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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)


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 polar conjugate of X(2963)
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(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) : :

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(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, 20991}, {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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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) : :

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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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) = {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) = 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) = 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(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(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(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) = 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): {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): {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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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) = 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(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(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(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(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(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    {1,21838}, {2,668}, {38,3121}, {39,42}, {43,2275}, {75,6379}, {76,21223}, {244,22173}, {292,3961}, {518,16584}, {519,21877}, {672,20228}, {726,21345}, {982,6377}, {984,21827}, {1011,2241}, {1449,2276}, {1500,17018}, {1574,4651}, {1575,4541}, {2092,16778}, {2229,17135}, {2238,17053}, {2886,16592}, {3681,21830}, {3741,16606}, {3778,20462}, {3971,20363}, {4022,6375}, {8624,21750}, {16058,16781}, {16746,16887}, {17165,21327}, {18172,20255}, {20859,20870}, {21330,22171} : :

X(22199) lies on these lines:


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(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(22258) = X(23)-vertex conjugate of X(23)
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(22259) = vertex conjugate of X(524) and X(525)
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 image 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(2257) = reflection of X(8798) in X(14363)
X(2257) = 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(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(22258) = isogonal conjugate of X(14360)
X(22258) = anticomplement X(13140)
X(22258) = X(3455)-cross conjugate of X(25)
X(22258) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14360}, {2, 16563}, {75, 2930}, {662, 18310}, {14210, 15899}
X(22258) = barycentric product X(6)X(13574)
X(22258) = 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) = 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) = {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 : : >br> 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) : :

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) = 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(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 polar conjugate of X(1212)
X(22343) = isotomic conjugate of polar conjugate of X(20229)


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(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(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 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(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) = {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(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, 20991}, {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(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(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(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(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(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}, {86, 10510}


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(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(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(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(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(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(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) : :

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(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(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(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(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(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(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(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) = 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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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) = 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(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.

underbar

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) = {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) = {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 : :

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(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) = {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) = {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) = {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) = {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) = {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(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(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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = 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) = {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) = {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) = {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) = {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) = {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(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(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) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (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}, {18613,20991}, {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) = {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, 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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = {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) = 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) = 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) = {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) = 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(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) = 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(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(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(20991)


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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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)

underbar



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(23113) = isogonal conjugate of polar conjugate of X(21272)


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(23112) = isogonal conjugate of polar conjugate of X(21271)


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(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) = 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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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) = {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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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(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 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(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(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(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(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(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(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(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(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(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(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(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(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)