {14995,22254}

X(23342) = reflection of X(21906) in X(11052)
X(23342) = isotomic conjugate of the isogonal conjugate of X(5118)
X(23342) = X(9150)-Ceva conjugate of X(99)
X(23342) = X(i)-cross conjugate of X(j) for these (i,j): {887, 3231}, {9148, 538}
X(23342) = X(i)-isoconjugate of X(j) for these (i,j): {661, 729}, {798, 3228}, {886, 4117}
X(23342) = X(i)-Hirst inverse of X(j) for these (i,j): {2, 14607}, {385, 5468}, {5108, 9182}
X(23342) = cevapoint of X(i) and X(j) for these (i,j): {538, 9148}, {887, 3231}
X(23342) = crosspoint of X(99) and X(9150)
X(23342) = trilinear pole of line {538, 3231}
X(23342) = crossdifference of every pair of points on line {512, 1084}
X(23342) = crosssum of X(512) and X(888)
X(23342) = barycentric product X(i)*X(j) for these {i,j}: {76, 5118}, {99, 538}, {670, 3231}, {799, 2234}, {4590, 9148}
X(23342) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 3228}, {110, 729}, {538, 523}, {887, 1084}, {888, 3124}, {1645, 23099}, {2234, 661}, {3231, 512}, {4590, 9150}, {5118, 6}, {5468, 14608}, {6786, 3569}, {9148, 115}, {14609, 9178}
X(23342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 385, 14898), (4590, 17941, 5467), (5468, 14607, 2421), (6189, 6190, 14607)

X(23343) =  K229 MOSES IMAGE OF X(100)

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

X(23343) lies on the cubics K229 and K635 and on these lines: {1,6}, {100,190}, {513,1026}, {523,2397}, {645,1634}, {660,876}, {765,3573}, {898,5381}, {899,19945}, {1282,16561}, {1308,2748}, {3196,8301}, {4069,4553}, {4363,4413}, {4370,8299}, {4422,21320}, {4567,5467}, {4585,6163}, {5380,9178}, {8053,17336}, {17350,20990}

X(23343) = isogonal conjugate of the anticomplement X(14434)
X(23343) = X(21)-beth conjugate of X(16501)
X(23343) = X(899)-zayin conjugate of X(513)
X(23343) = X(i)-Ceva conjugate of X(j) for these (i,j): {898, 100}, {5381, 6} X(23343) = X(i)-cross conjugate of X(j) for these (i,j): {890, 3230}, {891, 899}, {4526, 536} X(23343) = X(i)-isoconjugate of X(j) for these (i,j): {244, 898}, {514, 739}, {649, 3227}, {889, 3248}, {1015, 4607}, {5381, 21143} X(23343) = X(238)-Hirst inverse of X(1023)
X(23343) = X(1026)-line conjugate of X(513)
X(23343) = cevapoint of X(i) and X(j) for these (i,j): {890, 3230}, {891, 899}, {3994, 14430}
X(23343) = crosspoint of X(100) and X(898)
X(23343) = trilinear pole of line {899, 3230}
X(23343) = crossdifference of every pair of points on line {513, 1015}
X(23343) = crosssum of X(i) and X(j) for these (i,j): {513, 891}, {1646, 8027}
X(23343) = barycentric product X(i)*X(j) for these {i,j}: {100, 536}, {101, 6381}, {190, 899}, {651, 4009}, {660, 4465}, {662, 3994}, {668, 3230}, {765, 4728}, {891, 1016}, {898, 13466}, {3768, 7035}, {4526, 4998}, {4564, 14430}, {4567, 14431}, {4601, 14404}, {4604, 4937}, {4606, 4706}, {5378, 14433}, {5381, 14434}, {5383, 14426}, {6632, 19945}
X(23343) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 3227}, {536, 693}, {692, 739}, {765, 4607}, {890, 1015}, {891, 1086}, {899, 514}, {1016, 889}, {1252, 898}, {1646, 764}, {3230, 513}, {3768, 244}, {3994, 1577}, {4009, 4391}, {4465, 3766}, {4526, 11}, {4706, 4801}, {4728, 1111}, {4937, 4791}, {6381, 3261}, {14404, 3125}, {14426, 21138}, {14430, 4858}, {14431, 16732}, {14437, 1647}, {14440, 22107}, {14445, 22106}, {19945, 6545}

X(23344) =  K229 MOSES IMAGE OF X(101)

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

X(23344) lies on the cubic K229 and these lines: {3,23157}, {6,31}, {59,2283}, {63,16504}, {100,4585}, {101,692}, {109,6014}, {255,15625}, {523,2398}, {595,16493}, {765,3573}, {901,4638}, {919,6017}, {993,16506}, {1023,3251}, {1621,16494}, {3689,23202}, {3915,16492}, {4236,4840}, {4436,4579}, {4570,5467}, {5053,16694}, {5248,16500}, {6594,11712}, {8616,16495}, {9026,20780}

X(23344) = isogonal conjugate of X(6548)
X(23344) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 6548}, {3218, 513}, {21372, 649}
X(23344) = X(i)-Ceva conjugate of X(j) for these (i,j): {59, 17455}, {901, 101}, {1252, 1017}, {6551, 1252}, {9268, 6}
X(23344) = X(i)-cross conjugate of X(j) for these (i,j): {1017, 1252}, {1960, 902}
X(23344) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6548}, {2, 1022}, {81, 4049}, {88, 514}, {100, 6549}, {106, 693}, {244, 4555}, {513, 903}, {649, 20568}, {679, 900}, {901, 1111}, {905, 6336}, {1019, 4080}, {1086, 3257}, {1168, 4453}, {1320, 3676}, {1647, 4618}, {1797, 17924}, {2226, 3762}, {2403, 8056}, {3120, 4622}, {3122, 4634}, {3125, 4615}, {3261, 9456}, {3669, 4997}, {4591, 16732}, {4674, 7192}, {5376, 6545}, {8752, 15413}
X(23344) = cevapoint of X(i) and X(j) for these (i,j): {902, 1960}, {4895, 21805}
X(23344) = crosspoint of X(i) and X(j) for these (i,j): {101, 901}, {1252, 6551}
X(23344) = trilinear pole of line {902, 1017}
X(23344) = crossdifference of every pair of points on line {514, 1086}
X(23344) = crosssum of X(i) and X(j) for these (i,j): {2, 20042}, {513, 3960}, {514, 900}, {519, 4422}, {1086, 6550}, {1647, 6545}
X(23344) = barycentric product X(i)*X(j) for these {i,j}: {1, 1023}, {6, 17780}, {44, 100}, {58, 4169}, {59, 1639}, {101, 519}, {109, 2325}, {110, 3943}, {145, 2429}, {163, 3992}, {190, 902}, {644, 1319}, {651, 3689}, {662, 21805}, {668, 2251}, {678, 3257}, {692, 4358}, {765, 1635}, {813, 4432}, {825, 4439}, {900, 1252}, {901, 4370}, {1016, 1960}, {1017, 4555}, {1110, 3762}, {1262, 4528}, {1317, 5548}, {1331, 8756}, {1404, 3699}, {1415, 4723}, {1783, 5440}, {1877, 4587}, {1897, 22356}, {1978, 9459}, {2149, 4768}, {2415, 3052}, {3251, 5376}, {3285, 3952}, {3911, 3939}, {3977, 8750}, {4120, 4570}, {4557, 16704}, {4564, 4895}, {4567, 4730}, {4588, 4908}, {4600, 14407}, {4622, 21821}, {4627, 4819}, {4638, 8028}, {4700, 8694}, {4702, 8693}, {4727, 8652}, {4969, 8701}, {6079, 20972}, {6335, 23202}, {6544, 9268}, {7012, 14418}, {7045, 14427}, {15742, 22086}
X(23344) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6548}, {31, 1022}, {42, 4049}, {44, 693}, {100, 20568}, {101, 903}, {519, 3261}, {649, 6549}, {678, 3762}, {692, 88}, {902, 514}, {1017, 900}, {1023, 75}, {1110, 3257}, {1252, 4555}, {1404, 3676}, {1635, 1111}, {1647, 23100}, {1960, 1086}, {2251, 513}, {2429, 4373}, {3052, 2403}, {3285, 7192}, {3689, 4391}, {3939, 4997}, {3943, 850}, {3992, 20948}, {4120, 21207}, {4169, 313}, {4557, 4080}, {4567, 4634}, {4570, 4615}, {4730, 16732}, {4895, 4858}, {5440, 15413}, {6065, 4582}, {6066, 5548}, {8750, 6336}, {9459, 649}, {14407, 3120}, {14418, 17880}, {14436, 4475}, {17455, 4453}, {17780, 76}, {20972, 4927}, {21805, 1577}, {22086, 1565}, {22356, 4025}, {23202, 905}
X(23344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (692, 3939, 4557), (4570, 17943, 5467)

X(23345) =  K229 MOSES IMAGE OF X(106)

Barycentrics    a^2 (a+b-2 c) (b-c) (a-2 b+c) : :
X(23345) = X[1960] - 3 X[4491], 9 X[4057] - 10 X[8656]

X(23345) lies on the cubic K229, the conic {{A,B,C,X(12),X(6)}}, and on these lines: {1,513}, {6,649}, {56,4057}, {58,3733}, {86,4833}, {87,16495}, {88,659}, {106,1960}, {292,3572}, {514,996}, {522,10912}, {523,1222}, {660,876}, {667,2163}, {834,2334}, {874,889}, {891,4792}, {900,1120}, {901,4638}, {903,3226}, {1027,16507}, {1126,6371}, {1168,6550}, {1220,4581}, {1438,8658}, {1459,3445}, {1482,3667}, {2226,8661}, {2316,22108}, {2827,6095}, {3835,17313}, {3837,4997}, {4049,4778}, {4361,21211}, {4591,5467}, {4622,17929}, {4674,21385}, {4840,5331}, {9001,16504}, {14437,16672}, {17378,20295}

X(23345) = isogonal conjugate of X(17780)
X(23345) = X(i)-beth conjugate of X(j) for these (i,j): {3737, 1}, {5548, 2284}
X(23345) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 17780}, {649, 44}, {1018, 1635}, {1100, 1023}
X(23345) = X(i)-Ceva conjugate of X(j) for these (i,j): {901, 106}, {1168, 244}, {2226, 1015}, {4555, 88}, {4638, 6}
X(23345) = X(i)-cross conjugate of X(j) for these (i,j): {1015, 2226}, {1960, 649}, {3310, 3669}, {3937, 10428}, {6085, 513}, {8661, 1015}
X(23345) = cevapoint of X(i) and X(j) for these (i,j): {649, 1960}, {667, 21758}, {1015, 8661}
X(23345) = crosspoint of X(i) and X(j) for these (i,j): {88, 4555}, {106, 901}
X(23345) = trilinear pole of line {649, 1015}
X(23345) = crossdifference of every pair of points on line {44, 519}
X(23345) = crosssum of X(i) and X(j) for these (i,j): {44, 1960}, {519, 900}, {522, 3036}, {1635, 21805}, {1639, 3689}, {2325, 4528}, {3992, 4768}, {6544, 8028}
X(23345) = barycentric product X(i)*X(j) for these {i,j}: {1, 1022}, {6, 6548}, {58, 4049}, {88, 513}, {101, 6549}, {106, 514}, {244, 3257}, {649, 903}, {667, 20568}, {679, 1635}, {693, 9456}, {764, 5376}, {900, 2226}, {901, 1086}, {1015, 4555}, {1019, 4674}, {1168, 3960}, {1320, 3669}, {1357, 4582}, {1358, 5548}, {1417, 4391}, {1459, 6336}, {1647, 4638}, {1797, 7649}, {2087, 4618}, {2316, 3676}, {2401, 14260}, {2403, 3445}, {2441, 4373}, {3120, 4591}, {3121, 4634}, {3122, 4615}, {3125, 4622}, {3733, 4080}, {4025, 8752}, {6545, 9268}, {10015, 10428}
X(23345) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17780}, {31, 1023}, {42, 4169}, {88, 668}, {106, 190}, {244, 3762}, {512, 3943}, {513, 4358}, {514, 3264}, {649, 519}, {650, 4723}, {661, 3992}, {663, 2325}, {667, 44}, {798, 21805}, {901, 1016}, {903, 1978}, {1015, 900}, {1022, 75}, {1318, 4582}, {1320, 646}, {1417, 651}, {1459, 3977}, {1635, 4738}, {1797, 4561}, {1919, 902}, {1960, 4370}, {1977, 1960}, {1980, 2251}, {2170, 4768}, {2226, 4555}, {2316, 3699}, {2441, 145}, {3063, 3689}, {3121, 4730}, {3122, 4120}, {3248, 1635}, {3250, 4439}, {3257, 7035}, {3271, 1639}, {3310, 1145}, {3445, 2415}, {3733, 16704}, {3960, 1227}, {4049, 313}, {4378, 4506}, {4394, 4487}, {4591, 4600}, {4622, 4601}, {4674, 4033}, {4775, 4908}, {4790, 4742}, {4832, 4819}, {4834, 4727}, {4979, 4975}, {5548, 4076}, {6085, 16594}, {6548, 76}, {6549, 3261}, {8027, 2087}, {8632, 4432}, {8660, 20972}, {8752, 1897}, {9268, 6632}, {9456, 100}, {10428, 13136}, {14260, 2397}, {14936, 4528}, {16944, 4585}, {20568, 6386}, {20981, 4434}, {21143, 1647}, {21758, 214}, {21832, 4783}, {22096, 22086}, {22383, 5440}
X(23345) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17780}, {2, 1023}, {44, 190}, {59, 4768}, {81, 4169}, {99, 21805}, {100, 519}, {101, 4358}, {109, 4723}, {110, 3992}, {644, 3911}, {646, 1404}, {651, 2325}, {660, 4432}, {662, 3943}, {664, 3689}, {666, 14439}, {668, 902}, {678, 4555}, {692, 3264}, {765, 900}, {901, 4738}, {1016, 1635}, {1018, 16704}, {1252, 3762}, {1275, 14427}, {1293, 4487}, {1319, 3699}, {1332, 8756}, {1492, 4439}, {1639, 4564}, {1743, 2415}, {1783, 3977}, {1877, 4571}, {1897, 5440}, {1960, 7035}, {1978, 2251}, {2087, 6632}, {2429, 18743}, {3257, 4370}, {3285, 4033}, {3903, 4434}, {4120, 4567}, {4448, 5378}, {4528, 7045}, {4600, 4730}, {4601, 14407}, {4604, 4908}, {4606, 4700}, {4614, 4819}, {4615, 21821}, {4618, 8028}, {4742, 8694}, {4895, 4998}, {4975, 8701}, {5376, 6544}, {5379, 14429}, {5381, 14437}, {5382, 14425}, {5383, 14408}, {5388, 14436}, {6079, 17460}, {6335, 22356}, {6386, 9459}

X(23346) =  K229 MOSES IMAGE OF X(109)

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

X(23346) lies on the cubic K229 and these lines: {6,41}, {59,2283}, {109,692}, {523,2406}, {651,14315}, {1362,17455}, {1416,9456}, {1813,4557}, {5467,17942}

X(23346) = X(14733)-Ceva conjugate of X(109)
X(23346) = X(6139)-cross conjugate of X(1055)
X(23346) = cevapoint of X(1055) and X(6139)
X(23346) = crosspoint of X(109) and X(14733)
X(23346) = crossdifference of every pair of points on line {522, 1146}
X(23346) = crosssum of X(i) and X(j) for these (i,j): {522, 6366}, {527, 17044}
X(23346) = X(5011)-zayin conjugate of X(650)
X(23346) = X(i)-isoconjugate of X(j) for these (i,j): {522, 1156}, {650, 1121}, {693, 4845}, {2291, 4391}, {3261, 18889}
X(23346) = barycentric product X(i)*X(j) for these {i,j}: {59, 1638}, {100, 6610}, {101, 1323}, {108, 6510}, {109, 527}, {651, 1155}, {664, 1055}, {934, 6603}, {1262, 6366}, {1275, 6139}, {1308, 15730}, {1461, 6745}, {4564, 14413}, {7128, 14414}
X(23346) = barycentric quotient X(i)/X(j) for these {i,j}: {109, 1121}, {1055, 522}, {1155, 4391}, {1323, 3261}, {1415, 1156}, {6139, 1146}, {6603, 4397}, {6610, 693}, {14413, 4858}

X(23347) =  K229 MOSES IMAGE OF X(112)

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

X(23347) lies on the cubic K229 and these lines: {6,25}, {110,9064}, {112,1576}, {250,4230}, {523,2409}, {1304,5502}, {1990,9407}, {2407,3233}, {2967,6593}, {3542,5877}, {7473,16237}, {8749,9142}, {10098,10423}

X(23347) = isogonal conjugate of X(34767)
X(23347) = cevapoint of X(i) and X(j) for these {i,j}: {1495, 9409}, {3284, 14396}, {9407, 14398}
X(23347) = crosspoint of X(112) and X(1304)
X(23347) = crosssum of X(i) and X(j) for these {i,j}: {30, 23583}, {520, 8552}, {525, 9033}, {1650, 23616}
X(23347) = trilinear pole of line X(1495)X(9408)
X(23347) = crossdifference of every pair of points on line X(525)X(15526)
X(23347) = isogonal conjugate of the anticomplement X(14401)
X(23347) = X(i)-Ceva conjugate of X(j) for these (i,j): {1304, 112}, {4240, 2420}
X(23347) = X(i)-cross conjugate of X(j) for these (i,j): {9409, 1495}, {14398, 1990}
X(23347) = barycentric product X(i)*X(j) for these {i,j}: {4, 2420}, {6, 4240}, {25, 2407}, {30, 112}, {99, 14581}, {107, 3284}, {110, 1990}, {162, 2173}, {163, 1784}, {250, 1637}, {648, 1495}, {811, 9406}, {1304, 3163}, {3233, 8749}, {5379, 14399}, {5994, 6111}, {5995, 6110}, {6331, 9407}, {8750, 18653}, {9408, 16077}, {14254, 14591}, {14345, 15384}, {14398, 18020}, {14560, 14920}, {14583, 14590}
X(23347) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2394}, {30, 3267}, {32, 14380}, {112, 1494}, {1495, 525}, {1576, 14919}, {1637, 339}, {1650, 23107}, {1784, 20948}, {1974, 2433}, {1990, 850}, {2173, 14208}, {2207, 18808}, {2407, 305}, {2420, 69}, {2489, 12079}, {2631, 17879}, {3284, 3265}, {4240, 76}, {9406, 656}, {9407, 647}, {9408, 9033}, {9409, 15526}, {11589, 14638}, {14398, 125}, {14574, 18877}, {14581, 523}, {14583, 14592}
X(23347) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2394}, {74, 14208}, {75, 14380}, {304, 2433}, {326, 18808}, {525, 2349}, {656, 1494}, {1304, 17879}, {1577, 14919}, {2159, 3267}, {2632, 16077}, {4592, 12079}, {18877, 20948}

X(23348) =  K229 MOSES IMAGE OF X(691)

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

X(23348) lies on the cubics K147 and K229, and on these lines: {6,110}, {671,14995}, {691,5467}, {892,17941}, {2407,5466}, {2408,4226}, {2421,9186}, {5380,17944}, {8371,9182}, {9171,9181}

X(23348) = isogonal conjugate of X(34763)
X(23348) = crossdifference of every pair of points on line X(690)X(23992)
X(23348) = perspector of conic {{A,B,C,X(691),PU(62)}}
X(23348) = cevapoint of X(17964) and X(17993)
X(23348) = trilinear pole of line {2502, 9181}
X(23348) = crosssum of X(i) and X(j) for these {i,j}: {523, 9183}, {690, 33921}
X(23348) = X(17993)-cross conjugate of X(17964)
X(23348) = X(110)-Hirst inverse of X(111)
X(23348) = X(111)-daleth conjugate of X(20998)
X(23348) = X(i)-isoconjugate of X(j) for these (i,j): {896, 9180}, {2642, 18823}
X(23348) = barycentric product X(i)*X(j) for these {i,j}: {99, 17964}, {110, 17948}, {111, 9182}, {249, 18007}, {543, 691}, {662, 17955}, {671, 9181}, {892, 2502}, {4590, 17993}
X(23348) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 9180}, {691, 18823}, {2502, 690}, {9171, 1648}, {9181, 524}, {9182, 3266}, {17948, 850}, {17955, 1577}, {17964, 523}, {17993, 115}, {18007, 338}
X(23348) = {X(5467),X(9178)}-harmonic conjugate of X(691)

X(23349) =  K229 MOSES IMAGE OF X(739)

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

X(23349) lies on the cubic K229, and on these lines: {6,667}, {31,1919}, {81,3733}, {739,890}, {898,5381}, {2162,21007}, {3227,18825}, {4491,20332}, {5467,17939}

X(23349) = isogonal conjugate of the anticomplement X(1646)
X(23349) = X(898)-Ceva conjugate of X(739)
X(23349) = X(890)-cross conjugate of X(667)
X(23349) = cevapoint of X(667) and X(890)
X(23349) = crosspoint of X(739) and X(898)
X(23349) = trilinear pole of line {667, 1977} X(23349) = crossdifference of every pair of points on line {536, 6381}
X(23349) = crosssum of X(i) and X(j) for these (i,j): {536, 891}, {3994, 4728}, {8031, 14434}
X(23349) = X(4063)-zayin conjugate of X(899)
X(23349) = X(i)-isoconjugate of X(j) for these (i,j): {99, 3994}, {100, 6381}, {190, 536}, {664, 4009}, {668, 899}, {891, 7035}, {1016, 4728}, {1978, 3230}, {4465, 4562}, {4597, 4937}, {4600, 14431}, {4607, 13466}, {4998, 14430}
X(23349) = barycentric product X(i)*X(j) for these {i,j}: {513, 739}, {667, 3227}, {889, 1977}, {898, 1015}, {3248, 4607}, {5381, 8027}
X(23349) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 6381}, {667, 536}, {739, 668}, {798, 3994}, {890, 13466}, {1919, 899}, {1977, 891}, {1980, 3230}, {3063, 4009}, {3121, 14431}, {3227, 6386}, {3248, 4728}, {3249, 19945}, {21762, 14426}

X(23350) =  K229 MOSES IMAGE OF X(842)

Barycentrics    a^2 (b-c) (b+c) (a^2 b^2-b^4+a^2 c^2-c^4) (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) (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(23350) lies on the cubic K229, and on these lines: {6,526}, {250,4230}, {262,14223}, {297,18311}, {523,868}, {684,5968}, {842,7418}, {2799,14356}, {3447,21525}, {9139,9178}

X(23350) = isogonal conjugate of X(34761)
X(23350) = crossdifference of every pair of points on line X(542)X(23967)
X(23350) = X(i)-isoconjugate of X(j) for these (i,j): {293, 7473}, {1910, 14999}, {2247, 2966}
X(23350) = barycentric product X(i)*X(j) for these {i,j}: {325, 14998}, {511, 14223}, {842, 2799}, {868, 5649}, {3569, 5641}
X(23350) = barycentric quotient X(i)/X(j) for these {i,j}: {232, 7473}, {511, 14999}, {842, 2966}, {868, 18312}, {2491, 5191}, {3569, 542}, {8430, 16092}, {14223, 290}, {14998, 98}, {17994, 6103}

X(23351) =  K229 MOSES IMAGE OF X(2291)

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

X(23351) lies on the cubic K229, and on these lines: {6,663}, {9,3900}, {19,18344}, {55,657}, {57,513}, {284,21789}, {333,7253}, {673,885}, {926,2316}, {1751,23289}, {2195,18889}, {2291,6139}

X(23351) = X(14733)-Ceva conjugate of X(2291)
X(23351) = X(6139)-cross conjugate of X(663)
X(23351) = cevapoint of X(663) and X(6139)
X(23351) = crosspoint of X(2291) and X(14733)
X(23351) = trilinear pole of line {663, 14936}
X(23351) = crossdifference of every pair of points on line {527, 1323}
X(23351) = crosssum of X(i) and X(j) for these (i,j): {527, 6366}, {1155, 1638}
X(23351) = X(514)-zayin conjugate of X(1155)
X(23351) = X(i)-isoconjugate of X(j) for these (i,j): {100, 1323}, {190, 6610}, {527, 651}, {653, 6510}, {658, 6603}, {664, 1155}, {934, 6745}, {1055, 4554}, {1638, 4564}, {4998, 14413}, {6366, 7045}
X(23351) = barycentric product X(i)*X(j) for these {i,j}: {514, 4845}, {522, 2291}, {650, 1156}, {663, 1121}, {693, 18889}, {1146, 14733}, {3887, 15734}
X(23351) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 1323}, {657, 6745}, {663, 527}, {667, 6610}, {1121, 4572}, {1156, 4554}, {1946, 6510}, {2291, 664}, {3063, 1155}, {3271, 1638}, {4845, 190}, {8641, 6603}, {8645, 15730}, {14733, 1275}, {14936, 6366}, {18889, 100}

leftri

K635 Moses images: X(23352)-X(23355)

rightri

Let P be a point of the circumconic with perspector X(1); that is, the circumconic with barycentric equation a y z + b z x + c x y = 0. Points on this conic include X(i) for i = 88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052, 20332, as well as vertices of the Honsberger triangle (see X(7670) and the inner Conway triangle (see X(11677). Let Q be the trilinear pole of the line X(6)P, and let X be the Q-Ceva conjugate of P, so that also, X = crosspoint of P and Q. Then X lies on the cubic K635. The following four examples of K635 Moses images were contributed by Peter Moses, September 19, 2018. See also the preambles just before X(23097) and X(23342).

X(23352) =  K635 MOSES IMAGE OF X(88)

Barycentrics    a (a-2 b-2 c) (a+b-2 c) (b-c) (a-2 b+c) : :
\ X(23352) = X[1635] - 3 X[1769]

X(23352) lies on the cubic K635 and these lines: {1,513}, {45,4893}, {88,14315}, {106,14422}, {522,4049}, {900,903}, {1635,1769}, {2403,4977}, {3257,3573}, {3679,4777}, {4653,4833}, {4800,4945}

X(23352) = X(4893)-zayin conjugate of X(44)
X(23352) = X(i)-isoconjugate of X(j) for these (i,j): {44, 4604}, {89, 1023}, {519, 4588}, {902, 4597}, {1635, 5385}, {2163, 17780}, {3911, 5549}
X(23352) = crossdifference of every pair of points on line {44, 214}
X(23352) = crosssum of X(4777) and X(6702)
X(23352) = barycentric product X(i)*X(j) for these {i,j}: {45, 6548}, {88, 4777}, {106, 4791}, {513, 4945}, {514, 4792}, {901, 4957}, {903, 4893}, {1022, 3679}, {4049, 4653}, {4080, 4833}, {4752, 6549}, {4775, 20568}
X(23352) = barycentric quotient X(i)/X(j) for these {i,j}: {45, 17780}, {88, 4597}, {106, 4604}, {901, 5385}, {2177, 1023}, {4770, 3943}, {4775, 44}, {4777, 4358}, {4791, 3264}, {4792, 190}, {4814, 2325}, {4825, 4908}, {4833, 16704}, {4893, 519}, {4931, 3992}, {4944, 4723}, {4945, 668}, {6548, 20569}, {9456, 4588}

X(23353) =  K635 MOSES IMAGE OF X(162)

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

X(23353) lies on the cubic K635 and these lines: {1,19}, {107,110}, {108,109}, {653,692}, {885,14776}, {1895,14529}, {2283,7012}, {3573,5379}, {4579,6335}

X(23353) = X(851)-zayin conjugate of X(656)
X(23353) = X(9391)-cross conjugate of X(851)
X(23353) = X(i)-Hirst inverse of X(j) for these (i,j): {108, 109}
X(23353) = cevapoint of X(i) and X(j) for these (i,j): {851, 9391}
X(23353) = trilinear pole of line {851, 1430}
X(23353) = crossdifference of every pair of points on line {656, 3269}
X(23353) = crosssum of X(i) and X(j) for these (i,j): {656, 9391}
X(23353) = X(i)-isoconjugate of X(j) for these (i,j): {296, 522}, {521, 1937}, {525, 2249}, {652, 1952}, {1942, 8062}, {1945, 6332}, {1949, 4391}
X(23353) = barycentric product X(i)*X(j) for these {i,j}: {1, 1981}, {25, 15418}, {108, 1944}, {109, 1948}, {162, 8680}, {190, 1430}, {243, 651}, {648, 851}, {653, 1936}, {664, 2202}, {1020, 15146}, {1783, 5088}, {1951, 18026}
X(23353) = barycentric quotient X(i)/X(j) for these {i,j}: {108, 1952}, {243, 4391}, {450, 17899}, {851, 525}, {1415, 296}, {1430, 514}, {1936, 6332}, {1951, 521}, {1981, 75}, {2202, 522}, {5088, 15413}, {7360, 15416}, {8680, 14208}, {9391, 15526}, {15418, 305}

X(23354) =  K635 MOSES IMAGE OF X(190)

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

X(23354) lies on the cubic K635 and these lines: {1,2}, {100,932}, {646,3888}, {660,874}, {668,891}, {726,21140}, {876,4562}, {1016,3573}, {1018,4427}, {4033,4553}, {4124,4358}, {4600,17941}, {7243,17090}, {17142,18040}, {17233,21278}, {17475,20532}, {20533,20554}

X(23354) = anticomplement of the isogonal conjugate of X(5378)
X(23354) = X(668)-daleth conjugate of X(3952)
X(23354) = X(i)-zayin conjugate of X(j) for these (i,j): {1575, 649}, {18793, 659}, {20669, 20979}
X(23354) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {660, 149}, {765, 17794}, {813, 4440}, {1016, 20345}, {4562, 150}, {4583, 21293}, {5378, 8}, {7035, 20554}, {7077, 17036}
X(23354) = X(i)-Ceva conjugate of X(j) for these (i,j): {660, 3952}, {1016, 17475}, {8709, 190}
X(23354) = X(i)-cross conjugate of X(j) for these (i,j): {3837, 726}, {6373, 1575}
X(23354) = cevapoint of X(i) and X(j) for these (i,j): {726, 3837}, {812, 20530}, {1575, 6373}
X(23354) = crosspoint of X(i) and X(j) for these (i,j): {190, 8709}, {668, 4562}
X(23354) = trilinear pole of line {726, 1575}
X(23354) = crossdifference of every pair of points on line {649, 1977}
X(23354) = crosssum of X(i) and X(j) for these (i,j): {649, 6373}, {667, 8632}
X(23354) = X(i)-isoconjugate of X(j) for these (i,j): {513, 727}, {649, 20332}, {667, 3226}, {875, 3253}, {3248, 8709}, {3733, 18793}
X(23354) = barycentric product X(i)*X(j) for these {i,j}: {190, 726}, {646, 1463}, {668, 1575}, {670, 21830}, {765, 20908}, {1016, 3837}, {1978, 3009}, {4033, 18792}, {4562, 17793}, {4583, 17475}, {4600, 21053}, {4639, 20681}, {6386, 21760}, {6632, 21140}, {8709, 20532}
X(23354) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 20332}, {101, 727}, {190, 3226}, {644, 8851}, {726, 514}, {1016, 8709}, {1018, 18793}, {1463, 3669}, {1575, 513}, {3009, 649}, {3570, 3253}, {3837, 1086}, {6373, 1015}, {17475, 659}, {17793, 812}, {18792, 1019}, {20532, 3837}, {20663, 8632}, {20671, 6373}, {20681, 21832}, {20750, 22384}, {20759, 22092}, {20777, 22383}, {20785, 1459}, {20908, 1111}, {21053, 3120}, {21140, 6545}, {21760, 667}, {21830, 512}, {22092, 3937}
{X(668),X(3799)}-harmonic conjugate of X(3952)

X(23355) =  K635 MOSES IMAGE OF X(20332)

Barycentrics    a^2 (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(23355) lies on the cubic K635, the circumconic {{A,B,C,X(1),X(6)}}, and on these lines: {1,667}, {6,1919}, {58,16695}, {86,3253}, {87,513}, {106,727}, {292,8632}, {659,3226}, {898,8709}, {1019,16744}, {1120,8851}, {2279,8657}, {3573,5378}, {4491,20332}

X(23355) = X(3737)-beth conjugate of X(87)
X(23355) = X(649)-zayin conjugate of X(1575)
X(23355) = X(8709)-Ceva conjugate of X(20332)
X(23355) = X(i)-cross conjugate of X(j) for these (i,j): {659, 3733}, {6373, 649}
X(23355) = X(i)-isoconjugate of X(j) for these (i,j): {100, 726}, {190, 1575}, {660, 17793}, {668, 3009}, {765, 3837}, {799, 21830}, {1252, 20908}, {1463, 3699}, {1978, 21760}, {3952, 18792}, {4562, 17475}, {4567, 21053}, {4583, 20663}, {4589, 20681}, {6335, 20785}, {6373, 7035}
X(23355) = cevapoint of X(i) and X(j) for these (i,j): {649, 6373}, {667, 8632}
X(23355) = crosspoint of X(8709) and X(20332)
X(23355) = trilinear pole of line {649, 1977}
X(23355) = crossdifference of every pair of points on line {726, 1575}
X(23355) = crosssum of X(i) and X(j) for these (i,j): {726, 3837}, {812, 20530}, {1575, 6373}
X(23355) = barycentric product X(i)*X(j) for these {i,j}: {513, 20332}, {514, 727}, {649, 3226}, {1015, 8709}, {1019, 18793}, {3253, 3572}, {3669, 8851}
X(23355) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 20908}, {649, 726}, {667, 1575}, {669, 21830}, {727, 190}, {1015, 3837}, {1919, 3009}, {1977, 6373}, {1980, 21760}, {3122, 21053}, {3226, 1978}, {6373, 20532}, {8632, 17793}, {8851, 646}, {18793, 4033}, {20332, 668}, {21143, 21140}, {22096, 22092}

X(23356) =  X(99)X(11176)∩X(110)X(4590)

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

X(23356) lies on these lines: {99,11176}, {110,4590}, {351,670}, {468,6331}, {5152,20998}

X(23357) =  ISOGONAL CONJUGATE OF X(338)

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

X(23357) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Brocard axis (centered at X(23584)). (Randy Hutson, October 15, 2018)

X(23357) lies on these lines: {2,18020}, {23,232}, {39,14366}, {50,3289}, {110,647}, {112,6753}, {187,249}, {237,10317}, {476,12077}, {669,1576}, {691,10562}, {850,9514}, {1101,19622}, {1625,2623}, {2149,2150}, {2420,2436}, {2501,7471}, {2525,17708}, {3233,6587}, {4590,7779}, {4630,17938}, {5012,5661}, {5475,7578}, {6660,22121}, {9380,14587}, {9696,18334}, {14961,15388}

X(23357) = isogonal conjugate of X(338)
X(23357) = isotomic conjugate of X(23962)
X(23357) = X(2986)-vertex conjugate of X(6531)
X(23357) = cevapoint of X(i) and X(j) for these (i,j): {3, 1634}, {6, 1625}, {32, 1576}, {110, 5012}, {163, 2150}, {9233, 14574}, {11672, 14966}
X(23357) = crosspoint of X(249) and X(250)
X(23357) = trilinear pole of line {1576, 14270}
X(23357) = crossdifference of every pair of points on line {868, 5489}
X(23357) = crosssum of X(i) and X(j) for these (i,j): {115, 125}, {1648, 5099}
X(23357) = barycentric square of X(110)
X(23357) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 338}, {16565, 661}
X(23357) = X(i)-cross conjugate of X(j) for these (i,j): {6, 14586}, {32, 1576}, {160, 99}, {184, 110}, {571, 112}, {1501, 4630}, {3135, 925}, {6752, 1303}, {9233, 14574}, {10316, 4558}, {11135, 5994}, {11136, 5995}, {11672, 14966}
X(23357) = X(i)-isoconjugate of X(j) for these (i,j): {1, 338}, {2, 1109}, {4, 20902}, {10, 16732}, {11, 6358}, {12, 4858}, {19, 339}, {37, 21207}, {63, 2970}, {75, 115}, {76, 2643}, {85, 4092}, {92, 125}, {158, 15526}, {264, 3708}, {274, 21043}, {286, 21046}, {304, 8754}, {310, 21833}, {312, 1365}, {313, 3125}, {321, 3120}, {349, 4516}, {393, 17879}, {512, 20948}, {514, 4036}, {523, 1577}, {561, 3124}, {594, 1111}, {656, 14618}, {661, 850}, {662, 23105}, {668, 21131}, {693, 4024}, {799, 8029}, {823, 5489}, {826, 18070}, {868, 1821}, {1084, 1928}, {1086, 1089}, {1441, 21044}, {1969, 20975}, {2052, 2632}, {2165, 17881}, {2501, 14208}, {2616, 18314}, {2618, 15412}, {2972, 6521}, {2973, 3949}, {2996, 17876}, {3261, 4705}, {3700, 4077}, {3942, 7141}, {4064, 17924}, {4086, 7178}, {4602, 22260}, {6328, 20941}, {6335, 21134}, {6535, 16727}, {6757, 8287}, {8736, 17880}, {8818, 17886}, {8901, 14213}, {12079, 14206}, {20565, 21824}
X(23357) = barycentric product X(i)*X(j) for these {i,j}: {1, 1101}, {3, 250}, {5, 14587}, {6, 249}, {32, 4590}, {58, 4570}, {59, 60}, {99, 1576}, {101, 4556}, {109, 4636}, {110, 110}, {112, 4558}, {162, 4575}, {163, 662}, {184, 18020}, {552, 6066}, {593, 1252}, {670, 14574}, {691, 5467}, {757, 1110}, {758, 9274}, {765, 849}, {827, 1634}, {933, 23181}, {1262, 7054}, {1333, 4567}, {1397, 6064}, {1415, 4612}, {1437, 5379}, {1511, 15395}, {1625, 18315}, {2149, 2185}, {2150, 4564}, {2175, 7340}, {2206, 4600}, {2245, 9273}, {2421, 2715}, {2966, 14966}, {3003, 18879}, {3447, 14366}, {4076, 7342}, {4565, 5546}, {4576, 4630}, {5994, 17403}, {5995, 17402}, {6061, 7339}, {6065, 7341}, {9145, 11636}, {10411, 14560}, {10420, 15329}, {14570, 14586}, {15388, 20806}, {15460, 15461}, {17938, 17941}, {17939, 17944}, {17940, 17943}
X(23357) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 339}, {6, 338}, {25, 2970}, {31, 1109}, {32, 115}, {47, 17881}, {48, 20902}, {58, 21207}, {110, 850}, {112, 14618}, {163, 1577}, {184, 125}, {237, 868}, {249, 76}, {250, 264}, {255, 17879}, {512, 23105}, {560, 2643}, {577, 15526}, {662, 20948}, {669, 8029}, {692, 4036}, {849, 1111}, {1101, 75}, {1110, 1089}, {1333, 16732}, {1397, 1365}, {1501, 3124}, {1576, 523}, {1625, 18314}, {1634, 23285}, {1918, 21043}, {1919, 21131}, {1974, 8754}, {2149, 6358}, {2150, 4858}, {2175, 4092}, {2200, 21046}, {2205, 21833}, {2206, 3120}, {4556, 3261}, {4558, 3267}, {4570, 313}, {4575, 14208}, {4590, 1502}, {6056, 7068}, {6066, 6057}, {7335, 1367}, {7342, 1358}, {9233, 1084}, {9247, 3708}, {9274, 14616}, {9426, 22260}, {10316, 127}, {14560, 10412}, {14567, 1648}, {14570, 15415}, {14574, 512}, {14575, 20975}, {14585, 3269}, {14586, 15412}, {14587, 95}, {14966, 2799}, {17104, 17886}, {18020, 18022}, {18902, 2086}, {19627, 2088}

X(23358) =  MIDPOINT OF X(3) AND X(2917)

Barycentrics    a^2 (a^14-3 a^12 b^2+a^10 b^4+5 a^8 b^6-5 a^6 b^8-a^4 b^10+3 a^2 b^12-b^14-3 a^12 c^2+5 a^10 b^2 c^2+a^8 b^4 c^2-5 a^6 b^6 c^2+4 a^4 b^8 c^2-4 a^2 b^10 c^2+2 b^12 c^2+a^10 c^4+a^8 b^2 c^4+2 a^6 b^4 c^4-3 a^4 b^6 c^4-a^2 b^8 c^4+5 a^8 c^6-5 a^6 b^2 c^6-3 a^4 b^4 c^6+4 a^2 b^6 c^6-b^8 c^6-5 a^6 c^8+4 a^4 b^2 c^8-a^2 b^4 c^8-b^6 c^8-a^4 c^10-4 a^2 b^2 c^10+3 a^2 c^12+2 b^2 c^12-c^14) : :
X(23358) = 3 X[2917] - X[9920], 3 X[3] + X[9920], X[7691] + 2 X[10282], X[10274] - 3 X[11202], 3 X[5946] - 2 X[11262], X[12307] + 5 X[17821], 5 X[17821] - X[17824], 3 X[549] - 2 X[20376

See Antreas Hatzipolakis, Peter Moses and Angel Montesdeoca, Hyacinthos 28295 and Hyacinthos 28296.

X(23358) lies on these lines: {3,161}, {5,18428}, {22,22802}, {24,3574}, {26,5448}, {54,186}, {110,5562}, {182,9977}, {195,11935}, {539,9932}, {549,20376}, {550,13293}, {578,973}, {1147,1154}, {1899,12254}, {2070,15800}, {2777,13564}, {2883,7555}, {2888,10298}, {3357,7525}, {3410,9833}, {3515,12242}, {3519,21394}, {5876,6759}, {5946,11262}, {6000,7512}, {6644,6689}, {7492,20427}, {7503,18376}, {7575,20424}, {8995,9682}, {10115,19150}, {10117,14862}, {10192,12107}, {11430,11808}, {11597,22815}, {11802,18475}, {12307,17821}, {12359,12893}, {15646,22962}, {15712,23300}, {15750,19468}, {15761,19479}, {18570,22804}

X(23358) = midpoint of X(i) and X(j) for these {i,j}: {3, 2917}, {12307, 17824}
X(23358) = reflection of X(6145) in X(14076)
X(23358) = {X(7488),X(10282)}-harmonic conjugate of X(13289)
X(23358) = Kosnita-isogonal conjugate of X(5)

X(23359) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23359) lies on these lines: {31, 56}, {63, 23361}, {159, 1626}, {859, 1707}, {988, 23206}, {1444, 16678}, {1619, 16681}, {1621, 18619}, {2110, 20845}, {3666, 3827}, {8822, 23381}, {16679, 18611}, {23363, 23380}, {23364, 23379}, {23367, 23390}

X(23360) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23360) lies on these lines: {31, 1403}, {159, 1626}, {1470, 23383}, {16678, 23363}, {23371, 23390}

X(23361) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23361) lies on these lines: {1, 859}, {3, 10}, {6, 41}, {8, 4216}, {9, 10882}, {11, 13724}, {21, 5263}, {22, 3006}, {35, 7428}, {36, 3216}, {46, 23206}, {55, 10448}, {57, 20617}, {58, 20986}, {63, 23359}, {65, 22345}, {78, 4557}, {100, 15625}, {101, 22126}, {155, 11249}, {159, 1631}, {160, 7520}, {195, 22765}, {228, 2646}, {255, 14529}, {333, 1610}, {399, 2779}, {404, 1220}, {405, 19863}, {519, 19251}, {523, 20222}, {758, 22458}, {851, 7354}, {855, 6284}, {899, 4191}, {910, 1107}, {988, 1773}, {1001, 20876}, {1036, 4471}, {1125, 4245}, {1155, 22344}, {1329, 19513}, {1455, 22341}, {1464, 23154}, {1470, 10834}, {1498, 3428}, {1621, 7419}, {1630, 4269}, {1682, 4271}, {1698, 16374}, {1740, 3286}, {1764, 22299}, {2075, 2907}, {2594, 16980}, {2886, 9840}, {2929, 20838}, {2931, 19478}, {2948, 6763}, {3035, 19514}, {3086, 19256}, {3360, 20674}, {3511, 8301}, {3601, 15624}, {3616, 19245}, {3624, 19241}, {3634, 19261}, {3679, 19254}, {3828, 19252}, {4210, 5303}, {4999, 13731}, {5016, 20847}, {5248, 23364}, {5251, 16287}, {5550, 19291}, {5584, 16936}, {5835, 18235}, {5881, 15623}, {6366, 23220}, {7420, 11012}, {8071, 10037}, {8185, 11334}, {9937, 22659}, {10269, 15805}, {10475, 20967}, {10966, 15494}, {12635, 20760}, {14793, 20842}, {15267, 21147}, {16357, 16828}, {16414, 20108}, {19239, 19864}, {19250, 19862}, {19253, 19878}, {19259, 19858}, {19262, 19843}, {19292, 19877}

X(23361) = isogonal conjugate of isotomic conjugate of X(20245)
X(23361) = polar conjugate of isotomic conjugate of X(23131)
X(23361) = tangential-isogonal conjugate of X(3145)

X(23362) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23362) lies on these lines: {51, 23619}, {20864, 22199}, {21757, 23450}, {22343, 23412}, {23414, 23424}, {23415, 23451}, {23442, 23534}, {23636, 23660}

X(23363) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23363) lies on these lines: {3, 2792}, {99, 670}, {109, 692}, {320, 20878}, {1633, 6516}, {1761, 16559}, {8053, 23379}, {16678, 23360}, {23359, 23380}

X(23364) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23364) lies on these lines: {595, 17104}, {1621, 16872}, {3185, 8616}, {5248, 23361}, {8053, 23380}, {16678, 23360}, {23359, 23379}

X(23365) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23365) lies on these lines: {22, 1269}, {159, 1626}, {5301, 20999}

X(23366) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23366) lies on these lines: {159, 1626}, {23401, 23405}

X(23367) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23367) lies on these lines: {22, 16681}, {2916, 8053}, {16682, 23373}, {20999, 23396}, {23359, 23390}

X(23368) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23368) lies on these lines: {22, 23374}, {75, 16678}, {1621, 1623}, {20840, 20988}

X(23369) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23369) lies on these lines: {3, 10454}, {22, 157}, {75, 16678}, {759, 859}, {814, 7255}, {1617, 17189}, {1626, 4184}, {2933, 4225}, {3286, 5078}, {4653, 11334}

X(23370) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23370) lies on these lines: {1, 3286}, {3, 8301}, {21, 16683}, {35, 20475}, {55, 20985}, {274, 16684}, {1011, 5311}, {1107, 2223}, {1610, 16872}, {1621, 16691}, {1631, 20833}, {1964, 2176}, {4184, 17150}, {4436, 17143}, {4557, 16552}, {5132, 16476}, {5267, 8618}, {5283, 20990}, {9494, 16695}, {15624, 21384}, {16678, 16681}, {16823, 20875}, {18619, 23379}, {20358, 22060}

X(23371) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23371) lies on these lines: {2, 16683}, {3, 17031}, {6, 31}, {22, 16681}, {199, 20873}, {3112, 16684}, {3223, 8616}, {3741, 8618}, {4184, 17150}, {8267, 20045}, {13588, 16693}, {15588, 20999}, {23360, 23390}

X(23372) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23372) lies on these lines: {75, 16678}, {1621, 18619}, {18610, 18613}

X(23373) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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 + 2 a b^5 c - b^6 c - a^5 c^2 + b^5 c^2 + a^2 c^5 + 2 a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(23373) lies on these lines: {1626, 3188}, {16678, 23389}, {16681, 23378}, {16682, 23367}

X(23374) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23374) lies on these lines: {1, 18195}, {6, 22199}, {22, 23368}, {55, 19338}, {86, 1621}, {404, 5263}, {595, 3286}, {1001, 16287}, {1631, 8266}, {1634, 16872}, {11110, 23383}, {16682, 23379}, {20992, 23404}

X(23375) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23375) lies on these lines: {3, 16484}, {6, 23432}, {86, 1621}, {87, 8616}, {183, 20875}, {1631, 20878}, {3286, 3915}, {4188, 20470}, {5263, 23383}, {15621, 17277}, {16681, 23385}, {16682, 23388}, {16690, 23404}, {16876, 23379}

X(23376) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23376) lies on these lines: {36, 14190}, {859, 8053}, {16678, 16681}, {16692, 16695}, {23392, 23398}

X(23377) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23377) lies on these lines: {993, 8053}, {16678, 16681}

X(23378) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23378) lies on these lines: {3, 21240}, {157, 14017}, {16678, 18619}, {16681, 23373}

X(23379) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23379) lies on these lines: {55, 4022}, {75, 16678}, {1486, 1617}, {8053, 23363}, {16681, 16682}, {16876, 23375}, {17278, 20470}, {18619, 23370}, {23359, 23364}

X(23380) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23380) lies on these lines: {3, 10}, {48, 2276}, {75, 16678}, {846, 3185}, {3736, 20986}, {3772, 20470}, {4184, 16872}, {8053, 23364}, {16681, 23373}, {16876, 23385}, {23359, 23363}, {23388, 23391}

X(23381) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23381) lies on these lines: {1, 18177}, {3, 10472}, {19, 25}, {21, 5263}, {75, 16678}, {157, 1631}, {760, 23167}, {1043, 1610}, {1738, 16453}, {1740, 16877}, {1918, 2183}, {3713, 12329}, {3724, 17872}, {4000, 20470}, {4436, 16682}, {5369, 21777}, {8822, 23359}, {9798, 12514}, {16680, 18611}, {16688, 23392}

X(23382) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^10 b + a^9 b^2 - a^2 b^9 - a b^10 + a^10 c + a^8 b^2 c - a^2 b^8 c - b^10 c + a^9 c^2 + a^8 b c^2 - 2 a^5 b^4 c^2 - 2 a^4 b^5 c^2 + a b^8 c^2 + b^9 c^2 - 2 a^5 b^2 c^4 + 2 a^2 b^5 c^4 - 2 a^4 b^2 c^5 + 2 a^2 b^4 c^5 - a^2 b c^8 + a b^2 c^8 - a^2 c^9 + b^2 c^9 - a c^10 - b c^10) : :

X(23382) lies on these lines: {16683, 23339}, {20993, 23383}

X(23383) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23383) lies on these lines: {1, 859}, {3, 142}, {4, 15622}, {6, 2200}, {8, 19245}, {10, 4245}, {12, 13724}, {21, 16678}, {25, 225}, {31, 56}, {35, 16453}, {36, 7428}, {51, 2594}, {55, 2654}, {197, 13737}, {198, 4258}, {354, 22345}, {474, 19864}, {551, 19251}, {580, 692}, {674, 3682}, {676, 23220}, {851, 6284}, {855, 7354}, {993, 23391}, {999, 15654}, {1066, 8679}, {1104, 1402}, {1376, 3831}, {1403, 17054}, {1410, 1456}, {1451, 2187}, {1470, 23360}, {1612, 18610}, {1617, 1661}, {1621, 4225}, {1624, 13739}, {1626, 7742}, {1698, 19241}, {1730, 22300}, {1829, 8758}, {2178, 3053}, {2304, 3207}, {2333, 8608}, {2360, 20986}, {2933, 8069}, {2975, 7419}, {3085, 19256}, {3145, 20988}, {3286, 16690}, {3304, 14969}, {3338, 23206}, {3437, 7669}, {3616, 4216}, {3624, 16374}, {3634, 19250}, {3679, 19293}, {3811, 4557}, {3816, 19513}, {3847, 19546}, {3874, 22458}, {3915, 23404}, {3941, 18615}, {4191, 5217}, {4871, 16059}, {5259, 16287}, {5263, 23375}, {5266, 20990}, {5436, 10434}, {5587, 15623}, {5691, 15626}, {5937, 6626}, {6690, 13731}, {6691, 19514}, {7420, 10902}, {8193, 19869}, {9454, 23443}, {9780, 19291}, {10200, 19550}, {10483, 13744}, {11110, 23374}, {14667, 20832}, {14964, 22126}, {15571, 17733}, {15625, 16414}, {19248, 19878}, {19252, 19883}, {19261, 19862}, {19275, 19872}, {19759, 20992}, {20993, 23382}

X(23383) = isogonal conjugate of isotomic conjugate of X(17220)

X(23384) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23384) lies on these lines: {1, 2916}, {22, 17150}, {1621, 16705}, {2915, 8301}, {16678, 18616}

X(23385) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23385) lies on these lines: {3, 238}, {22, 20878}, {5989, 20873}, {8844, 23219}, {9025, 22381}, {16678, 16689}, {16681, 23375}, {16876, 23380}

X(23386) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23386) lies on these lines: {45, 55}, {993, 8053}, {1486, 14723}, {1631, 2932}, {4436, 16678}, {16681, 23387}

X(23387) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23387) lies on these lines: {55, 16672}, {859, 8053}, {1324, 1631}, {16681, 23386}

X(23388) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23388) lies on these lines: {1, 18181}, {1621, 1623}, {1631, 8266}, {1738, 20470}, {1769, 4491}, {4436, 16678}, {8053, 23363}, {16682, 23375}, {16873, 23402}, {18610, 23397}, {18613, 23339}, {23380, 23391}, {23392, 23394}

X(23389) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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 - a b^4 c^2 + b^5 c^2 - a^2 b c^4 - a b^2 c^4 + a^2 c^5 + 2 a b c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(23389) lies on these lines: {8638, 20839}, {16678, 23373}

X(23390) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23390) lies on these lines: {31, 19561}, {2930, 8053}, {23359, 23367}, {23360, 23371}

X(23391) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23391) lies on these lines: {3, 3636}, {55, 9345}, {86, 1621}, {993, 23383}, {1376, 3741}, {1617, 1631}, {3445, 5204}, {4423, 16373}, {16679, 16878}, {23380, 23388}

X(23392) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23392) lies on these lines: {1, 16680}, {3, 8301}, {36, 20875}, {55, 11998}, {859, 16693}, {993, 8053}, {999, 1486}, {1960, 9259}, {1964, 7083}, {3941, 18610}, {4574, 9016}, {8638, 20839}, {16681, 23393}, {16688, 23381}, {17435, 22310}, {23376, 23398}, {23388, 23394}

X(23393) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23393) lies on these lines: {1, 3286}, {3, 20475}, {6, 23427}, {21, 16691}, {993, 16683}, {1107, 20990}, {2223, 15621}, {2975, 16693}, {4436, 17144}, {4557, 21384}, {16678, 16689}, {16969, 20992}, {18613, 23407}

X(23394) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23394) lies on these lines: {3, 2789}, {99, 670}, {106, 1960}, {523, 7481}, {659, 8660}, {669, 4367}, {890, 21343}, {3733, 8672}, {3808, 21003}, {4057, 4778}, {4164, 9359}, {21791, 23572}, {23388, 23392}

X(23395) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23395) lies on these lines: {22, 16681}, {21006, 23399}

X(23396) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23396) lies on these lines: {6, 31}, {3223, 16690}, {16678, 16689}, {18610, 23398}, {20999, 23367}

X(23397) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23397) lies on these lines: {86, 1621}, {269, 1617}, {390, 20470}, {18610, 23388}

X(23398) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23398) lies on these lines: {1, 16681}, {23, 385}, {55, 199}, {86, 1621}, {101, 2388}, {292, 3747}, {740, 8301}, {804, 5991}, {1486, 8424}, {2076, 16365}, {3726, 8628}, {3930, 4557}, {4093, 5147}, {7453, 15621}, {8299, 20470}, {16679, 17469}, {17731, 20474}, {18610, 23396}, {23376, 23392}

X(23398) = isogonal conjugate of isotomic conjugate of anticomplement of X(3747)
X(23398) = isogonal conjugate of isotomic conjugate of anticomplementary conjugate of X(39367)

X(23399) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23399) lies on these lines: {523, 2073}, {768, 23093}, {1011, 4079}, {2352, 21348}, {3566, 23400}, {4184, 17159}, {21006, 23395}

X(23400) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23400) lies on these lines: {55, 798}, {513, 2078}, {669, 7253}, {1621, 17217}, {3566, 23399}, {3733, 8053}, {16692, 16695}, {20981, 20992}

X(23401) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23401) lies on these lines: {522, 1324}, {669, 4467}, {16692, 16695}, {23366, 23405}, {23388, 23392}

X(23402) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23402) lies on these lines: {1, 23378}, {3, 8299}, {6, 23646}, {25, 8735}, {55, 3735}, {157, 1486}, {692, 13006}, {906, 2876}, {1324, 20875}, {1631, 2932}, {2915, 16681}, {2936, 16678}, {4455, 7669}, {5172, 8618}, {5938, 16693}, {8638, 20839}, {16873, 23388}, {21382, 22310}

X(23402) = isogonal conjugate of isotomic conjugate of X(21293)
X(23402) = polar conjugate of isotomic conjugate of X(23137)

X(23403) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23403) lies on these lines: {6, 23658}, {669, 4467}, {814, 7255}, {2254, 3733}, {8654, 16751}, {21006, 23395}

X(23403) = isogonal conjugate of isotomic conjugate of X(21305)
X(23403) = polar conjugate of isotomic conjugate of X(23149)

X(23404) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23404) lies on these lines: {1, 18174}, {2, 11}, {31, 18613}, {56, 106}, {748, 15621}, {902, 20470}, {1769, 4491}, {1979, 8632}, {3722, 4557}, {3748, 20967}, {3915, 23383}, {4191, 21000}, {4694, 23169}, {5204, 8688}, {5248, 19531}, {8054, 16492}, {8616, 16678}, {10310, 12517}, {15507, 17724}, {16690, 23375}, {20992, 23374}

X(23405) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23405) lies on these lines: {21006, 23395}, {23366, 23401}

X(23505) = X(38275)-Ceva conjugate of X(513)

X(23406) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23406) lies on these lines: {22, 1602}, {23, 21009}, {4436, 16876}, {16685, 16875}, {16686, 16877}, {16692, 16695}

X(23407) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

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

X(23407) lies on these lines: {1, 21}, {2, 2223}, {3, 16823}, {35, 16825}, {42, 16476}, {55, 239}, {75, 4436}, {82, 5301}, {86, 3941}, {100, 4384}, {105, 21511}, {213, 17127}, {238, 869}, {274, 4184}, {405, 16830}, {748, 2664}, {894, 20992}, {902, 16497}, {980, 7191}, {1001, 14621}, {1011, 3757}, {1107, 3744}, {1376, 16815}, {1617, 7176}, {1964, 16690}, {2141, 3681}, {2267, 4579}, {2296, 16705}, {3230, 9463}, {3434, 14021}, {3661, 8299}, {3705, 8731}, {3766, 8638}, {3870, 21384}, {3920, 5283}, {4068, 17393}, {4447, 17244}, {4557, 17335}, {4640, 20358}, {4687, 20990}, {5015, 16290}, {5205, 16373}, {5263, 16050}, {5266, 16289}, {5284, 16831}, {7081, 16058}, {9941, 20590}, {10436, 16688}, {15485, 18794}, {15624, 17277}, {16678, 16681}, {16679, 17394}, {16969, 18278}, {17018, 20963}, {17333, 21320}, {17526, 19853}, {18613, 23393}, {21760, 23475}

X(23408) = (name pending)

Barycentrics    2 a^16-7 a^14 (b^2+c^2)-(b^2-c^2)^6 (b^4-3 b^2 c^2+c^4)+4 a^12 (b^4+3 b^2 c^2+c^4)+2 a^2 (b^2-c^2)^4 (3 b^6-b^4 c^2-b^2 c^4+3 c^6)+2 a^10 (8 b^6-b^4 c^2-b^2 c^4+8 c^6)+a^8 (-35 b^8+3 b^6 c^2-6 b^4 c^4+3 b^2 c^6-35 c^8)-a^4 (b^2-c^2)^2 (18 b^8+4 b^6 c^2-b^4 c^4+4 b^2 c^6+18 c^8)+a^6 (33 b^10-21 b^8 c^2+5 b^6 c^4+5 b^4 c^6-21 b^2 c^8+33 c^10) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28297.

X(23408) lies on this line: {2,3}

X(23409) = MIDPOINT OF X(5) AND X(13163)

Barycentrics    2 a^10+15 a^2 b^2 c^2 (b^2-c^2)^2-3 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)-2 a^6 (b^4-b^2 c^2+c^4)-a^4 (-4 b^6+17 b^4 c^2+17 b^2 c^4-4 c^6) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28297.

X(23409) lies on these lines: {2,3}, {195,7693}, {1209,20193}, {10112,13364}

X(23409) = midpoint of X(i) and X(j), for these {i, j}: {5,13163}, {3856,9825}

X(23410) = MIDPOINT OF X(2) AND X(13490)

Barycentrics    2 a^10-2 a^6 (b^2-c^2)^2+10 a^2 b^2 c^2 (b^2-c^2)^2-3 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)+2 a^4 (2 b^6-7 b^4 c^2-7 b^2 c^4+2 c^6) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28297.

X(23410) lies on these lines: {2,3}, {143,524}, {542,5462}, {1147,5476}, {1199,9143}, {1503,13363}, {3564,16776}, {5447,19924}, {11180,18951}, {11591,11745}, {11645,11695}, {12134,15026}, {12241,15465}, {13338,16310}, {16266,20423}

X(23410) = midpoint of X(i) and X(j), for these {i, j}: {2,13490}, {428,549}
X(23410) = reflection of X(i) in X(j), for these {i, j}: {547,10128}, {10691,10124}

X(23411) = MIDPOINT OF X(546) AND X(9825)

Barycentrics    2 a^10+16 a^2 b^2 c^2 (b^2-c^2)^2-3 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)-2 a^6 (b^4+c^4)+4 a^4 (b^6-4 b^4 c^2-4 b^2 c^4+c^6) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28297.

X(23411) lies on these lines: {2,3}, {156,18583}, {3564,10095}, {5892,16621}, {6146,14845}, {6689,15448}, {9820,19130}, {11264,13364}, {11451,16659}, {15026,18914}, {15028,16658}

X(23411) = midpoint of X(i) and X(j), for these {i, j}: {546,9825}, {12811,13163}

X(23412) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23412) lies on these lines: {184, 1475}, {6467, 20462}, {21757, 23443}, {22343, 23362}, {23415, 23526}, {23416, 23417}, {23418, 23438}, {23419, 23437}, {23424, 23454}

X(23413) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23413) lies on these lines: {13366, 23621}, {22199, 23418}, {22343, 23362}, {23429, 23454}

X(23414) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23414) lies on these lines: {31, 4253}, {39, 2309}, {244, 20880}, {386, 23659}, {700, 18050}, {1015, 23493}, {1125, 22172}, {2275, 4161}, {3122, 16604}, {3271, 23652}, {3840, 22189}, {4493, 18055}, {20859, 20866}, {20862, 23637}, {21757, 23446}, {22199, 23416}, {22343, 23443}, {23362, 23424}, {23420, 23441}, {23423, 23432}, {23431, 23436}, {23448, 23449}, {23456, 23457}, {23519, 23558}, {23554, 23564}, {23626, 23646}

X(23415) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23415) lies on these lines: {1, 20460}, {31, 1475}, {42, 20457}, {55, 20459}, {81, 18723}, {105, 1200}, {672, 8616}, {1149, 1613}, {1197, 1201}, {1334, 1621}, {1400, 1914}, {2162, 7248}, {2260, 21793}, {2280, 2347}, {9315, 10980}, {9454, 18613}, {20462, 20667}, {21757, 23470}, {22199, 22343}, {23362, 23451}, {23412, 23526}, {23430, 23440}, {23457, 23475}, {23524, 23565}, {23538, 23579}

X(23416) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23416) lies on these lines: {39, 20457}, {1475, 20978}, {2275, 3271}, {20460, 20864}, {20462, 20866}, {20863, 23649}, {22199, 23414}, {23412, 23417}, {23436, 23452}, {23437, 23461}, {23457, 23462}

X(23417) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23417) lies on these lines: {1, 21757}, {6, 3750}, {31, 23660}, {42, 20669}, {213, 17127}, {238, 1197}, {902, 20965}, {1613, 15485}, {1621, 21760}, {1977, 3720}, {1979, 17122}, {2308, 3747}, {3051, 3230}, {17123, 21792}, {21838, 23470}, {22199, 22343}, {23412, 23416}, {23419, 23441}, {23425, 23451}, {23445, 23446}, {23447, 23530}

X(23418) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23418) lies on these lines: {6, 2196}, {6139, 20958}, {22199, 23413}, {22343, 23437}, {23412, 23438}

X(23419) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23419) lies on these lines: {20665, 23538}, {20959, 23623}, {22199, 23413}, {22343, 23438}, {23412, 23437}, {23417, 23441}, {23525, 23544}

X(23420) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23420) lies on these lines: {304, 22413}, {3937, 17170}, {6467, 20963}, {22199, 23436}, {23412, 23416}, {23414, 23441}, {23427, 23437}

X(23421) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23421) lies on these lines: {1843, 23624}, {22343, 23362}, {23450, 23545}

X(23422) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23422) lies on these lines: {2393, 23625}, {22343, 23362}, {23466, 23471}

X(23423) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23423) lies on these lines: {1475, 20861}, {3124, 17065}, {3662, 18066}, {3778, 8629}, {22199, 23428}, {22343, 23476}, {23414, 23432}, {23424, 23431}, {23433, 23451}

X(23424) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23424) lies on these lines: {20859, 23626}, {22343, 23446}, {23362, 23414}, {23412, 23454}, {23423, 23431}, {23450, 23460}

X(23425) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23425) lies on these lines: {20864, 22199}, {20961, 23627}, {23417, 23451}, {23432, 23439}

X(23426) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23426) lies on these lines: {20669, 20974}, {20864, 22199}, {20962, 23628}, {23450, 23579}, {23469, 23476}

X(23427) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23427) lies on these lines: {1, 4704}, {6, 23393}, {39, 20464}, {42, 4253}, {194, 18194}, {1107, 3248}, {2275, 23652}, {2309, 20963}, {3009, 16552}, {3720, 17499}, {7032, 21384}, {16975, 23493}, {22199, 23414}, {23420, 23437}, {23447, 23456}, {23549, 23566}

X(23428) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23428) lies on these lines: {894, 1977}, {4699, 21787}, {17049, 21755}, {17350, 23538}, {18170, 21762}, {20964, 20965}, {21352, 21759}, {22199, 23423}, {22343, 23475}, {23548, 23571}

X(23429) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23429) lies on these lines: {1215, 3248}, {20965, 23629}, {21757, 21838}, {23362, 23414}, {23413, 23454}, {23446, 23450}

X(23430) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23430) lies on these lines: {20864, 22199}, {23412, 23416}, {23415, 23440}, {23456, 23565}

X(23431) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23431) lies on these lines: {22199, 23452}, {23414, 23436}, {23423, 23424}, {23631, 23636}

X(23432) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23432) lies on these lines: {6, 23375}, {2309, 23632}, {3778, 20868}, {22199, 22343}, {23414, 23423}, {23425, 23439}, {23444, 23451}, {23525, 23548}

X(23433) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23433) lies on these lines: {39, 2309}, {87, 1015}, {20868, 23643}, {22199, 22343}, {23423, 23451}, {23437, 23476}, {23523, 23579}

X(23434) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23434) lies on these lines: {22199, 23414}, {22343, 23449}, {23456, 23462}, {23464, 23465}

X(23435) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23435) lies on these lines: {22199, 23414}, {22343, 23448}

X(23436) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23436) lies on these lines: {22199, 23420}, {23414, 23431}, {23416, 23452}, {23619, 23635}

X(23437) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23437) lies on these lines: {3778, 20862}, {20864, 22199}, {20866, 23631}, {21746, 23636}, {22343, 23418}, {23412, 23419}, {23414, 23423}, {23416, 23461}, {23420, 23427}, {23433, 23476}

X(23438) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23438) lies on these lines: {39, 23476}, {321, 2170}, {20460, 23447}, {20864, 22199}, {22343, 23419}, {23412, 23418}, {23414, 23431}, {23451, 23455}

X(23439) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23439) lies on these lines: {39, 51}, {22199, 23423}, {23362, 23414}, {23425, 23432}, {23447, 23476}

X(23440) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23440) lies on these lines: {1, 23420}, {7, 3937}, {51, 1400}, {75, 22413}, {2310, 22187}, {3271, 7032}, {3794, 6646}, {20460, 20864}, {20985, 21746}, {22199, 23423}, {23415, 23430}, {23451, 23461}, {23456, 23524}, {23460, 23462}, {23546, 23571}

X(23441) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23441) lies on these lines: {1, 20460}, {41, 1206}, {1402, 20459}, {20864, 22199}, {20963, 20967}, {23414, 23420}, {23417, 23419}

X(23442) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23442) lies on these lines: {20968, 23641}, {23527, 23547}

X(23443) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23443) lies on these lines:

X(23444) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23444) lies on these lines: {42, 181}, {1213, 3122}, {2209, 2245}, {3764, 4272}, {7148, 21024}, {16589, 22172}, {20870, 20975}, {20970, 23659}, {21838, 23462}, {22199, 23423}, {22343, 23414}, {23432, 23451}, {23448, 23468}, {23473, 23532}, {23639, 23643}

X(23445) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23445) lies on these lines: {1, 23423}, {20861, 20963}, {23417, 23446}

X(23446) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23446) lies on these lines: {2308, 11205}, {21757, 23414}, {22343, 23424}, {23417, 23445}, {23429, 23450}, {23462, 23532}

X(23447) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23447) lies on these lines: {1, 21838}, {3, 6}, {8, 2229}, {10, 16606}, {72, 16584}, {213, 22061}, {292, 5293}, {442, 16592}, {893, 1046}, {978, 2238}, {1015, 1107}, {1193, 21753}, {1213, 16604}, {1500, 3997}, {1573, 19858}, {2292, 3121}, {2653, 20456}, {3159, 20688}, {3670, 6377}, {3678, 21830}, {3778, 22381}, {4647, 22184}, {7117, 23640}, {15985, 16887}, {17448, 21024}, {20460, 23438}, {20467, 20982}, {20966, 20971}, {22343, 23414}, {23417, 23530}, {23427, 23456}, {23439, 23476}, {23551, 23579}

X(23448) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23448) lies on these lines: {42, 3271}, {21838, 23552}, {22199, 23451}, {22343, 23435}, {23414, 23449}, {23444, 23468}

X(23449) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23449) lies on these lines: {42, 2183}, {21838, 23553}, {22343, 23434}, {23414, 23448}

X(23450) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23450) lies on these lines: {1015, 3124}, {21757, 23362}, {22343, 23454}, {23421, 23545}, {23424, 23460}, {23426, 23579}, {23429, 23446}, {23451, 23470}, {23452, 23456}

X(23451) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23451) lies on these lines: {3271, 8645}, {22199, 23448}, {22343, 23418}, {23362, 23415}, {23417, 23425}, {23423, 23433}, {23432, 23444}, {23438, 23455}, {23440, 23461}, {23450, 23470}, {23453, 23468}, {23456, 23458}

X(23452) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23452) lies on these lines: {8638, 20974}, {22199, 23431}, {23416, 23436}, {23450, 23456}

X(23453) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23453) lies on these lines: {20974, 20975}, {23451, 23468}

X(23454) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23454) lies on these lines: {20976, 23648}, {22343, 23450}, {23412, 23424}, {23413, 23429}

X(23455) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23455) lies on these lines: {42, 20459}, {55, 23649}, {1201, 21779}, {22199, 22343}, {23438, 23451}, {23539, 23565}

X(23456) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23456) lies on these lines: {1, 23416}, {58, 12835}, {1015, 1960}, {1357, 1358}, {3022, 4162}, {4014, 17761}, {22343, 23435}, {23414, 23457}, {23427, 23447}, {23430, 23565}, {23434, 23462}, {23440, 23524}, {23450, 23452}, {23451, 23458}, {23554, 23560}

X(23457) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23457) lies on these lines: {1, 4704}, {39, 42}, {330, 2998}, {1015, 23652}, {1201, 16476}, {2275, 20464}, {3009, 21384}, {3248, 17448}, {23414, 23456}, {23415, 23475}, {23416, 23462}, {23560, 23564}

X(23458) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23458) lies on these lines: {1475, 23656}, {1960, 23650}, {8640, 23563}, {23451, 23456}

X(23459) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23459) lies on these lines: {20977, 23651}, {23414, 23424}, {23463, 23464}

X(23460) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23460) lies on these lines: {2, 23652}, {39, 42}, {43, 7032}, {1920, 23508}, {3248, 16606}, {3741, 23493}, {21757, 21838}, {23424, 23450}, {23440, 23462}

X(23461) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23461) lies on these lines: {20978, 23653}, {22199, 22343}, {23416, 23437}, {23440, 23451}

X(23462) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23462) lies on these lines: {1, 23414}, {2, 22172}, {42, 51}, {291, 3685}, {672, 3747}, {1575, 3122}, {2085, 3970}, {2227, 3912}, {2276, 3778}, {3123, 20335}, {3271, 20464}, {3741, 22167}, {3831, 22189}, {3840, 22214}, {8640, 23464}, {9055, 20598}, {17760, 23478}, {21838, 23444}, {22199, 22343}, {23416, 23457}, {23434, 23456}, {23440, 23460}, {23446, 23532}, {23530, 23534}, {23531, 23563}

X(23463) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23463) lies on these lines: {2451, 23654}, {23459, 23464}

X(23464) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23464) lies on these lines: {320, 350}, {663, 6373}, {669, 2451}, {3271, 23560}, {8640, 23462}, {9297, 17458}, {23434, 23465}, {23459, 23463}

X(23465) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23465) lies on these lines: {2309, 8630}, {8641, 20980}, {22343, 23472}, {23434, 23464}

X(23466) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23466) lies on these lines: {6373, 23656}, {8640, 23469}, {23422, 23471}, {23434, 23464}, {23451, 23456}

X(23467) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23467) lies on these lines: {669, 20981}, {890, 3063}, {1107, 22226}, {2309, 3221}, {8630, 20980}, {8640, 23472}, {9491, 21763}, {23434, 23464}

X(23468) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23468) lies on these lines: {1, 23436}, {3271, 20975}, {23444, 23448}, {23450, 23452}, {23451, 23453}

X(23469) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23469) lies on these lines: {8630, 20868}, {8640, 23466}, {20974, 23470}, {20980, 23656}, {23426, 23476}, {23459, 23463}

X(23470) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23470) lies on these lines: {1015, 1977}, {1331, 2241}, {6377, 17477}, {16781, 22148}, {20460, 23559}, {20974, 23469}, {21757, 23415}, {21838, 23417}, {22199, 23538}, {23443, 23523}, {23450, 23451}, {23525, 23530}, {23543, 23565}, {23554, 23573}, {23560, 23562}

X(23471) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23471) lies on these lines: {3221, 23658}, {23422, 23466}, {23459, 23463}

X(23472) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23472) lies on these lines: {6, 1919}, {649, 854}, {657, 5029}, {1100, 17458}, {1449, 21389}, {1459, 9011}, {1924, 23531}, {1980, 23655}, {3737, 4893}, {4164, 4449}, {8632, 20980}, {8640, 23467}, {8643, 9032}, {14438, 21189}, {17379, 21191}, {17381, 21262}, {22343, 23465}

X(23473) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23473) lies on these lines: {1, 23414}, {38, 17760}, {39, 42}, {145, 12782}, {194, 10453}, {511, 1193}, {698, 4022}, {726, 3702}, {730, 17751}, {982, 7187}, {1107, 21700}, {1468, 5145}, {2275, 3056}, {2309, 22389}, {3061, 3116}, {3117, 20684}, {3616, 22172}, {3794, 3865}, {4871, 22190}, {20460, 20864}, {20663, 23640}, {23427, 23447}, {23444, 23532}, {23530, 23563}, {23531, 23534}

X(23474) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23474) lies on these lines: {20977, 20984}, {22199, 23423}, {23434, 23464}

X(23475) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23475) lies on these lines: {1, 21757}, {31, 2241}, {869, 20669}, {1197, 16476}, {1977, 21352}, {16826, 20332}, {18170, 21790}, {20985, 23660}, {21760, 23407}, {22199, 23414}, {22343, 23428}, {23415, 23457}

X(23476) =  (A,B,C,X(75); A',B',C',X(1)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

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

X(23476) lies on these lines: {39, 23438}, {20456, 20861}, {20868, 20974}, {22199, 23414}, {22343, 23423}, {23417, 23445}, {23426, 23469}, {23433, 23437}, {23439, 23447}

X(23477) =  X(1)X(5)∩X(4)X(14503)

Barycentrics    a b c (2 a^4-2 a^3 b-a^2 b^2+2 a b^3-b^4-2 a^3 c+4 a^2 b c-2 a b^2 c-a^2 c^2-2 a b c^2+2 b^2 c^2+2 a c^3-c^4)-(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) Sqrt[a b c (a b c-(a+b-c) (a-b+c) (-a+b+c))] : :
X(23477) = R X[1]-(R-Sqrt[R (R-2 r)]) X[5]

See also X(23517).

See Alexandr Skutin and Peter Moses, Hyacinthos 28312.

X(23477) lies on these lines: {1,5}, {4,14503}, {8,14504}, {10,3307}, {30,1381}, {946,3308}, {1382,15325}, {1737,2446}

X(23477) = reflection of X(23517) in X(5)
X(23477) = X(23517)-of-Johnson-triangle

X(23478) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23478) lies on these lines: {1, 21}, {75, 23498}, {304, 2227}, {561, 18832}, {1966, 17872}, {2085, 14210}, {3116, 18156}, {3721, 17470}, {6376, 20711}, {17448, 20598}, {17760, 23462}, {20590, 21318}, {21345, 23483}, {23481, 23484}, {23486, 23489}, {23501, 23507}

X(23479) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23479) lies on these lines: {1, 11325}, {42, 65}, {18671, 20361}, {21345, 23480}, {23484, 23485}, {23492, 23493}

X(23480) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23480) lies on these lines: {2594, 21319}, {21345, 23479}

X(23481) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23481) lies on these lines: {37, 65}, {75, 23502}, {3125, 21257}, {4118, 20596}, {21238, 21951}, {21345, 23492}, {23478, 23484}, {23486, 23490}, {23487, 23510}, {23493, 23496}

X(23482) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23482) lies on these lines: {75, 23483}, {2293, 2650}, {3726, 17452}, {21345, 23484}

X(23483) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23483) lies on these lines: {75, 23482}, {354, 1201}, {3869, 20358}, {17451, 20361}, {21345, 23478}, {23493, 23505}

X(23484) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23484) lies on these lines: {6, 2294}, {75, 23500}, {2170, 18194}, {17447, 20361}, {21345, 23482}, {23478, 23481}, {23479, 23485}

X(23485) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23485) lies on these lines: {1, 2}, {75, 23493}, {76, 23652}, {194, 22343}, {330, 2998}, {960, 20356}, {1909, 20464}, {3360, 23538}, {3691, 20457}, {4022, 17448}, {20590, 22197}, {20963, 23551}, {21345, 23478}, {23479, 23484}, {23495, 23496}

X(23486) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23486) lies on these lines: {38, 1755}, {75, 23495}, {561, 2643}, {21345, 23509}, {23478, 23489}, {23481, 23490}

X(23487) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23487) lies on these lines: {3721, 4137}, {21345, 23496}, {23481, 23510}, {23499, 23501}

X(23488) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23488) lies on these lines: {1, 6}, {75, 21345}, {192, 23632}, {312, 21895}, {1218, 21264}, {1575, 22316}, {3121, 20891}, {3840, 6375}, {21080, 22199}, {21330, 22218}, {23478, 23481}, {23498, 23500}

X(23489) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23489) lies on these lines: {2236, 17445}, {4117, 20889}, {21345, 23508}, {23478, 23486}

X(23490) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23490) lies on these lines: {37, 38}, {21345, 23478}, {23481, 23486}

X(23491) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23491) lies on these lines: {31, 1820}, {1580, 17442}, {1821, 6508}, {2170, 23538}, {3223, 17891}, {21345, 23482}, {23478, 23486}, {23501, 23502}

X(23492) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23492) lies on these lines: {1, 20794}, {75, 23482}, {20863, 21271}, {21345, 23481}, {23479, 23493}

X(23493) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23493) lies on these lines: {1, 87}, {2, 3223}, {8, 20464}, {10, 18793}, {21, 741}, {31, 172}, {42, 2229}, {75, 23485}, {171, 904}, {213, 6378}, {274, 18826}, {292, 1967}, {663, 875}, {869, 2258}, {1015, 23414}, {1042, 1284}, {1107, 22343}, {1402, 3747}, {1420, 7153}, {1575, 20971}, {1911, 2329}, {1973, 3010}, {2268, 16524}, {2296, 3720}, {3112, 18091}, {3248, 17448}, {3510, 17752}, {3721, 4128}, {3741, 23460}, {4116, 17750}, {4365, 17144}, {16969, 20992}, {16975, 23427}, {21345, 23501}, {21762, 23564}, {23479, 23492}, {23481, 23496}, {23483, 23505}, {23499, 23510}

X(23493) = isogonal conjugate of X(33296)

X(23494) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23494) lies on these lines: {902, 1962}, {4137, 21337}, {21345, 23481}, {23478, 23486}

X(23495) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23495) lies on these lines: {1, 82}, {75, 23486}, {23485, 23496}

X(23496) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23496) lies on these lines: {42, 2240}, {21345, 23487}, {23481, 23493}, {23485, 23495}

X(23497) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23497) lies on these lines: {1, 1424}, {11, 20255}, {55, 869}, {75, 23482}, {144, 145}, {497, 21281}, {2310, 20594}, {3208, 4531}, {3271, 4051}, {4513, 7077}, {4517, 4520}, {14923, 20863}, {21874, 21883}

X(23498) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23498) lies on these lines: {38, 17393}, {75, 23478}, {518, 2292}, {2227, 20932}, {4016, 17459}, {21345, 23481}, {23488, 23500}

X(23499) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23499) lies on these lines: {2611, 4128}, {23487, 23501}, {23493, 23510}, {23500, 23504}

X(23500) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23500) lies on these lines: {75, 23484}, {2170, 2643}, {3758, 17451}, {23488, 23498}, {23499, 23504}

X(23501) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23501) lies on these lines: {37, 65}, {21345, 23493}, {23478, 23507}, {23487, 23499}, {23491, 23502}

X(23502) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23502) lies on these lines: {2, 3721}, {75, 23481}, {1613, 3868}, {3009, 3726}, {3125, 20340}, {3727, 21352}, {21345, 23478}, {23491, 23501}

X(23503) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23503) lies on these lines: {44, 513}, {1577, 18271}, {1924, 8640}, {18197, 21763}

X(23504) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23504) lies on these lines: {2642, 2643}, {23499, 23500}

X(23505) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23505) lies on these lines: {244, 4117}, {2170, 21762}, {23483, 23493}

X(23506) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23506) lies on these lines: {1, 8640}, {649, 4879}, {669, 4083}, {1621, 1980}, {1962, 21350}, {4782, 8655}, {17018, 20983}

X(23507) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23507) lies on these lines: {75, 23481}, {192, 3721}, {2309, 3727}, {3959, 21299}, {17475, 18674}, {21345, 23482}, {23478, 23501}, {23488, 23498}

X(23508) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23508) lies on these lines: {75, 23485}, {1386, 21352}, {1920, 23460}, {1921, 23652}, {21345, 23489}, {23478, 23481}

X(23509) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23509) lies on these lines: {518, 20590}, {4118, 17448}, {21345, 23486}, {23478, 23481}, {23485, 23495}

X(23510) =  (A,B,C,X(75); A',B',C',X(75)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(1)

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

X(23510) lies on these lines: {65, 21318}, {21345, 23479}, {23481, 23487}, {23493, 23499}

X(23511) =  X(1)X(2)∩X(6)X(5437)

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

X(23511) lies on these lines: {1,2}, {5,18506}, {6,5437}, {9,3752}, {40,19517}, {44,3928}, {46,10899}, {57,1122}, {63,3973}, {72,8951}, {165,238}, {171,16469}, {210,3677}, {223,3911}, {226,4859}, {241,2124}, {269,5435}, {312,17151}, {329,4862}, {474,1453}, {518,5573}, {748,4512}, {940,16667}, {968,17125}, {982,5223}, {988,5234}, {992,15479}, {1054,1707}, {1104,5438}, {1111,20921}, {1191,1706}, {1279,3158}, {1376,7290}, {1420,7963}, {1427,16572}, {1435,1783}, {1465,1723}, {1575,16970}, {1616,2136}, {1699,1738}, {1716,2951}, {1724,3182}, {1739,2093}, {1750,5400}, {1757,18193}, {2324,3772}, {2331,17917}, {3073,10270}, {3243,4849}, {3246,4421}, {3305,4850}, {3361,5247}, {3452,4000}, {3646,3931}, {3663,18228}, {3666,3731}, {3715,4003}, {3729,17490}, {3740,7174}, {3749,16487}, {3751,10980}, {3875,18743}, {3929,17595}, {4255,5436}, {4310,21060}, {4328,5226}, {4387,4706}, {4413,5269}, {4417,17282}, {4452,8055}, {4640,15601}, {4888,9776}, {4902,5905}, {4907,17604}, {5266,21542}, {5440,16485}, {5540,21370}, {5717,17582}, {5743,17306}, {5785,11031}, {6678,19372}, {6769,19512}, {6927,9121}, {7280,11350}, {7322,17599}, {7381,18514}, {7382,18513}, {7988,17064}, {9535,10442}, {9575,16605}, {10563,14923}, {10856,21363}, {11260,15839}, {11523,17054}, {14555,17272}, {16475,17122}, {16700,18186}, {16736,18164}, {17056,20195}, {17123,17594}, {17160,20942}, {17720,20196}

X(23511) = X(i)-Ceva conjugate of X(j) for these (i,j): {269, 1}, {4452, 2136}, {5435, 57}
X(23511) = X(21896)-cross conjugate of X(4452)
X(23511) = X(i)-isoconjugate of X(j) for these (i,j): {6, 6553}, {9, 2137}, {55, 8051}
X(23511) = cevapoint of X(i) and X(j) for these (i,j): {6, 11506}, {1743, 7963}
X(23511) = crosspoint of X(i) and X(j) for these (i,j): {651, 5382}, {658, 7035}
X(23511) = crossdifference of every pair of points on line {649, 4162}
X(23511) = crosssum of X(657) and X(3248)
X(23511) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 2951}, {7, 63}, {27, 412}, {56, 1740}, {57, 1}, {81, 411}, {85, 1760}, {174, 40}, {266, 1742}, {273, 1748}, {278, 920}, {279, 1445}, {365, 170}, {366, 10860}, {507, 845}, {508, 1766}, {509, 165}, {555, 16551}, {651, 100}, {658, 3732}, {664, 190}, {934, 651}, {1170, 7676}, {1414, 662}, {1434, 17277}, {3669, 1052}, {4146, 21375}, {4565, 2617}, {4625, 799}, {4626, 658}, {7370, 978}, {7371, 57}
X(23511) = X(1459)-gimel conjugate of X(1707)
X(23511) = X(1462)-he conjugate of X(1743)
X(23511) = X(i)-zayin conjugate of X(j) for these (i,j): {56, 1743}, {614, 1}, {1104, 6}, {1122, 57}, {1427, 16572}, {20978, 43}
X(23511) = barycentric product X(i)*X(j) for these {i,j}: {1, 4452}, {7, 2136}, {57, 8055}, {75, 1616}, {86, 21896}, {92, 23089}, {269, 6552}, {7035, 17071}
X(23511) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6553}, {56, 2137}, {57, 8051}, {1616, 1}, {2136, 8}, {4452, 75}, {6552, 341}, {8055, 312}, {17071, 244}, {21896, 10}, {23089, 63}
X(23511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6048, 4882), (1, 16569, 8580), (2, 2999, 1), (2, 3687, 17284), (2, 4384, 18229), (2, 5256, 17022), (2, 17012, 5287), (2, 17020, 5256), (6, 16602, 5437), (42, 10582, 1), (43, 5272, 1), (57, 4383, 1743), (57, 16610, 8056), (200, 614, 1), (614, 899, 200), (978, 1722, 1), (995, 9623, 1), (1103, 3086, 1), (1201, 4853, 1), (1616, 21896, 2136), (1743, 8056, 57), (2999, 17022, 5256), (3666, 7308, 3731), (3751, 17063, 10980), (4383, 16610, 57), (5247, 11512, 3361), (5256, 17020, 2999), (5256, 17022, 1), (5393, 5405, 14986)

X(23512) =  X(1)X(11233)∩X(2)X(3)

Barycentrics    a^6+a^5 b-a^2 b^4-a b^5+a^5 c+3 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^2 b^2 c^2-2 a^2 b c^3+2 b^3 c^3-a^2 c^4+a b c^4-a c^5-b c^5 : :

X(23512) lies on these lines: {1,11233}, {2,3}, {6,9535}, {40,11679}, {56,10465}, {57,10444}, {86,10478}, {165,18229}, {171,516}, {190,21375}, {312,1766}, {314,1764}, {332,4417}, {333,573}, {497,1460}, {515,3687}, {517,1999}, {572,2051}, {651,23131}, {940,10446}, {962,5711}, {1043,10454}, {1397,9554}, {1427,5088}, {1746,17277}, {1754,5156}, {1763,3732}, {1897,20243}, {2635,22421}, {3198,7360}, {4872,21621}, {5783,18228}, {5786,9534}, {7365,17170}, {10436,10888}, {10860,12717}, {12565,17022}, {12610,19786}, {17080,17134}

X(23512) = crosssum of X(i) and X(j) for these (i,j): {798, 3270} X(23512) = X(i)-aleph conjugate of X(j) for these (i,j): {21, 1740}, {99, 651}, {274, 1445}, {314, 63}, {333, 1}, {645, 100}, {799, 3732}, {811, 653}, {1043, 2951}, {4560, 1052}, {4612, 2617}, {4625, 658}, {7058, 411}, {7257, 190}, {14089, 651}
X(23512) = X(15411)-zayin conjugate of X(798)
X(23512) = barycentric product X(i)*X(j) for these {i,j}: {75, 1610} X(23512) = barycentric quotient X(i)/X(j) for these {i,j}: {1610, 1}, {15267, 1254} X(23512) = X(i)-aleph conjugate of X(j) for these (i,j): {21, 1740}, {99, 651}, {274, 1445}, {314, 63}, {333, 1}, {645, 100}, {799, 3732}, {811, 653}, {1043, 2951}, {4560, 1052}, {4612, 2617}, {4625, 658}, {7058, 411}, {7257, 190}, {14089, 651}
X(23512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6999, 19542), (2, 19645, 6996), (3, 4, 1010), (3, 2050, 2), (3, 19273, 3523), (3, 19541, 11358), (411, 6836, 7513), (573, 13478, 333), (1746, 21363, 17277), (1763, 18750, 3732), (3151, 7560, 8613), (3522, 19278, 3), (16440, 16441, 4225)

X(23513) =  MIDPOINT OF X(5587) AND X(15173)

Barycentrics    3 a^5 b^2-2 a^5 b c+3 a^5 c^2-3 a^4 b^3-a^4 b^2 c-a^4 b c^2-3 a^4 c^3-6 a^3 b^4+10 a^3 b^3 c-4 a^3 b^2 c^2+10 a^3 b c^3-6 a^3 c^4+6 a^2 b^5-2 a^2 b^4 c-4 a^2 b^3 c^2-4 a^2 b^2 c^3-2 a^2 b c^4+6 a^2 c^5+3 a b^6-8 a b^5 c-3 a b^4 c^2+16 a b^3 c^3-3 a b^2 c^4-8 a b c^5+3 a c^6-3 b^7+3 b^6 c+9 b^5 c^2-9 b^4 c^3-9 b^3 c^4+9 b^2 c^5+3 b c^6-3 c^7 : :
X(23513) = X[4]+2*X[6713], X[10]+2*X[16174], X[100]-7*X[3090], X[104]+5*X[3091], 2*X[140]+X[22938], X[149]+11*X[5056], X[153]-13*X[5068], 4*X[547]-X[6174], 5*X[631]+X[10724], X[946]+2*X[6702], 2*X[1125]+X[6246], X[1145]-4*X[9956], X[1320]+5*X[5818], X[1482]+2*X[3036], X[1537]-4*X[9955], 5*X[1656]-2*X[3035], 5*X[1698]+X[14217], 7*X[3624]-X[12119], 7*X[3832]-X[10728], 4*X[3850]-X[22799], 7*X[3851]-X[10742], 11*X[3855]+X[12248], 13*X[5067]-X[13199], 5*X[5071]+X[10707], 11*X[5072]+X[12773], 13*X[5079]-X[12331], X[5887]+2*X[12736], X[5948]-4*X[10276], 5*X[10595]+X[12531], X[10767]+5*X[15059],X[10768]+5*X[14061],X[11715]+2*X[19925],X[12690]+2*X[22935],2*X[13226]+X[16128],2*X[13464]+X[15863],5*X[18493]+X[19914]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28311.

X(23513) lies on these lines: {1,5}, {2,5840}, {3,3847}, {4,6713}, {10,16174}, {30,21154}, {100,3090}, {104,3091}, {140,22938}, {149,5056}, {153,5068}, {381,2829}, {516,6882}, {517,17533}, {528,5055}, {547,6174}, {631,10724}, {946,6702}, {1125,6246}, {1145,9956}, {1320,5818}, {1482,3036}, {1532,18857}, {1537,9955}, {1656,3035}, {1698,14217}, {2800,3817}, {2802,10175}, {3070,13977}, {3071,13913}, {3560,10090}, {3624,12119}, {3816,6980}, {3825,6842}, {3829,5790}, {3832,10728}, {3850,22799}, {3851,10742}, {3855,12248}, {4187,11231}, {4193,5657}, {4996,6920}, {5067,13199}, {5071,10707}, {5072,12773}, {5079,12331}, {5187,11249}, {5731,6941}, {5770,11023}, {5848,14561}, {5887,12736}, {5948,10276}, {6705,6841}, {6830,9779}, {6879,12775}, {6911,10058}, {6912,18861}, {6917,12764}, {6929,13273}, {6931,10598}, {6946,17100}, {6958,10893}, {6959,10896}, {6968,10269}, {6971,7681}, {6973,10589}, {6981,10591}, {10265,12617}, {10595,12531}, {10767,15059}, {10768,14061}, {11230,17530}, {11715,19925}, {12047,12832}, {12515,12705}, {12690,22935}, {13226,16128}, {13464,15863}, {17605,20118}, {18493,19914}

X(23513) = midpoint of X(5587) and X(16173)
X(23513) = complement of X(34474)
X(23513) = centroid of X(3)X(4)X(11)
X(23513) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,11,119}, {5,5901,3614}, {5,10593,355}, {11,3614,1317}, {11,10956,5533}, {80,8227,11729}, {1387,10593,11}, {1656,10738,3035}, {3035,10738,10993}, {5533,7951,10956}, {6931,10598,11248}, {6968,10584,10269}, {6973,10589,22758}, {6981,10591,11499}, {7741,8068,11}, {9955,12619,1537}

X(23514) =  COMPLEMENT OF X(21166)

Trilinears    3cos(A)+cos(3*A)+(6-cos(2*A)+2cos(4*A))*Cos(B-C)+(cos(A)+3*cos(3*A))*cos(2*(B-C))+(3*cos(2*A)-2)*cos(3*(B-C)) : :
Barycentrics    2 a^6 b^2-5 a^4 b^4+6 a^2 b^6-3 b^8+2 a^6 c^2+2 a^4 b^2 c^2-4 a^2 b^4 c^2+10 b^6 c^2-5 a^4 c^4-4 a^2 b^2 c^4-14 b^4 c^4+6 a^2 c^6+10 b^2 c^6-3 c^8 : :
X(23514) = X[3]-4*X[6722], X[4]+2*X[6036], X[98]+5*X[3091], X[99]-7*X[3090], X[113]+2*X[15359], 2*X[140]+X[22515], X[147]-13*X[5068],X[148]+11*X[5056], 2*X[230]+X[13449], X[355]+2*X[11725], 2*X[546]+X[12042], 4*X[547]-X[2482], 2*X[620]-5*X[1656], 5*X[631]+X[10723],X[671]+5*X[5071], 7*X[3832]-X[10722], 8*X[3850]+X[10991], 7*X[3851]-X[6033], 11*X[3855]+X[9862], 13*X[5067]-X[13172], 11*X[5072]+X[12188],13*X[5079]-X[13188], X[5477]-4*X[18583], 5*X[5818]+X[7983], X[6034]+X[10516], X[6249]+2*X[9478], X[6781]-4*X[14693], 7*X[7989]-X[9864], 5*X[8227]-2*X[11724], X[9167]-2*X[15699], X[11632]+5*X[19709], X[11710]+2*X[19925], 4*X[11737]-X[22566], 2*X[14120]+X[16188],5*X[15081]+X[15342], X[15357]-4*X[20304], 7*X[15703]-4*X[22247],2*X[19662]+X[20423]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28311.

X(23514) lies on these lines: {2,9734}, {3,6722}, {4,6036}, {5,39}, {30,5215}, {98,3091}, {99,3090}, {113,15359}, {140,22515}, {147,5068}, {148,5056}, {230,13449}, {355,11725}, {381,2794}, {542,3545}, {543,5055}, {546,12042}, {547,2482}, {620,1656}, {631,10723}, {671,5071}, {3023,7173}, {3027,3614}, {3070,13967}, {3832,10722}, {3850,10991}, {3851,6033}, {3855,9862}, {5025,22712}, {5067,13172}, {5072,12188}, {5079,13188}, {5171,14063}, {5477,18583}, {5818,7983}, {5965,14568}, {6034,10516}, {6249,9478}, {6781,14693}, {7989,9864}, {8227,11724}, {9167,15699}, {11632,19709}, {11710,19925}, {11737,22566}, {14120,16188}, {15081,15342}, {15357,20304}, {15703,22247}, {19662,20423}

X(23514) = midpoint of X(i) and X(j) for these {i,j}: {2,14639}, {3545,9166}, {6034,10516}
X(23514) = reflection of X(i) in X(j) for these {i,j}: {9167,15699}, {9880,14639}
X(23514) = complement of X(21166)
X(23514) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,14061,6036}, {5,115,114}, {99,3090,6721}, {381,5461,6055}, {620,6321,10992}, {1656,6321,620}, {8227,13178,11724}

X(23515) =  COMPLEMENT OF X(15035)

Trilinears    cos(A)-(3+2*cos(2*A))*cos(B-C)+3*cos(A)*cos(2(B-C)) : :
Barycentrics    2 a^8 b^2-3 a^6 b^4-3 a^4 b^6+7 a^2 b^8-3 b^10+2 a^8 c^2-2 a^6 b^2 c^2+5 a^4 b^4 c^2-14 a^2 b^6 c^2+9 b^8 c^2-3 a^6 c^4+5 a^4 b^2 c^4+14 a^2 b^4 c^4-6 b^6 c^4-3 a^4 c^6-14 a^2 b^2 c^6-6 b^4 c^6+7 a^2 c^8+9 b^2 c^8-3 c^10 : :
X(23515) = X[3567]+X[12219], 7*X[3832]-X[10721], 5*X[3843]+X[20127], 7*X[3851]-X[7728], 11*X[3855]+X[12244], 5*X[3858]+X[14677], 13*X[5067]-X[12383], 11*X[5070]+X[12902], 5*X[5071]+X[9140], 11*X[5072]+X[10620], X[5095]-4*X[18583], 4*X[5159]-X[10564], 4*X[5461]-X[11656], X[5480]+2*X[6698], X[5562]+2*X[12236], X[5609]-10*X[12812], X[5655]-3*X[15046], 5*X[5818]+X[7984], X[5891]+2*X[12099], X[7722]-7*X[15043], 7*X[7989]-X[12368], 5*X[8227]-2*X[11723], 4*X[8254]-X[14049], 2*X[9956]+X[12261], 13*X[10303]-7*X[15036], X[10575]+2*X[12133], X[10625]-4*X[13416], X[11005]+5*X[14061], X[11693]-4*X[15699], X[11709]+2*X[19925], 2*X[11793]+X[11800], X[11804]+2*X[13565], 2*X[11806]+X[12825], X[12270]-13*X[15028], X[12281]+11*X[15024], X[13358]+2*X[14128], X[14094]-19*X[15022], X[14448]-10*X[15026], 5*X[19709]+X[20126]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28311.

X(23515) lies on these lines: {2,14644}, {3,6723}, {4,6699}, {5,113}, {52,11746}, {74,3091}, {110,569}, {114,15359}, {140,10113}, {146,5068}, {265,1656}, {355,11735}, {381,2777}, {389,7723}, {399,5079}, {403,14915}, {511,2072}, {541,3545}, {542,5050}, {546,12041}, {547,5642}, {568,10255}, {631,10733}, {1209,10170}, {1352,15117}, {1511,3628}, {1539,3850}, {1986,5462}, {2771,10157}, {2781,14845}, {2931,7395}, {3024,7173}, {3028,3614}, {3043,13434}, {3070,13969}, {3448,5056}, {3525,15044}, {3526,12121}, {3544,15054}, {3567,12219}, {3832,10721}, {3843,20127}, {3851,7728}, {3855,12244}, {3858,14677}, {5067,12383}, {5070,12902}, {5071,9140}, {5072,10620}, {5095,18583}, {5159,10564}, {5181,13162}, {5449,11459}, {5461,11656}, {5480,6698}, {5562,12236}, {5609,12812}, {5640,7577}, {5655,15046}, {5818,7984}, {5891,12099}, {5943,10628}, {6090,14852}, {6642,19457}, {6804,12319}, {7401,18933}, {7503,12893}, {7505,11750}, {7506,13289}, {7507,15473}, {7514,22109}, {7529,10117}, {7579,19130}, {7583,13979}, {7584,13915}, {7722,15043}, {7989,12368}, {8227,11723}, {8254,14049}, {8976,19051}, {9956,12261}, {10024,16836}, {10303,15036}, {10539,13198}, {10575,12133}, {10625,13416}, {11005,14061}, {11693,15699}, {11709,19925}, {11793,11800}, {11804,13565}, {11806,12825}, {12270,15028}, {12281,15024}, {12901,17928}, {13358,14128}, {13951,19052}, {14094,15022}, {14448,15026}, {15072,16868}, {15362,19924}, {16168,21315}, {17814,19456}, {19709,20126}

X(23515) = complement of X(15035)
X(23515) = midpoint of X(i) and X(j) for these {i,j}: {2,14644}, {4,15055}, {381,15061}
X(23515) = reflection of X(i) in X(j) for these {i,j}: {15055,6699}, {16111,15055}, {16222,5943}
X(23515) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,7687,12295}, {4,6699,16111}, {4,15059,6699}, {5,125,113}, {5,20304,125}, {5,20396,15063}, {110,3090,12900}, {110,15025,15081}, {113,125,16003}, {125,15063,10264}, {140,10113,16163}, {265,1656,5972}, {546,12041,13202}, {3090,15081,110}, {3628,11801,1511}, {6723,7687,3}, {8227,13211,11723}, {9826,15738,11562}, {10264,20396,125}, {11746,12358,52}, {15088,20304,5}

X(23516) =  X(5)X(128)∩X(930)X(3090)

Trilinears    cos(B-C)*(4+2cos(4*A)-cos(2*(B-C))+cos(2*A)*(-2+6*cos(2*(B-C)))) : :
Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (2 a^12-8 a^10 b^2+15 a^8 b^4-20 a^6 b^6+20 a^4 b^8-12 a^2 b^10+3 b^12-8 a^10 c^2+14 a^8 b^2 c^2-4 a^6 b^4 c^2 -15 a^4 b^6 c^2+26 a^2 b^8 c^2-13 b^10 c^2+15 a^8 c^4-4 a^6 b^2 c^4+8 a^4 b^4 c^4-14 a^2 b^6 c^4+25 b^8 c^4-20 a^6 c^6-15 a^4 b^2 c^6-14 a^2 b^4 c^6 -30 b^6 c^6+20 a^4 c^8+26 a^2 b^2 c^8+25 b^4 c^8-12 a^2 c^10-13 b^2 c^10+3 c^12) : :
X(23516) = 4*X[5]-X[128], X[930]-7*X[3090], X[1141]+5*X[3091], 5*X[1656]-2*X[13372], 2*X[3850]+X[12026], 11*X[5056]+X[11671], 13*X[5079]-X[13512]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28311.

X(23516) lies on these lines: {5,128}, {930,3090}, {1141,3091}, {1656,13372}, {3327,7173}, {3614,7159}, {3850,12026}, {5056,11671}, {5079,13512}, {7529,15959}, {13383,15366}

X(23516) = {X(5),X(137)}-harmonic conjugate of X(128)

X(23517) =  X(1)X(5)∩X(4)X(14504)

Barycentrics    a b c (2 a^4-2 a^3 b-a^2 b^2+2 a b^3-b^4-2 a^3 c+4 a^2 b c-2 a b^2 c-a^2 c^2-2 a b c^2+2 b^2 c^2+2 a c^3-c^4)+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) Sqrt[a b c (a b c-(a+b-c) (a-b+c) (-a+b+c))] : :
X(23517) = R X[1]-(R+Sqrt[R (R-2 r)]) X[5]

See also X(23477).

See Alexandr Skutin and Peter Moses, Hyacinthos 28312.

X(23517) lies on these lines: {1,5}, {4,14504}, {8,14503}, {10,3308}, {30,1382}, {946,3307}, {1381,15325}, {1737,2447}

X(23517) = reflection of X(23477) in X(5)
X(23517) = X(23477)-of-Johnson-triangle

X(23518) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b + c) (a^6 - a^4 b^2 - a^2 b^4 + b^6 - a^4 b c + 2 a^2 b^3 c - b^5 c - a^4 c^2 - 2 a^2 b^2 c^2 - b^4 c^2 + 2 a^2 b c^3 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4 - b c^5 + c^6) : :

X(23518) lies on these lines: {2, 11507}, {4, 3556}, {5, 15507}, {6, 17904}, {10, 321}, {65, 860}, {997, 4202}, {1837, 11105}, {2887, 3682}, {4295, 5125}, {5721, 20306}, {13725, 19843}, {17555, 18391}, {17867, 23674}

X(23519) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 - b^2 - c^2) (a^3 b^2 - a b^4 - 2 a^3 b c - a^2 b^2 c + b^4 c + a^3 c^2 - a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - b^2 c^3 - a c^4 + b c^4) : :

X(23519) lies on these lines: {184, 2210}, {217, 1404}, {21757, 23412}, {22344, 22383}, {22370, 23134}, {23414, 23558}, {23524, 23525}, {23545, 23547}

X(23520) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^5 b^2 - 2 a^3 b^4 + a b^6 - 2 a^5 b c - a^4 b^2 c + 3 a^3 b^3 c + 2 a^2 b^4 c - 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 + 3 a^3 b c^3 + a^2 b^2 c^3 + 2 a b^3 c^3 + b^4 c^3 - 2 a^3 c^4 + 2 a^2 b c^4 - a b^2 c^4 + b^3 c^4 - a b c^5 + a c^6 - b c^6) : :

X(23520) lies on these lines: {21757, 23412}, {23534, 23558}

X(23521) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (4 a^3 + a^2 b + 3 b^3 + a^2 c - 3 b^2 c - 3 b c^2 + 3 c^3) : :

X(23521) lies on these lines: {1, 1441}, {7, 1837}, {75, 3701}, {322, 20050}, {496, 3007}, {1111, 4896}, {3085, 15956}, {3673, 4346}, {3914, 23670}, {4420, 17160}, {16666, 16732}, {17067, 20905}, {18698, 19862}, {18815, 21453}

X(23522) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a - b - c) (a^2 b^2 - a b^3 - 2 a^2 b c + b^3 c + a^2 c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(23522) lies on these lines: {6, 1423}, {41, 21760}, {213, 1692}, {1911, 9468}, {3051, 9449}, {20460, 23551}, {20665, 21757}, {23523, 23544}

X(23523) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^2 + a b^3 - 2 a^2 b c - 2 a b^2 c - b^3 c + a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + a c^3 - b c^3) : :

X(23523) lies on these lines: {1, 21757}, {6, 3208}, {8, 20669}, {31, 18758}, {213, 5007}, {238, 21759}, {1197, 16466}, {1201, 1977}, {2162, 21214}, {3915, 21760}, {17752, 20332}, {23433, 23579}, {23443, 23470}, {23522, 23544}, {23535, 23543}, {23547, 23573}, {23561, 23578}

X(23524) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (b^2 - 3 b c + c^2) : :

X(23524) lies on these lines: {1, 4704}, {6, 292}, {31, 5042}, {42, 16667}, {87, 239}, {192, 9359}, {604, 2210}, {894, 18194}, {1201, 16469}, {1449, 2309}, {1475, 20667}, {1740, 17121}, {1743, 3009}, {2275, 23659}, {3056, 20456}, {3747, 21785}, {3758, 18170}, {4499, 4941}, {4860, 17477}, {7184, 17367}, {7202, 20274}, {8540, 20753}, {17754, 20464}, {20665, 21757}, {23415, 23565}, {23440, 23456}, {23519, 23525}, {23546, 23560}, {23561, 23566}

X(23525) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^2 + a b^3 - 2 a^2 b c - 2 a b^2 c - b^3 c + a^2 c^2 - 2 a b c^2 + a c^3 - b c^3) : :

X(23525) lies on these lines: {1, 21757}, {6, 979}, {10, 20669}, {213, 7296}, {595, 21760}, {978, 2162}, {1193, 1977}, {1197, 1203}, {4065, 17475}, {16468, 21759}, {23419, 23544}, {23432, 23548}, {23470, 23530}, {23519, 23524}, {23546, 23547}

X(23526) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 - b^2 - c^2) (b^3 + a b c - b^2 c - b c^2 + c^3) : :

X(23526) lies on these lines: {1, 23420}, {31, 184}, {48, 23231}, {63, 7015}, {77, 3937}, {125, 17047}, {326, 22413}, {1899, 4388}, {2876, 3056}, {2900, 3169}, {3010, 20667}, {3685, 3869}, {20753, 20777}, {22343, 23544}, {23075, 23125}, {23124, 23193}, {23412, 23415}, {23519, 23524}, {23543, 23546}

X(23526) = isogonal conjugate of polar conjugate of X(3959)

X(23527) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^5 b^2 - a b^6 - 2 a^5 b c + a^4 b^2 c - b^6 c + a^5 c^2 + a^4 b c^2 + a b^4 c^2 + b^5 c^2 + a b^2 c^4 + b^2 c^5 - a c^6 - b c^6) : :

X(23527) lies on these lines: {1501, 21744}, {21757, 23412}, {22343, 23424}, {23442, 23547}

X(23528) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (-a + b + c) (-a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c + 2 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(23528) lies on these lines: {1, 17860}, {4, 8}, {10, 23541}, {20, 20220}, {75, 280}, {78, 1229}, {312, 7080}, {1089, 6736}, {1441, 17858}, {1836, 5906}, {2975, 10538}, {3616, 20905}, {3701, 6735}, {3872, 4968}, {3914, 23542}, {4358, 5552}, {4359, 6349}, {4391, 23104}, {4647, 4847}, {4717, 6743}, {4858, 12053}, {4861, 6742}, {11681, 23101}, {17880, 20880}, {17888, 23675}, {18697, 20895}, {23529, 23668}, {23669, 23690}

X(23529) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (a - b - c) (a^3 b - a^2 b^2 + a b^3 - b^4 + a^3 c + 2 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 + 2 b^2 c^2 + a c^3 - c^4) : :

X(23529) lies on these lines: {8, 193}, {10, 774}, {37, 4081}, {142, 21931}, {281, 4319}, {354, 21924}, {480, 17281}, {594, 3059}, {612, 7046}, {1146, 14100}, {1486, 8756}, {1826, 22273}, {2310, 20262}, {2321, 2340}, {2345, 4012}, {3012, 17073}, {3914, 17860}, {4073, 6735}, {4688, 6067}, {4847, 17874}, {4907, 23058}, {12723, 21915}, {17872, 23536}, {18690, 23677}, {23528, 23668}

X(23530) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 - a b^4 - 2 a^3 b c + a^2 b^2 c - b^4 c + a^3 c^2 + a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 + b^2 c^3 - a c^4 - b c^4) : :

X(23530) lies on these lines: {1, 20460}, {35, 20459}, {1400, 7031}, {23417, 23447}, {23462, 23534}, {23470, 23525}, {23473, 23563}

X(23531) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 - a b^4 - 2 a^3 b c + a^2 b^2 c - b^4 c + a^3 c^2 + a^2 b c^2 + b^3 c^2 + b^2 c^3 - a c^4 - b c^4) : :

X(23531) lies on these lines: {1, 20460}, {36, 20459}, {609, 1400}, {672, 14964}, {859, 9454}, {1924, 23472}, {2093, 9315}, {2183, 2251}, {3500, 17753}, {4251, 23640}, {5398, 9447}, {22343, 23434}, {23439, 23447}, {23462, 23563}, {23473, 23534}

X(23532) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b^2 - 4 a b c - b^2 c + a c^2 - b c^2) : :

X(23532) lies on these lines: {1, 4704}, {6, 3009}, {31, 21785}, {42, 1449}, {81, 1178}, {87, 4393}, {869, 16667}, {872, 16668}, {899, 17121}, {1015, 23659}, {1100, 2309}, {1201, 16475}, {1964, 16666}, {2300, 2308}, {4991, 18792}, {9359, 17319}, {17379, 18194}, {20228, 20985}, {21757, 23546}, {23412, 23415}, {23444, 23473}, {23446, 23462}, {23551, 23560}

X(23533) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (-a b^3 + 2 a^2 b c + a b^2 c + b^3 c + a b c^2 - a c^3 + b c^3) : :

X(23533) lies on these lines: {1, 21838}, {39, 17017}, {171, 6377}, {238, 21827}, {1015, 1194}, {1386, 16584}, {1500, 17011}, {2229, 17150}, {2308, 8620}, {3121, 17469}, {3229, 20985}, {3589, 16587}, {5263, 22184}, {8054, 20965}, {16592, 17061}, {21757, 23578}

X(23534) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 - 2 a^3 b c + a^2 b^2 c - a b^3 c + 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) : :

X(23534) lies on these lines: {4251, 20663}, {5007, 20964}, {21757, 23547}, {22343, 23414}, {23362, 23442}, {23462, 23530}, {23473, 23531}, {23520, 23558}, {23549, 23554}

X(23535) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^2 - b^4 - a^2 b c - b^3 c + a^2 c^2 + 4 b^2 c^2 - b c^3 - c^4) : :

X(23535) lies on these lines: {1, 20460}, {40, 20459}, {595, 1438}, {774, 2170}, {1015, 17114}, {1755, 16781}, {2179, 16502}, {3361, 9315}, {3500, 7176}, {3915, 5364}, {23519, 23524}, {23523, 23543}

X(23536) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c - 2 a^2 b c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4 : :

X(23536) lies on these lines: {1, 224}, {2, 988}, {3, 1072}, {4, 614}, {6, 10404}, {8, 1738}, {10, 38}, {11, 1883}, {12, 3752}, {28, 5322}, {31, 1777}, {36, 1076}, {42, 21620}, {56, 225}, {57, 5230}, {65, 1086}, {79, 5315}, {226, 1193}, {239, 17680}, {244, 1210}, {278, 4320}, {354, 1834}, {379, 1468}, {386, 13407}, {388, 4000}, {443, 612}, {452, 16020}, {515, 3924}, {516, 3915}, {519, 5300}, {595, 1770}, {726, 3710}, {748, 12572}, {750, 12436}, {899, 21075}, {908, 978}, {946, 1201}, {960, 3782}, {964, 1125}, {982, 6734}, {995, 12047}, {999, 1070}, {1104, 7354}, {1149, 12053}, {1191, 1836}, {1220, 16706}, {1279, 6284}, {1329, 16610}, {1458, 5930}, {1463, 23154}, {1616, 12701}, {1626, 2218}, {1722, 3436}, {1785, 3086}, {1837, 17054}, {1838, 4198}, {1842, 22654}, {1877, 18961}, {2292, 3663}, {2475, 7191}, {2478, 5272}, {2999, 5290}, {3008, 12527}, {3216, 21077}, {3290, 5254}, {3333, 11269}, {3338, 5292}, {3701, 17674}, {3757, 4201}, {3944, 21214}, {3953, 10916}, {3987, 10915}, {4193, 5121}, {4255, 17718}, {4361, 10371}, {4646, 15888}, {4695, 6736}, {4719, 5718}, {4850, 5530}, {4862, 12526}, {4947, 8421}, {5015, 17678}, {5046, 7292}, {5266, 11112}, {5484, 16824}, {5573, 9581}, {5710, 5880}, {5717, 17017}, {5721, 12675}, {5793, 17290}, {5795, 17067}, {5835, 7263}, {7290, 9579}, {7968, 10911}, {7969, 10910}, {9597, 16968}, {10527, 17064}, {12699, 16483}, {15048, 16601}, {16062, 19798}, {17023, 17686}, {17690, 20045}, {17869, 17888}, {17871, 20320}, {17872, 23529}, {22197, 23636}, {23664, 23686}, {23677, 23689}

X(23537) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4 : :

X(23537) lies on these lines: {1, 224}, {2, 7283}, {3, 3772}, {4, 990}, {5, 3752}, {6, 8757}, {7, 387}, {8, 17184}, {10, 75}, {27, 58}, {30, 1104}, {31, 1770}, {35, 3011}, {36, 16049}, {37, 8728}, {40, 1072}, {42, 13407}, {43, 21077}, {46, 5230}, {57, 225}, {63, 1714}, {72, 3782}, {79, 1203}, {115, 14873}, {141, 5295}, {145, 19824}, {169, 5286}, {226, 386}, {239, 1330}, {244, 1099}, {278, 1448}, {321, 4202}, {442, 3666}, {443, 975}, {474, 17720}, {495, 4646}, {516, 595}, {519, 5100}, {536, 3695}, {553, 3017}, {614, 1370}, {849, 18653}, {908, 3216}, {910, 5305}, {942, 1086}, {946, 995}, {978, 3944}, {982, 10916}, {988, 17064}, {993, 19844}, {1010, 1125}, {1070, 3333}, {1071, 5721}, {1076, 15803}, {1089, 19835}, {1126, 7247}, {1191, 12699}, {1193, 3120}, {1210, 1785}, {1212, 15048}, {1279, 15171}, {1437, 5137}, {1453, 9579}, {1612, 7411}, {1698, 19822}, {1737, 21935}, {1836, 16466}, {1839, 16470}, {1842, 3220}, {1847, 3668}, {2049, 4657}, {2363, 5620}, {2475, 5262}, {2476, 4850}, {2549, 16968}, {2650, 11551}, {2901, 3912}, {2999, 9612}, {3008, 12572}, {3178, 4970}, {3244, 19830}, {3338, 11269}, {3452, 17749}, {3454, 3687}, {3616, 19823}, {3617, 19826}, {3625, 19831}, {3626, 19820}, {3634, 17593}, {3635, 19828}, {3636, 19829}, {3662, 10449}, {3664, 4658}, {3670, 6734}, {3672, 4208}, {3679, 19819}, {3702, 4442}, {3739, 4205}, {3741, 19787}, {3755, 21620}, {3797, 17673}, {3814, 19839}, {3822, 5530}, {3824, 17056}, {3825, 5121}, {3828, 19797}, {3831, 19810}, {3838, 4719}, {3840, 19803}, {3891, 5300}, {3924, 10572}, {3927, 17276}, {3946, 5717}, {3987, 6735}, {4187, 16610}, {4255, 11374}, {4256, 13411}, {4298, 9363}, {4306, 5930}, {4328, 5290}, {4358, 17674}, {4359, 5051}, {4361, 5814}, {4415, 5044}, {4642, 10039}, {4854, 6051}, {4871, 19801}, {4968, 4972}, {4999, 17070}, {5146, 5324}, {5179, 5254}, {5266, 17061}, {5267, 17512}, {5484, 16821}, {5711, 5880}, {5722, 17054}, {6684, 17734}, {9780, 19825}, {9843, 19802}, {10106, 15955}, {10198, 17594}, {10200, 11512}, {11019, 19790}, {11108, 17278}, {11573, 18178}, {12701, 16483}, {13740, 16706}, {14007, 17322}, {15978, 17761}, {16602, 17527}, {16970, 17732}, {17023, 19281}, {17189, 18650}, {17301, 17528}, {17872, 23664}, {17875, 17877}, {17888, 23555}, {19812, 19862}, {19816, 20340}, {19832, 19878}, {23542, 23661}, {23665, 23666}

X(23538) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^2 - 2 a^2 b c + a^2 c^2 + b^2 c^2) : :

X(23538) lies on these lines: {1, 21757}, {2, 1977}, {6, 31}, {43, 20669}, {87, 3223}, {238, 1613}, {748, 21001}, {899, 21780}, {1189, 23643}, {1197, 16468}, {1376, 1979}, {1397, 1915}, {1580, 16502}, {2170, 23491}, {2175, 14153}, {2176, 3051}, {2275, 15373}, {3112, 4363}, {3271, 3981}, {3360, 23485}, {3683, 16525}, {4383, 21792}, {7295, 10329}, {8616, 21760}, {11490, 23629}, {11688, 21785}, {17126, 20965}, {17277, 21787}, {17350, 23428}, {18194, 21345}, {20665, 23419}, {22199, 23470}, {22439, 23652}, {23415, 23579}

X(23539) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (3 a b^2 - 8 a b c + b^2 c + 3 a c^2 + b c^2) : :

X(23539) lies on these lines: {1, 4704}, {44, 3009}, {87, 16816}, {889, 3226}, {899, 3510}, {1402, 1404}, {2309, 16666}, {7032, 16670}, {8640, 23467}, {20331, 20464}, {21757, 23553}, {23455, 23565}, {23560, 23566}

X(23540) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (3 a b^2 - 10 a b c - b^2 c + 3 a c^2 - b c^2) : :

X(23540) lies on these lines: {1, 4704}, {42, 678}, {3009, 16670}, {21757, 23552}

X(23541) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^5 b - a^4 b^2 - a b^5 + b^6 + a^5 c - 2 a^4 b c + a^3 b^2 c - a^2 b^3 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 - a^2 b 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(23541) lies on these lines: {1, 23518}, {2, 11}, {8, 1897}, {10, 23528}, {40, 117}, {124, 4551}, {297, 20556}, {343, 17135}, {355, 11105}, {394, 6327}, {517, 860}, {962, 5125}, {1361, 3869}, {1809, 10527}, {1846, 3436}, {3914, 17862}, {4202, 19861}, {4318, 17923}, {11064, 21282}, {14304, 23678}, {17904, 22131}

X(23542) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (a - b - c) (a^5 b + a^4 b^2 - a b^5 - b^6 + a^5 c + 2 a^4 b c + a^3 b^2 c + a^2 b^3 c + 2 a b^4 c + b^5 c + a^4 c^2 + a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 + a^2 b 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(23542) lies on these lines: {1, 23518}, {2, 1259}, {5, 22458}, {6, 5906}, {7, 5125}, {8, 3891}, {10, 17862}, {12, 21320}, {78, 4202}, {318, 1235}, {860, 942}, {938, 17555}, {3914, 23528}, {4357, 5051}, {4429, 7080}, {5081, 5262}, {5722, 11105}, {8229, 22345}, {10527, 13725}, {23537, 23661}

X(23543) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (-b^3 + a b c + b^2 c + b c^2 - c^3) : :

X(23543) lies on these lines: {1, 21838}, {6, 9017}, {9, 21827}, {31, 3121}, {32, 21750}, {39, 5256}, {43, 292}, {57, 6377}, {87, 21387}, {614, 1015}, {1197, 7032}, {1475, 3117}, {2229, 3187}, {3772, 16592}, {3923, 21345}, {4011, 20363}, {4362, 16606}, {5364, 21760}, {16058, 16524}, {17017, 23632}, {20456, 20684}, {20665, 21757}, {23470, 23565}, {23523, 23535}, {23526, 23546}, {23564, 23566}

X(23544) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b - c) (a^2 b^2 + a b^3 - b^3 c + a^2 c^2 + 2 b^2 c^2 + a c^3 - b c^3) : :

X(23544) lies on these lines: {1, 20460}, {8, 9}, {41, 1914}, {56, 20674}, {65, 20459}, {893, 1193}, {1107, 2170}, {1475, 9575}, {2300, 21743}, {3747, 20967}, {6210, 20667}, {7987, 9315}, {17448, 20785}, {20036, 21387}, {22343, 23526}, {23419, 23525}, {23522, 23523}

X(23545) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^3 b^2 + a b^4 - 2 a^3 b c - a^2 b^2 c - b^4 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + b^2 c^3 + a c^4 - b c^4) : :

X(23545) lies on these lines: {6, 22370}, {1977, 20978}, {21757, 21838}, {23421, 23450}, {23519, 23547}, {23522, 23523}

X(23546) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^4 (a b^3 - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3) : :

X(23546) lies on these lines: {1, 23428}, {6, 190}, {31, 7104}, {213, 9490}, {1977, 7032}, {2300, 3051}, {3009, 21759}, {3169, 21753}, {9449, 21743}, {17053, 21755}, {21757, 23532}, {22199, 23548}, {22343, 23566}, {23440, 23571}, {23522, 23523}, {23524, 23560}, {23525, 23547}, {23526, 23543}

X(23546) = polar conjugate of isotomic conjugate of X(23519)

X(23547) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^5 (a b^4 - a b^3 c - b^4 c - a b c^3 + a c^4 - b c^4) : :

X(23547) lies on these lines: {6, 6376}, {39, 22427}, {8619, 21751}, {21757, 23534}, {22343, 23549}, {23442, 23527}, {23443, 23554}, {23519, 23545}, {23523, 23573}, {23525, 23546}

X(23548) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b^2 + c^2) (a b^3 + 2 a^2 b c - a b^2 c + b^3 c - a b c^2 + a c^3 + b c^3) : :

X(23548) lies on these lines: {1, 23423}, {672, 2300}, {732, 6646}, {3124, 17049}, {3688, 8041}, {17475, 23642}, {22199, 23546}, {23428, 23571}, {23432, 23525}, {23446, 23462}

X(23549) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (b^2 + c^2) (a^3 b^2 - 2 a^3 b c - a^2 b^2 c - a b^3 c + 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) : :

X(23549) lies on these lines: {1500, 2309}, {1580, 5299}, {21757, 23414}, {22343, 23547}, {23427, 23566}, {23432, 23525}, {23534, 23554}

X(23550) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^4 (a - b - c) (a^2 b^3 - a b^4 - 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 + a^2 c^3 + a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(23550) lies on these lines: {6, 3212}, {41, 1922}, {9449, 21751}, {23522, 23523}

X(23551) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (b + c) (a^2 b^2 - 2 a^2 b c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2) : :

X(23551) lies on these lines: {1, 23428}, {2, 6}, {42, 21759}, {87, 2229}, {1100, 21762}, {1977, 2309}, {2092, 20467}, {16589, 20669}, {20460, 23522}, {20963, 23485}, {20970, 20971}, {21757, 21838}, {23447, 23579}, {23532, 23560}

X(23552) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a - b - c) (3 a b^2 - 4 a b c - b^2 c + 3 a c^2 - b c^2) : :

X(23552) lies on these lines: {1, 23470}, {43, 4274}, {44, 4434}, {194, 21224}, {213, 1017}, {4984, 22086}, {20671, 23644}, {21757, 23540}, {21838, 23448}, {22343, 23553}

X(23553) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - 2 b - 2 c) (3 a b^2 - 2 a b c + b^2 c + 3 a c^2 + b c^2) : :

X(23553) lies on these lines: {43, 44}, {194, 16816}, {213, 902}, {2225, 19587}, {20671, 23645}, {21757, 23539}, {21838, 23449}, {22343, 23552}

X(23554) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (b - c)^2 (b + c) (a^3 b + a^3 c - a^2 b c - a b^2 c - a b c^2 - b^2 c^2) : :

X(23554) lies on these lines: {1015, 9427}, {2086, 16613}, {2238, 10027}, {3121, 4117}, {21757, 23558}, {22209, 22212}, {23414, 23564}, {23443, 23547}, {23456, 23560}, {23470, 23573}, {23534, 23549}

X(23555) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b + c) (-a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :
Barycentrics    a + (a + b + c) cos(B - C) : :
X(23555) = X[4647] + 2 X[6757]

X(23555) lies on these lines: {1,91}, {8,79}, {10,201}, {58,1733}, {75,17206}, {92,3559}, {191,14206}, {321,21077}, {347,18698}, {740,3811}, {1109,2292}, {1125,4858}, {1441,12609}, {1826,22005}, {3085,3743}, {3262,18697}, {3702,20886}, {4968,20887}, {5249,23555}, {6763,20879}, {7235,10974}, {17647,23661}, {17866,17877}, {21147,24806}, {23668,23690}

X(23555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1441, 17869, 12609), (17866, 17887, 17877)
X(23555) = X(664)-Ceva conjugate of X(1577)
X(23555) = barycentric product X(i)X(j) for these {i,j}: {274, 21696}, {321, 12047}, {523, 18740}
X(23555) = barycentric quotient X(i)/X(j) for these {i,j}: {12047, 81}, {18740, 99}, {21696, 37}
.

X(23556) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b + c) (a^6 - a^4 b^2 - a^2 b^4 + b^6 + 2 a^4 b c + 2 a^2 b^3 c - a^4 c^2 - 2 a^2 b^2 c^2 - b^4 c^2 + 2 a^2 b c^3 - a^2 c^4 - b^2 c^4 + c^6) : :

X(23556) lies on these lines: {10, 37}, {1400, 21016}, {2333, 21044}, {16580, 16607}, {16609, 18588}

X(23557) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b + c) (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c + a^2 b^3 c - a^4 c^2 - b^4 c^2 + a^2 b c^3 - a^2 c^4 - b^2 c^4 + c^6) : :

X(23557) lies on these lines: {10, 321}, {21437, 23684}

X(23558) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^5 b^2 - 2 a^3 b^4 + a b^6 - 2 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 + a^5 c^2 - a^4 b c^2 + 2 a^3 b c^3 - 2 a^3 c^4 + 2 a^2 b c^4 - a b c^5 + a c^6 - b c^6) : :

X(23558) lies on these lines: {21757, 23554}, {23414, 23519}, {23520, 23534}

X(23559) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^2 + a b^3 - 2 a^2 b c - 2 a b^2 c - b^3 c + a^2 c^2 - 2 a b c^2 + 6 b^2 c^2 + a c^3 - b c^3) : :

X(23559) lies on these lines: {1, 21757}, {145, 20669}, {1001, 21759}, {1191, 1197}, {20460, 23470}

X(23560) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^4 (b - c)^2 (a b + a c - 3 b c) : :

X(23560) lies on these lines: {890, 1977}, {3271, 23464}, {21757, 23540}, {22343, 23561}, {23456, 23554}, {23457, 23564}, {23470, 23562}, {23524, 23546}, {23532, 23551}, {23539, 23566}

X(23561) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^3 - a^2 b^2 c - 2 a b^3 c - a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3) : :

X(23561) lies on these lines: {1, 6}, {21757, 23532}, {21759, 21762}, {22343, 23560}, {23523, 23578}, {23524, 23566}

X(23562) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (b - c) (a^3 b - a b^3 + a^3 c - 5 a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 - b^2 c^2 - a c^3 + b c^3) : :

X(23562) lies on these lines: {6, 14408}, {8632, 20228}, {23470, 23560}

X(23563) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 - 2 a b^4 - 2 a^3 b c + a^2 b^2 c + a b^3 c - 2 b^4 c + 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 - 2 a c^4 - 2 b c^4) : :

X(23563) lies on these lines: {6, 14408}, {8632, 20228}, {23470, 23560}

X(23564) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^3 b^4 - a^2 b^4 c - 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 + a^2 b^2 c^3 + 2 a b^3 c^3 + b^4 c^3 + a^3 c^4 - a^2 b c^4 - a b^2 c^4 + b^3 c^4) : :

X(23564) lies on these lines: {1, 6}, {87, 3224}, {9490, 21759}, {21757, 23534}, {21762, 23493}, {23414, 23554}, {23457, 23560}, {23543, 23566}

X(23565) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a^2 b^2 - 5 a^2 b c - b^3 c + 2 a^2 c^2 + 2 b^2 c^2 - b c^3) : :

X(23565) lies on these lines: {1, 21757}, {6, 3749}, {31, 23622}, {200, 20669}, {614, 1977}, {1197, 16469}, {2162, 5272}, {6377, 9315}, {23415, 23524}, {23430, 23456}, {23455, 23539}, {23470, 23543}

X(23566) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^3 - a^2 b^2 c - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3) : :

X(23566) lies on these lines: {1, 21757}, {9, 21759}, {31, 32}, {238, 18278}, {239, 20332}, {1580, 16782}, {1924, 8640}, {1977, 3009}, {3224, 21387}, {3230, 9266}, {3747, 21760}, {22343, 23546}, {23427, 23549}, {23524, 23561}, {23539, 23560}, {23543, 23564}

X(23567) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^5 b - a^3 b^3 + a^5 c + a^4 b c - a^2 b^3 c - a b^4 c - 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 b c^4 + b^2 c^4) : :

X(23567) lies on these lines: {647, 21761}, {8640, 23458}, {21196, 22093}

X(23568) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (2 a^3 - a^2 b - a b^2 - a^2 c + a b c + b^2 c - a c^2 + b c^2) : :

X(23568) lies on these lines: {31, 23655}, {649, 2308}, {663, 22383}, {8640, 23467}, {21757, 23575}

X(23569) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^3 b - 2 a^2 b^2 + a^3 c - a^2 b c + a b^2 c - 2 a^2 c^2 + a b c^2 + b^2 c^2) : :

X(23569) lies on these lines: {1, 23458}, {649, 6363}, {659, 23650}, {672, 23656}, {8640, 23467}, {20981, 22384}, {23470, 23560}

X(23570) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (2 a^3 - a b^2 + b^2 c - a c^2 + b c^2) : :

X(23570) lies on these lines: {669, 2308}, {1980, 6373}, {8633, 20980}, {8640, 23467}

X(23571) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c)^2 (b + c) (2 a^3 - a b^2 - b^2 c - a c^2 - b c^2) : :

X(23571) lies on these lines: {1977, 3124}, {21838, 23448}, {23428, 23548}, {23440, 23546}, {23450, 23451}, {23456, 23554}

X(23572) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(23572) lies on these lines: {39, 23657}, {239, 514}, {663, 1912}, {667, 6373}, {1015, 23573}, {1919, 16695}, {1924, 23472}, {2484, 3805}, {2524, 3221}, {3249, 4401}, {8630, 23655}, {8632, 22383}, {8640, 23458}, {21791, 23394}, {23426, 23469}, {23434, 23464}

X(23573) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^4 (b - c)^2 (a^2 b^2 - 2 a b^2 c + a^2 c^2 - 2 a b c^2 + b^2 c^2) : :

X(23573) lies on these lines: {6, 4595}, {1015, 23572}, {23470, 23554}, {23523, 23547}

X(23574) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^3 b^3 + a^2 b^3 c + 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) : :

X(23574) lies on these lines: {1, 23503}, {649, 891}, {669, 21763}, {8640, 23458}, {23459, 23463}, {23632, 23658}

X(23575) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^4 (b - c) (a^2 b^2 - a b^3 - 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) : :

X(23575) lies on these lines: {6, 3835}, {39, 22445}, {649, 3051}, {21757, 23568}, {21760, 23655}

X(23576) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^3 (a^2 b^3 - a^2 b^2 c - 2 a b^3 c - a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - 2 a b c^3 - b^2 c^3) : :

X(23576) lies on these lines: {1, 6}, {2309, 21762}, {7032, 21759}, {20665, 21757}, {22343, 23546}, {23532, 23551}

X(23577) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (-3 a b^3 + 2 a^2 b c + 3 a b^2 c - b^3 c + 3 a b c^2 - 3 a c^3 - b c^3) : :

X(23577) lies on these lines: {1, 21838}, {292, 5524}, {896, 3121}, {1015, 3291}, {2229, 17162}, {4663, 16584}, {6377, 18201}, {8640, 23467}, {16592, 17070}

X(23578) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^2 + a b^3 - 4 a^2 b c - 2 a b^2 c - b^3 c + a^2 c^2 - 2 a b c^2 + a c^3 - b c^3) : :

X(23578) lies on these lines: {1, 4704}, {42, 1051}, {171, 8851}, {1149, 7290}, {1193, 7032}, {1386, 3248}, {1449, 3208}, {1468, 2053}, {3009, 16468}, {3214, 17121}, {3915, 21785}, {21757, 23533}, {23523, 23561}, {23525, 23546}

X(23579) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^2 + a b^3 - 2 a^2 b c - 2 a b^2 c + b^3 c + a^2 c^2 - 2 a b c^2 + a c^3 + b c^3) : :

X(23579) lies on these lines: {1, 4704}, {6, 41}, {8, 87}, {42, 3097}, {238, 1149}, {291, 8851}, {511, 20456}, {518, 3248}, {672, 21760}, {899, 16704}, {1015, 20669}, {1201, 16468}, {1740, 3214}, {1757, 3009}, {1911, 2340}, {1924, 23472}, {1964, 4663}, {2309, 4649}, {3685, 9359}, {3720, 19717}, {3751, 7032}, {3831, 17178}, {3999, 17477}, {6685, 18192}, {8054, 17449}, {17474, 23660}, {21757, 22199}, {23415, 23538}, {23426, 23450}, {23432, 23525}, {23433, 23523}, {23447, 23551}

X(23580) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 2 a^3 b c + 2 a^2 b^2 c + b^4 c + 2 a^2 b c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + b c^4 + c^5) : :

X(23580) lies on these lines: {1, 17860}, {318, 10573}, {321, 3679}, {3262, 21411}, {4299, 20220}, {4397, 23685}, {17867, 17887}

X(23581) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + a^2 b^2 c - b^4 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - a c^4 - b c^4 + c^5) : :

X(23581) lies on these lines: {1, 17877}, {75, 78}, {85, 318}, {321, 1930}, {908, 20235}, {1231, 6734}, {1441, 17858}, {2002, 7131}, {2140, 21429}, {3263, 3710}, {12609, 18693}, {14213, 17864}, {16747, 17167}, {17682, 21602}, {17861, 18692}, {17889, 17901}, {20236, 21414}, {20239, 20905}, {20435, 21420}, {20901, 21413}

leftri

Points associated with barycentric squares of lines (inscribed ellipses): X(23582) - X(23594), X(23962)-X(24041)

rightri

Several weeks before the appearance of this preamble on September 29, 2018, Clark Kimberling and Peter Moses, and independently, Randy Hutson, developed the notion of (pointwise) squares and cubes of points on lines, and they computed examples found below. Centers X(23582)-X(23594) were contributed by Kimberling and Moses, and X(23962)-X(24041) by Hutson. The latter are in two groups: X(23962)-X(23992) (barycentrc squares) and X(23993)-X(24041) (trilinear) See the preambles just before X(23962) and X(23993).

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC, not on a sideline BC, CA, AB. Let L be the trilinear polar of P, so that L meets the sidelines in 0 : q : -r, -p : 0 : r, p : -q : 0. The barycentric squares of these points are the points A' = 0 : q^2 : r^2, B' = p^2 : 0 : r^2, C' = p^2 : q^2 : 0. The perspector of A'B'C' and ABC is the barycentric square P^2 = p^2 : q^2 : r^2, so that P^2 is the perspector of the inellipse that is the locus of squares of points on L. The center of the ellipse is the point p^2 (q^2 + r^2) : q^2 (r^2 + p^2) : r^2 (p^2 + q^2), which is the X(2)-crosspoint of P^2, as well as the complement of the isotomic conjugate of P^2. The ellipse is here named the barycentric square of L.

The appearance of (L,i,j,k) in the following list means that L is the trilinear polar of X(i), and the associated inellipse has perspector X(j) and center X(k):

(line at infinity = X(30)X(513), 2, 2, 2)
(Euler line = (X(2)X(3), 648, 23582, 23583)
(Brocard axis = X(3)X(6), 110, 23357, 23584)
(X(1)X(3), 651, 1262, 23585)
(orthic axis = X(230)X(232), 4, 393, 3767)
(Lemoine axis = X(187)X(237), 6, 32, 8265)
(de Longchamps line= X(325)X(523), 76, 1502, 626)
(Gergonne line = X(241)X(514), 7, 279, 4000)
(Soddy line = X(1)X(7), 658, 23586, 23587)
(Nagel line = X(1)X(2), 190, 1016, 4422)
(Fermat axis = X(6)X(13), 476, 23588, 23589)
(van Aubel line = X(4)X(6), 655,23590,23591)
(X(1)X(5), 655,23592,23593)
(antiorthic axis = X(44)X(513), 1, 6, 39)
(X(1)X(4), 653, 23984, 23982)
(X(1)X(6), 100, 1252, 23988)
(X(2)X(6), 99, 4590, 620)
(X(115)X(125), 523, 115, 23991)

More generally, if m is a regular collineation, then the locus of m(P^2), as P traverses L, is an inellipse of the triangle m(A)m(B)m(C)..

X(23582) =  PERSPECTOR OF BARYCENTRIC SQUARE OF EULER LINE

Barycentrics    (a-b)^2 (a+b)^2 (a-c)^2 (a+c)^2 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 : :
Barycentrics    sec^2 A csc^2(B - C) : :
Barycentrics    1/(tan B - tan C)^2 : :
Barycentrics    1/(sin 2B - sin 2C)^2 : :
Barycentrics    1/((b^2 - c^2)^2 (b^2 + c^2 - a^2)^2) : :
X(23582) = 2 X[648] + X[16077]

X(23582) is the trilinear pole of line X(107)X(110), which is the tangent to the Steiner circumellipse at X(648), and the polar, with respect to the polar circle, of X(125) Line X(107)X(110) is also the locus of the trilinear pole of the tangent at P to hyperbola {{A, B, C, X(2), P}}, as P moves on the Euler line. It is also the locus of the trilinear poles of tangents to the inellipse centered at X(23582). (Randy Hutson, September 29, 2018)

X(23582) lies on the hyperbola {{A,B,C,PU(2))}} and these lines: {30,250}, {99,20580}, {107,691}, {112,2966}, {162,21761}, {249,297}, {447,4570}, {512,685}, {525,648}, {687,15352}, {1625,18831}, {1968,10684}, {2326,7128}, {2404,2407}, {2421,2442}, {4235,5649}, {4590,15014}, {11547,18879}

X(23582) = isogonal conjugate of X(3269)
X(23582) = isotomic conjugate of X(15526)
X(23582) = isotomic conjugate of the polar of X(32230)
X(23582) = barycentric square of X(648)
X(23582) = antitomic conjugate of X(15351)
X(23582) = polar conjugate of X(125)
X(23582) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3269}, {21381, 822}
X(i)-cross conjugate of X(j) for these (i,j): {2, 648}, {3, 18831}, {4, 6528}, {20, 99}, {22, 4577}, {30, 16077}, {132, 877}, {232, 685}, {250, 18020}, {393, 107}, {401, 2966}, {577, 110}, {858, 892}, {1370, 670}, {1968, 112}, {2052, 16813}, {2071, 18878}, {2475, 18026}, {3151, 190}, {3152, 664}, {3163, 4240}, {11547, 15352}, {14165, 15459}, {14807, 15165}, {14808, 15164}, {17907, 6331}, {18679, 653}
X(23582) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3269}, {3, 3708}, {6, 2632}, {19, 2972}, {31, 15526}, {32, 17879}, {41, 1367}, {48, 125}, {63, 20975}, {71, 18210}, {115, 255}, {122, 2155}, {163, 5489}, {184, 20902}, {201, 7117}, {213, 17216}, {228, 4466}, {326, 3124}, {339, 9247}, {394, 2643}, {520, 661}, {523, 822}, {525, 810}, {577, 1109}, {604, 7068}, {647, 656}, {798, 3265}, {906, 21134}, {1102, 2971}, {1146, 7138}, {1364, 2171}, {1365, 2289}, {1437, 21046}, {1562, 19614}, {1650, 2159}, {2170, 7066}, {2197, 7004}, {2616, 17434}, {2631, 14380}, {2638, 6354}, {2970, 4100}, {3049, 14208}, {3120, 3990}, {3122, 3998}, {3125, 3682}, {3690, 3942}, {3937, 3949}, {4024, 23224}, {4055, 16732}, {4064, 22383}, {4079, 4131}, {4091, 4705}, {4092, 7125}, {6507, 8754}, {18604, 21043}, {21044, 22341}
X(23582) = X(14908)-vertex conjugate of X(16081)
X(23582) = cevapoint of X(i) and X(j) for these (i,j): {2, 648}, {3, 1625}, {4, 112}, {6, 1624}, {99, 315}, {107, 393}, {110, 577}, {162, 2326}, {2479, 2480}, {3163, 4240}, {14590, 14920}
X(23582) = trilinear pole of line {107, 110}
X(23582) = barycentric product X(i)*X(j) for these {i,j}: {4, 18020}, {99, 107}, {110, 6528}, {112, 6331}, {162, 811}, {249, 2052}, {250, 264}, {286, 5379}, {393, 4590}, {648, 648}, {662, 823}, {685, 877}, {687, 16237}, {1118, 6064}, {1857, 7340}, {2396, 20031}, {2407, 15459}, {4230, 22456}, {4240, 16077}, {4558, 15352}, {4563, 6529}, {4600, 8747}, {4601, 5317}, {4620, 8748}, {14570, 16813}, {14615, 15384}
X(23582) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2632}, {2, 15526}, {3, 2972}, {4, 125}, {6, 3269}, {7, 1367}, {8, 7068}, {19, 3708}, {20, 122}, {25, 20975}, {27, 4466}, {28, 18210}, {30, 1650}, {59, 7066}, {60, 1364}, {75, 17879}, {86, 17216}, {92, 20902}, {99, 3265}, {107, 523}, {110, 520}, {112, 647}, {158, 1109}, {162, 656}, {163, 822}, {186, 16186}, {249, 394}, {250, 3}, {264, 339}, {270, 7004}, {393, 115}, {523, 5489}, {648, 525}, {685, 879}, {687, 15421}, {811, 14208}, {823, 1577}, {877, 6333}, {933, 23286}, {1093, 2970}, {1096, 2643}, {1101, 255}, {1118, 1365}, {1249, 1562}, {1304, 14380}, {1625, 17434}, {1629, 7668}, {1826, 21046}, {1857, 4092}, {1897, 4064}, {2052, 338}, {2189, 7117}, {2207, 3124}, {2420, 1636}, {3146, 13611}, {3267, 23107}, {4230, 684}, {4235, 14417}, {4240, 9033}, {4556, 4091}, {4563, 4143}, {4567, 3998}, {4570, 3682}, {4590, 3926}, {5317, 3125}, {5379, 72}, {6056, 7065}, {6064, 1264}, {6331, 3267}, {6524, 8754}, {6528, 850}, {6529, 2501}, {6530, 868}, {7012, 201}, {7115, 2197}, {7335, 1363}, {7340, 7055}, {7649, 21134}, {8747, 3120}, {8748, 21044}, {8884, 8901}, {11251, 13212}, {14587, 19210}, {14590, 8552}, {15352, 14618}, {15384, 64}, {15459, 2394}, {15742, 3695}, {16237, 6334}, {16813, 15412}, {17907, 127}, {18020, 69}, {18604, 16730}, {20031, 2395}, {23347, 9409}

X(23583) =  CENTER OF BARYCENTRIC SQUARE OF EULER LINE

Barycentrics    2 a^8-2 a^6 b^2-a^4 b^4+b^8-2 a^6 c^2+4 a^4 b^2 c^2-2 b^6 c^2-a^4 c^4+2 b^4 c^4-2 b^2 c^6+c^8 : :
Barycentrics    cos^2 B sin^2(C - A) + cos^2 C sin^2(A - B) : :
Barycentrics    (c^2 - a^2)^2 (c^2 + a^2 - b^2)^2 + (a^2 - b^2)^2 (a^2 + b^2 - c^2)^2 : :
X(23583) = 3 X[2] + X[648], 5 X[2] - X[1494], 5 X[648] + 3 X[1494], X[648] - 3 X[3163], X[1494] + 5 X[3163], X[287] - 5 X[3618], 3 X[1494] - 5 X[15526], 3 X[3163] + X[15526]

X(23583) is the center of the hyperbola, H, which is the locus of perspectors of circumconics centered at points on the Euler line, or equivalently, the locus of the X(2)-Ceva conjugate of P, as P moves on the Euler line. H passes through X(2), X(6), X(216), X(233), X(1196), X(1249), X(1560), X(3162), X(3163), and the vertices of the medial triangle. H is the complement of hyperbola {{A, B, C, X(2), X(69)}}, and is tangent to the Euler line at X(2). (Randy Hutson, September 29, 2018)

X(23583) lies on these lines: {2,648}, {3,9530}, {5,542}, {6,15595}, {125,9512}, {141,20204}, {287,3618}, {297,3284}, {338,3018}, {402,5972}, {441,1990}, {577,17907}, {620,2492}, {1249,6389}, {1316,8754}, {1576,2794}, {1632,2847}, {3506,19137}, {3589,14767}, {4422,15252}, {5422,18883}, {10314,16989}

X(23583) = midpoint of X(i) and X(j) for these {i,j}: {2, 3163}, {6, 15595}, {297, 3284}, {441, 1990}, {648, 15526}
X(23583) = complement X[15526]
X(23583) = crossdifference of every pair of points on line {7669, 9409}
X(23583) = crosssum of X(6) and X(3269)
X(23583) = X(2)-daleth conjugate of X(648)
X(23583) = X(2)-waw conjugate of X(525)
X(23583) = X(i)-complementary conjugate of X(j) for these (i,j): {107, 21253}, {162, 127}, {163, 122}, {250, 18589}, {1101, 6389}, {1576, 16595}
X(23583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 648, 15526), (3163, 15526, 648)

X(23584) =  CENTER OF BARYCENTRIC SQUARE OF BROCARD AXIS

Barycentrics    a^4 (a^4 b^4-2 a^2 b^6+b^8+a^4 c^4-2 a^2 c^6+c^8) : :

X(23584) lies on these lines: {2,6331}, {6,23181}, {39,14389}, {110,248}, {125,5661}, {216,6103}, {647,5972}, {7664,7806}, {8290,16951}, {14966,23217}

X(23585) =  CENTER OF BARYCENTRIC SQUARE OF LINE X(1)X(3)

Barycentrics    a^2 (a^4 b^2-2 a^2 b^4+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^2-2 a b^3 c^2+b^4 c^2+2 a^2 b c^3-2 a b^2 c^3-2 a^2 c^4+2 a b c^4+b^2 c^4-2 b c^5+c^6) : :

X(23585) lies on these lines: {2,6335}, {3,8750}, {6,1813}, {101,22145}, {241,8607}, {292,2277}, {647,16599}, {2275,7131}, {2808,22084}, {3752,16586}, {4000,17053}, {7117,16560}, {13006,16578}

X(23586) =  PERSPECTOR OF BARYCENTRIC SQUARE OF SODDY LINE

Barycentrics    (a-b)^2 (a-c)^2 (a+b-c)^4 (a-b+c)^4 : :

X(23586) lies on these lines: {479,7339}, {658,7658}, {934,8641}, {1275,10025}, {1323,7045}, {3676,4626}, {4569,17896}

X(23586) = isogonal conjugate of X(35508)
X(23586) = trilinear pole of line X(934)X(1633)
X(23586) = cevapoint of X(i) and X(j) for these {i,j}: {6, 934}, {279, 4626}, {479, 4617}
X(23586) = barycentric square of X(658)

X(23587) =  CENTER OF BARYCENTRIC SQUARE OF SODDY LINE

Barycentrics    2 a^6-2 a^5 b+5 a^4 b^2-8 a^3 b^3+2 a b^5+b^6-2 a^5 c-8 a^4 b c+8 a^3 b^2 c+12 a^2 b^3 c-6 a b^4 c-4 b^5 c+5 a^4 c^2+8 a^3 b c^2-24 a^2 b^2 c^2+4 a b^3 c^2+7 b^4 c^2-8 a^3 c^3+12 a^2 b c^3+4 a b^2 c^3-8 b^3 c^3-6 a b c^4+7 b^2 c^4+2 a c^5-4 b c^5+c^6 : :

X(23587) lies on these lines: {141,7358}, {3756,4000}, {7658,15252}

X(23588) =  PERSPECTOR OF BARYCENTRIC SQUARE OF FERMAT AXIS

Barycentrics    (a-b)^2 (a+b)^2 (a-c)^2 (a+c)^2 (a^2-a b+b^2-c^2)^2 (a^2+a b+b^2-c^2)^2 (a^2-b^2-a c+c^2)^2 (a^2-b^2+a c+c^2)^2 : :

X(23588) lies on these lines: {476,1637}, {3018,23357}, {3163,15395}

X(23588) = isotomic conjugate of X(23965)
X(23588) = barycentric square of X(476)
X(23588) = barycentric product X(36839)*X(36840)

X(23589) =  CENTER OF BARYCENTRIC SQUARE OF FERMAT AXIS

Barycentrics    2 a^12-4 a^10 b^2+6 a^8 b^4-6 a^6 b^6+a^4 b^8+b^12-4 a^10 c^2+2 a^6 b^4 c^2+8 a^4 b^6 c^2-2 a^2 b^8 c^2-4 b^10 c^2+6 a^8 c^4+2 a^6 b^2 c^4-16 a^4 b^4 c^4+2 a^2 b^6 c^4+7 b^8 c^4-6 a^6 c^6+8 a^4 b^2 c^6+2 a^2 b^4 c^6-8 b^6 c^6+a^4 c^8-2 a^2 b^2 c^8+7 b^4 c^8-4 b^2 c^10+c^12 : :
Barycentrics    (csc^2 B)(sin^2(C - A))(1 + 2 cos 2B)^2 + (csc^2 C)(sin^2(A - B))(1 + 2 cos 2C)^2 : :

X(23589) lies on these lines: {2,6035}, {1637,22104}, {3163,14611}, {10689,15544}

X(23590) =  PERSPECTOR OF BARYCENTRIC SQUARE OF VAN AUBEL LINE

Barycentrics    (a-b)^2 (a+b)^2 (a-c)^2 (a+c)^2 (a^2+b^2-c^2)^4 (a^2-b^2+c^2)^4 : :
Barycentrics    (tan^2 A)/(tan B - tan C)^2 : :

X(23590) lies on these lines: {107,6587}, {1971,1990}, {2501,6529}

X(23590) = isogonal conjugate of isotomic conjugate of X(34538)
X(23590) = isotomic conjugate of X(23974)
X(23590) = polar conjugate of the isotomic conjugate of X(32230)
X(23590) = barycentric square of X(107)

X(23591) =  CENTER OF BARYCENTRIC SQUARE OF VAN AUBEL LINE

Barycentrics    2 a^12-2 a^10 b^2+5 a^8 b^4-8 a^6 b^6+2 a^2 b^10+b^12-2 a^10 c^2-8 a^8 b^2 c^2+8 a^6 b^4 c^2+12 a^4 b^6 c^2-6 a^2 b^8 c^2-4 b^10 c^2+5 a^8 c^4+8 a^6 b^2 c^4-24 a^4 b^4 c^4+4 a^2 b^6 c^4+7 b^8 c^4-8 a^6 c^6+12 a^4 b^2 c^6+4 a^2 b^4 c^6-8 b^6 c^6-6 a^2 b^2 c^8+7 b^4 c^8+2 a^2 c^10-4 b^2 c^10+c^12 : :

X(23591) lies on these lines: {2,20313}, {1249,1301}, {3767,6388}, {6587,6716}

X(23592) =  PERSPECTOR OF BARYCENTRIC SQUARE OF LINE X(1)X(5)

Barycentrics    (a-b)^2 (a-c)^2 (a+b-c)^2 (a-b+c)^2 (a^2-a b+b^2-c^2)^2 (a^2-b^2-a c+c^2)^2 : :

X(23592) lies on these lines: {655,10015}, {2222,8648}

X(23592) = barycentric square of X(655)

X(23593) =  CENTER OF BARYCENTRIC SQUARE OF LINE X(1)X(5)

Barycentrics    2 a^8-4 a^7 b+6 a^5 b^3-5 a^4 b^4+2 a^2 b^6-2 a b^7+b^8-4 a^7 c+12 a^6 b c-10 a^5 b^2 c-4 a^4 b^3 c+10 a^3 b^4 c-6 a^2 b^5 c+4 a b^6 c-2 b^7 c-10 a^5 b c^2+20 a^4 b^2 c^2-10 a^3 b^3 c^2+6 a^5 c^3-4 a^4 b c^3-10 a^3 b^2 c^3+8 a^2 b^3 c^3-2 a b^4 c^3+2 b^5 c^3-5 a^4 c^4+10 a^3 b c^4-2 a b^3 c^4-2 b^4 c^4-6 a^2 b c^5+2 b^3 c^5+2 a^2 c^6+4 a b c^6-2 a c^7-2 b c^7+c^8 : :

X(23593) lies on these lines: {2,13136}, {44,908}, {4422,6332}

X(23594) =  X(514)X(1921)∩X(1577)X(21263)

Barycentrics    b^2*c^2*(a + b)*(b - c)*(a + c)*(a*b^2 + a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(23594) lies on the cubic K1074 and these lines: {514, 1921}, {1577, 21263}, {8714, 18155}

X(23594) = X(213)-isoconjugate of X(785)
X(23594) = barycentric product X(i)*X(j) for these {i,j}: {310, 784}, {693, 10471}
X(23594) = barycentric quotient X(i)/X(j) for these {i,j}: {86, 785}, {784, 42}, {2978, 1918}, {10458, 692}, {10471, 100}

X(23595) =  X(514)X(3064)∩X(3960)X(17925)

Barycentrics    b*c*(b - c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a^2*b) + b^3 - a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) : :

X(23595) lies on the cubic K1074 and these lines: {514, 3064}, {3960, 17925}, {7649, 21188}

X(23595) = X(i)-isoconjugate of X(j) for these (i,j): {101, 1794}, {219, 15439}, {906, 943}, {1175, 4574}, {1331, 2259}
X(23595) = barycentric product X(i)*X(j) for these {i,j}: {693, 1838}, {1841, 3261}, {1865, 7199}, {5249, 17924}
X(23595) = barycentric quotient X(i)/X(j) for these {i,j}: {34, 15439}, {513, 1794}, {942, 1331}, {1838, 100}, {1841, 101}, {1859, 3939}, {1865, 1018}, {2260, 906}, {2294, 4574}, {5249, 1332}, {6591, 2259}, {6734, 4571}, {7649, 943}

X(23596) =  X(10)X(514)∩X(824)X(3773)

Barycentrics    (b - c)*(b^2 - a*c)*(a*b - c^2)*(b^2 + b*c + c^2) : :

X(23596) lies on the cubic K1074 and these lines: {10, 514}, {824, 3773}, {830, 875}, {1019, 7255}, {1577, 1930}, {3864, 4486}, {20485, 20506}

X(23596) = crossdifference of every pair of points on line {1914, 1933}
X(23596) = barycentric product X(i)*X(j) for these {i,j}: {334, 1491}, {335, 824}, {693, 3864}, {1934, 3805}, {3250, 18895}, {3261, 3862}, {3661, 4444}, {4122, 18827}, {4475, 4583}, {4522, 7233}
X(23596) = X(i)-isoconjugate of X(j) for these (i,j): {238, 825}, {789, 14599}, {1492, 1914}, {2210, 4586}, {5384, 8632}
X(23596) = barycentric quotient X(i)/X(j) for these {i,j}: {291, 1492}, {292, 825}, {334, 789}, {335, 4586}, {660, 5384}, {788, 2210}, {824, 239}, {876, 985}, {984, 3573}, {1491, 238}, {3250, 1914}, {3661, 3570}, {3805, 1580}, {3862, 101}, {3864, 100}, {4122, 740}, {4444, 14621}, {4475, 659}, {4486, 4366}, {4522, 3685}, {4951, 4693}, {8630, 18892}

X(23597) =  X(101)X(668)∩X(514)X(659)

Barycentrics    (a^2 + a*b + b^2)*(b - c)*(a^2 - b*c)*(a^2 + a*c + c^2) : :

X(23597) lies on the cubic K1074 and these lines: {101, 668}, {514, 659}, {812, 4164}, {1577, 1924}, {3766, 4107}

X(23597) = midpoint of X(4107) and X(4375)
X(23597) = crossdifference of every pair of points on line {2276, 3116}
X(23597) = X(i)-isoconjugate of X(j) for these (i,j): {100, 3862}, {101, 3864}, {292, 3799}, {660, 2276}, {813, 984}, {869, 4562}, {1911, 3807}, {1922, 4505}, {3094, 8684}, {3250, 5378}, {3774, 4589}
X(23597) = barycentric product X(i)*X(j) for these {i,j}: {239, 4817}, {659, 870}, {812, 14621}, {985, 3766}, {3113, 3808}
X(23597) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 3799}, {239, 3807}, {350, 4505}, {513, 3864}, {649, 3862}, {659, 984}, {812, 3661}, {870, 4583}, {985, 660}, {1492, 5378}, {3716, 3790}, {4010, 3773}, {4124, 4522}, {4375, 3797}, {4448, 4439}, {4817, 335}, {7212, 16603}, {8632, 2276}, {14621, 4562}, {22384, 3781}

X(23598) =  X(2)X(514)∩X(80)X(900)

Barycentrics    (a - 2*b - 2*c)*(a + b - 2*c)*(b - c)*(a - 2*b + c) : :
X(23598) = 5 X[1022] - 2 X[2403], X[2403] - 10 X[4049], X[1022] - 4 X[4049], X[2403] - 5 X[6548]
X(23598) lies on the cubic K1074 and these lines: {2, 514}, {80, 900}, {88, 4508}, {106, 9093}, {3570, 4555}, {3679, 4777}, {3762, 20568}, {4671, 4791}

X(23598) = reflection of X(i) in X(j) for these {i,j}: {1022, 6548}, {6544, 21198}, {6548, 4049}
X(23598) = X(1022)-Hirst inverse of X(4379)
X(23598) = crossdifference of every pair of points on line {902, 17455}
X(23598) = X(i)-isoconjugate of X(j) for these (i,j): {44, 4588}, {89, 23344}, {902, 4604}, {1023, 2163}, {1319, 5549}, {1960, 5385}, {2251, 4597}
X(23598) = barycentric product X(i)*X(j) for these {i,j}: {75, 23352}, {88, 4791}, {514, 4945}, {693, 4792}, {903, 4777}, {1022, 4671}, {3257, 4957}, {3679, 6548}, {4049, 5235}, {4767, 6549}, {4893, 20568}
X(23598) = barycentric quotient X(i)/X(j) for these {i,j}: {45, 1023}, {88, 4604}, {106, 4588}, {903, 4597}, {1022, 89}, {2177, 23344}, {2316, 5549}, {3257, 5385}, {3679, 17780}, {4770, 21805}, {4774, 4434}, {4775, 902}, {4777, 519}, {4791, 4358}, {4792, 100}, {4800, 4432}, {4814, 3689}, {4893, 44}, {4931, 3943}, {4944, 2325}, {4945, 190}, {4948, 4753}, {4951, 4439}, {4957, 3762}, {23345, 2163}, {23352, 1}

X(23599) =  X(514)X(7216)∩X(1019)X(1434)

Barycentrics    b*c*(b - c)*(-a + b - c)*(a + b - c)*(-a*b + b^2 - a*c - 2*b*c + c^2) : :

X(23599) lies on the cubic K1074 and these lines: {514, 7216}, {1019, 1434}, {1111, 3323}, {3676, 4905}

X(23599) = X(i)-isoconjugate of X(j) for these (i,j): {101, 10482}, {692, 6605}, {1174, 3939}, {6606, 14827}
barycentric product X(i)*X(j) for these {i,j}: {85, 21104}, {693, 10481}, {1088, 6362}, {1233, 3669}, {1418, 3261}, {3676, 20880}, {4077, 17169}, {7178, 16708}
barycentric quotient X(i)/X(j) for these {i,j}: {142, 644}, {354, 3939}, {513, 10482}, {514, 6605}, {1088, 6606}, {1229, 6558}, {1233, 646}, {1418, 101}, {2488, 1253}, {3669, 1174}, {3676, 2346}, {3925, 4069}, {4847, 4578}, {6362, 200}, {6608, 480}, {10481, 100}, {10581, 6602}, {16708, 645}, {16713, 7259}, {17169, 643}, {18164, 5546}, {20880, 3699}, {21104, 9}, {21127, 220}

X(23600) =  X(1)X(2)∩X(69)X(222)

Barycentrics    (a-b-c) (a^2-b^2-c^2) (a^3+a^2 b+a b^2+b^3+a^2 c+2 a b c-b^2 c+a c^2-b c^2+c^3) : :

X(23600) lies on the cubic K1073 and these lines: {1,2}, {69,222}, {278,322}, {281,312}, {319,19795}, {329,1763}, {345,3694}, {440,1260}, {464,1259}, {1211,3713}, {1376,5800}, {3436,19645}, {3684,5802}, {3998,6350}, {5564,19794}, {5687,19542}, {8897,17170}, {16608,18141}, {17296,20266}, {17861,20237}, {19785,20895}

X(23600) = barycentric product X(i)*X(j) for these {i,j}: {69, 2551}, {312, 10319}
X(23600) = barycentric quotient X(i)/X(j) for these {i,j}: {2551, 4}, {10319, 57}

X(23601) =  X(1)X(41)∩X(222)X(1814)

Barycentrics    a (a-b-c)^2 (a^2+b^2-a c-b c) (a^2-b^2-c^2) (a^2+b^2-2 b c+c^2) (a^2-a b-b c+c^2) : :

X(23601) lies on the cubic K1073 and these lines: {1,41}, {222,1814}, {281,14942}, {7124,17170}

X(23601) = X(i)-isoconjugate of X(j) for these (i,j): {241, 1041}, {1037, 5236}, {1876, 7131}
X(23601) = barycentric product X(i)*X(j) for these {i,j}: {1040, 14942}, {1814, 6554}, {6559, 7289}
X(23601) = barycentric quotient X(i)/X(j) for these {i,j}: {1040, 9436}, {2082, 5236}, {2195, 1041}, {4319, 1861}, {7083, 1876}, {7124, 241}

X(23602) =  X(1)X(21)∩X(222)X(1444)

Barycentrics    a (a+b) (a-b-c) (a+c) (a^2-b^2-c^2) (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(23602) lies on these lines: {1,21}, {222,1444}, {278,8822}, {281,333}, {573,1817}, {1812,2193}

X(23602) = barycentric product X(i)*X(j) for these {i,j}: {1812, 5712}, {2185, 8896}
X(23602) = barycentric quotient X(i)/X(j) for these {i,j}: {8896, 6358}

X(23603) =  X(1)X(7)∩X(85)X(281)

Barycentrics    (a+b-c)^2 (a-b+c)^2 (a^2-b^2-c^2) (a^3-a^2 b+a b^2-b^3-a^2 c+2 a b c+b^2 c+a c^2+b c^2-c^3) : :

X(23603) lies on the cubic K1073 and these lines: {1,7}, {85,281}, {169,14256}, {219,20618}, {222,1814}

X(23603) = X(348)-beth conjugate of X(77)
X(23603) = X(i)-isoconjugate of X(j) for these (i,j): {33, 949}, {3423, 7079}
X(23603) = barycentric product X(i)*X(j) for these {i,j}: {348, 948}, {2263, 7182}, {2550, 7056}
X(23603) = barycentric quotient X(i)/X(j) for these {i,j}: {222, 949}, {948, 281}, {2263, 33}, {2550, 7046}, {7053, 3423}, {16054, 2322}
X(23603) = {X(7),X(3160)}-harmonic conjugate of X(3100)

X(23604) =  X(1)X(224)∩X(19)X(46)

Barycentrics    (b+c) (a^3+a^2 b+a b^2+b^3-a^2 c-2 a b c-b^2 c-a c^2-b c^2+c^3) (a^3-a^2 b-a b^2+b^3+a^2 c-2 a b c-b^2 c+a c^2-b c^2+c^3) : :

X(23604) lies on the cubics K109, K434, and on K1075 and these lines: {1,224}, {3,2218}, {19,46}, {27,1770}, {37,442}, {158,1737}, {596,4847}, {759,4278}, {1479,18589}, {1739,12616}, {1836,16471}, {2214,5880}, {3120,3682}, {3173,7702}, {4456,18785}, {5310,7465}

X(23604) = isogonal conjugate of X(1780)
X(23604) = X(i)-cross conjugate of X(j) for these (i,j): {71, 226}, {18674, 92}
X(23604) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1780}, {28, 11517}, {29, 3215}, {58, 3811}, {81, 2911}, {110, 15313}, {284, 1708}, {1172, 3173}, {1175, 14054}, {1333, 17776}, {2328, 4341}
X(23604) = cevapoint of X(656) and X(3120)
X(23604) = barycentric product X(i)*X(j) for these {i,j}: {10, 15474}, {1577, 13397}
X(23604) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1780}, {10, 17776}, {37, 3811}, {42, 2911}, {65, 1708}, {71, 11517}, {73, 3173}, {661, 15313}, {1409, 3215}, {1427, 4341}, {2294, 14054}, {13397, 662}, {15474, 86}, {16732, 17877}
X(23604) = {X(1738),X(1838)}-harmonic conjugate of X(1714)

X(23605) =  X(1)X(1447)∩X(3)X(3502)

Barycentrics    a (a^2 b-a b^2-a^2 c-a b c-b^2 c-a c^2+b c^2) (a^2 b+a b^2-a^2 c+a b c-b^2 c+a c^2+b c^2) (a^3-b^3+a b c-c^3) : :

X(23605) lies on the cubics K422 and K1075 and these lines: {1,1447}, {3,3502}, {4,3494}, {98,3507}, {182,6210}, {511,7166}, {1929,18788}, {2108,2115}, {5999,8926}, {6234,8924}, {7351,8841}

X(23605) = isogonal conjugate of X(8926)
X(23605) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8926}, {3512, 17754}, {4334, 7281}, {7261, 21010}
X(23605) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8926}, {4645, 20917}, {17798, 17754}, {19554, 21010}, {20715, 21101}
X(23605) = trilinear product X(5018)*X(7220)

leftri

Points associated with pointwise barycentric cubes of lines: X(23606-X(23616)

rightri

This preamble and centers X(22606)-X(23616) were contributed by Clark Kimberling and Peter Moses, September 23, 2018.

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC, not on a sideline BC, CA, AB. Let L be the trilinear polar of P, and let X be a point on L. The locus of a point X^3 (barycentric cube) is a cubic curve, here named the pointwise barycentric cubic of L.

Let f(p,q,r,x,y,z) = q9r9x3 + 3p9q3r3yz(q3z+r3y). An equation for the cubic follows:

f(p,q,r,x,y,z) + f(q,r,p,y,z,x) + f(r,p,q,z,x,y) - 21 (p q r)3x y z = 0.

An equivalent equation is (p^3 x + q^3 y + r^3 z)^3 - 27 p^3 q^3 r^3 x y z = 0.

If L = X(2)X(3) (the Euler line), then P = X(648), and the cubic passes through the points X(i) for i = 2, 3081, 6524, 23606, 23607, 23608, 23609.

If L is the line at infinity, then P = X(2), and the cubic is K656, which passes through X(i) for i = 2, 3081, 6545, 8017, 8028, 8029, 8030, 8031, 8032, 23610, 23611, 23612, 23613, 23614, 23615, 23616.

If L = X(1)X(2), then P = X(190), and the cubic passes through X(i) for i = 2, 31, 5423, 6535, 6632, 6652, 8028.

If L = X(1)X(6), then P = X(99), and the cubic passes through X(i) for i = 2, 1501, 4176, 6628, 8030.

If L = X(1)X(7), then P = X(664), and the cubic passes through X(i) for i = 2, 479, 6507, 6602, 7366, 7369

If L = X(6)X(75), then P = X(789), and the cubic passes through X(i) for i = 561, 1501, 6652, 7369.

More generally, if m is a regular collineation, then the locus of m(P^3), as P traverses L, is a cubic.

X(23606) =  Barycentric cube of X(3)

Barycentrics    a^6 (a^2-b^2-c^2)^3 : :
Barycentrics    sin^3 2A : :

X(23606) lies on these lines: {3,54}, {4,2055}, {6,6641}, {22,2967}, {25,10313}, {51,3284}, {110,6638}, {154,1576}, {184,418}, {216,13366}, {237,10316}, {394,426}, {417,1092}, {852,9306}, {2351,20975}, {3148,23115}, {3292,6509}, {3462,17401}, {6090,6617}, {7494,7774}, {11459,15781}, {13557,18531}

X(23606) = isogonal conjugate of the isotomic conjugate of X(1092)
X(23606) = X(i)-Ceva conjugate of X(j) for these (i,j): {577, 14585}, {10420, 1636}, {19210, 577}
X(23606) = crosspoint of X(i) and X(j) for these (i,j): {577, 1092}, {15958, 23357}
X(23606) = crossdifference of every pair of points on line {12077, 14618
X(23606) = crosssum of X(i) and X(j) for these (i,j): {4, 11547}, {125, 14618}, {338, 23290}, {1093, 2052}
X(23606) = polar conjugate of the isogonal conjugate of X(36433)
X(23606) = barycentric product X(i)X(j) for these {i,j}: {1, 4100}, {3, 577}, {6, 1092}, {31, 6507}, {32, 3964}, {48, 255}, {69, 14585}, {97, 418}, {184, 394}, {212, 7125}, {216, 19210}, {219, 7335}, {222, 6056}, {228, 18604}, {326, 9247}, {560, 1102}, {571, 16391}, {603, 2289}, {822, 4575}, {906, 23224}, {1437, 3990}, {1501, 4176}, {1790, 4055}, {2193, 22341}, {2972, 23357}, {3289, 17974}, {3926, 14575}, {4143, 14574}, {5562, 14533}, {14379, 15905}, {15958, 17434}
X(23606) = X(i)-isoconjugate of X(j) for these (i,j): {2, 6521}, {19, 18027}, {75, 1093}, {76, 6520}, {92, 2052}, {158, 264}, {393, 1969}, {561, 6524}, {823, 14618}, {1096, 18022}, {1577, 15352}, {6529, 20948}, {8794, 14213}
X(23606) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 18027}, {31, 6521}, {32, 1093}, {184, 2052}, {217, 13450}, {255, 1969}, {394, 18022}, {418, 324}, {560, 6520}, {577, 264}, {1092, 76}, {1102, 1928}, {1501, 6524}, {1576, 15352}, {3964, 1502}, {4100, 75}, {6056, 7017}, {6507, 561}, {7335, 331}, {9247, 158}, {14533, 8795}, {14574, 6529}, {14575, 393}, {14585, 4}, {19210, 276}, {23103, 23107}
X(23606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1993, 13409), (97, 5012, 3), (184, 577, 418), (394, 426, 2972), (2055, 14152, 4)

X(23607) =  Barycentric cube of X(5)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)^3 : :

X(23607) lies on these lines: {53, 418}, {3459, 13621}

X(23608) =  Barycentric cube 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)^3 : :

X(23608) lies on these lines: {20, 154}, {2060, 2131}

X(23609) =  Barycentric cube of X(21)

Barycentrics    a^3*(a + b)^3*(a - b - c)^3*(a + c)^3 : :

X(23609) lies on these lines: {60, 4267}, {7054, 8021}

X(23610) =  Barycentric cube of X(512)

Barycentrics    a^6 (b^2 - c^2)^3 : :

X(23610) lies on the cubic K656 and these lines: {2, 512}, {351, 9429}, {669, 881}, {688, 11205}, {1501, 9426}, {1645, 23099}, {2086, 22260}, {3060, 3221}, {8584, 9009}

X(23610) = Danneels point of X(512)
X(23610) = X(2)-of-cevian-triangle-of-X(512)

X(23611) =  Barycentric cube of X(511)

Barycentrics    a^6 (a^2 b^2 + a^2 c^2 - b^4 - c^4)^3 : :

X(23611) lies on the cubic K656 and these lines: {2, 51}, {184, 14966}, {237, 14251}, {351, 1636}, {1501, 23606}, {1976, 17970}

X(23611) = Danneels point of X(511)
X(23611) = X(2)-of-cevian-triangle-of-X(511)

X(23612) =  Barycentric cube of X(518)

Barycentrics    a^3 (a b + a c - b^2 -c^2)^3 : :

X(23612) lies on the cubic K656 and these lines: {2, 210}, {31, 6602}, {55, 2284}, {105, 7077}, {672, 3252}, {756, 17435}, {926, 1635}, {1362, 4712}, {3675, 20683}

X(23613) =  Barycentric cube of X(520)

Barycentrics    (cos A)^6 (b^ - c^2)^3 : :

X(23613) lies on the cubic K656 and this line: {2, 520}

X(23614) =  Barycentric cube of X(521)

Barycentrics    (cos A)^3 (cos B - cos C)^3 : :

X(23614) lies on the cubic K656 and these lines: {2, 521}, {1946, 2188}, {3900, 11189}, {8021, 23090}

X(23615) =  Barycentric cube of X(522)

Barycentrics    (b - c)^3 (b + c - a)^3 : :

X(23615) lies on the cubic K656 and these lines: {2, 522}, {33, 663}, {200, 3239}, {210, 3900}, {430, 661}, {514, 1699}, {652, 7082}, {926, 4120}, {1639, 14392}, {1857, 3064}, {2400, 3676}, {3700, 6608}, {4024, 21807}, {4163, 5423}, {6544, 11124}, {15280, 16892}

X(23616) =  Barycentric cube of X(525)

Barycentrics    (b^2 - c^2)^3 (b^2 + c^2 - a^2)^3 : :

X(23616) lies on the cubic K656 and these lines: {2, 525}, {324, 850}, {394, 3265}, {459, 2501}, {520, 3917}, {523, 1853}, {1636, 14417}, {1650, 5489}, {2525, 17434}, {2799, 14391}, {4143, 4176}, {6368, 11197}, {9007, 15533}

X(23616) = isotomic conjugate of polar conjugate of X(5489)

X(23617) =  CEVAPOINT OF X(6) AND X(9)

Barycentrics    a/((b-c)^2+a(b+c)) : :

Let P = X(23617). Let Pa be the isotomic conjugate of P with respect to AX(6)X(9), and define Pb and Pc cyclically. Then ABC and PaPbPc are perspective in X(13622). (Angel Montesdeoca, September 23, 2018)

X(23617) lies on these lines: {2,1407}, {6,145}, {9,604}, {31,200}, {37,9456}, {44,1333}, {81,4358}, {100,2347}, {281,608}, {391,2255}, {651,17353}, {739,8706}, {894,1462}, {1280,14523}, {1376,9309}, {2162,5276}, {2203,4183}, {2221,17740}, {2330,14100}, {3306,5575}, {3758,20946}, {4691,21267}, {5437,16666}, {5750,7110}, {16667,21896}, {22166,22266}

X(23617) = isogonal conjugate of X(3752)
X(23617) = trilinear pole of line X(667)X(3900) (the polar of X(9) wrt circumcircle)

X(23618) =  CEVAPOINT OF X(7) AND X(9)

Barycentrics    1/((b+c-a)(a^2(b+c)-2a(b-c)^2+(b-c)^2(b+c))) : :

Let P = X(23618). Let Pa be the isotomic conjugate of P with respect to AX(7)X(9), and define Pb and Pc cyclically. Then ABC and PaPbPc are perspective in X(3255). (Angel Montesdeoca, September 23, 2018)

X(23618) lies on these lines: {7,480}, {9,23062}, {85,728}, {144,220}, {527,10509}, {664,14100}, {1847,7079}, {3062,9312}, {3663,9440}, {4691,21267}, {22166,22266}

X(23618) = isogonal conjugate of X(1200)
X(23618) = trilinear pole of line X(3676)X(4105)

X(23619) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^2 b^2 - b^4 - a^2 b c + b^3 c + a^2 c^2 + b c^3 - c^4) : :

X(23619) lies on these lines: {6, 9247}, {10, 14963}, {19, 1745}, {39, 23438}, {42, 23620}, {51, 23362}, {73, 2333}, {92, 18027}, {213, 20982}, {581, 2304}, {993, 22447}, {1475, 20974}, {1953, 12047}, {2170, 21935}, {2179, 8608}, {2200, 2594}, {3778, 23626}, {3954, 20684}, {4020, 8679}, {4456, 22061}, {16980, 22070}, {17181, 18161}, {20863, 23631}, {20964, 23641}, {23436, 23635}

X(23619) = isogonal conjugate of isotomic conjugate of X(20305)
X(23619) = polar conjugate of isotomic conjugate of X(22069)

X(23620) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^2 - b^2 - c^2) (a^2 + b^2 - 2 b c + c^2) : :

X(23620) lies on these lines: {6, 1245}, {19, 4295}, {39, 20967}, {42, 23619}, {48, 4303}, {63, 3926}, {65, 2333}, {71, 72}, {73, 2200}, {125, 6537}, {169, 12609}, {184, 1475}, {213, 1042}, {218, 2183}, {573, 12520}, {610, 1044}, {758, 4456}, {1334, 3954}, {1473, 7124}, {1851, 2082}, {1899, 3691}, {2245, 21874}, {2260, 16466}, {2304, 4300}, {2503, 21951}, {3010, 20667}, {3269, 9560}, {3611, 20683}, {3682, 20727}, {3784, 22127}, {3827, 17442}, {6210, 6776}, {6467, 20963}, {7083, 16502}, {7117, 22344}, {7193, 23150}, {7289, 17170}, {11573, 22126}, {18671, 18732}, {20752, 23154}, {20812, 22350}, {22070, 22345}, {22097, 23151}, {23622, 23637}, {23623, 23636}, {23626, 23648}

X(23620) = isogonal conjugate of isotomic conjugate of X(18589)
X(23620) = polar conjugate of isotomic conjugate of X(22057)

X(23621) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^4 - a^2 b^2 - a^2 b c + b^3 c - a^2 c^2 - 2 b^2 c^2 + b c^3) : :

X(23621) lies on these lines: {6, 2179}, {39, 23622}, {40, 2304}, {41, 4642}, {42, 23619}, {46, 48}, {65, 2200}, {101, 3754}, {758, 22061}, {1973, 2266}, {4251, 4868}, {13366, 23413}, {18162, 21231}, {23629, 23648}

X(23621) = isogonal conjugate of isotomic conjugate of X(21231)
X(23621) = polar conjugate of isotomic conjugate of X(22394)

X(23622) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b - a^2 b^2 + a^3 c - 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(23622) lies on these lines: {6, 9432}, {31, 23565}, {39, 23621}, {41, 9593}, {42, 20455}, {101, 1054}, {213, 6377}, {890, 9299}, {1155, 9454}, {1403, 5364}, {1571, 2304}, {3306, 9310}, {3310, 21742}, {6139, 20958}, {14936, 20662}, {18723, 21232}, {20460, 22344}, {20976, 20981}, {23620, 23637}

X(23622) = isogonal conjugate of isotomic conjugate of X(21232)
X(23622) = polar conjugate of isotomic conjugate of X(22399)

X(23623) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b - c) (a^3 b + a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c - b^3 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - b c^3) : :

X(23623) lies on these lines: {1, 41}, {39, 23621}, {42, 23637}, {1055, 10475}, {1402, 1475}, {1572, 2304}, {2268, 5250}, {2300, 21743}, {18724, 21233}, {20959, 23419}, {20963, 20967}, {23620, 23636}

X(23623) = isogonal conjugate of isotomic conjugate of X(21233)
X(23623) = polar conjugate of isotomic conjugate of X(22400)

X(23624) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^4 b^2 - b^6 - a^4 b c + b^5 c + a^4 c^2 - b^4 c^2 + 2 b^3 c^3 - b^2 c^4 + b c^5 - c^6) : :

X(23624) lies on these lines: {6, 17453}, {39, 23197}, {42, 23619}, {213, 21749}, {306, 14963}, {1843, 23421}, {15523, 20727}, {16607, 18727}, {20982, 21750}

X(23624) = isogonal conjugate of isotomic conjugate of X(16607)
X(23624) = polar conjugate of isotomic conjugate of X(22402)

X(23625) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^4 b^2 - b^6 - a^4 b c + b^5 c + a^4 c^2 - a^2 b^2 c^2 - b^4 c^2 + 2 b^3 c^3 - b^2 c^4 + b c^5 - c^6) : :

X(23625) lies on these lines: {42, 23619}, {2393, 23422}, {4062, 14963}, {8635, 20981}, {18728, 21234}

X(23625) = isogonal conjugate of isotomic conjugate of X(21234)
X(23625) = polar conjugate of isotomic conjugate of X(22403)

X(23626) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (b^4 - b^3 c + b^2 c^2 - b c^3 + c^4) : :

X(23626) lies on these lines: {6, 1917}, {42, 20969}, {1923, 9018}, {3778, 23619}, {4118, 17211}, {4456, 8625}, {18168, 21235}, {20727, 23664}, {20859, 23424}, {20982, 23652}, {23414, 23646}, {23620, 23648}

X(23626) = isogonal conjugate of isotomic conjugate of X(21235)
X(23626) = polar conjugate of isotomic conjugate of X(22404)

X(23627) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^3 - b^5 + a b^3 c + b^3 c^2 + a^2 c^3 + a b c^3 + b^2 c^3 - c^5) : :

X(23627) lies on these lines: {39, 23438}, {1107, 14963}, {1193, 20982}, {1573, 20727}, {11263, 17443}, {20961, 23425}, {20963, 20974}, {23632, 23639}

X(23627) = isogonal conjugate of isotomic conjugate of X(21236)
X(23627) = polar conjugate of isotomic conjugate of X(22405)

X(23628) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^3 - b^5 - a b^3 c + b^3 c^2 + a^2 c^3 - a b c^3 + b^2 c^3 - c^5) : :

X(23628) lies on these lines: {6, 1324}, {39, 23438}, {672, 5164}, {1574, 20727}, {1575, 14963}, {8637, 20861}, {11813, 17444}, {20863, 23646}, {20962, 23426}

X(23628) = isogonal conjugate of isotomic conjugate of X(21237)
X(23628) = polar conjugate of isotomic conjugate of X(22406)

X(23629) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^2 b^2 - a^2 b c + a^2 c^2 + b^2 c^2) : :

X(23629) lies on these lines: {1, 668}, {6, 1923}, {8, 4116}, {39, 20457}, {42, 213}, {274, 3510}, {386, 16476}, {519, 4161}, {718, 1237}, {904, 5291}, {1207, 22409}, {1911, 5277}, {1964, 3216}, {2230, 20888}, {2309, 20669}, {3678, 4093}, {3754, 4128}, {3778, 23619}, {5312, 16468}, {6196, 17143}, {8625, 22061}, {11490, 23538}, {18170, 21238}, {20464, 20963}, {20965, 23429}, {20969, 20982}, {23621, 23648}

X(23629) = isogonal conjugate of isotomic conjugate of X(21238)
X(23629) = polar conjugate of isotomic conjugate of X(22409)

X(23630) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 + a^2 b^3 - a b^4 - b^5 - a^2 b^2 c + 2 a b^3 c - b^4 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - a c^4 - b c^4 - c^5) : :

X(23630) lies on these lines: {6, 22654}, {39, 23438}, {51, 1475}, {181, 5021}, {511, 21384}, {672, 16980}, {1212, 8679}, {1401, 16583}, {2262, 4292}, {3125, 17114}, {3271, 16502}, {3691, 3917}, {4020, 16588}, {6467, 20963}, {17435, 17442}, {17451, 23154}, {20962, 23649}

X(23630) = isogonal conjugate of isotomic conjugate of X(21239)
X(23630) = polar conjugate of isotomic conjugate of X(22410)

X(23631) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b^4 - b^5 + b^4 c + a c^4 + b c^4 - c^5) : :

X(23631) lies on these lines: {6, 9448}, {39, 20860}, {3764, 20975}, {3778, 23635}, {4022, 20819}, {8628, 21744}, {20859, 23424}, {20863, 23619}, {20866, 23437}, {23431, 23636}

X(23631) = isogonal conjugate of isotomic conjugate of X(17047)
X(23631) = polar conjugate of isotomic conjugate of X(22411)

X(23632) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b^3 + b^3 c + a c^3 + b c^3) : :

X(23632) lies on these lines: {2, 330}, {6, 2205}, {37, 17165}, {38, 8620}, {39, 42}, {75, 17486}, {192, 23488}, {292, 3920}, {518, 21814}, {672, 2300}, {716, 20889}, {726, 21327}, {749, 1100}, {893, 3218}, {1011, 16502}, {1015, 3720}, {1185, 2274}, {1193, 21753}, {1197, 17187}, {1206, 3736}, {1575, 4651}, {1914, 4184}, {2229, 3741}, {2309, 23432}, {3006, 18905}, {3121, 3666}, {3778, 8629}, {3989, 21827}, {3995, 20363}, {4358, 21902}, {4392, 20284}, {15569, 21820}, {16696, 16707}, {16713, 16721}, {17017, 23543}, {17135, 17448}, {17147, 21345}, {17184, 18904}, {17208, 18171}, {17756, 20012}, {20011, 20691}, {20457, 20966}, {20459, 20965}, {23574, 23658}, {23627, 23639}

X(23632) = isogonal conjugate of isotomic conjugate of X(21240)
X(23632) = polar conjugate of isotomic conjugate of X(22412)

X(23633) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b^2 - 2 b^3 + b^2 c + a c^2 + b c^2 - 2 c^3) : :

X(23633) lies on these lines: {1, 4787}, {6, 9459}, {39, 2309}, {42, 2183}, {320, 17449}, {669, 2451}, {674, 3009}, {902, 2245}, {1149, 3792}, {1201, 4259}, {1400, 22426}, {2223, 20984}, {2308, 4749}, {3123, 20358}, {3271, 20456}, {3688, 22172}, {4443, 21352}, {5165, 21747}, {5168, 22356}, {17157, 20081}, {17792, 21330}, {18173, 21241}, {20865, 20975}, {20962, 23644}, {22448, 23443}

X(23633) = isogonal conjugate of isotomic conjugate of X(21241)
X(23633) = polar conjugate of isotomic conjugate of X(22414)

X(23634) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a b^2 - b^3 + 2 b^2 c + 2 a c^2 + 2 b c^2 - c^3) : :

X(23634) lies on these lines: {39, 2309}, {42, 3271}, {902, 4274}, {1964, 8610}, {2177, 4266}, {2276, 4787}, {2278, 8626}, {2293, 20964}, {3056, 13330}, {3707, 21805}, {4161, 7032}, {4277, 23659}, {6007, 21352}, {13410, 20863}, {18174, 21242}

X(23634) = isogonal conjugate of isotomic conjugate of X(21242)
X(23634) = polar conjugate of isotomic conjugate of X(22415)

X(23635) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^4 - b^6 + b^4 c^2 + a^2 c^4 + b^2 c^4 - c^6) : :

X(23635) lies on these lines: {2, 3186}, {3, 6403}, {4, 3164}, {6, 157}, {25, 22240}, {39, 6467}, {51, 800}, {141, 3001}, {160, 566}, {193, 3095}, {216, 237}, {262, 9307}, {264, 18024}, {381, 2971}, {427, 13409}, {523, 3613}, {570, 2393}, {577, 8541}, {1974, 5158}, {2269, 22169}, {2967, 9308}, {2972, 5094}, {3003, 9969}, {3313, 22062}, {3778, 23631}, {6389, 14003}, {6391, 10983}, {6660, 19121}, {7716, 20897}, {7800, 22424}, {8266, 9019}, {9155, 11188}, {11574, 14096}, {12272, 20794}, {13342, 15004}, {13481, 18575}, {15851, 19118}, {18114, 19130}, {18175, 21243}, {21746, 23646}, {23436, 23619}

X(23635) = isogonal conjugate of isotomic conjugate of X(21243)
X(23635) = polar conjugate of isotomic conjugate of X(22416)

X(23636) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b^3 - b^4 + b^3 c + a c^3 + b c^3 - c^4) : :

X(23636) lies on these lines: {2, 17082}, {6, 1626}, {7, 21218}, {38, 20684}, {39, 23438}, {42, 20455}, {51, 20459}, {1015, 22200}, {1107, 20727}, {1201, 21813}, {1401, 16588}, {1469, 5364}, {2170, 3914}, {2275, 3981}, {2309, 6467}, {3119, 20205}, {3778, 8629}, {3782, 20593}, {4386, 20729}, {5249, 17451}, {5322, 19554}, {16721, 22064}, {17046, 17177}, {20462, 20866}, {20861, 22199}, {21746, 23437}, {22197, 23536}, {23362, 23660}, {23431, 23631}, {23620, 23623}

X(23636) = isogonal conjugate of isotomic conjugate of X(17046)
X(23636) = polar conjugate of isotomic conjugate of X(22064)

X(23637) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b - c) (a b^3 + b^4 - b^3 c + a c^3 - b c^3 + c^4) : :

X(23637) lies on these lines: {6, 2933}, {10, 2170}, {39, 23438}, {41, 386}, {42, 23623}, {43, 2082}, {181, 1475}, {213, 9561}, {218, 9567}, {672, 970}, {1193, 9547}, {1334, 1682}, {1575, 20727}, {2092, 2347}, {3061, 4167}, {3687, 20684}, {3691, 9564}, {3778, 23631}, {4426, 22447}, {5530, 17451}, {6735, 20594}, {7117, 20460}, {9560, 20982}, {18177, 21244}, {20861, 20864}, {20862, 23414}, {20974, 23649}, {23620, 23622}

X(23637) = isogonal conjugate of isotomic conjugate of X(21244)
X(23637) = polar conjugate of isotomic conjugate of X(22418)

X(23638) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b - c) (a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3) : :

X(23638) lies on these lines: {1, 5943}, {6, 181}, {37, 375}, {39, 23438}, {41, 6186}, {42, 51}, {43, 511}, {44, 22276}, {55, 2316}, {65, 14557}, {78, 10544}, {184, 20958}, {200, 3056}, {209, 2264}, {210, 3686}, {333, 9564}, {373, 3720}, {386, 15654}, {674, 4849}, {756, 1573}, {899, 3917}, {958, 1682}, {970, 5247}, {982, 2810}, {1193, 16980}, {1196, 21760}, {1197, 5052}, {1220, 9565}, {1284, 21361}, {1329, 17182}, {1364, 11502}, {1376, 3030}, {1401, 3752}, {1402, 2183}, {1469, 2999}, {2082, 20683}, {2175, 10833}, {2551, 10480}, {2985, 8707}, {3057, 5795}, {3060, 3240}, {3452, 21334}, {3715, 7064}, {3819, 16569}, {4270, 15494}, {4516, 7069}, {4868, 15049}, {5432, 18191}, {5640, 17018}, {5752, 10822}, {6745, 20359}, {7074, 7083}, {9026, 21342}, {9309, 9778}, {14936, 20665}, {14973, 17362}, {17053, 21936}, {17114, 23154}, {20456, 20460}, {20457, 22199}, {20459, 23653}, {20859, 20860}

X(23638) = crosssum of X(2) and X(56)
X(23638) = crosspoint of X(6) and X(8)
X(23638) = isogonal conjugate of isotomic conjugate of X(1329)
X(23638) = polar conjugate of isotomic conjugate of X(22071)

X(23639) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a b^3 + b^4 + a c^3 + c^4) : :

X(23639) lies on these lines: {6, 2915}, {39, 51}, {213, 4456}, {325, 18167}, {442, 3125}, {712, 1228}, {1211, 3954}, {1230, 22036}, {2092, 2183}, {2275, 14963}, {3124, 16589}, {3230, 22076}, {3454, 3721}, {3735, 5051}, {3778, 23619}, {3981, 5283}, {17053, 22073}, {17211, 18179}, {20467, 20982}, {20862, 23646}, {23444, 23643}, {23627, 23632}

X(23639) = isogonal conjugate of isotomic conjugate of X(21245)
X(23639) = polar conjugate of isotomic conjugate of X(22420)

X(23640) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b - c) (a^2 b^2 + a b^3 + b^3 c + a^2 c^2 - 2 b^2 c^2 + a c^3 + b c^3) : :

X(23640) lies on these lines: {6, 41}, {9, 19582}, {21, 2269}, {39, 23438}, {63, 194}, {333, 3691}, {672, 5247}, {958, 1334}, {1107, 2170}, {1195, 4269}, {1817, 17209}, {2238, 22065}, {2280, 23443}, {3721, 20785}, {3778, 6467}, {4020, 16583}, {4251, 23531}, {7117, 23447}, {17696, 22370}, {20663, 23473}, {20864, 23660}, {20963, 20967}

X(23640) = isogonal conjugate of isotomic conjugate of X(21246)
X(23640) = polar conjugate of isotomic conjugate of X(22421)

X(23641) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^4 - b^4 - c^4) (a^4 + b^4 - 2 b^3 c + 2 b^2 c^2 - 2 b c^3 + c^4) : :

X(23641) lies on these lines: {1400, 21749}, {20964, 23619}, {20968, 23442}, {21744, 21751}

X(23641) = isogonal conjugate of isotomic conjugate of X(21247)
X(23641) = polar conjugate of isotomic conjugate of X(22422)

X(23642) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b^2 + c^2) (a^2 b^2 + b^4 + a^2 c^2 + c^4) : :

Let La be the radical axis of the circumcircle and reflected A-Neuberg circle, and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. A'B'C' is homothetic to ABC at X(251); also, X(23642) = X(6)-of-A'B'C'. (Randy Hutson, October 15, 2018)

X(23642) lies on these lines: {2, 8788}, {6, 22}, {39, 1843}, {69, 194}, {141, 6665}, {311, 5254}, {511, 14133}, {826, 13232}, {1194, 11574}, {3051, 3313}, {3117, 20819}, {3124, 3589}, {3618, 3981}, {5013, 6461}, {8623, 22078}, {9969, 20965}, {12055, 14972}, {17475, 23548}, {18182, 21248}, {20861, 20963}

X(23642) = isogonal conjugate of isotomic conjugate of X(21248)
X(23642) = polar conjugate of isotomic conjugate of X(22424)

X(23643) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b + a c - b c) (a b^3 - a b^2 c - b^3 c - a b c^2 + a c^3 - b c^3) : :

X(23643) lies on these lines: {6, 20467}, {9, 43}, {39, 20667}, {1189, 23538}, {1500, 4704}, {2228, 6375}, {3778, 20462}, {17350, 20671}, {20861, 20864}, {20862, 21746}, {20868, 23433}, {23444, 23639} X(23643) = isogonal conjugate of isotomic conjugate of X(21250)
X(23643) = polar conjugate of isotomic conjugate of X(22427)

X(23644) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a - b - c) (a b^2 + b^3 - 2 b^2 c + a c^2 - 2 b c^2 + c^3) : :

X(23644) lies on these lines: {39, 20974}, {42, 3271}, {902, 22371}, {1017, 3124}, {1797, 10761}, {3778, 23645}, {4120, 4895}, {4152, 4969}, {8650, 20455}, {20671, 23552}, {20962, 23633}, {20966, 20975}

X(23644) = isogonal conjugate of isotomic conjugate of X(121)
X(23644) = crosssum of X(2) and X(106)
X(23644) = crosspoint of X(6) and X(519)
X(23644) = polar conjugate of isotomic conjugate of X(22428)

X(23645) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - 2 b - 2 c) (2 a b^2 + 2 b^3 - b^2 c + 2 a c^2 - b c^2 + 2 c^3) : :

X(23645) lies on these lines: {39, 20962}, {42, 2183}, {3778, 23644}, {4274, 21747}, {20671, 23553}, {20970, 21754}

X(23645) = isogonal conjugate of isotomic conjugate of X(21251)
X(23645) = polar conjugate of isotomic conjugate of X(22429)

X(23646) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c)^2 (a b^2 - b^3 + a b c - b^2 c + a c^2 - b c^2 - c^3) : :

X(23646) lies on these lines: {6, 23402}, {39, 20860}, {116, 1491}, {667, 7117}, {1146, 4705}, {1565, 2530}, {2876, 13006}, {3022, 4775}, {3271, 20975}, {8638, 20974}, {8678, 11998}, {18181, 21252}, {20862, 23639}, {20863, 23628}, {21746, 23635}, {23414, 23626}

X(23646) = isogonal conjugate of isotomic conjugate of X(21252)
X(23646) = polar conjugate of isotomic conjugate of X(22432)

X(23647) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c)^2 (b + c) (a^2 b^2 - b^4 + a^2 b c - b^3 c + a^2 c^2 - b^2 c^2 - b c^3 - c^4) : :

X(23647) lies on these lines: {125, 21339}, {649, 22094}, {661, 7004}, {3124, 6377}, {20860, 20966}, {20974, 20975}

X(23647) = isogonal conjugate of isotomic conjugate of X(21253)
X(23647) = polar conjugate of isotomic conjugate of X(22433)

X(23648) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^4 - 2 a^2 b c + b^3 c - b^2 c^2 + b c^3) : :

X(23648) lies on these lines: {42, 20982}, {101, 4128}, {560, 4736}, {662, 5539}, {668, 4154}, {758, 8625}, {1015, 5147}, {2643, 5540}, {20463, 20668}, {20976, 23454}, {23620, 23626}, {23621, 23629}

X(23648) = isogonal conjugate of isotomic conjugate of X(21254)
X(23648) = polar conjugate of isotomic conjugate of X(22434)

X(23649) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a b - 3 b^2 + a c + 2 b c - 3 c^2) : :

X(23649) lies on these lines: {6, 5204}, {31, 5022}, {39, 42}, {55, 23455}, {244, 1212}, {672, 1201}, {899, 21384}, {902, 16502}, {1015, 1334}, {1149, 3730}, {1193, 4253}, {1468, 9605}, {2260, 5069}, {2276, 17474}, {2280, 5013}, {2308, 5021}, {3778, 20978}, {4189, 16779}, {5030, 5299}, {10459, 17754}, {17448, 20331}, {18186, 21255}, {20459, 22343}, {20863, 23416}, {20962, 23630}, {20974, 23637}

X(23649) = isogonal conjugate of isotomic conjugate of X(21255)
X(23649) = polar conjugate of isotomic conjugate of X(22435)

X(23650) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^2 - 2 a b - 2 a c + 3 b c) : :

X(23650) lies on these lines: {1, 14408}, {6, 3768}, {649, 6085}, {659, 23569}, {661, 3738}, {663, 9032}, {667, 6373}, {888, 3747}, {890, 9299}, {926, 4502}, {1635, 2316}, {1654, 21261}, {1960, 23458}, {3271, 8645}, {3287, 3808}, {4775, 9461}, {21143, 22108}

X(23650) = isogonal conjugate of isotomic conjugate of X(4928)
X(23650) = polar conjugate of isotomic conjugate of X(22437)

X(23651) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^2 b^2 - 2 b^4 - a^2 b c + 2 b^3 c + a^2 c^2 - b^2 c^2 + 2 b c^3 - 2 c^4) : :

X(23651) lies on these lines: {42, 21745}, {3778, 23619}, {8630, 20868}, {20977, 23459}

X(23651) = isogonal conjugate of isotomic conjugate of X(21256)
X(23651) = polar conjugate of isotomic conjugate of X(22438)

X(23652) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b + c) (a^2 b^2 - 2 a^2 b c + a^2 c^2 + b^2 c^2) : :

X(23652) lies on these lines: {1, 20464}, {2, 23460}, {10, 18793}, {31, 11490}, {39, 20667}, {42, 213}, {51, 20456}, {76, 23485}, {239, 6196}, {386, 2309}, {904, 5247}, {978, 7032}, {1015, 23457}, {1193, 16476}, {1921, 23508}, {2275, 23427}, {3248, 16604}, {3271, 23414}, {3510, 16827}, {3778, 22381}, {3780, 4161}, {4128, 21951}, {14823, 18194}, {20982, 23626}, {21101, 22185}, {22439, 23538}

X(23652) = isogonal conjugate of isotomic conjugate of X(21257)
X(23652) = polar conjugate of isotomic conjugate of X(22439)

X(23653) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3 + c^4) : :

X(23653) lies on these lines: {1, 21795}, {6, 1174}, {39, 42}, {51, 20974}, {57, 14936}, {354, 16588}, {614, 1015}, {800, 2260}, {1202, 1458}, {2275, 2999}, {3870, 6184}, {16721, 18164}, {17474, 22070}, {20459, 23638}, {20978, 23461}, {21746, 23437}

X(23653) = isogonal conjugate of isotomic conjugate of X(21258)
X(23653) = polar conjugate of isotomic conjugate of X(22440)

X(23654) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^4 b^2 - a^2 b^4 + a^4 b c - a^2 b^3 c + a^4 c^2 - a^2 b^2 c^2 + b^4 c^2 - a^2 b c^3 + 2 b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(23654) lies on these lines: {1946, 21761}, {2451, 23463}, {3736, 4369}, {8630, 20868}, {8646, 20981}

X(23654) = isogonal conjugate of isotomic conjugate of X(21259)
X(23654) = polar conjugate of isotomic conjugate of X(22441)

X(23655) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(23655) lies on these lines: {1, 3835}, {31, 23568}, {42, 649}, {521, 2254}, {652, 5075}, {661, 663}, {665, 4524}, {669, 2451}, {693, 3907}, {926, 7180}, {991, 15599}, {1459, 1491}, {1919, 21005}, {1980, 23472}, {3510, 9294}, {3736, 18200}, {4367, 22090}, {6373, 8640}, {6589, 9029}, {8630, 23572}, {8641, 20980}, {8643, 14404}, {8646, 20981}, {17018, 20295}, {17072, 21300}, {21760, 23575}

X(23655) = isogonal conjugate of isotomic conjugate of X(17072)
X(23655) = polar conjugate of isotomic conjugate of X(22443)

X(23656) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (-a b^3 + a^2 b c - a b^2 c + b^3 c - a b c^2 + b^2 c^2 - a c^3 + b c^3) : :

X(23656) lies on these lines: {661, 3716}, {667, 23657}, {669, 2451}, {672, 23569}, {830, 2084}, {1475, 23458}, {2254, 3572}, {3271, 8645}, {4502, 4526}, {6373, 23466}, {8635, 20981}, {20295, 21834}, {20980, 23469}

X(23656) = isogonal conjugate of isotomic conjugate of X(21261)
X(23656) = polar conjugate of isotomic conjugate of X(22444)

X(23657) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (-b + c) (a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(23657) lies on these lines: {39, 23572}, {649, 14991}, {661, 784}, {667, 23656}, {1491, 2084}, {3221, 9494}, {3835, 21836}, {8630, 20868}, {8637, 20861}, {17458, 21099}

X(23657) = isogonal conjugate of isotomic conjugate of X(21262)
X(23657) = polar conjugate of isotomic conjugate of X(22445)

X(23658) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (-b + c) (a^2 b^4 + a^2 b^3 c + a^2 b^2 c^2 - b^4 c^2 + a^2 b c^3 - b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(23658) lies on these lines: {6, 23403}, {661, 18155}, {3221, 23471}, {8630, 20868}, {8635, 20981}, {23574, 23632}

X(23658) = isogonal conjugate of isotomic conjugate of X(21263)
X(23658) = polar conjugate of isotomic conjugate of X(22446)

X(23659) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b^3 + 2 a b c + c^3) : :

X(23659) lies on these lines: {1, 22172}, {6, 560}, {31, 573}, {38, 4416}, {39, 20667}, {42, 51}, {44, 21035}, {75, 751}, {239, 256}, {244, 3664}, {291, 17120}, {386, 23414}, {511, 1193}, {516, 4642}, {519, 22167}, {524, 4022}, {674, 872}, {869, 3056}, {982, 17364}, {984, 17331}, {1015, 23532}, {1100, 3122}, {1125, 22174}, {1400, 20985}, {1918, 4271}, {2092, 2309}, {2183, 20964}, {2209, 4266}, {2228, 3589}, {2260, 20984}, {2269, 3747}, {2275, 23524}, {2277, 7032}, {2308, 20966}, {2310, 23668}, {2670, 3124}, {3009, 21796}, {3123, 3946}, {3244, 22214}, {3670, 17770}, {3686, 3728}, {3745, 21936}, {3758, 4446}, {3759, 4443}, {3779, 4787}, {3846, 17202}, {3879, 21330}, {4277, 23634}, {4283, 16477}, {4735, 16669}, {6744, 22219}, {8300, 19572}, {12782, 17350}, {15624, 20973}, {17065, 17379}, {20970, 23444}

X(23659) = isogonal conjugate of isotomic conjugate of X(3846)
X(23659) = polar conjugate of isotomic conjugate of X(22447)

X(23660) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^2 b^2 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2) : :

X(23660) lies on these lines: {1, 6}, {2, 1197}, {31, 23417}, {39, 2309}, {42, 20457}, {81, 799}, {83, 21759}, {87, 6196}, {171, 1979}, {748, 1185}, {871, 14621}, {1194, 22200}, {1475, 14758}, {1914, 4279}, {1966, 20179}, {2209, 2241}, {2275, 5145}, {3051, 3720}, {3403, 20172}, {3993, 17475}, {5135, 14599}, {13410, 20962}, {16604, 18792}, {17122, 21792}, {17123, 21779}, {17142, 22036}, {17474, 23579}, {18046, 18148}, {20859, 20961}, {20864, 23640}, {20985, 23475}, {23362, 23636}

X(23660) = isogonal conjugate of isotomic conjugate of X(21264)
X(23660) = polar conjugate of isotomic conjugate of X(22449)

X(23661) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (-a + b + c) (-2 a^4 - a^3 b + a^2 b^2 + a b^3 + b^4 - a^3 c - 2 a^2 b c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(23661) lies on these lines: {1, 17860}, {2, 280}, {7, 8}, {10, 1074}, {20, 92}, {21, 243}, {29, 3100}, {40, 20223}, {63, 1715}, {78, 321}, {345, 3701}, {442, 2968}, {443, 7046}, {950, 4858}, {1043, 3615}, {1062, 5136}, {1089, 6745}, {1099, 1125}, {1214, 20222}, {1829, 15971}, {2475, 5081}, {3146, 5342}, {4202, 23662}, {4359, 6734}, {4385, 7080}, {4647, 6737}, {4692, 6736}, {4696, 6735}, {5174, 6839}, {5731, 20220}, {17863, 18690}, {17872, 23529}, {17875, 17887}, {23537, 23542}

X(23662) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b + c) (-a^4 + b^4 + c^4) (-a^6 - a^4 b^2 + a^2 b^4 + b^6 + 2 a^4 b c - 2 a^2 b^3 c - a^4 c^2 + 2 a^2 b^2 c^2 - b^4 c^2 - 2 a^2 b c^3 + a^2 c^4 - b^2 c^4 + c^6) : :

X(23662) lies on these lines: {1441, 4972}, {2172, 2908}, {4202, 23661}

X(23663) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b + c) (a^4 + 2 a^2 b^2 + b^4 - 4 a^2 b c + 2 a^2 c^2 - 2 b^2 c^2 + c^4) : :

X(23663) lies on these lines: {10, 3728}, {1441, 3914}, {2209, 10445}, {3747, 8804}, {5254, 11997}, {17872, 23529}

X(23664) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^2 (b + c) (a^2 b^2 + b^4 - a^2 b c - b^3 c + a^2 c^2 - b c^3 + c^4) : :

X(23664) lies on these lines: {10, 3728}, {1441, 23666}, {3122, 16583}, {3747, 4456}, {3914, 17867}, {4516, 22178}, {16600, 22172}, {17869, 23672}, {17872, 23537}, {20727, 23626}, {23536, 23686}

X(23665) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b^2 + c^2) (-2 a^4 - a^2 b^2 + b^4 - a^2 c^2 - 2 b^2 c^2 + c^4) : :

X(23665) lies on these lines: {1, 17865}, {31, 75}, {38, 20883}, {1109, 17446}, {2618, 8061}, {14213, 17872}, {17445, 20902}, {17859, 17875}, {17863, 17873}, {17868, 17876}, {23537, 23666}

X(23666) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b + c) (b^2 + c^2) (-a^4 - a^2 b^2 + a^2 b c + b^3 c - a^2 c^2 - 2 b^2 c^2 + b c^3) : :

X(23666) lies on these lines: {10, 20234}, {1441, 23664}, {3778, 21423}, {3914, 17864}, {17866, 23682}, {17867, 23672}, {20235, 22069}, {23537, 23665}

X(23667) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a (a - b - c) (a^3 b^2 - a^2 b^3 + a b^4 - b^5 + a^2 b^2 c + b^4 c + a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 - a^2 c^3 + a c^4 + b c^4 - c^5) : :

X(23667) lies on these lines: {8, 192}, {607, 4336}, {2082, 2310}, {17872, 23529}

X(23668) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a (b+c) (a^2 b+b^3+a^2 c+2 a b c-b^2 c-b c^2+c^3) : :

X(23668) lies on these lines: {1, 1958}, {8, 192}, {37, 4433}, {38, 3875}, {42, 1824}, {244, 3946}, {284, 8772}, {523, 20504}, {612, 1962}, {756, 2321}, {869, 17452}, {872, 21801}, {1045, 1959}, {1253, 3747}, {1419, 2263}, {1441, 3914}, {1500, 21804}, {1834, 7235}, {1999, 17797}, {2092, 4516}, {2170, 2309}, {2294, 2643}, {2310, 23659}, {2784, 12725}, {3125, 4890}, {3263, 4967}, {3755, 3778}, {3930, 7237}, {4085, 4695}, {4111, 21810}, {4360, 17446}, {4424, 4780}, {17863, 23676}, {17876, 18692}, {23528, 23529}

X(23669) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (a b + a c - b c) (b^3 + a b c - b^2 c - b c^2 + c^3) : :

X(23669) lies on these lines: {8, 726}, {75, 18207}, {523, 4446}, {1441, 17889}, {3596, 6382}, {4362, 17797}, {4398, 21142}, {4941, 21138}, {5018, 21147}, {6377, 17053}, {7235, 10251}, {21085, 21095}, {23528, 23690}

X(23670) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (-2 a + b + c) (-2 a^3 + a^2 b + 3 b^3 + a^2 c - 3 b^2 c - 3 b c^2 + 3 c^3) : :

X(23670) lies on these lines: {1, 17888}, {1109, 3626}, {1441, 23671}, {3914, 23521}, {17874, 17876}

X(23671) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (-a + 2 b + 2 c) (-4 a^3 - a^2 b + 3 b^3 - a^2 c - 3 b^2 c - 3 b c^2 + 3 c^3) : :

X(23671) lies on these lines: {1441, 23670}, {3625, 4647}, {3914, 17895}

X(23672) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b - c)^2 (b + c) (-a^4 - a^2 b^2 - 3 a^2 b c + b^3 c - a^2 c^2 + 2 b^2 c^2 + b c^3) : :

X(23672) lies on these lines: {3914, 23674}, {17205, 17886}, {17864, 23680}, {17867, 23666}, {17869, 23664}, {17876, 17877}, {17888, 23686}

X(23673) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b - c)^2 (b + c) (a^4 - 2 a^2 b^2 + b^4 - 3 a^2 b c + b^3 c - 2 a^2 c^2 + b c^3 + c^4) : :

X(23673) lies on these lines: {75, 4563}, {523, 2969}, {17876, 17878}, {17877, 17881}, {17886, 21430}

X(23674) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b + c) (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^2 b^3 c - b^5 c - a^4 c^2 + a^2 b^2 c^2 + a^2 b c^3 - a^2 c^4 - b c^5 + c^6) : :

X(23674) lies on these lines: {2, 5546}, {10, 17864}, {75, 339}, {101, 21253}, {110, 21294}, {125, 150}, {664, 6739}, {1211, 5723}, {1654, 3015}, {3914, 23672}, {5224, 14616}, {14208, 18689}, {17867, 23518}, {20522, 21431}

X(23675) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c - 6 a^2 b c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4 : :

X(23675) lies on these lines: {1, 224}, {2, 9369}, {10, 244}, {31, 4298}, {56, 3011}, {142, 10459}, {145, 1738}, {225, 1319}, {226, 1201}, {388, 614}, {748, 12527}, {908, 21214}, {946, 1149}, {995, 13407}, {1072, 1385}, {1086, 3057}, {1104, 5434}, {1125, 5192}, {1191, 10404}, {1193, 21620}, {1279, 7354}, {1616, 1836}, {1739, 10915}, {1834, 17609}, {2650, 5542}, {3120, 12053}, {3304, 3772}, {3315, 5086}, {3333, 5230}, {3436, 5272}, {3445, 11376}, {3616, 13161}, {3749, 4190}, {3752, 15888}, {3756, 17606}, {3915, 4292}, {3924, 10106}, {3976, 6734}, {4322, 5930}, {4694, 10916}, {4853, 4859}, {5121, 11681}, {5252, 17054}, {5484, 16823}, {5552, 11512}, {5573, 9578}, {5794, 17597}, {7292, 20060}, {10529, 17064}, {10587, 17594}, {11019, 21935}, {12607, 16610}, {12701, 16486}, {16602, 21031}, {17125, 18250}, {17888, 23528}, {21342, 21677}

X(23676) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b - c)^2 (-a^3 - a b^2 - 2 a b c + 2 b^2 c - a c^2 + 2 b c^2) : :

X(23676) lies on these lines: {244, 1109}, {1086, 21045}, {1441, 23677}, {3914, 23521}, {3942, 21142}, {4459, 21138}, {17861, 17872}, {17863, 23668}, {17876, 17877}, {17888, 23678}, {17895, 23682}

X(23677) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 b^2 + a b^4 - 2 a^3 b c - a^2 b^2 c - b^4 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + b^2 c^3 + a c^4 - b c^4 : :

X(23677) lies on these lines: {1, 23669}, {142, 21210}, {244, 20905}, {1441, 23676}, {3914, 17863}, {17861, 23682}, {18690, 23529}, {20258, 20590}, {20456, 21084}, {23536, 23689}

X(23678) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b - c) (-a^4 + b^4 - 3 a^2 b c + 4 a b^2 c - b^3 c + 4 a b c^2 - 4 b^2 c^2 - b c^3 + c^4) : :

X(23678) lies on these lines: {2, 2804}, {522, 21052}, {4397, 17899}, {4453, 21433}, {4768, 21180}, {6089, 9134}, {14304, 23541}, {17888, 23676}

X(23679) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b + c) (a^4 - 3 a^2 b^2 + 2 b^4 + 5 a^2 b c - b^3 c - 3 a^2 c^2 - 2 b^2 c^2 - b c^3 + 2 c^4) : :

X(23679) lies on these lines: {75, 6390}, {1441, 12609}, {6332, 17894}, {17887, 23555}, {20912, 21434}

X(23680) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    (b + c) (a^4 b^2 + a^2 b^4 - 2 a^4 b c + a^4 c^2 - 3 a^2 b^2 c^2 - b^4 c^2 + 2 b^3 c^3 + a^2 c^4 - b^2 c^4) : :

X(23680) lies on these lines: {1, 23669}, {10, 21431}, {3914, 17867}, {17864, 23672}, {17871, 23682}, {20257, 21210}

X(23681) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 + a b^2 + 2 b^3 - 2 a b c - 2 b^2 c + a c^2 - 2 b c^2 + 2 c^3 : :

X(23681) lies on these lines: {1, 224}, {2, 2415}, {6, 4654}, {9, 3782}, {11, 5573}, {27, 17189}, {30, 16485}, {31, 4312}, {57, 1020}, {63, 4862}, {81, 4888}, {92, 1111}, {142, 17022}, {165, 3011}, {200, 1738}, {225, 1467}, {226, 2999}, {269, 278}, {306, 17151}, {312, 17282}, {321, 17284}, {329, 3008}, {333, 17274}, {345, 1266}, {614, 990}, {756, 1698}, {908, 23511}, {940, 6173}, {982, 5231}, {1104, 9579}, {1119, 18678}, {1201, 11522}, {1279, 9580}, {1423, 1730}, {1468, 4355}, {1743, 5905}, {1754, 5735}, {1834, 11518}, {1836, 7290}, {1999, 17298}, {2886, 3677}, {3158, 17724}, {3242, 21949}, {3339, 5230}, {3452, 17067}, {3475, 3755}, {3662, 11679}, {3670, 5705}, {3673, 21436}, {3752, 5219}, {3760, 21416}, {3826, 7322}, {3875, 18134}, {3915, 9589}, {3924, 5691}, {3925, 7174}, {3929, 17276}, {3944, 5272}, {3946, 5712}, {3973, 17781}, {3982, 4644}, {4310, 4847}, {4346, 5273}, {4360, 19830}, {4415, 7308}, {4653, 5333}, {4887, 9965}, {4904, 13567}, {4906, 11235}, {5236, 17903}, {5269, 5880}, {5271, 17184}, {5437, 17720}, {5737, 17235}, {5739, 16833}, {7557, 9612}, {9581, 17054}, {10436, 19786}, {10447, 19787}, {10888, 12610}, {10980, 11269}, {14213, 17885}, {16602, 20196}, {16752, 17182}, {16834, 17778}, {17056, 17301}, {17160, 19831}, {17267, 22034}, {17861, 17862}, {17871, 17888}, {18216, 21955}, {20320, 20322}

X(23682) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 b^2 + a b^4 - a^2 b^2 c + b^4 c + a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - b^2 c^3 + a c^4 + b c^4 : :

X(23682) lies on these lines: {1, 224}, {10, 3765}, {31, 379}, {38, 20880}, {226, 869}, {239, 1738}, {516, 3747}, {614, 6817}, {674, 1086}, {726, 20432}, {851, 2223}, {908, 2664}, {1107, 3925}, {1441, 17872}, {1575, 20486}, {1836, 2176}, {2239, 3008}, {2486, 8610}, {3009, 3120}, {3741, 20913}, {3772, 21010}, {4201, 16823}, {4300, 15970}, {4388, 16827}, {4968, 21020}, {13161, 16830}, {15320, 16685}, {16752, 20556}, {17050, 21352}, {17448, 21949}, {17861, 23677}, {17866, 23666}, {17871, 23680}, {17895, 23676}

X(23683) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b - c) (a^6 - 2 a^4 b^2 + a^2 b^4 - 3 a^4 b c + 2 a^2 b^3 c + b^5 c - 2 a^4 c^2 + 2 a^2 b^2 c^2 + 2 a^2 b c^3 - 2 b^3 c^3 + a^2 c^4 + b c^5) : :

X(23683) lies on these lines: {75, 3265}, {522, 693}, {850, 7253}, {6332, 17894}

X(23684) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b - c) (a^4 + a^2 b^2 - 2 a b^3 + a^2 b c - 2 a b^2 c + b^3 c + a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 + b c^3) : :

X(23684) lies on these lines: {312, 676}, {522, 21438}, {3766, 21439}, {3904, 4968}, {4385, 10015}, {4397, 14208}, {17888, 23676}, {21437, 23557}

X(23685) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (b - c) (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(23685) lies on these lines: {75, 905}, {92, 20296}, {321, 4391}, {514, 21438}, {1734, 4647}, {4397, 23580}, {6332, 17894}, {14349, 20629}, {15416, 17924}

X(23686) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a (b - c)^2 (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(23686) lies on these lines: {116, 244}, {1111, 3123}, {17888, 23672}, {23536, 23664}

X(23687) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a (b - c) (a^3 b - a^2 b^2 + a b^3 - b^4 + a^3 c - a^2 b c + a b^2 c - b^3 c - a^2 c^2 + a b c^2 + a c^3 - b c^3 - c^4) : :

X(23687) lies on these lines: {1, 16757}, {612, 6588}, {650, 8642}, {693, 1734}, {2509, 11934}, {2522, 6182}, {3900, 6591}, {3914, 17896}, {4041, 6590}, {4468, 13259}, {14208, 21174}

X(23688) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    a^3 b^2 + a b^4 + a^2 b^2 c - b^4 c + a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + b^2 c^3 + a c^4 - b c^4 : :

X(23688) lies on these lines: {1, 23669}, {38, 20895}, {726, 4710}, {1441, 17872}, {3262, 17446}, {3741, 21442}, {3914, 17860}, {3946, 21210}, {17470, 21246}, {17863, 23668}

X(23689) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (2 a^4 + a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c - a b^2 c + a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(23689) lies on these lines: {1, 1441}, {1125, 20236}, {1386, 16732}, {1733, 3663}, {3086, 17086}, {3673, 4008}, {3914, 17884}, {4359, 17591}, {17871, 19785}, {17872, 23537}, {23536, 23677}

X(23690) =  (A,B,C,X(1); A',B',C',X(1)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

Barycentrics    b c (a^3 b - a^2 b^2 + a b^3 + b^4 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + c^4) : :

X(23690) lies on these lines: {1, 1441}, {10, 20236}, {321, 3944}, {516, 1733}, {517, 7235}, {518, 16732}, {740, 3262}, {1090, 1737}, {1711, 20223}, {1738, 4858}, {2171, 12047}, {3263, 5988}, {3434, 17871}, {3914, 14213}, {4318, 18815}, {4359, 17596}, {4397, 23580}, {4442, 20887}, {4459, 15310}, {4972, 20886}, {7672, 10573}, {9902, 20436}, {17860, 17890}, {17862, 17889}, {17869, 17891}, {20911, 21443}, {23528, 23669}, {23537, 23665}

X(23691) =  X(1)-HIRST INVERSE OF X(78)

Barycentrics    a (a^4 b-a^3 b^2-a^2 b^3+a b^4+a^4 c-2 a^3 b c+a^2 b^2 c+2 a b^3 c-2 b^4 c-a^3 c^2+a^2 b c^2-a^2 c^3+2 a b c^3+a c^4-2 b c^4) : :

X(23691) lies on the cubic K1076 and these lines: {1,2}, {190,2635}, {649,6003}, {1332,1936}, {2318,4417}, {3717,4551}

X(23691) = X(1)-Hirst inverse of X(78)
X(23691) = X(1)-line conjugate of X(3924)
X(23691) = crossdifference of every pair of points on line {649, 3924}

X(23692) =  X(1)-HIRST INVERSE OF X(284)

Barycentrics    a (a+b) (a+c) (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+b^5 c-a^4 c^2+a^3 b c^2+a^3 c^3-2 b^3 c^3+b c^5) : :

X(23692) lies on the cubic K1076 and these lines: {1,19}, {73,270}, {110,1936}, {162,2635}, {656,3737}, {662,1818}, {1736,2341}, {2074,3465}, {2906,3468}, {4575,18653}

X(23692) = X(1)-Hirst inverse of X(284)
X(23692) = X(1)-line conjugate of X(2294)
X(23692) = crossdifference of every pair of points on line {656, 2294}
X(23692) = barycentric product X(i)*X(j) for these {i,j}: {1, 448}, {284, 16090}
X(23692) = barycentric quotient X(i)/X(j) for these {i,j}: {448, 75}, {16090, 349}

X(23693) =  X(1)-HIRST INVERSE OF X(219)

Barycentrics    a^2 (a-b) (a-c) (b^2 (a-c)^2 (-a+b-c) c (a+b+c) (-a^2+b^2-c^2)-(a-b)^2 b (a+b-c) c^2 (a+b+c) (a^2+b^2-c^2)) : :

X(23693) lies on the cubic K1076 and these lines: {1,6}, {78,1935}, {95,307}, {100,2635}, {109,6745}, {182,18162}, {212,329}, {513,2077}, {516,3939}, {527,13329}, {651,1818}, {765,1861}, {908,1331}, {1253,5698}, {1428,21320}, {1463,1470}, {1745,11517}, {3035,9364}, {3072,21077}, {3073,3811}, {3220,21362}, {3827,6211}, {3834,18634}, {4645,5552}, {5255,12607}, {5285,21361}, {6210,12329}, {9441,17768}, {10025,20778}, {10915,12618}, {11248,15310}

X(23693) = {X(908),X(1331)}-harmonic conjugate of X(1936)
X(23693) = X(1)-Hirst inverse of X(219)
X(23693) = X(1)-line conjugate of X(1108)
X(23693) = crossdifference of every pair of points on line {513, 1108}

X(23694) =  X(1)X(514)∩X(103)X(672)

Barycentrics    a^2 (a^2+b^2-a c-b c) (a^2-a b-b c+c^2) (a^4 b^2-2 a^3 b^3+2 a b^5-b^6+a^4 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 b^2 c^3+4 b^3 c^3-b^2 c^4+2 a c^5-c^6) : :

X(23694) lies on the cubic K1076 and these lines: {1,514}, {103,672}, {105,1458}, {673,2635}, {1814,1936}, {2078,2195}, {3100,10025}

X(23694) = X(518)-isoconjugate of X(2724)
X(23694) = X(103)-line conjugate of X(672)
X(23694) = barycentric product X(673)X(2808)
X(23694) = barycentric quotient X(i)/X(j) for these {i,j}: {1438, 2724}, {2808, 3912}

X(23695) =  X(1)-HIRST INVERSE OF X(283)

Barycentrics    a^2 (a^2-b^2) (a^2-c^2) (b^2 (a-c)^2 (-a+b-c) c (a+c) (-a^2+b^2-c^2)-(a-b)^2 b (a+b) (a+b-c) c^2 (a^2+b^2-c^2)) : :

X(23695) lies on the cubic K1076 and these lines: {1,21}, {415,4620}, {661,22382}, {662,2635}, {908,4575}, {1936,4558}

X(23695) = X(523)-isoconjugate of X(2714)
X(23695) = X(1)-Hirst inverse of X(283)
X(23695) = X(22382)-line conjugate of X(661)
X(23695) = barycentric product X(i)*X(j) for these {i,j}: {63, 425}, {662, 2798}
X(23695) = barycentric quotient X(i)/X(j) for these {i,j}: {163, 2714}, {425, 92}, {2798, 1577}

X(23696) =  X(1)X(514)∩X(102)X(105)

Barycentrics    a (a-b-c) (b-c) (a^2+b^2-a c-b c) (a^2-b^2-c^2) (a^2-a b-b c+c^2) : :

X(23696) lies on the conic {{A,B,C,X(1),X(3)}} and these lines: {1,514}, {3,905}, {77,1459}, {78,6332}, {102,105}, {219,521}, {241,2424}, {284,1024}, {294,2338}, {296,8677}, {513,1037}, {522,4319}, {650,949}, {884,1036}, {919,2728}, {1742,21173}, {1777,4905}, {2195,3738}

X(23696) = X(i)-zayin conjugate of X(j) for these (i,j): {109, 2254}, {652, 672}
X(23696) = X(10099)-cross conjugate of X(885)
X(23696) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2283}, {19, 1025}, {25, 883}, {34, 1026}, {65, 4238}, {100, 1876}, {101, 5236}, {108, 518}, {109, 1861}, {241, 1783}, {278, 2284}, {651, 5089}, {653, 672}, {664, 2356}, {918, 7115}, {1458, 1897}, {2223, 18026}, {2254, 7012}, {4559, 15149}, {8750, 9436}
X(23696) = trilinear pole of line {652, 7004}
X(23696) = crossdifference of every pair of points on line {672, 1876}
X(23696) = crosssum of X(1458) and X(2254)
X(23696) = barycentric product X(i)*X(j) for these {i,j}: {63, 885}, {69, 1024}, {105, 6332}, {294, 4025}, {304, 884}, {333, 10099}, {345, 1027}, {521, 673}, {522, 1814}, {652, 2481}, {666, 7004}, {905, 14942}, {919, 17880}, {1416, 15416}, {1946, 18031}, {2195, 15413}
X(23696) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1025}, {48, 2283}, {63, 883}, {105, 653}, {212, 2284}, {219, 1026}, {284, 4238}, {294, 1897}, {513, 5236}, {521, 3912}, {649, 1876}, {650, 1861}, {652, 518}, {663, 5089}, {673, 18026}, {884, 19}, {885, 92}, {905, 9436}, {919, 7012}, {1024, 4}, {1027, 278}, {1438, 108}, {1459, 241}, {1814, 664}, {1946, 672}, {2195, 1783}, {3063, 2356}, {3737, 15149}, {6332, 3263}, {7004, 918}, {7117, 2254}, {8611, 3932}, {10099, 226}, {14942, 6335}, {22091, 6168}, {22383, 1458}, {23189, 18206}

X(23697) = X(63)X(499)∩X(155)X(2990)

Barycentrics    (b+c-a)*(a^3+(b+c)*a^2-(b^2+c^2)*a-(b^2-c^2)*(b-c))*(a^6+2*c*a^5-(3*b^2+c^2)*a^4-4*c*(b^2-b*c+c^2)*a^3+(3*b^4-c^4-2*b*c^2*(b-2*c))*a^2+2*(b^2-c^2)^2*c*a-(b^2-c^2)^3)*(a^6+2*b*a^5-(b^2+3*c^2)*a^4-4*b*(b^2-b*c+c^2)*a^3-(b^4-3*c^4-2*b^2*c*(2*b-c))*a^2+2*(b^2-c^2)^2*b*a+(b^2-c^2)^3) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28341.

X(23697) lies on these lines: {63, 499}, {155, 2990}, {912, 6504}, {6505, 10052}

X(23698) = INFINITY POINT OF THE LINE X(4)X(99)

Barycentrics    2*a^8-4*(b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*b^2*c^2*a^2-(b^2-c^2)^4 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28341.

X(23698) lies on these lines: {2, 9734}, {3, 115}, {4, 99}, {5, 620}, {12, 15452}, {13, 5474}, {14, 5473}, {20, 98}, {30, 511}, {40, 13178}, {55, 13182}, {56, 13183}, {104, 10769}, {140, 6722}, {147, 3146}, {182, 2549}, {247, 1316}, {262, 11361}, {265, 15357}, {316, 9867}, {325, 13449}, {376, 671}, {381, 2482}, {382, 6033}, {485, 8997}, {486, 9739}, {487, 6230}, {488, 6231}, {546, 20399}, {547, 22247}, {549, 5461}, {550, 11623}, {575, 15048}, {576, 7737}, {618, 5479}, {619, 5478}, {631, 14061}, {641, 6251}, {642, 6250}, {944, 7983}, {946, 11711}, {962, 7970}, {1151, 8980}, {1152, 13967}, {1160, 22809}, {1161, 22810}, {1350, 11646}, {1351, 5477}, {1352, 10008}, {1385, 11725}, {1478, 10086}, {1479, 10089}, {1562, 17974}, {1569, 3095}, {1587, 19109}, {1588, 19108}, {1614, 3044}, {1657, 10991}, {1885, 12131}, {1916, 11257}, {2023, 13334}, {2080, 6781}, {2453, 5181}, {2936, 18534}, {3018, 5467}, {3023, 6284}, {3027, 7354}, {3058, 18969}, {3398, 7765}, {3455, 12083}, {3524, 9166}, {3529, 9862}, {3534, 11632}, {3543, 6054}, {3575, 5186}, {3627, 22505}, {3628, 15092}, {3830, 8724}, {3839, 23234}, {3845, 14160}, {4027, 6658}, {4297, 11599}, {4299, 10069}, {4302, 10053}, {4558, 8754}, {5026, 5480}, {5054, 14971}, {5055, 9167}, {5059, 5984}, {5066, 14162}, {5077, 19662}, {5085, 6034}, {5097, 18907}, {5149, 19130}, {5152, 11676}, {5182, 14853}, {5254, 13335}, {5355, 11842}, {5434, 12354}, {5469, 21157}, {5470, 21156}, {5471, 5615}, {5472, 5611}, {5476, 11159}, {5691, 9864}, {5864, 6777}, {5865, 6778}, {5870, 6320}, {5871, 6319}, {5976, 6248}, {5985, 15680}, {5986, 20062}, {5987, 20063}, {6228, 6229}, {6249, 8290}, {6459, 19056}, {6460, 19055}, {6699, 15359}, {6723, 11007}, {6771, 6772}, {6774, 6775}, {6776, 10754}, {7472, 16188}, {7802, 9991}, {7833, 22712}, {8356, 15819}, {8596, 11177}, {8782, 9873}, {9115, 20426}, {9117, 20425}, {9732, 12602}, {9733, 12601}, {9757, 9892}, {9758, 9894}, {9775, 14360}, {9834, 13176}, {9835, 13177}, {9838, 13184}, {9839, 13185}, {10352, 10358}, {10724, 10768}, {10733, 11005}, {11001, 12243}, {11500, 13173}, {11602, 22890}, {11603, 22843}, {11606, 12122}, {12041, 15535}, {12113, 13179}, {12114, 13180}, {12115, 13189}, {12116, 13190}, {12121, 18332}, {12177, 14928}, {12184, 12943}, {12185, 12953}, {12217, 12218}, {12303, 12979}, {12304, 12978}, {12383, 15342}, {12902, 15545}, {12910, 19474}, {12911, 19473}, {12972, 12985}, {12973, 12984}, {13075, 18974}, {13076, 18975}, {13081, 18988}, {13082, 18989}, {14033, 14561}, {14120, 16760}, {14538, 23004}, {14539, 23005}, {14568, 21445}, {14830, 15681}, {15687, 22566}, {15980, 18860}, {16001, 22513}, {16002, 22512}, {16163, 16278}, {18800, 20423}, {21158, 22510}, {21159, 22511}, {22501, 22591}, {22502, 22592}

X(23698) = isogonal conjugate of X(23700)
X(23698) = circumnormal-isogonal conjugate of X(10425)

X(23699) = INFINITY POINT OF THE LINE X(4)X(111)

Barycentrics    6*(6*R^2-SW)*S^4+(SA-SW)*(54*R^2*SA-9*SA*SW-SW^2)*S^2-2*SB*SC*SW^3 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28341.

X(23699) lies on these lines: {3, 126}, {4, 111}, {5, 6719}, {20, 1296}, {30, 511}, {104, 10779}, {113, 9129}, {376, 10717}, {381, 9172}, {382, 11258}, {944, 10704}, {946, 11721}, {1614, 3048}, {3146, 20099}, {3325, 7354}, {6019, 6284}, {6776, 10765}, {11818, 15563}, {15560, 18420}

X(23699) = isogonal conjugate of X(23701)

X(23700) = ISOGONAL CONJUGATE OF X(23698)

Barycentrics    (SB+SC)*(4*S^4+(6*SB-5*SW)*SB*S^2-SC*SA*SW^2)*(4*S^4+(6*SC-5*SW)*SC*S^2-SW^2*SA*SB) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28341.

X(23700) lies on the circumcircle and these lines: {3, 10425}, {4, 14384}, {98, 3566}, {99, 3564}, {107, 460}, {112, 1692}, {511, 3565}, {512, 3563}, {691, 1351}, {805, 9737}, {2715, 3053}, {2855, 9744}, {14265, 22456}

X(23700) = circumperp isogonal conjugate of X(10425)
X(23700) = circumcircle-antipode of X(10425)
X(23700) = trilinear pole of the line {6, 6132}
X(23700) = Λ(X(4), X(99))
X(23700) = Cundy-Parry Phi transform of X(14253)
X(23700) = Cundy-Parry Psi transform of X(14384)

X(23701) = ISOGONAL CONJUGATE OF X(23699)

Barycentrics    (a^10-(b^2+5*c^2)*a^8-(b^2-2*c^2)*(7*b^2+2*c^2)*a^6+(3*b^6+4*c^6+11*b^2*c^2*(b^2-2*c^2))*a^4+(b^2-c^2)*(6*b^6+5*c^6-b^2*c^2*(18*b^2+7*c^2))*a^2-(b^4-c^4)*(2*b^6+c^6-b^2*c^2*(6*b^2+c^2)))*(a^10-(5*b^2+c^2)*a^8+(2*b^2-c^2)*(2*b^2+7*c^2)*a^6+(4*b^6+3*c^6-11*b^2*c^2*(2*b^2-c^2))*a^4-(b^2-c^2)*(5*b^6+6*c^6-b^2*c^2*(7*b^2+18*c^2))*a^2+(b^4-c^4)*(b^6+2*c^6-b^2*c^2*(b^2+6*c^2)))*a^2 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28341.

X(23701) lies on the circumcircle and these lines: {112, 12593}, {1296, 2393}, {1499, 2373}

X(23702) = COMPLEMENT OF X(22261)

Barycentrics    (S^2+(8*R^2-SA-2*SW)*SA)*(4*S^2+(SB+SC)*(8*R^2-SA-3*SW)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28341.

X(23702) lies on these lines: {2, 22261}, {140, 5449}, {570, 5254}, {6640, 15454}

X(23702) = complement of X(22261)
X(23702) = complementary conjugate of X(5562)

X(23703) = X(1)X(3)∩X(100)X(109)

Barycentrics    a (a-b) (a-c) (2 a-b-c) (a+b-c) (a-b+c) : :

X(23703) lies on the cubic K407 and these lines: {1,3}, {20,18340}, {100,109}, {108,1293}, {222,4421}, {294,2291}, {603,8715}, {664,4597}, {678,14191}, {860,13998}, {901,2222}, {902,1647}, {934,6014}, {1018,1415}, {1054,1421}, {1399,3293}, {1411,4674}, {1777,11499}, {1788,6788}, {1807,12515}, {2163,16236}, {2284,14589}, {2398,21105}, {2720,2743}, {3658,3737}, {3722,5083}, {4242,7012}, {4552,4781}, {6718,23541}, {19515,22102}

X(23703) = isogonal conjugate of X(23838)
X(23703) = X(2222)-Ceva conjugate of X(4551)
X(23703) = X(i)-cross conjugate of X(j) for these (i,j): {1635, 3911}, {4895, 44}, {17460, 765}, {23344, 1023}
X(23703) = cevapoint of X(i) and X(j) for these (i,j): {44, 4895}, {902, 1635}, {3689, 14418}
X(23703) = crosspoint of X(i) and X(j) for these (i,j): {655, 664}
X(23703) = trilinear pole of line {44, 1319}
X(23703) = crossdifference of every pair of points on line {650, 2170}
X(23703) = crosssum of X(i) and X(j) for these (i,j): {513, 1769}, {650, 4895}, {654, 663}
X(23703) = X(i)-aleph conjugate of X(j) for these (i,j): {100, 6326}, {6733, 6127}
X(23703) = X(519)-beth conjugate of X(14584)
X(23703) = X(i)-zayin conjugate of X(j) for these (i,j): {44, 650}, {517, 513}, {1731, 652}, {1807, 900}, {2161, 654}, {2361, 4040}, {10703, 2827}, {12034, 1635}
X(23703) = X(i)-isoconjugate of X(j) for these (i,j): {8, 23345}, {9, 1022}, {11, 901}, {55, 6548}, {88, 650}, {106, 522}, {284, 4049}, {513, 1320}, {514, 2316}, {649, 4997}, {652, 6336}, {663, 903}, {679, 4895}, {900, 1318}, {1015, 4582}, {1086, 5548}, {1168, 3738}, {1417, 4397}, {1639, 2226}, {1797, 3064}, {2170, 3257}, {2320, 23352}, {2364, 23598}, {2441, 6557}, {2804, 10428}, {3063, 20568}, {3271, 4555}, {3737, 4674}, {3939, 6549}, {4080, 7252}, {4391, 9456}, {4516, 4622}, {4530, 4638}, {4591, 21044}, {6332, 8752}, {6551, 7336}, {9268, 21132}
X(23703) = barycentric product X(i)*X(j) for these {i,j}: {7, 1023}, {44, 664}, {57, 17780}, {59, 3762}, {85, 23344}, {100, 3911}, {108, 3977}, {109, 4358}, {190, 1319}, {214, 655}, {519, 651}, {653, 5440}, {658, 3689}, {668, 1404}, {900, 4564}, {902, 4554}, {927, 14439}, {934, 2325}, {1014, 4169}, {1262, 4768}, {1275, 4895}, {1317, 3257}, {1332, 1877}, {1414, 3943}, {1415, 3264}, {1420, 2415}, {1461, 4723}, {1635, 4998}, {1639, 7045}, {2251, 4572}, {3992, 4565}, {4551, 16704}, {4573, 21805}, {4585, 14584}, {4620, 4730}, {6516, 8756}, {18026, 22356}
X(23703) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 522}, {56, 1022}, {57, 6548}, {59, 3257}, {65, 4049}, {100, 4997}, {101, 1320}, {108, 6336}, {109, 88}, {214, 3904}, {519, 4391}, {604, 23345}, {651, 903}, {664, 20568}, {678, 1639}, {692, 2316}, {765, 4582}, {900, 4858}, {902, 650}, {1017, 4895}, {1023, 8}, {1110, 5548}, {1317, 3762}, {1319, 514}, {1404, 513}, {1405, 23352}, {1415, 106}, {1420, 2403}, {1635, 11}, {1877, 17924}, {1960, 2170}, {2087, 21132}, {2099, 23598}, {2149, 901}, {2251, 663}, {2325, 4397}, {2429, 3680}, {3251, 4530}, {3285, 3737}, {3669, 6549}, {3911, 693}, {3943, 4086}, {4169, 3701}, {4370, 4768}, {4551, 4080}, {4559, 4674}, {4564, 4555}, {4620, 4634}, {4700, 4811}, {4730, 21044}, {4895, 1146}, {4969, 4985}, {5298, 4978}, {5440, 6332}, {9459, 3063}, {14122, 21204}, {14407, 4516}, {14418, 2968}, {14425, 4939}, {14427, 4081}, {16704, 18155}, {17455, 3738}, {17780, 312}, {21805, 3700}, {21859, 4013}, {22086, 7004}, {22356, 521}, {22371, 14418}, {23202, 652}, {23344, 9}

X(23704) = X(1)X(6)∩X(101)X(1292)

Barycentrics    a (a-b) (a-c) (a-b-c) (2 a^2-a b+b^2-a c-2 b c+c^2) : :

X(23704) lies on the cubic K407 and these lines: {1,6}, {101,1292}, {644,3939}, {4521,17780}

X(23704) = isogonal conjugate of X(37626)
X(23704) = X(644)-daleth conjugate of X(3939)
X(23704) = X(2348)-zayin conjugate of X(513)
X(23704) = X(i)-isoconjugate of X(j) for these (i,j): {514, 1477}, {1280, 3669}, {1358, 6078}
X(23704) = trilinear pole of line {2348, 8647}
X(23704) = barycentric product X(i)*X(j) for these {i,j}: {100, 5853}, {190, 2348}, {644, 3008}, {668, 8647}, {1279, 3699}
X(23704) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 1477}, {1279, 3676}, {2348, 514}, {3939, 1280}, {5853, 693}, {8647, 513}

X(23705) = X(1)X(2)∩X(100)X(1293)

Barycentrics    a (a-b) (a-c) (a-b-c) (a b+b^2+a c-4 b c+c^2) : :

X(23705) lies on the cubic K407 and these lines: {1,2}, {100,1293}, {101,9104}, {646,3699}

X(23705) = isogonal conjugate of X(37627)
X(23705) = X(i)-Ceva conjugate of X(j) for these (i,j): {765, 17460}, {3257, 644}
X(23705) = X(i)-isoconjugate of X(j) for these (i,j): {513, 8686}, {1357, 6079}
X(23705) = barycentric product X(i)*X(j) for these {i,j}: {190, 3880}, {644, 1266}, {645, 4695}, {646, 1149}, {3699, 16610}, {4069, 16711}, {4582, 17460}, {5548, 20900}
X(23705) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 8686}, {644, 1120}, {1149, 3669}, {3880, 514}, {4587, 1811}, {4695, 7178}, {16610, 3676}

X(23706) = X(1)X(4)∩X(108)X(109)

Barycentrics    a (a-b) (a-c) (a+b-c) (a-b+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3) : :

X(23706) lies on the cubic K407 and these lines: {1,4}, {108,109}, {1875,14260}, {1897,4551}, {3737,7452}, {4242,7012}

X(23706) = isogonal conjugate of X(37628)
X(23706) = cevapoint of X(i) and X(j) for these (i,j): {1457, 1769}
X(23706) = trilinear pole of line {1875, 2183}
X(23706) = X(5379)-beth conjugate of X(4242)
X(23706) = X(i)-zayin conjugate of X(j) for these (i,j): {2183, 652}, {6905, 656}
X(23706) = X(7012)-Ceva conjugate of X(1845)
X(23706) = X(1769)-cross conjugate of X(1785)
X(23706) = polar conjugate of isotomic conjugate of X(24029)
X(23706) = X(i)-isoconjugate of X(j) for these (i,j): {104, 521}, {219, 2401}, {345, 2423}, {513, 1809}, {522, 1795}, {909, 6332}, {1309, 1364}, {1946, 18816}, {2342, 4025}, {2720, 2968}, {4391, 14578}, {4571, 15635}, {7117, 13136}, {14312, 15405}
X(23706) = barycentric product X(i)*X(j) for these {i,j}: {34, 2397}, {108, 908}, {190, 1875}, {226, 4246}, {273, 2427}, {517, 653}, {651, 1785}, {655, 1845}, {664, 14571}, {1457, 6335}, {1465, 1897}, {1783, 22464}, {1846, 3257}, {2183, 18026}, {2804, 7128}, {7012, 10015}
X(23706) = barycentric quotient X(i)/X(j) for these {i,j}: {34, 2401}, {101, 1809}, {517, 6332}, {653, 18816}, {1395, 2423}, {1415, 1795}, {1457, 905}, {1465, 4025}, {1785, 4391}, {1845, 3904}, {1846, 3762}, {1875, 514}, {2183, 521}, {2397, 3718}, {2427, 78}, {3310, 7004}, {4246, 333}, {6735, 15416}, {7012, 13136}, {10015, 17880}, {14571, 522}, {22464, 15413}

X(23707) = ISOGONAL CONJUGATE OF X(2635)

Trilinears    1/(2 sec A - sec B - sec C) : :
Barycentrics    a (2 a^4 b-2 a^3 b^2-2 a^2 b^3+2 a b^4-a^4 c+2 a^2 b^2 c-b^4 c+a^3 c^2-a^2 b c^2-a b^2 c^2+b^3 c^2+a^2 c^3+b^2 c^3-a c^4-b c^4) (a^4 b-a^3 b^2-a^2 b^3+a b^4-2 a^4 c+a^2 b^2 c+b^4 c+2 a^3 c^2-2 a^2 b c^2+a b^2 c^2-b^3 c^2+2 a^2 c^3-b^2 c^3-2 a c^4+b c^4) : :

X(23707) lies on the conic {{A,B,C,X(1),X(3)}}, the curves Q084 and K1076, and on these lines: {1,653}, {3,651}, {29,823}, {77,658}, {78,190}, {100,219}, {162,284}, {283,662}, {296,1155}, {332,799}, {412,3362}, {655,1807}, {1795,13329}

X(23707) = isogonal conjugate of X(2635)
X(23707) = cevapoint of X(i) and X(j) for these (i,j): {1, 2635}, {2637, 2638}
X(23707) = X(i)-cross conjugate of X(j) for these (i,j): {2635, 1}, {2637, 653}
X(23707) = trilinear pole of line {1, 652}
X(23707) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 2635}, {652, 2637}
X(23707) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2635}, {653, 2637}
X(23707) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2635}, {1946, 2637}
X(23707) = {X(2636),X(2638)}-harmonic conjugate of X(653)

X(23708) = X(1)X(5)∩X(2)X(5119)

Barycentrics    -a^4+a^3 b+3 a^2 b^2-a b^3-2 b^4+a^3 c-4 a^2 b c+a b^2 c+3 a^2 c^2+a b c^2 +4 b^2 c^2-a c^3-2 c^4 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28316.

X(23708) lies on these lines: {1,5}, {2,5119}, {4,21578}, {7,90}, {10,6931}, {35,474}, {36,1012}, {40,6958}, {46,499}, {55,7743}, {56,9955}, {57,1727}, {63,11813}, {65,18493}, {79,3361}, {140,12701}, {226,10072}, {377,1125}, {381,1319}, {392,1698}, {497,6854}, {498,6983}, {515,6968}, {516,6966}, {950,6984}, {956,5087}, {997,11680}, {999,17605}, {1000,3090}, {1210,6860}, {1385,10896}, {1388,18480}, {1420,3585}, {1478,3817}, {1482,17606}, {1519,1709}, {1656,3057}, {1737,5603}, {1770,7288}, {1836,15325}, {1898,13373}, {2098,9956}, {2646,9669}, {3149,14798}, {3306,10199}, {3419,3829}, {3476,3545}, {3576,3583}, {3586,17532}, {3601,4857}, {3616,6871}, {3679,5123}, {3814,3872}, {3825,19860}, {3890,7504}, {3895,21630}, {4293,9779}, {4294,5550}, {4299,18483}, {4302,6955}, {4311,12571}, {4333,5204}, {4861,5154}, {4870,15934}, {5010,9580}, {5048,5790}, {5056,10051}, {5126,12943}, {5231,5692}, {5274,6993}, {5427,16118}, {5433,12699}, {5440,11235}, {5445,7991}, {5554,10043}, {5561,9579}, {5563,9612}, {5691,21842}, {5805,15803}, {5820,16475}, {5903,11522}, {6666,19854}, {6675,16155}, {6887,7162}, {6906,7704}, {6913,22767}, {6918,11508}, {7742,18482}, {7982,18395}, {9654,20323}, {10085,10785}, {10090,16174}, {10175,12647}, {10527,21616}, {10529,21077}, {10573,13464}, {10624,19862}, {10940,12609}, {11009,15079}, {12953,13624}, {13407,14986}, {16153,16617}, {19875,20196}

X(23708) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,5,10827}, {1,7741,10826}, {1,7988,7951}, {5,496,10957}, {5,1387,5252}, {5,11376,1}, {11,5886,1}, {11,11729,10073}, {11,15950,5722}, {12,11373,1}, {355,5886,11729}, {496,11375,1}, {499,946,46}, {1012,22835,1699}, {1125,1479,3612}, {1387,5252,1}, {1837,5901,1}, {3086,12047,3338}, {3582,18393,57}, {3616,10591,10572}, {3624,9614,35}, {5204,22793,4333}, {5252,11376,1387}, {5603,10589,1737}, {5722,5886,15950}, {5722,15950,1}, {5901,10593,1837}, {7743,11230,55}, {7951,16173,1}, {9581,9624,1}, {10785,12608,10085}

X(23709) = X(3)X(95)∩X(1154)X(10606)

Trilinears    cos(B-C)*(-1-5*cos(4*A)-cos(6*A)+8*cos(A)cos(B-C)+6*cos(3*A)cos(B-C)+2*cos(5*A)*cos(B-C)-4*cos(2(B-C))-cos(2*A)*(3+2*cos(2(B-C)))) : :
Barycentrics    a^2 (-(b^2-c^2)^2+a^2 (b^2+c^2)) (a^16-5 a^14 (b^2+c^2)+b^2 c^2 (b^2-c^2)^4 (b^4+4 b^2 c^2+c^4)+3 a^12 (3 b^4+7 b^2 c^2+3 c^4)-a^10 (5 b^6+33 b^4 c^2+33 b^2 c^4+5 c^6)+a^8 (-5 b^8+23 b^6 c^2+38 b^4 c^4+23 b^2 c^6-5 c^8)-a^4 (b^2-c^2)^2 (5 b^8+7 b^6 c^2+8 b^4 c^4+7 b^2 c^6+5 c^8)+a^2 (b^2-c^2)^2 (b^10-b^8 c^2-4 b^6 c^4-4 b^4 c^6-b^2 c^8+c^10)+a^6 (9 b^10-7 b^8 c^2-14 b^6 c^4-14 b^4 c^6-7 b^2 c^8+9 c^10)) : :

See Tran Quang Hung, Angel Montesdeoca, and Ercole Suppa, Hyacinthos 28317 and Hyacinthos 28318.

X(23709) lies on these lines: {3,95}, {1154,10606}, {7395,12012}, {11197,17928}

X(23710) = ORTHIC AXIS INTERCEPT OF X(1)X(4)

Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)*(2*a^2-(b+c)*a-(b-c)^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23710) lies on these lines: {1, 4}, {25, 18613}, {108, 2078}, {230, 231}, {354, 1824}, {415, 648}, {429, 15888}, {517, 1835}, {519, 860}, {551, 5136}, {942, 1825}, {971, 6357}, {1060, 1074}, {1062, 1076}, {1420, 14257}, {1426, 3057}, {1430, 8750}, {1465, 15252}, {1758, 9357}, {1826, 16777}, {1830, 1876}, {1842, 11363}, {1861, 1897}, {1865, 3723}, {1874, 17724}, {1880, 17602}, {1936, 7012}, {3304, 4185}, {3746, 7414}, {4200, 11240}, {4654, 5733}, {4658, 14016}, {5231, 7046}, {6851, 9643}, {8606, 10902}, {10295, 11809}, {11518, 14018}

X(23710) = polar conjugate of X(1121)
X(23710) = polar conjugate of isotomic conjugate of X(527)
X(23710) = X(63)-isoconjugate of X(2291)
X(23710) = = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1068, 225), (1785, 1870, 1877), (1897, 17923, 1861), (3011, 3012, 8758), (8758, 16272, 3012)

X(23711) = ORTHIC AXIS INTERCEPT OF X(4)X(11)

Barycentrics    (a^2+b^2-c^2)*(2*a^5-2*(b+c)*a^4-(3*b^2-8*b*c+3*c^2)*a^3+3*(b^2-c^2)*(b-c)*a^2+(b-c)^4*a-(b^4-c^4)*(b-c))*(a^2-b^2+c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23711) lies on these lines: {4, 11}, {33, 17728}, {105, 4232}, {140, 15252}, {208, 11376}, {230, 231}, {318, 6691}, {406, 3304}, {451, 15888}, {528, 4242}, {1387, 1845}, {1785, 15325}, {1897, 3035}, {2968, 6717}, {3515, 14667}, {5094, 20621}, {5433, 7952}, {6713, 21664}, {7079, 13609}, {15253, 21841}

X(23712) = ORTHIC AXIS INTERCEPT OF X(4)X(13)

Barycentrics    SB*SC*(3*S^2+sqrt(3)*(3*SA-SW)*S-9*SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23712) lies on these lines: {4, 13}, {230, 231}, {462, 8754}, {471, 648}, {6111, 10295}, {8738, 17983}

X(23712) = polar conjugate of the isotomic conjugate of X(530)

X(23713) = ORTHIC AXIS INTERCEPT OF X(4)X(14)

Barycentrics    SB*SC*(3*S^2-sqrt(3)*(3*SA-SW)*S-9*SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23713) lies on these lines: {4, 14}, {230, 231}, {463, 8754}, {470, 648}, {6110, 10295}, {8737, 17983}

X(23713) = polar conjugate of the isotomic conjugate of X(531)

X(23714) = ORTHIC AXIS INTERCEPT OF X(4)X(15)

Barycentrics    SB*SC*(3*SA-sqrt(3)*S)*(3*SB+3*SC+2*sqrt(3)*S) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23714) lies on these lines: {4, 15}, {186, 6104}, {230, 231}, {299, 340}, {396, 463}, {397, 13367}, {403, 6107}, {470, 8737}

X(23715) = ORTHIC AXIS INTERCEPT OF X(4)X(16)

Barycentrics    SB*SC*(3*SA+sqrt(3)*S)*(3*SB+3*SC-2*sqrt(3)*S) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23715) lies on these lines: {4, 16}, {186, 6105}, {230, 231}, {298, 340}, {395, 462}, {398, 13367}, {403, 6106}, {471, 8738}

X(23716) = ISOGONAL CONJUGATE OF X(20428)

Barycentrics    (SB+SC)*(SB*S^2+SA*SC*(SW+2*sqrt(3)*S))*(SC*S^2+SB*SA*(SW+2*sqrt(3)*S)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23716) lies on this line: {184, 9736}

X(23716) = isogonal conjugate of X(20428)
X(23716) = complement of anticomplementary conjugate of X(36993)
X(23716) = anticomplement of the complementary conjugate of X(13350)

X(23717) = ISOGONAL CONJUGATE OF X(20429)

Barycentrics    (SB+SC)*(SB*S^2+SA*SC*(SW-2*sqrt(3)*S))*(SC*S^2+SB*SA*(SW-2*sqrt(3)*S)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28330.

X(23717) lies on this line: {184, 9735}

X(23717) = isogonal conjugate of X(20429)
X(23717) = complement of anticomplementary conjugate of X(36995)
X(23717) = anticomplement of the complementary conjugate of X(13349)

X(23718) = X(5)X(5007)∩X(6179)X(9764)

Barycentrics    (a^6+2 a^4 b^2+2 a^4 c^2-b^4 c^2-b^2 c^4) (2 a^6-a^4 b^2+b^6-a^4 c^2-6 a^2 b^2 c^2-b^4 c^2-b^2 c^4+c^6) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28331.

X(23718) lies on these lines: {5, 5007}, {6179, 9764}

X(23719) = X(54)X(74)∩X(216)X(8612)

Trilinears    7*cos(A)+37*cos(3*A)+(-6-107*cos(2*A)+7*cos(6*A))*cos(B-C)+(67*cos(A)+7*cos(3*A)-7*cos(5*A))*cos(2(B-C))+(1-7*cos(2*A)-7*cos(4*A))*cos(3(B-C)) : :
Barycentrics    a^2 (a^18 b^2-6 a^16 b^4+13 a^14 b^6-7 a^12 b^8-21 a^10 b^10+49 a^8 b^12-49 a^6 b^14+27 a^4 b^16-8 a^2 b^18+b^20+a^18 c^2-10 a^16 b^2 c^2+30 a^14 b^4 c^2 -39 a^12 b^6 c^2+30 a^10 b^8 c^2-41 a^8 b^10 c^2+70 a^6 b^12 c^2-65 a^4 b^14 c^2+29 a^2 b^16 c^2-5 b^18 c^2-6 a^16 c^4+30 a^14 b^2 c^4-54 a^12 b^4 c^4 +39 a^10 b^6 c^4+a^8 b^8 c^4-28 a^6 b^10 c^4+36 a^4 b^12 c^4-25 a^2 b^14 c^4+7 b^16 c^4+13 a^14 c^6-39 a^12 b^2 c^6+39 a^10 b^4 c^6-18 a^8 b^6 c^6 +7 a^6 b^8 c^6+9 a^4 b^10 c^6-19 a^2 b^12 c^6+8 b^14 c^6-7 a^12 c^8+30 a^10 b^2 c^8+a^8 b^4 c^8+7 a^6 b^6 c^8-14 a^4 b^8 c^8+23 a^2 b^10 c^8-40 b^12 c^8 -21 a^10 c^10-41 a^8 b^2 c^10-28 a^6 b^4 c^10+9 a^4 b^6 c^10+23 a^2 b^8 c^10+58 b^10 c^10+49 a^8 c^12+70 a^6 b^2 c^12+36 a^4 b^4 c^12-19 a^2 b^6 c^12 -40 b^8 c^12-49 a^6 c^14-65 a^4 b^2 c^14-25 a^2 b^4 c^14+8 b^6 c^14+27 a^4 c^16+29 a^2 b^2 c^16+7 b^4 c^16-8 a^2 c^18-5 b^2 c^18+c^20) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28332.

X(23719) lies on these lines: {54,74}, {216,8612}, {6759,10979}, {12162,18464}, {18383,18416}

X(23720) = (name pending)

Trilinears    -66*cos(A)+24*cos(3*A)-21*cos(5*A)+3*cos(7*A)+(9-84*cos(2*A)-cos(6*A))*cos(B-C)+(18*cos(A)-29*cos(3*A)+3*cos(5*A))*cos(2*(B-C))+(-12+33*cos(2*A) -9*cos(4*A))*cos(3*(B-C))+(3*cos(A)+cos(3*A))*cos(4*(B-C)) : :
Barycentrics    6 a^10-5 a^8 b^2+2 a^6 b^4+6 a^4 b^6-8 a^2 b^8-b^10-5 a^8 c^2-10 a^6 b^2 c^2+3 a^4 b^4 c^2+25 a^2 b^6 c^2-b^8 c^2+2 a^6 c^4+3 a^4 b^2 c^4-42 a^2 b^4 c^4 +2 b^6 c^4+6 a^4 c^6+25 a^2 b^2 c^6+2 b^4 c^6-8 a^2 c^8-b^2 c^8-c^10 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28332.

X(23720) lies on this line: {2,3}

X(23721) = (name pending)

Barycentrics    4 sqrt(3) a^10-7 sqrt(3) a^8 b^2-2 sqrt(3) a^6 b^4+8 sqrt(3) a^4 b^6-2 sqrt(3) a^2 b^8-Sqrt[3] b^10-7 sqrt(3) a^8 c^2+20 sqrt(3) a^6 b^2 c^2-8 sqrt(3) a^4 b^4 c^2 -8 sqrt(3) a^2 b^6 c^2+3 sqrt(3) b^8 c^2-2 sqrt(3) a^6 c^4-8 sqrt(3) a^4 b^2 c^4+20 sqrt(3) a^2 b^4 c^4-2 sqrt(3) b^6 c^4+8 sqrt(3) a^4 c^6-8 sqrt(3) a^2 b^2 c^6 -2 sqrt(3) b^4 c^6-2 sqrt(3) a^2 c^8+3 sqrt(3) b^2 c^8-sqrt(3) c^10-2 a^6 b^2 S+6 a^4 b^4 S-6 a^2 b^6 S+2 b^8 S-2 a^6 c^2 S+4 a^4 b^2 c^2 S+6 a^2 b^4 c^2 S-8 b^6 c^2 S +6 a^4 c^4 S+6 a^2 b^2 c^4 S+12 b^4 c^4 S-6 a^2 c^6 S-8 b^2 c^6 S+2 c^8 S : :
Barycentrics    Sqrt[3] (a^2+b^2-c^2) (a^2-b^2+c^2) (4 a^6-7 a^4 b^2+2 a^2 b^4+b^6-7 a^4 c^2+12 a^2 b^2 c^2-b^4 c^2+2 a^2 c^4-b^2 c^4+c^6)+8 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S^3 : : (Peter Moses, September 25, 2018; c.f. X(23722))

See Kadir Altintas and Ercole Suppa, Hyacinthos 28332.

X(23721) lies on this line: {2,3}

X(23722) = (name pending)

Barycentrics    4 sqrt(3) a^10-7 sqrt(3) a^8 b^2-2 sqrt(3) a^6 b^4+8 sqrt(3) a^4 b^6-2 sqrt(3) a^2 b^8-sqrt(3) b^10-7 sqrt(3) a^8 c^2+20 sqrt(3) a^6 b^2 c^2-8 sqrt(3) a^4 b^4 c^2 -8 sqrt(3) a^2 b^6 c^2+3 sqrt(3) b^8 c^2-2 sqrt(3) a^6 c^4-8 sqrt(3) a^4 b^2 c^4+20 sqrt(3) a^2 b^4 c^4-2 sqrt(3) b^6 c^4+8 sqrt(3) a^4 c^6-8 sqrt(3) a^2 b^2 c^6 -2 sqrt(3) b^4 c^6-2 sqrt(3) a^2 c^8+3 sqrt(3) b^2 c^8-sqrt(3) c^10+2 a^6 b^2 S-6 a^4 b^4 S+6 a^2 b^6 S-2 b^8 S+2 a^6 c^2 S-4 a^4 b^2 c^2 S-6 a^2 b^4 c^2 S+8 b^6 c^2 S -6 a^4 c^4 S-6 a^2 b^2 c^4 S-12 b^4 c^4 S+6 a^2 c^6 S+8 b^2 c^6 S-2 c^8 S : :
Barycentrics    Sqrt[3] (a^2+b^2-c^2) (a^2-b^2+c^2) (4 a^6-7 a^4 b^2+2 a^2 b^4+b^6-7 a^4 c^2+12 a^2 b^2 c^2-b^4 c^2+2 a^2 c^4-b^2 c^4+c^6)-8 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S^3 : : (Peter Moses, September 25, 2018; c.f. X(23721))

See Kadir Altintas and Ercole Suppa, Hyacinthos 28332.

X(23722) lies on this line: {2,3}

X(23723) = (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - b^4 c + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(23723) lies on these lines: {513, 676}, {514, 647}, {693, 21189}, {1769, 4077}, {2504, 3798}, {3265, 23887}, {4025, 8714}, {4040, 7192}, {4106, 23810}, {4765, 14837}, {7649, 21184}, {18155, 23785}, {21190, 23802}, {21204, 23803}

X(23724) = (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^5 - a^4 b + a^3 b^2 + a^2 b^3 - a^4 c + b^4 c + a^3 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + b c^4) : :

X(23724) lies on these lines: {1, 17896}, {226, 6588}, {513, 676}, {514, 2501}, {693, 17218}, {1459, 4077}, {3664, 4131}, {3960, 23803}, {4905, 20295}, {5249, 16757}, {6591, 23806}, {17094, 21184}, {23733, 23798}, {23786, 23812}

X(23725) = (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-2 a^5 - a^4 b + 3 a^3 b^2 + a^2 b^3 - a b^4 - a^4 c + b^4 c + 3 a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 - a c^4 + b c^4)

X(23725) lies on these lines: {513, 676}, {514, 12077}, {693, 3664}, {22383, 23806}, {23791, 23812}

X(23726) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + a^2 b^2 c - b^4 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - a c^4 - b c^4 + c^5) : :

X(23726) lies on these lines: {278, 3064}, {513, 23727}, {514, 652}, {523, 2254}, {918, 8611}, {1769, 21107}, {4468, 6350}, {20999, 23865}, {21102, 21117}, {21118, 23735}, {23751, 23759}

X(23727) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^2 + b^2 + c^2) (-a^3 + b^3 - b^2 c - b c^2 + c^3) : :

X(23727) lies on these lines: {513, 23726}, {514, 3064}, {650, 1427}, {652, 17094}, {676, 1459}, {9001, 16892}, {23729, 23732}, {23731, 23748}, {23735, 23763}

X(23728) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (2 a^5 - 3 a^3 b^2 - a^2 b^3 + a b^4 + b^5 + a^2 b^2 c - b^4 c - 3 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 - a^2 c^3 + a c^4 - b c^4 + c^5) : :

X(23728) lies on these lines: {513, 23726}, {14400, 23806}, {21104, 23730}, {23740, 23763}

X(23729) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (2 a^2 + a b + b^2 + a c - 2 b c + c^2) : :

X(23729) lies on these lines: {512, 20507}, {513, 11934}, {514, 3700}, {523, 4382}, {649, 1638}, {661, 6084}, {812, 3004}, {900, 16892}, {918, 20295}, {1639, 3835}, {2527, 24924}, {3676, 4790}, {3766, 18071}, {3776, 4785}, {4010, 4977}, {4025, 6008}, {4369, 4927}, {4380, 4773}, {4468, 4940}, {4762, 4841}, {4943, 21115}, {4979, 6545}, {6009, 17494}, {6590, 23813}, {8712, 14298}, {23727, 23732}, {23736, 23760}, {23751, 23780}, {23753, 23754}

X(23730) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (2 a^4 - 2 a^3 b - a^2 b^2 + b^4 - 2 a^3 c + 4 a^2 b c - 2 b^3 c - a^2 c^2 + 2 b^2 c^2 - 2 b c^3 + c^4) : :

X(23730) lies on these lines: {7, 514}, {193, 4025}, {513, 14100}, {649, 7289}, {650, 1418}, {657, 1445}, {661, 18635}, {885, 4778}, {1419, 3676}, {3239, 4869}, {4336, 4449}, {4644, 21202}, {6545, 9318}, {9001, 16892}, {17276, 23757}, {17365, 21133}, {17375, 25259}, {21104, 23728}, {21105, 23766}

X(23731) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (2 a^2 + 2 a b + b^2 + 2 a c + c^2) : :

X(23731) lies on these lines: {513, 16892}, {514, 4024}, {1211, 6546}, {3004, 4750}, {4010, 4977}, {4369, 14475}, {4375, 6545}, {4778, 21116}, {4984, 21196}, {21104, 23728}, {23727, 23748}

X(23732) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^2 + b^2 + c^2) (-a^3 b + a^2 b^2 - a b^3 + b^4 - a^3 c - 2 a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + c^4) : :

X(23732) lies on these lines: {514, 18344}, {523, 1459}, {3649, 4804}, {21104, 23747}, {21107, 21118}, {23727, 23729}, {23738, 23748}

X(23733) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-2 a^5 - a^4 b + 3 a^3 b^2 + a^2 b^3 - a b^4 - a^4 c + b^4 c + 3 a^3 c^2 + 6 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(23733) lies on these lines: {513, 23799}, {693, 3664}, {17132, 25271}, {20295, 21173}, {23724, 23798}, {23806, 24793}

X(23734) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^5 b^2 - a^4 b^3 - a b^6 + b^7 + a^4 b^2 c - b^6 c + 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 - a^4 c^3 - b^4 c^3 + a b^2 c^4 - b^3 c^4 + b^2 c^5 - a c^6 - b c^6 + c^7) : :

X(23734) lies on these lines: {513, 23726}, {17463, 23773}

X(23735) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a b^4 - b^5 + b^4 c + a c^4 + b c^4 - c^5) : :

X(23735) lies on these lines: {513, 23754}, {21114, 23742}, {21118, 23726}, {23727, 23763}, {23759, 23768}

X(23736) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + 2 a^2 b^2 c - a b^3 c - b^4 c + a^3 c^2 + 2 a^2 b c^2 + 4 a b^2 c^2 - a^2 c^3 - a b c^3 - a c^4 - b c^4 + c^5) : :

X(23736) lies on these lines: {514, 9404}, {523, 2254}, {23729, 23760}, {23743, 23749}

X(23737) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^2 - a^2 b^3 - a b^4 + b^5 + a b^3 c - b^4 c + a^3 c^2 - a^2 c^3 + a b c^3 - a c^4 - b c^4 + c^5) : :

X(23737) lies on these lines: {514, 654}, {523, 2254}, {6154, 6366}, {7004, 21139}

X(23738) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (a b - b^2 + a c + 4 b c - c^2) : :

X(23738) lies on these lines: {513, 4162}, {514, 1734}, {661, 3777}, {764, 4983}, {1019, 2832}, {3309, 4959}, {3669, 4724}, {3762, 23789}, {3801, 21115}, {4017, 4778}, {4040, 14413}, {4435, 4979}, {4462, 21052}, {4801, 4804}, {4977, 17420}, {21104, 21118}, {23732, 23748}, {23755, 23764}

X(23739) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(23739) lies on these lines: {513, 3056}, {2533, 21113}, {3766, 21960}, {21104, 21114}

X(23740) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (a^2 b^2 - a b^3 + a b^2 c + a^2 c^2 + a b c^2 + 2 b^2 c^2 - a c^3) : :

X(23740) lies on these lines: {513, 23751}, {2254, 4976}, {21118, 23726}, {23728, 23763}, {23754, 23759}

X(23741) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b - 2 a^2 b^3 + b^5 + a^4 c + 2 a^2 b^2 c - 4 a b^3 c + b^4 c + 2 a^2 b c^2 + 8 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 + b c^4 + c^5) : :

X(23741) lies on these lines: {514, 14298}, {523, 2254}, {3064, 3669}, {23727, 23729}

X(23742) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b^3 - 2 a b^4 + b^5 + 2 a b^3 c - 2 b^4 c + b^3 c^2 + a^2 c^3 + 2 a b c^3 + b^2 c^3 - 2 a c^4 - 2 b c^4 + c^5) : :

X(23742) lies on these lines: {514, 8641}, {3801, 16892}, {21104, 23761}, {21114, 23735}

X(23743) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b^2 - a b^3 + 2 a b^2 c - b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - a c^3 - b c^3) : :

X(23743) lies on these lines: {513, 11934}, {514, 3709}, {523, 20507}, {798, 6084}, {918, 20954}, {4762, 7178}, {21111, 21133}, {21114, 21118}, {23736, 23749}, {23752, 23760}

X(23744) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b^2 - a b^3 - 2 a^2 b c + 2 a b^2 c - b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - a c^3 - b c^3) : :

X(23744) lies on these lines: {513, 11934}, {522, 20507}, {523, 3728}, {665, 23798}, {693, 20508}, {812, 7178}, {918, 23794}, {3700, 20954}, {3709, 23810}, {4762, 21120}, {4927, 21191}, {6084, 20979}, {20510, 21118}, {21114, 23760}, {23748, 23781}

X(23745) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b - 3 a b^2 + 2 b^3 + a^2 c + 4 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3) : :

X(23745) lies on these lines: {513, 23758}, {514, 4895}, {2254, 2826}, {2310, 21139}, {3810, 25259}, {14413, 21201}, {21104, 21118}, {21105, 23057}, {23764, 23770}

X(23746) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (2 a^2 b - 3 a b^2 + b^3 + 2 a^2 c + 8 a b c - b^2 c - 3 a c^2 - b c^2 + c^3) : :

X(23746) lies on these lines: {513, 3057}, {514, 4814}, {21104, 21118}

X(23747) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^3 - a^2 b^4 - a b^5 + b^6 + a^2 b^3 c - b^5 c + 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 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(23747) lies on these lines: {514, 1946}, {3801, 16892}, {21104, 23732}

X(23748) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3 + c^4) : :

X(23748) lies on these lines: {513, 14100}, {514, 657}, {523, 2254}, {918, 4171}, {20507, 21119}, {21102, 21133}, {21114, 21118}, {23727, 23731}, {23732, 23738}, {23744, 23781}

X(23749) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b^2 - c^2) (-a^2 b^2 + b^4 - a b^2 c - b^3 c - a^2 c^2 - a b c^2 - b c^3 + c^4) : :

X(23749) lies on these lines: {514, 7252}, {523, 21107}, {2525, 4036}, {4988, 21141}, {16892, 21102}, {21104, 21114}, {21118, 23726}, {21120, 21124}, {23736, 23743}, {23755, 23781}

X(23750) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^4 + b^4 + c^4) (-a^5 + b^5 - b^4 c - b c^4 + c^5) : :

X(23750) lies on these lines: (none)

X(23751) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a^2 (b - c) (a b^2 + b^3 - b^2 c + a c^2 - b c^2 + c^3) : :

X(23751) lies on these lines: {6, 649}, {513, 23740}, {514, 17789}, {647, 3768}, {661, 1639}, {3835, 18134}, {4079, 4813}, {6363, 7180}, {9313, 23655}, {9404, 23650}, {17778, 20295}, {21118, 23754}, {23726, 23759}, {23729, 23780}

X(23752) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b^2 - c^2) (-a^2 b + b^3 - a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : :

X(23752) lies on these lines: {242, 514}, {513, 21118}, {522, 4707}, {523, 656}, {650, 4802}, {1090, 16732}, {1577, 4064}, {2517, 23877}, {4036, 4088}, {4040, 21179}, {4077, 24006}, {4560, 21187}, {4777, 7655}, {4841, 6587}, {4977, 21111}, {6075, 18210}, {14413, 21112}, {20504, 21131}, {21104, 21114}, {23743, 23760}, {23757, 23775}

X(23753) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (b^2 + c^2) (-a^3 b + a^2 b^2 - a b^3 + b^4 - a^3 c - 2 a^2 b c - b^3 c + a^2 c^2 - a c^3 - b c^3 + c^4) : :

X(23753) lies on these lines: {514, 5299}, {23729, 23754}

X(23754) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (b^2 + c^2) (-2 a^3 - a b^2 + b^3 - b^2 c - a c^2 - b c^2 + c^3) : :

X(23754) lies on these lines: {513, 23735}, {4977, 21125}, {21118, 23751}, {23729, 23753}, {23740, 23759}

X(23755) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b^2 - c^2) (-2 a^2 - a b + b^2 - a c - 2 b c + c^2) : :

X(23755) lies on these lines: {239, 514}, {513, 21118}, {523, 4729}, {525, 4024}, {661, 6587}, {1577, 4120}, {1946, 4378}, {2533, 4088}, {2785, 17166}, {3328, 18210}, {3566, 4804}, {3669, 21828}, {4379, 6332}, {4801, 21116}, {4833, 11125}, {4893, 14837}, {4977, 21121}, {6372, 21132}, {21120, 21127}, {23738, 23764}, {23749, 23781}

X(23756) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a b - a c + b c) (a^2 b^2 - a b^3 - 2 a^2 b c + b^3 c + a^2 c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(23756) lies on these lines: {918, 4462}, {3665, 4444}, {5518, 21051}, {16892, 21128}, {20510, 21118}, {21104, 23765}

X(23757) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (2 a - b - c) (b - c) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : :

X(23757) lies on these lines: {1, 522}, {37, 650}, {121, 4768}, {190, 9268}, {192, 25259}, {513, 3057}, {514, 1000}, {523, 2292}, {656, 1834}, {900, 1317}, {944, 3667}, {1145, 1769}, {2920, 4057}, {3159, 4064}, {3259, 3326}, {3672, 21202}, {3987, 21189}, {4000, 7658}, {4370, 23972}, {7649, 7952}, {14442, 17464}, {14475, 24873}, {17246, 21133}, {17276, 23730}, {21104, 23760}, {21118, 23758}, {23752, 23775}

X(23758) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (a - 2 b - 2 c) (b - c) (a^2 b - b^3 + a^2 c + a b c + b^2 c + b c^2 - c^3) : :

X(23758) lies on these lines: {513, 23745}, {514, 4667}, {523, 10015}, {1459, 4802}, {4774, 4777}, {6370, 25259}, {21118, 23757}

X(23759) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^3 (b + c) (-2 a^2 - a b + b^2 - a c + c^2) : :

X(23759) lies on these lines:{333, 514}, {513, 23763}, {764, 21134}, {6545, 21131}, {23726, 23751}, {23735, 23768}, {23740, 23754}, {23760, 23777}, {23761, 23764}

X(23760) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^3 (a^2 - 2 a b + b^2 - 2 a c + c^2) : :

X(23760) lies on these lines: {9, 514}, {344, 4468}, {513, 14100}, {522, 6601}, {693, 1229}, {1108, 3669}, {1445, 2402}, {3309, 15185}, {3676, 4000}, {3942, 21143}, {4686, 4777}, {21104, 23757}, {21114, 23744}, {21132, 21133}, {23729, 23736}, {23743, 23752}, {23759, 23777}, {23762, 23775}, {23764, 23766}

X(23761) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (a - b - c) (-b + c)^3 (a b - b^2 + a c + b c - c^2) : :

X(23761) lies on these lines: {11, 6545}, {55, 514}, {522, 4863}, {3676, 17728}, {4124, 21132}, {5326, 6544}, {5432, 6546}, {6548, 10589}, {17597, 21185}, {21104, 23742}, {21118, 24111}, {23759, 23764}

X(23762) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^3 (b + c) (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + c^4) : :

X(23762) lies on these lines: {284, 514}, {16732, 21131}, {21133, 21134}, {23760, 23775}

X(23763) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (-b + c) (2 a^5 - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 + b^5 + 2 a^2 b^2 c - b^4 c - 2 a^3 c^2 + 2 a^2 b c^2 - 2 a^2 c^3 + a c^4 - b c^4 + c^5) : :

X(23763) lies on these lines: {513, 23759}, {4449, 7073}, {23727, 23735}, {23728, 23740}

X(23764) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (3 a - b - c) (b - c)^3 : :

X(23764) lies on these lines: {8, 514}, {10, 21129}, {46, 4498}, {145, 2403}, {513, 3057}, {522, 3680}, {649, 2082}, {764, 1647}, {1022, 21201}, {1420, 8643}, {3987, 4905}, {4024, 22036}, {4448, 24099}, {6546, 9458}, {21118, 23765}, {23738, 23755}, {23745, 23770}, {23759, 23761}, {23760, 23766}

X(23765) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (b^2 - 3 b c + c^2) : :

X(23765) lies on these lines: {10, 514}, {513, 4162}, {659, 3669}, {667, 2832}, {814, 21222}, {891, 4905}, {1022, 4040}, {3309, 21343}, {3762, 23815}, {3803, 4367}, {3837, 4462}, {4784, 8712}, {4807, 23796}, {21104, 23756}, {21118, 23764}

X(23766) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (4 a^2 - 3 a b - b^2 - 3 a c + 4 b c - c^2) : :

X(23766) lies on these lines: {11, 21139}, {514, 4089}, {21105, 23730}, {23760, 23764}

X(23767) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 + a^2 b^2 c - 2 b^4 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5) : :

X(23767) lies on these lines: {21118, 23726}, {23771, 23772}

X(23768) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (a b^3 - a b^2 c - a b c^2 - b^2 c^2 + a c^3) : :

X(23768) lies on these lines: {513, 23740}, {514, 25128}, {523, 21438}, {650, 876}, {693, 8034}, {764, 3776}, {1491, 4841}, {2512, 4490}, {3004, 3777}, {4468, 21349}, {21104, 23756}, {23735, 23759}

X(23769) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - 3 a^2 b^2 + 3 a b^3 - b^4 + a^3 c + 6 a^2 b c - 3 a b^2 c + 4 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 6 b^2 c^2 + 3 a c^3 + 4 b c^3 - c^4) : :

X(23769) lies on these lines: {513, 11934}, {514, 4130}, {657, 6084}, {918, 23819}, {3676, 17410}, {4832, 7178}

X(23770) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b + b^3 + a^2 c - 2 a b c - b^2 c - b c^2 + c^3) : :

X(23770) lies on these lines: {2, 2977}, {11, 2969}, {105, 659}, {149, 900}, {325, 523}, {513, 11934}, {514, 3716}, {522, 3776}, {764, 2826}, {812, 4458}, {876, 7233}, {891, 10015}, {918, 4010}, {1638, 9508}, {1639, 4802}, {2254, 6545}, {2832, 21201}, {2879, 24476}, {3777, 6362}, {3801, 3910}, {4083, 7178}, {4088, 4728}, {4804, 16892}, {4809, 6009}, {4830, 13246}, {4913, 21212}, {6366, 21343}, {17464, 20504}, {18220, 24097}, {21204, 25380}, {23745, 23764}

X(23770) = isotomic conjugate of X(35574)

X(23771) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a (-b + c)^2 (a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - a^2 b^2 c - a^3 c^2 - a^2 b c^2 + 2 b^3 c^2 - a^2 c^3 + 2 b^2 c^3 + a c^4) : :

X(23771) lies on these lines: {3122, 7200}, {23767, 23772}

X(23772) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(23772) lies on these lines: {11, 2969}, {56, 653}, {75, 7209}, {116, 4092}, {244, 24136}, {513, 24840}, {514, 3271}, {522, 4014}, {523, 1086}, {784, 4403}, {1015, 21210}, {1111, 4516}, {1358, 4081}, {1362, 4605}, {1441, 17447}, {2310, 21139}, {3122, 20512}, {3123, 21140}, {3136, 21325}, {3675, 4858}, {3942, 4459}, {4367, 22096}, {4552, 21320}, {4953, 6362}, {7354, 12135}, {23767, 23771}

X(23773) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a b + a c - b c) (a b - b^2 + a c - c^2) : :

X(23773) lies on these lines: {11, 23776}, {192, 1423}, {1647, 16892}, {2310, 21139}, {3123, 21138}, {17463, 23734}, {23760, 23764}

X(23774) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^3 - a^2 b - a^2 c + a b c + b^2 c + b c^2) : :

X(23774) lies on these lines: {11, 23776}, {192, 1423}, {1647, 16892}, {2310, 21139}, {3123, 21138}, {17463, 23734}, {23760, 23764}

X(23775) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^3 (b + c) (a^3 - a^2 b - a b^2 + b^3 - a^2 c - a b c - a c^2 + c^3) : :

X(23775) lies on these lines: {21, 514}, {513, 17637}, {522, 6598}, {523, 21677}, {764, 18210}, {867, 6545}, {1834, 7178}, {3801, 4804}, {3887, 4707}, {21132, 21134}, {23752, 23757}, {23759, 23761}, {23760, 23762}

X(23776) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^2 b^2 - a b^3 + a^2 b c - a b^2 c + b^3 c + a^2 c^2 - a b c^2 - a c^3 + b c^3) : :

X(23776) lies on these lines: {11, 23773}, {244, 16892}, {7004, 21139}, {23767, 23771}

X(23777) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c)^3 (a^2 + a b + a c - 2 b c) : :

X(23777) lies on these lines: {312, 514}, {513, 21342}, {3175, 4106}, {4383, 4498}, {6545, 8042}, {23759, 23760}

X(23778) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^3 b^3 - a^2 b^4 + a^3 b^2 c + a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^3 c^3 - a b^2 c^3 - a^2 c^4 + b^2 c^4) : :

X(23778) lies on these lines: {17463, 23734}, {23767, 23771}

X(23779) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a^2 b^2 - 2 a b^3 + 2 b^4 + a^3 c + 2 a^2 b c + a b^2 c - b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 - 2 a c^3 - b c^3 + 2 c^4) : :

X(23779) lies on these lines: {2310, 21139}, {21104, 21114}, {21136, 21145}

X(23780) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a^2 b^2 + a^3 c + 4 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 + b c^3) : :

X(23780) lies on these lines: {513, 3056}, {514, 24290}, {21104, 21118}, {23729, 23751}

X(23781) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = CEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a b^3 + b^4 + 2 a^2 b c - b^3 c - a c^3 - b c^3 + c^4) : :

X(23781) lies on these lines: {513, 21114}, {514, 4435}, {812, 20513}, {900, 21133}, {7004, 21139}, {16892, 21128}, {20507, 21132}, {20980, 21182}, {21104, 21118}, {23729, 23753}, {23744, 23748}, {23749, 23755}

X(23782) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (a^4 b - b^5 + a^4 c + a^3 b c - a^2 b^2 c - a b^3 c - a^2 b c^2 + b^3 c^2 - a b c^3 + b^2 c^3 - c^5) : :

X(23782) lies on these lines: {10, 15416}, {513, 7254}, {663, 830}, {693, 21174}, {1519, 1769}, {4905, 20295}, {7253, 21189}, {8714, 23801}, {21109, 21118}, {23789, 23798}

X(23783) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^6 b + a^5 b^2 - a^2 b^5 - a b^6 + a^6 c + a^4 b^2 c - a^2 b^4 c - b^6 c + a^5 c^2 + a^4 b c^2 + a b^4 c^2 + b^5 c^2 - a^2 b c^4 + a b^2 c^4 - a^2 c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(23783) lies on these lines: {513, 676}, {514, 2485}, {15413, 23788}, {23874, 24718}

X(23784) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^6 b + a^5 b^2 - a^2 b^5 - a b^6 + a^6 c + a^4 b^2 c - a^2 b^4 c - b^6 c + 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 - a^2 b c^4 + a b^2 c^4 - a^2 c^5 + b^2 c^5 - a c^6 - b c^6) : :

X(23784) lies on these lines: {513, 676}, {514, 2492}, {2486, 23822}

X(23785) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a b^3 + a^3 c - b^3 c - a c^3 - b c^3) : :

X(23785) lies on these lines: {513, 23829}, {514, 798}, {521, 3879}, {522, 693}, {802, 3776}, {824, 22044}, {918, 3709}, {1638, 17066}, {2509, 17353}, {2786, 21194}, {3004, 8672}, {3261, 20520}, {4357, 15413}, {4397, 4967}, {8714, 20954}, {15419, 21173}, {18155, 23723}, {20517, 21178}, {20518, 21187}, {21210, 23826}, {23786, 23793}, {23794, 23810}

X(23786) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b - a b^4 + a^4 c - b^4 c - a c^4 - b c^4) : :

X(23786) lies on these lines: {513, 3776}, {514, 669}, {647, 23887}, {784, 4142}, {3741, 23799}, {4025, 8714}, {21204, 23818}, {23724, 23812}, {23785, 23793}

X(23787) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - b^4 c + a^3 c^2 + 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - b c^4) : :

X(23787) lies on these lines: {7, 3676}, {693, 21189}, {4025, 20954}, {4467, 20520}

X(23788) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (a + b) (b - c) (a + c) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : :

X(23788) lies on these lines: {88, 4049}, {162, 658}, {244, 1111}, {514, 16754}, {693, 21189}, {1019, 21188}, {1443, 1447}, {3261, 4025}, {3310, 10015}, {3741, 23817}, {4560, 14837}, {7199, 21183}, {7253, 7661}, {7658, 16751}, {14208, 25008}, {15413, 23783}, {18815, 21180}, {21297, 23810}

X(23789) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b - a b^2 + a^2 c + 2 a b c + b^2 c - a c^2 + b c^2) : :

X(23789) lies on these lines: {10, 514}, {513, 11813}, {519, 21302}, {551, 663}, {693, 4905}, {891, 4807}, {1125, 4040}, {1734, 4801}, {2254, 4151}, {3244, 4449}, {3663, 4406}, {3676, 20517}, {3762, 23738}, {3837, 4129}, {3840, 4379}, {4142, 21181}, {4374, 20525}, {4724, 19862}, {4794, 15808}, {6549, 24232}, {7192, 23791}, {7199, 23790}, {11019, 21183}, {16737, 17205}, {23782, 23798}

X(23790) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c + b^3 c - a c^3 + b c^3) : :

X(23790) lies on these lines: {10, 20949}, {513, 4357}, {514, 8061}, {522, 693}, {4785, 21191}, {4905, 15413}, {4967, 20906}, {7199, 23789}, {17023, 21007}, {17353, 21390}, {21099, 21261}

X(23791) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^2 - a^2 b^3 + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(23791) lies on these lines: {10, 17494}, {513, 3716}, {514, 3005}, {693, 3741}, {850, 20525}, {2525, 23887}, {3971, 21611}, {4025, 8714}, {7192, 23789}, {21204, 23805}, {23725, 23812}, {23823, 24192}

X(23792) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (a^3 b + a^2 b^2 - a b^3 - b^4 + a^3 c + a^2 b c - a b^2 c - b^3 c + a^2 c^2 - a b c^2 + 4 b^2 c^2 - a c^3 - b c^3 - c^4) : :

X(23792) lies on these lines: {513, 7203}, {693, 21189}, {1769, 3676}, {4106, 23800}, {4905, 20295}, {17896, 20520}

X(23793) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b - a^3 b^2 + a^2 b^3 - a b^4 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c - a^3 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(23793) lies on these lines: {693, 23811}, {2826, 7180}, {3676, 4905}, {8714, 18155}, {17494, 17594}, {21201, 21212}, {23785, 23786}

X(23794) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    b c (b - c) (-2 a^2 + a b + a c + b c) : :

X(23794) lies on these lines: {75, 4408}, {239, 4501}, {320, 350}, {514, 4502}, {522, 3766}, {798, 4380}, {812, 4391}, {900, 3261}, {918, 23744}, {1577, 4785}, {3667, 4374}, {4140, 4462}, {4728, 21191}, {4777, 20949}, {4790, 18154}, {4820, 20952}, {4962, 20907}, {8714, 23807}, {23785, 23810}, {23798, 23829}

X(23795) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (3 a^2 b - 3 a b^2 + 3 a^2 c + 2 a b c - b^2 c - 3 a c^2 - b c^2) : :

X(23795) lies on these lines: {10, 2254}, {513, 23809}, {514, 4730}, {519, 21222}, {551, 3960}, {693, 4905}, {900, 21630}, {2814, 5493}, {3122, 17205}, {3244, 3887}, {3716, 19862}, {4010, 23814}, {6372, 22320}, {21181, 21201}

X(23796) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (3 a^2 b - 3 a b^2 + 3 a^2 c + 4 a b c + b^2 c - 3 a c^2 + b c^2) : :

X(23796) lies on these lines: {513, 1125}, {514, 4770}, {693, 4905}, {3762, 21052}, {4791, 24720}, {4807, 23765}

X(23797) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^5 b - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^3 b c^2 + a b^3 c^2 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - a c^5 - b c^5) : :

X(23797) lies on these lines: {514, 810}, {693, 21174}, {4905, 23811}, {7649, 21188}, {8714, 18155}, {17924, 23537}, {20517, 21178}

X(23798) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a b^3 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c + 2 a b c^2 + 2 b^2 c^2 - a c^3 - b c^3) : :

X(23798) lies on these lines: {513, 3664}, {514, 3709}, {522, 4411}, {665, 23744}, {693, 21189}, {3663, 24002}, {3667, 3676}, {4905, 23819}, {8714, 20954}, {21179, 21202}, {21182, 21185}, {23724, 23733}, {23782, 23789}, {23794, 23829}, {24225, 24236}

X(23799) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^4 b - a^3 b^2 + a^2 b^3 + a b^4 - a^4 c + b^4 c - a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(23799) lies on these lines: {241, 514}, {513, 23733}, {693, 21189}, {850, 4025}, {3663, 17896}, {3664, 4131}, {3741, 23786}, {4847, 23687}, {7192, 21173}, {8714, 18155}, {21208, 24192}, {23807, 23829}, {23810, 23813}, {23824, 24194}

X(23800) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    a (b - c) (a^2 b - b^3 + a^2 c + a b c - c^3) : :

X(23800) lies on these lines: {1, 15313}, {36, 238}, {84, 23838}, {514, 656}, {521, 3669}, {522, 693}, {523, 1734}, {649, 16612}, {802, 4444}, {832, 4367}, {900, 5533}, {1394, 14812}, {1459, 3960}, {1491, 8672}, {1519, 1769}, {2457, 21102}, {2509, 21390}, {3309, 6129}, {3762, 20316}, {3777, 6371}, {4010, 23818}, {4064, 23875}, {4086, 17072}, {4106, 23792}, {4131, 7203}, {4551, 15632}, {4778, 17420}, {4858, 23820}, {6006, 6615}, {7649, 21188}, {7650, 8714}, {15413, 23829}, {20293, 21222}, {21348, 24290}, {23810, 23819}

X(23801) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 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 + b c^4) : :

X(23801) lies on these lines: {693, 21189}, {905, 4106}, {3663, 4025}, {4077, 20520}, {4656, 25259}, {4823, 14837}, {7658, 24175}, {8714, 23782}, {20295, 21173}, {21174, 21182}

X(23802) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^9 - a^8 b + a^5 b^4 + a^4 b^5 - a^8 c + b^8 c + a^5 c^4 - b^5 c^4 + a^4 c^5 - b^4 c^5 + b c^8) : :

X(23802) lies on these lines: {21190, 23723}

X(23803) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^3 b^2 - a^2 b^3 - a^3 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(23803) lies on these lines: {10, 20983}, {513, 3716}, {514, 850}, {726, 21350}, {786, 4842}, {838, 17072}, {894, 21392}, {1019, 17174}, {3960, 23724}, {4106, 23825}, {4129, 4813}, {4502, 24083}, {6373, 23301}, {8640, 24674}, {8714, 23805}, {14426, 25126}, {17197, 24194}, {21173, 23806}, {21204, 23723}

X(23804) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^5 b + a b^5 - a^5 c + 2 a^2 b^3 c + b^5 c + 2 a^2 b c^3 + 2 b^3 c^3 + a c^5 + b c^5) : :

X(23804) lies on these lines: {693, 21193}, {2786, 21194}, {4142, 21208}, {7265, 22011}, {20295, 23805}, {21192, 25381}

X(23805) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^4 b + 2 a^3 b^2 + a b^4 - a^4 c + b^4 c + 2 a^3 c^2 + 2 a b^2 c^2 + a c^4 + b c^4) : :

X(23805) lies on these lines: {244, 21212}, {513, 3776}, {4024, 22011}, {8714, 23803}, {16892, 17140}, {20295, 23804}, {21204, 23791}

X(23806) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^4 b + a^3 b^2 + a^2 b^3 - a b^4 - a^4 c + b^4 c + a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(23806) lies on these lines: {142, 4885}, {226, 650}, {514, 3064}, {516, 8641}, {693, 5249}, {905, 4106}, {1938, 3671}, {2254, 3667}, {3663, 25098}, {3817, 15283}, {4129, 14837}, {4500, 17758}, {6591, 23724}, {10006, 21635}, {11019, 15280}, {14077, 21620}, {14400, 23728}, {21173, 23803}, {22383, 23725}, {23733, 24793}

X(23807) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    b c (b - c) (-a^2 b^2 - a^2 c^2 + b^2 c^2) : :

X(23807) lies on these lines: {75, 4083}, {76, 21260}, {274, 667}, {321, 21836}, {514, 4374}, {693, 784}, {772, 21123}, {905, 14296}, {1930, 21440}, {3250, 21438}, {3766, 4025}, {8631, 24356}, {8714, 23794}, {16747, 17924}, {20910, 21191}, {21056, 22028}, {23799, 23829}

X(23808) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (3 a^3 b - 3 a b^3 + 3 a^3 c - 8 a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 + 2 b^2 c^2 - 3 a c^3 + b c^3) : :

X(23808) lies on these lines: {2, 23838}, {5, 3667}, {10, 522}, {142, 3835}, {513, 1125}, {514, 4364}, {693, 4357}, {764, 4778}, {900, 6702}, {1387, 3738}, {1643, 3707}, {1769, 3762}, {2254, 24183}, {2827, 6713}, {3616, 14812}, {3737, 17588}, {5051, 6615}, {7649, 11105}, {8714, 23809}, {21174, 24162}, {21189, 24178}, {21193, 24180}

X(23809) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-3 a^3 b + 3 a b^3 - 3 a^3 c + a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 + 2 b^2 c^2 + 3 a c^3 + b c^3) : :

X(23809) lies on these lines: {513, 23795}, {522, 14315}, {900, 3835}, {1734, 4443}, {3900, 14563}, {8714, 23808}, {24184, 24457}

X(23810) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a b^3 + a^3 c - 4 a^2 b c + 2 a b^2 c - b^3 c + 2 a b c^2 + 2 b^2 c^2 - a c^3 - b c^3) : :

X(23810) lies on these lines: {7, 3676}, {37, 514}, {75, 522}, {513, 3664}, {650, 6666}, {693, 4357}, {900, 20520}, {3663, 24457}, {3709, 23744}, {3945, 14812}, {4106, 23723}, {4526, 20507}, {6545, 24403}, {17260, 17494}, {21200, 24195}, {21201, 21202}, {21204, 21211}, {21297, 23788}, {23785, 23794}, {23799, 23813}, {23800, 23819}, {23814, 23816}

X(23811) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^4 b - a^3 b^2 + a^2 b^3 - a b^4 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c - a^3 c^2 - 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 - b c^4) : :

X(23811) lies on these lines: {38, 522}, {42, 514}, {614, 885}, {693, 23793}, {1647, 21201}, {1734, 25006}, {3741, 8714}, {3757, 17496}, {4025, 20518}, {4905, 23797}, {11019, 21183}, {17194, 21173}, {21209, 24186}

X(23812) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    2 a^3 + 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(23812) lies on these lines: {2, 17770}, {10, 2895}, {51, 2392}, {86, 4425}, {226, 5061}, {513, 3742}, {514, 8029}, {519, 17163}, {551, 5426}, {940, 25385}, {1125, 6536}, {1961, 21093}, {1962, 2796}, {2887, 4670}, {3120, 8025}, {3616, 12579}, {3664, 3741}, {3821, 19684}, {3980, 5712}, {4011, 4648}, {4297, 16124}, {4655, 19701}, {4683, 5333}, {4697, 17056}, {4703, 15668}, {4854, 5625}, {4892, 6703}, {7321, 17600}, {10180, 17768}, {11263, 17167}, {17379, 17889}, {17778, 21085}, {18139, 24295}, {23724, 23786}, {23725, 23791}, {23823, 24210}

X(23813) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (-a^2 - a b - a c + 4 b c) : :

X(23813) lies on these lines: {2, 2516}, {320, 350}, {514, 4940}, {650, 4382}, {812, 4394}, {850, 4145}, {900, 3676}, {1577, 8712}, {2526, 4804}, {3004, 4777}, {3239, 6084}, {3835, 4762}, {4025, 4926}, {4369, 6008}, {4379, 4790}, {4820, 16892}, {4897, 21183}, {6009, 11068}, {6590, 23729}, {23799, 23810}

X(23813) = anticomplement of X(2516)

X(23814) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b - 3 a b^2 + a^2 c + 2 a b c + b^2 c - 3 a c^2 + b c^2) : :

X(23814) lies on these lines: {8, 1022}, {10, 514}, {513, 1125}, {522, 596}, {551, 6161}, {693, 20888}, {946, 3667}, {1647, 21201}, {2827, 20418}, {2832, 25380}, {3244, 14421}, {3309, 13464}, {3665, 3676}, {3669, 10106}, {3835, 17758}, {4010, 23795}, {4049, 21132}, {4448, 19862}, {4701, 9260}, {4905, 12047}, {5270, 21301}, {6789, 23345}, {7192, 17210}, {8714, 23815}, {17169, 20295}, {19945, 21211}, {23810, 23816}

X(23815) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a b^2 - a b c - b^2 c + a c^2 - b c^2) : :

X(23815) lies on these lines: {512, 24720}, {513, 11813}, {514, 3837}, {693, 784}, {764, 4391}, {814, 3960}, {826, 3776}, {891, 17072}, {905, 19947}, {1019, 24719}, {1491, 4978}, {1577, 3777}, {2787, 3669}, {3762, 23765}, {3835, 6372}, {3900, 21627}, {4010, 4905}, {4083, 4807}, {4106, 23828}, {4142, 21204}, {4378, 21301}, {4462, 14431}, {4705, 4801}, {4818, 6367}, {4992, 6005}, {6362, 15280}, {8714, 23814}, {14349, 21146}

X(23816) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (3 a^2 + a b - 2 b^2 + a c - b c - 2 c^2) : :

X(23816) lies on these lines: {58, 86}, {69, 6789}, {1565, 3756}, {1647, 4089}, {2424, 15634}, {3120, 17198}, {3416, 19899}, {3663, 23869}, {23810, 23814}

X(23817) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (2 a^4 b + a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 a^4 c - 2 b^4 c + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - 2 a c^4 - 2 b c^4) : :

X(23817) lies on these lines: {351, 514}, {3122, 23820}, {3741, 23788}, {4025, 8714}, {4750, 21181}, {14417, 23887}

X(23818) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^2 b^3 - a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(23818) lies on these lines: {513, 3716}, {514, 23301}, {661, 21260}, {693, 784}, {4010, 23800}, {21204, 23786}

X(23819) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    b c (b - c) (5 a^2 - 2 a b + b^2 - 2 a c - 2 b c + c^2) : :

X(23819) lies on these lines: {320, 350}, {514, 4171}, {657, 812}, {918, 23769}, {3667, 24002}, {3766, 4397}, {3835, 21127}, {4380, 25009}, {4462, 6084}, {4885, 17410}, {4905, 23798}, {6182, 21301}, {23800, 23810}

X(23820) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^5 b - a^4 b^2 - a^3 b^3 + a^2 b^4 + 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^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 + b^2 c^4) : :

X(23820) lies on these lines: {514, 20975}, {693, 2310}, {3122, 23817}, {4092, 17072}, {4858, 23800}

X(23821) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^3 - 2 a^2 b + a b^2 - 2 a^2 c + 3 a b c + b^2 c + a c^2 + b c^2) : :

X(23821) lies on these lines: {10, 190}, {11, 1357}, {79, 1156}, {256, 2481}, {513, 2486}, {514, 4516}, {516, 1756}, {522, 16732}, {1111, 2310}, {2795, 22003}, {3122, 17205}, {3271, 17761}, {3551, 24230}, {3667, 4459}, {4553, 22031}, {4683, 4847}, {4778, 7202}, {4896, 11019}, {4905, 24234}, {10868, 18698}, {13576, 21362}, {17792, 22019}

X(23822) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^3 b + a^2 b^2 + a^3 c + a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 - b^2 c^2) : :

X(23822) lies on these lines: {10, 75}, {115, 116}, {1111, 3123}, {2486, 23784}, {3120, 23824}, {3122, 17205}, {23810, 23814}

X(23823) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^3 + a b^2 + 3 a b c + b^2 c + a c^2 + b c^2) : :

X(23823) lies on these lines: {86, 24227}, {513, 1086}, {514, 2643}, {527, 23682}, {784, 4403}, {1111, 21210}, {1738, 17770}, {2486, 16726}, {3120, 17197}, {3122, 17205}, {3125, 23774}, {3248, 17761}, {3914, 4667}, {4274, 11246}, {4516, 7200}, {7321, 24212}, {8714, 24225}, {23791, 24192}, {23812, 24210}

X(23824) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (a + b) (a + c) (a b + a c - b c) (b - c)^2 : :

X(23824) lies on these lines: {244, 1111}, {310, 4609}, {514, 3121}, {1647, 17198}, {1978, 22045}, {3120, 23822}, {3122, 23817}, {3741, 16739}, {3840, 16703}, {4025, 24186}, {4871, 18157}, {6377, 16742}, {16708, 17063}, {16887, 24172}, {23799, 24194}

X(23825) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b^2 + a^2 b^3 - 2 a^3 b c - 2 a^2 b^2 c + a^3 c^2 - 2 a^2 b c^2 + 4 a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(23825) lies on these lines: {2, 649}, {321, 514}, {513, 3741}, {693, 8042}, {812, 21894}, {3840, 8027}, {4106, 23803}, {6384, 20954}, {21204, 21211}

X(23826) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c)^2 (a^4 b^2 + a^3 b^3 + 2 a^4 b c + a^3 b^2 c + a^2 b^3 c + a^4 c^2 + 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) : :

X(23826) lies on these lines: {514, 1084}, {693, 3123}, {2486, 23784}, {3122, 23817}, {21210, 23785}

X(23827) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (3 a^3 b - 3 a b^3 + 3 a^3 c + 2 a^2 b c - b^3 c - 3 a c^3 - b c^3) : :

X(23827) lies on these lines: {514, 2642}, {522, 693}, {3122, 17205}, {3882, 17136}, {4357, 23829}, {21181, 21205}

X(23828) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (b - c) (a^3 b - a b^3 + a^3 c + 4 a^2 b c + b^3 c - a c^3 + b c^3) : :

X(23828) lies on these lines: {513, 4357}, {693, 4905}, {1019, 17174}, {4106, 23815}

X(23829) =  (A,B,C,X(2); A',B',C',X(513)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(514)

Barycentrics    (a + b) (a + c) (b - c) (a b - b^2 + a c - c^2) : :

X(23829) lies on these lines: {99, 109}, {239, 514}, {244, 1111}, {335, 2786}, {512, 4897}, {513, 23785}, {522, 4406}, {665, 918}, {693, 4905}, {2481, 4458}, {3004, 6372}, {3667, 17217}, {3676, 18033}, {3737, 15419}, {3879, 8674}, {3912, 24290}, {4151, 4467}, {4357, 23827}, {4444, 24287}, {4468, 16751}, {4736, 24038}, {7253, 17096}, {15413, 23800}, {17023, 24285}, {20295, 23804}, {23794, 23798}, {23799, 23807}

X(23830) =  X(1)X(6)∩X(190)X(2415)

Barycentrics    a (a-b) (a-c) (a^2 b-3 a b^2+a^2 c+2 b^2 c-3 a c^2+2 b c^2) : :

X(23830) lies on the cubic K085 and these lines: {1,6}, {190,2415}, {644,21343}, {4557,8660}

X(23831) =  X(1)X(21)∩X(100)X(1293)

Barycentrics    a (a-b) (a-c) (a^3-a^2 b-2 a b^2-a^2 c+3 a b c+b^2 c-2 a c^2+b c^2) : :

X(23831) lies on the cubic K085 and these lines: {1,21}, {100,1293}, {110,8690}, {662,23363}, {901,6079}, {4427,6089}

X(23831) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {4582, 21294}, {5376, 2893}, {5548, 21221}, {9268, 2475}
X(23831) = crosspoint of X(4564) and X(6635)
X(23831) = crossdifference of every pair of points on line {661, 16613}
X(23831) = crosssum of X(2170) and X(8661)

X(23832) =  X(1)X(3)∩X(100)X(190)

Barycentrics    a^2 (a-b) (a-c) (a b+b^2+a c-4 b c+c^2) : :

X(23832) lies on the cubics K085 and K723 and on these lines: {1,3}, {10,13744}, {11,19515}, {31,16493}, {100,190}, {101,6014}, {109,1293}, {649,2284}, {883,4897}, {901,4638}, {902,16501}, {1011,16500}, {1026,6161}, {1054,23404}, {1149,17109}, {1376,9458}, {2222,2743}, {3052,16492}, {3869,15625}, {3880,23205}, {6174,15507}, {8053,16494}, {8698,8701}, {9352,18613}, {11260,22376}, {12329,16504}, {13205,20999}, {16495,20992}

X(23832) = X(5376)-Ceva conjugate of X(6)
X(23832) = X(6085)-cross conjugate of X(1149)
X(23832) = X(i)-isoconjugate of X(j) for these (i,j): {244, 6079}, {513, 1120}, {522, 8686}, {1811, 7649}
X(23832) = cevapoint of X(1149) and X(6085)
X(23832) = crosspoint of X(i) and X(j) for these (i,j): {59, 6551}, {100, 901} X(23832) = trilinear pole of line {1149, 20972}
X(23832) = crossdifference of every pair of points on line {650, 1015}
X(23832) = crosssum of X(i) and X(j) for these (i,j): {11, 6550}, {512, 21894}, {513, 900}, {514, 4928}, {1647, 23764}
{X(100),X(4781)}-harmonic conjugate of X(4436)
X(23832) = X(643)-beth conjugate of X(17780)
X(23832) = X(519)-zayin conjugate of X(513)
X(23832) = barycentric product X(i)*X(j) for these {i,j}: {57, 23705}, {100, 16610}, {101, 1266}, {190, 1149}, {651, 3880}, {662, 4695}, {901, 16594}, {1016, 6085}, {1252, 4927}, {1332, 1878}, {3257, 17460}, {4555, 20972}, {4557, 16711}, {4591, 21041}, {6335, 23205}, {9268, 21129}
X(23832) = barycentric quotient X(i)/X(j) for these {i,j}: {101, 1120}, {906, 1811}, {1149, 514}, {1252, 6079}, {1266, 3261}, {1415, 8686}, {1878, 17924}, {3880, 4391}, {4695, 1577}, {6085, 1086}, {8660, 1015}, {16610, 693}, {17109, 1022}, {17460, 3762}, {20972, 900}, {23205, 905}, {23705, 312}

X(23833) =  X(1)X(2)∩X(3699)X(3837)

Barycentrics    (a-b) (a-c) (a^2 b^2+a b^3-4 a^2 b c+2 a b^2 c-b^3 c+a^2 c^2+2 a b c^2-2 b^2 c^2+a c^3-b c^3) : :

X(23833) lies on the cubic K085 and these lines: {1,2}, {3699,3837}, {3888,3952}

X(23833) = crossdifference of every pair of points on line {649, 16614}

X(23834) =  X(1)X(4401)∩X(2)X(4406)

Barycentrics    a (b-c) (2 a^2 b+2 a b^2-3 a^2 c-3 b^2 c+a c^2+b c^2) (3 a^2 b-a b^2-2 a^2 c-b^2 c-2 a c^2+3 b c^2) : :

X(23834) lies on the conic {{A,B,,C,X(1),X(2)}}, the cubics K085 and K090, and on these lines: {1,4401}, {2,4106}, {88,2441}, {649,8056}, {659,1280}

X(23835) =  X(1)X(4139)∩X(10)X(3667)

Barycentrics    a (b-c) (2 a^2 b+a b^2-b^3-a^2 c-3 a b c+b^2 c-a c^2+2 b c^2) (a^2 b+a b^2-2 a^2 c+3 a b c-2 b^2 c-a c^2-b c^2+c^3) : :

X(23835) lies on the cubic K085 and these lines: {1,4139}, {10,3667}, {37,4394}, {75,4897}, {4674,6085}

X(23836) =  X(1)X(3667)∩X(8)X(513)

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(23836) lies on the Feuerbach hyperbola, the cubic K085, and on these lines: {1,3667}, {7,4106}, {8,513}, {9,649}, {21,3733}, {80,2827}, {104,8686}, {314,7192}, {514,4900}, {522,3680}, {900,1120}, {901,6079}, {1000,3309}, {2254,9365}, {3572,4876}, {3738,12641}, {5559,6003}

X(23836) = X(6079)-anticomplementary conjugate of X(21290)
X(23836) = X(6079)-Ceva conjugate of X(1120)
X(23836) = X(1639)-cross conjugate of X(514)
X(23836) = isotomic conjugate of the anticomplement X(2087)
X(23836) = X(7253)-beth conjugate of X(3680)
X(23836) = cevapoint of X(i) and X(j) for these (i,j): {11, 6550}, {512, 21894}, {513, 900}, {514, 4928}, {1647, 23764}
X(23836) = crosspoint of X(i) and X(j) for these (i,j): {1120, 6079}
X(23836) = trilinear pole of line {650, 1015}
X(23836) = crossdifference of every pair of points on line {1149, 20972}
X(23836) = crosssum of X(1149) and X(6085)
X(23836) = X(i)-isoconjugate of X(j) for these (i,j): {56, 23705}, {100, 1149}, {101, 16610}, {109, 3880}, {110, 4695}, {692, 1266}, {765, 6085}, {901, 17460}, {1110, 4927}, {1331, 1878}, {1897, 23205}, {3257, 20972}, {7035, 8660}, {17109, 17780}
X(23836) = barycentric product X(i)*X(j) for these {i,j}: {514, 1120}, {1086, 6079}, {1811, 17924}, {4391, 8686}
X(23836) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 23705}, {513, 16610}, {514, 1266}, {649, 1149}, {650, 3880}, {661, 4695}, {900, 16594}, {1015, 6085}, {1086, 4927}, {1120, 190}, {1635, 17460}, {1647, 21129}, {1811, 1332}, {1960, 20972}, {1977, 8660}, {3762, 20900}, {4120, 21041}, {6079, 1016}, {6591, 1878}, {7192, 16711}, {8686, 651}, {22086, 22082}, {22383, 23205}

X(23837) =  X(513)X(979)∩X(1120)X(4491)

Barycentrics    a^2 (b-c) (a^3 b+2 a^2 b^2+a b^3-a^3 c-2 a^2 b c-2 a b^2 c-b^3 c-a^2 c^2+4 a b c^2-b^2 c^2) (a^3 b+a^2 b^2-a^3 c+2 a^2 b c-4 a b^2 c-2 a^2 c^2+2 a b c^2+b^2 c^2-a c^3+b c^3) : :

X(23837) lies on the conic {{A,B,C,X(1),X(6)}}, the cubic K085, and on these lines: {513,979}, {1120,4491}, {3445,4057}

X(23837) = X(3737)-beth conjugate of X(979)
X(23837) = trilinear pole of line {649, 16614}

X(23838) =  X(1)X(513)∩X(8)X(522)

Barycentrics    a (a+b-2 c) (a-b-c) (b-c) (a-2 b+c) : :
X(23838) = 3 X[1022] - 2 X[23345], X[23345] - 3 X[23352]

X(23838) lies on the Feuerbach hyperbola, the cubics K230 and K407, and on these lines: {1,513}, {4,2457}, {7,3676}, {8,522}, {9,650}, {21,3737}, {80,900}, {88,1156}, {90,11512}, {104,106}, {294,1024}, {314,18155}, {514,1000}, {521,3680}, {523,5559}, {663,2320}, {693,17274}, {901,2222}, {903,2481}, {941,4813}, {1027,3257}, {1320,3738}, {1389,6003}, {1476,4017}, {1643,16670}, {1797,8759}, {2298,4979}, {2403,4778}, {2804,12641}, {2997,17896}, {3309,3577}, {3716,4997}, {3887,4792}, {3900,4900}, {4040,15175}, {4057,7428}, {4391,4494}, {4526,4876}, {4905,7284}, {4943,5557}, {4977,13606}, {7661,10309}, {8674,13143}, {9001,16496}, {10305,21172}, {15446,21173}, {17333,17494}

X(23838) = reflection of X(i) in X(j) for these {i,j}: {1022, 23352}, {3737, 6615}, {14812, 1}
X(23838) = reflection of X(14812) in the line X(1)X(3)
X(23838) = isogonal conjugate of X(23703)
X(23838) = X(i)-Ceva conjugate of X(j) for these (i,j): {901, 4674}, {3257, 2316}, {6548, 1022}
X(23838) = X(i)-cross conjugate of X(j) for these (i,j): {3738, 3737}, {4895, 650}
X(23838) = X(1)-Hirst inverse of X(14190)
X(23838) = cevapoint of X(i) and X(j) for these (i,j): {513, 1769}, {650, 4895}, {654, 663}
X(23838) = crosspoint of X(903) and X(3257)
X(23838) = trilinear pole of line {650, 2170}
X(23838) = crossdifference of every pair of points on line {44, 1319}
X(23838) = crosssum of X(i) and X(j) for these (i,j): {44, 4895}, {902, 1635}, {3689, 14418}
X(23838) = X(i)-beth conjugate of X(j) for these (i,j): {21, 14812}, {7253, 8}
X(23838) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 23703}, {37, 1023}, {101, 1635}, {513, 484}, {650, 44}, {1168, 901}
X(23838) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23703}, {7, 23344}, {44, 651}, {56, 17780}, {57, 1023}, {59, 900}, {100, 1319}, {101, 3911}, {108, 5440}, {109, 519}, {190, 1404}, {214, 2222}, {653, 22356}, {655, 17455}, {664, 902}, {901, 1317}, {934, 3689}, {1145, 2720}, {1262, 1639}, {1331, 1877}, {1412, 4169}, {1414, 21805}, {1415, 4358}, {1461, 2325}, {1635, 4564}, {1813, 8756}, {1960, 4998}, {1983, 14628}, {2149, 3762}, {2251, 4554}, {2429, 5435}, {3285, 4552}, {3943, 4565}, {4528, 7339}, {4530, 4619}, {4559, 16704}, {4572, 9459}, {4620, 14407}, {4819, 5545}, {4895, 7045}, {5298, 8701}, {6099, 12832}, {6174, 14733}, {6551, 14027}, {7128, 14418}, {18026, 23202}
X(23838) = barycentric product X(i)*X(j) for these {i,j}: {8, 1022}, {9, 6548}, {11, 3257}, {21, 4049}, {88, 522}, {106, 4391}, {244, 4582}, {312, 23345}, {513, 4997}, {514, 1320}, {521, 6336}, {644, 6549}, {650, 903}, {663, 20568}, {679, 1639}, {693, 2316}, {901, 4858}, {1111, 5548}, {1168, 3904}, {1318, 3762}, {2170, 4555}, {2226, 4768}, {2320, 23598}, {2403, 3680}, {3737, 4080}, {4516, 4615}, {4530, 4618}, {4560, 4674}, {4622, 21044}, {5376, 21132}
X(23838) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 23703}, {9, 17780}, {11, 3762}, {41, 23344}, {55, 1023}, {88, 664}, {106, 651}, {210, 4169}, {513, 3911}, {521, 3977}, {522, 4358}, {649, 1319}, {650, 519}, {652, 5440}, {654, 214}, {657, 3689}, {663, 44}, {667, 1404}, {901, 4564}, {903, 4554}, {926, 14439}, {1022, 7}, {1146, 4768}, {1168, 655}, {1318, 3257}, {1320, 190}, {1417, 1461}, {1635, 1317}, {1639, 4738}, {1797, 6516}, {1946, 22356}, {2170, 900}, {2310, 1639}, {2316, 100}, {2441, 1420}, {3022, 14427}, {3063, 902}, {3119, 4528}, {3239, 4723}, {3257, 4998}, {3270, 14418}, {3271, 1635}, {3287, 4434}, {3680, 2415}, {3700, 3992}, {3709, 21805}, {3737, 16704}, {3900, 2325}, {3904, 1227}, {4041, 3943}, {4049, 1441}, {4391, 3264}, {4435, 4432}, {4474, 4506}, {4516, 4120}, {4521, 4487}, {4582, 7035}, {4622, 4620}, {4674, 4552}, {4765, 4742}, {4814, 4908}, {4895, 4370}, {4976, 4975}, {4979, 5298}, {4997, 668}, {5548, 765}, {6336, 18026}, {6548, 85}, {6591, 1877}, {8648, 17455}, {8752, 108}, {9456, 109}, {14427, 4152}, {14936, 4895}, {18344, 8756}, {20568, 4572}, {23345, 57}, {23352, 5219}

X(23839) =  X(7)X(517)∩X(57)X(934)

Barycentrics    a (a+b-c)^2 (a-b+c)^2 (a b-b^2+a c-3 b c-c^2) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28349.

X(23839) lies on these lines: {7,517}, {57,934}, {65,279}, {77,11529}, {85,3869}, {269,18421}, {664,3873}, {942,3160}, {994,3668}, {1088,4566}, {1159,1443}, {1170,2082}, {1323,5902}, {1442,15934}, {1446,3212}, {1462,1572}, {2809,7672}, {3339,7177}, {3340,4350}, {3868,9312}, {4328,9819}, {5543,9957}, {5903,10481}, {6516,9352}, {6604,14923}, {6767,7269}

X(23840) =  X(9)X(374)∩X(40)X(15288)

Barycentrics    a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-6 a^4 b c+6 a^3 b^2 c-4 a^2 b^3 c+a b^4 c+2 b^5 c-a^4 c^2+6 a^3 b c^2-4 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2-2 a^3 c^3-4 a^2 b c^3-2 a b^2 c^3-4 b^3 c^3+2 a^2 c^4+a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28349.

X(23840) lies on these lines: {9,374}, {40,15288}, {65,169}, {72,22011}, {101,354}, {198,18443}, {226,4904}, {1212,14110}, {1903,4846}, {2389,21867}, {2809,5728}, {3057,16601}, {3730,7957}, {3753,8074}, {5819,6001}, {6554,7686}

X(23841) =  MIDPOINT OF X(355) AND X(389)

Barycentrics    -a^2 (a^3 b^2+a^2 b^3-a b^4-b^5+a^2 b^2 c-b^4 c+a^3 c^2+a^2 b c^2+4 b^3 c^2+a^2 c^3+4 b^2 c^3-a c^4-b c^4-c^5) : :
X(23841) = X[1]-3*X[5943], 3*X[2]+X[16980], X[8]+3*X[51], X[40]+X[13598], X[52]+3*X[5790] ,X[145]-9*X[5640], X[355]+X[389], 9*X[373]-5*X[3616], 3*X[375]-X[960], 2*X[1125]-3*X[6688], X[1385]-2*X[11695], X[1483]-5*X[15026], 5*X[1698]-3*X[3819], 3*X[3060]+5*X[3617] ,7*X[3622]-15*X[11451], X[3679]+X[21849], X[3751]+X[14913], 3*X[3917]-7*X[9780], X[4297]-2*X[17704], 7*X[4678]+9*X[11002], X[5446]+X[5690] ,X[5562]-5*X[5818],3*X[5587]-X[5907],9*X[5650]-13*X[19877], 2*X[6684]-X[13348], 3*X[7967]-11*X[15024], 3*X[9730]+X[18525],7*X[9781]+X[12245],2*X[9956]-X[11793], 6*X[10219]-5*X[19862], 9*X[12045]-8*X[19878], X[12237]+X[12787], X[12238]+X[12788], 3*X[13570]-2*X[18483], 9*X[14845]-5*X[18493],3*X[16836]-X[18481]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28349.

X(23841) lies on these lines: {1,5943}, {2,16980}, {8,51}, {10,511}, {40,13598}, {52,5790}, {101,1126}, {145,5640}, {181,5247}, {182,9798}, {197,13323}, {355,389}, {373,3616}, {375,960}, {515,9729}, {517,5795}, {518,9822}, {674,4662}, {916,9947}, {942,2810}, {952,5462}, {958,970}, {993,15489}, {1125,6688}, {1385,11695}, {1469,1722}, {1483,15026}, {1698,3819}, {2390,10107}, {2392,3918}, {2551,10441}, {2807,19925}, {3060,3617}, {3271,5255}, {3295,4266}, {3622,11451}, {3679,21849}, {3686,9052}, {3751,14913}, {3812,8679}, {3917,9780}, {4245,5399}, {4297,17704}, {4663,8681}, {4678,11002}, {5260,22076}, {5293,10544}, {5302,22276}, {5446,5690}, {5562,5818}, {5587,5907}, {5650,19877}, {5752,9708}, {5844,10095}, {6000,18480}, {6684,13348}, {7967,15024}, {8192,10601}, {9730,18525}, {9781,12245}, {9956,11793}, {10219,19862}, {10459,20962}, {12045,19878}, {12237,12787}, {12238,12788}, {12410,17810}, {13570,18483}, {13754,18357}, {14845,18493}, {16836,18481}, {17757,18180}

X(23842) =  X(1)X(7335)∩X(56)X(11334)

Barycentrics    -a (a+b-c) (a-b+c) (2 a^3-a^2 b-2 a b^2+b^3-a^2 c+2 a b c-b^2 c-2 a c^2-b c^2+c^3) (a^4-a^3 b+a b^3-b^4-a^3 c+2 a^2 b c-a b^2 c-a b c^2+2 b^2 c^2+a c^3-c^4) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28349.

X(23842) lies on these lines: {1,7335}, {56,11334}, {65,11700}, {946,1319}

X(23843) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^5 - a^3 b^2 + a^2 b^3 - b^5 - a^3 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - c^5) : :

X(23843) lies on these lines: {1, 11334}, {3, 10}, {6, 9247}, {12, 13733}, {21, 23369}, {25, 225}, {35, 984}, {36, 20842}, {55, 976}, {56, 244}, {73, 14529}, {157, 14017}, {184, 2594}, {186, 16305}, {255, 8679}, {474, 19846}, {513, 1777}, {581, 20986}, {849, 7236}, {859, 8185}, {1329, 13732}, {1473, 11509}, {1486, 2385}, {1602, 1603}, {1631, 2915}, {1754, 22300}, {1973, 8608}, {2176, 21004}, {2354, 5301}, {2886, 19548}, {3185, 10902}, {3515, 8756}, {3733, 7163}, {4218, 9780}, {4557, 11517}, {5217, 16064}, {7742, 20470}, {8069, 10037}, {8193, 15621}, {8758, 11363}, {10267, 23846}, {11248, 23845}, {11337, 16678}, {11365, 18613}, {13738, 20989}, {14963, 20739}, {15177, 15623}, {20875, 23852}, {20990, 23856}

X(23843) = isogonal conjugate of isotomic conjugate of X(21270)
X(23843) = isogonal conjugate of polar conjugate of X(17902)
X(23843) = isogonal conjugate of anticomplement of X(36033)
X(23843) = polar conjugate of isotomic conjugate of X(22130)

X(23844) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(23844) lies on these lines: {1, 7428}, {3, 214}, {6, 2179}, {8, 12746}, {25, 1825}, {40, 3185}, {46, 20470}, {55, 976}, {65, 23383}, {72, 15621}, {73, 2390}, {100, 25253}, {160, 18612}, {184, 23360}, {197, 9911}, {198, 21801}, {221, 7138}, {484, 16453}, {517, 23361}, {519, 22458}, {855, 10950}, {859, 5903}, {942, 18613}, {1089, 4557}, {1191, 1403}, {1319, 22344}, {1324, 11849}, {1329, 15507}, {1473, 11510}, {1482, 15654}, {1626, 10267}, {2187, 14529}, {2933, 11248}, {3057, 22345}, {3159, 8715}, {3220, 3746}, {3295, 22654}, {3754, 4245}, {3913, 20760}, {4646, 20967}, {5264, 20990}, {5440, 15625}, {5693, 15623}, {6001, 15622}, {8608, 23620}, {8666, 23169}, {9798, 10679}, {10571, 23981}, {11194, 23085}, {12513, 20805}, {15071, 15626}, {23851, 23861}

X(23845) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b) (a - c) (a b + b^2 + a c - 2 b c + c^2) : :

X(23845) lies on these lines: {3, 214}, {6, 9432}, {25, 1830}, {31, 8054}, {38, 55}, {40, 23361}, {46, 23383}, {56, 20843}, {57, 18613}, {63, 15621}, {78, 15625}, {80, 13744}, {100, 190}, {101, 1293}, {109, 692}, {110, 901}, {165, 3185}, {197, 2823}, {244, 23404}, {484, 859}, {513, 4551}, {519, 23169}, {643, 1634}, {902, 20780}, {999, 11717}, {1155, 20470}, {1158, 15622}, {1319, 23205}, {1324, 12778}, {1331, 23344}, {1376, 3923}, {1403, 3052}, {1768, 15626}, {1772, 15906}, {2283, 8638}, {2390, 22350}, {2617, 4833}, {2835, 24025}, {2933, 3556}, {3035, 15507}, {3057, 22344}, {3145, 14882}, {3220, 5537}, {3286, 5143}, {3827, 9371}, {3913, 20805}, {4421, 20760}, {4588, 8697}, {4640, 22325}, {4995, 21319}, {5119, 23206}, {5132, 17601}, {5903, 7428}, {6014, 8694}, {8715, 22458}, {10306, 22654}, {11248, 23843}, {11849, 23850}, {12513, 23085}, {12702, 15654}, {16492, 17477}, {16679, 17126}, {17724, 24405}, {23067, 23703}

X(23845) = isogonal conjugate of isotomic conjugate of X(21272)
X(23845) = isogonal conjugate of polar conjugate of X(17906)
X(23845) = polar conjugate of isotomic conjugate of X(23113)

X(23846) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 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 + b c^4) : :

X(23846) lies on these lines: {1, 859}, {3, 214}, {6, 23623}, {8, 4557}, {25, 1831}, {48, 3285}, {55, 2933}, {65, 20470}, {78, 15621}, {197, 3295}, {198, 1953}, {228, 3057}, {595, 17104}, {1319, 22345}, {1510, 3733}, {1610, 1621}, {1626, 3556}, {1697, 15624}, {1790, 23360}, {2099, 13738}, {2217, 5248}, {2390, 4303}, {3754, 16414}, {3869, 16678}, {4068, 23381}, {4855, 15625}, {5250, 8053}, {5710, 20990}, {5903, 16453}, {6261, 15622}, {6326, 15623}, {8666, 22458}, {9454, 23544}, {9798, 16202}, {9959, 24434}, {10246, 15654}, {10267, 23843}, {10434, 15829}, {10950, 13724}, {11043, 17061}, {11194, 20805}, {11813, 19648}, {12513, 20760}, {12635, 23853}, {15888, 21319}, {16679, 16691}, {23370, 23861}

X(23846) = isogonal conjugate of isotomic conjugate of X(21273)
X(23846) = isogonal conjugate of polar conjugate of X(18677)
X(23846) = polar conjugate of isotomic conjugate of X(23114)

X(23847) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^7 + a^4 b^3 - a^3 b^4 - b^7 + a^4 c^3 + b^4 c^3 - a^3 c^4 + b^3 c^4 - c^7) : :

X(23847) lies on these lines: {6, 17453}, {22, 321}, {25, 3772}, {55, 976}, {1011, 1626}, {1726, 3220}, {2174, 2276}, {2176, 21774}, {2353, 18616}, {3419, 9798}, {3666, 11334}, {5101, 10829}, {10832, 22654}, {20987, 23365}, {21004, 21775}

X(23847) = isogonal conjugate of isotomic conjugate of X(17492)
X(23847) = polar conjugate of isotomic conjugate of X(23116)

X(23848) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^7 + a^4 b^3 - a^3 b^4 - b^7 + a^3 b^2 c^2 - a^2 b^3 c^2 + a^4 c^3 - a^2 b^2 c^3 + b^4 c^3 - a^3 c^4 + b^3 c^4 - c^7) : :

X(23848) lies on these lines: {6, 23625}, {22, 5695}, {23, 4442}, {55, 976}, {2254, 3733}, {5938, 18617}, {19596, 23366}

X(23848) = isogonal conjugate of isotomic conjugate of X(21274)
X(23848) = polar conjugate of isotomic conjugate of X(23117)

X(23849) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^5 - b^5 - c^5) : :

X(23849) lies on these lines: {6, 1917}, {22, 23339}, {55, 20994}, {1626, 23370}, {1631, 2915}, {2921, 23376}, {3556, 23861}, {9018, 9247}, {16682, 23378}, {18297, 20874}, {20833, 23402}, {21004, 23863}

X(23849) = isogonal conjugate of isotomic conjugate of X(21275)
X(23849) = polar conjugate of isotomic conjugate of X(23118)

X(23850) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^5 - a^3 b^2 + a^2 b^3 - 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 - c^5) : :

X(23850) lies on these lines: {1, 1283}, {3, 10}, {6, 23627}, {22, 3757}, {24, 242}, {25, 1838}, {35, 3961}, {55, 1782}, {56, 11334}, {404, 24988}, {500, 20986}, {692, 5399}, {1473, 11507}, {1478, 13733}, {1617, 11365}, {1623, 5253}, {1914, 21744}, {2828, 14703}, {2915, 16678}, {2922, 8424}, {3220, 10902}, {3556, 10267}, {3634, 16422}, {4218, 5260}, {5172, 7428}, {5204, 20842}, {5248, 12579}, {5329, 19762}, {6187, 24443}, {8053, 16119}, {8185, 13738}, {9590, 20838}, {11337, 16817}, {11849, 23845}, {13730, 18954}, {14453, 23537}, {14963, 23150}, {16453, 20989}, {17524, 23369}, {20831, 23383}, {20840, 20988}, {21004, 21008}, {22161, 23156}, {23370, 23862}

X(23850) = isogonal conjugate of isotomic conjugate of X(21276)
X(23850) = polar conjugate of isotomic conjugate of X(23119)

X(23851) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 - a^2 b^3 + a^3 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3) : :

X(23851) lies on these lines: {1, 20475}, {2, 16683}, {3, 8301}, {6, 1923}, {10, 8618}, {25, 1840}, {35, 238}, {36, 23393}, {55, 869}, {171, 23396}, {404, 16693}, {1107, 18758}, {1631, 2915}, {2053, 6187}, {2329, 15621}, {4020, 9016}, {4362, 20855}, {5248, 16292}, {5253, 16691}, {5277, 20990}, {8266, 16684}, {8424, 24729}, {8616, 14823}, {8671, 24047}, {9018, 23619}, {11490, 21793}, {20358, 22449}, {20878, 23407}, {20994, 21004}, {23844, 23861}

X(23851) = isogonal conjugate of isotomic conjugate of X(21278)
X(23851) = polar conjugate of isotomic conjugate of X(23121)
X(23851) = pole wrt circumcircle of line X(798)X(812)

X(23852) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^5 - a^4 b + a b^4 - b^5 - a^4 c + b^4 c + a c^4 + b c^4 - c^5) : :

X(23852) lies on these lines: {3, 8299}, {6, 9448}, {22, 16681}, {55, 3721}, {157, 1631}, {159, 16872}, {1284, 1486}, {1602, 23379}, {1617, 16691}, {1626, 3188}, {3148, 17798}, {6660, 23853}, {7742, 16693}, {8628, 21771}, {16678, 18619}, {20875, 23843}, {23370, 23389}

X(23852) = isogonal conjugate of isotomic conjugate of X(21280)
X(23852) = polar conjugate of isotomic conjugate of X(23123)

X(23853) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b - a b^3 + a^3 c - a^2 b c + a b^2 c - b^3 c + a b c^2 - a c^3 - b c^3) : :

X(23853) lies on these lines: {1, 3}, {6, 22199}, {8, 13738}, {11, 19540}, {22, 6360}, {25, 92}, {31, 3955}, {87, 8616}, {100, 4191}, {145, 4225}, {197, 4362}, {198, 3684}, {228, 3870}, {329, 15507}, {333, 859}, {388, 9840}, {405, 1220}, {496, 19543}, {497, 4192}, {499, 19549}, {518, 3185}, {595, 19762}, {851, 3434}, {947, 13346}, {958, 23383}, {1001, 16058}, {1011, 1621}, {1037, 7015}, {1056, 19262}, {1260, 10477}, {1376, 3741}, {1458, 3784}, {1486, 8424}, {1626, 20841}, {1631, 20473}, {1755, 22163}, {1999, 11350}, {2176, 16584}, {2178, 4386}, {2187, 7193}, {2300, 5120}, {2319, 20471}, {2550, 16056}, {3052, 3286}, {3085, 13731}, {3086, 19513}, {3149, 15623}, {3190, 9052}, {3421, 19256}, {3436, 13724}, {3724, 3938}, {3751, 20967}, {3871, 16451}, {3996, 5687}, {4184, 8025}, {4216, 20037}, {4245, 9708}, {4413, 16409}, {4419, 13097}, {4428, 8053}, {4504, 16695}, {5020, 6708}, {5263, 11358}, {5274, 19647}, {5284, 16373}, {5399, 5752}, {6660, 23852}, {7288, 19514}, {7416, 10446}, {7580, 14942}, {7677, 19649}, {8624, 21775}, {8666, 15654}, {9259, 21001}, {9669, 19648}, {9709, 10479}, {9798, 20836}, {9909, 20875}, {10589, 19546}, {10591, 19646}, {11688, 24349}, {12513, 23361}, {12635, 23846}, {15325, 19550}, {16405, 24552}, {16408, 19863}, {16872, 20794}, {17735, 21769}

X(23853) = complement of X(36855)
X(23853) = isogonal conjugate of isotomic conjugate of X(21281)
X(23853) = polar conjugate of isotomic conjugate of X(23125)
X(23853) = {X(55),X(56)}-harmonic conjugate of X(171)

X(23854) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a^3 - a^2 b + a b^2 - 2 b^3 - a^2 c + b^2 c + a c^2 + b c^2 - 2 c^3) : :

X(23854) lies on these lines: {3, 142}, {6, 9459}, {22, 3007}, {23, 385}, {25, 8756}, {36, 14190}, {55, 16672}, {674, 22356}, {902, 8610}, {1626, 23391}, {2183, 23344}, {2223, 21009}, {2293, 22357}, {2930, 20474}, {2937, 18119}, {3285, 8626}, {3286, 16801}, {3746, 4068}, {4497, 7083}, {5563, 16679}, {7669, 20877}, {8266, 23375}, {16678, 18661}, {16686, 17798}, {20989, 23858}

X(23854) = isogonal conjugate of isotomic conjugate of X(21282)
X(23854) = polar conjugate of isotomic conjugate of X(23126)

X(23855) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 - 2 a^2 b + 2 a b^2 - b^3 - 2 a^2 c + 2 b^2 c + 2 a c^2 + 2 b c^2 - c^3) : :

X(23855) lies on these lines: {3, 142}, {6, 23634}, {45, 55}, {56, 16694}, {984, 3746}, {999, 23377}, {1621, 1623}, {1995, 20875}, {2292, 3242}, {2293, 22356}, {3304, 16679}, {3941, 5563}, {4428, 20834}, {4429, 5047}, {4497, 20992}, {8641, 24457}, {8671, 24497}, {13615, 15621}, {15625, 16293}, {18613, 20835}

X(23855) = isogonal conjugate of isotomic conjugate of X(21283)
X(23855) = polar conjugate of isotomic conjugate of X(23127)

X(23856) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^9 + a^8 b - a b^8 - b^9 + a^8 c - 2 a^4 b^4 c + b^8 c - 2 a^4 b c^4 + 2 a b^4 c^4 - a c^8 + b c^8 - c^9) : :

X(23856) lies on these lines: {6, 23641}, {2178, 21774}, {2353, 21322}, {16687, 18610}, {20990, 23843}, {20993, 23382}, {21771, 21776}

X(23856) = isogonal conjugate of isotomic conjugate of X(21288)
X(23856) = polar conjugate of isotomic conjugate of X(23132)

X(23857) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^4 b^2 - a^2 b^4 - 2 a^4 b c + 2 a b^4 c + a^4 c^2 + a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 + 2 a b c^4 - b^2 c^4) : :

X(23857) lies on these lines: {3, 238}, {6, 20467}, {55, 24307}, {932, 25311}, {1030, 20855}, {1486, 20873}, {1631, 20473}, {2932, 11334}, {2933, 20872}, {4057, 17262}, {9025, 23086}

X(23857) = isogonal conjugate of cyclocevian conjugate of X(38247)
X(23857) = polar conjugate of isotomic conjugate of X(23134)

X(23858) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^4 - a^3 b + a b^3 - b^4 - a^3 c + 5 a^2 b c - 5 a b^2 c + b^3 c - 5 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 - c^4) : :

X(23858) lies on these lines: {3, 8}, {6, 23644}, {36, 9324}, {45, 55}, {88, 999}, {105, 11284}, {199, 7669}, {244, 3304}, {612, 4516}, {1054, 3987}, {1282, 5537}, {1631, 23859}, {1797, 2810}, {2177, 5168}, {2182, 3689}, {3145, 8715}, {3303, 3722}, {3746, 5293}, {3939, 22371}, {4421, 16064}, {8299, 16373}, {8301, 15571}, {8650, 20468}, {10117, 20851}, {10535, 22356}, {16842, 24542}, {16862, 24988}, {20989, 23854}

X(23858) = isogonal conjugate of isotomic conjugate of X(21290)
X(23858) = polar conjugate of isotomic conjugate of X(23135)

X(23859) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (4 a^4 + 2 a^3 b - 2 a b^3 - 4 b^4 + 2 a^3 c + 5 a^2 b c - 5 a b^2 c - 2 b^3 c - 5 a b c^2 + 4 b^2 c^2 - 2 a c^3 - 2 b c^3 - 4 c^4) : :

X(23859) lies on these lines: {3, 1698}, {6, 23645}, {55, 16672}, {1631, 23858}, {18755, 21782}

X(23859) = isogonal conjugate of isotomic conjugate of X(21291)
X(23859) = polar conjugate of isotomic conjugate of X(23136)

X(23860) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^7 - a^5 b^2 + a^2 b^5 - b^7 - 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 + a^2 c^5 + b^2 c^5 - c^7) : :

X(23860) lies on these lines: {6, 23647}, {199, 20871}, {2915, 2932}, {7669, 16873}, {14667, 23402}, {20839, 21004}

X(23860) = isogonal conjugate of isotomic conjugate of X(21294)
X(23860) = polar conjugate of isotomic conjugate of X(23138)

X(23861) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a - b) (a - c) (b + c) (a^2 + a b + b^2 + a c - b c + c^2) : :

X(23861) lies on these lines: {1, 20474}, {6, 23648}, {55, 21004}, {101, 512}, {110, 23390}, {523, 3732}, {643, 3573}, {758, 23398}, {1018, 4069}, {1252, 8664}, {1631, 4736}, {2292, 8852}, {2948, 8053}, {3556, 23849}, {3869, 16681}, {3878, 16689}, {4068, 24436}, {4128, 5147}, {6631, 25309}, {23370, 23846}, {23844, 23851}

X(23861) = isogonal conjugate of isotomic conjugate of X(21295)
X(23861) = polar conjugate of isotomic conjugate of X(23139)

X(23862) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (2 a^5 - a^3 b^2 + a^2 b^3 - 2 b^5 - a^3 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - 2 c^5) : :

X(23862) lies on these lines: {6, 23651}, {23, 23395}, {55, 21772}, {814, 7255}, {1631, 2915}, {20918, 23398}, {23370, 23850}, {23376, 23402}

X(23862) = isogonal conjugate of isotomic conjugate of X(21298)
X(23862) = polar conjugate of isotomic conjugate of X(23142)

X(23863) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 b^2 - a^2 b^3 + a^2 b^2 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3) : :

X(23863) lies on these lines: {3, 238}, {6, 23652}, {9, 18758}, {25, 2053}, {55, 869}, {56, 85}, {595, 11490}, {614, 22449}, {958, 19312}, {2238, 23212}, {3056, 22065}, {4423, 16354}, {16969, 20475}, {21001, 23396}, {21004, 23849}

X(23863) = isogonal conjugate of isotomic conjugate of X(21299)
X(23863) = polar conjugate of isotomic conjugate of X(23143)

X(23864) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a + b) (a - b - c) (b - c) (a + c) (a^2 b - a b^2 + a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(23864) lies on these lines: {3, 4369}, {6, 23654}, {55, 7252}, {523, 2073}, {650, 1946}, {661, 1011}, {669, 7253}, {814, 7255}, {824, 23093}, {3286, 18199}, {3733, 8646}, {3736, 23092}, {3737, 8641}, {4184, 7192}, {4191, 24924}, {4367, 22089}, {4560, 8638}, {4833, 8053}, {6590, 22388}, {22443, 23655}

X(23864) = isogonal conjugate of isotomic conjugate of X(21300)
X(23864) = polar conjugate of isotomic conjugate of X(23145)

X(23865) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^3 - 2 a^2 b + a b^2 - 2 a^2 c + a b c - b^2 c + a c^2 - b c^2) : :

X(23865) lies on these lines: {3, 15599}, {6, 23655}, {23, 385}, {25, 3064}, {55, 649}, {333, 25301}, {513, 2078}, {650, 8642}, {652, 9029}, {661, 8645}, {663, 6589}, {667, 3900}, {830, 22160}, {1001, 3835}, {1617, 3676}, {1621, 20295}, {2487, 9511}, {3286, 18200}, {3733, 8646}, {3757, 20952}, {4367, 8638}, {4428, 4785}, {4524, 22108}, {5737, 25128}, {8053, 16874}, {8611, 8635}, {8640, 21003}, {8654, 15621}, {8678, 21789}, {20999, 23726}

X(23865) = isogonal conjugate of isotomic conjugate of X(21302)
X(23865) = isogonal conjugate of anticomplement of X(38991)
X(23865) = polar conjugate of isotomic conjugate of X(23146)

X(23866) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^4 + a b^3 - 2 a^2 b c + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(23866) lies on these lines: {6, 23656}, {23, 385}, {667, 1734}, {1491, 8654}, {1769, 4491}, {2254, 3733}, {21003, 23401}

X(23866) = isogonal conjugate of isotomic conjugate of X(21303)
X(23866) = polar conjugate of isotomic conjugate of X(23147)

X(23867) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (b - c) (a^4 + a b^3 + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3) : :

X(23867) lies on these lines: {6, 23657}, {522, 1324}, {667, 1734}, {804, 5152}, {814, 7255}, {832, 3733}, {1491, 8636}, {2084, 8634}, {4367, 4453}

X(23867) = isogonal conjugate of isotomic conjugate of X(21304)
X(23867) = polar conjugate of isotomic conjugate of X(23148)

X(23868) =  (A,B,C,X(75); A',B',C',X(6)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(6)

Barycentrics    a^2 (a^3 - b^3 - a b c - c^3) : :

X(23868) lies on these lines: {1, 20836}, {3, 238}, {6, 560}, {9, 20678}, {19, 25}, {29, 6284}, {31, 199}, {41, 3779}, {48, 3056}, {56, 77}, {75, 1281}, {100, 17280}, {101, 3688}, {237, 16872}, {256, 1582}, {284, 21746}, {572, 3271}, {573, 2175}, {674, 2174}, {692, 4271}, {789, 9230}, {904, 1964}, {1001, 17322}, {1030, 2110}, {1283, 2652}, {1333, 4749}, {1376, 17289}, {1400, 19133}, {1428, 22390}, {1580, 24478}, {1610, 8240}, {2178, 21010}, {2183, 2330}, {2194, 4269}, {2246, 21039}, {2309, 5110}, {2317, 8540}, {2667, 23398}, {3207, 10387}, {3664, 5144}, {3773, 5687}, {4224, 21321}, {4265, 20470}, {4300, 20838}, {4413, 17385}, {4421, 17264}, {4423, 16352}, {4492, 7087}, {5132, 20872}, {5201, 23374}, {5217, 11344}, {5248, 25354}, {5263, 19312}, {5285, 20967}, {6541, 8715}, {7241, 7246}, {7291, 17447}, {8300, 24575}, {14370, 20994}, {16547, 21804}, {16679, 21009}, {16681, 18755}, {17788, 17797}, {18042, 25048}, {18162, 20358}, {19297, 20990}, {20999, 23379}

X(23868) = isogonal conjugate of X(7224)
X(23868) = polar conjugate of isotomic conjugate of X(23150)

X(23869) =  MIDPOINT OF X(1) AND X(6788)

Barycentrics    a^3 (b+c)-a^2 (-3 b^2+10 b c-3 c^2)+a (b^3+c^3)-(b^2-c^2)^2 : :
X(23869) = (r^2+13 r R-s^2) X(1) - 3 r (2 r-R) X(2)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28355.

X(23869) lies on these lines: {1,2}, {65,13756}, {244,21630}, {764,21201}, {946,3667}, {952,11717}, {1015,21090}, {1290,2718}, {2802,3756}, {3315,16173}, {4694,11813}, {11814,21087}, {12016,18240}, {18326,18493}

X(23869) = midpoint of X(1) and X(6788)
X(23869) = reflection of X(i) in X(j) for these {i, j}: {6789,1125}, {21087,11814}
X(23869) = X(133)-of-Fuhrmann-triangle

leftri

Points on the infinity line: X(23870-X(23888)

rightri

Contributed by Clark Kimberling and Peter Moses, September 27, 2018.

Suppose that a point in the plane of a triangle ABC is given by P = p : q : r (barycentrics). The infinite difference point of P is introduced here as the point given by D(P) = q - r : r - p : p - q.

X(23870) =  INFINITE DIFFERENCE POINT OF X(13)

Barycentrics    (b^2 - c^2)*(Sqrt[3]*(-a^2 + b^2 + c^2) + 2*S) : :

X(23870) lies on these lines: {2, 9201}, {13, 2394}, {14, 14223}, {30, 511}, {99, 5994}, {618, 5664}, {850, 20579}, {1649, 9194}, {3268, 6137}, {5460, 11627}, {6110, 6782}, {6669, 14566}, {9131, 13305}, {9147, 13304}, {9200, 9979}, {11078, 14446}

X(23870) = isogonal conjugate of X(5995)
X(23870) = isotomic conjugate of X(23895)
X(23870) = polar conjugate of X(36306)
X(23870) = pole wrt polar circle of trilinear polar of X(36306) (line X(4)X(13))

X(23871) =  INFINITE DIFFERENCE POINT OF X(14)

Barycentrics    (b^2 - c^2)*(Sqrt[3]*(-a^2 + b^2 + c^2) - 2*S) : :

X(23871) lies on these lines: {2, 9200}, {13, 14223}, {14, 2394}, {30, 511}, {99, 5995}, {619, 5664}, {850, 20578}, {1649, 9195}, {3268, 6138}, {5459, 11625}, {6111, 6783}, {6670, 14566}, {9131, 13304}, {9147, 13305}, {9201, 9979}, {11092, 14447}

X(23871) = isogonal conjugate of X(5993)
X(23871) = isotomic conjugate of X(23896)
X(23871) = polar conjugate of X(36309)
X(23871) = pole wrt polar circle of trilinear polar of X(36309) (line X(4)X(14))

X(23872) =  INFINITE DIFFERENCE POINT OF X(17)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2 + 2*Sqrt[3]*S) : :

X(23872) lies on these lines: {2, 20579}, {15, 15412}, {30, 511}, {623, 18314}, {648, 5995}, {2394, 12816}, {6116, 14618}, {6138, 9979}, {7684, 15451}, {9131, 22933}

X(23872) = isogonal conjugate of X(16806)

X(23873) =  INFINITE DIFFERENCE POINT OF X(18)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2 - 2*Sqrt[3]*S) : :

X(23873) lies on these lines: {2, 20578}, {16, 15412}, {30, 511}, {624, 18314}, {648, 5994}, {2394, 12817}, {6117, 14618}, {6137, 9979}, {7685, 15451}, {9131, 22888}

X(23873) = isogonal conjugate of X(16807)

X(23874) =  INFINITE DIFFERENCE POINT OF X(19)

Barycentrics    (b - c)*(-a^2 + b^2 + c^2)*(a^2 + b^2 + 2*b*c + c^2) : :

X(23874) lies on these lines: {30, 511}, {656, 4025}, {905, 20315}, {1459, 4064}, {2484, 6590}, {2509, 3239}, {4086, 21186}, {4088, 17418}, {4391, 7649}, {4985, 21185}, {7253, 14954}, {14837, 20316}, {17496, 20294}, {18160, 21178}

X(23874) = isogonal conjugate of X(32691)
X(23874) = crossdifference of every pair of points on line X(6)X(1245)

X(23875) =  INFINITE DIFFERENCE POINT OF X(35)

Barycentrics    (b - c)*(-a^2*b + b^3 - a^2*c - a*b*c + b^2*c + b*c^2 + c^3) : :

X(23875) lies on these lines: {30, 511}, {650, 21192}, {693, 7265}, {1019, 21392}, {1734, 4088}, {3239, 21188}, {3700, 4823}, {3716, 20517}, {3776, 22037}, {3960, 6332}, {4025, 14838}, {4064, 23800}, {4391, 4707}, {4791, 7178}, {14349, 16892}, {18004, 21260}

X(23876) =  INFINITE DIFFERENCE POINT OF X(36)

Barycentrics    (b - c)*(-a^2*b + b^3 - a^2*c + a*b*c + b^2*c + b*c^2 + c^3) : :

X(23876) lies on these lines: {10, 4522}, {30, 511}, {321, 4391}, {693, 4707}, {905, 3666}, {1577, 2610}, {1734, 4424}, {3700, 4791}, {3904, 4467}, {3960, 4025}, {4120, 21130}, {4823, 7178}, {4944, 21198}, {6332, 14838}, {14349, 21124}, {17147, 17496}

X(23877) =  INFINITE DIFFERENCE POINT OF X(41)

Barycentrics    (b - c)*(-(a*b^2) + b^3 - a*b*c - a*c^2 + c^3) : :

X(23877) lies on these lines: {30, 511}, {650, 4142}, {905, 4458}, {1491, 3801}, {1577, 4522}, {1734, 4707}, {2517, 23752}, {2530, 3776}, {3716, 21185}, {3904, 4449}, {4017, 20294}, {4064, 7650}, {4088, 4391}, {4147, 10015}, {4462, 21132}, {6050, 13246}, {7178, 17072}, {7662, 8045}, {14838, 20517}, {17924, 21108}, {20504, 23528}

X(23878) =  INFINITE DIFFERENCE POINT OF X(51)

Barycentrics    (b^2 - c^2)*(-a^4 + a^2*b^2 + a^2*c^2 + 2*b^2*c^2) : :

X(23878) lies on these lines: {2, 647}, {30, 511}, {598, 2394}, {671, 8430}, {879, 11179}, {905, 20907}, {1577, 4140}, {2485, 18314}, {2489, 4580}, {2525, 6563}, {2528, 19568}, {3050, 12151}, {3175, 3700}, {3267, 7799}, {4108, 8644}, {4524, 4685}, {5077, 10097}, {5466, 11167}, {5664, 15810}, {7610, 9175}, {7625, 8176}, {7630, 23285}, {7652, 7739}, {7817, 8574}, {9131, 13307}, {9148, 17414}, {9178, 9462}, {9404, 19723}, {9979, 13306}, {14480, 17708}, {14566, 14762}, {16229, 17994}

X(23878) = isogonal conjugate of X(26714)
X(23878) = isotomic conjugate of trilinear pole of line X(2)X(51)
X(23878) = isotomic conjungate of isogonal conjugate of X(3288)

X(23879) =  INFINITE DIFFERENCE POINT OF X(58)

Barycentrics    (b^2 - c^2)*(a*b + b^2 + a*c + b*c + c^2) : :

X(23879) lies on these lines: {30, 511}, {116, 3708}, {647, 8045}, {661, 7265}, {693, 23685}, {850, 1577}, {1019, 4467}, {3050, 22154}, {3267, 4509}, {3700, 4129}, {4122, 4705}, {4369, 21192}, {4500, 4823}, {4560, 17161}, {4707, 4838}, {4841, 22037}, {4978, 16892}, {7662, 20517}

X(23880) =  INFINITE DIFFERENCE POINT OF X(65)

Barycentrics    (b - c)*(-a + b + c)*(a^2 + a*b + a*c + 2*b*c) : :

X(23880) lies on these lines: {30, 511}, {650, 3975}, {693, 3669}, {905, 1577}, {1459, 17478}, {2526, 21301}, {3700, 6332}, {3904, 4820}, {3960, 4823}, {4025, 7178}, {4041, 4474}, {4140, 15416}, {4147, 4913}, {4148, 4765}, {4367, 7662}, {4449, 4804}, {4462, 17494}, {4791, 14838}, {4801, 21222}, {4939, 7208}, {4940, 14349}, {4976, 21120}, {6129, 7650}, {14837, 17069}, {16759, 21044}

X(23881) =  INFINITE DIFFERENCE POINT OF X(66)

Barycentrics    (b^2 - c^2)*(b^2 + c^2)*(-a^4 + b^4 + c^4) : :

X(23881) lies on these lines: {30, 511}, {2485, 16757}, {2525, 23285}, {4580, 10313}

X(23882) =  INFINITE DIFFERENCE POINT OF X(72)

Barycentrics    (b - c)*(-a^3 + a*b^2 + 2*a*b*c + 2*b^2*c + a*c^2 + 2*b*c^2) : :

X(23882) lies on these lines: {30, 511}, {650, 1577}, {663, 4804}, {667, 7662}, {693, 905}, {3261, 17899}, {3669, 4077}, {4106, 14349}, {4391, 17494}, {4500, 8045}, {4765, 14837}, {4791, 20317}, {4801, 17496}, {4815, 6129}, {4820, 7265}, {4823, 4885}, {4874, 6050}, {4913, 17072}, {4976, 7178}, {17069, 21188}

X(23883) =  INFINITE DIFFERENCE POINT OF X(79)

Barycentrics    (b - c)*(a + 2*b + 2*c)*(-a^2 + b^2 + b*c + c^2) : :

X(23883) lies on these lines: {30, 511}, {3700, 21192}, {4467, 7265}, {4820, 4823}, {4838, 4960}

X(23884) =  INFINITE DIFFERENCE POINT OF X(80)

Barycentrics    (a - 2*b - 2*c)*(b - c)*(a^2 - b^2 + b*c - c^2) : :

X(23884) lies on these lines: {30, 511}, {1022, 21115}, {1639, 10015}, {3762, 4080}, {3904, 3960}, {4049, 4928}, {4791, 4944}, {4893, 21130}, {4945, 23598}

X(23885) =  INFINITE DIFFERENCE POINT OF X(82)

Barycentrics    (b - c)*(b^2 + c^2)*(a^2 + b^2 + b*c + c^2) : :

X(23885) lies on these lines: {30, 511}, {8060, 21194}, {8061, 16892}

X(23886) =  INFINITE DIFFERENCE POINT OF X(87)

Barycentrics    (b - c)*(-(a*b) - a*c + b*c)^2 : :

X(23886) lies on these lines: {30, 511}, {75, 17458}, {192, 20979}, {594, 21262}, {1919, 4360}, {3835, 20906}, {3875, 20370}, {4057, 17318}, {4375, 7220}, {4393, 23472}, {4397, 4486}, {4932, 17159}, {20907, 21206}, {20908, 23685}, {21350, 22226}, {22316, 22322}

X(23887) =  INFINITE DIFFERENCE POINT OF X(101)

Barycentrics    (b - c)*(-a*b^2 + b^3 - a*c^2 + c^3) : :

X(23887) lies on these lines: {1, 3904}, {10, 10015}, {30, 511}, {647, 23786}, {676, 1125}, {905, 20517}, {1111, 3120}, {1331, 2398}, {1577, 21118}, {1638, 21181}, {2254, 4707}, {2525, 23791}, {2530, 3801}, {3265, 23723}, {3310, 13006}, {3626, 4528}, {3716, 21201}, {3762, 4088}, {3960, 4458}, {4036, 21111}, {4064, 4985}, {4086, 21102}, {4142, 14838}, {4404, 21119}, {4522, 4791}, {4712, 13259}, {4809, 14419}, {6332, 21185}, {8062, 21179}, {14417, 23817}, {15065, 18003}, {20294, 21189}, {23184, 23383}

X(23888) =  INFINITE DIFFERENCE POINT OF X(106)

Barycentrics    (b - c)*(-2*a + b + c)*(a*b + b^2 + a*c - b*c + c^2) : :

X(23888) lies on these lines: {30, 511}, {1022, 4453}, {1647, 4475}, {1797, 2403}, {3310, 3960}, {3762, 4120}, {3904, 21385}, {4049, 4927}, {4707, 21115}, {4809, 14421}, {4928, 21198}

X(23889) =  X(1)X(21)∩X(110)X(8691)

Barycentrics    a (a-b) (a+b) (a-c) (a+c) (2 a^2-b^2-c^2) : :

X(23889) lies on the cubic K228 and these lines: {1,21}, {110,8691}, {163,662}, {1022,4622}, {1023,4567}, {1026,17943}

X(23889) = isogonal conjugate of X(23894)
X(23889) = X(2642)-cross conjugate of X(896)
X(23889) = cevapoint of X(896) and X(2642)
X(23889) = trilinear pole of line {896, 922}
X(23889) = crossdifference of every pair of points on line {661, 2643}
X(23889) = crosssum of X(i) and X(j) for these (i,j): {661, 2642}, {896, 17467}
X(23889) = X(i)-aleph conjugate of X(j) for these (i,j): {99, 16563}, {892, 16568}, {14089, 16563}
X(23889) = X(i)-zayin conjugate of X(j) for these (i,j): {896, 661}, {897, 2642}, {16568, 798}
X(23889) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23894}, {2, 9178}, {4, 10097}, {6, 5466}, {25, 14977}, {67, 10561}, {98, 8430}, {111, 523}, {115, 691}, {512, 671}, {525, 8753}, {647, 17983}, {661, 897}, {669, 18023}, {690, 10630}, {843, 18007}, {892, 3124}, {895, 2501}, {923, 1577}, {1383, 23288}, {1637, 9139}, {1989, 9213}, {2395, 5968}, {2408, 21448}, {2433, 9214}, {2444, 5485}, {2492, 10415}, {3125, 5380}, {3569, 9154}, {3700, 7316}, {5547, 7178}, {9134, 15387}, {9180, 17964}, {14273, 15398}, {14618, 14908}, {14998, 16092}, {17414, 18818}, {17993, 18823}
X(23889) = barycentric product X(i)X(j) for these {i,j}: {1, 5468}, {63, 4235}, {75, 5467}, {99, 896}, {100, 6629}, {101, 16741}, {110, 14210}, {162, 6390}, {163, 3266}, {187, 799}, {190, 16702}, {468, 4592}, {524, 662}, {643, 7181}, {670, 922}, {811, 3292}, {1414, 3712}, {2642, 4590}, {4567, 4750}, {4584, 4760}, {4599, 7813}, {4600, 14419}, {4602, 14567}, {4603, 7267}, {4610, 21839}, {4614, 4831}
X(23889) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 5466}, {31, 9178}, {48, 10097}, {63, 14977}, {110, 897}, {162, 17983}, {163, 111}, {187, 661}, {351, 2643}, {524, 1577}, {662, 671}, {690, 1109}, {799, 18023}, {896, 523}, {922, 512}, {1101, 691}, {1576, 923}, {1755, 8430}, {2642, 115}, {3266, 20948}, {3292, 656}, {3712, 4086}, {4062, 4036}, {4235, 92}, {4570, 5380}, {4575, 895}, {4750, 16732}, {4831, 4815}, {5467, 1}, {5468, 75}, {6149, 9213}, {6390, 14208}, {6629, 693}, {7181, 4077}, {9181, 17955}, {14210, 850}, {14417, 20902}, {14419, 3120}, {14559, 2166}, {14567, 798}, {16702, 514}, {16741, 3261}, {17466, 9134}, {21839, 4024}, {23200, 810}

X(23890) =  X(1)X(3)∩X(101)X(651)

Barycentrics    a (a-b) (a-c) (a+b-c) (a-b+c) (2 a^2-a b-b^2-a c+2 b c-c^2) : :

X(23890) lies on the cubic K228 and these lines: {1,3}, {101,651}, {106,1462}, {109,14074}, {1018,6516}, {1023,1025}, {1055,1323}, {1262,1983}, {1308,14733}, {5290,19885}

X(23890) = X(i)-zayin conjugate of X(j) for these (i,j): {1155, 650}, {1776, 652}, {5572, 23351}, {15726, 513}
X(23890) = X(4564)-Ceva conjugate of X(15730)
X(23890) = X(14413)-cross conjugate of X(1323)
X(23890) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23351}, {514, 4845}, {522, 2291}, {650, 1156}, {663, 1121}, {693, 18889}, {1146, 14733}, {3887, 15734}
X(23890) = X(i)-Hirst inverse of X(j) for these (i,j): {241, 23703}
X(23890) = cevapoint of X(i) and X(j) for these (i,j): {1055, 14413}, {6603, 14414}
X(23890) = trilinear pole of line {1155, 6610}
X(23890) = crossdifference of every pair of points on line {650, 2310}
X(23890) = barycentric product X(i)X(j) for these {i,j}: {75, 23346}, {100, 1323}, {190, 6610}, {527, 651}, {653, 6510}, {658, 6603}, {664, 1155}, {934, 6745}, {1055, 4554}, {1638, 4564}, {4998, 14413}, {6366, 7045}, {6516, 23710}
X(23890) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 23351}, {109, 1156}, {527, 4391}, {651, 1121}, {692, 4845}, {1055, 650}, {1155, 522}, {1323, 693}, {1415, 2291}, {1638, 4858}, {6139, 2310}, {6174, 4768}, {6510, 6332}, {6610, 514}, {6745, 4397}, {14392, 4081}, {14413, 11}, {14414, 2968}, {23346, 1}
{X(1025),X(4564)}-harmonic conjugate of X(1023)

X(23891) =  X(1)X(2)∩X(100)X(4482)

Barycentrics    (a-b) (a-c) (a b+a c-2 b c) : :

X(23891) lies on the cubic K228 and these lines: {1,2}, {100,4482}, {101,9067}, {190,646}, {335,4674}, {514,4169}, {1016,1023}, {1022,4444}, {1145,4437}, {1981,15742}, {3760,4050}, {3761,4659}, {3799,4767}, {4465,13466}, {4568,21272}, {4738,17755}, {4986,21232}, {5541,17738}, {20533,21290}

X(23891) = isogonal conjugate of X(23892)
X(23891) = X(4607)-Ceva conjugate of X(190)
X(23891) = X(i)-cross conjugate of X(j) for these (i,j): {3768, 899}, {4728, 536}, {14430, 6381}
X(23891) = X(i)-Hirst inverse of X(j) for these (i,j): {239, 17780}, {6633, 9458}
X(23891) = cevapoint of X(i) and X(j) for these (i,j): {536, 4728}, {899, 3768}
X(23891) = crosspoint of X(190) and X(4607)
X(23891) = trilinear pole of line {536, 899}
X(23891) = crossdifference of every pair of points on line {649, 3248}
X(23891) = crosssum of X(649) and X(3768)
X(23891) = X(4607)-aleph conjugate of X(899)
X(23891) = X(i)-zayin conjugate of X(j) for these (i,j): {899, 649}, {7035, 4607}
X(23891) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23349}, {513, 739}, {667, 3227}, {889, 1977}, {898, 1015}, {3248, 4607}, {5381, 8027}
X(23891) = barycentric product X(i)X(j) for these {i,j}: {75, 23343}, {99, 3994}, {100, 6381}, {190, 536}, {664, 4009}, {668, 899}, {891, 7035}, {1016, 4728}, {1978, 3230}, {4465, 4562}, {4597, 4937}, {4600, 14431}, {4607, 13466}, {4998, 14430}
X(23891) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 23349}, {101, 739}, {190, 3227}, {536, 514}, {765, 898}, {890, 3248}, {891, 244}, {899, 513}, {1016, 4607}, {1646, 21143}, {3230, 649}, {3768, 1015}, {3994, 523}, {4009, 522}, {4465, 812}, {4526, 2170}, {4706, 4778}, {4728, 1086}, {4937, 4777}, {6381, 693}, {6632, 5381}, {7035, 889}, {13466, 4728}, {14404, 3122}, {14426, 3123}, {14430, 11}, {14431, 3120}, {14434, 19945}, {14437, 2087}, {19945, 764}, {23343, 1}
X(23891) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (668, 4595, 1018), (1016, 3570, 1023)

X(23892) =  X(1)X(649)∩X(6)X(667)

Barycentrics    a^2 (b-c) (2 a b-a c-b c) (a b-2 a c+b c) : :

X(23892) lies on the conic {{A,B,C,X(1),X(6)}}, the cubic K228, and on these lines: {1,649}, {6,667}, {56,8657}, {86,1019}, {87,4040}, {106,739}, {292,875}, {609,1919}, {813,898}, {870,4817}, {2279,8656}, {3226,3227}

X(23892) = isogonal conjugate of X(23891)
X(23892) = X(i)-aleph conjugate of X(j) for these (i,j): {889, 21389}, {898, 4040}, {3227, 5540}, {4607, 649}
X(23892) = X(3768)-cross conjugate of X(649)
X(23892) = cevapoint of X(649) and X(3768)
X(23892) = trilinear pole of line {649, 3248} X(23892) = crossdifference of every pair of points on line {536, 899} X(23892) = crosssum of X(i) and X(j) for these (i,j): {536, 4728}, {899, 3768}
X(23892) = X(i)-zayin conjugate of X(j) for these (i,j): {190, 3768}, {649, 899}
X(23892) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23343}, {100, 536}, {101, 6381}, {190, 899}, {651, 4009}, {660, 4465}, {662, 3994}, {668, 3230}, {765, 4728}, {891, 1016}, {898, 13466}, {3768, 7035}, {4526, 4998}, {4564, 14430}, {4567, 14431}, {4601, 14404}, {4604, 4937}, {4606, 4706}, {5378, 14433}, {5381, 14434}, {5383, 14426}, {6632, 19945}
X(23892) = barycentric product X(i)X(j) for these {i,j}: {75, 23349}, {244, 898}, {514, 739}, {649, 3227}, {889, 3248}, {1015, 4607}, {5381, 21143}
X(23892) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 23343}, {512, 3994}, {513, 6381}, {649, 536}, {663, 4009}, {667, 899}, {739, 190}, {898, 7035}, {1015, 4728}, {1919, 3230}, {1977, 3768}, {3122, 14431}, {3227, 1978}, {3248, 891}, {3249, 1646}, {3271, 14430}, {3768, 13466}, {4775, 4937}, {8027, 19945}, {8632, 4465}, {23349, 1}

X(23893) =  X(1)X(650)∩X(4)X(3064)

Barycentrics    a (a-b-c) (b-c) (a^2-2 a b+b^2+a c+b c-2 c^2) (a^2+a b-2 b^2-2 a c+b c+c^2) : :

X(23893) lies on the Feuerbach hyperbola, the cubic K228, and on these lines: {1,650}, {4,3064}, {7,514}, {8,3239}, {9,3900}, {21,1021}, {104,2291}, {294,4845}, {885,4530}, {1000,14330}, {1023,5377}, {1121,2481}, {1156,3887}, {3062,3309}, {3254,6366}, {3255,6362}, {4526,9365}, {4813,5556}, {17435,23838}

X(23893) = trilinear pole of line {650, 2310}
X(23893) = crossdifference of every pair of points on line {1155, 6610}
X(23893) = crosssum of X(i) and X(j) for these (i,j): {1055, 14413}, {6603, 14414}
X(23893) = X(1156)-aleph conjugate of X(9355)
X(23893) = X(650)-zayin conjugate of X(1155)
X(23893) = X(2170)-cross conjugate of X(15734)
X(23893) = orthocenter of X(4)X(7)X(9)
X(23893) = X(i)-isoconjugate of X(j) for these (i,j): {2, 23346}, {59, 1638}, {100, 6610}, {101, 1323}, {108, 6510}, {109, 527}, {651, 1155}, {664, 1055}, {934, 6603}, {1262, 6366}, {1275, 6139}, {1308, 15730}, {1461, 6745}, {1813, 23710}, {4564, 14413}, {7128, 14414}
X(23893) = barycentric product X(i)X(j) for these {i,j}: {75, 23351}, {522, 1156}, {650, 1121}, {693, 4845}, {2291, 4391}, {3261, 18889}
X(23893) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 23346}, {513, 1323}, {649, 6610}, {650, 527}, {652, 6510}, {657, 6603}, {663, 1155}, {1121, 4554}, {1156, 664}, {2170, 1638}, {2291, 651}, {2310, 6366}, {3022, 14392}, {3063, 1055}, {3270, 14414}, {3271, 14413}, {3287, 6647}, {3900, 6745}, {4845, 100}, {4895, 6174}, {14392, 6068}, {14413, 3321}, {14733, 7045}, {18344, 23710}, {18889, 101}, {22108, 15730}, {23351, 1}

X(23894) =  X(1)X(661)∩X(10)X(4024)

Barycentrics    a (b^2-c^2) (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) : :

X(23894) lies on the cubic K228 and these lines: {1,661}, {10,4024}, {37,4705}, {65,10097}, {75,1577}, {111,759}, {671,4444}, {897,2642}, {923,1910}, {1023,5380}, {4730,9278}

X(23894) = isogonal conjugate of X(23889)
X(23894) = isotomic conjugate of X(24039)
X(23894) = X(23894) = X(2642)-cross conjugate of X(661)
X(23894) = X(1)-Hirst inverse of X(17955)
X(23894) = cevapoint of X(i) and X(j) for these (i,j): {661, 2642}, {896, 17467}
X(23894) = trilinear pole of line {661, 2643}
X(23894) = crossdifference of every pair of points on line {896, 922}
X(23894) = crosssum of X(896) and X(2642)
X(23894) = X(i)-aleph conjugate of X(j) for these (i,j): {671, 16562}, {892, 798}, {897, 2640}
X(23894) = X(i)-zayin conjugate of X(j) for these (i,j): {661, 896}, {662, 2642}
X(23894) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23889}, {2, 5467}, {3, 4235}, {6, 5468}, {99, 187}, {100, 16702}, {101, 6629}, {110, 524}, {112, 6390}, {163, 14210}, {249, 690}, {250, 14417}, {323, 14559}, {351, 4590}, {468, 4558}, {648, 3292}, {662, 896}, {670, 14567}, {691, 2482}, {692, 16741}, {799, 922}, {805, 5026}, {827, 7813}, {907, 3793}, {1384, 2418}, {1576, 3266}, {1992, 2434}, {2407, 9717}, {2421, 5967}, {2709, 18800}, {2966, 9155}, {3712, 4565}, {4062, 4556}, {4567, 14419}, {4570, 4750}, {4627, 4831}, {5118, 14608}, {5477, 10425}, {5546, 7181}, {6331, 23200}, {6593, 17708}, {9115, 10409}, {9117, 10410}, {9132, 10552}, {17941, 18872}
X(23894) = barycentric product X(i)X(j) for these {i,j}: {1, 5466}, {19, 14977}, {75, 9178}, {92, 10097}, {111, 1577}, {523, 897}, {656, 17983}, {661, 671}, {691, 1109}, {798, 18023}, {850, 923}, {892, 2643}, {1821, 8430}, {2166, 9213}, {3120, 5380}, {4077, 5547}, {4086, 7316}, {8753, 14208}, {9180, 17955}
X(23894) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 5468}, {19, 4235}, {31, 5467}, {111, 662}, {512, 896}, {513, 6629}, {514, 16741}, {523, 14210}, {649, 16702}, {656, 6390}, {661, 524}, {669, 922}, {671, 799}, {798, 187}, {810, 3292}, {895, 4592}, {897, 99}, {923, 110}, {1577, 3266}, {1924, 14567}, {2642, 2482}, {2643, 690}, {3122, 14419}, {3124, 2642}, {3125, 4750}, {3708, 14417}, {4017, 7181}, {4041, 3712}, {4079, 21839}, {4516, 14432}, {4705, 4062}, {4750, 16733}, {4770, 4933}, {4822, 4831}, {5380, 4600}, {5466, 75}, {5547, 643}, {7316, 1414}, {8061, 7813}, {8430, 1959}, {8753, 162}, {9178, 1}, {10097, 63}, {10561, 16568}, {14908, 4575}, {14977, 304}, {17955, 9182}, {17983, 811}, {18023, 4602}, {21832, 4760}

X(23895) =  ISOGONAL CONJUGATE OF X(6137)

Barycentrics    (a^2-b^2) (-a^2+c^2) (Sqrt[3] (a^2+b^2-c^2)+2 S) (Sqrt[3] (a^2-b^2+c^2)+2 S) : :

X(23895) lies on the Steiner circumellipse and X(13)-Simmons circumconic, and on these lines: {2,18777}, {13,531}, {15,5916}, {99,5995}, {110,476}, {290,300}, {298,1494}, {299,5641}, {385,11081}, {396,1989}, {524,11078}, {530,11586}, {621,10217}, {754,14902}, {892,9206}, {1992,21466}, {2153,18827}, {2407,17402}, {3180,11080}, {3228,3457}, {5467,14185}, {5613,14356}, {8836,18122}, {9142,14181}

X(23895) = reflection of X(i) in X(j) for these {i,j}: {11078, 11537}, {11092, 396}
X(23895) = isogonal conjugate of X(6137)
X(23895) = X(14560)-vertex conjugate of X(17402)
X(23895) = cevapoint of X(i) and X(j) for these (i,j): {13, 20578}, {396, 523}
X(23895) = trilinear pole of line {2, 13}
X(23895) = antitomic conjugate of X(23745)
X(23895) = X(i)-cross conjugate of X(j) for these (i,j): {523, 11119}, {621, 18020}, {3180, 4590}, {11127, 249}, {14446, 14}, {19772, 23582}, {20578, 13}, {23283, 11118}
X(23895) = polar conjugate of isogonal conjugate of X(38414)
X(23895) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6137}, {14, 2624}, {15, 661}, {298, 798}, {470, 810}, {523, 2151}, {526, 2154}, {656, 8739}, {923, 9204}, {1094, 20578}, {2152, 23284}, {2643, 17402}, {6149, 20579}
X(23895) = barycentric product X(i)X(j) for these {i,j}: {13, 99}, {76, 5995}, {94, 17403}, {110, 300}, {299, 476}, {670, 3457}, {799, 2153}, {3266, 9206}, {4563, 8737}, {4590, 20578}, {5618, 11129}
X(23895) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6137}, {13, 523}, {14, 23284}, {16, 526}, {99, 298}, {110, 15}, {112, 8739}, {163, 2151}, {249, 17402}, {299, 3268}, {300, 850}, {395, 14447}, {476, 14}, {524, 9204}, {648, 470}, {1989, 20579}, {2152, 2624}, {2153, 661}, {3457, 512}, {4226, 6782}, {4240, 6110}, {5612, 8562}, {5618, 11080}, {5994, 11086}, {5995, 6}, {6138, 2088}, {8737, 2501}, {9206, 111}, {11080, 20578}, {11081, 6138}, {11537, 9200}, {14560, 3458}, {16806, 8603}, {17402, 11131}, {17403, 323}, {18777, 9201}, {20578, 115}

X(23896) =  ISOGONAL CONJUGATE OF X(6138)

Barycentrics    (a^2-b^2) (-a^2+c^2) (Sqrt[3] (a^2+b^2-c^2)-2 S) (Sqrt[3] (a^2-b^2+c^2)-2 S) : :

X(23896) lies on the Steiner circumellipse and X(14)-Simmons circumconic, and on these lines: {2,18776}, {14,530}, {16,5917}, {99,5994}, {110,476}, {290,301}, {298,5641}, {299,1494}, {385,11086}, {395,1989}, {524,11092}, {531,15743}, {622,10218}, {754,14903}, {892,9207}, {1992,21467}, {2154,18827}, {2407,17403}, {3181,11085}, {3228,3458}, {5467,14187}, {5617,14356}, {5619,10410}, {8838,18122}, {9142,14177}

X(23896) = reflection of X(i) in X(j) for these {i,j}: {11078, 395}, {11092, 11549}
X(23896) = isogonal conjugate of X(6138)
X(23896) = X(14560)-vertex conjugate of X(17403)
X(23896) = cevapoint of X(i) and X(j) for these (i,j): {14, 20579}, {395, 523}
X(23896) = trilinear pole of line {2, 14}
X(23896) = antitomic conjugate of X(23745)
X(23896) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 6138}, {3179, 2624}
X(23896) = X(i)-cross conjugate of X(j) for these (i,j): {523, 11120}, {622, 18020}, {3181, 4590}, {11126, 249}, {14447, 13}, {19773, 23582}, {20579, 14}, {23284, 11117}
X(23896) = polar conjugate of isogonal conjugate of X(38413)
X(23896) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6138}, {13, 2624}, {16, 661}, {299, 798}, {471, 810}, {523, 2152}, {526, 2153}, {656, 8740}, {923, 9205}, {1095, 20579}, {2151, 23283}, {2643, 17403}, {6149, 20578}
X(23896) = barycentric product X(i)X(j) for these {i,j}: {14, 99}, {76, 5994}, {94, 17402}, {110, 301}, {298, 476}, {670, 3458}, {799, 2154}, {3266, 9207}, {4563, 8738}, {4590, 20579}, {5619, 11128}
X(23896) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6138}, {13, 23283}, {14, 523}, {15, 526}, {99, 299}, {110, 16}, {112, 8740}, {163, 2152}, {249, 17403}, {298, 3268}, {301, 850}, {396, 14446}, {476, 13}, {524, 9205}, {648, 471}, {1989, 20578}, {2151, 2624}, {2154, 661}, {3458, 512}, {4226, 6783}, {4240, 6111}, {5616, 8562}, {5619, 11085}, {5994, 6}, {5995, 11081}, {6137, 2088}, {8738, 2501}, {9207, 111}, {11085, 20579}, {11086, 6137}, {11549, 9201}, {14560, 3457}, {15224, 4373}, {16807, 8604}, {17402, 323}, {17403, 11130}, {18776, 9200}, {20579, 115}

X(23897) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (-a^2 - a b + b^2 - a c - 3 b c + c^2) : :

X(23897) lies on these lines: {2, 23903}, {8, 10026}, {10, 115}, {12, 594}, {45, 1213}, {148, 6626}, {338, 349}, {442, 19584}, {661, 24119}, {3124, 21951}, {3661, 20337}, {3925, 21025}, {4037, 21674}, {4688, 8287}, {6625, 17731}, {8818, 17275}, {16044, 17277}, {16589, 25352}, {16592, 17448}, {16604, 16613}, {17056, 17316}, {17330, 17577}, {17337, 17541}, {23898, 23911}, {23900, 23906}, {23901, 23904}, {23909, 23914}, {23916, 23940}, {23928, 23934}, {23935, 23944}, {23936, 23939}, {24275, 24880}, {24902, 24956}

X(23898) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^4 b^2 + a^3 b^3 - 2 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 + 4 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 - 2 a^2 c^4 - a b c^4 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(23898) lies on these lines: {2, 23899}, {5, 21046}, {4212, 15377}, {17451, 21961}, {21011, 21670}, {23897, 23911}, {23901, 23910}, {23903, 23939}, {23929, 23938}

X(23899) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^2 + b^2 + c^2) (-a^4 - a^3 b + b^4 - a^3 c - a^2 b c - 2 b^2 c^2 + c^4) : :

X(23899) lies on these lines: {2, 23898}, {3, 21046}, {4, 9}, {21025, 21915}, {23904, 23905}, {23906, 23921}, {23907, 23920}, {23927, 23929}

X(23900) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (2 a^6 + 2 a^5 b - 3 a^4 b^2 - 3 a^3 b^3 + a b^5 + b^6 + 2 a^5 c + 2 a^4 b c - 3 a^3 b^2 c - 3 a^2 b^3 c + a b^4 c + b^5 c - 3 a^4 c^2 - 3 a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 - 3 a^3 c^3 - 3 a^2 b c^3 - 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(23900) lies on these lines: {2, 23898}, {140, 21046}, {21012, 21672}, {23897, 23906}

X(23901) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^2 b^2 - a b^3 + b^4 - 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 + c^4) : :

X(23901) lies on these lines: {2, 23927}, {10, 3685}, {141, 21043}, {594, 20679}, {3613, 15523}, {3661, 21728}, {4062, 17309}, {6646, 21089}, {17340, 21047}, {21330, 21725}, {23897, 23904}, {23898, 23910}, {23908, 23925}, {23909, 23917}, {23916, 23921}, {23918, 23935}, {23928, 23947}, {23929, 23932}, {23936, 23937}, {23943, 23944}

X(23902) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (a - b - c) (b + c) (a^3 + a^2 b - b^3 + a^2 c + a b c + b^2 c + b c^2 - c^3) : :

X(23902) lies on these lines: {2, 23904}, {9, 4092}, {10, 7235}, {144, 1654}, {197, 199}, {210, 8013}, {542, 2948}, {594, 4557}, {4046, 7172}, {4433, 6600}, {4516, 22176}, {6741, 24435}, {21023, 21231}, {23903, 23925}, {23921, 23934}

X(23903) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 + a^2 b + b^3 + a^2 c + a b c - b^2 c - b c^2 + c^3) : :

X(23903) lies on these lines: {2, 23897}, {6, 5046}, {10, 4037}, {37, 21029}, {145, 10026}, {148, 17103}, {220, 1834}, {346, 1213}, {671, 1509}, {1100, 8818}, {1577, 7264}, {1640, 3700}, {2082, 2503}, {2275, 16613}, {3124, 3959}, {3178, 21057}, {3679, 6537}, {3726, 13161}, {3727, 24210}, {3914, 21951}, {4854, 21965}, {4972, 21025}, {5051, 21024}, {5164, 5697}, {5254, 24512}, {5264, 9664}, {5309, 16783}, {5949, 16777}, {8287, 17301}, {13881, 19765}, {16592, 24217}, {17316, 20337}, {21090, 21802}, {23898, 23939}, {23902, 23925}, {23929, 23953}

X(23904) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 b - 2 a b^3 + b^4 + a^3 c + 4 a^2 b c + 3 a b^2 c - b^3 c + 3 a b c^2 - 2 a c^3 - b c^3 + c^4) : :

X(23904) lies on these lines: {2, 23902}, {142, 4092}, {3925, 21023}, {21049, 21927}, {21931, 21967}, {23897, 23901}, {23899, 23905}, {23928, 23943}, {23944, 23947}

X(23905) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (2 a^3 + 3 a^2 b + a b^2 + b^3 + 3 a^2 c + 4 a b c + a c^2 + c^3) : :

X(23905) lies on these lines: {1, 10026}, {2, 23897}, {10, 37}, {86, 6625}, {115, 1125}, {230, 19312}, {442, 20531}, {519, 6537}, {1107, 16613}, {1211, 6542}, {1220, 6543}, {1962, 21701}, {3124, 3727}, {3589, 16918}, {3884, 5164}, {5254, 13740}, {6656, 17245}, {8287, 17062}, {16826, 17056}, {21711, 21816}, {23899, 23904}, {23907, 23925}, {23911, 23939}, {23928, 23929}, {23931, 23932}

X(23906) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (2 a^5 - 3 a^3 b^2 + b^5 + 2 a^3 b c - a b^3 c - 3 a^3 c^2 + 4 a b^2 c^2 - b^3 c^2 - a b c^3 - b^2 c^3 + c^5) : :

X(23906) lies on these lines: {2, 23920}, {3700, 21948}, {21013, 21676}, {23897, 23900}, {23899, 23921}

X(23907) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (a - b - c) (b + c) (2 a^5 + 4 a^4 b + a^3 b^2 - 2 a^2 b^3 - 2 a b^4 - b^5 + 4 a^4 c + 10 a^3 b c + 6 a^2 b^2 c - a b^3 c + a^3 c^2 + 6 a^2 b c^2 + 8 a b^2 c^2 + b^3 c^2 - 2 a^2 c^3 - a b c^3 + b^2 c^3 - 2 a c^4 - c^5) : :

X(23907) lies on these lines: {2, 23921}, {21, 21044}, {1213, 2294}, {1962, 21965}, {23897, 23900}, {23899, 23920}, {23905, 23925}

X(23908) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (-a^2 + b^2 + c^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 - a c^3 - b c^3 + c^4) : :

X(23908) lies on these lines: {594, 21015}, {7085, 17233}, {23899, 23904}, {23901, 23925}, {23913, 23920}, {23924, 23928}

X(23909) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (-a^2 b^2 - a b^3 + b^4 + a^2 b c - 2 b^3 c - a^2 c^2 + b^2 c^2 - a c^3 - 2 b c^3 + c^4) : :

X(23909) lies on these lines: {2, 24367}, {15523, 20490}, {16886, 20655}, {21057, 21696}, {23897, 23914}, {23901, 23917}, {23910, 23916}, {23918, 23939}, {23928, 23931}

X(23910) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^2 b^4 - a b^5 + b^6 - a b^4 c - b^5 c - b^4 c^2 - a^2 c^4 - a b c^4 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(23910) lies on these lines: {2, 23932}, {5025, 21043}, {16894, 21681}, {21048, 21670}, {23898, 23901}, {23909, 23916}, {23938, 23946}

X(23911) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^4 b^2 + a^3 b^3 - 2 a^2 b^4 - a b^5 + b^6 + 2 a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c + a^4 c^2 + 2 a^3 b c^2 + 6 a^2 b^2 c^2 + 4 a b^3 c^2 - b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + 4 a b^2 c^3 + 2 b^3 c^3 - 2 a^2 c^4 - 2 a b c^4 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(23911) lies on these lines: {21018, 21682}, {23897, 23898}, {23905, 23939}, {23917, 23923}

X(23912) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^4 b^2 + a^3 b^3 - 2 a^2 b^4 - a b^5 + b^6 - b^5 c + a^4 c^2 + 2 a^2 b^2 c^2 - b^4 c^2 + a^3 c^3 + 2 b^3 c^3 - 2 a^2 c^4 - b^2 c^4 - a c^5 - b c^5 + c^6) : :

X(23912) lies on these lines: {115, 20595}, {1953, 21018}, {21019, 21683}, {23897, 23898}

X(23913) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^3 b - a^2 b^2 + a b^3 - a^3 c - 4 a^2 b c - 4 a b^2 c - a^2 c^2 - 4 a b c^2 - 2 b^2 c^2 + a c^3) : :

X(23913) lies on these lines: {2, 23928}, {10, 20360}, {335, 3728}, {442, 20488}, {594, 2294}, {1213, 21047}, {2292, 3842}, {2643, 3739}, {4647, 6541}, {20337, 21728}, {23897, 23901}, {23908, 23920}, {23934, 23943}

X(23914) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (a^4 + a^3 b - a^2 b^2 + a^3 c + 4 a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 - b^2 c^2 + b c^3) : :

X(23914) lies on these lines: {3125, 24169}, {20655, 21057}, {21021, 21684}, {23897, 23909}

X(23915) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^4 b^2 - a^3 b^3 + a^2 b^4 - 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 - 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) : :

X(23915) lies on these lines: {2, 23929}, {10, 18061}, {21022, 21685}, {21048, 21681}, {23898, 23901}, {23932, 23938}

X(23916) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 b^3 - 2 a b^5 + b^6 + 2 a^2 b^3 c + a b^4 c - 2 b^5 c + 2 a b^3 c^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(23916) lies on these lines: {21023, 21687}, {23897, 23940}, {23901, 23921}, {23909, 23910}

X(23917) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^3 b^2 - a^2 b^3 + a b^4 - 2 a^2 b^2 c - 2 a b^3 c + b^4 c - a^3 c^2 - 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 + a c^4 + b c^4) : :

X(23917) lies on these lines: {2, 23897}, {115, 3741}, {1213, 4972}, {3136, 15523}, {10026, 17135}, {20490, 21020}, {23901, 23909}, {23911, 23923}, {23930, 23939}

X(23918) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^3 b^2 - a^2 b^3 + a b^4 + 2 a^3 b c - 2 a b^3 c + b^4 c - a^3 c^2 - 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 - 2 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4) : :

X(23918) lies on these lines: {2, 23897}, {115, 3840}, {10026, 10453}, {16606, 16613}, {21025, 21688}, {23901, 23935}, {23909, 23939}

X(23919) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 b - a^2 b^2 - 3 a b^3 + 2 b^4 + a^3 c + 4 a^2 b c + 2 a b^2 c - 2 b^3 c - a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 - 3 a c^3 - 2 b c^3 + 2 c^4) : :

X(23919) lies on these lines: {2, 23937}, {21026, 21689}, {23897, 23901}, {23943, 23947}, {23948, 23949}

X(23920) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 b^2 - 2 a b^4 + b^5 + 2 a^2 b^2 c + a b^3 c - 2 b^4 c + a^3 c^2 + 2 a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3 - 2 a c^4 - 2 b c^4 + c^5) : :

X(23920) lies on these lines: {2, 23906}, {21029, 21687}, {23897, 23898}, {23899, 23907}, {23901, 23909}, {23908, 23913}

X(23921) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (a - b - c) (b + c) (a^3 b^2 + 2 a^2 b^3 - b^5 + a b^3 c + 2 b^4 c + a^3 c^2 - b^3 c^2 + 2 a^2 c^3 + a b c^3 - b^2 c^3 + 2 b c^4 - c^5) : :

X(23921) lies on these lines: {2, 23907}, {6, 21014}, {210, 21718}, {3061, 4193}, {10026, 21677}, {20654, 21030}, {23897, 23898}, {23899, 23906}, {23901, 23916}, {23902, 23934}, {23939, 23942}

X(23922) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (a - b - c) (b + c) (a^3 b + 2 a^2 b^2 - b^4 + a^3 c + a b^2 c + b^3 c + 2 a^2 c^2 + a b c^2 + b c^3 - c^4) : :

X(23922) lies on these lines: {2, 23902}, {8, 21}, {210, 3773}, {594, 20487}, {3452, 4092}, {3925, 7235}, {6675, 24640}, {20653, 21031}, {23897, 23898}, {24432, 24953}

X(23923) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (-a^3 b^2 - 2 a^2 b^3 + b^5 - a^2 b^2 c - 2 a b^3 c - b^4 c - a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - 2 a b c^3 - b^2 c^3 - b c^4 + c^5) : :

X(23923) lies on these lines: {10, 18755}, {16886, 21682}, {20654, 21019}, {21024, 21046}, {23897, 23909}, {23898, 23901}, {23911, 23917}

X(23924) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (a^4 + a^3 b - 2 a^2 b^2 - a b^3 + b^4 + a^3 c + 5 a^2 b c + a b^2 c - b^3 c - 2 a^2 c^2 + a b c^2 - a c^3 - b c^3 + c^4) : :

X(23924) lies on these lines: {2, 23902}, {12, 7363}, {226, 4092}, {354, 6741}, {594, 21688}, {4046, 10327}, {5554, 21677}, {7235, 21717}, {23897, 23909}, {23908, 23928}, {23946, 23947}

X(23925) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (a - b - c) (b + c) (a^4 b + 2 a^3 b^2 - a b^4 + a^4 c + 2 a^3 b c + 3 a^2 b^2 c + a b^3 c - 2 b^4 c + 2 a^3 c^2 + 3 a^2 b c^2 + 2 a b^2 c^2 + 2 b^3 c^2 + a b c^3 + 2 b^2 c^3 - a c^4 - 2 b c^4) : :

X(23925) lies on these lines: {594, 21011}, {23897, 23898}, {23901, 23908}, {23902, 23903}, {23905, 23907}

X(23926) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^4 + b^4 + c^4) (-a^6 - a^5 b + b^6 - a^5 c - a^4 b c - b^4 c^2 - b^2 c^4 + c^6) : :

X(23926) lies on these lines: {2980, 21011}

X(23927) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^4 + a^3 b + b^4 + a^3 c + a^2 b c - 2 b^2 c^2 + c^4) : :

X(23927) lies on these lines: {2, 23901}, {6, 21043}, {8, 21728}, {10, 192}, {42, 2165}, {3974, 8013}, {17262, 21047}, {17350, 21089}, {23899, 23929}, {23902, 23903}

X(23928) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    a (b + c)^2 (a^2 + a b + b^2 + a c - b c + c^2) : :

X(23928) lies on these lines: {1, 1326}, {2, 23913}, {8, 192}, {37, 2054}, {38, 3571}, {55, 199}, {86, 20538}, {244, 17045}, {523, 17246}, {594, 756}, {661, 22260}, {1500, 7237}, {1953, 16872}, {2309, 3727}, {2611, 17447}, {2667, 4016}, {3666, 17470}, {3778, 4424}, {4003, 17476}, {4026, 4642}, {4132, 17457}, {4646, 6042}, {4971, 24454}, {5263, 17872}, {16598, 17593}, {16706, 21254}, {17302, 21295}, {23897, 23934}, {23901, 23947}, {23902, 23903}, {23904, 23943}, {23905, 23929}, {23908, 23924}, {23909, 23931}, {24443, 24923}

X(23929) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    a^2 (b + c) (a^2 b^2 + a b^3 + b^4 + 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 + c^4) : :

X(23929) lies on these lines: {1, 20700}, {2, 23915}, {10, 75}, {42, 21008}, {5254, 21043}, {18757, 20472}, {21725, 24443}, {23898, 23938}, {23899, 23927}, {23901, 23932}, {23903, 23953}, {23905, 23928}

X(23930) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (-a^3 b - 2 a^2 b^2 + b^4 - a^3 c - 2 a^2 b c - 3 a b^2 c - b^3 c - 2 a^2 c^2 - 3 a b c^2 - 2 b^2 c^2 - b c^3 + c^4) : :

X(23930) lies on these lines: {2, 23901}, {10, 846}, {1211, 21043}, {8013, 20679}, {15523, 20488}, {17757, 20653}, {20483, 21675}, {21020, 21054}, {21028, 21682}, {21692, 21730}, {23897, 23909}, {23917, 23939}

X(23931) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (b^2 + c^2) (-a^4 - a^3 b - a b^3 + b^4 - a^3 c - 2 a^2 b c - 2 b^3 c + b^2 c^2 - a c^3 - 2 b c^3 + c^4) : :

X(23931) lies on these lines: {594, 21037}, {23905, 23932}, {23909, 23928}

X(23932) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (b^2 + c^2) (-2 a^4 - 2 a^3 b - a^2 b^2 - a b^3 + b^4 - 2 a^3 c - 2 a^2 b c - 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 + c^4) : :

X(23932) lies on these lines: {2, 23910}, {10, 82}, {6656, 21043}, {23901, 23929}, {23905, 23931}, {23915, 23938}

X(23933) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    a (a - b - c) (b + c) (a^3 b - b^4 + a^3 c + 4 a^2 b c + a b^2 c - b^3 c + a b c^2 + 4 b^2 c^2 - b c^3 - c^4) : :

X(23933) lies on these lines: {10, 17864}, {21031, 21039}, {23902, 23903}

X(23934) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c)^2 (-2 a^3 - 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(23934) lies on these lines: {2, 23901}, {10, 894}, {199, 20989}, {1213, 21043}, {3728, 21725}, {3932, 21674}, {7227, 21047}, {8013, 10026}, {23897, 23928}, {23902, 23921}, {23913, 23943}

X(23935) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a b - a c + b c) (a^3 b^2 + a^2 b^3 - a b^4 - 2 a^3 b c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 - 2 b^3 c^2 + a^2 c^3 - 2 b^2 c^3 - a c^4 + b c^4) : :

X(23935) lies on these lines: {594, 21941}, {20489, 21011}, {20654, 21040}, {23897, 23944}, {23901, 23918}

X(23936) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (2 a - b - c) (b + c) (a^3 b + 2 a^2 b^2 - b^4 + a^3 c - 2 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 + b c^3 - c^4) : :

X(23936) lies on these lines: {1, 8258}, {2, 23943}, {20653, 21041}, {23897, 23939}, {23901, 23937}

X(23937) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (a - 2 b - 2 c) (b + c) (a^3 b + 2 a^2 b^2 - b^4 + a^3 c + a^2 b c + 2 a b^2 c + b^3 c + 2 a^2 c^2 + 2 a b c^2 + b^2 c^2 + b c^3 - c^4) : :

X(23937) lies on these lines: {2, 23919}, {20653, 21042}, {23901, 23936}

X(23938) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b - c)^2 (b + c)^3 (-3 a^2 - a b + b^2 - a c - b c + c^2) : :

X(23938) lies on these lines: {10, 7983}, {11, 21944}, {115, 2643}, {4041, 21725}, {23898, 23929}, {23910, 23946}, {23915, 23932}, {23939, 23953}, {23940, 23943}

X(23939) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b - c)^2 (b + c) (a^3 - 2 a b^2 + b^3 - 3 a b c - 2 a c^2 + c^3) : :

X(23939) lies on these lines: {2, 23906}, {3120, 4024}, {21044, 21046}, {23897, 23936}, {23898, 23903}, {23905, 23911}, {23909, 23918}, {23917, 23930}, {23921, 23942}, {23938, 23953}, {23943, 23950}

X(23940) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b - c)^2 (b + c) (a^3 b - 2 a b^3 + b^4 + a^3 c + a^2 b c - 2 a b^2 c - 2 a b c^2 - b^2 c^2 - 2 a c^3 + c^4) : :

X(23940) lies on these lines: {2533, 21044}, {21045, 21046}, {23897, 23916}, {23938, 23943}

X(23941) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b - c)^2 (b + c)^3 (a^4 + a^3 b - 2 a^2 b^2 - a b^3 + b^4 + a^3 c + a^2 b c - a b^2 c - b^3 c - 2 a^2 c^2 - a b c^2 - a c^3 - b c^3 + c^4) : :

X(23941) lies on these lines: {661, 21054}, {8901, 21046}

X(23942) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 - a b^2 + 2 b^3 - a b c - 3 b^2 c - a c^2 - 3 b c^2 + 2 c^3) : :

X(23942) lies on these lines: {2, 23897}, {8, 115}, {661, 4051}, {2321, 21712}, {3621, 10026}, {4442, 21965}, {5839, 8818}, {5949, 17314}, {23921, 23939}

X(23943) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b - c)^2 (b + c) (-a^2 - 3 a b + b^2 - 3 a c + b c + c^2) : :

X(23943) lies on these lines: {2, 23936}, {2643, 8287}, {3120, 18004}, {23901, 23944}, {23904, 23928}, {23913, 23934}, {23919, 23947}, {23938, 23940}, {23939, 23950}

X(23944) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a b^3 - a^2 b c - 2 a b^2 c - 2 a b c^2 - b^2 c^2 + a c^3) : :

X(23944) lies on these lines: {2, 23913}, {8, 20360}, {10, 21810}, {75, 1581}, {321, 21688}, {523, 7263}, {740, 2294}, {4128, 18194}, {4359, 17470}, {4444, 22260}, {15349, 19843}, {17278, 21254}, {17447, 24165}, {21257, 21951}, {23668, 25124}, {23897, 23935}, {23901, 23943}, {23904, 23947}

X(23945) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (2*a^3+3*(b+c)*a^2-(3*b^2+4*b*c+3*c^2)*a-(b+c)*(b^2-3*b*c+c^2))*(b^2-c^2) : :

X(23945) lies on these lines: {4120, 21714}, {23939, 23943}, {24121, 24290}

X(23946) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^2 b^4 - 3 a^2 b^2 c^2 - a b^3 c^2 - a b^2 c^3 - b^3 c^3 + a^2 c^4) : :

X(23946) lies on these lines: {2, 23915}, {10, 3735}, {313, 21716}, {23897, 23935}, {23910, 23938}, {23924, 23947}

X(23947) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (a^3 b + b^4 + a^3 c + 2 a^2 b c - a b^2 c - b^3 c - a b c^2 - 2 b^2 c^2 - b c^3 + c^4) : :

X(23947) lies on these lines: {2, 23897}, {12, 3178}, {115, 3912}, {257, 312}, {333, 17685}, {524, 21221}, {525, 1577}, {536, 8287}, {1213, 17280}, {1266, 17058}, {1698, 3712}, {2901, 14873}, {3175, 16603}, {3932, 20657}, {4851, 8818}, {5949, 17243}, {6541, 20546}, {6542, 6543}, {16831, 17720}, {18635, 20171}, {23901, 23928}, {23904, 23944}, {23919, 23943}, {23924, 23946}

X(23948) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b^2 - c^2) (a^3 b + a^2 b^2 - a b^3 + a^3 c + 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) : :

X(23948) lies on these lines: {21051, 21715}, {21958, 23301}, {23919, 23949}, {23954, 24381}

X(23949) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b^2 - c^2) (a^4 - 2 a^2 b^2 + a b^3 - 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(23949) lies on these lines: {2, 23954}, {8611, 21052}, {23919, 23948}

X(23950) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b^2 - c^2) (a^3 b^2 + a^2 b^3 - a b^4 + 2 a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 + 2 a^2 b c^2 - 4 a b^2 c^2 + a^2 c^3 - 2 a b c^3 - a c^4 + b c^4) : :

X(23950) lies on these lines: {2, 24380}, {21053, 21722}, {23919, 23948}, {23939, 23943}

X(23951) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b^2 - c^2) (a^4 + a^3 b - a^2 b^2 + a^3 c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 + 3 b^2 c^2 + b c^3) : :

X(23951) lies on these lines: {2, 24381}, {661, 2533}, {4024, 10278}, {4036, 9508}, {23919, 23948}

X(23952) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b^2 - c^2) (a^3 b^2 + a^2 b^3 - a b^4 + a^3 b c + a^2 b^2 c - a b^3 c + b^4 c + a^3 c^2 + a^2 b c^2 - a b^2 c^2 + a^2 c^3 - a b c^3 - a c^4 + b c^4) : :

X(23952) lies on these lines: {21055, 21724}

X(23953) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    a (b - c)^2 (b + c) (a^3 + a^2 b + a b^2 + a^2 c + 3 a b c - 2 b^2 c + a c^2 - 2 b c^2) : :

X(23953) lies on these lines: {3122, 21043}, {23903, 23929}, {23938, 23939}

X(23954) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    a (b^2 - c^2) (a^3 - b^3 + a b c - 2 b^2 c - 2 b c^2 - c^3) : :

X(23954) lies on these lines: {2, 23949}, {512, 661}, {656, 4122}, {3932, 17069}, {23948, 24381}

X(23955) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = CEVIAN TRIANGLE OF X(10)

Barycentrics    (b + c) (-a^3 b - 2 a^2 b^2 - 2 a b^3 + b^4 - a^3 c - 2 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 - 2 a c^3 - b c^3 + c^4) : :

X(23955) lies on these lines: {2, 23902}, {10, 37}, {1211, 21728}, {4092, 4357}, {7081, 17315}, {23897, 23935}, {23901, 23928}, {23913, 23934}

X(23956) =  X(4)X(94)∩X(93)X(22823)

Trilinears    (3+4*cos(2*A))*(cos(A)-cos(B-C))*(1+2*cos(2*A)-4*cos(A)*cos(B-C)+2*cos(2*(B-C))) : :
Barycentrics    (a^2+b^2-c^2) (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2+c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (2 a^4-4 a^2 b^2+2 b^4-4 a^2 c^2+3 b^2 c^2+2 c^4) : :
Barycentrics    Csc[3 A] (8 Cos[A]-Sec[A]) Sin[A]^2 : :

See Antreas Hatzipolakis, Peter Moses, and Ercole Suppa Hyacinthos 28368 and Hyacinthos 28366.

X(23956) lies on these lines: {4,94}, {93,22823}, {275,1141}, {317,328}, {1989,3087}, {3520,5961}, {5627,10152}, {10733,15469}

X(23956) = {X(4),X(265)}-harmonic conjugate of X(6344)
X(23956) = X(6149)-isoconjugate of X(21400)
X(23956) = barycentric product X(94)X(21844)
X(23956) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 21400}, {21844, 323}

X(23957) =  X(4)X(94)∩X(1511)X(22823)

Trilinears    2*cos(5*A)-19*cos(B-C)-26*cos(2*A)*cos(B-C)-8*cos(4*A)*cos(B-C) +2*cos(3*A)*(5+4*cos(2*(B-C)))+2*cos(A)*(11+8*cos(2*(B-C)))-2*cos(3*(B-C))-4*cos(2*A)*cos(3*(B-C)) : :
Barycentrics    2 a^16-6 a^14 b^2+4 a^12 b^4+2 a^10 b^6-2 a^6 b^10-4 a^4 b^12+6 a^2 b^14-2 b^16-6 a^14 c^2+16 a^12 b^2 c^2-11 a^10 b^4 c^2-9 a^8 b^6 c^2+13 a^6 b^8 c^2+11 a^4 b^10 c^2-24 a^2 b^12 c^2+10 b^14 c^2+4 a^12 c^4-11 a^10 b^2 c^4+20 a^8 b^4 c^4-10 a^6 b^6 c^4-19 a^4 b^8 c^4+36 a^2 b^10 c^4-20 b^12 c^4+2 a^10 c^6-9 a^8 b^2 c^6-10 a^6 b^4 c^6+24 a^4 b^6 c^6-18 a^2 b^8 c^6+22 b^10 c^6+13 a^6 b^2 c^8-19 a^4 b^4 c^8-18 a^2 b^6 c^8-20 b^8 c^8-2 a^6 c^10+11 a^4 b^2 c^10+36 a^2 b^4 c^10+22 b^6 c^10-4 a^4 c^12-24 a^2 b^2 c^12-20 b^4 c^12+6 a^2 c^14+10 b^2 c^14-2 c^16 : :

See Antreas Hatzipolakis, Peter Moses, and Ercole Suppa Hyacinthos 28368 and Hyacinthos 28366.

X(23957) lies on these lines: {4,94}, {1511,22823}

X(23958) =  X(1)X(4744)∩X(2)X(7)

Barycentrics    a (2 a^2-2 b^2+3 b c-2 c^2) : :
Barycentrics    3-4 Cos[A] : :

See Antreas Hatzipolakis, Peter Moses, and Ercole Suppa Hyacinthos 28368 and Hyacinthos 28366.

X(23958) lies on these lines: {1,4744}, {2,7}, {8,3336}, {21,5708}, {23,1473}, {31,18201}, {40,3623}, {46,145}, {56,18419}, {72,17572}, {75,5372}, {81,89}, {84,17578}, {88,4383}, {100,4430}, {149,3474}, {165,3957}, {171,4392}, {189,21739}, {191,5550}, {222,1994}, {238,9335}, {244,4650}, {323,23140}, {404,3940}, {484,3241}, {499,1749}, {518,9352}, {750,7226}, {896,17063}, {938,15680}, {942,4189}, {982,17024}, {1155,3873}, {1376,4661}, {1407,1993}, {1454,3600}, {1621,4860}, {1707,7292}, {1768,9812}, {1788,20060}, {2095,6909}, {2320,5425}, {2403,4498}, {2975,5221}, {3052,3315}, {3060,3937}, {3075,18477}, {3337,3616}, {3338,3622}, {3522,5709}, {3550,17449}, {3579,3889}, {3666,10987}, {3832,18540}, {3868,4188}, {3916,16865}, {3927,17531}, {4000,16568}, {4253,21372}, {4359,5361}, {4393,20367}, {4641,14997}, {4757,21842}, {4850,16668}, {4858,14211}, {4973,5902}, {5057,17728}, {5068,7330}, {5180,10072}, {5211,20064}, {5220,9342}, {5265,7098}, {5422,22129}, {5439,16859}, {5535,5731}, {5536,9778}, {5770,6839}, {6510,17092}, {6762,20052}, {6763,9780}, {7004,9539}, {7085,7496}, {7171,15683}, {7191,18193}, {7293,7492}, {7419,23085}, {9782,19854}, {10056,16763}, {11010,20057}, {11246,11680}, {12704,20070}, {13243,19541}, {14986,17437}, {15650,17535}, {15934,17549}, {16704,17490}, {16816,18206}, {16948,17054}, {17018,17596}, {17577,18541}, {18391,20067}, {19245,23169}

X(23958) = X(3635)-zayin conjugate of X(9)
X(23958) = crossdifference of every pair of points on line {663, 4770}
X(23958) = barycentric product X(86)*X(4084)
X(23958) = barycentric quotient X(4084)/X(10)

X(23959) =  X(1)X(6246)∩X(8)X(10738)

Barycentrics    (2 a^5-3 a^4 b+a^3 b^2+a^2 b^3-3 a b^4+2 b^5-2 a^4 c+6 a^3 b c-6 a^2 b^2 c+6 a b^3 c-2 b^4 c-4 a^3 c^2+a^2 b c^2+a b^2 c^2-4 b^3 c^2+4 a^2 c^3-6 a b c^3+4 b^2 c^3+2 a c^4+2 b c^4-2 c^5) (2 a^5-2 a^4 b-4 a^3 b^2+4 a^2 b^3+2 a b^4-2 b^5-3 a^4 c+6 a^3 b c+a^2 b^2 c-6 a b^3 c+2 b^4 c+a^3 c^2-6 a^2 b c^2+a b^2 c^2+4 b^3 c^2+a^2 c^3+6 a b c^3-4 b^2 c^3-3 a c^4-2 b c^4+2 c^5) : :

See Antreas Hatzipolakis, Peter Moses, and Ercole Suppa Hyacinthos 28368 and Hyacinthos 28366.

X(23959) lies on the Feuerbach hyperbola and these lines: {1,6246}, {8,10738}, {21,11231}, {84,10728}, {952,1392}, {1000,10947}, {2320,6980}, {2800,5560}, {7319,12247}, {10090,15079}, {10698,21398}, {12119,20107}

X(23960) =  MIDPOINT OF X(2077) AND X(8148)

Barycentrics    a (2 a^6-6 a^5 b+12 a^3 b^3-6 a^2 b^4-6 a b^5+4 b^6-6 a^5 c+22 a^4 b c-19 a^3 b^2 c-14 a^2 b^3 c+25 a b^4 c-8 b^5 c-19 a^3 b c^2+42 a^2 b^2 c^2-19 a b^3 c^2-4 b^4 c^2+12 a^3 c^3-14 a^2 b c^3-19 a b^2 c^3+16 b^3 c^3-6 a^2 c^4+25 a b c^4-4 b^2 c^4-6 a c^5-8 b c^5+4 c^6) : :
X(23960) = X[36] - 3 X[10247], 2 X[6681] - 3 X[10283], 5 X[5048] - X[13528], X[5535] - 9 X[16200]

See Antreas Hatzipolakis, Peter Moses, and Ercole Suppa Hyacinthos 28368 and Hyacinthos 28366.

X(23960) lies on these lines: {1,3}, {3036,3814}, {5690,20107}, {6681,10283}

X(23960) = midpoint of X(2077) and X(8148)

X(23961) =  MIDPOINT OF X(3) AND X(36)

Barycentrics    a^2 (-2 a^5+2 a^4 (b+c)+a^3 (4 b^2-6 b c+4 c^2)+a^2 (-4 b^3+b^2 c+b c^2-4 c^3)+(b-c)^2 (2 b^3+b^2 c+b c^2+2 c^3)-2 a (b^4-3 b^3 c+3 b^2 c^2-3 b c^3+c^4)) : :
X(23961) = X[5]-2*X[6681], X[104]+3*X[13587], 2*X[140]-X[3814], 5*X[631]-X[5080], 7*X[3523]+X[20067], 3*X[3582]-X[10738], 3*X[4881]-X[6265], X[6882]+X[15326], X[22793]-2*X[22835]

See Antreas Hatzipolakis, Peter Moses, and Ercole Suppa Hyacinthos 28368 and Hyacinthos 28366.

X(23961) lies on these lines: {1, 3}, {5, 6681}, {20, 10598}, {24, 1878}, {30, 6713}, {104, 13587}, {140, 3814}, {182, 9037}, {214, 14988}, {355, 4188}, {404, 9956}, {515, 12619}, {535, 549}, {631, 5080}, {912, 22935}, {953, 8697}, {993, 5123}, {2771, 18861}, {3523, 20067}, {3582, 10738}, {3838, 6914}, {4299, 6958}, {4881, 6265}, {4996, 5440}, {5057, 6875}, {5146, 7501}, {5303, 6940}, {5450, 6924}, {5587, 18515}, {5840, 15325}, {5886, 6950}, {6882, 15326}, {6906, 9955}, {6942, 18481}, {6961, 10526}, {6970, 18516}, {6971, 10483}, {7288, 10525}, {7508, 10165}, {7743, 10058}, {11499, 19537}, {15446, 17606}, {16371, 22758}, {22793, 22835}

X(23961) = midpoint of X(i) and X(j) for these {i,j}: {3,36}, {1385, 10225}, {2077,22765}, {6882,15326}
X(23961) = reflection of X(i) in X(j) for these {i,j}: {5,6681}, {1385,18857}, {3814,140}, {5048,15178}, {22793,22835}
X(23961) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,10246,5010}, {3,16203,5217}, {3,22765,2077}, {36,2077,22765}, {36,5172,5126}, {5172,5204,36}, {5450,6924,18480}, {14792,14800,2646}, {15326,21154,6882}

leftri

Points associated with barycentric squares of lines (inscribed ellipses): X(23962)-X(23992)

rightri

Contributed by Randy Hutson, September 29, 2018

Continuing from the preamble before X(23582), let L be a line in the plane of ABC and P = p : q : r (barycentrics) be the trilinear pole of L. Let E be the inellipse that is the (pointwise) barycentric square of L. The vertex conjugate of the foci of E is the point a^2 p^2 : b^2 q^2 : c^2 r^2. Let U = u : v : w and X = x : y : z be points on L so that U^2 and X^2 lie on E. The trilinear pole of the tangent to E at U^2 is the barycentric product P*U = p*u : q*v : r*w, which lies on the trilinear polar of the Brianchon point (perspector) of E. The intersection of the tangents to E at U^2 and X^2 is the barycentric product U*X = u*x : v*y : w*z.

X(23962) = BARYCENTRIC SQUARE OF X(850)

Barycentrics    csc^2 A sin^2(B - C) : :
Barycentrics    (b^2 - c^2)^2/a^4 : :
Barycentrics    squared distance from A to Brocard axis : :

X(23962) lies on these lines: {2, 6331}, {23, 17984}, {76, 94}, {125, 850}, {264, 5169}, {290, 3448}, {339, 868}, {1502, 4609} et al

X(23962) = isogonal conjugate of X(23963)
X(23962) = isotomic conjugate of X(23357)
X(23962) = anticomplement of X(23584)
X(23962) = complement of isogonal conjugate of X(36198)
X(23962) = barycentric square of X(850)
X(23962) = barycentric product X(75)*X(23994)

X(23963) = BARYCENTRIC SQUARE OF X(163)

Barycentrics    a^6/(b^2 - c^2)^2 : :

X(23963) lies on these lines: {6, 250}, {50, 3289}, {249, 2076}, {1576, 3049}, {2211, 18374}, {2715, 3050} et al

X(23963) = isogonal conjugate of X(23962)
X(23963) = barycentric square of X(163)
X(23963) = vertex conjugate of foci of inellipse centered at X(23584) (the barycentric square of the Brocard axis)

X(23964) = BARYCENTRIC SQUARE OF X(162)

Barycentrics    tan^2 A csc^2(B - C) : :
Barycentrics    (sin^2 A)/(tan B - tan C)^2 : :
Barycentrics    (sin^2 A)/(sin 2B - sin 2C)^2 : :
Barycentrics    a^2/((b^2 - c^2) (b^2 + c^2 - a^2))^2 : :

X(23964) lies on these lines: {4, 1554}, {23, 232}, {107, 685}, {112, 647}, {249, 297}, {1495, 8744} et al

X(23964) = isogonal conjugate of X(15526)
X(23964) = isotomic conjugate of X(36793)
X(23964) = polar conjugate of X(339)
X(23964) = cevapoint of X(6) and X(112)
X(23964) = X(6)-cross conjugate of X(112)
X(23964) = barycentric square of X(162)
X(23964) = cevapoint of circumcircle-intercepts of Moses radical circle
X(23964) = trilinear pole of line X(112)X(1576) (the tangent to the circumcircle at X(112))
X(23964) = vertex conjugate of the foci of the inellipse centered at X(23583) (the barycentric square of the Euler line)
X(23964) = X(92)-isoconjugate of X(2972)

X(23965) = BARYCENTRIC SQUARE OF X(3268)

Barycentrics    (csc^2 A)(sin^2(B - C))(1 + 2 cos 2A)^2 : :
Barycentrics    squared distance from A to Fermat axis : :

X(23965) lies on these lines: {2, 6035}, {323, 7799}, {3258, 3268}, {5641, 14731}

X(23965) = isogonal conjugate of X(23966)
X(23965) = isotomic conjugate of X(23588)
X(23965) = anticomplement of X(23589)
X(23965) = barycentric square of X(3268)

X(23966) = ISOGONAL CONJUGATE OF X(23965)

Barycentrics    (sin^4 A)(csc^2(B - C))/(1 + 2 cos 2A)^2 : :

X(23966) lies on these lines: {6, 15395}, {14398, 14560}

X(23966) = isogonal conjugate of X(23965)
X(23966) = vertex conjugate of the foci of the inellipse centered at X(23589) (the barycentric square of the Fermat axis)

X(23967) = BARYCENTRIC SQUARE OF X(542)

Barycentrics    (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))^2 : :

X(23967) is the center of hyperbola {{A, B, C, X(2), X(476)}}, which is the locus of trilinears poles of lines parallel to the Fermat axis (i.e., lines passing through X(542)). The perspector of the hyperbola is X(542).

X(23967) lies on the Steiner inellipse and these lines: {2, 2966}, {6, 23992}, {30, 115}, {39, 18334}, {50, 8623}, {248, 9140}, {401, 8859}, {441, 524}, {476, 23598}, {523, 3163}, {525, 2482}, {647, 5642}, {1084, 3003} et al

X(23967) = midpoint of X(2) and X(2966)
X(23967) = reflection of X(35088) in X(2)
X(23967) = complement of X(5641)
X(23967) = perspector of circumparabola centered at X(542)
X(23967) = X(2)-Ceva conjugate of X(542)
X(23967) = barycentric square of X(542)
X(23967) = crossdifference of every pair of points on line X(842)X(7418) (the tangent to circumcircle at X(842))
X(23967) = Steiner-inellipse-antipode of X(35088)

X(23968) = TRILINEAR POLE OF LINE X(5191)X(23967)

Barycentrics    (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))/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2)) : :

Line X(5191)X(23967) is the tangent to the inellipse that is the barycentric square of the Fermat axis, at X(23967) (the barycentric square of X(542)).

Let P1 and P2 be the two points on the Fermat axis whose trilinear polars are parallel to the Fermat axis. P1 and P2 lie on the circumconic centered at X(23967) (hyperbola {{A, B, C, X(2), X(476)}}), and circle {{X(2), X(476), X(23969)}}. X(23968) is the barycentric product P1*P2.

X(23968) lies on these lines: {6, 13}, {476, 2395}, {2394, 2407}, {14560, 15475}, {14999, 18312}

X(23968) = barycentric product X(476)*X(542)
X(23968) = trilinear pole of line X(5191)X(23967)

X(23969) = INVERSE-IN-DAO-MOSES-TELV-CIRCLE OF X(476)

Barycentrics    a^2/((b^2 - c^2) ((a^2 - b^2 - c^2)^2 - b^2 c^2) (2 a^6 - b^6 - c^6 - 2 a^4 b^2 - 2 a^4 c^2 + a^2 b^4 + a^2 c^4 + b^4 c^2 + b^2 c^4)) : :

Let A', B', C' be the intersections of the Fermat axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(23969).

X(23969) lies on the circumcircle and these lines: {74, 2088}, {98, 1989}, {99, 5649}, {110, 14998}, {125, 14846}, {476, 1637}, {842, 2493}, {935, 16230}, {2373, 18883} et al

X(23969) = trilinear pole of line X(6)X(14560)
X(23969) = inverse-in-Dao-Moses-Telv-circle of X(476)
X(23969) = barycentric product X(476)*X(842) (circumcircle-X(2)-antipodes)

X(23970) = BARYCENTRIC SQUARE OF X(3239)

Barycentrics    (b - c)^2 (b + c - a)^4 : :
Barycentrics    squared distance from A to Soddy line : :

X(23970) lies on these lines: {2, 23587}, {6, 13138}, {312, 4437}, {346, 3699}, {594, 7101}, {1146, 7358} et al

X(23970) = isogonal conjugate of X(23971)
X(23970) = isotomic conjugate of X(23586)
X(23970) = anticomplement of X(23587)
X(23970) = barycentric square of X(3239)

X(23971) = BARYCENTRIC SQUARE OF X(934)

Barycentrics    a^2/((b - c) (b + c - a)^2)^2 : :

The trilinear pole of X(23971) passes through X(6614).

X(23971) lies on the Steiner inellipse and these lines: {934, 6129}, {1262, 6610}

X(23971) = isogonal conjugate of X(23970)
X(23971) = barycentric square of X(934)
X(23971) = vertex conjugate of foci of inellipse centered at X(23591) (the barycentric square of the Soddy line)

X(23972) = BARYCENTRIC SQUARE OF X(516)

Barycentrics    a^2 (a^2 - b^2 cos C - c^2 cos B)^2 : :
Barycentrics    (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c))^2 : :

X(23972) is the center of hyperbola {{A, B, C, X(2), X(658)}}, which is the locus of trilinear poles of lines parallel to the Soddy line (i.e. lines that pass through X(516)).

X(23972) lies on the Steiner inellipse, the inellipse centered at X(23587), and these lines: {1, 1146}, {2, 18025}, {6, 7}, {115, 118}, {220, 1783}, {594, 2331}, {676, 9502}, {1015, 16583} et al

X(23972) = complement of X(18025)
X(23972) = perspector of circumparabola centered at X(516)
X(23972) = X(2)-Ceva conjugate of X(516)
X(23972) = barycentric square of X(516)

X(23973) = TRILINEAR POLE OF LINE X(910)X(23972)

Barycentrics    a^2 (a^2 - b^2 cos C - c^2 cos B) / ((1 + cos A) (cos B - cos C)) : :
Barycentrics    (2*a^3-(b+c)*a^2-(b^2-c^2)*(b-c))*(a-b+c)^2*(-a+c)*(a+b-c)^2*(a-b) : :

Line X(910)X(23972) is the tangent to the inellipse that is the barycentric square of the Soddy line, at X(23972) (the barycentric square of X(516)).

Let P1 and P2 be the two points on the Soddy line whose trilinear polars are parallel to the Soddy line. P1 and P2 lie on the circumconic centered at X(23972) (hyperbola {{A, B, C, X(2), X(9057)}}), and circle {{X(2), X(9057), X(24016)}}. The midpoint of P1 and P2 is the tripolar centroid of X(658). X(23973) is the barycentric product P1*P2.

X(23973) lies on these lines: {1, 7}, {651, 653}, {883, 1275}, {934, 1633}, {13149, 14544}

X(23973) = barycentric product X(516)*X(658)
X(23973) = trilinear pole of line X(910)X(23972)
X(23973) = crossdifference of every pair of points on line X(657)X(3270)

X(23974) = BARYCENTRIC SQUARE OF X(3265)

Barycentrics    cot^2 A (tan B - tan C)^2 : :
Barycentrics    (b^2 - c^2)^2 (b^2 + c^2 - a^2)^4 : :
Barycentrics    squared distance from A to van Aubel line : :

X(23974) lies on these lines: {2, 20313}, {122, 3265}, {305, 1972}, {3926, 4563}

X(23974) = isogonal conjugate of X(23975)
X(23974) = isotomic conjugate of X(23590)
X(23974) = anticomplement of X(23591)
X(23974) = barycentric square of X(3265)

X(23975) = BARYCENTRIC SQUARE OF X(24019)

Barycentrics    (sin A tan A)^2 / (tan B - tan C)^2 : :
Barycentrics    (c^2-a^2)^2*a^2*(a^2-b^2)^2*(a^2-b^2+c^2)^4*(a^2+b^2-c^2)^4 : :

X(23975) lies on these lines: {6, 15384}, {1971, 1990}

X(23975) = isogonal conjugate of X(23974)
X(23975) = barycentric square of X(24019)
X(23975) = vertex conjugate of foci of inellipse centered at X(23591) (the barycentric square of the van Aubel line)

X(23976) = BARYCENTRIC SQUARE OF X(1503)

Barycentrics    (2 a^6 - b^6 - c^6 + (b^2 c^2 - a^4) (b^2 + c^2))^2 : :

X(23976) is the center of hyperbola {{A, B, C, X(2), X(107)}}, which is the locus of trilinear poles of lines parallel to the van Aubel line (i.e. lines passing through X(1503)).

X(23976) lies on the Steiner inellipse and these lines: {4, 32}, {6, 15526}, {107, 23591}, {187, 3184}, {800, 1084}, {1033, 15259} et al

X(23976) = complement of X(35140)
X(23976) = perspector of circumparabola centered at X(1503)
X(23976) = X(2)-Ceva conjugate of X(1503)
X(23976) = barycentric square of X(1503)

X(23977) = TRILINEAR POLE OF LINE X(16318)X(23976)

Barycentrics    (b^6 + c^6 - 2 a^6 + a^4 b^2 +a^4 c^2 - b^4 c^2 - c^4 b^2)/((b^2 - c^2) (b^2 + c^2 - a^2)^2) : :

Line X(16318)X(23976) is the tangent to the inellipse that is the barycentric square of the van Aubel line, at X(23976) (the barycentric square of X(1503)).

Let P1 and P2 be the two points on the van Aubel line whose trilinear polars are parallel to the van Aubel line. P1 and P2 lie on the circumconic centered at X(23976) (hyperbola {{A, B, C, X(2), X(107)}}). The midpoint of P1 and P2 is the tripolar centroid of X(107). X(23977) is the barycentric product P1*P2.

X(23977) lies on these lines: {4, 6}, {112, 1289}, {648, 2404}, {877, 23582}, {2445, 15639} et al

X(23977) = barycentric product X(107)*X(1503)
X(23977) = trilinear pole of line X(16318)X(23976)

X(23978) = BARYCENTRIC SQUARE OF X(4391)

Barycentrics    b^2 c^2 (b - c)^2 (b + c - a)^2 : :
Barycentrics    squared distance from A to line X(1)X(3) : :

X(23978) lies on these lines: {2, 6335}, {5, 7141}, {76, 4572}, {312, 3969}, {313, 1229}, {339, 2973}, {346, 646}, {394, 2988}, {2968, 14010} et al

X(23978) = isogonal conjugate of X(23979)
X(23978) = isotomic conjugate of X(1262)
X(23978) = anticomplement of X(23585)
X(23978) = barycentric square of X(4391)

X(23979) = BARYCENTRIC SQUARE OF X(109)

Barycentrics    a^4/((b - c) (b + c - a))^2 : :

X(23979) lies on these lines: {6, 7115}, {59, 5078}, {109, 6589}, {1262, 17966} et al

X(23979) = isogonal conjugate of X(23978)
X(23979) = barycentric square of X(109)
X(23979) = vertex conjugate of foci of inellipse centered at X(23585) (the barycentric square of line X(1)X(3))

X(23980) = BARYCENTRIC SQUARE OF X(517)

Barycentrics    a (-1 + cos B + cos C) (b (-1 + cos C + cos A) + c (-1 + cos A + cos B)) : :
Barycentrics    (sin^2 A)(-1 + cos B + cos C)^2 : :
Barycentrics    a^2 (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)^2 : :

X(23980) is the center of the circumconic that is the locus of trilinear poles of lines parallel to line X(1)X(3) (i.e. lines that pass through X(517)). This circumconic is a hyperbola that passes through X(2), X(6335), X(9058) and X(24029), and is the isogonal conjugate of line X(6)X(650) and the isotomic conjugate of line X(2)X(905).

X(23980) lies on the Steiner inellipse and these lines: {2, 18816}, {6, 101}, {9, 216}, {37, 1146}, {44, 8607}, {115, 119}, {198, 478}, {220, 15629}, {223, 1020}, {226, 1086}, {1211, 15526} et al

X(23980) = complement of X(18816)
X(23980) = perspector of circumparabola centered at X(517)
X(23980) = X(2)-Ceva conjugate of X(517)
X(23980) = barycentric square of X(517)

X(23981) = TRILINEAR POLE OF LINE X(1457)X(2183)

Barycentrics    a^2 (b^3 + c^3 - (a^2 + b c) (b + c) + 2 a b c)/((b - c)(b + c - a)) : :

Line X(1457)X(2183) is the tangent to the inellipse that is the barycentric square of line X(1)X(3), at X(23980) (the barycentric square of X(517)).

Let P1 and P2 be the two points on line X(1)X(3) whose trilinear polars are parallel to line X(1)X(3). P1 and P2 lie on the circumconic centered at X(23980) (hyperbola {{A, B, C, X(2), X(6335), X(9058)}}), and circle {{X(2), X(2720), X(9058)}}. X(23981) is the barycentric product P1*P2.

X(23981) lies on these lines: {1, 3}, {42, 20122}, {100, 108}, {109, 692}, {119, 1846}, {874, 4998}, {901, 2720}, {1293, 8059} et al

X(23981) = barycentric product X(517)*X(651)
X(23981) = trilinear pole of line X(1457)X(2183)
X(23981) = crossdifference of every pair of points on line X(650)X(1146)

X(23982) = CENTER OF BARYCENTRIC SQUARE OF LINE X(1)X(4)

Barycentrics    (cot^2 B)(cos C - cos A)^2 + (cot^2 C)(cos A - cos B)^2 : :
Barycentrics    (csc^2 B)(sec C - sec A)^2 + (csc^2 C)(sec A - sec B)^2 : :
Barycentrics    (c - a)^2 (c + a - b)^2 (c^2 + a^2 - b^2)^2 + (a - b)^2 (a + b - c)^2 (a^2 + b^2 - c^2)^2 : :

X(23982) lies on these lines: {2, 23983}, {6, 7}, {1104, 13568}, {3772, 13567} et al

X(23982) = complement of X(23983)

X(23983) = BARYCENTRIC SQUARE OF X(6332)

Barycentrics    (cot^2 A)(cos B - cos C)^2 : :
Barycentrics    (csc^2 A)(sec B - sec C)^2 : :
Barycentrics    (b - c)^2 (b + c - a)^2 (b^2 + c^2 - a^2)^2 : :
Barycentrics    squared distance from A to line X(1)X(4) : :

X(23983) lies on these lines: {2, 23982}, {64, 7219}, {69, 144}, {312, 343}, {345, 394}, {1086, 17878}, {1146, 23978}, {1364, 7068}, {2968, 3270}, {3695, 5562}, {6332, 16596}

X(23983) = isogonal conjugate of X(23985)
X(23983) = isotomic conjugate of X(23984)
X(23983) = anticomplement of X(23982)
X(23983) = barycentric square of X(6332)

X(23984) = PERSPECTOR OF BARYCENTRIC SQUARE OF LINE X(1)X(4)

Barycentrics    (tan^2 A)/(cos B - cos C)^2 : :
Barycentrics    (sin^2 A)/(sec B - sec C)^2 : :
Barycentrics    1/((b - c) (b + c - a) (b^2 + c^2 - a^2))^2 : :

X(23984) lies on these lines: {108, 1946}, {516, 1785}, {653, 14837}, {1262, 14953} et al

X(23984) = isogonal conjugate of X(35072)
X(23984) = isotomic conjugate of X(23983)
X(23984) = polar conjugate of X(2968)
X(23984) = cevapoint of X(6) and X(108)
X(23984) = X(6)-cross conjugate of X(108)
X(23984) = trilinear pole of line X(108)X(676) (the tangent to the circumcircle at X(108))
X(23984) = barycentric square of X(653)

X(23985) = BARYCENTRIC SQUARE OF X(108)

Barycentrics    (sin^2 A)(tan^2 A)/(cos B - cos C)^2 : :
Barycentrics    (sin^4 A)/(sec B - sec C)^2 : :
Barycentrics    a^2/((b - c) (b + c - a) (b^2 + c^2 - a^2))^2 : :

X(23985) lies on these lines: {108, 6588}, {910, 7115}, {1262, 14953} et al

X(23985) = isogonal conjugate of X(23983)
X(23985) = barycentric square of X(108)
X(23984) = X(63)-isoconjugate of X(3270)
X(23985) = vertex conjugate of foci of inellipse centered at X(23982) (the barycentric square of line X(1)X(4))

X(23986) = BARYCENTRIC SQUARE OF X(515)

Barycentrics    (2 a^4 - a^3 (b + c) - a^2 (b - c)^2 + a (b - c)^2 (b + c) - (b^2 - c^2)^2)^2 : :

X(23986) lies on the Steiner inellipse, the inellipse centered at X(23982), and these lines: {6, 281}, {57, 1020}, {115, 117}, {1015, 1108}, {2181, 8755} et al

X(23986) = complement of X(34393)
X(23986) = crosssum of X(6) and X(102)
X(23986) = crosspoint of X(2) and X(515)
X(23986) = barycentric square of X(515)
X(23986) = crossdifference of every pair of points on line X(102)X(8677)

X(23987) = TRILINEAR POLE OF LINE X(2182)X(8755)

Barycentrics    (2 a^4 - a^3 (b + c) - a^2 (b - c)^2 + a (b - c)^2 (b + c) - (b^2 - c^2)^2)/((b - c) (b + c - a) (b^2 + c^2 - a^2)) : :

Line X(2182)X(8755) is the tangent to the inellipse that is the barycentric square of line X(1)X(4), at X(23986) (the barycentric square of X(515)).

Let P1 and P2 be the two points on line X(1)X(4) whose trilinear polars are parallel to line X(1)X(4). P1 and P2 lie on the circumconic centered at X(23986) (hyperbola {{A, B, C, X(2), X(9056)}}). X(23987) is the barycentric product P1*P2.

X(23987) lies on these lines: {1, 4}, {108, 676}, {651, 1897}, {1309, 2405}, {2406, 7452} et al

X(23987) = barycentric product X(515)*X(653)
X(23987) = trilinear pole of line X(2182)X(8755)

X(23988) = CENTER OF BARYCENTRIC SQUARE OF LINE X(1)X(6)

Barycentrics    a^2 (a^2 b^2 + a^2 c^2 - 2 a b^3 - 2 a c^3 + b^4 + c^4) : :

The inellipse that is the barycentric square of line X(1)X(6) passes through X(6), X(32), X(220), X(1017), X(1500) and X(6184). The Brianchon point (perspector) of this inellipse is X(1252).

X(23988) lies on these lines: {2, 4554}, {6, 1331}, {9, 1937}, {11, 5701}, {37, 17724}, {39, 4850}, {55, 7123}, {100, 294}, {511, 2225}, {650, 3035}, {1015, 3315}, {1194, 16584}, {1212, 5723} et al

X(23988) = isotomic conjugate of polar conjugate of X(5185)
X(23988) = complement of X(23989)

X(23989) = BARYCENTRIC SQUARE OF X(693)

Barycentrics    (b - c)^2/a^2 : :
Barycentrics    squared distance from A to line X(1)X(6) : :

X(23989) lies on these lines: {2, 4554}, {11, 693}, {63, 7243}, {76, 1978}, {85, 18359}, {149, 2481}, {279, 331}, {310, 6650}, {321, 1233}, {394, 2989}, {561, 8024}, {850, 1109}, {1111, 3120}, {1565, 2969}, {1921, 3266} et al

X(23989) = isogonal conjugate of X(23990)
X(23989) = isotomic conjugate of X(1252)
X(23989) = anticomplement of X(23988)
X(23989) = barycentric square of X(693)

X(23990) = BARYCENTRIC SQUARE OF X(649)

Barycentrics    a^4 (a - b)^2 (a - c)^2 : :

X(23990) lies on these lines: {6, 59}, {101, 6586}, {594, 15742}, {692, 3063}, {765, 5384}, {919, 1618}, {1110, 2251}, {1252, 3285} et al

X(23990) = isogonal conjugate of X(23989)
X(23990) = barycentric square of X(649)
X(23990) = vertex conjugate of foci of inellipse centered at X(23988) (the barycentric square of line X(1)X(6))

X(23991) = CENTER OF BARYCENTRIC SQUARE OF LINE X(115)X(125)

Barycentrics    (b^2 - c^2)^2 (2 a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2) : :

The Brianchon point (perspector) of this inellipse is X(115). The vertex conjugate of the foci of this inellipse is X(3124).

X(23991) lies on these lines: {2, 4590}, {39, 18122}, {115, 523}, {125, 9151}, {230, 3284}, {325, 3291}, {524, 1570}, {620, 14588}, {688, 2679}, {924, 14113}, {1084, 2485} et al

X(23991) = reflection of X(14588) in X(620)
X(23991) = complement of X(4590)
X(23991) = crosspoint of X(2) and X(115)
X(23991) = crosssum of X(6) and X(249)
X(23991) = X(2)-Ceva conjugate of X(620)
X(23991) = center of circumconic that is locus of trilinear poles of lines passing through X(620)
X(23991) = perspector of circumconic centered at X(620)
X(23991) = involutary conjugate of QA-P29 wrt quadrangle ABCX(2)
X(23991) = trilinear pole wrt medial triangle of line X(115)X(125)

X(23992) = BARYCENTRIC SQUARE OF X(690)

Barycentrics    (b^2 - c^2)^2 (2 a^2 - b^2 - c^2)^2 : :

Let A'B'C' be the orthic triangle. X(23992) is the radical center of the Parry circles of AB'C', BC'A', CA'B'.

X(23992) is the center of hyperbola {{A, B, C, X(2), X(2770)}}, which is the locus of trilinear poles of lines parallel to line X(115)X(125) (i.e. lines passing through X(690)). This hyperbola is the isogonal conjugate of line X(6)X(110) and the isotomic conjugate of line X(2)X(99).

X(23992) lies on the Steiner inellipse, the inellipse centered at X(23991), and these lines: {2, 892}, {6, 23967}, {32, 14357}, {115, 523}, {125, 17416}, {126, 325}, {187, 524}, {230, 3163}, {385, 7664}, {468, 16103}, {620, 9182}, {647, 1084}, {843, 11006}, {888, 2679}, {1648, 1649}, {2682, 14443} et al

X(23992) = reflection of X(35087) in X(2)
X(23992) = isogonal conjugate of X(34539)
X(23992) = complement of X(892)
X(23992) = crosspoint of X(i) and X(j) for these {i,j}: {2, 690}, {523, 524}

X(23992) = crosssum of X(i) and X(j) for these {i,j}: {6, 691}, {110, 111}
X(23992) = trilinear pole of line X(14443)X(14444)
X(23992) = perspector of circumparabola centered at X(690)
X(23992) = X(2)-Ceva conjugate of X(690)
X(23992) = barycentric square of X(690)
X(23992) = crossdifference of every pair of points on line X(691)X(5467)
X(23992) = Steiner-inellipse-antipode of X(35087)

leftri

Points associated with trilinear squares of lines (inscribed ellipses): X(23993)-X(24041)

rightri

Contributed by Randy Hutson, September 29, 2018

The following discussion is analogous to the preambles just before X(23582) and X(23962).

Suppose that P = p : q : r (trilinears) is a point in the plane of a triangle ABC, not on a sideline BC, CA, AB. Let L be the trilinear polar of P, so that L meets the sidelines in 0 : q : -r, -p : 0 : r, p : -q : 0. The trilinear squares of these points are the points A' = 0 : q^2 : r^2, B' = p^2 : 0 : r^2, C' = p^2 : q^2 : 0. The perspector of A'B'C' is the trilinear square P^2, so that P^2 = p^2 : q^2 : r^2 is the perspector of the inellipse that is the locus of squares of points on L. The center of the ellipse is the point p^2(q^2 + r^2) : q^2(r^2 + p^2) : r^2(p^2 + q^2), which is the X(2)-crosspoint of P^2, as well as the complement of the isotomic conjugate of P^2. The ellipse is here named the trilinear square of L.

Let E be the inellipse that is the trilinear square of L. The vertex conjugate of the foci of E is the point a^2 p^2 : b^2 q^2 : c^2 r^2. Let U = u : v : w and X = x : y : z be points on L so that U^2 and X^2 lie on E. The trilinear pole of the tangent to E at U^2 is the trilinear product P*U = p*u : q*v : r*w, which lies on the trilinear polar of the Brianchon point (perspector) of E. The intersection of the tangents to E at U^2 and X^2 is the trilinear product U*X = u*x : v*y : w*z.

X(23993) = CENTER OF TRILINEAR SQUARE OF BROCARD AXIS

Barycentrics    csc B sin^2(C - A) + csc C sin^2(A - B) : :
Barycentrics    a^3 (c^3 (c^2 - a^2)^2 + b^3 (a^2 - b^2)^2) : :

The inellipse that is the trilinear square of the Brocard axis passes through X(31), X(255), X(849), X(1094), X(1095), X(1917) and X(23996). The Brianchon point (perspector) of this inellipse is X(1101).

X(23993) lies on these lines: {2, 23994}, {10, 16573}

X(23993) = complement of X(23994)

X(23994) = TRILINEAR SQUARE OF X(1577)

Trilinears    (csc^2 2A) (sin 2B - sin 2C)^2 : :
Trilinears    squared distance from A to Brocard axis : :
Barycentrics    csc A sin^2(B - C) : :
Barycentrics    b^3 c^3 (b^2 - c^2)^2 : :

X(23994) lies on these lines: {1, 336}, {2, 23993}, {75, 2166}, {326, 20571}, {561, 4602}, {1109, 21207}, {1577, 3708} et al

X(23994) = isogonal conjugate of X(23995)
X(23994) = isotomic conjugate of X(1101)
X(23994) = anticomplement of X(23993)
X(23994) = trilinear square of X(1577)
X(23994) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23994}, {31, 1101}, {163, 163}

X(23995) = TRILINEAR SQUARE OF X(163)

Trilinears    (sin^2 2A)/(sin 2B - sin 2C)^2 : :
Trilinears    sin^2 A csc^2(B - C) : :
Trilinears    a^4/(b^2 - c^2)^2 : :

X(23995) lies on these lines: {163, 810}, {922, 1101}

X(23995) = isogonal conjugate of X(23994)
X(23995) = trilinear square of X(163)
X(23995) = vertex conjugate of foci of inellipse centered at X(23993) (the trilinear square of the Brocard axis)
X(23995) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23994}, {1577, 1577}

X(23996) = TRILINEAR SQUARE OF X(511)

Trilinears    cos^2(A + ω) : :
Trilinears    a^2 (a^2 b^2 + a^2 c^2 - b^4 - c^4)^2 : :

X(23996) lies on the inellipse centered at X(10), the inellipse centered at X(23993), and these lines: {1, 1581}, {31, 4575}, {38, 1109}, {47, 1917}, {63, 1956}, {98, 9413}, {163, 255}, {244, 5249}, {293, 662}, {1111, 3670}, {1355, 7062}, {1733, 2227}, {1755, 9417} et al

X(23996) = trilinear square of X(511)

X(23997) = TRILINEAR POLE OF LINE X(1755)X(9417)

Trilinears    cos(A + ω) csc(B - C) : :
Trilinears    a^2 (a^2 b^2 + a^2 c^2 - b^4 - c^4)/(b^2 - c^2) : :

Line X(1755)X(9417) is the tangent to the inellipse that is the trilinear square of the Brocard axis, at X(23996) (the trilinear square of X(511)).

Let P1 and P2 be the two points on the Brocard axis whose trilinear polars are parallel to the Brocard axis. P1 and P2 lie on the circumconic centered at X(11672) (hyperbola {{A, B, C, X(2), X(110)}}), and circle {{X(2), X(110), X(2715)}}. X(23997) is the trilinear product P1*P2.

X(23997) lies on these lines: {1, 21}, {162, 799}, {163, 1983}, {2617, 3909} et al

X(23997) = trilinear pole of line X(1755)X(9417)
X(23997) = crossdifference of every pair of points on line X(661)X(1109)
X(23997) = trilinear product X(i)*X(j) for these {i,j}: {2, 14966}, {3, 4230}, {6, 2421}, {32, 2396}, {99, 237}, {110, 511}, {163, 1959}, {184, 877}, {232, 4558}, {240, 4575}, {249, 3569}, {250, 684}, {325, 1576}, {648, 3289}, {662, 1755}, {670, 9418}, {691, 9155}, {799, 9417}, {2211, 4563}, {2491, 4590}, {2966, 11672}, {5467, 5968}, {11672, 23996}
X(23997) = barycentric product X(i)*X(j) for these {i,j}: {1, 2421}, {31, 2396}, {48, 877}, {63, 4230}, {75, 14966}, {99, 1755}, {110, 1959}, {163, 325}, {232, 4592}, {237, 799}, {240, 4558}, {297, 4575}, {662, 511}, {2966, 23996}, {3569, 24041}

X(23998) = CENTER OF TRILINEAR SQUARE OF EULER LINE

Barycentrics    b (tan C - tan A)^2 + c (tan A - tan B)^2 : :
Barycentrics    b (c^2 - a^2)^2 (c^2 + a^2 - b^2)^2 + c (a^2 - b^2)^2 (a^2 + b^2 - c^2)^2 : :

The inellipse that is the trilinear square of the Euler line passes through X(75), X(158), X(255), X(1087), X(1097), X(1098), and X(1099).

X(23998) lies on these lines: {2, 2632}, {520, 3042}, {525, 15252}, {4458, 6370} et al

X(23998) = complement of X(2632)

X(23999) = PERSPECTOR OF TRILINEAR SQUARE OF EULER LINE

Trilinears    csc^2 2A csc^2(B - C) : :
Barycentrics    (csc A)/(tan B - tan C)^2 : :
Barycentrics    1/(a (b^2 - c^2)^2 (b^2 + c^2 - a^2)^2) : :

X(23999) lies on these lines: {425, 18020}, {811, 14208}

X(23999) = isotomic conjugate of X(2632)
X(23999) = polar conjugate of X(3708)
X(23999) = trilinear pole of line X(662)X(811)
X(23999) = trilinear square of X(648)
X(23999) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2632}, {48, 3708}, {647, 647}

X(24000) = TRILINEAR SQUARE OF X(162)

Trilinears    1/(tan B - tan C)^2 : :
Trilinears    1/(sin 2B - sin 2C)^2 : :
Trilinears    sec^2 A csc^2(B - C) : :
Trilinears    1/((b^2 - c^2)^2 (b^2 + c^2 - a^2)^2) : :

X(24000) lies on these lines: {162, 656}, {240, 1101}, {823, 24006}, {1784, 1955}, {24001, 24024} et al

X(24000) = isogonal conjugate of X(2632)
X(24000) = isotomic conjugate of X(17879)
X(24000) = trilinear pole of line X(162)X(163)
X(24000) = trilinear square of X(162)
X(24000) = vertex conjugate of foci of inellipse centered at X(23998) (the trilinear square of the Euler line)
X(24000) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2632}, {48, 20902}, {656, 656}
X(24000) = polar conjugate of X(20902)

This is the end of PART 12: Centers X(22001) - X(24000)

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)